sikSaKNnamuxkat;ebtugBRgwgedayEdk

Reinforced Concrete DESIGN

GkniBn M. Nadim Hassoun

Akthem Al-Manaseer

bkERbeday etg qay

viTasanCatiBhubeckeTskm<Ca
mhaviTalysMNg;sIuvil

qaM 2010

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

matika
Contents

I. esckIepIm (Introduction)

1>1> rcnasm<nBIebtugGarem: ................................................................................................1

1>2> KuNsm,ti nigKuNvibtirbs;ebtugGarem: ........................................................................1
1>3> bnk ............................................................................................................................2
1>4> karRbmUlbnk ..............................................................................................................4
1>5> karbMElgbnk ..............................................................................................................8
1>6> eRKOgbgMnrcnasm<nebtugGarem: ...............................................................................10
1>7> CMhannkarKNnaeRKOgbgMBIebtugGarem: ....................................................................11
II.lkNnebtugGarem:

2>1> ktaCH\TiBldl;ersIusg;ebtug ....................................................................................12

2>2> ersIusg;rgkarsgt; .....................................................................................................14
2>3> ersIusg;rgkarTaj ......................................................................................................15
2>4> ersIusg;rgkarkat; ........................................................................................................15
2>5> m:UDuleGLasicrbs;ebtug ............................................................................................16
2>5> m:UDuleGLasicrbs;ebtug ............................................................................................16
2>7> m:UDulnPaBrwg bm:UDulkmaMgkat; ..................................................................................16
2>9> bERmbRmYlmaDrbs;ebtug ...........................................................................................17
2>9>1> karrYmmaD ...............................................................................................................17
2>9>2> karrIkmaDedaykarekIneLIgnkemA ..........................................................................18
2>10> Creep .......................................................................................................................18
2>11> m:UEDlsRmab;TsSn_TaykarrYmmaD nig creep rbs;ebtug ............................................19
2>11>1> m:UEDl ACI 209....................................................................................................19
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2>11>2> m:UEDl B3 .............................................................................................................21

2>11>3> m:UEDl GL 2000 ...................................................................................................25
2>11>4> m:UEDl CEB 90......................................................................................................27
2>11>5> m:UEDl CEB 90-99 ................................................................................................30
2>12> m:as;maDebtug ..........................................................................................................41
2>12> RbePTEdkeRbIkgebtug .............................................................................................42
III.

viPaKFwmebtugGarem:rgkarBt;begag

Flexural Analysis of Reinforced Concrete Beam

3>1> karsnt; (Assumption)...............................................................................................44

3>2> RbePTnkar)ak;edaykarBt; nigEdnkMNt;sac;lUteFob ...............................................44
3>2>1> kar)ak;edaykarBt; .................................................................................................44
3>2>2> EdnkMNt;bERmbRmYlrageFobsRmab; tension-controlled section nig compressioncontrolled section.................................................................................................45

3>3> emKuNbnk ................................................................................................................47

3>4> emKuNkat;bnyersIusg; ..............................................................................................48
3>5> karEbgEckkugRtaMgsgt;smmUl .................................................................................48
3>6> srsEdkrgkmaMgTajnmuxkat;ctuekaNEkgrgkarBt; ...............................................50
3>6>1> balanced section ....................................................................................................51
3>6>2> PaKryEdkGtibrma ................................................................................................52
3>6>3> PaKryEdkGb,brma ..............................................................................................57
3>7> muxkat;lm .................................................................................................................57
3>8> bNMnEdk .................................................................................................................61
3>9> muxkat;ctuekaNEkgCamYyEdkrgkmaMgsgt; ...............................................................62
3>9>1> enAeBlEdksgt;eFVIkardl;cMNuc yield .....................................................................63
3>9>2> enAeBlEdksgt;eFVIkarmindl;cMNuc yield ................................................................68
3>10> viPaKmuxkat;GkSret T nigmuxkat;GIu I ........................................................................71
3>10>1> TTwgRbsiTPaB .....................................................................................................72
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3>10>2> muxkat;GkSret T RtUv)anKitCaragctuekaNEkg ....................................................73

3>10>3> viPaKmuxkat;ragGkSret T ....................................................................................74
3>11> TMhMnmuxkat;FwmGkSr T eka .................................................................................80
3>11> muxkat;GkSr L pab; ..................................................................................................81
karKNnaFwmebtugGarem:rgkarkac;begag
4>1> km<s;RbsiTPaBsRmab;Fwm nigkRmalxN ...................................................................82
4>2> muxkat;ctuekaNEkgCamYyEdkrgkarTaj ....................................................................82
4>3> KMlatEdk nigRsTab;karBarEdk ..................................................................................84
4>3>1> KMlatEdk ................................................................................................................84
4>3>2> RsTab;karBarEdk ...................................................................................................85
4>3>2> RsTab;karBarEdk ...................................................................................................86
4>3>4> km<s;Gb,brmarbs;muxkat;ebtug .............................................................................86
4>4>muxkat;ctuekaNEkgCamYyEdkrgkarsgt;......................................................................91
4>5> KNnamuxkat;GkSret T ..............................................................................................98
IV.

V.

viFIKNnaepSgeTot

Alternative Design Methods

5>1> esckIepIm (Introduction) .........................................................................................105

5>2> emKuNbnk (Load Factors) ......................................................................................105
5>3> emKuNkat;bnyersIusg; (Strength-Reduction Factor ) .......................................105
5>4>muxkat;ctuekaNEkgCamYyEdkrgkarTaj (Rectangular Sections with Tension
Reinforcement) .........................................................................................................108

5>5> muxkat;ctuekaNCamYynwgEdkrgkarsgt; (Rectangular Sections with Compression

Reinforcement) ........................................................................................................111

5>6> karKNnamuxkat;GkSret (Design of T-Section) .....................................................113

5>7> viFI strut and tie (Strut and Tie Method) ................................................................115
5>7>1> esckIepIm (Introduction)....................................................................................115
5>7>2> viFIsaRsKNnatam ACI (ACI Design Procedure) ...............................................118
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5>7>3> tRmUvkarsRmab;karKNna (Design Requirement) .................................................120

VI.

PaBdab nigsameRbH

6>1> sameRbHenAkgeRKOgbgMebtug (Deflection of Structural Concrete Members) .........123

6>2> PaBdabxN (Instantaneous Deflection) ...............................................................124
6>2>1> m:UDuleGLasic (Modulus of Elasticity) .............................................................125
6>2>2> pleFobm:UDuleGLasic (Modular Ratio) .............................................................125
6>2>3> m:Um:g;eRbH (Cracking moment) ...........................................................................125
6>2>4> m:Um:g;niclPaB (Moment of inertia) ....................................................................126
6>2>5> lkNrbs;muxkat; (Properties of sections) .......................................................130
6>3> PaBdabryeBlyUr (Long-term Deflection) ............................................................131
6>4> PaBdabGnuBaat (Allowable Deflection)..................................................................132
6>5> PaBdabEdlbNalmkBIbnSMbnk (Deflection Due to Combinations of Load) .........132
6>6> PaBdabenAkgGgt;rgkarBt; (Cracks in Flexural Members)....................................142
6>7> tRmUvkarrbs;bTdan ACI Code (ACI Code Requirement) ......................................146
VII.

RbEvgEdkbgb; bRbEvgEdkRCYs

7>1> esckIepIm .................................................................................................................153

7>2> karbegItkugRtaMgsit..................................................................................................153
7>2>1> PaBsitedaykarBt; ..............................................................................................153
7>2>2> karBiesaFsRmab;RbsiTPaBPaBsit........................................................................154
7>3> RbEvgbgb;sRmab;tMbn;Taj .......................................................................................156
7>3>1> RbEvgbgb;mUldan l ............................................................................................156
7>3>2> emKuN ACI Code sRmab;KNna l sRmab;srsEdkrgkarTaj ..........................158
7>3>3> rUbmnsRmYlsRmab; l ..........................................................................................159
7>4> RbEvgbgb;sRmab;tMbn;sgt; l ...............................................................................161
7>5> segbkarKNna l kgtMbn;Taj.................................................................................162
7>6> muxkat;eRKaHfak;enAkgGgt;rgkarBt; ..........................................................................165
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7>7> TMBk;; .........................................................................................................................169

7>6> kartEdk....................................................................................................................172
7>7> karbBab;Edk ............................................................................................................174
VIII.

kmaMgkat;TTwg nigkmaMgTajGgt;RTUg
Shear and Diagonal Tension

8>1> esckIepIm .................................................................................................................180

8>2> kugRtaMgkmaMgkat;enAkgFwmebtugGarem: ....................................................................180
8>3> kareFVIkarrbs;FwmedayKanEdkkmaMgkat;TTwg .............................................................183
8>4> \TiBlm:Um:g;eTAelIersIusg;kmaMgkat; ...........................................................................185
8>5> FwmmanEdkkmaMgkat;..................................................................................................188
8>6> tRmUvkarrbs; ACI Code sRmab;karKNnakmaMgkat;TTwg..........................................191
8>6>1> muxkat;eRKaHfak;sRmab;karKNnaersIusg;kmaMgkat;TTwgmFm
Critical section for nominal shear strength calculation........................................191

8>6>2> muxkat;EdkGb,brmasRmab;EdkkmaMgkat;TTwg ........................................................191

8>6>3> kmaMgkat;TTwgGtibrmaEdlTb;edayEdkkmaMgkat;TTwg V ....................................193
8>6>4> KMlatEdkkgGtibrma .............................................................................................193
8>6>5> ersIusg; yield rbs;EdkkmaMgkat;TTwg.....................................................................194
8>6>6> TMBk;rbs;Edkkg ....................................................................................................194
8>6>7> EdkkgenAEdlenAEk,rTRm .....................................................................................196
8>6>8> RbEvgRbsiTPaBrbs;EdkdgErk ..............................................................................196
8>7> karKNnaEdkkgbBar ...............................................................................................196
8>8> segbviFIsaRsKNnaEdkkgbBar..............................................................................198
8>9> kmaMgkat;TTwgEdlbNalBIbnkGefr.........................................................................204
8>10> kugRtaMgkmaMgkat;TTwgenAkgGgt;Edlmankm<s;ERbRbYl ..........................................208
8>11> Ggt;rgkarBt;CeRmAeRCA ..........................................................................................214
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IX.

kRmalxNmYyTis

9>1> RbePTkRmalxN ......................................................................................................231

9>2> karKNnankRmalxNtan;mYyTis ............................................................................232
9>3> EdnkMNt;kgkarKNnaEdlGnuelameTAtam ACI CODE .........................................235
9>4> EdksItuNPaB nigEdkrYmmaD.......................................................................................236
9>5> lMGitsrsEdk ..........................................................................................................237
9>8> karEbgEckbnkBIkRmalxNmYyTiseTAFwmTRm..........................................................238
9>9> RbBnkRmalxNrnUtmYyTis (One-Way joist Floor System) ...................................243
X.

ssrrgkmaMgcMGkS

10>1> esckIepIm ..............................................................................................................248

10>2> RbePTssr.............................................................................................................248
10>3> kareFVIkarrbs;ssrrgbnkcMGkS .............................................................................250
10>4> lkxNrbs; ACI Code.........................................................................................250
10>5> smIkarsRmab;KNna ...............................................................................................252
10>6> kmaMgTajcMGkS.......................................................................................................253
XI.

eRKOgbgMrgkarsgt; nigrgkarBt;

11>1> esckIepIm .............................................................................................................255

11>2> karsnt;sRmab;KNnassr .....................................................................................256
11>3> daRkamGnrkmbnk-m:Um:g; (Load-moment interaction diagram) ..........................257
11>4> karpl;nUvsuvtiPaB (Safety provisions)..................................................................259
11>5> Balanced condition muxkat;ctuekaN ..................................................................261
11>6> muxkat;ssreRkamGMeBIbnkcakpit (Column sections under eccentric loading)......264
11>7> ersIusg;rbs;ssrsRmab;kar)ak;edaykarTaj
(Strength of columns for tension failure) .............................................................265

11>8> ersIusg;rbs;ssrsRmab;kar)ak;edaykarsgt;

(Strength of columns for compression failure) ....................................................268

11>8>1> dMeNaHRsaysakl,g (Trial solution)................................................................269

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11>8>2> dMeNaHRsayviPaKcMnYn (Numerical Analysis Solution) ......................................271

11>8>3> dMeNaHRsayRbEhl (Approximate Solution) ...................................................273
11>9> ]TahrN_sRmab;daRkamGnrkm (Interaction Diagram Example) ........................275
11>10> ssrmuxkat;ctuekaNCamYyEdkxag (Rectangular columns with side bars) .......276
11>11> lTPaBRTbnkrbs;ssrmuxkat;mUl (Load Capacity of Circular Columns) .......278
11>11>1 lkxN Balanced ...........................................................................................280
11>11>2 ersIusg;rbs;muxkat;mUlsRmab;kar)ak;edaykarsgt;
Strength of circular column for compression failure ...........................................284

11>11>3 ersIusg;rbs;muxkat;mUlsRmab;kar)ak;edaykarTaj

Strength of circular column for tension failure....................................................285

11>12> karviPaK nigkarKNnassredayeRbIdaRkam

Analysis and Design of Column Using Charts ....................................................286

11>13> karKNnassreRkambnkcakpit
(Design of Columns under Eccentric Loading) ...................................................292

11>13>1 KNnassrsRmab;kar)ak;edaykarsgt;

(Design of Column for Compression Failure) .....................................................292

11>13>2 KNnassrsRmab;kar)ak;edaykarTaj

(Design of Column for tension Failure)...............................................................298

11>14> karBt;tamBIrTis (Biaxial Bending) ..................................................................300

11>15> ssrmUlCamYynwgkarrayEdkesIeRkamm:Um:g;Bt;BIrTis
Circular Columns with Uniform reinforcement Under Biaxial Bending.............303

11>16> ssrmuxkat;kaer nigctuekaNeRkamm:Um:g;Bt;BIrTis

(Square and Rectangular Columns under Biaxial Bending) ................................304

11>16>1> viFI; Bresler Reciprocal Method ......................................................................304

11>16>2> viFIExSvNbnk Bresler (Bresler Load Contour Method) .................................305
11>17> viFIExSvNbnk Parme (Parme Load Contour Method) ......................................305
11>18> smIkarp)ak; (Equation of failure surface) ........................................................312
XII.

ssrEvg

12>1> esckIepIm .............................................................................................................315

12>2> RbEvgssrRbsiTPaB (Effective Column Length) Kl ........................................315
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12>3> emKuNRbEvgRbsiTPaB (Effective Length Factor) K ............................................316

12>4> PaBrwgRkajrbs;Ggt; (Member Stiffness) EI .......................................................321
12>5> EdnkMNt;sRmab;pleFobrlas; (Limitation of The Slenderness Ratio) Kl / r ...321
12>5>1> eRKagGt;eyal (Nonsway Frames) ....................................................................321
12>6> viFIKNnabEnmm:Umg; (Moment-Magnifier Design Method)....................................323
12>6>1> esckIepIm (Introduction)....................................................................................323
12>6>2> m:Um:g;bEnmenAkgeRKagGt;eyal (Magnified Moments in Nonsway Frames) ..324
12>6>3> m:Um:g;bEnmenAkgeRKageyal (Magnified Moments in sway Frames)................325
u

XIII.

eCIgtag

FOOTINGS

13>1> esckIepIm (Introduction).......................................................................................335

13>2> RbePTeCIgtag (Types of Footings) .......................................................................336
13>3> karBRgaysm<aFdI (Distribution of soil pressure) .................................................339
13>4> karBicarNakgkarKNna (Design Consideration) ................................................341
13>4>1> TMhMeCIgtag (size of Footing).............................................................................341
13>4>2> kmaMgkat;mYyTis kmaMgkat;Fwm Vu1 One-Way Shear (Beam Shear)............342
13>4>3> kmaMgkat;BIrTis kmaMgpug Vu1 Two-Way Shear (Punching Shear)..............343
13>4>4> ersIusg;Bt; nigEdkeCIgtag (Flexural Strength and Footing Reinforcement)....345
13>4>5> lTPaBRTRTg;rbs;ssrenARtg;)at (Bearing Capacity of Column at Base) .......347
13>4>6> RbEvgEdkRCYs (Development Length of the Reinforcing bars) ......................349
13>4>7> sRmutDIepr:g;Esl karKNnaeCIgtagkglkxNlMnwg
Differential Settlement (Balanced Footing Design) ........................................349

13>5> eCIgtagebtugsuT (Plain Concrete Footings) ........................................................350

13>6> Combined Footings.............................................................................................362
13>7> eCIgtageRkambnkssrcakpit (Footings under Eccentric Column Loads) .........370
13>8> eCIgtageRkamm:Um:g;BIrTis (Footings under Biaxial Moment) ...............................372
13>9> kRmalxNelIdI (Slabs on Ground)........................................................................375
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13>10> eCIgtagenAelIssrRKwH (Footings on Piles) .........................................................379

XIV.

CBaaMgTb;

Retaining Wall

14>1> esckIepIm (Introduction)......................................................................................378

14>2> RbePTCBaaMgTb; (Types of Retaining Walls) .........................................................378
14>3> kmaMgenAelICBaaMgTb; (Forces on Retaining Walls) ...............................................380
14>4> sm<aFdIGkm nigsm<aFdIskm (Active and Passive Soil Pressures) ......................381
14>5> \TiBlnbnkbEnm (Effect of Surcharge) ..............................................................386
14>6> kmaMgkkitenAelI)atCBaaMgTb; (Friction on the Retaining Wall Base) ....................387
14>7> sanPaBlMnwgRbqaMgnwgkarRkLab; (Stability against Overturning) ......................388
14>8> smamaRtnCBaaMgTb; (Proportions of Retaining Walls).......................................389
14>9> tRmUvkarsRmab;KNna (Design Requirement).......................................................390
14>10> karbgrTwk (Drainage) ........................................................................................391
14>11> CBaaMgCan;eRkamdI (Basement Walls) .................................................................406
XV.

karKNnasMrab;kmaMgrmYl
Design for Torsion

15>1> esckIepIm (Introduction) .....................................................................................411

15>2> m:Um:g;rmYlenAkgFwm (Torsional Moments in Beams)..............................................413
15>3> kugRtaMgrmYl (Torsional Moments in Beams) .......................................................413
15>4> m:Um:g;rmYlenAkgmuxkat;ctuekaN (Torsional Moments in Rectangular Sections)....416
15>5> kmaMgpbrvagkmaMgkat; nigkmaMgrmYl (Combined Shear and Torsion)....................417
15>6> RTwsIkarrmYlsRmab;Ggt;ebtug (Torsion Theories for Concrete Members) ............418
15>6>1> Skew Bending Theory ......................................................................................418
15>6>2> Space Truss Analogy ........................................................................................419
15>7> ersIusg;rmYlnGgt;ebtugsuT (Torsional Strength of Plain Concrete Members)..422
15>8> karrmYlenAkgGgt;ebtugBRgwgedayEdk (Torsion in Reinforced Concrete
Memebers (ACI Code Procedure)) .................................................................423

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15>8>1> sBaaNTUeTA (General) .....................................................................................423

15>8>2> )a:ra:Em:RtFrNImaRtnkarrmYl (Torsional Geometric Parameters) ......................424
15>8>3> m:Um:g;rmYleFVI[eRbH Tcr (Cracking Torsional Moment T ) ................................425
15>8>4> m:Um:g;rmYllMnwg nwgm:Um:g;rmYlRtUvKa (Equilibrium Torsion and
cr

Compatibility Torsion) ....................................................................................428

15>8>5> karkMNt;nersIusg;m:Um:g;rmYl (Limitation of Tortional Moment Strength) .........429

15>8>6> muxkat;Rbehag (Hollow Section) .......................................................................431
15>8>7> EdkRTnug (Web Reinforcement).........................................................................431
15>8>8> EdkTb;karrmYlGb,brma (Minimum Torsional Reinforcement) .........................433
15>9> segbviFIsaRsKNnaeday ACI Code (Summary of ACI Code Procedures) ........433
XVI.

FwmCab; nigeRKag

Continuous Beams and Frames

16>1> esckIepIm (Introduction) ......................................................................................445

16>2> m:Um:g;GtibrmaenAkgFwmCab; (Maximum Moment in Continuous Beams)...............446
16>2>1> eKalkarN_viPaK (Basic Analysis) ......................................................................446
16>2>2> karGnuvtkardak;bnk (Loading Application)......................................................446
16>2>3> m:Um:g;viCmanGtibrma nigGb,brmaenAkgElVg
Maximum and Minimum Positive Moments within a Span ...........................447

16>2>4> m:Um:g;GviCmanGtibrmaenAelITRm (Maximum Negative Moments at Supports) ...448

16>2>5> m:Um:g;enAkgFwmCab; (Moments in Continuous Beams) .......................................449
16>3> eRKagsMNg;GKar (Building Frames) ....................................................................453
16>4> Portal Frames ........................................................................................................455
16>4>1> cugsnak;BIr (Two Hinged Ends) ........................................................................455
16>4>2> cugbgb;BIr (Two Fixed Ends) ............................................................................456
16>5> eRKagTUeTA (General Frames) ................................................................................458
16>6> karKNnasnak;rbs;eRKag (Design of Frame Hinges)...........................................460
16>6>1> Mesnager hinges ...............................................................................................460
16>6>2> Considre hinges ..............................................................................................462
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16>7> esckIepImBIkarKNnasanPaBkMNt; (Introduction to Limit Design) ....................473

16>7>1> lkNTUeTA (General) .....................................................................................473
16>7> 2> KMnitnkarKNnasanPaBkMNt; (Limit Design Concept) ...................................474
16>7> 3> eKalkarN_nsnak;)asic (Plastic Hinge Concept)...........................................475
16>8> emkanicnkar)ak; (The Collapse Mechanism) .......................................................477
16>9> eKalkarN_nkarKNnasanPaBkMNt; (Principles of Limit Design) .......................477
16>10> Ednx<s; nigEdnTabnemKuNbnk (Upper and Lower Bounds of Load Factors) ..479
16>11> karviPaKsanPaBkMNt; (Limit Analysis)..............................................................479
16>12> mMurgVilrbs;snak;)asic (Rotation of Plastic Hinges) ..........................................484
16>12>1> RbEvgsnak;)asic (Plastic Hinge Length) ......................................................484
16>12>2> emKuNEbgEckkMeNag (Curvature Distribution Factor ) .................................486
16>12>3> snsSn_nPaBsVit (Ductilty Index ) ...............................................................487
16>12>4> mMurgVilEdlRtUvkar (Required Rotation) ..........................................................488
16>12>5> lTPaBTb;mMurgVil (Rotation Capacity Provided) .............................................488
16>13> segbviFIsaRskgkarKNnasanPaBkMNt;
(Summary of Limit Design Procedure)...........................................................491

16>14> karEbgEckm:Um:g;eLIgvijnm:Um:g;GviCmanenAkgFwmCab;

Moment Distribution of Negative Moments in Continuous Beams................495

XVII.

karKNnakRmalxNBIrTis

Design of Two-Way Slabs

17>1> esckIepIm (Introduction) ....................................................................................507

17>2> RbePTkRmalxNBIrTis (Types of Two-Way Slabs).............................................507
17>3> kareRCIserIsRbBnkRmalxNebtugEdlmanlkNesdkic
Economical Choice of Concrete Floor Systems..............................................511

17>4> eKalKMnitkgkarKNna (Design Concept) ..............................................................512

17>5> ceRmokelIssr nigceRmokkNal (Column and Middle Strips) ............................514
17>6> kRmas;kRmalGb,brmaedIm,IkMritPaBdab
(Minimum Slab Thickness to Control Deflection) ..............................................517

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17>7> ersIusg;kMlaMgkat;TTwgrbs;kRmalxN (Shear Strength of Slabs)...........................524

17>7>1> kRmalxNBIrTisEdlRTedayFwm (Two-Way Slabs Supported on Beams) ........524
17>7>2> kRmalxNBIrTisEdlKanFwm (Two-Way Slabs without Beams)........................525
17>7>3> EdkkmaMgkat;TTWgenAkgkRmalxNBIrTisEdlKanFwm
Shear Reinforcement in Two-Way Slabs without Beams ...............................525

17>8> karviPaKkRmalxNBIrTisedayviFIKNnaedaypal;

Analysis of Two-Way Slabs by the Direct Design Method ............................530

17>8>1> karkMNt; (Limitations).......................................................................................530

17>8>2> m:Um:g;saTicemKuNsrub (Total Factored Static Moment).....................................530
17>8>3> karEbgEckm:Um:g;tambeNaykgkRmalxN
(Longitudinal Distribution of Moment in Slabs).............................................532

17>8>4> karEbgEckm:Um:g;tamTTwgkgkRmalxN

(Transverse Distribution of Moment in Slabs)................................................535

17>8>5> karpl;rbs; ACI sRmab;\TiBlrbs;KMrUnkardak;bnk

(ACI Provisions for Effects of Pattern Loading).............................................539

17>8>6> karlMGitsrsEdk (Reinforcement Details) ......................................................540

17>8>7> viFIPaBrwgRkajEdlRtUv)anEktRmUvsRmab;ElVgcug
(Modified Stiffness Method for End Spans) ...................................................540

17>8>8> segbviFIKNnaedaypal; (Summary of the Direct Design Method (DDM)) ....543

17>9> m:Um:g;KNnaenAkgssr (Design Moments in Column)...........................................567
17>10> karbMElgm:Um:g;minesIeTAkgssr
(Transfer of Unbalanced Moments to Columns).............................................569

17>10>1> karbMElgm:Um:g; (Transfer of Moment) ............................................................569

17>10>3> kugRtaMgkmaMgkat;EdlbNalBI M f (Shear Stress Due to) M f ...................570
17>11> kRmalxN Waffle (Waffle Slabs) ...................................................................573
17>12> viFIeRKagsmmUl (Equivalent Frame Method)....................................................592
XVIII.

CeNIr

Stairs

18>1> esckIepIm (Introduction) ......................................................................................606

18>2> RbePTCeNIr (Types of Stairs)..............................................................................607
T.Chhay

xii

Contents

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

XIX.

esckIENnaMBIebtugkugRtaMg

Introduction to Prestressed Concrete

19>1> ebtugeRbkugRtaMg (Prestressed Concrete) ...............................................................634

19>1>1> eKalkarN_nkareFVIeRbkugRtaMg (Principles of Prestressing) ...............................634
19>1>2> karGnuvteRbkugRtaMgedayEpk (Partial Prestressing) ..........................................642
19>1>3> karcat;cMNat;fak;Ggt;rgkarBt;ebtugeRbkugRtaMg
(Classification of Prestressed Concrete Flexural Members) ...........................646

19>2> smar nigtRmUvkarsRmab;beRmIbRmas;

(Material and Serviceability Requirement) .....................................................647

19>2>1> ebtug (Concrete) ...............................................................................................647

19>2>2> EdkeRbkugRtaMg (Prestressing Steel) ...................................................................648
19>2>3> EdkBRgwg (Reinforcing Steel) ...........................................................................649
19>3> kMhatbg;eRbkugRtaMg (Loss of Prestress)................................................................649
19>3>1> Lump-sum losses .............................................................................................649
19>3>2> kMhatbg;edaysar (Elastic Shortening of Concrete) ........................................650
19>3>3> kMhatbg;edaysarkarrYmmaD (Loss Due to Shrinkage) .....................................651
19>3>4> kMhatbg;edaysar creep rbs;ebtug ....................................................................652
19>3>5> kMhatbg;edaysar Relaxation rbs;Edk .............................................................653
19>3>6> kMhatbg;edaysarkmaMgkkit (Loss Due to Friction)..........................................653
19>3>7> kMhatbg;edaysar Anchor set ...........................................................................655
19>4> viPaKGgt;rgkarBt;begag (Analysis of Flexural Members)....................................660
19>4>1> kugRtaMgEdlbNalBIlkxNmanbnk niglkxNKanbnk
Stresses Due to Loaded and Unloaded condition...........................................660

19>4>2> EdnkMNt;sl (Kern Limits) ...............................................................................661

19>4>3> karkMNt;tmncMNakpit (Limiting Values of Eccentricity) ...........................662
19>4>4> tmkMNt;mkmaMgeRbkugRtaMgenAeBlepr
(Limiting Values of the Prestessing Force at Transfer) ..................................664

19>5> KNnaGgt;rgkarBt;begag (Design of Flexural Members).....................................673

matika

xiii

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

19>5>1> sBaaNTUeTA (General)........................................................................................673

19>5>2> muxkat;ctuekaN (Rectangular Sections) ............................................................675
19>5>3> muxkat;Edlmansab (Flanged Sections) ............................................................677
19>5>4> EdkBRgwgrgeRbkugRtaMg (Nonprestressed Reinforcement) ..................................677
19>6> m:Um:g;eRbH (Cracking Moment) ...............................................................................680
19>7> PaBdab (Deflection) ..............................................................................................683
19>8> KNnasRmab;kmaMgkat;TTwg (Design for Shear) ....................................................685
19>8>1> viFIcMbg (Basic Approach)..................................................................................686
19>8>2> ersIusg;kmaMgkat;Edlpl;edayebtug (Shear Strength Provided by Concrete) ....686
19>8>3> EdkkmaMgkat; (Shear Reinforcement).................................................................689
19>8>4> EdnkMNt; (Limitation) ........................................................................................690
19>9> KNnaCMhandMbUgnGgt;ebtugeRbmugRtaMgrgkarBt;
Preliminary Design of Prestressed Concrete Flexural Members.....................695

19>9>1> rUbrag nigTMhM (Shapes and Dimensions) .............................................................695

19>9>2> kmaMgeRbkugRtaMg nigRkLapEdk (Shapes and Dimensions) ...............................695
19>10> kugRtaMgbkxagcug (End-Block Stresses) ............................................................698
19>10>1> Ggt;rgkugRtaMgTajmun (Pretensioned Members) .............................................698
19>10>2> Ggt;rgkugRtaMgTajeRkay (Post-tensioned Members) .....................................699
]bsm<n

T.Chhay

.........................................................................................................................702

xiv

Contents

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

I.

esckIepIm

1>1> rcnasm<nBIebtugGarem:
ebtug Cafsib,nimitEdlekIteLIgedaysarkarpSMKansmarCaeRcIndUcCa fbMEbk xSac; sIum:gt Twk
nigeBlxHmanTwkfaMKImI. smarTaMgenHRtUv)anlaybBalKaedaysmamaRtRtwmRtUv Tuk[kkkayeTACa
ebtug. ebtugCasmarEdleFVIkar)anlnwgkarsgt;.
edIm,I[smareFVIBIebtugGacTb;nwgkarBt;)an luHRtaEtvaRtUv)anBRgwgedaysrsEdk eBlenaH
eK[eQaHvafa ebtugGarem:. EdkCasmarEdleFVIkar)anlnwgkarTaj.
CaTUeTAkgkarKNnarcnasm<nBIebtugGarem:mYy KWeKRtUvGnuvteLIgCaBIrCMhanFM
- kMNt;nUvRbePTbnkepSgEdlGnuvtmkelIrcnasm<n edayeRbInUvviFIsaRsRtwmRtUvkgkarviPaK
eRKOgbgM.
- kMNt;nUvTMhMmuxkat;EdknRKb;rcnasm<nTaMgGs; [manlkNesdkic edayBicarNaelIsuvtiPaB
sirPaBkareRbIR)as; nigtYnaTIrbs;eRKOgbgM.
ebtugGarem: CasmarmYyEdleKeRbIR)as;CaTUeTAenAkgkarKNnaGKarRKb;RbePT. vaCasmarEdl
pMeLIgedaysmarBIrRbePTKW ebtug nigEdk. eRKOgbgMEdleFVIBIebtugGarem: RtUv)aneRbIR)as;sRmab;RTnUv
bnkeRcInRbePT.
1>2> KuNsm,ti nigKuNvibtirbs;ebtugGarem:
ebtugGarem: CasmarEdlRtUv)aneKeRbIR)as;y:agTUlMTUlaysRmab;RbePTCaeRcInneRKOgbgM.
vaRtUv)aneKykmkeRbobeFobCamYynwgEdk sRmab;lkNesdkic nigkarRbtibti.
KuNsm,tirbs;ebtugGarem:RtUv)ansegbdUcxageRkam
- vamanersIusg;x<s;kgkarsgt; (high compressive strength)
- vaTb;Tl;nwgePIg)anlCagEdk
- vamanGayukalEvgCamYynwgkarEfTaMTab
- sRmab;eRKOgbgMxHdUcCa TMnb;Twk ssr nigeCIgtag vaCasmarEdlmanlkNesdkicx<s;
- vaGacRtUv)aneKeRbIsRmab;kMNt;rUbragtamtRmUvkar nigeFVICaeRKOgbgMcak;eRscCamYynwgPaB
dabGb,brma.
esckIepIm

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

KuNvibtirbs;ebtugGarem:RtUv)ansegbdUcxageRkam
- vamanersIusg;TabkgkarTajRbEhlCamYyPaKdb;nersIusg;rgkarsgt;
- vaRtUvkarkarlay karcak; karEfTaM EdlTaMgenHsuTEtCH\TiBldl;ersIusg;rbs;va
- Bum<EdleRbIsRmab;cak;ebtugmantmx<s;. tmBum<esIresInwgtmebtug.
- vamanersIusg;rgkarsgt;TabCagEdkRbEhlCadb;dg EdlCaehtueFVI[muxkat;ssrFMsRmab;
GKareRcInCan;.
- vaekItmansameRbH EdlbNalmkBIkarrYmmaD nigkarGnuvtrbs;bnkGefr.
1>3> bnk
eRKOgbgMrcnasm<nRtUv)anKNnaedIm,IRTnUvbnkCak;lak;.
bnkKWCakmaMgTaMgLayNaEdlGnuvtmkelIrcnasm<n Edlpl;nUvsmamaRtmuxkat;rbs;va. CaTU
eTAbnkRtUv)anEckecjCa bnkefr bnkGclt nigbnkGefr bnkclt.
bnkefr (Dead load) EdlrYmmanTmn;rbs;rcnasm<n Tmn;pal;rbs;va nigTmn;smarepSgeTot
EdlmanenAelIeRKOgbgM dUcCa \dkar:U )ayGr ek,g CBaaMg >>>. bnkefrRtUv)anKNnaCamYynwgkRmitRtwm
RtUvnTMhMrbs;smar nigTmn;maDrbs;va.
bnkGefr (Live load) CabnkTaMgLayNaEdlminEmnCabnkefr. vaCaRbePTbnkEdlmanlkN
minnwg brBay bclt. vaGacGnuvtmkelIeRKOgbgMyWt Pam bBar bedk ehIyGaMgtg;sIuetrbs;vaGac
ERbRbYleTAtameBlevla. CaTUeTAbnkGefrrYmman
- bnkEdlekItBITmn;mnusS eRKOgsgarwm nig]bkrN_smar
- kmaMgpbEdl)anBIxl; nig)anBIbERmbRmYlsItuNPaB
- Tmn;rbs;RBilRbsinebIman
- sm<aFrbs;Twk bdImkelICBaaMgTb;dI
- Tmn;rbs;cracrN_mkelIs<an
- kmaMgpbDINamicEdl)anBIbnkclt bbnkb:HTgic rBayEpndI bkarpH
bnkRBilGacERbRbYlcenaHBI 0.5kN / m 2kN / m GaRsyelIGakasFatutamtMbn;.
bnkxl;GacERbRbYlcenaHBI 0.75kN / m 1.5kN / m GaRsynwgel,nxl;. sm<aFxl;n
eRKOgbgM F RtUv)anKNnatamrUbmnxageRkam
2

F = 0.00064CsV 2

Edl
T.Chhay

- sm<aFxl; kN / m

Introduction

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

- el,nxl; m / s
C - emKuNrUbragrbs;GKar enAcenaHBI 1.2 1.3

taragTI1 bnkBRgayesI
Design live load

Occupancy

Content

Assembly hall

Fixed seats
Movable seats
Operating rooms
Private rooms
Guest rooms
Public rooms
Balconies
Private house and apartments
Public rooms
Classrooms
Corridors
Reading rooms
Stack rooms
Offices
Lobbies

Hospital
Hotel

Housing
Institution
Library
Office building
Stairs
Storage warehouses

kN / m 2

2.9
4.8
2.9
1.9
1.9
4.8
4.8
1.9
4.8
1.9
4.8
2.9
7.2
2.4
4.8
4.8
4.8
12.0
4.8

Light
Heavy

Yard and terraces

taragTI2 dg;sIuet nigdg;sIuetxnic

Material

Specific gravity

Building material
Brick
Cement, Portland, loose
Cement, Portland, set
Earth, dry, packed
Sand or gravel, dry, packed
Sand or gravel, wet
Liquids
Oils
Water (at 4 o c )
Ice
Metal and mineral
Aluminum
Copper
Iron
Lead
Steel, rolled
Limestone or marble

esckIepIm

1.8-2.0
2.7-3.2
-

Density
kg / m 3

1924
1443
2933
1523
1600-1924
1892-1924

0.9-0.94
1
0.88-0.92

930
1000
898

2.55-2.75
9.0
7.2
11.38
7.85
2.5-2.8

2645
8913
7214
11380
7855
2645
T.Chhay

mhaviTalysMNg;sIuvil

NPIC

Sandstone
Shale or slate
Normal weight concrete
Plain
Reinforced or prestressed

2.2-2.5
2.7-2.9

2356
2805

2.2-2.4
2.3-2.5

2324
2405

1>4> karRbmUlbnk
edIm,IQaneTAdl;karKNnamuxkat;EdkEdlRtUvkar enAkgmuxkat;eRKOgbgM CadMbUgeyIgRtUvRbmUl
bnkTaMgGs;EdlmanGMeBIeTAelIeRKOgbgMenaH. karRbmUlbnkKWeyIgRtUvRbmUlBIelIcuHmkeRkam )annyfa
dMbUgeyIgRbmUlbnkelIdMbUl belIkRmalxN bnab;mkRbmUlbnkmkelIFwm rYcehIyRbmUlbnkmkelI
ssr nigcugeRkayRbmUlbnkmkelIRKwH.
bnkTMnajEckCabIRbePTKW
- bnkBRgayelIp Edlmanxat (kN / m ) bnkenHmanGMeBIelIkRmalxN bdMbUl. bnkTaMgenHrYm
man bnkGefr nigbnkefr rYmbBalTaMgTmn;pal;xn.
- bnkBRgayelIRbEvg Edlmanxat (kN / m) bnkenHmanGMeBIelIFwm. bnkTaMgenHrYmmanbnkEdl
)anbBanBIkRmalxNmkelIFwmtamsameRbH 45 bnkrbs;CBaaMg nigbnkpal;xn. bnkBRgayelI
RbEvgmanragCa ctuekaNEkg ctuekaNBay nigRtIekaN.
- bnkcMcMNuc Edlmanxat (kN ) bnkenHmanGMeBIelIssr nigRKwH. bnkTaMgenHrYmmanbnkEdl)an
bBanBIkRmalxNmkelIFwmehIybnbBanmkelIssr nigbnkpal;xn.
eRkABIbnkTMnajTaMgenH enAmanbnkedkeTot. eRkamGMeBInbnkEdl)anGnuvteTAelIrcnasm<n
eyIgGackMNt;)an nUvkmaMgkat;TTwg m:Um:g;Bt; >>>.
]TahrN_ eKmanpHmYyEdlmankMBs;1Can; 4m ehIymanTMhMdUcbgajkgrUb
k> cUrkMNt;bnkEdlmanGMeBIelIkRmalxN S
x> cUrkMNt;bnkEdlmanGMeBIelIFwm B1
K> cUrkMNt;bnkEdlmanGMeBIelIFwm B2
X> cUrkMNt;bnkEdlmanGMeBIelIssr C
dMeNaHRsay
k> kMNt;bnkEdlmanGMeBIelIkRmalxN S
bnkEdlmanGMeBIelIkRmalxNrYmmanbnkefr nigbnkGefr
bnkefrman
2

T.Chhay

Introduction

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

B2 (2030)

B1(2045)

C (2020)

Slab beam column plane

Tile 1cm
Mortar 4cm
Concrete 12cm
Plastering 2cm
B2

Section A-A
Tile 1cm
Mortar 4cm
Concrete 12cm
Plastering 2cm
B1

Section B-B

esckIepIm

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

- kar:UkRmas; 1cm = 0.01 22 = 0.22kN / m

- )ayGrkRmas; 4cm = 0.04 22 = 0.88kN / m
- ebtug Tmn;pal; self weight kRmas; 12cm = 0.12 24 = 2.88kN / m
- kMe)arbUkkRmas; 2cm = 0.02 20 = 0.40kN / m
bnkGefr EdlmanGMeBIelIkRmalxNsRmab;pH tamtaragxagelI L = 1.9kN / m
xageRkamCataragbgajBIbnkEdlmanGMeBIelIkRmalxN S KitkgmYyktap
RbePTbnk
eQaHbnk
bnk (kN / m )
kar:U
0.22
0.88
)ayGr
bnkefr Dead
load (D)
ebtugkRmalxN
2.88
0.4
kMe)arbUk
bnkGefr Live
1.9
bnkGefr
Load (L)
bnksrub
D+L
6.28
2

x> kMNt;bnkEdlmanGMeBIelIFwm B1
Fwm B1 RTkRmalxN S tamTisEvg
+ bnkpal;xn = 0.45 0.2 24 = 2.16kN / m CabnkrayragctuekaN
+ bnkCBaaMg\d 10cm= (4 0.45) 0.1 20 = 7.1kN / m CabnkrayragctuekaN
+ bnkkRmalxN = 4.38 (2 0.1) = 8.32kN / m CabnkrayragctuekaNBay
+ bnkGefr = 1.9 (2 0.1) = 3.61kN / m CabnkrayragctuekaNBay
xageRkamCataragbgajBIbnkEdlmanGMeBIelIFwm B1 KitkgmYyktaRbEvg
RbePTbnk
eQaHbnk
bnk (kN / m) ragBRgaybnk
2.16
bnkpal;xn
ctuekaN
bnkCBaaMg
7.1
bnkefr D
ctuekaN
16.64
bnkkRmalxN
ctuekaNBay
bnkGefr L
bnkclt
7.22
ctuekaNBay
T.Chhay

Introduction

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

B2 (2030)

B1(2045)

K> kMNt;bnkEdlmanGMeBIelIFwm B2
Fwm B2 RTkRmalxN S tamTisxI
+ bnkpal;xn = 0.3 0.2 24 = 1.44kN / m CabnkrayragctuekaN
+ bnkCBaaMg\d 10cm= (4 0.3) 0.1 20 = 7.4kN / m CabnkrayragctuekaN
+ bnkkRmalxN = 4.38 (2 0.1) = 8.32kN / m CabnkrayragRtIekaN
+ bnkGefr = 1.9 (2 0.1) = 3.61kN / m CabnkrayragRtIekaN
xageRkamCataragbgajBIbnkEdlmanGMeBIelIFwm B2 KitkgmYyktaRbEvg
RbePTbnk
eQaHbnk
bnk (kN / m) ragBRgaybnk
1.44
bnkpal;xn
ctuekaN
bnkCBaaMg
7.4
bnkefr D
ctuekaN
16.64
bnkkRmalxN
RtIekaN
bnkGefr L
bnkclt
7.22
RtIekaN
X> kMNt;bnkEdlmanGMeBIelIssr C

esckIepIm

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

B2 (2030)

C (2020)

B1(2045)

bnkssrCan;elI = [(4-0.12)0.20.2]24=3.72kN
+ bnkkRmalxNCan;elI S = 4.3846=105.12kN
+ bnkCBaaMgenAelIFwm B1 = (6-0.2)(4-0.45)0.120= 41.18kN
+ bnkCBaaMgenAelIFwm B 2 = (4-0.2)(4-0.3)0.120=28.12kN
+ bnkFwm B1 = (6-0.2)0.2(0.45-0.12)24=9.19kN
+ bnkFwm B 2 = (4-0.2)0.2(0.3-0.12)24=3.28kN
+ bnkGefr = 1.9(4-0.2)(6-0.2) = 41.88kN
dUcenHbnksrubEdlmanGMeBIelIssr C = 232.5kN
+

1>5> karbMElgbnk
edaysarbnkBRgayelIFwmmanragepSg EdlnaM[eyIgBi)akkgkaredaHRsay dUcenHeyIgeRbI
viFanbMElgbnkenaH[eTACaragctuekaNEkg.
k> karbMElgBIragRtIekaN mkragctuekaNEkg
q

T.Chhay

qe

Introduction

viTasanCatiBhubeckeTskm<Ca
qe =

Department of Civil Engineering

2
q
3

x> karbMElgbnkBIragctuekaNBaymkragctuekaNEkg
q

qe

4 a
qe = q[1 ( ) 2 ]
3 L

K> karRbmUlbnkBIelIkRmalxNmkelIFwm
edIm,IgayRsYl kgkarbMElgbnkBIelIkRmalxNmkelIFwm
eyIgGaceRbInUvrUbmnxageRkam
S - RbEvgtamTisxI
L - RbEvgtamTisEvg
m=

S
L

C
45o

45o

45o

45o

B
L

sRmab;FwmxI BC
bnkenAelIkRmal = w KitCa kN / m
RkLapbnkmkelIFwm = S4 KitCa m
2

esckIepIm

T.Chhay

mhaviTalysMNg;sIuvil

bnkmkelIFwm =

NPIC

Sw
Sw

2
3

KitCa kN / m

sRmab;FwmEvg AB
bnkenAelIkRmal = w KitCa kN / m
RkLapbnkmkelIFwm = SL2 S4 = S4 ( 2 mm ) KitCa m
2

bnkmkelIFwm =

Sw 3 m 2
(
)
3
2

KitCa kN / m bnkBRgayesI

1>6> eRKOgbgMnrcnasm<nebtugGarem:
ebtugGarem:CasmarEdlRtUveKRbIR)as;esIrRKb;GKar rYmbBalTaMgGKarmYyCan; nigeRcInCan;. rcna
sm<nnGKarEdleFVIBIebtugGarem:rYmman
- kRmalxN ( slab ) CaeRKOgbgMbnHedk EdlrgnUvbnkbBar kdUcCabnkedk. CaTUeTAkRmas;
rbs;kRmalxNmanTMhMtUcCagqayebIeRbobeFobeTAnwgbeNay bTTwgrbs;kRmal.
- Fwm ( beam ) CaeRKOgbgMEdlmanRbEvgEvg edk beRTt CamYynwgTTwg nigCeRmAkMNt;mYy.
tYnaTIcMbgrbs;FwmKWRTbnkmkBIkRmalxN nigxnvapal;.
- ssr ( column ) CaeRKOgbgMdsMxan; EdlmantYnaTIRTbnkEdlmkBIFwm bkRmalxN. vaGac
RbQmnwgkmaMgcMpit nigkmaMgcMpitrYmbBalm:Um:g;.
- eRKag ( frame ) CaeRKOgbgMEdlpSMeLIgedayman Fwm nigssr bpSMeLIgedayman kRmalxN Fwm
nigssr. vaGacCaeRKagsaTickMNt; beRKagsaTicminkMNt;.
- RKwH ( footing ) rYmman eCIgtag bRKwHCab; EdlRTssr nigbBanbnkeTAdIedaypal;.
- CBaaMg ( wall ) CaeRKOgbgMbnHbBar EdlRbQmnwgkmaMgTMnaj kdUcCakmaMgedk kgkrNIdUcCa
CBaaMgeRkamdI.

T.Chhay

10

Introduction

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

1>7> CMhannkarKNnaeRKOgbgMBIebtugGarem:
eRkayeBlEdlbg;sabtkm RtUv)anseRmcTTYlyk visVkrcab;epImkMNt;nUvRbBnrcnasm<ndl
mYy edayFananUvsuvtiPaB nigsirPaBrbs;GKar. KMnitnkarbegItKMrUrcnasm<nepSg kRtUv)anykmk
BicarNa edIm,ITTYl)anlkNesdkicx<s; edayQrelImUldansmar niglkxNdI. CaTUeTA lTpl
TaMgenHTTYl)anBI
- begItCaKMrUrcnasm<neRKOgbgMEdlRTRTg;bnk
- kMNt;nUvRbePTepSgKanbnkEdlmanGMeBIelIGKar
- viPaKrcnasm<neRKOgbgMedayeRbIkMuBTr bedayd edIm,IkMNt;nUvm:Um:g;Gtibrma kmaMgkat;TTwg
kmaMgrmYl kmaMgtamGkS nigkmaMgepSgeTot
- kMNt;smamaRtneRKOgbgM nigkMNt;nUvbrimaNEdkEdlRtUvkar
- begItnUvbg;rcnasm<n nigkarlMGitlkNbeckeTsedIm,IeFVIkarsagsg;

esckIepIm

11

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

II.

lkNnebtugGarem:

2>1> ktaCH\TiBldl;ersIusg;ebtug
CaTUeTAebtugCasmar EdlekItBIkarrYmpSMeLIgeday fbMEbk xSac; sIum:gt nigTwk EtkgkrNIxH Rb
ePTepSgnTwkfaMKImIkRtUv)anbEnmedIm,IehtuplepSg. smarTaMgenH)anlaypSMKa rYcRtUv)ancak;cUl
kgBum< Tuk[kkrwgedIm,I[ersIusg;rbs;vaekIneLIgCabnbnab;. ersIusg;rbs;ebtugvaGaRsyeTAelIktaCa
eRcIn ehIyvaGacERbRbYlsRmab;karplitdUcKa. ktacMbgsMxan;EdlCH\TiBldl;ersIusg;ebtugman
- pleFobTwkelIsIum:gt (water-cement ratio) vaCaktaEdlsMxan;CageK EdlCH\TiBleTAelIersIusg;rbs;ebtug. edIm,IseRmcnUvGIuRdakmrbs;sIum:gt pleFobTwkelIsIum:gt EdlRtUvkarcM)ac;RtUvesInwg 0.25
KitCam:as;;. pleFobTwkelIsIum:gt EdlmantmFMCag besInwg 0.35 edayKanlayTwkfaMKImI RtUv)aneRbI
sRmab;ebtugedIm,ImUlehtugayRsYleFVIkar. enAeBlEdlpleFobTwkelIsIum:gtmantmkan;EtFM enaHersIusg;ebs;ebtugkan;EtmantmtUc.
- lkN nigsmamaRtnsmasPaBnebtug ebtugCakarrYmpSMeLIgeday fbMEbk xSac; sIum:gt nig
Twk. karekIneLIgnUvbrimaNsIum:gt nigkareRbIR)as;nUvfbMEbkEdlmanlkNksNan eFVI[ersIusg;ebtug
ekIneLIg. TwkfaMKImIiBiesskRtUv)anbEnmedIm,IplitnUvebtugEdlmanKuNPaB nigersIusg;tamtRmUvkar.
- viFIlay nigkarEfTaM kareRbIR)as;nUvm:asIunlayebtug nigryeBlRtwmRtUvkgkarlaysuTEtCYy
sRmalnUv\TiBlEdlb:HBal;dl;ersIusg;ebtug. dUcKa kareRbIR)as;nUvm:asIunbgab;ebtug (vibrator) edIm,IeFVI
[ebtughab;ENn CYy[PaKryrnes<atkgebtugxiteTArktmGb,brma. pleFobrnes<at 5% eFVI[
ersIusg;ebtugFak;cuH 30% . karEfTaMebtugkCaktamYydsMxan;Edlman\TiBlelIersIusg;rbs;ebtugEdr.
TaMgsMeNIm nigsItuNPaBsuTEtman\TiBledaypal;eTAelIGIuRdatkmrbs;sIum:gt. ryeBlnkarmansM
eNImkan;EtEvg ersIusg;kan;EtekIneLIg. RbsinebIsItuNPaBkgGMLgeBlnkarEfTaMekInx<s;CagsItuNPaBkgGMLgeBlnkarcak;ebtug CalTplebtugmanersIusg;tamtRmUvkarmun 28f .
- Gayuebtug ersIusg;rbs;ebtugekIneLIgtamGayurbs;va ehIyGIuRdakmrbs;sIum:gtenAEtbnrab;Ex.
kgkarGnuvt ersIusg;ebtugRtUv)ankMNt;edaykareFVIBiesaFn_eTAelIsMNakKMrUragsIuLaMg bKUb enAGayu 7f
nig28f. CaTUeTA ersIusg;rbs;ebtugenAGayu 28fesInwg 1.3dgeTA 1.7dgnersIusg;ebtugenAGayu 7f .
sRmab;sIum:gtBrElnFmta ersIusg;ekIneLIgtameBledayeFobnwgersIusg;enAGayu 28fRtUv)anbgajkg
taragxageRkam

T.Chhay

12

Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Gayu
pleFobersIusg;

Department of Civil Engineering

0.67

14

28

0.86

Ex

1.17

Ex

1.23

qaM

1.27

qaM

1.31

qaM

1.35

- lkxNnkardak;bnk ersIusg;ebtugrgkarsgt; RtUv)ankMNt;edaykarBiesaFkarsgt;bMEbksM

NakKMrUragsIuLaMg bKUbkgryeBlBIr bInaTI. eRkamGMeBInbnkGciny_rab;qaM ersIusg;ebtugRtUv)ankat;
bnyRbEhl 30% . eRkamGMeBInbnkGciny_ 1fersIusg;ebtug)at;bg;RbEhl 10% . kgkrNIEdl
rcnasm<nrgnUvbnkdEdl GMeBIDINamic nigGMeBIb:HTgic vaRtUv)aneKykmkBicarNakgkarKNnaeRKOgbgM
ebtugGarem:.
- rUbrag nigxatrbs;sMNakKMrU xatTUeTArbs;sMNakKMrUebtugEdleKeRbIsRmab;TsSn_TaynUversIusg;
ebtugGarem:KW ragsIuLaMgEdlmanGgt;pit 15cm nigkm<s; 30cm nigragKUbEdlmanRTnug 15cm . enAeBl
EdleKyksMNakKMrUragsIuLaMgmkeFVIBiesaFn_EdlmanxatepSgKa eKsegteXIjfaxatkan;EtFM ersIusg;kan;
EtFak;. xageRkamCataragbgajBIersIusg;rbs;sMNakKMrUebtugbTdanEdlmankm<s;esInwgBIrdgGgt;pit
eFobnwgxat 15cm 30cm .
xatsIuLaMg cm

ersIusg;eFob

5 10
10 20
15 30
20 40
30 60
45 90
60 120
90 180

1.09
1.06
1.00
0.96
0.91
0.86
0.84
0.82

eBlxHsMNakKMrUragsIuLaMgbTdanminRtUv)anykmkeFIVkarBiesaF. pleFobkm<s;elIGgt;pitrbs;
sMNakKMrUkan;EtFM ersIusg;kan;EtTabtamkarkt;sMKal;BIkareFVIBiesaFn_. edIm,ITTYl)annUversIusg;smmUl
eTAnwgersIusg;sMNakKMrUbTdaneKRtUvKuNnwgemKuNkMEn EdlmanbgajkgtaragxageRkam
pleFobkm<s;elIGgt;pit
emKuNkMEnersIusg;
ersIusg;eFobnwgsMNakKMrUbTdan
lkNnebtugGarem:

1.75

1.5

1.25

1.1

0.75

0.5

1.00

0.98

0.96

0.93

0.90

0.87

0.7

0.5

1.00

1.02

1.04

1.06

1.11

1.18

1.43

2.00

13

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

TMnak;TMngersIusg;nsMNakKMrUragsIuLaMg nigragKUbsRmab;ersIusg;rgkarsgt;epSg bgajkg

taragxageRkam
ersIusg;rgkarsgt;
N / mm

15.5

20

24.5

27

34.5

37

41.5

45

51.5

0.77

0.76

0.81

0.87

0.91

0.93

0.94

0.95

0.96

0.96

pleFobersIusg;
sMNaksIuLaMgelIsMNakKUb
CaTUeTAeKsnt;

f 'c (cylinder ) = 0.85 f 'c (cube) = 1.1 f 'c ( prisme)

2>2> ersIusg;rgkarsgt;
kgkarKNna eK)ansnt;faebtugCasarFatuEdlrgnUvkugRtaMgsgt; minEmnkugRtaMgTaj. dUecH
ersIusg;rgkarsgt;CalkNvinicy KuNPaBrbs;ebtug. sMNakKMrUsRmab;ykmkeFVIkarBiesaFmanTRmg;Ca
sIuLaMg KUb nigRBIs.
sMNakKMrUEdlmanTRmg; KUbEdlmanRTnug 15cm b 20cm RtUv)aneKeRbIenAcRkPBGg;eKs
GaLWm:g; nigEpkxHrbs;GWr:ub.
sMNakKMrURBIs RtUv)aneRbIenARbeTs)araMg rusSI nigRbeTsdTeTot ehIyCaTUeTAmanmuxkat;
7 7 30 b 10 10 50 edayeKykvaeTAsgt;epk.
muneKykvaeTABiesaF sMNakKMrURtUv)aneKEfTaMedaysMeNIm nigykeTABiesaFenAGayu 28 feday
GnuvtbnkekIneLIgrhUtdl;vaEbk. karEbkrbs;ebtugGacekIteLIgedaysar kugRtaMgTaj kugRtaMgkat;
kugRtaMgsgt; bbnSMnkugRtaMgTaMgenH.

T.Chhay

14

Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

karEbkrbs;sMNakKMrUebtugmanbIy:agdUcxageRkam
a. eRkamGMeBInkmaMgsgt;tamGkS sMNakKMrUGacEbkedaykmaMgkat;
b. sMNakKMrU)anEbkedaymanlkNCaCYr dUcCaeRcok. karEbkEbbenH ekIteLIgEtcMeBaHebtug
EdlmanersIusg;x<s;
c. ekIteLIgedaysarbnSMnkmaMgkat; nigkmaMgeRcok.
2>3> ersIusg;rgkarTaj
ebtugCasmarRsYy EdlvamanminGacRbQmnwgkugRtaMgTaj)an ehIyvaRtUv)aneKykmkBicarNa
sRmab;bBaaeRbH kmaMgkat; nigkmaMgrmYl.
karBiesaFedaypal;elIersIusg;rgkarTajminGaceFVIeTA)an edaykartRmg;min)anl nigbBaa
]bkrN_sRmab;cab;. dUecHkarBiesaFedayminpal; edIm,IQaneTAkMNt;ersIusg;rgkarTajman karBiesaF
edayeRcok nigkarBiesaFedaykarkac;bM)ak;.
karBiesaFedayeRcok RtUv)aneFVIelIsMNakKMrUragsIuLaMgbTdan 15 30cm . sMNakKMrURtUv)andak;
epk rYceRbIkmaMgsgt;elIkaRTnab;rhUtdl;eRcokEbkCaBIrbMENk.
karBiesaFedaykac;bM)ak; RtUv)aneFVIeLIgelIsMNakKMrUragRBIs 15 15 70cm dak;KgelITRmEdl
mancmay 60cm BIKa ehIyeKeRbIbnkedIm,Ikac;bM)ak;BIr Edlmancmay 10cm BIGkS.
CaTUeTAersIusg;rgkarTajmantmesInwg 10% nersIusg;rgkarsgt; mannyfaersIusg;rgkarTaj
tUcCagersIusg;rgkarsgt; 10 dg.
2>4> ersIusg;rgkarkat;
kmaMgkat;suTkRmnwgCYbRbTHenAkgeRKOgbgMebtugGarem:Nas; eRBaHCaTUeTAvaEtgEtekIteLIgeRkam
GMeBInkmaMgtamGkS. eRKOgbgMEdlrgnUvkmaMgkat;suTKW)ak;CaBIrcMENk. dUecHeRKOgbgMebtugRtUvEtrwg
RKb;RKan;edIm,ITb;nwgkmaMgkat;.
ersIusg;rgkarkat;RtUv)anKitedaymantmFMCag 20% 30% nersIusg;rgkarTaj besInwg 12%
nersIusg;rgkarsgt;.

lkNnebtugGarem:

15

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

2>5> m:UDuleGLasicrbs;ebtug
m:UDuleGLasic CalkNemkanicdsMxan;mYyrbs;ebtug Edl)anBIkarBiesaFkarsgt;elIsMNakKMrU
ragsIuLaMg. m:UDuleGLasic)anBIpleFobrvagkugRtaMg nigbERmbRmYlrageFob.
Ec =

unit stress
unit strain

xageRkamCarUbmnsamBasRmab;karKNna m:UDuleGLasicrbs;ebtug
Ec = 0.043w1.5 f 'c

Edl

- m:as;maDrbs;ebtug
f ' - ersIusg;rgkarsgt;sMNakKMrUsIuLaMg
kgkrNI w = 2320kg / m eK)an
w

Ec = 4700 f 'c

2>6> emKuNBrsug bpleFobBrsug

emKuNBrsug CapleFobrvagbERmbRmYlragtamTTwg nigbERmbRmYltambeNay. emKuNenH
ERbRbYlcenaHBI 0.15 0.20 . emKuNBrsugRtUv)aneKeRbIsRmab;rcnasm<nsaTicminkMNt;. CaTUeTA
sRmab;ebtug eK)ankMNt;ykemKuNBrsugmFmEdlmantmesI 0.18 .
2>7> m:UDulnPaBrwg bm:UDulkmaMgkat;
m:UDulnPaBrwgrbs;ebtugRtUv)ankMNt;edaysmIkarxageRkam
Gc =

Ec
2(1 + )

RbsinebI = 16 G

= 0.43Ec

2>8> pleFobm:UDul
pleFobm:UDul n CapleFobrvagm:UDuleGLasicrbs;Edk nigm:UDuleGLasicrbs;ebtug.
n=

Es
Ec

edaym:UDuleGLasicrbs;Edkmantmefr E = 200000MPa
ehIym:UDuleGLasicrbs;ebtugGaRsynwgersIusg;rgkarsgt; E
42
n=
Edl f ' KitCa MPa .
f'
s

= 4700 f 'c

T.Chhay

16

Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

2>9> bERmbRmYlmaDrbs;ebtug
ebtugTTYlnUvkarERbRbYlmaDkgxNeBlkkrwg. RbsinebIva)at;sMeNImedaysarrMhYt varYmmaD.
b:uEnRbsinebIvakkrwgkgTwk vanwgrIkmaD. ktankarERbRbYlmaDebtug naM[ekItmankarERbRbYlsMeNIm
RbtikmKImIrvagsIum:gt nigTwk bERmbRmYlsItuNPaB nigrgnUvbnk.
2>9>1> karrYmmaD
bERmbRmYlmaDebtugEdlrwg minesInwgmaDTwkEdl)an)at;bg;enaHeT. rMhYtnTwkesrIbegItnUvkarrYm
maDdtictYcbMputesIrEtKan. edaysarEtebtugbnkarkkrwg TwkRtUv)anhYt ehIymaDnTwksIum:gtERbRbYl
bNal[ebtugrYmmaD.
ktaCaeRcInEdlCH\TiBldl;karrYmmaDrbs;ebtug bNalmkBIbERmbRmYllkxNsMeNIm
- brimaNsIum:gt nigTwk enAkgsmamaRtnkarlayebtug kalNabrimaNsIum:gt nigTwkkan;Et
eRcIn karrYmmaDkan;EtxaMg.
- smasPaB nigPaBm:g;pg;rbs;sIum:gt sIum:gtersIusg;eLIgx<s;qab; nigsIum:gtkemAtic mankarrYm
maDxaMgCagsIum:gtBrElnFmta. sIum:gtEdlmanlkNm:t;pg;Cag rIkmaDCageRkamlkxN
sMeNIm.
- RbePT brimaN nigcMNat;fak;fbMEbk TMhMnfbMEbkkan;EttUc rYmmaDkan;EtxaMg. brimaNfbM
Ebkkan;EteRcIn rYmmaDkan;Etkat;bny.
- TwkfaMKImI TwkfaMKImIEdlCYy[ebtugkan;Etrav eFVI[ebtugkan;EtrYmmaD.
- brimaNEdkkgebtug edaysarkarrYmmaDekItelIeRKOgbgMebtugGarem: kugRtaMgTaj)anekItelI
ebtug ehIymantmesIKanwgkugRtaMgsgt;EdlekItmanelIEdk. kugRtaMgTaMgenHRtUv)anbEnm
ehIykayeTACabnk. dUecH sameRbHGacekItmanenAkgebtug enAeBlNaEdleKeRbIPaKry
EdkeRcIn. karBRgayEdk)anl naM[mankarEbgEckkugRtaMgTajkgebtug)anl CYykat;
bnybERmbRmYlkugRtaMgxagkg.
CaTUeTA ebtugrYmmaDxaMgkgGMLgeBlnkarcab;epImkarkkrwgdMbUg b:uEneRkaymkkarrYmmaDenH
RtUv)ankat;bny. tmnkarrYmmaDsrubsRmab;ebtugFmtaERbRbYlBI 200 10 700 10 . RbsinebI
eRKOgbgMmanTRmbgb;sgag kugRtaMgTajekIneLIgRbEhl 10MPa . RbsinebIebtugRtUv)anEfrkSasMeNIm
Cab;lab;kgryeBlmYyc,as;las;eRkayeBlcak; karrYmmaDRtUv)ankat;bny. dUecHvaCakarsMxan;Nas;
EdleKRtUvkarEfTaMebtugsRmab;ryeBlmYymin[ticCag 7 f.
6

lkNnebtugGarem:

17

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

2>9>2> karrIkmaDedaykarekIneLIgnkemA
ebtugrIkmaDCamYynwgkarekIneLIgnkemA ehIyrYmmaDCamYynwgkarfycuHnkemA. emKuNlUt
mannyfaeRKOgbgMmYymanRbEvg 30m lUt)an 10mm
kemArbs;ebtugmantmRbEhl = 10
sRmab;bERmbRmYlkemA 33 c . RbsinebIeRKOgbgMEdlmanTRmsgag ehIymin)anBRgwgedayEdk kugRtaMg
RbEhl 7MPa GacekIteLIg.
sRmab;rcnasm<nebtugGarem:EdlmanRbEvgEvg eKRtUveFVItMNsRmab;karrIkmaD expansion joint
ral;RbEvg 30m 60m mYy. tMNenHmanTMhMRbEhl 25mm .
5 mm

2>10>

Creep

KWCakMhUcRTg;RTayrbs;ebtugeRkamvtmannbnkGciny_. kMhUcRTg;RTayenHekIneLIgeTA
tamryeBlnkardak;bnk.
ebtugCasmareGLas)asic. edaycab;epImCamYynwgkugRtaMgtUc ebtugekItmanbERmbRmYlrag
eFob)asicbEnmBIelIbERmbRmYlrageFobeGLasic.
rUb 2>5 bgajBIsIuLaMgebtugrgbnk. kMhUcRTg;RTayxNKW EdlesInwgpleFobkugRtaMgelIm:U
DuleGLasic. RbsinebIkugRtaMgenHenArkSaGMLg ryeBlmYy vanwgekItmanbERmbRmYlrageFobbEnm
EdlbNalBI creep. RbsinebIeKdkbnkenHecj bERmbRmYlrageFobeGLasic nwgRtLb;eTArkPaB
edImvij ehIybERmbRmYlrageFobedaysar creep xHkRtLb;eTArkPaBedImEdr EtvaenArkSabERmbRmYlrag
eFob)asicGciny_ dUcbgajkgrUb. enAkgkrNIenH = (1 ) Edl CaGRtankarRtLb;eTArk
sPaBedImvijrbs;bERmbRmYlrageFobedaysar creep srub. mantmcenaH 0.1 nig 0.2 .
ktaEdlCH\TiBldl; creep rbs;ebtugrYmman
- kRmitkugRtaMg creep ekIneLIgCamYynwgkarekIneLIgnkugRtaMg.
- ryeBlnkardak;bnk creep ekIneLIgCamYynwgryeBlnkardak;bnk. RbEhlCa 80% n
creep ekItmankgGMLgeBlbYnExdMbUg ehIy 90% ekIteLIgeRkayryeBlRbEhlBIrqaM.
- ersIusg; nigGayurbs;ebtug creep nwgmantmtUcCagRbsinebIvargbnkenAGayucas;.
- lkxNbriyakas creep RtUv)ankat;bnyCamYynwgkarekIneLIgnsMeNIm.
- kRmitnkardak;bnk creep ekIneLIgCamYynwgkMeNInnkRmitnkardak;bnk.
- PaKryEdk nigkarBRgaysrsEdkenAkgGgt;ebtugGarem: creep nigmantmtUcCagsRmab;Ggt;
EdlmanPaKryEdkx<s; nigmankarBRgaysrsEdk)anl.
- TMhMnm:as;ebtug creep fycuHCamYynwgkarekIneLIgnTMhMrbs;sMNakKMrU.
Creep

T.Chhay

18

Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

- RbePT PaBm:t;pg; nigbrimaNsIum:gt brimaNsIum:gtCH\TiBly:agxaMgdl; creep cugeRkayrbs;

ebtug. Creep edaysarsIum:gtmantmFMCag creep rbs;ebtug 15 dg.
- pleFobTwkelIsIum:gt creep ekIneLIgCamYynwgkarekIneLIgnpleFobTwkelIsIum:gt.
- RbePT nigcMNat;fak;fbMEbk fbMEbkEdlmanTMhMdUcKanwgeFVI[ebtugENnl ehIyvanwgkat;bny
creep.
- RbePTnkarEfTaMebtug karEfTaMebtugedaycMhayTwkEdlmansItuNPaBx<s; kdUcCakareRbIR)as;
nUvTwkfaMKImIRbePT plasticizer nwgkat;bnybrimaN creep.
Creep minRtwmEtekItmanenAkgGgt;rgkarsgt;b:ueNaHeT b:uEnvakekItmanenAkgGgt;rgkarTaj
Ggt;rgkarBt; nigGgt;rgkarrmYl. pleFobn creep enAkgGgt;rgkarTajelI creep enAkgGgt;rgkar
sgt;FMCagmYysRmab;BIrs)ah_dMbUg b:uEnpleFobenHnwgfycuHsRmab;ryeBlyUr.
2>11> m:UEDlsRmab;TsSn_TaykarrYmmaD nig creep rbs;ebtug
2>11>1> m:UEDl ACI 209
viTasanebtugGaem:ricENnaMm:UEDl ACI 209. m:UEDlenHRtUv)anbegIteLIgdMbUgenAkgqaM 1970.
m:UEDlenHRtUv)aneRbICaeRcInqaMkgkarDIsajrcnasm<nebtug. m:UEDlenHmanlkNsamBakgkareRbI Etva
minsUvsuRkit.

karKNnakarrYmmaD

eKGacKNnakarrYmmaDEdleRbIm:UEDl ACI 209 RbsinebIeKsal;)a:ra:Em:Rt niglkxNdUcteTA viFI

EfTaM ebtugEfTaMedaysMeNIm bedaycMhayTwk relative humidity H RbePTsIum:gt rUbragsMNakKMrU
bERmbRmYlrageFobedaykarrYmmaDcugeRkay Gayurbs;ebtugeRkayeBlcak; t Gayurbs;ebtugEdl
cab;epImst CaTUeTAeKykGayuenAcugbBab;nkarEfTaM t .
bERmbRmYlrageFobedaysarkarrYmmaDRtUv)ankMNt;dUcxageRkam
shu

s (t ) =

Edl

(t tc ) K K
ss sh shu
b + (t tc )

Gayurbs;ebtugeRkayeBlcak; f
t = Gayurbs;ebtugEdlcab;epImst f
b = tmefrkgkarkMNt;bERmbRmYlrageFobedaysarkarrYmmaD GaRsynwgviFIEfTaMebtug b = 35
sRmab;ebtugEdlEfTaMedaysMeNIm nig b = 55 sRmab;ebtugEdlEfTaMedaycMhayTwk
K = emKuNkMENrUbrag nigTMhMsRmab;karrYmmaD
t=
c

ss

lkNnebtugGarem:

19

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

emKuNkMEN relative humidity sRmab;karrYmmaD

bERmbRmYlrageFobedaysarkarrYmmaDcugeRkay = 780 10 (mm / mm ) sRmab;ebtug
EdlEfTaMedaysMeNIm nigedaycMhayTwk
eKGackMNt;emKuNkMENrUbrag nigTMhMsRmab;karrYmmaDdUcxageRkam
K sh =

shu

V
K ss = 1.14 0.0035
S

maDrbs;sMNakKMrU (mm )
S = RkLaprbs;sMNakKMrU (mm )
emKuNkMEN relative humidity sRmab;karrYmmaDKW
1.40 0.01H sRmab; 40% H 80%
K =
3.00 0.03H sRmab; 81% H 100%
Edl H = relative humidity KitCa %

Edl

V=

sh

karKNna creep

eKGacKNnabERmbRmYlrageFobGaRsybnksrub
xN t CamYynwgkugRtaMg efr tamrUbmnxageRkam

ic

(t , t0 )

enAxN t rbs;ebtugEdlrgbnkenA

ic (t , t0 ) = i (t ) + c (t , t0 )

bERmbRmYleGLasicedImenAeBlrgbnk
(t , t ) = bERmbRmYledaysar creep enAxN t t

Edl

i (t0 ) =
c

i (t0 ) =

Ecmt 0

c (t , t0 ) =

Edl

Ecmt 0

Cc (t )

m:UDuleGLasicenAGayunkardak;bnk KitCa MPa

C (t ) = emKuN creep enAxN t
CaTUeTA bERmbRmYlGaRsybnksrubRtUv)anbgajedayCab;Tak;TgnwgGnuKmn_ creep J (t, t ) EdltMNag
[bERmbRmYlrageFobGaRsybnksrubenAxN t EdlekIteLIgedaykugRtaMgefrktaEdleFVIGMeBItaMgBI
xN t .
Ecmt 0 =
c

J (t , t0 ) =

1 + Cc (t )
Ecmt 0

Ecmt 0 = 0.043( )

3/ 2

Edl
T.Chhay

f 'c (t0 )

Tmn;maDebtug (kg / m )
3

20

Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca
f 'c (t0 ) =

ersIusg;rgkarsgt;rbs;ebtugmFmenAGayuEdlrgbnk

f 'c (t0 ) = f cm 28

Edl

Department of Civil Engineering

t0
b + ct0

ersIusg;rgkarsgt;rbs;ebtugenAGayu 28 f MPa
b nig c CatmefrEdlmanbgajenAkgtaragxageRkam.
f cm 28 =

RbePTsIum:gt

ebtugEdlEfTaMedaysMeNIm

b=4

c = 0.85

b =1

c = 0.95

II

b = 2.30

c = 0.92

b = 0.70

c = 0.98

eKGackMNt;emKuN creep
Cc (t ) =

Edl

ebtugEdlEfTaMedaycMhayTwk

Cc (t )

dUcxageRkam

t 0.60
Ccu K ch K ca K cs
10 + t 0.60

emKuN creep cugeRkay = 2.35

= emKuNkMEN relative humidity sRmab; creep
= emKuNkMENGayuenAeBlrgbnk
= emKuNkMENrUbrag nigTMhMsRmab; creep

Ccu =
K ch
K ca
K cs

emKuNkMEN
viFItMEhTaM
EfTaMedaysMeNIm
EfTaMedaycMhayTwk

t0

K ca

K ch

K cs

40%

N/A

N/A

N/A

40%

1.25(t0 )

1.27 0.0067 H

1.14 0.0035(V / S )

40%

1.13(t0 )

1.27 0.0067 H

1.14 0.0035(V / S )

40%

N/A

N/A

N/A

0.118

0.095

2>11>2> m:UEDl B3

karKNnakarrYmmaD
)a:ra:Em:RtEdlRtUvkarsRmab;KNnabERmbRmYlrageFobedaysarkarrYmmaDEdleRbIm:UEDl B3 KW
ersIusg;rgkarsgt;mFmrbs;ebtugenAGayu 28f lkxNtMEhTaM RbePTsIum:gt relative humidity
brimaNTwkkgebtug nigrUbragrbs;sMNakKMrU.
lkNnebtugGarem:

21

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

eKGac)a:n;sanbERmbRmYlrageFobedaykarrYmmaDedayeRbIsmIkarxageRkam
s (t ) = ( shu )(K h )S (t )

bERmbRmYlrageFobedaykarrYmmaDcugeRkay
K = relative humidity sRmab;karrYmmaD
S (t ) = GnuKmn_eBlsRmab;karrYmmaD
eKGacKNnabERmbRmYlrageFobedaykarrYmmaDedayeRbIsmIkarxageRkam
= [0.019(w) ( f ) + 270] 10
Edl = emKuNkMENRbePTsIum:gt
= emKuNkMENlkxNtMEhTaM
w = brimaNTwk kg / m
f
= ersIusg;rgkarsgt;rbs;ebtugenAGayu 28 f
emKuNkMEN CaGnuKmn_eTAnwgRbePTsIum:gt
RbePTsIum:gt
Edl

shu =
h

0.28

2.1

shu

cm 28

cm 28

1.00

II

0.85

III

1.10

emKuNkMEN CaGnuKmn_eTAnwgRbePTntMEhTaM
RbePTntMEhTaM
EfTaMedaysMeNIm
EfTaMedayTwk b H = 100%
RKbkgGMLgtMEhTaM
2

GnuKmn_sMeNImsRmab;karrYmmaD
sMeNIm

0.75
0.85

1.10

Kh

1 (H / 100)

H 98%

H = 100%

0 .2

eFVIGaMgETb:ULasglIeNEGr

98% H 100%

T.Chhay

22

Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

eKGackMNt;GnuKmn_eBlsRmab;karrYmmaD S (t ) tamsmIkarxageRkam
t tc
Tsh

S (t ) = tanh

Edl

GayuebtugeRkayeBlcak; f
t = GayuebtugEdlcab;epImst f
T = karrYmmaDBak;kNal (shrinkage half-time)
t=
c

sh

Tsh = 0.085(tc )

0.08

( f cm 28 )0.25 [2 K s (V / S )]2

Edl K = emKuNkMENrUbragmuxkat;. eKsnt; K = 1 RbsinebIRbePTGgt;minRtUv)ankMNt;. tarag

xageRkambgajBItmrbs; K .
emKuNkMEN K CaGnuKmn_eTAnwgrUbragmuxkat;
rUbragmuxkat;
K
kRmal
1.00
sIuLaMg
1.15
RBIskaer
1.25
EsVr
1.30
KUb
1.55
s

karKNna creep
GnuKmn_ creep RtUv)an[edaysmIkarxageRkam
J (t , t0 ) = q1 + C0 (t , t0 ) + Cd (t , t0 , tc )

Edl

bERmbRmYlrageFobxN
C (t , t ) = GnuKmn_sRmab; creep EdlpSMeLIgeday aging viscoelastic, nonaging viscoelastic nig
aging flow.
C (t , t , t ) = GnuKmn_sRmab; drying creep.
q1 =
0

q1 =

Edl

0.6
Ecm 28

Ecm 28 =

m:UDuleGLasicrbs;ebtugenAGayu 28 f

Ecm 28 = 4735 f cm 28

eKGackMNt;GnuKmn_ C (t, t ) dUcxageRkam

0

lkNnebtugGarem:

23

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

C0 (t , t0 ) = q2Q(t , t0 ) + q3 ln 1 + (t t0 )

Edl

q2 =

0.1

]+ q ln tt
4

)a:ra:Em:Rt aging viscoelastic

Q(t , t0 ) = binomial integral

)a:ra:Em:Rt nonaging viscoelastic

q = )a:ra:Em:Rt aging flow
t = Gayurbs;ebtugenAeBlrgbnk f
q3 =
4

q2 = 185.4(c )

0. 5

Edl
Edl

( f cm 28 )0.9 106

brimaNsIum:gt kg / m
3

1 / r (t 0 )

Q (t )r (t 0 )
Q(t , t0 ) = Q f (t0 )1 + f 0 r (t 0 )
Z (t , t0 )
1
Q f (t0 ) =
2/9
4/9
0.086(t0 ) + 1.21(t0 )

ln 1 + (t t0 )
Z (t , t0 ) =
t0

0.1

r (t0 ) = 1.7(t0 )

+8

0.12

w
q3 = 0.29q2
c
a
q4 = 20.3
c

0.7

10 6

eKGacKNnaGnuKmn_sRmab; drying creep dUcxageRkam

Cd (t , t0 , tc ) = q5 e 8 H (t ) e 8 H (t 0 )

Edl

q5 =
q5 =

Edl

)a:ra:Em:Rt drying creep

0.757 shu 106

shu =

0.6

f cm 28

bERmbRmYlrageFobedaykarrYmmaDcugeRkay

H
H (t ) = 1 1
S (t )
100

H
H (t0 ) = 1 1
S (t0 )
100

S (t ) = tanh

S (t0 ) = tanh

T.Chhay

t tc
Tsh

t0 tc
Tsh

24

Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca
Tsh = 0.085(tc )

0.08

Department of Civil Engineering

( f cm 28 )0.25 [2 K s (V / S )]2

2>11>3> m:UEDl GL 2000

karKNnakarrYmmaD
)a:ra:Em:RtEdlRtUvkarsRmab;KNnabERmbRmYlrageFobedaysarkarrYmmaDEdleRbIm:UEDl GL 2000
KWersIusg;rgkarsgt;mFmrbs;ebtugenAGayu 28f f relative humidity H Gayurbs;ebtugenAeBl
cab;epImrYmmaD t RbePTsIum:gt nigrUbragrbs;sMNakKMrU.
eKGacKNnabERmbRmYlrageFobedaykarrYmmaDedayeRbIsmIkarxageRkam
cm 28

s (t ) = shu (h ) (t )

Edl

bERmbRmYlrageFobedaykarrYmmaDcugeRkay
(h ) = emKuNkMENsRmab;\TiBlrbs;sMeNIm
(t ) = emKuNkMENsRmab;\TiBlneBl
eKGackMNt; edaysmIkarxageRkam
shu =

shu

1/ 2

30

f cm 28

shu = (900)K

Edl

K=

10 6

tmefrEdlGaRsynwgRbePTsIum:gt
= ersIusg;rgkarsgt;mFmrbs;ebtugenAGayu 28 f

f cm 28

tmefr K EdlCaGnuKmn_eTAnwgRbePTsIum:gt
RbePTsIum:gt

I
1.00

II
0.75

III

1.15

eKGackMNt;emKuNkMEN (h) dUcxageRkam

H
(h ) = 1 1.18

100

lkNnebtugGarem:

25

T.Chhay

mhaviTalysMNg;sIuvil

Edl

NPIC

eKGacKNnaemKuNkMEN (t ) tamrUbmnxageRkam
H = relative humidity

1/ 2

t tc

(t ) =
2
(
)
t

t
+
0
.
12
V
/
S
c

GayuebtugeRkayeBlcak; f
t = Gayurbs;ebtugenAeBlcab;epImrYmmaD
V / S = pleFobmaDelIRkLap mm

Edl

t=
c

karKNna creep
GnuKmn_ creep pSMeLIgedayBIrEpkKW bERmbRmYlrageGLasic nigbERmbRmYlrageday creep.
J (t , t0 ) =

Edl

1
Ecmt 0

(t , t0 )
Ecm 28

m:UDuleGLasicrbs;ebtugenAeBldak;bnk
E
= m:UDuleGLasicrbs;ebtugenAGayu 28 f
(t ,t ) = emKuN creep
Ecmt 0 =
cm 28

Ecmt 0 = 3500 + 4300 f cmt 0

Edl

f cmt0 =

ersIusg;rgkarsgt;Fmmrbs;ebtugenAeBlrgbnk. eKGackMNt;vadUcxageRkam

f cmt 0 = f cm 28

t3/ 4
a + bt 3 / 4

emKuN a nig b Tak;TgeTAnwgRbePTsIum:gtdUcbgajkgtaragxageRkam.

emKuN a nig b EdlCaGnuKmn_eTAnwgRbePTsIum:gt
RbePTsIum:gt
I
II
III

a
2.8
3.4
1.0

b
0.77
0.72
0.92

eKGacKNnaemKuN creep (t, t ) dUcxageRkam

0

(t t )0.3 7 0.5 t t 0.5

0
0
+
+ 2.5 1 1.086h 2
(t , t0 ) = (tc )2
0.3

(t t0 ) + 14 t0 t t0 + 7

RbsinebI t
T.Chhay

= tc

t t0

2
t t + 0.12(V / S )
0

enaH (t ) = 1
c

26

Properties of Reinforced Concrete

0. 5

viTasanCatiBhubeckeTskm<Ca

RbsinebI t

Department of Civil Engineering

enaH

> tc

0. 5


t0 tc

(tc ) = 1
2
t0 tc + 0.12(V / S )

0.5

H = relative humidity

h = H / 100

2>11>4> m:UEDl CEB 90

karKNnakarrYmmaD
)a:ra:Em:RtEdlRtUvkarsRmab;KNnabERmbRmYlrageFobedaysarkarrYmmaDEdleRbIm:UEDl CEB 90
KWersIusg;rgkarsgt;mFmrbs;ebtugenAGayu 28f f relative humidity H Gayurbs;ebtugenAeBl
cab;epImrYmmaD t RbePTsIum:gt nigrUbragrbs;sMNakKMrU.
eKGacKNnabERmbRmYlrageFobedaykarrYmmaDedayeRbIsmIkarxageRkam
cm 28

s (t , tc ) = cs s (t , tc )
0

Edl

emKuN notional shrinkage

(t , t ) = emKuNEdlBNnaBIkarekItmankarrYmmaDCamYynwgeBl
emKuN notional shrinkage KW
cs =
0

cs = s ( f cm 28 ) RH
0

Edl

s ( f cm 28 ) =

emKuNersIusg;ebtugnkarrYmmaD

s ( f cm 28 ) = 160 + 10( sc ) 9

RH =

f cm 28
6
10
10

emKuN relative humidity nemKuN notional shrinkage EdlmanbgajenAkgtaragxag

eRkam.
emKuN
sMeNIm

RH

RH

40% H < 90%

1.55 arh

H 99%
0.25

sc =
f cm 28

emKuNEdlGaRsynwgRbePTsIum:gt EdlmanbgajenAkgtaragxageRkam
= ersIusg;rgkarsgt;mFmrbs;ebtugenAGayu 28 f

lkNnebtugGarem:

27

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

emKuN
RbePTsIum:gt
kkrwgyWt
kkrwgFmta belOn
kkrwgelOn
sc

H
=1

100

arh

sg;darGWr:ub

sg;darGaemric

sc

SL

II

RS

III

karekItmankarrYmmaDGaRsynwgeBlRtUv)ankMNt;tamsmIkarxageRkam
sc (t tc ) =

Edl

(t tc )
2
0.56(he / 4 ) + (t tc )

Gayurbs;ebtug f
t = Gayurbs;ebtugenAeBlcab;epImkarrYmmaD
h = kRmas;RbsiTPaBedIm,IKitbBalpleFobmaDelIRkLap
eKGackMNt;kRmas;RbsiTPaB h dUcxageRkam
t=
c

2 Ac
u

he =

Edl

muxkat;rbs;Ggt;eRKOgbgM mm
u = brimaRtrbs;Ggt;eRKOgbgMEdlb:HCamYynwgbriyakas mm
Ac =

karKNna creep
GnuKmn_ creep tMNag[kugRtaMgsrubGaRsynwgbERmbRmYlrageFobkgmYyktakugRtaMg. eKGac
kMNt;vadUcxageRkam.
J (t , t0 ) =

Edl

1
Ecmt 0

(t , t0 )
Ecm 28

m:UDuleGLasicenAGayuEdlrgbnk
E
= m:UDuleGLasicenAGayu 28 f
(t ,t ) = emKuN creep
Ecmt 0 =
cm 28

Ecmt 0 = Ecm 28e

T.Chhay

28

0.5 S 1
t

28

Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca
S=

Department of Civil Engineering

emKuNGaRsynwgRbePTsIum:gt ehIyGackMNt;BItaragxageRkam

emKuN S CaGnuKmn_nRbePTsIum:gt
RbePTsIum:gt
sg;darGWr:ub
SL
kkrwgyWt
R
kkrwgFmta belOn
RS
kkrwgelOn
Ecm 28 = 215003

sg;darGaemric

II

0.38

0.25

III

0.20

f cm 28
10

eyIgGackMNt;emKuN creep (t,t ) BIsmIkarxageRkam

0

(t , t0 ) = 0 c (t , t0 )

Edl

emKuN notional creep

(t ,t ) = smIkarEdlBNnaBIkarekItman creep GaRsynwgeBleRkayeBlrgbnk

0 =
c

0 = RH ( f cm 28 ) (t0 )

Edl

RH =

emKuNsMeNImbriyasnemKuN notional creep Edl[eday

RH = 1 +

1 H / 100
0.163 he / 4

( f cm 28 ) =

emKuNersIusg;ebtugnemKuN notional creep Edl[eday

( f cm 28 ) =

5.3
f cm 28 / 10

(t0 ) =

emKuNGayuebtugenAeBlrgbnknemKuN notional creep Edl[eday

(t0 ) =

1
1 + t00.2

smIkarEdlBNnaBIkarekItman creep GaRsynwgeBleRkayrgbnk (t,t ) RtUv)anKNnaedayeRbI

smIkarxageRkam
0

t t0

c (t , t0 ) =
H + t t0

0. 3

H = 1.5he 1 + (0.012 H )18 + 250 1500

lkNnebtugGarem:

29

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

2>11>5> m:UEDl CEB 90-99

m:UEDl CEB 90-99 RtUv)anEkERbBIm:UEDl CEB90.

karKNnakarrYmmaD
enAkgm:UEDlfIenH karrYmmaDsrubrYmmanbgMnkarrYmmaDeday autogenous nig edaysarst. enA
kgebtug high-performance karrYmmaDeday autogeneous mantmFM ehIyeKRtUvBicarNavakgkar)a:n;
RbmaNkarrYmmaD. viFIfIenHpl;nUvkar)a:n;RbmaNkarrYmmaDnebtugFmta nigebtug high-performance
manlkNsuRkit.
eKGacKNnabERmbRmYlrageFobedaykarrYmmaDsrubedayeRbIsmIkarxageRkam
s (t , tc ) = as (t ) + ds (t , tc )

Edl

enAxN t
(t, t ) = drying shrinkage enAxN t
eKGacKNna autogenous shrinkage dUcxageRkam
as (t ) = autogenous shrinkage
ds

as (t ) = as ( f cm 28 ) as (t )
0

Edl

emKuN notional autogenous shrinkage

(t ) = GnuKmn_edIm,IBNnaeBlevlaEdlekItman autogenous shrinkage
eKGacKNnaemKuN notional autogenous shrinkage ( f ) dUcxageRkam
as ( f cm 28 ) =
0

as

as 0

f cm 28 / 10
10 6
6 + f cm 28 / 10

as ( f cm 28 ) = as
0

Edl

cm 28

2.5

emKuNGaRsynwgRbePTsIum:gt
= 800 sRmab;sIum:gtkkrwgyWt
= 700 sRmab;sIum:gtkkrwgFmta belOn
= 600 sRmab;sIum:gtersIusg;x<s;kkrwgelOn
f
= ersIusg;rgkarsgt;mFmrbs;ebtugenAGayu 28 f MPa
eKGacKNnaGnuKmn_ (t ) edayeRbIsmIkarxageRkam
as =

cm 28

as

as (t ) = 1 e 0.2 (t )

Edl

0.5

Gayurbs;ebtug f
eKGackMNt; drying shrinkage
t=

ds

(t, tc ) edaysmIkarxageRkam

da (t , tc ) = ds ( f cm 28 ) RH (H ) ds (t tc )
0

Edl
T.Chhay

ds ( f cm 28 ) =
0

emKuN notional drying shrinkage

30

Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

emKuNEdlKitBI\TiBlrbs; relative humidity mkelI drying shrinkage

(t t ) = GnuKmn_edIm,IBNnaeBlevlaEdlekItman drying shrinkage
eKGacKNnaemKuN notional drying shrinkage ( f ) BIsmIkarxageRkam
RH (H ) =
ds

ds ( f cm 28 ) = (220 + 110 ds1 )e

0

Edl

2
ds
f cm 28

/ 10

]10

ds 0

cm 28

emKuNEdlGaRsynwgRbePTsIum:gt
= 3 sRmab;sIum:gtkkrwgyWt
= 4 sRmab;sIum:gtkkrwgFmta bkkrwgelOn
= 6 sRmab;sIum:gtersIusg;x<s;kkrwgelOn
= emKuNEdlGaRsynwgRbePTsIum:gt
= 0.13 sRmab;sIum:gtkkrwgyWt
= 0.11 sRmab;sIum:gtkkrwgFmta bkkrwgelOn
= 0.12 sRmab;sIum:gtersIusg;x<s;kkrwgelOn
eKGacKNnaemKuN (H ) tamrUbmnxageRkam

H
1.551
sRmab; 10% H 99%
=
100

0.25
sRmab; H 99%

ds1 =

ds 2

RH

s1

RH

s1

Edl

H = Ambient relative humidity (%)

emKuNEdlKitBIkarkkrwgedayxngenAkgebtug high-performance
eyIgGackMNt;vadUcxageRkam
s1 =

35

s1 =
f cm 28

0.1

1.0

eKGacKNnaGnuKmn_
Edl

(t tc )

ds (t tc ) =
2

(
)
(
)
0
.
56
h
/
4
t
t
+

e
c

0.5

GayuebtugenAeBlcab;epImst f
2A
= TMhM notional rbs;Ggt; mm Edl A CaRkLapmuxkat; mm nig u CabrimaRt
h =
u
rbs;Ggt;Edlb:HCamYynwgbriyakas mm

tc =

karKNna creep
lkNnebtugGarem:

31

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

kugRtaMgsrubGaRsynwgbERmbRmYlrageFobkgmYyktankugRtaMg EdleKGacehAm:ageTotfa
compliance creep bGnuKmn_ creep. eKGackMNt;vaedayeRbIsmIkarxageRkam
J (t , t0 ) =

Edl

Ecmt 0

(t , t0 )
Ecm 28

m:UDuleGLasicenAGayurgbnk MPa
E
= m:UDuleGLasicenAGayu 28 f
(t ,t ) = emKuN creep
Ecmt0 =
cm 28

Ecmt 0 = Ecm 28e

S

0.5 S

28
T

CaemKuNGaRsynwgRbePTsIum:gt nigersIusg;sgt; ehIyeKGackMNt;vaBItaragxageRkam

emKuN S CaGnuKmn_eTAnwgRbePTsIum:gt nigersIusg;rgkarsgt;

RbePTsIum:gt
f (MPa )
ersIusg;FMkkrwgelOn
60
kkrwgFmta nigelOn
kkrwgyWt
RKb;RbePT
> 60
cm 28

Ecm 28 = 215003

S
0.20
0.25
0.30
0.20

f cm 28
10

eKGacKNnaemKuN creep (t,t ) BIsmIkarxageRkam

0

(t , t0 ) = 0 c (t , t0 )

Edl

emKuN notional creep

(t ,t ) = smIkarEdlBNnaBIkarekItman creep CamYynwgeBleRkayeBlrgbnk

0 =
c

0 = RH ( f cm 28 ) (t0 )
RH =

emKuN relative humidity nemKuN notional creep

RH = 1 +

Edl

1 H / 100
1 2
0.163 he / 4

35
1 =

f cm 28

0.7

35

0.2

2 =

f cm 28
T.Chhay

32

Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca
( f cm 28 ) =

emKuNersIusg;ebtugnemKuN notional creep

( f cm 28 ) =

Edl

Department of Civil Engineering

5.3
f cm 28 / 10

(t0 ) =

emKuNGayuebtugenAeBlrgbnknemKuN notional creep

(t0 ) =

1
1 + t00.2

+
1
t0 = t 0 ,T
0.5
1. 2
2 + t 0 ,T

Gayurbs;ebtugenAeBlrgbnk f
t = Gayurbs;ebtugenAeBlrgbnkEdlEktRmUveTAtamsItuNPaBrbs;ebtug
sRmab; T = 20 C / t RtUvKanwg t
= emKuNEdlGaRsynwgRbePTsIum:gt
= 1 sRmab;sIum:gtkkrwgyWt
= 0 sRmab;sIum:gtkkrwgFmta belOn
= 1 sRmab;sIum:gtersIusg;FMkkrwgelOn
eKGackMNt;smIkarEdlBNnakarekItman creep CamYynwgeBleRkayeBlrgbnk (t,t ) eday
eRbIsmIkarxageRkam
t0 =
0 ,T

0 ,T

t t0

c (t , t0 ) =
H + t t0

0. 3

H = 1.5he 1 + (0.012 H )18 + 250 3 1500 3

35

f cm 28

0.5

3 =

]TahrN_ 2>1 KNnabERmbRmYlragedaysarrYmmaD nig creep compliance sRmab;sMNakKMrUebtugEdl

[xageRkam. eRbIm:UEDl ACI 209.
eK[ sMeNIm H = 75%
he = 2V / S = 2 Ac / u = 76mm
f cm 28 = 45.2 MPa
w = 207.92kg / m3
w / c = 0.46
a / c = 3.73

t = 35

lkNnebtugGarem:

33

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

f
= 8 f

t0 = 28
tc

= 2405kg / m3

sIum:gtRbePT III
ebtugEfTaMedaysMeNIm
dMeNaHRsay
karKNnakarrYmmaD
s (t ) =

(t tc ) K K
ss sh shu
b + (t tc )

shu = 780 106 mm / mm

edayebtugEfTaMedaysMeNIm b = 35
V / S = 38mm

V
K ss = 1.14 0.0035 = 1.14 0.0035(38) = 1.007
S

sRmab; H = 75%
K sh = 1.40 0.01H = 1.40 0.01(75) = 0.65
(t tc ) K K
s (t ) =
ss sh shu
b + (t tc )
(35 8) (1.007 )(0.65)(780 10 6 ) = 222.3 10 6 mm / mm
=
35 + (35 8)

karKNna creep
J (t , t0 ) =

1 + Cc (t )
Ecmt 0

kMNt; E
edaysarsIum:gtRbePT III nigebtugEfTaMedaysMeNIm b = 2.30 / c = 0.92
cmt 0

f 'c (t0 ) = f cm 28

t0
28
= 45.2
= 45.1MPa
b + ct0
2.3 + 0.92 28

Ecmt 0 = 0.043( )

3/ 2

f 'c (t0 ) = 0.043(2405)

3/ 2

45.1 = 34058.8MPa

kMNt; C (t )
c

Ccu = 2.35
K ch = 1.27 0.0067(H ) = 1.27 0.0067(75) = 0.767
K ca = 1.25(t0 )

0.118

T.Chhay

= 1.25(28)

0.118

= 0.844

34

Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

K cs = 1.14 0.0035(V / S ) = 1.14 0.0035(38) = 1.007

t 0.60
350.60
C
K
K
K
=
2.35 0.767 0.844 1.00 = 0.702
cu ch ca cs
10 + t 0.60
10 + 350.60
1 + Cc (t ) 1 + 0.702
1
=
= 49.9 10 6
J (t , t0 ) =
Ecmt 0
MPa
34058.8

Cc (t ) =

]TahrN_ 2>2 edayeRbIm:UEDl B3 KNnakarrYmmaD nigGnuKmn_ creep sRmab;sMNakKMrUEdl[enAkg

]TahrN_ 2>1.
dMeNaHRsay
KNnakarrYmmaD
s (t ) = ( shu )(K h )S (t )

kMNt;
edaysIum:gtRbePT III = 1.10
edaysarebtugEfTaMgedaysMeNIm = 1.0
= [0.019(w) ( f )
+ 270] 10
= (1.10 )(1.0 )[0.019(207.92 ) (45.2 )
+ 270] 10 = 827 10
kMNt; K
H
75
sRmab; H = 75% enaH K = 1 100
=1
= 0.578

100
kMNt; S (t )
K = 1.0 edaysareyIgmindwgBIRbePTGgt;
shu

0.28

2.1

shu

cm 28

0.28

2.1

mm / mm

0.08

( f cm 28 )0.25 [2 K s (V / S )]2

0.08

(45.2)0.25 [2(1.0)(38)]2 = 160.3

Tsh = 0.085(tc )

= 0.085(8)
S (t ) = tanh

35 8
t tc
= tanh
= 0.389
160.3
Tsh

s (t ) = ( shu )(K h )S (t ) = (827 106 )(0.578)(0.389) = 185.9 106 mm / mm

KNna creep
J (t , t0 ) = q1 + C0 (t , t0 ) + Cd (t , t0 , tc )

kMNt; q

Ecm 28 4735 f cm 28 = 4735 45.2 = 31833.9 MPa

lkNnebtugGarem:

35

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

0.6
0.6
1
=
= 18.85 10 6
Ecm 28 31833.9
MPa

q1 =

kMNt; C (t,t )
0

c=

w
207.92
=
= 452kg / m3
w/ c
0.46

q2 = 185.4(c )

0. 5

( f cm 28 )0.9 106 = 185.4(452 )0.5 (45.2)0.9 106

= 127.6 106
Q f (t0 ) =
Z (t , t0 ) =

0.086(t0 )

1
1
=
= 0.182
4/9
2/9
4/9
+ 1.21(t0 )
0.086(28) + 1.21(28)

2/9

ln 1 + (t t0 )
t0

0. 1

r (t0 ) = 1.70(t0 )

0.12

] = ln[1 + (35 28) ] = 0.150

0.1

28

+ 8 = 1.7(28)

0.12

Q f (t0 )r (t 0 )
Q(t , t0 ) = Q f (t0 )1 +
r (t 0 )
Z (t , t0 )

+ 8 = 10.54

1 / r (t 0 )

0.18210.54
= 0.1821 +
10.54
0.150

1 / 10.54

= 0.148

w
4
q3 = 0.29q2 = 0.29(127.6 10 0.6 )(0.46) = 1.66 10 6
c
a
q4 = 20.3
c

0.7

10 6 = 20.3(3.73)

0. 7

C0 (t , t0 ) = q2Q (t , t0 ) + q3 ln 1 + (t t0 )

0.1

10 6 = 8.08 10 6

]+ q ln tt
4

= (127.6 10 6 )(0.148) + (1.66 10 0.6 )ln 1 + (35 28)

= 22.01 10 6

0. 1

]+ (8.08 10 )ln 35

28
6

1
MPa

KNna C (t, t , t )
d

q5 =

0.757 shu 106

f cm 28

0.6

0.757 827 106

0.6

45.2

= 297.5 10 6

S (t ) = 0.389

S (t0 ) = tanh

28 8
t0 tc
= tanh
= 0.339
160.3
Tsh

75
H
H (t ) = 1 1
0.389 = 0.903
S (t ) = 1 1
100

100

75
H
H (t0 ) = 1 1
0.339 = 0.915
S (t0 ) = 1 1
100

100

Cd (t , t0 , tc ) = q5 e 8 H (t ) e 8 H (t 0 )
= (297 10 6 ) e 8 0.903 e 80.915 = 2.43 10 6

T.Chhay

36

1
MPa
Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

J (t , t0 ) = q1 + C0 (t , t0 ) + Cd (t , t0 , tc )
= (18.85 10 6 ) + (22.01 10 6 ) + (2.43 10 6 ) = 43.3 10 6

1
MPa

]TahrN_ 2>3 edayeRbIm:UEDl GL 2000 KNnakarrYmmaD nigGnuKmn_ creep sRmab;sMNakKMrUEdl[

enAkg]TahrN_ 2>1.
dMeNaHRsay
KNnakarrYmmaD
s (t ) = shu (h ) (t )

KNna
edaysIum:gtRbePT III K = 1.15
shu

1/ 2

shu

30

= (900)K
f cm 28

1/ 2

30
10 6 = (900)(1.15)

45.2

10 6 = 843.2 10 6 mm / mm

KNna (h)
4

75
H
= 0.627
= 1 1.18
100
100

(h ) = 1 1.18

KNna (t )
1/ 2

t tc

(t ) =
2
t tc + 0.12(V / S )

1/ 2

35 8

=
2
35 8 + 0.12(38)

= 0.367

s (t ) = shu (h ) (t ) = (843.2 106 )(0.627 )(0.367 ) = 194 106 mm / mm

KNna creep

1
(t , t0 )
+
Ecmt 0
Ecm 28

J (t , t0 ) =

KNna E nig E
t = 28 f E
cmt 0

cm 28

cmt 0

= Ecm 28

Ecm 28 = 3500 + 4300 f cm 28 = 3500 + 4300 45.2 = 32409.3MPa

KNna (t,t )
0

t0 = 28 > tc = 8

0.5


t0 t c

(tc ) = 1
2
t0 tc + 0.12(V / S )

0.5

0.5


28 8

= 1
2
28 8 0.12(38)

0. 5

= 0.824

h = H / 100 = 75 / 100 = 0.75

lkNnebtugGarem:

37

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

0. 5
(t t )0.3 7 0.5 t t 0.5

t t0
2
0
0
+

+ 2.5 1 1.086h
(t , t0 ) = (tc )2
0.3
t t + 0.12(V / S )2
(t t0 ) + 14 t0 t t0 + 7
0

0. 5
(35 28)0.3 7 0.5 35 28 0.5

35 28
2

= 0.824 2
+
+ 2.5 1 1.086(0.75)
0.3
2
(35 28) + 14 28 35 28 + 7
35 28 + 0.12(38)

= 0.773

J (t , t0 ) =

Ecmt 0

(t , t0 )
Ecm 28

1
0.773
1
+
= 54.7 10 6
MPa
32409.3 32409.3

]TahrN_ 2>4 edayeRbIm:UEDl CEB 90 KNnabERmbRmYlragedaykarrYmmaD nigGnuKmn_ creep sRmab;

sMNakKMrUEdl[enAkg]TahrN_ 2>1.
dMeNaHRsay
KNnakarrYmmaD
s (t , tc ) = cs s (t , tc )
0

KNna

cs 0

sc = 8

s ( f cm 28 ) = 160 + 10( sc ) 9

f cm 28
6
10
10

45.2

6
6
= 160 + 10(8) 9
10 = 518.4 10 mm / mm
10

sRmab; H = 75%
RH = 1.55 arh
arh = 1 (H / 100 )3 = 1 (75 / 100 )3 = 0.578

RH = 1.55 arh = 1.55 0.578 = 0.896

cs = s ( f cm 28 )( RH ) = (518.4 10 6 )( 0.896 ) = 464.2 10 6 mm / mm
0

KNna

sc

(t tc )

he =

2 Ac
= 76mm
u

(t tc )
=
2
0.56(he / 4 ) + (t tc )

sc (t tc ) =

(35 8)
= 0.343
2
0.56(76 / 4 ) + (35 8)

s (t , tc ) = ( cs ) s (t tc ) = ( 464.2 10 6 )(0.343) = 159.3 106 mm / mm

0

KNna creep
J (t , t0 ) =
T.Chhay

1
Ecmt 0

(t , t0 )
Ecm 28
38

Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

KNna E nig E
t = 28 f E
cmt 0

cm 28

cmt 0

Ecm 28 = 215003

KNna (t,t )

= Ecm 28

f cm 28
45.2
= 215003
= 35548MPa
10
10

1 75 / 100
1 H / 100
=1+
= 1.586
3
0.163 76 / 4
0.16 he / 4

RH = 1 +

( f cm 28 ) =
(t0 ) =

5.3
f cm 28 / 10

5.3
= 2.49
45.2 / 10

1
1
=
= 0.488
0. 2
0.1 + t0
0.1 + 280.2

0 = RH ( f cm 28 ) (t0 ) = (1.586)(2.49)(0.488) = 1.927

H = 1.5he 1 + (0.012 H )18 + 250 = 1.5(76 ) 1 + (0.012 75)18 + 250 = 379 1500
t t0

c (t , t0 ) =
H + t t0

0. 3

35 28
=

379 + 35 28

(t , t0 ) = 0 c (t , t0 ) = 1.927 0.3 = 0.578

1
(t , t0 )
1
0.578
J (t , t0 ) =

Ecmt 0

Ecm 28

35548

35548

0.3

= 0.3

= 44.4 10 6

1
MPa

]TahrN_ 2>5 edayeRbIm:UEDl CEB 90 fI cUrKNnabERmbRmYlragedaykarrYmmaD nigGnuKmn_ creep

sRmab;sMNakKMrUEdl[enAkg]TahrN_ 2>1.
dMeNaHRsay
KNnakarrYmmaD
s (t , tc ) = as (t ) + ds (t , tc )

KNna

as

(t )

as = 600

sRmab;sIum:gtersIusg;x<s;kkrwgelOn
2. 5

f
/ 10
10 6
as 0 ( f cm 28 ) = as cm 28
6
+
/
10
f
cm 28

2.5

45.2 / 10
6
6
= 600
10 = 72.6 10 mm / mm
6 + 45.2 / 10

as (t ) = 1 e 0.2(t ) = 1 e 0.2(35 ) = 0.694

0.5

0.5

as (t ) = as ( f cm 28 ) as (t ) = ( 72.6 106 )(0.694) = 50.4 106 mm / mm

0

lkNnebtugGarem:

39

T.Chhay

mhaviTalysMNg;sIuvil

KNna

ds

NPIC

(t, tc )

sRmab;sIum:gtersIusg;x<s;kkrwgelOn
= 0.12 sRmab;sIum:gtersIusg;x<s;kkrwgelOn

ds1 = 6
ds 2

ds ( f cm 28 ) = (220 + 110 ds1 )e

0

ds0 f cm 28 / 10

] 10

= (220 + 110 6 )e 0.12 (45.2 ) / 10 106 = 511.6 10 6 mm / mm

35

f cm 28

0. 1

35
=

45.2

s1 =

0.1

= 0.97 1.0

sRmab; 40% < H = 75% < 99%(0.97) = 96.5%

100

RH = 1.551

75 3

1
.
55
=

= 0.896

100

(t tc )

ds (t tc ) =
2

0.56(he / 4 ) + (t tc )

0. 5

(35 8)

=
2
0.56(76 / 4 ) + (35 8)

0.5

= 0.343

ds (t , tc ) = ds ( f cm 28 ) RH (H ) ds (t tc )
0

= 511.6 10 6 ( 0.896 )(0.343) = 157.2 10 6 mm / mm

s (t , tc ) = as (t ) + ds (t , tc ) = ( 50.4 106 ) + ( 157.2 106 ) = 207.6 106 mm / mm

KNna creep
J (t , tc ) =

1
Ecmt 0

KNna E nig E
t = 28 f E

(t , tc )
Ecm 28

cm 28

cmt 0
0

cmt 0

Ecm 28 = 215003

= Ecm 28

f cm 28
45.2
= 215003
= 35548MPa
10
10

KNna (t,t )
0

35
1 =

f cm 28

0.7

0.2

35
=
45.2

0. 7

= 0.836

35
35
=
2 =
= 0.950

45.2
f cm 28
1 H / 100
1 75 / 100

RH = 1 +
1 2 = 1 +
0.836 0.950 = 1.415
3
3
0.16 76 / 4

0.16 he / 4
5.3
5.3
( f cm 28 ) =
=
= 2.49
f cm 28 / 10
45.2 / 10
T.Chhay

0.2

40

Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

+ 1 = 2.8
+ 1 = 32.5 0.5
t0 = t0 ,T
1.2
1. 2
2 + 28

2 + t0 ,T

1
1
(t0 ) =
=
= 0.475
0.2
0.1 + t0
0.1 + 32.50.2

0 = RH ( f cm 28 ) (t0 ) = 1.419 2.49 0.475 = 1.674

35
3 =

f cm 28

0.5

35
=
45.2

0. 5

= 0.880

H = 1.5he 1 + (0.012 H )18 + 250 3

= 1.5 76 1 + (0.012 75) + 250 0.88 = 351 1500 0.88 = 1320

18

t t0

H + t t0

c (t , t0 ) =

0.3

35 28
=

351 + 35 28

(t , t0 ) = 0 c (t , t0 ) = 1.678 0.307 = 0.515

1
(t , t0 )
1
0.515
J (t , t0 ) =

Ecmt 0

Ecm 28

35548

35548

0.3

= 0.307

= 42.6 10 6

1
MPa

2>12> m:as;maDebtug
m:as;maDrbs;ebtugFmtakkrwgEdleRbIR)as;sRmab;GKar nigrcnasm<nepSgeTotGaRsyeTA
elIkarlay TMhM nigcMNat;fak;fbMEbk pleFobTwkelIsIum:gt nigersIusg;rbs;ebtug. xageRkamCam:as;maD
ebtugEdleKGacykmkBicarNa
- m:as;maDebtugsuTEdleRbIfbMEbkTMhM 20mm ERbRbYlcenaHBI 2320 2400kg / m . sRmab;
ebtugEdlmanersIusg;ticCag 280kg / cm = 28MPa tm 2320kg / m GacRtUv)anykmkeRbI. sRmab;
ebtugEdlmanersIusg;FMCagenH m:as;maDebtugmantm 2400kg / m GacRtUv)anykmkeRbI.
- m:as;maDebtugsuTEdleRbIfbMEbkTMhM 100mm 150mm ERbRbYlcenaHBI 2400 2560kg / m .
tmmFmesInwg 2480kg / m GacRtUv)anykmkeRbI.
- m:as;maDebtugGarem: EdlmanPaKryEdk 0.7% 1.5% mantmRbEhl 2400kg / m . sRmab;
ebtugGarem:EdlmanPaKryEdleRcInCagenH m:as;maDrbs;ebtugRtUv)aneKsnt;RbEhl 2500kg / m .
- m:as;maDebtugRsalERbRbYlBI 320 1440kg / m
- m:as;maDebtugFn;ERbRbYlBI 3200 4300kg / m
3

lkNnebtugGarem:

41

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

2>13> RbePTEdkeRbIkgebtug
EdkEdleRbIenAkgebtug manCaeRcInRbePTEdlRtUv)ancat;cMNat;fak;dUcxageRkam
- EdkmUl (round bars) EdkmUlRtUv)aneRbIy:agTUlMTUlayenAkgebtug. Ggt;pitrbs;EdkmUlEdl
eKGacrk)anenAelITIpSarmancab;BI 6mm dl; 36mm bUkrYmCamYynwgGgt;pitBiessBIrRbePTeTot
KW 45mm nig 57mm . GaRsyedaypxagrbs;va eKEbgEckvaCaBIrRbePTeTotKW Edkrelag
(plain bars) nigEdkfaMgGMeBA (deformed bars). CaTUeTA EdkrelagRtUv)aneKeRbICaEdkEdlman
tYnaTITI2 (secondary reinforcement) bCaEdkkg (stirrups and ties). EdkfaMgGMeBARtUv)aneK
pliteLIgCaBiessedIm,IbegInPaBsitrvagebtug nigsrsEdk ehIykat;bnynUvTTwgsameRbHrbs;
ebtugkgtMbn;rgkmaMgTaj.
Ggt;pitrbs;Edkrelag Gacvas;)anedayRsYl EtsRmab;EdkfaMgGMeBA Ggt;pitrbs;vaeKkMNt;
edayykGgt;pitrbs;rgVg;Edk Edl)anBImuxkat;rbs;va. pxageRkArbs;EdkfaMgGMeBAmansNan
CaeRcInRbePTEdlmanbgajkgrUbxageRkam.
- EdksMNaj; (Welded fabrics and mats) EdksMNaj;manCaeRcInesrIeTAtamRbEvgEdkTTwg nig
EdkbeNay. CaTUeTAvaRtUv)aneKpSaredaydak;EdkbeNay nigEdkTTwg[EkgKaRtg;kEngkat;
Ka.
- kabeRbkugRtaMg (Prestressed concrete wires and strands) kabeRbkugRtaMgCaRbePTEdkBiess
EdlmanersIusg;x<s;. kabEdlersIusg;rgkmaMgTajx<s;manGgt;pitCaTUeTA 5mm nig 7mm RtUv
)aneKykmkbegItCakabeRbkugRtaMg EdlmankabxageRkAR)aMmYyBTCMuvij kabmYyEdlmanGgt;
pitFMCag. ersIusg;rbs;kabeRbkugRtaMgGtibrmamantm 1750MPa b 1890MPa .
m:UDuleGLasicrbs;srsEdk eTaHrgkarTajkI rgkarsgt;kI eKkMNt;yk E = 2 10 MPa . em
KuNlUtkemArbs;srsEdkeKyk = 10 mm / c dUcemKuNlUtkemArbs;ebtugEdr.
5

T.Chhay

42

Properties of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

muxkat;Edk (mm)

pmuxkat;Edk (cm )

Tmn; (kg / m)

brimaRt (cm)

RB6
RB8
DB12
DB16
DB20
DB25
DB28
DB32

0.28
0.64
1.13
2.01
2.84
4.91
6.16
8.04

0.222
0.499
0.888
1.58
2.23
3.85
4.83
6.31

1.89
2.83
3.77
5.03
5.97
7.86
8.80
10.06

lkNnebtugGarem:

43

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

III.

viPaKFwmebtugGarem:rgkarBt;begag

Flexural Analysis of Reinforced Concrete Beam

3>1> karsnt;
(Assumption)
ebtugGarem:CasmarminEmnrUbFatusac;mYy BIeRBaHvaekIteLIgedaysmarBIrRbePTKW ebtug nig
Edk. dUecH eRKOgbgMebtugGarem:EdlkMNt;edayersIusg;cugeRkay RtUvkMNt;tamkarsnt;xageRkam
- bERmbRmYlrageFobrbs;ebtug nigEdkRtUvEtmantmdUcKa mannyfaPaBsitrvagebtug nigEdk
mantmRKb;RKan;.
- bERmbRmYlrageFobrbs;ebtugRtUvEt smamaRteTAnwgcmayBIGkSNWt
- m:UDuleGLasicrbs;Edk RtUvEtyk E = 2 10 MPa . kugRtaMgrbs;EdkkgtMbn;eGLasicRtUvEt
mantmesIplKuNrvagbERmbRmYlrageFobCamYynwgm:UDuleGLasic.
- muxkat;Rtg;enAEtRtg;eRkayeBlrgkarBt;
- ersIusg;Tajrbs;ebtugRtUv)anecal BIeRBaHersIusg;TajebtugmantmtUcCagersIusg;sgt;dl;eTA
10 dg ehIysameRbHrbs;ebtugRtUvsnt;faKan\TiBl nigm:ageTotmuneBleRbH muxkat;ebtugTaMgmUlman
RbsiTPaBkgkarTb;nwgm:Um:g;xageRkA.
- bERmbRmYlrageFobGtibrmarbs;ebtugkRmitRtwm 0.003
- rUbragnkarBRgaykugRtaMgsgt;rbs;ebtug snt;manragctuekaNEkg
5

3>2> RbePTnkar)ak;edaykarBt; nigEdnkMNt;sac;lUteFob

3>2>1> kar)ak;edaykarBt;
eRKOgbgMrgkarBt; Gac)ak;edaybIkrNIGaRsyeTAnwgPaKryEdkEdl)andak;enAkgmuxkat;ebtug
- EdkGaceTAdl;cMNuc yield munebtugeFVIkardl;ersIusg;Gtibrma. kgkrNIenH kar)ak;bNalmkBI
sac;lUteFobrbs;EdkmantmFMCag besI 0.005 . muxkat;manbrimaNEdktic ehIyRtUv)aneK[eQaH fa
tension-controlled section .
- EdkGaceTAdl;cMNuc yield GMLgeBlEdlebtugeFVIkardl;ersIusg;GtibrmaEdr. muxkat;RtUv)aneK
[eQaHfa balanced section .
- ebtugGacEbkmuneBlEdlEdkeFVIkardl;cMNuc yield bNalmkBIPaKryEdkeRcInenAkgmuxkat;.
kgkrNIenHebtug)aneFVIkardl;ersIusg ;Gtibrma ehIymansac;lUteFobGtibrma 0.003 Edr b:uEnkugRtaMg
T.Chhay

44

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

rbs; EdkmantmticCagersIusg;KNna Edl f < f . sac;lUteFobrbs;EdkmantmtUcCag besI

0.002 . muxkat; enHRtUv)aneK[eQaHfa compression-controlled section.
eK)ansnt;faebtugEbkedaysarkmaMgsgt; enAeBlEdlbERmbRmYlrageFobrbs;ebtugmantm
0.003 EtkgkarBiesaFtmenHERbRbYlBI 0.0025 0.004 .
kgkarKNnaFwm eKeRCIserIsyk tension-controlled section eday[EdkeFVIkardl;ersIusg;
KNna (design strength) munebtugEbk. sameRbHrbs;ebtugrIkFMeLIg EdlCasBaaRbkasGasnmuneBl
ebtugEbk ehIyrcnasm<n)ak;Ebk.
kgkarKNnaFwmEdleRCIserIsyk compression-controlled section nig balanced section
ebtugEbkPam rcnasm<n)ak;EbkmYyrMBicedayKankarRbkasGasn. kareRCIserIsmuxkat;EbbenHRtUv
)aneCosvag.
s

3>2>2> EdnkMNt;bERmbRmYleFobsRmab; tension-controlled section nig compression-controlled

section

karKNnapl;[cMeBaHkarKNnaebtugGarem:sRmab; tension-controlled section b compression

controlled section. muxkat;TaMgBIr RtUv)ankMNt;edaybERmbRmYlrageFobrgkarTajsuT (net tension
strain (NTS)). elIsBIenHlkxNBIreTot)anekItKW balanced strain condition nig transition region
condition. lkxNTaMgbYnenHRtUv)ankMNt;dUcxageRkam
- Compression-controlled section Camuxkat;EdlbERmbRmYlrageFobrgkarTajsuT (NTS)
sRmab;EdkrgkarTajEpkxageRkAbMputmantmtUcCagbERmbRmYlrageFobrgkarsgt; enAeBlEdlbERm
bRmYlrageFobrbs;ebtugrgkarsgt;mantmesI 0.003 . krNIenHekIteLIgCaTUeTAcMeBaHssrEdlrgbnk
tamGkS nigm:Um:g;.
- Tension-controlled section Camuxkat;EdlbERmbRmYlrageFobrgkarTajsuT (NTS) sRmab;
EdkrgkarTaj EpkxageRkAbMputmantmFMCag besI 0.005 kgkrNIEdlebtugmanbERmbRmYlrageFob
dl;EdnkMNt; 0.003 .
- muxkat;EdlbERmbRmYlrageFobrgkarTajsuT (NTS) sRmab;EdkrgkarTajEpkxageRkAbMput
mantmsitenAcenaH compression-controlled section nig tension-controlled section KWenAcenaH
0.002 0.005 Camuxkat; transition region
viPaKebtugGarem:rgkarBt;begag

45

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

- Balanced strain condition ekItmanenAkgmuxkat; enAeBlEdlbERmbRmYlrageFobEdktMbn;Taj

mantmesI = Ef kgxNEdlebtugrgkarsgt;manbERmbRmYlrageFobmantmesI 0.003 .
y

Section condition

Concrete strain

Steel strain

Note ( f y = 400MPa)

Compression-controlled

0.003

t f y Es

t 0.002

Tension-controlled

0.003

Transition region

0.003

t 0.005
f y E s t 0.005

t 0.005
0.002 t 0.005

Balanced strain

0.003

s = f y Es

s = 0.002

Transition region

0.003

0.004 t < 0.005

0.004 t < 0.005

T.Chhay

46

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

3>3> emKuNbnk
bnkEdlmanGMeBIelIeRKOgbgMRtUv)anKuNCamYynwgemKuNbnk edIm,IkarBarkar)ak;Pam nigpl;
nUvkarKNnamYyEdlmanlkNesdkic. emKuNbnkGaRsynwgRbePTbnk nigkarbnSMbnk. emKuNbnk
sRmab;bnkGefr KW 1.6 emKuNbnksRmab;bnkefr KW 1.2 . dUecHkarbnSMbnksRmab;bnkGefr nigbnk
efrKW
U = 1 .2 D + 1 .6 L

Edl

- bnkKNna (ultimate load)

L - bnkGefr
D - bnkefr

viPaKebtugGarem:rgkarBt;begag

47

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

3>4> emKuNkat;bnyersIusg;
emKuNkat;bnyersIusg; mantmtUcCag 1. emKuNkat;bnyersIusg;GaRsynwgRbePTn
eRKOgbgM
= 0.90
- sRmab;muxkat;rgkarTaj
- sRmab;muxkat;rgkarsgt;
= 0.70
k> CamYyEdkkgvN
x> CamYyEdkkgdac;
= 0.65
- sRmab;ebtugsuT
= 0.55
- sRmab;kmaMgkat; nigkmaMgrmYl
= 0.75
= 0.65
- sRmab;RTnab;enAelIebtug
= 0.75
- sRmab;KMrU strut and tie
3>5> karEbgEckkugRtaMgsgt;smmUl
kugRtaMgEbgEckkgebtugrgkarsgt;enAxNeBl)ak;RtUv)ansnt;famanragctuekaNEkg ctuekaNBay ExSekag)a:ra:bUl bragNamYyepSgeTotGaRsyedaykaryl;RBmKaenAeBleFVIBiesaFn_.
enAeBlEdlFwmerobnwg)ak; srsEdk)aneFVIkardl;cMNuc yield mun RbsinebImuxkat;enaHman
brimaNEdktic (under-reinforced section) ehIykgkrNIenHsrsEdkeFVIkardl;kugRtaMgKNna (design
stress). EtRbsinebImuxkat;manEdkeRcIn enaHebtugnwgEbkmun ehIybERmbRmYlrageFobRtUv)ansnt;faesI
0.003 .
kmaMgsgt; C ekItmanenAkgtMbn;sgt; ehIykmaMgTaj T ekItmanenAtMbn;TajEdlsitenARtg;nIv:U
Edk. eKsal;TItaMgnkmaMg T BIeRBaHvamanGMeBIRtYtsIuKanwgGkSTIRbCMuTmn;rbs;Edk. TItaMgrbs;kmaMg
C eKGacsal;)an luHNaEteKsal;maDntMbn;sgt; ehIyeBlenaHeKGackMNt;)annUvTItaMgTIRbCMuTmn;
)an. RbsinebIeKsal;TItaMgrbs;kmaMgTaMgBIr enaHeKGackMNt;nUvRbEvgdXas; EdlCacmayBIkmaMgTaj
mkkmaMgsgt;.
RbsinebIebtugEbk enAeBlbERmbRmYlrageFob ' = 0.003 ehIyRbsinebIEdkeFVIkardl;cMNuc
yield f = f enaHmuxkat;Ca balanced section. kmaMgsgt; C RtUv)ansMEdgedaymaDnbkkugRtaMg
EdlmanragminksNanekItmanelIpctuekaNqt bc .
c

T.Chhay

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Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

maDnkugRtaMgsgt;snt;esI C = bc( f ' )

Edl f ' CakugRtaMgmFmnbkkugRtaMgminksNan
TItaMgrbs;kmaMgsgt; KWmancmay z BIsrsEpkxagelIbMputEdlGaccat;TukCaEpkncmay c
cmayBIsrsEpkxagelImkGkSNWt.
1

z = 2c

sRmab;ebtugEdlmanersIusg; f 'c 28MPa .

RtUv)ankat;bnyeday 0.04 ral; 7 MPa sRmab;ebtugEdlmanersIusg; f ' c > 28MPa .
= 0.425 sRmab;ebtugEdlmanersIusg; f ' 28MPa .
RtUv)ankat;bnyeday 0.025 ral; 7 MPa sRmab;ebtugEdlmanersIusg; f ' c > 28MPa .
edIm,IsRmYldl;karKNnakmaMgkgnmuxkat; enaH ACI code )anyknUvkugRtaMgEbgEckkgmuxkat;
ragctuekaNEkgEdlmantm 0.85 f ' BRgayesIelItMbn;sgt;smmUl EdlxNedaybnat;RsbnwgGkSNWt
EdlmanRbEvg a = c .
= 0.85 sRmab;ebtugEdlmanersIusg; f ' 28MPa .
f ' 28
= 0.85 0.05(
) sRmab;ebtugEdlmanersIusg; 28MPa < f ' 56 MPa .
7
= 0.65 sRmab;ebtugEdlmanersIusg; f ' > 56MPa .
sRmab;muxkat;ragctuekaNEkg RkLaptMbn;sgt;mantmesI ba ehIytmkugRtaMgBRgayesIKW
0.85 f ' Edlpl;nUvmaDkugRtaMgsrubesInwg 0.85 f ' ab ehIyRtUvKanwgkmaMgsgt; C . sRmab;muxkat;
epSgBIragctuekaNEkg kmaMgsrubesInwgplKuNRkLaptMbn;sgt;CamYynwg 0.85 f ' .
1 = 0.72
1

viPaKebtugGarem:rgkarBt;begag

49

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NPIC

3>6> srsEdkrgkmaMgTajnmuxkat;ctuekaNEkgrgkarBt;
PaKryEdkenAkgmuxkat;ebtugkglkxN balanced RtUv)aneK[eQaHfa
EdlCapleFobrvagmuxkat;Edk A nigmuxkat;RbsiTPaB bd

balanced steel ratio b

b =

Edl

As
bd

- TTwgmuxkat;eRKOgbgMtMbn;sgt;
d - cmayBIsrsEpkxageRkAbMputmkTIRbCMuTmn;EdkrgkmaMgTaj km<s;RbsiTPaB
smIkarlMnwgBIr EdlCaeKalkarN_kgkarviPaK nigKNnaeRKOgbgMehIymantmRKb;muxkat; nigRKb;
RbePTbnkKW
- kmaMgsgt;RtUvmantmesIkmaMgTaj C = T
- ersIusg;m:Um:g;Bt;xagkg M esIeTAnwgplKuNrvagkmaMgsgt; bkmaMgTajCamYynwgdXas;
M = C (d z ) = T (d z ) nig M = M Edl emKuNkat;bnyersIusg;
b

T.Chhay

50

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

kareRbIR)as;nUvsmIkarTaMgenHRtUv)anBnl;sRmab;muxkat;ragctuekaNEkgCamYyEdktMbn;Taj. mux
kat;GacCa balanced section muxkat;EdlmanEdktic muxkat;EdlmanEdkeRcIn GaRsyedaykareRbIR)as;nUv
PaKryEdk.
3>6>1> balanced section

CMhanTI1 BIdaRkambERmbRmYlrageFob eyIg)an

cb
0.003
=
fy
d cb
Es
c
0.003
b =
fy
d
0.003 +
Es

edayCMnYs E
cb = (

= 200000 MPa

600
)d
600 + f y

CMhanTI2 BIsmIkarlMnwg eyIg)an

C = T 0.85 f 'c ab = As f y

a=

As f y
0.85 f 'c b

Edl a - CaRbEvgbkrgkarsgt; mantmesInwg c

edaysarvaCa balanced section dUecHPaKryEdkRtUv)aneRbIKW
1 b

b =

As
bd

As = b bd

viPaKebtugGarem:rgkarBt;begag

51

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CMnYs A eTAkgsmIkarxagelI
s

0.85 f 'c ab = bbdf y

b =

0.85 f 'c a 0.85 f 'c

=
( 1cb )
f yd
f yd

CMnYstm c

=(

b = 0.851

f 'c
600
(
)
f y 600 + f y

600
)d
600 + f y

eTAkgsmIkarxagelI eyIg)an

CMhanTI3 BIsmIkarlMnwgnm:Um:g;xagkg eyIg)an

M n = C (d z ) = T (d z )

sRmab;muxkat;ragctuekaNEkg cmay z = a2
a
a
M n = C (d ) = T (d )
2
2

sRmab;muxkat; balanced section bmuxkat;EdlmanbrimaNEdktic

T = As f y

dUecH M = A f (d a2 )
m:Um:g;kgxagelIEdl)anKNna RtUvkat;bnyedayemKuN
n

M n = As f y (d

As f y

1.7 f 'c b

smIkarenH sresredayCab;GBaat
M n = f y bd (d

bdf y

1.7 f 'c b

) = f y bd 2 (1

f y
1.7 f 'c

eyIgGacsresrsmIkarxagelIenHCa
M n = Ru bd 2

Edl R = f (1 1.7ff ' )

pleFobrvagRbEvgbkkugRtaMgsgt;smmUl a nig km<s;RbsiTPaBnmuxkat; d
y

f y
a
=
d 0.85 f 'c

3>6>2> PaKryEdkGtibrma
PaKryEdkGtibrma EdlGaceRbIenAkgmuxkat;ebtugEdlmanEtEdkrgkmaMgTaj QrelIeKal
karN_bERmbRmYlrageFobsuTenAkgEdkrgkmaMgTaj PaKryEdk balanced nigersIusg;rbs;Edk.
max

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Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

TMnak;TMngrvagPaKryEdkenAkgmuxkat; nigbERmbRmYlrageFobsuT
0.003 +

fy

Es
=
b 0.003 + t

sRmab;

f y = 414 MPa

0.003 +
t

=(

fy
Es

) 0.003

nigsnt; f y / Es = 0.00207

- sRmab;muxkat;rgkmaMgTaj 0.005 snt; = 0.005 b dc 0.375

d - cmayBIsrsEpkxageRkAbMputeTAGkSEdkTajCYrTI1
t

0.00507
=
b
0.008

kgkrNIEdl =

max

max = 0.63375 b

PaKryEdkenHeFVI[FwmmanlkNyWtRKb;RKan;munnwg)ak;
Casegb
sRmab; f = 276MPa = 0.5474
y

max

f y = 345MPa max = 0.5905 b

f y = 517 MPa max = 0.6983 b

sRmab;muxkat;rgkmaMgTaj = 0.9
- sRmab;muxkat;enAkgtMbn; transition region snt;
b 0.6 > dc > 0.375
viPaKebtugGarem:rgkarBt;begag

53

= 0.004

minRtUvtUcCag 0.004

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

0.00507
=
b
0.007

kgkrNIEdl =

max t

max t = 0.724 b

sRmab;muxkat;enAkgtMbn; transition region

= 0.65 + ( t 0.002)(

< 0.9

250
)
3

]TahrN_1 sRmab;muxkat;dUcbgajkgrUb
k> kMNt;muxkat;Edk balanced section
x> muxkat;EdkGtibrmaEdlGnuBaateday ACI Code sRmab; tension-controlled section nigsRmab;muxkat;
enAkgtMbn; transition region
K> TItaMgGkSNWt nigRbEvgbkkugRtaMgsgt;sRmab;muxkat;rgkmaMgTaj
smtikm f ' = 28MPa nig f = 400MPa
y

dMeNaHRsay
k> kMNt;muxkat;Edk balanced section
b = 0.851

f 'c
600
(
)
f y 600 + f y

eday f ' = 28MPa

c

b = 0.852

f y = 400 MPa

nig

= 0.85

28
600
(
) = 0.030345
400 600 + 400

muxkat;EdkEdldak;kgmuxkat;ebtugedIm,I)anlkxN balanced KW
T.Chhay

54

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

Asb = b bd = 0.030345 40 65 = 78.897cm 2

x> muxkat;EdkGtibrmasRmab; tension controlled section

0.003 +

max = (

fy
Es

0.003 + t

sRmab;

max =

) b

= 0.005
0.005
b = 0.625 b = 0.625 0.030345 = 0.019
0.008

As max = b maxbd = 0.019 40 65 = 49.4cm 2

sRmab; = 0.9

muxkat;EdkGtibrmasRmab;muxkat;kgtMbn; transition region

0.003 +

max = (

sRmab;

fy
Es

0.003 + t
t

max =

) b

= 0.004
0.005
b = 0.714 b = 0.714 0.030345 = 0.0217
0.007

As max = b maxbd = 0.0217 40 65 = 56.42cm 2

sRmab; = 0.817

K> TItaMgGkSNWt nigRbEvgbkkugRtaMgsgt;sRmab; tension-controlled section

amax =

As max f y

0.85 f 'c b

49.4 400
= 20.76cm
0.85 28 40

cmayBIsrsEpkxagelImkGkSNWtKW
c=

20.76
= 24.42cm
0.85

]TahrN_2 kMNt;ersIusg;m:Um:g;KNna nigTItaMgGkSNWtnmuxkat;ctuekaNEkgdUcbgajkgrUbxageRkam.

RbsinebIeKeRbIEdk 3DB30 ersIusg;ebtug f ' = 20MPa nig f = 400MPa
y

viPaKebtugGarem:rgkarBt;begag

55

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mhaviTalysMNg;sIuvil

NPIC

dMeNaHRsay
muxkat;Edk 3DB30 A = 21.195cm
21.195
PaKryEdkeRbIR)as;kgebtug = bdA = 30
= 0.0128
55
PaKryEdk balanced kgebtug = 0.85 ff ' ( 600600+ f ) = 0.021675
2

400
= 16.62cm
RbEvgbkkugRtaMgsgt; a = 0.85A ff ' b = 021.85.195
20 30
s

TItaMgGkSNWt

16.62
c=
=
= 19.55cm
1 0.85
a

bERmbRmYlrageFobEdksuT

0.003 +

t = (

b
tension controlled section = 0.9

fy
Es

) 0.003 = 0.0055 > 0.005

ersIusg;m:Um:g;xagkgKNna
a
2

M n = As f y (d ) = 0.9 21.195 400 (55

16.62
) 10 3 = 356.25kN .m
2

]TahrN_3 kMNt;ersIusg;m:Um:g;KNna nigTItaMgGkSNWtnmuxkat;ctuekaNEkgdUcbgajkgrUbxagelI.

EteKeRbIEdk 3DB32 vij ersIusg;ebtug f ' = 20MPa nig f = 400MPa
dMeNaHRsay
muxkat;Edk 3DB32 A = 24.1152cm
PaKryEdkeRbIR)as;kgebtug = bdA = 2430.1152
= 0.0146
55
PaKryEdk balanced kgebtug = 0.85 ff ' ( 600600+ f ) = 0.021675
c

400
= 18.91cm
RbEvgbkkugRtaMgsgt; a = 0.85A ff ' b = 240.85.1152
20 30
s

TItaMgGkSNWt

18.91
c=
=
= 22.25cm
1 0.85

bERmbRmYlrageFobEdksuT

0.003 +

t = (

fy
Es

) 0.003 = 0.0044 < 0.005

muxkat;enAkgtMbn; transition region = 0.65 + ( 0.002)( 250

) = 0.85
3
ersIusg;m:Um:g;KNna M = A f (d a2 ) = 0.85 24.1152 400 (55 162.62 ) 10
sRmab;muxkat;rgkmaMgTaj = 0.005

= 373.43kN .m

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56

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca
max =

Department of Civil Engineering

0.005
b = 0.625 b = 0.625 0.021675 = 0.01355
0.008

As max = maxbd = 0.01355 30 55 = 22.3575cm 2 < 24.1153cm 2

400
= 17.535cm
RbEvgbkkugRtaMgsgt; a = 0.85A ff ' b = 220.85.3575
20 30
s

17.535
a
M n = As f y (d ) = 0.9 22.3575 400 (55
) 10 3 = 372.11kN .m
2
2

eyIgeXIjfa tmnersIusg;mantmesIresIKa EdleKGacTTYlyk)an.

3>6>3> PaKryEdkGb,brma
RbsinebIm:Um:g;Gnuvtn_mkelIFwmmantmtUc ehIyTMhMnmuxkat;FMCagGVIEdlRtUvkarsRmab;Tb;Tl;nwg
m:Um:g; enaHkarKNnanwgbgaj[eXIjmuxkat;EdktUc bkKan. RbsinebImindak;srsEdk Fwmrgm:Um:g;nwgrg
kar)ak;Pam. ACI Code kMNt;nUvmuxkat;EdkGb,brma A
f'
1.4
A
=
b d nig
b d
f
4f
s min

s min

sRmab;krNIFwmragGkSr T EdlsabrgkmaMgTaj enaHmuxkat;EdkRtUvyktmtUcCageKrvagsmIkar

xagelI nigxageRkam
As min =

Edl

f 'c
bw d
2 fy

sRmab;muxkat;ragctuekaNEkg
CaTTwgsab

bw = b
bw

3>7> muxkat;lm
muxkat;EdlmanlkNlm RbsinebIersIusg;m:Um:g;kgnmuxkat;FMCag besIm:Um:g;xageRkA
M M . viFIsaRsGacsegbdUcxageRkam
- KNnam:Um:g;xageRkAEdlGnuvtmkelIeRKOgbgM M
n

M u = 1.2M D + 1.6M L

- KNna M sRmab;muxkat;EdlsrsEdkrgkmaMgTaj
+ RtYtBinitfa < <
A f
+ kMNt; a =
nigRtYtBinit sRmab;
0.85 f ' b
n

min

max

kMNt; M

a
= As f y (d )
2

viPaKebtugGarem:rgkarBt;begag

57

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

- RbsinebI M

Mu

enaHmuxkat;manlkNlm

]TahrN_4 eKmanFwmTRmbgb;mYyEdlmanRbEvg 2.5m . FwmenHmanmuxkat;ragctuekaNEkgdUcbgaj

kgrUb. FwmRTbnkefr EdlrYmmanbnkpal;xnrbs;vasrub 22kN / m nigbnkGefr 13kN / m . edayeRbI
f ' = 28MPa nig f = 400MPa cUrepgpat;fa FwmenHmansuvtiPaBRKb;RKan;kgkarRTbnkxagelI.
y

dMeNaHRsay
bnkKNna
Wu = 1.2 D + 1.6 L = 1.2 22 + 1.6 13 = 47.2kN / m

m:Um:g;KNna
M u = Wu

L2
2.52
= 47.2
= 147.5kN .m
2
2

muxkat;Edk

As = 11.3982cm 2
T.Chhay

58

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

PaKryEdkenAkgmuxkat;ebtug
=

As 11.3982
=
= 0.012256
bd 20 46.5

PaKryEdk balanced

f 'c
600
(
) = 0.030345
f y 600 + f y

b = 0.851

RbEvgbkkugRtaMgsgt;
a=

As f y
0.85 f 'c b

11.3982 400
= 9.578cm
0.85 28 20

TItaMgGkSNWt
c=

9.578
= 11.268cm
0.85

bERmbRmYlrageFobEdksuT
0.003 +

t = (

fy
Es

) 0.003 = 0.00938 > 0.005 tension-controlled section = 0.9

ersIusg;m:Um:g;xagkgKNna
a
2

M n = As f y (d ) = 0.9 11.3982 400 (46.5

9.578
) 10 3 = 171.155kN .m
2

muxkat;manlkNRKb;RKan;
]TahrN_5 eKmanFwmmuxkat;mYymanRbEvg 6m . FwmenHmanmuxkat;dUcbgajkgrUb. edayeRbI
f ' = 20MPa nig f = 400 MPa kMNt;bnkGefrrayesIGnuBaat. FwmenHminmanbnkefrGVIeRkABITmn;
xnvaeT.
M n > M u

viPaKebtugGarem:rgkarBt;begag

59

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

dMeNaHRsay
Tmn;pal;rbs;Fwm
WD = 30 52.5 10 4 24 = 3.78kN / m

muxkat;Edk
As = 14.71875cm 2

RbEvgbkkugRtaMgsgt;
a=

As f y
0.85 f 'c b

14.71875 400
= 17.32cm
0.85 20 20

PaKryEdkeRbIR)as;enAkgmuxkat;ebtug
=

As 14.71875
=
= 0.009345
bd 30 52.5

PaKryEdk balanced kgebtug

f 'c
600
(
) = 0.021675
f y 600 + f y

b = 0.851

bERmbRmYlrageFobEdksuT
0.003 +

t = (

fy
Es

) 0.003 = 0.0086 > 0.005 tension-controlled section = 0.9

ersIusg;m:Um:g;xagkgKNna
a
2

M n = As f y (d ) = 0.9 14.71875 400 (52.5

eday[ M = M
m:ageTot M = 1.2M
u

17.32
) 10 3 = 232.3kN .m
2

+ 1.6 M L

3.78 6 2
W
) + 1.6( L 6 2 ) = 20.412 + 7.2WL
8
8
232.3 20.412
WL =
= 29.43kN / m
7.2
232.3 = 1.2(

]TahrN_6 RtYtBinitmuxkat;dUcbgajkgrUbxageRkam edIm,ITb;Tl;nwg

m:Um:g;KNna 41kN.m . edayeRbI f ' = 20MPa nig f = 235MPa .
dMeNaHRsay
muxkat;Edk
c

As = 3.3912cm2

PaKryEdkeRbIR)as;enAkgmuxkat;ebtug
T.Chhay

60

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

As 3.3912
=
= 0.00377
bd 20 45

PaKryEdkGb,brmaeRbIR)as;enAkgmuxkat;ebtug
min = max(

f 'c 1.4
, ) = max(0.004756,0.00596) = 0.00596
4 fy fy

< min

As min = 0.00596 20 52.5 = 6.258

dUecHeKRtUveRbIEdk 3DB18 A = 7.63cm

RbEvgbkkugRtaMgsgt;
s

a=

As f y
0.85 f 'c b

> 6.258cm 2

7.63 235
= 5.274cm
0.85 20 20

ersIusg;m:Um:g;xagkgKNna
a
2

M n = As f y (d ) = 0.9 7.63 235 (45

5.274
) 10 3 = 68.4kN .m
2

M n > M u

dUecH Edk 3DB18 RKb;RKan;edIm,ITb;Tl;nwgm:Um:g;KNnaxageRkA .

3>8> bNMnEdk
enAeBlEdlkarKNnamuxkat;EdkRtUvkarsRmab;ebtugmanbrimaNeRcIn ]TahrN_enAeBlEdl
RtUv)aneRbI eBlenaHeKBi)akkgkarBRgayEdkeTAkgmuxkat;ebtug.
ACI Code )anGnuBaat[EdkbeNayGacdak;CabNMEdl manTRmg;dUcbgaykgrUb. bNMnEdk
cab;BIbYnGaceFVIeTA)anedaymanEdkkgBTCMuvij. kareFVIbNMEdkkgenHkGacRbRBwteTA)ansRmab;ssr.

max

viPaKebtugGarem:rgkarBt;begag

61

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

bNMnEdk RtUv)ancat;TukCaEdkmYyedImsRmab; kMNt;KMlatEdk nigkRmas;karBarebtug. Ggt;pit

nEdkeTalRtUv)anbMEbkBIRkLapsmmUlrbs;bNMEdk.

segb karkMNt;EdkrgkmaMgTajsRmab;muxkat;ctuekaNEkg
1> kMNt;PaKryEdkeRbIR)as;enAkgebtug = bdA
2> kMNt;PaKryEdk balanced = 0.85 ff ' ( 600600+ f ) nigPaKryEdkGtibrma
s

0.003 +

Es
) b
0.008
f 'c 1.4
= max(
, )
4 fy fy

max = (
min

fy

sRmab;muxkat;rgkmaMgTaj. dUcKa kMNt;PaKryEdkGb,brma

3> RbsinebI < < kMNt; a = 0.85A ff ' b / c / nig = 0.9 . RbsinebI <
PaKryEdkEdleRbIR)as;kgebtugminRKb;RKan; eTaHCay:agNaPaKryEdkEdleRbIR)as;kgebtug
RtUvEt . RbsinebI enaH < 0.9 .
4> kMNt;ersIusg;m:Um:g;xagkgKNna M = A f (d a2 )
s

min

max

min

min

max

3>9> muxkat;ctuekaNEkgCamYyEdkrgkmaMgsgt;
enAkgmuxkat;ebtug muxkat;EdkEdlTb;nwgm:Um:g;Bt; RtUv)ankMNt;ecjBIbnkxageRkAEdlmanGMeBI
elIeRKOgbgM edayeFVIy:agNa[ersIusg;m:Um:g;xagkgFMCag besInwgm:Um:g;xageRkA. b:uEnenAeBlEdlmux
kat;ebtug TTwg nigkm<s;RbsiTPaB mantmtUcenaH RtUv)aneRbI. RbsinebIm:Um:g;xageRkAFMCag
ersIusg;m:Um:g;xagkg enaHbrimaNEdksgt; nigEdkTajRtUv)anbEnm.
Edksgt;RtUv)aneRbI enAeBlEdlmuxkat;ebtugRtUv)ankMNt;edaymUlehtusabtkm. pl
RbeyaCn_rbs;Edksgt;KW kat;bnyPaBdabryeBlyUr nigedIm,IgayRsYldak;Edkkg.
muxkat;EdkDubmanBIrkrNIEdleKRtUvBicarNa GaRsyeTAnwgkareFVIrbs;Edkdl;cMNuc yield bGt;.
max

T.Chhay

62

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

3>9>1> enAeBlEdksgt;eFVIkardl;cMNuc yield

m:Um:g;xagkgGacRtUv)anEckecjCaBIr dUcbgajkgrUb M Cam:Um:g;EdlekItBIkmaMgsgt;rbs;ebtug

nigkmaMgTajsmmUlrbs;Edk A sRmab;muxkat;eKal. M Cam:Um:g;bEnmEdlekItBIkmaMgsgt;enAkg
Edksgt; A' nigkmaMgTajenAkgEdkrgkmaMgTajbEnm A .
u1

s1

u2

s2

m:Um:g; M Cam:Um:g;Edl)anBImuxkat;sRmab;EdkrgkarTajeKal
u1

T1 = Cc As1 f y = 0.85 f 'c ab

a=

As1 f y
0.85 f 'c b

a
M u1 = As1 f y (d )
2
0.003 +

fy

A
nigtUcCag besI = ( 0.008E ) sRmab;[muxkat;
karkMNt; M RtUv[ < bd
rgkarTajeKal.
BicarNaelIm:Um:g; M edaysnt;fa muxkat;Edkrgkarsgt; A' eFVIkardl;cMNuc yield
s

s1

u1

max

u2

viPaKebtugGarem:rgkarBt;begag

63

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

M u 2 = As 2 f y (d d ' )
M u 2 = A' s f y (d d ' )

- CacmayBIsrsEpkxageRkAbMputeTAGkSEdkrgkarsgt;
kgkrNIenH A = A' begItnUvkmaMgesIKa EtTisedApyKa
m:Um:g;srub esInwgplbUknm:Um:g; M nig M
d'

s2

u1

u2

a
2

M n = M u1 + M u 2 = [ As1 f y (d ) + A' s f y (d d ' )]

muxkat;EdksrubEdleRbIsRmab;karTajCaplbUknbrimaNEdk A nig A
dUecH A = A + A = A + A'
s1

s1

s2

s1

s2

As1 = As A' s
( A A's ) f y
a= s
0.85 f 'c b

dUecHeK)an M

a
= [( As A' s ) f y (d ) + A' s f y (d d ' )]
2
fy
0 . 003 +
Es
1 = ( ' ) max = b (
)
0 . 008
n

(1)
nigeyIgman
sRmab; f = 414MPa enaH ( ' ) 0.63375 / = 0.9 nig = 0.005
kar)ak;rbs;FwmbNalmkBIEdksrubrgkarTajeFVIkardl;cMNuc yield ehIykarEbkPamrbs;ebtugRtUv)an
eCosvag.
RbsinebI = ( ' ) > enHmuxkat;sitenAtMbn; transition region Edl
y

max

0.003 +

fy

Es
)
0.007

( ' ) max,t = b (

enaH

kgkrNIenH < 0.9 sRmab; M nig = 0.9 sRmab; M

u1

eK)an M = [( A A' ) f (d a2 )] + 0.9 A' f (d d ' )

cMNaMfa ( A A' ) bd
enAkgtMbn;sgt; kmaMgEdkrgkarsgt;KW C = A' ( f
edayKitfapebtugEdlCMnYsedaypEdk A' enaH
n

u2

max,t

0.85 f 'c )

T = As f y = Cc + C s = 0.85 f 'c ab + A' s ( f y 0.85 f 'c )

As f y A' s f y + 0.85 f 'c A' s = 0.85 f 'c ab

eday 0.85 f ' ab = A

c

T.Chhay

s1

fy
64

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

As f y A' s f y + 0.85 f 'c A' s = As1 f y

EckGgTaMgBIrnwg bdf
' (1 0.85

f 'c
) = 1
fy

dUecH ' (1 0.85 ff ' )

Edl

As1
max
bd

0.003 +

max

= b (

fy

Es
)
0.008

(2)

PaKryEdkrgkarTajsrubGtibrma EdleRbIenAkgmuxkat;ctuekaNEkg enAeBlEdlEdkrgkar

sgt;eFVIkardl;cMNuc yield
Max = ( max + ' )

mannyfa muxkat;EdkrgkarTajsrubeRbIenAkgmuxkat;ctuekaN enAeBlEdkrgkarsgt;eFVIkardl;

cMNuc yield MaxA = bd ( + ' )
edIm,I[dwgfa Edkrgkarsgt;eFVIkardl;cMNuc yield eyIgRtUvBinitbERmbRmYlrageFob eday[
s

's y =

max

fy
Es

tamrUbxagelI eyIg)an
c
=
d'

0.003
0.003

fy

600
600 f y

Es
600
)d '
c=(
600 f y

eyIgman A f = 0.85 f ' ab

b:uEn A = A A' nig = '
s1 y

s1

viPaKebtugGarem:rgkarBt;begag

65

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

dUecHeyIg)an ( A A' ) f
s

= 0.85 f 'c ab

( ' )bdf y = 0.85 f 'c ab

( ' ) = 0.85(

f 'c a
)( )
fy d

eday a = c = ( 600600 f
1

)d '

dUecH ( ' ) = 0.85 ( ff ' )( dd' )( 600600 f

)=K

RbsinebI ( ' ) K enaHEdkrgkarsgt;eFVIkardl;cMNuc yield.

eyIgeXIjfa enAeBlEdlbrimaNEdkrgkarTajeKal A ekIneLIg enaH T nig C kmantmkan;
EtFMEdr ehIyGkSNWtnwgFak;cuH eBlenaHbERmbRmYlrageFobrbs;Edkrgkarsgt;kekIneLIg rhUtdl;cMNuc
yield.
s1

T.Chhay

66

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

]TahrN_7 FwmctuekaNEkg EdlmanTTwg 30cm nigkm<s;RbsiTPaB d = 60cm . EdkrgkarTajman

6DB 28 teRmobCaBIrCYr Edkrgkarsgt;man 2 DB 22 . kMNt;ersIusg;m:Um:g;xagkgRbsinebIeKeRbI
f ' = 28MPa nig f = 400MPa .
c

dMeNaHRsay
muxkat;EdkrgkarTaj A = 36.93cm PaKryEdkrgkarTaj = 3036.9360 = 0.02052
muxkat;Edkrgkarsgt; A' = 7.6cm PaKryEdkrgkarsgt; ' = 307.660 = 0.0042
muxkat;EdkrgkarTajeKal A = 29.33cm PaKryEdkrgkarTajeKal = 3029.3360 = 0.01629
2

s1

K = 0.851 (

28 6
600
600
f 'c d '
= 0.01517
)( )(
) = 0.852
400 60 600 400
f y d 600 f y

eday ( ' ) K enaHEdkrgkarsgt;eFVIkardl;cMNuc yield

sRmab; f ' = 28MPa nig f = 400MPa = 0.030345
eday ( ' ) < = 0.9
ersIusg;m:Um:g;xagkg
c

max

= 0.019

max

a
2

M n = [( As A' s ) f y (d ) + A' s f y (d d ' )]

a=

( As A's ) f y
0.85 f 'c b

a = 16.43cm

M n = 0.9[29.33 400 (60

16.43
) + 7.6 400 (60 6)] 10 3 = 694.5kN .m
2

viFIm:ageTot epgpat;faetIEdkrgkarsgt;eFVIkardl;cMNuc yield benA

c=

a
16.43
=
= 19.33cm
0.85 0.85

viPaKebtugGarem:rgkarBt;begag

67

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

sac;lUteFobEdkrgkarsgt; ' = c c d ' 0.003 = 1919.33.33 6 0.003 = 0.00207

sac;lUteFobrbs;Edk = 0.002
eday ' > Edkrgkarsgt;eFVIkardl;cMNuc yield
s

(60 + 6) 19.33
d c
t = ( t )0.003 =
0.003 = 0.007 > 0.005
19.33
c
c 19.33
=
= 0.322 < 0.375
d
60

b
muxkat;EdkrgkarTajsrub

RtwmRtUv

MaxAs = bd ( max + ' ) = 30 60 (0.019 + 0.0042) = 41.76cm 2 > As

3>9>2> enAeBlEdksgt;eFVIkarmindl;cMNuc yield

dUckarbkRsayxagelI RbsinebI ( ' ) < 0.85 ( ff ' )( dd' )( 600600 f

)=K

enaHEdksgt;eFVIkarmindl;cMNuc yield eT. enHbgajfa RbsinebI ( ' ) < K EdkrgkarTajeFVI

kardl;cMNuc yield mun ebtugmanbERmbRmYlrageFobGtibrma 0.003 ehIyEdkrgkarsgt;keFVIkarmindl;
cMNuc yield Edr. pleFob d ' c kan;EtFM mannyfakalNaeKdak;Edkrgkarsgt;enACitGkSNWt enaHsac;
lUteFobrbs;Edkrgkarsgt;kan;EttUc.
RbsinebIEdksgt;eFVIkarmindl;cMNuc yield dMeNaHRsayTUeTAGaceFVIeTA)anedayQrelIeKalkarN_
saTic.
' s = 0.003(

eday[ C

c d'
)
c

f ' s = E s ' s = 600(

c d'
)
c

= 0.85 f 'c 1cb

C s = A' s ( f ' s 0.85 f 'c ) = A' s [600(

edaysar T = A f
s

= Cc + C s

enaH

As f y = 0.85 f 'c 1cb + A' s [600(

c d'
)0.85 f 'c ]
c

c d'
)0.85 f 'c ]
c

(0.85 f 'c 1b)c 2 + [(600 A' s ) (0.85 f 'c A' s ) As f y ]c 600 A' s d ' = 0

smIkarenHmanTRmg; A c
eRkayeBlKNna c
KNna f ' = 600( c c d ' )

+ A2 c + A3 = 0

KNna a = c KNna C A' [600( c c d ' )0.85 f ' ] nigKNna

1

Cc = 0.85 f 'c 1cb

T.Chhay

68

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

a
2

M n = [Cc (d ) + C s (d d ' )]

enAeBlEdksgt;eFVIkarmindl;cMNuc yield/
ctuekaNEkgKW
MaxAs = max bd + A' s

nigEdkTajsrubRtUvkarsRmab;muxkat;

f 's < f y

f 's
' f 's
= bd ( max +
)
fy
fy

edayEckGgTaMgBIrnwg bd eyIg)anPaKryEdk
Max =

MaxAs
f'
max + ' s
bd
fy

b ( ' ff ' )
s

max

kgkrNIenH a = A 0f.85 fA'' bf '

s

a
2

M n = [( As f y A' s f ' s )(d ) + A' s f ' s (d d ' )]

segb viFIsaRsviPaKmuxkat;CamYyEdkrgkarsgt;
1> kMNt; / ' / ( ' ) dUcKakMNt; /
2> kMNt; K = 0.85 ( ff ' )( dd' )( 600600 f )
max

min

3> RbsinebI ( ' ) K enaHEdkrgkarsgt;eFVIkardl;cMNucyar f ' = f . RbsinebI

( ' ) < K enaHEdkrgkarsgt;eFVIkarmindl;cMNuc yield f ' < f .
4> RbsinebIEdkrgkarsgt;eFVIkardl;cMNuc yield
k> BinitemIl ( ' ) b 0.005 / eRbI = 0.9
x> kMNt; a = ( A0.85 Af'' )bf
s

max
s

min

K> kMNt; M = [( A A' ) f (d a2 ) + A' f (d d ' )]

X> muxkat;EdkrgkarTajGtibrma A EdlGaceRbIenAkgmuxkat;KW
n

MaxAs = bd ( max + ' ) As

5> RbsinebIEdkrgkarsgt;eFVIkarmindl;cMNuc yield

k> KNnacmayGkSNWt c edayeRbIsmIkar T = C
x> kMNt; f ' = 600( c c d ' )

+ Cc

viPaKebtugGarem:rgkarBt;begag

69

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

K> RtYtBinit ( ' ff ' ) b MaxA EdlGaceRbIenAkgmuxkat; RtUvEtFMCagbesI A

s

max

Edl)aneRbI
MaxAs = bd ( max + '

f 's
) As
fy

X> kMNt; a = A 0f.85 fA'' bf ' b a = c

s

g> kMNt; M = [( A f A' f ' )(d a2 ) + A' f ' (d d ' )]

]TahrN_8 kMNt;ersIusg;m:Um:g;kgnmuxkat;dUcbgajkgrUb edayeRbI f ' = 35MPa / f = 400MPa . eK
eRbIEdkrgkarsgt; 3DB25 Edl A' = 14.72cm nigEdlrgkarTaj 6DB30 Edl A = 42.39cm .
n

dMeNaHRsay
A'
14.72
kMNt; = bdA = 3542.3957 = 0.02125 / ' = bd
=
= 0.00738 / ( ' ) = 0.01387
35 57
eday f ' = 35MPa = 0.85 0.05( f ' 728 ) = 0.85 0.05( 35 7 28 ) = 0.8
35 6.5
600
kMNt; K = 0.85 ( ff ' )( dd' )( 600600 f ) = 0.85 0.8( 400
)( )(
) = 0.020355
57 600 400
s

eday ( ' ) < K enaHEdkrgkarsgt;eFVIkarmindl;cMNuc yield

f 'c
600
(
) = 0.0357
f y 600 + f y

b = 0.851
max =

0.005
0.0357 = 0.02231
0.008

muxkat;rgkarTaj = 0.9
kMNt;cmayGkSNWt c
C = 0.85 f ' ab eday a = c = 0.8c C = 0.85 35 0.8c 350 = 8330c
( ' ) < max

T.Chhay

70

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

C s = A' s ( f ' s 0.85 f 'c )

c d'
c 65
c 65
f ' s = 600(
) C s = 1472[600(
) 0.85 35] = 883200(
) 43792
c
c
c

eday

T = As f y = 4239 400 = 1695600 N

1695600 = 8330c + 883200(

c 65
) 43792
c

8330c 2 856192c 57408000 = 0

c = 149mm = 14.9cm
a = 0.8 14.9 = 11.92cm
c d'
14.9 6.5
f ' s = 600(
) f ' s = 600
= 339MPa
c
14.9

kMNt;
kMNt; C = 0.85 f ' ab C = 0.85 35 119.2 350 = 1241170N = 1241.17kN
kMNt; C = A' ( f ' 0.85 f ' ) C = 1472(339 0.85 35) = 455216 N = 455.216kN
edIm,IkMNt;ersIusg;m:Um:g;kg eKRtUvKitm:Um:g;eFobGkSEdkTaj A
c

a
2

M n = [Cc (d ) + C s (d d ' )] = 0.9[1241.17(0.57

0.1192
) + 455.216(0.57 0.065)]
2

M n = 863.38kN .m

RtYtBinit ( ' ff ' )

s

max

(0.02125 0.00738

kMNt;muxkat;EdkTajGtibrma MaxA = bd (
s

MaxAs = 35 57(0.02231 + 0.00738

max

+ '

339
) = 0.015 < 0.02231
400
f 's
)
fy

339
) = 56.99cm 2 > 42.39cm 2
400

RtwmRtUv

RtwmRtUv

14.9
c
=
= 0.25 < 0.375
d t 57 + 9 6.5
d c
t = t
0.003 = 0.009 > 0.005
c

muxkat;rgkarTaj

3>10> viPaKmuxkat;GkSret T nigmuxkat;GIu I

CaFmtakRmalxN nigFwmRtUv)aneKcak;CamYyKa edIm,IbegItCaeRKOgbgMEtmYy (monolithic
structure). kRmalxNmankRmas;esIgCagFwm. eRkamGMeBInkugRtaMgBt; EpknkRmalxNEdlCaEpk
rbs;FwmrgnUvkugRtaMgsgt;GaRsyeTAelITItaMgGkSNWt. EpknkRmalxNEdleFVIkarCamYyFwmRtUv)aneK
[eQaHfa sab (flange) EdlbgajkgrUbedayp bt . EpknFwmEdlenAsl; Edlbgajedayp
(h t )b RtUv)aneK[eQaHfa RTnug (stem b web).
w

viPaKebtugGarem:rgkarBt;begag

71

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

sRmab;muxkat;GkSr I mansabBIr KWsabrgkarsgt;EdlcUlrYmeFVIkar nigsabrgkarTajEdlKanRb

siTPaB BIeRBaHvaenABIeRkamGkSNWt ehIyEdlminRtUv)aneKykvamkKit. dUecH karviPaK nigkarKNna
Fwmmuxkat; I manlkNdUcKanwgFwmmuxkat; T .

3>10>1> TTwgRbsiTPaB
sRmab;muxkat;GkSr T EdlsabmanRbEvgEvg kugRtaMgsgt;manragCa):ar:abUl EdltmGtibrma
sitenAelIFwm ehIytmGb,brmasitenAcmay x BImuxrbs;Fwm. ehIykugRtaMgkERbRbYlBIsrsEpkxag
elIsab mksrsEpkxageRkamsab BIGtibrmamkGb,brma.

tmbERmbRmYlenHGaRsyeTAnwgTItaMgGkSNWt.
kugRtaMgsmmUl CakugRtaMgBRgayesImanGMeBIelITTwgsabsmmUl b . TTwgRbsiTPaB b RtUv)an
eKkMNt;edayGnuKmneTAnwg
e

T.Chhay

72

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

RbelaHElVg s
TTwgRTnug b
TMnak;TMngrvagkRmas;kRmalxN nigkm<s;srubrbs;Fwm
lkxNTRmrbs;Fwm samBa bCab;
lkxNbnk BRgayesI bcMcMNuc
pleFobrvagRbEvgFwmcenaHm:Um:g;sUn nigTTwgRTnug nigcmayrvagRTnug
1

)ankMNt;nUvTTwgRbsiTPaBedaykMNt;yktmGb,brmansmIkarxageRkam
L
-b =
Edl L CaRbEvgFwm
4
- b = 16t + b Edl t kRmas;kRmalxN nig b TTwgRTnug
- b = b Edl b cmayBIcenaHGkSkRmalxN
muxkat;ragGkSr T bmuxkat;ragGkSr I GacRtUvviPaKCaragctuekaNEkg bragGkSr T GaRsyelITI
taMgGkSNWt.
ACI Code
e

3>10>2> muxkat;GkSret T RtUv)anKitCaragctuekaNEkg

kgkrNIenH km<s;nbkkugRtaMgsmmUl a sitenAkgsab a t begIt)anCapkugRtaMgsgt;esI
nwg b a . muxkat;ebtugBIeRkamGkSNWtRtUv)aneKsnt;faKanRbsiTPaB ehIymuxkat;RtUv)aneKKitfaman
EdkrgkarTaj Edl)anBnl;BIxagelI edayRKan;EtCMnYs b eday b .
dUecH a = 0.85A ff' b
e

nig M

a
= As f y (d )
2

viPaKebtugGarem:rgkarBt;begag

73

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

RbsinebI km<s; a ekIneLIgeday a = t enaH M

kgkrNIenH t = 0.85A ff' b b A = 0.85 ff ' b t
s

t
= As f y (d )
2

sRmab;karviPaKenH A A nig
s

s max

0.005

3>10>3> viPaKmuxkat;ragGkSret T
kgkrNIenH GkSNWtsitenAelIRTnug. EpkxHrbs;ebtugenAkgRTnugmanRbsiTPaBkgkarTb;Tl;
nwgm:Um:g;xageRkA.
kmaMgsgt; C = 0.85 f ' [b t + b (a t )]
TItaMgrbs; C sitenAelITIRbCMuTmn;rbs;pragGkSr T enAcmay z BIsrsEpkxageRkAbMput.
c

T.Chhay

74

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

karviPaKmuxkat;ragGkSr T manlkNRsedogKanwgkarviPaKmuxkat;ebtugEdlEdkrgkarsgt;
edaycat;Tukpebtug (b b )t smmUleTAnwgEdksgt; A' . karviPaKenHEckecjCaBIrEpkdUcbgajkg
rUbxageRkam
e

- muxkat;eKalragctuekaNEkg b d nigmuxkat;Edk A . kmaMgsgt; C

a
T = A f ehIyRbEvgdXas; (d ) .
2
w

s1

s1

= 0.85 f 'c abw

nigkmaMg

viPaKebtugGarem:rgkarBt;begag

75

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

- muxkat;Edlmansabebtugsgxag 2 [(b b )t ] / 2 begIt)anCakmaMgsgt;edayKuNCamYy

t
0.85 f ' nigRbEvgdXas;esInwg (d ) . RbsinebI A Camuxkat;EdkTajEdlbegItkmaMgesInwg
2
kmaMgsgt;EdlbegItedayebtugsabsgxag dUecH A = 0.85 f ' ft (b b )
e

sf

sf

muxkat;Edksrub A EdleRbIkgmuxkat;GkSr T KW A = A + A
b A = A A
muxkat;GkSr T sitkgsanPaBlMnwg dUecH C = T / C = T nig C = C + C
BicarNaelIsmIkar C = T sRmab;muxkat;eKalctuekaNEkg eK)an
A f = 0.85 f ' ab b ( A A ) f = 0.85 f ' ab
dUecH a = (0A.85 fA' b) f
cMNaMfa b RtUv)aneRbIedIm,IkMNt; a .
ersIusg;nm:Um:g;kgnmuxkat;CaplbUknm:Um:g;BIr M nig M
s1

sf

sf

s1

s1

sf

= T1 + T2 + T

sf

u1

u2

M n = M u1 + M u 2
a
a
M u1 = As1 f y (d ) = ( As Asf ) f y (d )
2
2
( As Asf ) f y
As1 = As Asf
a=
0.85 f 'c bw
t
M u 2 = Asf f y (d )
2
a
t
M n = [( As Asf ) f y (d ) + Asf f y (d )]
2
2

nig

Edl

BicarNaelImuxkat;RTnug b d / sac;lUteFobsuT GackMNt;BI a / c nig d dUcxageRkam

RbsinebI c = a nig d = h 6.5cm bnab;mk = 0.003 (c cd )
sRmab;muxkat;rgkarTajenAkg RTnug/ 0.005 .
karKNnaersIusg;m:Um:g;kgsRmab;muxkat;GkSr T bmuxkat;GkSr I
GacKNnaedayeRbIsmIkarxagelI EteKcaM)ac;RtUvRtYtBinitlkxNxageRkam
- PaKryEdkTajsrubeFobRkLapRbsiTPaBRTnugRtUvFMCag besI
w

min

w =

As
min
bw d

min =

f 'c
4 fy

1.4
fy

- RtYtBinit bERmbRmYlrageFobsuTFMCag besI

T.Chhay

76

0.005

sRmab;muxkat;rgkarTaj
Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

- muxkat;EdkGtibrma MaxA enAkgmuxkat;GkSr T RtUvEtFMCag besI muxkat;EdkEdl)aneRbI A

sRmab;muxkat;rgkarTaj CamYy = 0.9
s

MaxAs = Asf ( flange) + max (bw d )( web)

1
[0.85 f 'c t (b bw )] + max (bw d )
fy

MaxAs =

PaKryEdkeFobnwgRTnug

Asf
As
( max +
)
bw d
bw d

w f max

smIkarTUeTAsRmab;KNna MaxA enAkgmuxkat;GkSr T enAeBl a > t GackMNt;tam

s

C = 0.85 f 'c [(be bw )t + abw ]

= 0.375 sRmab;RTnug
sRmab; = 0.003 nig = 0.005 / dc = 0.0030.003
+ 0.005
dUecH a = c = 0.375 d
muxkat;EdkGtibrmaesInwg Cf
c

dUecH MaxA = 0.85 ff ' [(b b )t + 0.375 b d ]

c

1 w

segb viFIsaRsviPaKmuxkat;GkSret T bGkSrGil L pab;

1> kMNt;TTwgRbsiTPaB b nigkMNt; /

2> kMNt; a = 0.85A ff' b
3> RbsinebI a < t enaHmuxkat;eFVIkarCaragctuekaNEkg
- kMNt; M = A f (d a2 )
cMNaMfa c = a nig = 0.003 (c cd ) 0.005 sRmab;muxkat;rgkarTaj = 0.9
max

min

- RtYtBinit

As
min
bw d

- MaxA = f1 [0.85 f ' t (b b )] +

s

max

(bw d ) As

4> RbsinebI a > t enaHmuxkat;eFVIkarCaragGkSret

k> kMNt; A = 0.85 f ' ft (b b )
c

sf

viPaKebtugGarem:rgkarBt;begag

77

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

x> kMNt; a = ( A0.85 Af'' )bf

K> RtYtBinit eFobnwgRkLapRTnug
Edl = bAd nig = bA d
s

max

sf

bRtYtBinit

f'
MaxAs = 0.85 c [(be bw )t + 0.3751bw d ] As
fy

X> kMNt; a = (0A.85 fA' b) f

s

sf

/ sRmab; = 0.9

g> kMNt; M = [( A A ) f (d a2 ) + A f (d 2t )]
]TahrN_9 FwmebtugGarem:EdlmanRbEvg 4.5m ehIymanKMlatBImYyeTAmYyRbEvg 2m . FwmenHRTkM
ralxNEdlmankRmas; 10cm . kMNt;nUversIusg;m:Um:g;kgrbs;FwmkNal. eKeRbI f ' = 20MPa nig
f = 400 MPa .
n

sf

sf

dMeNaHRsay
kMNt;TTwgRbsiTiPaB
450
L
be = min{16t + bw ; ; b} = min{16 10 + 25;
;200} = 112.5cm
4
4
T.Chhay

78

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

kMNt;km<s;bkkugRtaMg
A f
/ A = 14.72cm
a=
0.85 f ' b
s

a=

14.72 400
= 3.08cm < t
0.85 20 112.5

dUecHeyIgRtUvKNnaCaragctuekaNEkgEdlmanTTwg b = 112.5cm
PaKryEdkGb,brma = 4 ff ' 1f.4 = 0.0035
e

min

min

PaKryGtibrma

max

= 0.625 0.85 1

PaKryEdkeFobnwgpRkLaRTnug

f 'c
fy

600
) = 0.01355
600 + f y

As
14.72
=
= 0.01472 > 0.0035
bw d 25 40

= 3.62cm
TItaMgGkSNWt c = a = 30..08
85
1

bERmbRmYlrageFobsuTrbs;Edk = 0.003( d c c ) = 0.003( 403.623.62 ) = 0.03 > 0.005 = 0.9

KNna M = A f (d a2 ) = 0.9 1472 400(400 302.8 ) = 203807232N .mm = 203.81kN .m
epgpat;muxkat;Gtibrma MaxA = f1 [0.85 f ' t (b b )] + (b d ) A
t

max

RtwmRtUv
]TahrN_10 KNnaersIusg;m:Um:g;kgnmuxkat;GkSr T dUcbgajkgrUb edayeRbI
f = 400 MPa .
MaxA = 37.22cm 2 > As

f 'c = 25MPa

nig

dMeNaHRsay
eK[ b = b = 90cm / b
e

= 25cm d = 43cm

viPaKebtugGarem:rgkarBt;begag

nig A

79

= 36.93cm 2

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

KNna a = 0.85A ff' b = 036.85.93 25 400

= 7.72cm > t
90
eday a > t sikSaCaragGkSr T
KNna A = 0.85 f ' ft (b b ) = 24.17cm
s

sf

As1 = As Asf = 12.76cm 2

epgpat;

As1 f y

a ( web) =

0.85 f 'c bw
a ( web)
c=
= 11.29cm

12.76 400
= 9.6cm
0.85 25 25

d t = 52 6.5 = 45.8cm

t = 0.003(

dt c
) = 0.00917 > 0.005 = 0.9
c

RtYtBinit A
KNna M

s min

= min bw d = 0.0035 25 43 = 3.76cm 2 < 36.93cm 2

RtwmRtUv

a
t
= [( As Asf ) f y (d ) + Asf f y (d )]
2
2
96
70
M n = 0.9[(3693 2417)400(430 ) + 2417 400(430 )
2
2
n

M n = 519172920 N .mm = 519.173kN .m

3>11> TMhMnmuxkat;FwmGkSr T eka

eBlxH FwmGkSr T eka RtUv)aneRbIedIm,IbEnmprgkarsgt;. muxkat;enHRtUv)aneKeRbIsRmab;Fwm
EdleKcak;TukCamun.
ACI Code )anENnaMnUvTMhMmuxkat;sRmab;GkSr T ekadUcxageRkam
- kRmas;sab t RtUvFMCag besIBak;kNalTTwgRTnug b
- TTwgsrubrbs;sab b RtUvEttUcCag besIbYndgTTwgRTnug b
w

T.Chhay

80

Flexural Analysis of Reinforced Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

3>11> muxkat;GkSr L pab;

Fwmmuxkat;GkSr L pab;CaFwmEdlRTkRmalxNEpkxageKbMput. TTwgRbsiTPaBrbs;muxkat;enHRtUv
)ankMNt;nUvtmGb,brmansmIkarxageRkam
- (b b ) 12L
- (b b ) 6t
- (b b ) 2l
Edl L - RbEvgFwm
l - KMlatFwm
e

viPaKebtugGarem:rgkarBt;begag

81

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

IV.

karKNnaFwmebtugGarem:rgkarkac;begag

4>1> km<s;RbsiTPaBsRmab;Fwm nigkRmalxN

Fwmmankm<s;x<s; bTabGaRsyeTAnwgRbEvgrbs;Fwm nigbnkxageRkAEdlvaRtUvRT. edIm,Iepgpat;
CamYYyPaBdab xageRkamenHCarUbmnsRmab;kMNt;km<s;FwmGaRsy nwgRbEvgFwm.

kRmalxN
mYyTis

L/20

L/24

L/28

L/10

Fwm

L/16

L/18.5

L/21

L/8

TTwgFwmRtUv)ankMNt;ecjBIkm<s;rbs;Fwm edaykMNt;enAcenaH 13 d 12 d .
4>2> muxkat;ctuekaNEkgCamYyEdkrgkarTaj
xageRkamCarUbmnsRmab;karKNnamuxkat;rgkarTaj
b = 0.851

f 'c
600
(
)
f y 600 + f y

0.003 +

max = b (

sRmab;

fy

Es
)
0.008

f y = 400 MPa

max = 0.625 b

sRmab;ebtugEdlmanersIusg; f ' 28MPa .

f ' 28
= 0.85 0.05(
) sRmab;ebtugEdlmanersIusg; 28MPa < f ' 56MPa .
7
= 0.65 sRmab;ebtugEdlmanersIusg; f ' > 56MPa .
tmnbERmbRmYlrageFobrbs;Edkkan;EtFM bgajBIPaBsVitrbs;ebtugGarem:kan;EtFM )annyfaenA
eBlmuxkat;ebtugkan;EtFMCamYyPaKryEdkticPaBsVitkan;EtFM pymkvijenAeBlmuxkat;ebtugkan;EttUcCa
ehIy

= 0.85

T.Chhay

82

Flexural Design on Reinforced Concrete Beams

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

mYyPaKryEdkkan;EtFMenaHPaBsVitrbs;ebtugkan;EttUc. xageRkamCataragbgajBIPaKryEdksMNUmBr
GaRsyersIusg;ebtug nigersIusg;Edk.
taragTI1 PaKryEdksMNUmBr
s

f y (MPa)

f 'c ( MPa)
20

235
400
400
500
400
500

28
35

% s
1.4
1.2
1.4
1.2
1.6
1.4

bERmbRmYlrageFobEdksuT edIm,IeGaymuxkat;rgkarTajRtUv)ankMNt;FMCag 0.005 enaH

= 0.9 .
250
= 0.65 + ( 0.002)(
) bERmbRmYlrageFobEdksuT enAcenaH 0.004 0.005 .
3
smIkarKNnaersIusg;m:Um:g;kgmanTRmg;dUcxageRkam
t

M n = M u = Ru bd 2

Edl R = f (1 1.7ff ' ) = R

Edl = 0.9 sRmab; tension-controlled section
nig < 0.9 sRmab;muxkat;enAkgtMbn; transition region
dUcenH M = M = A f (d 1.A7 f f' b )
y

dUcKa M = M = f bd ( 1.7ff ' )

kgkrNIEdleyIgsal;m:Um:g;KNnaxageRkA nigsal;ersIusg;smar eyIgenAsl;GBaatbIeTotEdl
minTan;sal; kgenaHmanTTwgFwm b km<s;RbsiTPaB d nigPaKryEdk . dMeNaHRsayGaceFVIeTA)anluH
RtaEteyIgRtUveFVIkarsnt;cMeBaHGBaatBIr. CaTUeTA RtUv)ansnt; edayeRbI nigdUcKa b kRtUv)an
snt;.
xageRkamCaviFIsaRskgkaredaHRsayedaysal; M / f ' nig f
- RbsinebI RtUv)ansnt; edayeGayenAcenaH nig enaHeyIgGackMNt;
2

max

1
2

Ru = f y (1

f y

1.7 f 'c

max

1
2

karKNnaFwmebtugGarem:rgkarkac;begag

83

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

bd 2 =

Mu
Ru

CaTUeTApleFob db ERbRbYlBI 2 3 eKniymyk 2

dUcenH eKGackMNt; b nig d
dUcKa eyIg)an A = bd
- RbsinebI b nig d RtUv)aneGay enaHPaKryEdkEdlRtUvkarkMNt;tamrUbmnxageRkam
s

0.85 f 'c
4M u
(1 1
)
fy
1.7f 'c bd 2

As = bd

- RbsinebI b RtUv)ansnt;bEnmBIelI enaHeKRtUv

KNna R = f (1 1.7ff ' )
y

KNna d =

Mu
Ru b

RbsinebI db = 2 enaH d =

2M u
Ru b

As = bd

4>3> KMlatEdk nigRsTab;karBarEdk

4>3>1> KMlatEdk

T.Chhay

84

Flexural Design on Reinforced Concrete Beams

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

srsEdkRtUv)anteRmobedaymanKMlaty:agticbMputesInwgGgt;pitEdk b:uEnminRtUvtUcCag 25mm

edIm,IeGayeKGacbgab;ebtug)any:aggayRsYlkgeBlcak;ebtug. KMlatEdksRmab;karteRmobEdkbBar
EdlmaneRcInCagmYyRsTab; minRtUvmantmtUcCag 25mm Edr. RbsinebIKMlatEdkmanTMhMtUcenaHl,ay
ebtugminGacBTCMuvijEdk)anleT.
4>3>2> RsTab;karBarEdk
RsTab;karBarEdk Casac;ebtugEdlenAcenaHpxageRkA nigprbs;Edk. eKcaM)ac;RtUvkarRsTab;kar
BarEdkeRBaHvamanplRbeyaCn_bYny:ag
- edIm,IetagsitsrsEdkeTAnwgebtugEdleFVIeGaysmarTaMgBIreFVIkarCamYyKa. \TiBlrbs;PaB
sitGaRsyeTAnwgkRmas;RsTab;karBar.
- edIm,IkarBarsrsEdkeTAnwgERcHsIuEdk.
- edIm,IkarBarkar)at;bg;ersIusg;EdkEdlbNalmkBIkemA. kRmas;RsTab;karBar 20mm GacTb;
Tl; nwgePIgeqH)an 1em:ag.
- sRmab;yandan eragcRk cMNtrfyn RsTab;karBarbEnmRtUv)aneKdak;BIelIkRmalxNEfmeTot
edIm,IkarBarkarswkercrwlEdlbNalmkBIcracrN_.
kRmas;RsTab;karBarEdkGaRsyeTAnwgmCdanEdleRKOgbgMenaHsitenA. xageRkamCatarag
bgajBIkRmas;RsTab;karBarGb,brma
mCdan
kRmas;karBarEdk (mm)
ebtugcak;pal;nwgdI
75
ebtugcak;pal;nwgdI bhalxl;
+ Ggt;pitEdkFMCag 16mm
50
+ Ggt;pitEdktUcCag 16mm
35
ebtugmincak;pal;nwgdI bminhalxl;
- kRmalxN CBaaMg
+ Ggt;pitEdkFMCag 36mm
35
+ Ggt;pitEdktUcCag 36mm
20
- Fwm ssr
35
- kRmalekag
karKNnaFwmebtugGarem:rgkarkac;begag

85

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

Ggt;pitEdkFMCag 20mm
+ Ggt;pitEdktUcCag 20mm
+

20

15

4>3>3> TTwgGb,brmarbs;muxkat;ebtug
smIkarTUeTAedIm,IkMNt;TTwgGb,brmarbs;muxkat;ebtugGacsresrdUcxageRkam
b = nD + (n 1) s + 2(Edkkg ) + 2(RsTab;karBarEdk )
Edl n - cMnYnEdkbeNay
D - Ggt;pitEdkEdlFMCageK
s - KMlatEdk
min

4>3>4> km<s;Gb,brmarbs;muxkat;ebtug
km<s;rbs;muxkat;ebtugRtUv)ankMNt;edayGaRsynwgRsTab;Edk.
- EdkmYyRsTab; h = d + D2 + 50mm
- EdkBIrRsTab; h = d + D + 60mm
]TahrN_1 kMNt;muxkat;Edk nigmuxkat;ebtugedIm,ITb;Tl;nwgm:Um:g;KNna 490kN .m edayeRbIPaKryEdk
Gtibrma sRmab;muxkat;rgkarTaj. smtikm f ' = 20MPa nig f = 400MPa .
1

max

dMeNaHRsay
eday f ' = 20MPa /
c

b = 0.851
T.Chhay

f y = 400MPa 1 = 0.85

nig = 0.9 sRmab; tension-controlled section

f 'c
600
(
)
f y 600 + f y
86

Flexural Design on Reinforced Concrete Beams

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

20
600
(
) = 0.021675
400 600 + 400
fy
0.003 +
Es
max = b (
) = 0.625 b = 0.01355
0.008
max f y
0.01355 400
Ru (max) = max f y (1
) = 0.9 0.01355 400(1
) = 4.1MPa
1.7 f 'c
1.7 20

b = 0.85 2

bd 2 =

Mu
490 10 6
=
= 119.5 10 6 mm 3
Ru (max)
4.1

dUcenHsRmab;karsnt; b / kMNt; d nig A = bd

b = 200mm / d = 773mm / A = 20.95cm
s

/
/
b = 300mm / d = 630mm / A = 25.61cm 6DB 25
b = 400mm / d = 546.5mm / A = 29.62cm
* kareRCIserIskm<s;RbsiTPaBGaRsynwgktaxageRkam
- km<s;bnb; TTwgrbs;muxkat;tUcpl;nUvkm<s;FwmFM Edlkat;bnylMhkm<s;. elIsBIenH FwmeRCA
cegtkat;bnyersIusg;m:Um:g;edaykarxUcRTg;RTayxag lateral deformation.
- brimaN nigkarBRgaysrsEdk FwmcegtRtUvkarsrsEdkeRcInCagmYyRsTab; dUcenHvabegIt
km<s;Fwm.
- kRmas;CBaaMg RbsinebIbksIum:gt_RtUv)aneRbI TTwg b RtUv)aneRCIserIsesInwgkRmas;CBaaMg.
sRmab; GKarCBaaMgxagRkas;CagCBaaMgkg.
* kareRCIserIsmuxkat;EdkGaRsynwgktaxageRkam
- karteRmobEdkRKb;RKan;enAkgmuxkat; CaTUeTA mYyRsTab; bBIrRsTab; nigbMeBjtamlkxN ACI
Code sRmab;KMlatEdkGb,brma.
- pmuxkat;EdkeRCIserIsRtUvmantmEk,rbMputpsrsEdktRmUvkar
dUcenHeyIgeRCIserIsyk b = 300mm / d = 630mm / A = 25.61cm 6 @ DB25 RtUvteRmobBIrRsTab;
kMNt;km<s;Fwm h = d + D + 60mm = 630 + 25 + 60 = 715mm
dUcenHyk h = 750mm d = 665mm
b = 250mm d = 691.5mm As = 23.42cm 2
2

]TahrN_2 edaHRsay]TahrN_1 edayeRbI RbEhl 1% nig b = 35cm

karKNnaFwmebtugGarem:rgkarkac;begag

87

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NPIC

dMeNaHRsay
eday f ' = 20MPa /

f y = 400MPa max = 0.01355

Ru = f y (1

bd 2 =

f y

1.7 f 'c

) Ru = 0.9 0.01 400(1

sRmab; tension-controlled section

0.01 400
) = 3.1765MPa
1.7 20

M u 490 106
=
= 154257830mm3
3.1765
Ru

sRmab; b = 35cm d = 665mm

As = 0.01 35 66.5 = 23.275cm 2

edayeRCIserIs 4DB28 mYyRsTab; A = 24.62cm

epgpat;TTwg b = nD + (n 1)s + 95mm edayyk s = D
b = 7 D + 95mm = 7 28 + 95 = 291mm < 35cm epgpat;
kMNt;km<s;Fwm h = d + D2 + 50mm = 665 + 282 + 50 = 729mm yk h = 75cm
edayEdkEdldak;mantmFMCagEdkKNna dUcenHeyIgGacbnykm<s;FwmBI 75cm 72cm
75
sRmab;karERbRbYlkm<s;Fwm A = 23.275( 72
) = 24.24cm < 24.62cm
2

min

min

min

d = 720 64 = 656mm

RtYtBinitersIusg;m:Um:g;
24.62
= 0.0098 < max tension-controlled section
35 72
As f y
24.62 400
a=
= 16.55cm
=
0.85 f 'c b 0.85 20 35

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Department of Civil Engineering

a
2

M n = As f y (d ) = 0.9 2462 400 (656

RtYtBinitsac;lUteFobEdksuT
t = (

0.005

165.5
) = 508.083 106 N .mm = 508.083kN .m > 490MPa
2

) 0.003

20
600
(
) = 0.021675
400 600 + 400

0.0098
=
= 0.452
b 0.021675
0.005
t =
0.003 = 0.008 > 0.005
0.452
a
16.55
c=
=
= 19.47
0.85 0.85
c 19.47
=
= 0.3 < 0.375
dt
65.6

b = 0.85 2

epgpat;

m:ageTot

epgpat;

]TahrN_3 kMNt;muxkat;EdksRmab;muxkat;eGayxageRkam b = 25cm nig h = 50cm EdlTb;nwgm:Um:g;

KNna M = 200kN .m . smtikm f ' = 28MPa nig f = 400MPa .
u

dMeNaHRsay
edaysnt;Edk DB25 mYyRsTab;
d = 500 70 = 430mm = 43cm

eday

f 'c = 28MPa

f y = 400 MPa 1 = 0.85 b = 0.851

karKNnaFwmebtugGarem:rgkarkac;begag

89

f 'c
600
(
)
f y 600 + f y

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

28
600
(
) = 0.030345
400 600 + 400
f
0.003 + y
Es
max = b (
) = 0.625 b = 0.019
0.008
0.85 f 'c
4M u
=
(1 1
)
fy
1.7f 'c bd 2

b = 0.852

edaysnt; = 0.9

0.85 28
4 200 106
(1 1
) = 0.01356 < max
400
1.7 0.9 28 250 4302

BitR)akd

tension-controlled section

As = bd = 0.01356 25 43 = 14.58cm2
DB 25 As 25 = 4.9cm 2
A
n= s =3
As 25

dUcenH A = 3DB25 = 14.7cm

]TahrN_4 kMNt;muxkat;EdkcaM)ac;sRmab; b = 35cm RbsinebIvaRbQmnwgm:Um:g;KNna M

smtikm f ' = 28MPa nig f = 400MPa .
dMeNaHRsay
sRmab; f ' = 28MPa / f = 400MPa / = 0.85
= 0.030345 / = 0.019
sRmab;muxkat;rgkarTaj = 0.9
edayeRbI = 0.019
c

= 425kN .m

max

max

Ru (max) = max f y (1
bd 2 =

max f y

1.7 f 'c

) = 0.9 0.019 400(1

0.019 400
) = 5.75MPa
1.7 28

425 106
Mu
=
= 73.913 106 mm3
5.75
Ru (max)

sRmab; b = 35cm d = 46cm

As = 0.019 35 46 = 30.59cm 2

edayeRbIEdk 32 4edIm 4DB32 = 32.154cm

EdkRtUv)anteRmobmYyCYr b = nD + (n 1)s + 95mm
edayyk D = s b = 7 D + 95mm = 7 32 + 95 = 319mm < 350mm RtwmRtUv
km<s;Fwm h = d + D2 + 50mm = 460 + 322 + 50 = 526mm yk 53cm
2

min

min

T.Chhay

90

Flexural Design on Reinforced Concrete Beams

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

BiPakSa
edaysarsrsEdkeRbIR)as; 32.154cm FMCagsrsEdktRmUvkar 30.59cm
2

32.154
= 0.02 > max
35 46

bERmbRmYlrageFobEdksuT

=(

0.005

) 0.003 = 0.0046 > 0.004

muxkat;sitkgtMbn; transition region

250
) = 0.867 < 0.9
3
As f y
32.154 400
a=
=
= 15.44cm
0.85 f 'c b 0.85 28 35
a
154.4
M n = As f y (d ) = 0.867 3215.4 400(460
) = 426.86 106 N .mm = 426.86kN .m 425kN .m
2
2
= 0.65 + ( t 0.002)(

eyIgeXIjfa sRmab;bMErbMrYlmuxkat;Edk 32.154 30.59 = 1.546cm RbEhl 5% enaHersIusg;m:Um:g;man

tmRbEhlKa edaysar fycuH. dUcenHkarKNnamuxkat;edayeGaymuxkat;rgkarTaj = 0.9 man
lkNesdkic.
2

4>4> muxkat;ctuekaNEkgCamYyEdkrgkarsgt;
RbsinebIm:Um:g;KNnamantmFMCagersIusg;m:Um:g;kg enaHmuxkat;RtUv)armuxkat;EdkbEnmsRmab;tMbn;
rgkarTaj nigrgkarsgt;. srsEdkrgkarsgt;pl;nUvkmaMgsgt;bEnmeTAelIkmaMgsgt;rbs;ebtug.
k> edaysnt;EdkrgkarTajmanmYyRsTab;
viFIsaRsKNnasRmab;muxkat;ctuekaNEkgCamYyEdkrgkarsgt; enAeBleKsal; M / f ' / b / d
nig d ' mandUcxageRkam
- KNnaPaKryEdk balanced nigPaKryEdkGtibrma edayeRbIsmIkarxageRkam
u

max

b = 0.851

nig

f 'c
600
(
)
f y 600 + f y

0.003 +
max

s1

Ru (max) = max f y (1

fy

Es
)
0.008

= b (

kMNt;muxkat;EdkGtibrmasRmab;rgkarTaj A
- KNna R edayeRbI ( = 0.9)
u (max)

= maxbd

max

max f y

1.7 f 'c

- KNnaersIusg;m:Um:g;kgEdlekItedaysarmuxkat;EdkrgkarTaj M edayeRbI R
u1

u (max)

M u1 = Ru (max)bd 2

karKNnaFwmebtugGarem:rgkarkac;begag

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NPIC

RbsinebI M
+ RbsinebI M
+

0.85 f 'c
fy

enaHRtUvkarEdkrgkarsgt;
> M enaHminRtUvkarEdkrgkarsgt;eT. KNnaPaKryEdk tamrUbmn
4M
(1 1
) nigKNna A = bd .
1.7f ' bd
u1

< Mu

u1

- KNna M = M M CaersIusg;m:Um:g;EdlekItBIEdkrgkarsgt;
- KNna A BI M = A f (d d ' ) nigbnab;mkKNna A = A + A
- KNnakugRtaMgenAkgEdkrgkarsgt;dUcxageRkam
c d'
+ KNna f ' = 600(
) f
c
+ bKNna ' BIdaRkambMErbMrYlrageFob nig f ' = ' E . RbsinebI ' = enaHEdkrgkar
sgt; yield ehIy f ' = f .
+ KNna A' BI M = A f ' (d d ' ) .RbsinebI f ' = f enaH A' = A . EtebI f ' < f
enaH A' > A ehIy A' = A ( ff ' ) .
- eRCIserIsmuxkat;EdksRmab; A nig A' edIm,IeGaysmlmnwgTTwg b . CaTUeTA Edk A eRcIn
teRmobCaBIrRsTab; A' eRcInteRmobCamYyRsTab;.
- KNna h = d + 65mm sRmab;EdkrgkarTajmYyRsTab; nig h = d + 90mm sRmab;Edkrgkar
TajBIrRsTab;. RtYtBinitfa [ ' ( ff ' )] < edayeRbI d fI bRtYtBinit
u2

s2

u1

u2

s2

s1

s2

u2

s2

s2

s2

s2

max

As (max) = bd [ max + ' (

f 's
)] As
fy

A'
eday = bdA nig '= bd
. karRtYtBinitenHmincaM)ac;eT
s

RbsinebI RtUv)aneRbIenAkgmuxkat;eKal.
- RbsinebIcaM)ac; eKRtUvKNnaersIusg;m:Um:g;nmuxkat;cugeRkay M ehIyeRbobeFobCamYy M
edayeGay M M .
- RtYtBinitbERmbRmYlrageFobEdksuT = ( d c c )0.003 0.005
x> edaysnt;EdkrgkarTajmanBIrRsTab;
kgkrNIEdkrgkarTajmanBIrRsTab; eKGacsnt;fa d = h 90mm nig
d = h 65mm = d + 25mm . eKmanviFIBIry:agkgkarkMNt;muxkat;srsEdk
- vIFITI1 edaysnt; sac;lUteFobEdksuTenAnIv:UTIRbCMuTmn;EdkTajesI 0.005 b = 0.005
enAnIv:U d . kgkrNIenH bERmbRmYlrageFobEdksuT sRmab;srsEdkRsTab;eRkambMputman
tmFMCag 0.005 .
max

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92

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viTasanCatiBhubeckeTskm<Ca
t = (

Department of Civil Engineering

dt c
)0.003 > 0.005
c

. dUcenHeyIgedaHRsaydUckrNIxagelI EdkrgkarTajmanmYyRs

Tab;.
- vIFITI2 edaysnt; = 0.005 enAnIv:UEdkRsTab;eRkameK d . kgkrNIenH bERmbRmYlrag
eFobEdkeFobenAnIv:UTIRbCMuTmn;EdkTajmantmtUcCag 0.005 = ( d c c )0.003 < 0.005
EdlenAEtGacTTYlyk)an. dMeNaHRsaytamviFITI2enH segbdUcxageRkam
3
+ KNna d = h 65mm / c = ( )d nig a = c
8
+ KNnakmaMgsgt;enAkgebtug C = 0.85 f ' ab = T = A f
A
kMNt; A . KNna M = A f (d a2 ) . = bd
/ = 0.9
+ KNna M = M M edaysnt; d '= 65mm
+ KNna A BI M = A f (d d ' ) eday f ' = f . muxkat;EdkrgkarTajsrub
t

s1 y

s1

s1

u1

u2

u1

s2

s1 y

u2

s2

As = As1 + As 2

KNnakugRtaMgenAkgEdkrgkarsgt;dUcxageRkam
c d'
KNna f ' = 600(
) f
c
bKNna ' BIdaRkambMErbMrYlrageFob nig f ' = ' E . RbsinebI ' = enaHEdkrg
karsgt; yield ehIy f ' = f .
KNna A' BI M = A f ' (d d ' ) . RbsinebI f ' = f enaH A' = A . EtebI
f
f ' < f enaH A' > A ehIy A' = A ( ) .
f'
]TahrN_5 eKmanFwmmYymanmuxkat; b = 25cm nigkm<s; h = 55cm EdlRtUvRTnUvm:Um:g;Bt;KNna
M = 300kN .m . KNnasrsEdktRmUvkar. smtikm f ' = 20 MPa nig f = 345MPa .
dMeNaHRsay
- kMNt;ersIusg;m:Um:g;EdlekItBIEdkrgkarTajCaeKalsRmab;lkxN tension-controlled section
sRmab; f ' = 20MPa nig f = 345MPa / = 0.85
+

u2

s2

s2

s2

s2

b = 0.851

600
20
600
f 'c
(
) = 0.852
(
) = 0.0266
345 600 + 345
f y 600 + f y

fy
345
0.003 +
Es
200000 ) = 0.0157
max = b (
) = 0.0266(
0.008
0.008
f
0.0157 345
Ru (max) = max f y (1 max y ) = 0.9 0.0157 345(1
) = 4.1MPa
1.7 f 'c
1.7 20
0.003 +

karKNnaFwmebtugGarem:rgkarkac;begag

93

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mhaviTalysMNg;sIuvil

NPIC

edaysnt;EdkrgkarTajmanBIrRsTab;
d = h 90mm = 550 90 = 460mm
M n = Ru (max)bd 2 = 4.1 250 460 2 = 216.89 106 N .mm = 216.89kN .m < M u = 300 MPa

dUcenHmuxkat;enHRtuvkarEdkrgkarsgte; dIm,ITb;Tl;nwgm:Um:g;EdlenAsl;.
- KNna A / M nig M
s1

u1

u2

As1 = maxbd = 0.0157 25 46 = 18.06cm 2

M u1 = M n = 216.89kN .m
M u 2 = M u M u1 = 300 216.89 = 83.11cm 2

- KNna A nig A' Edl)anBI M edaysnt; d '= 6cm

s2

u2

M u 2 = As 2 f y (d d ' )
As 2 =

Mu2
83.11 106
=
= 669mm2 = 6.69cm2
f y (d d ' ) 0.9 345 (460 60)

EdkrgkarTajsrub A = A + A = 18.06 + 6.69 = 24.75cm

Edkrgkarsgt; A' = A = 6.69cm eFVIkardl; yield
- RtYtBinitEdksgt;eFVIkardl; yield
s

s1

s2

fy

s2

345
= 0.001725
200000 200000
As1 f y
1806 345
a=
=
= 146.6mm
0.85 f 'c b 0.85 20 250
a 146.6
c=
=
= 172.47mm
1 0.85
172.47 60
's = (
) 0.003 = 0.00196 > y
172.47

y =

eday

eFVIkardl; yield

- RtYtBinit
edayeyIgeRbI nig R mkeRbIsRmab;edaHRsay edayeGaybERmbRmYlrageFobEdksuTenAnIv:UTIRbCMu
Tmn;Edk = 0.005 . dUcenHeyIgRtUvkMNt; sRmab;EdkenARsTab;eRkameKbMput.
t

max

dt = 550 60 = 490mm
490 172.47
d c
t = ( t
)0.003 = 0.0055 > 0.005
)0.003 = (
172.47
c

RtwmRtUv

dUcenH EdktRmUvkarsRmab;karTaj A = 24.75cm eRbI 5DB25

sRmab;karsgt; A' = 6.69cm eRbI 2DB22
2

T.Chhay

94

Flexural Design on Reinforced Concrete Beams

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

]TahrN_6 eKmanFwmmYymanmuxkat; b = 30cm nigkm<s; h = 50cm EdlRtUvRTnUvm:Um:g;Bt;KNna

M = 400kN .m . KNnasrsEdktRmUvkar. smtikm f ' = 28MPa nig f = 400MPa .
dMeNaHRsay manBIrviFIsaRs
vIFITI1
- kMNt;ersIusg;m:Um:g;Gtibrmanmuxkat;EdlmanEtEdkrgkarTajCaeKal
sRmab; f ' = 28MPa / f = 400MPa / = 0.85
= 0.030345 / = 0.019
sRmab; tension-controlled section = 0.9
u

max

Ru (max) = max f y (1

max f y

1.7 f 'c

) = 0.9 0.019 400(1

0.019 400
) = 5.75MPa
1.7 28

edaysnt; d = h 90 = 500 90 = 410mm EdkmanBIrRsTab;

M u1 = Ru bd 2 = 5.75 300 4102 = 290 106 N .mm < 290MPa

dUcenHRtUvkarEdksgt;
- KNna A , M , A nig A
s1

u2

s2

As1 = maxbd = 0.019 30 41 = 23.37cm 2

M u 2 = M u M u1 = 400 290 = 110kN .m

M u 2 = As 2 f y (d d ' )
As 2 =

edaysnt; d '= 60mm

110 106
= 873mm 2 = 8.73cm 2
0.9 400 (410 60)

karKNnaFwmebtugGarem:rgkarkac;begag

95

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mhaviTalysMNg;sIuvil

NPIC

muxkat;EdkTajsrub A = A + A = 23.37 + 8.73 = 32.1cm

eRbIEdk 5DB30
- RtYtBinitEdkrgkarsgt;eFVIkardl; yield bGt;
s

s1

' K = 0.851

s2

600
f 'c d '
( )(
)
f y d 600 + f y

600
28 60
) = 0.00444
)(
(
400 410 600 + 400
A
23.37
' = s1 =
= 0.019 < K
bd 30 41
K = 0.852

Edkrgkarsgt;eFVIkarmindl; yield eT dUcenH

- KNna f '

f 's < f y

f 's = 600(

c d'
) fy
c

kMNt; c BI A

s1

= 23.37cm2

As1 f y
2337 400
=
= 130.92mm
0.85 f 'c b 0.85 28 300
a 130.92
c=
=
= 154.02mm
1
0.85
154.02 60
f 's = 600(
) = 366.3MPa f y
154.02
a=

- KNna A' BI M
s

A's =

u2

= A's f 's (d d ' )

110 106
= 953mm 2 = 9.53cm 2
0.9 366.3 ( 410 60)

eRbIEdk 2DB25

eyIgeXIjva eKarBtamlkxN
( '

f 's
) max
fy

- RtYtBinitbERmbRmYlrageFobEdksuT enAnIv:UEdkRsTab;xageRkam eRBaHeyIgsnt;fa bERmbRmYlrag

eFobEdksuT enAnIv:UTIRbCMuTmn;Edk
t

dt = 500 60 = 440mm
d c
440 154.02
t = ( t
)0.003 = (
)0.003 = 0.0056 > 0.005
c
154.02

T.Chhay

96

RtwmRtUv

Flexural Design on Reinforced Concrete Beams

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

vIFITI2
edayeRbIEdkrgkarTajBIrRsTab; nigsac;lUteFobEdksuTenARsTab;eRkambMput
- KNna d = 500 60 = 440mm BIdaRkambERmbRmYlrageFob

= 0.005

0.003
0.003
c
=
=
= 0.375
dt 0.003 + t 0.008
c = 0.375dt = 165mm
a = 0.85c = 140.25mm

- kmaMgsgt;kgebtug
C1 = 0.85 f 'c ab = 0.85 28 140.25 300 = 1001385 N = 1001.385kN

sRmab;muxkat;manEdkrgkarTajCaeKal
eday C = T A = Tf = 1001385
= 2503mm
400
1

s1

= 25.03cm 2

d = 500 90 = 410mm
140.25
a
M u1 = As1 f y (d ) = 0.9 2503 400 (410
) = 306.25 106 N .mm = 306.25kN .m
2
2
A
25.03
1 = s1 =
= 0.0203
bd 30 41

- eday M

> M u1

dUcenHmuxkat;RtUvkarEdkrgkarsgt;

M u 2 = 400 306.25 = 93.75kN .m

karKNnaFwmebtugGarem:rgkarkac;begag

97

T.Chhay

mhaviTalysMNg;sIuvil
As 2 =

NPIC

Mu2
93.75 106
=
= 744mm2 = 7.44cm 2
0.9 f y (d d ' ) 0.9 400 (410 60)

muxkat;EdkrgkarTajsrub A = A + A = 25.03 + 7.44 = 32.47cm eRbIEdk 5DB30

- RtYtBinit Edkrgkarsgt;eFVIkardl; yield bGt;
2

K = 0.852

Edkrgkarsgt;eFVIkarmindl; yield eT dUcenH

f 's < f y

c d'
165 60
) = 600(
) = 381.82MPa
c
165

KNna A' BI M
s

A's =

s2

28 60
600
(
)(
) = 0.00444
400 410 600 + 400

( ' ) = 1 < K

f 's = 600(

s1

u2

Mu2
93.75 106
=
= 779mm 2 = 7.79cm2
0.9 f 's (d d ' ) 0.9 381.82 (410 60)

eRbIEdk 2DB25

- RtYtBinitersIusg;m:Um:g;kg
As = 5DB30 = 35.325cm 2
A's = 2 DB 25 = 9.81cm2
As1 = As A's = 25.515cm 2
a
M n = [ As1 f y (d ) + A's f 's (d d ' )]
2
140.25
M n = 0.9[2551.5 400 (410
) + 981 381.82 (410 60)] = 430.18 106 N .mm
2
d c
410 165
s = (
)0.003 = (
)0.003 = 0.0045
c
165

eyIgeXIjfa

4>5> KNnamuxkat;GkSret T
kgkarKNnamuxkat;GkSr T edaysal;m:Um:g;KNna M / kRmas;sab t nigTTwgsab b . kRmas;
rbs;RTnug b ERbRbYlBI 20cm 50cm . GBaatBIreTotEdlRtUvkarKNnaKW km<s;RbsiTPaB d nigmux
kat;Edk A . eKmanBIkrNIEdlCYbRbTH
- enAeBleKsal; d
+ RtYtBinitfa muxkat;eFVIkarCaragctuekaNEkg bGkSret T edaysnt; a = t
KNnaersIusg;m:Um:g;sRmab;sabTaMgmUl
u

t
2

M nf = (0.85 f 'c )bt (d )

RbsinebI M > M enaH a > t KNnaCaragGkSret T. RbsinebI M < M enaH a < t KNna
CaragctuekaNEkg.
0.85 f '
4M
+ RbsinebI a < t enaHKNna =
(1 1
) / KNna A = bd .
1.7f ' bd
f
u

nf

epgpat;
T.Chhay

min

nf

.
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Department of Civil Engineering

RbsinebI a > t enaHKNna A sRmab;Epksabsgxag >>>

sf

Asf = 0.85 f 'c (b bw )t / f y

t
M u 2 = Asf f y (d )
2

m:Um:g;EdlTb;edayRTnugKW
M u1 = M u M u 2

KNna edayeRbI M / b nig d

1

1 =

u1

0.85 f 'c
4 M u1
(1 1
)
fy
1.7f 'c bw d 2

nigKNna A = b d
muxkat;Edksrub A = A + A
bnab;mkRtYtBinit A A dUcKa RtUvRtYtBinit
s1

1 w

s1

sf

s max

A
min
bw d

RbsinebI a = t enaH A = 0.85f f ' bt

c

- enAeBleKminsal; d nig A karKNnaRtUveFVItamviFIsaRsxageRkam

+ snt; a = t nigKNnabrimaNEdksrub A EdlRtUvkarsRmab;Tb;nwgkmaMgsgt;kgsab
TaMgmUl bt
s

sft

Asft =

0.85 f 'c bt
fy

kMNt; d BI A nig a = t tamrUbmnxageRkam

sft

t
M u = Asft f y (d )
2

RbsinebI eKyktam d KNnarkeXIjenaH A = A nig h = d + 65mm sRmab;

EdkmanmYyRsTab; b h = d + 90mm sRmab;EdkmanBIrRsTab;.
RbsinebI eKyk d fIFMCag d KNnaenaHmuxkat;eFVIkarCaragctuekaNEkg.
ehIy = 0.85f f ' (1 1 1.74fM' bd ) / KNna A = bd .

sft

RbsinebI eKyk d fItUcCag d KNnaenaHmuxkat;eFVIkarCaragGkSr T. ehIy

muxkat;EdkcugeRkay A FMCag A . kgkrNIenH eKRtUveFVIdUckrNIxagelIedIm,IkMNt;
muxkat; A .

21

sft

karKNnaFwmebtugGarem:rgkarkac;begag

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]TahrN_7 eKmanFwmmYymanGkSr T EdlmanRTnug b = 25cm RbEvgsab b = 100cm kRmas;sab

t = 10cm nigkm<s;RbsiTPaB d = 37cm . kMNt;muxkat;EdkcaM)ac; edIm,ITb;nwgm:Um:g; M = 375kN .m .
smtikm f ' = 20MPa nig f = 400MPa .
dMeNaHRsay
- RtYtBinitGkSNWt edaysnt; a = t = 10cm
w

t
2

M n = 0.85 f 'c bt (d ) = 0.9 0.85 20 1000 100(370

eday M > M muxkat;manlkNCactuekaNEkg

- kMNt;muxkat;EdkrgkarTaj
n

100
) = 489.6 106 N .mm
2

0.85 f 'c
4M u
0.85 20
4 375 106
(1 1
)
(
1
1
) = 0.00845
=

1.7f 'c bd 2
400
1.7 0.9 20 1000 3702
fy

As = bd = 0.00845 100 37 = 31.265cm 2

edayeRbI 5DB30 = 35.325cm

35.325
- RtYtBinit = bAd = 25
= 0.0382 >
37
2

min

1.4
= 0.0035
400

0.85 f 'c
As max =
[(be bw )t + 0.3751bw d ] = 44.41cm 2 > As
fy
As f y
35.325 400
a=
=
= 8.31cm
0.85 f 'c b 0.85 20 100
37 9.78
a 8.31
c=
=
= 9.78cm
s = 0.003(
) = 0.00835 > 0.005
9.78
1 0.85

dUcKa

nig

]TahrN_8 RbBnkRmalxNdUcbgajkgrUb EdlpMeLIgedaykRmalxNEdlmankRmas; t = 8cm EdlRT

edayFwmRbEvg L = 430cm EdlmanKMlatBIKa l = 300cm KitBIGkSmkGkS. FwmmanRTnug b = 35cm
w

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Department of Civil Engineering

nigkm<s;RbsiTPaB d = 47cm . kMNt;muxkat;EdkcaM)ac;edIm,ITb;nwgm:Um:g;KNna M

smtikm f ' = 20MPa nig f = 400MPa .

= 575kN .m

dMeNaHRsay
- kMNt;RbEvgsab
16t + bw
16 80 + 350 = 1630mm
L

4300
be = min
= min
= 1075mm
4

4
3000mm

dUcenH b = 1075mm
- RtYtBinitTItaMgGkSNWt edaysnt; a = t
e

t
2

M n = 0.85 f 'c bt (d )
M n = 0.9 0.85 20 1075 80(470

80
) = 565.794 N .mm = 565.794kN .m < 575kN .m
2

eday M < M muxkat;manlkNCaragGkSr T dUcenH a > t .

- kMNt;muxkat;EdksrubsRmab;Tb;Tl;CamYykmaMgsgt;kgsab
n

Asf =

0.85 f 'c (b bw )t 0.85 20 (1075 350)80

=
= 2456 mm 2
fy
400

t
80
M u 2 = Asf f y (d ) = 0.9 2456 400(470 ) = 380.188 106 N .mm
2
2
0.85 f 'c
4 M u1
1 =
(1 1
)
fy
1.7f 'c bw d 2

1 =

0.85 20
4 194.812 106
(1 1
= 0.0077
400
1.7 0.9 20 350 4702

karKNnaFwmebtugGarem:rgkarkac;begag

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enaH A = b d = 0.0077 35 47 = 12.67cm

muxkat;EdkTajsrub A = 12.67 + 24.56 = 37.23cm
eRbI 6DB30 BIrRsTab;
- km<s;srubrbs;muxkat; h = 470 + 90 = 560mm
2

s1

1 w

0.85 f 'c
[(be bw )t + 0.3751bwd ] = 50.87cm 2 > As
fy
As f y
42.39 400
t a =
=
= 9.278cm
0.85 f 'c b 0.85 20 107.5
a 9.278
c=
=
= 10.91cm
1 0.85
As max =

- RtYtBinit /

d t = 56 6 = 50cm
50 10.91
) = 0.0107 > 0.005
t = 0.003(
10.91

]TahrN_9 sRmab;RbBnkRmalxNmYy EdlmanTTwgsabRbEvg b = 122cm TTwgRTnug b = 40 ehIy

kRmalxNmankRmas; t = 10cm . KNnamuxkat;GkSr T edIm,IrgnUvm:Um:g;KNna M = 1100kN .m .
smtikm f ' = 20MPa nig f = 400MPa .
e

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Department of Civil Engineering

dMeNaHRsay
- edaysarminsal;km<s;RbsiTPaB eyIgeGay a = t
kMNt;muxkat;Edk A sRmab;sabTaMgmUl
sft

Asft =

0.85 f 'c bt 0.85 20 122 10

= 51.85cm 2
=
fy
400

eGay M

KNna

t
= Asft f y (d )
d
2
Mu
t
1100 106
100
+
= 639.3mm
d =
+ =
Asft f y 2 0.9 5185 400 2
u

eyIgeXIjfa RbsinebI d = 63.93cm enaH A = A

- RbsinebI d > 63.93cm / eyIgsnt;yk d = 67cm enaH a < t muxkat;manlkNCaragctuekaNEkg
PaKryEdkRtUv)anKNnatamrUbmnxageRkam
s

sft

0.85 f 'c
4M u
(1 1
)
1.7f 'c bw d 2
fy

0.85 20
4 1100 106
(1 1
= 0.006
400
1.7 0.9 20 1220 6702

enaHmuxkat;Edk A = bd = 0.006 122 67 = 49.04cm

- RbsinebI d < 63.93cm / eyIgsnt;yk d = 60cm enaH a > t muxkat;manlkNCaragGkSret T .
2

Asf =

0.85 f 'c t (b bw ) 0.85 20 10 (122 40)

= 34.85cm 2
=
fy
400

t
100
M u 2 = Asf f y (d ) = 0.9 3485 400(600
) = 690 106 N .mm
2
2
M u1 = 1100 690 = 410kN .m

sRmab;muxkat;EdkrgkmaMgTajeKal b

Ru =

= 40cm d = 60cm

nig M

u1

= 410kN .m

M u1
410
=
= 2847.2 kN m 2 = 2.85MPa
2
bw d
0.4 0.60 2

1 =

0.85 f 'c
4 Ru
0.85 20
4 2.85
(1 1
)=
(1 1
) = 0.0088
1.7f 'c
400
1.7 0.9 20
fy

As1 = 1bwd = 0.0088 40 60 = 21.12cm2

As = As1 + Asf = 34.85 + 21.12 = 55.97cm 2

eRbIEdk 7 DB32 = 56.27cm

- RtYtBinit

a=

As1 f y
0.85 f 'c bw

21.12 400
= 12.42cm
0.85 20 40

karKNnaFwmebtugGarem:rgkarkac;begag

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a
12.42
=
= 14.6cm
0.85 0.85
dt = 63cm

c=

d c
t = 0.003 t
= 0.0099 > 0.005 tension-controlled section
c

- KNnamuxkat;EdksrubGtibrma
As max = 0.85

f 'c
20
[(be bw )t + 0.3751bw d ] = 0.85
[(1220 400)100 + 0.375 0.85 400 600)]
fy
400

As max = 6736mm 2 = 67.36cm 2 > 56.27cm 2

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Department of Civil Engineering

V.

viFIKNnaepSgeTot

Alternative Design Methods

5>1> esckIepIm (Introduction)

enAkgemeronTI3 nigTI4 karviPaK nigkarKNnaGgt;ebtugBRgwgEdkRtUv)anBnl;edayQrelIeKal
karN_Edlpl;[eday ACI Code 318-05. viFIKNnadTRtUv)anbgajenAkg]bsm<n Appendix B n
ACI Code edayeyageTAtamemKuNbnkEdl[enAkg]bsm<n Appendix C. viFIKNnaepSgeTotenH
KWCaeKalkarN_nkarviPaK nigkarKNnaenAkg ACI Code 318-99. vamanlkNRsedogKaxHeTAnwg viFI
Edl)anBnl;BImun elIkElgEtvaeRbIemKuNbnk nigemKuNkat;bnyersIusg; xusKa. smIkarviPaK nig
smIkarKNnaeKalEdlmanenAkgemeronmun nwgRtUv)aneRbIenATIenH. enAeBleKeRbI Appendix B edIm,I
KNna eKRtUvCMnYsnUvGVIEdlRtUvKaenAkg Code enaH.
5>2> emKuNbnk (Load Factors)
RbsinebIersIusg;TRmUvkar required strength RtUv)antageday U ehIykmaMgxl; nigkmaMgrBaydIRtUv
)antageday W nig E erogKa enaHtam ACI Code, Appendix C ersIusg;TRmUvkar U KYrEtCatmEdlFM
CageKkgcMeNambnSMbnkxageRkam
1> sRmab;krNIbnkefr bnkGefr nigbnkxl;
U = 1 .4 D + 1 .7 L

U = 0.75(1.4 D + 1.7 L) + (1.6W

(5-1a)

b 1.0E)

(5-1b)

b 1.0E )
(5-1c)
2> enAeBlbnkxl; W minRtUv)ankat;bnyedayemKuNTisedA directionality factor 1.3W Gac
RtUv)aneRbICMnYs 1.6W . enAeBlEdlbnkrBaydIRtUv)anQrenAelIbnkeFVIkar service forces enaH
1.4 E GacRtUv)aneRbICMnYs[ 1.0 E .
3> kgkrNIEdlbnksm<aFdI H RtUv)anbBaleTAkgkarKNna
U = 0.9 D + (1.6W

U = 1 .4 D + 1 .7 L + 1 .7 H

(5-2a)

enAeBlEdlbnkefr D nigbnkGefr L kat;bny\TiBlrbs; H enaH

U = 0 .9 D + 1 .7 H

(5-2b)

sRmab;bnSMbnkn D / L b H
U = 1 .4 D + 1 .7 L

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NPIC

4> RbsinebITmn; nigbnksm<aFEdl)anmkBIsarFaturav F RtUv)anbBaleTAkgkarKNna

U = 1.4 D + 1.7 L + 1.4 F

(5-3a)

enAeBlEdlbnkefr D nigbnkGefr L kat;bny\TiBlrbs; F enaH

U = 0 .9 D + 1 .4 F

(5-3b)

sRmab;bnSMbnkn D / L b F
U = 1 .4 D + 1 .7 L

sm<aFbBarnsarFaturavKYrRtUv)anKitCabnkefr.
5> enAeBlEdl\TiBlTgic impact effects RtUv)anrab;bBal enaHvaRtUv)anKitbBaleTAkgbnkGefr.
6> enAeBlEdl structural effects T nsRmut differential settlement, creep, karrYmmaD
(shrinkage) b bNrsItuNPaB mantmFM vaKYrRtUv)anrab;bBaleTAkgbnSMbnkn
U = 0.75(1.4 D + 1.4T + 1.7 L)
U = 1.4 D + 1.4T

(5-4a)
(5-4b)

smIkar (5-1a) RtUv)aneRbICaTUeTA. emKuNbnkefresInwg 1.4 nigemKuNbnkGefresInwg 1.7 .

sRmab;bnkefr nigbnkGefrcMcMNuc PD nig PL enaHbnkcMcMNucemKuN PU = PD + PL dUcKa
M U = M D + M L Edl M D nig M L m:Um:g;bnkefr nigm:Um:g;bnkGefrerogKa.
5>3>emKuNkat;bnyersIusg; (Strength-Reduction Factor )
ersIusg; nominal strength nmuxkat;RtUv)ankat;bnyedayemKuN edIm,IKitsRmab;kar)at;bg;
ersIusg;enAkgsmardtictYc small adverse variations in material strength karplitEdleFIVeLIgedayd
artisanry TMhMxat karRKb;RKg nigkRmitnkarRtYtBinit. emKuN CaEpkmYynemKuNsuvtiPaB.
bTdan ACI Code, Section C.3 (Appendix C) kMNt;nUvtmxageRkamedIm,IeRbIR)as;
= 0.90
- sRmab;muxkat;rgkarTaj
- sRmab;muxkat;rgkarsgt;
k> CamYyEdkkgvN
= 0.70
= 0.65
x> CamYyEdkkgFmta
- sRmab;kmaMgkat; nigkmaMgrmYl
= 0.75
= 0.65
- sRmab;RTnab;enAelIebtug
- sRmab;karBt;enAelIebtugsuT benAelIebtugEdlmanbrimaNEdkGb,brma 1.4 / f y = 0.65

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sRmab;muxkat;EdlsitenAkgtMbn; transition region rvagmuxkat;rgkarTaj tension-controlled

section nigmuxkat;rgkarsgt; compression-controlled section enaH GacnwgekIneLIgCabnat;rhUtdl;
0 .9 .
eKkGaceRbIemKuNkat;bnyersIusg; sRmab;ssr bmuxkat;Edlman t < 0.005 edayvaERb
RbYleTAtamkrNIxageRkam
1> enAeBlEdl Pu = Pn 0.1 f 'c Ag enaH = 0.7 sRmab;ssrEdkkgFmta nig = 0.75
sRmab;EdkkgvN. krNIekIteLIgCaTUeTAsRmab;muxkat;rgkarsgt; compression control.
Ag Camuxkat;eBj.
2> rvagtm 0.1 f 'c Ag b Pn mYyNaEdltUcCag nigsUn ehIy Pu sitenAkgtMbn;Taj
tension control zone nig FMCag 0.7 b 0.75 . ACI Code, Section C3.2 kMNt;fa
sRmab;Ggt;Edlman f y minFMCag 400MPa CamYyEdksIuemRTI nigCamYycmayrvagEdk
rgkarsgt; nigkarTaj (d d ' ) minRtUvticCag 0.7h h =km<s;srubrbs;muxkat; nig
d = h d s enaHtMl RtUv)anekIneLIgCabnat;eTArk 0.9 .
sRmab;tMbn; transition region, RtUv)ankMNt;edayviFan linear interpolation rvag 0.7 b
0.75 nig 0.9 . rUb 5>1 bgajBIbMErbMrYlrbs; sRmab;Edk 400 MPa . smIkarbnat;mandUcxageRkam
= 0.57 + 67 t
sRmab;muxkat;EdkkgFmta
(5-5)
= 0.65 + 50 t
sRmab;muxkat;EdkkgvN
(5-6)
m:agvijeTot enAkgtMbn; transition region GacRtUv)ankMNt;CaGnuKmn_eTAnwg (dt / c)
sRmab;Edk 400MPa dUcxageRkam
d
= 0.37 + 0.20 t sRmab;muxkat;EdkkgFmta
(5-7)
c
dt

sRmab;muxkat;EdkkgvN
Edl c Cakm<s;GkSNWtenAersIusg; normal strength.
= 0.50 + 0.15

viFIKNnaepSgeTot

107

(5-8)

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NPIC

5>4> muxkat;ctuekaNEkgCamYyEdkrgkarTaj

(Rectangular Sections with Tension

Reinforcement)

BIkarviPaKnmuxkat;ctuekaNEkgEdkrgkarTaj smIkarxageRkamRtUv)anbMEbk Edl

KitCa MPa
b = 0.851

f 'c

nig

fy

f 'c 600
f y 600 + f y

RbsinebIPaKryEdkGtibrmaRtUv)ankMNt; 0.75b enaH

max = 0.75b = 0.63751

f 'c 600
f y 600 + f y

(5-9)

enHbgajfa max = 0.75b FMCag max = 0.634b Edl)an[enAkgemeronTI3 sRmab;Edk

400 MPa .
sRmab; f ' 28MPa
c

max = 0.542

f 'c 600
f y 600 + f y

(5-10)

sRmab;ebtugEdlmanersIusg; f ' 28MPa .

f ' 28
) sRmab;ebtugEdlmanersIusg; 28MPa < f ' 56MPa .
= 0.85 0.05(
7

1 = 0.85

1
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Department of Civil Engineering

sRmab;ebtugEdlmanersIusg; f ' > 56MPa .

PaKryEdknmuxkat; balanced section b nigPaKryEdkGtibrmaGnuBaat max GacRtUv)an
KNnasRmab;tmepSgKan f 'c nig f y dUcbgajenAkgtarag 5>1. PaKryEdkKNnaEdlesIeLIgsRmab;
max kRtUv)anbgajenAkgtarag 5>1.
1 = 0.65

taragTI5>1 PaKryEdkEdlRtUv)anesIreLIg

f y (MPa)

f 'c ( MPa)
20

235
400
400
500
400
500

28
35

% s
1.4
1.2
1.4
1.2
1.4
1.2

smIkarm:Um:g;KNnaRtUv)anbMEbkenAkgemeronmunmanTRmg;dUcxageRkam
M n = M u = Ru bd 2

Edl

(3-21)

f y

= Rn
Ru = f y 1
1.7 f 'c

(3-22)

nig = 0.9 . sRmab;muxkat;rgkarTaj (tension-controlled section) / t 0.005

M n = M u = As f y d

dUcKa

As f y

1.7 f 'c b

M n = M u = f y bd 2 d

(3-19a)

f y

1.7 f 'c

(3-20)

eyIgeXIjfasRmab;eRkABIm:Um:g;emKuN M u / f 'c / f y eKmanGBaatbIenAkgsmIkarenHKW b / d nig

. dUcenHeKminGacedaHRsaysmIkarenH)aneT Tal;EtGBaatBIrRtUv)ansnt;. CaTUeTA eKeRcInsnt;
edayeRbI max nig b kRtUv)ansnt;Edr. edayQrelIkarBiPakSaBIxagedIm krNIxageRkamRtUv)anbegIt
eLIgenAeBl M u / f 'c / f y RtUv)ansal;
1> RbsinebI RtUv)ansnt; enaH Ru GacRtUv)anKNnaBIsmIkar (3-22) Edl[ bd 2 = M u / Ru .
GkKNnaGaceRbI rhUtdl; max EdlbegItmuxkat;ebtugEdkrgkarTajGb,brma. RbsinebI
eRbI min vanwgbegItmuxkat;ebtugGtibrma. RbsinebI b RtUv)ansnt;bEnmBI elI enaH d Gac
RtUv)anKNnadUcxageRkam
d=

Mu
Ru b

(5-11)

RbsinebI d / b = 2 enaH d = 3 (2M u / Ru ) nig b = d / 2 bgittmeTArktmEdlFM.

viFIKNnaepSgeTot

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2> RbsinebI d nig b RtUv)an[ enaHPaKryEdkRtUvkar GacRtUv)anKNnaedaysmIkar (3-20)

eKTTYl)an
3> = 0.85f f 'c 1 1 1.74fM' ubd 2
(5-12)
c

0.85 f 'c
2 Ru
=
1 1

fy
0.85 f 'c
y

nig As = bd
Ca]TahrN_/ RbsinebI M u = 275.72kN .m / b = 300mm / d = 450mm / f 'c = 20MPa nig
f y = 400 MPa enaH = 0.0154 BIsmIkar (5-12) nig As = bd = 0.0154 300 450 = 2079mm 2 enA
eBlEdleK[ b nig d eKKYrEtBinitemIlfaetIeKRtUvkarEdkrgkarsgt; bGt; eRBaHEt d tUc. eKGacedaH
Rsayva)andUcxageRkam
k> KNna max nig Ru,max = max f y [1 (max f y / 1.7 f 'c )]
x> KNna M n,max = Ru,maxbd 2 = ersIusg;m:Um:g;Gtibrmarbs;muxkat;EdkrgkarTaj.
K> RbsinebI M u < M n,max enaHvaminRtUvkarEdkrgkarTajeT. KNna nig As BIsmIkar (512)

X> RbsinebIeKsal; nig b KNna Ru

f y

Ru = f y 1
1 .7 f ' c

KNna d BIsmIkar (5-11)

Mu
nig As = bd
d=
Rb
u

]TahrN_TI1

kMNt;muxkat;EdkcaM)ac;sRmab;muxkat;EdlmanTTwg b = 250mm nigkm<s;srub d = 700mm rUbTI5>2

RbsinebIvargnUvm:Um:g;emKuNxageRkA 312kN.m . eK[ f 'c = 28MPa nig f y = 400MPa .

dMeNaHRsay

1> snt;eRbIEdk DB25 mYyRsTab; epgpat;enAeBleRkay d = 700 50 = 650mm .

2> RtYtBinitemIlfaetImuxkat;RtUvkarEdksgt; bGt;. eRbobeFobersIusg;m:Um:g;KNnanmuxkat; eday
eRbI max CamYym:Um:g;KNna. sRmab; f 'c = 28MPa nig f y = 400MPa / max = 0.02276 .

max f y
= 6.63MPa
Ru = max f y 1
1.7 f 'c

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ersIusg;m:Um:g;KNnanmuxkat;ebtugEdkrgkarTajKW
M n, max = Ru / maxbd 2 = 6.63 250 650 2 10 6 = 700.3kN .m > 312kN .m

dUcenH < max enaHvaCamuxkat;EdlmanEtEdkrgkarTaj.

3> KNna BIsmIkar (5-12) edIm,ITTYl)an = 0.0089 / As = bd = 0.0089 250 650
= 1446mm 2 eRbIEdk 3DB 25 (As = 1472mm 2 ). muxkat;cugeRkayRtUv)anbgajenAkgrUbTI 5>2.
4> epgpat; t
1472 400
= 98.96mm
0.85 28 250
a
c=
= 116.4mm
0.85
d c
t = t
0.003 = 0.0137 > 0.005
c
a=

= 0.9

5>5> muxkat;ctuekaNCamYynwgEdkrgkarsgt;

(Rectangular Sections with Compression

Reinforcement)

muxkat;ebtugEdkrgkarTaj singly reinforced section EdlmanersIusg;m:Um:g;GtibrmaenAeBlEdl

max rbs;EdkRtUv)aneRbI. RbsinebIm:Um:g;emKuNFMCagersIusg;m:Um:g;kg krNImuxkat;RtUv)ankMNt; enaHeK
RtUvkarmuxkat;EdkDub doubly reinforced section edaybEnmEdkTaMgenAkgtMbn;sgt; nigtMbn;Taj. viFI
saRssRmab;KNnamuxkat;ctuekaNEkgCamYyEdksgt; enAeBlEdleKsal; M u / f 'c / b / d nig d '
Rtuv)ansegbenAkgemeronTI4. karEdlxusKamanEtmYyKW max = 0.75b RtUv)aneRbIenAkgkarKNnaenH
max = 0.63751

viFIKNnaepSgeTot

f 'c 600
f y 600 + f y

(5-9)
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dUcKa RtUvepogpat; t 0.005 sRmab; = 0.9 .

]TahrN_TI2 muxkat;FwmRtUv)ankMNt;eday b = 300mm nigkm<s;srub h = 500mm ehIyrgnUvm:Um:g;emKuN
M u = 447.5kN .m . kMNt;muxkat;EdkcaM)ac;edayeRbI f 'c = 28MPa nig f y = 400MPa . eyagtamrUb
5>3.

dMeNaHRsay

1> kMNt;ersIusg;m:Um:g;KNnanmuxkat;EdkeTal. snt; = 0.018 . dUcenH Ru = 5.5MPa .

sRmab; EdkBIrRsTab; d = 500 90 = 410mm
M u1 = Ru bd 2 = 5.5 300 410 2 10 6 = 277.4kN .m

m:Um:g;KNnaKW M u = 447.5kN .m > 277.4kN .m dUcenHeKRtUvkarEdksgt;

2> KNna As1 / M u 2 / As 2 nig As
As1 = bd = 0.018 300 410 = 2214mm 2
M u 2 = M u M u1 = 447.5 277.4 = 170.1kN .m
M u 2 = As 2 f y (d d ' )

snt; d ' = 50mm

170.1 10 6 = 0.9 As 2 400(410 50)

As 2 = 1312.5mm 2

6DB28
3> epgpat;PaB yield rbs;Edkrgkarsgt;. Edkrgkarsgt; yield RbsinebI
As = As1 + As 2 = 2214 + 1312.5 = 3526.5mm 2

f 'c d ' 600

f y d 600 f y
28 50 600
K = (0.85) 2

= 0.0185
400 410 600 400
A
2214
' = s1 =
= 0.018 < K
bd 300 410

' K = 0.851

dUcenH Edkrgkarsgt;Gt; yield

T.Chhay

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4> KNna f 's f 's = 600[(c d ') / c] f y /

kMNt; As1 As1 = 2214mm2
As1 f y

a=

2214 400
= 124mm
0.85 28 300

0.85 f 'c b
124
a
c=
=
= 145.9mm
1 0.85
145.9 50
= 394.4 MPa < 400MPa
f 's = 600
145.9

5> KNna A's BI M u 2 = A's f 's (d d ' )

170.1 106 = 0.9 A's 394.4(410 50)

dUcenH A's = 1331mm2 bKNna A's BI A's = As 2 ( f y / f 's ) = 1331mm2 3DB25

6> epgpat;
dt c
0.003
c

t =

d t = h d ' = 500 50 = 450mm

450 145.9
0.003 = 0.006 > 0.005
145.9
c 145.9
=
= 0.324 < 0.375 (OK)
450
dt

t =

= 0.9

7> epgpat; M n cugeRkay/ As = 3694.5mm2 / A's = 1472.6mm2 /

As1 = 2221.9mm 2 / a = 124.5mm nig c = 146.5mm
124.5

M n = 2221.9 400 410

+ 1472.6 394.4(410 50 ) = 518.15kN .m
2

epgpat; t / dt = 450mm
dt c
0.003 = 0.006 > 0.005
c

t =

= 0.9
M n = 0.9 518.15 = 466.3kN .m > 447.5kN .m

5>6> karKNnamuxkat;GkSret (Design of T-Section)

kgkarKNnamuxkat;GkSret enAeBlEdleKsal;m:Um:g;emKuN M u kRmas;sab T TTwg b RtUv)an
kMNt;BIkarKNnakRmalxN ehIykarkMNt;rbs; ACI Code sRmab;TTwgsabRbsiTPaB b RtUv)an[enAkg
emeronTI3. kRmas;RTnug bw GacRtUv)ansnt;edayERbRbYlBI 200 500mm TMhMEdlRtUv)aneRbIKWsitenA
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cenaH 300 400mm . GBaatBIrRtUvkarkMNt;CacaM)ac;KW d nig As . CMhannkarKNnaRtUv)ansegbenA

kgemeronTI4.
]TahrN_TI3 muxkat;FwmGkSret RtUv)anbgajenAkgrUbTI4 manTTwgRTnug bw = 250mm TTwgsab b = 1m
kRmas;sab t = 100mm nigkm<s;RbsiTPaB d = 370mm . kMNt;muxkat;EdkcaM)ac;RbsinebIm:Um:g;emKuN
420kN.m . eK[ f 'c = 28MPa nig f y = 400MPa .

dMeNaHRsay

1> KNnaTItaMgTItaMgGkSNWt Edlmuxkat;GacmanragctuekaN. snt;km<s;rbs;bksgt;

a = 100mm Edl a = t = 100mm enaH

t
2

M n = (0.85 f 'c )bt d = 685.44kN .m > 420kN .m

m:Um:g;KNnaEdlsabebtugGacRT)anFMCagm:Um:g;emKuNEdlmanGMeBIelIva. dUcenH muxkat;eFVIkar

manragctuekaN.
2> kMNt;muxkat;EdkTaj edayKitmuxkat;manragctuekaNEdl b = 1000mm
Ru =

Mu
bd

420000000
1000 370 2

= 3.06MPa

BIsmIkar (5-12) sRmab; Ru = 3.07MPa nig = 0.0092

As = bd = 0.0092 1000 370 = 3404mm 2

eRbI 6DB28 / As = 3694.5mm2 BIrRsTab;

3> epgpat;fa w = As / bwd min / w = 3404 /(250 375) = 0.0363 > min = 0.00333
4> epgpat; t = dt c c 0.003 dt = 375mm
a=

T.Chhay

3404 400
= 57.21mm
0.85 28 1000

c=

114

57.21
= 67.3mm
0.85
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= 0.9

t = 0.0135 > 0.005

5>7> viFI

strut and tie

5>8> 5>7>1> esckIepIm

(Strut and Tie Method)

(Introduction)

/ ENnaMnUvviFImYyepSgeTot eRkABIviFIEdl)anBnl;BIxagedImenAkgem
eronTI 3. viFIenHeK[eQaHfa strut and tie model. viFIepSgenHRtUv)anGnuvty:agmanRbsiTPaBenAkg
tMbn;Edldac; discontinuity enAkgeRKOgbgM dUcCatMbn;TRm tMbn;EdlbnkGnuvt btMbn;Edlmuxkat;FrNI
maRtpas;brPamdUcCa brackets nig portal frames. enAkgtMbn;TaMgenH muxkat;rabesIminrkSaenArabesI
eRkayeBlrgkarBt; dUcGVIEdl)ansnt;enAkgemeronTI3 ehIyvaRtUv)aneK[eQaHfa tMbn; D (D-region)
rUbTI5>5 a. tMbn;epSgeTotebs;Fwmsg;da RTwsIbTFwmmUldan nigTMnak;TMng linear strain relationship
RtUv)anGnuvt. tMbn;TaMgenHRtUv)aneK[eQaHfa tMbn; B (B-region) rUbTI5>5 a.
edayQrelIeKalkarN_ St. Venant PaBdac;KaenAkgkarEbgEckkugRtaMgenAkgtMbn; D Edl
bNalmkBIragFrNImaRt blkxNbnkbgajfakugRtaMgbNalmkBIbnktamGkS nigm:Um:g;Bt; kar
BRgaykugRtaMgesIrEtmanlkNCabnat;enAcmayRbEhlnwgkm<s; h rbs;Ggt;BIcMNucdac; rUbTI5>5 b nig
c. RbsinebItMbn; D BIrCan;Ka bCYbKa BYkvaGacRtUv)anKitCatMbn; D EtmYy. pleFobrvagRbEvgGtibrma
nigkm<s;esInwg 2 EdlbegItmMuGb,brma 26.5o rvag strut and tie bRbEhl 25o .
enAkgKMrU strut and tie rUbTI5>6 cMNucEdlkmaMgbICYbKaenAtMN D RtUv)aneK[eQaHfa cMNuc
node nigmaDebtugEdlenACMuvijcMNuc node RtUv)anehAfatMbn;cMNuc nodal zone. kmaMgEdlmanGMeBIenA
elIcMNuc node GacERbRbYleTAtamkmaMgTaj nigkmaMgsgt;nbnSMepSg dUcCa C C C / C C T /
C T T / T T T rUbTI 5>7. rUbTI5>8 bgajBIRbePTtMbn;cMNuc typical nodal zone sRmab;kar
GnuvtbnkepSg cMENkrUbTI 5>9 bgajBI extended nodal zone sRmab;srsEdkmYy beRcInRsTab;.
ACI Code, Appendix A

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KMrU strut and tie

NPIC

(Strut and Tie Model)

KMrU strut and tie GacRtUv)anbgajedayKMrU truss CamYynwgkmaMgeFVIGMeBIenAelIcMNucepSg. \Lv

BicarNanUv truss EdkEdl)anbgajenAkgrUbTI 5>10. edaysarEtvamanlkNsIuemRTI RbtikmenAcMNuc
A nig B esIKa R A = RB = 20kN nigBIlMnwgntMN A nig D kmaMgTajenAkg AB = 20kN enAeBlEdl
kmaMgsgt;enAkg AD b BD = 28.3kN . Ggt; AB RtUv)anKitCa tie cMENk AD nig BD RtUv)ancat;Tuk
Ca strut. kmaMgenAkgGgt;epSgeTotesIsUn. edayeRbobeFob truss enHCamYyFwmbtugenAkgrUbTI 5>6a
eyIgGaceXIjfaRkLapPaKeRcInn ACD nig BED nigRkLapEdlenABIxageRkam nodal zone D min
manRbsiTPaB nigeFVIkarCa filler. kmaMgenAkg strut sRmab;lkxNbnkenH FMCagkmaMgenAkg tie. kg
krNIenH vamanRkLapebtugRKb;RKan;edIm,IeFVIkarCa strut rUbTI5>6a. eKRtUvkarCacaM)ac;nUvsrsEdk
edIm,IeFVIkarCa tie sRmab; AB . karcgPab;dRtwmRtUvrbs; tie mansarsMxan;Nas;sRmab;karKNnaRbkb
edaysuvtiPaB. karcgPab;KYreFVIeLIgenAtMbn; nodal zone.
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5>7>2> viFIsaRsKNnatam ACI (ACI Design Procedure)

edayQrelI ACI Code, Section A.2 karKNnatMbn; D-region rab;bBalnUvCMhanxageRkam
- kMNt; nigbMEbknUvtMbn;nImYy
- kMNt;kmaMgpbEdlmanGMeBIelIEdntMbn; D-region nImYy
- eRCIserIsKMrU truss edIm,IbBankmaMgpbenAkgtMbn; D-region. GkSn strut nig tie KYrRtYtsIuKaCa
mYynwgtMbn;sgt; compression field nigtMbn;Taj tension field.
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- kMNt;TTwgRbsiTPaBrbs; struts nig nodal zones edayQrelIersIusg;ebtug ersIusg;Edk nigKMrU

truss Edl)aneRCIserIs.
- epgpat;lkxNeFVIkar serviceability condition EdleyageTAtamtRmUvkarrbs; ACI Code.
PaBdabrbs;Fwmx<s; deep beam GacRtUv)anKNnaedayeRbIkarviPaKeGLasic elastic analysis.
lkxNRKb;RKgsameRbHn ACI Code, Section 10.6.4 KYrRtUv)anepgpat;edaysnt;fa tie
RtUv)aneRsabenAkgRBIsebtug eyagtam RA.4.2 .

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5>7>3> tRmUvkarsRmab;karKNna
(Design Requirement)
tRmUvkarKNnasRmab; struts nig tie GacRtUv)ansnt;dUcxageRkam
1> KNna struts, ties nigtMbn; nodal zone
Fn Fu

Edl

(5-13)

kmaMgenAkg struts, ties nigtMbn; nodal zone Edl)anBIbnkemKuN

Fn = ersIusg; nominal strength rbs; struts, ties nigtMbn; nodal zone
= 0.75 sRmab;TaMg struts nig tie
2> ersIusg;rbs; struts ersIusg;sgt; nominal compressive strength rbs; struts EdlKanEdk
beNay Fns KYrEttUcCagtm Fns enAcugTaMgBIrrbs; struts
Fu =

Fns = f ce Acs

Edl

(5-14)

RkLapmuxkat;enAcugmagrbs; struts
f ce = ersIusg;sgt;RbsiTPaBrbs;ebtugEdltUcCagenAkg struts b nodal zone.

Acs =

f ce = 0.85 s f 's

Edl

T.Chhay

(5-15)

sRmab; struts manrUbragCaRBIs

s = sRmab; struts EdlTTwgRtg;muxkat;kNalGgt;FMCag TTwgenAcMNuc node
(bottle-shaped struts) CamYybrimaNEdkRKb;;RKan;edIm,ITb;nwg kugRtaMgTaj
tamTTwg.
s = 0.6 dUcGVIEdl)anerobrab;xagelI edayKanbrimaNEdkRKb;;RKan;edIm,ITb;nwg
s =

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kugRtaMgTajtamTTwg = 1.0 sRmab;ebtugTmn;Fmta normal-weight

concrete, 0.85 sRmab;ebtugxSac;Tmn;Rsal sand-lightweight concrete nig
0.75 sRmab;ebtugTmn;RsalTaMgGs; lightweight concrete.
s = 0.4 sRmab; struts enAkgGgt;Taj bsab
s = 0.6 sRmab;krNIepSgeTotTaMgGs;
3> EdkExVg struts rUbTI5>11 sRmab; f 'c 35MPa tm s = 0.75 GacRtUv)aneRbIRbsinebI
GkSrbs; struts RtUv)anExVgedayRsTab;Edk

Asi
sin i 0.003
bs si

Edl

(5-16)

RkLapmuxkat;EdksrubenAKMlat si enAkgRsTab;TI i Edlkat; strut enAmMu

i CamYyGkSrbs; strut.
si = KMlatEdkenAkgRsTab;TI i Edlkat; strut enAmMu i CamYyGkSrbs; strut .
bs = TTwgGgt;
1 = mMurvagGkSrbs; strut nigr)arenAkgRsTab;TI i nr)arEdlkat;Kaeday strut.
Asi =

4> Edkrgkarsgt;enAkg struts Edkrgkarsgt;GacRtUv)aneRbIedIm,IbegInersIusg;rbs; strut

Fns = f ce Acs + A's f 's

Edl
viFIKNnaepSgeTot

(5-17)

ersIusg;n strut BRgwgedayEdkbeNay

A's = RkLapnEdksgt;enAkg strut

Fns =

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NPIC

kugRtaMgenAkg A's f 's = f y sRmab; 400 500MPa

5> ersIusg;rbs; tie ersIusg; nominal strength n tie, Fnt KW
f 's =

Fnt = Ats f y + Atp ( f se + f p )

Edl

(5-18)

RkLapEdkminrgeRbkugRtaMgenAkg tie
Atp = RkLapEdkeRbkugRtaMg
f se = kugRtaMgRbsiTPaBeRkayeBl)at;bg;enAkgEdkrgeRbkugRtaMg
f p = karbegInkugRtaMgeRbkugRtaMgEdlbNalmkBIbnkemKuN
Atp = 0 sRmab;Ggt;minrgeRbkugRtaMg
Ats =

( f se + f p ) f py

(5-19)

eKGacGnuBaat[yk f p = 400MPa sRmab; bonded prestressed reinforced b

f p = 70MPa sRmab; unbonded prestressed reinforced . dUcKa EdnkMNt;x<s;nkarGnuvt
sRmab;TTwgrbs; tie GacRtUv)anykdUcxageRkam
wt , max = Fnt /( f cebs )

(5-20)

6> ersIusg;rbs;tMbn; nodal zones ersIusg; nominal compression strength ntMbn; nodal zones
Fnn KYrEtesI
Fnn = f ce Anz

(5-21)

RkLapxagebs; nodal zone bmuxkat;rbs; nodal zone EdlEkgeTAnwg

kmaMgpbenAelImuxkat;
7> Confinement enAkgtMbn; nodal zones: y:agehacNas;Edk confinement RtUv)anpl;[enA
kgtMbn; nodal zone nig\TiBlrbs;vaRtuv)anKaMRTedaykarBiesaF nigkarviPaKenaHkugRtaMg
rgkarsgt;RbsiTPaBKNnaenAelIpntMbn; nodal zone EdlbNalmkBIkmaMg strut nigkmaMg
tie minKYrelIsBI tmxageRkam
Edl

Anz =

f ce = 0.85 n f 'c

Edl

(5-22)

enAkgtMbn; nodal zone EdlPab;eday strut b bearing areas bTaMgBIr

C C C node.
n = 0.8 enAkgtMbn; nodal zone Edlf<k;Pab; tie mYy C C T node.
n = 0.6 enAkgtMbn; nodal zone Edlf<k;Pab; tie BIr beRcIn C T T node.
karGnuvtnviFI strut and tie method sRmab;Fwmx<s;manenAkgemeronTI8 ]TahrN_TI6.
T.Chhay

n = 1.0

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VI.

PaBdab nigsameRbH

6>1> sameRbHenAkgeRKOgbgMebtug (Deflection of Structural Concrete Members)

Ggt;ebtugrgkarBt;RtUv)anKNnasRmab;suvtiPaB nigkareRbIR)as; (serviceability) . Ggt;Edl)an
KNnaedayeKarBtamsmIkar nigkarkMNt;EdlEcgedaybTdan ACI Code nwgmansuvtiPaB. dUcenH dUc
karBnl;kgemeronmun TMhMGgt;nImYyEdl)ankMNt; kdUcbrimaNEdkcaM)ac;edIm,IrkSanUvlTPaBm:Um:g;kg
esI bFMCagm:Um:g;xageRkA. enAeBlEdl TMhMmuxkat;cugeRkayRtUv)anKNna FwmRtUv)anepgpat;sRmab;lk
xNbeRmIbRmas; (serviceability) dUcCasameRbH nigPaBdab. edIm,IkarBarnUvsameRbH nigPaBdabelIslb;
eKcaM)ac;manPaBrwgRkajrbs;Ggt;RKb;RKan;.
kareRbIR)as; ACI Code provision Edl)anBicarNanUvTMnak;TMng nonlinear relationship rvag
stress nig strain enAkgebtug )anpl;CalTplnUvmuxkat;tUcCagmuxkat;EdlKNnadayRTwsIeGLasic. bT
dan ACI Code ,Section 9.4 TTYlsal;nUvkareRbIR)as;EdkEdlman yield strength rhUtdl; 560MPa nig
kareRbIR)as;nUvebtugersIusg;x<s;. kareRbInUvebtug nigEdkersIusg;x<s;pl;CalTplnUvmuxkat;tUcCag ehIy
karbnynUvPaBrwgRkajTb;nwgkarBt;rbs;Ggt; )an[PaBdabekInFMCag.
PaBdabEdlGacGnuBaatRtUv)ankMNt;edayktaepSgdUcCa RbePTGKar rUbragrbs;GKar vtman
rbs;Bidan nigCBaaMg karxUcxatrMBwgTukEdlbNalBIPaBdabelIslb; nigRbePT nigTMhMrbs;bnkclt.
bTdan ACI Code , Section 9.5 kMNt;bBaak;kRmas;Gb,brmasRmab;Ggt;rgkarBt;mYyTis oneway flexural members nigkRmalxNmYyTis dUcbgajkgtaragTI 1. tmTaMgGs;kgtaragKWsRmab;
Ggt;EdlminRT bEdlminPab;eTAnwgCBaaMgxN bsMNg;epSgeTotEdlTMngeFVI[xUcxatedayPaBdabFM.

taragTI1 kRmas;Gb,brmarbs;Fwm nigkRmalxNmYyTis L RbEvgElVg

Ggt;

kRmalxNtan;mYyTis
Fwm bkRmalxN
ribbed mYyTis
PaBdab nigsameRbH

Yield
Strength

f y (MPa)

TRmsamBa

cugmagCab; cugsgagCab;

Cantilever

275

L / 25

L / 30

L / 35

L / 12.5

350

L / 22

L / 27

L / 31

L / 11

400

L / 20

L / 24

L / 28

L / 10

275

L / 20

L / 23

L / 26

L / 10

350

L / 18

L / 20.5

L / 23.5

L/9

400

L / 16

L / 18.5

L / 21

L/8

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NPIC

kRmas;Gb,brmaEdlbgajenAkgtaragTI1 RtUv)aneRbIsRmab;Ggt;EdlplitBIebtugTmn;Fmta
normal-weight concrete, W = 2320kg / m 3 nigsRmab;EdkEdlman yield strength dUcbgajkgtarag.
tmRtUv)anEktRmUvsRmab;krNIebtugTmn;Rsal b yield strength rbs;EdkxusBI 400MPa dUcxageRkam
- sRmab;rbtugTmn;RsalEdlmanTmn;maDsitkgcenaH 1400kg / m3 1900kg / m3 tmenAkg
taragsRmab; f y = 400MPa RtUvKuNnwg (1.65 0.0003125Wc ) b:uEnminRtUvtUcCag 1.09 . Edl
Wc Cam:as;maDebtug.
- sRmab; yield strength rbs;EdkxusBI 400MPa tmenAkgtaragsRmab; f y = 400MPa RtUvKuN
nwg (0.4 + f y / 689.5) .
6>2> PaBdabxN (Instantaneous Deflection)
PaBdabrbs;Ggt;ekIteLIgCacMbgedaysarbnkefrbUknwgbnkGefrmYyEpk bTaMgGs;. PaBdab
EdlekIteLIgPambnab;BIGnuvtbnkeK[eQaHfa PaBdabPam (immediate deflection) bPaBdabxN
(instantaneous deflection) . eRkambnkEdlGnuvtsitesr PaBdabnwgekIneLIgKYr[kt;sMKal;CamYynwg
eBl. eKmanviFIepSgsRmab;KNnaPaBdabenAkgeRKOgbgMsaTickMNt; nigsaTicminkMNt;. karKNnaPaB
dab instantaneous deflection KWQrelI elastic behavior nGgt;rgkarBt;. PaBdabeGLasic CaGnuKmn_eTAnwgbnk W ElVg L m:Um:g;niclPaB I nigm:UDuleGLasicrbs;smar E
ML2
WL3
WL

=
=
= f
K

EI
EI
EI

(6-1)

Edl W = bnksrubenAelIElVg ehIy nig K CaemKuNEdlGaRsyeTAnwgdWeRknPaBbgb;rbs;

TRm bERmbRmYlm:Um:g;niclPaBtambeNayElVg nigkarBRgaybnk. ]TahrN_ PaBdabGtibrmanbnk
BRgayesIenAelIFwmTRmsamBaKW
=

5WL3
5wL4
=
384 EI 384 EI

(6-2)

Edl W = bnksrubenAelIElVg = wL bnkBRgayesIelImYyxatRbEvg ElVg. PaBdabrbs;Fwm

CamYynwglkxNbnk niglkxNTRmepSg edayCab;GnuKmn_eTAnwgbnk ElVg nig EI RtUv)an[enAkg
]bsm<n C nigenAkgesovePAviPaKeRKOgbgM.
edaysareKsal; W nig L enaHeKRKan;EtKNnam:UDuleGLasic E nigm:Um:g;niclPaB I nGgt;eb
tug bPaBrwgRkajTb;karBt;rbs;Ggt; EI .
6>2>1> m:UDuleGLasic (Modulus of Elasticity)
bTdan ACI Code , Section 8.5 kMNt;bBaak;fa eKGacykm:UDuleGLasicrbs;ebtug Ec
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sRmab;tm Wc enAcenaH 1400kg / m3 eTA 2500kg / m3

(6-3)
Ec = 4780 f 'c
sRmab;ebtugTmn;Fmta normal-weight concrete Wc = 2320kg / m3
CaTUeTAm:UDuleGLasicRtUv)ankMNt;edayebtugsMNakKMrYsIuLaMgrgbnkkgryeBlxI short-term
loading of concrete cylinder. enAkgGgt;BitR)akd creep EdlbNalmkBIbnkEdlGnuvtmkelICab;
lab; y:agehacNas;kmanbnkefr man\TiBleTAelIm:UDulenAEpkrgkarsgt;nGgt;. sRmab;Epkrgkar
Taj m:UDulenAkgEpkTajRtUv)ansnt;dUcKanwgm:UDulenAkgEpksgt; enAeBlEdltm stress mantmtUc.
enAeBl stress FM m:UDuleGLasicfycuHCaxaMg. elIsBIenH m:UDuleGLasicERbRbYltambeNayElVgEdl
bNalmkBIbERmbRmYlnm:Um:g;nigkmaMgkat;TTwg.
6>2>2> pleFobm:UDuleGLasic (Modular Ratio)
pleFobm:UDuleGLasic n = Es / Ec EdlRtUv)aneRbIenAkgkarbMElgRkLap dUcEdl)anBnl;enA
kgemeronTI2 EpkTI10. vaRtUv)anKitCacMnYnKt; EtminGactUcCag 6 . ]TahrN_ enAeBlEdl
f 'c = 17.5MPa
enaH n = 10
enaH n = 9
f 'c = 20MPa
f 'c = 30MPa
enaH n = 8
enaH n = 7
f 'c = 35MPa
sRmab;ebtugTmn;Fmta (normal-weight concrete) n GacykesI 42 / f 'c .
6>2>3> m:Um:g;eRbH (Cracking moment)
dMeNIrkarnkar)ak;rbs;FwmTRmsamBargbnkRtUv)anBnl;enAkgkfaxN 3>3. enAeBlbnk
GnuvtntUc ekItmanm:Um:g;tUc nig stress enAsrsrgkarTajeRkAbMputnwgmantmtUcCagm:UDuldac;
(modulus of rupture) rbs;ebtug f r = 0.623 f 'c . RbsinebIeKbegInbnkrhUtdl;kugRtaMgTaj (tensile
stress) xiteTACitkugRtaMgdac;mFm f r enaHsameRbHnwgcab;epImekIteLIg. RbsinebIkugRtaMgTajFMCag f r
muxkat;nwgeRbH ehIykrNImuxkat;eRbHnwgcab;epImekIteLIg. enHmannyfaeKmanbIkrNIRtUvBicarNa
- enAeBlEdlkugRtaMgTaj ft tUcCag f r muxkat;Gt;eRbHTaMgmUlRtUv)anBicarNaedIm,IKNnalkN
rbs;muxkat;. enAkgkrNIenH m:Um:g;niclPaB I g RtUv)aneRbI I g = bh3 / 12 Edl bh = muxkat;ebtugTaMg
mUl.
- enAeBlEdlkugRtaMgTaj ft esInwg fr = 0.623 f 'c sameRbHnwgcab;epImekItman ehIym:Um:g;Edl
begItnUvkugRtaMgenHRtUv)aneK[eQaHfa m:Um:g;eRbH (cracking moment) . edayeRbIrUbmnkugRtaMgBt;
Ig
c
f r = M cr
M cr = f r
b
(6-4)
I
c
Ec = 0.043Wc1.5 f 'c

PaBdab nigsameRbH

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Edl fr = 0.623 f 'c / I g = m:Um:g;niclPaBmuxkat;leBj nig c = cmayBIGkSNWteTAsrsrgkar

TajEpkxageRkAbMput. ]TahrN_ sRmab;muxkat;ctuekaNEkg I g = bh3 / 12 nig c = h / 2
- enAeBlEdlm:Um:g;xageRkAFMCagm:Um:g;eRbH (cracking moment) M cr enaHkrNImuxkat;eRbHnwgcab;
epImekIteLIg ehIyebtugenAkgtMbn;rgkarTajRtUv)anecal. muxkat;eRbHbMElg (transformed cracked
section) RtUv)aneRbIedIm,IKNnam:Um:g;niclPaBeRbH (cracking moment of inertia) I cr edayeRbIRkLap
ebtugenAkgtMbn;sgt; nigRkLapEdkbMElg (transformed steel area) nAs .
]TahrN_6>1 muxkat;ebtugctuekaNEkgRtUv)anBRgwgedayEdk 3DB28 enAkgmYyCYr ehIymanTTwg
300mm nigkm<s;srub 650mm cMENk d = 585mm rUb 6>1. cUrKNnam:UDuldac; (modulus of
rupture) f r / m:Um:g;niclPaBmuxkat;leBj I g nigm:Um:g;eRbH M cr edayeRbI f 'c = 28MPa nig
f y = 400MPa .

dMeNaHRsay

1> m:UDuldac; modulus of rupture fr = 0.623 f 'c = 3.28MPa

2> m:Um:g;niclPaBnmuxkat;leBj I g = bh3 / 12 = 300(650)3 / 12 = 6.87 109 mm4
3> m:Um:g;eRbH M cr = f r I g / c = 3.28 6.87 109 / 325 = 69.3kN.m

6>2>4> m:Um:g;niclPaB (Moment of inertia)

edayKuNm:Umg;niclPaBeTAnwgm:UDuleGLasiceyIgTTYl)anPaBrwgRkajTb;nwgkarBt;rbs;Ggt;.
eRkambnktUc m:Um:g;GtibrmaEdlekItmannwgmantmtUc ehIy tensile stress enAsrsrgkarTajxageRkA
bMputnwgmantmtUcCagm:UDuldac;rbs;ebtug. enAkgkrNIenH muxkat;eRbHbMElgeBj (gross transformed
cracked section) nwgmanRbsiTPaBkgkarpl;nUvPaBrwgmaM. enAxNbnkeFVIkar bbnkFM sameRbHenAtMbn;
TajEdl)anmkBIkarBt;nwgekIteLIg. enARtg;muxkat;eRbH TItaMgGkSNWtsitenAx<s; b:uEnenARtg;muxkat;
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kNalcenaHsameRbHenAelIbeNayFwm TItaMgGkSNWtsitenATab Ek,rEdkrgkarTaj. manEtTItaMgTaMg

BIrenHeT Edlmuxkat;eRbHbMElgmanRbsiTPaBkgkarkMNt;PaBrwgRkajrbs;Ggt;. dUcenH m:Um:g;niclPaB
RbsiTPaBERbRbYly:agxaMgtambeNayElVg. enATItaMgm:Um:g;Bt;Gtibrma enaHEpkebtugeRbHenAtMbn;rgkar
TajminRtUv)anKitkgkarKNnam:Um:g;niclPaB. enARtg;cMNucrbt; kugRtaMgtUc ehIymuxkat;TaMgmUlGacnwg
mineRbH. sRmab;sanPaBenH nigenAkgkrNIsRmab;FwmEdlmanFwmERbRbYl dMeNaHRsaydRtwmRtUvman
lkNsKsaj.
rUb 6>2 a bgajBIExSekagbnk-PaBdabrbs;FwmebtugEdlRtUv)aneFVIBiesaFn_rhUtdl;)ak;. Fwm
RtUv)anRTedayTRmsamBa manRbEvg 17 ft nigrgnUvbnkcMcMNucBIrEdlmancmayBIKa 5 ft edaysIuemRTInwg
GkS. FwmRtUv)anrgnUvkardak;bnkBIrCMu TImYy ExSekag cy1 ExSekagbnk-PaBdabCabnat;Rtg;eLIgelI
dl;bnk P = 1.7 K enAeBlEdlFwmcab;epImekItmansameRbH. bnat; a bgajnUvTMnak;TMngbnk-PaBdab
edayeRbIm:Um:g;niclPaBsRmab;muxkat;bMElgGt;eRbH. eKGacemIleXIjfaPaBdabBitR)akdrbs;FwmeRkam
bnkEdltUcCagbnkeRbH edayQrelImuxkat;Gt;eRbHsac;mYy homogeneous cracked section mantm
Ek,reTAnwgPaBdabEdl)anmkBIkarKNna bnat; a . ExSekag cy1 bgajnUvExSekagPaBdabBitR)akd
enAeBlEdlbnkekIneLIgrhUtdl;Bak;kNalbnkcugeRkay (ultimate load). CRmal (slop) rbs;ExSekag
enARKb;nIv:UbnkTaMgGs; mantmtUcCagCRmalrbs;bnat; a edaysarEtsameRbHekItman ehIyEpkEdl
eRbHnmuxkat;ebtug)ankat;bnynUvPaBrwgRkajrbs;Fwm. bnab;mkbnkRtUv)andkecjPaBdabEdlenA
esssl; (residual deflection) RtUv)anGegteXIjmanenAkNalElVgFwm. enAeBlmansameRbHkarsnt;
BIkarRbRBwteTArbs;muxkat;Gt;eRbHeRkambnktUcminRtUv)anykmkGnuvt.
sRmab;karGnuvtbnkCMuTIBIr PaBdab ExSekag c )anekIneLIgkgkRmitFMCagbnat; a edaysar
EtersIusg;rbs;srsebtugTajRtUv)an)at;bg;. enAeBlbnkekIneLIg TMnak;TMngrvagbnk nigPaBdabRtUv
)anbgajedayExSekag cy2 . RbsinebIbnkenACMuTImYyekIneLIgdl;bnkcugeRkay enaHExSekag cy1 nwgcab;
ykKngExSekag cy2 enARtg;bnkRbEhl 0.6Pu . ExSekag c bgajBIdMeNIrkarBitR)akdrbs;FwmsRmab;
karbEnmbnk bkardkbnk.
bnat; b bgajBITMnak;TMngrvagbnk nigPaBdabedayQrelImuxkat;bMElgeRbH. eKeXIjfa PaB
dabEdlKNnaedayQrelIeKalkarN_hwg xusKaBIPaBdabBitR)akd. rUb 6>2 b bgajBIbERmbRmYlPaB
rwgRkaj EI rbs;FwmCamYynwgkarekIneLIgnm:Um:g;. bTdan ACI Code, section 9.5 ENnaMnUvsmIkaredIm,I
kMNt; m:Um:g;niclPaBRbsiTPaBEdleRbIenAkgkarKNnaPaBdabenAkgm:Um:g;rgkarBt;. m:Um:g;niclPaB
PaBdab nigsameRbH

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RbsiTPaBEdl[eday ACI Code (Eq. 9.8) edayQrelIsmIkarEdlesIedayelak Branson nwgRtUv)an

KNnadUcxageRkam
3
M 3
M
I e = cr I g + 1 cr I cr I g
Ma
Ma

Edl

(6-5)

m:Um:g;niclPaBRbsiTPaB
fr I g

M cr = m:Um:g;eRbH/
Y
Ie =

(6-6)

m:UDuldac;rbs;ebtug fr = 0.623 f 'c

(6-7)
M a = m:Um:g;KanemKuNGtibrmarbs;Ggt;enAxNEdlPaBdabkMBugRtUv)anKNna
I g = m:Um:g;niclPaBrbs;muxkat;ebtugleBjeFobGkSTIRbCMuTmn; edayecalEdl
I cr = m:Um:g;niclPaBnmuxkat;bMElgeRbH
Yt = cmayBIGkSTIRbCMuTmn;edayecalEdkeTAprgkarTaj.
xageRkamCakarkMNt;edaybTdan
1> sRmab;FwmCab; m:Um:g;niclPaBRbsiTPaBGacRtUv)anyktmmFmnm:Um:g;niclPaBnmuxkat;
Edlmanm:Um:g;viCman nigGviCmanFMCageK.
2> sRmab;ebtugTmn;Rsal m:UDuldac; fr edIm,IeRbIenAkgsmIkar (6-6) esInwg
f ct
f
Edl
f 'c
(6-8a)
f r = 0.623 ct
0.556
0.556
Edl fct CaersIusg;TajedaykarbMEbk (splitting tensile strength). enAeBlEdleKGt;sal;
f ct enaH f r GacRtUv)anKitdUcxageRkam
fr =

f r = 0.465 f 'c

(6-8b)

sRmab;ebtugeFVIBIxSac;Tmn;Rsal (sand-lightweight concrete)

f r = 0.532 f 'c

(6-8c)

3> sRmab;Ggt;RBIs Ie GacyktmEdlTTYl)anBIsmIkar (6-5) enAkNalElVgsRmab;FwmTRm

samBa nigFwmCab; nigenATRmsRmab;Fwm cantilever (ACI Code, section 9.5.2) .
cMNaMfa Ie Edl)anmkBIkarKNnatamsmIkar (6-5) pl;nUvtmEdlsitenAcenaHm:Um:g;nicl
PaBleBj gross moment of inertia I g nig m:Um:g;niclPaBeRbH cracked moment of inertia
M
I cr edayGnuKmn_eTAnwgkRmitnpleFob cr . Ggt;ebtugEdlmanbrimaNEdkeRcIn Gacnwg
M
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man Ie xiteTACit Icr b:uEnsRmab;Ggt;EdlmanRTnug flanged member Gacnwgman Ie xiteTA

Cit I g .

4> sRmab;FwmCab; tmRbEhlntmmFmn Ie sRmab;Ggt;RBIs bminRBIs edIm,IeFVI[lTpl

kan;EtRbesIreLIgmandUcxageRkam
(6-9)
- sRmab;FwmEdlmancugsgagCab; Average Ie = 0.7 I m + 0.15( Ie1 + Ie2 )
- sRmab;FwmEdlmancugmagCab; Average Ie = 0.85I m + 0.15(Icon )
(6-10)
PaBdab nigsameRbH

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Edl I m = Ie enAkNalElVg Ie1, I e2 = Ie enAcugFwmsgag nig Icon = Ie enARtg;cug

Cab;. Ie GacyktmmFmn Ie enARtg;muxkat;Edlmanm:Um:g;viCman nigGviCmanFM. eKKYr
eRbI moment envelope kgkarKNnatmviCman nigGviCmann Ie . enAkgkrNIEdlFwmrg
bnkcMpitFM manEt Ie kNalElVgKYrRtUv)aneRbI.
6>2>5> lkNrbs;muxkat; (Properties of sections)
edIm,IKNnam:Um:g;niclPaBmuxkat;eBj gross section nigmuxkat;eRbH cracked section eKcaM)ac;RtUv
KNnacmayBIsrsrgkarsgt; compression fiber eTAGkSNWt neutral axis x b kd .
1> m:Um:g;niclPaBeBj I g ecalmuxkat;EdkTaMgGs;enAkgebtug
a. sRmab;muxkat;ctuekaNEkgEdlmanTTwg b nigkm<s;srub h enaH I g = bh3 / 12 .
b. sRmab;muxkat; T-section EdlmanTTwgsab b TTwgRTnug bw nigkRmas;RTnug t KNna y
cmayBIGkSTIRbCMuTmn;eTAcMNucx<s;bMputrbs;sab
bt 2

+ b (h t ) (h t )
2 + t
2 w

y=
bt + bw (h t )

(6-11)

bnab;mkeTotKNna
2
bt 3
t ( y t )3 (h y )

+ bt y + bw
Ig =
+ bw
2
3
3

12

(6-11a)

2> m:Um:g;niclPaBeRbH Icr yk x = cmayBIGkSNWtmksrsrgkarsgt;eRkAbMput x = kd

a. muxkat;ctuekaNEkgCamYyEtnwgEdkrgkarTaj As
i. KNna x BIsmIkarxageRkam
bx 2
nAs (d x) = 0
2

(6-12)

KNna Icr = bx3 / 3 + nAs (d x)2

b. muxkat;ctuekaNEkgCamYyEdkrgkarTaj As nigEdkrgkarsgt; A's
bx 2
i. KNna x
+ (n 1) A's ( x d ' ) nAs (d x) = 0
2
ii. KNna I cr = (bx 3 / 3) + (n 1) A's ( x d ' ) 2 + nAs (d x) 2
c. T-section CamYyEdkrgkarTaj As
(x t )2 nA (d x) = 0
t
i. KNna x
bt ( x ) + bw
s
2
2
ii.

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(6-13)
(6-13a)

(6-14)

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KNna Icr = bt12 + bt x 2t + bw (x 3 t ) + nAs (d x)2

3

ii.

(6-14a)

6>3> PaBdabryeBlyUr (Long-term Deflection)

PaBdabrbs;Ggt;ebtugEdkBRgwgbnekIneLIgeRkambnkEdlGnuvtsitesr ebIeTaHbICavayWtebI
eRbobeFobCamYyeBlkeday. karrYmmaD nig creep CamUlehtueFVI[ekItmanPaBdabbEnmEdleK[eQaH
fa PaBdabryeBlyUr long-term deflection. varg\TiBlCacMbgBIsItuNPaB sMeNIm Gayurbs;ebtug
enAxNnkardak;bnk karEfTaMebtug brimaNEdkrgkarsgt; nigdg;sIuetbnkefr. bTdan ACI Code,
Section 9.5 esIfa y:agehacNas;tmEdlTTYledaykarviPaKdRtwmRtUvmYy KWPaBdabryeBlyUrbEnm
sRmab;Ggt;rgkarBt;ebtugFmta nigebtugTmn;RsalKYr)an TTYlBIkarKuNPaBdabPamCamYyemKuN
=

Edl

(6-15)

1 + 50 '

emKuNsRmab;PaBdabbEnmEdlbNalmkBI\TiBlryeBlyUr
'= A's / bd sRmab;muxkat;enAkNalElVgrbs;FwmTRmsamBa bFwmCab;
bsRmab;muxkat;enATRm rbs;Fwm cantilever .
= emKuNGaRsyeBl time-dependent factor
sRmab;bnkefrEdlGacnwgykdUcbgajkg taragTI 6>2.
emKuN RtUv)aneRbIedIm,IKNnaPaBdabEdlbNalmkBIbnkefr nigEpkxHnbnkGefrEdlnwg
sitenAzitezrsRmab;ryeBlmYyRKb;RKan;edIm,IbegItnUvPaBdabGaRsynwgeBlmYyKYr[kt;sMKal;. em
KuN CaGnuKmn_eTAnwglkNsmar EdlsMEdgeday niglkNmuxkat; sMEdgeday (1 + 50 ' ) .
enAkgsmIkar (6-15) \TiBlrbs;Edkrgkarsgt;KWTak;TgeTAnwgRkLapebtugCagTak;TgeTAnwgpleFob
rvagEdksgt;elIEdkTaj.
ACI Code Commentary, section 9.5 bgajnUvExSekagedIm,IKNna sRmab;ryeBlticCag 60
Ex. tmTaMgenHRtUv)anKNnadUcbgajkgtaragTI 6>2.
PaBdabsrubesInwgplbUkPaBdabPam nigPaBdabbEnmryeBlyUr. ]TahrN_ PaBdabbEnm
ryeBlyUrsrubrbs;FwmrgkarBt;CamYy ' = 0.01 enAryeBl 5 qaMesInwgplKuN CamYynwgPaBdab
Pam Edl = 2 /(1 + 50 0.01) = 1.33 .
=

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tarag 6>2 emKuNGaRsyeBlsRmab;PaBdabryeBlyUr

ryeBl Ex

12

24

36

48

60

0 .5

1 .0

1 .2

1 .4

1 .7

1 .8

1 .9

2 .0

6>4> PaBdabGnuBaat (Allowable Deflection)

GaRsyeTAtambTdan ACI Code, Section 9.5 PaBdabminRtUvelIstmdUcxageRkam
- L /180 sRmab;PaBdabPamEdlbNalmkBIbnkGefrsRmab;dMbUlerobesIEdlminRTGgt;Edl
TMngeFVIxUcxat.
- L / 360 sRmab;PaBdabPamEdlbNalmkBIbnkGefrsRmab;kRmalxNEdlminRTGgt;Edl
TMngeFVIxUcxat.
- L / 480 sRmab;EpknPaBdabsrubEdlekIteLIgeRkayBIkarPab;Ggt; dUcCaplbUkPaBdabry
eBlyUrEdlbNalmkBIbnkefrTaMgGs; nigPaBdabPamEdlbNalmkBIbnkGefrbEnm
sRmab;kRmalxN bdMbUlEdlRTGgt;EdlTMngnwgeFVI[xUcxat.
- L / 240 sRmab;EpknPaBdabsrubEdlekIteLIgeRkayBIGgt;RtUv)anPab; sRmab;kRmalxN b
dMbUlEdlminRTGgt;EdlTMngnwgeFVI[xUcxat.
6>5> PaBdabEdlbNalmkBIbnSMbnk (Deflection Due to Combinations of Load)
RbsinebIFwmrgnUvbnkeRcInRbePT BRgayesI BRgayminesI bbnkcMcMNuc brgnUvm:Um:g;cug enaH
PaBdabGacRtUv)anKNnasRmab;bnk bkmaMgtamRbePTmYyEdlGnuvtmkelIFwmdac;edayELkBIKa ehIy
PaBdabsrubRtUvKNnaedayviFItRmYtpl superposition. enHmannyfaPaBdabmYyRtUveFVIplbUknBVnCa
mYyKaedIm,ITTYl)anPaBdabsrub. PaBdabrbs;bnkeRkambnkmYyRtUv)anbgajenAkgtarag 6>3.

]TahrN_6>2 KNnaPaBdabxNkNalElVgsRmab;FwmTRmsamBadUcbgajkgrUb 6>3 EdlRTnUvbnk

efrBRgayesI 5.85kN / m nigbnkGefr 8.75kN / m rYmnwgbnkefrcMcMNuc 22.25kN enARtg;kNal

ElVg. eK[ f 'c = 28MPa / f y = 400MPa / b = 330mm nig d = 530mm nigkm<s;srub 630mm
n = 8 .

dMeNaHRsay

1> epgpat;km<s;Gb,brmaGaRsyeTAtambTdan ACI Code tarag 6>1

km<s;srubGb,brma = 16L = 12200
= 762.5mm
16

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edaysarkm<s;srubEdleRbIR)as; 630mm < 762.5mm dUcenH eKRtUvkarepgpat;PaBdab.

PaBdab nigsameRbH

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2> PaBdabenAkNalElVgEdlbNalmkBIbnkBRgayKW
1 =

5wL4
384 Ec I e

PaBdabenAkNalElVgEdlbNalmkBIbnkcMcMNucKW
2 =

PL3
48 Ec I e

edaysarEteKsal; w / P nig L dUcenHeKRtUvKNnam:UDuleGLasic

Ec nigm:Um:g;niclPaBRbsiTPaB I e .
3> m:UDuleGLasicrbs;ebtugKW
Ec = 4780 28 = 25293.4MPa

4> m:Um:g;niclPaBRbsiTPaB
3
M 3
M cr

Ie =
I g + 1 cr I cr I g

Ma
Ma

kMNt;tmTaMgGs;enAGgxagsaM
wL2 PL (8.75 + 5.85)12.22 22.25 12.2
+
=
+
= 339.5kN .m
8
4
8
4
bh3 330(630)3
Ig =
=
= 6.88 109 mm 4
12
12
fr I g
h
f r = 0.623 f 'c = 0.623 28 = 3.3MPa
M cr =
Yt = = 315mm
Yt
2
Ma =

eday

M cr =

3.3 6.88 109

= 72.1kN .m
315

m:Um:g;niclPaBnmuxkat;bMElgeRbH Icr RtUv)anKNnadUcxageRkam

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Department of Civil Engineering

kMNt;TItaMgGkSTIRbCMuTmn;sRmab;muxkat;eRbHedaysmIkarm:Um:g;nmuxkat;bMElgeFobGkSTIRbCMu
Tmn;eTA 0 yk x = kd = cmayBIGkSTIRbCMuTmn;
bx 2
nAs (d x) = 0
2

n=

Es
=8
Ec

165 x 2 + 24328 x 12893840 = 0

I cr =

As = 3041mm 2

x = 215.38mm

bx
330(215.38)3
+ nAs (d x) 2 =
+ 8 3041(530 215.38) 2 = 3.5 109 mm 4
3
3
3

CamYynwgGgTaMgGs;Edl)anKNna eyIgTTYl)an
3
72.1 3
72.1
9
9
9
4
Ie =
6.88 10 + 1
3.5 10 = 3.53 10 mm
339
.
5
339.5

5> KNnaPaBdabBIbnkepSg
5wL4
1 bNalmkBIbnkrayesI =
384 E I

c e

1 =
2

5(8.75 + 5.85) 12200

= 47.2mm
384 25293.4 3.53 109
PL3
=
48 Ec I e
4

bNalmkBIbnkcMcMNuc

2 =

22250 122003
= 9.43mm
48 25293.4 3.53 109

PaBdabPamsrub = 1 + 2 = 47.2 + 9.43 = 56.63mm

6> eRbobeFobtmEdl)anmkBIkarKNnaCamYyPaBdabGnuBaat
PaBdabPamEdl)anmkBIbnkGefrBRgayesI 8.75kN .m esInwg 8.75 47.2 /(8.75 + 5.85) =
28.29mm . RbsinebIGgt;CaEpkmYyrbs;kRmalxNEdlminRT bPab;eTAnwgCBaaMgxN bGgt;epSgeTot
EdlGacTMngeFVI[xUcxatedaysarPaBdabFM enaHPaBdabPamGnuBaatEdlbNalmkBIbnkGefresInwg
L
12200
=
= 33.9mm > 28.29mm
360
360

EtRbsinebIGgt;CaEpknkRmalxNdMbUl nigRsedogKaeTAGVIEdl)anerobrab;xagelI enaHPaBdab

L 12200
PamGnuBaatEdlbNalmkBIbnkGefrKW 180
=
= 67.8mm > 28.29mm . tmGnuBaatTaMgBIrFM
180
CagPaBdabBitR)akd 28.29mm EdlbNalmkBIbnkGefrBRgayesI.

PaBdab nigsameRbH

135

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NPIC

]TahrN_6>3 kMNt;PaBdabryeBlyUr long-term deflection rbs;FwmenAkg]TahrN_6>2 RbsinebIem

KuNGaRsyeBl time-dependent factor esInwg 2.0 .

dMeNaHRsay

1> bnkEdlGnuvtGciny_bNal[manPaBdabryeBlyUrKWekItmkBIbnkGefr EdlrYmmanbnk

efrBRgayesI 5.85kN / m nigbnkefrcMcMNuc 22.25kN EdlGnuvtn;enAkNalElVg.
PaBdabbNalmkBIbnkBRgayesI = 5.85 47.2 /(8.75 + 5.85) = 18.9mm
PaBdabCaGnuKmn_bnat;eTAnwgbnk w ehIytmepSgeTot L / Ec / Ie dUcKa
PaBdabbNalmkBIbnkcMcMNuc = 9.43mm
PaBdabsrubEdlbNalmkBIbnkzitezr sustained load = 18.9 + 9.43 = 28.33mm
2> sRmab;PaBdabryyUrbEnm PaBdabPamRtUv)anKuNnwgemKuN
=

1 + 50 '

2
1+ 0

kgkrNIenH A's = 0 dUcenH = 2.0

PaBdabryeBlyUrbEnm = 2 28.33 = 56.66mm
3> PaBdabryeBlyUrsrubCaplbUkrvagPaBdabPamCamYynwgPaBdabryeBlyUrbEnm
56.63 + 56.66 = 113.29mm

4> PaBdabbNalmkBIbnkefrCamYynwgPaBdabryeBlyUrbEnmEdlbNalmkBIkarrYmmaDnig
creep KW 28.33 + 56.66 = 85mm

]TahrN_6>4 KNnaPaBdabxN nigPaBdabenAryeBl 1qaM enAcugTMenrrbs;Fwm cantilever dUcbgaj

kgrUbTI 6>4. FwmmanRbEvg 6.1m nigRTbnkefrBRgayesI 5.85kN / m bnkGefrBRgayesI 5.85kN / m

bnkefrcMcMNuc PD = 13.35kN enAcugTMenr nigbnkGefrcMcMNuc PL = 17.8kN EdlGnuvtenAcmay 3.05m
BITRmbgb;. eK[ f 'c = 28MPa / f y = 400MPa / b = 300mm / d = 550mm / nigkm<s;muxkat;srub
635mm srsEdkrgkarTajKW 6 DB 25 nigEdkrgkarsgt; 2 DB 25 .

dMeNaHRsay

1> km<s;Gb,brma L8 = 6100

= 762.5mm > 635mm dUcenHeKRtUvepgpat;PaBdab.
8
PaBdabGtibrmarbs;Fwm cantilever KWsitenAcugTMenr. PaBdabenAcugTMenrKWdUcxageRkam
PaBdabEdlbNalmkBIbnkBRgay
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1 =

Department of Civil Engineering

wL4
8 EI

PaBdabEdlbNalmkBIbnkefrcMcMNucenAcugTMenr
PD L3
3EI

2 =

PaBdabGtibrmaenAcugTMenrEdlbNalmkBIbnkGefrcMcMNucenA a = 3.05m BIcugbgb;

3b
P a2
PL a 3
3 = L (3L a )
b
(1 + )
6 EI
3EI
2a
2> m:UDuleGLasicrbs;ebtugTmn;RsalKW
Ec = 4780 28 = 25293.4 MPa

3> m:Um:g;GtibrmaenAcugbgb;
Ma =

5.85 6.12
wL2
+ 6.1PD + 3.05 PL =
+ 6.1 13.5 + 3.05 17.8 = 245.5kN .m
2
2

4> m:Um:g;niclPaBeBj Etebtug

Ig =

5>

M cr

bh3 300 6353

=
= 6.4 109 mm 4
12
12
f r I g 0.623 28 6.4 109
=
=
= 66.5kN .m
635
Yt
2

6> kMNt;TItaMgGkSNWt bnab;mkkMNt;m:Um:g;niclPaBnmuxkat;bMElgeRbH. Kitm:Um:g;nmuxkat;

eFob GkSTIRbCMuTmn;nigdak;[vaesIsUn. eRbI n = 8 edIm,IKNnamuxkat;bMElgrbs; As nig eRbI
(n 1) = 7 sRmab;KNnamuxkat;bMElgrbs; A's . yk kd = x
b

(x )2 + (n 1) A'
2

s ( x d ' ) nAs ( d

x) = 0

150 x 2 + 30434 x 13370440 = 0

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sRmab;muxkat;enH x = 213.875mm
I cr =

b 3
x + (n 1) A's ( x d ' ) 2 + nAs (d x) 2 = 3.8 109 mm 4
3

7> m:Um:g;niclPaBRbsiTPaBKW

3
M 3
M cr
I g + 1 cr I cr I g
I e =
M
Ma
a

3
66.5 3
66.5
9
9
9
Ie =
6
.
4

10
+
1

3.8 10 = 3.85 10
245
.
5
245
.
5

8> KNnaPaBdabeRkambnkepSgEdlmanGMeBIelIFwm
11.7 6100 4
1 bNalBIbnkBRgayesI 11.7 kN / m =
= 20.8mm
8 25293.4 3.85 109
1 bNalBIbnkefr = 10.4mm
13500 61003
2 bNalBIbnkefrcMcMNucenAcugTMenr =
= 10.5mm
3 25293.4 3.85 109
3 bNalBIbnkGefrcMcMNucenA 3.05m BIcugbgb;
3 =

17800 3050 2 (3 6100 3050)

= 4.3mm
6 25293.4 3.85 109

PaBdabPamsrub = 1 + 2 + 3 = 20.8 + 10.5 + 4.3 = 35.6mm

9> PaBdabbEnmryeBlyUr esInwgplKuNrvagPaBdabPamCamYynwgemKuN
sRmab;ryeBl 1qaM = 1.4
982
A's
=
= 0.006
bd 300 550
1.4
=
= 1.077
1 + 50 0.006

'=

PaBdabPamsrub s EdlbNalmkBIbnkzitezr sustained load enATIenHmanEtbnkefr

5.85kN / m nig PD = 13.35kN enAcugTMenr
s = 10.4 + 10.5 = 20.9mm

PaBdabryeBlyUrbEnm = 1.077 20.9 = 22.5mm

10> PaBdabryeBlyUrsrubCaplbUkrvagPaBdabPamCamYynwgPaBdabryeBlyUrbEnm
EdlekItBIkarrYmmaD nig creep .
Total = 35.6 + 22.5 = 58.1mm

]TahrN_6>5 KNnaPaBdabxNenAkNalElVgrbs;FwmenAkgrUb 6>5 EdlFwmenHmanRbEvg 9.8m .

FwmenHRtUv)anbnedayTRmCaeRcInEdlmanRbEvgElVgepSgKa. daRkamm:Um:g;Bt;nigmuxkat;rbs;FwmenA

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kNalElVg nigTRmkRtUv)anbgaj. FwmenHRTbnkefrBRgayesI 61.3kN / m nigbnkGefr 52.5kN / m .

eK[ f 'c = 20MPa / f y = 400MPa nig n = 9.2 .
m:Um:g;enAkNalElVg M D = 260kN .m
M ( D + L ) = 650kN .m
m:Um:g;enATRmxageqVg A M D = 242.75kN .m M ( D + L) = 569.5kN .m
m:Um:g;enATRmxagsaM B M D = 293kN .m
M ( D + L ) = 735kN .m

dMeNaHRsay

1> Fwm AB rgnUvm:Um:g;viCmanEdleFVI[dabcuHeRkamenAkNalElVg nigm:Um:g;GviCmanenAcugTaMgBIr

EdleFVI[dabeLIgelIenAkNalElVg. dUcEdl)anBnl;BIxagedIm PaBdabCaGnuKmn_eTAnwg
m:Um:g;niclPaBRbsiT Ie . enAkgFwmCab; tmrbs; Ie EdlRtUv)aneRbICatmmFmsRmab;tMbn;m:U
m:g;viCman nigm:Um:g;GviCman. dUcenHmuxkat;bInwgRtUv)anBicarNa muxkat;enAkNalElVg nigmuxkat;
enATRmTaMgBIr.
2> KNna Ie sRmab;RkLapeBjnmuxkat;TaMgGs; kd = 356mm nig I g = 5.2 1010 mm4 dUcKa
f r = 0.623 f 'c = 2.8MPa nig Ec = 4780 f 'c = 21376.8MPa sRmab;RKb;muxkat;. tmrbs;
kd / I cr nig M cr sRmab;muxkat;eRbH/ I e sRmab;Etbnkefr edayeRbI M a sRmab;bnkefr nig
I e sRmab;bnkefr nigbnkGefr edayeRbI M a sRmab;bnkefr nigbnkGefrRtUv)anKNna
nigbegItCa taragdUcxageRkam
I e mm 4
I e mm 4
4
muxkat;
kd mm
I cr mm M cr kN.m
bnkGefr DL + LL
kNalElVg 169.8
2.12 1010
226
4.14 1010
2.25 1010
TRm A
281
409
1.52 1010
5.2 1010
2.88 1010
TRm B
320
1.84 1010
409
5.2 1010
2.42 1010
cMNaMfa enAeBlEdlFwmrgEtbnkefr ehIypleFob M cr / M a FMCag 1.0 enaH Ie esInwg I g .
3> kMNt;tmmFmrbs; Ie BIsmIkar (6.9)
I e1 (average) = 0.7(2.25 1010 ) + 0.15(2.88 + 2.42 )1010 = 2.37 1010 mm 4
sRmab;bnkefr nigbnkGefr
I e (average for end sections) = 0.5(2.88 + 2.42)1010 = 2.65 1010 mm 4

I e 2 (average) = 0.5 2.25 1010 + 2.65 1010 = 2.45 1010 mm 4

sRmab;Etbnkefr
PaBdab nigsameRbH

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I e (average for end sections) = 5.2 1010 mm 4

I e3 (average) = 0.5 4.14 1010 + 5.2 1010 = 4.67 1010 mm 4

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viTasanCatiBhubeckeTskm<Ca

4> KNnaPaBdabPamenAkNalElVg
5wL4
1 bNalmkBIbnkBRgay =
384 EI
2

Department of Civil Engineering

cuHeRkam
2

M AL
bNalmkBIm:Um:g;enAcMNuc A / M A = 16
eLIgelI
EI
e
2

M BL
bNalmkBIm:Um:g;enAcMNuc B / M B = 16
eLIgelI
EI e
PaBdabsrub = 1 2 3
PaBdabsRmab;bnkefrBRgayesI 61.3kN / m edayyk M A ( DL) = 242.75kN .m /
M B ( DL) = 293kN .m nig I e3 = 4.67 1010 mm 4 CMnYseTAkgsmIkarxagelI eyIgTTYl)an
= 7.4 1.5 1.8 = 4.1mm cuHeRkam
PaBdabEdlbNalBIbnSMbnkefr nigbnkGefrRtUv)anKNnaedayykbnkefrbUknwgbnkGefr
= 113.8kN / m / M A ( DL + LL ) = 569.5kN .m / M B ( DL + LL) = 735kN .m nig
3

I e 2 = 2.45 1010 mm 4

cuHeRkam
PaBdabPamEdlbNalBIEtbnkGefrKW 11.2 4.1 = 7.1mm cuHeRkam.
RbsinebIkarkMNt;PaBGnuBaatiKW L / 480 = 9800 / 480 = 20.4mm enaHmuxkat;KWRKb;RKan;.
cMNucmYycMnYnEdlTak;TgnwglTpl
1> RbsinebIeKeRbIEt Ie nmuxkat;kNalElVg Ie = 2.25 1010 mm4 enaHPaBdabEdlbNalmkBI
bnkefr nigbnkGefrRtUv)anKNnaeday KuNtmEdlTTYl)ankgCMhanTI4 CamYynwgpl
eFobntm Ie TaMgBIr
2.45 1010
bnkefr + bnkGefr = 11.2
= 12.2mm
2.25 1010
PaBxusKamantmtUc RbEhl 8% .
2> RbsinebI Ie1 mFm RtUv)aneRbI Ie1 = 2.37 1010 mm4 enaH
2.37 1010
bnkefr + bnkGefr = 11.2
= 11.8mm . PaBxusKamantmtUcRbEhl
2.25 1010
5% .
3> edIm,IPaBgayRsYleKGacRbI Ie nmuxkat;kNalElVg elIkElgEteKRtUvkartmCak;lak;.
= 26.1 6.5 8.4 = 11.2mm

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6>6> sameRbHenAkgGgt;rgkarBt; (Cracks in Flexural Members)

karsikSaBIkarkekItsameRbH kareFVIkarrbs;sameRbHenAeBlbnkekIneLIg nigkarRtYtBinitsameRbH
mansarsMxan;Nas;sRmab;karKNnaeRKOgbgMebtugEdkBRgwgdCak;lak;mYy. enAkgGgt;rgkarBt; sam
eRbH ekIteLIgeRkambnkeFVIkar working load nigedaysarebtugexSayTb;nwgkarTaj enaHEdkRtUv)andak;
enAtMbn;TajEdleRbH edIm,ITb;nwgkarTajEdlbgeLIgedaykmaMgxageRkA.
sameRbHedaykarBt;ekIteLIgenAeBlEdlkugRtaMgenAsrsrgkarTajxageRkAeKbMputmantmelIsBI
m:UDuldac;rbs;ebtug (modulus of rupture of concrete) . CamYynwgkareRbIR)as;EdkBRgwgEdlmanersIusg;
x<s; (high-strength reinforcing bars) sameRbHGacnwgekItmaneRcInenAelIGgt;ebtugGarem:. CamYynwgkar
eRbIR)as;EdkrgkarTajx<s; high-tensile steel manRbeyaCn_CaeRcIn b:uEneKminGacecosputBIsameRbHEdl
eKmincg;)anenaHeT. sameRbHEdlmanTMhMFM )anGnuBaat[manERcHsIuEdk beFVI[mankarRCabTwkEdleFVI
[)at;bg;esaPNPaBrbs;eRKOgbgM.
sameRbHekItmanenAelIebtug enAeBlEdlrntUcEdlminGackMNt;)anekItmanenAkgFwmebtugEdlCa
lTplnkugRtaMgTajxagkg. kugRtaMgkgTaMgenHGacnwgbNalmkBIkrNImYy beRcInCagmYynkrNIxag
eRkam
- kmaMgxageRkA dUcCa kmaMgTajtamGkSedaypal; kmaMgkat;TTwg m:Um:g;Bt; bm:Um:g;rmYl
- karrYmmaD
- creep
- karrIkmaDxagkgEdlCalTplBIkarpas;brlkNrbs;smasFatupSMebtug
CaTUeTA sameRbHRtUv)anEbgEckCaBIrRbePTFMKW sameRbHbnab;bnSM (secondary cracks) nig
sameRbHcMbg (main cracks).
6>6>1> sameRbHbnab;bnSM (secondary Cracks)
sameRbHbnab;bnSM CasameRbHdtUcekItmanenAdMNak;kaldMbUgnkareRbH EdlekIteLIgedaykar
rIkmaDxagkg nigkarRbTajKansmasFatuebtug nigedaykugRtaMgEdl)anBIkarBt;tUcbNalmkBITmn;
pal;rbs;Ggt; nigbnkefrdTeTot. sameRbHmanbIRbePTKW
- sameRbHedaykarrYmmaD (shrinkage cracks) CasameRbHsMxan; BIeRBaHvaCH\TiBlelITRmg;ragrbs;
kareRbHEdlekIteLIgedaybnkenAkgGgt;rgkarBt;. enAeBlEdlBYkvaekIteLIg BYkvabegItnUv
KngdexSayenAkgebtug. enAeBlEdlbnkRtUv)anGnuvtsameRbHcab;epImelcecjrUbragenAelI
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muxkat;EdlexSayCageK dUcCatambeNayEdkBRgwg. cMnYn sameRbHEdlekIteLIgRtUv)ankMNt;

edaybrimaNnkarrYmmaDrbs;ebtug nigvtmannkarbgb;TRm. eKBi)akkgkarRKb;RKgsameRbH
edaykarrYmmaDenHNas;.
- sameRbHbnab;bnSMedaykarBt; (secondary flexural cracks) CaTUeTAvamanKMlatFM ehIysam
eRbHmYyminman\TiBlkgkarbegItsameRbHdTeToteT. vaRtUv)anrMBwgnwgekItmaneRkambnktUc
dUcCabnkefr. enAeBlbnkRtUv)anGnuvtCabnbnab;eTAelIFwmsamBa kugRtaMgTajekItmanenA
srsxageRkam ehIyenAeBlEdlvaFMCagkugRtaMgTajEdlekItBIkarBt;rbs;ebtug sameRbHcab;
epImekIteLIg. vacab;epImrIkFMCabnbnab; niglatsnwgeTAkan;GkSNWt. vaCakarBi)akkgkarTsSn_
TaynUvmuxkat;EdlsameRbHbnab;bnSMcab;epImekItman BIeRBaHebtugminEmnCasmarrUbFatusac;
mYy (homogeneous material) nigsmarrUbFatuesIsac; (isotropic material).
elak Salinger nigelak Billing )an):an;RbmaNkugRtaMgEdkmunkareRbHRtwmEtBIRbEhl 42MPa
eTA 49MPa . TMhMeRbHdMbUgRtUv)anrMBwgRbEhl 0.025mm enAsrsTajxageRkAbMputrbs;ebtug.
enAeBlsameRbHRtUv)anbegIt kugRtaMgTajrbs;ebtugenAmuxkat;eRbHfycuHdl; 0 ehIysrsEdk
TTYlnUvkmaMgTajTaMgGs;. enAkgxNenH vaekItmankarsNkrvagsrsEdk nigebtugedaysar
Etebtug nigEdkmansac;lUtxusKa niglatsnwgeTAdl;muxkat;Edlebtug nigEdkman strain esIKa.
rUbTI6>6 bgajBIKMrUnkarEbgEckkugRtaMgrvagsameRbHenAkgGgt;eRkamkmaMgTajtamGkS.

PaBdab nigsameRbH

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- sameRbHbnab;bnSMERcH (corrosion secondary cracks) ekIteLIgenAeBlsMeNImmansarFatuKImIdUc

CasUdmkrY (sodium chloride NaCl ) kabUnDIGuksIut (carbon dioxide CO2 ) nig]snGuksIuEsn
RCabcUleTAkgpebtug ehIybegItCaERcHsIuEdk. bnSMGuksIutEdlbegItedaykarxUcxatrbs;Edk
RKbdNb;nUvbrimaNeRcInCagEdk nigCRmujsm<aFemkanicEdlbnlatsnwgsameRbHkan;EtFM.
sameRbHRbePTenHmaneRKaHfak;EdlGaceFVI[eRKOgbg)M ak;. kar)ak;dMbUlenA Muskegan enArd
Michigan kg qaM 1955 bNalmkBIERcHsIuEdk RtUv)anraykarN_eday Shermer. karrIkraldal
nsameRbH nigkarpHebtug (spalling of concrete) enAelIs<an San Mateo-Hayward kgGMLg
eBl 7qaM enArd California RtUv)anraykarN_eday Stratfull. sameRbHedayERcHGacRtUv)an
bBab;edayeRbInUvviFIsaRssagsg;RtwmRtUv nigebtugEdlmanKuNPaBx<s;.
6>6>2> sameRbHcMbg (Main Cracks)
sameRbHcMbg (main cracks) ekIteLIgenAdMNak;kalbnab;BIsameRbHbnab;bnSM (secondary
cracks). vaRtUv)anbegIteLIgedayPaBxusKan strain enAkgEdk nigebtugenAelImuxkat;eRbH. karRbRBwt
eTArbs;sameRbHcMbgpas;brenABIrdMNak;kalepSgKa. enAeBlEdlEdkrgkugRtaMgTajtUc cMnYnsameRbH
ekIneLIg b:uEnTMhMsameRbHenAmanTMhMtUc EtenAeBlEdlkugRtaMgTajekIneLIg va)aneTAdl;dMNak;kal
lMnwg. enAeBlEdlkugRtaMgekIneLIgkan;Etx<s; dMNak;kalTIBIrnkareRbHekIteLIg ehIyTMhMeRbHrIkFM b:uEn
BuMmankarekIneLIgnUvcMnYnsameRbHKYr[kt;sMKal;eT. CaTUeTAsameRbHmYy bBIrcab;epImBRgIkTMhMFMCagsam
eRbHdT EdlbegItCasameRbHeRKaHfak; rUbTI 6>7.
sameRbHenAkgFwm nigeRKOgbgMrgkarTaj Edl)ansikSaedayGkGegtkarTsSn_TayBITMhMsam
eRbH nigkarRKb;RKgsameRbHCacMENknbBaaEdlRtUvsikSa.
TaMgenHRtUv)anBiPakSaenATIenH CamYynigtRmUvkarrbs;bTdan ACI Code.
- ebIeyagtamRTwsIsameRbHEdlmaneRsc TMhMsameRbH (crack width) nigKMlatsameRbH (crack
spacing) GaRsyeTAelIeRcInkta EdlrYmmanPaKryEdk karBRgayEdkenAkgmuxkat;ebtug kug
RtaMgBt;rbs;EdkenAbnkeFVIkar service load kRmas;ebtugkarBarEdk niglkNrbs;sarFatupSM
ebtug. smIkarepSgsRmab;TsSn_TayTMhM nigKMlatsameRbHenAkgGgt;ebtugGarem:RtUv)an
bgajenAkgsnisiTsIGMBIkarkekItkaretagsit nigkarkekItsameRbHenAkgebtugBRgwgedayEdk
Symposium on Bond and Crack Formation in Reinforced Concrete enA Stockholm RbeTs
Sweden kgqaM 1957. elak Chi nigelak Kirstein )anbgajsmIkarsRmab;TMhMeRbH nigKMlat
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sameRbHCaGnuKmn_eTAnwgRkLapRbsiTPaBrbs;ebtugEdlenACMuvijEdk RkLapebtugmUlEdl
manGgt;pitesIGgt;EdkbYndgRtUv)aneRbIedIm,IKNnaTMhMeRbH. smIkardTeTotRtUv)anbgajenA
TsvtSr_eRkay eTot.

elak Gergely nigelak Lutz bgajnUvrUbmnxageRkamsRmab;kMNt;TMhMeRbH

W = 11f s 3 Ad c 10 6

(6-16)

Edl / A nig f s RtUv)ankMNt;BIxagedIm nig dc = kRmas;karBarebtugEdlRtUv)anvas;BIsrs

TajEpkxageRkAbMputeTAGkSEdkCitCageK. tm GacRtUv)anykRbEhl 1.2 sRmab;Fwm nig 1.35
sRmab;kRmalxN. cMNaMfa f s KitCa MPa nig W KitCa mm .
pleFobrvagTMhMsameRbHGtibrmaelITMhMsameRbHmFmRtUv)anrkeXIjsitenAcenaH 1.5 nig 2.0
Edl)anraykarN_edayGkGegtCaeRcIn. tmmFmeKGaceRbI 1.75 .
PaBdab nigsameRbH

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- TMhMeRbHEdlGacGnueRKaH (tolerable crack width) eKminGaceCosputBIkarkekItsameRbHenAelI

Ggt;ebtugGarem:)aneT. eTaHbICamankarsikSa nigkarsagsg;y:agykcitTukdak;y:agNakeday
kvaenAEtkekItnUvsameRbHb:unsrssk;Edr. CaTUeTAsameRbHRtUv)anvas;enAelIprbs;ebtug b:uEn
Cak;EsgsameRbHmantaMgBInIv:UEdkem:H EdlCakEngekItmanERcHsIuEdk. tRmUvkaresaPNPaBCH
\TiBldl;TMhMsameRbHGnuBaat. sameRbHEdlGacemIleXIjedayEPkTeTmanTMhMRbEhl
0.15mm vaGaRsyeTAnwg texture rbs;pebtug. tmepSgsRmab;sameRbHGnuBaatenAnIv:UEdk
RtUv)anesIeLIgedayGkGegtCaeRcIn EdlvasitenAcenaH 0.25 nig 0.40mm sRmab;Ggt;xagkg
nigsitenAcenaH 0.15 nig 0.25mm sRmab;Ggt;xageRkA. TMhMsameRbH 0.40mm sRmab;Ggt;xag
kg nig 0.32mm sRmab;Ggt;xageRkA eRkamlkxNstGacRtUv)anGnueRKaH.
- karRKb;RKgsameRbH (crack control) eRcInekItmanenAelIGgt;ebtugGarem:EdleRbIEdkersIusg;x<s;
(high-strength steel) . sameRbHkan;EtFMekItmaneRkambnkeFVIkar edaysarEtkugRtaMgGnuBaat
x<s;. karRKb;RKgsameRbHGaRsyeTAnwgTMhMsameRbHGnuBaat . vaCakarRbesIrEdlmansameRbH
tUcEteRcIn CaCagmansameRbHFMEttic. sameRbHbnab;bnSM (secondary crack) RtUv)ankat;
bnyedaykarRKb;RKgnUvbrimaNkMe)areRbgTaMgGs; pleFobTwkelIsIum:gt PaBRCabTwkrbs;f
bMEbknigebtug kRmitnkarEfTaMebtug niglkxNbgk;cug (end-restraint condition).
ktaEdlTak;TgnwgkarRKb;RKgsameRbHcMbg KWkugRtaMgEdk PaBetagsitrbs;ebtug kar
BRgayEdk Ggt;pitEdk PaKryEdk kRmas;karBarebtug niglkNrbs;sarFatupSMebtug. edIm,I
CYykat;bnyTMhMsameRbH eKRtUveFVI[ktaEdl)anerobrab;xagelIy:agehacmYymanlkNkan;Et
l.
6>7 tRmUvkarrbs;bTdan ACI Code (ACI Code Requirement)
edIm,IRKb;RKgsameRbHenAkgGgt;ebtugGarem: bTdan ACI Code, chapter 10 kMNt;nUvtRmUvkarxag
eRkam
1> manEtEdk deformed bar RtUv)anGnuBaat[eRbICaEdkem (main reinforcement)
2> EdkrgkarTajKYrEtRtUv)anBRgayenAtMbn;TajGtibrma (ACI Code, section 10.6.3)
3> enAeBlEdlsabrbs;muxkat;siteRkamkarTaj EpknEdkemKYrRtUv)anBRgayenAkgcMeNamtm
EdltUcCageKkgcMeNamTTwgsabRbsiTPaB (effective flange width) nigmYyPaKdb;nElVg
one-tenth of the span (ACI Code, section 10.6.6).
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4> ersIusg;KNna (design yield strength) rbs;EdkminKYr[elIsBI 560MPa (ACI Code, section
9.4).
5> KMlatGtibrma s BIEdkEdlmancmayeTApebtugrgkarTajCitCageK sRmab;FwmebtugGarem: nigkM
ralxNmYyTisRtUv)ankMNt;
s = 105000 / f s 2.5Cc

(6-17)

b:uEnminRtUvFMCag 300(280 / f s )
Edl f s = kugRtaMgKNnaenAkgEdkeRkambnkeFVIkarRtUv)ankMNt;edaypleFobrvagm:Um:g;
KanemKuN nigplKuNrvagRkLapEdk nigRbEvgdXas;kg f s = M /( As jd ) . m:ag
vijeTot eKGaceRbI f s = 2 f y / 3 edaysarEtkareRbItmRbEhlrbs; jd = 0.87d .
Cc = kRmas;ebtugkarBarEdk (clear cover) BIprgkarTajCitbMputeTApEdkrgkar
Taj.
s = KMlatKitBIGkSnEdkrgkarTajEdlCitCageKeTApebtugrgkarTaj.
karkMNt;enHRtUv)aneRbIsRmab;EtFwmebtugGarem: nigkRmalxNmYyTisEdlsitenAkglk
xNbrisanFmta. karkMNt;KMlatminTak;TgnwgTMhMEdkeT EdlvanaM[mankareRbIR)as;TMhMEdktUc
CagedIm,IbMeBjlkxNKMlat. sRmab;krNIFwmebtugGarem:EdleRbIEdk f y = 420MPa nig
Cc = 50mm . enaHKMlatGtibrma s RtUv)anKNnadUcxageRkam
snt; f s = 0.66 420 = 280MPa
s = (105000 / 280 ) 2.5 50 = 250mm

EdldUcKaeTAnwg 300(280 / 280) = 300mm

6> enAkgbTdanBImun karRKb;RKgsameRbHRtUv)anQrelIemKuN Z EdlkMNt;dUcxageRkam
Z = f s 3 Ad c 31kN / mm sRmab;Ggt;enAxagkg
(6-18)
Z 26kN / mm
sRmab;Ggt;enAxageRkA
(6-19)
Edl f s = kugRtaMgBt;enAbnkeFVIkar nigGacykesI 2 f y / 3 / A nig dc CatMbn;TajRbsiTPaB
rbs;ebtug nigkRmas;ebtugkarBarEdk erogKa. smIkarenHQrelIsmIkar (6-16) edaysnt;TMhMsameRbH
RtUv)ankMNt;Rtwm 0.40mm sRmab;Ggt;enAxagkg nig 0.32mm sRmab;Ggt;enAxageRkA. edIm,ITTYl)an
Z tUc eKRtUvbnykRmas;karBarebtug. kRmas;ebtugminRtUvelIsBI 50mm .
7> Edkxag skin reinforcement Ask sRmab;Fwmkm<s;eRCA Edlmankm<s;RbsiTPaB d 900mm
srsEdkesIgKYrRtUv)anbEnmenAEk,rpbBarkgtMbn;TajedIm,IRKb;RKgsameRbHenAelIRTnug
PaBdab nigsameRbH

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BIelIEdkem. bTdan ACI Code, section 10.6.7 )anbriyayBIEdkbEnmEdlCaEdkxag skin

reinforcement Ask . EdkxagKYrRtUv)anBRgayesIelIpTaMgsgagsRmab;cmay d / 2 Ek,rEdk
rgkarTajbMput.
KMlat Ssk rvagEdkbeNaynEdkxagminKYrFMCagtmEdltUcCageKkgcMeNamtm d / 6 /
300mm nig 1000 Ab /(d 750) Edl Ab CaRkLapnmuxkat;EdkmYy KitCa mm 2 rUbTI6>8.
S sk

d /6

= min
300
1000 A /( d 750)
b

Edkxag kGacRtUv)anKitbBalkgkarKNnaersIusg; (strength) edayQrelIkarKNnakugRtaMg

stress nigbERmbRmYlrageFob (strain). elIsBIenH EdkxagsrubenApTaMgsgagnGgt; minKYrelIsBIBak;
kNalmuxkat;EdkrgkarTajsrub As eT.
eyagtamrUbTI 6>8 RbsinebI b = 400mm / d = 950mm nigeRCIserIsEdk DB10 Ab =
78mm 2 CaEdkxag enaHKMlatrbs;Edk DB10 EdlCaEdkbeNayKYryktmtUcCageKkgcMeNam
300mm / 950 / 6 = 158mm nig (1000 78) /(950 750) = 390mm . dUcenH eyIgeRCIserIsykKMlat
150mm . EdkxagKYrRtUv)anBRgayedayEdk DB10 kgKMlat 150mm elIkm<s; d / 2 sRmab;xagmYy
rUb TI 6>8. RkLapsrubEdkxagsRmab;pTaMgBIr = 6 78 = 468mm2 EdltUcCag
0.5 As = 0.5 4021 = 2010.5mm 2 .
taragxageRkampl;nUvKMlatGtibrmaenAelIpmYy Ssk RkLapEdkxagGb,brmamYy Ab .

km<s;Fwm d / mm
KMlatGtibrma Ssk / mm
Ab Gb,brma/ mm 2

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800

900

1000

1100

1200

1300

1400

1500

150

150

150

175

200

200

225

250

7 .5

22.5

37.5

61.25

90

110

146.25

187.5

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]TahrN_6>6 muxkat;FwmTRmsamBaRtUv)anbgajenAkgrUb 6>9

a.
b.
c.

epgpat;muxkat;EdktamtRmUvkarrbs; ACI Code

kMNt;TMhMeRbHEdlnwgGacekItman
epgpat;emKuN Z tamsmIkar (6-18)
eK[ f 'c = 28MPa / f y = 400MPa nigEdkkg DB10 .

dMeNaHRsay

1> rUb 6>9 muxkat; a

a. sRmab; 3DB 28 / As = 1847mm 2 / kRmas;ebtugkarBarEdk Cc = 60 28 / 2 = 46mm / snt;
f s = 0.66 f y = 0.66 400 = 264MPa / KMlatGtibrma
s = 105000 / 264 2.5 46 = 282.7 mm EdltUcCag 300(280 / 264) = 318.2mm .
KMlatEdlpl;[ = 0.5(300 60 60) = 90mm edayKitBIGkSEdkeTAGkSEdk EdltUcCag
282.7 mm .
b. sRmab;muxkat;enH d c = 60mm . RkLapebtugrgkarTajRbsiTPaBsRmab;EdkmYyKW
A = 300(2 60 ) / 3 = 12000mm 2

KNnaTMhMsameRbHtamsmIkar

(6-16)

W = 11f s 3 Ad c 10 6

edayGgt;enHCaFwmenaH = 1.2 nig

f s = 240MPa

W = 11 1.2 2403 12000 60 10 6 = 0.28mm

PaBdab nigsameRbH

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c.

NPIC

EdltUcCag 0.40mm nig 0.32mm sRmab;Ggt;xagkg nigsRmab;Ggt;xageRkA.

tamsmIkar (6-18)
Z = f s 3 Ad c = 2403 12000 60 10 3 = 21.5kN / mm

EdltUcCag 31kN / mm nig 26kN / mm sRmab;Ggt;xagkg nigsRmab;Ggt;xageRkA.

2> rUb 6>9 muxkat; b

a. karKNnaKMlatEdkmanlkNRsedogKacMNuc a xagelI.
b. sRmab;muxkat;enH d c = 60mm ehIyEdkRtUv)andak;CaBIrRsTab;. TIRbCMuTmn;rbs;EdkKW 90mm
BIsrsxageRkam. RkLapebtugrgkarTajRbsiTPaBsRmab;EdkmYyKW
A = 300(2 90 ) / 6 = 9000mm 2

W = 11 1.2 2403 9000 60 10 6 = 0.26mm

c.

RKb;RKan;

tamsmIkar (6-18)
Z = f s 3 Ad c = 2403 9000 60 10 3 = 19.5kN / mm

karBiPakSa

RKb;RKan;

eyIgGaceXIjfaKMlat s enAkgsmIkar (6-17) CaGnuKmn_nkugRtaMgkgEdkTaj bCaGnuKmn_eday

minpal;nbERmbRmYlrageFobenAkgEdkTaj f s = Es s ehIy Es sRmab;EdkesI 2.1105 MPa .
dUcKaKM latkGaRsynwgkRmas;ebtugkarBarEdk Cc . karekIneLIgnUvkRmas;ebtugkarBar vanwgkat;bny
KMlat EdlvaminTak;TgeTAnwgTMhMEdkEdleRbIenAkgmuxkat;eT.

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enAkg]TahrN_enH TMhMsameRbHEdlrMBwgTukRtUv)anKNnaedaysmIkar (6-16) edIm,Ipl;[nisiSt

nigvisVkrnUvtRmUvkarRKb;RKgsameRbH nigTMhMsameRbH. CaTUeTATMhMsameRbHRtUv)anvas;enAelIFwmEdl
eFVIkarBiesaFn_enAkgmnIrBiesaFn_ bkenAkgeRKOgbgMCak;EsgEdlsiteRkambnk EdleFVI[mansameRbH
eRKaHfak;ekIteLIgenAkgFwm nigkgkRmalxN ehIyEdlRtUvkarkarBiesaFn_. RbsinebITMhMsameRbHEdl
)anvas;mun nigeRkaykardak;bnkFMCag yield strain rbs;Edk enaHEdkemnwgmanlkN)asic nig KanRb
siTPaB. eKGacvas;TMhMeRbHCak;Esg)anedaybnat;sRmab;vas;sameRbHEdleKGacrkTij)an. eRkABI
kugRtaMgEdk nigkRmas;ebtugkarBarEdk W enAGaRsynwgktaTIbIKW A EdlCaRkLapebtugrgkarTajBTCMu
vijEdkrgkarTajmYy.
]TahrN_6>7 KNnaFwmTRmsamBaEdlmanRbEvg 7.3m RTnUvbnkefrBRgayesI 21.9kN / m nigbnk
GefrBRgayesI 17.2kN / m . eRCIserIsmuxkat;EdkRKb;RKan; nigepgpat;KMlatEdkedIm,IbMeBjlkxN
ACI Code. eK[ b = 400mm / f 'c = 28MPa / f y = 400MPa PaKryEdk 0.8% nigkRmas;ebtugkar
BarEdk Cc = 50mm .

dMeNaHRsay

1> sRmab;PaKryEdk = 0.8% / Ru = 2.7MPa = 0.9 . m:Um:g;emKuNxageRkAKW

M u = wL2 / 8 nig w = 1.2 21.9 + 1.6 17.2 = 53.8kN / m
M u = 53.8 7.32 / 8 = 348.624kN .m
M u = Ru bd 2 d = 568mm

As = 0.008 400 568 = 1817.6mm 2

eRbI 3DB28 As = 1847.3mm2 mYyCYr. km<s;srubrbs;muxkat;Fwm h = 650mm .

CaTUeTA
d = 650 50 28 / 2 = 586mm
rUbTI6>10
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2> KNnabnkeFVIkar (service load) nigm:Um:g;eFVIkar (service moment)

w = 21.9 + 17.2 = 39.1kN / m

M u = 39.1 7.32 / 8 = 260.5kN .m

3> KNnakm<s;GkSNWt kd nigRbEvgdXas; jd tamsmIkar (6-12)

b(kd ) 2 / 2 nAs (d kd ) = 0 n = 8
kd = 174.4mm

As = 1847.3mm 2 d = 586mm

jd = d kd / 3 = 527.9mm

cMNaMfa eKGacyktmRbEhlrbs;
4> KNnakugRtaMg f s

j = 0.87

j = 527.9 / 586 = 0.9

kgkrNIEdl kd minRtUv)anKNna.

M = As f s jd
f s = 260.5 106 /(1847.3 527.9) = 267 MPa

5> KNnaKMlat s tamrUbmn (6-17)

s = 105000 / 267 2.5 50 = 268.3mm

EdltUcCag 300(280 / 267) = 314.6mm

KMlatEdl[ = 0.5(400 64 64) = 136mm < 230.8mm

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VII.

RbEvgEdkbgb; bRbEvgEdkRCYs

7>1> esckIepIm
lkNbnSIKarvagEdk nigebtug enAkgeRKOgbgMebtugGarem:QrelIeKalkarN_ PaBsitrvagsmar
TaMgBIreRkayebtugkkrwg. RbsinebIsrsEdkEdlmanmuxkat;mUlRtg;RtUv)anbgb;enAkgebtug caM)ac;eKRtUv
mankmaMgmYyRKb;RKan;edIm,ITajsrsEdkenaHecjBIebtug. RbsinebIRbEvgbgb;nsrsEdkenAkgebtug
manRbEvgEvgRKb;RKan; enaHsrsEdkGaceFVIkardl;cMNuc yield edayTukRbEvgxHrbs;srsEdkenAkg
ebtug. kmaMgsitGaRsyeTAnwgPaBkkitrvagEdk nigebtug. kmaMgkkitTTYl\TiBlCacMbgbNalBIPaB
eRKImnpxagrbs;srsEdk karlayebtug karRskmaD nigkRmas;ebtugeRsabEdk. EdkfaMgGMeBA
(deformed bar) pl;nUvPaBsitlCagEdkmUl. ersIusg;ebtugkan;EtFM PaBsitkan;EtFM. kRmas;ebtugeRsab
Edkkan;EtRkas; kugRtaMgPaBsitrbs;Edkkan;EtFM.
CaTUeTA ersIusg;PaBsitTTYl\TiBlBIktaxageRkam
- ersIusg;rbs;Edk f / RbEvgbgb;kan;EtEvg sRmab; f kan;EtFM
- KuNPaB nigersIusg;sgt;rbs;ebtug f ' / kalNa f ' kan;EtFM enaHRbEvgbgb;kan;xI
- muxkat;Edk KMlat nigTItaMgrbs;EdkenAkgmuxkat;ebtug. EdkbeNayEdldak;kgKMlatbBar
FMCag 30cm manersIusg;sitTabCag GaRsyedaykarrYmmaD nigkarRskrbs;ebtugGMLgeBl
dMeNIrkarkkrwgrbs;ebtug. dUcKa KMlatEdkkan;EtFM pl;nUversIusg;sitkan;EtFM eRBaHvapl;nUv
pebtugRKb;RKan;CMuvijEdk.
- kRmas;ebtugeRsabEdk. kRmas;kan;EttUc bg[mankareRbH.
- karxb;EdkbeNayedayEdkTTwg. karxb;EdkbeNayedayEdkTTwg bEdkkgRKb;RKan;
karBarkarpHEbkebtugCuMvijEdkbeNay.
y

7>2> karbegItkugRtaMgsit
7>2>1> PaBsitedaykarBt;
eKmanFwmmYyRbEvg dx rgnUvbnkBRgayesI. eday[m:Um:g;EdlekIteLIgenAEpkmagCa M nig
mageTotCa M ehIy M mantmFMCag M . m:Um:g;nwgbegItnUvkmaMgsgt; nigkmaMgTajxagkg. eday
M > M enaH T > T dUcKa C > C .
1

RbEvgEdkbgn; bRbEvgEdkRCYs

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enAmuxkat;NamYyenaH T = Mjd
Edl jd CadXas;
T1 T2 = dT =

dM
jd

b:uEn T = T + u Odx
Edl u CakugRtaMgsitmFm
O CaplbUkbrimaRtnmuxkat;r)arenAEpkrgkarTaj
dUcenH T T = u Odx = dM
jd
1

1
dM

dx jd O
dM
=V
dx
V
u =
jd O

u =

Eteday

(7.1)

m:ageTotedIm,IsRmYlkgkarKNna j RtUv)ansnt;esInwg 0.87

dUcenH u = 0.87Vd O
edayersIusg;nPaBsit RtUv)ankat;bnyedayemKuN = 0.85 enaH
Un =

V
0.87 d O

(7.2)

7>2>2> karBiesaFsRmab;RbsiTPaBPaBsit
karBiesaFedIm,IkMNt;lTPaBkugRtaMgsitGaceFVIeLIgedayeRbI pullout test rUbTI 7>2. kar
BiesaFenH kMNt;lTPaBsitnpEdkRbePTepSgedayeFobeTAnwgRbEvgbgb;. kugRtaMgTajEbkEckesI
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CMuvijsrsEdkenARtg;muxkat;Cak;lak; nigERbRbYltambeNayRbEvgbgb; nigenARtg; radial distance

BIprbs;srsEdk. b:uEnkarBiesaFenHmin)ansMEdgBIPaBsitRbsiTPaBenAkgprbs;srsEdkenAkg
Ggt;rgkarBt;eT eRBaHkugRtaMgERbRbYltamkm<s;rbs;muxkat;ebtug.
RbePTTIBIrnkarBiesaFGacGnuvt)ancMeBaHEdkbgb; rUbTI 7>3. enAkgkarBiesaFenH eKbegIn
kmaMg P bnicmg ehIyeKkt;RtacMnYn KMlat nigTMhMsameRbH. kugRtaMgsitERbRbYltambeNayRbEvg
srsEdkcenaHsameRbH. bERmbRmYlrageFobrbs;srsEdkmantmGtibrmaenARtg;muxkat;eRbH ehIy
fycuHenARtg;muxkat;kNalrvagmuxkat;eRbH.
eKkGaceFVIkarBiesaFeTAelIGgt;rgkarBt; edIm,IsikSaBIRbsiTPaBnPaBsittambeNayprbs;Edk
Taj. karviPaKkugRtaMgsitrbs;srsEdkRtUv)anbgajrYcmkehIyenAkgkfaxN 7>2>1.
RbEvgbgb; (development length) CaRbEvgrbs;srsEdknImYy edIm,IbegIt yield strength eBj
eljedaymin[rbUtedaysarersIusg;sit (bond strength). RbsinebIsrsEdkminmanRbEvgbgb;RKb;RKan;
enaHkugRtaMgsitenAkgtMbn;Tajrbs;FwmnwgmantmFM ehIyvanwgeFVI[mansameRbH nigrEhkkRmas;ebtug
karBarEdkCMuvijEdkrgkarTaj. RbsinebIkarrEhkebtugbndl;cugsrsEdk Fwmnwg)ak;. cMNaMfa
KMlattUcrvagEdkrgkarTaj nigkRmas;ebtugkarBarEdkenAEpkxag nig)atrbs;FwmmantmtUcnwgkat;bny
ersIusg;sitrbs;srsEdk.

RbEvgEdkbgn; bRbEvgEdkRCYs

155

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7>3> RbEvgbgb;sRmab;tMbn;Taj
7>3>1> RbEvgbgb;mUldan l
RbsinebIsrsEdkbgb;kgebtugrgkmaMgTaj T enaHkmaMgTajenaHRtUvTb;edaykugRtaMgsitrvag
srsEdk nigebtug. kmaMgTajGtibrmaRtUvmantmesInwg A f Edl A muxkat;srsEdk. kmaMgenH
d

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Department of Civil Engineering

RtUv)anTb;edaykmaMgkg U Ol Edl U kugRtaMgsitcugeRkaymFm l RbEvgbgb;Edk nig O brimaRt

srsEdk (D) . kmaMgTaMgBIrenHRtUvEtesIKaedIm,ImanlMnwg.
n

As f y = U nOld
ld =

As f y

(7.3)

U nO

eday A = d4 nig O = d
2
b

ld =

db f y

(7.4)

4U n

mannyfa RbEvgbgb;GaRsynwg muxkat;Edk ersIusg;rbs;Edk nigkugRtaMgsitEdlCaGnuKmn_eTA

nwg f ' .
c

U n = 0.5 f 'c (
ld =

c 1
)
db 2

0 .5 f y

(7.5)

f 'c ( c 1 )
db 2

eday[ c = 1.5d
ld =

db

0.5 f y
f 'c

db

(7.6)

smIkarRbEvgbgb;sRmab;tMbn;Taj Edl)anBIkareFVIBiesaFn_ ehIyRtUv)ansRmYledayKNkmkarKW

ld =

9 fy

db
10 f 'c ( c + K tr )
db
c + K tr
2.5
db

(7.7)

Edl
emKuNTItaMgsrsEdk
emKuNRTnab;Edk
emKuNTMhMEdk
emKuNTmn;ebtug
c KMlatEdk bRTnab;ebtugkarBarEdk
c RtUvmantmtUckgcMeNamkrNITaMgBIrxageRkam
- cmaytUcCageK Edlvas;BIpebtugmkGkSEdk
- mYydgknHnKMlatEdk EdlKitBIGkSmkGkS
RbEvgEdkbgn; bRbEvgEdkRCYs

157

T.Chhay

mhaviTalysMNg;sIuvil
K tr

NPIC

snsSn_EdkTTwg

K tr =

Atr f yt
10 sn

muxkat;EdkTTwg
f ersIusg;EdkTTwg
s KMlatEdkGtibrmarbs;EdkTTwgKitBIGkSmkGkS
n cMnYnEdkTTwg
cMNaM 1 (c + K ) / d minKYrFMCag 2.5 edIm,IFanasuvtiPaBdl;kardac;edaykarrbUt.
2 tmrbs; f ' minRtUvFMCag 8.3MPa .
3 K RtUv)aneKsnt;mantmesIsUnedIm,IsRmYldl;karKNna.
Atr

yt

tr

tr

7>3>2> emKuN ACI Code sRmab;KNna l sRmab;srsEdkrgkarTaj

1 emKuNTItaMgsrsEdk
= 1.3 sRmab;EdkEdldak;manKMlatx<s;Cag 30cm BIEdkbgb;
= 1 sRmab;krNIepSgeTot
2 emKuNRTnab;Edk
= 1.5 sRmab;Edklab epoxy-coating ehIymanRTnab;ebtugkarBarEdktUcCag 3d nigman
KMlatEdktUcCag 6d
= 1.2 sRmab;Edklab epoxy-coating krNIepSgeTot
= 1 sRmab;EdkminmanlabfaMkarBar
3 emKuNTMhMEdk
= 0.8 sRmab;EdkEdlmanGgt;pittUcCag besI 19mm
= 1 sRmab;EdkEdlmanGgt;pitFMCag besI 22mm
d

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Department of Civil Engineering

4 emKuNTmn;ebtug
= 1.3 sRmab;ebtugRsal RbsinebIeKsal;ersIusg;TajedaykarrEhkrbs;ebtugTmn;Rsal
f eRbI = 0.56 f ' / f b:uEn 1
= 1 sRmab;ebtugFmta
5 ACI Code GnuBaat[eRbI K = 0 eTaHbICamanvtmanEdkTTwgkeday. enAkgkrNIenH
ct

ct

tr

fy

9
ld
=
d b 10

f 'c ( c / d b )

tmrbs; f ' minRtUvFMCag 8.3MPa .

6 R CaemKuNkat;bnyEdlbNalBIbrimaNEdkelIs. ACI Code, section 12.2.5 GnuBaat[
kat;bny l edayemKuN R enAeBlEdlEdkenAkgGgt;rgkatBt;elIstRmUvkarnkarviPaK
elIkElgEtkEngEdleKRtUvkarCaBiessnUvkarf<k; bRbEvgbgb;sRmab; f bsrsEdkRtUv)an
sikSaKNnaedayBicarNa\TiBlrBaydI.
A (tRmUvkar )
R =
A (pl[
; )
7 sRmab;RKb;krNITaMgGs; RbEvgbgb; l minRtUvtUcCag 300mm .
c

7>3>3> rUbmnsRmYlsRmab; l

GnuBaat[eRbIrUbmnsRmYlsRmab;KNnapleFob l / d . vaQrelI
PaBCak;EsgEdlkrNIsagsg;bcb,neRbIKMlat nigRsTab;ebtugkarBarEdkCamYynwgEdkxb; (confining
reinforcement) dUcCaEdkkg (stirrup and tie) EdleFVI[tm (c + K ) 1.5d . elIsBIenH karBiesaF
bgajfa eKGackat;bgyRbEvgbgb; l 20% sRmab;EdkEdlmanGgt;tUcCag besI19mm . edayQrelI
karsnt;TaMgenH nigsnt;fa (c + K ) = 1.5d enaHsmIkar 7.7 RtUv)ansRmYldUcxageRkam
1 sRmab;srsEdkEdlmanGgt;pitFMCag besI 22mm
ACI Code, section 12.2.2

tr

tr

f
ld 3
= y
f'
db 5
c

(7.8)

sRmab;srsEdkEdlmanGgt;pittUcCagbesI 19mm
f
ld 12
= y
f'
d b 25
c

(7.9)

pleFob l / d enAkgsmIkar 7.9 esInwg 80% npleFob l / d enAkgsmIkar 7.8. eKeRbI

smIkarTaMgenHkgkrNIEdleKCYblkxNmYykgcMeNamlkxNTaMgBIrxageRkam
d

RbEvgEdkbgn; bRbEvgEdkRCYs

159

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mhaviTalysMNg;sIuvil

NPIC

k KMlatBIpEdkeTApEdkEdlRtUvbgb; bRCYsminRtUvtUcCag d / RsTab;ebtugkarBarEdkmin

RtUvtUcCag d ehIyEdkkgEdldak;elIRbEvg l minRtUvtUcCagtmGb,brmaEdl[eday
kUd.
x KMlatBIpEdkeTApEdkEdlRtUvbgb; bRCYsminRtUvtUcCag 2d / RsTab;ebtugkarBarEdk
minRtUvtUcCag d .
2 sRmab;krNIdTeTot tmrbs; l / d enAkgsmIkar 7.8 nig 7.9 RtUvKuNnwg 1.5 edIm,IeFVI[va
smmUleTAnwgsmIkar 7.7.
smIkarTaMgenHmanlkNgayRsYlkgkareRbIR)as;sRmab;lkxNTUeTAEdlBak;BnnwgkarsikSa
KNna nigkarsagsg;. ]TahrN_ sRmab;RKb;eRKOgbgMebtugTmn;Fmta ( = 1.0) srsEdkEdlminlab
epoxy-coating ( = 1.0 ) EdkEdlmanGgt;pitFMCag besI 22mm ( = 1.0) enaHsmIkar 7.8 kayCa
b

ld 3 f y
=
d b 5 f 'c

(7.10)

eKeRbIsmIkarenH RbsinebIvaeKarBlkxN k nig x b:uEnsRmab;krNIdTeTot eKRtUvKuN l

1.5 b

/ db

ld
9 f
= y
d b 10 f 'c

CamYynwg
(7.11)

sRmab;lkxNdUcKa nigsRmab;srsEdkEdlmanGgt;pittUcCag besI 19mm smIkar 7.9 kayCa

ld 12 f y
=
d b 25 f 'c

(7.12)

eKeRbIsmIkarenH RbsinebIvaeKarBlkxN k nig x b:uEnsRmab;krNIdTeTot eKRtUvKuN l

1.5 b
ld 18 f y
=
d b 25 f 'c

/ db

CamYynwg
(7.13)

kgkarsikSaKNna nigkarsagsg;GKarebtugGarem: eKniymeRbI f ' = 28MPa nig f

ebIeKCMnYstmTaMgenHeTAkgsmIkarxagelI enaH
smIikar 7.10 kayCa l = 45d
Ggt;pitEdk 22mm
smIikar 7.11 kayCa l = 67.5d
Ggt;pitEdk 22mm
smIikar 7.12 kayCa l = 36d
Ggt;pitEdk 19mm
smIikar 7.13 kayCa l = 54d
Ggt;pitEdk 19mm
c

T.Chhay

= 400MPa

. Rbsin

(7.10a)
(7.11a)

(7.12a)

(7.13a)

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Department of Civil Engineering

tmepSgeTotrbs;pleFob l / d RtUv)anbgajenAkgtarag 7>1. tarag 7>2 bgajBIRbEvgbgb; l

sRmab;srsEdkepSg enAeBl f = 400MPa nig f ' = 20MPa nig 30MPa sRmab;krNIeKarB
lkxN nigkrNIepSgeTot.
d

tarag 7>1 tmrbs; l

sRmab;tmepSgn

/ db

f 'c

nig f EdkrgkarTaj
y

f y = 235MPa

f 'c

19mm

(MPa )

f y = 400MPa
22mm

19mm

22mm

eKarB krNIepSg eKarB krNIepSg eKarB krNIepSg eKarB krNIepSg

lkxN eTot lkxN eTot lkxN eTot lkxN eTot

21
28
35
42

24.6
21.3
19.1
17.4

36.9
32.0
28.7
26.1

30.7
26.6
23.8
21.8

46.1
39.9
35.7
32.7

41.9
36.3
32.5
29.6

62.9
54.5
48.9
44.4

52.4
45.4
40.6
37.0

78.6
68.1
60.9
55.5

tarag 7>2 RbEvgbgb; l (mm) sRmab;EdkrgkarTaj nig f = 400MPa ( = = = 1.0)

RbEvgbgb; l (mm) sRmab;EdkrgkarTaj
Ggt;pitrbs;
elxsMKal;
f ' = 20MPa
f ' = 30MPa
srsEdk
rbs;Edk
eKarB
krNIepSg
eKarB
krNIepSg
(mm )
lkxN
eTot
lkxN
eTot
y

10M
15M
20M
25M
30M
35M

11.3
16.0
19.5
25.2
29.9
35.7

485
687
1046
1352
1605
1915

728
1031
1569
2028
2408
2873

396
561
854
1104
1310
1564

594
842
1281
1656
1965
2346

7>4> RbEvgbgb;sRmab;tMbn;sgt; l
RbEvgbgb;sRmab;tMbn;sgt;RtUv)aneKKitfamanRbEvgxICagRbEvgbgb;sRmab;tMbn;Taj eRBaHkmaMg
xH RtUv)anbBaneTAebtugedaysrsEdk nigm:ageTotvaminmansameRbHenAtMbn;TMBk;. RbEvgbgb; l
sRmab;tMbn;sgt;RtUv)ankMNt;eday
dc

dc

ldc =

0.24db f y
f 'c

RbEvgEdkbgn; bRbEvgEdkRCYs

0.043d b f y

(7.14)

161

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

ehIyvaminRtUvmantmtUcCag 20cm . eKGackat;bnyRbEvgbgb; l edayKuNnwg R = A tRmUvkar

/ A Edlpl;[. sRmab;Ggt;rgkarsgt;ebtugGarem:EdlmanEdkkgvNGgt;pitFMCag 6mm nigKMlat
tUcCag besI 100mm eKGacKuN l nsmIkar 7.14 eday R = 0.75 . CaTUeTA l = l (R b R )
200mm . tarag 7>3 nig 7>4 [tmrbs; l / d enAeBl f = 400 .
dc

dc

sl

dc

tarag 7>3 tmrbs; l

ldc / d b = 0.24 f y /
f y (MPa )

/ db

dc

sl

sRmab;tmepSgrbs;

f 'c

nig f Edkrgkarsgt;/ l Gb,brma= 200mm

y

dc

f 'c 0.043 f y

f 'c (MPa )

235
400

21

28

32

12.3
21

10.6
18.1

10.1
17.2

tarag 7>4 RbEvgbgb;mUldan l (mm) sRmab; Edkrgkarsgt; ( f = 400MPa)

elxsMKal;
Ggt;pitrbs;
RbEvgbgb; l (mm) enAeBl
21(MPa )
28(MPa )
rbs;Edk
srsEdk (mm)
dc

dc

10M
15M
20M
25M
30M
35M

11.3
16.0
19.5
25.2
29.9
35.7

237
336
410
529
628
750

f 'c =

204
290
353
456
541
646

32(MPa )

200
275
335
4.33
514
614

7>5> segbkarKNna l kgtMbn;Taj

edaysnt;karsagsg;Fmta (c + K ) / d 1.5 .
1 RbsinebIeKeKarBlkxNmYykgcMeNamlkxNTaMgBIrxageRkam
k KMlatBIpEdkeTApEdk d / RsTab;ebtugkarBarEdk d ehIyKMlatEdkkgminRtUvtUcCagtm
Gb,brmaEdl[edaykUd.
x KMlatBIpEdkeTApEdk 2d / RsTab;ebtugkarBarEdk minRtUvtUcCag d . enaH
sRmab;EdkEdlmanGgt;pitFMCag besI 22mm / dl = 53 ff '
(7.8)
d

tr

sRmab;EdkEdlmanGgt;pittUcCag besI 19mm / dl

d
b

f
12
y
f'
25
c

(7.9)

2 sRmab;krNIepSgeTot KuNpleFobxagelInwg 1.5 .

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3 cMNaMfa f ' 69MPa ehIy 1.7 tmrbs; , nig RtUv)anBnl;BIxagedIm.

4 sRmab;EdkkBaM eTaHbICakgtMbn;TajkI kgtMbn;sgt;kI eKRtUvbegIn l 20% sRmab;kBaMEdkbI nig 33%
sRmab;kBaMEdkbYn. kBaMEdkmYyRtUv)aneKBicarNaCaEdkeTalEdlmanGgt;pit nigRkLapsmmUleTA
nwgRkLapsrubrbs;EdkTaMgGs;kgmYykBaM. eKeRbIGgt;pitsmmYlenHedIm,IRtYtBinitKMlat nigRsTab;ebtugkarBarEdk.
c

]TahrN_ 7>1 rUb 7>5 bgajBImuxkat;FwmebtugGarem:TRmsamBaEdlman 4DB25 EdlhuBTedayEdk

kg DB10 @150 . kMNt;RbEvgbgb;rbs;EdkRbsinebIFwmenHplitBIebtugTmn;Fmta ehIyEdkminmanlab
epoxy. f ' = 21MPa nig f = 400 MPa .
c

dMeNaHRsay
1 RtYtBinitlkxNKMlat nigRsTab;ebtugkarBarEdk
k d = 25mm
x RsTab;ebtugkarBarEdk = 60 12.5 = 47.5mm > d
K KMlatrvagsrsEdk = 300 3 120 25 = 35mm > d
X EdkbeNayRtUv)ancgxb;edayEdkkg DB10 . dUcenH FwmenHeKarBtamlkxN. enaH
3 f
l
=
sRmab;srsEdkEdlmanGgt;pit > 22mm
(7.8)
d
5 f'
b

2 tmrbs;emKuN sRmab;EdkxageRkam = 1.0 / sRmab;Edkminmanlab epoxy = 1.0 nigsRmab;

ebtugTmn;Rsal = 1.0 . ehIyBinitemIlfa f ' = 21 = 4.58MPa < 8.3MPa .
c

RbEvgEdkbgn; bRbEvgEdkRCYs

163

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mhaviTalysMNg;sIuvil

NPIC

ld
3 400
=
= 52.4
d b 5 21

dUcenH l = 52.4 25 = 1310mm yk 1350mm . cMNaMfa RbsinebIeKeRbIrUbmnTUeTAsRmab;KNna l

smIkar 7.7 edaysnt; K = 0 enaH
d

/ db

tr

3 f y
ld
=
d b 5 f 'c (cb / d b )

(7.8)

enAkg]TahrN_enH = = = 1.0 .
ehIy c = cmaytUcCageKBIGkSrbs;EdkeTApebtugEdlCitbMput (c ) bBak;kNalnKMlatBIGkSeTAGkS
srsEdk (c ) .
0.5(300 120 )
c = 60mm
c =
= 30mm lub
3
(c + K ) / d = 30 / 25 = 1.2 < 1.5 / dUcenH eRbI (c + K ) / d = 1.5 . dUcenH l / d = 3 f / 5 f ' dUc
CMhanTI 2 nig l = 1350mm .
cMNaMfa RbsinebIEdkminRtUv)ancgxb;edayEdkkg tmrbs; l RtUvKuNnwg 1.5 s = 35mm < 2d =
50mm .
b

tr

tr

]TahrN_ 7>2 eFVI]TahrN_ 7>1 eLIgvij RbsinebIFwmenHplitBIebtugfbMEbkTmn;Rsal srsEdklab

faM epoxy ehIy A tRmYvkartamkarviPaKKW 1800mm .
dMeNaHRsay
1 tmrbs;emKuN = 1.0 EdkeRkam/ = 1.5 EdklabfaM/ = 1.3 ebtugfTmn; nig R = A
tRmUvkar /( A pl;[ ) = 1800 / 1962.5 = 0.92 . tmrbs; = 1.5 edaysarRTnab;ebtugkarBarEdktUc
Cag 3d = 75mm . RtYtfa = 1.0 1.5 = 1.5 < 1.7 .
f
2 dl = 3R5
sRmab;srsEdk > 22mm
f'
2

3 0.92(1.5)(1.3)(400)
=
= 94
5 21

ld = 2350mm

]TahrN_ 7>3 ssrebtugGarem:man 8DB32 EdlRtUvbgtcUlkgeCIgtag. kMNt;RbEvgbgb;EdlcaM)ac;

sRmab;bgb;cUlkgRKwH. eRbI f ' = 28MPa nig f = 400MPa .
dMeNaHRsay
RbEvgbgb;kgtMbn;sgt;KW
c

T.Chhay

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viTasanCatiBhubeckeTskm<Ca
ldc =

0.24d b f y
f 'c

Department of Civil Engineering

0.043d b f y

0.24 25 400
= 455mm
28

lub
l Gb,brmaKW 0.043 25 400 = 430mm b:uEnvaminGactYcCag 200mm eT. edaysarvaminmanemKuN
dTeTotdUcenH l = 455mm .
ldc =

dc

7>6> muxkat;eRKaHfak;enAkgGgt;rgkarBt;
muxkat;eRKaHfak;sRmab;RbEvgbgb;EdkenAkgGgt;rgkarBt;KW
enARtg;cMNuckugRtaMgGtibrma
enARtg;cMNucEdlEdkbeNayrgkarTajRtUv)anbBab; bBt;
enARtg;prbs;TRm
enARtg;cMNucrbt;Edlm:Um:g;brsBaa
muxkat;eRKaHfak;sRmab;KMrUFwmCab;rgbnkBRgayesIRtUv)anbgajenAkgrUbTI 7>6. muxkat; nig
RbEvgbgb;RtUv)anBnl;dUcxageRkam
1 muxkat;eRKaHfak;sRmab;Edkm:Um:g;GviCmanmancMnYnbI muxkat;TI ! KWenARtg;pTRm Edlm:Um:g;
GviCman kdUcCakugRtaMgmantmGtibrma. eKRtUvRtYtBinitRbEvgbgb;cMnYnBIrKW x nig x .
muxkat; @ Camuxkat;EdlcMENknEdkm:Um:g;GviCmanxHRtUv)anbBab;. edIm,IbegItkmaMgTaj
eBjelj srsEdkRtUv)anbgtcmay x muneBlEdlbBab;va. enAeBlcMENknsrsEdk
RtUv)anbBab; srsEdkEdlenAsl;begItkugRtaMgGtibrma.
muxkat; # enARtg;cMNucrbt;. eKRtUvbgtsrsEdkenHcmay x BImuxkat; # x RtUvEtFMCag
besInwgkm<s;RbsiTPaB d b 12d b 1 / 16 nRbEvgElVgcenaHpssrxagkg edayykmYy
NaEdlmantmFMCageK. eyagtam ACI Code, section 12.12.3 y:agehacNas;k 1 / 3 n
brimaNsrsEdksrubEdlpl;[sRmab; m:Um:g;GviCmanenARtg;TRmRtUvbgtcmay x BIcMNuc
rbt;.
2 muxkat;eRKaHfak;sRmab;Edkm:Um:g;viCmanmancMnYnbI muxkat;TI $ Camuxkat;Edlmanm:Um:g;viCman
kdUcCakugRtaMgGtibrma. eKRtUvRtYtBinitRbEvgbgb;cMnYnBIrKW x nig x . RbEvg x CaRbEvg
bgb; l EdlkMNt;eday ACI Code, section 12.11 nigdUckarbgajBIxagelI. RbEvg x FMCag
besInwg d b 12d .

RbEvgEdkbgn; bRbEvgEdkRCYs

165

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

muxkat; % Camuxkat;EdlcMENknEdkm:Um:g;viCmanxHRtUv)anbBab;. edIm,IbegItkmaMgTaj

eBjelj srsEdkRtUv)anbgtcmay x . srsEdkEdlenAsl;nwgbegItkugRtaMgGtibrma
edaysarcMENknsrsEdkRtUv)anbBab; . eyagtam ACI Code, section 12.11.1 enARtg;p
TRm muxkat; ! y:agehacNas;k 1/ 4 nsrsEdkm:Um:g;viCmanenAkgGgt;Cab;RtUvbgtcUl
eTAkgTRm. sRmab;Ggt;samBa y:agehacNas;k 1 / 3 n brimaNsrsEdksrubEdlpl;[
sRmab;m:Um:g;viCmanRtUvbgtcUlTRm.
enARtg;cMNucrbt; muxkat; ^ RtUvGnuvtedayeKarBtam ACI Code, section 12.11.3.
2

]TahrN_ 7>4
FwmCab;mankarlMGitsrsEdkdUcbgajkgrUbTI 7>7. m:Um:g;Bt;GtibrmaviCman nigGviCmankRtUv)anbgaj.
eyIgRtYvRtYtBinitRbEvgbgb;enARKb;muxkat;eRKaHfak;TaMgGs;. eK[ f ' = 21MPa / f = 276MPa / b =
300mm / d = 460mm nigRbEvgElVg L = 7.2m .
c

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166

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Department of Civil Engineering

dMeNaHRsay
muxkat;eRKaHfak;KW (1) enARtg;prbs;TRmsRmab;EdkrgkarTaj nigEdkrgkarsgt; muxkat; ! (2) enARtg;
cMNucEdlEdkTajRtUv)anbBab;enAkgElVg muxkat; @ nig % (3) enARtg;cMNucrbt; muxkat; # nig ^ nig
(4) enARtg;kNalElVg muxkat; $.
1 BIrUbTI 7>7 RbEvgbgb;sRmab;Edkm:Um:g;GviCmanKW 3DB28 RtUv)anbBab;enARtg;cmay x = 1.35m BI
prbs;TRm b:uEnEdkbIeTotRtUv)anbgtdl;cmay 1.8m BIpTRm.
k kMNt;RbEvgbgb;sRmab;EdkxagelI
RbEvgEdkbgn; bRbEvgEdkRCYs

167

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

Ggt;pitEdkm:Um:g;GviCman d = 28mm
RsTab;ebtugkarBarEdk = 60 282 = 46mm > d
cmayBIpeTApEdk = 300 2 120 28 = 62mm > 2d
dUcenH dl = 35f 'f sRmab; d 22mm
b

eday = 1.3 sRmab;EdkxagelI/ = 1.0 / = 1.0

enaH l = 3(15.3)21276 28 = 1315mm
x epgpat;RbEvgbgb; x = 1350mm > l
(O.K)
K RbEvgbgb; x EdlbgtBIcMNucEdlminRtUvkar 3DB28
d

x2 = max(d ;12d b ) = 460mm

X epgpat;RbEvgbgb; x

x4 = 1800 450 = 1350mm > ld

(O.K)

g ecjBIcMNucrbt; muxkat; # eKRtUvbgtbrimaNEdky:agtic 1 / 3 nbrimaNEdksrubcmayy:ag

tic d b 12d b L / 16 BIcMNucrbt;enH. EdkEdlbgthYscMNucrbt;KW 3DB28 dUcenHRKb;RKan;.
epgpat; x = x = 460mm
eyIgman 1800 1000 = 800mm > 460mm (O.K)
2 Edkrgkarsgt;enARtg;pTRm muxkat; ! KWEdkEdlmanmuxkat; 25mm RbEvgbgb; x esInwg
0.24d f
0.24 25 276
l =
=
= 361mm yk 370mm
f'
21
b

dc

Gb,brma = 0.043d f = 0.043 25 276 = 297mm

b:uEnvaminRtUvtUcCag 200mm . l Edlpl;[ = 380mm > 370mm O.K.
3 RbEvgEdkbgb;sRmab;srsEdkm:Um:g;viCman 3DB25 RtUv)andak;cmay 1.8m BIkNalElVg ehIy
EdkEdlenAsl;RtUv)anbgtdl;TRm. RbEvgbgb; x BIkNalElVgKW
l = 3 276 28 / (5 21 ) = 1011mm < 1800mm
O.K.
x = max(d ;12d ) = 460mm RKb;RKan;
eKGackMNt;TItaMgCak;EsgkgkarbBab;EdkenAkgElVgedayeRbIdaRkamm:Um:g;ersIusg; (momentresistance diagram) EdlnwgBnl;enAkgkfaxN 7>9.
ldc

dc

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7>7> TMBk;; ACI Code, section 12.5 nig 7.1

TMBk;RtUv)aneRbIenAxagcugr)ar enAeBlEdlRbEvgbgb;Rtg;EdlvaRtUvkarmantmtUcCagRbEvgbgb;
caM)ac;. )annyfa r)arEdlmanTMBk;xagcug RtUvkarRbEvgbgb;xICagr)arRtg;. Ggt;pitGb,brmarbs;BMnt;
Edlvas;BIpxagkgrbs;TMBk; D ERbRbYleTAtamGgt;pitrbs;r)ar
- sRmab; DB10 DB25 / D = 6d
- sRmab; DB28 DB36 / D = 8d
- sRmab; DB43 DB58 / D = 10d
RbEvgbgb;sRmab;r)arEdlmanTMBk;RtUv)ankMNt;edayrUbmnxageRkam
b

lhb =

CaTUeTA l

dh

0.24 f y
f 'c

db

0.24 f y
f 'c

db

Edl
emKuNRTnab;Edk
= 1.2 sRmab;Edklab epoxy-coating
= 1 sRmab;EdkminmanlabfaMkarBar
emKuNTmn;ebtug
= 1.3 sRmab;ebtugRsal
= 1 sRmab;ebtugFmta
RbEvgbgb;rbs;r)arEdlmanTMBk;RtUv)anKuNCamYyemKuNsRmab;krNIxageRkam
- r)arEdlmanGgt;pittUcCag 36mm EdlmanTMBk;Bt;mMu 90 Bak;edayEdkkgbBar bedkEdl
manKMlatminFMCag 3d RbEvgbgb;RtUvKuNCamYy 0.8 .
- r)arEdlmanGgt;pittUcCag 36mm EdlmanRTnab;karBarEdkxagFMCag 6cm RbEvgbgb;RtUv
KuNCamYy 0.7 . eKeRbIemKuNdUcKasRmab;TMBk; 90 enAeBlkRmas;ebtugkarBarEdkenAxag
TMBk;FMCag 50mm .
- r)arEdlmanGgt;pittUcCag 36mm EdlmanTMBk;Bt;mMu 180 Bak;edayEdkkgbBarehIyman
KMlatminFMCag 3d RbEvgbgb;RtUvKuNCamYy 0.8 .
- enAeBlminRtUvkareRbIEdkTMBk; eKGacKuNRbEvgbgb;sRmab;EdkelIsedayemKuNpleFob
o

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tRmUvkar )
A (pl[
; )
- enAeBleKeRbIEdkTMBk;mankRmas;ebtugkarBarEdkxagelI xageRkam nigxag tUcCag 60mm
enARtg;cugminCab;rbs;Ggt; eKRtUvdak;EdkkgedayKMlattUcCag besI 3d . elIsBIenH
eKminGaceRbIemKuN 0.8 )aneT.
RbEvgbgb; l sRmab;EdkfaMgGMeBArgkarTajminRtUvtUcCag max(8d / 150mm) . cMNaMfa TMBk;
minmanRbsiTPaBsRmab;Edkrgkarsgt;eT.
karlMGitBITMBk; 90 nig 180 RtUv)anbgajenAkgrUbTI 7>8. eKRtUvkarTMhMEdlbgajkgrUbedIm,I
karBarkarrEhk nigkarpat;ebtug. rUbTI 7>9 a bgajBIkarlMGitBITMBk; enARtg;cugminCab;CamYynwgkRmas;
ebtugkarBarEdktUcCag 60mm EdlGackarBarkarpat;ebtug. kareRbIEdkkgbiTCitKWcaM)ac;sRmab;karsikSa
KNnadRtwmRtUv. rUbTI 7>9 b nig c bgajBIkardak;EdkkgbBar nigEdkkgedk. rUbTI 7>10 bgajBIkar
BRgaykugRtaMgtambeNayTMBk; 90 eRkamGMeBIkmaMgTaj P .
Rs =

As (

dh

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]TahrN_ 7>5 KNnaRbEvgbgb;EdlRtUvkarsRmab;EdkxagelIEdlmanGgt;pit 25mm rbs;Fwmkugsul

dUcbgajenAkgrUbTI 7>11 EdlbgtcUleTAkgssrRbsinebIsrsEdkenaH
k Rtg;
x manTMBk; 90 enAxagcug
K manTMBk; 180 enAxagcug
o

RbEvgEdkbgn; bRbEvgEdkRCYs

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NPIC

EdkRtUv)ancgxb;edayEdkkgEdlmanGgt;pit DB10 @150 ehIymankRmas;ebtugkarBarEdk 40mm

nigKMlatBIpEdkeTApEdk 50mm . eRbI f ' = 28MPa nig f = 400MPa .
c

dMeNaHRsay
k EdkRtg; EdkmanGgt;pit d = 25mm . edaysarKMlatBIpEdkeTApEdk = 2d nigkRmas;ebtugkar
BarEdkFMCag d ehIyEdkRtUvxb;edayEdkkg dUcenHvaeKarBlkxN k nig x. eKGaceRbIsmIkar
7.10 edIm,IKNna l . l = 1134mm sRmab;EdkxagelI = 1.3 dUcenH l = 1474mm .
x EdkEdlmanTMBk; 90 sRmab;EdkEdlmanGgt;pit d = 25mm RbEvgbgb; l = 0.24d f / f '
= 454mm . edaysarminmankarEktRmUv enaH l = 454mm > 8d = 200mm
karlMGitRtUv)anbgajenAkgrUbTI 7>11. eKmineRbIemKuN = 1.3 sRmab;Ggt;EdlmanTMBk;eT.
c EdkEdlmanTMBk; 180 RbEvgbgb; l = 454mm dUckarKNnaxagelI. edaysarminmankarEktRmUv
enaH l = 454mm > 8d = 200mm karlMGitRtUv)anbgajenAkgrUbTI 7>11.
b

dh

hb

hb

dh

7>8> kartEdk
enAeBlEdlEdkmanRbEvgxI eKcaM)ac;RtUvtvaedaydak;bRBasKanUvRbEvgmYysmRsbedIm,IbBankug
RtaMgnPaBsitBIr)armYyeTAr)armYy. kartEdkGaceFVIeLIgedaykartbRBasmux nigkarpSa. kartEdk
edaykarbRBasminGaceFVIeTA)aneT sRmab;r)arEdlmanGgt;pitFMCag 36mm . sRmab;tMNpSaRtUveFVIeLIg
eday[TwkbnSamanersIusg;esI 125% nersIusg;rbs;Edk.

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kartEdkminRtUv)anGnuBat[eFVIenARtg;muxkat;Edlmanm:Um:g;GtibrmaeT. elIsBIenH eKkRtUveCos

vagkartEdkRtg;TItaMgEtmYykEng edIm,IeCosvagkarFak;exSayrbs;muxkat;ebtug kareTIsETgEdk nigkar
Bi)akcak;ebtug.
k> kartEdkenAtMbn;rgkarTaj l
RbEvgtEdk l = l RtUv)aneRbIenAeBlEdlmuxkat;EdkticCagBak;kNalRtUv)ant nigmuxkat;Edk
eRbIR)as;esI2dgmuxkat;RtUvkarTaMgGs;elIRbEvgEdlt. RbEvgtEdk l = 1.3l RtUv)aneRbIenAeBlNa
EdlkartEdkeFVIeLIgenATItaMgEtmYy EtCasMNUmBr eKminRtUvtEdkenAkEngdUcKaeLIy. sRmab;RKb;krNI
RbEvgtEdkminRtUvmantmtUcCag 30cm .
x> kartEdkenAtMbn;rgkarsgt; l
RbEvgtEdkenAtMbn;sgt; l RtUvmantmFMCag besIRbEvgbgb;enAtMbn;rgkarsgt;.
l = 0.0724 f d
sRmab; f 400MPa
l = (0.13 f 24)d sRmab; f > 400MPa
sRmab;RKb;krNI RbEvgtEdkminRtUvmantmtUcCag 30cm . RbsinebIebtugEdlykmkeRbIman
ersIusg;tUcCag 20MPa enaHRbEvgtEdkRtUvKuNnwg 1.3 . sRmab;ssrEdlmanEdkkgragCarWusr enaH
RbEvgtEdkRtUvKuNnwg 0.75 EtsRmab;EdkkgFmtaRtUvKuNnwg 0.83 b:uEnminRtUvtUcCag 30cm .
st

st

st

sc

sc

sc

sc

]TahrN_ 7>6 KNnaRbEvgbRBassRmab; 6DB25 EdkrgkarTajxageRkam CaBIrCYr KMlatBIpEdkeTA

pEdk 60mm nigkRmas;karBarebtug 40mm sRmab;krNIxageRkam
k enAeBleKtbRBasEdkbIedIm nig ( A pl;[ ) /( A tRmYvkar ) > 2
x enAeBleKtbRBasEdkbYnedIm nig ( A pl;[ ) /( A tRmUvkar ) < 2
s

RbEvgEdkbgn; bRbEvgEdkRCYs

173

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NPIC

K enAeBleKtbRBasRKb;srsEdkTaMgGs;Rtg;TItaMgEtmYy. eK[ f ' = 35MPa nig f = 400MPa .

dMeNaHRsay
k sRmab; d = 25 ehIy = = = = 1.0 smIkar 7.8/ 35 = 5.9MPa < 8.3MPa
l = 40.6d BIeRBaHvaeKarBlkxN k nig x. l = 40.6 25 = 1015mm . sRmab; ( A pl;[ ) /
( A tRmYvkar ) > 2 / l = 1.0l = 1015mm > 300mm Gb,brma. EdkEdlRtUvbRBastUcCagEdksrub
Bak;kNal.
x l = 1015mm dUckarKNnaxagelI. edaysar ( A pl;[ ) /( A tRmUvkar ) < 2 / l = 1.3l = 1320mm
EdlFMCag 300mm .
K l = 1320mm > 300mm .
c

st

st

st

]TahrN_ 7>7 KNnaRbEvgbRBassRmab; DB32 Edkrgkarsgt;enAkgssrEdleRbIEdkkgFmta enA

eBl f ' = 35MPa nigenAeBl k f = 400MPa x f = 550MPa .
dMeNaHRsay
k sRmab; DB = 32 / l = 0.24 400
32 = 520mm . RtYtBinit l 0.0724(32)(400) = 927 mm .
35
dUcenH l = 930mm .
x l = 520mm dUckarKNnaxagelI. RtYtBinit l (0.13 550 24)32 = 1520mm .
dUcenH l = 1550mm .
c

db

sc

sc

db

sc

sc

7>9> karbBab;Edk
ersIusg;m:Um:g;enAkgFwmCaGnuKmn_eTAnwgkm<s;RbsiTiPaB d / TTwgFwm b / nigmuxkat;Edk A sRmab;
karEdleKsal;ersIusg;Edk f nigersIusg;ebtug f ' . sRmab;FwmEdlmanTTwg nigkm<s;efr enaHmuxkat;
EdkERbRbYleTAtamm:Um:g;Bt;EdlmanGMeBItambeNayFwm. dUcenHeKcaM)ac;RtUvbBab;Edk enAkEngNaEdl
eKminRtUvkarvaedIm,ITb;Tl;nwgkugRtaMgBt;. sRmab;krNIxH dUcCaFwmCab; EdksRmab;m:Um:g;viCman RtUv)aneK
Bt;eLIgelIedaymMu 45 edIm,Ipl;CaEdkrgkmaMgTajsRmab;m:Um:g;GviCmanenAelITRm.
ersIusg;m:Um:g;rbs;FwmrgkarTajenARKb;muxkat;TaMgGs;RtUv)ankMNt;eday
s

a
M u = As f y (d )
2

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dXas; (d a2 ) ERbRbYlBImuxkat;mYyeTAmuxkat;mYytambeNayFwm dUckarERbRbYlbrimaNsrs

EdkEdr eTaHCay:agenHkeday eyIgeXIjfabERmbRmYldXas;mantUcbMput ehIyminmantmxusKaeRcInBI
muxkat;m:Um:g;GtibrmaeT. dUcenH eKGacsnt;faersIusg;m:Um:g;sRmab;muxkat;smamaRtnwgkmaMgTaj bmux
kat;srsEdk edayKitTaMgRbEvgTMBk;.
edIm,IkMNt;TItaMgnkarbBab;Edk eKRtUvKUrdaRkamm:Um:g;EdlekItBIkmaMgxageRkAmuneK. daRkam
ersIusg;m:Um:g;RtUv)anKUrenAelIRkaPicEtmYy edIm,IbgajTItaMgsrsEdkxHEdlminRtUvkarRbEvgEvgCag
tRmUvkar. ersIusg;m:Um:g;cugeRkaysRmab;srsEdkmYysrsKW
a
M ub = Asb f y (d )
2
As f y
a=
0.85 f 'c b

Edl

- muxkat;EdkmYysrs
cMNucRbsBVrvagbnat;ersIusg;m:Um:g; CamYydaRkamm:Um:g;xageRkA bgajBITItaMgEdleyIgGacbBab;
Edk)an. b:uEneKcaM)ac;RtUvbEnmRbEvgEdkedIm,IPaBsitCamYyebtug. ACI Code )ankMNt;faRKb;Edk
EdlbBab;RtUvbEnmRbEvgesInwgkm<s;RbsiTPaBrbs;Fwm b 12d edayyktmmYyNaEdlFMCageK
elIsBIenH ACI Code )ankMNt;fay:agticNas; 1 / 3 nsrsEdksRmab;m:Um:g;viCmanRtUvbgseTAkg
TRmsRmab;FwmTRmsamBa.
Asb

RbEvgEdkbgn; bRbEvgEdkRCYs

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edIm,IgayRsYlkgkarbgajBIkarbBab;Edk rUb 7>13 bgajFwmTRmsamBargbnkBRgayesI mux

kat;rbs;va nigdaRkamm:Um:g;. ExSekagm:Um:g;Bt;manrag)a:ra:bYlEdlmanm:Um:g;GtibrmaenAkNalElVg
272kN .m . edaysarFwmBRgwgeday 4 DB 25 enaHersIusg;m:Um:g;rbs;srsEdkmYyedImKW
a

M ub = Asb f y d
2

As f y
1963 350
a=
=
= 128.3mm
0.85 f 'c b 0.85 21 300
128.3 6

M ub = 0.9 491 350 540

10 = 73.6kN .m
2

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dYcenH ersIusg;m:Um:g;rbs;EdkbYnsrsKW 294.4kN .m EdlFMCagm:Um:g;xageRkA 272kN .m . Rbsin

ebIRbsinebIeyIgKUrdaRkamm:Um:g;[RtUvmaRtdanenAelIExSeKal A A eyIgeXIjfaeKGacbBab;srsEdk
TImYyenARtg;cMNuc a srsEdkTIBIrenARtg;cMNuc b nigsrsEdkTIbIenARtg;cMNuc c ehIysrsEdkTIbYn
enARtg;TRmcug A . cMNucTaMgenHCaTItaMgsRmab;bBab;EdktamRTwsI. b:uEn eKcaM)ac;begItEpknersIusg;xH
rbs;srsEdkedayPaBsit dUckarBnl;BIxagelI. ACI Code kMNt;[EdkEdlbBab;TaMgGs;bneday
RbEvgEdlesInwgkm<s;RbsiTPaB d rbs;Fwm b 12d edayyktmmYyNaEdlFMCageK BIcMNucbBab;tam
RTwsI a, b, nig c . Code, section 12.11.1 kkMNt;Edrfa y:agehacNas;1 / 3 nEdkm:Um:g;viCmanRtUv)an
bncUleTAkgTRmsRmab;FwmsamBa. dUcenH sRmab;]TahrN_Edlerobrab;enATIenH eKRtUvbgtEdkBIrsrs
eTAkgTRm ehIydaRkamersIusg;m:Um:g;Edl)anbgajenAkgrUbTI 7>13 RtUvekabdaRkamm:Um:g;xageRkARKb;
cMNucTaMgGs;. srsEdknImYyTTYl)anlTPaBTb;Tl;nwgbnkeBjeljenARtg;cmay l BIcugrbs;va.
sRmab;FwmCab; eKRtUvBt;srsEdkenARtg;cMNucEdlRtUvkar nigeRbIvaedIm,ITb;Tl;nwgm:Um:g;GviCman
enARtg;tMN. y:agehacNas;keKRtUvbnEdk1 / 3 nEdkm:Um:g;GviCmanTaMgGs;Edlpl;[sRmab;m:Um:g;GviCmanBIcMNucrbt;cmayEdlesInwgkm<s;RbsiTPaB d , b12d b L / 16 edayyktmmYyNaEdlFMCageK.
eKkeRbI bent bar edIm,ITb;Tl;nwgEpkxHrbs;kugRtaMgkmaMgkat;enAkgFwmpgEdr. daRkamersIusg;
m:Um:g;sRmab;FwmCab;KMrURtUv)anbgajenAkgrUbTI 7>14.
]TahrN_ 7>8 sRmab;FwmTRmsamBaEdlbgajenAkgrUbTI 7>15 cUrsikSaKNnaFwmEdlrgbnkKNna
xageRkA nigsg;daRkamersIusg;m:Um:g;. dUcKa bgajBITItaMgEdlGacbBab;Edk. eRbI b = 250mm / pl
eFobEdkKW 0.018 / f ' = 21MPa / f = 280MPa .
dMeNaHRsay
sRmab; = 0.018 / R = 3.90MPa ehIy M = R bd
eday M = 176.8kN.m
enaH d = 426mm
yk h = 500mm
b

As = 0.018 250 440 = 1980mm 2

eRbI 4DB25
km<s;RbsiTPaBCak;Esg d = 500 440 = 60mm
a=

1963.5 280
= 123.2mm
0.85 21 250

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ersIusg;m:Um:g;sRmab;EdkmYyedIm
123.2 6
a

M ur = As f y d = 0.9 491 280 440

10 = 46.8MPa
2
2

ersIusg;m:Um:g;sRmab;EdkbYnedIm
M ur = 187.2kN .m

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daRkamersIusg;m:Um:g;RtUv)anbgajenAkgrUbTI 7>15. cMNaMfa eKGacBt;srsEdk bbBab;srsEdkenA

cmay 440mm b 12d edayykmYyNaEdlFMCageK bnab;BIcMNucEdleKminRtUvkarsrsEdk tam
RTwsI. RbEvgbgb; l sRmab; d = 25mm KW 36.7d = 920mm . cMNucbBab;EdkTImYy nigTIBIrenARtg;
cMNuc A nig B b:uEncMNucCak;EsgKWenARtg;cMNuc A' nig B' cmay 450mm BIcMNuc A nig B . BI A' RbEvg l
= 920mm fyeRkayRtUv)aneFVIedIm,IbegItdaRkamersIusg;m:Um:g;. cugrbs;EdkEdlenAsl;BIreTotRtUvbgt
cYleTAkgTRm Rtg;cMNuc C ' . CaTUeTA eKmineRbIEdkEdlbBab;enAkgElVgRtg;cMNuc A' nig B' Ca bent bar
edIm,ITb;Tl;kmaMgkat;eT.
b

RbEvgEdkbgn; bRbEvgEdkRCYs

179

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VIII.

kmaMgkat;TTwg nigkmaMgTajGgt;RTUg
Shear and Diagonal Tension

8>1> esckIepIm
enAeBlEdlFwmTRmsamBargnUvbnk m:Um:g;Bt; nigkmaMgkat;TTwgnwgekIteLIgelIRbEvgFwm. edIm,IRT
nUvbnkenHedaysuvtiPaB FwmRtUv)aneFVIkarKNnaeLIgedIm,ITb;nwgkmaMgTaMgBIrRbePTenH. karKNnasRmab;
karBt;RtUv)aneFVIeLIgdMbUgeK edIm,IkMNt;muxkat;Fwm nigEdkemcaM)ac; dUcEdl)anENnaMBIemeronmun.
bnab;mkFwmRtUv)anKNnaedIm,ITb;nwgkmaMgkat;TTwg. kgkrNIEdlEdkkgminRtUv)andak; enaHFwm
nwg)ak;edaykmaMgkat;TTwg. kar)ak;edaykmaMgkat;TTwg ekIteLIgedaymanPaBdabtUc nig)at;bg;nUv
lkNyWt ehIymin)anRbkasGasnenAmuneBl)ak;eT. sRmab;kar)ak;edaykarBt;begag ekIteLIgeday
karekIneLIgnUvPaBdabbnicmg nigmansameRbH dUcenHvamankarpl;sBaaRbkasGasnmuneBl)ak;cug
eRkay. karKNnasRmab;kmaMgkat;TTwg RtUv)aneFVIeLIgedIm,IFananUvkar)ak;edaykmaMgkat;TTwgekIteLIg
eRkaykar)ak;edaykarBt;begag.

8>2> kugRtaMgkmaMgkat;enAkgFwmebtugGarem:
rUbmnTUeTAsRmab;kugRtaMgkmaMgkat;TTwgenAFwmrUbFatusac;mYy (homogenous beam) karBRgay
kugRtaMg
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=

Edl

Department of Civil Engineering

*>!

VQ
Ib

- kmaMgkat;srubenAmuxkat;EdlRtUvsikSa
Q - m:Um:g;saTiceFobGkSNWtnmuxkat;EdlxNedaybnat;EdlRtUvsikSakugRtaMgkmaMgkat;
I - m:Um:g;niclPaBnmuxkat;eFobGkSNWt
b - TTwgFwmnmuxkat;EdlRtUvsikSakugRtaMgkmaMgkat;
karBRgaykugRtaMgm:Um:g;Bt; nigkmaMgkat;TTwgGaRsyeTAnwgRTwsIbTeGLasicsRmab;muxkat;Fwm
ctuekaN dUcbgajkgrUbTI8>2.
kugRtaMgm:Um:g;Bt;
V

f =

Mc
I

Edl kugRtaMgkmaMgkat;enARKb;cMNucTaMgGs;RtUv)anKNnatamrUbmnkugRtaMgkmaMgkat;
=

VQ
Ib

kugRtaMgkmaMgkat;GtibrmasitenAelIGkSNWt nigmantmesI 1.5v kmaMgkat;TTwgmFm

Edl = bhV . ExSekagkmaMgkat;TTwgmanrag):ar:abUl.
a

sRmab;FwmebtugssEdkrgkarTaj (singly reinforced concrete beam) karBRgaykugRtaMg kmaMg

kat;TTwgenAelIGkSNWtmanrag):ar:abUl. enAeRkamGkSNWt kugRtaMgkmaMgkat;TTwgGtibrmamantmefr
ehIy)anrkSatmenHRtwmnIv:UEdkrgkarTaj BIeRBaHvaKankarpas;brkmaMgTajBIcMNucGkSNWtehIym:ag
eTotkmaMgTajkgebtugRtUv)anecal. kugRtaMgkmaMgkat;TTwgmantmesIsUn enAeRkamnIv:UEdk rUbTI8> 3.
kmaMgkat; nigkmaMgTajGgt;RTUg

181

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NPIC

sRmab;ebtugsrsEdkEdlmanEdkrgkarsgt; nigmuxkat;GkSret karBRgaykugRtaMgkmaMgkat;

TTwgRtUv)anbgajkgrUbTI8>3. eyIgsegteXIjfa kmaMgkat;TTwgesIrEtTaMgGs; RtUv)anTb;edayRTnug
EdlsabTb;nwgPaKrytUcbMput. sRmab;karGnuvtesIrTaMgGs; eKecalnUvlTPaBTb;kmaMgkat;TTwgrbs;
sab.

eyagtamrUbTI8>1 edayykFwmmYykMNat;tUc dx mkviPaK eyIgeXIjfa m:Um:g;Bt;enAcugsgxagn

kMNat; M nig M minmantmesIKaeT. edaysar M < M dUcenHedIm,IrkSalMnwgsRmab;kMNat; dx kmaMg
sgt; C RtUvmantmFMCag C rUbTI8>4. dUcenHkugRtaMgkmaMgkat;TTwg v ekItmanenAelImux kat;edk
a a1 b b b1 rUbTI8>4 a. kugRtaMgkmaMgEkg (normal stresses) nigkugRtaMgkmaMgkat;TTwg (shear
stresses) enAelIGgt;tUcenAkRmitnIv:U a a1 b b b1 RtUv)anbgajenAkgrUbTI8>4 b. cMNaMfa kugRtaMg
kmaMgEkg (normal stresses) enAnIv:UnGkSNWtKW 0 b:uEnkmaMgkat;TTwgmantmGtibrma. kmaMgkat;TTwg
edkesInwgkmaMgkat;TTwgbBar dUcbgajenAkg rUbTI8>4 b. enAeBlEdl kugRtaMgkmaMgEkgmantm
esIsUn btUc enaHkrNIkmaMgkat;TTwgsuTGacekItman. kgkrNIenH kugRtaMgTajGtibrma f t manGMeBI
tammMu 45o rUbTI8>4 c.
1

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kugRtaMgTajsmmUleTAnwgkugRtaMgem dUcbgajkgrUbTI 8>4 d. kugRtaMgemenHRtUv)aneKehAfa kug

RtaMgTajGgt;RTUg. enAeBlkugRtaMgTajGgt;RTUgmantmesIersIusg;Tajrbs;ebtug sameRbHGgt;RTUg
ekIteLIg. karviPaKy:agsegbenHBnl;BIKMnitnkmaMgTajGgt;RTUg nigsameRbHGgt;RTUg. kareFVIkarCak;
EsgmanlkNsKsajCag ehIyvaTTYl\TiBlBIktaepSg. sRmab;bnSMnGMeBInkmaMgkat;TTwg nig
kmaMgEkgenAcMNucNamYyenAkgFwm kmaMgTajGgt;RTUg (principal stresses) Gtibrma nigGb,brma f p
RtUv)an[edaysmIkarxageRkam
2

fp =

Edl

f
f
+ v2
2
2

*>@

GaMgtg;sIuetnkugRtaMgEkgEdlbNalmkBIkarBt;
v = kugRtaMgkmaMgkat;
kar)ak;edaykmaMgkat;TTwgenAkgFwmebtugPaKeRcInTMngCaekIteLIgenAkEngEdlkmaMg
kat;TTwgGtibrma CaTUeTAenAEk,rTRmnGgt;. PstagTImYynkar)ak;EdlKYr[PyxayKWkarekIteLIgnUv
sameRbHGgt;RTUg.
f =

8>3> kareFVIkarrbs;FwmedayKanEdkkmaMgkat;TTwg
ebtugexSaykgkarTaj ehIyFwmGac)ak;RbsinebImuxkat;EdkminRtwmRtUvRtUv)anpl;[. kugRtaMg
TajekItmanenAkgFwmbNalmkBIkmaMgTajtamGkS kmaMgBt; kmaMgkat;TTwg kmaMgrmYl bbnSMnbnk
TaMgenH. TItaMgnsameRbHenAkgFwmebtugGaRsynwgTisedAnkugRtaMgem (principal stresses). bnSMnkug
RtaMgkmaMgEkg normal stress nigkugRtaMgkmaMgkat;TTwg begIt)ankmaMgTajtamGgt;RTUg (diagonal
tension) GtibrmaEdlsitenARbEhlcmay d BImuxnTRm.
kareFVIkarrbs;FwmebtugGarem:edaymanbKanEdkkmaMgkat;TTwg RtUv)anBiesaFeRkamGMeBInkarekIn
eLIgnbnkdUc)anerobrab;enAkgemeronTI3. enAkgkarBiesaFFwm sameRbHbBarEdlekItBIkarBt;ekIteLIg
enAelImuxkat;Edlmanm:Um:g;Bt;Gtibrma enAeBlEdlkugRtaMgTajenAkgebtugelIsBIm:UDuldac; (module of
rupture) rbs;ebtug b f r = 0.623 f 'c . sameRbHeRTtenAkgRTnugekItmanenAkgdMNak;kalbnab;enATI
taMgEk,rTRm.

kmaMgkat; nigkmaMgTajGgt;RTUg

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NPIC

sameRbHeRTtEdlekItmanenAkgFwmEdlminTan;)aneRbHBIdMbUg CaTUeTARtUv)aneK[eQaHfa sam

eRbHkmaMgkat;RTnug (web-shear crack). RbsinebIsameRbHeRTtcab;epImenABIelIsameRbHEdlekItBIkar
Bt;EdlmanRsab; ehIyrIksayenAkgFwm enaHsameRbHRtUv)aneK[eQaHfa sameRbHkmaMgkat;rgkarBt;
(flexural-shear crack) rUbTI8>5. sameRbH web-shear crack ekItmanenAkgFwmEdlmankmaMgkat;FM nig
T.Chhay

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m:Um:g;Bt;tUcenAkgRTnugesIg. vaCasameRbHminFmtaehIyGacekItmanenAEk,rcMNucrbt;nFwmCab; bEk,r

TRmnFwmsamBa.
sameRbH flexural-shear crack CaRbePTsameRbHFmtaEdleKGaceXIjmanenAelIFwm. dMbUgsam
eRbH flexural crack ekItmanbBarenAelIFwm bnab;mksameRbHeRTtcab;epImekItmanBIelIkMBUlnsameRbH
flexural crack enAeBlEdlkugRtaMgkmaMgkat;TTwgekItmanenAkgtMbn;enaH. enAkugtMbn;EdlmankugRtaMg
kmaMgkat;TTwgFM FwmRtUvEtBRgwgedayEdkkg stirrup bEdkBt; (bent bar) edIm,IeFVI[FwmmanlkNsVit
(ductile) Edlmin)ak;b dac;. edIm,IeCosvagkar)ak;edaykmaMgkat; munkar)ak;edaykarBt; emKuNsuvtiPaB
FMRtUv)anpl;[edIm,ITb;nwgkar)ak;edaykmaMgkat;TTwg. ACI Code kMNt;emKuNkat;bnyersIusg;
= 0.75 sRmab;kmaMgkat;TTwg.
ersIusg;kmaMgkat;TTwgenAkgGgt;ebtugGarem:RtUv)anekIteLIgedaybnSMnkmaMgemkanicxageRkam
rUbTI8>5
- ersIusg;kmaMgkat;TTwgnebtugminTan;eRbH Vz
- kareprkmaMgkat;TTwgrvagGnrp (interface shear transfer) Va EdlbNalmkBIkarbgaMKarvag
fbMEbktambeNaypdKRKatrbs;sameRbH
- GMeBIrbs;Fr (arch action)
- GMeBIEdkf<k; (dowel action) Vd EdlbNalmkBIersIusg;nr)arEdkbeNayeTAkmaMgkat;TTwg
tamTTwg (transverse shearing force)
bEnmeTAelIkmaMgTaMgenH EdkkmaMgkat;TTwg (shear reinforcement) begInersIusg;kmaMgkat;TTwg
Vs edayGaRsyeTAelIGgt;pit nigKMlatEdkkgEdleRbIenAkgGgt;ebtug. RbsinebIEdkkmaMgkat;TTwg
minRtUv)andak;enAkgFwmragctuekaN enaHsmamaRtnkmaMgkat;TTwgEdlTb;edaykmaMgemkanicepSgKW
BI 20% eTA 40% eday Vz / BI 35% eTA 50% eday Va / BI 15% eTA 25% eday Vd .
8>4> \TiBlm:Um:g;eTAelIersIusg;kmaMgkat;
sRmab;FwmTRmsamBaeRkamGMeBIbnkBRgayesI muxkat;kNalElVgrgnUvm:Um:g;Bt;FM nigkmaMgkat;
TTwgtUc besIsUn EdlpyBImuxkat;enAEk,rTRmEdlm:Um:g;Bt;mantmtUc kmaMgkat;TTwgmantmFM rUbTI
8>1. kmaMgkat;TTwg nigm:Um:g;mantmFMenAEk,rTRmkNalsRmab;FwmCab;. enATItaMgEdlkmaMgkat;TTwg
FM nigm:Um:g;Bt;tUc enaHvanwgmankareRbHedaykarBt;tictYc ehIykugRtaMgmFm v = V / bd . kugRtaMgkmaMg
TajtamGgt;RTUgCakugRtaMgEdleRTtedaymMuRbEhl 45o rUbTI8>4 c . sameRbHGgt;RTUgGacrMBwgfanwg
ekItman enAeBlEdlkugRtaMgkM;laMgTajGgt;RTUgEdlsitenAEk,rtMbn;GkSNWtxiteTACit belIsersIusg;
kmaMgkat; nigkmaMgTajGgt;RTUg

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NPIC

kmaMgTajrbs;ebtug. CaTUeTA ersIusg;kmaMgkat;TTwgcugeRkay (ultimate shear strength) ERbRbYlcenaHBI

0.29 f 'c nig 0.42 f 'c . eRkayBIkarBiesaFn_eTAelIFwmCaeRcInGMBIkmaMgkat;TTwg nigkmaMgTajtam
Ggt;RTUg eK)anrkeXIjfaenAkgtMbn;EdlmankmaMgkat;TTwgFM nigm:Um:g;Bt;tUc enaHsameRbHkmaMgTaj
Ggt;RTUgRtUv)anbegIteLIgenAeBlkmaMgkat;TTwgmFm
V = 0.29 f ' b d
*>#
Edl bw CaTTwgRTnugmuxkat;GkSret bTTwgmuxkat;ctuekaN
d Cakm<s;RbsiTPaBrbs;Fwm
enATItaMgEdlkmaMgkat;TTwg nigm:Um:g;Bt;mantmFM enaHsameRbHedaykarBt; (flexural crack)
RtUv)anekIteLIgdMbUg. enAdMNak;kalTImYy sameRbHxHBt;kgTisedAGgt;RTUgenAeBlEdlkugRtaMgkmaMg
TajGgt;RTUg EdlsitenABIcugxagelInsameRbHTaMgenaHFMCagkugRtaMgkmaMgTajrbs;ebtug. RbsinebIeK
[m:Um:g;FMmanGMeBIelIFwm sRmab;muxkat;EdlmanbrimaNEdkRKb;RKan; enaHkmaMg nominal shear force enA
eBlEdlsameRbHekItmanRtUv)an[dUcxageRkam
*>$
Vcr = 0.16 f 'c bw d
cr

T.Chhay

c w

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tmenHKWtUcCagtmEdl[edaysmIkar *># eRcInCagBak;kNalenAeBlEdlm:Um:g;Bt;mantm

tUc. enHmannyfam:Um:g;Bt;FMkat;bnytmrbs;kugRtaMgkmaMgkat;TTwgenAeBlEdlsameRbHekIteLIg.
smIkarxageRkamRtUv)anesIeLIgedIm,ITsSn_TaynUvkugRtaMg nominal shear stress enAeBlEdlsamGgt;
RTUgRtUv)anrMBwgfaekItman

V d
V
vcr =
*>%
= 0.16 f 'c + 17.2 w u 0.29 f 'c
b d
M

ACI Code, Section 11.3.2

)anyksmIkarenHsRmab;KNnakmaMg nominal ultimate shear force

EdlTb;edayebtug
*>^
Edl w = As / bwd / d Cakm<s;RbsiTPaBrbs;muxkat;Fwm/ bw CaTTwgRTnugsRmab;muxkat;GkSr
et bTTwgnmuxkat;ctuekaN Vu nig M u CakmaMgkat;TTwgcugeRkay (ultimate shearing force) nigm:Um:g;
Bt;cugeRkay (ultimate bending moment) EdlekIteLIgkgeBldMNalKaenAelImuxkat;sikSa.
tmn Vu d / M u minRtUvFMCag 1.0 sRmab;smIkar *>^. RbsinebI M u mantmFMenAkgsmIkar
*>^ enaHtYTIBIrnwgmantmtUcesIEtesIsUn enaH vc xiteTArk 0.16 f 'c . RbsinebI M u mantmtUc enaH
tYTIBIrnwgmantmFM ehIytm 0.29 f 'c lub. eRkABIsmIkar *>^ ACI Code, Section 11.3.1GnuBaat[
KNnaersIusg;kmaMgkat;rbs;ebtugdUcxageRkam
Vc = (0.17 f 'c )bw d
*>&
1> kgkrNIsRmab;kmaMgsgt;tamGkS N u
V d
*>*
Vc = (0.16 f 'c + 17.2 w u )bw d 0.29 f 'c bw d
M
Vc = (0.16 f 'c + 17.2 w

Vu d
)bw d 0.29 f 'c bw d
Mu

Mm

Edl

4h d
= M u Nu

8
A
w = s
bw d

km<s;srubrbs;Fwm
Vu d / M u GacFMCag 1.0 b:uEn Vc minRtUvFMCag
h=

Vc = bw d (0.29 f 'c ) 1 +

*>(

Nu
3.45 Ag

Ag CaRkLapTaMgGs; (gross section) KitCa mm 2

Edl
m:agvijeTot Vc GacRtUv)anKNnaeday

kmaMgkat; nigkmaMgTajGgt;RTUg

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NPIC

Vc = bw d (2 + 0.145

*>!0

Nu
) f 'c
Ag

2> kgkrNIsRmab;kmaMgTajtamGkS N u
Vc = bw d (2 + 0.58

Edl
RbsinebI

*>!!

Nu
) f 'c
Ag

mantmGviCmansRmab;kmaMgTaj
Vc GviCman enaH Vc RtUv)anykesIsUn.
Nu

8>5> FwmmanEdkkmaMgkat;
EdkEdleRbIedIm,ITb;nwgkmaMgkat; EdleKeRbIGacmaneRcInRbePTepSg
a. Edkkg EdleKeRbIedaydak;EkgeTAnwgEdkbeNay Edkem bRtUv)aneKdak;eRTt mMuEdl
eKniymeRbI 45 . EdkkgEdleKniymeRbImanmuxkat; DB10 nig DB12 .
b. EdkBt; EdlCaEpkmYyrbs;EdkbeNayEdleKBt;eLIg enAkEngEdleKbBab; edaymMu
30 nig 60 CaTUeTA 45 .
c. bnSMrvagEdkkg nigEdkBt;
d. sMNaj;Edk CamYynwgsMNaj;EkgeTAnwgGkS
e. EdkkgvN EdleKeRbIsRmab;ssr
ersIusg;kmaMgkat;TTwgrbs;FwmebtugGarem:RtUv)anbegIneLIgedaykareRbInUvEdkkmaMgkat;TTwg.
munnwgekItnUvsameRbHGgt;RTUg EdkkmaMgkat;TTwgCYyersIusg;kmaMgkat;TTwgtictYcbMput. eRkayeBlEdl
sameRbHkmaMgkat;TTwgekIteLIg EdkkmaMgkat;TTwgbegInersIusg;kmaMgkat;rbs;Fwm ehIykmaMgkgmg
eTotEdlekIteLIgenAmuxkat;eRbH. enAeBlbrimaNEdkkmaMgkat;TTwgtUc kar)ak;EdlekIteLIgedaysar
EdkenARTnugeFVIkardl; yield GacnwgekIteLIg b:uEnRbsinebIbrimaNEdkkmaMgkat;TTwgFM enaHkar)ak;eday
shear-compression failure GacnwgekIteLIg TaMgenHCaGVIEdleyIgKYreCosvag.
ebtug Edkkg (stirrups) nigEdkdgErk (bent bars) eFVIGMeBIrYmKaedIm,ITb;nwgkmaMgkat;TTwg.
edaysarersIusg;rgkarsgt;x<s; ebtugedIrtYCaGgt;rgkarsgt;Ggt;RTUgnRbBnFwm cMENkEdkkgedIrtYCa
Ggt;rgkarTajbBar. kmaMgsgt;Ggt;RTUg kdUcCabgMkmaMgbBarrbs;va mantmesInwgkmaMgTajenAkg
Edkkg. EdkBt;dgErk (bent-up reinforcement) kedIrtYdUcCaGgt;TajenAkg truss Edr rUbTI *>^.
o

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CaTUeTA karcUlrYmrbs;EdkkmaMgkat;TTwgeTAkgersIusg;kmaMgkat;TTwgrbs;FwmebtugGarem:
GacRtUv)anBiBNnadUcxageRkam
- vaTb;Tl;EpkxHnkmaMgkat;TTwg/ V
- vabegInGaMgtg;sIuetnkmaMgkat;rvagGnrp/ Va rUbTI *>%/ edayTb;Tl;nwgkarrIkFMnsam
eRbHeRTt (inclined crack).
- vabegInnUvkmaMgf<k; (dowel force)/ Vd rUbTI *>%/ enAkgEdkbeNay
- GMeBITb; (confining action) rbs;EdkkgeTAelIebtugGacbegInersIusg;rbs;va
- GMeBITb; (confining action) rbs;EdkkgeTAelIebtugbegInnUvsmtPaBrgVilnsnak;)asic
(rotation capacity of plastic hinge) EdlekItmanenAkgeRKOgbgM indeterminate structure
eRkambnkcugeRkay nigbegInRbEvgEdl yield GacekItmanenAelIva.
eday V CaersIusg;kmaMgkat;rbs;muxkat;ebtugGarem:enaH
s

kmaMgkat; nigkmaMgTajGgt;RTUg

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*>!@

Vn = Vc + Vs

- ersIusg;kmaMgkat;)anBIebtug
V - ersIusg;kmaMgkat;)anBIEdk
RbsinebI V kmaMgkat;Edl)anBIbnkxageRkAenaH
Edl

Vc
s

*>!#

Vu Vn = (Vc + Vs )

V = 1.2V + 1.6V
nig = 0.75
Edl
V RtUv)anKNnaedaykarviPaK truss rUbTI *>&. sRmab;sameRbH 45 nigesrInEdkkg b
EdkdgErk. kmaMgkat;bBar V esInwgplbUkbgMkmaMgbBarnkmaMgTajEdlekItmanenAkgEdkeRTt
V = nA f sin
*>!$
Edl
-muxkat;kat;EdkkmaMgkat;CamYyKMlat s
A
f
- ersIusg;EdkkmaMgkat;
u

yt

yt

eday ns = aa + a a
1

1 2

d = a1a4 = aa1tg 45o

d = a1a4 = aa2tg

BIRtIekaN aa a
BIRtIekaN aa a
1 4

1 2

ns = d (cot 45o + cot ) = d (1 + cot )

d
n = (1 + cot )
s
A f d
A f d
Vs = v yt sin (1 + cot ) = v yt (sin + cos )
s
s

eKTTYl)an
sRmab;krNIEdkkgbBar
T.Chhay

*>!%

= 90o
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A f d
b
s=
s
V
sRmab;krNIEdkkgbBar = 45
A f d
A f d
s = 1 .4
b
V = 1.4
V
s
sRmab;krNIEdkdgErkEtmYy bRkumEdkenAmYykEng
V = A f sin
b A = f Vsin
Vs =

Av f yt d

yt

*>!^

yt

yt

*>!&

yt

*>!*

yt

sRmab;
Av = 1.4

= 45o

*>!(

Vs
f yt

8>6> tRmUvkarrbs;

ACI Code

sRmab;karKNnakmaMgkat;TTwg

8>6>1> muxkat;eRKaHfak;sRmab;karKNnaersIusg;kmaMgkat;TTwgmFm
Critical section for nominal shear strength calculation

GnuBaat[ykmuxkat;eRKaHfak;sRmab;karKNnaersIusg;kmaMgkat;
mFmenAcmay d BIpmuxnTRm. karENnaMenHQrenAelIPaBCak;EsgEdlsameRbHeRTtdMbUgeKTMngCa
ekIteLIgenAelIFwmRtg;cmay d BITRmEdleRcInelcecjenAeBleFVIBiesaFn_. muxkat;eRKaHfak;enHRtUv)an
GnuBaat enAkglkxNEdlRbtikmTRmbBankmaMgsgt;eTAkgtMbn;cug bnkRtUv)anGnuvtenAelI benAEk,r
kMBUlnGgt;ehIyKanbnkcMcMNucGnuvtenAcenaHpnTRm nigTItaMgnmuxkat;eRKaHfak;. bTdankkMNt;Edr
faEdkkmaMgkat;TTwgRtUv)andak;enAcenaHpnTRm nigcmay d .
ACI Code, Section 11.1.3

8>6>2> muxkat;EdkGb,brmasRmab;EdkkmaMgkat;TTwg
vtmanrbs;EdkkmaMgkat;TTwgenAkgFwmebtugTb;Tl;nwgkarrIkraldalnsameRbHeRTt. m:agvij
eTot PaBsVit (ductility) ekIneLIg ehIyva)anRbkasGasnmuneBl)ak;. RbsinebIKanEdkkmaMgkat;TTwg
enaHFwmmanlkNRsYyehIy)ak;edaymin)anR)ab;mun. dUcenH muxkat;EdkkmaMgkat;TTwgRtUv)ankMNt;
eday Code. ACI Code, Section 11.5.5 tRmUvEdkkgTaMgGs;[manRkLapEdkkmaMgkat;TTwgGb,brma
Av esInwg
b s
b s
Av = 0.062 f 'c w 0.35 w
*>@0
f
f
yt

yt

Edl bw CaTTwgnRTnug nig s CaKMlatrbs;Edkkg. eKRtUvkarbrimaNEdkkmaMgkat;TTwgGb,brmaenAeBlEdl V > 0.5V elIkElg

u

kmaMgkat; nigkmaMgTajGgt;RTUg

191

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NPIC

- kRmalxN nigeCIgtag
- rnUtebtug
- FwmEdlmankm<s;tUcCag max{250mm; 2.5 dgkRmas;nsab; ; 1.5 nTTwgRTnug }
RbsinebI 0.062 f 'c = 0.35 enaH f 'c = 31.9MPa . enHmanyfaenAeBlEdl f 'c < 32MPa enaH
tmGb,brma Av = 0.35bw s / f yt manlkNlub EtenAeBlEdl f 'c 32MPa enaHtmGb,brma A =
0.062 f ' b s / f manlkNlub. karekIneLIgnUvRkLapEdkkmaMgkat;TTwgsRmab; f 'c 32MPa
KWedIm,IkarBarCamunnUvkar)ak;edaykmaMgkat;TTwg (shear failure )Pam enAeBlEdlekItmansameRbHeRTt.
vaCakarGnuvtmYydFmtakgkardMeLIgkRmas;kRmalxN kRmas;eCIgtag bkm<s;Fwmrak;edIm,IbegIn
lTPaBTb;Tl;nwgkmaMgkat;TTwg. EdkkgGacnwgKan\TiBlenAkgGgt;rak; edaysartMbn;rgkarsgt;rbs;
vamankm<s;tUcEmnETn nigminmanTMBk;RKb;RKan;EdlRtUvkarsRmab;Edkkg. sRmab;FwmEdlminrak; eKmin
RtUvkarEdkkmaMgkat;TTwgenAeBlEdl Vu < 0.5Vc .
RkLapEdkkmaMgkat;TTwgGb,brmaGacnwgRtUv)anTTYledayeRbIEdkkg DB10 dak;enAKMlatGtibrma S max . RbsinebI f yt = 400MPa ehIyEdkkg DB10 manragGkSr U eCIgBIrRtUv)aneRbI enaH
smIkar *>@0kayeTACa
Av f yt
Av f yt

S max =
*>@!
0.35b
(0.062 f ' )b
v

yt

sRmab; f 'c < 32MPa / S max = 157 400 / 0.35bw = 179400 / bw

*>@@
sRmab; f 'c = 32MPa / S max = 179000 / bw
sRmab; f 'c = 35MPa / S max = 171200 / bw
sRmab; f 'c = 42MPa / S max = 156250 / bw
RbsinebIeKeRbIEdk DB12 manragGkSr U enaH
sRmab; f 'c < 32MPa / S max = 258250 / bw
*>@#
sRmab; f 'c = 32MPa / S max = 25750 / bw
sRmab; f 'c = 35MPa / S max = 246450 / bw
sRmab; f 'c = 42MPa / S max = 224950 / bw
RtUvcgcaMfa S max minRtUvFMCag 600mm b d / 2 eT.
taragTI 1 pl;nUv S max edayQrelIsmIkar *>@@ nig *>@#. KMlatcugeRkayKYrEtRtUvrMkil
eTArktmEdltUc. ]TahrN_ S max = 515mm kayeTACa S max = 500mm .

taragTI1> tmrbs; S
T.Chhay

max

= Av f yt / 0.35bw = 60cm
192

. enAeBlEdl

f yt = 400MPa

nig

f 'c < 32MPa

Shear and Diagonal Tension

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

bw (cm)

25

30

35

40

45

50

55

60

bw

S max (cm) DB10

60

55

50

40

35

35

30

25

179400 / bw

S max (cm) DB12

60

60

60

60

55

50

45

40

258250 / bw

8>6>3> kmaMgkat;TTwgGtibrmaEdlTb;edayEdkkmaMgkat;TTwg V
edIm,IkarBarkar)ak; shear-compression failure EdlebtugGacEbkedaykugRtaMgkmaMgkat;TTwgFM
nigkugRtaMgkmaMgsgt;enAkgtMbn;eRKaHfak; enABIelIkMBUlnsameRbHGgt;RTUg ACI Code, Section
11.5.6.8, tRmUv[ V 0.67 f ' b d . RbsinebI V > 0.67 f ' b d enaHeKRtUvtMelIgmuxkat;ebtug.
edayQrenAelIkarkMNt;enH
RbsinebI f 'c = 20MPa enaH Vs 3bwd b Vs / bwd 3MPa
RbsinebI f 'c = 28MPa enaH Vs 3.5bwd b Vs / bwd 3.5MPa
RbsinebI f 'c = 35MPa enaH Vs 4bwd b Vs / bwd 4MPa
s

c w

8>6>4> KMlatEdkkgGtibrma
edIm,IFanafasameRbHGgt;RTUgRtUvkat;Edkkgy:agehacmYy enaH ACI Code, Section 11.5.4 tRmUv
fa KMlatrvagEdkkgminKYrelIs d / 2 b 600mm RbsinebI V 0.33 f ' b d edayQrelIkarsnt;fa
sameRbHGgt;RTUgekItmantammMu 45o niglatsnwgtamcmayedkRbEhlcmay d . enAkgtMbn;kmaMgkat;
TTwgFM Edl Vs > 0.33 f 'c bwd KMlatEdkkgGtibrmacenaHEdkkgminRtUvFMCag d / 4 . karkMNt;enHcaM
)ac;edIm,IFana[sameRbHGgt;RTUgkat;Edkkgy:agehacbI. enAeBlEdl V > 0.67 f ' b d karkMNt;
nKMlatGtibrmaminRtUv)anGnuvt ehIyTMhMrbs;muxkat;ebtugKYrRtUv)andMeLIg.
karkMNt;TIBIrsRmab;KMlatGtibrmanEdkkg kGacTTYl)anBIlkxNmuxkat;EdkkmaMgkat;TTwg
Gb,brma. Av Gb,brma RtUv)anTTYlenAeBlKMlat s Gb,brma smIkar *>@!.
karkMNt;TIbIsRmab;KMlatGtibrmaesInwg 600mm enAeBlEdl V 0.33 f ' b d nigesInwg
300mm enAeBlEdl 0.33 f 'c bw d < Vs 0.67 f 'c bw d . tmtUcCageKnKMlatGtibrmaRtUv)anyk
mkeRbI. tRmUvkarKMlatGtibrman ACI Code
FanaKMlatCitKarbs;EdkkgedIm,Icab;EdkrgkarTajbeNay enAkgFwm
edayehtuenHva)anbegInlTPaBbgb;Edkrbs;va Vd rUbTI *>%.
s

c w

kmaMgkat; nigkmaMgTajGgt;RTUg

193

c w

c w

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

8>6>5> ersIusg; yield rbs;EdkkmaMgkat;TTwg

ACI Code, Section 11.5.2. tRmUv[ersIusg; design yield strength rbs;EdkkmaMgkat;TTwg
minKYrelIs 420MPa . mUlehtuEdlenABIeRkaykarsMercenHKWedIm,IkMNt;TMhMsameRbHEdlbNalmkBI
kmaMgTajGgt;RTUg nigedIm,IFanafaEKmrbs;sameRbHenArkSapb:HCitKaedIm,IbegInkmaMgbBannkmaMg
kat;rvagGnrp Va rUbTI *>%. sRmab;srsEdkfaMgGMeBAsRmab;pSar (welded deformed wire fabric)
ersIusg; design yield strength minKYrelIs 560MPa .
8>6>6> TMBk;rbs;Edkkg
tRmUveGayEdkkmaMgkat;TTwgRtUv)andak;enAEk,rsrsrgkar
sgt;eRkAbMput nigsrsrgkarTajeRkAbMputtamEtGaceFVIeTA)an CamYynwgtRmUvkarrbs; code sRmab;
kRmas;karBarEdk edaysarEt enAeBlEdlbnkEdlmanGMeBIenAelIFwmxiteTACitbnkcugeRkay (ultimate
load) sameRbHkM laMgTaj edaykarBt; (flexural tension crack) bnayy:ageRCAcUleTAkgFwm. dUcKa
edIm,I[EdkkgTTYl)annUversIusg; yield eBj vaRtUvkarTMBk;Edll. enAeBlEdlbnkEdlmanGMeBIelIFwm
xiteTACitbnkcugeRkay (ultimate load) kugRtaMgenAkgEdkkg)aneTAdl;kugRtaMg yield rbs;va enAcMNuc
EdlsameRbH Ggt;RTUgkat;cMEdkkgenaH. tRmUvkarrbs; ACI Code sRmab;TMBk;Edkkg/ Section 12.13
dUcxageRkam
- karBt;nImYyenAkgEpkCab;nEdkkgGkSr U Fmta bBhuEdkkgGkSr U KYrBTCMuvijEdk
beNay (ACI Code, Section 12.13.3) emIlrUbTI *>*a.
- Code GnuBaat[eRbInUvTMBk; standard 90o / 135o b 180o CMuvijEdkbeNysRmab;Edkkg
DB16 . RbsinebIEdkkg DB19 / DB 22 nig DB 25 CamYynwg f yt > 280MPa enaH Code,
Section 12.13.2 tRmUvTMBk; standard bUknwgRbEvgbgb; 0.17 d b f yt / f 'c cenaHBak;kNal
km<s;Fwm nigEpkxageRkAnTMBk;. RbsinebIEdkRtUvBt;edaymMu 90o RbEvgBntminRtUvtUcCag
12d b . sRmab;Edk DB16 bEdkkgTMhMtUcCagenH RbEvgBntKW 6d b (ACI Code, Section
7.1) emIlrUbTI *>*.
- RbsinebIEdkkgGkSr U DubRtUv)aneRbIedIm,IpMCaEdkkgbiTCit RbEvgRCYs (lap length) minRtUv
tUcCag 1.3ld (ACI Code, Section 12.13.5) emIlrUbTI *>*c .
- srsEdkEdlpSar (welded wire fabric) RtUv)aneRbIsRmab;EdkkmaMgkat;TTwgenAkg]sSahkmplitTukmun (precast industry) . TMBk;lMGitRtUv)anpl;[enAkg ACI Code, Section
12.13.2.3 nigenAkgesckIBnl; (commentary) rbs;va.
ACI Code, Section 11.5.2.

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- EdkkgbiTCitRtUv)anpl;[sRmab;FwmEdlrgnUgkmaMgrmYl (ACI Code, Section 7.11).

- FwmEdlenABTCMuvijeRKOgbgMRtUveRbIEdkkgbiTCitedIm,IrkSa structural integrity rbs;Ggt;
(ACI Code, Section 7.13.2.2).

kmaMgkat; nigkmaMgTajGgt;RTUg

195

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

8>6>7> EdkkgenAEdlenAEk,rTRm
kMNt;faEdkkgkmaMgkat;TTwgEdlpl;[enAcenaHpTRm nigmuxkat;
eRKaHfak; (critical section) EdlsitenAcmay d BITRmKYrRtUv)anKNnasRmab;kmaMgkat;TTwg Vu dUcKa
enAnwgmuxkat;eRKaHfak;. vaCakarGnuvtFmtaedaydak;EdkkgTImYyenAcmay s / 2 BIpnTRm Edl s
CaKMlatEdlKNnaedaysmIkar *>!^ sRmab; Vu enAmuxkat;eRKaHfak;.
ACI Code, Section 11.1.3

8>6>8> RbEvgRbsiTPaBrbs;EdkdgErk
manEtbIPaKbYnRtg;cMNuckNalnEpkeRTtnEdkbeNayRtUv)anKitfamanRbsiTPaBsRmab;Edk
kmaMgkat;TTwg. enHmannyfa KMlatGtibrmarbs;EdkdgErkKW 0.75(d d ' ) . BIrUbTI *>( RbEvgRbsiTPaB
rbs;EdkdgErkKW 0.75(d d ' ) / sin 45o = 0.75(1.414)(d d ' ) = 1.06(d d ' ) . KMlatGtibrma s esInwg
cmayedkEdl)anBIkarTMlak;cMeNalEkgnRbEvgRbsiTPaBEdkdgErk. dUcenH
S max = 1.06(d d ' ) cos 45o b S max = 1.06(d d ' )0.707 = 0.75(d d ' )
8>7> karKNnaEdkkgbBar
eKRtUvkarEdkkg (stirrup) enAeBlEdl Vu > 12 Vc . EdkkgGb,brmaRtUv)aneRbIenAeBlEdl
1 V < V < V . kgkrNIenHeKeRbIEdkkg DB10 EdlRtUv)andak;nUvKMlatGtibrma. enAeBlEdl
c
u
c
2
Vu > Vc eKRtUvkardak;EtEdkkgCamYyKMlattUcCagKMlatGtibrma ehIyGacRtUv)anKNnaedayeRbI
smIkar *>!^ S = Av f yt d / Vs .
EdkkgEdlRtUv)aneRbICaTUeTAenAkgmuxkat;ebtugCaEdkkg DB10 nig DB12 GkSr U eCIgBIr
CamYynwg f yt = 400MPa . RbsinebI DB10 RtUv)aneRbIenaH smIkar*>!^kayCa
S Av f yt 157 400 62800
=
=
=
*>@$
d
Vs
Vs
Vs
RbsinebI DB12 RtUv)aneRbIenaH
S Av f yt 226 400 90400
=
=
=
*>@%
d
Vs
Vs
Vs
pleFobKMlatEdkkgelIkm<s;RbsiTPaB d rbs;Fwm GaRsynwg Vs . tmn S / d sRmab;tm
epSgKan Vs enAeBlEdl f yt = 400MPa RtUv)an[enAkgtaragTI2 nigtaragTI3 sRmab;Edk DB10 nig
Edk DB12 erogKa. tmdUcKaRtUv)anbgajCadaRkamdUcenAkgrUbTI 8>10 nigrUbTI 8>11.

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Department of Civil Engineering

taragTI2> pleFob S / d sRmab;tm V f

s

Vs (kN )

125.6 142.7 190.3 237.9

0.5

S /d

0.44

yt

251.2

0.33 0.264

0.25

taragTI3> pleFob S / d sRmab;tm V f

s

Vs (kN )
S /d

= 400MPa S / d = 62800 / Vs

yt

DB10

285.5 330.5 380.6 418.7 475.8

0.22

0.19 0.165

= 400MPa S / d = 90400 / Vs

0.15 0.132

592.5

0.106

DB12

180.8

225

265

310

361.6

445

490.0

535

665

775

850

0.5

0.40

0.34

0.29

0.25

0.20

0.18

0.17

0.14

0.12

0.11

kmaMgkat; nigkmaMgTajGgt;RTUg

197

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

tamtarag nigdaRkamxagelIeyIgGacsnidan)anfa
- RbsinebIeKeRbI DB10 / S = d / 2 enAeBlEdl Vs 125.6kN . enAeBlEdl Vs ekIneLIg
S / d fycuHtamExSekageTArktm 0.132 enAeBl Vs = 475.8kN . RbsinebIKMlatGb,brma
RtUv)ankMNt;Rtwm 75mm enaH d 568mm . enAeBlEdl Vs > 251.2kN enaH S d / 4 .
- RbsinebIeKeRbI DB12 / S = d / 2 enAeBlEdl Vs 180.8kN . enAeBlEdl Vs ekIneLIg
S / d fycuHtamExSekageTArktm 0.14 enAeBl Vs = 665kN . RbsinebIKMlatGb,brma
RtUv)ankMNt;Rtwm 75mm enaH d 535mm . enAeBlEdl Vs > 361.6kN enaH S d / 4 .
- RbsinebIeKeRbIEdkkgGkSr U f yt = 280MPa enaHeKRtUvKuN S / d edaytm 7 /10 bCaTUeTA
f yt / 400 .
8>8> segbviFIsaRsKNnaEdkkgbBar
CMhankgkarKNnaEdkkgbBarsRmab;kmaMgkat;TTwg edayeyagtam ACI Code GacRtUv)an
segbdUcxageRkam
f. kMNt;kmaMgkat;KNna V BIbnkEdlGnuvtmkelIeRKOgbgM. kmaMgkat;KNnaRKITicEdl
RtUvykmksikSasitenARbEvg d BImuxnTRm.
Vd
g. kMNt; V = 0.17 f ' b d b V = (0.16 f ' + 17.2
)b d 0.29 f ' b d
M
u

c w

c w

bnab;mkKNna 12 V

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h.

Department of Civil Engineering

k> RbsinebI V < 12 V muxkat;minRtUvkarEdkkg

x> RbsinebI 12 V < V V eRbImuxkat;EdkkgGb,brma
K> RbsinebI V > V muxkat;Edkkg RtUvKNnadUcxageRkam
kMNt;kmaMgkat;EdlTb;eday Edkkg
u

i.

k.

Vu Vc

Vs =
j.

kMNt; V = 0.33 f ' b d nig V = 0.67 f ' b d = 2V . RbsinebI V > V

dMeLIgmuxkat;.
kMNt;KMlatrbs;Edk s = A Vf d
kMNt;KMlatEdkGtibrmaEdlGnuBaateday ACI Code. KMlatEdkGtibrmaCatmtUcbMput
n s nig s
k> s = d2 60cm RbsinebI V V = 0.33 f ' b d
d
s = 30cm RbsinebI V < V V
4
A f
x> s = 3 Ab f 16
b f'
c1

c w

c2

c w

c1

c2

yt

l.

c1

yt

c1

c w

c2

yt

k> RbsinebI s < s eRbI s

x> RbsinebI s > s eRbI s
n. ACI Code min)ankMNt;nUvKMlatGb,brmaeT. eRkamlkxNFmta KMlatGb,brma S
RtUv)ansnt;ykesInwg 75mm sRmab; d 50cm nigmanKMlatGb,brmaesInwg 100mm
sRmab;FwmeRCA (deep beam) . RbsinebI S mantmtUcenaH eKGactMeLIgmuxkat;Edkkg
beRbIEdkkgeCIgeRcIn rUbTI 8>8.
o. sRmab;muxkat;mUl RkLapEdleRbIsRmab;KNna
Vc = plKuNGgt;pitCamYykm<s;RbsiTPaB d / Edl d = 0.8 nGgt;pit/ ACI Code,
Section11.3.3 .
]TahrN_ 1 FwmTRmsamBamanmuxkat;ctuekaN b = 30cm / d = 55cm nig h = 60cm ehIyRtUv)anBRgwg
eday 4DB25 . epgpat;faetImuxkat;enHRKb;RKan; bGt;sRmab;kmaMgkat;TTwgemKuN (ultimate shear
force) xageRkam. RbsinebIvaminRKb;RKan; cUrKNnaEdkkmaMgkat;TTwgkgTRmg;CaEdkkgGkSr U . eday
eRbI f 'c = 28MPa nig f yt = 400MPa .
m.

kmaMgkat; nigkmaMgTajGgt;RTUg

max

max

max

199

T.Chhay

mhaviTalysMNg;sIuvil

k> Vu = 50kN

NPIC

x> Vu = 110kN

dMeNaHRsay

K> Vu = 240kN

X> Vu = 345kN

g> Vu = 570kN

CaTUeTA bw = b = 300mm / d = 550mm nig

Vc = (0.17 f 'c )bd = 0.75(0.17 28 )300 550 10 3 = 111.3kN
1 V
c
2

= 55.65kN

(
Vc 2 = (0.67

) (
)
f 'c )bd = 576kN

Vc1 = 0.33 f 'c bd = 0.33 28 300 550 10 3 = 288kN

k>
x>

/ muxkat;RKb;RKan; edayminRtUvkarEdkkmaMgkat;TTwg.
Vu = 110kN > 12 Vc / b:uEnvatUcCag Vc = 111.3kN . eday Vs = 0 dUcenH muxkat;RtUvkarEdk
kmaMgkat;TTwgGb,brma. eRbI DB10 CaEdkkgGkSr U enAKMlatGtibrma.
Vu = 50kN < 12 Vc = 55.565kN

Av = 2 10 2

= 157mm 2

KMlatGtibrmaCatmtUcCageKkgcMeNam
S 2 = d / 2 = 275mm yk 250mm lub
S 3 = Av f yt / 0.35bw = 157 400 /(0.35 300) = 598mm

yk 550mm beRbItaragTI1

S 4 = 600mm

dUcenHeRbIEdkkg DB10 @ 250mm

K> Vu = 240kN > Vc / RtUvkarEdkkmaMgkat;TTwg. karKNnaGaceFVIeLIgCaCMhanxageRkam
KNna Vs = (Vu Vc ) / = (240 111.3)/ 0.75 = 171.6kN
edaysar Vs < Vc1 enaH S max = d / 2 600mm
eRCIserIs DB10 CaEdkkgGkSr U nigKNnaKMlatRtUvkaredayQrelI Vs
Av f yt d 157 400 550
S1 =
=
= 201mm yk 200mm
Vs
171600
KNnaKMlatGtibrma S 2 = 250mm / S3 = 550mm nig S 4 = 600mm dUcenH S max = 250mm
edaysar S = 200mm < S max = 250mm
dUcenHeRbIEdkkg DB10 @ 200mm
X> Vu = 345kN > Vc /RtUvkarEdkkmaMgkat;TTwg.
KNna Vs = (Vu Vc ) / = (345 111.3) / 0.75 = 311.6kN
edaysar Vs > Vc1 enaH S max = d / 4 300mm yk 125mm
edaysar Vc1 < Vs < Vc2 enaHeKGaceRbIEdkkg edaymincaM)ac;dMeLIgmuxkat;ebtug.
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Department of Civil Engineering

eRCIserIs DB10 CaEdkkgGkSr U nigKNnaKMlatRtUvkaredayQrelI Vs

Av f yt d 157 400 550
S1 =
=
= 110mm yk 100mm
Vs
311600
KNnaKMlatGtibrma S 2 = d / 4 = 137.5mm yk 125mm / S3 = 550mm nig S 4 = 300mm
dUcenH S max = 125mm
edaysar S = 100mm < S max = 125mm
dUcenHeRbIEdkkg DB10 @100mm
g> Vu = 570kN > Vc / RtUvkarEdkkmaMgkat;TTwg.
KNna Vs = (Vu Vc ) / = (570 111.3)/ 0.75 = 611.6kN
edaysar Vs > Vc2 enaHmuxkat;minRKb;RKan;. eKRtUvkardMeLIgTMhMrbs;muxkat;mYy bkTaMgBIr.
cMNaM taragTI 2 nigrUbTI 8>10 kGacRtUv)aneRbIedIm,IKNnaKMlat S sRmab; K> nig X> )anpgEdr.
1> sRmab; K> Vs = 171.6kN BIrUbTI 8>10 btaragTI 2> sRmab;EdkkgGkSr U DB10 eyIg
TTYl)an S / d = 0.37 dUcenH S1 = 203.5mm EdltUcCag d / 2 = 250mm . cgcaMfa S max
EdlQrelI Vs KW d / 2 minEmn d / 4 eT. dUcKaBItaragTI 1> eyIgTTYl)an
S 3 = Av f yt / 0.35bw = 550mm .
2> sRmab; X> Vs = 311.6kN / S / d = 0.18 enaH S1 = 100mm / Vs = 311.6kN > 251.2kN enaH
S max = d / 4 RtUv)aneRbI.
]TahrN_2 FwmTRmsamBaEdlmanRbEvg 5.2m nigmanRbEvgcenaHssr (clear span) 4.9m edayRTnUv
bnkBRgayesIefr 65kN / m nigbnkBRgayesIGefr 55kN / m . TMhMrbs;Fwm nigsrsEdkRtUv)anbgaj
enAkgrUbTI 8>12. epgpat;muxkat;sRmab;kmaMgkat;TTwg nigKNnaEdkkmaMgkat;TTwgcaM)ac;. eK[
f 'c = 20 MPa nig f y = 400MPa .

dMeNaHRsay

eK[ bw = 350mm / d = 580mm

1> KNnakmaMgkat;TTwgemKuN (ultimate shear) BIbnkxageRkA
bnkBRgayesIemKuN = 1.2 65 + 1.6 55 = 166kN / m
166 4.9
Vu enABImuxpTRm =
= 406.7 kN
2
KNna Vu enAcmayBImuxpnTRm = 406.7 0.58 166 = 310.42kN
kmaMgkat; nigkmaMgTajGgt;RTUg

201

T.Chhay

mhaviTalysMNg;sIuvil

2> KNna Vc :

NPIC

Vc = (0.17 f 'c )bw d = 0.75 0.17 20 350 580 10 3 = 115.75kN

1 V
c
2

= 57.87 kN

KNna Vc1 = 0.33 f 'c bwd = 0.33 20 350 580 10 3 = 299.6kN

KNna Vc2 = 2Vc1 = 599.2kN
3> eday Vu > Vc dUcenHmuxkat;RtUvkarEdkkmaMgkat;TTwg. cmay x' EdlenARtg;cmayenHmuxkat;
ebtugminRtUvkarEdkkmaMgkat;TTwg enA 12 Vc KW
406.7 57.87 4.9
x' =
= 2.10m

406.7

4> KNna Vs = (Vu Vc )/ = (310.42 115.75)/ 0.75 = 259.56kN . edaysar Vs < Vc1 enaH
S max = d / 2 600mm RtUv)anBicarNa beyagtamrUbTI 8>10 btaragTI2 Vs > 251.2kN .
5> KNnaEdkkg eRCIserIsEdkkgGkSr U DB10 / Av = 157mm 2 . KNna S1 edayQrelI
Vs = 259.56kN / S1 = Av f yt d / Vs = 140mm yk 125mm byk S / d = 0.24 BItaragTI2
bBIrUbTI 8>10
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Department of Civil Engineering

6> KNnaKMlatGtibrma S 2 = d / 2 = 580 / 2 = 290mm yk 250mm /

S 3 = Av f yt / 0.35bw = 500mm beRbItaragTI1/ S 4 = 600mm . dUcenH S max = 250mm .
7> edaysar S1 = 125mm < S max = 250mm eRbI DB10 @125mm
8> KNna Vs sRmab;KMlatGtibrma 250mm
Vs =

As f yt d
s

157 400 580 3

10 = 145.7 kN
250

Vs = 109.3kN
Vc + Vs = 115.75 + 109.3 = 225kN

cmay x1 EdlenARtg;cmayenHmuxkat;GaceRbIKMlat s = 250mm

406.7 225 4.9
x1 =
= 1.09m

406.7 2

edaysar x1 mantmtUc eRbI s = 125mm sRmab;cmayFMCag besI 1.09m . cMNaMfa

RbsinebI x1 Evg KMlatenAcenaH 150mm eTA 250mm GacRtUv)anbEnm.
9> EdkkgRtUv)anBRgaydUcxageRkam
dak;EdkkgTI1enAcmay S / 2 BImuxpnTRm
EdkkgTImYyenA S / 2 = 125 / 2 = 62.5mm yk 50mm
R)aMbYnEdkkgmanKMlat S = 125mm = 1125mm
srub 1175mm > 1090mm
bYnEdkkgmanKMlat S = 250mm = 1000mm
srub 2175mm < 2450mm
cMnYnEdkkgsrubsRmab;FwmKW 2(1 + 9 + 4) = 28 . karBRgayEdkkgRtUv)anbgajenAkgrUbTI 8>13
kmaMggkat;TTwgEdl)anKNnaRtUv)anbgajenAkgrUbTI 8>12.
10> dak;Edkkg DB12 cMnYnBIredIm enABIelImuxkat;FwmedIm,IedIrtYrCaEdkkgBr.

kmaMgkat; nigkmaMgTajGgt;RTUg

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8>9> kmaMgkat;TTwgEdlbNalBIbnkGefr
enAkg]TahrN_TI2 TaMgbnkefr nigbnkGefrRtUv)ansnt;faBRgayesIeBjtambeNayFwm Edl
begIt)ankmaMgkat;TTwgsUnenAkNalElVg. CaFmta bnkefrBitCaBRgayeBjelIbeNayFwm Etbnk
GefrGacGnuvteBj bkGnuvtEtEpkxHrbs;Fwm EdltRmUv[mankmaMgkat;TTwgGtibrmaekIteLIgenAkNal
ElVg bmuxkat;kMNt;NamYy. rUbTI 8>14 a bgajBIFwmTRmsamBaCamYynwgbnkBRgayesIGnuvteBj
beNayFwm. kmaMgkat;TTwgERbRbYlCaragbnat;tambeNayFwm CamYynwgkmaMgkat;TTwgGtibrmaenATRm
A.
kgkrNIEdlbnkGefr W2 = 1.6WL kmaMgkat;TTwgGtibrmamanGMeBIenARtg;TRm A enAeBlEdl
W2 GnuvteBjElVgFwm rUbTI 8>14 a . kmaMgkat;TTwgGtibrmaekItmanenAkNakElVgRbsinebIbnkGefr
RtUv)andak;EtBak;kNalFwm BC rUbTI 8>14 b EdlbegIt)an Vu enAkNalElVgesInwg W2 L / 8 . dUc
enH kmaMgkat;TTwgKNnaRtUv)anbegIteLIgedaykarbEnmkmaMgkat;TTwgGtibrmaEdlbNalmkBIbnk
Gefr EdlRtUv)andak;enAelIRbEvgepSgnElVg eTAelIkmaMgkat;TTwgGefr rUbTI 8>14 c . vaCakar
GnuvtFmtaedayKitkmaMgkat;TTwgGtibrmaRtg;TRm A esInwg Wu L / 2 = (1.2WD + 1.6WL ) L / 2 / b:uEn Vu
enAkNalElVgesI W2 L / 8 = (1.6WL ) L / 8 CamYybnat;Rtg;ERbRbYltambeNay AC nig CB dUcbgajenA
kg rUbTI 8>14 d. karKNnasRmab;kmaMgkat;TTwgenAkkrNIenHnwgGnuvtdUcKanwgkarBnl;kg]TahrN_2.
RbsinebI karerobrab;xagelIenHGnuvteTAFwmkg]TahrN_2 enaH Vu enATRm A = 406.7kN nig Vu enA
kNalElVg = (1.6 55)4.9 / 8 = 53.9kN .

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]TahrN_3 Fwm cantilever RbEvg 3m manmuxkat;ctuekaNEkg nigRTnUvbnkemKuNBRgayesI nigcMcMNuc

bnkpal;xnRtuv)anrab;bBalrYc dUcbgajenAkgrUb 8>15. edayeRbI f 'c = 28MPa nig

KNnaEdkkmaMgkat;TTwgcaM)ac;sRmab;dak;kgFwmTaMgmUledayeyagtam ACI Code.

kmaMgkat; nigkmaMgTajGgt;RTUg

205

f y = 400MPa

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dMeNaHRsay

1> KNnakmaMgkat;TTwgtambeNayFwmEdlbNalmkBIbnkxageRkA
Vu enATRm = 80 3 + 89 + 36 = 365kN
510
Vud enAcmay d = 365 80
= 351.4kN
3000
Vu enAcmay 1.2m xageqVg = 365 80 1.2 = 269kN
Vu enAcmay 1.2m xagsaM = 269 89 = 180kN
Vu enAcugTMenr = 36kN
daRkamkmaMgkat;TTwgRtUv)anbgajenAkgrUbTI 8>15.
2> KNna Vc
Vc = (0.17 f 'c bd ) = 0.75(0.17 28 )300 510 10 3 = 103.2kN
1 V
c
2

= 51.6kN

edaysar Vud > Vc muxkat;ebtugRtUvkarEdkkmaMgkat;TTwg. KNna

Vc1 = 0.33 f 'c bd = (0.33 28 )300 510 10 3 = 267.2kN
Vc 2 = 2Vc1 = 534.4kN
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cmay x EdlenARtg;enHmuxkat;minRtUvkarEdkkmaMgkat;TTwg enA 12 Vc = 51.6kN EdlRtVv)an

vas;BITRm A .
180 51.6
x = 1.2 +
1.8 = 2.8m
180 36

200mm BIcugTMenr. dUcKa x1 sRmab; Vc KW 2.16m BI A 840mm BIcugTMenr

3> Epk AC kmaMgkat;TTwgKNna Vu = Vud = 351.4kN . KNna Vs = (Vu Vc ) /
= (351.4 103.2) / 0.75 = 330.9kN . edaysarEt Vc1 < Vs < Vc 2 enaH S max d / 4 RtUv)an
BicarNa bepgpat;edayeRbIrUbTI 8>10.
4> KNnaEdkkg eRCIserIsEdkkgGkSr U DB10 / Av = 157mm 2 . KNna S1 QrelI Vs
S1 =

Av f yt d
Vs

157 400 510

= 100mm
330900

eRbI 100mm bTTYl s / d = 0.19 BIrUb 8>10.

5> KNnaKMlatGtibrma S 2 = d / 4 = 510 / 4 = 127.5mm yk 125mm
Av f yt
157 400
S3 =
= 550mm BItaragTI1 sRmab; b = 300mm
=
0.35b
0.35 300
w

S 4 = 300mm

dUcenH S max = 125mm

6> eday S = 100mm < S max = 125mm dUcenHeRbIEdkkg DB10 @100mm
7> enAcMNuc C / kmaMgkat;TTwgKNna Vu = 269kN > Vc enaH Vs = (269 103.2)/ 0.75 = 221kN .
S1 = Av f yt d / Vs = 145mm
Vs = 221kN < Vc1 = 267.2kN

S 2 = d / 2 = 255mm

b 250mm

enaH S1 = 145mm b 125mm

8> edaysarKMlat 125mm nig 100mm mantmEk,rKa eRbIEdkkg DB10 @100mm sRmab;Epk
AC .
9> Epk BC
S1 = 145mm < S 2

A. Vu = 180kN > Vc
Vs = (180 103.2) / 0.75 = 102.4kN < Vc1 = 267.2kN
B. S1 = Av f yt d / Vs = 157 400 510 / 102400 = 313mm

btUcCag S3 = 550mm nig S 4 = 600mm .

yk S max = 250mm . eRbIEdkkg DB10 @ 250mm sRmab;Epk BC .

C. S 2 = d / 2 = 510 / 2 = 255mm

kmaMgkat; nigkmaMgTajGgt;RTUg

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10> karBRgayEdkkgedayvas;ecjBITRm A dak;EdkkgTImYyenA S2 = 50mm

12 100 = 1200mm
7 250 = 1750mm

srub 3000mm
karBRgayEdkkgRtUv)anbgajenAkgrUbTI 8>16. EdkkgsrubmancMnYn 20 .

8>10> kugRtaMgkmaMgkat;TTwgenAkgGgt;Edlmankm<s;ERbRbYl
edaysarEtkugRtaMgkmaMgkat;TTwg v CaGnuKmn_nkm<s;RbsiTPaB d dUcenHkugRtaMgkat;TTwgERbRbYl
tambeNayFwmebtugBRgwgedayEdkCamYynwgkm<s;ERbRbYl. enAkgFwmEbbenH rUbTI 8>17 eKBicarNa
elIFatuGnntUc dx . kmaMgsgt; C enAelImuxkat;NamYyesInwgm:Um:g;Eck[dXas; b C = M / y . edrIevTI
mYyn C KW
dC =

ydM Mdy
y2

RbsinebI C1 FMCag C2 enaH C1 C2 = dC = vbdx

ydM Mdy

dM M
2 dy
y
y
y
1 dM M dy
v=

yb dx by 2 dx
vbdx =

edaysar y = jd / dM / dx esInwgkmaMgkat;TTwg V nig d ( jd ) / dx CaCRmal (slope)/

V
M d
V
M

( jd ) nig v =
v=

tan

2
2
bjd
bjd
dx

b( jd )

T.Chhay

b( jd )

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Edl V nig M CakmaMgkat;TTwg nigm:Um:g;xageRkA ehIy CamMuCRmalnpmYyrbs;FwmeTAnwgpmYy

eTotrbs;Fwm. sBaabUkRtUv)aneRbIenAeBlEdlkm<s;fycuHm:Um:g;ekIneLIg b:uEnsBaadkRtUv)aneRbIenAeBl
km<s;ekIneLIgehIym:Um:g;kekIneLIg. rUbmnenHRtUv)aneRbIenAeBlEdlmMuCRmaltUc EdlmMu 30o .

TRmg;samBansmIkar *>@^GacRtUvbegIteLIgedaysMrYl j enaHeyIgTTYl)an

v=

*>@&

V
M
2 tan
bd bd

sRmab;viFIKNnaersIusg; smIkarxageRkamGacRtUv)aneRbI
vu =

*>@*

Vu
M

tan
bd bd 2

sRmab;kmaMgkat;TTwg
*>@(
rUbTI 8>18 bgajBIFwm cantilever CamYynwgbnkcMcMNucenAcugTMenr. m:Um:g;nigkm<s; d ekIneLIgkg
TisedAeTArkTRm. kgkrNIenH sBaadkRtUv)aneRbIenAsmIkar *>@& *>@* nig*>@(. dUcKa sBaadk
RtUv)aneRbIsRmab;muxkat; t enAkgFwmTRmsamBadUcbgaj ehIysBaabUkRtUv)aneRbIsRmab;muxkat; Z
Edlm:Um:g;ekIneLIgenAeBlEdlkm<s;fycuH.
enAkgkrNICaeRcIn karERbRbYlkm<s;rbs;FwmekItmanenAelIEpkrbs;FwmEdlenAEk,rTRm rUbTI
8>18.
Vn = Vu

Mu
tan
d

kmaMgkat; nigkmaMgTajGgt;RTUg

209

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karBesaFeTAelIFwmCamYykm<s;ERbRbYlbgajfa FwmEdlmankm<s;FMenATRmCaTUeTA)ak;edaysar
kmaMgkat;TTwgsgt;. FwmEdlmankm<s;tUcenATRmCaTUeTA)ak;edaysarPaBKanesrPaB EdlbNalmkBI
karraldalnsameRbHemenAkgFwmeLIgelI ehIybnab;mksameRbHenaHraldaltamTisedkenAelImuxkat;
FwmEpkxagelI. karBiesaFk)anbgajEdrfa sRmab;FwmEdlmankm<s;ERbRbYl rUbTI 8>18 CamYynwgmMu
eRTt RbEhl 10o nigrgnUvkmaMgkat;TTwg nigkmaMgBt; ersIusg;kmaMgkat;TTwgrbs;ebtug VCV GacRtUv
)anKNnaeday
*>#0
VCV = Vc (1 + tan )
Edl VCV = ersIusg;kmaMgkat;TTwgrbs;FwmCamYynwgkm<s;ERbRbYl

V d
Vc = 0.16 f 'c + 17.2 w u bw d (0.29 f 'c )bw d
ACI Code Eq.11.6
M

T.Chhay

mMuEdlbegIteLIgedayTisrbs;Edk. vaRtUv)anKitfaviCmansRmab;FwmEdlmankm<s;tUc
enATRm nigGviCmansRmab;FwmEdlmankm<s;FMenATRm rUbTI 8>18
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km<s;RbsiTPaBrbs;FwmenATRm
ACI Code Eq.11.3 CasmIkarRtUv)ansRmYl nigGaceRbIedIm,IKNna Vc
*>#!
Vc = (0.17 f 'c )bw d
]TahrN_4 KNnaFwm cantilever dUcbgajenAkgrUbTI 8>19 eRkamGMeBIbnkemKuN. FwmenHkm<s;srub
enAcugTMenr 300mm ehIyekIneLIgeTArkTRm. edayeRbIPaKryEdk = 1.5% / f 'c = 28MPa /
f y = 400MPa nig b = 250mm .
ds =

dMeNaHRsay

1> M u TRm = 36.5 2.52 / 2 + 62 2.5 = 269kN.m

2> sRmab; = 1.5% / Ru = 4.72MPa
d=

M
=
Ru b

269 10 6
= 477.5mm
4.72 250

As = 0.015 250 477.5 = 1790mm 2

eRbIEdk 3DB28 yk d = 490mm / h = 550mm .

3> KNnasRmab;kmaMgkat;TTwg kmaMgkat;TTwgGtibrmaenATRmKW 62 + 36.5 2.5 = 153.25kN .

edaysarmuxkat;FwmERbRbYl m:Um:g;RtUv)anBicarNakgkarKNnakmaMgkat;TTwg. edaysarkm<s;
FwmekIneLIgCamYym:Um:g;ekIneLIg sBaadkRtUv)aneRbIenAkgsmIkar *>@*
kmaMgkat; nigkmaMgTajGgt;RTUg

211

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mhaviTalysMNg;sIuvil
vu =

NPIC

Vu
Mu

(tan )
bd bd 2

edIm,Irk tan / yk d enAcugTMenresI 490mm nig d enAKl;TRmesI 240mm

tan =

490 240
= 0.1
2500
153250
269 10 6
=

0.1 = 1.07 MPa

0.75 250 490 0.75 250 490 2

TRm
4> kugRtaMgkmaMgTTwgenAcugTMenrKW Vu / bd M u = 0
vu

vu =

62000
= 1.38MPa
0.75 250 240

5> enAcmay d = 490mm BImuxpnTRm km<s;RbsiTPaBKW 441mm BIrUbFrNImaRt

Vu = 153.25 36.5 0.49 = 135.4kN
2

Mu

enAcmay 490mm BITRm= 62 2.01 + 36.5 2.012

vu =

= 198.4kN.m

135.4 103
198.4 10 6 0.1

= 1.09MPa
0.75 250 441 0.75 250 4412

6> enAkNalElVg 1.25m BITRm

d = 365mm

Vu = 153.25 36.5 1.25 = 107.6kN

1.25 2
= 106kN .m
2
107.6 10 3
106 10 6 0.1
vu =

= 1.15MPa
0.75 250 365 0.75 250 365 2
M u = 62 1.25 + 36.5

dUcKa enAcmay 1.9m BITRm 0.6m BIcugTMenr

d = 300mm

Vu = 83.9kN

M u = 43.8kN

vu = 1.23kN

enAcmay 2.2m BITRm 0.3m BIcugTMenr

d = 270mm

Vu = 73kN

M u = 20.2kN

vu = 1.29kN

tmTaMgGs;enHRtUv)anbgajenAkgrUbTI 8>20
7> kugRtaMgkmaMgkat;TTwgedayebtugKW
0.17 28 = 0.9 MPa

kugRtaMgkmaMgkat;TTwgGb,brmaEdlRtUvTb;edayEdkkmaMgkat;TTwg
vus = 1.38 0.9 = 0.48MPa

vu nig vus RtUv)anekIneLIgedaypleFob 1/ kgsmIkar 8>28

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8> eRCIserIsEdkkg DB10 EdlmanEdkBIr

Av = 2 78.5 = 157mm 2
Av f yt 157 400
S
=
=
= 523mm
v s bw 0.48 250
d
S max
= 245mm
120mm
2
Av f yt
157 400
=
S max
Av =
= 718mm
0.35bw 0.35 250

caM)ac;

sRmab;

eTA

enAcugTRm

sRmab;Gb,brma
9> epgpat;KMlatGtibrma (d / 2) : vus 0.33

f 'c

0.33 f 'c = 0.33 28 = 1.74MPa > 0.48MPa

10> karBRgayEdkkg cmayBIcugTMenr

EdkkgcMnYnmYymancmay 50mm = 50mm
EdkkgcMnYndb;mancmay120mm = 1200mm
EdkkgcMnYnbImancmay175mm = 525mm
EdkkgcMnYnbI;mancmay 200mm = 600mm
srub = 2375mm
dUcenHenAsl;cmay 125mm BIpnmuxTRm.

kmaMgkat; nigkmaMgTajGgt;RTUg

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8>11> Ggt;rgkarBt;CeRmAeRCA
Ggt;rgkarBt;KYrRtUv)anKNnaCaFwmeRCARbsinebIpleFobn clear span ln Edlvas;BIpmuxTl;Ka
rbs;TRm rUbTI 8>21 elIkm<s;srub h mantmtUcCag 4 (ACI Code, Section 11.8) . Ggt;KYrrgnUvbnk
enAelIpEdlQmnwgpnTRm EdlGaceFVI[ strut rgkarsgt;GacbegIteLIgenAcenaHbnk nigTRm rUbTI
8>22. RbsinebIbnkGnuvtenA)at bpxagrbs;FwmeRCA smIkarKNnakmaMgkat;TTwgsRmab;FwmFmta
Edl)an[BIxagmuxKYrRtUv)aneRbI. Ca]TahrN_ FwmeRCAKWCaFwmElVgxIEdlRTbnkFn;/ CBaaMgbBareRkam
bnkTMnajEpndI (gravity load), shear wall, nigkRmalxNrgnUvbnkedk.
niymnyrbs;Ggt;rgkarBt;eRCAkRtUv)anbgajenAkg ACI Code, Section 10.7.1. vabgajfa
Ggt;rgkarBt;EdlmanpleFob ln / h < 4 nigtMbn;rgbnkcMcMNucsitenAcmayBIrdgnkm<s;rbs;Ggt;BI
pnTRmRtUv)ancat;TukCaGgt;rgkarBt;eRCA. FwmEbbenHKYrRtUv)anKNnaedayKitnUvkarBRgay nonlinear
T.Chhay

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nkugRtaMg nigkarPat;xag (lateral buckling) rUbTI 8>22 a.

rUbTI 8>22 a bgajBIkarBRgaykugRtaMgeGLasic enARtg;muxkat;kNalElVgnFwmeRCA nigrUbTI

8>22 b bgajBIExSekagkugRtaMgemenAkgFwmeRCAEdlrgbnkenApxagelI (top-load deep beam). ExS
Cab;bgajBI karBRgaykugRtaMgTaj ExSdac;bgajBIkarBRgaykugRtaMgsgt;. eRkambnkFn; sameRbH
bBareRTtekItmanenAkgebtugkgTisedAEkgnwgkugRtaMgTajem ehIyesIrEtRsbeTAnwgExSKngdac; rUbTI
8>22 c . dUcenH eKRtUvkarTaMgEdkedk nigEdkbBaredIm,ITb;nwgkugRtaMgem. elIsBIenH EdkrgkarBt;
edaykarTaj (tensile flexural reinforcement) RtUv)andak;enARbEhlmYyPaKR)aMenA)atrbs;FwmtamKng
kugRtaMgTaj rUbTI 8>22b . CaTUeTA karviPaKFwmeRCAmanlkNsKsaj nigGacGnuvtedayeRbIKMrU truss
bedIm,ITTYl)anlTplkan;EtsuRkiteKeRbIviFI finite element bviFIRsedogKa. edIm,IgayRsYlkgkarKNna
kmaMgkat;TTwgnFwmeRCA eKGacGnuvttamCMhanEdl)anerobrab;xageRkam
1> muxkat;eRKaHfak; (critical section) RbsinebImuxkat;eRKaHfak;sRmab;KNnakmaMgkat;TTwg
enAkgFwmeRCAEdlRTbnkbBarGnuvtenApxagelIrbs;Fwm sitenAcmay X BIpnTRm enaHcM
gay X GacRtUv)ankMNt;dUcxageRkam rUbTI 8>23
a. sRmab;FwmeRCAEdlRTbnkBRgayesI X = 0.15l n / Edl l n = clear span .
b. sRmab;bnkcMcMNuc X 1 = 0.5a1 TRmxageqVg b X 2 = 0.5a 2 TRmxagsaM rUbTI
8>23/ Edl a1 nig a2 esInwg shear span Ek,rTRmnImYy. Shear span
CacmayBIbnkcMcMNuceTApnTRm.
enAkgRKb;krNITaMgGs; cmay X / X 1 nig X 2 dac;xatminRtUvFMCagkm<s;RbsiTPaB d .
kmaMgkat; nigkmaMgTajGgt;RTUg

215

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2> ersIusg;kmaMgkat;TTwgGtibrma Vn ersIusg;kmaMgkat;TTwgGtibrma Vn sRmab;Ggt;rgkar

Bt;eRCAminKYrmantmFMCagtmxageRkam = 0.75
*>#@ a
sRmab; ldn < 2 / Vn = 0.67 f 'c bwd
sRmab; 2 ldn 5 / Vn = 0.05510 + ldn f 'c bwd
*>#@ a

byk Vn = 0.83 f 'c bwd

*>##
krNITaMgBIr manEcgenAkg ACI Code, Section 11.8.3. RbsinebI Vu > Vn enaHeKRtUvdMeLIg
muxkat;ebtug.
3> a. ersIusg;kmaMgkat;TTwgrbs;ebtug Vc ersIusg;kmaMgkat;Fmta (nominal shear strength)
Vc rbs;ebtugGacRtUv)anKNnadUcxageRkam
*>#$
Vc = 0.17 f 'c bw d
Vc enHRsedogKanwgersIusg;kmaMgkat;TTwgebtugsRmab;FwmFmta dUcenAkgEpkxagmuxnem
eronenH.
b. m:agvijeTot eKGaceRbIsmIkarmYyepSgeTotEdlmanTak;Tgnwgm:Um:g;emKuN nigkmaMgkat;
TTwgemKuNenAmuxkat;eRKaHfak;

2.5M u
V d
0.16 f 'c + 17.2 w u bw d
Vc = 3.5
*>#%
V d
M

kmaMgkat; nigkmaMgTajGgt;RTUg

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NPIC

b:uEn Vc minKYrelIsBI 0.5 f 'c bwd

tm (3.5 2.5M u / Vu d ) minKYrFMCag 2.5 nigminKYrtUcCag 1. tmn M u nig Vu
RtUv)anykenARtg;muxkat;KNnaeRKaHfak;. ersIusg;kmaMgkat;FMnsmIkar *>#$ RtUv)aneRbICamYyKMnitfa
sameRbHEdltUcesIrEtemIlmineXIjGacekItmanenAkgFwmeRCA nigGacGueRKaH)an. sameRbHGaccab;epIm
ekItmanenARbEhlmYyPaKbInbnkemKuN.
4> EdkkmaMgkat;TTwg enAeBlkmaMgkat;TTwgemKuN Vu > Vc eKRtUvdak;EdkkmaMgkat;TTwg
Edlcat;Tukfa Vu = (Vc + Vs ) b Vs = (Vu Vc ) / . CMhannkarKNnamandUcxageRkam
a. kMNt; Vs kmaMgTb;edayEdkkmaMgkat;TTwg Vs RtUv)ankMNt;BIsmIkarxageRkam
A 1 + l n / d Avh 11 l n / d
*>#^
Vs = v
f y d

+
S
S
12
12

Edl Av = RkLapsrubnEdkkmaMgkat;TTwgbBarEdlmanKMlat S v ehIyEkgeTAnwgEdk

emrgkarTajedaykarBt;npxagTaMgBIrrbs;Fwm
Avh = RkLapsrubnEdkkmaMgkat;TTwgedkEdlmanKMlat S h RsbnwgEdkemrgkar
TajedaykarBt;npxagTaMgBIrrbs;Fwm
b. KMlatEdkkmaMgkat;TTwgKW
KMlatQrGtibrma S v d5 300mm
KMlatedkGtibrma S h d5 300mm
c. EdkkmaMgkat;TTwgGb,brma RkLapEdkkmaMgkat;TTwgbBarKW Av = 0.0025bw S v .
RkLapEdkkmaMgkat;TTwgedkKW Avh = 0.0015bw S h .
d. EdkkmaMgkat;TTwgRtUvkarenARtg;muxkat;eRKaHfak;KYrRtUv)anlatsnwgeBjRbEvg nigkm<s;
rbs;FwmeRCA.
e. sRmab;FwmeRCACab; EdkkmaMgkat;TTwgdUcKaKYrRtUv)aneRbIenARKb;ElVg RbsinebIElVgTaMgenaH
manRbEvgesIKaCamYybnkRsedogKa.
5> EdkrgkarBt;nFwmeRCA dMeNIrRbRBwteTAnkarBt;rbs;FwmeRCAKWmanlkNsKsaj nigTam
TarkarviPaKkugRtaMg nigbMErbMrYlrageFobtamlkN nonlinear tamkm<s;rbs;Fwm. sRmab;kar
KNnadMbUg viFIdsRmYlxageRkamGacRtUv)aneRbI
M n = As f y ( y )

Edl y = dXas; = (d a / 2) . edaysartm (d a / 2) mankarBI)akkgkarKNna/ d

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Xas; y GacRtUv)anKNnaRbEhlykesInwg 0.6h sRmab; ln / h = 1 nigmantm

esInwg 0.8h sRmab; ln / h = 2 . viFanRtzan (Linear interpolation) GacRtUv)an
eRbIedIm,IKNna y enAeBl ln / h ERbRbYlcenaH 1.0 nig 2.0 . dUcenH
M
*>#&
As = u
yf
y

tmn As minGactUcCagEdkrgkarBt;Gb,brmaEdlRtUvkarsRmab;FwmFmtaEdlnwg[enAeBl
bnab; edaysnt; d = 0.9h .
0.25 f 'c
1.38
As Gb,brma =
bw d
bw d
*>#*
f
f
y

GgTIBIrlub enAeBlEdl f 'c < 30MPa . cMNaMfa f y nig f 'c KitCa MPa .
EdkrgkarTajedaykarBt; (flexural tension reinforcement) KYrdak;enA h / 4 eTA h / 5 n
Fwm nigKYrmanKMlatRKb;RKan;tambeNay)attMbn;Taj. EdkrgkarTajKYrEtf<k;eTAkgTRm
[)anl.
sRmab;karviPaK nigkarKNnaEdlmanlkNsuRkwt nigsRmab;FwmeRCACab; viFI nonlinear
Edl manlkNht;ct;GacRtUv)aneRbIedIm,IbrimaNdRtwmRtUv nigkarBRgaynEdkrgkarTaj.
]TahrN_5 FwmeRCATRmsamBamanElVgRbEvg 4.2m man clear span RbEvg ln = 3.6m km<s;srub
h = 2.5m nigTTwg b = 0.4mm . FwmeRCARTedaybnkeFVIkarefBRgayesI 600kN / m rYmbBalbnkpal;
xn nigbnkGefr 320kN / m enAelIEpkxagelInFwm. KNnasrsEdkrgkarBt; nigEdkkmaMgkat;TTwg
sRmab;FwmenH edayeRbI f 'c = 28MPa nig f y = 400MPa rUbTI8>24.

dMeNaHRsay

1> KNnasRmab;EdkTb;nwgm:Um:g;
Wu = 1.2WD + 1.6WL = 1.2 600 + 1.6 320 = 1232kN / m
Wu L2 1232 4.2 2
Mu =
=
= 2716.56kN .m
8
8
l n 3 .6
=
= 1.44
h 2 .5

kMNt;dXas;/ y . sRmab; ln / h = 1 / y = 0.6d nigsRmab; ln / h = 2 / y = 0.8d dUcenHsRmab;

ln / h = 1.44 / y = 0.688d eday interpolation = 0.688 0.9 2.5 = 1.55m edaysnt;
d = 0 .9 h
As =

Mu
2716.56 10 6
=
= 4868mm 2
yf y 0.9 1550 400

kmaMgkat; nigkmaMgTajGgt;RTUg

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edaysar f 'c < 30MPa

1.38
1.38
As Gb,brma =
bw d =
400 2250 = 3105mm 2
f
400
y

dUcenH As = 4868mm 2 lb;. eRbI 10DB25 4909mm 2 EdlmanR)aMedImenApmYy edayBRgay

kgkm<s; h / 5 = 500mm EdlCatMbn;TajnFwm. KMlatrbs;Edk = 500 / 5 = 100mm . EdkTaMgenH
KYrbgb;cUleTAkgTRm[)anl.
2> KNnasRmab;kmaMgkat;TTwg
k> KNna Vu nig M u enAcmay x = 0.15ln = d BImuxpnTRm
0.15l n = 0.15 3.6 = 0.54m < 2.25m
3.6
Vu = 1232
1232 0.54 = 1552.3kN
2
0.54 2
M u = 1232 3.6 0.54 1232
= 1017.9kN .m
2
Mu
1017.9
=
= 0.29
Vu d 1552.3 2.25

KNna

x> KNna Vc
3 .5 2 .5

Mu
= 3.5 2.5(0.29 ) = 2.775 > 2.5
Vu d

dUcenH eRbI 2.5 . enAkgkrNIenH KNna M u / Vu d edIm,IeRbIkgkarKNna Vc

2.5 = 3.5 2.5M u /(Vu d )

Mu
Vu
= 0.4
= 2.5
Vu d
Mud
A
4909
w = s =
= 0.00545
bw d 400 2250

Vc = 2.5 0.16 28 + (17.2 0.00545 2.5) 400 2250 10 3 = 2432.2kN

Vc 0.5 f 'c bw d = 0.5 28 400 2250 = 2381.2kN

dUcenH Vc = 2381.2kN lb;. Vc = 1785.9kN

K> KNna Vs = (Vu Vc ) / edaysar Vc = 1785.9kN > 1552.3kN enaH Vs = 0
dUcenHeKRtUvkarEdkkmaMgkat;TTwgGb,brma.
X> KNnaEdkkmaMgkat;TTwg
edaysnt;eRbIEdk DB12 sRmab;dak;enApsgagTaMgtamTisedAedk nigTisedAQrenaH
Av = Avh = 2 12 2

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= 226mm 2

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KMlatGtibrmaGnuBaatrbs;EdkQr nigEdkedk
S v = S vh =

d 2250
=
= 450mm > 300mm
5
5

yk Sv = S nh = 300mm

EdkQrGb,brma
Av Gb,brma= 0.0025 400 300 = 300mm 2 > 226mm 2
bBaHKMlatEdkkgbBar Sv = 400 226
= 226mm
0.0025
EdkedkGb,brma
Avh Gb,brma= 0.0015 400 300 = 180mm 2 < 226mm 2
dUcenH eRbI DB12 @ 200 sRmab;TisbBar
nig DB12 @ 300 sRmab;Tisedk
3> RbsinebIeyIgeRbI Vc = 0.17 f 'c bw d enaH Vc = 0.17 28 400 2250 10 3 = 809.6kN nig
Vc = 607.2kN < 1552.3kN . dUcenH eKRtUvkarEdkkmaMgkat;TTwg.
Vs =

1552.3 607.2
= 1260.1kN
0.75

edaysnt;eRbIEdk DB12 sRmab;dak;enApsgagTaMgtamTisedAedk nigTisedAQrenaH

Av = Avh = 2 12 2

= 226mm 2

edaysnt;faKMlatrbs;EdkTaMgBIrTisedAesInwg Sv = S h = S nig ln / d = 3.6 / 2.25 = 1.6

A 1 + l n / d Avh 11 l n / d
Vs = v
+

f y d

S v 12 S h 12
226 1 + 1.6 226 11 1.6
1260100 =
400 2250
+

S 12 S 12

yk S = 150mm EdltUcCag Sv Gtibrma nig S h Gtibrma. eRbI S = 150mm sM

rab;TaMgKMlatedk nigKMlatQr.
Av Gb,brma= 0.0025 400 150 = 150mm 2 < 226mm 2
Avh Gb,brma= 0.0015 400 150 = 90mm 2 < 226mm 2
dUcenH eRbI DB12 @150mm enAelIpTaMgBIrTaMgTisedAedk nigTisedAQr. sMNaj;EdkpSarGac
RtUv)aneRbIedIm,ICMnYskarBRgayEdkEdlCadMeNaHRsaymYyEdlmanlkNsnSMsMcCag. kar
BRgayEdkenAkgmuxkat;ebtugRtUv)anbgajenAkgrUbTI 8>24.
S = 161mm

kmaMgkat; nigkmaMgTajGgt;RTUg

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]TahrN_6 FwmeRCA strut and tie

FwmeRCATRmsamBaEdlman clear span = 3.6m km<s;srub = 1.8m nigTTwg = 450mm . FwmenHRT

ssrkaerEdlmanRCug = 450mm enAkNalElVgEdlRTnUvbnkefr = 1335kN nigbnkGefr = 1070kN .
KNnaFwmenHedayeRbIviFI strut and tie. eK[ f 'c = 28MPa nig f y = 400MPa rUbTI8>25.

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dMeNaHRsay

1> KNnabnkemKuN rUbTI 8>25

Tmn;Fwm = 4.5 1.8 0.45 25 = 91.1kN
edaysarTmn;pal;rbs;FwmmantmtUcebIeFobCamYybnkcMcMNucenAkNalElVg dUcenHbEnmvaeTA
kgbnkGefrcMcMNucEdlmanGMeBIenAkNalElVg
Pu = 1.2 D + 1.6 L = 1.2(1335 + 91.1) + 1.6 1070 = 3423.32kN
R A = RB = 1711.66kN

2> epgpat;faetIFwmenHeRCAtamkarEcgrbs; ACI Code, Section 11.8: clear span ln = 3.6m nig
h = 1.8m ehIy ln / d = 2 < 4 dUcenH FwmenHCaFwmeRCA.
3> KNnaersIusg;kmaMgkat;TTwgGtibrmanmuxkat;Fwm
yk Vu enARtg; A = R A = 1771.66kN nigsnt;yk d = 0.9h = 0.9 1.8 = 1.62m
Vn = 0.83 f 'c bw d = 0.83 28 450 1620 10 3 = 2743.1kN

OK

Vn = 0.75 2743.1 = 2057.3kN > Vu

4> eRCIserIsKMrU truss

eRCIserIsKMrU truss RtIekaN. snt;facMNuc node eFVIGMeBIsitenAGkSTRm nigenAcmay 150mm BI
EKm)at bEKmkMBUlFwm rUbTI8>26. KMrU strut and tie pMeLIgedayGgt; tie AB mYynigGgt;
strut BIr AD nig DB . dUcKa RbtikmenARtg;cMNuc A nigcMNuc B nigbnk Pu Rtg;cMNuc D
tMNagCa strut bBar.
RbEvg strut Ggt;RTUg AD = 1.52 + 2.0252 = 2.52m
yk CamMurvagGgt; strut nigGgt; tie enaH
1.5
enaH = 36.5o > 26o OK
tan =
= 0.7407
2.025
5> KNnakmaMgenAkgGgt; truss
kmaMgsgt;enAkgGgt; strut AD = FAD = FBD = 1711.66 21..525 = 2875.6kN
kmaMgTajenAkgGgt; tie AB = FAB = 2875.6 22..025
= 2311kN
52
6> KNnaersIusg;RbsiTPaB f ce . snt;EdkTb; (confining reinforcement) RtUv)andak;edIm,IkarBar
kmaMgbMEbk (splitting force). Ggt; strut AD nig DB tMNag[Ggt;rgkarsgt;ragdb (bottleshape compression member) dUcenH s = 0.75 .
kmaMgkat; nigkmaMgTajGgt;RTUg

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f ce = 0.85 s f 'c = 0.85 0.75 28 = 17.85MPa

Ggt; strut bBarenAcMNuc A / B nig D manmuxkat;esI (uniform section) dUcenH s = 1.0

f ce = 0.85 s f 'c = 0.85 1 28 = 23.8MPa

tMbn;cMNuc (nodal zone) D mankmaMg C C C dUcenH s = 1.0 . ersIusg;RbsiTPaBenA nodal

zone D KW
f ce = 0.85 s f 'c = 0.85 1 28 = 23.8MPa

edaysarEtGgt; strut AD nig DB Pab;eTAcMNucepSgeTot enaH

zone TaMgGs;.

f ce = 17.85MPa

lb;elI nodal

7> KNna nodal zone

k> KNna nodal zone enAcMNuc A snt;fakmaMgn nodal zone mankugRtaMgdUcKaKW 17.85MPa
ehIypEkgeTAnwgkmaMgEdlRtUvKa
Fn Fu
b f ce Acs Fu
Edl = 0.75 sRmab; strut, tie nig node.
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RbEvgnpedk ab rUbTI8>27 a esInwg

Fu /(f ce b) = 1711.66 103 /(0.75 17.85 450) = 284mm
2311
ac = 284
= 383mm
1711.66
2875.6
bc = 284
= 477mm
1711.66

BIFrNImaRt RbEvg
dUcKa RbEvg
TIRbCMuTmn;rbs; nodal zone sitenA 383 / 2 = 191.5mm BI)atnFwm eyIg)ansnt; 150mm
x> KNna nodal zone enAcMNuc D rUbTI8>27 b
RbEvgnpedk de = 3423.32 103 /(0.75 17.85 450) = 568mm
2875.6
RbEvgn df = ef = 568 3423
= 477mm
.32
RbEvgrbs; fg = 477 2 ( 568
) 2 = 383mm
2
dUcenH TIRbCMuTmn;n nodal zone sitenA 383 / 3 = 128mm BIpxagelIrbs;FwmeyIg)ansnt;
150mm
8> KNnaEdkQr nigEdkedk
k> EdkQr mMurvagEdkQr nigGgt; strut KW 53.5o BIrUbTI 8>27 a. eRbIEdk DB16 EdlmanKMlat

300mm / As = 2 16 2 = 402mm 2 eCIgBIr/ sin 53.5o = 0.804

4
( Asi / bs S ) sin i = 402 /(450 300) 0.804 = 0.0024

x> Edkedk mMurvagEdkedk nigGgt; strut KW 36.5o BIrUbTI 8>27 a. eRbIEdk DB16 EdlmanKMlat

300mm / As = 2 16 2 = 402mm 2 eCIgBIr/ sin 36.5o = 0.595

4
( Asi / bs S ) sin i = 402 /(450 300) 0.595 = 0.0018

K> ( Asi / bs S ) sin i srub = 0.0024 + 0.0018 = 0.0042 > 0.003

9> KNnaGgt; tie edk AB
k> KNna As
Fu = As f y

OK

As = 2311 10 3 / (0.75 400) = 7703mm 2

eRbI 12DB30 As = 8482mm 2 dak;CabICYrdUcbgajkgrUbTI 8>27 c.

x> KNnaRbEvgf<k; (anchorage length) RbEvgf<k;RtUv)anvas;BIcMNuccab;BI nodal zone
rUbTI8>28.
tan 36.5 o = 190 / x

kmaMgkat; nigkmaMgTajGgt;RTUg

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x = 257 mm

RbEvgTMBk;Edlman = 257 + 142 + 225 35 kRmas;ebtugkarBarEdk = 589mm

RbEvgTMBk;caM)ac;sRmab;Edk DB30 KW 47.5 30 = 1425mm > 589mm
dUcenHeRbITMBk; 90o cgPab;CamYyEdkssr
(

l dh = 0.02 e f y d b /

e = = 1.0

f 'c
d b = 30

l dh = (0.24 400 )30 / 28 = 544mm < 589mm

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]TahrN_7 eKmanFwmTRmsamBamYyEdlmanRbEvg 6m clear

rgnUvbnkBRgayefr
47.5kN / m nigbnkBRgayGefr 25kN / m . FwmenHmanmuxkat; b = 35cm nig d = 55cm . FwmenHBRgwg
edayEdk 4DB25 BRgaymYyCYr. cUrkMNt;nUvmuxkat;caM)ac;sRmab;kmaMgkat;TTwg. smtikm
f ' = 28MPa nig f = 280MPa .
c

span

yt

kmaMgkat; nigkmaMgTajGgt;RTUg

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dMeNaHRsay

k> bnkemKuN bnkKNna

1.2 D + 1.6 L = 1.2 47.5 + 1.6 25 = 97 kN / m

x> kmaMgkat;TTwgKNnaenARtg;muxxagssr
Vu = 97

6
= 291kN
2

K> kmaMgkat;TTwgKNnaenAcmay d BIpxagssr

Vud = 291 (0.55 97) = 237.65kN

X> ersIusg;kmaMgkat;TTwgEdlTb;edayebtug
Vc = 0.17 f 'c bd = 0.17 28 350 550 = 173.2kN

Vc = 130kN
1
Vc = 65kN
2

g> kmaMgkat;TTwgEdlEdkRtUvTb;
Vs =

Vu Vc

237.65 (0.75 173.2)

= 143.5kN
0.75
1
Vc = 65kN
2

c> cmayBImuxpssrmk
x' =

291 65
3 = 2.33m
291

q> KNnaEdkkg
1> eRCIserIsEdk RB10 EdkkgmaneCIgBIr
Av = 2 78.5mm 2 = 157mm 2
A f d 157 280 550
s1 = v yt =
= 168.5mm < 600mmm
Vs
143.5 103

dUcenHykKMlat 160mm RtYtBinitKMlatGtibrma

d 550
=
= 275mm
2
2
3A f
3 157 280
s3 = v yt =
= 376.8mm
b
350
s2 =

s1 < s2 < s3

2> RtYtBinitKMlatGtibrmaesI

d
4

Vc1 = 0.33 f 'c bd = 0.33 28 350 550 = 336.1kN

Vc 2 = 2Vc1 = 2 336.1kN = 672.2kN
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eday V < V dUcenHKMlatEdkGtibrmakRmitRtwm s = d2 = 275mm

C> kmaMgkat;TTwgEdlTb;edayEdksRmab;KMlat s = d2 = 275mm
s

max

c1

max

Vs ( for smax = 275mm) =

Av f yt d
smax

157 280 550

= 87.9kN
275

Vs = 0.75 87.9kN = 65.94kN

cmayBImuxTRm eTAdl;EdkkgEdlmanKMlat s
x1 =

max

= 275mm

291 (130 + 65.94)

3 = 0.98m
291

dUcenH sRmab; 0.98m BImuxnTRm eRbIEdkkg RB10 KMlat 160mm nigsRmab;EpkenAsl;

eRbIEdkkg Gb,brma KMlatGtibrma
Q> karBRgayEdkkg
Edkkg1 manKMlat 2s = 80mm
Edkkg6 manKMlat 160mm = 960mm
srub 1040mm = 1.04m > 0.98m
Edkkg6 manKMlat 270mm = 1620mm
srub 2660mm = 2.66m < 3m
nigEdkkgcugeRkay (3 2.66) = 0.34m
srubEdkkgTaMgGs;EdlRtUveRbIsRmab;RbEvgFwm 6m man 28kg.

kmaMgkat; nigkmaMgTajGgt;RTUg

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IX.

kRmalxNmYyTis

9>1> RbePTkRmalxN
eRKOgbgMkRmalxNebtugRtUv)ansagsg;eLIgedIm,Ipl;nUvprabesI CaTUeTAmanTisedAedk.
kRmalxNGacRTedayCBaaMg bedayFwmebtugsrsEdkEdlCaTUeTARtUv)ancak;kgeBldMNalKaCamYy
kRmalxN (cast monolithically with the slab) bedayFwmEdk bedayssr bedaydI. CaTUeTAkRmas;
kRmalxNman tmtUcebIeRbobeFobCamYynwgRbEvgElVg rUbTI 9>1.
kRmalxNebtugenAkgGaKarGacEbgEckdUcxageRkam
a. kRmalxNmYyTis (one-way slab) RbsinebIkRmalxNEdlRTedayRCugBIrQmKa enaHvanwgekag
b dabkgTisedAEkgnwgRCugnTRm. GMeBIrbs;rcnasm<nKWmanEtmYyTis ehIybnkEdlRTeday
kRmal xNenAkgTisedAEdldabtUc. kRmalxNRbePTenHRtUv)aneK[eQaHfa kRmalxNmYy
Tis rUb TI 9>1a. RbsinebIkRmalxNRtUv)anRTedayRCugbYn ehIypleFobRCugEvgelIRCugxIFM
Cag besInwg 2 enaHbnkPaKeRcIn RbEhl 95% beRcInCagenHRtUv)anRTenAkgTisedAxI ehIy
GMeBImYyTis RtUv)anBicarNasRmab;karGnuvtn_TaMgGs; rUb TI 9>1b. RbsinebIkRmalxNRtUv)an
eFVIeLIgedayebtugBRgwgedayEdkedayKanrnxl; (no void) enaHeK[eQaHfa kRmalxNmYy
Tistan; (one-way solid slab) . rUb TI 9>1 c, d nig e bgajBIbg;BuH nigkarBRgayEdk.
b. RbBnkRmalxNrnUtmYyTis (one-way joist floor slab) kRmalxNRbePTenHeKk[eQaHfa
ribbed slab. vapSMeLIgedaykRmalxNEdlCaTUeTAmankRmas;BI 50mm eTA 100mm RTeday
rnUtebtugGarem: (ribs or joist). CaTUeTArnUtmanragsc nigmanKMlatesIKaminelIs 750mm .
rnUt RtUv)anRTedayFwmEdlsitenAelIssr. KMlatrvagrnUtGacbegIt)anedaykareRbIBum<Edk b
Bum<eFVIBI fiberglass rUbTI9>2. enAkgkRmalxNrnUtxH KMlatrvagrnUtGacRtUv)anbMeBjeday filler
edIm,IeFVI[kRmalxNEpkxageRkamrabesI.
c. RbBnkRmalxNBIrTis (two-way floor system) enAeBlEdlkRmalxNRTedayRCugbYn ehIy
pleFobnRCugEvgelIRCugxItUcCag 2 enaHkRmalnwgdabedaykMeNagDubelITisedATaMgBIr. bnk
kRmalxNRtUv)anbBankgTisedABIreTAFwmTaMgbYnEdlBTCMuvijkRmalxN eyagtamemeronTI
17. RbePTepSgeTotrbs;RbBnkRmalxNBIrTisman flat plate floor, flat slab nig waffle slab
EdlkRmalxNTaMgenHnwgBnl;kgemeronTI17. kgemeronenHeyIgniyayEtBIRbBnkRmalxN
mYyTis.
kRmalxNmYyTis

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9>2> karKNnankRmalxNtan;mYyTis
RbsinebIkRmalebtugcak;kgkRmas;esIedayKanRbehag eKGacehAva)anfa kRmaltan;. enAkg
kRmalxNmYyTis (one-way slab) pleFobnRbEvgEvgrbs;kRmalxNelIRbEvgsIrbs;vaFMCag 2.
bnkrbs;kRmalesIrEtTaMgGs;RtUv)anbBaneTATisedAxI ehIykRmalxNGaccat;TukdUcCaFwm. ceRmok
kRmalxNkta CaTUeTAmanRbEvg 1m RtUv)ancat;TukCaFwmctuekaN. FwmenHmanTTwg1m CamYynwgkm<s;
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esInwgkRmas;kRmalxN nigRbEvgElVgesInwgcmayrvagTRm. dUcenHkRmalxNmYyTispSMeLIgedayFwm

ctuekaNCaeRcInCab;Ka rUbTI9>1.

RbsinebIkRmalxNmanEtmYyElVg ehIyvasitenAelITRmedayesrI enaHm:Um:g;viCmanGtibrma M

sRmab;bnkBRgayesI w KW M = (wL2 ) / 8 Edl L KWCaRbEvgElVgrvagTRm. RbsinebIkRmalxNRtUv)an
sagsg;eLIgkgeBldMNalKaCamYy bCab;enAelITRmCaeRcIn enaHm:Um:g;viCman nigGviCmanRtUv)anKNna
edaykarviPaKeRKOgbgM (structural analysis) bedayviFIemKuNm:Um:g; (moment coefficient) sRmab;Fwm
Cab;. ACI Code, Section 8.3 GnuBaat[eRbInUvemKuNm:Um:g; nigemKuNkmaMgkat;enAkgkrNIkRmalxN
man eRcInElVgesIresIKa rUbTI 9>3. krNIenHGaceRbI)ansRmab;ElVgBIrEdlenACab; benAsgagEvgCag
ElVgEdlenA kNal bElVgCab;enaHticCag 20% . sRmab;bnkBRgayesI bnkGefrktaminKYrFMCagbnk
efrkta 3bI eT. RbsinebIlkxNTaMgenHminRtUv)anbMeBjeTenaH eKRtUvkarkarviPaKeRKOgbgM (structural
analysis). enAkgkarviPaKeRKOgbgM m:Um:g;Bt;GviCmanenARtg;GkSrbs;TRmRtUv)anKNna. tmEdlRtUv
BicarNaenAkgkarKNnaKWm:Um:g;GviCmanenABImuxpnTRm. edIm,ITTYl)antmenH eKRtUvdktmm:Um:g;
GtibrmaenAGkSTRmeday Vb / 3 Edl V CakmaMgkat;TTwgEdl)anBIkarviPaKeRKOgpM nig b KWCaTTwgrbs;
TRm.
Vb
M f enABImuxpnTRm = M c enARtg;GkSnTRm
(>!
3
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NPIC

bEnmBIelIm:Um:g; kmaMgTajGgt;RTUg nigRbEvgf<k;nEdk KYrEtRtUv)anepgpat;sRmab;karKNnad

RtwmRtUv.
lkxNEdlemKuNm:Um:g;sRmab;FwmCab; nigkRmalCab;Edl[enAkgrUbTI 9>3 KYrRtUv)aneRbIGacRtUv
segbdUcxageRkam
a. ElVgmanRbEvgesIesIKa ElVgEvg 1.2 ElVgxI
b. bnkRtUvBRgayesI
c. pleFob bnkGefr / bnkefr RtUvtUcCagbesInwg 3 .
d. sRmab;kRmalxNEdlmanElVgtUcCagbesI 3m m:Um:g;Bt;GviCmanenABImuxpnTRmesInwg
1 w l2.
12 u n
e. sRmab;TRm A EdlminCab; nigmanPaBTMenr enaHemKuNm:Um:g;enARtg;cMnuc A KWesIsUn
nigRtg;cMnuc B = + 111
f. kmaMgkat;TTwgenARtg;cMnuc C = 1.15wu lu / 2 nigenABImuxpTRmdTeTotTaMgGs;esInwg
1
2

wu lu

g. M u =

T.Chhay

emKuN wu lu2 / Edl ln = clear span.

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9>3> EdnkMNt;kgkarKNnaEdlGnuelameTAtam ACI CODE

EdnkMNt;xageRkamRtUv)ankMNt;eday ACI Code
a. ceRmokkRmalxNEdlmanTTwg 1m RtUv)ansnt;.
b. kRmas;Gb,brmankRmalxNmYyTisEdleRbIEdk 400 MPa Gnuelamtam ACI Code, Table 9.5a
sRmab;kRmalxNtan;mYyTis nigsRmab;kRmalxNrnUtmYyTisKYresInwgtmdUcxageRkam
- sRmab;TRmsamBa kRmalxNtan; (solid slab)/ h = L / 20 kRmalxNrnUt (ribbed slab),
h = L / 16
- sRmab;kRmalxNEdlCab;mag kRmalxNtan;/ h = L / 24 kRmalxNrnUt ,
h = L / 18.5
- sRmab;kRmalxNEdlCab;sgag kRmalxNtan; / h = L / 28 kRmalxNrnUt ,
h = L / 21
- sRmab;kRmalxN cantilever: kRmalxNtan; / h = L / 10 kRmalxNrnUt , h = L / 8
- sRmab; f y xusBI 400 MPa enaHtmTaMgGs;xagelIRtUvKuNnwg 0.4 + 1.5 10 3 f y Edl f y Kit
Ca MPa .
kRmas;Gb,brmaenHRtUv)aneRbITal;EtkarKNnaPaBdabbgajfakRmas;tUcCagenHGac
eRbIedayKanplb:HBal;Rtlb;mkvij.
c. eKRtUvepgpat;PaBdabenAeBlTRmkRmalxNEdlPab;eTAnwgsMNg;TMngCargeRKaHfak;xaMgeday
PaBdabFM. PaBdabkMNt;RtUv)ankMNt;eday ACI Code, Table 9.5b .
d. eKcUlciteRCIserIskRmas;kRmalxNedaytMeLIg bbnymg 10mm .
e. eKRtUvepgpat;kmaMgkat;TTwg ebIeTaHbIvaminmantmFMkeday.
f. kRmas;ebtugkarBarEdk (concrete cover) enAkgkRmalxNebtugminRtUvtUcCag 20mm
sRmab;pkRmalxNEdlRbQmnwgkarswk bb:Hpal;nwgdI. kgkrNIenH d = h 20mm d b / 2
eyagtamrUbTI 9>1 d .
g. sRmab;eRKagkRmalxNEdlmankRmas;esI enaHbrimaNEdkGb,brmaenAkgTisedArbs;kRmalxN
minRtUv tUcCagbrimaNEdktRmUvkarsRmab;karrYmmaD nigTb;nwgsItuNPaB (shrinkage and
temperature reinforcement) eT (ACI Code, Section 7.12).
h. EdkemminRtUvmanKMlatFMCagkRmas;kRmalxNbIdg 3h bFMCag 450mm eLIy (ACI Code,
Section 7.6.5).
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i.

j.

NPIC

eKRtUveRbIRbBnEdkRtg;TaMgenAEpkxagelI nigEpkxageRkamrbs;kRmalxNCab;. eBlxHeKkGac

eRbIRbBnEdkcRmuH EdkRtg; nigEdkBt;pgEdr.
bEnmBIelIEdkem eKkRtUvdak;EdkEdlEkgnwgEdkemEdr. EdkbEnmenHeKehAfa Edkrg
(secondary, distribution) EdkrYmmaD bEdksItuNPaB (shrinkage or temperature
reinforcement).

9>4> EdksItuNPaB nigEdkrYmmaD

CaTUeTAebtugrYmmaDenAeBlEdlTwkkMe)arsIum:g;trwg ehIyeK)anKitTunCamunnUvbrimaNBitR)akdn
karrYmmaD. RbsinebIkRmalxNRtUv)aneKdak;[cltedayesrIenAelITRmrbs;va enaHvanwgrYjedIm,ITTYlnUv
karrYmmaD. b:uEnkRmalxN bGgt;epSgeTotRtUv)anPab;y:agrwgeTAnwgGgt;dTeTotneRKOgbgM EdlbegIt
nUvkRmitnkarTb;y:agBitR)akdenAcug. CalTpleKTTYl)ankugRtaMgTajEdleKsal;faCakugRtaMgrYm
maD. karfycuHnUvkugRtMgEdlekItBIsItuNPaB nigkugRtaMgEdlekItBIkarrYmmaD)aneFVI[ekItmansameRbH
b:unsrssk; (hairline crack). srsEdkEdkdak;enAkgkRmalxNKWedIm,ITb;Tl;nwgkarrYjxI nigkarral
dalnsameRbH. edaysarEtebtugrYmmaD srsEdkrgnUvkarsgt;.
EdksRmab;karkugRtaMgrYmmaD nigkugRtaMgsItuNPaBEdlEkgnwgEdkem RtUvdak;enAkgkRmalxN
EdlEdkemRtUv)andak;EtmYyTis. ACI Code, Section 7.12.2 kMNt;nUvGRtaEdkGb,brmadUcxageRkam
- sRmab;kRmalxNEdleRbIEdk 280MPa b 350MPa enaH = 0.2%
- sRmab;kRmalxNEdleRbIEdk 400MPa bEdkpSar bEdksMNaj;pSar enaH = 0.18%
KankrNIEdlEdkRtUv)andak;manKMlatFMCag 5dgkRmas;kRmalxN bFMCag 450mm eT.
sRmab;EdkrYmmaD nigEdksItuNPaB kRmas;ebtugTaMgmUl h EdlRbQmnwgkarrYmmaDRtUveRbIedIm,I
KNnaRkLapEdk. ]TahrN_ RbsinebIkRmalxNmankRmas;srub h = 150mm nig f y = 400MPa enaH
RkLapEdkEdlRtUvkarsRmab;TTwgkRmalxN 1m KW As = 1000 150 0.0018 = 270mm 2 . KMlatEdk
S GacKNnadUcxageRkam
1000 Ab
S=
(>@
As
Edl Ab = RkLapEdkeRCIserIs
As = RkLapEdkKNna
]TahrN_ RbsinebIeKeRbIEdk DB10 Ab = 78.5mm 2 enaH S = 1000 78.5 / 270 = 290mm
yk 250mm . RbsinebIeKeRCIserIsEdk DB12 Ab = 113mm 2 enaH S = 1000 113 / 270 = 418mm
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yk 400mm . EdkenHmantYnaTICaEdkrgEdlRtUv)andak;EkgnwgEdkemEdlRtUv)anKNnaedaykarviPaK
edaykarBt; (flexural analysis).
9>5> lMGitsrsEdk
enAkgkRmalxNmYyTisCab; RkLapEdkemRtUv)anKNnaBIRKb;eRKaHfak;TaMgGs; TaMgenAkNalElVg
nigenATRm. kareRCIserIssrsEdk niglMGitsrsEdkGaRsyeTAnwgRkLapEdk KMlatRtUvkar nigRbEvg
TMBk;. eKmanRbBnrayEdkBIrRbePT.
enAkgRbBnEdkRtg; rUbTI9>4 EdkRtg;RtUv)aneRbIsRmab;dak;enAEpkxagelI nigEpkxageRkamnRKb;
kRmalxNTaMgGs;. karcMNayelIfvikar nigeBlevlasRmab;plitEdkRtg; RtUvkarticCagkgkarplitEdk
Bt; dUcenHRbBnEdkRtg;RtUv)anTUlMTUlayCagenAkgsMNg;.
enAkgRbBnEdkBt; b trussed EdkRtg; nigEdkBt;RtUv)andak;qas;KaenAkgkRmalxN. TItaMgBt;
RtUv)anepgpat;sRmab;karBt; kmaMgkat;TTwg nigtRmUvkarRbEvgf<k;. sRmab;bnkFmtaenAkgsMNg;
bg;lMGitsrsEdksRmab;ElVgxag nigElVgkNalnkRmalxNtan;mYyTisRtUv)anbgajenAkgrUbTI 9>4.

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NPIC

9>6> karEbgEckbnkBIkRmalxNmYyTiseTAFwmTRm
enAkgRbBnkRmalxNmYyTis bnkBIkRmalxNRtUv)anbBaneTAFwmTRmtamcugEvgnkRmalxN.
Fwm bBanbnkrbs;vaeTAssrTRm.

BIrUbTI 9>5 eyIgeXIjfa Fwm B2 RTbnkBIkRmalxNEk,rBIr. edaycat;TukCaFwmRbEvg 1m bnkbBan

eTAFwmesInwgRkLapnceRmokTTwg 1m nigbeNay S KuNnwgGaMgtg;sIuetbnkEdlmanGMeBIelIkRmal
xN.
bnkenHbegItCabnkBRgayesIenAelIFwm
UB = US S

bnkBRgayesIenAelIFwmxag B1 esInwgBak;kNalbnkenAelIFwm B2 edaysarEtvaRTkRmalxNEt

mag.
bnkenAelIssr C4 esInwgRbtikmBIFwm B2 Ek,rBIr
bnkenAelIssr C4 = U B L = U S LS
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bnkenAelIssr C3 esInwgBak;kNalbnkenAelIssr C4 edaysarEtvaRTkRmalxNEtmag.

dUcKacM eBaHbnkenAelIssr C2 nig C1 KW
bnkenAelIssr C2 = U S L2 S = bnkenAelIssr C3
bnkenAelIssr C1 = U S L2 S2
BIkarviPaKenH eyIgeXIjfassrnImYyRTbnkBIkRmalxNEdlenACMuvijssrenaH edayKitrhUtdl;
GkSnkRmalxNEk,r L / 2 tamTisEvg nig S / 2 tamTisxI.
karBRgaybnkBIkRmalxNBIrTiseTAFwmTRm nigssrnwgBnl;enAkgemeronTI17.

]TahrN_TI1 KNnaersIusg;m:Um:g;KNnarbs;kRmalxNEdlmankm<s;srub h = 175mm nigBRgwgeday

Edk DB20 EdlmanKMlat S = 175mm . eRbI

f 'c = 20MPa

dMeNaHRsay

nig

f y = 400MPa

1> KNnakm<s;RbsiTPaB d
d
d = h 20mm b emIlrUb9>1 d
2
d = 175 20mm

20
= 145mm
2

2> KNna As mFmEdlpl;sRmab;TTwgkRmalxN 1m . RkLapnEdk DB20 KW Ab = 314mm 2

As =

1000 314
= 1794mm 2 / m
175

3> eRbobeFobPaKryEdkEdleRbICamYynwg max nig min . sRmab; f 'c = 20MPa nig

f y = 400MPa enaH max = 0.01374 nig min = 0.00333 .
eRbIR)as; = 1794 /(1000 140) = 0.0128 EdlRKb;RKan; = 0.9 .
4> KNna M n = As f y (d a / 2)
a = As f y /(0.85 f ' c b) = 1794 400 /(0.85 20 1000) = 42.2mm

M n = 0.9 1794 400 (140 42.2 / 2) = 76.79kN .m

]TahrN_TI2 kMNt;bnkGefrBRgayesIGnuBaatEdlGacGnuvtenAelIkRmalxNn]TahrN_TI1 RbsinebI

ElVgkRmalxNmanRbEvg 4.9m sitenAcenaHTRmsamBa nigRTnUvbnkefrBRgayesI KanbBalbnkpal;

xn 4.8kN / m 2 .

dMeNaHRsay

1> ersIusg;m:Um:g;KNnankRmalxNKW 76.79kN.m elIkRmalxNTTwg 1m

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NPIC

M u = M n = 76.79 =

Wu L2 Wu (4.9) 2
=
8
8

bnkBRgayesIemKuNKW Wu = 25.6kN / m 2
2> Wu = 1.2D + 1.6L
D = 4.8 + 0.175 1 25 = 9.2kN / m 2
L = 9.1kN / m 2

]TahrN_TI3 KNnakRmalxNTRmsamBaRbEvg 3.6m edIm,IRTbnkefrBRgayesI minrab;bBalbnk

pal;xn 5.75kN / m 2 nigbnkGefrBRgayesI 4.8kN / m 2 . eRbI

kMNt;rbs; ACI Code.

f 'c = 20MPa

f y = 400MPa

nigkar

dMeNaHRsay

1> snt;kRmas;kRmalxN. sRmab; f y = 400MPa kRmas;Gb,brmaedIm,IRKb;RKgPaBdabKW L / 20

= 180mm . snt;kRmas;srub h = 175mm nigsnt; d = 150mm RtUvepgpat;enAeBleRkay.
2> KNnabnkemKuN bnkkRmalxN = 0.175 25 = 4.4kN / m 2
Wu = 1.2 D + 1.6 L = 1.2 (4.4 + 5.75) + 1.6 4.8 = 19.86kN / m 2

sRmab;kRmalxNTTwg 1m / M u = Wu L2 / 8
19.86 3.6 2
Mu =
= 32.2kN .m
8

3> KNna As sRmab; M u = 32.2kN .m / b = 1m nig d = 0.15m / Ru = M u / bd 2 = 1.43MPa /

= 0.0042 < max = 0.01374 / = 0.9 .
As = bd = 0.0042 1000 150 = 630mm 2

eRCIserIsEdk DB12 As = 113mm 2 / nig S = 1000 113 / 630 = 179mm

epgpat; d BitR)akd/ d = 175 20 6 = 149mm GacTTYlyk)an.
dUcenH S = 175mm nig As = 645mm 2
4> epgpat;ersIusg;m:Um:g;nmuxkat;cugeRkay
645 400
= 15.2mm
0.85 f 'c b 0.85 20 1000
a
15.2

M n = As f y d = 0.9 645 400150

= 33kN .m > M u = 32.2kN .m
2
2

a=

As f y

5> KNnaEdkrgEdlEkgnwgEdkem. sRmab;

f y = 400 MPa

min = 0.0018
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Ash = 0.0018 1000 175 = 315mm 2

eRCIserIsEdk DB12 / Ab = 113mm 2 / S = 1000 113 / 315 = 358.7mm

dUcenHeRbI DB12 @ 350mm sMeab;Edkrg.
6> epgpat;kmaMgkat;TTwgcaM)ac; Vu enAcmay d BITRmKW 19.86(1.8 0.15) = 32.8kN
Vc = 0.17 f 'c bd = 0.75 0.17 20 1000 150 10 3 = 85.5kN

dUcenHmuxkat;RKb;RKan;.
7> muxkat;cugeRkay h = 175mm / Edkem DB12 @175mm nigEdkrg DB12 @ 350mm .
1
Vc = 42.75kN > Vu = 32.8kN
2

]TahrN_TI4 muxkat;kRmalxNtan;mYyTisCab;RtUv)anbgajkgrUbTI 9>6. kRmalxNenHRTedayFwm

EdlmanRbEvg 7.2m . bnkGefrmanGMeBIelIkRmalxN)anmkBIbnkpal;xnbUknwg 3.7kN / m 2 ehIy

bnkGefrKW 6.2kN / m 2 cenaHTRmsamBa. KNnakRmalxNCab; nigKUrnUvbg;lMGitsrsEdk. eK[
f 'c = 20 MPa / f y = 280 MPa .

dMeNaHRsay

1> kRmas;Gb,brmankRmalxNTI1KW L / 30 edaysarcugmagCab; nigcugmageTotGt;Cab;.

cmayrvagGkSFwmGaccat;TukCaElVg L = 3.6m . sRmab; f y = 280MPa
kRmas;srubGb,brma 30L = 3600
= 120mm
30
kRmas;srubGb,brmasRmab;ElVgkNal 35L = 3600
= 103mm
35
snt;kRmas;rbs;kRmalxNesIKaRKb;ElVg h = 125mm EdlFMCag 120mm . dUcenH eKmincaM)ac;
epgpat;PaBdab.
2> KNnabnk nigm:Um:g;enAelIceRmokkta
bnkGefr=bnkpal;xn + 3.7 = 0.125 25 + 3.7 = 6.8kN / m 2
bnkemKuN U = 1.2D + 1.6L = 1.2 6.8 + 1.6 6.2 = 18.1kN / m 2
RbEvg clear span KW 3.35m . m:Um:g;caM)ac;sRmab;ElVgTImYyenAelITRmEdlmanElVgCab;esInwg
UL2 / 10
Mu =

U (3.35)2
= 20.3kN .m
10

3> snt; = 1.4% enaH Ru = 3.12MPa tmenHtUcCag max = 0.0169 nigFMCag min = 0.005
= 0.9
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d=

NPIC

Mu
20.3 10 6
=
= 81mm
Ru b
3.12 1000

As = bd = 0.014 1000 81 = 1134mm 2

eRCIserIs DB16
kRmas;srub = d + 162 + 20 = 109mm
eRbIkRmas;kRmalxN 125mm dUckarsnt;xagelI
km<s; d EdleRbIBitR)akd = 125 20 8 = 97mm
4> m:Um:g; nigsrsEdkRtUvkarenARtg;muxkat;dTeTot edayeRbI d = 97mm mandUcxageRkam
TItaMg emKuNm:Um:g; M u (kN .m) Ru = M u / bd 2 (MPa) (%) As (mm 2 ) Edk nigKMlat
1
24
A
0.00500
485
DB12@200
8.46
tUc
B

1
+ 14

14.51

1.54

0.00646

627

DB16@300

1
10

20.31

2.16

0.00928

900

DB16@200

1
11

18.47

1.96

0.00836

811

DB16@200

1
+ 16

12.70

1.35

0.00561

545

DB16@200

karBRgaysrsEdkRtUv)anbgajenAkgrUbTI 9>7.
5> kmaMgkat;TTwgekItmanenAkRmalxNxageRkAnTRmTI2/ muxkat; C
6> Vu Rtg; C = 1.15ULn / 2 = 1.15 182.1 3.55 = 36.95kN
7> Vc = 0.17 f 'c bd = 0.75 0.17 20 1000 97 103 = 55.3
lTplenHGacTTYlyk)an. cMNaMfa kardak;srsEdkkmaMgkat;TTwgGb,brmaenAeBlEdl Vu > 12 Vc
minRtUv)anGnuvtsRmab;kRmalxNeT ACI Code, Section 11.5.5.

]TahrN_TI5 kMNt;bnkemKuNBRgayesIenAelIFwmkNalEdlRTkRmalxNn]TahrN_TI 4. dUcKa

KNnabnkcMGkSenAelIssrkNal eyagtambg;TUeTAnrUbTI9>5.

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dMeNaHRsay

1> bnkemKuNBRgayesIkgmYyEm:RtRbEvgenAelIFwmkNalesInwgbnkemKuNBRgayesIenAlIkRmal
xNKuNnwg S EdlCaRCugxIrbs;kRmalxN. dUcenH
U Fwm= U kRmalxN S = 18.1 3.8 = 68.8kN / m
bnkrbs;FwmKYrRtUv)anKitbBal. RbEvgrbs;FwmKW 7.2m
kMNt;km<s;srub h = 20L 0.8 = 720.2 0.8 = 0.288m yk 300mm
kRmas;kRmalxNKW 125mm dUcenHkm<s;nRTnugrbs;FwmKW 300 125 = 175mm
bnkemKuNnRTnugFwm = 0.175 0.25 25 1.2 = 1.3kN / m
bnkBRgayesIsrubelIFwm = 68.8 + 1.3 = 70.1kN / m
2> bnkenAleIFwmkNal
Pu = 70.1 7.2 = 504.72kN / m

9>7> RbBnkRmalxNrnUtmYyTis (One-Way joist Floor System)

RbBnkRmalxNrnUtmYyTispSMeLIgedaykRmalxNRbehag (hollow slab) CamYykm<s;srubFMCag
kRmas;rbs;kRmalxNtan;mYyTis. kRmalxNenHmanlkNsnSMsMcsRmab;GKarEdlmanbnktUc nig
manElVgEvg dUcCa salaeron mnIreBT nigsNaKar. bnkenAkgtMbn;TajKan\TiBldUcenH tMbn;enHRtUv
)anTuk[RbehagrvagrnUt bRtUv)anbMeBjCamYysSmarEdlmanTmn;Rsal edIm,Ikat;bnyTmn;pal;rbs;
kRmalxN.
viFIsaRskgkarKNna nigtRmUvkarnkRmalxNrnUt (ribbed slab) GnuvtdUckarKNnaFwmmuxkat;ctu
ekaN nigFwmmuxkat;GkSret EdlmanenAkgemeronTI3. cMnucxageRkamRtUv)anGnuvtsRmab;KNnakRmal
xNrnUtmYyTis.
1 rnUt (rib) manragscxageRkam nigmanKMlatdUcKaRbEhl 400mm eTA 750mm . Rbehag
RtUv)anbegItrUbragedayeRbIBum<EdlmanTTwg 500mm nigCeRmA 150mm eTA 500mm GaRsy
eTAelItRmUvkarnkarKNna. kMeNInbTdanrbs;km<s;KWmg 50mm .
2 rnUtminRtUvmanTTwgtUcCag100mm nigminRtUvmanCeRmAFMCagTTwg 3.5 dg. KMlat clear
spacing rvagrnUtminRtUvFMCag 750mm (ACI Code, Section 8.11).
3 ersIusg;kmaMgkat;TTwgEdlpl;edayebtugsRmab;rnUtGacykesInwg 10% FMCagersIusg;kmaMg
kat;TTwgEdlpl;edayebtugsRmab;Fwm. mUlehtucMbgKWbNalmkBIGnrkmrvagkRmalxN
nigrnUtEdlmanKMlatCitKa (ACI Code, Section 8.11.8).
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mhaviTalysMNg;sIuvil

NPIC

4 CaTUeTAkRmalxNenABIelIrnUtmankRmas;BI 50mm eTA 100mm nigmanEdkGb,brma EdkrYm

maD. kRmas;enHminRtUvtUcCag 1 / 12 n clear span rvagrnUt b 50mm (ACI Code, Section
8.11.6).
5 emKuN ACI sRmab;KNnam:Um:g;enAkgkRmalxNCab; GacRtUv)aneRbIsRmab;KNnakRmalxN
rnUtCab;.
6 eKmanEdnkMNt;bEnmmYycMnYn EdlRtUv)ansegbdUcxageRkam
- TTwgGb,brmarbs;rnUtKW mYyPaKbInkm<s;srub b 100mm edayykmYyNaEdlFMCag.
- EdkrgenAkgkRmalxNenAkgTisedAEkgnwgrnUtminKYrtUcCagbrimaNEdkrYmmaD btUcCag
1 / 5 nRkLapEdkemenAkgrnUt.
- EdkrgEdlRsbnwgrnUtKYrdak;enAkgkRmalxN ehIyKMlatrbs;vaminRtUvmancmayFMCag
Bak;kNalnKMlatrvagrnUt.
- RbsinebIbnkGefrenAelIkRmalxNrnUttUcCag 3kN / m 2 ehIyRbEvgrbs;rnUtFMCag 5m
enaHrnUtTTwgrg (secondary transverse rib) RtUv)andak;enAkNalElVg TisedArbs;vaKW
EkgnwgTisedArbs;rnUtem nigBRgwgedayEdkEdlmanbrimaNesInwgbrimaNEdkenAkgrnUt
em. EdkxagelIrbs;vaminRtUvtUcCagBak;kNalEdkemenAkgtMbn;Taj. rnUtrgmannaTI
BRgwgkRmalxN (floor stiffener) .
- RbsinebIbnkGefrFMCag 3kN / m 2 ehIyrnUtemmanRbEvgERbRbYlcenaHBI 4m eTA 7m eK
RtUvdak;rnUtrgmYydUcbgajxagelI. RbsinebIrnUtemmanRbEvgFMCag 7m y:agehacNas;
rnUtrgBIrRtUv)andak;CamYynwgsrsEdkdUckarENnaMxagelI.
]TahrN_TI6 kMNt;kRmalxNrnUtmYyTisenAxagkgCamYynwgTinnyxageRkam RbEvgrnUt 6m TRm
samBa bnkefr edaymin)anbBalbnkpal; 0.75kN / m 2 bnkGefr 4.1kN / m 2 / f 'c = 28MPa nig
f y = 400 MPa .

dMeNaHRsay

1> KNnakRmalxN snt;kRmas;kRmalxNxagelIesInwg 50mm EdlPab;CamYynwgrnUtEdlman

clear span 500mm . sSmaredIm,IbMeBj (filler) minRtUv)aneRbI. bnkpal;rbs;kRmalxNKW
0.050 25 = 1.25kN / m 2 .
bnkefrsrub D = 1.25 + 0.75 = 2.kN / m 2
U = 1.2 D + 1.6 L = 1.2 2 + 1.6 4.1 = 9kN / m 2

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UL2
12
9 0.5 2
Mu =
= 0.19kN .m
12

Mu =

kRmalxNRtUv)ancak;Cab;CamYynwgrnUt

edaycat;Tukm:Um:g;enAkgkRmalxNRtUvRTedayebtugsuT
enaHersIusg;TajedaykarBt;GnuBaatrbs;ebtugKW f t = 0.415
sg; = 0.55 / f t = 0.415 28 = 2.2MPa .
ersIusg;TajrgkarBt; = McI = f t

f 'c

CamYynwgemKuNkat;bnyersIu

bh 3 1000 50 3
=
= 10.42 10 6 mm 4
12
12
h
c = = 25mm
2
I
10.42 10 6
M = f t = 0.55 2.2
10 6 = 0.5kN .m > M u = 0.19kN .m
c
25
I=

muxkat;RKb;RKan;

EdkrYmmaD As = 0.0018 1000 50 = 90mm 2

Edk DB10 manKMlat 300mm RtUv)andak;EkgnwgTisedArbs;rnUt. sMNaj;EdkpSar (welded
wire fabric) manlkNsnSMsMcsRmab;krNIbrimaNEdktic.
eRbIEdk DB10 manKMlat 300mm sRmab;dak;RsbnwgTisedArnUt edayEdkmYyedImRtUv)andak;enA
BIelIrnUtnImYy nigEdkmYyeTotRtUv)andak;enAcenaHkNalrnUtBIr.
2> KNnam:Um:g;enAkgrnYt
km<s;Gb,brma = 20L = 206 = 0.3m
km<s;srubrbs;rnUtnigkRmalxNKW 250 + 50 = 300mm
snt;TTwgrnUtEpkxageRkamKW 100mm nigEpkxagelI nIv:U)atkRmalxNKW 150mm rUbTI9>8.
TTwgmFmrbs;rnUtKW 125mm
cMNaMfa kMeNInnTTwgrbs;rnUtedayeRbIBum<cltEdlmanpleFob 1 / 12 edk 1 nigQr 12
Tmn;rnUt= 0.125 0.25 25 = 0.78kN / m
rnUtRTnUvbnkmkBIkRmalxNEdlmanTTwg 600mm bUknwgbnkpal;rbs;va
U = 9 0.6 + 1.2 0.78 = 6.35kN / m
UL2 6.35 6 2
Mu =
=
= 28.6kN .m
8
8

3> KNnarnUt km<s;srubrbs;rnUtKW 300mm .

snt;eRbIEdk DB16 nigkRmas;ebtugkarBarEdk 20mm
kRmalxNmYyTis

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km<s;RbsiTPaB d = 300 20 8 = 272mm

BinitemIlersIusg;nsab snt;muxkat; tension-controlled section/ = 0.9
t

Edl C = 0.85 f 'c bt

M n sab = C d
2

50

M u = 0.9(0.85 28 600 50) 272 10 6 = 158.7kN .m

2

ersIusg;m:Um:g;ebs;sabFMCagm:Um:g;Gnuvtn_ dUcenHrnUtmannaTICamuxkat;ctuekaNEkgEdlmanTTwg
b = 600 mm nigkm<s;nbkrgkarsgt;smmUl a KWtUcCagkRmas;kRmalxN 50mm .

a
2

M n = As f y d = As f y (d

As f y
1.7 f ' c b

As 400

28.6 10 6 = 0.9 As 400 272

1.7 28 600

As = 297mm 2
297 400
= 8.32mm < 50mm
a=
0.85 28 600

eRbIEdk 2DB16 sRmab;rnUtmYy (402mm 2 )

As min = 0.0033bw d = 0.0033 125 272 = 112.2mm 2 < 297mm 2

BinitemIl
=

297
= 0.018 < max = 0.019
600 272

EdlCamuxkat; tension-controlled section/ = 0.9 .

4> KNnakmaMgkat;TTwgenAkgrnUt ersIusg;kmaMgkat;TTwgGnuBaatnRTnugrnUtKW
Vc = (1.1) 0.17 f 'c bw d = 0.75 1.1 0.17 28 125 272 10 3 = 25.2kN

kmaMgkat;TTwgemKuNenAcmay d BITRmKW
Vu = 6.35 (3 0.272) = 17.3kN < Vc = 25.2kN

dUcenHEdkkgGb,brmaRtUv)aneRbI ehIykrNIenHEdk DB12 RtUv)anbEnmenAkgkRmalxNEpk

xagelIedIm,ITb;Edkkg. vaCakarRbesIreBsinebIeKbEnmrnUtrgmYyenAkNalElVgEkgnwgrnUtem
ehIymanbrimaNEdkdUcKanwgrnUtem edIm,IeFVI[kRmalxNmanlkNrwgmaM.

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X.

ssrrgkmaMgcMGkS

10>1> esckIepIm
ssrCaeRKOgbgMdsMxan;sRmab;RTnUvbnksgtc; MGkS ehIymanpleFobssrelIRCugxagEdltUc
CageK FMCag 3 . sRmab;sMNg;ebtugGarem: Fwm kRmal nigssrRtUv)ancak;kgeBlEtmYy (monolithic)
EdleFVI[manm:Um:g;xH manGMeBIelIssredaysarkarTb;enAxagcug. elIsBIenHeTot ssrsRmab;GaKar
eRcInCan; kartRmg;ssrtamTisQr minGaceFVIeTA)an\texaHenaHeT EdlbegIt[mankmaMgcakpiteday
eFobeTA nwgGkSssr. kmaMgcakpitbegIt[manm:Um:g;elIssr. dUcenH ssrEdlrgnUvbnktamGkSsuTmin
GacekItmanenAkgsMNg;ebtugGarem:eT. EteKGacsnt;fa ssrrgkmaMgcMGkS kalNavamancgaycakpit
e tUc EdlmantmRbEhl 0.1h . h CaCeRmAsrubrbs;ssr nig e CacgaycakpitBIGkSssr. edayeb
tugmanersIusg;sgt;x<s; ehIyvaCasmarEdlmantmminf dUcenHvaRtUv)aneKeRbIedIm,IKNnasRmab;eRKOg
bgMrgkmaMgsgt; edaymanlkNesdkicx<s;.
10>2> RbePTssr
ssrRtUv)anEbgEckedayQrelIeKalkarN_epSgdUcxageRkam
i. QrelIeKalkarN_bnk ssrcat;cMNat;fak;dUcxageRkam
+ ssrrgbnkcMGkS EdlbnkRtUv)ansnt;manGMeBIenAcMGkSnmuxkat;ssr
+ ssrrgbnkcakpit EdlbnkmanGMeBIenAcgay e BIGkSnmuxkat;ssr. cgay e GacsitenA
tamGkS x b GkS y EdlbegIt)anCam:Um:g;tamGkS x b GkS y .
+ ssrrgbnkcakpitBIrTis EdlbnkRtUv)anGnuvtenAcMNucNamYynmuxkat;ssr ehIyEdl
begIt)anCam:Um:g;tamGkS x nigGkS y .
ii. QrelIeKalkarN_RbEvgssrcat;cMNat;fak;dUcxageRkam
+ ssrxI Edlssr)ak;edaykarEbkebtug bEdkeFVIkardl;cMnuc yield eRkamlTPaBRTRTg;bnk
eBjrbs;ssr
+ ssrEvg EdlkarPat; nigPaBrlas;RtUv)anykmkBicarNa dUcenHlTPaBRTRTg;bnkrbs;ssr
RtUv)ankat;bnyebIeFobnwgssrxI
iii. QrelIeKalkarN_rUbragnmuxkat; muxkat;ssrGacmanragCa kaer ctuekaNEkg mUl >>>> brUbrag
cg;)anNamYy.
iv. QrelIeKalkarN_EdkkgTTwg ssrRtUv)anEbgEckCa
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ssrEdlmanEdkkgdac;BIKa EdlmantYnaTIhMuBTEdkbBarem. EdkkgCaTUeTAmanKMlatdUc

KaeBjmYykm<s;ssr.
+ ssrEdlmanEdkkgvN EdlmantYnaTIhMuBTEdkbBarem ehIyCYybegInPaByWtrbs;ssrmun
eBl)ak;. CaTUeTA Edkkg nigEdkkgvN RtUv)aneKeRbIedIm,IkarBarPaBrlas;rbs;ssr kugRtaMgEdk
bBarEdlekItBIkarPat; nigkarpHEbknebtugkarBarsrsEdk.
v. QrelIeKalkarN_viPaKeRKag ssrCaEpkmYyneRKagEdlmanCnl;RbqaMgnwgkareRTtcMehog nig
CaEpkmYyneRKagEdlminmanCnl;RbqaMgnwgkareRTtcMehog. cnl;Gac)anmkBIkareRbICBaaMgeb
tugGarem: bcnl;BIeRKag. sRmab;eRKagmancnl; ssrCaeRKOgbgMEdleRbIsRmab;Tb;nwgbnkTMnaj
ehIyCBaaMgebtugGarem:RtUv)aneKeRbIedIm,ITb;nwgkmaMgtamTisedk nigbnkxl;. eRKagKancnl;
ssrCaeRKOgbgMEdleRbIedIm,ITb;nwgbnkTMnajpg nigbnkxl;pg dUcenHlTPaBRTRTg;rbs;ssr
RtUv)ankat;bny.
vi. QrelIeKalkarN_smar ssrGacCassrebtugGarem: ssrebtugeRbkugRtaMg ssrsmas.
ssrebtugGarem:RtUv)aneKeRbICaTUeTAsRmab;sMNg;GKar.
+

ssrrgkmaMgcMGkS

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10>3> kareFVIkarrbs;ssrrgbnkcMGkS
enAeBlEdlbnkmanGMeBIcMGkSnssrxIebtugGarem: ebtugRtUv)anBicarNa[eFVIkarCalkNeGLa
sic enAeBlEdlvargnUvkugRtaMgTabEdlmantmRbEhl f ' . RbsinebIbnkEdlmanGMeBIenAelIssrxit
eTArkersIusg;Gtibrmarbs;va enaHebtugxiteTArkersIusg;Gtibrma ehIyEdkkxiteTArkersIusg; yield f .
lTPaBRTRTg;bnkcugeRkaymFmrbs;ssr GacsresrdUcxageRkam
1
3

Po = 0.85 f 'c An + Ast f y

Edl A nig A CaRkLapebtugsuT nigRkLapEdk

n

st

An = Ag Ast

Camuxkat;rgGMeBI
kar)ak;rbs;ssr ekIteLIgBIrRbePTepSgKa GaRsyeTAnwgkareRbIR)as;Edkkg nigEdkkgvN.
sRmab;ssrEdkkg ebtug)ak;edaypHEbkpat;ecjeRkA EdkembBar)ak;edayPat;ecjcenaHEdkkg dUcenH
ssr)ak;ekIteLIgPam. sRmab;ssrEdkkgvN)ak;edaymankarbgajnUvrMhUr EdlekItmannUvkMhUcRTg;
RTayEdlGacKit)an munnwgkar)ak;eBjelj.
Ag

10>4> lkxNrbs; ACI Code

ACI Code )anbgajnUvlkxNmYycMnYndUcxageRkamkgkarKNnaeRKOgbgMrgkarsgt;
- sRmab;ssrrgkarsgt;cMGkS kdUccakpit ACI Code )ankMNt;ykemKuNkat;bnyersIusg;
= 0.65 sRmab;ssrEdlmanEdkkgdac; nig = 0.7 sRmab;ssrEdlmanEdkkgvN. tmxusKa
0.05 bgajGMBIPaBsVitrbs;ssrEdlmanEdkkgvN. emKuNkat;bnyersIusg;sRmab;ssrmantmtUc
CagsRmab;eRKOgbgMrgkarBt; nigkMmaMgkat;TTwg enHedaysarEtersIusg;rbs;ssrrgkarsgt;cMGkSGaRsy
CacMbgeTAnwgersIusg;rbs;ebtug EdlpyBIeRKOgbgMrgkarBt;EdlTTYl\TiBlticCagBIersIusg;ebtug CaBi
esskgkrNI tension failure section. elIsBIenHeTot rgnUvkarbMEbkFatul,ayebtugeRcInCagkrNIFwm.
ssrRtUvcak;bBarx<s; nigBum<cegt b:uEnebtugkgFwmRtUv)ancak;kgBum<rak; edk. dUcenHkar)ak;rbs;ssr
enAkgeRKagRtUv)aneKBicarNaeRcInCagFwmkRmal.
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- PaKryEdkemGb,brma 1% nigGtibrmaKW 8% nmuxkat;ssr. muxkat;EdkGb,brmaRtUv)an

dak;sRmab;Tb;nwgkarBt;EdlvaGacekItmaneLIg nigCYykat;bny\TiBl creep nigkarrYmmaDrbs;ebtug
eRkamGMeBInkugRtaMgsgt;. sRmab;karGnuvt vaCakarBi)akbMputkgkarBRgayEdkEdlmanmuxkat;eRcInCag
8% eTAkgssr BIeRBaHeKRtUvkarKMlatEdksmrmedIm,I[ebtugeRcotcUlcenaHEdk.
- y:agehacNas;EdkbYnedImRtUv)andak;sRmab;ssrctuekaN nigssrmUlEdlmanEdkkgdac;
nigEdkR)aMmYyedImsRmab;ssrmUlEdlmanEdkkgvN. sRmab;muxkat;ssrepSgeTot EdkRtUv)andak;RKb;
RCugTaMgGs;. EdkminRtUv)andak;edaymanKMlatFMCag 15cm . kgkrNIEdkmanKMlatFMCag 15cm eKRtUv
dak;EdkkgbEnm. RsTab;ebtugkarBarEdkGb,brma 40mm .

- PaKryKitCamaDnEdkkgvNelImaDrbs;slssr RtUv)ankMNt;
s

A
f'
s = 0.45( g 1) c
Ac
f yt

Edl

Ag

Ac
f yt

ssrrgkmaMgcMGkS

muxkat;ssr
muxkat;ssrKitRtwmEpkxageRkAEdkkgvN
ersIusg;Edkkg
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- muxkat;EdkkgvNGb,brma 9.5mm nigmanKMlatcenaH 2.5cm 7.5cm . kartEdkeFVIeLIg

edaypSarehIyRbEvgtEdksRmab;faMgGMeBA 48d nigFMCag 30cm sRmab;Edkrelag 72 nigFMCag 30cm .
- Ggt;pitnEdkkgdac;Gb,brma 9.5mm RtUv)aneRbIedIm,IrMuBTEdkbBarEdlmanGgt;pittUcCag
32mm nigsRmab;EdkbBarEdlmanGgt;pitFMCag 32mm EdkkgRtUvmanGgt;pit 12mm .
- KMlatEdkkgminRtUvmantmFMCag
48

min 16d
b

Edl Ggt;pitEdkkg/ d Ggt;pitEdkbBar/ b RCugEdlmanTMhMtUcCageKnmuxkat;ssr.

10>5> smIkarsRmab;KNna
bnkEdlssrGacRT)anRtUv)ankMNt;edaysmIkar
Po = 0.85 f 'c An + Ast f y

EtCaTUeTA ssrminEdlrgbnkcMGkSeT vaEtgEtrgbnkcakpitbnic dUcenH ACI Code )anKuNnwg

emKuN 0.8 sRmab;ssrEdlmanEdkkgdac; nig 0.85 sRmab;ssrEdlmanEdkvN.
sRmab;ssrEdlmanEdkkgdac; bnkemKuNEdlssrGacRT)ankMNt;eday
P = P = 0.8[0.85 f ' ( A A ) + A f ] nig = 0.65
sRmab;ssrEdlmanEdkkgvN emKuNEdlssrGacRT)ankMNt;eday
P = P = 0.85[0.85 f ' ( A A ) + A f ] nig = 0.7
smIkarxagelITaMgBIrGacsresry:agsegbdUcxageRkam
u

st

st

st

st

Pu = Pn = k[0.85 f 'c Ag + Ast ( f y 0.85 f 'c )]

Edl = 0.65 nig k = 0.8 sRmab;ssrEdlmanEdkkgdac; nig = 0.7 nig k = 0.85 sRmab;
ssrEdlmanEdkkgvN.
smIkarxagelIRtUv)aneRbIedIm,IKNnalTPaBRTRTg;rbs;ssrrgbnkcMGkS.
RbsinebIpleFobEdk = AA enaHsmIkarxagelIkayeTACa
st

Pu = Pn = kAg [0.85 f 'c + g ( f y 0.85 f 'c )]

smIkarxagelIenHvij RtUv)aneRbIedIm,IKNnamuxkat;ssr edaysnt;PaKryEdlenAcenaH

1% 8% .

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10>6> kmaMgTajcMGkS
ebtugnwgminmansameRbHeT ebIvargnUvkugRtaMgEdlmantmTabCagersIusg;rgkarTajrbs;va kg
krNIenH TaMgebtug nigEdkTb;nwgkugRtaMgTaj. b:uEnenAeBlEdl kmaMgTajmantmFMCagersIusg;rgkar
Tajrbs;va RbEhl 101 nersIusg;rgkarsgt;rbs;ebtug sameRbH)anekItmanenAelImuxkat;ebtug ehIy
kmaMgTajTaMgGs; RtUvTb;Tl;edayEdk. bnkFmtaEdlGgt;TTYl)anKWCaersIusg;Tajrbs;Edk
Tn = Ast f y
Tu = Ast f y

sRmab;karTajcMGkS
sRmab;Ggt;EdlrgkarTaj ebtugeRbH ehIyr)arEdkrgnUvbnkTajTaMgGs; ebtugmantYnaTICaGk
karBarePIg nigERcHsIuEdk. sRmab;GagskTwk eKRtUvykcitTukdak;eTAelIebtug edaymin[vamansameRbH
eRkamGMeBInkmaMgTajEdlbNalmkBIsm<aFTwk.
]TahrN_10>1 KNnassrxImuxkat;ctuekaNEdlmanEdkkgdac; edIm,IRTbnkemKuNcMpit
P = 1765kN . edayeRbI f ' = 30MPa / f = 400MPa / TTwgssr b = 30cm nigPaKryEdk = 2% .
dMeNaHRsay
bnkemKuNEdlssrGacssrGacRT)an
= 0.9

Pu = kAg [0.85 f 'c + g ( f y 0.85 f 'c )]

sRmab;ssrEdlmanEdkkgdac; = 0.65 nig k = 0.8

Pu = 0.65 0.85 Ag [0.85 f 'c + g ( f y 0.85 f 'c )]

kMNt;muxkat;ssr
Ag =

Pu
0.65 0.8 [0.85 f 'c + g ( f y 0.85 f 'c )]

edaybnkRtUvkarRTnUvbnkcMpit P = 1765kN nigssrRtUvmanPaKryEdk

u

Ag =

= 2%

1765 10
= 102887 mm 2
0.65 0.8 [0.85 30 + 0.02(400 0.85 30)]
3

eday b = 30cm
a=

102887
= 343mm
300

yk a = 350mm = 35cm A = 105000mm

muxkat;Edk A = 0.02 102887 = 2057mm
ykEdk DB22 cMnYn 6 edIm A = 2280mm

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RtYtBinitbnkKNnaEdlssrGacRT)an
Pu = 0.65 0.8[0.85 f 'c ( Ag Ast ) + Ast f y ]
Pu = 0.65 0.8[0.85 30 (105000 2280) + 2280 400] 103 = 1836kN > 1765kN

OK !

ykEdkkgmanGgt;pit 10mm
KMlatEdkkg
48
48 10
480

min 16d = min 16 22 = min 352 = 300

b
300
300

dUcenHeRbIEdkkg DB10 @ 300 .

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eRKOgbgMrgkarsgt; nigrgkarBt;

11>1> esckIepIm
eRKOgbgMbBarCaEpkmYyrbs;eRKagsMNg; EdlrgkmagM sgt; nigm:Um:g;. kmaMgTaMgenH )anBIkmaMg
xageRkAdUcCa bnkefr bnkGefr nigbnkxl;. kmaMgRtUv)ankMNt;eday karKNnaedayd bedaykMuBTr
EdlQrelIeKalkarN_saTic nigviPaKeRKOgbgM (structural analysis). Ca]TahrN_ sRmab;rUb (1) bgaj
faeRKagQrelITRm hinged BIr EdlrgbnkemKuNBRgayesIenAelIGgt; BC. daRkamm:Um:g;Bt;RtUv)anKUr
enAEpkxagTaj. kMNat;ssr AB nig CD rgnUvkmaMgsgt; nigm:Um:g;Bt;. pleFobrvagm:Um:g;Bt; nigkmaMg
sgt;RtUv)an[eQaHfa cmaycakpit e Edl e = MP . e CacmayBITIRbCMuTmn;)asic(plastic centroid) n
muxkat;eTAcMNucnbnkmanGMeBI. TIRbCMuTmn;)asic (plastic centroid) RtUv)anTTYledaykarkMNt;TItaMg
kmaMgpbbegItedaysrsEdk nigebtugedaysnt;kugRtaMgsgt;sRmab;Edk f nigkugRtaMgsgt;sRmab;
ebtug 0.85 f ' . sRmab;muxkat;sIuemRTI TIRbCMuTmn;)asic (plastic centroid) RtYtsIuKaCamYyTIRbCMuTmn;rbs;
muxkat;. sRmab;muxkat;minsIuemRTI TIRbCMuTmn;)asic (plastic centroid) RtUv)ankMNt;edayeRbIm:Um:g;eFob
nwgGkS arbitrary axis.
n

rUbTI1 eRKagTRm pin BIrCamYynwgdaRkamm:Um:g;

]TahrN_11>1 kMNt;TIRbCMuTmn;)asic (plastic centroid) nmuxkat;dUcbgajkgrUbTI2. smtikm
f 'c = 28MPa

nig

f y = 400 MPa

eRKOgbgMrgkarsgt; nigrgkarBt;

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rUbTI2 TIRbCuMTmn;)asic (P.C)nmuxkat;

dMeNaHRsay

!> kugRtaMgsgt;sRmab;ebtugRtUv)ankMNt;yk 0.85 f '

F = kmaMgkgrbs;ebtug = 0.85 f ' A
c

= (0.85 28) 350 500 = 4165kN

sitenAelITIRbCMuTmn;nmuxkat;ebtug enAcmay 250mm BIGkS A A

@> kmaMgenAkgsrsEdk
Fc

Fs1 = As1 f y = 4

282

400 = 985.2kN
4
282
Fs 2 = As 2 f y = 2
400 = 492.6kN
4

#> kMNt;m:Um:g;eFob A A
x=

(4165 250) + (985.2 65) + (492.6 435)

= 233.85mm
4165 + 985.2 + 492.6

dUcenH TIRbCMuTmn;)asic (plastic centroid) RtUvsitenAelIcmay 233.85mm BIGkS A A

$> RbsinebI A = A muxkat;sIuemRTI dUcenH x = 250mm BIGkS A A .
s1

s2

11>2> karsnt;sRmab;KNnassr
GaRsytam ACI Code EdnkMNt;sRmab;karKNnassrkMNt;dUcxageRkam
!> bERmbRmYlrageFob (strain) enAkgebtug nigEdk RtUvsmamaRteTAnwgcmayBIGkSNWt.
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@> RtUvEtbMeBjlkxN smIkarlMnwgnkmaMg nigPaBRtUvKanbERmbRmYlrageFob (strain

compatibility).
#> bERmbRmYlrageFobrbs;ebtugrgkarsgt;EdleRbIR)as;GtibrmaKW 0.003 .
$> ersIusg;rbs;ebtugrgkarTajGacRtUvecal.
%> kugRtaMgenAkgEdkKW f = E f .
^> bkkugRtaMgGackMNt;manragctuekaNCamYykugRtaMg 0.85 f ' BRgayBIRbEvg a = c . Edl
c CacmayBIGkSNWt nig
s

0.85

f ' 28
1 = 0.85 0.05( c
)
7

0.65

f 'c 28MPa

sRmab;ebtugEdlmanersIusg; 28MPa < f ' 56MPa

c

f 'c > 56 MPa

11>3> daRkamGnrkmrvagbnk nigm:Um:g; (Load-moment interaction diagram)

enAeBlEdlbnktamGkSRtUv)anGnuvtmkelIssrxI krNIdUcxageRkamGacekIteLIg edayGaRsy
eTAnwgTItaMgGnuvtbnkedayeFobeTAnwg TIRbCMuTmn;)asic (plastic centroid).
kmaMgsgt;tamGkS P CakmaMgsgt;tamGkSEdlmantmFGM nuvtenAelITIRbCMuTmn;)asic (plastic
centroid) e = 0 nig M = 0 . kar)ak;rbs;ssr ekIteLIgedayebtugEbk nigEdkeFVIkardl; yielding.
vaRtUv)ansMEdgeday P enAelIExSekag.
!> Maximum nominal axial load P : CakrNIEdlkmaMgtamGkSGnuvteTAelImuxkat;CamYy cM
gaycakpit eccentricity Gb,rma. tam ACI Code, P = 0.80P sRmab;ssrEdkkgdac; tie
= 0.85 P sRmab;ssrEdlmanEdkkgdUcrWusr spirally reinforced column . kar)ak;
column nig P
ekIteLIgedayebtugEbk nigEdkeFVIkardl; yielding.
@> Compression failure: CakrNIEdlbnktamGkSFMGnuvtenAcmaycakpittUc. bnktamGkSkg
krNIenHmantmERbRbYlBI tmGtibrma P = P eTAtmGb,rma P = P (balanced load). s
r)ak;edayebtugEbkenAEpkrgkarsgt;CamYYynwgbERmbRmYlrageFob strain = 0.003 cMENkkugRtaMg
kgEdk EpkrgkarTaj KWtUcCag yield strength f < f . kgkrNIenH P > P nig e < e .
#> Balanced condition P : lkxNenHekItmaneLIgenAeBlEdl bERmbRmYlrageFobrgkarsgt;
(compression strain) enAkgebtugmantmesI 0.003 ehIybERmbRmYlrageFobrbs;EdkrgkarTajman
tm = Ef . kar)ak;rbs;ebtugekIteLIgdMNalKanwgEdk yield. m:Um:g;EdlekItedaysarbnkenH
o

n max

n max

n max

n max

eRKOgbgMrgkarsgt; nigrgkarBt;

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RtUv)aneKehAfa balanced moment M cmaycakpitRtUv)aneKehAfa balanced eccentricity

M
.
e =
P
b

rUbTI3 a daRkamGnrkmbnk-m:Um:g;
$> Tension failure: CakrNIekItmanenAxNEdl bnktamGkStUc nigcMNakpitFM ehIyEdlman
m:Um:g;FM. muneBl)ak; kugRtaMgTajekItmanenAelIEpkdFMnmuxkat; bNal[EdkrgkarTaj yield muneBl
ebtugEbk. enAeBl)ak; bERmbRmYlrageFobrbs;EdkrgkarTajmantmFMCagbERmbRmYlrageFob yield
ehIybERmbRmYlrageFobenAkgebtugesI 0.003 . krNIenHekItmanBI Balanced condition eTAdl;
pure flexure P < P nig e > e .
%> Pure flexure: muxkat;kgkrNIenHrgm:Um:g;Bt; M Edl P = 0 . kar)ak;dUcKanwg kar)ak;rbs;
FwmrgkarBt;. cMNakpitRtUv)ansnt;fa Gnn.
y

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rUbTI3 b muxkat;ssrEdlbgajBITItaMgbnk P sRmab;lkxNbnkepSg

n

11>4> karpl;nUvsuvtiPaB (Safety provisions)

!> emKuNbnksRmab; bnk gravity nigbnkxl;
U = 1 .4 D

eRKOgbgMrgkarsgt; nigrgkarBt;

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U = 1 .2 D + 1 .6 L
U = 1.2 D + 1.6 L + 0.8W
U = 1.2 D + 1.0 L + 1.6W
U = 0.9 D + 1.6W

bnSMbnkemKuNEdlmantmsFMCageKRtUv)anykmkeRbIR)as;sRmab;karKNna.
@> emKuNkat;bnyersIusg; eRbIsRmab;KNnassrGaRsyeTAnwgkrNIxageRkam
k> enAeBl P = P 0.1 f ' A eBlenaH = 0.65 sRmab;ssrEdkkgdac; (tie
column) ehIy = 0.7 sRmab;muxkat;ssrEdlmanEdkkgdUcrWusr (spirally reinforced column) .
krNIenHssrRtUv)anrMBwgfa)ak;edaykarsgt;.
u

rUbTI4 tmemKuN
x> muxkat;EdlbERmbRmYlrageFobrgkarTajsuT (net tensile strain) sRmab;ersIusg;
Fmta (nominal strength) enAkgEdkrgkarTajeRkAeKbMput KWsitenAcenaH 0.005 nig 0.002 (transition
region) ERbRbYlCalkNbnat;cenaH 0.9 nig 0.65 b 0.7 .
sRmab;muxkat;EdlmanEdkkgdUcrWusr spiral section
1
200
5
(11-1)
= 0.7 + ( 0.002)(
) b = 0.7 + 0.2

c/d 3
3
t

sRmab;muxkat;epSgeTot
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= 0.65 + ( t 0.002)(

250
)
3

b = 0.65 + 0.25 c /1d

5

3

(11-2)

K> enAeBl P = 0 kgkrNIrgkarBt;suT = 0.9 sRmab;muxkat; tension-controlled

section nigERbRbYlBI 0.9 nig 0.65 b 0.7 enAkgtMbn; transition region.
u

11>5> Balanced condition muxkat;ctuekaN

Balanced condition ekItmanenAkgmuxkat;ssrenAeBlEdl bnkEdlGnuvtmkelImuxkat;ssr
Edlman nominal strength begItbERmbRmYlrageFobesI 0.003 enAkgsrsrgkarsgt;rbs;ebtug
nigbERmbRmYlrageFobesI = Ef enAkgr)arEdkrgkarTajkgeBldMNalKa. enHKWCakrNIBiessEdl
GkSNWt GacRtUv)ankMNt;BI strain diagram edaysal;tmFMbMput. enAeBlEdlbnkcakpitmantmFMCag
P enaHeKehAmuxkat;enaHfa compression control. pymkvijeKehAfa tension control .
y

karviPaK balanced column section GacRtUv)anBnl;dUcxageRkam

!> yk c CacmayBIsrsrgkarsgt;qaybMputmkGkSNWt. BI strain diagram
b

cb (balanced)
=
d

eday E
Cb =

0.003
0.003 +

(11-3)

fy
Es

= 200000MPa
600d
600 + f y

km<s;bkrgkarsgt;smmUl (equivalent compressive block)

eRKOgbgMrgkarsgt; nigrgkarBt;

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600
d
ab = 1cb =
600 + f 1
y

0.85

f ' 28
)
1 = 0.85 0.05( c
7

0.65

Edl

(11-4)
f 'c 28MPa

sRmab;ebtugEdlmanersIusg; 28MPa < f ' 56MPa

c

f 'c > 56 MPa

@> BIsmIkarlMnwg plbUkkmaMgtamTisedkesIsUn

Pb Cc C s + T = 0

Edl

nig T = A f

Cc = 0.85 f 'c ab

(11-5)

enAeBlEdlEdkrgkarsgt;eFVIkardl; yield
c d'
f ' = 600
f pymkvij
c

Cs = A' ( f 's 0.85 f 'c )

f 's = f y
s

Pb = 0.85 f 'c ab + A's ( f 's 0.85 f 'c ) As f y

(11-6)

#> cMNakpit e RtUv)anvas;BI plastic centroid nig e' RtUv)anvas;BITIRbCMuTmn;nEdkrgkarTaj.

e' = e + d " sRmab;krNIrnH e' = e + d " Edl d " CacmayBITIRbCMuTmn;)asc
i eTATIRbCMuTmn;Edkrgkar
Taj. e RtUv)anKNnaedayKitm:Um:g;Rtg; plastic centroid
b

a
d " ) + C s (d d ' d " ) + Td "
2
a
Pb eb = 0.85 f 'c ab(d d " ) + A' ( f 's 0.85 f 'c )(d d ' d " ) + As f y d "
2

Pb eb = Cc (d

(11-7)

b
cMNakpit balanced eccentricity
eb =

(11-8)

Mb
Pb

(11-9)

sRmab;muxkat;minEmnctuekaN eyIgeRbIviFIsaRsdUcKakgkarviPaK edayKitRkLapBitR)akdrbs;

ebtugrgkarsgt;.
emKuNkat;bnyersIusg; sRmab; balanced condition CamYy f = 400MPa RtUv)ansnt;yk 0.65
b 0.7 . enHedaysar = = Ef = 0.002 .
]TahrN_11> 2 kMNt;kmaMgsgt; balanced compressive force P rYckMNt; e nig M sRmab;
muxkat;bgajkgrUb. eK[ f ' = 27MPa nig f = 400MPa .
y

dMeNaHRsay

!> sRmab; balanced condition bERmbRmYlrageFobenAkgebtugKW 0.003 ehIybERmbRmYlrageFob

enAkgEdk
y =

T.Chhay

400
= 0.002
200000
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@> TItaMgGkSNWt
cb =

600
d = 0.6 500 = 300mm
600 + f y

rUbTI6 balanced condition

ab = 1cb = 0.85 300 = 255mm

@> RtYtBinit Edkrgkarsgt;/ BI strain diagram

's

0.003

c d ' 300 50
=
c
300

's = 0.0025 > y

dUcenHEdkrgkarsgt; yield

c d"
f 's = 600
fy
c
300 50
f 's = 600
= 500MPa > 400 MPa
300

bRtYtBinittam

dUcenH f ' = f = 400MPa

$> KNnakmaMgmanGMeBImkelImuxkat;
s

Cc = 0.85 f 'c ab = 0.85 27 255 350 = 2048.3kN

Ts = As f y = 282 400 = 985.2kN
C s = A's ( f y 0.85 f 'c ) = 28 2 (400 0.85 27) = 928.7 kN

%> KNna P nig e

e

Pb = Cc + Cs T = 2048.3 + 928.7 985.2 = 1991.8kN

a
M b = Pb eb = Cc (d d " ) + C s (d d ' d " ) + Td "
2
255
M b = 1991.8eb = 2048.3(500
225) + 928.7(500 50 225) + 985.2 225 = 732.8kN .m
2
732.8
eb =
= 0.368m
1991.8

^> sRmab; balanced condition = 0.65

eRKOgbgMrgkarsgt; nigrgkarBt;

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Pb = 0.65 1991.8 = 1294.67kN

M b = 0.65 732.8 = 476.32kN .m

11>6> muxkat;ssreRkamGMeBIbnkcakpit (Column sections under eccentric loading)

sRmab;krNIBIr enAeBlEdlmuxkat;ssr)ak;edaykarsgt; bkarTaj smIkarlMnwgmUldanBIrGac
RtUv)aneRbIsRmab;viPaKssrEdlrgbnkcakpit.

rUbTI7 krNITUeTA muxkat;ctuekaNEkg

!> plbUkkmaMgtamGkSedk btamGkSQRtUvesIsUn
@> plbUkm:Um:g;eFobnwgGkSNamYyRtUvesIsUn
eyagtamrUb eKGacsresrsmIkarTaMgBIrxagelIdYcxageRkam
(11-10)
!> P C C + T = 0
Edl C = 0.85 f ' ab
C = A' ( f ' 0.85 f ' )
RbsinebIEdkrgkarsgt; yield enaH f ' = f
RbsinebIEdkrgkarTaj yield enaH f = f
T=A f
@> Kitm:Um:g;Rtg;cMNuc A
n

a
Pn e'Cc (d ) Cs (d d ' ) = 0
2

Edl
b

Pn =

e' = e + d "

e' = e + d

d " CacmayBITIRbCMuTmn;)asiceTATIRbCMuTmn;rbs;EdkrgkarTaj
sRmab;muxkat;ssrEdlmanEdksIuemRTI

1
a

Cc ( d ) + C s ( d d ' )

2
e'

Kitm:Um:g;Rtg; C
T.Chhay

h
2

(11-11)

(11-12)

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a
a
a

Pn e'(d ) T (d ) C s ( d ' ) = 0
2
2
2

a
a
T (d ) + Cs ( d ' )
2
2
Pn =
a
(e'+ d )
2

RbsinebI A = A' ehIy

s

f s = f 's = f y

(11-13)

(11-14)

enaH

As f y (d d ' ) As f y (d d ' )
=
a
h a
e'+ d
e +
2
2 2
h a
Pn (e + )
2 2
As = A's =
f y (d d ' )

Pn =

(11-15)

(11-16)

11>7> ersIusg;rbs;ssrsRmab;kar)ak;edaykarTaj (Strength of columns for tension failure)

enAeBlEdlssrrgbnkcakpitCamYynwgcMNakpit e FM enaHeKrMBwgfassrnwg)ak;edaykarTaj.
ssr)ak;edayEdkeFIVkardl; yield ehIyebtugEbkenAeBlEdl strain rbs;EdkFMCag ( = f / E ) .
kgkrNIenH nominal strength P nwgmantmtUcCag P bkcMNakpit e = M / P FMCag balanced
eccentricity e . edaysarkgkrNIxHeKmankarBi)akkgkarTsSn_TayfavaCamuxkat; tension control b
compression control enaHeKGacsnt;fa tension failure GacekIteLIgenAeBl e > d . karsnte; nHGac
eFVIeLIgenAeBleRkay.
smIkarlMnwgTUeTA
P C C + T = 0 nig
a
P e'C (d ) C (d d ' ) = 0 GacRtUv)aneRbIR)as;sRmab;KNna nominal strength
2
rbs;ssr.
!> sRmab;kar)ak;edaykarTaj EdkrgkarTaj yield f = f . snt;fakugRtaMgEdkrgkarsgt;
f' = f .
@> KNna P = C + C T
Edl C = 0.85 f ' ab
y

Cs = A's ( f y 0.85 f 'c )

T = As f y

#> KNna P edayKitm:Um:g;Rtg; A

n

a
Pn e' = Cc (d ) + Cs (d d ' )
2

eRKOgbgMrgkarsgt; nigrgkarBt;

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Edl e' = e + d " b e' = e + d h2 enAeBl A = A'

$> BICMhan @ nig # eyIg)an
s

Cc + C s T =

1
a

Cc ( d ) + C s ( d d ' )

e'
2

vaCasmIkarTIdWeRkTI2 EdlmanGBaat a . CMnYstm C / C nig T ehIyedaHRsayrk a .

%> eRkayBICMnYs C / C nig T smIkardWeRkTI2 Gacsresry:agsRmYldUcxageRkam
c

Aa 2 + Ba + C = 0

Edl

A = 0.425 f 'c b
B = 0.85 f 'c b(e'd ) = 2 A(e'd )
C = A's ( f 's 0.85 f 'c )(e'd + d ' ) As f y e'
a=

B B 2 4 AC
2A

RbsinebI f ' 0.85 f ' < 0 RtUvykvaesI 0 .

^> CMnYs a eTAkgsmIkarCMhan @edIm,ITTYl P . m:Um:g; M kMNt;tam M = P e
&> RtYtBinitemIlfaetIEdkrgkarsgt; yield dUckarsnt; bGt;. RbsinebI ' enaH Edkrgkar
sgt; yield . pymkvij f ' = E ' . GnuvtCMhan @ dl;% mgeTot. ' = [(c d ' ) / c]0.003 /
f
nig c = a / .
=
E
*> RtYtBinitfamuxkat;Ca tension control . Tension control enAeBlNa e > e b P < P .
(> Net tensile strain enAkgmuxkat; CaFmtaFMCag limit strain sRmab; compression-controlled
section 0.002 . dUcenHtmnemKuNkat;bnyersIusg; ERbRbYlcenaHBI 0.65 b 0.70 nig 0.90 .
1
5
smIkar = 0.7 + ( 0.002)( 200
) b = 0.7 + 0.2
sRmab;muxkat;EdlmanEdkkgdUcrWusr
c/d 3
3
s

spiral section

nig = 0.65 + (

0.002)(

250
)
3

b = 0.65 + 0.25 c /1d

5

3

sRmab;muxkat;epSgeTot

RtUv)aneRbIsRmab;KNnarkemKuNkat;bnyersIusg; .
]TahrN_11>3 kMNt; nominal compressive strength P sRmab;muxkat;Edl[dUckgrUbxageRkam
RbsinebI e = 500mm .
n

dMeNaHRsay

!> eday e = 500mm > d = 485mm snt;famuxkat;)ak;kglkxN tension failure condion controls

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EdlRtUveFVIkarRtYtBinitenAeBleRkayeTot. Strain rbs;EdkrgkarTaj GacFMCag dUcenHeyIg

ykkugRtaMg f . edaysnt;faEdkrgkarsgt; yield f ' = f EdlRtUvRtYtBinitenAeBleRkay.
s

rUbTI8 ]TahrN_TI3 kar)ak;edaykarTaj Tension failure

@> BIsmIkarlMnwg P = C + C T
Edl C = 0.85 f ' ab = 0.85 27 350a = 8.03akN
n

Cs = A's ( f y 0.85 f 'c ) = 4

28 2
(400 0.85 27) = 928.68kN
4

T = As f y = 28 2 400 = 985.2kN

Pn = 8.03a + 928.68 985.2 = 8.03a 56.52

#> Kitm:Um:g;Rtg; A

Pn =

1
a

Cc ( d ) + C s ( d d ' )

2
e'

edayTIRbCMuTmn;)asic plastic centroid sitenAelITIRbCMuTmn;nmuxkat; d "= 210mm .

e' = e + d " = 500 + 210 = 710mm
1
a

8.03a (485 ) + 928.68(485 65) = 0.0056a 2 + 5.49a + 549.36

Pn =

710
2

$> pMsmIkar 1 nig 2 eyIg)an

0.0056a 2 + 2.54a 605.88 = 0
a = 172.74mm

%> P = 8.03 172.74 56.52 = 1330.58kN

n

M n = 1330.58 0.5 = 665.29kN .m

^> RtYtBinitfa Edkrgkarsgt; yield bGt;

eRKOgbgMrgkarsgt; nigrgkarBt;

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172.74
= 203.22mm
0.85
203.22 65
400
's =
0.003 = 0.00204 > y =
= 0.002
203.22
200000
c=

dUcenHEdksgt; yield

RtYtBinit strain enAkgEdkTaj

485 203.22
0.003 = 0.00416 > y
203.22

s =

RbsinebIEdksgt;Gt; yield eRbI f ' = ' E rYceFVIkarKNnaeLIgvij.

&> KNna eday = 0.00416 muxkat;sitenAkgtMbn; transition region
s

250
= 0.83
3
Pn = 0.83 1330.58 = 1104.38kN
M n = 0.83 665.29 = 552.19kN .m

= 0.65 + ( t 0.002)

11>8> ersIusg;rbs;ssrsRmab;kar)ak;edaykarsgt; (Strength of columns for compression failure)

RbsinebIbnkGnuvtn_sgt; P FMCagbnk balanced force P bcMNakpit e = MP tUcCag e enaH
ssrnwgrMBwgfaRtUv)ak;edaykarsgt;. kgkrNI compression controls ehIy strain rbs;ebtugnwgmantm
0.003 Edl strain rbs;EdkmantmtUcCag . PaKeRcInrbs;muxkat;ssrnwgrgkarsgt;. GkSNWtxit
eTArkEdkrgkarTaj edaybegInmuxkat;sgt; dUcenHcmayeTAGkSNWt c > c .
edaysareKBi)akkgkarTsSn_TaynUvmuxkat;ssrfa tension failure b compression failure
eK)ansnt;fa enAeBl e < 2d3 enaHssr)ak;eday compression failure EdlRtUvepgpat;enAeBleRkay.
edIm,IKNna nominal load strength P eKeRbIeKalkarN_saTic. karviPaKmuxkat;ssrsRmab;kar)ak;eday
n

rUbTI9 daRkam strain enAeBl compression controls

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karsgt; compression failure eKGaceRbIsmIkar P C C + T = 0 nigsmIkar

a
P e'C (d ) C (d d ' ) = 0 nigdMeNaHRsaymYykgcMeNamdMeNaHRsayxageRkam.
2
n

11>8>1> dMeNaHRsaysakl,g (Trial solution)

dMeNaHRsayenHRtUv)ansegbdUcCMhanxageRkam
!> KNnacmayeTAGkSNwtsRmab;muxkat; balanced section c

cb =

600d t
600 + f y

(11-3)

@> kMNt; P edayeRbIlkxNlMnwg

n

Pn = Cc + Cs T

(11-10)

#> KNna P edayKitm:Um:g;Rtg;EdkrgkarTaj A

n

a
Pn e' = Cc (d ) + Cs (d d ' )
2

Edl
b

(11-11)

kgkrNITUeTA
h
e' = e + d enAeBl A = A'
2
e' = e + d "

Cc = 0.85 f 'c ab
Cs = A's ( f 's 0.85 f 'c )
T = As f s

$> edaysnt;tm c > c KNna a = c . snt;

%> KNna f
b

f 's = f y

d c
f s = s Es = 600 t
fy
c

^> CMnYstmEdlrkeXIjeTAkgsmIkarCMhan @ nigCMhan # edIm,Irk P nig P . RbsinebI

P P eRCIsyktmtUcCageK bmFmPaKn P nig P . EtebI P mantmxusKaqayBI P eK
RtUvsnt; c b a fI ehIyeFVIkarKNnaeLIgvijcab;epImBICMhan $ rhUtdl; P P . eKGacTTYlyk)an
ebI P nig P xusKa 1% .
&> epgpat;fa Edkrgkarsgt; yield edayKNna ' = 0.003[(c d ') / c] ehIyeRbobeFobCamYy
f
= E . enAeBlEdl ' Edkrgkarsgt; yield RbsinebImindUcenaHeT f ' = ' E b
n1

n1

n2

n1

n2

n1

n2

n1

n1

n2

n2

n2

c d'
f s = 600
fy
c

*> epgpat;fa e < e b P > P sRmab; compression failure.

b

eRKOgbgMrgkarsgt; nigrgkarBt;

269

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NPIC

(> sRmab; compression controlled section CaTUeTA net tensile strain enAkgmuxkat;tUcCag
0.002 . dUcenH emKuNkat;bnyersIusg; = 0.65 b 0.70 sRmab;ssrEdleRbIEdkkgvN.
t

]TahrN_11>4 kMNt; nominal compressive strength P sRmab;muxkat;Edl[RbsinebI e = 254mm .

n

rUbTI10 ]TahrN_TI4 Compression controls

dMeNaHRsay
!> edaysar e = 254mm < 23d = 333.33mm . snt; compression failure. karsnt;enHRtUvepgpat;enA
eBleRkay. KNnacmayeTAGkSNWtsRmab; balanced section c :
b

cb =

600 500
600d t
=
= 300mm
600 + f y 600 + 400

@> BIsmIkarlMnwg
Pn = Cc + Cs T

Edl

(11-10)

Cc = 0.85 f 'c ab = 0.85 27 a 350 = 8.03akN

C s = A's ( f y 0.85 f 'c ) = 28 2 (400 0.85 27) = 928.68kN

edaysnt;Edkrgkarsgt; yield karsnt;enHRtUvepgpat;enAeBleRkay

T = A f = 28 f = 2.46 f kN f < f
2

Pn = 8.03a + 928.68 2.46 f s

#> Kitm:Um:g;Rtg; A

Pn =

a
1

Cc ( d ) + C s ( d d ' )

e'
2

(11-11)

TIRbCMuTmn;)asicsitenAelITIRbCMuTmn;rbs;muxkat; d "= 225mm

e' = e + d " = 254 + 225 = 479 mm

1
a

8.03a 500 + 928.8(500 50 ) = 8.38a 0.0084a 2 + 872.57

Pn =

479
2

T.Chhay

270

Members in Compression and Bending

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Department of Civil Engineering

$> edaysnt; c = 342mm EdlmantmFMCag c

= 300mm

a = 0.85 342 = 290.7 mm

CMnYstm a eTAkgsmIkarkgCMhanTIBIrxagelIeyIg)an
Pn1 = 8.38 290.7 0.0084 290.7 2 + 872.57 = 2598.78kN

%> KNna f BIdaRkam strain enAeBlEdl c = 340mm

s

500 342
600 = 277.19 MPa
342
f
277.19
s = t = s =
= 0.00139
Es 200000

fs =

^> edayCMnYs a = 290.7mm nig

f s = 277.19 MPa

eTAkgsmIkarCMhanTImYyedIm,IKNna P

n2

Pn 2 = 8.03 290.7 + 928.68 2.46 277.19 = 2581.11kN

eday P nig P mantmxusKamindl; 1% dUcenHeyIgyk P = 2581.11kN

n1

n2

M n = Pn e = 2581.11 0.254 = 655.6kN .m

&> epgpat;fa Edkrgkarsgt; yield BIdaRkam strain

's =

342 50
0.003 = 0.00256 > y = 0.002
342

dUcenH Edkrgkarsgt; yield dUckarsnt;.

*> P = 2581.11kN > P = 1991.8kN ehIy e = 254mm < e = 368mm bgajfavaCamuxkat;
compression control dUckarsnt;. cMNaM eKGaceFVIkarsakl,gKNnaedIm,I[ P nig P mantm
kan;EtesIka.
(> KNna
n

n1

d t = d = 500mm

c = 342mm
500 342
= 0.003
= 0.00139 < 0.002
342

enAnIv:UedkrgkarTaj

n2

enaH = 0.65

Pn = 0.65 2581.11 = 1677.72kN

M n = 0.65 655.6 = 426.14kN .m

11>8>2> dMeNaHRsayviPaKcMnYn (Numerical Analysis Solution)

enAeBl compression control karviPaKssrGaceFVIeTA)anedaykat;bnykarKNnamkRtwmsmIkar
dWeRkTI3 EdlmanTRmg; Aa + Ba + Ca + D = 0 rYcedaHRsayrktm a edayviFIcMnYn numerical method
b a GacTTYl)anBIm:asIunKitelx. BIsmIkarlMnwg
3

Pn = Cc + Cs T = (0.85 f 'c ab) + A's ( f y 0.85 f 'c ) As f s

eRKOgbgMrgkarsgt; nigrgkarBt;

271

(11-10)
T.Chhay

mhaviTalysMNg;sIuvil

NPIC

Kitm:Um:g;Rtg;EdkTaj A

Pn =

a
a
1
1

Cc (d ) + Cs (d d ' ) = 0.85 f 'c ab(d ) + A's ( f y 0.85 f 'c )(d d ' )

2
2
e'
e'

(11-11)

BIdaRkam strain
d c
s = t
0.003 =
c

(d

)
0.003

kugRtaMgenAkgEdkTajKW
f s = s Es = 200000 s =

600
( 1d a )
a

edayCMnYstm f eTAkgsmIkar (11-10) nigedaHRsaysmIkar (11-10) nig (11-11) eRkayBI

sRmYlrYc eyIgTTYl)an
s

0.85 f 'c b 3
2

a + [0.85 f 'c b(e' d )]a + [ A's ( f y 0.85 f 'c )(e'd + d ' ) + 600 As e' ]a 600 As e' 1d = 0
c

enHCasmIkardWeRkTI3 EdlmanTRmg; Aa
Edl A = 0.852f ' b

+ Ba 2 + Ca + D = 0

B = 0.85 f 'c b(e' d )

C = A's ( f y 0.85 f 'c )(e' d + d ' ) + 600 As e'
D = 600 As e' 1d

enAeBlEdleKKNna)antm A / B / C nig D enaH a GacRtUv)anKNnaedayviFIsakl,g b

TTYl)anedaypal;BIm:asIunKitelx. dMeNaHRsaysmIkardWeRkTI3 GacTTYl)anedayeRbIviFI NewtonRaphson . viFIenHmanRbsiTPaBkgkaredaHRsayrkbsrbs; f ( x ) = 0 . vaTak;TgnwgbeckeTssamBa
ehIyeKqab;TTYl)ancemIyedayeFVItamCMhanxageRkam
!> [ f (a) = Aa + Ba + Ca + D nigKNna A / B / C nig D
@> KNnaedrIevTImYyn f (a) f ' (a) = 3 Aa + 2Ba + C
#> edaysnt;tmdMbUg a KNnatmbnab;
3

f ( ao )
a1 = ao
f ' ( ao )

$> edayeRbItm a KNna a dUcsmIkarxagelI

1

f (a1 )
a2 = a1
f ' (a1 )

T.Chhay

272

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viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

%> Gnuvtn_nUvviFIenHrhUtdl;)antmsuRkitmYy a a . kgkrNIviPaKssrenAeBl compression

control tm a EtgEtFMCag a . dUcenH eKcab;epImCamYy a = a ehIyGnuvtsmIkarxagelIBIrdgedIm,I
TTYl)ancemIy.
n

n 1

]TahrN_11>5 eFVI]TahrN_TI11>4eLIgvijedayeRbIviFI numerical analysis

dMeNaHRsay

!> KNna A / B / C nig D nigkMNt;

A=

f (a)

0.85 f 'c b 0.85 27 350

=
= 4016.25
2
2

B = 0.85 f 'c b(e'd ) = 0.85 27 350(479 500 ) = 168682.5

C = A's ( f y 0.85 f 'c )(e' d + d ' ) + 600 As e'
C = 28 2 (400 0.85 27)(479 500 + 50) + 600 28 2 479
C = 734800328.08
D = 600 As e' 1d = 600 282 479 0.85 500 = 300844190383.4
f (a ) = 4016.25a 3 168682.5a 2 + 734800328.08a 300844190383.4

@> KNnaedrIevTI1
f ' (a ) = 12048.75a 2 337365a + 734800328.08

#> [ a

= ab = 255mm

a1 = 255

sRmab;muxkat; balanced section c

= 300mm

nig a

= 255mm

f (255)
= 295.39
f ' (255)

$> nigKNna a
2

a2 = 295.39

f (295.39)
= 292.4mm
f ' (295.39)

tmrbs; a mantmRsedognwg a enAkg]TahrN_TI3. CMnYstm a eTAkgsmIkar (11-10) b (11-11)

eKTTYl)an P = 2594.66kN
n

11>8>3> dMeNaHRsayRbEhl (Approximate Solution)

smIkar approximate RtUv)anesIeLIgeday Whitney edIm,IedaHRsayrk nominal compressive
strength sRmab;ssrxIenAeBl compression control.
Pn =

A's f y
bhf 'c
+
3he
e
+ 1.18
+ 0.5
2
d
(d d ' )

eRKOgbgMrgkarsgt; nigrgkarBt;

(11-17)

273

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mhaviTalysMNg;sIuvil

NPIC

smIkarxagelIenHGaceRbIeTA)ansRmab;EtssrEdlmansrsEdksIuemRTIteRmobEtmYyRsTab;
ehIyRsbeTAnwgGkSnkarBt;.
smIkar approximate TI2 RtUv)anesIeLIgeday Hsu
1. 5

Pn Pb M n

+
Po Pb M o

Edl

= 1.0

Pn nominal axial strength

Pb , M b nominal load

(11-18)

nmuxkat;ssr

nig nominal moment nmuxkat; balanced section

M n = nominal bending moment = Pn e

enAeBl e=0 P = 0.85 f ' ( A

A = gross area nmuxkat; = bh
A = muxkat;EdkbeNayminEmnkugRtaMgsrub

Po = nominal axial load

Ast ) + Ast f y

st

]TahrN_6 kMNt; nominal compressive strength P sRmab;muxkat;Edl[dUckg]TahrN_TI4

n

edaysmIkar !!-!& nig !!-!*edayeRbInUvcMNakpitdUcKa e = 254mm rYceRbobeFobcemIy.

dMeNaHRsay

!> dMeNaHRsaytamsmIkar Whitney

k> lkNnmuxkat; b = 350mm / h = 550mm / d = 500mm / d ' = 50mm / A' = 2463mm
nig (d d ' ) = 450mm
x> GnuvtsmIkar Whitney
Pn =

350 550 27
2463 400
+
= 2745.15kN
3 550 254
254
1
.
18
+
+
0
.
5
500 2
450

Pn = 0.65 2745.15 = 1784.35kN

K> P EdlKNnaedaysmIkar Whitney CatmEdlminsnSMsMcenAkg]TahrN_enH ehIytm

P = 2745.15kN KWFMCagtmsuRkit P = 2581.11kN EdlKNnaedaysmIkarsaTickg]TahrN_TI4.
@> dMeNaHRsaytamsmIkar Hsu
k> sRmab; balanced condition P = 1991.8kN nig M = 732.8kN ]TahrN_TI2
x> P = 0.85 f ' ( A A ) + A f = 0.85 27 (550 350 2 2463) + 2 2463 400
n

st

st

Po = 6275.22kN

K>
T.Chhay

Pn 1991 .8
0254 Pn
+

6275 .22 1991 .8 732 .8

274

1 .5

=1
Members in Compression and Bending

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

edayKuNnwg 1000 ehIyedaHRsayrk P

0.23346 Pn + 0.00654 Pn1.5 = 1465

edaykarsakl,gtm P = 2587.65kN EdlmantmRbEhl 2581.11kN EdlKNnaedaysa

n

Tic.
11>9> ]TahrN_sRmab;daRkamGnrkm (Interaction Diagram Example)
enAkg]TahrN_TI11>2 bnk balanced load P , M nig e RtUv)anKNnasRmab;muxkat;dUckgrUb
TI6 e = 368mm . dUcKa enAkg]TahrN_TI3 nigTI4 load capacity sRmab;muxkat;dUcKaRtUv)anKNna
sRmab;krNIenAeBl e = 500mm tension failure nigenAeBl e = 254mm compression failure.
tmTaMgenHnwgRtUvbgajenAkgtaragTI1.
edIm,IKUrdaRkamGnrkmrvagbnk nigm:Um:g; tmepSgn P nig M RtUv)anKNnasRmab;tm
e epSg Edl e ERbRbYlBI e = 0 eTA e = Gtibrma sRmab;krNIm:Um:g;Bt;suT pure moment enAeBl
P = 0 . daRkamGnrkmrvagbnk nigm:Um:g;RtUv)anbgajkgrUbTI11. bnk P = 4078.90kN Cabnk
cMGkStamRTwsI enAeBl e = 0 . Et ACI Code GnuBaatbnkGb,brmaRtwmEt 0.8P = 3263.12kN Edl
b

no

no

taragTI1taragKNnasegb
eRKOgbgMrgkarsgt; nigrgkarBt;

rUbTI11daRkamGnrkmrvagbnk nigm:Um:g;
275

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mhaviTalysMNg;sIuvil

NPIC

RtUvKanwgcMNakpwtGb,brma. cMNaMfa sRmab;kar)ak;edaykarsgt; compression failure e < e nig

P > P ehIysRmab;kar)ak;edaykarTaj (tension failure) e > e nig P < P . krNI e = Gtibrma
ssrrgnUv m:Um:g;Bt;suTdUckrNI Fwm.
b

11>10> ssrmuxkat;ctuekaNCamYyEdkxag (Rectangular columns with side bars)

enAkgmuxkat;ssrxH EdkRtUv)aneKdak;BRgaytamRCugTaMgGs;. EdkxagRtUv)andak;tamkm<s;nmux
kat;edaybEnmeTAelIEdkTaj nigEdksgt; A nig A' ehIyRtUv)aneKkMNt;[eQaH A rUbTI12.
kgkrNIenH viFIsaRskgkarKNnaEdl)anBnl;rYcmkehIyGacRtUv)anGnuvt edayKitBicarNabEnmkar
pas;br strain tamkm<s;nmuxkat; nigTMnak;TMngkmaMgenAkgEdkxagnImYyeTAkgtMbn;sgt; btMbn;Taj
nmuxkat;. kmaMgTaMgenHRtUv)anbUkbEnmeTAelI C C nig T edIm,IkMNt; P smIkarmanragdUcxag
eRkam
s

ss

Pn = Cc + Cs T

(11-10a)

]TahrN_TI11> 7 Bnl;BIkarKNnaenH. cMNaMfa RbsinebIEdkxagsitenAEk,rGkSNWt rUb12 b

strain nigkmaMg enAkgEdkmantmtUcNas;EdleKGacecal)an. cMENkEdkEdlsitenAEk,r A nig A'
man tmFMKYrsm nigCYybegInlTPaBRTRTg;bnknmuxkat;.
s

rUbTI12 EdkxagenAkgmuxkat;ctuekaNEkg

]TahrN_11>7 kMNt;bnk P m:Um:g; M nigcMNakpit e sRmab;muxkat;bgajkgrUbTI13. edayeRbI

b

f 'c = 28MPa
T.Chhay

nig

f y = 400 MPa

276

Members in Compression and Bending

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Department of Civil Engineering

dMeNaHRsay muxkat; balanced section RtUv)anKNnadUcKanwg]TahrN_TI11>2Edr. eK[ b = h =

/ d = 485mm nig d ' = 65mm . A = A' = 5 324
6 DB32 3DB32 sRmab;mag.
1> KNnacmayeTAGkSNWt

550mm

= 4021.24mm 2 (5DB32 )

/ nigEdkxag

600
d = 600 485 = 291mm
cb =
600 + f t 600 + 400
y

ab = 0.85cb = 0.85 291 = 247.35mm

2> KNnakmaMgenAkgebtug nigEdk tamryrUb 13 a . enAtMbn;sgt;

Cc = 0.85 f 'c ab = 0.85 28 247.35 550 = 3237.81kN
Cs = A's ( f 's 0.85 f 'c )

enAnIv:U 65mm
c d'
291 65
f 's1 = 600
= 600
= 466MPa > 400MPa
c
291

dUcenH

f 's = 400MPa

Cs1 = 4021.24(400 0.85 28) = 1512.79kN

enAnIv:U 170mm
c d'
291 170
f 's 2 = 600
= 600
= 249.48MPa
c
291
32 2
(249.48 0.85 28) = 363kN
Cs 2 = 2
4

enAnIv:U 275mm
c d'
291 275
f 's 3 = 600
= 600
= 32.99MPa
c
291
32 2
(32.99 0.85 28) = 14.78kN
Cs 3 = 2
4

enAkgtMbn;Taj nIv:U 380mm

s4 =

380 291
0.003 = 917.53 10 6
291

f s 4 = 200000 917.53 10 6 = 183.5MPa

T1 = 2

32 2
4

(183.5) = 295.16kN

T2 = 4021.24 400 = 1608.5kN

eRKOgbgMrgkarsgt; nigrgkarBt;

277

T.Chhay

mhaviTalysMNg;sIuvil

3> KNna

NPIC

Pb = Cc + C s T

Pb = 3237.81 + (1512.79 + 363 + 14.78) (295.16 + 1608.5) = 3224.72kN

4> KNnam:Um:g;Rtg;TIRbCMuTmn;)asic
M b = 3237.81 151.325 + 1512.79 210 + 363 105 + 295.16 105 + 1608.5 210
M b = 1214.54kN .m
M
1214.54
eb = b =
= 0.377m
Pb 3224.72

5> KNna sRmab; balanced section

= y = 0.002 = 0.65

Pn = 0.65 3224.72 = 2096.07kN

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Department of Civil Engineering

niig M b = 0.65 1214.54 = 749.45kN .m

]TahrN_11>8 edaHRsay]TahrN_TI11>7 eLIgvijenAeBlEdl e = 152mm .

dMeNaHRsay

enaHvaCalkxN)ak;edaykarsgt; (compression failure

condition). snt; c = 399.5mm edaykarsakl,g nig a = 399.5 0.85 a = 339.58mm
rUbTI13 b .
2> KNnakmaMgenAkgebtug nigEdk
1> edaysar

e = 152mm < eb = 326mm

Cc = 0.85 28 339.58 550 = 4445.1kN

dUcKanwgkrNI balanced

f s1 = 400 MPa

nig

f s 2 = 344.68MPa
f s 3 = 186.98MPa
f s 4 = 29.29MPa
f s 5 = 128.41MPa

3> KNna

Cs1 = 1512.79kN

nig
nig
nig
nig

Cs 2 = 516.13kN
Cs 3 = 262.48kN
Cs 4 = 8.83kN
T = 516.37kN

Pn = Cc + Cs T = 6228.96kN

M n = Pn e = 6228.96 152 = 946.8kN .m

4> KNna Pn edayKitm:Um:g;Rtg; As

1
a

Cc (d ) + C s1 (d d ' ) + C s 2 (d d ' s) + C s 3 (d d '2s) + Cs 4 (d d '3s )

2
e'

550
h
= 362mm
e' = e + d = 152 + 485
2
2
Pn =

s = 105mm

KMlatrvagEdkxag efr sRmab;]TahrN_enH

339.58

+ 1512.79(485 65) + 516.13(485 65 105)

1 4445.1 485
Pn =
2

362
+ 262.48(485 65 2 105) + 8.83(485 65 3 105)

Pn = 6230kN

5> KNna
d t = d = 485mm

c = 399.5mm

enAnIv:UEdkTaj = 0.03(dt c) / c = 0.03(485 399.5) / 399.5 = 0.00064

eday t < 0.002 enaH = 0.65
t

Pn = 0.65 6228.96 = 4048.8kN

eRKOgbgMrgkarsgt; nigrgkarBt;

279

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

M n = 0.65 946.8 = 615.42kN .m

cMNaM RbsinebIEdkxagminRtUv)anKit enaH

Pb = 3142.1kN

enA e = 152mm = 4592.23 + 1512.79 422.48 = 5682.54kN

RbsinebIeKKitEdkxagenaH Pb ekIneLIgRbEhl 2.6% nig Pn ekIneLIgRbEhl 9.6% .
Pn

11>11> lTPaBRTbnkrbs;ssrmuxkat;mUl (Load Capacity of Circular Columns)

11>11>1 lkxN Balanced
tmnbnk balanced load Pn nigm:Um:g; balanced moment M n sRmab;muxkat;mUlGacRtUv)ankM
Nt;edayeRbIsmIkarlMnwgdUckrNImuxkat;cuekaNpgEdr. srsrEdkenAkgmuxkat;rgVg;EdlRtUv)anteRmob
eTAtamcmayBIGkSTIRbCMuTmn;)asicERbRbYl KWGaRsyeTAnwgcMnYnEdkenAkgmuxkat;. bBaacMbgKWrkkm<s;
bksgt; a nigkugRtaMgenAkgsrsEdk. ]TahrN_xageRkamBnl;BIkarviPaKmuxkat;eRkamlkxN
balanced condition. nitiviFIdUcKaGacRtUv)aneRbIedIm,IviPaKmuxkat;sRmab; tension control b compression
control.
]TahrN_11>9 kMNt;bnk balanced load Pn nig m:Um:g; balanced moment M n sRmab;ssrmuxkat;rgVg;
EdkkgvNGgt;pit 400mm CamYynwg 8DB28 dUcbgajkgrUbTI14. eK[ f 'c = 28MPa nig
Fy = 400MPa .

dMeNaHRsay

1> edaysarEtEdksIuemRTInwgGkS A A Edlkat;tamTIRbCMuTmn;rgVg; enaHTIRbCMuTmn;)asicsitenAelI

GkSenaH.
2> kMNt;TItaMgTIRbCMuTmn;GkSNWt
d t = 329.34mm

y =

fy
Es

cb
0.003
600
=
=
dt 0.003 + y 600 + f y
cb =

600
329.34 = 197.6mm
600 + 400

ab = 167.96mm

3> kMNt;lkNrbs;ceRmokrgVg; circular segment rgVg;TI15

RkLapceRmokrgVg; = r 2 ( sin cos )
T.Chhay

280

(-19)

Members in Compression and Bending

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

TItaMgTIRbCMuTmn; x BITIRbCMuTmn; 0
2 (r sin 3 )
x=

(-20)

3 sin cos
Z =rx

r cos = (r a )

(-21)

a
cos = 1
r

(-22)

167.96
cos = 1
= 0.16
200

/
nig = 1.41rad
RkLapceRmokrgVg; = 2002 (1.41 0.16 0.987)
= 80.79o sin = 0.987

= 50083.2mm 2
200 0.987 3
2
x =
= 102.39mm
3 (1.41 0.987 0.16 )
Z = 200 102.39 = 97.61mm

4> kMNt;kmaMgsgt; Cc
Cc = 0.85 f 'c RkLapceRmokrgVg;
= 0.85 28 50083.2 = 1192kN

vaeFVIGMeBIenA 102.39mm BITIRbCMuTmn;ssr

5> KNna strain, stress nig kmaMgenAkgEdkrgkarTaj nigEdkrgkarsgt;.
kMNt;bERmbRmYlrageFob strain BIdaRkambERmbRmYlrageFob.
sRmab; T1
= y = 0.002
T1 = 2

282
4

f s = f y = 400MPa

400 = 492.6kN

sRmab; T2
s3 =

55.98
y = 8.5 10 4
131.74

f s 3 = 8.5 10 4 200000 = 170MPa

T2 = 2

282
4

170 = 209.36kN

sRmab; Cs1
s1 =

126.94
0.003 = 1.93 103
197.6

f s1 = 1.93 10 3 200000 = 386MPa < 400 MPa

eRKOgbgMrgkarsgt; nigrgkarBt;

281

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Cs1 = 2

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282
(386 0.85 28) = 446.05kN
4

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sRmab; Cs2
s2 =

51.18
0.003 = 7.77 10 4
197.6

f s 2 = 7.77 104 200000 = 155.4MPa

cs 2 = 2

282
(155.4 0.85 28) = 162.07kN
4

6> kugRtaMgenAkgEdkrgkarsgt;RtUv)ankat;bny edIm,IKitenAkgebtugEdlCMnYsedayEdk.

kmaMg balanced KW Pb = Cc + Cs T
Pb = 1192 + (446.05 + 162.07) (492.6 + 209.36) = 1098.16kN

sRmab;muxkat; balanced t = 0.002 nig = 0.65

Pb = 713.8kN

7> Kitm:Um:g;Rtg;TIRbCMuTmn;)asic GkS A A kat;tamGkSTIRbCMuTmn; sRmab;kmaMgTaMgGs;

M b = Pb eb = [Cc 102.39 + Cs1 129.34 + Cs 2 53.58 + T1 129.34 + T2 53.58)
M b = 263.36kN .m

M b = 171.18kN .m
eb =

263.36
= 239.8mm
1098.16

eRKOgbgMrgkarsgt; nigrgkarBt;

283

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NPIC

11>11>2 ersIusg;rbs;muxkat;mUlsRmab;kar)ak;edaykarsgt;
Strength of circular column for compression failure

muxkat;ssreRkamkmaMgcMNakpit GacRtUv)anviPaKtamCMhandUcmuxkat; balanced Edr. vaTTYl

)anedaykarsnt; C > Cb b a > ab nigKNnakmaMgenAkgebtug nigEdkenATItaMgepSgKaedIm,IkMNt;
Pn1 = Cc + Cs T . dUcKa M n GacRtUv)anKNnaedayKitm:Um:g;Rtg;TIRbCMuTmn;)asic TIRbCuMTmn;rbs;
muxkat; ehIykMNt; Pn2 = Men . RbsinebItm Pn1 nig Pn2 minRbhak;RbEhlKaeTenaH snt; C b a fI
ehIyeFVIkarKNnaeLIgvij emIlcMNucTI8. tmxusKarvag Pn1 nig Pn2 sitenArgVg; 1% . muxkat;Ca
compression controls enAeBl e < eb b Pn > Pb .
sRmab;]TahrN_ RbsinebIvaTamTarkMNt;ersIusg;rbs;muxkat;ssrenAkg]TahrN_TI9 enAeBlEdl
e = 150mm Pn GacRtUv)anKNnaedayCMhandUcnwg]TahrN_TI 9.
1> eday e = 150mm tUcCag eb = 239.8mm lkxN)ak;edaykarsgt; compression failure
condition ekIteLIg.
2> snt; c = 225mm edaykarsan > Cb = 197.6 nig a = 191.25mm
3> KNna x = 89.63mm / Z = 110.37mm RkLapceRmokrgVg; = 59332.97mm2
4> -5> KNnakmaMg Cc = 1412.125kN / Cs1 = 463.29kN Cs 2 = 228.73kN / T1 = 342.66kN /
T2 = 93.84kN

6> KNna Pn1 = Cc + Cs T = 1667.64kN

7> Kitm:Um:g;Rtg;GkSssr TIRbCMuTmn;)asic
M n = 248.1kN .m
M
Pn 2 = n = 1653.97 mm
e

EdlmantmRbEhl Pn1 tmxusKaRbEhl 1% . dUcenH

Pn = 1653.97kN

cMNaM RbsinebIEdkkgrbs;ssrCaEdkkgvNdUcrWusrenaH = 0.70 .

smIkartmRbEhl approximate equation sRmab;karKNna Pn sRmab;muxkat;mUl enAeBl
compression controls RtUv)anesIeLIgedayelak Whitney
Pn =

Edl

T.Chhay

Ag f ' c

9.6he
+ 1.18

2
(0.8h + 0.67 Ds )

Ast f y
3e

+ 1
Ds

(11-23)

RkLapmuxkat; gross area

h = Ggt;pitmuxkat;
Ag =

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Ggt;pitmuxkat;ssrEdlvas;BITIRbCMuTmn;EdkmageTATIRbCMuTmn;EdkmageTot
Ast = RkLapmuxkat;EdkbBar
e = cMNakpitEdlvas;BITIRbCMuTmn;)asic
Ds =

]TahrN_11>10 KNnaersIusg;kmaMgsgt; nominal P sRmab;muxkat;sRmab;]TahrN_TI11>9 edayeRbI

n

smIkar Whitney RbsinebIcMNakpit e = 150mm

dMeNaHRsay

1> e = 150mm tUcCag eb = 239.8mm . tamkarKNnadUceBlmun bgajfamuxkat;ssrCamuxkat;

compression controls.
2> edayeRbIsmIkar Whitney
h = 400mm

Ag =

h2 =

400 2 = 125663.7 mm 2

D s = 400 120 = 280mm

As = 8
Pn =

28 2
4

= 4926mm 2

125663.7 28

9.6 400 150

+
1
.
18

2
(0.8 400 + 0.67 280 )

4926 400
= 1785.94kN
3 150
+ 1

280

3> M n = Pn e = 1785.94 0.15 = 267.89kN .m

tm Pn enAeBlenHFMCagtm Pn = 1653.97kN EdlKNnaenAeBlmunedaysaTic.
11>11>3 ersIusg;rbs;muxkat;mUlsRmab;kar)ak;edaykarTaj
Strength of circular column for tension failure

kar)ak;edaykarTajsRmab;ssrmUlenAeBlbnkRtUv)anGnuvtn_enARtg;cMNakpit e > eb b
Pn < Pb . enAkgkrNIenH muxkat;ssrGacRtUv)anviPaKtamCMhandUckarviPaKmuxkat; balanced nigdUc
kg]TahrN_TI8. karviPaKRtUv)aneFVIeLIgedaysnt; C < Cb b a < ab rYcehIyGnuvttamCMhanBnl;
kgkfaxN11>1. cMNaMfa edaysarEtsrsrEdkRtUv)anteRmobedaymancenaHefrtambrimaRtmuxkat;
rgVg; enaHEdkTaj As Edlpl;[GacmantmtUc ehIylTPaBRTbnkkkayCamantmtUc. dUcenH eK)an
ENnaM[eCosvagkareRbIR)as;muxkat;mUlsRmab;krNIkar)ak;edaykarTaj tension failure.
eRKOgbgMrgkarsgt; nigrgkarBt;

285

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NPIC

11>12> karviPaK nigkarKNnassredayeRbIdaRkam

Analysis and Design of Column Using Charts

karviPaKmuxkat;ssrEdl)anBnl;BIxagedImKWQrelIeKalkarN_saTic. sRmab;karviPaK bkar

KNnassrCaCMhandMbUg daRkam btaragBiessGacRtUv)aneRbIedIm,IkMNt; Pn nig M n sRmab;muxkat;
Edl[ nigkarKNnamuxkat;EdkcaM)ac;sRmab; Pu nig M u Edl[. daRkam nigtaragenHRtUv)ane)aH Bum<
pSayedayviTasanebtugGaemric American Concrete Institute (ACI) viTasanebtugBRgwgedayEdk
Concrete Reinforcing Steel Institute (CRSI) nigsmaKmsIum:gt_BrEln Porland Cement Association
(PCA). karKNnassrcugeRkayRtUvEteFVIeLIgedayQrelIsmIkarsaTic edaykarKNnaedayd beday
kmviFIkMuBTr. kareRbIdaRkam ACI RtUv)anbgajenAkg]TahrN_xageRkam. daRkamRtUv)anbgajkgrUb
TI16 nigrUbTI17.TinnyTaMgenHRtUv)ankMnt;sRmab;muxkat;ssrdUcbgajenARCugxagelIEpkxagsaMn
tarag.

]TahrN_11 kMNt;srsrEdkcaM)ac;sRmab;ssr

dUcbgajenAkgrUbTI 18 a
edIm,IRTnUvbnkemKuN 2150kN nigm:Um:g;emKuN 440kN.m . ssrmanTTwg 350mm nigbeNaysrub
h = 500mm . eRbI f 'c = 28MPa / f y = 400MPa .
short tied column

dMeNaHRsay

440
1> cMNakpit e = MP u = 2150
= 204.65mm
u
yk d = 500 60 = 440mm
380
h = 500 120 = 380mm enaH =
= 0.76
500
2> eday e = 204.65 < d snt;famuxkat;)ak;edaykarsgt; (compression-controlled section)
CamYynwg = 0.65

2150
= 3307.7 kN
0.65
440
Mn =
= 676.9kN .m
0.65
Pn
3307.7 10 3
= 0.675
Kn =
=
f ' c Ag 28 350 500
Pn =

nig

e
204.65
Rn = K n = 0.675
= 0.276
h
500

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287

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eRKOgbgMrgkarsgt; nigrgkarBt;

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289

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3> BItaragkgrUbTI16 sRmab; = 0.7 / = 0.048 dUcKasRmab; = 0.8 / = 0.043

eday interpolation sRmab; = 0.76 / = 0.045
As = 0.045 500 350 = 7875mm 2

eRbI 10DB32 (As = 8042.48mm 2 )/ R)aMedImenAtamRCugxI. eRbIEdkkg DB10 @ 350mm

rUbTI18 a

]TahrN_12 eRbItaragedIm,IkMNt;bnkersIusg; P rbs;ssrxIdUcbgajkgrUbTI 18 b EdlGnuvt

n

enAcmaycMNakpit e = 305mm . eRbI

f ' c = 35MPa

dMeNaHRsay
A.

nig

f y = 400 MPa

lkNrbs;muxkat; H = 600mm / h = 600 120 = 480mm cmayrvagEdlTaj

32 2
4
=
= 0.03
600 350
8

B.

480
nigEdk sgt;. = 600
= 0.8 ehIy
eday e < d / snt;vaCa compression-controlled section.
yk t = 0.002 / ff s = 1.0 ehIy = 0.65 BItaragkgrUbTI 17 eKTTYl)an
y

K n = 0.36 =

Pn
35 600 350

dUcenH Pn = 2646kN
eRKOgbgMrgkarsgt; nigrgkarBt;

291

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C.

Epgpat;karsnt;sRmab;muxkat; compression controlled

sRmab; K n = 0.36
Rn = K n

D.

NPIC

e
= 0.183
h

BIdaRkameyIgTTYl)an = 0.019 < 0.03 / dUcenH Pn > 2646kN edIm,I)an = 0.03

karsakl,gelIkTI2 yk t = 0.0015 / f s = 0.0015 200000 = 300MPa
f s 300
=
= 0.75
f y 400

= 0.03

K n = 0.43

Pn = 0.43 35 600 350 = 3160.5kN

E.

Epgpat;karsnt; sRmab; K n = 0.43 / Rn = K n he = 0.219

BItarag = 0.03 dUcGVIEdl[
dUcenH Pn = 3160.5kN
Pn = 0.65 3160.5 = 2054.3kN nig M n = 626.6kN .m
tamkarviPaK Pn = 2027kN mantmRbEhlKanwgkarKNnaedayeRbItarag.

11>13> karKNnassreRkambnkcakpit (Design of Columns Under Eccentric Loading)

karKNnassrmanlkNsKsajCagkarviPaKssr edaysarEtbnkxageRkA Pu nigm:Um:g; M u
Casmtikm ehIyeKRtUvkarkMNt;nUvGBaatCaeRcIndUcCa b / h / As / A's CamYynwgkarkMNt;rbs; ACI
Code. vaCakarGnuvtTUeTA edaysnt;dMbUgnUvmuxkat;ssr ehIykMNt;brimaNmuxkat;EdkRtUvkar. Rbsin
ebIGkKNnaRtUvkardUrmuxkat;EdkKNna enaHmuxkat;ssrkRtUv)anEkERbeTAtamenaHEdr. ]TahrN_xag
eRkambgajBIkarKNnassr.
11>13>1 KNnassrsRmab;kar)ak;edaykarsgt; (Design of Column for Compression Failure)
sRmab; compression failure eKniymeRbI As = A's sRmab;muxkat;ctuekaN. cMNakpit e = MP u .
u
edayQrenAelItmrbs; e eKman2krNIRtUv)anbegIteLIg
1> enAeBlEdl e 100mm krNIcMNakpitGb,brmaGacekItman EdlGaceKNnaedayeRbIrUbmn
Pu = Pn = K [0.85 f ' c Ag + Ast ( f y 0.85 f ' c )] Edl = 0.65 nig K = 0.80 sRmab;ssrEdl
manEdkkgdac; nig = 0.70 nig K = 0.85 sRmab;ssrEdlmanEdkvN sUmemIl]TahrN_kg
emeronssrrgkmaMgcMGkS. sRmab;krNIepSgBIenH GkKNnaGacGnuvttamkrNITI2. krNIbnk
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enHRbRBwteTAsRmab;ssrGKarCan;eRkamnGKareRcInCan; Edlm:Um:g; M u )anmkBIRbBnmYyCan; nig

Pu )anmkBIbnkRKb;Can;EdlmanGMeBIenABIelIva.
2> tMbn; compression failure KWtMbn;EdlsitenAcenaHGkSQreTAbnat; balanced load dUcbgaj
kgrUbTI 3 nigrUbTI11. kgkrNIenH muxkat; bh GacRtUv)ansnt; ehIybnab;mkmuxkat;EdkRtUv)an
KNnasRmab; Pu nig M u Edl[. CMhannkarKNnaRtUv)ansegbdUcxageRkam
k> snt;muxkat;kaer bctuekaN bh rYckMNt; d / d ' nig e = MP u
u

x> edaysnt; As = A's KNna A's BIsmIkar

A' s f y
bhf 'c
+
Pn =
3he
e
+ 1.18
+ 0.5
2
(d d ' )
d

edayeRbI

TMhMmuxkat;Edl)ansnt; nig = 0.65 sRmab;ssrEdleRbIEdkkgFmta. yk As = A's

rYceRCIserIsmuxkat;RKb;RKan;. kMNt;muxkat;BitR)akdEdleRbIsRmab; As nig A's . m:ag
vijeToteKGaceRbIdaRkam ACI.
K> epgpat;fa 1% g = As bh+ A's 8% . RbsinebI g mantmtUc kat;bnymuxkat;
snt; b:uEnBRgIkmuxkat;RbsinebIeKcg;)anmuxkat;EdktUc.
X> epgpat;PaBRKb;RKan;rbs;muxkat;cugeRkayedayKNna Pn BIsmIkarsaTic dUcBnl;
kg]TahrN_xagedIm. Pn Pu .
g> kMNt;EdkkgcaM)ac;.
rUbmnRbhak;RbEhl approximate formula y:agsamBa sRmab;kMNt;muxkat;ssrdMbUg bh bPaK
ryEdksrub total steel retio g KW
(11-24)
Pn = K c bh 2 b
Pu = Pn = K c bh 2
Edl K c mantmdUcbgajkgtaragTI2 nigbgajkgrUbTI19 sRmab;Edk f y = 400MPa nig
As = A' s . xatrbs; K c KW kN / m 3 .

taragTI2 tmrbs; K f
c

g (% )
1%
4%
8%

= 400 MPa

Kc
f 'c (28MPa)
24817
37574
54675

eRKOgbgMrgkarsgt; nigrgkarBt;

f 'c (35MPa)
30246
43003
60103

293

f 'c (42 MPa)

35286
48044
65144

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

K c ( kN/m )
69000
64000
59000
54000
49000
44000
39000

f'c=28MPa
f'c=35MPa
f'c=42MPa

34000
29000
24000
1

8
g (%)

rUbTI19 tm K c nig g
tm K c RtUv)anbgajkgtaragTI2 CatmRbhak;RbEhl niggayRsYleRbIedaysar K c ekIneLIg
mg 5429 sRmab;karekIneLIgrbs; f 'c mg 7 . sRmab;muxkat;dUcKa enAeBlEdlcMNakpit e = MP u ekIn
u
eLIg Pn fycuH dUcenH K c fycuH. dUcenH tm K c sMEdgbnk Pn enAelIdaRkamGnrkmcenaH 0.8Pno nig
Pb dUcbgajkgrUbTI 3 nigTI 11.
Linear interpolation GacRtUv)aneRbI. ]TahrN_ K c = 46124.5 sRmab; g = 6% nig
f 'c = 28MPa . CMhankgkarKNnamuxkat;ssrGacRtUv)ansegbdUcxageRkam
1> snt;muxkat;dMbUgsRmab;muxkat;ssr bh
2> KNna K c = (bhPu 2 )
3> kMNt; g BItaragTI 2 sRmab; f 'c Edl[
4> kMNt; As = A's = g2bh rYceRCIserIsEdkbBar nigEdkkg.
5> kMNt; Pn nmuxkat;cugeRkaytamsmIkarsaTic dMeNaHRsayCak;lak;. tmn Pn KYrEt
mantmFMCagbesI Pu . RbsinebImindUecaHeT EktRmUv bh b g .
m:agvijeTot RbsinebIeKcg;)anPaKryEdksrubCak;lak; ]TahrN_ g = 6% bnab;mkGnuvtdUc
xageRkam
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1> snt; g dUcEdlTamTar nigbnab;mkKNna e = MP u

u
2> edayQrelI f 'c nig g Edl[/ kMNt; K c BItaragTI2
3> KNna bh 2 = PKu bnab;mkeRCIserIs b nig h . GnuvtCMhan 4 nig 5 eLIgvij.
c
eKKYrEtepgpat;fa 1% g 8% . dUcKa epgpat;fa c
600d t
Edl)anmkBIkarKNnatamsaTicmantmFMCag cb = 600
sRmab; compression failure .
+f
y

]TahrN_13 kMNt;muxkat;EdkTaj nigmuxkat;Edksgt;sRmab;ssrEdleRbIEdkkgFmtamanmuxkat;

400 600

edIm,IRTbnk Pu = 3470kN nig M u = 530kN .m . edayeRbI

dMeNaHRsay

f 'c = 28kN

nig

f y = 400 MPa

530
1> KNna e = MP u = 3470
= 152.74mm . eyIgman h = 600mm yk d = 550mm nig d ' = 50mm
u

edaysar e < 23 d = 366.67mm snt;fa compression failure.

2> snt; As = A's . kMNt;tmdMbUgrbs; A's tamrUbmn
Pn =

A' s f y
bhf 'c
+
3he
e
+ 1.18
+ 0.5
2
(d d ' )
d
P 3470
Pn = u =
= 5338.5kN

0.65

(11-17)

sRmab;

A' s = 4271.8mm 2 = As

eRbIEdk DB32 n = 6 edIm

6 DB32 = 4825.5mm 2 sRmab; As nig A' s rUbTI 20

eRKOgbgMrgkarsgt; nigrgkarBt;

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4825.5
3> g = 2400
= 4% EdlvaRtUvEttUcCag 0.08 nigFMCag 0.01
600
4> epgpat;muxkat;edaysmIkarsaTictamCMhannkarKNnakg]TahrN_TI4 eKTTYl)an
a = 430.18mm / c = 506.09mm / Cc = 4095.32kN
C s = 4825.5(400 0.85 28) = 1815.35kN
550 506.09
d c
f s = 600
= 52.06 MPa
= 600
506.09
c
T = As f s = 4825.5 52.06 = 251.21kN
Pn = Cc + C s T = 5659.5kN > 5338.5kN

cMNaMfa RbsinebI Pn < Pu cUrdMeLIg As nig A's rYceFVIkarKNnaeLIgvij.

5> epgpat; Pn edayeRbIsmIkar Pn = e1' Cc d a2 + Cs (d d ' ) Edl e' = e + d h2

eyIgTTYl)an Pn = 5659kN
6> sRmab;muxkat; balanced section
600
cb =
600 + f y

d t = 600 550 = 330mm

1000

edaysarEt c = 506.09mm > cb = 330mm vaCakrNI compression failure dUckarsnt;.

7> edayeRbIEdkkgmanGgt;pit 10mm
KMlatEdkkg
480
48 10
48

min 16d = min 16 32 = min 512 = 400

400
400
b

dUcenHeRbIEdkkg DB10 @ 400 .

]TahrN_14 eFVI]TahrN_TI13 eLIgvijedayeRbIsmIkar 11-24

dMeNaHRsay
1> muxkat;ssrEdl[ 400 600
2> kMNt; K c BIsmIkar 11-24
kN
3> K c = bhPu 2 = 0.65 3470
= 37073 3
2
0 .4 0 .6
m

kN
4> BItaragTI 2 brUbTI19 sRmab; K c = 37073 mm
f 'c = 28MPa eday interpolation
3
1
eyIgTTYl)an g = 1 + (37073 24817) 375744 24817
= 3.88%
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5> KNna As = A's = bh / 2 = 0.0388(400)(600)/ 2 = 4656mm 2

eRbIEdk DB32 n = 6 edIm
6> 6DB32 = 4825.5mm 2
7> kMNt; Pu edayeFVItamCMhan 4-7 sRmab;]TahrN_TI13. Pn = 5659kN > Pn = 5338.5kN
dUcenHmuxkat;EdkRKb;RKan;
8> RbsinebImuxkat;minRKb;RKan; b Pn < Pn tMeLIgmuxkat; As nig A's rYceFVIkarepgpat;eLIgvij
edIm,ITTYl)antmEk,r.

]TahrN_15 KNnamuxkat;ssrctuekaNEkgedIm,IRTbnk P

nig M u = 630kN .m
= 400MPa nig b = 450mm .
= 3150kN

CamYynwgPaKryEdksrub g RbEhl 4% . eRbI f ' = 28MPa / f

dMeNaHRsay
630
1> KNna e = MP u = 3150
= 0.2m . snt; compression failure ( = 0.65 ) RtUvepgpat;enA
u
eBleRkay ehIy As = A's
2> sRmab; = 4% nig f 'c = 28MPa enaH K c = 37574 taragTI2
3> KNna bh 2 BIsmIkar (-24): Pu = K cbh 2 b 3150 = 0.65(37574)(0.45)h 2 dUcenH
h = 0.535m dUcenHyk h = 550mm .
KNna As = A's = 0.04(4502 550) = 4950mm 2 . eRCIserIs 5DB36
( As = 5089.4mm 2 ) dUcbgajkgrUbTI 21. eRbIEdkkg DB12 @ 450 .
4> epgpat; muxkat;cugeRkayedaykarviPaK RsedogKanwg]TahrN_TI4 eyIgTTYl)an
a = 327.8mm / c = 385.65mm / Cc = 0.85 f 'c ab = 3510.7 kN / f ' s = 400MPa /
d c
C s = A' s ( f y 0.85 f 'c ) = 1914.6kN / f s = 600
= 146.79 MPa / nig T = 747 kN
c
dUcenH Pn = Cc + Cs T = 4678.3kN ehIy Pu = Pn = 3041kN < 3150kN
edaysarmuxkat;minRKb;RKan; eyIgRtUvdMeLIgmuxkat;Edk bmuxkat;ebtug rYceFVIkarepgpat;eLIg
vij. yk h = 600mm rUbTI21.
5> sRmab;muxkat; balanced section
600
cb =
d t = 318mm < c = 436.32mm d = 530mm
600 + 400
dUcenH vaCa compression failure dUckarsnt;.
c

eRKOgbgMrgkarsgt; nigrgkarBt;

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11>13>2 KNnassrsRmab;kar)ak;edaykarTaj (Design of Column for tension Failure)

kar)ak;edaykarTaj (tension failure) ekItmanenAeBl Pn < Pu bkcMNakpit e > eb dUckarBnl;
enAkgEpkTI7. kgkarKNnassr Pu nig M u RtUv)an[ ehIyvaTamTarnUvkarkMNt;muxkat;ssr nigmux
kat;Edk. vaGacRtUv)ansnt; dUckarENnaM favaCa tension control enAeBlNa
530 sMrab;
M
h < 600mm
. kgkrNIenH muxkat;ssrGacRtUv)ansnt; ehIybnab;mk As nig
>
P
600 sMrab;
h 600mm
u

RtUv)ankMNt;. daRkam ACI GacRtUv)aneRbIedIm,IKNna g sRmab;muxkat;Edl[CamYynwg As = A's .

cMNaMfa ERbRbYlcenaH 0.65(0.7) nig 0.9 dUckarBnl;kgEpkTI 4.
enAeBl tension controls EdkTaj yields b:uEnEdksgt;Gac yields nigmin yields. karsnt;dMbUg
f ' s = f y nig As = A' s . smIkar (-16) kgEpkTI 6 GacRtUv)aneRbIedIm,IKNnatmdMbUgrbs; As nig
A' s .
A' s

h a

Pn e +
2 2
As = A' s =
f y (d d ')

(11-16)

edaysar a minRtUv)andwgenAeLIy snt; a = 0.4d nig Pu = Pn bnab;mk

As = A' s =

Pu (e 0.5h + 0.2d )
f y (d d ' )

(11-25)

muxkat;ssrcugeRkayKYrRtUv)anepgpat;edaysmIkarsaTicedIm,Ibgajfa P
TI16 Bnl;BIviFIsaRskgkarKNnaenH.

T.Chhay

298

Pu

. ]TahrN_

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enAeBlbnk P mantmtUcNas;ebIeRbobeFobCamYynwgm:Um:g; M TMhMrbs;muxkat;GacRtUv)ankM

Nt;edayeRbIEt M EtmYy)anehIy edaysnt;fa P = 0 . muxkat;cugeRkayKYrRtUv)anepgpat;eday
smIkarsaTic. krNIenHekIteLIgsRmab;eRKagGKarmYyCan; bBIrCan; EdlGKarenaHRtUv)aneKeRbIsRmab;eFVI
CasaltaMgBiBN bkGKarTaMgLayNaEdlmanlkNdUcKaenaH. sRmab;krNIenH A' GacRtUv)ansnt;
[mantmtUcCag A .
karKNnay:aglMGitsRmab;saltaMgBiBNkm<s;mYyCan;Edlmansnak;BIrRtUv)anBnl; enAkgCMBUkTI 16
FwmCab; nigeRKag.
u

]TahrN_16 kMNt;srsrEdkcaM)ac;sRmab;ssrragctuekaNEkg

FmtaRTbnk P

= 1140kN

nig M

= 850kN .m

. eRbI

nig

f ' c = 28MPa

400 560

EdlmanEdkkg

f y = 400 MPa

dMeNaHRsay
1> KNna e = MP

850
= 0.7456m
1140

. yk d = 560 60 = 500mm . edaysar

bedaysar e > d snt;fassrenH)ak;edaykrNI tension failure

enaH = 0.9 RtUvepgpat;enAeBleRkay.
2> snt; A = A' nig f ' = f nigeRbIsmIkar (-25) edIm,IkMnt; A nig A' . eday P = 1140kN /
e = 745.6mm / h = 560mm / d = 500mm / nig d ' = 60mm
Mu
= 745.6mm > 530mm
Pu

As = A' s =

1140 10 3 (745.6 0.5 560 + 0.2 500 )

= 4070.71mm 2
0.9 400(500 60 )

eRKOgbgMrgkarsgt; nigrgkarBt;

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eRbI 5DB32 (4021.24mm ) sRmab; A nig A' . rUbTI22

(4021.24) = 0.0359 EdltUcCag 0.08 nigFMCag 0.01 .
3> epgpat; = 2400
560
4> epgpat;kareRCIserIsmuxkat;edaysmIkarsaTic EdlkarKNnaRsedogKanwg]TahrN_TI3
a. kMNt;tmrbs; a edayeRbIsmIkarTUeTA Aa + Ba + C = 0 CamYynwg
h
e' = e + d = 965.6mm / A = 0.425 f ' b = 4760 / B = 2 A(e' d ) = 4432512 /
2
C = A' ( f 0.85 f ' )(e' d + d ') A f e' = 758040793 . eKTTYl)an a = 147.62mm
nig c = a / 0.85 = 173.67mm .
c d'
173.67 60
b. epgpat; f ' f ' = 600
= 600
= 392.71MPa
c
173.67
c. KNna a eLIgvij
2

C = A' s ( f ' s 0.85 f ' c )(e' d + d ') As f ' s e' = 773454123.3

eKTTYl)an a = 150.25mm nig c = 176.77mm

.77 60
epgpat; f ' f ' = 600 c c d ' = 600 176176
= 396.34 MPa
.77
KNna C = 0.85 28 150.25 400 = 1430.38kN

d.

C s = A' s ( f ' s 0.85 f ' c ) = 4021.24(396.34 0.85 28) = 1498.07 kN

T = As f y = 4021.24 400 = 1608.5kN
Pn = C c + C s T = 1319.95kN

e.

5> KNna = 0.003 d c c = 0.0055 edaysarEt = 0.0055 > 0.005 enaH = 0.9

6> P = 0.9 1319.95 = 1187.95kN > 1140kN muxkat;RKb;RKan;

t

11>14> karBt;tamBIrTis (Biaxial Bending)

karviPaK nigkarKNnassreRkamGMeBIbnkcakpitEdl)anBiPakSakngmk CakrNIkarBt;mYyTis.
enHmannyfa P GnuvtenAelIGkS y rUbTI23 begIt)anbnSMnkmaMgcMGkS P nigm:Um:g;Bt;CMuvijGkS x
esInwg M nx = Pn e y b P GnuvtenAelIGkS x rUbTI24 CamYynwgcMNakpit e begIt)anbnSMnkmaMgcM
GkS P nigm:Um:g;Bt; M ny = Pn e x .
n

T.Chhay

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RbsinebIbnk P GnuvtenAkEngNamYyEdlmancmay e BIGkS x nigcmay e BIGkS y enAeBl

enaHmuxkat;ssrnwgrgnUvbnSMbnkEdlman bnkcMpit P m:Um:g;Bt;CMuvijGkS x KW M nx = Pn e y nigm:Um:g;Bt;
CMuvijGkS y KW M ny = Pn e x rUbTI25.kgkrNIenHmuxkat;ssrrgnUvkarBt;tamBIrTis. vaminEmngay
RsYleTenAeBlEdleKeRbIeKalkarN_saTicedIm,IviPaK nigKNnamuxkat;ssrsRmab;krNIenH. GkSNwtsit
enAelIRCugEdlkat;GkS x nigGkS y ehIyeKRtUvkarkarKNnadEvgedIm,IkMNt;TItaMgrbs;GkSNWtbERm
bRmYlrageFob strain RkLapebtugrgkarsgt; nigkmaMgkgrYmTaMgcMNucEdlvaGnuvt. dUcenH eKcaM)ac;
RtUvbegItnUvdMeNaHRsaydmansamBamYyedIm,IKNnanUvlTPaBRTRTg;rbs;ssrbnkcMpit nigm:Um:g;Bt;
BIrTis. rUbmn Edl)anbegIteLIgsRmab;ssrrgm:Um:g;BIrTis Tak;TgeTAnwgersIusg;rbs;vaTb;nwgkarBt;CMuvij
GkSemnImYy.
rUbTI26 bgajBIExSekagGnrGMeBIbnk-m:Um:g;kglMh 3TMhM sRmab;ersIusg;m:Um:g;BirTisrbs;ssr
rgkmaMgcMGkS. esrInExSekagGnrGMeBIbnk-m:Um:g;mYyTis EdlmankaMKUsBIGkS Pn )anbegItCapnExS
ekag GnrGMeBIbnk-m:Um:g;BIrTis. ExSekag M ox bgajnUvExSekagGnrGMeBIsRmab;m:Umg;mYyTisCMuvijGkS x
nigExSekag M oy bgajnUvExSekagGnrGMeBIsRmab;m:Umg;mYyTisCMuvijGkS y . bg;enARtg;bnkcMGkSefr Pn
bgajBIExSvNnm:Um:g;Bt; M n CMuvijGkSepSg.
ssrEdlmanmuxkat;epSgRtUv)aneRbIR)as;edIm,ITb;Tl;nwgbnkcMGkS nigmU:m:g;Bt;BIrTis. muxkat;
ssrmUl kaer bctuekaNEkgGacRtUv)aneKeRbIR)as;CamYynwglTPaBTb;nwgm:Um:g;CMuvijGkS x nigGkS y Edl
mantmesIKa bminesIKa.
n

eRKOgbgMrgkarsgt; nigrgkarBt;

301

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NPIC

302

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Department of Civil Engineering

11>15> ssrmUlCamYynwgkarrayEdkesIeRkamm:Um:g;Bt;BIrTis
Circular Columns With Uniform reinforcement Under Biaxial Bending

ssrmUlCamYynwgkarrayEdkEdlmanlkNksNan manlTPaBTb;nwgm:Um:g;dUcKaRKb;Tis. Rb
sinssrmUlRbQmnwgm:Um:g;BIrTisCMuvijGkS x nigGkS y enaHm:Um:g;mYyTissmmUl M u GacRtUv)anKNna
edayeRbIsmIkarxageRkam
Mu =

nig e =
Edl

(M ux )2 + (M uy )2
(e x )2 + (e y )2

= Pu e

(11-26)

Mu
Pu

(11-27)

m:Um:g;emKuNCMuvijGkS x
M uy = Pu e x m:Um:g;emKuNCMuvijGkS y
M u = Pu e m:Um:g;emKuNmYyTissmmUlnmuxkat;EdlekIteday M ux nig M uy
sRmab;ssrmUl eKcaM)ac;eRbIEdky:agtic6edIm ehIyEdkTaMgenaHRtUv)anBRgayesIenAkgmuxkat;

enaH.

M ux = Pu e y

]TahrN_17 ssrmUl

kMNt;lTPaBRTbnk P rbs;ssrragmUlEdlmanGgt;pit 500mm CamYyEdk 10DB32 enAeBl

Edl e x = 100mm nig e y = 150mm . eRbI f 'c = 28MPa nig f y = 400MPa .
dMeNaHRsay
1> KNnacMNakpitEdlsmmUleTAnwgkareFVIGMeBItammYyTisedayeRbIrUbmn (11-27)
n

e = e x2 + e 2y = 100 2 + 150 2 = 180.28mm

2> kMNt;lTPaBRTbnkrbs;ssredayQrelIcMNakpit e = 180.28mm . edaHRsaydUc]TahrN_

TI 9 eKTTYl)an
d = 430.7 mm
a = 250.75mm
c = 295mm edaysakl,g
C s = 1188.35kN T = 662.09kN
Pn = C c + C s T = 2871.75kN

C c = 2345.49kN

3> sRmab;lkxN balanced condition

cb =

600
600
dt =
430.7 = 258.42mm
600 + 400
1000

c = 295mm > cb

eRKOgbgMrgkarsgt; nigrgkarBt;

EdlCakrNI)ak;edaykarsgt; (compression failure)

303

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NPIC

11>16> ssrmuxkat;kaer nigctuekaNeRkamm:Um:g;Bt;BIrTis

(Square and Rectangular Columns under Biaxial Bending)

11>16>1> viFI; Bresler Reciprocal Method

ssrkaer bctuekaNrgnUvm:Um:g;Bt;CMuvijGkSemrbs;vamantmminesIKa RtUvkarbrimaNEdkxusKa
sRmab;TisedAmYy. viFIsaRsRbEhlsRmab;viPaKmuxkat;EbbenHRtUv)anbegIteLIgedayelak Boris
Bresler ehIyRtUv)aneK[eQaHfa viFI Bresler Reciprocal Method. tamryviFIenH lTPaBRTbnkrbs;
ssreRkamGMeBIm:Um:g;Bt;2TisGacRtUv)anKNnaedayeRbIsmIkarxageRkam
b

1
1
1
1
=
+

Pu Pux Puy Puo

(11-28)

1
1
1
1
=
+

Pn Pnx Pny Pno

(11-29)

Edl

= lTPaBRTbnkemKuNeRkamGMeBIm:Um:g;BIrTis
Pux = lTPaBRTbnkemKuNtamGkSmYyenAeBlbnkeFVIGMeBIenAcMNakpit e nig e x = 0
Puy = lTPaBRTbnkemKuNtamGkSmYyenAeBlbnkeFVIGMeBIenAcMNakpit e x nig e y = 0
Puo = bnkcMpitemKuNenAeBlEdl e y = e x = 0
Pu

Pn =

Pu

Pnx =

Pux

Pny =

Puy

Pno =

Puo

lTPaBRTbnktamTismYy Pnx / Pny / Pno GacRtUv)anKNnaedayeRbIsmIkarnigviFIsaRsEdl)an

ENnaMBIxagmuxkgemeronenH. bnab;mk vaRtUv)anCMnYseTAkgsmIkarTI (-29) edIm,IKNna Pn .
smIkar Bresler mannysRmab;RKb;krNITaMgGs;enAeBlNaEdl Pn 0.10Pno . enAeBlEdl
Pn < 0.10 Pno kmaMgtamGkSGacRtUvecal ehIymuxkat;GacRtUv)anKNna dUcGgt;rgnUvm:Um:g;Bt;BIrTissuT
edayeRbIsmIkarxageRkam
b

M ux M uy
+
1.0
Mx
My

(11-30)

M nx M ny
+
1.0
M ox M oy

(11-31)

Edl
= m:Um:g;KNnaCMuvijGkS x
M uy = Pu e x = m:Um:g;KNnaCMuvijGkS y
M x nig M y = lTPaBTb;m:Um:g;CMuvijGkS x nigGkS y

M ux = Pu e y

M nx =

T.Chhay

M ux

M ny =

M uy

M ox =

Mx

304

M oy =

My

Members in Compression and Bending

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smIkar Bresler minRtUv)anENnaM[eRbIenAeBlEdlmuxkat;rgnUvbnkTajtamGkSeT.

11>16>2> viFIExSvNbnk Bresler (Bresler Load Contour Method)
sRmab;viFIenH p)ak; failure surface Edl)anbgajenAkgrUbTI 26 RtUv)ankat;edaybg;ntmefr
P pl;nUvtmnmU:m:g; M nig M . CaTUeTA rUbmnsRmab;viFIenHKW
n

nx

M
M nx

+ ny
M
M ox
oy

ny

= 1.0

(11-32)

elak Bresler )anbgajfa niTsSn GacmantmdUcKasRmab;tYTaMgBIrnsmIkarenH (1 = 2) .

m:agvijeTot Kat;)anbBaak;fatm ERbRbYlcenaH 1.15 nig 1.55 ehIyvaGacRtUv)ansnt;esI 1.50
sRmab;ssrmuxkat;ctuekaNEkg. sRmab;muxkat;kaer ERbRbYlBI 1.5 eTA 2 ehIytmmFm = 1.75
RtUv)aneKeRbIsRmab;karKNnaGnuvtn_. enAeBlEdsrsEdkRtUv)anBRgaymanlkNdUcKasRmab;muxTaMg
bYnrbs;ssrkaer enaHeKGacsnt;yk = 1.5 .
1. 5

M
M nx

+ ny
M
M ox
oy
1.5

= 1.0

(11-33)

bTdanGg;eKs British Code )ansnt; = 1.0; 1.33; 1.67 nig 2.0 enAeBlEdlpleFob 1.1PP
esInwg 0.2; 0.4; 0.6 nig 0.8 erogKa.
u

uo

11>17> viFIExSvNbnk Parme (Parme Load Contour Method)

viFIExSvNbnk load contour methode EdlesIeLIgeday smaKmn_sIum:gtBrELn Porland
Cement Association (PCA) CaviFImYyEdlbegIteLIgedayeRbIviFIExSvNbnk Bresler. enAkgviFIenH Edl
eKGac ehAm:ageTotfa Parme load contour methode cMNuc B enAelIExSvNbnk nbg;edkenARtg;bnk
efr P dUcbgajenAkgrUbTI 26 RtUv)ankMNt;y:agdUcenHfa pleFobnlTPaBTb;m:Um:g;BIrTis M nig
M esI nwgpleFobnlTPaBTb;m:Um:g;mYyTis M nig M .
M
M
M
M
b
=
=
=
M
M
M
M
n

nx

ny

ox

nx

ox

nx

ny

ny

oy

ox

oy

oy

tm EdlmanenAkgrUbTI 27 bgajnUvEpkefrEdllTPaBTb;m:Um:g;tamTisnImYyGacRtUv)an
GnuBaati[GnuvtmkelImuxkat;ssrkgeBldMNalKa.

eRKOgbgMrgkarsgt; nigrgkarBt;

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sRmab;karKNnaGnuvtn_ ExSvNbnk load contour Edl)anbgajenAkgrUbTI27 GacRtUveRbIedIm,I

KNnaedaykarRbhak;RbEhledayeRbIbnat;Rtg; AB nig BC . CRmal (slope) rbs;bnat; AB KW
(1 ) / nigCRmal (slope) rbs;bnat; BC KW /(1 ) . dUcenHenAeBl
M
M
M
M 1
enaH
(11-34)
=1
>
+

M
M
M
M
ehIyenAeBl

ny

nx

ny

oy

ox

oy

M ny
M oy

<

M nx
M ox

enaH

M nx
M ox

nx

M 1
+ ny
M oy
Pn
Po
ox

= 1

(11-35)

tmBitR)akdrbs; GaRsyeTAnwgpleFob sMPar niglkNnmuxkat;. sRmab;ssrEdl

RTnUvbnkRsal nwgERbRbYlBI 0.55 eTA 0.7 . tmmFm = 0.65 GacRtUv)aneRbIsRmab;karKNna.
enAeBlEdlsrsEdkRtUv)anBRgaymanlkNdUcKaRKb;pTaMgbYnrbs;muxkat;ctuekaN enaHpl
eFob MM mantmRbhak;RbEhlnwg bh Edl b nig h CaTTwg nigkm<s;srubrbs;muxkat;ctuekaNerogKa.
ox

oy

edayCMnYspleFobenHeTAkgsmIkarTI (-34) nig (-35) eKTTYl)an

b 1
M ny + M nx
M oy
h
h 1
M nx + M ny
M ox
b
h
= 0.65
= 1.5
b

sRmab;
T.Chhay

(11-36)
(11-37)

nig

306

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M oy M ny + 0.36M nx

(11-38)

M ox M nx + 0.8M ny

(11-39)

BIkarbgajenH eyIgeXIjfasmIkarsKsajEdlmanlkNpal;sRmab;KNnassreRkambnkcM
GkS nigm:Um:g;BIrTisminGaceFVIeTA)aneLIy. dUcenHGkKNnaKYrmanbTBiesaFn_RKb;RKan;edIm,IsannUvmuxkat;
dMbUgedayeRbInUvtm P / M nig M CamYynigsmIkarsRmab;mYyTis nigbnab;mkepgpat;PaBRKb;RKan;
rbs;muxkat;edayeRbIsmIkarsRmab;m:mU :g;BIrTis bedayeRbIkMuBTr.
]TahrN_18 ssrxIEdleRbIEdkkgFmtamanmuxkat; 400 600 CamYynwg 8DB32 BRgaydUc
bgajkgrUbTI 28. kMNt;bnkKNna design load P enAelImuxkat; RbsinebIvaeFVIGMeBIenA e = 205mm
nig e = 305mm . eRbI f ' = 35MPa / f y = 400MPa nig smIkar Bresler Reciprocal equation.
dMeNaHRsay
1> kMNt;lTPaBRTbnkmYyTis P tamGkS x enAeBl e = 305mm . kgkrNIenH b = 400mm nig
h = 600mm / d = 540mm / d '= 60mm nig A = A' = 2412.74mm .
dMeNaHRsayGaceFVIeTAtamCMhandUckg]TahrN_TI 2 nigTI4sRmab;lkxNmuxkat; balanced
condition nigmuxkat; compression-control condition.
a. sRmab; balanced condition
n

nx

ny

nx

600
d = 600 540 = 324mm
cb =
600 + f
600 + 400
y

ab = 0.8 324 = 259.2mm

= 0 .8

enAeBl

f 'c = 35MPa

Cc = 0.85 f 'c ab = 3084.48kN

324 60
f 's = 600
= 488.89MPa
324

dUcenH

Cs = A's ( f y 0.85 f 'c ) = 893.32kN

T = As f y = 965.1kN

f 's = f y = 400MPa

Pbx = Cc + Cs T = 3012.7 kN

= 0.65 sRmab; = 0.002

sRmab; e = 305mm < d = 540mm / snt;famuxkat;sitkglkxN compression failure
ehIy GnuvttamCMhankg]TahrN_TI4 edIm,ITTYl)an a = 265.69mm nig
a
c=
= 332.1mm > c = 324mm . dUcenH muxkat;Camuxkat; compression control.
0.8
epgpat;
c d'
f ' = 600
= 491.60MPa > f dUcenH eyIgyk f ' = f = 400MPa
c
Pbx = 0.65Pbx = 0.65 3012.7 = 1958.25kN

b.

eRKOgbgMrgkarsgt; nigrgkarBt;

307

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NPIC

epgpat; f = 600 d c c = 375.59MPa < f

KNnakmaMg C = 0.85 f ' ab = 3161.71kN
s

Cs = A's ( f y 0.85 f 'c ) = 893.32kN

T = As f s = 906.2kN
Pnx = Cc + Cs T = 3148.83kN > Pbx
failure

dUcenHvaCakrNI compression

dUckarsnidan.

d c
0.003 = 0.00188 < 0.002 = 0.65
c

t =

Pnx = 2046.74kN
c.

Kitm:Um:g;Rtg; A edayeRbIsmIkar (-11)

s

d "= 240mm
e' = 545mm

1
a
Pnx = Cc d + Cs (d d ') = 3148.80kN
e'
2

2> kMNt;lTPaBRTbnkmYyTis P tamGkS y enAeBl e = 205mm . kgkrNIenH h = 400mm nig

b = 600mm / d = 340mm / d ' = 60mm nig A = A' = 2412.74mm .
dMeNaHRsaynwgeFVIeLIgedayeRbIsmIkarsaTic dUcesckIBnl;kgCMhanxagelI.
a. sRmab; balanced condition
ny

600
d = 600 340 = 204mm
cb =
600 + f
600 + 400
y

ab = 0.8 204 = 163.2mm

= 0 .8

enAeBl

f 'c = 35MPa

Cc = 0.85 f 'c ab = 2913.12kN

204 60
f 's = 600
= 423.5MPa
204

dUcenH

Cs = A's ( f y 0.85 f 'c ) = 893.32kN

T = As f y = 965.1kN

f 's = f y = 400MPa

Pby = Cc + Cs T = 2841.34kN

= 0.65 sRmab; = 0.002

sRmab; e = 205mm < d = 340mm / snt;famuxkat;sitkglkxN compression failure
ehIyGnuvttamCMhankg]TahrN_TI4 edIm,ITTYl)an a = 166mm nig c = 0a.8 = 207.5mm
> c = 204mm . dUcenH muxkat;Camuxkat; compression control. epgpat;
c d'
f ' = 600
= 426.5MPa > f dUcenH eyIgyk f ' = f = 400MPa
c

Pby = 0.65Pby = 0.65 2841.34 = 1846.9kN

b.

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epgpat; f = 600 d c c = 383MPa < f

KNnakmaMg C = 0.85 f ' ab = 2963.3kN
s

Cs = A's ( f y 0.85 f 'c ) = 893.32kN

T = As f s = 924.22kN
Pny = Cc + Cs T = 2932.4kN > Pby
failure

dUcenHvaCakrNI compression

dUckarsnidan.

d c
0.003 = 0.0019 < 0.002 = 0.65
c

t =

Pny = 1906.1kN
c.

Kitm:Um:g;Rtg; A edayeRbIsmIkar (-11)

s

d "= 140mm
e' = 345mm

1
a
Pny = Cc d + Cs (d d ') = 2932.5kN
e'
2

3> kMNt;bnkcMpit P tamRTwsI

no

Pno = 0.85 f 'c Ag + Ast ( f y 0.85 f 'c )

= 0.85 35 (400 600 ) + 6434(400 0.85 35) = 9522.2kN

Pno = 6189.4kN

4> edayeRbIsmIkar Bresler (11-28) nigKuNCamYy 1000 eyIg)an

1000
1000
1000
1000
=
+

= 0.8516
2046.74 1906.1 6189.4
Pu
P
Pu = 1174.3kN
Pn = u = 1806.6kN
0.65

nig

cMNaM
1> smIkarRbEhl approximate equation btaragExSekag ACI GacRtUv)aneRbIedIm,IkMNt; P nig
P . edaysarEtsmIkar Bresler pl;cUvdMeNaHRsayEdlmantmRbhak;RbEhl enaHeKKYr
EteRbIviFIsaRskgkaredaHRsayEdlmanlkNsuRkitdUcEdl)anGnuvtkg]TahrN_xagelIenH
edIm,IkMNt; P nig P . dMeNaHRsayRbhak;RbEhlCaeRcIn pl;nUvlTplminsuRkit. eKGac
rk)annUvkmviFIkMuBTrEdlQrelIeKalkarN_smIkarsaTic ehIyeKGaceRbIvaedIm,Iepgpat;nUv
lTplrbs;eyIg.
nx

ny

nx

eRKOgbgMrgkarsgt; nigrgkarBt;

ny

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2> enAkg]TahrN_TI 18 xagelI RkLapEdkenARKb;RCugRtUv)aneRbIBIrdg edaymgeRbIedIm,I

KNna P nigmgeToteRbIedIm,IKNna P . dMeNaHRsayEdlmanlkNsnSMsMc KweRbIEdk
EdlsitenARCugEtBak;kNalsRmab;TismYy Edl A = A' = 1608.5mm enaHvanwgkat;bny
tmrbs; P nig P .
ny

nx

nx

ny

]TahrN_19 kMNt;bnkKNna nominal design load P sRmab;ssrmuxkat;dUc]TahrN_TI18

n

edayeRbIviFIExSvNbnk Parme. emIlrUbTI29.

dMeNaHRsay
1> snt; = 0.65 . lTPaBRTbnktammYyTissRmab;TisedA x nig y RtUv)anKNnaenA
kg]TahrN_TI18.
P = 3148.83kN
P = 2932.4kN
2> P = 2046.74kN P = 1906.1kN
lTPaBTb;m:Um:g;rbs;muxkat;CMuvijGkS x
ux

uy

nx

ny

M ox = Pnx e y = 3148.83 0.305 = 960.4kN .m

lTPaBTb;m:Um:g;rbs;muxkat;CMuvijGkS x
M oy = Pny ex = 2932.4 0.205 = 601.14kN .m

3> eday P Ca nominal design load enaH nominal design moment enAelImuxkat;CMuvijGkS x KW
n

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M nx = Pn e y = 0.305Pn

ehIy nominal design moment enAelImuxkat;CMuvijGkS y KW

M ny = Pn ex = 0.205Pn

4> RtYtBinit MM

ny

>

oy

0.250 Pn 0.305Pn
>
601.14
960.4

M nx
M ox

b 3.4110

Pn > 3.17 10 4 Pn

dUcenHeRbIsmIkar (-34)
5>

0.250 Pn 0.305Pn 1 0.65

+

=1
601.14
960.4 0.65

eyIgTTYl)an

Pn = 1953.125kN
Pu = Pn = 1269.5kN

( = 0.65)

eyIgeXIjfa P EdlKNnaedaysmIkar Parm FMCag P EdlKNnaedaysmIkar Bresler

RbEhl 8% .
u

eRKOgbgMrgkarsgt; nigrgkarBt;

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11>18> smIkarp)ak; (Equation of failure surface)

smIkarTUeTAsRmab;viPaK nigKNnassrebtugxIEdkkgFmtamuxkat;ctuekaNEkgRtUv)anesIeLIg
edayelak Hsu. smIkarRtUv)ansnt;tMNag[p)ak; failure surface nigdaRkamGnrkm interaction
diagram rbs;ssrEdlrgnUvm:Um:g;BIrTis nigbnkcMGkS dUcbgajkgrUbTI26. bnkcMGkSGacCakmaMgsgt;
bkmaMgTaj.
1.5

M
Pn Pb M nx

+
+ ny
M
Po Pb M bx
by
1.5

Edl

= 1.0

(11-40)

ersIusg;Tb;kmaMgtamGkS nominal axial strength viCmanenAeBlrgkarsgt; nigGviC

manenAeBlrgkarTaj sRmab;cMNakpitEdl[.
P = bnktamGkS nominal axial load viCmanenAeBlrgkarsgt; nigGviCmanenAeBlrgkar
Taj sRmab;cMNakpitsUn.
P = bnksgt;tamGkS nominal axial compressive load enAkglkxN balanced strain.
, M = m:Um:g;Bt; nominal bending moment CMuvijGkS x nig y erogKa.
, M = m:Um:g;Bt; nominal balanced bending moment CMuvijGkS x nig y erogKa enAlkxN
balanced strain condition.
edIm,IeRbIsmIkar (11-40) RKb;tYTaMgGs;RtUvmansBaaviCman. tmrbs; P KW
Pn =

M nx
M bx

ny

by

Po = 0.85 f 'c ( Ag Ast ) + Ast f y

(11-41)

bnk nominal balanced load, P / nigm:Um:g; nominal balanced moment, M = P e Edl)an[

enAkgsmIkar (11-6) nig (11-7) erogKa sRmab;muxkat;CamYynwgEdkrgEtkarTaj nigsgt;b:ueNaH.
sRmab;muxkat;epSgeTot tmenHGacTTYl)anedayeRbIeKalkarN_smIkarsaTic.
cMNaMfa smIkarp)ak; failure surface kGaceRbIsRmab;m:Um:g;mYyTisEdlbgajenA kgdaRkam
Gnrkm (interaction diagram) pgEdr. kgkrNIenH GgTIbInwgRtUvlb;ecalenAeBl e = 0 nigGgTIBIrRtUv
)anlb;ecalenAeBl e = 0 .
enAeBl e = 0 manEtm:Um:g;Bt;CMuvijGkS x
b

b b

1. 5

Pn Pb M nx

Po Pb M bx

= 1.0

(11-42)

vaCasmIkar (11-18) EdlmanBIxagedIm enAeBl e

Pn Pb M ny

+
M

P
P
b
o
by

T.Chhay

=0

manEtm:Um:g;Bt;CMuvijGkS y

1. 5

= 1 .0

(11-43)

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Gnuvtn_smIkar (11-42) eTAkg]TahrN_TI 2 nigTI 4. P = 1991.8kN M = 732.8kN .m

28
28
e = 254mm nig P = 0.85 27 (550 350
8) + (
8 400) = 6275.23kN
4
4
b

bx

Pn 1991.8
0.254 Pn
+

6275.23 1991.8 732.8

1.5

=1

0.02764 Pn1.5 + Pn = 6275.23

Pn = 2603.5kN

EdlvamantmRbEhlnwg P EdlKNnaedaykarviPaK.
n

]TahrN_20 kMNt;bnkKNna nominal design load P sRmab;ssrmuxkat;dUc]TahrN_TI18

n

edayeRbIsmIkarp)ak; (equation of failure surface).

dMeNaHRsay
1> KNna P = 0.85 f ' ( A A ) + A f
o

st

st

Po = 0.85 35(600 400

32 2
32 2
8) + 8
400 = 9522.18kN
4
4

2> KNna P nig M edayeRbIsmIkar (11-8) nig (11-9) tamGkS x nig y erogKa
A. tamGkS x
d = 540mm
b

600d t
600 540
cbx =
=
= 324mm
600 + f y 600 + 400

abx = 0.8 324 = 259.2mm

c d'
f 's = 600
= 488.89MPa
c
d "x = 240mm

f 's = 400MPa

As = A's = 2412.74mm 2

Pbx = 0.85 f 'c abx b + A's ( f y 0.85 f 'c ) As f y

= 0.85 35 259.2 400 + 2412.74(400 0.85 35) 2412.74 400 = 3012.7kN

a

M bx = 0.85 f 'c abx b d bx d "x + A's ( f y 0.85 f 'c )(d d 'd "x ) + As f y d "x
2

= 973.28kN .m
B.

tamGkS y
cby =

d t = 340mm

d " y = 140mm

As = A's = 2412.74mm 2

600d t
600 340
=
= 204mm
600 + f y 600 + 400

aby = 0.8 204 = 163.2mm

c d'
f 's = 600
= 423.53MPa
c

eRKOgbgMrgkarsgt; nigrgkarBt;

313

f 's = 400MPa

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NPIC

Pby = 0.85 35 163.2 600 + 2421.74(400 0.85 35) 2412.74 400 = 2841.34kN
a

M by = 0.85 f 'c aby h d by d " y + A's ( f y 0.85 f 'c )(d d 'd " y ) + As f y d " y
2

= 605.09kN.m

3> KNnabnk nominal balanced load P sRmab;m:Um:g;BirTis

bb

tan =

M ny
M nx

Pbx Pby
90

Pn ex 205
=
= 0.672
Pn e y 305

= 33.9o

3012.7 2841.34
P
= o b o
o
90
90 33.9

Pb
90o

Pb = 106.8kN
Pbb = Pby + Pb = 2841.34 + 106.8 = 2948.14kN

4> KNna P BismIkarp)ak; failure surface

n

0.205Pn
Pn 2948.14 0.305Pn

+
+

605.09
9522.18 2948.14 973.28
1.5

1.5

= 1.0

Pn + 0.0775Pn1.5 = 9522.18

edaykarsakl,g eyIgTTYl)an P = 2094.3kN . edaysar P < P vaCakrNI)ak;eday

karTaj tension failure sRmab;karBt;BIrTis. dUcenHeyIgyk P = 9522.18kN
edIm,IrkSaGg TImYyviCman.
n

bb

Pn 2948.14

0.305Pn
0.205Pn

+
+
9522.18 2948.14 973.28
605.09
1.5

1. 5

= 1.0

Pn 0.147 Pn1.5 = 9522.18

nig P = 1177.5kN
cMNaM ersIusg;RTbnk P rbs;ssrmuxkat;ctueekaNEdl)anKNnaedayeRbIsmIkar Bresler
reciplocal equation (11-18) viFI Parme method (11-19) nigviFI Hsu method (11-20) edIm,I
TTYl)an P = 1174.3kN / 1269.5kN nig 1177.5kN erogKa. eyIgeXIjfa viFI Parme
method pl;nUvtmFMCageKsRmab;]TahrN_enH.
Pn = 1811.5kN

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XII.

ssrEvg

12>1> esckIepIm
sRmab;karKNnassrxIEdl)anBnl;enAkgBIremeronxagedIm )ansnt;fa karPat; (buckling) kar
rYjxIeGLasic (elastic shortening) nigm:Um:g;TIBIr (secondary moment) EdlbNalmkBIPaBdabtamTTwg
(lateral deflection) man\TiBlCaGb,brmaeTAelIersIusg;cugeRkay (ultimate strength) rbs;ssr dUcenH
ktaTaMgenH minRtUv)anrab;bBaleTAkgdMeNIrkarnkarKNnaeT. b:uEn sRmab;ssrEvg ktaTaMgGs;enHRtUv
EtykmkBicarNa. RbEvgbEnmnwgbNal[mankarkat;bnyersIusg;rbs;ssr edayERbRbYlCamYynwg
km<s;RbsiTPaB nigTTwgrbs;muxkat; pleFobrlas; (slenderness ratio) niglkxNcugssr.
ssrEdlman slenderness ratio FMnwgkat;bnylTPaBRTRTg;rbs;ssry:agxaMg Et slenderness
tUcmannyfassrxI ehIykarkat;bnyersIusg;GacnwgminKYr[cab;GarmN_. pleFobrlas;
ratio
(slenderness ratio) KWCapleFobrvagkm<s;ssr l CamYynwgkaMniclPaB radius of gyration r Edl
r = I / A kgenaH I Cam:Um:g;niclPaBnmuxkat; moment of inertia of the section nig A CaRkLap
muxkat;.
sRmab;muxkat;ctuekaNEdlmanTTwg b nigkm<s; h rUbTI 1 I = bh / 12 nig A = bh dUcenH
r = 0.288h bedaytmRbEhl r = 0.3h . dUcKa I = b h / 12 nig r = 0.288b b r = 0.3b .
sRmab;ssrmUlCamYynwgGgt;pit D enaH I = I = D / 64 nig A = D / 4 dUcenH r = r = 0.25D .
CaTUeTA ssrGacRtUv)anBicarNa dUcteTA
1> EvgCamYynwg slenderness ratio FM RtUvkarRbBnBRgwg b shear wall.
2> EvgCamYynwg slenderness ratio lmEdlbg[mankarkat;bnyersIusg;ssr enaHRbBnBRgwg
GacnwgminRtUvkar Etkarkat;bnyersIusg;RtUvEtBicarNa.
3> xIEdl slenderness ratio tUcEdlbNal[mankarkat;bnyersuIsg;scesIg. kmaMgkat;Gac
RtUv)anecal dUcerobrab;BIemeronmun.
2

12>2> RbEvgssrRbsiTPaB (Effective Column Length) Kl

pleFobrlas; (slenderness ratio) l / r GacRtUv)anKNnay:agsuRkitenAeBlEdlRbEvgRbsiTPaB
rbs;ssr Kl RtUv)aneRbI. RbEvgRbsiTPaBenHGnuKmn_eTAnwgBIrktaFM
1> RbEvgKanTRm (unsupported length) l sMEdgnUvkm<s;minKitTRmrbs;ssrrvagBIrkRmal
xN. vaRtUv)anvas;Ca clear distance rvagkRmalxN Fwm
u

ssrEvg

315

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NPIC

bGgeRKOgbgMEdlpl;nUvTRmxagdl;ssr. enAkgRbBnkRmalxN flat slab CamYynwg

column capital km<s; unsupported height rbs; ssrRtUv)anvas;BIpxagelIrbs;kRmalxN
xageRkameTA)atrbs; column capital. RbsinebI ssrRtUv)anRTCamYyFwmEdlmankm<s;x<s;
tamTismYyCagtamTismYyeTot enaH l KYrEtKNnatamTisTaMgBIr tamTis x nig y
nmuxkat;ssr. lTplEdlFMCagRtUv)anBicarNakgkarKNna.
2> emKuNRbEvgRbsiTPaB K bgajnUvpleFobncmayrvagcMNucnm:Um:g;sUnenAkgssr nig
km<s;KanTRmrbs;ssrkgTisedAmYy. ]TahrN_ RbsinebIRbEvgKanTRm (unsupported
height) rbs;ssr TRmsnak; (hinged) enAcugsgag Edlkareyalxag (sidesway) RtUv)an
Tb; KW l nigcMNucm:Um:g;sUnenAcug nigKl;ssr EdlenAcugTRmRtIekaN (hinged) enaHemKuN
K = l / l KWesInwg 1. RbsinebIssr manTRmbgb; (fixed) enAcugsgag ehIykareyalxag
(sidesway) RtUv)anTb; cMNucrbt; cMNucm:Um:g;sUn sitenA l / 4 BIcugTRm. dUcenH K =
0.5l / l = 0.5 rUbTI2 edIm,IKNnatmdRtwmRtUvrbs; K krNI cMbgBIrRtUv)anBicarNa.
- enAeBleRKagbgMEdlpMeLIgedayFwm nigssrRtUv)anBRgwgedayCBaaMg shear wall
RbBnBRgwgrwg (rigid bracing) bTRmxagEdl)anmkBIeRKagbgMenACab;nwgva. cugrbs;
ssrnwgsitenATItaMgdEdl EdlkarrMkilxagrbs;tMNRtUv)ankarBar. CaTUeTAsRmab;
eRKagBRgwg tmrbs; K KWtUcCag besInwg 1. ACI code, section 10.12 esI[eRbI
K = 1.
- enAeBleRKagbgMminRtUv)anBRgwg vanwgGaRsyeTAnwgPaBrwgRkaj (stiffness) rbs;Fwm
nigssr edIm,ITb;nwgPaBdabxag. edaysarkarrMkilrbs;tMNminRtUv)ankarBar eRKag
egakeTAtamTisrbs;bnkxag. tmrbs; K sRmab;ssr nigeRKagRtUv)an[enAkg
rUbTI2 edayBicarNakrNITaMgBIr KWenAeBlkareyalxag sidesway RtUv)ankarBar nig
minRtUv)ankarBar.
u

12>3> emKuNRbEvgRbsiTPaB (Effective Length Factor) K

RbEvgRbsiTPaBrbs;ssrGacRtUv)anKNnaedayeRbIdaRkam alignment chart kgrUbTI3. edIm,Irk
emKuNRbEvgRbsiTPaB K dMbUgeKcaM)ac;RtUvKNnarkemKuNTb; restraint factor nig enAxagcugnig
Kl;ssrerogKa Edl
A

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rbs;ssr
(12-1)
EI / l rbs;Fwm
Edl l = RbEvgKitBIGkSeTAGkSntMNrbs;eRKag
l = RbEvgElVgKitBIGkSeTAGkSntMNrbs;eRKag
TaMgBIrsitenAkgbg;Bt;. emKuN enAxagcugKYrEtrYmbBalTaMgssr nigFwmEdlCYbKaenARtg;
tMN. sRmab;TRmsnak; hinged end KWGnn nigGacsnt;esI 10 . sRmab;TRmbgb; fixed end KWsUn
nigGacsnte; sI 1. tmsnt;TaMgenHGaceRbI)anedaysarEtenAkgeRKagbgMebtugGarem:Kansnak;Kankkit
l\texaH bTRmbgt;l\texaHenaHEdr.
dMeNIrkarrk K KWKNna sRmab;cugssr nig sRmab;Kl;ssr. dak; nig eTAkgda
Rkam alignment chart nrUbTI3 rYcPab;cMNucTaMgBIredaykat;ExSkNal EdlbgajBItm K . daRkam
BIrEdlmanlkNRsedogKaRtUv)anbgaj mYysRmab;eRKagBRgwg Edlkareyalxag (sidesway) RtUv)an
karBar nigmYyeTotsRmab;eRKagFmta Edlkareyalxag (sidesway) minRtUv)ankarBar. karbegItdaRkam
enHKWQrelIkarsnt;fa
- eRKagbgMpMeLIgedayeRKagctuekaNsIuemRTI
- m:Um:g;Bt;Fwm)anEckmkssredayTak;TgnwgPaBrwgRkajrbs;va
- ssrTaMgGs;TTYlnUvbnkFMenAeBlCamYyKa
edIm,ICMnYsnUvkareRbIdaRkam alignment chart EdlbgajkgrUbTI3 ACI Code Commentary )an
esInUvsmIkarsRmYldUcxageRkamsRmab;KNnaemKuNRbEvgRbsiTPaB K .
=

EI / lc

ssrEvg

317

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1> sRmab;Ggt;rgkarsgt;EdlmankarBRgwg tmrbs; K GacRtUv)anyktmtUcCageKkg

cMeNamsmIkarTaMgBIrxageRkam
K = 0.7 + 0.05( A + B )
K = 0.85 + 0.05 min

(12-2)
(12-3)

Edl nig Catmrbs; enAcugsgagrbs;ssr nig CatmtUcbMputntmTaMgBIr.

2> sRmab;Ggt;rgkarsgt;EdlKankarBRgwgEtRtUv)anTb;enAcugsgag tmrbs; K Gacsnt;dUc
xageRkam
sRmab; < 2 / K = 2020 1 +
(12-4)
sRmab; 2 / K = 0.9 1 +
(12-5)
Edl CatmmFmrbs; enAcugsgagGgt;rgkarsgt;.
3> sRmab;Ggt;rgkarsgt;KankarBRgwgmanTRmsnak; hinged enAcugmag enaH K GacRtUv)an
snt;dUcxageRkam
A

min

K = 2 + 0.3

(12-6)

Edl CatmenAcugEdlmankarTb;.

]TahrN_ 1 edayeRbInUvsmIkarxagedIm cUrkMNt;emKuNRbEvgRbsiTPaB K sRmab;Ggt;rgkarsgt;enA

kgeRKagCamYynwglkxNxageRkam
1> eRKagRtUv)anBRgwgedIm,ITb;nwgkareyalxag (sidesway) ehIy = 2.0 nig = 3.0 enAcug
xagelI nigxageRkamrbs;Ggt;.
2> eRKagminRtUv)anBRgwgedIm,ITb;nwgkareyalxag (sidesway) eT ehIy = 2.0 nig = 3.0 .
Ggt;RtUv)anbgb;enAcugsgag.
3> eRKagminRtUv)anBRgwgedIm,ITb;nwgkareyalxag (sidesway) eT ehIy = 0.0 TRmsnak; nig
= 3 .0 .
A

dMeNaHRsay

1> BIsmIkar (-2) nig (-3)

K1 = 0.7 + 0.05(2 + 3) = 0.95 < 1.0

K 2 = 0.85 + 0.05(2) = 0.95 < 1.0

eRCIserIsyknUvtmtUcCageKkgcMeNam K nig K . kgkrNIenH K = 0.95 .

1

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319

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2> tmmFmrbs;

NPIC

= (2 + 3) / 2 = 2.5

. eday

>2

eRbIsmIkar (12-5)

K = 0.9 1 + 2.5 = 1.684

3> BIsmIkar (12-6)

K = 2 + 0.3(3) = 2.9

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12>4> PaBrwgRkajrbs;Ggt; (Member Stiffness) EI

PaBrwgRkajrbs;Ggt;eRKagesInwgplKuNrvagm:UDuleGLasic E CamYynwgm:Um:g;niclPaBnmuxkat;
I . tmn E nig I sRmab;ebtugGarem:GacRtUv)anKNnadUcxageRkam
1> m:UDuleGLasicrbs;ebtugRtUv)anBnl;kgemeronTI2. bTdan ACI Code [smIkarxag eRkam
E = 0.043w
f'
b E = 4780 f '
sRmab;ebtugTmn;Fmta. cMENkm:UDuleGLasicrbs;EdkKW E = 2.110 MPa .
2> sRmab;Ggt;ebtugGarem: m:Um:g;niclPaB I ERbRbYltambeNayrbs;Ggt; GaRsyeTAnwgkRmit
eRbH nigPaBryEdkEdl)aneRbIR)as;.
edIm,IkMNt;nUvemKuN EI RtUvEt)ankMNt;sRmab;Fwm nigssr. dUcenH EI GacRtUv)ankM
Nt;dUcxageRkam ACI Code, section 10.11.1
sRmab;Fwm
I = 0.35 I
sRmab;ssr
I = 0.70 I
I = 0.70 I
sRmab;CBaaMgKaneRbH
sRmab;CBaaMgmaneRbH
I = 0.35 I
sRmab;kRmalxN (flat plate nig flat slab) I = 0.25I
Edl I Cam:Um:g;niclPaBsRmab;muxkat;ebtugeBjeFobGkSkat;tamTIRbCMuTmn; edayecal
Edk.
3> RkLapmuxkat; A = A RkLapmuxkat;eBj gross-sectional area
4> m:Um:g;niclPaBKYrEtRtUv)anEckeday (1 + ) enAeBlEdlbnkxagefr (sustained lateral load)
manGMeBIelIeRKagbgM bsRmab;epgpat;sirPaB stability check Edl
1.5

d =

maximum factored sustained axial load

total factored axial load

12>5> EdnkMNt;sRmab;pleFobrlas; (Limitation of The Slenderness Ratio) Kl / r

12>5>1> eRKagGt;eyal (Nonsway Frames)
bTdan ACI Code, section 10.12 ENnaMnUvEdnkMNt;xagRkamrvagssrxI nigssrEvgenAkgeRKag
BRgwg Gt;eyal nonsway
1> \TiBlrbs;PaBrlas; slenderness GacRtUv)anecal ehIyssrGacRtUv)anKNnaedayKitCassrxI
enAeBlEdl
u

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Klu
12 M 1
34
r
M2

(12-7)

Edl M nig M Cam:Um:g;emKuNenAcugssr ehIy M

1

> M1

2> pleFob MM RtUvcat;Tukfa viCmanRbsinebIGgt;RtUv)anenAkgkMeNageTal single curvature

nigGviCmansRmab;kMeNagDub double curvature dUcbgajkgrUbTI4.
3> tY (34 12M / M ) KYrminRtUvFMCag 40.
4> RbsinebIm:Um:g;ssremKuN (factored column moment) esIsUn b e = M / P < e tmrbs;
M KYrEtRtUv)anKNnaedayeRbIcMNakpitGb,brma
1

min

emin = (15.24 + 0.03h)

M 2 = Pu (15.24 + 0.03h)

(12-8)
(12-9)

Edl M Cam:Um:g;Gb,brma. m:Um:g; M KYrEtRtUv)anBicarNaedayeFobnwgGkSnImYyrbs;ssrdac;

edayELkBIKa. tm K GacRtUv)ansnt;esInwg 1.0 sRmab;eRKagBRgwg braced frame elIkElgEt
vaRtUv)anKNnaedayQrelIkarviPaK EI .
2

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12>5>2> eRKageyal (sway Frames)

enAkgGgt;rgkarsgt;minBRgwgTb;nwgkareyalxag sidesway \TiBlrbs;pleFobrlas;
slenderness ratio GacecalenABlEdl
Klu
< 22
r

(ACI Code, section 10.13)

(12-10)

12>5>3> pleFobrlas;FM (High slenderness ratio)

enAeBlEdlGgt;rgkarmYydac;edayELkenAkgeRKagmanpleFobrlas; slenderness ratio
Kl / r > 100 viFIm:Um:g; magnifier (moment magnifier method) rbs; ACI Code minGacRtUv)aneRbI
ehIykarviPaK rigorous dWeRkTIBIr rigorous second-order RtUv)aneRbICMnYsvij. Etmuxkat;GacRtUv)an
dMeLIgedIm,Ikat;bnypleFob Kl / r . tM;l 100 bgajBIkarBiesaFCak;EsgcMNat;fak;x<s; (ACI
Code, section 10.10.5) .
u

12>6> viFIKNnabEnmm:Umg; (Moment-Magnifier Design Method)

12>6>1> esckIepIm (Introduction)
CMhandMbUgkgkarKNnam:Um:g;enAkgssrEvgKWkMNt;faetIeRKagEdlKNna CaeRKakBRgwg bmin
BRgwgTb;nwg sidesway . RbsinebImanGgBRgwgxag dUcCa shear walls nig shear trusses bssrmanPaBrwg
RkajTTwg lateral stiffness efr enaHPaBdabTTwg lateral deflection mantmtUc ehIy\TiBlrbs;vaeTAelI
ersIusg;ssrktUcEdr. eKGacsnt;faeRKagbgMenAkgmYyCan;RtUv)anBRgwgRbsinebI
P
Q = u o 0.05
Vuslc

(-11)

Edl Pu nig V CabnkbBarsrub nigkmaMgkat; erogKa ehIy PaBdabeFobdWeRkTImYy (firstorder relative deflection) rvagkMBUl nig)atrbs;Can;EdlbNalmkBI V . RbEvg l CaRbEvgrbs;Ggt;rg
karsgt;enAkgeRKagbgM edayvas;BIGkSeTAGkSrbs;tMNrenAkgeRKag.
CaTUeTA Ggt;rgkarsgt;GacrgnUgPaBdabTTwg lateral deflection EdlbNalmkBIm:Um:g;TIBIr
(secondary moment). RbsinebIm:Um:g;TIBIr M ' RtUv)anbEnmeTAelIm:Um:g;EdlGnuvtelIssr M enaHm:Um:g;
cugeRkayKW M = M + M ' . viFIRbEhl (approximate method) sRmab;kMNt;m:Um:g;cugeRkay M KWCakar
KuNm:Um:g; M edayemKuNEdleKehAfa emKuNbEnmm:Um:g; (magnifying moment factor) ehIyemKuNenH
RtUvEt FMCagbesInwg 1.0 . b M = M nig 1.0 . m:Um:g; M RtUv)anTTYlBIkarviPaKeRKageGLasic
us

us

max

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eday eRbIbnkemKuN ehIyvaCam:Um:g;GtibrmaEdlmanGMeBIenAcugssr benAkgssr RbsinebIbnkxagman

vtman.
RbsinebI\TiBl P RtUv)anykmkBicarNa vanwgcaM)ac;RtUvEteRbIkarviPaKdWeRkTIBIr edIm,IKitBI
TMnak;TMng nonlinear relationship rvagbnk PaBdabTTwg nigm:Um:g;. eKGaceRbIkmviFIkMuBTredIm,IedaHRsay
va. bTdan ACI Code GnuBaati[eRbIkarviPaKssrdWeRkTImYy bdeW RkTIBIr. karviPaKssrdWeRkTIBIr RtUv)an
tRmUv[eRbIenAeBlEdl Klu / r > 100 . viFIKNnam:Um:g;bEnmrbs; ACI Code CaviFIsRmYlsRmab;KNna
emKuNbnkbEnmTaMgeRKagBRgwg nigeRKagminBRgwg.
12>6>2> m:Um:g;bEnmenAkgeRKagGt;eyal (Magnified Moments in Nonsway Frames)
\TiBlrbs;pleFobrlas; slenderness ratio Klu / r enAkgGgt;rgkarsgt;neRKagBRgwgGac
RtUv)anecalRbsinebI Klu / r 34 12M1 / M 2 dUcbgajenAkgEpk 5>1 . RbsinebI Kl / r >
34 12 M / M enaH\TiBlPaBrlas;RtUv)anBicarNa. dMeNIrkarkMNt;emKuNbEnm ns enAkgeRKagmin
eyalGacRtUv)ansegbdUcxageRkam (ACI Code, section 10.12)
1> kNt;faeRKagCaeRKagBRgwgTb;nwg sidesway bGt; rYckMNt;RbEvgKanTRm lu nigemKuNRbEvg
RbsiTPaB K K RtUv)ansnt;[esI 1.0
2> KNnaPaBrwgRkajrbs;Ggt; EI edayeRbIsmIkar
u

EI =

0.2 Ec I g + Es I se

(12-12)

1 + d

bsmIkarEdlsRmYlCag
EI =

0.4 Ec I g

EI = 0.25Ec I g

Edl

(12-13)

1 + d

sRmab;

d = 0.6

(-1214)

Ec = 4780 f 'c
Es = 2.1 105 MPa

m:Um:g;niclPaBnmuxkat;ebtugtamGkSNamYyEdleyIgBicarNaedayecal As
I se = m:Um:g;nicalPaBnmuxkat;EdkeFobGkSTIRbCMuTmn;rbs;muxkat;ebtug
Ig =

d =

1.2 D
(sustained)
maximum factored axial sustained load
=
1.2 D + 1.6 L
maximum factored axial load

cMNaMfa d xagelICapleFobEdlFab;KNnam:Um:g;bEnmenAkgssrEdlbNalmkBIbnk
sustained .
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smIkar (-13) nig (-14) manlkNsuRkitticCagsmIkar (-12) . elIsBIenH smIkar (12-14)

TTYl)anedaysnt; d = 0.6 CMnYskgsmIkar (-13) .
3> kMNt;bnk Euler buckling/ Pc
Pc =

2 EI

(12-15)

(Klu )2

eRbItmrbs; EI / K nig lu dUcKNnaBICMhan 1> nigCMhan 2>.

4> KNnatmnemKuN Cm edIm,IeRbIenAkgsmIkarnemKuNm:Um:g;bEnm moment-magnifier
factor. sRmab;Ggt;BRgwgedayKanbnkxag transverse load
Cm = 0.6 +

0.4 M 1
0.4
M2

(12-16)

Edl M1 / M 2 viCmanRbsinebIssrRtUv)anBt;kgkMeNageTal. sRmab;Ggt;CamYybnkxag

enA cenaHTRm Cm KYrRtUv)anykesInwg 1.0 .
5> KNnaemKuNm:Um:g;bEnm ns
ns =

Cm
1.0
1 ( Pu / 0.75Pc )

(12-17)

Edl Pu CabnkemKuN nig Pc nig Cm RtUv)anKNnaBIxagelI.

6> KNnaGgt;rgkarsgt;edayeRbIbnkemKuNtamGkS Pu BIkarviPaKeRKagdRtwmRtUv nigm:Um:g;
bEnm magnified moment M c EdlKNnadYcxageRkam
M c = ns M 2

(12-18)

Edl M 2 Cam:Um:g;emKuNEdlFMCagEdlekItBIbnk EdllTplmineyal. sRmab;eRKagBRgwg

Tb;nwg sidesway emKuNeyalKW s = 0 . enAkgeRKagGt;eyal nonsway frame PaBdab
TTwgRtUv)anrMBwg[tUcCagbesInwg H /1500 Edl H Cakm<s;srubrbs;eRKag.
12>6>3> m:Um:g;bEnmenAkgeRKageyal (Magnified Moments in sway Frames)
\TiBlrbs;PaBrlas;GacRtUv)anecalenAkgeRKageyal sway frame KanBRgwg unbraced
enAeBlEdl Klu / r < 22 . karKNnaemKuNbEnm magnification factored s sRmab;eRKageyal Kan
BRgwg RtUv)ansegbdUcxageRkam (ACI Code, Section 10.13)
1> kMNt;faeRKagCaeRKagKanBRgwgTb;nwg sidesway bGt; rYckMNt;RbEvgKanTRm lu nigemKuN
RbEvgRbsiTPaB K EdlGacTTYlBIsmIkar (12-4) (12-5) nig (12-6) bdaRkamrUbTI3.
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2-4> KNna EI / Pc nig Cm dUc[kgsmIkar (12-12) dl; (12-16). cMNaMfa d edIm,IKNna EI

KW CapleFobrvagkmaMgkat;TTwgefremKuNGtibrma maximum factored sustained shear
tamCan; nigkmaMgkat;TTwgemKuNsrubenAkgCan;enaH.
5> KNnaemKuNm:Um:g;bEnm moment-magnifier factor/ s
s =

1
1.0
1 ( Pu / 0.75 Pc )

(12-19)

Edl s 2.5 nig Pu CaplbUkbnkbBarTaMgGs;enAkgmYyCan; nig Pc CaplbUk

bnksRmab;ssrEdlTb;nwgkareyal sway enAkgmYyCan;. dUcKa
sM s =

M 2s
Ms
1 ( Pu / 0.75 Pc )

(12-20)

Edl M s Cam:Um:g;emKuNxagcugbNalmkBIbnkEdlbegItkareyalEdlTTYlyk)an.
6> KNnam:Um:g;cugbEnm M1 nig M 2 enAxagcugGgt;rgkarsgt;Etg dUcxageRkam
M 1 = M 1ns + s M 1s
M 2 = M 2ns + s M 2 s

(12-21)
(12-22)

Edl M1ns nig M 2ns Cam:Um:g;EdlTTYlBIlkxNGt;eyal b:uEn M1s nig M 2s Cam:Um:g;Edl

TTYl)anBIlkxNeyal. RbsinebI M 2 > M1 BIkarviPaKeRKag enaHkarKNnam:Um:g;bEnmKW
M c = M 2 ns + s M 2 s

(12-23)

m:Um:g;cug M1 nig M 2 enAkgsmIkar (-21) (-22) nig (-23) manm:Um:g;Gt;eyal bUknigm:Um:g;

eyalbEnm CamYynwglkxNEdl
lu
<
r

35
Pu / f 'c Ag

(12-24)

enAkgkrNIenH Ggt;rgkarsgt;KYrEtRtUv)anKNnasRmab;bnkemKuNtamGkS Pu nig M c .

b:uEnkgkrNIEdl
lu
>
r

35
Pu / f 'c Ag

(12-25)

Ggt;rgkarsgt;KYrEtRtUv)anKNnasRmab; Pu nigm:Um:g;Gt;eyalbEnm ns M 2 bUkCamYy

nwgm:Um:g;eyalbEnm s M 2 CamYynwgm:Um:g;KNna M c = ns M 2ns + s M 2s . krNIenHGac
ekItmansRmab;ssrEvg slender column CamYynwgbnktamGkSFM enAeBlEdlm:Um:g;Gtibrma
ekItmanenAcenaHcugssr nigminenAxagcug.
bTdan ACI Code, section 10.13.4 GnuBaatnUvviFIepSgeTotsRmab;karKNna s M s n
smIkar (12-20) edayeRbIsnsSn_sirPaB stability index Q Edl[kgsmIkar (12-11) Edl
s 1.5
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sM s =

Department of Civil Engineering

Ms
Ms
1 Q

(-26)

]TahrN_ 2 muxkat;ssrdUcbgajkgrUbTI5 RTbnktamGkS P

nigm:Um:g; M D = 157kN .m
EdlbNalmkBIbnkefr nigbnktamGkS PL = 490kN nigm:Um:g; M L = 126kN .m EdlbNalmkBIbnk
clt. ssrCaEpkrbs;eRKagBRgwg nigmankMeNageTaltamGkSem. RbEvgKanTRmrbs;ssrKW lc = 5.8m
ehIym:Um:g;enAcugTaMgsgagrbs;ssrmantmsIKa. epgpat;muxkat;ssredayeRbI f 'c = 28MPa nig
f y = 400MPa .
D

= 605kN

dMeNaHRsay

1> KNnabnkcugeRkay ultimate load

Pu = 1.2 PD + 1.6 PL = 1.2 605 + 1.6 490 = 1510kN
M u = 1.2M D + 1.6M L = 1.2 157 + 1.6 126 = 390kN .m
M
390
e= u =
= 258.3mm
Pu 1510

2> RtYtBinitemIlfaetIssrEvgbxI. edaysareRKagRtUv)anBRgwg snt; K = 1.0

r = 0.3h = 0.3 550 = 165mm nig lu = 5.8m
Klu 5800
=
= 35.15
r
165

sRmab;ssrBRgwg RbsinebI Klu / r 34 12M1 / M 2 \TiBlnPaBrlas;GacRtUv)anecal.

edaysarssrekagedaykMeNageTal enaH M1 / M 2 viCman. dUcenH
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34 12

NPIC

M1
= 34 12 = 22
M2

edaysar Klu / r = 35.15 > 22 enaH\TiBlnPaBrlas;RtUv)anBicarNa.

3> KNna EI BIsmIkar (-12)
A. KNna E
c

Ec = 4780 f 'c = 4780 28 = 25293.4MPa

Es = 2.1 105 MPa
B.

m:Um:g;niclPaBKW

350(550)
= 4852.6 106 mm 4
12
3

Ig =

As = A's =

4 282
= 2463mm 2
4

550 120
6
4
I se = 2 2463
= 227.7 10 mm
2

pleFobm:Um:g;GefrKW
d =
C.

1.2 605
= 0.48
1510

PaBrwgRkajKW
EI =

0.2 Ec I g + Es I se
1 + d
0.2 25293.4 4852.6 106 + 2.1 105 227.7 106
= 48.9 1012 N .mm 2
1 + 0.48

4> KNna P

EI
2

Pc =

(Klu )2

2 48.9 1012
(5800) 2

= 14346.72kN

5> KNna C BIsmIkar (-16)

m

Cm = 0.6 + 0.4

M1
0.4
M2

= 0.6 + 0.4(1) = 1.0

6> KNnaemKuNm:Um:g;bEnmBIsmIkar (12-17)

ns =

Cm
1
=
= 1.16
1 ( Pu / 0.75Pc ) 1 [1510 /(0.75 14346.72]

7> KNnam:Um:g;KNna design moment nigbnkKNna design load edaysnt; = 0.65

1510
= 2323kN
0.65
390
Mn =
= 600kN .m
0.65
Pn =

KNna M
T.Chhay

= 1.16 600 = 696kN .m

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696
KNnacMNakpit
e=
= 300mm
2323
8> kMNt;ersIusg; nominal load strength nmuxkat;edayeRbI e = 300mm edayeRbIsmIkar (11-4)
Pn = 8.33a + 926.58 2.46 f s
h
550
e' = e + d = 300 + 490
= 515mm
2
2

1
a

Pn =
8.33a 490 + 926.58(490 60 )

515
2

3 2
= 7.93a 8.1 10 a + 773.65

(I)

(II)

BIsmIkar (I) nig (II) eyIgTTYl)an a = 267mm / f = 338MPa nig P = 2319.2kN .

edaysarEtersIusg;bnk load strength P = 2319.2kN nigbnktRmUvkar required load
P = 2323kN mantmRbhak;RbEhlKa enaHmuxkat;RtUv)ancat;TukfaRKb;RKan;. RbsinebI
muxkat;minRKb;RKan; RtUvdMeLIgmuxkat;Edk.
9> epgpat;tmsnt;
s

a = 267 mm

d t = 490mm

c = 314.12mm

dt c
0.003 = 0.00168 < 0.002
c

t =

dUcenH = 0.65

]TahrN_ 3 epgpat;PaBRKb;RKan;rbs;ssrkg]TahrN_TI2 RbsinebIRbEvgKanTRm (unsupported

length) lu = 3m

dMeNaHRsay

. kMNt;bnk nominal load GtibrmaenAelIssr.

1> bnkEdlGnuvtKW P = 2323kN nig M = 600kN .m

2> epgpat;PaBEvgxIrbs;ssr l = 3m / r = 0.3 550 = 165mm nig K = 1.0 eRKagRtUv)anBRgwg
Tb;nwgkareyalxag sidesway .
n

Klu 3000
=
= 18.2
r
165

epgpat; Kl

/ r = 34 12 M 1b / M 2b

34 12(1) = 22
Klu
= 18.2 < 22
r

eday
enaH \TiBlnPaBrlas;Gacecal)an.
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3> kMnt;lTPaBRTbnk nominal load ebs;ssrxI dUcBnl;enAkg]TahrN_TI4 nemeronTI11

eRKOgbgMrgkarsgt; nigkarBt;;. eyIgTTYl)an Pn = 2574.9kN sRmab; e = 258.3mm
EdlFMCagbnkcaM)ac; Pn = 2323kN .

]TahrN_ 4 epgpat;PaBRKb;RKan;rbs;ssrkg]TahrN_TI2 RbsinebIeRKagminRtUv)anBRgwgTb;nwgeyal

xag sidesway emKuNbgb;cug end-restraint factor KW A = 0.8 nig B = 2 ehIyRbEvgKanTRm

unsupported length KW lu = 4850mm .

dMeNaHRsay

1> kMNt;tm K BIdaRkam alignment chart rUbTI3 sRmab;eRKagminBRgwg. Pab;tm A = 0.8

nig B = 2 kat;ExS K Rtg; K = 1.4 .
Klu 1.4 4850
=
= 41.15
r
165

2> sRmab;eRKagKanBRgwg RbsinebI Klu / r 22 ssrGacRtUv)anKNnadUcssrxI.

edaysarEttm Klu / r = 41.15 > 22 eKRtUvEtKitBI\TiBlnPaBrlas;.
3> KNnaemKuNm:Um:g;bEnm ns eK[ Cm = 1 / K = 1.4 / EI = 48.9 1012 N .m2
BI]TahrN_TI2 nig
Pc =

2 EI

(Klu )2

ns =

2 48.9 1012

(1.4 4850)2

= 10468.1kN

Cm
1
=
= 1.24
1510
Pu

1
0.75 10468.1
0.75 Pc

4> BI]TahrN_TI2 Pu = 1510kN nig M u = 390kN .m b Pn = 2323kN nig M n = 600kN .m

m:Um:g;KNna M c = 1.24 600 = 744kN .m dUcenH
e=

ns M n
Pn

744
= 320.3mm
2323

5> epgpat;PaBRKb;RKan;rbs;ssrxIsRmab; Pn = 2323kN / M c = 744kN .m nig e = 320.3mm .

viFIsaRskgkaredaHRsayRtUv)anBnl;kg]TahrN_TI4 n emeronTI 11 eRKOgbgMrgkarsgt;
nigkarBt;.
6> BI]TahrN_TI4 n emeronTI11 eRKOgbgMrgkarsgt; nigkarBt; eyIg)an
Pn = 8.33a + 926.58 2.46 f s
h
550
e' = e + d = 320.3 + 490
= 535.3mm
2
2
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Pn =

Department of Civil Engineering

1
a

8.33a 490 + 926.58(490 60 )

535.3
2

= 7.625a 0.00778a 2 + 744.31

eyIgTTYl)an a = 259.86mm
dUcenH c = 305.7mm nig Pn = 2200kN . lTPaBRTbnkrbs;ssr Pn = 2200kN tUc
CagbnkEdlRtUvRT Pn = 2323kN . dUcenHmuxkat;minRKb;RKan;.
7> begInmuxkat;EdkBI 4DB28 eTA 4DB30 ehIyeFVIkarKNnaepgpat;eLIgvij enaHeyIg
TTYl)an Pn = 2335kN / t < 0.002 nig = 0.65 .

]TahrN_ 5 KNnassrkaer:xagkgsRmab;Can;TImYynGKarkariyaly8Can;. km<s; clear height nCan;

TImYyKW 4.9m nigkm<s;sRmab;Can;dTeTotKW 3.4m . GKarenHman 24RbGb; rUbTI6 ehIyssrminRtUv

)anBRgwgTb;nwgkareyalxag sidesway. bnkEdlGnuvtmkelIssrxagkgCan;TImYy bNalmkBITMnaj
EpndI nigxl;dUcxageRkam
= 1690 kN
bnkefrtamGkS
bnkGefertamGkS = 623kN
bnkxl;tamGkS = 0kN
m:Um:g;bnkefr
= 43.4kN.m xagelI 73.2kN.m xageRkam
= 27.1kN.m xagelI 48.8kN.m xageRkam
m:Um:g;bnkGefr
m:Um:g;bnkxl;
= 67.8kN.m xagelI 67.8kN.m xageRkam
EI / l sRmab;Fwm
= 40 106 kN.mm
eRbI f 'c = 35MPa / f y = 400MPa nigtRmUvkarrbs;bTdan ACI Code. snt;fa bnkEdlman
GMeBIelIssrxageRkAesI 2 / 3 nssrxagkg ehIybnkEdlmanGMeBIelIssrRtg;RCugesI 1 / 3 nssrxag
kg ehIy d = 0.55 .

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dMeNaHRsay

1> KNnabnkemKuNedayeRbIkarpSMbnk. sRmab;bnkTMnajEpndI

Pu = 1.2 D + 1.6 L = 1.2 1690 + 1.6 623 = 3024.8kN
M u = M 2ns = 1.2 M D + 1.6 M L = 1.2 73.2 + 1.6 48.8 = 165.92kN .m

sRmab;bnkTMnajEpndI nigbnkxl;
Pu = (1.2 D + 0.5L + 1.6W ) = 1.2 1690 + 0.5 623 + 0 = 2339.5kN
M uns = M 2ns = 1.2 M D + 1.6 M L = 1.2 73.2 + 1.6 48.8 = 165.92kN .m
M us = M 2 s = 1.6 M w = 1.6 67.8 = 108.48kN .m

bnSMbnkepSgeTotEdlminsMxan;
Pu = 0.9 D + 1.6W = 0.9 1690 + 1.6 0 = 1521kN
M 2 = 0.9 M D = 1.2 73.2 = 87.84kN .m

M 2 s = 1.6 M w = 1.6 67.8 = 108.48kN .m

M
M
165.92
e = u = 2ns =
= 54.85mm
Pu
Pu
3024.8
emin = (15.24 + 0.03h) = 15.24 + 0.03 460 = 29.04mm < 54.85mm

2> eRCIserIsmuxkat;dMbUgrbs;ssredayEpkelIbnSMbnkTMnajEpndIedayeRbItaragbdaRkam.
eRCIserIsmuxkat;ssr 460 460 CamYynwgEdk DB32 cMnYn4edIm rUbTI7.

3> epgpat; Klu / r

Ig =

460 4
= 37.3 108 mm 4
12

Ec = 28278.9 MPa

sRmab;ssr I = 0.7 I g
sRmab;ssrEdlmankm<s; 4.9m
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EI 0.7 37.3 108 28278.9

=
= 15.1 109 N .mm
lc
4900

sRmab;ssrEdlmankm<s; 3.4m
EI 0.7 37.3 108 28278.9
=
= 21.7 109 N .mm
lc
3400

sRmab;Fwm EI g / lb = 40 109 N .mm / I = 0.35I g nig

EI / lb = 0.35 40 109 = 14 109 N .mm

(EI / lc ) = (15.1 + 21.7) 10 = 1.3

(top) = (bottom) =
9
9

(EI / lb )

2 14 10

BItarag alignment chart K = 1.4 sRmab;eRKagKanBRgwg nig K = 0.8 sRmab;eRKag

BRgwg.
Klu 1.4 4900
= 49.7
=
0.3 460
r

EdlFMCag 22 nigtUcCag 100 . dUcenH eKRtUvBicarNaBI slenderness ratio.

4> epgpat; lu / r = 4900 /(0.3 460) = 35.5
35
35
=
= 54.8
Pu / f 'c Ag
3024800 /(35 460 460)

(-24)

edaysarEt lu / r < 54.8 m:Um:g; nonsway moment mincaM)ac;bEnm.

5> KNna Pc
Ec = 28278.9MPa
Ig =

Es = 2.1 105 MPa

2

460 4
= 37.3 108 mm 4
12

I se =

4 32 2 340
6
4

= 93 10 mm
4 2

d = 0.55
EI =
EI =

0.2 Ec I g + E s I se
1 + d
0.2 28278.9 37.3 108 + 2.1 105 93 106
= 26.2 1012 N .mm 2
1 + 0.55

edIm,IKNna s / d = 0 enaH EI = 1.55 26.2 1012 = 40.63N .mm2

2 EI 2 26.2 1012
Pc =
=
= 16827.86kN
BRgwg
( Kl ) 2
(0.8 4900) 2
u

Pc =

EI
2

( Klu ) 2

2 40.63 1012
(1.4 4900) 2

= 8521.15kN

KanBRgwg

sRmab;mYyCan;enAkgGKar eKmanssrxagkg 14 ssrxageRkA 18 nigssrkac;RCug 4 .

1
3

2
3

Pu = 14(2339.5) + 18( 2339.5) + 4( 2339.5) = 63946.3kN

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2
3

Pc = 14(8521.15) + 22( 8521.15) = 244273kN

s =

1
= 1.54
63946.3
1 (
)
0.75 244273

EdlFMCag 1 nigtUcCag 2.5 smIkar (-19)

M c = M 2 ns + s M 2 s = 165.92 + 1.54 108.48 = 333kN .m

6> bnkKNnaKW Pu = 2339.5kN nig M c = 333kN .m

e=

333
= 142.34mm
2339.5

emin = (15.24 + 0.03h) = 15.24 + 0.03 460 = 29.04mm < e

tamkarviPaK sRmab; e = 142.34mm nig A = 1608.5mm2 = 0.65

lTPaBRTbnkrbs; ssrmuxkat; 460 460 KW Pn = 2348.1kN nig M n = 334.2kN .m
dUcenHmuxkat;KWRKb;RKan;. dMeNaHRsaymanlkNRsedogKaeTAnwg]TahrN_TI4 kg
emeroneRKOgbgMrgkarsgt; nigkarBt;. tm a = 242.86mm /
c = 303.57 mm / f s = 190.6 MPa / f ' s = 400 MPa / Pb = 1676.8kN nig
eb = 218mm .
400 303.57
= 0.65
t = 0.003
= 0.00095 < 0.002 /
303.57

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XIII.

eCIgtag

FOOTINGS

13>1> esckIepIm (Introduction)

eCIgtagebtugGarem:CaGgt;neRKOgbgMEdlRtUv)aneRbIedIm,IRTssr nigCBaaMg nigmannaTI
bBan ehIynigBRgaybnkrbs;vaeTAdI. karKNnaKWQrelIkarsnt;faeCIgtagmanlkNrwg dUcenH
bERmbRmYlsm<aFdIEdlenABIeRkameCIgtammanlkNCabnat; (linear). eKTTYl)ansm<aFdIman
lkNesIenAeBlEdlbnkssrRtYtsIuKaCamYynwgTIRbCMuTmn;rbs;eCIgtag. ebIeTaHbICakarsnt;enH
Gac TTYlyk)ansRmab;eCIgtagrwg Etkarsnt;enHkayeTACaminsUvsuRkitenAeBlEdleCIgtagkay
eTACamanlkNTn; (flexible) xaMg. karKNnadRtwmRtUvrbs;eCIgtagtRmUv[
- minRtUvFMCaglTPaBRTRTg;bnkrbs;dI.
- RtUveCosvagsRmutFM sRmutDIepr:g;Esl (differential settlement) bmMurgVil.
- RtUvmanemKuNsuvtiPaBRKb;RKan;Tb;nwgkarrGil (sliding) nigb karRkLab; (overturning).
RbePTeCIgtagEdlmanlkNTUeTAbMputenAkgsMNg;RtUv)aneRbIKW single footing nig wall
footing rUbTI13>1 nig 13>2. enAeBlEdlbnkssrRtUv)anbBaneTAdIedayeCIgtag enaHdIrg
karsgt;. brimaNnsRmutGaRsynwgktaCaeRcIn dUcCa RbePTdI GaMgtg;sIuetbnk CeRmABIeRkamnIv:U
dI RbePTeCIgtag. RbsinebIrcnasm<nEtmYymaneCIgtagxusKa enaHvanwgmansRmutxusKa ehIykug
RtaMgfInwgekItmanenAkgeRKOgbgM. sRmutDIepr:g;EslFMnwgbg[Ggt;enAkgsMNg;EdlminEmnCa
eRKOgbgMrgeRKaHfak; bEpkxHEdlrgplb:HBal;Gac)ak;.
CaTUeTA bnkbBarRtUv)andak;enAcMTIRbCMuTmn;rbs;eCIgtag. RbsinebIkmaMgGnuvtn_pbmin
RtYtsIunwgTIRbCMuTmn;rbs;eCIgtag enaHm:Um:g;Bt;ekItman. kgkrNIenH sm<aFenABIeRkameCIgtagmag
FMCag sm<aFenABIeRkameCIgtagmageTot.
RbsinebIlTPaBRTRTg;bnkrbs;dIxusKaenAeRkameCIgtagepSgKa enaHsRmutDIepr:g;Eslnwg
ekItman ]TahrN_ RbsinebIeCIgtagnGKarxHQrelIdI xHeTotQrelIf. enAkgkrNIEbbenH
eKKYreFVIkarEcksMNg;enHCaBIrEpkedaytMN edIm,IGnuBaat[vaRsutedaykraCBIKa.
CeRmAeCIgtagBIeRkamnIv:UdIKWCaktadsMxan;kugkarKNnaeCIgtag. CeRmAKYrRtUv)ankMNt;BIkar
BiesaFdI Edl)anpl;nUvBtmandKYr[TukcitBIlTPaBRTRTg;rbs;dIdmansuvtiPaBeTAtamRsTab;dI
eCIgtag

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epSgKaBIeRkamnIv:UdI. r)aykarN_BIkarBiesaFdIkMNt;nUvlTPaBRTRTg;GnuBaatedIm,IeRbIkgkarKNna.
enAtMbn;RtCak;EdlmanFak;RBil (frost action) GacnwgekIneLIg bfycuH. dUcenHeKcaM)ac;RtUvdak;
eCIgtagBIeRkam freezing depth edIm,IeCosvagkarmanclna.
13>2> RbePTeCIgtag (Types of Footings)
eKmaneCIgtagCaeRcInRbePTEdleRbIsRmab;RTssr bCBaaMg. eCIgtagEdleKeRbICaTUeTAman
dUcxageRkam
- Wall footings RtUv)aneRbIedIm,IRT structural wall EdlRTbnkBIkRmalxN bedIm,IRT
nonstructural walls. vamanTTwgkMNt; nigmanRbEvgCab;KaenABIeRkamCBaaMg rUbTI13>1. Wall
footings GacmankRmas;EtmYy Gacmanfak; bGacmanCRmal.

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b single footings RtUv)aneRbIedIm,IRTssreTal rUbTI13>2. vaGacmanragkaer

ctuekaNEkg brgVg;. mgeTot eCIgtagmankRmas;esI manfak; bmanCRmal. vaCaRbePTeCIgtag
EdlmanlkNsnSMsMcbMput nigvaRtUv)aneRbIenAeBlssrmanKMlatqay. RbePTEdleKcUlcit
eRbICageKKWmanragkaer bctuekaNEkgCamYykRmas;esI.
- Combined footing rUbTI13>3CaTUeTARTssrBIr bssrbIEdlminenACYrEtmYy. ragrbs;
eCIgtagkgbg;GacCactuekaNEkg bctuekaNBay GaRsynwgbnkssr. Combined footings
RtUv)aneRbIenAeBlssrBIrenACitKaEdleCIgtag single footings minGaceRbI)an benAeBlssrmYy
sitenAelI benAEk,r property line.
-

eCIgtag

Isolated,

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rUbTI13>4 rYmmaneCIgtag single footings BIrPab;edayFwm

mYy b strap mYy nigRTssreTalBIr. vaRtUv)aneRbIenAeBlEdleCIgtagmYyRTssrcakpit ehIy
eCIgtagmYysitenAEk,rmancmayKMlattUc. RbePTeCIgtagenHCMnYs combined footings nigeBlxH
manlkNsnSMsMcCag.
- Continuous footings rUbTI13>5 RTnUvssrmYyCYrEdlmancMnYncab;BIbIeLIg. vamanTTwg
kMNt; nigCab;KaBIeRkamssrTaMgGs;.
-

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b mat foundations rUbTI13>6 pSMeLIgedayeCIgtagmYy CaTUeTARtUv)andak;enABI

eRkamtYGKarTaMgmUl nigRTssrGKar. vaRtUv)aneRbIenAeBl
a. lTPaBRTRTg;rbs;dItUc
b. bnkssrFM
c. Single footings minGaceRbI
d. Piles minRtUv)aneRbI
e. sRmutDIepr:g;EslRtUv)ankat;bnytamryRbBnRKwHTaMgmUl
- Pile caps rUbTI13>7 CakRmalxNRkas;EdlRtUv)aneRbIedIm,IcgPab;RkumssrRKwHCamYy
Ka nigedIm,IRT nigbBalbnkssreTAssrRKwH.
-

Raft

13>3> karBRgaysm<aFdI (Distribution of soil pressure)

rUbTI 13>8 bgajBIeCIgtagEdlRTssreTal. enAeBlbnkssr P RtUv)andak;enABIelITI
RbCMuTmn;eCIgtag enaHsm<aFesIRtuv)ansnt;ekItmanenAelIpdIBIeRkampeCIgtag. b:uEnkarBit kar
BRgaysm<aFdIBitR)akdminesIeT vaGaRsynwgktaCaeRcIn CaBiessGaRsynwgFatupSMrbs;dI
(composition of the soil) nigkRmitTn;rlas; (degree of flexibility) rbs;eCIgtag.
eCIgtag

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]TahrN_ karBRgaysm<aFenAelIdIEdlKanPaBsit (cohesionless soil) bdIxSac;enABIxag

eRkameCIgtagrwg (rigid footing) RtUv)anbgajenAkgrUbTI 13>9. sm<aFmantmGtibrmaenABIxag
eRkamGkSeCIgtag ehIyfycuHenAxagcugeCIgtag. dIEdlKanPaBsit)ancltecjBIcugneCIgtag
begIt[mankarkat;bnysm<aF b:uEnsm<aFekIneLIgenAmMGkSedIm,IbMeBjlkxNlMnwg. RbsinebI
eCIgtagsitenABIelIdIsit (cohesive soil) dUcCadI\d sm<aFenABIxageRkamcugeCIgtagFMCagenARtg;
GkSrbs;eCIgtag rUbTI 13>10. dI\dEk,rcugRKwHmanPaBsitxaMgCamYynwgdI\denAEk,rEdlBTCMu
vijeCIgtag begIt)anCakarBRgaysm<aFminesI.
CaTUeTAsm<aFdIRTRTg;GnuBaat qa RtUv)ankMNt;ecjBIkarBiesaFn_dI. sm<aFdIRTRTg;GnuBaat
enHERbRbYleTAtamRbePTdI BItmx<s;bMputsRmab;RsTab;f (rocky bed) eTAtmTabbMputsRmab;Rs
Tab;dIl,b; (silty soil). ]TahrN_ qa sRmab; sedimentary rock esI 1450kN / m 2 / sRmab;
compacted gravel esI 400kN / m 2 / sRmab; well-graded compacted sand esI 300kN / m 2 / nig
sRmab; silty-gravel esI 150kN / m 2 .

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eyagtamrUbTI 13>8 enAeBlEdlbnk P manGMeBI EpkneCIgtagBIeRkamssrrkklnwg

RsutcuHeRkam. eCIgtagnwgyknUvTRmg;ExSRtg;esI EdlbegItCasm<aFbukeLIgelIvijedIm,ITb;Epkn
eCIgtag. EpknImYymannaTICa cantilever nigRtUv)anKNnasRmab;TaMgm:Um:g;Bt; nigkmaMgkat;TTwg.
karKNnaeCIgtagnwgRtUv)anBnl;y:aglMGitenAeBleRkay.
13>4> karBicarNakgkarKNna (Design Consideration)
eCIgtagRtUv)anKNnaedImI,RTbnkssr nigbBanbnkssreTAdIy:agsuvtiPaB. viFIsaRskg
karKNnaRtUvEtykersIusg;caM)ac;xageRkammkBicarNa
- RkLapeCIgtagQrelIlTPaBRTRTg;dIGnuBaat
- kmaMgkat;mYyTis one-way shear
- kmaMgkat;BIrTis two-way shear bkmaMgpug punching shear
- m:Um:g;Bt; nigbrimaNEdkRtUvkar
- lTPaBRTRTg;rbs;ssrenA)atrbs;va nigEdk dowel caM)ac;
- RbEvgf<k;rbs;Edk
- sRmutDIepr:g;Esl
ersIusg;caM)ac;TaMgenHnwgRtUvBnl;enAkgEpkxageRkam.
13>4>1> TMhMeCIgtag (size of Footing)
RkLapeCIgtagGacRtUv)anKNnaBIbnkxageRkABitR)akd m:Um:g; nigbnkKanemKuNEdl
minFMCaglTPaBRTRTg;rbs;dI. CaTUeTA sRmab;bnkQr
load (including self - weight)
RkLapeCIgtag = total service
!#>!
allowable soil pressure, q
a

b
Edl bnkeFVIkarsrubCabnkKNnaKanemKuN. enAeBlEdlRkLapRtUv)ankMNt;sm<aFdI
emKuNRtUv)anTTYledayEckbnkemKuN Pu = 1.2D + 1.6L edaykLapeCIgtag. kareFVIEbb
enHedIm,IKNnaeCIgtagtamviFIKNnaersIusg; (strength design method) .
P (total )
Area =
qa

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qu =

NPIC

!#>@

Pu
area of footing

sm<aFdIGnuBaat qa RtUv)anTTYlBIkarBiesaFdI nigQrelIlkxNbnkeFVIkar (service load) .

13>4>2> kmaMgkat;mYyTis kmaMgkat;Fwm Vu1 One-Way Shear (Beam Shear)
sRmab;eCIgtagCamYynwgkmaMgBt;mYyTis muxkat;eRKaHfak;sitenAcmay d BIpmuxrbs;
ssr. kmaMgTajGgt;RTUgenAmuxkat; m m kgrUbTI13>11 GacRtUv)anBinitemIldUcGVIEdl)aneFVI
sRmab;Fwm. kmaMgkat;TTwgGnuBaatenAkgkrNIenHesInwg
2
Vc = f ' c bd
= 0.75
!#>#
12
Edl b = TTwgrbs;muxkat; m m . kmaMgkat;TTwgemKuNenAmuxkat; m m GacRtUv)an
KNnadUcxageRkam
L C

!#>$
Vu1 = qu b d
2 2

RbsinebIEdkkmaMgkat;TTwgminRtUv)aneRbI enaH d GacRtUv)anKNna edaysnt; Vu = Vc

12Vu1
!#>%
d=
2 f ' b
c

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13>4>3> kmaMgkat;BIrTis kmaMgpug Vu1 Two-Way Shear (Punching Shear)

kmaMgkat;BIrTisCargVas;nkmaMgTajGgt;RTUgEdlbNalmkBI\TiBlnbnkssrenAelI
eCIgtag. sameRbHeRTt GacekIteLIgenAkgeCIgtagenAcmay d / 2 BImuxpssrenARKb;RCug. eCIg
tagnwg)ak;enAeBlEdlssrBayamTMlHu EpkneCIgtag rUbTI13>12.
ACI Code, Section 11.12.2 GnuBaatersIusg;kmaMgkat;TTwg Vc enAkgeCIgtagEdlKanEdk
kmaMgkat;TTwgsRmab;GMeBIkmaMgkat;BIrTis esInwgtmtUcCageKbMputn
4
Vc1 =
f 'c bo d
!#>^
12

4 f ' c bo d
Vc 2 = 2 +
12

d
f ' c bo d
Vc3 = s + 2
bo
12

!#>&
!#>*

pleFobRCugEvgelIRCugxIrbs;ssr
bo = brimaRtnmuxkat;eRKaHfak;EdlenAcmay d / 2 BIRkLapbnk muxkat;ssr
emIlrUbTI 13>12
d = km<s;RbsiTPaBrbs;eCIgtag
sRmab;tm Vc1 nig Vc2 eyIgeXIjfa Vc1 lub tUcCag Vc2 enARKb;eBlEdl 2 b:uEn
Vc 2 lub tUcCag Vc1 enARKb;eBlEdl > 2 . enHbgaj[eXIjfakmaMgkat;TTwgGnuBaat Vc
RtUv)ankat;bnysRmab;eCIgtagEvgxaMg. bERmbRmYlsm<aFdIBitR)akdtambeNayRCugEvgCamYynwg
karekIneLIgn . sRmab;rUbragdTeToteRkABIctuekaN KWCapleFobrvagTMhMEvgCageKbMputn
RkLapbnkRbsiTPaBkgTisedAEvg elITTwgFMCageKbMputkgTisedAxI EkgnwgTisedAEvg.
sRmab;smIkar !#>* s RtUv)ansnt;esI 40 sRmab;ssrenAkNal/ esI 30 sRmab;ssr
enAxag nigesI 20 sRmab;ssrenARCug. ersIusg;kmaMgkat;TTwgrbs;ebtug Vc3 bgajBI\TiBln
kMeNIn bo elI d . sRmab;pleFobx<s;n bo / d enaH Vc3 nwglub.
Edl

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QrelItmTaMgbIxagelIrbs; Vc km<s;RbsiTPaB d EdlcaM)ac;sRmab;kmaMgkat;BIrTisKW)an

mkBItmEdlFMCageKkgcMeNamrUbmnxageRkam = 0.75
3Vu 2
Edl 2
!#>(
d1 =
f' b
c o

d1 =

d2 =

12Vu 2

4
2 + f ' c bo

12Vu 2

Edl > 2

!#>!0
!#>!!

s + 2 f ' c bo
bo

kmaMgkat;TTwgBIrTis Vu 2 nigkm<s;RbsiTPaB d caM)ac; RbsinebIEdkkmaMgkat;TTwgminRtUv

)aneRbIGacRtUv)anKNnadUcxageRkam eyagtamrUbTI 13>12
a. snt; d
b. kMNt; bo bo = 4(c + d ) sRmab;ssrkaer EdlRCugesI c .
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bo = 2(c1 + d ) + 2(c 2 + d )

sRmab;ssrctuekaN EdlmanRCugesInwg c1 nig

c2

c.

d.

kmaMgkat;TTwg Vu 2 eFVIGMeBIenAelImuxkat;EdlmanbeNay bo = 4(c + d ) b bo = 2(c1

+ d ) + 2(c 2 + d ) nigkm<s; d . muxkat;enHrgnUvkmaMgbBarcuHeRkam Pu nigkmaMg
bBareLIgelI qu smIkar !#>@. dUcenH
Vu 2 = Pu qu (c + d ) 2
sRmab;ssrkaer
!#>!@a
Vu 2 = Pu qu (c1 + d )(c 2 + d ) sRmab;ssrctuekaNEkg
!#>!@b
kMNt; d FMbMput n d1 nig d 2 . RbsinebI d EdlrkeXIjminmantmEk,rtm
d Edl)ansnt; eFVIkarsnt;nigedaHRsayeLIgvij.

13>4>4> ersIusg;Bt; nigEdkeCIgtag (Flexural Strength and Footing Reinforcement)

muxkat;eRKaHfak;sRmab;m:Um:g;Bt;ekItmanenABImuxpssr muxkat; n n / rUbTI 13>13.
eKRtUvBinitemIlm:Um:g;Bt;enAelITisnImYyrbs;eCIgtag nigdak;brimaNEdkRKb;RKan;. sRmab;eCIg
tagkaer nigssrkaer m:Um:g;Bt;mantmesIKaTaMgBIrTis. edIm,IkMNt;muxkat;EdkcaM)ac; eKRtUveRbI
km<s;rbs;eCIgtagTaMgBIrTis. edaysarEtEdkenAkgTismYysitenAelIEdkenAkgTismYyeTot
km<s;RbsiTPaB d ERbRbYleTAtamGgt;pitrbs;EdleRbI. tmmFmrbs; d GacRtUv)anykmk
eRbI. tmsRmab;karGnuvt d RtUv)ansnt;esI h 110mm .
CaerOy km<s;RbsiTPaBrbs;eCIgtagRtUv)anRKb;RKgedaykmaMgkat;TTwg EdlcaM)ac;
RtUvkarkm<s;eCIgtagFMCagkm<s;eCIgtagEdlRtUvkaredaym:Um:g;Bt;. EdkBRgwgkgTisedAnImYyGac
RtUv)anKNnaenAkgkrNIGgt;rgkarBt;dUcxageRkam
As f y

M u = As f y d
!#>!#
1.7 f ' b

m:ageTot PaKryEdk GacRtUv)ankMNt;dUcxageRkam smIkar $>@

=

0.85 f ' c
2 Ru
1 1

f y
(0.85 f ' c )

!#>!$

Edl Ru = M u / bd 2 . enAeBlEdl Ru RtUv)ankMNt; kRtUv)anTTYlBIsmIkar !#>!$.

tRmUvkarpleFobEdkGb,brmaenAkgGgt;rgkarBt;KWesInwg 1.4 / f y enAeBl
f 'c < 31MPa nigesInwg f ' c / 4 f y enAeBl f 'c 31MPa . b:uEn ACI Code, Section 10.5
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bgajfasRmab;kRmalxNEdlmankRmas;esIRkLapGb,brma nigKMlatGtibrmanEdkenAkgTis
edAnkarBt;KYrRtUvkarEdkrYmmaD nigEdksItuNPaB. tRmUvkarEdkGb,brmaelIkeRkayenHmantm
tUcehIypleFobEdkGb,brmaEdlmantmFMCagRtUv)anesIeLIg EtvaminKYrFMCag 1.4 / f y .

EdkenAkgeCIgtagmYyTis nigeCIgtagBIrTisRtUvEtBRgayeBjTTwgeCIgtag. kgkrNI

eCIgtagctuekaNBIrTis ACI Code, Section 15.4.4 kMNt;faenAkgTisedAEvg EpknEdksrub s l
RtUv)anEbgEckesIenAelITTwgrbs;eCIgtag. enAkgTisedAxI PaKryBitR)akdrbs;EdksrubenAkgTis
edAenHRtUvEt)andak;esIenAkg bandwidth esInwgRbEvgnRCugxIneCIgtagGaRsynwg
2
!#>!%
s =
+1
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Edl

Department of Civil Engineering

!#>!^
Bandwidth RtUvEtsitenAkNalnGkSrbs;ssr rUbTI13>14. EdkEdlenAsl;enA
kgTisedAxIRtUvEtBRgayesIBIxageRkA Bandwidth . PaKryEdkenAsl;enHminKYrticCagEdkrYmmaD
nigEdksItuNPaB.
enAeBlEdlssreRKOgbgMGMBIEdk bssr\dRtUv)aneRbI enaHmuxkat;eRKaHfak;sRmab;
m:Um:g;Bt;enAkgeCIgtagRtUv)anykenAcenaHBak;kNal nigEKmCBaaMg\d nigcenaHBak;kNalp
ssr nigEKmn steel base plate (ACI Code, Section 15.4.2).
=

long side of footing

short side of footing

(Bearing Capacity of Column at Base)

13>4>5> lTPaBRTRTg;rbs;ssrenARtg;)at
bnkmkBIssrmanGMeBIelIeCIgtagenAKl;rbs;ssr EdlmanRkLapesInwgRkLapmux
kat;ssr. kmaMgsgt;RtUv)anbBanmkssredaypal;eday bearing enAelIebtug. kmaMgEdlman
GMeBIenAelIebtugenARtg;Kl;nssrminRtUvelIslTPaBRTRTg;rbs;ebtugEdlkMNt;eday ACI Code,
Section 10.17
lTPaBRTRTg; N1 = (0.85 f 'c A1 )
!#>!&
Edl = 0.65 nig A = RkLap bearing nssr.
tmnlTPaBRTRTg;Edl[ edaysmIkar 13.17 GacRtUv)anKuNedayemKuN A / A 2.0
sRmab; bearing enAelIeCIgtagenA eBlpEdlRTFMCagRkLapbnkenARKb;RCug. A enATIenHCa
RkLapnEpkneCIgtagEdlrUb FrNImaRtdUcKa ehIyRtYtsIuKaCamYynwgRkLapbnk rUbTI
13>15. edaysarEt A > A enaH emKuN A / A nwgFMCagmYy EdlbgajfalTPaBRTRTg;
GnuBaatekIneLIgedaysarTRmxagnRkLapeCIgtagEdlBTCMuvijKl;ssr. lTPaBRTRTg;kMEN
modified bearing strength KW
A
!#>!*
N 2 = (0.85 f ' c A1 ) 2 2 (0.85 f ' c A1 )
A
1

RbsinebIbnkemKuN P FMCag N b N EdkRtUv)andak;edIm,IbBankmaMgelIs. eKGac

TTYlva)anedaydak;Edk dowel bEdkssrRtUv)andak;bgscUleTAkgeCIgtag. kmaMgelIsKW
Pex = Pu N1 ehIyRkLaprbs;Edk dowel KW Asd = ( Pex / f y ) 0.005 A1 Edl A CaRkLap
rbs;muxkat;ssr. y:agehacNas;EdkbYnedImRtUv)aneRbIbYnedImenARCugTaMgbYnrbs;ssr.
u

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RbsinebIkmaMgemKuNtUcCag N b N enaHeKdak;EdkGb,brma. ACI Code, Section

15.8.2 bgajfaRkLapEdkGb,brmarbs;Edk dowel y:agehacNas;esInwg 0.005 Ag nigminRtUv
ticCagbYnedIm Edl A CaRkLapeBj (gross section) rbs;muxkat;ssr. EdkGb,brmakRtUv)an
eRbIEdrenAeBlEdlkmaMgemKuNFMCag N b N . Edk dowel RtUv)andak;enARCugTaMgbYnrbs;ssr
ehIybgscUleTATaMgkgssr nigTaMgkgeCIgtag. Ggt;pitEdk dowel minKYrFMCagEdkbBarenAkg
ssr 4mm . tRmUvkarenHcaM)ac;edIm,IFanakareFVIkardlrvagssr nigeCIgtag. EdkbBasrbs;Edk
dowel RtUv)anBinitemIledIm,IkMNt;nUvkarbBandlnkmaMgsgt;eTAkgeCIgtag.
1

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Department of Civil Engineering

13>4>6> RbEvgEdkRCYs (Development Length of the Reinforcing bars)

muxkat;eRKaHfak;sRmab;RtYtBinitRbEvgEdkbgb;dUcKasRmab;m:Um:g;Bt;.
RbEvgbgb;sRmab; Edksgt;RtUv)an[enAkgemeronTI7
l dc =

0.24 f y d b

f 'c

b:uEnvaenHminRtUvtUcCag 0.044d b f y 20cm . sRmab;tmepSgeToteyag tamemeronTI7.

13>4>7> sRmutDIepr:g;Esl karKNnaeCIgtagkglkxNlMnwg
Differential Settlement (Balanced Footing Design)

CaTUeTAeCIgtagRTnUvbnkdUcxageRkam
- bnkefrEdl)anmkBIeRKagxageRkam (substructure) nigeRKagxagelI
(superstructure) .
- bnkGefrEdlGnuvtmkelI
- Tmn;rbs;smarEdleRbIsRmab;cak;bMeBj
- bnkxl;
eCIgtagnImYyenAkgGKarRtUv)anKNnaedIm,IRTbnkGtibrmaEdlGacekItmanenAelIssr
Edl)anBIkarbnSMbnkdeRKaHfak;bMput edayeRbIsm<aFRTRTg;rbs;dIGnuBaat.
bnkefr nigRbEhlCabnkGefrmYyEpktUc EdleKehAfa bnkGefrFmta (usual live
load) GacGnuvtCab;Gt;dac;mkelIeRKagsMNg;. bnkGefrEdlenAsl;epSgeTot GacekItmanyUr
mg ehIymanGMeBImkelIeRKagsMNg;EpkxHEtb:ueNaH EdlbNal[manbnkepSgKaenAelIssr.
dUcenH sm<aFdIenABIeRkameCIgtagepSgKa nwgERbRbYlGaRsynwgbnkenAelIssr ehIysRmutDIepr:g;
EslnwgekItmanenABIeRkameCIgtagepSgKansMNg;EtmYy. edaysarsRmutedayEpk (partial
settlement) minGaceCosput enaHbBaabEgVr[sRmutDIepr:g;Eslrbs;sMNg;CagGt;n[)an.
brimaNnsRmutDIepr:g;EslGaRsynwgPaBxusKankRmitENnrbs;dI/ kRmas;smarEdlGacsgt;
)anenABIeRkamnIv:UeCIgtag nigPaBrwgRkajn combined footing nigsMNg;xagelI. sRmutDIepr:g;
EslFMFIV[ebtugeRbH nigeFVI[xUcxatdl;kargarbegIy (cladding or finishing) CBaaMgxN nigBi
dan.

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sRmutDIepr:g;EslGacsMEdgedaykarrmUlrbs;eRKag (angular distortion of structure).

Bjerrum bgajfakarEdkkMNt;eRKaHfak;rbs;karrmYlsRmab;lkxNxHERbRbYlcenaHBI 1 / 600 eTA
1 / 150 GaRsynwgkarxUcxatEdlnwgekItmanenAelIrcnasm<n.
sRmab;karGnuvtn_ eKGacsnt;fasm<aFdIeRkambnkefrmantmesIKasRmab;RKb;eCIgtag
dUcenH vaeFVI[manbnkesI. bnkefr bbnkFmta GacRtUv)ansnt;esInwgplbUkrvagbnkGefr nig
PaKrynbnkGefrEdlekItmanCaerOyenAelIeRKagsMNg;. KankrNIEdllTPaBRTRTg;dIGnuBaat
FMCagbnkefrbUknwgbnkGefrGnuBaatsRmab;eCIgtagnImYyeT. ]TahrN_TI13>4 Bnl;BIviFIsaRs
sRmab;KNnaRkLapeCIgtagEdlykmkKitBicarNaBI\TiBlrbs;sRmutDIepr:g;Esl.
13>5> eCIgtagebtugsuT (Plain Concrete Footings)
eCIgtagebtugsuTGacRtUv)aneRbIedIm,IRTCBaagM \d bbnkRsal nigbBalbnkTaMgenaHeTAdI
Edlva. ACI Code GnuBaat[eRbIeCIgtagebtugsuTenAelIdI RbsinkugRtaMgKNnaminRtUvFMCagkrNI
xageRkam
- kugRtaMgrgkarBt;GtibrmasRmab;karTajRtUvEttUcCagbesInwg 5 f 'c / 12 Edl
= 0.55
- kugRtaMgGtibrmaenAkgkmaMgkat;TTWgmYyTis (beam action)RtUvtUcCagbesInwg f 'c / 9
Edl = 0.55
- kugRtaMgGtibrmaenAkgGMeBIBIrTisKW
1
2
2.66
+
f ' c
f 'c
Edl = 0.55
12
9 9
- ersIusg;kmaMgsgt;GtibrmaminKYrFMCagkarkMNt;rbs;lTPaBRTRTg;ebtugGnuBaat. f 'c
nebtugsuTminKYrtUcCag 17MPa .
- kRmas;Gb,brmarbs;eCIgtagebtugsuTminKYrtUcCag 20cm .
- muxkat;eRKaHfak;sRmab;m:Um:g;Bt;KWsitenAprbs;ssr bCBaaMg.
- muxkat;eRKaHfaksRmab;GMeBInkmaMgkat;TTwgmYyTis nigkmaMgkat;TTwgBIrTisKWsitenAcmay
d nig d / 2 BIpnssr bCBaaMg erogKa. eTaHbICaeCIgtagebtugsuTminRtUvkarEdkkeday
kvaCakarEdlRbesIkgkardak;srsEdkrYmmaDsRmab;TisTaMgBIrrbs;eCIgtag.
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Department of Civil Engineering

- kugRtaMgEdlbNalBIbnkemKuNRtUv)anKNnaedaysnt;BRgayCaragbnat;enAkgebtug.
- km<s;RbsiTPaB d RtUv)anKitfaesInwgkm<s;srubdk[ 75mm .
- sRmab;karBt; nigkmaMgkat;mYyTis eRbImuxkat;eBj bh b:uEnsRmab;kmaMgkat;BIrTis eRbI
b h edIm,IKNna Vc .
o

]TahrN_ 13>1 KNnaeCIgtagebtugBRgwgedayEdkedIm,IRTCBaaMgebtugEdlmanTTwg 50cm Edl

RTbnkefr 379.5kN / m EdlrYmbBalbnkpal;rbs;CBaaMg nigRTnUvbnkGefr 292kN / m . )at

rbs;eCIgtagsitenACeRmA 1.8m BInIv:UdI. eK[ f ' = 28MPa / f y = 400MPa nigsm<aFdIGnuBaat
0.24 MPa = 240kN / m .
c

dMeNaHRsay

1> KNnasm<aFdIRbsiTPaB. snt;km<s;srubrbs;eCIgtagKW 50cm . Tmn;rbs;eCIgtagKW

0.5 25 = 12.5kN / m . edaysnt;Tmn;maDrbs;dIesInwg 16kN / m
enaHTmn;dIbMeBjBIelIeCIgtagKW 16 (1.8 0.5) = 20.8kN / m .
bnkCBaaMgEdlbgb;enAkgdIKW 25 0.5 (1.8 0.5) /1m = 16.25kN / m .
bnkCBaaMgebtugbgt;kgdIKitCadI 16 0.5 (1.8 0.5) /1m = 10.4kN / m .
sm<aFdIRbsiTPaBenA)atrbs;eCIgtagKW
240 12.5 20.8 16.25 + 10.4 = 200.85kN / m .
2> KNnaTTwgrbs;eCIgtagsRmab;CBaaMgRbEvg 1m
TTwgrbs;eCIgtag = bnksrub sm<aFdIRbsiTPaB
2

total loal
effective soil pressure
379.5 + 292
=
= 3.34m
200.85

width of footing =

yk 3.4m
3> sm<aFdIsuT= bnkemKuN TTwgeCIgtag sRmab;RbEvg 1m
Pu = 1.2 D + 1.6 L = 1.2 379.5 + 1.6 292 = 922.6kN / m
922.6
(net upward pressure) q u =
= 271.4kN / m 2
3.4

sm<aFdIsuT
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4> epgpat;km<s;snt;sRmab;tRmUvkarkmaMgkat;TTwg. kRmas;karBarEdksRmab;eCIgtagKW

75mm ehIysnt;eRbIEdk DB 25 enaH d = 500 75 12 = 413mm .
muxkat;eRKaHfak;sRmab;kmaMg kat;TTwgmYyTisKWsitenAcmay d BIpnBaaMg
c
500 6
B
3400
Vu = q u d = 271.4
413
10 = 0.28kN
2
2
2
2

kmaMgkat;TTwgmYyTisGnuBaat= 0.17 28 = 0.9

Vu
280
=
= 415mm
d EdlRtUvkar =
0.17 f ' b 0.75 0.9 1
c

km<s;srubKW 415 + 75 + 12 = 502mm b 500mm . d Cak;EsgKW 500 75 12 = 413mm

dUckarsnt;.
cMNaMfa eKnwgRtUvkarkarsakl,gmYycMnYnedIm,Isnt; nigKNnatm d [mantmEk,r.
5> KNnam:Um:g;Bt; nigEdk. muxkat;eRKaHfak;KWsitenApnCBaaMg
2

1 B c
271.4 3.4 0.5

qu =

= 285.3kN .m
2 2 2
2 2
2
M
285.3 10 6
Ru = u2 =
= 1.67 MPa
bd
1000 413 2

Mu =

sRmab; Ru = 1.67MPa /
=

0.85 f ' c
fy

f ' c = 28MPa

nig

f y = 400MPa

enaHPaKryEdk

0.85 28

2 Ru
2 1.67
=
1 1
1 1
= 0.0048
(0.85 f ' c )
0.9 0.85 28
400

PaKryEdkGb,brmasRmab;Ggt;rgkarBt;
min =

1.4 1.4
=
= 0.0035
f y 400

PaKryEdkrYmmaDKW 0.18% sRmab;

dUcEdl)an KNna

f y = 400MPa

. dUcenHeRbI = 0.0048

As = 0.0048 1000 413 = 1982.4mm 2 / m

As = 2182mm 2
eRbIEdk DB25 EdlmanKMlat 225mm
6> RtYtBinitRbEvgEdkbgb;sRmab;Edk DB25
l d = 48d b = 48 25 = 1200mm
eyagtamemeronTI 7
Edl[
ld =
T.Chhay

B c
3400 500
75 =

75 = 1375mm
2
2
2 2
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7> KNnaEdkrgkgTisedAbeNayeCIgtag As = 0.0018 1000 500 = 900mm 2 / m

eRbIEdk DB16 KMlat 200mm A = 1005mm . bg;lMGitRtUv)anbgajenAkgrUbTI
13>16.
2

]TahrN_ 13>2 KNnaeCIgtag single footing edIm,IRTssrebtugeRbIEdkkgFmtasitenAxagkg

EdleRbI 8DB28 . ssrenHRTnUvbnktamGkSGefrKanemKuN 1090kN nigbnkcMGkSefrKanemKuN

890kN . )atrbs;eCIgtagsitenACeRmA 1200mm BInIv:UdI ehIysm<aFdIGnuBaatKW 240kN / m 2 .
eK[ f 'c = 28MPa / f y = 400MPa .

dMeNaHRsay

1> KNnasm<aFdIRbsiTPaB. snt;kRmas;srubrbs;eCIgtag 600mm . Tmn;eCIgtagKW

0.6 25 = 15kN / m . bnkdIenABIelIeCIgtag snt;Tmn;maDrbs;dIesInwg 16kN / m KW
2

0.6 16 = 9.6kN / m 2

sm<aFdIRbsiTPaBKW = = 240 15 9.6 = 215.4kN / m

2> KNnaRkLaprbs;eCIgtag
bnkeFVIkar = D + L = 1090 + 890 = 1980kN
1980
RkLapeCIgtag = 215
= 9.2m
.4

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RCugrbs;eCIgtagKW = 3.03m
dUcenHeRbI = 3m rUbTI 13>17
3> sm<aFdIsuT= bnkemKuN RkLapeCIgtag
Pu = 1.2 D + 1.6 L = 1.2 1090 + 1.6 890 = 2732kN
2732
(net upward pressure) q u =
= 303.6kN / m 2
3 3

=
sm<aFdIsuT
4> epgpat;km<s;snt;EdlbNalmkBIkmaMgkat;TTwgBIrTis. RbsinebIeKGt;eRbIEdkkmaMgkat;
TTwg enaHkmaMgkat;TTwgBIrTiskMNt;km<s;caM)ac;rbs;eCIgtag. sRmab;km<s;eCIgtagsrub
snt; 600mm KNna d eTAGkSTIRbCMuTmn;nEdkRsTab;xagelIedIm,Idak;enAkgeCIgtagTaMg
BIrTis. eRbIEdk DB15 sRmab;KNna d .
d = 600 75 37 = 488mm

vaCakarRbesIrEdlsnt; d = h 112
bo = 4(c + d ) = 4(450 + 488) = 3752mm
c + d = 450 + 488 = 938mm

Vu 2 = Pu q u (c + d )2 = 2732 303.6(0.450 + 0.488)2 = 2464.9kN

3Vu 2
3 2464900
d1
=
=
= 496.6mm = 1
0.75 28 3752
f ' c bo
12Vu 2
12 2464900
=
d2
=
= 275.8mm

40 488

sd
0.75
+ 2 28 3752

+ 2 f ' c bo

3752

bo

caM)ac;

smIkar !#>(

caM)ac;

= 40 sRmab;ssrenAkNal edaysar d > d snt; dUcenHeyIgRtUveFVIkarKNnasak

l,grhUtdl; d nig d mantmRbEhlKa. eRkayBIkarKNnamkeyIgTTYl)an
h = 610mm / d = 498mm / d = 495.5mm . dUcenH km<s;snt;elIkeRkay h = 610mm
RKb;RKan;.
5> RtYtBinitkm<s;EdlbNaledaysarGMeBIkmagM kat;mYyTis muxkat;eRKaHfak;enAcmay d BI
pssr
cmayBIcugeCIgtag = L2 2c d = 777mm
1

Vu1 = 303.6 0.777 3 = 707.7 kN

km<s;caM)ac;sRmab;kmaMgkat;TTwgmYyTisKW
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d=

Department of Civil Engineering

Vu1
0.75 0.17 f ' c b

707700
0.75 0.17 28 3000

= 350mm < 498mm

6> KNnaEdk nigm:Um:g;Bt;. muxkat;eRKaHsitenApssr. cmayBIcugeCIgtagKW

L c
= 1500 225 = 1275mm
2 2
2

1 L c
1
q u b = 303.6 1.275 2 3 = 740.3kN .m
2 2 2
2
6
M
740.3 10
Ru = u2 =
= 0.995MPa
bd
3000 498 2

Mu =

BIsmIkar !#>!$ eyIg)an = 0.0028

As = bd = 0.0028 3000 498 = 4183.2mm 2

Gb,brma EdkrYmmaD = 0.0018 3000 498 = 2689.2mm < 4183.2mm

A Gb,brma EdkrgkarBt; = 0.0033 3000 498 = 4930.2mm
dUcenH eyIgyk As = 4930.2mm 2 . eRbI 13DB22 As = 4941.7mm 2 KMlat rbs;EdkKW
s = (3000 150) / 12 = 237.5mm enAkgTiwedATaMgBIr.
7> RtYtBinitkugRtaMgRT (bearing stress)
a. ersIusg;RT (bearing strength) N enA)atrbs;ssr A1 = 450 450mm 2 KW
2

As

N1 = (0.85 f ' c A1 ) = 0.65 0.85 28 450 450 10 3 = 3132.7 kN

b.

ersIusg;RT (bearing strength) N enApxagelIrbs;eCIgtag A

2

N 2 = N1

= 3 3m 2

KW

A2
2N 1
A1

A2 = 9000000mm 2

A1 = 202500mm 2

9000000
= 6.67 > 2
202500

dUcenH N 2 = 2 N1 = 6265.4kN . edaysar Pu = 2732kN < N1 enaHkugRtaMgRTKWRKb;

RKan;. RkLapGb,brmanEdk dowel caM)ac;KW 0.005 A = 0.005 450 450
= 1012.5mm . cMnYnEdkGb,brmaKW 4 dUcenHeRbIEdk DB 25 dak;enARCugTaMgbYnrbs;
ssr.
RbEvgbgb;rbs;Edk dowel enAkgkmaMgsgt;
1

c.

l dc =

eCIgtag

0.24d b f y
f 'c

0.24 25 400
28

= 453.5mm

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Gb,brma = 0.044d b f y = 440mm > 200mm . dUcenHeRbIEdk dowel DB25 4edIm

RbEvg 455mm dak;kgssrnigeCIgtag. cMNaMfa l dc < d = 498mm RKb;RKan;.
8> RbEvgbgb;rbs;EdkemenAkgeCIgtagsRmab; DB22 KW
l d = 48 22 = 1056mm eyagtamem eronTI 7. RbEvgbgb;BitR)akdenAkgeCIgtagKW
l = L / 2 c / 2 75 = 1500 225 75 = 1200mm .
bg;lMGitrbs;eCIgtagRtUv)anbgajenAkgrUbTI 13>17.
l dc

]TahrN_ 13>3 KNnaeCIgtagragctuekaNsRmab;ssrn]TahrN_TI13>2 RbsinebIRCugmYy

rbs;eCIgtagRtUv)ankMNt;esI 2.6m .
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Department of Civil Engineering

dMeNaHRsay

1> viFIsaRsKNnasRmab;eCIgtagctuekaNmanlkNRsedogKanwgkarKNnaeCIgtagkaer
eday KitBicarNaGMeBIrbs;kmaMgmkelIeCIgtagtmaTisnImYydac;edayELkBIKa.
2> BI]TahrN_mun RkLapcaM)ac;rbs;eCIgtagKW 9.2m
beNayeCIgtag = 92..26 = 3.54m
dUcenHyk 3.6m rUbTI 13>18. TMhMeCIgtagKW 2.6 3.6m
3> Pu = 2732kN . dUcenH sm<aFdIsuT (net upward pressure) KW
2

qu =

2732
= 291.9kN / m 2
2.6 3.6

4> RtYtBinitkm<s;EdlbNalBIkmaMgkat;TTwgmYyTis. muxkat;eRKaHfak;sitenAcmay d BIp

rbs;ssr. sRmab;TisedAEvg
L c

Vu1 = d q u b
2 2

3.6 0.45

0.498 291.9 2.6 = 817.4kN

2
2

kmaMgkat;TTwgenHlb;

sRmab;TisedAxI V = 606.3kN
mineRKaHfak;
6Vu1
6 817.4 10 3
d caM)ac; =
=
= 475mm
f ' b 0.75 28 2600
u1

EdleRbI = 498mm > 475mm

5> RtYtBinitGMeBIkmaMgkat;TTwgBIrTis kmaMgkat;pug (punching shear). muxkat;eRKaHfak;
enAcmay d / 2 BIpssrTaMgbYnRCug.
d

bo = 4(450 + 498) = 3792mm

(c + d ) = 450 + 498 = 948mm

=

3.6
= 1.38 < 2
2.6

eRbI Vu =

f ' c bo d / 3

Vu 2 = Pu q u (c + d ) 2 = 2732 291.9 0.948 2 = 2469.7 kN

3Vu 2
3 2469700
d1 =
=
= 492mm
f ' c bo 0.75 28 3792
d 2 = 271.5mm

eCIgtag

357

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

6> KNnaEdkenAkgTisedAEvg. muxkat;eRKaHfak;enAnwgpnTRm. cmayBIcugneCIgtagKW

L c 3600 450
=

= 1575mm
2 2
2
2
1
M u = 291.9 1.575 2 2.6 = 941.3kN .m
2
M
941.3 10 3
Ru = u2 =
= 1.46MPa
bd
2.6 0.498 2

GnuvttamsmIkar !#>!$ eyIgTTYl)an = 0.0042

As = 0.0042 2600 498 = 5438.2mm 2

Gb,brma EdkrYmmaD = 0.0018 2600 610 = 2855mm 2

As Gb,brma EdkrgkarBt; = 0.0033 2600 498 = 4273mm 2
eRbI As = 5438.2mm 2 . eRCIserIs 10DB28 As = 6157.5mm 2 KMlatEdkKW
As

S = (2600 150 ) / 9 = 272mm

7> KNnaEdkenAkgTisedAx.I cmayBIpssreTAcugeCIgtagKW

2600 450

= 1075mm
2
2
1
M u = 291.9 1.075 2 3.6 = 607.2kN .m
2
M
607.2 10 3
Ru = u2 =
= 0.68MPa
bd
3.6 0.498 2

GnuvteTAkgsmIkar !#>!$ eK)an = 0.0019

As = 0.0019 3600 498 = 3406mm 2

Gb,brma EdkrYmmaD = 0.0018 3600 610 = 3953mm 2

As Gb,brma EdkrgkarBt; = 0.0033 3600 498 = 5916mm 2
tmrbs; As EdlRtUveRbIRtUvEtFMCagbesI 3953mm 2 . eRbI 18DB20 As = 5655mm 2
As

s =

2
2
=
= 0.84
+ 1 3 .6
+1
2 .6

cMnYnEdkenAkgceRmokRbEvg 2.6m KW 18 0.84 = 15 edIm.

cMnYnEdkenAsl;sRmab;RCugmag KW (18 15) / 2 = 2 edIm. dUcenHdak;Edk 15DB20
enAkgceRmokRbEvg 2.6m ehIydak; 2DB20 As = 628mm 2 enAelIRbEvg 0.5m
enAelIRCugnImYysgagceRmok 2.6m . cMnYnEdksrubKW 19 edIm. enAkg]TahrN_enH
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358

Footings

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

EdkRtUv)anBRgayedayKMlatesIKa elIRbEvg 3.6m / S = (3600 150) / 18 = 192mm .

bg;lMGitsrsrEdkRtUv)anbgajenAkg rUbTI !#>!*.
8> RtYtBinitkugRtaMgRTenA)atssr dUcEdl)anBnl;enAkg]TahrN_mun. eRbIEdk dowel
DB 25 cMnYn 4 edIm.
9> RbEvgbgb;rbs;Edkem l d = 960mm sRmab; DB20 nig l d = 1344mm sRmab; DB28 .
3600 450

l d Edldak; TisEvg =

75 = 1500mm
2
2

l d Edldak; TisxI = 1000 mm

]TahrN_ 13>4 KNnaRkLapeCIgtagcaM)ac;sRmab;sRmutesIKa karKNnaeCIgtagEdlman

lMnwg (balanced footing design) RbsinebIbnkGefrFmta (usual live load) esI 20% sRmab;
eCIgtag

359

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NPIC

RKb;eCIgtagTaMgGs;. eCIgtagrgnUvbnkefr nigbnkGefrdUcbgajenAkgtaragxageRkam. sm<aF

dIsuTGnuBaat (allowable net soil pressure) esI 287kN / m 2 .
elxeCIgtag
bnkefr
bnkGefr

535kN

800kN

620kN

845kN

935kN

670kN

980kN

890kN

755kN

1070kN

dMeNaHRsay

1> kMNt;eCIgtagEdlmanpleFobbnkGefrelIbnkefrFMCageK. enAkg]TahrN_enH eCIgtag

elx 3 manpleFob 1.44 FMCagpleFobdT.
2> KNnabnkFmtasRmab;RKb;eCIgtagTaMgGs;. bnkFmtaKWCabnkefr nigEpknbnkGefr
EdlmanGMeBIrCaTUeTAenAelIeRKagsMNg;. enAkg]TahrN_enH
bnkFmta = D.L + 0.2( L.L)
tmnbnkFmtaRtUv)anbgajenAkgtaragxageRkam.
3> kMNt;RkLaprbs;eCIgtagEdlmanpleFob L.L / D.L
D.L + L.L
620 + 890
RkLapeCIgtagelx 3 = allowable
=
= 5.26m 2
soil pressure
287
sm<aFdIFmtaeRkameCIgtagelx 3 KW
Usual load
798
=
= 151.7kN / m 2
Area of footing 5.26

4> KNnaRkLapcaM)ac;sRmab;eCIgtagnImYyedayEckbnkFmtarbs;vaedaysm<aFdIenAeRkam
eCIgtagelx 3 . RkLaprbs;vaRtUvbgajenAkgtaragxageRkam. ]TahrN_ eCIgtagelx 1
RkLapcaM)ac;rbs;vaesI 669 / 151.7 = 4.41m 2 .
5> KNnasm<aFdIGtibrmaeRkameCIgtagnImYy
D+L
q max =
287kN / m 2 sm<aFdIGnuBaat
area

T.Chhay

360

Footings

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

elxeCIgtag
Dead load
Live load

bnkFmta = D.L + 0.2(L.L) (kN )

load
RkLapcaM)ac; = Usual
(m 2 )
151.7
D+L
sm<aFdIGtibrma = area (kN / m 2 )

1.25

1.225

1.44

0.89

1.14

669

996

798

996

1149

4.41

6.57

5.26

6.57

7.57

273.2

270.9

2.87

243.5

264.9

]TahrN_ 13>5 KNnaeCIgtagebtugsuTedIm,IRTCBaaMgebtugEdlmankRmas; 400mm . bnkEdl

manGMeBIelICBaaMgrYmmanbnkefr 233.5kN / m rYmbBalbnkpal;rbs;CBaaMgrYc nigbnkGefr

146kN / m . )atrbs;eCIgtagsitenACeRmA 1200mm BInIv:Ud.
I eK[ f 'c = 20MPa nigsm<aFdI
GnuBaat 240kN / m 2 .

dMeNaHRsay

1> KNnasm<aFdIRbsiTPaB. snt;km<s;srubeCIgtagKW 710mm

Tmn;rbs;eCIgtag = 0.71 24 = 17kN / m 2
edaysnt;Tmn;maDdIesI 16kN / m 3 enaHTmn;rbs;dIenAelIeCIgtagesI (1.2 0.71) 16
= 7.84kN / m 2 .
sm<aFdIRbsiTPaBKW 240 17 7.84 = 215.16kN / m 2 .
2> KNnaTTwgeCIgtagsRmab;CBaaMgRbEvg 1m b = 1m
total load
effective soil pressure
233.5 + 146
=
= 1.76m
215.16

width of footing =

yk 1.8m rUbTI 13>19

3> U = 1.2D + 1.6L = 1.2 233.5 + 1.6 146 = 513.8kN / m

sm<aFdIsuT (net upward pressure) esI qu = 513.8 / 1.8 = 285.4kN / m 2
4> RtYtBinitkugRtaMgBt;. muxkat;eRKaHfak;KWsitenApCBaaMg. sRmab;RbEvg1m nCBaaMg
nigeCIgtag
2

Mu =

eCIgtag

1 L c
1
1 .8 0 .4
qu = 285.4

= 69.9kN .m
2
2 2 2
2
2
361

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mhaviTalysMNg;sIuvil

NPIC

kMNt;km<s;RbsiTPaB d esI 710 75 = 635mm edaysnt;fa)at 75mm KanRbsiTPaB.

Ig =

bd 3 1000 6353
=
= 2.13 1010 mm 4
12
12
M c 69.9 10 6 635
fr = u =

= 1.04MPa
I
2.13 1010 2

kugRtaMgTajedaykarBt;KW
kugRtaMgTajedaykarBt;GnuBaatKW 0.415 f 'c = 0.415 0.55 20 = 1.02MPa Ek,r
5> RtYtBinitkugRtaMgkmaMgkat;TTwg muxkat;eRKaHfak;sitenAcmay d = 635mm BIpCBaaMg
1.8 0.4

L c

Vu = qu d = 285.4

0.635 = 18.55kN
2
2

2 2

3
f ' c bd 0.55 20 1000 635 10
Vc =
=
= 173.5kN
9
9

dUcenH muxkat;manlkNRKb;RKan;. vaCakarRbesIrEdleRbIbrimaNEdkGb,brmasRmab;TaMg

BIrTis.

13>6>

Combined Footings

enAeBlEdlssrmanTItaMgsitenAEk,rRBM enaHEpkneCIgtag single footing GaclycUl

eTAkgRBMGkCitxag. edIm,IeCosvagsanPaBEbbenH eKGacdak;ssrenAelIRCugEKmneCIgtagbegIt
)anCabnkcakpit. eKminGaceRbIdMeNaHRsayenHsRmab;lkxNxH nigeBlxHvaCadMeNaHRsay
T.Chhay

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Footings

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

EdlminsnSMsMc. karKNnadlGacTTYledalbBaleCIgtagCamYynwgeCIgtagssrxagkgEdlsit
enACitCageK begIt)anCa combined footing . TIRbCMuTmn;rbs; combined footing RtYtsIuKaCamYy
nwgkmaMgpbnssrTaMgBIr.
krNIepSgeTot Edl combined footing caM)ac;RtUv)aneRbIKWenAeBlEdldImanlkNexSay
ehIyeCIgtagssrmYyKgelIeCIgtagenAEk,r. ragrbs;eCIgtag combined footing GacCactuekaN
Ekg bctuekaNBay rUbTI 13>20. enAeBlEdlbnkssrxagenAEk,rRBMmantmFMCagbnkssr
kg eCIgtagmanragctuekaNBayGacRtUv)aneRbIedIm,IrkSaTIRbCMuTmn;eCIgtagenAelIExSCamYynwgbnk
pbnssrTaMgBIr. CaTUeTA eKcUlciteRbIeCIgtagragctuekaNEkg.

beNay nigTTwgrbs;eCIgtagRtUv)aneRCIserIsedaykMeNInmg 75mm EdlGacbNal[

sm<aFBRgayesIenAeRkameCIgtagERbRbYltUc. sRmab;sm<aFBRgayesIeLIgelI (uniform upward
pressure) eCIgtagnwgdabdUcbgajenAkgrUbTI 13>21. ACI Code, Section 15.10 min)anpl;nUvviFI
KNnalMGitsRmab; combined footing eT. CaTUeTA karKNnaQrelIkarviPaKeRKOgbgM (structural
analysis).
viFIdsamBankarKNnaKWcat;TukeCIgtagdUcFwmenAkgTisedAEvg EdlRTnUvsm<aFBRgayesI
eLIgelI (uniform upward pressure) qu . sRmab;TisedATTwg eKsnt;fabnkssrRtUv)anBRgay
eBjTTwgenABIeRkamssrEdlesInwgTTwgssrbUknwg d nRCugnImYy. m:agvijeTot bnkssrman
GMeBIelIFwmenAeRkamssrCamYynwgeCIgtagEdlmanTTWgGtibrma c + 2d nigbeNayesInwgRCugxI
rbs;eCIgtag rUbTI 13>22. eKkGaceRbITTwgEdlmantmtUcCagrhUtdl; c + d .
eCIgtag

363

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NPIC

]TahrN_ 13>6 KNnaeCIgtag combined footing ragctuekaNEkgedIm,IRTssrBIrdUcbgajkg

rUbTI 13>23. ssrxagelx I manmuxkat; 400 400mm nigRT D.L. 800kN nig L.L 535kN .
ssrxagkgelx II manmuxkat; 500 500mm nigRTbnk D.L. 1115kN nig L.L 625kN . sm<aF
dIGnuBaatKW 240kN / m 2 ehIy)atrbs;eCIgtagsitenACeRmA 1.5m BInIv:UdI. KNnaeCIgtagedayeRbI
f 'c = 28MPa / f y = 400MPa nigviFIKNnaersIusg; ACI.
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364

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viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

dMeNaHRsay

1> kMNt;TItaMgkmaMgpbnbnkssr. Kitm:Um:g;CMuvijGkSnssrxagelx I

(1115 + 625) 4900
x=
= 2772.7m BIssr I
(1115 + 625) + (800 + 535)
cmayBIkmaMgpbmkExSRBMKW 2772.7 + 600 = 3372.7mm . RbEvgrbs;eCIgtagKW
3372.7 2 = 6745.4mm yk 6750mm . kgkrNIenHkmaMgpbrbs;ssrRtYtsIuKaCamYy
nwgkmaMgpbrbs;sm<aFdIEdlmanGMeBIelIeCIgtag.
2> kMNt;RkLaprbs;eCIgtag. snt;km<s;eCIgtagsrub 920mm d = 920 115 = 805mm
bnkeFVIkarsrub = 800 + 535 + 1115 + 625 = 3075kN Total actual (working)load
sm<aFdIfI = 240 0.92 25 0.58 16 = 207.72kN / m 2 New upward pressure
snt;Tmn;maDdI 16kN / m 3
3075
RkLapcaM)ac; = 207
Required area
= 14.8m 2
.72
.8
TTwgeCIgtag = 14
= 2.19m Width of footing
6.75
ykTTwgeCIgtagesI 2.2m . dUcenHTMhMeCIgtagKW 6.75 2.2m RkLap 14.85m 2
3> kMNt;sm<aFdIemKuN factored soil pressure edayeRbIbnkemKuN
Pu1 ssrelx I = 1.2 800 + 1.6 535 = 1816kN
Pu 2 ssrelx II = 1.2 1115 + 1.6 625 = 2338kN
sm<aFdIemKuNsuTKW qu = (1816 + 2338) / 14.85 = 279.73kN / m 2 net factored soil
pressure
4> KUrdaRkamkmaMgkat;TTwgemKuNsRmab;FwmEdlmanRbEvg L = 6750mm
EdlenAelIssrBIr nigrgnUvsm<aFdI 279.73 2.2 = 615.4kN / m kgRbEvgeCIgtag 1m
Vu enApxageRkArbs;ssr I = 615.4 (0.6 0.2) = 246.16kN
Vu enApxagkgrbs;ssr I = 1816 615.4(0.6 + 0.2 ) = 1323.68kN
Vu enApxageRkArbs;ssr II = 615.4 (1.25 0.25) = 615.4kN
Vu enApxagkgrbs;ssr II = 2338 615.4 (1.25 + 0.25) = 1414.9kN
5> KUrdaRkamm:Um:g;emKuNedayKiteCIgtagCaFwmEdlman L = 6750mm RTedayssrBIr.
sm<aFdIBRgayesI 615.4kN / m

eCIgtag

365

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mhaviTalysMNg;sIuvil

M u1

NPIC

enApxageRkAssr I = 615.4 0.24

= 49.2kN .m

enApxagkgssr II = 615.4 12 = 307.7kN .m

m:Um:g;GtibrmaekItmanenAkmaMgkat;TTwgesIsUn
615.4
(2.15 + 0.2 + 0.6)2
M u GtibrmaKNnaBIssr I = 1816(2.15 + 0.2)
2
M u2

= 1589.84kN.m

Mu

GtibrmaKNnaBIssr II = 2338(2.3 + 0.25) 6152 .4 (2.3 + 0.25 + 1.25)2

= 1518.7 kN.m

m:Um:g;EdlKNnaBIRCugTaMgBIrneCIgtagRtUvmantmRbhak;RbEhlKa. kgkrNIenH eyIgyk

M u max = 1589.84kN.m . bERmbRmYltmrbs;m:Um:g;GaRsyCacMbgCamYynwgRbEvg nig
TTwgrbs;eCIgtag.
6> RtYtBinitkm<s;sRmab;kmaMgkat;mYyTis. kmaMgkat;TTwgGtibrmaekItmanenAcmay
d = 805mm BIpssrxagkg II rUbTI 13>23.
Vu1 = 1414.9 615.4 0.805 = 919.5kN
Vu1
919500
d=
=
= 619.5mm
0.17 f 'c b 0.75 0.17 2200 28

km<s;RbsiTPaBEdl[KW d = 805mm > 619.5mm dUcenHeCIgtagmanlkNRKb;RKan;.

7> RtYtBinitkm<s;sRmab;kmaMgTTwgBIrTis kmaMgkat;pug. sRmab;ssrxagkg
bo = 4(c + d ) = 4(500 + 805) = 5220mm

(c + d ) = (500 + 805) = 1305mm

kmaMgkat;TTwg Vu 2 enAmuxkat; d / 2 BIRKb;RCugrbs;ssresI

Vu 2 = Pu 2 qu (c + d ) 2 = 2338 279.73 1.305 2 = 1861.6kN
3Vu 2
3 1861600
d=
= 269.6mm < 805mm
=
f 'c bo 0.75 28 5220

ssrxageRkARtUv)anRtYtBinitnigbgajfaminmaneRKaHfak;.
8> RtYtBinitkm<s;sRmab;m:Um:g; nigkMNt;srsEdkcaM)ac;enAkgTisEvg
m:Um:g;Bt;Gtibrma = 1589.84kN .m
T.Chhay

366

Footings

viTasanCatiBhubeckeTskm<Ca
Ru =

Mu
bd 2

Department of Civil Engineering

1589.84 10 6
2200 805 2

= 1.115MPa

GnuvtkgsmIkar !#>!$ PaKryEdkKW = 0.0032 0.0033( min )

As = 0.0033 2200 805 = 5844.3mm 2

Gb,brma rYmmaD = 0.0018 2200 920 = 3643.2mm 2

As = 5844.3mm 2 lub. eRbI 10 DB 28 As = 6157.5mm 2
150 (concrete cover)
KMlatEdk = 22009(no.
= 228mm
of spacing)
EdkRtUv)andak;bgtenAcenaHssrenAEpkxagelIeCIgtagCamYykRmas;karBarEdk 75mm .
dak;EdkGb,brmaenA)atrbs;eCIgtagedIm,ITb;nwgm:Um:g;viCman. RtYtBinitRbEvgbgb; ld BIp
ssr.
EdkGb,brmaKW As = 3643.2mm 2 . eRbI 8DB25 As = 3927mm 2 .
RbEvgEdkbgb;caM)ac;sRmab;EdkemxagelIKW 1.3ld = 1.3 48 d b = 1747mm RtUv)andak;
ecjBIcMNucm:Um:g;Gtibrma. RbEvgEdkbgb;Edldak;enARtg;ssrTaMgBIrKWRKb;RKan;.
9> sRmab;EdkenAkgTisedAxI. karKNnam:Um:g;Bt;enAkgTisedAxI TisedATTwg manlkN
dUcKakgkrNI single footing. EdkenAeRkamssrnImYy RtUv)andak;enAkgceRmokTTwg
(bandwidth) EdlesInwgplbUkTTwgssrnigkm<s;RbsiTPaB d BIrdg rUbTI !#>@$.
a. EdkenABIeRkamssr I
As

Bandwidth = 400 (column width) + 400 (on exterior side of column) + 805
= 1605mm

eRbI 1.6m . sm<aFdIeLIgelIsitenAkgTisedAxIeRkamssr I KW

Pu1
1816
=
= 825.5kN / m
width of footing
2.2
2200 400

= 900mm
2
2
825.5 2
Mu
I =
0.9 = 334.3kN.m
2
M
334.3 10 6
Ru = u2 =
= 0.32MPa
bd
1600 805 2

cmayBIcugTMenreTApssrKW
enApnssr

PaKryEdk tUcCagPaKryEdkGb,brmasRmab;EdkrYmmaD 0.0018 .

As min = 0.0018 1600 920 = 2649.6mm 2

eCIgtag

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NPIC

eRbI 6DB25 As = 2945.2mm 2 dak;enAkg bandwidth RbEvg 1.6m .

b.

EdkenABIeRkamssr II

Bandwidth = 500 + 805 + 805 = 2110mm

eRbI 2100mm . sm<aFdIeLIgelIsiTenAkgTisedAxIeRkamssr II KW

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368

Footings

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

Pu 2
2338
=
= 1062.7kN / m
width of footing
2.2
2200 500

= 850mm
2
2
1062.7
Mu
II
=
0.85 2 = 383.9kN.m
2
6
M
383.9 10
Ru = u2 =
= 0.28MPa
bd
2100 805 2

cmayeTApssrKW
enApnssr kgTisedAxI

PaKryEdk tUcCagPaKryEdkGb,brmasRmab;EdkrYmmaD 0.0018 .

As min = 0.0018 2100 920 = 3477.6mm 2

eRbI 8DB25 As = 3927mm 2 dak;enAkg bandwidth RbEvg 2.1m BIeRkamssr

II dUcbgajenAkgrUb !#>@# nig !#>@$. RbEvgbgb;rbs;Edk DB 25 enAkgTisedAxI
KW 1.2m .

eCIgtag

369

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NPIC

13>7> eCIgtageRkambnkssrcakpit (Footings under Eccentric Column Loads)

enAeBlEdlssrbBanEtkmaMgtamGkS enaHeKGacKNnaeCIgtagEdlmanbnkeFVIGMeBIcMTIRb
CMuTmn;rbs;eCIgtag EdlbegIt)anCasm<aFBRgayesIeRkameCIgtag. b:uEn enAkgkrNIxHssrbBan
kmaMgcMGkS nigm:Um:g;Bt; dUckgkrNIeCIgtageRKagbgb;cug (fixed-end frame). sm<aF q EdlekIt
manenAelIdInwgminBRgayesI EdlGackMNt;BIsmIkarxageRkam
P Mc
!#>!(
0
q=
A
I
Edl A nig I CaRkLap nigm:Um:g;niclPaBneCIgtag erogKa. eKmanlkxNdIxusKaEdl
GaRsynwgtm P nig M / nig sm<aFdIGnuBaat. lkxNKNnaxusKaRtUv)anbgajenAkgrUbTI 13>
25 nigRtUv)ansegbdUcxageRkam
1> enAeBl e = M / P < L / 6 sm<aFdImanragctuekaNBay
P Mc
P 6M
!#>@0
q max = +
=
+ 2
A
I
LB
q min

BL
P Mc
P 6M
=
=

A
I
LB BL2

!#>@!

2> enAeBl e = M / P = L / 6 sm<aFdImanragctuekaNEkg

P 6M 2 P
+
=
LB BL2 LB
P 6M
=0=

LB BL2

q max =

b LBP = 6M2
BL
3> enAeBlEdl e > L / 6 sm<aFdImanragRtIekaN
q min

!#>@@
!#>@#

L y L
= e
3
2
3x
P = q max B
2
2P
4P
q max =
=
3xB 3B( L 2e)
x=

!#>@$
4> enAeBleCIgtagRtUv)anrMkilcmay e BIGkSnssredIm,IbegItsm<aFdIBRgayesIenA
eRkameCIgtag. m:Um:g;GtibrmaekItmanenAmuxkat; n n
M
M = M ' Hh nig e =
P

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13>8> eCIgtageRkamm:Um:g;BIrTis (Footings under Biaxial Moment)

enAkgkrNIxH eCIgtagGacrgnUvkmaMgtamGkS nigm:Um:g;Bt;BIrTisCMuvijGkS x nigGkS y dUc
krNIm:asIunscEdlbgVilxn)an 360o . enaHeCIgtagRtUv)anKNnasRmab;bnkeRKaHfak;.
tamrUbTI 13>26 RbsinebIbnkcMGkS P manGMeBIenAcmay ex BIGkS y nig e y BIGkS x .
enaH
M x = Pe y
nig M y = Pex
eCIgtag

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P M xc y M y cx
+
+
A
Ix
Iy

sm<aFdIenARCug ! KW

q max =

enARCug @ KW

q2 =

P M xc y M y cx

+
A
Ix
Iy

enARCug # KW

q3 =

P M xc y M y cx

A
Ix
Iy

enARCug $ KW

q4 =

P M xc y M ycx
+

A
Ix
Iy

cMNaMfa kugRtaMgxageRkAminRtUvFMCagsm<aFRTRTg;rbs;dI nigm:ageTotdIminGacrgkmaMg

Taj)aneT Edl q 0 .

]TahrN_ 13>7 ssrctuekaNEdlmanmuxkat; 300 600mm dUcbgajkg rUbTI 13>27 a rgnUv

bnkcMGkS PD = 980kN nigm:Um:g; M D = 244kN .m EdlbNalBIbnkefr nigbnkcMGkS
PL = 734kN nigm:Um:g; M L = 190kN .m EdlbNalBIbnkGefr. )atrbs;eCIgtagsitenACeRmA
1.5m BInIv:UdI ehIysm<aFRTRTg;dIGnuBaatKW 240kN / m 2 . KNnaeCIgtagedayeRbI f 'c = 28MPa /
f y = 400MPa .

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dMeNaHRsay

eCIgtagrgnUvbnkcMGkS nigm:Um:g;
P = 980 + 734 = 1714 kN
M = 244 + 190 = 434kN .m
M
434
e=
=
= 0.253m
P 1714

yk 250mm
cMNakpitKW
eKGacKNnaeCIgtagtmaviFIsaRsBIrKW
viFIsaRsTI1 rMkilGkSeCIgtagcmay e = 250mm BIGkSssr. kgkrNIenH sm<aFdIRtUv)an
cat;TukfaBRgayesIenABIeRkameCIgtag rUbTI 13>27 b.
viFIsaRsTI2 GkSrbs;eCIgtag nigGkSrbs;ssrRtYtsIuKa. kgkrNIenH sm<aFdImanragCa
RtIekaN bctuekaNBay rUbTI 13>27 c ehIytmGtibrma nigGb,brmaRtUv)anKNnadUcbgaj
enAkgrUbTI 13>27.
karGnuvtnviFIsaRsTaMgBIrsRmab;]TahrN_TI 13>7 GacnwgBnl;y:agsegbdUcxageRkam
1> sRmab;viFIsaRsTI1 snt;km<s;eCIgtag 500mm d = 420mm nigsnt;Tmn;maDdIesI
16kN / m 3

sm<aFdIeLIgelIsuTKW 240 0.5 25 1 16 = 211.5kN / m 2

1714
RkLapeCIgtag = 211
= 8.1m 2
.5
snt;TTwgeCIgtagKW 2.7m enaHeCIgtagmanbeNay 8.1 / 2.7 = 3m . dUcenHeyIgeRCIserIs
eCIgtagEdlmanTMhM 2.7 3m nigssrmanTItaMgcakpitdUcbgajkgrUbTI 13>27 d. GkS
eCIgtagmancmay 250mm BIGkSssr.
2> \LvviFIsaRsKNnamanlkNRsedogKanwgkarKNnaeCIgtageTal (single footing).
RtYtBinitkm<s;eCIgtagsRmab;GMeBIkmaMgkat;TTwgmYyTis nigkmaMgkat;TTwgBIrTis. kMNt;m:U
m:g;Bt;enApssrsRmab;TisxI nigTisEvg. edaysarcMNakpitrbs;eCIgtag muxkat;eRKaH
fak;nwgsitenAxageqVgnpssr dUcbgajkgrUbTI 13>27 d. cmayeTAcugeCIgtagKW
1450mm

Pu = 1.2 D + 1.6 L = 1.2 980 + 1.6 734 = 2350.4kN

2350.4
= 290.2kN / m 2
qu =
2.7 3

eCIgtag

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m:Um:g;Gtibrma M u = 290.2 2.7 1.452

= 823.7kN .m

enAelITTwg 2.7m

enAkgTisedAxI M u = 290.2 3 1.22 = 626.8kN.m

BinitelIgvijkm<s;eCIgtagEdl)ansnt;RbsinebIcaM)ac; nigeRCIserIsbrimaNEdkcaM)ac;enA
kgTisTaMgBIrrbs;eCIgtag dUcEdl)anBnl;enAkg]TahrN_GMBIeCIgtageTal (single
footing).

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3> sRmab;viFIsaRsKNnaTI2 KNnaRkLaprbs;eCIgtagtamviFIdUcEdl)anBnl;enAkgviFI

saRsTI1 bnab;mkKNnasm<aFdIGtibrma nigeRbobeFobvaCamYynwgsm<aFdIGnuBaatEdleRbI
bnkFmta
bnksrub P = 1714kN
TMhMrbs;eCIgtag = 2.7 3m
edaysarcMNakpitKW e = 250mm < L / 6 = 3000 / 6 = 500mm karBRgaysm<aFdIeLIgelI
manragCactuekaNBay. KNnasm<aFdIGtibrma nigGb,brma
q max =

P 6M
1714
6 434
+ 2 =
+
= 318.8kN / m 2 > 211.5kN / m 2
2
LB BL
2.7 3 2.7 3

eCIgtagKansuvtiPaB. sakl,g 2.8 4m RkLap = 11.2m 2

P 6M
1714
6 434
+ 2 =
+
= 211.2kN / m 2 < 211.5kN / m 2
LB BL
2.8 4 2.8 4 2
P 6M
1714
6 434
=
2 =

= 94.9kN / m 2
2
LB BL
2.8 4 2.8 4

q max =
q min

5> KNnasm<aFeLIgelIemKuNedayeRbIbnkemKuN bnab;mkKNnam:Um:g; nigkmaMgkat;TTwg dUc

Bnl;kg]TahrN_mun.
13>9> kRmalxNelIdI (Slabs on Ground)
kRmalxNebtugEdldak;pal;elIdIGacnwgrgnUv
- bnkBRgayesIenAelIprbs;va EdlbegIt)anCakmaMgkgtUc
- bnkcMcMNuc bbnkBRgayminesIEdlbegItm:Um:g;Bt; nigkmaMgkat;. kugRtaMgkmaMgTajekIt
eLIg ehIysameRbHnwgekItmanEpkxHrbs;kRmalxN.
kugRtaMgTajCaTUeTAekItmanedaybnSMn
- karrYj (contraction)EdlbNalmkBIsItuNPaB nigkarrYmmaD EdlbgaMgedaykmaMgkkit
rvagkRmalxN nig subgrade begItCakugRtaMgTaj.
- karxUcragmuxkat;rbs;kRmalxNedaykarmYl (warping of the slabe).
- lkxNnkardak;bnk
- sRmut
eCIgtag

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tMN contraction joint RtUv)anbegIteLIgedIm,Ikat;bnykugRtaMgTajenAkgkRmalxN.

tMN expansion joint RtUv)andak; sRmab;kRmalxNesIgEdlmankRmas;rhUtdl; 250mm .
kRmalxNCan;eRkamdIneRKagsMNg;GKarsak;enA GacCaebtugGarem:EdlmankRmas;BI
100mm eTA 150mm EdlBRgwgedaysMNaj;Edk. sRmab;XaMg kRmalxNGacmankRmas;BI
150mm eTA 300mm GaRsyelIbnkenAelIkRmalxN. kRmalxNCan;eRkamdIRtUv)anKNna
edIm,ITb;nwgsm<aFdI eLIgelI nigsm<aFTwk. RbsinebIkRmalxNsitenAelIdIEdlmanlkNsirPaB
bENn (incompressible soil) enaHsRmutDIepr:g;EslRtUv)anecal. kgkrNIenH kRmas;kRmal
xNGacGb,brmaRbsinebIKanRsTab;TwkeRkamdI. RbsinebIsRmutDIepr:g;EslFM enaHkRmalxN
RtUv)anKNnaCa stiff raft foundation.
(Footings on Piles)
13>10> eCIgtagenAelIssrRKwH
enAeBlEdlRsTab;dImanlkNTn;enAkgkRmas;dRkas; ehIylTPaBRTRTg;rbs;vatUc eK
nwgminENnaM[dak;eCIgtagedaypal;enAelIdIeT. vaCakarRbesIrkgkarbBankmaMgtamssrRKwHeTA
RsTab;deRCAEdlmanlkNrwgRKb;RKan;edIm,IRTbnk bbegItkmaMgkkitRKb;RKan;CMuvijpxagssr
RKwH.
ssrRKwHCaeRcInRbePTRtUv)aneRbIsRmab;RKwH. kareRCIserIsssrRKwHGaRsyelIlkxNdI/
vtmanRsTab;TwkeRkamdI/ tYrnaTIrbs;ssrRKwH nigfvikar. ssrRKwHGaceFVIBI ebtug Edk beQI.
CaTUeTA eCIgtagssrRKwH (pile cap) mansarsMxan;kgkarEbgEckkmaMgBIssreTAk,al
ssrRKwH. eCIgtagssrRKwH (pile cap)RtUvmanTMhMRKb;RKan;edIm,IQrelIssrRKwH. eCIgtagssrRKWH
(pile cap) RtUv)anKNnadUcFwmEdledkenAcenaHssrRKwH nigRTbnkcMcMNucBIssr. enAeBlEdl
ssrRtUv)anRTedayssrRKwHBIr eCIgtagssrRKwHGacRtUv)anKNnadUc truss ebtugGarem:ragRtI
ekaN.
ACI Code, Section 15.2 bgajfakarKNnam:Um:g; nigkmaMgkat;TTwgsRmab;eCIgtagelIssr
RKwHRtUv)anQrelIkarsnt;faRbtikmmkBIssrRKwHsitenAcMGkSssrRKwH. RkLapeCIgtag bcMnYn
ssrRKwHRtUv)ankMNt;BIbnk nigm:Um:g;KanemKuN.

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kRmas;ebtugGb,brmasRmab;eCIgtagssrRKwHRtUv)ankMNt;Rtwm 300mm (ACI Code,

Section 15.7) . sRmab;karKNnassrRKwH nigeCIgtagssrRKwHlMGit sUmGanesovePAvisVkmRKwH
(books on foundation engineering) .

eCIgtag

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XIV.

CBaaMgTb;

Retaining Wall

14>1> esckIepIm (Introduction)

CBaaMgTb;CaeRKOgbgMEdlRtUv)aneRbIedIm,Ipl;lMnwgdl;dI bsmarepSgeTot nigedIm,IkarBarva
BICRmalFmCatirbs;va. kgnyenH CBaaMgTb;rkSadIEdlmankRmitkm<s;xusKaedaypTaMgBIrrbs;va.
smarEdlRtUv)anTb;enAnIv:Ux<s;CagGnuvtkmaMgeTAelICBaaMgTb;EdlGacbNal[vaRkLab;
(overturning) b)ak; (failure). CBaaMgTb;RtUv)aneRbIenAkgsMNg;s<anCa abutment/ enAkgsMNg;Ca
basement walls/ nigenAkg embankments. vakRtUv)aneRbIedIm,ITb;sarFaturav dUcCaGagTwk nigGag
sewage-treatment tank.
14>2> RbePTCBaaMgTb; (Types of Retaining Walls)
CBaaMgTb;RtUv)ancat;cMNat;fak;dUcxageRkam eyagtamrUb 14>1
- Gravity walls CaTUeTArYmmanebtugsuT b\d edayGaRsyeTAelITmn;pal;rbs;vaedIm,I
pl;lMnwgRbqaMgkarrujnsmarEdlRtUvTb;. CBaaMgenHmanTMhMFMEdlkugRtaMgTajminekIt
manenAkgebtug b\dbNalmkBIkmaMgrujelICBaaMgeT. km<s;sRmab;Gnuvtkg gravity
walls minRtUvFMCag 3m .
- Semigravity walls Ca gravity walls Edlman)atFMedIm,Ipl;lTPaBlMnwgrbs;CBaaMg nig
edIm,IkarBarkarekIteLIgnkugRtaMgTajenAelI)at. brimaNEdkdtictYcRtUveRbIenAkg)at b
enAkgCBaaMgenAeBlxH edIm,Ikat;bnymuxkat;dFMrbs;CBaaMg.
- Cantilever retaining wall CaCBaaMgebtugGarem:EdlCaTUeTARtUv)aneRbIEdlmankm<s;BI
2.4m eTA 6m . vaCaRbePTFmtabMputneRKagsMNg;Tb; edaysarlkNesdkic nig
lkNsamBarbs;va. CBaaMg cantilever retaining wall CaeRcInRbePTRtUv)anbgajkgrUb
14>1.
- Counterfort retaining walls mankm<s;x<s;Cag 6m begItm:Um:g;Bt;FMenA)atnCBaaMgTb;
EdleFVI[karKNnanCBaaMgKanlkNesdkic.
dMeNaHRsaymYyenAkgkrNIenHKWkarbEnmnUvCBaaMgTTwg (counterfort)
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EdlcgPab;CBaaMgeTAnwg)at. Counterfort mannaTICaGgt;rgkarTajEdlRTCBaaMgbBar.

karKNnaCBaaMgenHmanlkNesdkicedaysarEtCBaaMgRtUv)anKitCakRmalxNCab;Edl
manKMlattamTisedkcenaH counterfort.
kRmalxNRtUv)anKNnadUckRmalxNEdlRTenAelITRmbIRCug rUbTI 14>1 h .

CBaaMgTb;

manTRmg;RsedogKanwg nounterfort wall Edr b:uEnkgkrNIenH

CBaaMgTTwgmanTItaMgsitenAmuxmageTotEdleyIgemIleXIj ehIyCBaaMgTTwgenHmannaTICa
Ggt;rgkarsgt; rUbTI 14>1 i . CBaaMgEbbenHmanlkNesdkicenAeBlEdlvamankm<s;
x<s;Cag 6m . vaminmankareBjniymeT edaysarEteKemIleXIjCBaaMgTTwg (buttress).
Buttressed retaining wall

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CaCBaaMgTb;enAEpkxagelIEdlRtUveRbIedIm,IRTkRmals<an. CBaaMgRtUv)an
snt;fabgb;enAEpkxageRkamenAnwg)at nigRTedayTRmsamBaenAEpkxagelI.
- Basement walls Tb;sm<aFdIBIpmagrbs;CBaaMg niglatsnwgtamTisQrBIkRmalxNCan;
eRkamdIeTAkRmalxNCan;TImYy. CBaaMgGacRtUv)ansnt;bgb;enA)at nigRTedayTRmsamBa
bTb;edayEpk (partially restrained) enAEpkxagelI.
Bridge abutment

14>3> kmaMgenAelICBaaMgTb; (Forces on Retaining Walls)

CaTUeTA CBaaMgTb;rgnUvbnkTMnajEpndI nigsm<aFdIEdlbNalmkBIsmarTb;enAelICBaaMg.
bnkTMnajEpndIEdlbNalmkBITmn;rbs;smarRtUv)ankMNt;edaykarKNnaedaypal; niggay
RsYl. GaMgtg;sIuet nigTisedArbs;sm<aFdIenAelICBaaMgTb;GaRsyelIRbePT niglkNdIEdlRtUv
)anTb; nigktaepSgeTotEdleKminGacKNna)anRtwmRtUvdUcbnkTMnajEpndIeT. eKGackMNt;sm<aF
dImanGMeBIelICBaaMgTb;tamRTwsI nigviFIsaRsEdlmanBnl;kgmuxviCaemkanicdI. sanPaBlMnwgrbs;
CBaaMgTb; nig\TiBlnkmaMgDINamiceTAelICBaaMgkmanEcgkgmuxviCaenHEdr.
dIRtsuH (granular material) dUcCaxSac; manlkNemkanicxusBIdIsit (cohesive material)
dUcCadI\d bBIbnSMnRbePTdITaMgBIr. eTaHCaGaMgtg;sIuetsm<aFdIenAelICBaaMgTb;manlkNsKsaj
eK)ansnt;karBRgaysm<aFdImkelICBaaMgTb;manlkNCabnat;. GaMgtg;sIuetsm<aFekIneLIg
smamaRtCamYynwgCeRmA ehIytmrbs;vaCaGnuKmn_eTAnwgkm<s;CBaagM / Tmn;dI nigRbePTdI. GaMg
tg;sIuetdI p enACeRmA h BIeRkampdIRtUv)anKNnadUcxageRkam
p = Cwh
!$>!
Edl w CaTmn;maDdI nig C CaemKuNEdlGaRsyeTAnwglkNrUbrbs;dI. tmrbs;emKuN
C ERbRbYlBI 0.35 sRmab;dIRtsuH (loose granular soil) dUcCaxSac; eTA 1.0 sRmab;dIsit (cohesive
soil) dUcCadI\desIm. RbsinebICBaaMgTb;RtUv)ansnt;rwgdac;xat enaHkrNIsm<aFdIenAesomnwgekIt
eLIg. eRkamsm<aFdI CBaaMgGacdab brMkiledaybrimaNtictYcBIdI ehIysm<aFdIskmnwgekIteLIg
dUcbgajenAkgrUbTI 14>2. RbsinebICBaaMgrMkileq<aHeTAdI enaHsm<aFdIGkmnwgekItman. TaMgsm<aF
dIskm nigsm<aFdIGkmRtUv)ansnt;faERbRbYlCabnat;CamYynwgCeRmACBaaMg rUbTI 14>2. sRmab;
dIRtsuH (granular) st/ Gt;sit (noncohesive) RtUv)ansnt;fasm<aFBRgaymanragCabnat; (linear)
dIsit bdIxSac;EqtTwkmanlkNepSg KWsm<aFBRgaymanlkNminEmnbnat; (nonlinear). dUc
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enH CaTUeTAeKeRbIdIRtsuHCasmarcak;bMeBjedIm,Ipl;nUvdaRkamsm<aFbnat; nigm:ageTotpl;

lTPaBkgkarbgrTwkBICBaaMgxageRkay.
sRmab;sm<aFbnat; GaMgtg;sIuetsm<aFskm nigsm<aFGkmRtUv)anKNnadUcxageRkam
Pa = C a wh
!$>@
!$>#
Pp = C p wh
Edl Ca nig C p CaemKuNnsm<aFskm nigsm<aFGkmerogKa.

14>4> sm<aFdIGkm nigsm<aFdIskm (Active and Passive Soil Pressures)

RTwsIdsamBaCaeKBIrEdlRtUv)aneRbICaTUeTAkgkarKNnasm<aFdIKW Rankine nig Coulomb.
a. sRmab;viFIrbs; Rankine CBaaMgTb;RtUv)ansnt;fa yield RKb;RKan;edIm,IbegItnUvsanPaBlM
nwg)asic enAkgdMudIenApCaBaaMg. dIEdlenAesmenArkSakgsanPaBeGLasic. RTwsIenH
GnuvtCacMbgcMeBaHdIKanPaBsit (cohesiveless)/ dIEdlminGacsgt;)an (incompressible)/
dIEdlmanlkNsac;mYy (homogeneous) nigminKitnUvkmaMgkkitrvagdI nigCBaaMg. sm<aF
dIskmenACeRmA h manGMeBIelICBaaMgTb;edayQrelIRTwsI Rankine RtUv)ankMNt;dUcxag
eRkam
1 sin

Pa = C a wh = wh
!$>$
1 + sin

CBaaMgTb;

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NPIC

1 sin

C a =
1 + sin

mMukkitkgrbs;dI tarag !$>!

wh 2 1 sin

sm<aFskmsrub

Ha =
2 1 + sin
=

tarag !$>! tmn w nig

!$>%

RbePTdIcak;bMeBj

Tmn;maD kg / m 3

mMukkitkg

Soft clay

1440 1920

0 o 15 o

Medium clay

1600 1920

15 o 30 o

Dry loose silt

1600 1920

27 o 30 o

Dry dense silt

1760 1920

30 o 35 o

Loose sand and gravel

1600 2100

30 o 40 o

Dense sand and gravel

1920 2100

25 o 35 o

Dry loose sand, graded

1840 2100

33o 35 o

Dry dense sand, graded

1920 2100

42 o 46 o

kmaMgpb H a GnuvtGMeBIenAkm<s; h / 3 BI)at rUbTI !$>@. enAeBlEdldIrgbnkbEnmeday

mMu EdlpMCamYyGkSedk enaH
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cos cos 2 cos 2

C a = cos

2
2
cos + cos cos

!$>^

nig H a = Ca wh2
kmaMgpb H a eFVIGMeBIenAkm<s; h / 3 nigeRTtedaymMu eTAnwgbnat;edk rUbTI 14>3. tm
C a EdlsMEdgedaysmIkar !$>^ sRmab;tmepSgn nig RtUv)anbgajenAkgtarag
!$>@.
Pa = C a wh

tarag !$>@ tmrbs; C

mMukkitkg

28 o

30 o

32 o

34 o

36 o

38 o

40 o

0o

0.361

0.333

0.307

0.283

0.260

0.238

0.217

10 o

0.380

0.350

0.321

0.294

0.270

0.246

0.225

20 o

0.461

0.414

0.374

0.338

0.306

0.277

0.250

25 o

0.573

0.494

0.434

0.385

0.343

0.307

0.275

30 o

0.

0.866

0.574

0.478

0.411

0.358

0.315

sm<aFdIGkmekItmanenAeBlEdlCBaaMgTb;rkM il nigsgt;dI. sm<aFdIGkmenAkm<s; h' enA

elICBaaMgTb;CamYynwgdIcak;bMeBjtamTisedkRtUv)ankMNt;dUcxageRkam
1 + sin

!$>&
Pp = C p wh = wh
1 sin
Edl

1 + sin 1
=
C p =
1 sin C a

sin

!$>*
sm<aFGkmsrubKW H p = wh2 11 + sin

kmaMgpb H p manGMeBIenA h' / 3 BI)at rUbTI 14>2. enAeBldIrgnUvbnkbEnmedaymMu

2

Pp = C p wh

CBaaMgTb;

cos 2 cos 2
2
2
cos cos cos

eTAnwgbnat;edk enaH C p = cos cos +

nig

H p = Cp

!$>(

wh 2
2

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manGMeBIenA h' / 3 ehIyeRTtedaymMu eTAnwgbnat;edk rUbTI 14>4. tm C p Edl

sMEdgedaysmIkar !$>( sRmab;tmepSgn nig RtUv)anbgajenAkgtarag !$>#.
tmn nig w ERbRbYleTAtamRbePTdIcak;bMeBj. tamkarENnaM tmFmtarbs; nig
w RtUv)an[enAkgtarag !$>!.
Hp

tarag !$># tmrbs; C

mMukkitkg

28 o

30 o

32 o

34 o

36 o

38 o

40 o

0o

2.77

3.00

3.25

3.54

3.85

4.20

4.60

10 o

2.55

2.78

3.02

3.30

3.60

3.94

4.32

20 o

1.92

2.13

2.36

2.61

2.89

3.19

3.53

25 o

1.43

1.66

1.90

2.14

2.40

2.68

3.00

30 o

0.

0.87

1.31

1.57

1.83

2.10

2.38

b.

T.Chhay

sRmab;RTwsIrbs; Coulomb/ sm<aFdIskmRtUv)ansnt;faCalTplndMudI (wedge of soil)

rGilRbqaMgmkelIpCBaaMgTb;. dUcenHRTwsI Coulomb KWsMedAeTARTwsI wedge. enAeBl
EdlkmaMgkkitRtUv)anKitenAkgCBaaMgTb; eK)ansnt;faprGilCabgerobesI EtkarBitva
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manlkNekagbnic. kMhusenAkgkarsnt;enHRtUv)anecalenAkgkarKNnasm<aFdIskm.
smIkar Coulomb edIm,IKNnasm<aFdIskm nigsm<aFdIGkmmandUcxageRkam
sm<aFdIskmKW Pa = Ca wh
cos 2 ( )
Edl Ca =
!$>!0 a
2

sin ( + )sin ( )
cos 2 cos( + )1 +

cos( + ) cos( )

mMukkitkgrbs;dI
= mMunpsm<aFdImkGkSQr
= mMukkittambeNaypCBaaMg mMurvagdInigebtug
= mMunbnkbEnm (surcharge) eTAGkSedk
sm<aFdIskmsrubKW
Edl

H a = Ca

wh 2
h
= Pa
2
2

enAeBlEdlpCBaaMgmanlkNbBarRtg; enaH = 0 o nigRbsinebI = enaH Ca

ehIysmIkar !$>!0 a kayeTACasmIkar !$>^ rbs; Rankine.
sm<aFdIGkmKW Pp = C p wh
cos 2 ( + )
Edl C p =
!$>!0 a
2

sin ( + )sin ( + )
cos 2 cos( )1

cos( + ) cos( )

c.

CBaaMgTb;

tmrbs; nig w ERbRbYleTAtamRbePTdIcak;bMeBj. tamkarENnaM tmFmtarbs;

nig w RtUv)an[ enAkgtarag !$>!.
enAeBlEdldIEqtTwk rnes<atrbs;dIeBjeTAedayTwk EdlbegItCasm<aFGIuRdUsaTic. enAkg
krNIenHTmn;maDdIEdllicTwk (submerged unit weight)RtUv)aneRbI. Tmn;maDdIlicTwk Ca
Tmn;maDdIEdl)andkTmn;maDTwk. \TiBlrbs;sm<aFTwkGIuRdUsaTicRtUvEtKitenAkgkar
KNnaCBaaMgTb;EdlRbQmnwgRsTab;Twkx<s; nigdIlicTwk. eKGaceRbImMukkitkgEdl)an
bgajenAkgtarag !$>!.

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14>5> \TiBlnbnkbEnm (Effect of Surcharge)

bnkepSgnbnkRtUv)andak;enAelIpndIcak;bMeBjBIxageRkayCBaaMgTb;. RbsinebIbnk
CabnkBRgayesI km<s;smmUlrbs;dI hs RtUv)ansnt;famanGMeBIeTAelICBaaMgedIm,IbegInsm<aF.
sRmab;CBaaMgdUcbgajenAkgrUbTI 14>5 sm<aFedkEdlbNalBIbnkbEnmmantmefreBjkm<s;
rbs;CBaaMgTb;.

hs =

Edl

!$>!!

ws
w

km<s;smmUlrbs;dI
ws = sm<aFbnkelIs
w = Tmn;maDdI
2
sm<aFsrubKW H a = H a1 + H a 2 = Ca w h2 + hhs
hs =

!$>!@

enAkgkrNIbnkBRgayesIedayEpkmanGMeBIenAcmaymYyBICBaaMgTb; enaHvamanEtsm<aFn
bnkbEnmsrubmYyEpkman\TiBleTAelICBaaMg rUbTI !$>^.
vaCakarGnuvtn_TUeTAedaysnt;fakm<s;RbsiTPaBnbnkbEnmKW h' Edlvas;BIcMnuc B eTA)at
rbs;CBaaMgTb;. bnat; AB pMmMu 45o CamYyGkSedk.
kgkrNIbnkkg;LanmanGMeBIenAcmaymYyCBaaMg bnkRtUv)anBRgayelIpkMNt;mYy Edl
CaTUeTAkMNt;eday specification EdleKTTYlsal;dUcCa AASHTO nig AREA.
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14>6> kmaMgkkitenAelI)atCBaaMgTb; (Friction on the Retaining Wall Base)

bgMkmaMgtamTisedknkmaMgTaMgGs;manGMeBIelICBaaMgTb;)anrujCBaaMgtamTisedAedk.
)atCBaaMgTb;RtUvEtFMRKb;RKan;edIm,ITb;nwgkarrGilrbs;CBaaMg. emKuNkmaMgkkitEdlRtUv)aneRbIKWCa
emKuNkmaMgkkitndIenAelICBaaMgsRmab; coarse granular soils nigCaersIusg;kmaMgkat;TTwgndI
sit. emKuNkmaMgkkit EdlGacTTYlyksRmab;RbePTdIepSgmandUcxageRkam
- Coarse-grained soils without silt, = 0.55
- Coarse-grained soils with silt, = 0.45
- Silt, = 0.35
- Sound rock, = 0.60
kmaMgkkitsrub F enAelI)atedIm,ITb;nwgkarrGilKW
!$>!#
F = R + H p
Edl = emKuNkmaMgkkit
R = kmaMgbBarmanGMeBIelI)at
H p = kmaMgTb;Gkm
emKuNsuvtiPaBRbqaMgnwgkarrGilKW
emKuNsuvtiPaB = HF = RH+ H p 1.5
!$>!$
ah
ah
Edl H ah CabgMkmaMgedknsm<aFskm H a . emKuNsuvtiPaBRbqaMgnwgkarrGilminKYr
tUcCag 1.5 RbsinebIkmaMgTb;Gkm H p RtUv)anecal nigminKYrtUcCag 2.0 RbsinebI H p RtUv)an
ykmkKitBicarNakgkarKNna.
CBaaMgTb;

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14>7> sanPaBlMnwgRbqaMgnwgkarRkLab; (Stability against Overturning)

bgMkmaMgedknsm<aFskm H a nwgeFVI[CBaaMgTb;RkLab;eFobcMnuc 0 rUbTI 14>7. m:U
m:g;eFVI[RkLab; (overturning moment) M 0 = H a h / 3 . Tmn;rbs;ebtug nigdIbegItCam:Um:g;Edl
eFVI[manlMnwg (balancing moment or tightening moment) edIm,ITb;Tl;nwgm:Um:g;eFVI[RkLab;.

m:Um:g;eFVI[manlMnwgsRmab;krNICBaaMgdUcbgajkgrUbTI 14>7 KW
M b = w1 x1 + w2 x 2 + w3 x3 = wx

emKuNsuvtiPaBRbqaMgnwgkarRkLab;KW
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emKuNsuvtiPaB = MM b = H wsh 2.0

0

!$>!%

emKuNsuvtiPaBenHminRtUvKYrtUcCag 2.0 eT.

kmaMgpbnkmaMgTaMgGs;manGMeBIelICBaaMgTb; R A RbsBVenA)atCBaaMgTb;Rtg;cMnuc C rUb
TI 14>7. CaTUeTA cMnuc C minRtYtsIuKaCamYyGkSrbs;)atCBaaMg dUcenHvabegIt)anCakmaMgcakpit
enAelIeCIgtag. vaCakarRbesIredayrkSacMnuc C enAkgEpkkNalnRbEvgmYyPaKbIn)atCBaaMg
edIm,IeFVI[)atCBaaMgTaMgmUlsiteRkamsm<aFdI. krNIeCIgtageRkambnkcakpitRtUv)anBnl;kg
emeronTI 13.
14>8> smamaRtnCBaaMgTb; (Proportions of Retaining Walls)
karKNnaCBaaMgTb;cab;epImCamYynwgkarmuxkat;sakl,gEdlmanTMhMRbhak;RbEhl.
bnab;mkmuxkat;snt;enaHRtUv)anepgpat;sanPaBlMnwg nigPaBRKb;RKan;rbs;eRKOgbgM. xageRkam
Cac,ab;EdlGaceRbIedIm,IkMNt;xatRbhak;RbEhlnEpkepSrbs;CBaaMgTb; cantilever

CBaaMgTb;

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a.

NPIC

km<s;rbs;CBaaMg km<s;srubrbs;CBaaMgesInwgPaBxusKankRmitkm<s;bUknwg 900mm eTA

EdlCakarKNnaCeRmAnCRmabTwkkk (frost penetration depth) sRmab;rdPaK

xageCIgrbs;RbeTsGaemrik.
kRmas;CBaaMg GaMgtg;sIuetnsm<aFekIneLIgeTAtamCeRmArbs;CBaaMg ehIytmGtibrma
rbs;vasitenA)atCBaaMg. dUcenH m:Um:g;Bt;Gtibrma nigkmaMgkat;TTwgGtibrmaenAkgCBaaMg
ekItmanenAnwg)atrbs;va. kRmas;CBaaMgenA)atRtUv)ankMNt;esInwg1 / 12 eTA1 / 10 nkm<s;
CBaaMg h . kRmas;rbs;CBaaMgenAkMBUlRtUv)ansnt; 200mm nig 300mm . edaysarCBaaMg
Tb;RtUvKNnasRmab;Tb;sm<aFdIskm edIm,IeFVI[manPaBdabrbs;CBaaMgtUc vaCakarRbesIr
edIm,IeFVI[pCBaaMgmanCRmalGb,brmakgkRmit 25mm kgkm<s; 1m . sRmab;CBaaMgTab
Edlmankm<s; 3m eKGacykkRmas;CBaaMgefr)at.
RbEvgeCIgtag karkMNt;dMbUgsRmab;RbEvg)atesInwg 2 / 5 eTA 2 / 3 nkm<s;CBaaMg h .
kRmas;rbs;)at kRmas;)atxageRkamRtUv)ankMNt;esIkRmas;CBaaMgenA)at EdlesInwg1 / 12
eTA 1 / 10 nkm<s;CBaaMg. kRmas;Gb,brmaRtUv)ankMNt;RbEhl 300mm . )atCBaaMgGac
mankRmas;esI bscenAcugTaMgsgag Edlmanm:Um:g;Bt;sUn.
smamaRtrbs;CBaaMg cantilever retaining wall RtUv)anbgajenAkgrUbTI 14>8.
1200mm

b.

c.
d.

14>9> tRmUvkarsRmab;KNna

(Design Requirement)

ACI Code, Chapter 14

pl;nUvviFIsaRssRmab;KNnalTPaBRTRTg;rbs;CBaaMg. tRmUvkar

cMbgdUcxageRkam
a. kRmas;Gb,brmanCBaaMgKW 1 / 25 nkm<s; bnRbEvgCBaaMg ykmYyNaEdltUcCageK b:uEn
minRtUvtUcCag 100mm .
b. RkLapGb,brmanEdkedkenAkgCBaaMgKW 0.0025bh Edl bh CaRkLapebtugCBaaMg.
tmenHRtUv)ankat;bnyRtwm 0.002bh RbsinebIeKeRbIEdk DB16 bEdkEdlmanmuxkat;
tUcCagenH nigman f y 400MPa . sRmab;EdksMNaj;pSar EdkmUl bEdkfaMgGMeBA Rk
LapEdkGb,brmaKW 0.002bh .

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c.

d.

e.

Department of Civil Engineering

RkLapGb,brmanEdkQrKW 0.0015bh b:uEnvaRtUv)ankat;bnyRtwm 0.0012bh RbsinebI

Edk DB16 bEdkEdlmanGgt;pittUcCagenHCamYynwg f y 400MPa RtUv)aneRbI. sRmab;
EdksMNaj;pSar EdkmUl bEdkfaMgGMeBA brimaNEdkGb,brmaKW 0.0012bh .
KMlatGtibrmanEdkbBar nigEdkedkKWtmtUcCageKkgcMeNam 450mm bkRmas;CBaaMg
bIdg.
RbsinkRmas;CBaaMgFMCag 250mm enaHEdkbBar nigEdkedkKYrEtdak;BIrRsTab;RsbKanwg
pxagkg nigxageRkAdUcxageRkam
sRmab;pCBaaMgxageRkA y:agtic1 / 2 nbrimaNEdk As b:uEnminRtUveRcInCag 2 As / 3 KYr
mankRmas;karBarEdkGb,brma 50mm b:uEnminRtUvFMCag 1 / 3 nkRmas;CBaaMg. enHeday
sar pCBaaMgxageRkARbQmnwglkxNGakasFatuepSgKa nigbERmbRmYlsItuNPaB.
sRmab;pCBaaMgxagkg EdktRmUvkarenAkgTisedAnImYyKYrmankRmas;karBarEdkGb,brma
20mm b:uEnminRtUvFMCag 1 / 3 nkRmas;CBaaMg.
EdkGb,brmaenAkgeCIgtagCBaaMg eyagtam ACI Code, Section 10.5.3 KWfaEdktRmUvkar
sRmaab;EdkrYmmaD nigEdksItuNPaB Edlfa 0.0018bh enAeBl f y = 400MPa nig
0.0020bh enAeBl f y = 280 MPa b f y = 350 MPa . edaysarbrimaNEdkGb,brma
mantmtUceBk enAkgkarGnuvteKRtUvdMeLIgvadl;brimaNEdkGb,brma As sRmab;karBt;
f'
1.4
=
b d nig
b d
!$>!^
A
4f
f
c

s min

14>10> karbgrTwk (Drainage)

sm<aFdIEdl)anerobrab;BIxagedImmin)anrab;bBalsm<aFGIuRdUsaTic. RbsinebITwkRbmUlpMenA
xageRkayCBaaMgTb; sm<aFTwkRtUvEtrab;bBalkgkarKNna. p bRsTab;TwkeRkamdIGacRCabenAkg
dIcak; nigbegItkrNIdIlicTwk. edIm,IeCosvagsm<aFGIuRdUsaTic karbgrTwkKYrRtUvdak;enABIeRkay
CBaaMg. RbsinebIdIGt;manPaBsitbgrTwk)anlRtUv)aneRbIsRmab;cak;bMeBj CBaaMgGacRtUv)anKNna
sRmab;E sm<aFdIsuT. RbBnkarbgrTwkGacrYmmanmYy bbnSMnkarerobrab;xageRkam
- rnRbehag (weep holes) enAkgCBaaMgTb;manGgt;pit 100mm bFMCagenH nigmanKMlat
RbEhl 1.5m enAkNalGkSedk nigGkSQr rUbTI 14>9 a .
CBaaMgTb;

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- TIbRbehag (perforated pipe) EdlmnaGgt;pit 200mm dak;tambeNaynCBaaMgBTCMu

vijedayxSac; bfbMEbk rUbTI 14>9 b .
- labpCBaaMgxagcak;bMeBjCamYy asphalt edIm,IkarBarkarRCabTwk.
- viFIepSgeTotedIm,IbgrTwk

]TahrN_14>1 muxkat;sakl,grbs;CBaaMgTb;ebtugsuT semigravity RtUv)anbgajenAkgrUbTI

14>10. CBaaMgenHcaM)ac;RtUvkarRtYtBinitsuvtiPaBTb;nwgkarRkLab;/ karrGil nigRTRTg;nUvsm<aFBI

eRkameCIgtag. eK[ Tmn;maDdIcak;bMeBj 17.6kN / m3 / mMukkitkg = 35o / emKuNkkitrvag
ebtug nigdI = 0.5 / sm<aFdIGnuBaatKW 120kN / m 2 nig f 'c = 20MPa .

dMeNaHRsay

1> edayeRbIsmIkar Rankine

Ca =

1 sin 1 0.574
=
= 0.271
1 + sin 1 + 0.574

sm<aFGkmenAEpkxagmuxKWsRmab;km<s; 300mm EdlCatmtUcGacecal)an.

Ha =

C a wh 2 0.271 17.6 3.32

=
= 26kN
2
2

eFVIGMeBIenAcmay h / 3 = 1.1m BI)at.

2> m:Um:g;eFVI[RkLab; (overturning moment) KW M 0 = 26 1.1 = 28.6kN .m
3> KNnam:Um:g;eFVI[manlMnwg (balancing moment) M b edayKiteFobcMnuc 0 rUbTI 14>10
Ha

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CBaaMgTb;

Department of Civil Engineering

393

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

Tmn; (kN )

dXas; (m)

m:Um:g; (kN .m)

w1 = 0.3 3 24 = 21.6
1
w2 = 3 (1.05 0.3) 24 = 27
2

0. 4

8.64

0 .8

21.6

0 .8

9.22

1.05

20.79

1.45

23

w3 = 0.3 1.6 24 = 11.52

1
w4 = 3 (1.05 0.3) 17.6 = 19.8
2
w5 = 0.3 3 17.6 = 15.84

w = R = 95.76kN

M b = M = 83.25kN

4> emKuNsuvtiPaBRbqaMgnwgkarRkLab;KW 83.25 / 28.6 = 2.91 > 2.0

5> kmaMgRbqaMgkarrGil F = R = 0.5 95.76 = 47.88kN
emKuNsuvtiPaBTb;nwgkarrGilKW F / H a = 47.88 / 26 = 1.84 > 1.5
6> KNnasm<aFdIeRkam)at
a.
cmayrbs;kmaMgpbBIcMnuc 0
x=

b.

c.

M b M 0 83.25 28.6
=
= 0.57 m
95.76
R

cMNakpitKW e = 0.8 0.57 = 0.23m . kmaMgpb

R manGMeBIenAkgEpkkNalnmYyPaKbIRbEvgeCIgtag nigmancMNakpit
e = 0.23m BIGkSrbs;)at rUbTI 14>10.
sRmab;eCIgtagRbEvg 1m nigRbEvgeCIgtagRbsiTPaB 1.6m
m:Um:g;niclPaB I = 11.63 /12 = 0.34m 4 / RkLap A = 1.6m 2
sm<aFdIenAcugTaMgBIrrbs;eCIgtagKW q1, q2 = R / A Mc / I .
m:Um:g; M KW R e = 95.76 0.23 = 22kN .m nig c = 0.8m
95.76 22 0.8
+
= 111.6kN
1.6
0.34
95.76 22 0.8
q2 =

= 8.1kN
1.6
0.34

q1 =

7> RtYtBinitkugRtaMgBt;enAkgebtugRtg;cMnuc A
a.
sm<aFdIRtg;cMnuc A BIFrNImaRtKW
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1.35
q A = 8.1 +
(111.6 8.1) = 95.4kN
1.6
b.

m:Um:g; M A RtUv)anKNnaenARtg; A bNalmkBIkugRtaMgctuekaNEkg nigkugRtaMg

RtIekaN
MA =

c.

95.4 0.25 2 16.2 0.25

2
+
0.25 = 3.32kN .m
2
2
3

kugRtaMgBt;enAkgebtugKW

Mc 3.32 0.15
=
10 3 = 0.221MPa

3
I
2.25 10
h 0.3
c= =
= 0.15m
I = 1 0.33 / 12 = 2.25 10 3 m 4
2
2

d.

Edl
nig
m:UDuldac;rbs;ebtug (modulus of rupture of concrete) KW
0.623 f 'c = 2.8MPa > 0.221MPa

.8
emKuNsuvtiPaBRbqaMgnwgkareRbHKW 02.221
= 12.67 . dUcenH
muxkat;manlkNRKb;RKan;. eKmincaM)ac;esInUvmuxkat;epSgeTotedIm,IRtYtBiniteT.

]TahrN_14>2 KNnaCBaaMgTb; cantilever edIm,ITb;dIRcaMgkm<s; 5m . enAelIdIEdlTb;manbnk

bEnm 9.25kN / m 2 . eK[ Tmn;maDdIcak;bMeBj 18.5kN / m3 / mMukkitkg = 35o / emKuNkkit

rvagebtug nigdI = 0.5 / emKuNkkitrvagRsTab;dI = 0.7 / lTPaBRTdIGnuBaat 200kN / m 2 /
f 'c = 20MPa nig f y = 400MPa .

dMeNaHRsay

1> kMNt;TMhMrbs;CBaaMgTb;edayeRbITMnak;TMngRbhak;RbEhlEdl)anbgajenAkgrUbTI 14>8

a. km<s;CBaaMg edaybgb;CBaaMgTb; 900mm enaHkm<s;rbs;CBaaMgTb; h = 5 + 0.9 = 5.9m
b. kRmas;)at snt;kRmas;)atesI 0.08h = 0.08 5.9 = 0.472m yk 0.45m . km<s;
rbs;tYCBaaMg 5.45m
c. RbEvgeCIgtag RbEvgeCIgtagERbRbYlcenaH 0.4h nig 0.67h . edaysnt;yktm
mFm enaHRbEvgeCIgtagesI 0.53h = 0.53 5.9 = 3.127m yk 3.2m . RbEvg)atenABI
muxtYCBaaMgERbRbYlcenaH 0.17h nig 0.125h . snt;RbEvg)atxagmuxesI
0.17 5.9 = 1m .

CBaaMgTb;

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d.

NPIC

kRmas;CBaaMg kRmas;Gtibrmarbs;CBaaMgsitenAKl;EdlERbRbYlcenaH 0.08h eTA

0.1h . eRCIserIskRmas;CBaaMgGtibrmaesInwgkRmas;eCIgtagKW 0.45m .
kRmas;Gb,brmarbs;CBaaMg sitenAkMBUlrbs;CBaaMg edaykMNt;yk 0.3m .
CRmalGb,brmarbs;pCBaaMgKW 25mm / m . sRmab;CBaaMgEdlmankm<s;
5.45m enaHRbEvgscesInwg 5.45 0.025 = 0.136m EdltUcCagRbEvgscEdlman
0.45 0.3 = 0.15m . TMhMrbs;CBaaMg Edl)anesIeLIgRtUv)anbgajenAkg
rUbTI14>11.

2> tamsmIkar Rankine

Ca =

1 sin 1 0.574
=
= 0.271
1 + sin 1 + 0.574

3> emKuNsuvtiPaBRbqaMgnwgkarRkLab;GacRtUv)ankMNt;dUcxageRkam
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KNnakmaMgKanemKuNBitR)akdEdlmanGMeBIelICBaaMgTb;. dMbUg kMNt;nUvral;kmaMg

EdleFVI[CBaaMgRkLab;
w
9.25
hs EdlbNalBIbnkelIs = s =
= 0 .5 m
w 18.5

a.

p1 = C a whs = 0.271 18.5 0.5 = 2.5kN / m 2

p 2 = C a wh = 0.271 18.5 5.9 = 29.58kN / m 2
5.9
H a 2 = 2.5 5.9 = 14.75kN
=
= 2.95m
2
1
5.9
H a 2 = 29.58 5.9 = 87.26kN
=
= 1.97m
2
3

dXas;
dXas;

b.

m:Um:g;EdleFVI[RkLab; (overturning moment) KW

14.75 2.95 + 87.26 1.97 = 215.4kN.m

m:Um:g;eFVI[manlMnwgEdlRbqaMgnwgkarRkLab; emIlrUbTI 14>12

c.

kmaMg (kN )

dXas; (m)

m:Um:g; (kN .m)

w1 = 0.3 5.45 25 = 40.875

1
w2 = 0.15 5.45 25 = 10.22
2

1.3

53.14

1 .1

11.24

w3 = 3.2 0.45 25 = 36

1.6

57.6

w4 = 1.75 5.45 18.5 = 176.44

2.375

419

w = R = 263.5kN

M = 541kN .m

541
emKuNsuvtiPaBRbqaMgnwgkarRkLab;KW 215
= 2.51 > 2
.4
4> KNnasm<aFdIeRkameCIgtag. Kitm:Um:g;eFobcMnuc 0 rUbTI 14>12 edIm,IkMNt;TItaMg kmaMgpb
R nkmaMgbBar
x=

M Hy = balancing M overturning M
R

541 215.4
263.5

3.2
= 1.236m >
= 1.066m
3

cMNakpitKW e = 3.2 / 2 1.236 = 0.364m . kmaMgpb R manGMeBIenAkgEpkkNaln

RbEvgmYyPaKbIneCIgtag nigmancMNakpit e = 0.476m BIGkSrbs;eCIgtag. sRmab;
RbEvgbeNay 1m rbs;eCIgtag
RkLap = 1 3.2 = 3.2m 2
CBaaMgTb;

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m:Um:g;niclPaB I = 1 312.2

= 2.73m 4

R R e C 263.5 263.5 0.364 1.6

+
=
+
= 138.56kN / m 2 < 200kN / m 2
A
I
3.2
2.73
R R e C 263.5 263.5 0.364 1.6
=

q2 =
= 26.13kN / m 2
3.2
2.73
A
I

q1 =

sm<aFdIKWRKb;RKan;

5> KNnaemKuNsuvtiPaBRbqaMgnwgkarrGil. emKuNsuvtiPaBGb,brmaRtUvesI 1.5 .

kmaMgeFVI[rGil = H a1 + H a2 = 14.75 + 87.26 = 102kN
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kmaMgTb;kMu[rGil = R = 0.5 253.5 = 126.75kN

emKuNsuvtiPaBRbqaMgnwgkarrGil = 126102.75 = 1.24 < 1.5
kmaMgTb;RbqaMgnwgkarrGilEdlmanmin)anpl;suvtiPaBRKb;RKan;RbqaMgnwgkarrGileT. enA
kgkrNIenH key RtUv)anbegItedIm,IbegItsm<aFGkmFMRKb;RKan;edIm,ITb;nwgkmaMgdFMEdleFVI
[rGil. tYnaTIepSgeTotrbs; key KWpl;nUvRbEvgbgb;RKb;RKan;sRmab;Edk dowel rbs;tY
CBaaMg. dUcenH key RtUv)andak;enATItaMgEdlpxagmuxrbs;vamancmay 150mm BIpxag
eRkayrbs;CBaaMg rUbTI14>13. enAkgkarKNnasm<aFGkm dIenABIelIeCIgtagEpkxag
muxRtUv)anecalkgkm<s; 300mm . dUcenHenAkg]TahrN_enHkm<s;dIKItBI)ateCIgtagesI
600mm . snt; key mankm<s; t = 450mm nigTTwg b = 450mm .
1 + sin
1
1
=
=
= 3.69
1 sin C a 0.271
1
1
H p = C p w(h'+t )2 = 3.69 18.5(0.6 + 0.45)2 = 37.63kN
2
2

Cp =

\LvkarrGilGacnwgekItmanenAelIp AC / CD nig EF rUbTI14>13. prGil AC

CaprGilrvagRsTab;dInigRsTab;dIEdlmanemKuNkmaMgkkitkg = tan = tan 35o = 0.7 .
cMENkprGil CD nig EF CaprGilrvagebtugnigRsTab;dIEdlmanemKuNkmaMgkkitkg
esI 0.5 dUcEdl)an[enAkgsmtikmrbs;]TahrN_.
ersIusg;kmaMgkkitKW F = 1R1 + 2 R2 .
138.56 + 92.89
R1 = Rbtikmn AC =
1.3 = 150.4kN
2

R2 = R R1 = 263.5 150.4 = 113.1kN

92.89 + 26.13
R2 =
CDF =
1.9 = 113.1kN
2

Rbtikmn

F = 0.7 150.4 + 0.5 113.1 = 161.83kN

F + H p = 161.83 + 37.63 = 199.5kN

kmaMgTb;srubKW
.5
161.83
emKuNsuvtiPaBRbqaMgnwgkarrGilKW 199
= 1.96 < 1.5 b
= 1.59 < 1.5
102
102
emKuNsuvtiPaBFMCag 1.5 EdlRtUv)anENnaMenAeBlEdlsm<aFGkmRbqaMgnwgkarrGilmin
RtUv)anKitbBal.

CBaaMgTb;

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6> KNnatYCBaaMg. karKNnanGgt;ebtugGarem:GaceFVI)anedayeRbI ACI Code alternative

design method, appendix B and C.
a. Edkem (main reinforcement) kmaMgtamTisedkEdlGnuvtmkelICBaaMgRtUv)anKNna
edayeRbIemKuN 1.6 . muxkat;eRKaHfak;sRmab;m:Um:g;Bt;KWsitenARtg;)atnCBaaMg
km<s; h = 5.45m
KNnakmaMgemKuN
P1 = 1.6(C a whs ) = 1.6(0.271 18.5 0.5) = 4.01kN / m 2
P2 = 1.6(C a wh ) = 1.6(0.271 18.5 5.45) = 43.72kN / m 2
5.45
H a1 = 4.01 5.45 = 21.85kN
=
= 2.725m
2
1
5.45
H a 2 = 43.72 5.45 = 119.14kN
=
= 1.82m
2
3

dXas;

dXas;

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enA)atrbs;CBaaMg = 21.85 2.725 + 119.14 1.82 = 276.4kN .m

kRmas;srubEdleRbIKW 450mm / b = 1m nig
Mu

d = 0.45 (concrete cover) (half the bar diameter) = 0.387m

Ru =

Mu
bd 2

PaKryEdk

276.4 10 6

= 1.85MPa
1000 387 2
0.85 f 'c
2 Ru
=
1 1
f y
0.85 f 'c

= 0.0058

As = 0.0058 1000 387 = 2244.6mm 2

eRbI DB25 KMlat 200mm As = 2454.4mm 2 . EdkbBarGb,brma As eyagtam

ACI Code, Section 14.3 KW
As = 0.0015 450 1000 = 675mm 2 < 2454.4mm 2

edaysarEtm:Um:g;fycuHtamkm<s;rbs;CBaaMg As KYrRtUv)ankat;bnyeTAtamm:Um:g;Edl
ekItmanenAelICBaaMg. CakarGnuvtn_ eKKYreRbI As bKMlatmYy
sRmab;Bak;kNalCBaaMg xageRkam nig As bKMlatmYyepSgeTot
sRmab;Bak;kNalCBaaMgxagelI. edIm,IKNna m:Um:g;enAkm<s;Bak;kNalCBaaMg
2.725m BIkMBUl
P1 = 1.6(C a whs ) = 1.6(0.271 18.5 0.5) = 4.01kN / m 2
P2 = 1.6(C a wh ) = 1.6(0.271 18.5 2.725) = 21.86kN / m 2
2.725
H a1 = 4.01 2.725 = 10.93kN
=
= 1.36m
2
1
2.725
H a 2 = 21.86 2.725 = 29.78kN
=
= 0.91m
2
3

dXas;

dXas;

enAkm<s;Bak;kNal = 10.93 1.36 + 29.78 0.91 = 41.96kN .m

kRmas;srubenAkm<s;Bak;kNalKW 300 +2 450 = 375mm / b = 1m nig
Mu

d = 0.375 0.05 0.013 = 0.312m

M
41.96 10 6
Ru = u2 =
= 0.43MPa
bd
1000 312 2
0.85 f 'c
2 Ru
=
1 1
f y
0.85 f 'c

PaKryEdk

= 0.0013

As = 0.0013 1000 312 = 405.6mm 2

CBaaMgTb;

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As min = 0.0015 1000 375 = 562.5mm 2 > 405.6mm 2

b.

eRbIEdk DB12 KMlat 200mm As = 565.5mm 2 .

EdksItuNPaB nigEdkrYmmaD EdkedkGb,brmaenA)atCBaaMgGaRsynwg ACI Code,
Section 14.3 KW
As min = 0.002 1000 450 = 900mm 2

sRmab;mYyPaKbIBI)at/ edaysnt;eRbIEdk DB16 btUcCagenH.

As min = 0.002 1000 400 = 800mm 2

sRmab;BIPaKbIBIkMBUl. edaysar pxagmuxrbs;CBaaMgrgnUvbERmbRmYlsIuNPaB

dUcenH eKeRbI mYyPaKBIreTAmYyPaKbInEdkedk enApxageRkArbs;CBaaMg nigdak;nUv
brimaNdUcKaenApxagkg.
0.5 As = 0.5 900 = 450mm 2

c.

d.

eRbIEdk DB12 KMlat 200mm As = 565.5mm 2 enATaMgpxagkg nigpxageRkA

rbs;CBaaMg. eRbIEdk DB12 KMlat 300mm edIm,IRTEdkedkrYmmaD nigEdksItuNPaB.
Edk dowel sRmab;EdkbBar RbEvgf<k;rbs;Edk DB25 eTAkgeCIgtagRtUvEtman
RbEvg y:agticbMput 550mm . eRbIRbEvgEdkbgb; 600mm eTAkgeCIgtag nig key
BIeRkamtY CBaaMg.
KNnasRmab;kmaMgkat;TTwg muxkat;eRKaHfak;sRmab;kmaMgkat;TTwgsitenAcmay
d = 0.387 mm BI)attYCBaaMg. enARtg;muxkat;enH cmayBIkMBUlesI
5.45 0.387 = 5.063m

P1 = 1.6(C a whs ) = 1.6(0.271 18.5 0.5) = 4.01kN / m 2

P2 = 1.6(C a wh ) = 1.6(0.271 18.5 5.063) = 40.61kN / m 2
H a1 = 4.01 5.063 = 20.3kN
1
H a 2 = 40.61 5.063 = 102.8kN
2
H = H a1 + H a 2 = 123.1kN

Vc = 0.17 f 'c bd = 0.75 0.17 20 1000 387 10 3 = 220.7kN > 123.1kN

7> KNnaeCIgtagEpkxageRkayCBaaMg emKuNbnk 1.2 RtUv)aneRbIsRmab;KNnam:Um:g;Bt; emKuN

nigkmaMgkat;TTwgEdlekItBIdIcak;bMeBjnigebtug. cMENkemKuNbnk 1.6
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RtUv)aneRbIsRmab;bnkbEnm.
sm<aFdIeLIgelIRtUv)anecaledaysarvanwgkat;bny\TiBlndIcak;bMeBj nigebtugeTAelI
eCIgtagEpkxageRkayCBaaMg. eyagtamrUbTI 14>12 bnksrubenAelIeCIgtagEpkxag
eRkayrbs;CBaaMgKW
Vu = 1.2(5.45 1.75 18.5 + 0.45 1.75 25) + 1.6 0.5 1.75 18.5 = 261.3kN
1.75
Mu
= Vu
= 228.64kN .m
2

enARtg;pCBaaMg
CaTUeTAmuxkat;eRKaHfak;sRmab;kmaMgkat;TTwgsitenAcmay d BIpCBaaMgenAeBlEdlkmaMg
RbtikmbegItkmaMgsgt;enAtMbn;xagcugrbs;Ggt;. kgkrNIenH muxkat;eRKaHfak;RtUv)anKit
faekItmanenAnwgpCBaaMg edaysarEtkmaMgTaj minEmnkmaMgsgt;ekItmanenAkgebtug.
Vu = 261.3kN
2
2
Vc =
f 'c bd = 0.75
20 1000 362 10 3 = 202.4kN < 261.3kN
12
12

edaysar Vc < Vu = 216.3kN enaHmuxkat;RtUv)andMeLIgedayGRta 261.3 / 202.4 bkEdk

kmaMgkat;TTwgRtUv)andak;.
261.3
d caM)ac; =
362 = 467 mm
202.4
kRmas;eCIgtagsrubcaM)ac; = 467 + 75 + 13 = 555mm
ykkRmas;eCIgtagesI 560mm nig d = 472mm
Ru =

228.64 10 6
1000 472 2

= 1.03MPa

= 0.0031

As = 1463.2mm 2

EdkrYmmaDGb,brma As = 0.0018 1000 560 = 1008mm 2

EdkrgkarBt;Gb,brma As = 0.0033 1000 472 = 1557.6mm 2
eRbIEdk DB20 manKMlat 200mm As = 1570.8mm 2 . RbEvgEdkbgb;sRmab;
DB 20 esI nwg 1.4l d = 1100mm .
dUcenHEdkRtUv)andak;bgscUleTAkgEpkxagmuxrbs;eCIgtageday RbEvg 1100mm .
EdkrYmmaD igEdksItuNPaBsRmab;TisedAEvgmincaM)ac;RtUv)andak;enAkgeCIgtagEpkxagmux
nigEpkxageRkayrbs;eCIgtageT. b:uEnCakarGnuvtn_ eKRtUvdak;brimaNEdkGb,brmaenAkg
TisedAenH edayeRbI DB12 @ 300 .
CBaaMgTb;

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8> KNnaeCIgtagEpkxagmuxCBaaMg Epkxagmuxrbs;eCIgtageFVIkardUcCaFwm cantilever Edl

rgnUvsm<aFdIeLIgelICamYynwgbnkemKuN 1.6 edaysarvaekIteLIgCadMbUledaykmaMgxag.
muxkat;eRKaHfak;sRmab;kmaMgkat;TTwgsitenAcmay d BIpxagmuxrbs;CBaaMg edaysarEt
RbtikmenAkgTisedAnkmaMgkat;TTwgbegItkmaMgsgt;enAkgEpkxagmuxrbs;eCIgtag.
eyagtamrUbTI 14>13 Epkxagmuxrbs;eCIgtagrgnUvsm<aFeLIgelIedaysardI nigsm<aFcuH
eRkamedaysarbnkpal;rbs;kRmaeCIgtagEpkxagmux.
138.56 + 120
Vu = 1.6
0.528 1.2(0.56 25)0.528 = 100.3kN
2

vamantmtUcCag Vc = 263.8kN .

12
2
103.43 2
M u = 1.6
1 + (138.56 103.43) 1 0.5 1.20.56 25 = 93.1kN .m
2
3
2

Ru =

Mu
bd 2

93.1 10 6
1000 472 2

= 0.42 MPa

= 0.00125

As = 0.00125 1000 472 = 590mm 2

EdkGb,brmasItuNPaB nigrYmmaD As = 0.0018 1000 560 = 1008mm 2

EdkGb,brmargkarBt; As = 0.0033 1000 472 = 1557.6mm 2
eRbIEdk DB20 manKMlat 200mm dUcKanwgEpkxagmuxCBaaMg. RbEvgEdkbgb;sRmab;Edk
DB 20 esInwg 650mm . bgsEdkcUleTAkgEpkxageRkayCBaaMgRbEvg 650mm . bg;srs
EdklMGitRtUv)anbgajenAkgrUbTI 14>14.
9> Shear keyway rvagCBaaMg nigeCIgtag enAkgkarsagsg;CBaaMgTb; eCIgtagRtUv)ancak;mun
bnab;mkeTIbEpkxagelIRtUv)ancak;bn. Construction joint RtUv)aneRbIenA)atCBaaMg. p
tMNRtUv)anykmanragCa keyway dUcbgajenAkgrUbTI 14>15 bRtUvTuk[manlkN
KeRKIm rUbTI 14>14. tMNRtUvmanlTPaBbBankmaMgkat;TTwgBItYCBaaMgeTAeCIgtag.
10> enAkgkarKNnaenH eKcaM)ac;RtUvdak;RbBnbgrTwkBIdIcak;bMeBj[)anRtwmRtUv BIeRBaHsm<aFdI
EdleRbIKWsRmab;dIcak;bMeBjEdlbgrTwkecj. Weep holes KYrRtUv)andak;enAkgCBaaMg Edl
manGgt;pit 100mm manKMlat 1.5m tamTisedAedk nigTisedAbBar.

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14>11> CBaaMgCan;eRkamdI (Basement Walls)

CBaaMgCan;eRkamdIRtUv)aneKsnt;fasitenAcenaHkRmalxNCan;eRkamdI nigkRmalxNCan;
pal;dI. eKmanBIrkrNIkgkarKNnaCBaaMgCan;eRkamdI.
TImYy enAeBlEdlCBaaMgRtUv)ansg;enABIelIkRmalxNCan;eRkamdI CBaaMgnwgrgnUvsm<aFdI
tamTisedk nigminrgbnkbBareRkABIbnkpal;xn. kgkrNIenH CBaaMgmannaTICa cantilever ehIy
brimaNEdkRKb;RKan;KYrRtUv)andak;sRmab;karKNnaCBaaMg cantilever. krNIenHGaceCosvag)an
eda dMeLIgkRmalxNCan;eRkamdI nigkRmalxNCan;TImYymunnwgcak;bMeBjdI.
TIBIr enAeBlEdlkRmalxNCan;TImYy nigkRmalxNCan;epSgeTotRtUv)ansg; ehIysMNg;
RtUv)andak;bnkeBj enaHCBaagM Can;eRkamdImannaTICaCBaaMg propped cantilever EdlrgnUvsm<aFdI
kdUcCasm<aFbBar.
sRmab;mMukkitkg = 35o enaHemKuNsm<aFskm Ca = 0.271 . sm<aFsIedkenAnwg)atKW
Pa = C a wh . sRmab; w = 17.6kN / m 3 nigCBaaMgmankm<s;mFm h = 3m enaH
Pa = 0.271 17.6 3 = 14.31kN / m 2

32
= 21.46kN / m
2

nCBaaMg
H a eFVIGMeBIenARtg; h / 3 = 3 / 3 = 1m BI)at. sm<aFbEnmRbEhl 9.6kN / m 2 RtUv)anKitfa
manenAelIdIxageRkayCBaaMg enaH
H a = 0.271 17.6

hs =

9.6
= 0.55m
17.6

Ps = C a whs = 0.271 17.6 0.55 = 2.62kN / m 2

nCBaaMg
H s nbnkbEnmeFVIGMeBIenA h / 2 = 1.5m BI)at.
kgkarKNnaelIkmun eK)ansnt;fadIcak;bMeBjmanlkNst b:uEnsRmab;krNIenHeKcaM)ac;
RtUvKitBIsm<aFTwkenABIeRkayCBaaMg. sm<aFTwkGtibrmaekItmanenAeBlkm<s;CBaaMgTaMgmUlrbs;
CBaaMgCan;eRkamdIrgnUvsm<aFTwk. Pw = 10 3 = 30kN / m 2
32
wh 2
Hw =
= 10 = 45kN / m nCBaaMg
2
2
sm<aFGtibrmaminGacmanvtmanenABIeRkayCBaaMgCab;rhUteT. dUcenH RbsinebIdIesImyUr
mg enaHeKGacyksm<aFTwkRtwm 50% mkKit.
H s = (Ca whs )h = 2.62 3 = 7.86kN / m

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Pw 30
=
= 15kN / m 2
2
2
H
H ' w = w = 22.5kN / m
2

nCBaaMg
H ' w eFVIGMeBIenA h / 3 = 3 / 3 = 1m BI)at. TwkGacRtUv)ankarBarBIkarRbmUlpMRbqaMgnwgCBaaMg
edayerobcMkarbgrenAEpkxageRkamnCBaaMg.
bEnmBIelIkarerobcMRbBnbgr CBaaMgEpkxageRkARtUvlabfaMkarBarCMrabTwk (waterproofing
or damp-proofing membrane) . ACI Code, Section 14.5
kMNt;fakRmas;Gb,brmanCBaaMgeRkam dIEpkxageRkA nigRKwHrbs;vaKW 190mm . CaTUeTA kRmas;
Gb,brmarbs;CBaaMgRTesInwgtmtUcCageK kgcMeNam 1/ 25 nkm<s; b 1/ 25 RbEvgCBaaMg b
100mm .

]TahrN_14>3 kMNt;kRmas; nigbrimaNEdkcaM)ac;sRmab;CBaaMgTb;Can;eRkamdIdUcbgajkgrUbTI

14>16. eK[ Tmn;maDdIcak;bMeBj 17.6kN / m3 / mMukkitkg = 35o /

f y = 400MPa .

f 'c = 20MPa

nig

dMeNaHRsay

1> CBaaMgmanElVgtamTisQr nigRtUv)ancat;TukfamanTRmbgb;enA)at nigTRm propped

enAkMBUl. edaycat;TukRbEvgElVg L = 2.925m dUcbgajkgrUbTI 14>16. sm<aFxag
epSgEdlmanGMeBIelICBaaMgkgRbEvg 1m mandUcteTA
sm<aFEdlekItBIsm<aFdI Pa = 14.31kN / m nig H a = 21.46kN
sm<aFEdlekItBIsm<aFTwk Pw = 15kN / m nig H w = 22.5kN
sm<aFEdlekItBIbnkelIs Ps = 2.62kN / m nig H s = 7.86kN
H a nig H w KWbNalmkBIbnkBRgayrayRtIekaN b:uEn H s )anBIbnkBRgayesI. BI
rUbTI 14>16 nigedayeRbIemKuNm:Um:g;nFwm propped EdlrgbnkRtIekaN nigbnk
BRgayesI CamYynwgemKuNbnk = 1.6 (ACI Code, Appendix C)
M u = 1.6(H a + H w )

L
L
+ 1.6 H s
7.5
8
2.925
2.925
= 1.6(21.46 + 22.5)
+ 1.6 7.86
= 32.03kN.m
7.5
8

CBaaMgTb;

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21.46 + 22.5 7.86 32.03

+
= 18.78kN
RB = 1.6

3
2 2.925

R A = 1.6(21.46 + 22.5 + 7.86 ) 18.78 = 64.13kN

m:Um:g;viCmanGtibrmaEdlekItmanenAmuxkat;EdlmankmaMgkat;sUn
14.31 + 15 x
Vu = 18.78 1.6(2.62 x ) 1.6
=0

2.925 2
2

x = 1.291m

2.62
12.94 1.2912
1.2912 +
M c = 18.78 1.291 1.6
= 15kN .m
2

2
3

2> edaysnt; = 0.01 enaH Ru = 3MPa

d=

Mu
30.03 10 6
=
= 100mm
Ru b
3 1000

kRmas;srub = 100 + 40 + 6 = 146mm . eRbIkRmas; 190mm dUcenH d = 144mm

Ru =

Mu
bd 2

30.03 10 6
1000 144 2

= 1.45MPa

PaKryEdk = 0.0045 nig As = 0.0045 1000 144 = 648mm 2

As Gb,brma = 0.0015bh = 0.0015 1000 190 = 285mm 2
As Gb,brmargkarBt; = 0.0033 1000 144 = 475.2mm 2
eRbIEdk DB16 manKMlat 300mm As = 670mm 2
3> sRmab;m:Um:g;viCman M c = 15kN .m
Ru =

15 10 6
1000 144 2

= 0.72 MPa

= 0.0022

As = 0.0022 1000 144 = 316.8mm 2 < 670mm 2

eRbIEdk DB12 manKMlat 250mm As = 452mm 2

4> m:Um:g;sUnekItmanenAcmay 2.288m BIkMBUl nig 0.637m BI)at. RbEvgRCYsrbs;Edk
DB16 KW 360mm tMbn;sgt;. dUcenH dak;Edkem DB16 RbEvg 637 mm + 360mm
= 997 mm b 1000mm BI)at rYcbnab;mkeRbIEdk DB12 KMlat 300mm enApxageRkA.
sRmab;pxagkg eRbIEdk DB12 KMlat 250mm .

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5> brimaNEdkbeNay eRbIPaKryEdkGb,brma = 0.002 ACI Code, Section 14.3

b As = 0.002 1000 190 = 380mm2 . eRbI DB12 KMlat 300mm enApTaMgsgag.
6> RbsinebIm:Um:g;Bt;enA)atnCBaaMgmantmFM enaHeKcaM)ac;RtUvkarkRmas;kRmalxN
CBaaMgFM ]TahrN_ 300mm bFMCagenH. enAkgkrNIxHeKGaceRbI haunch dUcbgaj
kgrUbTI 14>17. eKeFVIEbbenHedIm,Ikat;bnykRmas;CBaaMg edaysarEtm:Um:g;RtUv)an
KNnaenAelI muxkat;BIelI haunch Etmg.
7> kRmalxNCan;eRkamdIGacmankRmas;FMCagkRmas;CBaaMg nigGacRtUv)anBntecjeRkA
hYs pCBaaMgRbEhl 250mm bFMCagenHRbsinebIcaM)ac;.
CBaaMgTb;

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XV.

karKNnasRmab;kmaMgrmYl

Design for Torsion

15>1> esckIepIm (Introduction)

kugRtaMgrmYlekItmanenAkgmuxkat;FwmenAeBlEdlm:Um:g;manGMeBIRsbeTAnwgmuxkat;enaH.
m:Um:g;rmYleFVI[Ggt;vil nigmansameRbHenAelIprbs;va CaTUeTAEtgekItmanenAelImuxkat;mUl.
edIm,IbgajkugRtaMgrmYl eKGnuvtkmaMgrmYl T Fwm cantilever muxkat;mUlEdleFVIBI elastic
homogenous material dUcbgajkgrUbTI 15>1. kmaMgrmYlnwgeFVI[Fwmvil. cMNuc B clteTA
cMNuc B' enAxagcugrbs;Fwm b:uEncugmageTotrbs;FwmRtUv)anbgb;. mMu RtUv)aneKehAfa mMurmYl
(angle of twist). bg; AO' OB nwgdUrrageTACarag AO'OB' . edaysnt;fa Ggt;enArkSaRbEvgrbs;
vadEdl enaH shear strain KW
=

BB' r
=
L
L

Edl L CaRbEvgrbs;Fwm nig r CakaMrbs;muxkat;rgVg;.

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enAkgeRKOgbgMebtugGarem: Ggt;nwgrgm:Um:g;rmYlenAeBlGgt;enaHekagenAkgbg;/ RT
cantilever slab/ mannaTICa spandrel beam (end beam)/ bCaEpkrbs;CeNIrvil.
Ggt;eRKOgbgMGacrgnUvEtkmaMgrmYlsuT benAkgkrNICaeRcIn vargCamYyKakgeBlEtmYy
nUvkmaMgkat;TTwg nigm:Um:g;Bt;. ]TahrN_TI15>1 bgajBIkmaMgepSgEdlGacGnuvtmkelImuxkat;
epSgKanFwm cantilever.

]TahrN_TI15>1

KNnakmaMgEdlmanGMeBIenAmuxkat; !/ @ nig # nFwm cantilever EdlbgajenAkgrUbTI 15>2. Fwm

rgnUvkmaMgbBar P1 = 67kN / kmaMgedk P2 = 53.5kN EdleFVIGMeBIenAcMNuc C nigbnkedk
P3 = 89kN EdlGnuvtenAcMNuc B nigEkgeTAnwgTisedArbs;kmaMg P2 .

dMeNaHRsay

yk N = kmaMgEkg (normal force)/ V = kmaMgkat; (shear force)/ M = m:Um:g;Bt; (bending

moment)/ T =m:Um:g;rmYl (torsional moment). kmaMgTaMgGs;RtUv)anbgajenAkgtaragxageRkam
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muxkat; N (kN )
V y (kN )
M x (kN .m) M y (kN .m)
Vx (kN )
T (kN .m)
!
0
180.9
144.45
53.5
67
0
@ 53.5 sgt;
0
144.45
89
67
180.9
# 53.5 sgt; 241.2
464.85
89
67
180.9
RbsinebI P1 / P2 nig P3 CabnkemKuN Pu = 1.2PD + 1.6PL enaHral;tmenAkgtaragCakmaMg
KNnaemKuN.
15>2> m:Um:g;rmYlenAkgFwm

(Torsional Moments in Beams)

dUcbgajenAkgrUbTI 15>1 kmaMgGacGnuvtenAelIeRKagsMNg;GKar edayeFVI[manm:Um:g;

rmYl. RbsinebIkmaMgcMcMNuc P GnuvtenARtg;cMNuc C enAelIeRKag ABC dUcbgajenAkgrUb 15>3
a vabegItm:Um:g;rmYl T = PZ enAkgFwm AB Rtg;cMNuc D . enAeBl D sitenAkNalElVgnFwm
AB enaHm:Um:g;rmYlKNnaenAkgkMNat; AD esInwgm:Um:g;rmYlKNnaenAkgkMNat; DB besInwg
1
T . RbsinebIkRmal cantilever slab RtUv)anRTedayFwm AB rUbTI 15>3 b enaHkRmalxN
2
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begItm:Um:g;rmYlBRgayesI mt tambeNayFwm AB . m:Um:g;rmYlBRgayesIenH KWekItBIbnkenAelI

ceRmokTTwkmYyktarbs;kRmalxN. RbsinebI S CaTTwgn cantilever slab nig w CabnkenA
elIkRmalxN kN / m 2 enaH mt = wS 2 / 2 kN .m / m nFwm AB . m:Um:g;rmYlKNnaGtibrma
enAkgFwm AB KW T = mt L / 2 EdlGnuvtenARtg;cMNuc A nig B . krNIbnkepSgeTotRtUv)anbgaj
enAkgtarag 15>1. CaTUeTA daRkamm:Um:g;rmYlenAkgFwmmanrag nigmantmdUcKanwgdaRkamkmaMg
kat;TTwgsRmab;FwmEdlrgnUvkmaMg mt nig T .
15>3> kugRtaMgrmYl (Torsional Moments in Beams)
edayBicarNaelIFwm cantilever Edlmanmuxkat;mUl rUbTI15>1 enAeBlEdlm:Um:g;rmYl T
manGMeBIelIFwm vanwgbegIt[mankmaMgkat;TTwg dV EkgeTAnwgkaMrbs;muxkat;. BIlkxNlMnwgm:Um:g;
rmYlxageRkARtUv)anTb;edaym:Um:g;rmYlxagkgEdlmantm T esIKaEtTisedApyKa . RbsinebI dV
CakmaMgkat;TTwgeFVIGMeBIelIp dA rUbTI 15>4 enaHGaMgtg;sIuetnkmaMgrmYlKW
T = rdV

edayyk v CakugRtaMgkmaMgkat;TTwenaH
nig T = rvdA
dV = vdA
kmaMgkat;TTwgeGLasicGtibrmaekItmanenApxageRkArbs;muxkat;rgVg;Rtg;kaM r CamYynwg
kRmas; dr enaHkmaMgrmYlGacRtUv)ankMNt;edayKitm:Um:g;eFobnwgcMNuc 0 sRmab;RkLapkg
dT = (2rdr )vr

Edl 2rdr CaRkLapkg nig v CakugRtaMgkmaMgkat;TTwgenAkgkg. dUcenH

!%>!

T = (2rdr )vr = 2r 2 dr
R

sRmab;muxkat;RbehagEdlmankaMxagkg R1 /
!%>@

T = 2r 2 dr
R1

sRmab;muxkat;tan; edayeRbIsmIkar !%>! nig v = vmax r / R

R
R
v r
2
T = 2r 2 max dr =
vmax r 3dr

0
0
R
R
R4
2
=
= vmax R 3
vmax
4 2
R

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!%>#
m:Um:g;niclPaBb:UElrnmuxkat;rgVg;KW J = R 4 / 2 . dUcenH kugRtaMgkmaMgkat;GacRtUv)an
sresrCaGnuKmn_nm:Um:g;niclPaBb:UElrdUcxageRkam
TR
!%>$
vmax =
J
vmax =

2T

R 3

15>4> m:Um:g;rmYlenAkgmuxkat;ctuekaN (Torsional Moments in Rectangular Sections)

karKNnakugRtaMgenAkgGgt;manmuxkat;minmUlEdlrgbnkrmYlminsamBadUckarKNna
sRmab;muxkat;mUleT. b:uEn lTplEdlTTYlBIRTwsIeGLasic (theory of elasticity) bgajfakug
RtaMgkmaMgkat;TTwgGtibrmasRmab;muxkat;ctuekaNEkgGacRtUv)ankMNt;dUcxageRkam
T
!%>%
vmax = 2
x y
Edl

kmaMgrmYlEdlGnuvt
x = RCugxIrbs;muxkat;ctuekaN
y = RCugEvgrbs;muxkat;ctuekaN
= emKuNEdlGaRsynwgpleFobn y / x tmrbs;vaRtUv)an[enAkgtarag
xageRkam.
T=

y/x

1 .0

1 .2

1 .5

2 .0

4. 0

10

0.208

0.219

0.231

0.246

0.282

0.312

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kugRtaMgkmaMgkat;TTwgGtibrmaekItmanenAtamGkSnRCugEvg y rUbTI 15>5.

sRmab;Ggt;EdlekItBIkarpMnmuxkat;ctuekaNEkg dUcCamuxkat;GkSr L / T nig I tm
GacRtUv)ansnt;faesInwg 1/ 3 ehIymuxkat;GacRtUv)anEckecjCamuxkat;ctuekaNCaeRcInEdlman
RCugEvg yi nigRCugxI xi . kugRtaMgkmaMgkat;TTwgGacRtUv)anKNnaBI
3T
vmax =
!%>^
x2 y
i i
Edl xi2 y i CatmEdl)anBIplbUkmuxkat;ctuekaNEkgtUc. enAeBlEdl

y / x 10

eK

GaceRbIsmIkarsRmYlxageRkam
v max =

!%>&

3T

x y1 0.63

15>5> kmaMgpbrvagkmaMgkat; nigkmaMgrmYl (Combined Shear and Torsion)

enAkgkrNIGnuvtn_CaeRcIn Ggt;eRKOgbgMGacrgnUvTaMgkmaMgkat; nigkmaMgrmYlCamYyKa.
kugRtaMgkmaMgkat;GacnwgekItmanenAkgmuxkat;CamYynwgkugRtaMgkmaMgkat;mFm = v1 enAkgTis
edAnkmaMgkat; V rUbTI 15>6 a. kmaMgrmYl T begItkugRtaMgrmYlenAelIRKb;RCugrbs;muxkat;
ctuekaN ABCD rUbTI 15>6 a CamYynig v3 > v2 . karBRgaykugRtaMgcugeRkayRtUv)anTTYlBI
karbUkbBalnUv\TiBlnkugRtaMgkmaMgkat; nigkugRtaMgrmYl edIm,IbegIttmGtibrmaesI v1 + v3 enA
karKNnasRmab;kmaMgrmYl

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elIRCug CD b:uEnRCug AB nwgmankugRtaMgcugeRkayesI v1 v3 . RCug AD nig BC nwgrgEtkugRtaMg

rmYl v2 . muxkat;RtUvd)anKNnasRmab;kugRtaMgGtibrma v = (v1 + v3 ) .

15>6> RTwsIkarrmYlsRmab;Ggt;ebtug (Torsion Theories for Concrete Members)

eKmanviFICaeRcInsRmab;viPaKGgt;ebtugBRgwgedayEdkEdlrgkarrmYl brgkarrmYl karBt;
nigkarkat;kgeBlEtmYy. CaTUeTAviFIKNnasMGageTAelIRTwsIeKalBIrKW the skew bending theory
nig space truss analogy.
15>6>1> Skew Bending Theory
viFIeKalrbs; skew bending theory EdlENnaMeday Hsu CaviFIEdlsikSakar)ak;nmuxkat;
ctuekaNedaykarrmYlEdlekItedaykarBt;eFobGkSRsbeTAnwgpnmuxkat; y FMCag nigeRTteday
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mMu 45o eTAnwgGkSEvgnFwm rUbTI 15>7. QrelIviFIsaRsenH m:Um:g;rmYlGb,brma Tn GacRtUv)an

KNnadUcxageRkam

x2 y
f
Tn =
3 r

!%>*

Edl f r KWm:UDuldac;rbs;ebtug. f r RtUv)ansnt;esInwg 5 f 'c / 12 enAkgkrNIenH Edl

RtUv)aneRbobeFobCamYy 7.5 f 'c /12 EdlTTYleday ACI Code sRmab;KNnaPaBdabenAkgFwm.
kmaMgrmYlTb;edayebtugsMEdgdUcxageRkam
1 2
Tc =
!%>(
x y f 'c
x
nigkmaMgTb;karrmYledayEdkTb;karrmYlKW
1 ( x1 y1 At f y )
!%>!0
Ts =
s
dUcenH Tn = Tc + Ts Edl Tn lTPaBTb;m:Um:g;rmYl nominal nmuxkat;.
15>6>2> Space Truss Analogy

karKNnasRmab;kmaMgrmYl

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viFIsaRsn space truss analogy KWQrelIkarsnt;falTPaBTb;Tl;karrmYlrbs;ebtugGar

em:muxkat;ctuekaNRtUv)anKitecjEtBIEdknigebtugEdlBTCMuvijEdkb:ueNaH. kgkrNIenH muxkat;
thin-wall RtUv)ansnt;mannaTICa space truss rUbTI 15>8. ceRmokebtugvNeRTtcenaHsameRbH
Tb;kmaMgsgt; b:uEnEdkbeNayenARCug nigEdkkgTb;nwgkmaMgTajEdlekItedaym:Um:g;rmYl.
kareFVIkarrbs;FwmebtugGarem:EdlrgkarrmYlsuTGacbgajedayRkaPicnTMnak;TMngrvagkar
rmYlnigmMurmYl dUcbgajenAkgrUbTI15>9. eyIgemIleXIjfa munnwgeRbH ebtugTb;nwgkugRtaMgrmYl
nigEdkswgEtKanrgkugRtaMg. eRkayeBleRbH kareFVIkarrbs;FwmCalkNeGLasicminGacGnuvt)an
dUcenHmMurmYlekIteLIgPam EdlekIneLIgrhUtdl;lTPaBTb;Tl;m:Um:g;rmYlekItman. karkMNt;Edl
manlkNRbhak;RbEhlnlTPaBTb;karrmYlsRmab;muxkat;eRbHGacnwgsMEdgdUcxageRkam
A f
Tn = 2 t s x1 y1
!%>!!
s
Edl At = neCIgmagrbs;Edkkg
s = KMlatEdkkg
x1 nig y1 = RbEvgxI nigRbEvgEvg KitBIGkSeTAGkSnEdkkgbiTCit bBIEdkenARCug.

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smIkarmunecalnUvlTPaBTb;karrmYlrbs;ebtug. Mitchell nig Collins ENnaMnUvsmIkarxag

eRkamedIm,IKNnamMurmYlkgmYyktaRbEvg
P P ( tan ) 2 d
+
= o l + h h
!%>!@

2A
tan
P
sin

Edl

l =

bERmbRmYlrageFob (strain) enAkgEdkbeNay (longitudinal reinforcing

steel)

bERmbRmYlrageFobenAkgEdkkg (hoop steel)

d = bERmbRmYlrageFobebtugGgt;RTUgenARtg;TItaMgnkmaMgpbnrMhUrkmaMgkat;
h =

(shear flow)

brimaRtrbs;EdkkgKitRtwmGkSEdk

P
= mMunkmaMgsgt;Ggt;RTUg = ( d + l ) / d + h h
P
Ph =

RkLapEdlBTCMuvijedaykmaMgkat; b
= torque / 2q Edl q = rMhUrkmaMgkat;
Po = brimaRtnKngrMhUrkmaMgkat; brimaRtrbs; Ao
smIkarmMurmYlxagelImanlkNRsedogKanwgsmIkarmMukMeNagkgkarBt; rUbTI 15>10
Ao =

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c + s

!%>!#
Edl c nig s CabERmbRmYlrageFobenAkgebtug nigEdk erogKa. smIkardsamBaRtUv)an
bkRsayeday Solanki edIm,IkMNt;lTPaBTb;nwgkarrmYlsuTrbs;FwmebtugGarem: edayQrelI
space truss analogy dUcxageRkam
= curvature =

As f sy
Tu = (2 Ao )
Po

Edl

Ah f hy

!%>!$

/ nig s RtUv)anBnl;BIxagelI
As f sy = kmaMg yield nEdkbeNayTaMgGs;
Ah f hy = kmaMg yield nEdkkg
ACI Code )anTTYlykRTwsIenHedIm,IKNnaGgt;eRKOgbgMebtugEdlrgkarrmYl bkarrmYl
nigkarkat;enAkgviFIsaRsdsRmYl.
Ao Po

15>7> ersIusg;rmYlnGgt;ebtugsuT (Torsional Strength of Plain Concrete Members)

Ggt;ebtugrgkarrmYlCaTUeTARtUv)anBRgwgedayEdkTb;nwgkarrmYlBiess. kgkrNIEdlkug
RtaMgrmYlmantmtUc nigRtUvkarKNnasRmab;Ggt;ebtugsuT kugRtaMgkmaMgkat; vtc GacRtUv)an
kMNt;edayeRbIsmIkar !%>^
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vtc =

3T

x2 y

Department of Civil Engineering

f 'c
2

nigmMurmYlKW = 3TL / x3 yG / Edl T Cam:Um:g;rmYlEdlGnuvtmkelImuxkat; tUcCagm:Um:g;

rmYlEdleFVI[eRbH nig G KWCam:UDulkmaMgkat; nigGacRtUv)ansnt;esInwg 0.45 dgnm:UDuleGLasicrbs;ebtug Ec Edl G = 2135 f 'c . kmaMgkat;TTwgeFVI[eRbHedaysarkarrmYl (torsional
cracking shear) vc enAkgebtugsuTGacRtUv)ansnt;esI 0.5 f 'c . dUcenH sRmab;muxkat;ctuekaN
ebtugsuT
2
Tc =
x y f 'c
!%>!%
12
nigsRmab;muxkat;EdlpSMeLIgedayctuekaNEkgeRcIn

Tc =
f 'c x 2 y
!%>!^
12
15>8> karrmYlenAkgGgt;ebtugBRgwgedayEdk

(Torsion in Reinforced Concrete Memebers

(ACI Code Procedure))

15>8>1> sBaaNTUeTA (General)

dMeNIrkarKNnasRmab;karrmYlmanlkNRsedogKaeTAnwgkmaMgkat;TTwgedaykarBt;. enA
eBlEdlm:Um:g;rmYlemKuNGnuvtenAelImuxkat;FMCaglTPaBTb;m:Um:g;rmYlkgrbs;ebtugGacTb;)an
enaHsameRbHEdlekItedaykarrmYl (torsional crack) ekIteLIg dUcenHEdkTb;karrmYl (torsional
reinforcement) kgTRmg;CaEdkkgbiTCit (closed stirrup or hoop reinforcement) RtUv)andak;.
bEnmBIelIEdkkgbiTCit EdkbeNaykRtUv)andak;enAtamRCugrbs;Edkkg nigRtUv)anBRgayy:ag
lenACMuvijmuxkat;. TaMgEdkkgbiTCit nigEdkbeNaymansarsMxan;Nas;kgkarTb;nwgkmaMgTaj
Ggt;RTUgEdlbNaymkBIkmaMgrmYl EdkEtmYyRbePTnwgKanRbsiTPaBeTebIKanEdkmYyRbePT
eTot. EdkkgRtUvEtbiTCit edaysarkugRtaMgrmYlekItmanenARKb;RCugrbs;muxkat;.
EdkcaM)ac;sRmab;karrmYlRtUv)anbEnmelIEdkcaM)ac;sRmab;kmaMgkat; sRmab;karBt; nig
kmaMgtamGkS. EdkEdkcaM)ac;sRmab;karrmYlRtUv)andak;edIm,IeFVI[ersIusg;m:Um:g;rmYlrbs;muxkat;
Tn FMCagbesInwgm:Um:g;rmYlemKuN Tu EdlRtUv)anKNnaBIbnkemKuN.
Tn Tu
!%>!&
karKNnasRmab;kmaMgrmYl

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enAeBleKRtUvkarEdkTb;karrmYl ersIusg;m:Um:g;rmYl Tn RtUv)anKNnaedaysnt;kmaMg

rmYl Tu TaMgGs; RtUv)anTb;edayEdkkg nigEdkbeNayCamYynwgersIusg;Tb;karrmYlrbs;ebtug
Tc = 0 . kgeBlCamYyKa ersIusg;kmaMgkat;EdlTb;edayebtug vc RtUv)ansnt;enAdEdledayKan
karERbRbYledaysarvtmanrbs;ersIusg;rmYl.
15>8>2> )a:ra:Em:RtFrNImaRtnkarrmYl (Torsional Geometric Parameters)

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enAkg ACI Code, Section 11.6 karKNnasRmab;karrmYlKWQrenAelI space truss analogy

dUcbgajenAkgrUbTI 15>8. eRkayeBlEdlsameRbHedaykarrmYlekIteLIg karrmYlRtUv)anTb;
edayEdkkgbiTCit EdkbeNay nigersIusg;kmaMgsgt;Ggt;RTUgrbs;ebtug. sac;ebtugenAxageRkA
EdkkgkayeTACaKanRbsiTPaB nigRtUv)anecalenAkgkarKNna. RkLapBTCMuvijedayGkSnEdk
kgbiTCitxageRkAbMput RtUv)ankMNt;eday Aoh pqUtenAkgrUbTI 15>11. edaysarGgdTeTot
RtUv)aneRbIenAkgsmIkarKNna vakRtUv)anENnaMCadMbUgenATIenHedIm,ICYy[karyl;nUvsmIkarman
lkNgayRsYl. BIrUbTI 15>11 GgEdl[RtUv)ankMNt;dUcxageRkam
Acp = RkLapmuxkat;ebtugEdlBTCMuvijedaybrimaRtxageRkAnmuxkat;ebtug
Pcp = brimaRtnmuxkat;ebtugTaMgmUl Acp
Aoh = RkLapEdlBTCMuvijedayGkSnEdkrgkarrmYlTTwgbiTCitxageRkAbMput pqUtkgrUbTI
15>11
Ao = RkLapEdlBTCMuvijedayKngrMhUrkmaMgkat;TTwg nigGacykesInwg 0.85 Aoh
Ph = brimaRtebtugrbs;EdkrgkarrmYlTTwgbiTCitxageRkAbMput
= mMunkmaMgsgt;Ggt;RTUgcenaH 30 o eTA 60 o bGacykesInwg 45o sRmab;Ggt;ebtugGarem:
sRmab;muxkat;GkSr T nig L TTwgRbsiTPaBnsabmagRtUv)ankMNt;esInwgkm<s;FwmEdl
sitenABIelI bBIeRkamkRmalxN edayykmYyNaEdlFMCag b:uEnminRtUvFMCag 4 dgkRmas;kRmal
xNeT ACI Code, Sections 11.6.1 and 13.2.4.
15>8>3> m:Um:g;rmYleFVI[eRbH Tcr (Cracking Torsional Moment T )
m:Um:g;eFVI[eRbHeRkamm:Um:g;rmYlsuT Tcr GacRtUv)anTajecjedayCMnYsmuxkat;BitR)akd
munnwgeRbH CamYynwg thin-walled tube smmUl t = 0.75 Acp / Pcp / CamYynwgRkLapEdlBTCMuvij
edayGkSCBaaMg A0 = 2 Acp / 3 . enAeBlEdl kugRtaMgTajGtibrma kugRtaMgem mantmesI
f 'c / 3 sameRbHnwgekItman ehIyCaTUeTAm:Um:g;rmYl T esInwg
T = 2 Aot
!%>!*
Edl = kugRtaMgkmaMgkat;edaykarrmYl = f 'c / 3 sRmab;sameRbHedaykarrmYl.
CMnYs eday f 'c / 3
cr

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Tcr =

NPIC

2
f 'c Acp
= Tn
3 Pcp

nig

!%>!(

Tu = Tcr

edaysnt;fam:Um:g;rmYltUcCagbesInwg Tcr / 4 nwgmineFVI[mankarkat;bnyersIusg;Tb;karBt;

bTb;kmaMgkat;enAkgGgt;nrcnasm<n ACI Code, Section 11.6.1 GnuBaat[ecalnUv\TiBlm:Um:g;
rmYlenAkgGgt;ebtugGarem:enAeBlEdlm:Um:g;rmYlemKuN Tu Tcr / 4 b
2
f 'c Acp
!%>@0
Tu
= Ta
12 P

cp

enAeBlEdl Tu FMCagtmenAkgsmIkar !%>@0 Tu TaMgGs;RtUv)anTb;edayEdkkgbiTCit

nigEdkbeNay. m:Um:g;rmYl Tu RtUv)anKNnaBImuxkat;EdlmanTItaMgRtg;cmay d BIpnTRm nig
Tu = Tn Edl = 0.75 .

]TahrN_TI15>1

sRmab;muxkat;bIEdlbgajenAkgrUbTI 15>12 nigQrelIkarkMNt; ACI Code cUrkMNt;

m:Um:g;eFVI[eRbH Tcr
b. m:Um:g;rmYlemKuNGtibrma Tn EdlGacGnuvtelImuxkat;nImYyedaymineRbIEdkRTnugTb;kar
rmYl.
a.

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snt;

f 'c = 28MPa

DB12

f y = 400 MPa

/ kRmas;ebtugkarBarEdk 40mm nigeRbIEdkkg

dMeNaHRsay
1>

a.

muxkat; !
mU:m:g;eFVI[eRbH Tcr GacRtUv)anKNnaBIsmIkar !%>!(
Tcr =

2
f 'c Acp
3 Pcp

sRmab;muxkat;enH Acp = xo yo RkLapmuxkat;TaMgmUl Edl xo = 400mm nig

yo = 610mm

Acp = 400 610 = 244000mm 2

Pcp =

brimaRtnmuxkat;ebtugTaMgmUl

= 2( xo + yo ) = 2(400 + 610) = 2020mm

Tcr = 0.75
b.

Tn

GnuBaatEdlGacGnuvtedaymineRbIEdkTb;karrmYlRtUv)anKNnaBIsmIkar !%>@0

Ta =

2>
a.

28 244000 2
= 39kN .m
3 2020

Tcr
4

39
= 9.75kN .m
4

muxkat; @
dMbUgKNna Acp nig Pcp sRmab;muxkat;enH nigGnuvtsmIkarTI !%>!( edIm,IKNna
Tcr . edaysnt;sabRtUv)andak;CamYyEdkkgbiTCit
enaHsabRbsiTPaBEdlRtUveRbIenA RCugmagnRTnugesInwg $dgkRmas;sab b
4(100 ) = 400mm = hw = 400mm
Acp = web area + area of effective flanges
Acp = 500 350 + 2 100 400 = 255000mm 2
Pcp = 2(b + h ) = 2(350 + 2 400 + 500) = 3300mm

Tcr = 0.75

karKNnasRmab;kmaMgrmYl

28 255000 2
= 26kN .m
3 3300

427

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

cMNaM RbsinebIsabRtUv)anecal ehIyEdkTb;karrmYlRtUv)andak;EtenAkgRTnug

enaH
Acp = 350 500 = 175000mm 2
Pcp = 2(350 + 500) = 1700mm

Tcr = 23.8kN .m
b.

Tn

GnuBaatEdlGacGnuvtedaymineRbIEdkTb;karrmYl

Ta =

3>
a.

Tcr
4

26
= 6.5kN .m
4

muxkat; 3
snt;sabRtUv)andak;EdkkgbiTCit RbEcgRbsiTPaBesInwg
hw = 370mm < 4 150 = 600mm
Acp = 350 520 + 370 150 = 237500mm 2
Pcp = 2(b + h) = 2(350 + 370 + 520) = 2480mm

Tcr = 0.75

28 237500 2
= 30kN .m
3 2480

cMNaM RbsinebIsabRtUv)anecal enaH

Acp = 350 520 = 182000mm 2
Pcp = 2(350 + 520) = 1740mm

Tcr = 25.2kN .m
b.

Tn

GnuBaat Tn = T4cr = 304 = 7.5kN .m

15>8>4> m:Um:g;rmYllMnwg nwgm:Um:g;rmYlRtUvKa (Equilibrium Torsion and Compatibility Torsion)

kgkarviPaKeRKOgbgMGgt;ebtug kmaMgepSgEdlGnuvtrYmman kmaMgEkg (normal force)/
m:Um:g;Bt; (bending moment)/ kmaMgkat; (shear force) nigm:Um:g;rmYl Edl)anBnl;enAkg]TahrN_
TI 15>1. karKNnaGgt;ebtugGarem:KWQrelIkar)ak;rbs;Ggt;GMeBIrbs;bnkemKuN. sRmab;Ggt;
saTicminkMNt; (statically indeterminate member) karEbgEckm:Um:g;mgeTot (redistribution of
moments) ekItmanmuneBl)ak; dUcenHm:Um:g;KNnaGacnwgRtUv)ankat;bny b:uEn sRmab;Ggt;saTickM

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Design for Torsion

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Department of Civil Engineering

Nt; (statically determinate member) dUcCaFwmsamBa (simple beam) bFwm cantilever Kankar
EbgEckm:Um:g;mgeTotekIteLIgeT.
enAkgkarKNnaGgt;Edlrgm:Um:g;rmYl eKmanBIrkrNIEdlGacGnuvtbnab;BIkareRbH.
!> krNIm:Um:g;rmYllMnwg (equilibrium torsion case) ekItmanenAeBlm:Um:g;rmYlEdlRtUvkar
sRmab;eRKOgbgMsitkgsanPaBlMnwg ehIy Tu minGacRtUv)ankat;bnyedaykarEbg
EckeLIgvijrbs;m:Um:g;eT dUckrNIFwmTMrsamBa. kgkrNIenHEdkTb;rmYlRtUv)andak;
edIm,ITb;RKb; Tu . rUbTI 15>13 FwmEdlenAEKmRTkRmalxN cantilever EdlKankar
EbgEckm:Um:g;mgeTotekItman.
@> krNIm:Um:g;rmYlRtUvKa (compatibility torsion case) ekItmanenAeBlm:Um:g;rmYl Tu Gac
RtUv)ankat;bnyedaykarEbgEckkmaMgkgmgeTotbnab;BIeRbH enAeBlEdlPaBRtUvKa
nkMhUcRTg;RTay (compatibility of deformation) RtUv)anrkSa enAkgGgt;eRKOgbgM.
rUbTI 15>14 bgajBI]TahrN_sRmab;krNIenH EdlFwmTTwgBIrmanGMeBIelIFwmEKmbegIt
m:Um:g;rmYl. mMurmYlFMekItman enAeBlsameRbHedaykarrmYlelcecj Edlpl;nUvkar
bgEckbnkdFMenAkgeRKOgbgM. vanwgeTAdl;m:Um:g;rmYlEdleFVI[eRbH Tcr eRkamGMeBI
nbnSM karBt; karkat; nigkarrmYl enAeBlEdlkugRtaMgem (principle stress) mantm
RbEhl f 'c / 3 . enAeBlEdl Tu > Tcr m:Um:g;rmYlesInwg Tcr smIkar !%>!( Edl
Gacsnt;ekItmanenARtg;muxkat;eRKaHfak;enACitpnTRm.
ACI Code kMNt;m:Um:g;rmYlKNnaesInwgtmtUcCageKn Tu Edl)anBIbnkemKuN b Tcr
BIsmIkar !%>!(.
15>8>5> karkMNt;nersIusg;m:Um:g;rmYl (Limitation of Tortional Moment Strength)
ACI Code,Section 11.6.3 kMNt;TMhMmuxkat;edaysmIkarxageRkamBIr
!> sRmab;muxkat;tan;
2

Vu Tu Ph
Vc + 2
+

bw d 1.7 Aoh
bw d 3

karKNnasRmab;kmaMgrmYl

429

f 'c

!%>@!

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

@> sRmab;muxkat;Rbehag

Vu Tu Ph
Vc + 2

bw d 1.7 Aoh
bw d 3

f 'c

!%>@@

Edl Vc = f 'c bwd / 6 = ersIusg;kmaMgkat;sRmab;ebtugTMgn;Fmta. Ggt;dTeTotRtUv)an

kMNt;enAkgEpk 8>2.
karkMNt;enHKWQrelIPaBCak;EsgEdlfaplbUknkugRtaMgEdlbNalBIkmaMgkat; nigm:Um:g;
rmYl GgxageqVg minRtUvFMCagkugRtaMgEdleFVI[eRbHbUknwg 2 f 'c / 3 . krNIdUcKaRtUv)anGnuvt
edIm,IKNnakmaMgkat;edayKanm:Um:g;rmYlenAkgemeronTI 8. eKRtUvkarkarkMNt; (limitation) edIm,I
kat;bnysameRbH nigedIm,IkarBarEbkpebtugEdlbNalmkBIkugRtaMgkmaMgkat;TTwgeRTt nig
m:Um:g;rmYleRTt.

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Department of Civil Engineering

15>8>6> muxkat;Rbehag (Hollow Section)

bnSMnkugRtaMgkmaMgkat; nigkugRtaMgm:Um:g;rmYlenAkgmuxkat;RbehagRtUv)anbgajenAkgrUb
15>6 EdlkRmas;CBaaMgRtUvOansnt;faefr. enAkgmuxkat;RbehagxH
kRmas;CBaaMgGacERbRbYlCMuvij brimaRt. sRmab;krNIenH smIkar !%>@@
RtUv)ankMNt;enATItaMgEdlGgxageqVgmantmGtibrma. cM NaMfa enAnwgsabxagelI
nigsabxageRkam CaTUeTAkugRtaMgkmaMgkat;RtUv)anecal. CaTUeTA Rbsin
ebIkRmas;CBaaMgnmuxkat;Rbehag t tUcCag Aoh / Ph enaHsmIkar !%>@@ kayCa
V 2

Vu Tu Ph

+
c +
f 'c
!%>@#
1.7 A t
3
b d
b d

oh

(ACI Code, Section 11.6.3)

15>8>7> EdkRTnug (Web Reinforcement)

dUcEdl)anBnl;rYcehIy viFI ACI Code sRmab;KNnaGgt;Edlrgm:Um:g;rmYlKWQrelI space
truss analogy enAkgrUbTI 15>8. bnab;BIkareRbHedaykarrmYl eKRtUvkarEdkBIrRbePTedIm,ITb;nwg
m:Um:g;rmYlEdlGnuvt Tu KW EdkTTwg (transverse reinforcement) At enAkgTRmg;CaEdkkgbiTCit
nig EdkbeNay (longitudinal reinforcement) Al . ACI Code )anbgajnUvsmIkarxageRkamedIm,I
KNna At nig Al
!> EdkkgbiTCit At EdlGacKNnadUcxageRkam
2 Ao At f yt cot
!%>@$
Tn =
s
nig = 0.75
Edl Tn = Tu
At = RkLapneCIgmYyrbs;EdkkgbiTCit
f yt = ersIusg;yal (yield strength) rbs; At At 400MPa
s = KMlatEdkkg
Ao nig RtUv)ankMNt;enAkgEpk 8>2. smIkar !%>@$ GacRtUv)ansresrdUcxageRkam
At
Tn
=
!%>@%
s 2 A f cot
o yt

RbsinebI = 45o enaH cot = 1.0 nigRbsinebI

karKNnasRmab;kmaMgrmYl

431

f yt = 400 MPa

enaHsmIkar !%>@% kayCa

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

!%>@^
Edl Tn KitCa N .mm . KMlatEdkkg s minRtUvFMCagntmtUcCageKkgcMeNam Ph / 8 nig
300mm . sRmab;muxkat; RbehagrgkarrmYl cmayEdlvas;BIGkSnEdkkgeTApxagkgrbs;
CBaaMgminRtUvtUcCag 0.5 Aoh / Ph .
@> EdkbeNaybEnm Al EdlcaM)ac;sRmab;karrmYlminKYrtUcCagtmxageRkam
A f yt 2
!%>@&
cot
Al = t Ph
f
s
At
Tn
=
s 800 Ao

Rbsin = 45 nig
!%>@& kayCa
o

f yt = f y = 400MPa

sRmab;TaMgEdkkg nigEdkbeNay enaHsmIkar

A
A
Al = t Ph = 2 t ( x1 + y1 )
s
s

!%>@*
Ph RtUv)ankMNt;enAkgEpk 8>2. cMNaMfa EdkEdlcaM)ac;sRmab;karrmYlKYrRtUv)anbEnmBI
elI EdlEdlcaM)ac;sRmab;kmaMgkat; m:Um:g;Bt; nigkmaMgtamGkSEdleFVIGMeBIrYmKaCamYykmaMgrmYl.
karkMNt;epSgeTotsRmab;EdkbeNay Al mandUcxageRkam
a.
Ggt;pitEdktUcbMputsRmab;EdkbeNayKW DB10 bKMlatEdkkgelI 24 b s / 24
edayykmYyNaEdlmantmtUcCageK.
b.
EdkbeNayKYrRtUv)anBRgayCMuvijbrimaRtrrbs;EdkkgCamYyKMlatGtibrma
300mm .
c.
EdkbeNayKYrEtdak;enAkgEdkkg y:agehacNas;kdak;EdkenARKb;mMurbs;Edkkg.
EdkEdldak;enAnwgmMurbs;EdkkgRtUv)aneKrkeXIjfamanRbsiTPaBkgkarbegItersIu
sg;m:Um:g;rmYl nigkgkarkarBarsameRbH.
d.
EdkTb;m:Um:g;rmYlRtUvdak;enAcmay (bt + d ) BIcMNucEdlRTwsIRtUvkar Edl bt
CaTTwgn Epkrbs;muxkat;EdlmanEdkkgTb;kmaMgrmYl.
15>8>8> EdkTb;karrmYlGb,brma (Minimum Torsional Reinforcement)
enAkEngNaEdlEdkTb;karrmYlGb,brmaRtUvkar EdkTb;karrmYlGb,brmaRtUv)ankMNt;dUc
xageRkam (ACI Code, Section 11.6.5)
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Department of Civil Engineering

!> EdkkgbiTCitGb,brmasRmab;bnSMnkmaMgkat;TTwg nigkarrmYl emIlEpk 8>6

0.35bw s
sRmab; f 'c < 31MPa
Av + 2 At
f
yt

b s
Av + 2 At 0.063 f 'c w
f yt

sRmab;

f 'c 31MPa

!%>@(
Edl Av = RkLapeCIgTaMgBIrrbs;EdkkgEdlkMNt;)anBIkmaMgkat;
At = RkLapeCIgEtmYyrbs;EdkkgEdlkMNt;BIm:Um:g;rmYl
s = KMlatEdkkg
f yt = ersIusg;yal (yield strength) rbs;Edkkg 400 MPa
KMlatEdkkg s minKYrFMCagtmtUcCagkgcMeNam Ph / 8 nig 300mm . KMlatenHRtUvkar
edIm,IRKb;RKgsameRbH.
@> RkLapEdksrubGb,brmarbs;EdkbeNayTb;karrmYl
5 f 'c Acp At f yt

Ph
!%>#0
Al min =
f
f
s

Edl At / s minRtUvyktUcCag 173bw / f yt .

Al Gb,brmaenAkgsmIkar !%>#0 RtUv)ankMNt;edIm,Ipl;nUvGRtaGb,brmanmaDEdkTb;
kMmaMgrmYlelImaDebtug mantmRbEhl 1% sRmab;ebtugGarem:EdlrgkmaMgrmYlsuT.
15>9> segbviFIsaRsKNnaeday ACI Code (Summary of ACI Code Procedures)
viFIsaRsKNnasRmab;bnSMkmaMgkat;TTwg nigkmaMgrmYlGacRtUv)ansegbdUcxageRkam
!> KNnakmaMgkat;TTwgemKuN Vu nigm:Um:g;rmYlemKuN Tu BIkmaMgEdlGnuvtmkelIeRKOg
bgM. tmeRKaHfak;sRmab;kmaMgkat;TTwg nigkmaMgrmYlKWsitenARtg;muxkat;EdlmancM
gay d BIprbs;TMr.
@> a. eKRtUvkarEdkkmaMgkat;TTwgenAeBl Vu > Vc / 2 Edl Vc = f 'c bw d / 6 .
b. EdkTb;karrmYlRtUvkarenAeBlEdl
2
f 'c Acp
!%>@0
Tu >
12 P

cp

RbsinebIEdkRTnugRtUvkarGnuvtviFIsaRsxageRkam.
karKNnasRmab;kmaMgrmYl

433

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

#> KNnasRmab;kmaMgkat;TTwg
a. KNnaersIuisg;kmaMgkat; nominal Edlpl;edayebtug Vc . kMNt;kmaMgkat;TTwg
EdlTb;edayEdkRTnug
V Vc
Vs = u
Vu = Vc + Vs b

b. eRbobeFob Vs Edl)anKNnaCamYynwgtmGnuBaatGtibrma 2 f 'c bw d / 3 eyag

tam ACI Code. RbsinebI Vs tUcbnkarKNna EtpymkvijtMeLIgTMhMmuxkat;rbs;
ebtug.
c. EdkRTnugkmaMgkat;TTwgRtUv)anKNnadUcxageRkam
Av =

Edl

Vs s
f yt d

RkLapneCIgTaMgBIrrbs;Edkkg
s = KMlatEdkkg
EdkkmaMgkat;TTwgkgmYyktaRbEvgKW
Av =

Av
V
= s
s
f yt d
d.

RtYtBinit Av / s Edl)anKNnaCamYynwg Av / s Gb,brma

b
Av
(min) = 0.063 f 'c w
f yt
s

0.35 bw
f yt

Gb,brma RtUv)ankMNt;edaybTdaneRkambnSMnGMeBIrbs;kmaMgkat;TTwg nigkM

laMgrmYlRtUv)an[enAkgCMhanTI5
$> KNnasRmab;karrmYl
a. RtYtBinitfaetIm:Um:g;rmYlemKuN Tu begItm:Um:g;rmYllMnwg (equilibrium torsion)
bm:Um:g;rmYlRtUvKa (compatibility torsion). sRmab; equilibrium torsion eRbI Tu .
sRmab; compatibility torsion m:Um:g;rmYlKNnaKWtmtUcCageKn Tu BIbnkemKuN
nig
2
f 'c Acp
Tu 2 =
!%>!(
3 P
Av

T.Chhay

cp

434

Design for Torsion

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b.

c.

Department of Civil Engineering

RtYtBinitfaetITMhMnmuxkat;RKb;RKan;bGt;. vaTTYl)anedayeRbIsmIkar !%>@!

sRmab;muxkat;tan; bsmIkar !%>@@ sRmab;muxkat;Rbehag. RbsinebItmenAGg
xageqVgFM Cag (Vc / bwd + 2 f 'c / 3) enaHbegInmuxkat; pymkvijKNnabn.
sRmab;muxkat; Rbehag RtYtBinitfaetIkRmas;CBaaMg t tUcCag Aoh / Ph bGt;.
RbsinebIvatUcCageRbI smIkar !%>@# pymkvijeRbIsmIkar !%>@@.
kMNt;EdkkgbiTCitcaM)ac;BIsmIkar !%>@%
At
Tn
!%>@%
=
s 2 A f cot
o yt

minRtUvtUcCag 173bw / f yt . ehIy mMu Gacsnt;esI 45o / Tn = Tu / nig

= 0.75 .
snt; Ao = 0.85 Aoh = 0.85(x1 y1 )
Edl x1 nig y1 CaTTwg nigkm<s;rbs;muxkat;KitBIGkSeTAGkSEdkkg emIlrUb TI
!%>!!. sRmab; = 45o nig f y = 400MPa
At
Tn
!%>@^
=
s 800 Ao
KMlatGnuBaatGtibrma s KWtmtUcCageKn 300mm b Ph / 8 .
kMNt;EdkbeNaybEnmBIsmIkar !%>@&
A f yt 2
cot
!%>@& a
Al = t Ph
f
s
At / s

d.

EtminRtUvtUcCag
5 f 'c Acp
Al min =
12 f y

A f yt
t P

s h fy

!%>@& b

!%>@*
sRmab; = 45o nig f yt = 400MPa enaH Al = ( At / s )Ph
EdkbeNayTb;karrmYlKYrmanGgt;pity:agticesIKMlatEdkkgelI 24 b s / 24 b:uEn
minRtUvtUcCag DB10 . EdkbeNayRtUvdak;enAkgEdkkgbiTCitCamYyKMlatGtibrma
esI 300mm . y:agehaceKRtUvdak;EdkmYyedImenARKb;mMurbs;Edkkg. CaTUeTAmYy
PaKbIn Edk Al RtUv)anbEnmeTAelIEdkTaj mYyPaKbIenABak;kNalkm<s;rbs;mux
kat; nigmYyPaKbIeTotenAEpksgt;.
%> kMNt;RkLapsrubnEdkkgbiTCitEdlbNalBI Vu nig Tu .
karKNnasRmab;kmaMgrmYl

435

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

Avt = ( Av + 2 At )

!%>@(

0.35bw s
f yt

eRCIserIsEdkkgbiTCitsmrmCamYyKMlat s EdlmantmtUcCageKkgcMeNam 300mm

nig Ph / 8 .
EdkkgKYrRtUv)andak;enAcmay (bt + d ) eRkaycMNucEdlRTwsIRtUvkar Edl bt = TTwgn
muxkat;EdlTb;nwgkmaMgrmYl.

]TahrN_15>3 (Equilibrium Torsion)

kMNt;brimaNEdkRTnugcaM)ac;sRmab;muxkat;ctuekaNEkgdUcbgajenAkgrUbTI 15>15.
muxkat;rgnUvkM laMgkat;emKuN Vu = 213.5kN nigkmaMgrmYllMnwg (equilibrium torsion)
Tu = 41kN .m enATItaMg Edlmancmay d BIpnTMr. eK[ f 'c = 28MPa nig f y = 400MPa .

dMeNaHRsay

CMhanxageRkambgajBIviFIsaRskgkarKNna
1> kmaMgKNnaKW Vu = 213.5kN nig Tu = 41kN .m
2> a. EdkTb;kmaMgkat;RtUvkarenAeBl Vu > Vc / 2 .
Vc =

T.Chhay

f 'c bd =

0.75
28 (400)(520) 10 3 = 137.6kN
6

436

Design for Torsion

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Department of Civil Engineering

Vu = 213.5kN >

b.

Vc
2

= 68.8kN

eKRtUvkarEdkTb;kmaMgkat;.
eKRtUvkarEdkTb;karrmYlenAeBl
2
f 'c Acp
Tu >
= Ta
12 Pcp

Acp = xo yo = 400 580 = 232000mm 2

Pcp = 2( xo yo ) = 2(400 + 520) = 1840mm
0.75 28 (232000 )2 6
Ta =
10 = 9.7 kN .m
12 1840

Tu = 41kN .m > 9.7 kN .m

EdkTb;kmaMgrmYlRtUvkarcaM)ac;. cMNaMfa RbsinebI Tu tUcCag 9.7kN.m enaHEdkTb;kar

rmYlnwgminRtUvkar b:uEnEdkTb;kmaMgkat;RtUvkar.
3> KNnakmaMgkat;TTwg
a. Vu = Vc + Vs / Vs = 101.2kN
b. Vs (max) =
c.

2
3

f 'c bd =

2
28 (400)(520) = 733.8kN > Vs
3

Av
V
101.2 10 3
= s =
= 0.5mm 2 / m
s
f y d 400 520

eCIgBIr
eCIgmYy

Av
= 0.25mm 2 / m
2s

4> KNnasRmab;karrmYl
a. kmaMgrmYlKNna Tu = 41kN .m . KNnalkNmuxkat; edaysnt;kRmas;ebtugkarBar
Edk 40mm nigeRbIEdkkg DB12
x1 = 400 2(40 + 6 ) = 308mm
y1 = 580 2(40 + 6 ) = 488mm

CakarGnuvtn_ eKGacsnt; x1 = b 90mm nig y1 = h 90mm

Aoh = x1 y1 = 308 488 = 150304mm 2
Ao = 0.85 Aoh = 127758.4mm 2
Ph = 2(x1 + y1 ) = 2(308 + 488) = 1592mm

karKNnasRmab;kmaMgrmYl

437

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

nig cot = 1.0

RtYtBinitPaBRKb;RKan;rbs;muxkat;edayeRbIsmIkar !%>@!
= 45o

b.

Vu Tu Ph

Vc + 2

bw d 1.7 Aoh
bw d 3

Vc = 137.6kN

f 'c

nig Vc = 183.5kN
2

137600 41000000 1592

Left hand side =
= 1.82 MPa
+
400 520 1.7 150304 2
2

183500
Right hand side = 0.75
+
28 = 3.3MPa > 1.82 MPa
400 520 3

c.

muxkat;RKb;RKan;
kMNt;EdkkgbiTCitcaM)ac;EdlbNalBIkarrmYlBIsmIkar !%>@%
At
Tn
=
s 2 Ao f yt cot
Tn =

Tu

41
= 54.7 kN .m
0.75

cot = 1.0

At
54.7 10 6
=
= 0.535mm 2 / m
s 2 127758.4 400

d.

Ao = 127758.4mm 2

eCIgmYy

kMNt;EdkbeNaybEnmBIsmIkarTI !%>@&
A f yt 2
Al = t Ph
cot
s f y
At
= 0.535
Ph = 1592mm
s

f yt = f y = 400MPa

cot = 1.0

Al = 0.535 1592 = 851.72mm 2

A f yt
t Ph
12 f y
s f y
At
Acp = 232000mm 2
= 0.535
f yt = f y = 400MPa
s
5 28 (232000 )
Al (min) =
(0.535)(1592) = 427 mm 2
12 400

Al (min) =

5 f 'c Acp

lb;
5> kMNt;RkLapEdkkgsrub
A
A A
a. sRmab;eCIgmYyrbs;Edkkg vt = t + v
s
s 2s
Al = 851.72mm 2

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438

Design for Torsion

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b.

c.

Department of Civil Engineering

EdkkgEdlcaM)ac; Avt = 0.535 + 0.25 = 0.785mm 2 / m eCIgmYy

eRbIEdk DB12 RkLapmuxkat;rbs;EdkkgsRmab;eCIgmYyKW 113mm 2
113
spacing of stirrups =
= 144mm yk 140mm
0.785
KMlatGtibrma s = P8h = 1592
= 199mm b 300mm mYyNaEdltUcCag.
8
KMlatEdleRbIKW 140mm < 199mm
0.35bw 0.35 400
Avt / s Gb,brma =
=
= 0.35mm 2 / m < 0.785mm 2 / m
f
400
yt

6> edIm,IrkkarBRgayEdkbeNay cMNaMfa Al srub = 851.72mm 2 . eRbImYyPaKbIenAEpkxag

elI b 851.72 / 3 = 283.9mm 2 edIm,IbEnmenAkgEdkrgkarsgt; A's . eRbImYyPaKbIdak;enA
EpkxageRkam edIm,IbEnmBIelIEdkrgkarTaj nigEdkmYyPaKbIeTotdak;enAkm<s;Bak;
kNal.
a. RkLapEdksrubenAEpkxagelIesI 226 + 283.9 = 509.9mm 2 . eRbI 3DB16
As = 603mm 2
b. RkLapEdksrubenAEpkxageRkamesI 3078.8 + 283.9 = 3362.7mm 2 . eRbI 3DB 28
nig 2DB32 enARCugmMu As srub = 3455.8mm 2
Al srubEdleRbI = (603 226 ) + (3455.8 3078.8) = 754mm 2
c. enAkm<s;Bak;kNal eRbIEdk 2DB12 As = 226mm 2
bg;srsEdklMGitRtUv)anbgajenAkgrUbTI 15>15. KMlatEdkbeNayesInwg
230mm EdltUcCagKMlatEdkGtibrmaEdlRtUvkar 300mm 2 . Ggt;pitEdkkg DB12 Edl
eRbIFMCagGgt;pitGb,brma DB10 bKMlatEdkkgelI 24 s / 24 = 5.8mm .

]TahrN_15>4 (Compatibility Torsion)

edaHRsay]TahrN_TI 15>3 eLIgvij RbsinebIkmaMgrmYlemKuNCa compatibility torsion.

dMeNaHRsay

eyagtamdMeNaHRsaykg]TahrNITI 15>3
!> kmaMgKNnaKW V u = 213.5kN nig compatibility torsion Tu = 41kN .m
@> CMhan (a) nig (b) dUcKaenAkg]TahrN_TI 15>3. eKRtUvkarEdkRTnug.
karKNnasRmab;kmaMgrmYl

439

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NPIC

#> CMhan (c) KWdUcKa.

$> KNnasRmab;kmaMgrmYl
edaysar compatibility torsion Tu = 41kN.m enaH Tu KNnaRtUvtUvCag 41kN.m b Tcr
RtUv)an[enAkgsmIkar !%>!(
2
f 'c Acp 0.75 28 232000 2 6

10 = 38.7 kN .m
=
1840
3 Pcp
3

Tcr =

edaysarEt Tcr < Tu / eRbI Tu = 38.7kN.m . GnuvteLIgvijRKb;CMhanenAkg]TahrN_TI

15>3 edayeRbI Tu = 38.7kN.m edIm,IkMNt;famuxkat;RKb;RKan;.
At
eCIgmYy
= 0.5mm 2 / m
s
Al = 0.5 1592 = 796mm 2

eRbI Al = 852mm 2 > Al (min)

%> Avt caM)ac; = 02.5 + 0.5 = 0.75mm 2 / m eCIgmYy
s=

113
= 150.6mm
0.75

eRbI 150mm . eRCIserIsEdkbeNay nigEdkkgdUckg]TahrN_TI 15>3.

]TahrN_15>5 (L-section with Equilibrium Torsion)

FwmxagnRbBnkRmalxNrbs;GKardUcbgajenAkgrUbTI 15>16. muxkat;enAcmay d BIpnTRm

rg Vu = 235kN nig equilibrium torque Tu = 27kN.m . KNnaEdkRTnugcaM)ac;edayeRbI
f 'c = 28MPa nig f y = 400MPa sRmab;RKb;EdkEdleRbIenAkgFwm.

dMeNaHRsay

1> kmaMgKNnaKW Vu = 235kN nig Tu = 27kN.m

2> a. EdkTb;kmaMgkat;RtUvkarenAeBl Vu > Vc / 2
Vc =
Vu >

f 'c

Vc
2

bw d =

0.75 28
350 455 10 3 = 105.3kN
6

= 52.65kN

eKRtUvkarEdkkmaMgkat;TTwg
b. RtYtBinitfaetIEdkTb;karrmYlRtUvkarbGt;. snt;fasabcUlrYmkgkarTb;karrmYl RbEvg
T.Chhay

440

Design for Torsion

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

sabRbsiTPaBKW hw = 380mm < 150 4 = 600mm .

xo = 350mm
nig yo = 530mm
Acp = (350 530) + (150 380) = 242500mm 2
Pcp = 2(730 + 530) = 2520mm

BIsmIkar !%>@0

Ta =

242500 2 6
0.75
10 = 7.7kN .m
28
2520
12

Tu > Ta

muxkat;RtUvkarEdkTb;karrmYl.
3> KNnaEdkTb;kmaMgkat;TTwg
a.

Vu = Vc + Vs

235 = 105.3 + 0.75Vs

b.
c.

Vs = 173kN
2
Vs (max) =
f 'c bw d = 561.8kN > Vs
3
Av
V
173000
= s =
= 0.95mm 2 / m
s
f y d 400 455

eCIgBIr

Av 0.95
=
= 0.475mm 2 / m
2s
2

4> KNnaEdkTb;karrmYl Tu = 27kN.m

a. KNnalkNmuxkat;edaysnt; kRmas;ebtugkarBarEdk 40mm nigEdkkg DB12 .
RTnug x1 = 350 (2 40) 12 = 258mm y1 = 530 (2 40) 12 = 438mm
sab x1 = 380mm EdkkghYscUleTAkgRTnug
y1 = 150 92 = 58mm

Aoh = (58 380 ) + (258 438) = 135044mm 2

Ao = 0.85 Aoh = 114787.4mm 2
Ph = 2(58 + 380) + 2(258 + 438) = 2268mm

= 45o
b.

cot = 1.0

RtYtBinitPaBRKb;RKan;rbs;muxkat;edayeRbIsmIkarTI !%>@! Vu = 235kN /

Vc = 105.3kN / Vc = 140.4kN / Tu = 27kN .m

karKNnasRmab;kmaMgrmYl

441

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2
6
235000 27 10 2268
Left hand side =
= 2.47 MPa
+
350 455 1.7 135044 2
140400 2

28 = 3.3MPa
Right hand side = 0.75
+
350 455 3

c.

muxkat;manlkNRKb;RKan;
kMNt;EdkkgbiTCitedIm,ITb;karrmYl At / s BIsmIkar !%>@%
At
Tn
27 10 6
= 0.392mm 2 / m
=
=
s
2 Ao f yt cot 0.75 2 114787.4 400

d.

KNnaEdkbeNaybEnmBIsmIkar !%>@* sRmab;

f 'c = 400 MPa

eCIgmYy
nig cot = 1.0

A
Al = t Ph = 0.392 2268 = 889mm 2
s
Al min

BIsmIkar !%>#0 KW

Al min =

5 28 242500
889 = 447.7mm 2
12 400

karcUlrYmrbs;sabRtUv)anecaledaysarTTYl)anlTplxusKatictYc nigtmBlkm
ticCag.
5> kMNt;RkLapmuxkat;EdkkgbiTCit
A
A A
a. sRmab;eCIgmYy vt = t + v
s
s 2s
muxkat;cM)ac; Avt = 0.392 + 0.475 = 0.867mm 2 / m eCIgmYy
eRCIserIsEdk DB12 As = 113mm 2
KMlatEdkkg = 0113
eRbI 125mm
= 130mm
.867
P
2268
b. KMlatEdkGtibrma s max = h =
= 283.5mm . eRbI s = 125mm dUckarKNna.
8
8
Avt 0.35bw 0.35 350
c.
=
=
= 0.31mm 2 / m < 0.867mm 2 / m dUcenHeRbI
s
f
400
yt

DB12 @125

6> kMNt;karBRgayrbs;EdkbeNay. Al srubKW 889mm 2 . eRbImYyPaKbI b

889 / 3 = 296.3mm 2 enAEpkxagelI EpkkNal nigEpkxageRkam.
a. brimaNEdksrubenAEpkxagelI = 628.3 + 296.3 = 924.6mm 2 eRbI 3DB 20
As = 942.5mm 2
T.Chhay

442

Design for Torsion

viTasanCatiBhubeckeTskm<Ca
b.

c.

Department of Civil Engineering

brimaNEdksrubenAEpkxagelI = 2463 + 296.3 = 2759.3mm 2 eRbI 5DB28

As = 3078.8mm 2
Al srubEdleRbI = (942.5 628.3) + (3078.8 2463) = 930mm 2
eRbIEdk 2DB12 enABak;kNalkm<s; As = 226mm 2 . bg;EdklMGitRtUv)anbgajenA
kgrUbTI 15>16. KMlatEdkbeNayKW 190mm < 300mm . Ggt;pitrbs;EdkkgKW
12mm EdlFMCagGgt;pitEdk DB10 bKMlatEdkkgelI 24 s / 24 = 5.2mm .
bEnmEdkbeNay DB12 enARKb;mMurbs;EdkkgenAkgRTnugFwm nigsabFwm.

karKNnasRmab;kmaMgrmYl

443

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T.Chhay

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444

Design for Torsion

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

XVI.

FwmCab; nigeRKag

Continuous Beams and Frames

16>1> esckIepIm (Introduction)

GKarebtugBRgwgedayEdkpMeLIgedayGgt;eRKOgbgMCaeRcInRbePTdUcCa kRmalxN (slab) Fwm
(beam) ssr (column) nigeCIgtag (footing) . Ggt;eRKOgbgMTaMgenHGacRtUv)ancak;dac;eday
ELkBIKa dUcCakRmalebtugcak;eRsc (precast concrete slab) Fwmcak;eRsc nigssrcak;eRsc eday
manbnSl;TuksrsEdkedIm,IPab;KaBIGgt;eRKOgbgmM YyeTAGgt;eRKOgbgMmYyeTot edim,IeFVI[vakay
Carcnasm<nEtmYy (monolithic structure). Ggt;Edlcak;eRscRtUv)anKNnadUcGgt;eRKOgbgMenA
elITRmsamBa eTaHbICaRbePTnPaBCab;xHRtUv)anpl;[enAEpkcugrbs;va. sRmab;Ggt;Edlman
lkNEtmYy (monolithic member) PaBCab;nGgt;nImYyRtUv)anpl;[ EdleFVI[Ggt;eRKOg
bgMRtUv)anviPaKdUcrcnasm<nsaTicminkMNt; (statically indeterminate structure) .
karviPaK nigkarKNnakRmalxNmYyTisCab;RtUv)anbgajenAkgemeronTI 9 ehIyemKuNKNna
(design coefficient) nigsrsEdklMGitRtUv)anbgajenAkgrUbTI 9>8 nig 9>9. enAkgRbBnkRmal
xNmYyTis bnkkRmalxNRtUv)anbBanmkFwmTRm dUcbgajenAkgrUb 16>1 a . RbsinebIbnk
emKuNenAelIkRmalxNKW wu enaHbnkBRgayesIenAelIFwm AB nig BC kgmYyktaRbEvgKW
wu s bUknwgTmn;pal;rbs;Fwm. bnkBRgayesIenAelIFwm DE nig EF KW wu s / 2 bUknwgTmn;pal;
rbs;Fwm. bnkenAelIssr B esInwg wu Ls b:uEnbnkenAelIssr E / A nig D KW ws Ls / 2 /
ws Ls / 2 nig ws Ls / 4 erogKa.
enAkgkRmalxNBIrTisEdlRTedayFwmenARCugTaMgbYn bnkkRmalxNRtUv)anbBaneTAFwmBI
RkLapcMNuHEdlBTedaybnat; 45o dUcbgajenAkgrUbTI 16>1 b. EpkxHnbnkkRmalxN
RtUv)anbBaneTAFwmEvg AB / BC / DE nig EF BIRkLapctuekaNBay b:uEnbnkkRmalxNEdl
enAsl;RtUv)an bBaneTAFwmxI AD / BE nig CF BIRkLapRtIekaN. sRmab;kRmalxNkaer bnk
RtUv)anbBaneTAFwm EdlBTCMuvijBIRkLapRtIekaN. FwmenAxagkgTTYlbnkBIRCugTaMgBIr b:uEnFwm
xagrgbnkEtmYyxag b:ueNaH. edaysarFwmTaMgBIrTisRtUv)ancak;ebtugkgeBlCamYyKanwgkRmal
xN enaHvaRtUv)anviPaKCa FwmCab;saTicminkMNt; (statically indeterminate continuous beam).
FwmbBanbnkbneTAssr. bnkenAelIssr B esInwg wu Ls enAeBlEdl bnkenAelIssr E / A
FwmCab; nigeRKagCab;

445

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

nig D KW ws Ls / 2 / ws Ls / 2 nig ws Ls / 4 erogKa.

RkLapcMNuHsRmab;ssrnImYyRtUv)anKitBIGkSrbs;ElVgEdlenA Ek,rRKb;Tis.

16>2> m:Um:g;GtibrmaenAkgFwmCab; (Maximum Moment in Continuous Beams)

16>2> 1> eKalkarN_viPaK (Basic Analysis)
CaTUeTA karKNnam:Um:g;Bt; nigkmaMgkat;TTwgenAkgFwmCab;ebtugGarem:KWQrelIRTwsIeGLasic
(elastic theory). enAeBlmuxkat;ebtugGarem:RtUv)anKNnaedayeRbIviFIKNnaersIusg; (strength
design method) lTplEdlTTYl)annwgmineqIytbCamYynwgkarviPaKeGLasic (elastic analysis)
eT. b:uEn ACI Code min)anbBallkxNsRmab;karKNna)asic (provision for a plastic design)
b karKNnasanPaBkMNt; (limit state design) neRKOgbgMCab;ebtugGarem: RKan;EtGnuBaatkarEbg
Eckm:Um:g;eLIgvij (moment redistribution) dUcEdlnwgBnl;enAkgemeronenH.
16>2> 2> karGnuvtkardak;bnk (Loading Application)
m:Um:g;Bt;enARtg;cMNucNamYyenAkgFwmCab; vaminGaRsyEtnwgbnkEdlmanTItaMgenAelIFwmdUcKa
enaHeT b:uEnvakGaRsynwgbnkEdlsitenAelIFwmdTeTotEdr. kgkrNIbnkefr RKb;FwmTaMgGs;RtUv
T.Chhay

446

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

Etdak;bnkkgeBldMNalKa edaysarEtbnkefrmanTItaMg nigtmkMNt;. kgkrNIbnkclt b

bnkGefrEdlekIteLIgmgmal EbbbTnkardak;bnkRtUv)anBicarNay:gNaedIm,IkMNt;m:Um:g;GtibrmaenATItaMgeRKaHfak;. ExS\TiBlGacRtUv)aneRbIedIm,IkMNt;TItaMgrbs;bnkGefredIm,IKNnam:Um:g;
Gtibrma nigm:Um:g;Gb,brma. b:uEnenAkgemeronenH k,ndsamBaEdlQrelIExSekagbnk-PaBdab
(load-deflection curve) RtUv)aneRbIedIm,IkMNt;EbbbTnkardak;bnkEdlbegItm:Um:g;Gtibrma.
16>2> 3> m:Um:g;viCmanGtibrma nigGb,brmaenAkgElVg
Maximum and Minimum Positive Moments within a Span

FwmCab; nigeRKagCab;

447

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NPIC

m:Um:g;Bt;viCmanGtibrmaenAkgFwmTRmsamBaEdlrgbnkBRgayesI w KWsitenAkNalElVg Ca
2
mYynwg M = wl8 . RbsinebIcugrbs;vamYy bTaMgBIrRtUv)anbn karTb;enAcugEdlCab;nwgbegItm:Um:g;
GviCmanenAelITRm ehIyTItaMgnbnkviCmanGtibrmaGacpas;brbnicBIkNalElVg. ragdabnFwm
Cab;sRmab;kardak;bnkEtelIElVgEtmYy RtUv)anbgajenAkgrUbTI 16>2 a . PaBdabcuHeRkambgaj
fam:Um:g;viCman ehIyPaBdabeLIgelIbgajm:Um:g;GviCman. RbsinebIRKb;ElVgEdldabcuHeRkamRtUv)an
dak;bnk enaHbnknImYynwgbegInm:Um:g;viCmanenAkgElVg AB rUbTI 16>2 d . dUcenHedIm,IKNna
m:Um:g;viCmanGtibrmaenAkgElVgNamYy bnkGefrRtUv)andak;enAelIElVgenaH nigenAelIElVgqas;TaMg
sgag. m:Um:g;bnkGefremKuNEdlKNnadUcBnl;BImunRtUv)anbEnmBIelIm:Um:g;bnkGefremKuNenA
elImuxkat;dUcKaedIm,ITTYl)anm:Um:g;viCmanGtibrma.
daRkamm:Um:g;Bt;Edl)anBIbnkBRgayesIenAelIElVg AB RtUv)anbgajenAkgrUbTI 16>2 b.
PaBdab nigm:Um:g;Bt;fycuHy:agelOnCamYynwgcmayBIElVg AB Edlrgbnk. dUcenH edIm,IsRmYl
karviPaKFwmCab; m:Um:g;enAkgElVgNamYyGacRtUv)anKNnaedayBicarNaEtFwmEdlrgbnk AB nig
FwmBIreTotEdlenAsgagva nigsnt;TRmbgb;enAcugEdlqayTaMgsgag rUbTI 16>2 c .
RbsinebIElVgEdlCab;nwgElVg AB RtUv)andak;bnk ExSekagdabRtUv)anbgajenAkgrUb 16>2 e .
PabdabenAkgElVg AB nwgeLIgelI ehIym:Um:g;GviCmannwgRtUv)anbegItenAkgFwm AB . m:Um:g;GviCmanenHnwgRtUv)anbEnmeTAelIm:Um:g;viCmanEdl)anBIbnkefredIm,ITTYl)anm:Um:g;Bt;cugeRkay. dUcenH
edIm,IKNnam:Um:g;viCmanGb,brma bm:Um:g;GviCmanGtibrma enAkgElVg AB bnkGefrRtUv)andak;
enAElVgEk,rnwgElVg AB nigRKb;ElVgqas;nwgElVgEdlrgbnk rUbTI 16>2 e .
16>2> 4> m:Um:g;GviCmanGtibrmaenAelITRm (Maximum Negative Moments at Supports)
kgkrNIenH eKcaM)ac;kMNt;m:Um:g;GviCm anGtibrmaenAelITRmNamYy dUcCaTRm A rUbTI 16>3.
enAeBlEdlElVg AB RtUv)andak;bnk m:Um:g;GviCmanRtUv)anbegItenAelITRm A . dUcKa bnknElVg
AF knwgbegItm:Um:g;GviCmanenAelI A Edr. dUcenH edIm,IKNnam:Um:g;GviCmanGtibrmaenAelITRmNa
mYy bnkGefrRtUv)andak;enAelIElVgEk,rTaMgBIr nigenAelIRKb;ElVgqas;TaMgsgag rUbTI 16>3.

T.Chhay

448

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

enAkgkarviPaKeRKOgbgM (structural analysis) nFwmCab; RbEvgElVgRtUv)anKitBIGkSeTAGkS

TRmEdlRTedayTRmcugkaMbiT (knife-edge support). enAkgkarGnuvt TRmRtUv)aneFVIeLIgedayman
TMhMRKb;RKan;edIm,ITTYlbnkEdlbBanedayFwm CaTUeTAm:Um:g;eFVIGMeBIenAnwgpnTRm. edIm,IKNna
m:Um:g;enAnwgpnTRm eKGacdkedaym:Um:g;EdlesInwg Vu c / 3 BIm:Um:g;emKuNenAGkSrbs;TRm Edl Vu
KWCakmaMg kat;TTwgemKuN nig c CaTTwgssr.
16>2> 5> m:Um:g;enAkgFwmCab; (Moments in Continuous Beams)
FwmCab; nigeRKagCab;GacRtUv)anviPaKedayeRbIviFIRbhak;RbEhl (approximate method) bkm viFIkMuBTr. viFIepSgeTotdUcCa viFIbnk nigbMlas;TI (displacement and force method) nkarviPaK
EdlQrelIkarKNnanm:aRTIsnPaBrwgRkaj nigPaBrlas; (stiffness and flexibility matrices)
GacRtUvykmkeRbI. viFI slope deflection nigviFI moment-distibution kGacRtUv)aneRbI. viFITaMgenH
manBnl;enAkgesovePAEdlniyayBIkarviPaKeRKOgbgM (structural analysis) nFwm nigeRKag.
ACI Code, Section 8.3 pl;nUvemKuNRbhak;RbEhlsRmab;KNnam:Um:g;Bt; nigkmaMgkat;TTwgenAkg
Fwm nigkRmalxNCab;. emKuNTaMgenHRtUv)an[enAkgemeronTI 9. m:Um:g;EdlTTYl)anBIemKuN
ACI nwgmantmFMCagm:Um:g;EdlTTYl)anBIkarKNnaviPaKbnic. eKRtUveKarBtamkarkMNt;Edlman
EcgenAkgkareRbIR)as;emKuNTaMgenaH.

]TahrN_TI 16>1

RbBnkRmalxN-Fwm Edl)anbgajenAkgrUbTI 16>4 RTnUvbnkGefrBRgayesI 6.2kN / m 2

nigbnkefrEdlrYmmanbnkpal;rbs;kRmalxNbUknwg 3.8kN / m 2 . edayeRbIemKuNm:Um:g; ACI
KNnaFwmCab;enAxagkg nigKUrmuxkat;lMGit. eK[ f 'c = 28MPa / f y = 400MPa / TTWgFwm
b = 300mm / muxkat;ssr 300 300mm nigkRmas;kRmalxN 125mm .

FwmCab; nigeRKagCab;

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dMeNaHRsay

1> KNnakRmalxN kRmalxNmannaTICakRmalxNmYyTis edaysarpleFobRCugEvgelI

RCugxIFMCag 2 . karKNnanRbePTkRmalxNCab;RtUv)anBnl;enAkg]TahrN_TI 9>4.
2> bnkenAelIkRmalxN
bnkefr Dead load = 0.125 25 + 3.8 = 6.925kN / m 2
bnkGefr Live load = 6.2kN / m 2
bnkemKuN Ultimate load (wu ) = 1.2 6.925 + 1.6 6.2 = 18.23kN / m 2
bnkenAelIFwm Fwmxagkg ABC RTbnkkRmalxNTaMgsgag
CamYynwgTTwgkRmalxNsrub 3.6m
bnkemKuNenAelIFwm
Factored load on beam = 3.6 18.23 + 1.2(self - weight of beam web )

km<s;rbs;FwmGacRtUv)ankMNt;edayeRbIemKuNnkRmas;FwmGb,brmadUcEdl)anbgajenA
kgtarag]bsm<n B.6. sRmab; f y = 420MPa km<s;FwmGb,brmarbs;FwmTI1 AB KW

T.Chhay

450

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

. snt;km<s;Fwmsrub 550mm km<s;RTnugFwmKW

550 125 = 425mm . dUcenHbnkemKuNenAelIFwm ABCD KW
L / 18.5 = 7.2 / 18 = 0.389m

wu = 3.6 18.23 + 1.2(0.425 0.3) 25 = 69.45kN / m

yk wu = 69.5kN / m
3> m:Um:g;enAkgFwm ABC emKuNm:Um:g;RtUv)anbgajenAkgrUbTI 9>8. FwmCab;Kaman 5 ElVg
ehIysIuemRTIKaeFobGkSkNalRtg;cMNuc D . dUcenH eKcaM)ac;KNnaEtFwmBak;kNal
ABCD BIeRBaHFwmBak;kNaleTotnwgmanTMhMehIynwgbrimaNEdkdUcKa. edaysarElVg
AB nig BC minesIKaEtmanpleFob 7.9 / 7.2 mantmtUcCag 1.2 enaHemKuNm:Um:g; ACI
GacGnuvt)ansRmab;FwmenH. elIsBIenHRbEvg clear span mFmEdlenAEk,rRtUv)aneRbI
edIm,IKNnam:Um:g;GviCmanenAelITRm.
m:Um:g;enAmuxkat;eRKaHfak;RtUv)anKNnadUcxageRkam rUbTI 16>4
M u = coefficient wu ln2

TItaMg
emKuNm:Um:g;
M u kN .m

1
16

206.8

@
+

1
14

236.35

1
10

365.3

$
+

1
16

250.9

1
11

364.9

^
+

1
16

250.9

4> kMNt;TMhMFwm nigsrsEdk

a. m:Um:g;GviCGtibrmaKW 365.3kN .m . edayeRbI max = 0.016 enaH Ru = 5MPa / = 0.9
d=

Mu
365.3 10 6
=
= 493.5mm
Ru b
5 300

sRmab;EdkmYyRsTab; km<s;FwmsrubKW 493.5 + 60 = 553.5mm yk 600mm

d EdleRbIR)as;BitR)akdKW 600 60 = 540mm
As = 0.016 300 540 = 2592mm 2

eRbI 4DB30 mYyRsTab;

FwmCab; nigeRKagCab;

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NPIC

cMNaMfa km<s;FwmEdleRbIenATIenHKW 600mm EdlFMCagkm<s;Fwmsnt;EdlesInwg 550mm

edIm,IkMNt;Tmn;rbs;Fwm. edaysarbnkbEnmmantmtUcGacecal)an eKmincaM)ac;eFVIkar
KNnaeLIgvijeT.
b. muxkat;enATRmeFVIGMeBICamuxkat;ctuekaNEkgCamYyEdkTajenAkgsab.
EdkcaM)ac;enAkgTRm mandUcxageRkam
TItaMg
!
#
%

c.

M u (kN .m)

206.8

365.3

364.9

Ru ( MPa)

2.36

4.17

4.17

(%)

0.74

1.39

1.39

As (mm 2 )

1199

2252

2252

cMnYnEdk DB30

sRmab;muxkat;GkSr T enAkNalElVg M u = 250.9kN.m . sRmab; a = 25mm

nigTTwgsab
b f = 1.8m
As =

Mu
250.9 10 6
=
= 1321.2mm 2
a 0.9 400(540 12.5)

f y d
2

As f y
1321.2 400
a a=
=
= 12.34mm
0.85 f 'c b 0.85 28 1800

RtYtBinit
CamYy a = 12.34mm eKTTYl)an As = 1305.5mm 2 . dUcenHeRbIEdk 3DB25
As = 1472.6mm 2 sRmab;RKb;muxkat;kNalElVgTaMgGs;. bg;EdklMGitRtUv)anbgaj
enAkgrUbTI 16>5.
5> KNnaFwmsRmab;kmaMgkat;TTwg dUcEdl)anBnl;enAkgemeronTI 8.
6> RtYtBinitPaBdab nigsameRbH dUcEdl)anBnl;enAkgemeronTI 9.

T.Chhay

452

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viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

16>3> eRKagsMNg;GKar (Building Frames)

eRKagGKarKWCaRbBneRKOgbgMbITMhM kglMh (three-dimensional structural system) EdlpM
eLIgeday Ggt;Rtg;Edlsagsg;CamYyKa (built monolithically) nigmantMNrwg. eRKagGacman
mYyElVg nigmankm<s;mYyCan; dUcCa portal frame nig gable frame dUcbgajenAkgrUbTI 16>6 a b
GacpSM eLIgedayeRcInElVg nigeRcInCan; dUcbgajenAkgrUbTI 16>6 b. RKb;Ggt;rbs;eRKagRtUv)an
cat;Tuk faCab;sRmab;TisTaMgbI ehIyssrcUlrYmCamYynwgFwmedIm,ITb;bnkxageRkA. eRkABIkarkat;
bnym:Um:g; edaysarPaBCab; eRKagGKarcg;)ankarBRgaybnkkan;EtesIkan;Etl. \TiBlnbnk
xag dUcCabnkxl; nigbnkrBaydI RtUv)anBRgayenAelIeRKagTaMgmUl edIm,IbegInsuvtiPaBrbs;va.
sRmab;karKNna viFIRbhak;RbEhl (approximate method) GacRtUv)aneRbIedaysnt;RbBneRKag
BIrTMhM kgbg;.
eRKagEdlrgnUvRbBnbnkGacRtUv)anviPaKedayviFIeRKagsmmUl (equivalent frame method).
enAkgviFIenH karviPaKtamCan;KWRtUveFVIeLIgedaysnt;facugssrEdlenAqayBIelI nigBIeRkamnIv:U
kRmalxNRtUv)anbgb; rUbTI 16>7. CaTUeTA karviPaKRtUv)aneFVIeLIgedayeRbIviFIEbgEckm:Um:g;
(moment-distribution method).

FwmCab; nigeRKagCab;

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NPIC

enAkgkar TMhMkRmal (panel) cmaycenaHssr/ cMnYnCan; nigkm<s;nCan;nImYyRtUv)ansal;BI

eRBaHBYkvaRtUv)anpl;[edaykarKNnasabtkm ehIyRbBnbrikaRtUv)anKitrYcCaeRsc. TMhMssr
nigTMhMFwmRtUv)aneFVIkar)a:n;sanCamun ehIypleFobPaBrwgRkajrbs;vaEdlQrelImuxkat;ebtug
RtUv)aneRbI. enAeBlm:Um:g;RtUv)ankMNt; eKRtUvRtYtBinitmuxkat;Edl)ansnt;BImun nigeFVIkarEktRmUv
RbsinebIcaM)ac;. karviPaKEdlmanlkNsuRkitCagGaceFVIeLIgedayeRbIkmviFIkMuBTrEdlRtUv)an
ENnaMsRmab;karviPaK eRKOgsaTicminkMNt;EdlmanGBaateRcIn. viFInkarviPaKRtUv)anBNnaenAkg
esovePACaeRcInEdlniyayBIkarviPaKeRKOgbgM.

T.Chhay

454

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viTasanCatiBhubeckeTskm<Ca

16>4>

Department of Civil Engineering

Portal Frames

pSMeLIgedayFwmebtugGarem:EdlrwgRtUv)ancak;CamYynigTRmssrrbs;va. tMN
rvagFwm nigssrRtUv)anBicarNabgb;rwgCamYynwgplbUkm:Um:g;enARtg;tMNesIsUn. Portal frames
RtUv)aneRbIenAkgsMNg;salEdlmanElVgFM (large-span halls)/erag (sheds)/ s<an nigs<ankat;
RClgPM (viaducts). Ggt;EpkxagelIrbs;eRKagGacedk (portal frames) beRTt (gable frames)
rUb TI 16>8. eCIgeRkamrbs;eRKagGacCaTRmbgb; bTRmsnak;.
Portal frames saTicminkMNt;GacRtUv)anviPaKedayviFIEbgEckm:Um:g; bviFIdTeTotEdleRbI
edIm,IviPaKeRKagsaTicminkMNt;. Ggt;rbs;eRKagRtUv)anKNnasRmab;m:Um:g; kmaMgkat;TTwg nigkmaMg
Ekg b:uEneCIgtagRtUv)anKNnaedIm,IRTkmaMgEdlmanGMeBIenA)atssr.
Fwm nigssrrbs;eRKagGacmanTMhMesI bERbRbYl dUcbgajenAkgrUbTI 16>8. kmaMgenAkg
portal frames EdlmanmYyElVgehIymanmuxkat;esIGacRtUv)anKNnadUcxageRkam
Portal frames

16>4> 1> cugsnak;BIr (Two Hinged Ends)

kmaMgenAkgGgt;rbs; portal frames Edlmancugsnak;BIrGacRtUv)anKNnaedayeRbIsmIkar
xageRkam rUbTI 16>9
sRmab;krNIbnkBRgayesIenAelIGgt; BC / yk
I
h
K = 3 + 2 2
I1 L

nig I 2 = m:Um:g;niclPaBrbs;ssr nigFwm

h nig L = km<s; nigRbEvgElVgrbs;eRKag
m:Um:g;Bt;enAtMN B nig C KW
Edl

I1

M B = MC =

wL2
4K
2

m:Um:g;viCmanGtibrmaenAkNalElVg BC = wL8 + M B
kmaMgRbtikmtamTisedkenARtg;cMNuc A KW H A = M B / h = H D
kmaMgRbtikmtamTisQrenARtg;cMNuc A KW V A = wL / 2 = VD
sRmab;bnkBRgayesIenAelIEtBak;kNalFwm BC rUbTI 16>9 b
FwmCab; nigeRKagCab;

455

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

M B = MC =

HA = HD =

wL2
8K

MB
h

V A = 3wL / 8

nig

VD = wL / 8

16>4> 2> cugbgb;BIr (Two Fixed Ends)

kmaMgenAkgGgt;rbs; portal frames Edlmancugbgb;BIrGacRtUv)anKNnaedayeRbIsmIkar
xageRkam rUbTI 16>10
sRmab;krNIbnkBRgayesIenAelIGgt; BC / yk
I
h
K1 = 2 + 2
I1 L
M B = MC =

T.Chhay

wL2
4K
456

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

MB
2
wL2
M (midspan) =
+ MB
8
3M A
HA = HD =
2

MA = MD =

nig V A = VD = wL2
sRmab;bnkBRgayesIenAelIEtBak;kNalFwm BC
I
h
K 2 = 1 + 6 2
I1 L
2
1
wL 1

MA =
8 3K1 8 K 2
MC =

wL2 2
1

8 3K1 8 K 2

HA = HD =
VA =

FwmCab; nigeRKagCab;

MB =

1
wL2 2

+
8 3K1 8 K 2

MD =

wL2 1
1

8 3K1 8 K 2

1
wL2

8
K1 h

wL
VD
2

nig

VD =

457

wL
1
1

8 4 K 2

T.Chhay

mhaviTalysMNg;sIuvil

16>5> eRKagTUeTA

T.Chhay

NPIC

(General Frames)

458

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

rUbragcMbgrbs;eRKagKWtMNrwg EdlPab;FwmdMbUledk beRTteTAnwgGgt;eRKOgbgMEdlRTva.

PaBCab;rvagGgt;nwgEbgEckm:Um:g;Bt;EdlmanCab;CamYynwgRbBnbnkeTAGGt;eRKOgbgMepSgeTot
eTAtampleFobnPaBrwgRkajrbs;va. eRKagGacRtUv)ancat;cMNat;fak;dUcxageRkam
- eRKagsaTickMNt; (statically determinate frame) rUbTI 16>11 a
- eRKagsaTicminkMNt; (statically indeterminate frame) rUbTI 16>12
- eRKagsaTicminkMNt; CamYynwgGgt;cMNg (statically indeterminate frame with tie) rUbTI
16>13

FwmCab; nigeRKagCab;

459

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NPIC

16>6> karKNnasnak;rbs;eRKag (Design of Frame Hinges)

RbePTdsMxan;nsnak;EdleRbIenAkgeRKagebtugKW Mesnager hinges, Considre hinges
nig lead hinges.
16>6>1> Mesnager hinges
CaTUeTA kmaMgEdlmanGMeBIenAelIsnak;KWkmaMgedk H nigkmaMgbBar P . kmaMgpbn
kmaMgTaMgBIr R RtUv)anbBaneTAeCIgtagtamryEdkExVg A nig B dUcEdl)anbgajenAkgrUbTI
16>14. PaBeRTtrbs;r)ar A nig B eTAnwgGkSedkERbRbYlBI 30o eTA 60o CamYynwgcmay
Gb,brma a Edlvas;BIEpkeRkameKrbs;eRKagssr esInwg 8D Edl D CaGgt;pitrbs;Edk
eRTt. cenaH y rvageRKagssr nigEpkxagelIrbs;eCIgtagERbRbYlcenaHBI 25mm eTA 1.3h' Edl
h' Ca TTwgrbs;muxkat;ebtugenAnIv:Usnak;. km<s;cenaHsRmab;karGnuvtn_ERbRbYlBI 50mm eTA
100mm . karvilrbs;cugeRKagRtUv)aneFVIeLIgedaysnak; ehIyCaTUeTAcenaHRtUv)anbMeBjeday
bituminous cork nigsmarEdlmanlkNbt;EbnRsedogKa. Bitumen karBar cork kMu[xUceday
sarkarb:HCamYydI. EdkExVg A nig B rgnUvkugRtaMgsgt;EdlminRtUvFMCag 1/ 3 nersIusg;yal
(yield strength) rbs;Edk f y eRkambnkeFVIkar (service load) b 0.55 f y eRkambnkemKuN
T.Chhay

460

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

. kugRtaMgtUcRtUv)ansnt;edaysarkarvilenAsnak;cg;Bt;Edk nigegItkugRtaMgBt;TI
BIr (secondary flexural stresses). CaTUeTA eKcaM)ac;RtUvrkSakugRtaMgsgt;[tUclmRbesIrCagKNna
kugRtaMgTIBIr. RkLapEdk A nig B RtUv)anKNnadUcxageRkam
RkLapmuxkat;Edk A As1 = 0.55R1 f
!^>!
(factored load)

RkLapmuxkat;Edk B As2 = 0.55R2 f

!^>@

Edl R1 nig R2 CabgMnkmaMgpb R enAkgTisedAnEdkeRTt A nig B edayeRbIbnkem

KuN. CaTUeTA bgMkmaMg R1 nig R2 RtUv)anTTYledaysmIkarsaTicxageRkam
H + R2 sin = R1 sin
nig R2 = R1 sinH
!^># a
dUcKa (R1 + R2 )cos = Pu
dUcenH R1 = cosPu R2 = cosPu R1 sinH
H
1 Pu
+

2 cos sin

!^># b
Edksnak;eRTtbBankmaMgrbs;vatamrycMNgenAtamRbEvgEdkbgb;eTAkgssreRKag nig
eCIgtag. dUcenH EdkeRTtbeBajkmaMgFak;ecjeRkA EdlRtUvTb;edayEdkcMNg. EdkcMNgRtUv)an
dak;elIcmay a = 8D Ggt;pitEdkEdlFMCageKkgcMeNam A nig B TaMgenAkgssr nigeCIg
tag. kmaMgFak;ecj (bursting force) GacRtUv)ankMNt;dUcxageRkam
P
Ha
F = u tan +
!^>$
2
0.85d
RbsinebIkarcUlrYmrbs;ebtugRtUv)anecal enaHRkLapmuxkat;rbs;EdkcMNg Ast EdlcaM)ac;
Tb;kmaMg F KW
F
F
!^>%
Ast =
=
f
0.85 f
R1 =

kugRtaMgenAkgEdkcMNgkGacRtUv)ankMNt;dUcxageRkam
Pu
Ha
tan +
0.85d 0.85 f
f s ( tie) = 2
y
0.005ab + Ast ( tie)

Edl

!^>^

RkLpmuxkat;Edkkgkgcmay a = 8D
d = km<s;RbsiTPaBrbs;muxkat;ssr
Ast =

FwmCab; nigeRKagCab;

461

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mhaviTalysMNg;sIuvil

NPIC

TTwgrbs;muxkat;ssr
snak;RbePTenHRtUv)aneRbIsRmab;kmaMglm nigRtUv)ankMNt;edaycMnYnEdkeRTtGtibrma
EdlGacRtUv)andak;enAkgTTwgssr.
b=

16>6>2> Considre hinges

PaBxusKarvag Considre hinges nig Mesnager hinges KWfakmaMgEkg Pu RtUv)ansnt;fa
vaRtUv)anbBaneTAeCIgtagedaykUnssrEdkkgvN (spirally reinforced short columns) mYybeRcIn
Edldak;bBaleTAkgeCIgtag b:uEnkmaMgedk H RtUv)ansnt;Tb;edayEdkeRTt A nig B rUbTI
!^>%. lTPaBRTbnkrbs;kUnssrEdkkgvNGacRtUv)ankMNt;edayeRbIsmIkar !^>& edayecalem
KuN 0.85 sRmab;cMNakpitGb,brma.
Pu = Pn = 0.70[0.85 f 'c (Ag Ast ) + Ast f y ]
!^>&
T.Chhay

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Department of Civil Engineering

Edl Ag CaRkLapmuxkat;rbs;muxkat;snak;ebtug b bh' / nig Ast CaRkLapmuxkat;Edk

beNayenAkgEdkkgvN. EdkcMNgRtUv)andak;enAkgssrrhUtdl;cmayEdlesInwgRCugEvgrbs;
muxkat;ssr h .

16>6>3> Lead hinges

eBlxH Lead hinges RtUv)aneRbIenAkgeRKagssrebtugGarem:. enAkgsnak;RbePTenH CaTU
eTA lead plate EdlmankRmas;BI 20mm eTA 25mm RtUv)anebIedIm,IbBankmaMgtamGkS Pu eTAeCIg
tag. kmaMgedk H RtUv)anTb;edayEdkbBarEdldak;enARtg;GkSrbs;ssr nigRtUv)andak;bBal
eTAkgssr rUbTI 16>16. enARtg;)atrbs;ssr kmaMgEkg Pu minRtUvFMCagersIusg;Tb;
(0.85 f 'c A1 ) EdlkMNt;eday ACI Code, Section 10.15 Edl = 0.65 nig A1 = bh' . RkLa
pmuxkat;EdkbBarKW As = H / 0.6 f y Edl H = kmaMgedkemKuN.

FwmCab; nigeRKagCab;

463

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NPIC

]TahrN_16>2

salmYymanTMhM 25.2 12m RtUv)anRKbedaykRmalebtugGarem:EdlRTeday portal frame Edl

mansnak;enATRm. Portal frame manKMlatBIGkSeTAGkS 3.6m rUbTI 16>17. km<s;eRKagKW
4.5m ehIyKanssrenAkgsaleT. bnkefrenAelIkRmalxNKW)anBIbnkpal;bUknwg 3.6kN / m 2
Edl)anmkBIkargarbegIyrbs;dMbUl. bnk efrenAelIkRmalxNKW 4kN / m 2 . KNnaeRKagenA
xagkgeday eRbI f 'c = 28MPa nig f y = 400MPa sRmab;eRKag ehIymuxkat;ssrmanTTwg
b = 400mm .

dMeNaHRsay

karKNnaeRKagemnGKarrYmmandUcxageRkam
- KNnakRmalxNmYyTis
- viPaK Portal frame
- KNnaFwmeRKagEdlbNalmkBIm:Um:g;
- KNnaFwmeRKagEdlbNalmkBIkmaMgkat;
- KNnassr
- KNnasnak;
- KNnaeCIgtag
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464

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

1> kRmalxNmYyTis kRmas;Gb,brmarbs;kRmalxNTI1KW L / 30 edaysarcugmagCab;

nigcugmageTotminCab; tarag B.6 enAkg]bsm<n B.
km<s;Gb,brma = 3600
= 120mm
30
karKNnakRmalxNGnuvttamCMhanEdlmanenAkg]TahrN_9>5.
2> viPaK Portal frame enAxagkg
a. bnkenAelIkRmalxNKW
bnkefrenAelIkRmalxN = 3.6 + 0.12 25 = 6.6kN / m 2
bnkemKuNenAelIkRmalxN = 1.2 6.6 + 1.6 4.0 = 14.32kN / m 2
b. kMNt;bnkenAelIeRKag eRKagxagkgRTbnkBIkRmalxN 3.6m
bEnmBIelIbnkpal;rbs; FwmeRKag. snt;km<s;rbs;FwmKW
L / 24 = 12000 / 24 = 500mm . edayeRbIkm<s;BIeRkam kRmalxN 400mm
enaHkm<s;srubrbs;FwmKW 520mm .
bnkefrBIbnkpal;rbs;Fwm = 0.4 2 25 = 4kN / m
bnkemKuNsrubenAelIeRKag = 14.32 3.6 + 1.2 4 = 56.352kN / m
FwmCab; nigeRKagCab;

465

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c.

NPIC

dUcenH wu = 56.5kN / m
kMNt;m:Um:g;niclPaBnmuxkat;Fwm nigmuxkat;ssr. FwmeFVIkarCalkNmuxkat;GkSr T .
TTwgRbsiTPaBrbs;kRmalxNEdleFVIskmPaBCamYynwgFwmKWCatmtUcCageKn 1/ 4
n RbEvgElVg = 12 / 4 = 3m / 16hs + bw = 16 0.12 + 0.4 = 2.32m nig 3.6m . yk
b = 2.32m . TIRbCMuTmn;nmuxkat;BIsrsxagelIeKKW
y=

2320 120 60 + 400 400 320

= 154.89mm
2320 120 + 400 400

2320
400
120 3 + 2320 120 94.89 2 +
+ 400 2 165.112
I b (beam) =

12
12

= 9336 10 6 mm 4

vaCakarGnuvtTUeTAedayKitm:Um:gn; iclPaBRbhak;RbEhlrbs;FwmGkSr T esInwg 2 dg

nm:Um:g;niclPaBrbs;muxkat;ctuekaNEkgEdlmankm<s;srubCaplbUkrvagkm<s;RTnugFwm
nigkRmas;kRmalxN
I b (beam) = 2

400 520 3
= 9374 10 6 mm 4
12

sRmab;Fwmxag m:Um:g;niclPaBRbhak;RbEhlKW I = 1.5 bh3 /12 .

snt;muxkat;ssresI 400 500mm EdlmanTTwgdUcTTwgFwm
I c (column) =
d.

400 500 3
= 4167 10 6 mm 4
12

KNnaemKuN K
9374 10 6
I
h
4500
= 4.68
K = 3 + 2 b = 3 + 2
+
6
12000
4167
10

L Ic

tamrUbTI 16>17 nigsRmab;bnkBRgayesI wu = 56.5kN / m enAelIGgt; BC

M B = MC

wu L2
56.5 12 2
= 434.6kN .m
=
=
4K
4 4.68

m:Um:g;viCmanGtibrmaenAkNalElVg BC esInwg
wu

L2
56.5 12 2
+ MB =
434.6 = 582.4kN / m
8
8

kmaMgRbtikmedkenARtg;cMNuc A KW
HA = HD =

M B 434.6
=
= 96.6kN
h
4.5

kmaMgRbtikmbBarenARtg; A KW
T.Chhay

466

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Department of Civil Engineering

wu L
+ weight of column
2
56.5 12
VA =
+ 0.4 0.5 4.5 25 = 361.5kN
2

V A = VD =

e.

f.

daRkamm:Um:g;Bt;RtUv)anbgajenAkgrUbTI 16>17.
edIm,IBicarNa\TiBl sideway enAelIeRKag bnkGefrRtUv)andak;enAelIEtBak;kNalFwm
BC ehIym:Um:g;RtUv)anKNnaenARtg;muxkat;eRKaHfak;. krNIenHmineRKaHfak;enAkg]TahrN_enHeT.
kmaMgkat;TTWgGtibrmaenAcugTaMgBIrrbs;Fwm BC ekItmanenAeBlFwmRtUv)andak;bnkCa
mYynwgbnkemKuN wu b:uEnkmaMgkat;TTwgGtibrmaekItmanenAeBlEdlFwmRtUv)andak;
bnkCamYynwgEtBak;kNalbnkGefr nigbnkefreBj
56.5 12
= 339kN
2
L
12
Vu (at midspan) = W1 = 1.7 4 3.6 = 36.72kN
8
8

Vu (at support ) =

g.
h.

kmaMgtamGkSenAkgssrnImYyKW V A = VD = 361.5kN
ykcMNucm:Um:g;sUnenAkgFwm BC sitenAcmay x BI B / enaH
x
x2
M B = wu L wu
2
2
x
x2
434.6 = 56.5 12 56.5
2
2
x = 1.46m

x 2 12 x + 15.384 = 0

BITRm B

3> KNnaFwm BC
a. KNnamuxkat;eRKaHfak;enAkNalElVg. M u = 582.4kN .m / TTwgRTnugKW bw = 400mm
TTwgsabKW b = 2320mm nig d = 520 90 = 430mm snt;eRbIEdkBIrRsTab;.
RtYtBinitfaetImuxkat;eFVIkarCaragmuxkat;ctuekaNCamYynwgTTwgRbsiTPaB
b = 2320mm bGt;? snt; a = 25mm enaH
582.4 10 6
= 3874.9mm 2
a
25
f y (d ) 0.9 400 430
2
2

As f y
3874.9 400
a=
=
= 28mm < 120mm
0.85 f 'c b 0.85 28 2320
As =

FwmCab; nigeRKagCab;

Mu

467

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

snt;mantmRbhak;RbEhl a KNna. muxkat;eFVIkarCamuxkat;ragctuekaN dUcenH

eRbIEdk 6DB28 . RtYtBinit bmin edIm,Idak;EdkmYyCYr
a

bmin = 11 28 + 2 10 + 80 = 408mm > 400mm

dak;EdkBIrCYr/ bgajenAkgrUbTI 16>18.

b. KNnamuxkat;eRKaHfak;enARtg; B M u = 434.6kN .m / b = 400mm nig d = 520 60
= 460mm sRmab;EdkmYyRsTab;. kRmalxNrgkarTaj dUcenHEdkRtUv)andak;enAkg
muxkat;EpkxagelI.
Ru =

Mu
bd 2

434.6 10 6
400 460 2

= 5.135MPa

= 0.0166 < max = 0.018

As = 0.0166 400 460 = 3054.4mm 2

eRbI 5DB28 kgmYyCYr.

4> KNnaFwm BC edaysarkmaMgkat;TTwg
a. muxkat;eRKaHfak;enAcmay d BIpssrCamYycmayBIGkSssr 250 + 460 = 710mm
dUcenH
Vu (at distance d) = 339 - 56.5 0.71 = 298.9kN
b.

ersIusg;kmaMgkat;Edlpl;edayebtug
Vc =

f 'c bw d =

0.75
28 400 460 10 3 = 121.7 kN
6

kmaMgkat;Edlpl;edayEdkRTnug (web reinforcement) KW

Vs = Vu Vc = 298.9 121.7 = 177.2kN
Vs =
c.

d.

177.2
= 236.3kN
0.75

eRCIserIsEdkkg DB12 nig Av = 2 122 4 = 226mm 2

460
dUcenH s = AvVf y d = 226236 400
= 176mm yk 175mm
.3 10 3
s
KMlatGtibrmanEdkkg DB12 KW
d
s max = = 230mm
yk 225mm
2
400
b smax = 0A.35v fby = 0226
= 646mm
.35 400
w

T.Chhay

468

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

RtYtBinitsRmab;KMlatGtibrman d2 Vs 13
Vs

f 'c bw d

28
400 460 10 3 = 324.5kN
3

tm Vs = 236.3 < 324.5kN . dUcenHeRbI smax = 225mm .

V ' s (for s max = 220mm) =

Av f y d
s

226 400 460 3

10 = 184.8kN
225

V ' s = 0.75 184.8 = 138.6kN

cmayBIpssreTATItaMgEdl smax = 225mm RtUv)aneRbIesInwg 1.56m

e. karBRgayEdkkg
EdkkgTImYymanKMlat s / 2 = 75mm
7 EdkkgmanKMlat 175mm = 1225mm
19 EdkkgmanKMlat 225mm = 4275mm srub 5575mm
cmayBIpssreTAGkSrbs;FwmesInwg 6000mm 250mm = 5750mm . BRgayEdkkg
dUcKasRmab;Bak;kNalFwmEdlenAsl; nigdak;EdkkgmYyeTotenAkNalElVg. cMnYnEdk
kgsrubKW 55 kg.
5> KNnamuxkat;ssrenAcMNuc B M u = 434.6kN .m / Pu = 339kN / b = 400mm nig
h = 500mm

a.

snt;eRkamGMeBIbnkEdl[eRKagnwgminrgnUv sideway enaH\TiBlrbs;PaBrlas;

(slenderness ration) GacRtUv)anecal nigssrGacRtUv)anKNnaCassrxIenAeBl
KLu
12 M 1
emIlEpk 12>5
< 34
4
M2
M1 = 0
nig M 2 = 434.6kN .m
yk K = 0.8 rUbTI 12>2 Lu = 4.5 0.252 = 4.24m nig r = 0.3 500 = 150mm enaH
KLu
4.24
= 0.8
= 22.6 < 34
r
0.15

RbsinebI K RtUv)ansnt;esInwg 1.0 enaH

KLu
= 28.3 < 34
r

b.

dUcenH/ karKNnaGgt;CassrxI.
dMeNIrkarKNnaRsedogKanwg]TahrN_TI 11>16 nig 11>3.

FwmCab; nigeRKagCab;

469

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

.6
cMNakpit (e) = MP u = 434
= 1.282m
339
u
vaCacMNakpitFM nigvaRtUv)ansnt;Camuxkat;enAkgtMbn; transition region/ < 0.9
d = 500 60 = 440mm

c.

edaysar e = 1.282m KWFMCag d qayNas;/ kMNt;tmRbhak;RbEhl As nig A's EtBI

M u nigbnab;mkRtYtBinitmuxkat;cugeRkayedaysaTic Edl)anBnl;enAkg]TahrN_
11>3. sRmab; M u = 434.6kN .m / b = 400mm / h = 500mm nig d = 440mm /
Ru = M u / bd 2 = 434.6 10 6 / 400(440 )2 = 5.61MPa

nig As = bd = 3220.8mm 2
eRCIserIs 3DB28 nig 2DB32 As = 3456mm 2 .
eRCIserIs A's = As / 3 = 3220.8 / 3 = 1074mm 2 nig 3DB22 A's = 1140mm 2 rUbTI
16>18. enAeBlcMNakpit e mantmFM/ vaCakarGnuvtFmta A's = As / 3 b As / 2
CMnYs[ As = A's .
RtYtBinitlTPaBnmuxkat;cugeRkayedayeRbI As = 3456mm 2 nig A's = 1140mm 2
Rsedognwg]TahrN_ 11>3 GaRsyeTAtamCMhanxageRkam
= 0.0183

d.

i.

Pn = Cc + C s T
Cc = 0.85 f 'c ab = 0.85 28 400a = 9520a
C s = A' s ( f ' s 0.85 f 'c ) = 1140(400 0.85 28) = 428868kN
T = As f y = 3456 400 = 1382400kN
Pn = 9520a + 428868 1382400 = 9520a 953532

ii.

(I)

Kitm:Um:g;eFobRtg; As
Pn =

1
a
Cc d + C s (d d ')

e'
2

Edl d " sitenAcmay As eTATIRbCMuTmn;nmuxkat;. TIRbCMuTmn;ekItman

enAcmay 276.68mm BIsrssgt;xageRkAbMput nig d " = d x = 163.32mm
e' = e + d "

e' = 1282 + 163.32 = 1445.32mm

1
a

Pn =
9520a 440 + 428868(440 60 )

1445.32
2

= 3.3a 2 + 2898.2a + 112757

T.Chhay

470

(II)

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca
iii.

iv.

Department of Civil Engineering

KNnasmIkar I nig II eKTTYl)an a = 150mm nig Pn = 474.5kN .

epgpat; f 's = 600(c d ')/ c f y / c = a / 0.85 = 176.47mm / enaHeyIgTTYl)an
f ' s = 600(176.47 60 ) / 176.47 = 396 MPa EdlmantmEk,r 400 MPa dUc
Edl)ansnt;kgkarKNna. eRCIserIsEdkkg DB10 EdlmanKMlat 400mm .
eptpat; d t = 440mm
dt c
440 176.47
0.003 =
0.003 = 0.0045
176.47
c

t =

250
= 0.858
3

= 0.65 + ( t 0.002)

Pn = 0.858 474.5 = 407 kN > 339kN

muxkat;RKb;RKan;.
6> epgpat;muxkat;ssrenAkm<s; 2.25m Bak;kNalBI A . M u = 434.6 / 2 = 217.3kN .m
Pu = 339 + 11.25 = 350.25kN

eRbIEdk As = 3DB28 nig A's = 3DB22 . tamviFIedaHRsayRsedogKakgCMhan 5>

Pn = 632.8kN > 350.25kN Edk DB32 Gacpac;)an EtRtUvBntRbEvgbgb;BIeRkam
Bak;kNalkm<s;ssr.
7> KNnasnak;enARtg;cMNuc A M u = 0 / H = 96.6kN / Pu = 361.5kN
a. eRCIserIssnak; Mesnager hinge . edayeRbIsmIkar 16.3 a nig 16.3b/ R1 = 305.3kN
nig R2 = 112.1kN eyagtamrUbTI 16>19 CamYynwg = 30o
As1 =

R1
305300
=
= 1387.7mm 2
0.55 f y 0.55 400

eRCIserIs 3DB25 As = 1472.6mm 2

As 2 =

R2
112100
=
= 509.5mm 2
0.55 f y 0.55 400

eRCIserIs 2DB22 As = 760mm 2 . teRmobEdkExVgedaydak; DB25 nigbnab;mkdak;

DB 22 dUcEdlbgajenAkgrUbTI 16>19 beRbI 5DB 25

FwmCab; nigeRKagCab;

471

T.Chhay

mhaviTalysMNg;sIuvil

b.

NPIC

EdkcMNgTTwgKYrdak;tambeNaycmay a = 8D = 8 25 = 200mm enAkgssr nigeCIg

tag. kmaMgFak;KW
F=

Pu
Ha
tan +
2
0.85d

= 30 o d = 440mm

T.Chhay

nig a = 200mm
472

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca
F=

Department of Civil Engineering

96.6 200
361.5
tan 30 o +
= 156kN
2
0.85 440
156000
=
= 458.8mm 2
0.85 400

RkLapmuxkat;EdkcMNg
RbsinebIEdkcMNgbiTCit DB10 eCIgBIr RtUv)anerIs enaHRkLapmuxkat;KW 157mm 2 .
.8
cMnYnEdkcMNgKW 458
= 2.9 . eRbIEdkcMNg 3 kgEdlmanKMlat 200 / 2 = 100mm
157
dUcbgajenAkgrUbTI 16>19.
8> KNnaeCIgtag RbsinebIkm<s;eCIgtagRtUv)ansnt;esI h' enaHkmaMgEdlmanGMeBIelIeCIgtag
CakmaMgEkg P nigm:Um:g; M = H / h' . sMBaFdIRtUv)anKNnaBIsmIkar !#>!$ BIemeronTI 13
q=+

P Mc

allowable soil pressure

A
I

viFIsaRsKNnaeCIgtagmanlkNRsedogKaeTAnwg]TahrN_TI 13>7.

16>7> esckIepImBIkarKNnasanPaBkMNt;
16>7>1> lkNTUeTA (General)
FwmCab; nigeRKagCab;

(Introduction to Limit Design)

473

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

karKNnasanPaBkMNt;neRKOgbgMmanbICMhandUcxageRkam
- karkMNt;bnkKNnaemKuN EdlTTYledayKuNbnkefr nigbnkGefrCamYynwgemKuNbnk.
ACI Code TTYlnUvemKuNdUcEdl[enAkgemeronTI3.
- viPaKeKOgbgMeRkamGMeBIbnkemKuN (ultimate or factor load) edIm,IkMNt;m:Um:g;emKuN nig
kMmaMgemKuNenAeBlEdleKOgbgMdYl b)ak;. viFInkarviPaKenHRtUv)aneRbIsRmab;karKNna
eRKOgbgMEdk EtsRmab;eRKOgbgMebtugGarem: ACI Code min)anTTYlykviFIenHTaMgGs;eT
edaysarkarxVHPaBsVit (ductility) rbs;Ggt;ebtugGarem:. ACI Code GnuBaatEtEpknkar
EbgEckeLIgvijnUvm:Um:g;edayEpk (partial redistribution of moment) enAkgrcnasm<neday
QrelIPaKryEdl)anmkBIkarBiesaFn_ EdlnwgBnl;kgemeronenH.
- KNnaGgt;nImYyneRKOgbgMedaysnt;fava)ak;edaym:Um:g;emKuN nigkmaMgemKuNEdl)an
mkBIkarviPaKeRKOgbgM. viFIenHRtUv)aneRbIR)as;CaTUeTAsRmab;karKNnaebtugGarem: ehIy
ACI Code GnuBaatnUvkareRbIR)as;viFIKNnalTPaBRTRTg; (strength design method)
dUcEdl)an Bnl;kgemeronkngmk.
16>7> 2> KMnitnkarKNnasanPaBkMNt; (Limit Design Concept)
karKNnasanPaBkMNt;enAkngebtugGarem:sMedAelIkarEbgEckm:Um:g;eLIgvijEdlekItman
enAelIeRKOgbgMTaMgmUl dUcEdlEdkBRgwgenAmuxkat;eRKaHfak;eFVIkardl; yield strength rbs;va.
ersIusg;cugeRkay (ultimate strength) rbs;eRKOgbgMekIneLIgenAeBlmuxkat;rbs;vaekIneLIg. eTaH
bICakaryal (yielding) rbs;EdkeFVI[manPaBdabFM EdlKYrEteCosvageRkambnkeFVIkar (service
load) eRKOgbgMsaTicminkMNt;mindYlb)ak;enAeBlEdlEdkBRgwgnmuxkat;TImYyyal. elIsBIenH
ersIusg;EdlbMrugTuky:ageRcInsitenAcenaHkaryaldMbUg nigkar)ak;rbs;eRKOgbgM.
kgkarKNnaeRKOgbgMEdk BakkarKNna)asic (plastic design) RtUv)aneRbIedIm,Ibgajkar
pas;brkgkarEbgEckm:Um:g;enAkgeRKOgbgMdUcCa steel fiber enARtg;muxkat;eRKaHfak; rgkugRtaMg
rhUtdl;ersIsg;yal. karekIneLIgnUvkugRtaMgtamkm<s;rbs;muxkat;EdkeRkamkarekIneLIgrbs;bnk
RtUv)anbgajenAkgrUbTI 16>20. Portal frame RtUv)anBiesaFn_edIm,IGegtlTPaBRTRTg;karvil
rbs;snak;)asicrbs;ebtugGarem:. b:uEn ACI Code tRmUv[mankarsikSaRsavRCavCaeRcIneTotmun
nwgTTYlykkarKNnasanPaBkMNt;.
T.Chhay

474

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

16>7> 3> eKalkarN_nsnak;)asic (Plastic Hinge Concept)

ExSkMeNag rbs;Ggt;ekIneLIgCamYynwgm:Um:g;Bt; M . sRmab;FwmebtugEdlmanbrmaN
Edktic daRkamm:Um:g;-ExSkMeNag nigExSekagbnk-PaBdab RtUv)anbgajenAkgrUbTI 16>21. Fwm
EdlmanbrimaNEdk balanced nigbrimaNEdkeRcInminRtUv)anGnuBaat[eRbIeday ACI Code eT
edaysarva)ak;edaykarpHEbkebtug nigbgajnUvRbeLaHExSkMeNagtUcenAeBlm:Um:g;emKuN rUbTI
16>22.
EpksMxan;rbs;daRkamExSrkMeNag-m:Um:g;enAkgrUbTI 16>21 KWsitenAcenaHcMNuc B nig C
Edlm:Um:g; M u rkSatmefrsRmab;RbeLaHntm dFM. kgkarviPaKsanPaBkMNt; ExSekagExSkM
eNag-m:Um:g;GacRtUv)ansnt;manTRmg; idealized form dUcbgajenAkgrUbTI 16>23 EdlExSkM
eNag enAcenaH B nig C RtUv)ansnt;faefr edIm,IbegItrUbragsnak;)asic. edaysarebtugCa
smarRsYy CaTUeTAeK)anKitnUvEdnkMNt;edal[Ggt;)ak;enARtg;ExSkMeNagGtibrmaRtg; C .
elak Cranston )anbgajfaeRKagebtugGarem:EdlKNnaCaTUeTA GacmanlTPaBTb;nwgkar
vil)anFM ehIyExSkMeNagGtibrmaenARtg;cMNuc C nwgminGaceTAdl; Tal;EteRKag)ak;sin. dUcenH
enAeBlGgt;RTm:Um:g;esInwgm:Um:g;emKuN M u rbs;va ExSkMeNagbnekIneLIgcenaHcMNuc B nig C
edayKankarERbRbYlm:Um:g; edIm,IbegItsnak;)asic. karekIneLIgnExSkMeNagGnuBaat[EpkdT
eTotneRKOgbgMsaTicminkMNt;edIm,ITTYlbnkbEnm.

FwmCab; nigeRKagCab;

475

T.Chhay

mhaviTalysMNg;sIuvil

T.Chhay

NPIC

476

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

16>8> emkanicnkar)ak; (The Collapse Mechanism)

kgkarKNnasanPaBkMNt; Ggt;ebtugGarem:GaceTAdl;ersIusg;cugeRkay (ultimate
strength) enAeBlvaCitnwgrlM b)ak;. Ggt;dYlrlMenAeBlEdlcMnYnsnak;)asicRKb;RKan;bMElgva
[cUleTACaFwmemkanicnkar)ak;. cMnYn caM)ac;nsnak;)asic n GaRsyeTAnwgdWeRk redundancy r
rbs;eRKOgbgM. TMnak;TMngrvag n nig r edIm,IbegItFwmemkanicnkar)ak;KW
n = 1+ r

]TahrN_ enAkgFwmTRmsamBaKanGBatielIseT Edl r = 0 . dUcenH FwmkayeTACaKansirPaB

ehIy)ak;enAeBlEdlsnak;)asicmYyekItmanenARtg;muxkat;nm:Um:g;Gtibrma dUcbgajenAkgrUbTI
16>24 a. karGnuvtn_eTAelIFwm nigeRKagkRtUv)anbgajenAkgrUbTI 16>24.
16>9> eKalkarN_nkarKNnasanPaBkMNt; (Principles of Limit Design)
eRkambnkeFVIkar (working load) karEbgEckm:Um:g;enAkgeRKagsaTicminkMNt;KWQrelIRTwsI
eGLasic ehIyeRKOgbgMTaMgmUlenAEtrkSaenAkgEdneGLasic. enAkgkarKNnasanPaBkMNt; enA
eBlEdlvaeTAdl;emkanicnkarrlM karEbgEckm:Umg: ;enAeBl)ak;edaybnkemKuN xusKaBIkarEbg
EckedayQrelIRTwsIeGLasic. karxusKaenHbgaj[eXIjnUvkarEbgEckm:Um:g;eLIgvij (moment
redistribution).
sRmab;karKNnasanPaBkMNt;manny lkxN 4 RtUvEteKarBdac;xat
- lkxNemkanicnkar)ak; (Mechanism condition) snak;)asicRKb;RKan;RtUv)anbegIteLIg
edIm,IbMElgeRKOgbgMTaMgmUl bEpkxHrbs;eRKOgbgMeTACaGgt;emkanicnkar)ak;.
- lkxNlMnwg (Equilibrium condition) karEbgEckm:Um:g;RtUvEtmanlMnwgCamYynwgbnkEdl
Gnuvt.
- lkxNyal (Yield condition) m:Um:g;emKuNminRtUvelIsersIusg;m:Um:g;enARKb;TItaMgrbs;
eRKOgbgM.
- lkxNrgVil (Rotation condition) snak;)asicRtUvEtmanlTPaBTb;rgVil (rotation
capacity)RKb;RKan; edIm,IGnuBaatkarekItmanFwmemkanicnkar)ak;.

FwmCab; nigeRKagCab;

477

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

manEtlkxNbIdMbUgeTEdlGnuvtkgkarKNna)asic edaysarlTPaBTb;rgVilmanenAkg
smarsVitdUcCaEdkrYceTAehIy. lkxNTIbYnbegInEdnkMNt;bEnmeTAelIkarKNnakMNt;nGgt;eb
tugGarem:edayeRbobeFobCamYykarKNna)asic (plastic design).

T.Chhay

478

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

16>10> Ednx<s; nigEdnTabnemKuNbnk (Upper and Lower Bounds of Load Factors)

eRKOgbgMEdlCitdYlrlMRtUvEtmancMnYnsnak;)asicRKb;RKan;edIm,IbMElgvaeTACaGgt;emkanic
nkar)ak;. sRmab;TItaMgNamYynsnak;)asicenAelIeRKOgbgM bnkEdleFVI[rlMGacRtUv)anKNna
EdlGacFMCag besIbnkBit. edaysarbnkKNnaminGacFMCagbnkEdleFVI[rlMBitsRmab;eRKOg
bgM enaHviFIsaRsKNnabgajnUvEdnsIueNm:aTic bEdkx<s; (upper or kinematic bound) rbs;bnk
EdleFVI[rlMBit. dUcenHRKb;Ggt;emkanicnkarrlMEdlGacekItmanRtUv)aneFVIkarGegt M u EdltUc
CageK nwgekIteLIgedaybnkBit. elak Horne )anBnl;Ednx<s;edaysnt;eRKOgbgMCaGgt;emka
nicnkar)ak; nigbnab;mkKNnakmnxageRkA We EdleFVIedaybnkxageRkA nigkmnkg Wi EdleFVI
enAsnak;)asic. RbsinebI We = Wi enaHbnkEdlGnuvtFMCag besIbnkEdleFVI[)ak;.
RbsinebIdaRkamm:Um:g;enARtg;cMNucNamYyRtUv)anbegItedIm,IbRgb;lMnwgsaTiceRkamGMeBIbn
kGnuvtenAeBl)ak; enaHbnkEdlGnuvttUcCag besIbnkEdleFVI[)ak;Bit. sRmab;daRkamm:Um:g;
epSgeTot eKGacTTYlbnkemKuNepSgeTot. tmEdlFMCagnEdntUc bEdnsaTic (lower or static
bound) RtUv)anTTYlenAeBlm:Um:g;enAmuxkat;CaeRcInsRmab;daRkamm:Um:g;Edlsnt;xiteTACitm:Um:g;
EdleFVI[rlM. elak Horne )anBnl;EdnTabedaysnt;karEbgEckm:Um:g;xusKaedIm,ITTYlnUveRKOg
bgMEdlsitenAkgsanPaBlMnwgCamYybnkxageRkA nigeKarBlkxNyalenAelIeRKOgbgMTaMgmUl.
kgkrNIenH bnkxageRkAtUcCag besIbnkEdleFVI[rlM.
16>11> karviPaKsanPaBkMNt; (Limit Analysis)
sRmab;karviPaKedaydMeNIrkarKNnasanPaBkMNt; viFIBIrGacRtUv)aneRbI viFIkmnCak;Esg
(virtual work method) nigviFIlMnwg (equilibrium method). sRmab;viFI virtual work kmnRtUv)an
eFVI edaybnkemKuN Pu b wu edIm,IbegItPaBdab virtual deflection Edl[ RtUv)andak;
[esIkmnEdlRsUbykenAsnak;)asic. kmnxageRkAEdleFVIedaybnkKW We = (wu ) b
(Pu ) . kmnEdlRsUbykedaysnak;)asicKWkmn = Wi = (M u ) .

]TahrN_16>3

FwmEdlbgajenAkgrUbTI 16>25 RTnUvbnkcMcMNucenAkNalElVg. KNnam:Um:g;EdleFVI[)ak;enA

muxkat;eRKaHfak;.
FwmCab; nigeRKagCab;

479

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dMeNaHRsay
1>

T.Chhay

FwmCaGgt;saTicminkMNt; 1 dWeRk ehIycMnYnsnak;EdlRtUvkaredIm,IbMElgFwmeTACaFwm

emkanicnkar)ak;KW n = 1 + 1 = 2 snak;)asic enARtg;cMNuc A nig C . snak;)asicTI
mYyekItmanenARtg;cMNuc A ehIy FwmeFVIkarCaFwmTRmsamBarhUtdl;vaxiteTArk
emkanicnkar)ak;.

480

Continuouse Beams and Frames

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2>

Department of Civil Engineering

RbsinebImMurgVil ekItmanenAsnak;)asicenAcugbgb; A enaHmMurgVilenARtg;cMNuc

C = 2 . PaBdabenARtg;cMNuc C eRkambnkKW (L / 2 ) rUbTI 16>25.
L
We = external work = Pu = Pu

Wi = internal work = M u = M u1 ( ) + M u 2 (2 )

RbsinebImuxkat;BIrenARtg; A nig C manTMhM nigbrimaNEdkdUcKa enaH

M u1 = M u 2 = M u nig Wi = 3M u . eday[ We = Wi
PL
L
M u1 + 2 M u 2 = Pu = 3M u
ni
g
Mu = u
6
2

]TahrN_16>4

KNnam:Um:g;EdleFVI[)ak;enAmuxkat;eRKaHfak;sRmab;FwmEdlbgajenAkgrUbTI 16>26 EdlbNal

edaybnkBRgayesI wu .

dMeNaHRsay
1>
2>

cMnYcsnak;)asicKW 2 .
sRmab;PaBdabenARtg; C = 1.0 mMurgVilenARtg;cMNuc A A = 1 / a nig B = 1 / b ehIy
c = A + B =

3>

a+b L
=
ab
ab

kmnxageRkAKW We = wu = wu 12 L = w2u L
kmnxagkgKW Wi = M u = M u1 A + M u 2 C
1
1 1
+ M u2 +
a
a b
M
M
2M
wu = u1 + u 2 + u 2
L a
a
La

= M u1

dak;[ We nig Wi
RbsinebIm:Um:g;TaMgsgagesIKa enaH
wu =

FwmCab; nigeRKagCab;

2M u
L

2
1 2M u 2 L a
a + (L a ) = L a (L a )

481

!^>(
!^>!0

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mhaviTalysMNg;sIuvil

T.Chhay

NPIC

482

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

4>

Department of Civil Engineering

edIm,IKNnaTItaMgsnak;)asicenARtg; C EdlbegIttmGb,brmanbnkEdleFIV[)ak;
wu eFVIedrIevsmIkarTI !^>( eFobnwg a nigdak;[vaesIsUn
wu
=0
a

RbsinebI M u1

M
M
M u 2
=0
2u1 + u2 2
a
a
(L a )2

2
1
= M u2 = M u
=
2
a
( L a )2

enaH

b
a = L(2 2 ) = 0.586 L
BIsmIkar !^>!0 bnkEdleFVI[)ak;KW wu = 11.66(M u / L2 ) nigm:Um:g;EdleFVI[)ak; KW
M u = 0.0858wu L2 . RbtikmenAcMNuc A KW 0.586 wu L nig RbtikmenARtg;cMNuc B
KW 0.414wu L .
enAkgviFIlMnwg lMnwgrbs;Fwm brbs;Ggt;dac;edayELkrbs;FwmRtUv)ansikSaeRkambnk
EdlbgajenAeBl)ak;. edIm,IbgajkarviPaKedayviFIenH ]TahrN_BIxagmuxTaMgBIr
RtUv)aneFVIeLIgvijenATIenH.

]TahrN_16>5

sRmab;FwmEdlbgajenAkgrUbTI 16>25 kMNt;m:Um:g;EdleFVI[)ak;edayeRbIviFIlMnwg.

dMeNaHRsay

snak;)asicBIrRtUv)anbegItenAcMNuc A nig C . edayeyagtamrUbTI 16>25 e

kmaMgRbtikmenAcMNuc A KW (Pu / 2) + (M u1 / L) nig RbtikmenAcMNuc B KW (Pu / 2) (M u1 / L)
KitlMnwgnFwm BC nigKitm:Um:g;eFobcMNuc C /
Pu M u1 L

= M u 2
L 2
2
L
M u1 + 2M u 2 = Pu
2

EdlRsedogKanwgsmIkar !^>#. enAeBl M u1 = M u 2 = M u / enaH

3M u = Pu

L
2

M u = Pu

FwmCab; nigeRKagCab;

L
2

483

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NPIC

]TahrN_16>6

KNnam:Um:g;EdleFVI[)ak; (collapse moment) sRmab;FwmEdlbgajenAkgrUbTI 16>26 edayviFI

lMnwg.

dMeNaHRsay
1>

snak;)asicBIrRtUv)anbegItenARtg;cMNuc A nig C . eyagtamrUbTI 16>26 d /

kmaMgRbtikmenA Rtg; A = wu (L / 2) + (M u1 / L) nigkmaMgRbtikmenARtg;
B = wu (L / 2) (M u1 / L ) . bnkenAelI BC KW wu b EdlmanGMeBIenA b / 2 BI B Edl
b = L a . edayKitlMnwgnkMNat; BC nigKitm:Um:g; eFobcMNuc C /
b
L M

wu u1 b (wu b ) = M u 2
2
2
L

RbsinebI M u1 = M u 2 = M u enaH
b
(L b ) = M u (1 + b ) = M u (2 L a )
2
L
L
2M u (2 L a )
wu =

L
a (L a )
wu

EdlRsedogKanwgsmIkar !^>$
Mu =

2>

wu L a(L a )

(2 L a )
2

TItaMg a GacRtUv)ankMNt;dUcBImun Edl a = 0.586L / M u = 0.0858wu L2 nig

wu = 11.66(M u / L2 ).

16>12> mMurgVilrbs;snak;)asic (Rotation of Plastic Hinges)

16>12>1> RbEvgsnak;)asic (Plastic Hinge Length)
karsnt;rbs;RTwsIeBlfamMurgVilminEmneGLasic (inelastic rotation) nebtugekItmanenA
cMNucm:Um:g;GtibrmaenAeBlEdlEpkdTeTotrbs;Ggt;eFVIkarCalkNeGLasic. tamBit mMurgVil
)asic (plastic rotation) ekItmanenATaMgsgagnmuxkat;m:Um:g;GtibrmaelIRbEvgkMNt;mYy. RbEvg
enHRtUv)aneK[eQaHfa RbEvgsnak;)asic l p . RbEvgsnak;)asic l p CaGnuKmn_eTAnwgkm<s;
RbsiTPaB d nigcmayBImuxkat;nm:Um:g;x<s;bMputeTAcMNucm:Um:g;sUn (contra-flexure) .

T.Chhay

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Department of Civil Engineering

eyagtamrUbTI 16>27 a RbEvg l p / 2 tag[RbEvgsnak;)asicenAelIEpkmagnGkS rbs;

TRm. M u nig u Cam:Um:g;emKuN nigkMeNagemKuNenARtg;muxkat;eRKaHfak; Edl M y nig y Ca
m:Um:g; nigkMeNagenARtg;cMNucyaldMbUg (first yield). kMeNag)asicenARtg;muxkat;eRKaHfak; p esI
(u y ) ehIylTPaBTb;rgVilesInwg ( p l p ).

RbEvgsnt;rbs;snak;)asicRtUv)ansnt;famanRbEvgRbEhlkm<s;RbsiTPaB.
Corley )anesInUvsmIkarxageRkamsRmab;RbEvgsmmUlnsnak;)asic
z
l p = 0.5d + 0.2 d
d

!^>!!
Edl z = cmaynmuxkat;eRKaHfak;eTAcMNucm:Um:g;sUn nig d = km<s;RbsiTPaBrbs;muxkat;.
Mattock )anesInUvTRmg;dsmBa
l p = 0.5d + 0.05 z
!^>!@
FwmCab; nigeRKagCab;

485

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mhaviTalysMNg;sIuvil

NPIC

karBiesaFCaeRcInenAelIebtugGarem:Cab;bgajfa l p GacRtUv)ansnt;esInwg 1.06d . vak)an

bgajEdrfaRbEvgnsnak;)asicenAkgFwmebtugGarem:Cab;EdlmanTMBk;enAxagcugekIneLIgCamYynwg
nwgkarekIneLIgnbrimaNEdk fibers nigEdkemGaRsynwgsmIkarxageRkam
l p = (1.06 + 0.13 s )d
!^>!#
Edl = PaKryEdkemenAkgmuxkat; nig s = PaKryn fibers EdkedaymaD/
0 s 1.2 . ]TahrN_ RbsinebI = 1% nig s = 0.8% enaH l p = 1.164d .
16>12>2> emKuNEbgEckkMeNag

T.Chhay

(Curvature Distribution Factor )

486

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

emKuNsMxan;mYyeTotEdlTak;TgkgkarKNnamMurgVil)asicKWemKuNEbgEckkMeNag . kM
eNagtambeNaysnak;)asicERbRbYly:agFM ehIyenAkgkarKNnamMurgVilemKuNenHminRtUv)anKit
EdleFVI[karKNnasnak;)asicmantmFM. eyagtamrUbTI 16>27 RkLapqUt ABC tag[mMurgVil
minEmneGLasicEdlGacekItmanenAsnak;)asic b:uEnRkLapGt;qUt EBF tag[karcUlrYmn
lkNeGLasicrbs;mMurgVilelIRbEvgrbs;Ggt;. RkLapqUt ABC GacRtUv)ansnt; esInwgplKuN
rvag nigRkLapsrub ABCD enAkgRbEvgsnak;)asic l p / 2 EdlenAelIRCugmag rbs;muxkat;
eRKaHfak;. emKuNEbgEckkMeNag CapleFobnmMurgVil)asicBit pc elI l p Edl CakMeNag
nig l p CaRbEvgnsnak;)asic. tm sitenAcenaH 0.5 nig 0.6 . karBiesaF)anbgajfa
GacRtUv)ansnt;esInwg 0.56 . enAeBlEdlEdk fibers EdlmanTMBk;cugRtUv)aneRbIenA kgFwmebtug
tm fycuHGaRsyeTAnwgsmIkarxageRkam
= 0.56 0.16 s
!^>!$
Edl s CaPaKry fibers Edk/ 0 s 1.2% . karfycuHnemKuNEbgEckkMeNagnebtugEdl
eRbIEdk fibers min)anbgajfalTPaBTb;rgVilfycuHeT. kMeNag)asicrbs;ebtugEdleRbIEdk fibers
EtgEtFMCagkMeNag)asicrbs;ebtugEdlmineRbIEdk fibers. rUbTI 16>28 bgajBIkarEbgEckkM
eNagtambeNayRbEvgsnak;)asic. RkLap ABC1 tag[mMurgVil)asicsRmab;ebtugEdlmin
manEdk fibers/ = 0.56 b:uEnRkLap ABC 2 nig ABC3 tag[mMurgVil)asicsRmab;ebtugEdl
manEdk fibers 0.8% nig 1.2% erogKa.
16>12>3> snsSn_nPaBsVit (Ductilty Index )
pleFobnkMeNagemKuNnigkMeNagyaldMbUgRtUv)aneK[eQaHfasnsSn_nPaBsVit/
= u / y . snsSn_nPaBsVitrbs;ebtugGarem:ERbRbYlcenaH 4 nig 6 . RbsinebIEdk fibers
RtUv)aneRbIenAkgFwmebtugGarem: enaHsnsSn_nPaBsVitekIneLIgtamsmIkarxageRkam
' = (1.0 + 3.8 s )
!^>!%
Edl = pleFobkMeNagemKuNelIkMeNagyaldMbUg
' = snsSn_nPaBsVitnebtugEdlmanEdk fibers
s = PaKrynEdk fibers edaymaD/ 0 s 1.2%
FwmCab; nigeRKagCab;

487

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mhaviTalysMNg;sIuvil

NPIC

16>12>4> mMurgVilEdlRtUvkar (Required Rotation)

eKRtUvkarmMurgVilnsnak;)asicenAeRKOgbgMminkMNt;ebtugGarem:edIm,IGnuBaat[snak;)asic
ekItman ehIyeRKOgbgMEdlxiteTArkemkanicnkar)ak;GacRtUv)ankMNt;eday slope deflection BI
smIkarxageRkam. sRmab;kMNat; AB cenaHsnak;)asicNIr mMurgVilenARtg; A KW
L
A =
[2(M A M FA ) + (M B M FB )]
!^>!^
6 Ec I
Edl M A nig M B = m:Um:g;emKuNenARtg;cMNuc A nig B erogKa
M FA nig M FB = m:Um:g;bgb;cugeGLasicenARtg;cMNuc A nig B
EC = m:UDuleGLasicrbs;ebtug = 4730 f 'c
I = m:Um:g;niclPaBnmuxkat;eRbH emeronTI 5
16>12>5> lTPaBTb;mMurgVil (Rotation Capacity Provided)
snak;)asicrgkarTajKMrUenARtg;muxkat;TRm nigmuxkat;kNalElVgneRKagRtUv)anbgajenA
kgrUbTI 16>29.lTPaBTb;mMurgVilGaRsyCacMbgeTAnwg
- Ultimate strain capacity rbs;ebtug 'c EdlGacRtUv)ansnt;esI 0.003 b 0.0035 .
- RbEvg l p EdlenAelIRbEvgenHsnak;)asiceFVIkardl;yal.
RbEvgenHRtUv)ansnt;mantmRbEhlkm<s;RbsiTPaBnmuxkat;Edlsnak;)asicekItman
lp = d
- km<s;rbs;bksgt; c enAkgebtugenAeBl)ak;enARtg;muxkat;nsnak;)asic. mMurgVil
nsnak;)asicRtUv)anKNnatamsmIkarxageRkam
pI p
=
!^>!&
c
Edl p KWCakMeNInn strain enAkgebtugEdlvas;BI initial yielding rbs;EdkenAkgmuxkat;
rUbTI 16>29 c
p = 'c c1 = 0.0035 c1

RbsinebI l p = d ehIypleFob c / d esInwg 0.5 /

=

T.Chhay

(0.0035 c1 )d
d

0.0035 c1

488

Continuouse Beams and Frames

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Department of Civil Engineering

BIRtIekaNbERmbRmYlrageFob (strain triangle) rUbTI 16>29/

c f y d f y

=
=
d c E s d d E s 1

c1 = y

Edl

ersIusg;yalrbs;Edk nig E s = m:UduleGLasicrbs;Edk = 2 105 MPa . dUcenH

fy
0.0035 c1 0.0035
!^>!*
=

E s (1 )
sRmab;Edk f y = 280MPa nigedayeRbItmGtibrman = 0.50 enaH
fy =

min =

0.0035
280

= 4.2 10 3 rad
5
0 .5
2 10 (1 0.5)

sRmab;Edk
min =

f y = 400MPa

nig max = 0.44 enaH

0.0035
280

= 4.38 10 3 rad
5
0.44
2 10 (1 0.44)

FwmCab; nigeRKagCab;

489

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

Edl)anKNnaenATIenH)anBIEtmag ehIymMurgVilGnuBaatsrubenARtg;snk;)asicesInwg
2 min b 2 . BitGacRtUv)anKNnadUcxageRkam
eKman a = 1c nig 1 = 0.85 sRmab; f 'c 28MPa
min

c=

As f y

(0.85)2 f 'c b

f y
As f y
c
=
=
0.5
d 0.72 f ' c bd 0.72 f ' c

!^>!(

Edl = As / bd . eKnwgTTYl)an max enAeBlEdleRbI max .

RbsinebIlTPaBTb;rgVilminRKb;RKan; eKGacbegInmuxkat; bkat;bnyPaKryEdkedIm,I
TTYl)an c tUc/ tUc nig FM. RbsinebIeKeRbIEdkkgvN bERmbRmYlrageFobemKuNEdleFVI
[EbkebtugnwgekIneLIgx<s;dl; 0.012 .
sRmab;snak;)asicsgt; dUcenAkgssr
pl p
!^>@0
=
h
Edl h = km<s;srubrbs;muxkat; nig l p = RbEvgEdl yielding ekItman. enAkgsnak;sgt; l p
ErbRbYlcenaH 0.5h nig h .
enAeBlebtugrgkugRtaMgemKuNdl; f 'c / c = 0.002 dUcenH p = 'c 0.002
= 0.0035 0.002 = 0.0015 CamMurgVilGb,brmaenARCugmag. dUcenH
min =

0.0015 0.5h
= 0.00075rad
h

CamYynwgEdkkgvN/ GacnwgekIneLIgdl;
max = (0.012 0.002)

0.5h
= 0.005rad
h

tm 'c = 0.012 CatmEdlFMNas; dUcenHtmtUcCagenHGacRtUv)aneRbICamYyEdkkgvN.

RbsinebImindUcenHeT eKGaceRbImuxkat;epSgeTot.
enAkgFwmCab;ebtugGarem:EdlmanEdk fibers mMurgVil)asicGacRtUv)anKNnadUcxageRkam
fy

0.0035

!^>@!
p =

E (1 )

Edl

= (4.3 + 2.24 s 0.00625 f y + 4.17 s )

= 0.56 0.16 s

T.Chhay

490

!^>@@
!^>!$
Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

ersIusg;yalrbs;Edk
E s = m:UDleGLasicrbs;Edkem
= PaKryEdkem
s = PaKryEdk fibers KitmaD
smIkar !^>@! Bgajy:agc,as;famMurgVil)asicnebtugGarem:EdleRbIEdk fibers KWGaRsyeTA
nwg PaKryEdkem/ PaKryEdk fibers nigersIusg;yalrbs;va. karbegInersIusg;yalkat;bnymMu
rgVil)asic. smIkar !^>@! krab;bBalnUv\TiBlnRbEvgsnak;)asicenAelImMurgVilEdr.
xageRkamCaTRmg;EdlRtUv)amsRmYledIm,IKNnamMurgVil)asic
0.003
!^>@#
p =

]TahrN_ RbsinebI s = 0 nig f y = 400MPa enaH p1 = 0.00289 / nigRbsinebI s = 1% /
= 1.5% nig f y = 400MPa enaH p 2 = 0.01235 / . enHmannyfa lTPaBTb;mMurgVilrbs;
FwmebtugGarem:nwgekIneLIgRbEhlbYndgRbsinebIeKeRbIEdk fibers 1% .
fy =

16>13> segbviFIsaRskgkarKNnasanPaBkMNt; (Summary of Limit Design Procedure)

- KNnabnkemKuNedayeRbIemKuNbnkEdl[enAkgemeronTI 3 wu = 1.2D + 1.6L
- kMNt;Ggt;emkanicnkar)ak;/ snak;)asic/ nigm:Um:g;emKuN M u
- KNnamuxkat;eRKaHfak;edayeRbIviFIKNnaersIusg; (strength design method)
- kMNt;mMurgVilcaM)ac;nsnak;)asic
- KNnalTPaBTb;mMurgVilenARtg;muxkat;nsnak;)asic. lTPaBTb;mMurgVilRtUvEtFMCagmMurgVil
caM)ac;.
- RtYtBinitemKuNRbqaMg yielding rbs;Edk nigsameRbHFM EdlCapleFobrvag
M u / elastic moment at service load

- epgpat;PaBdab nigsameRbHeRkamGMeBIbnkeFVIkar
- RtYtBinitemIlfaEdkkmaMgkat;TTwgRKb;RKan;RtUv)andak;enARKb;muxkat;

FwmCab; nigeRKagCab;

491

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NPIC

]TahrN_16>7

FwmdUcEdlbgajenAkgrUbTI 16>30 RtUv)anbgb;enAcugTaMgsgag nigRTbnkBRgayesIemKuN

80kN / m nigbnkcMcMNucemKuN 213.5kN . KNnaFwmedayeRbIdMeNIrkarKNnasanPaBkMNt;.
eK[ b = 350mm / f 'c = 21MPa nig f y = 280MPa .

dMeNaHRsay

1> bnkBRgayesIemKuN wu = 80kN / m . bnkcMcMNucemKuN Pu = 213.5kN / m .

2> snak;)asicekItmanenARtg;cMNuc A / B nig C EdlbNal[manemkanicnkar)ak;dUc
bgajenAkgrUbTI 16>30. edayeRbIviFIkmnCak;Esg (virtual work method) nigsnt;PaB
dabktaenARtg;cMNuc C enaHkmnxageRkAesInwg
We = 213.5 1 + 80 3.7 = 509.5kN .m

kmnxagkgEdlRsUbedaysnak;)asicKW
Wi = M u (at A) + M u (at B) + 2 M u (at C)
4M u
= 4 M u =
3 .7

eday[ We = Wi enaH M u = 471.3kN .m . tamkarviPaKTUeTA[

Mu =

wu L2
L
7 .4 2
7 .4
+ Pu = 80
+ 213.5
= 471.3kN .m
16
8
16
8

3> KNnamuxkat;eRKaHfak;enARtg;cMNuc A / B nig C sRmab; M u = 471.3kN .m . BItaragenA

kg]bsm<n B nigsRmab; f 'c = 21MPa / f y = 280MPa CamYynwgPaKryEdk
= 0.013 eK)an Ru = 2.95MPa max = 2.04%
M u = Ru bd 2
d=

471.3 10 6
= 676mm
2.95 350

km<s;srubrbs;FwmKW h = 676 + 60 = 736mm yk 740mm

As = bd = 0.013 350 676 = 3075.8mm 2

eRbIEdk 5DB28 mYyRsTab; As = 3079mm 2 /

bmin = 9 28 + 2 10 + 70 = 342mm < 350mm
As f y
3079 280
=
a=
= 138mm
0.85 f 'c b 0.85 21 350

T.Chhay

492

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca
c=

Department of Civil Engineering

a
= 162.35mm
0.85

4> mMurgVilcaM)ac;rbs;snak;)asicKW
a. A =

c 162.35
=
= 0.24
d
676

L
[2(M A M FA ) + (M B M FB )]
6 Ec I

E c = 4730 f ' c = 4730 21 = 21676MPa

E s = 200000MPa
b.

kMNt;m:Um:g;cugbgb;enARtg; A nig B edayeRbIbnkemKuN

M FA = M FB =

c.

wu L2
L
7 .4 2
7 .4
+ Pu = 80
+ 213.5
= 562.55kN .m
12
8
12
8

m:Um:g;)asic M A = M B = 471.3kN .m
m:Um:g;niclPaBnmuxkat;eRbHGacRtUv)anKNnadUcxageRkam
I cr = b

x3
+ nAs (d x )2
3

Edl x CacmayBIsrsrgkarsgt;eTAGkSNWt (kd ) . edIm,IkMNt; x emIlemeronTI

6/ x = 257.4mm nig I cr = 68.44 108 mm 4
d. mMurgVilcaM)ac;Gb,brma edaycat;Tukm:Um:g;TaMgGs;enARtg; A nig B GviCman/ enaH
7400

A =

6 21676 68.44 10
= 0.00227 rad

[2( 471.3 + 562.55) + ( 471.3 + 562.55)] 10 6

5> lTPaBTb;mMurgVilKW
A =

0.0035

fy

E s (1 )

0.0035
280

0.24
200000(1 0.24)

= 0.0127rad > 0.00227rad

lTPaBTb;mMurgVilesIRbEhl 5.5 dgnmMurgVilcaM)ac; enHbgajfamuxkat;RKb;RKan;.

6> RtYtBinitpleFobrvagm:Um:g;emKuNelIm:Um:g;eGLasiceRkambnkeFVIkar
wL2
L 50 7.4 2 133.4 7.4
MA = MB =
+P =
+
= 351.6kN .m
12
8
12
8

138
a
6
M n = As f y d = 0.9 3079 280 676
10 = 471kN .m
2
2

471
= 1.34
yielding
351.6

pleFobKW
enARtg;TRm.
FwmCab; nigeRKagCab;

EdltMNag[emKuNsuvtiPaBRbqaMgnwg

493

rbs;Edk

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

7> RtYtBinitPaBdabGtibrmaEdlbNalmkBIbnkeFVIkar enAkNalElVg ykbnkeFVIkar

BRgayesI w = 50kN / m nig P = 133.4kN enaH
1 =

wL4
384 EI

sRmab;bnkcMcMNucenAkNalElVg

T.Chhay

494

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca
2 =

Department of Civil Engineering

PL3
192 EI

PaBdabsrubKW
=

50 7400 4

384 21676 68.44 108

4.53
1
=
=
L 7400 1633

133400 7400 3
192 21676 68.44 108

= 4.53mm

EdlCapleFobtUcEmnETn
8> brimaNEdkkmaMgkat;TTwgRKb;RKan;RtUv)andak;edIm,IeCosvagkar)ak;edaykmaMgkat;TTwg
EdlGacekItman.
16>14> karEbgEckm:Um:g;eLIgvijnm:Um:g;GviCmanenAkgFwmCab;
Moment Distribution of Negative Moments in Continuous Beams

karEbgEckm:Um:g;GviCmaneLIgvijenAkgGgt;rgkarBt;Cab;KWQrelIsac;lUteFobsuT (net
tensile strain NTS) t sRmab;TaMgebtugGarem: nigebtugeRbkugRtaMg. rUbTI 16>31 bgajlImIt
GnuBaatkgkarEbgEckm:Um:g;eLIgvij. vabgajfaPaKrym:Um:g;GviCmanekIneLIg bfycuHenAelITRm
rbs;FwmCab; q' EdlKNnaedayRTwsIeGLasicminRtUvFMCag 1000 t % CamYytmGtibrma 20% .
karEbgEckm:Um:g;eLIgvijRtUv)anGnuBaatEtenAeBlEdl t 0.0075 EdlbgajfavamanPaBsVit
RKb;RKan;enARtg;muxkat;Edlm:Um:g;RtUv)ankat;bny. enAeBl t < 0.0075 karEbgEckm:Um:g;eLIgvij
minRtUv)anGnuBaateLIy. m:Um:g;GviCmanEdl)anEktMrYvRtUv)aneRbIsRmab;KNnam:Um:g;viCmanEdl)an
EktRmUv (ACI Code, Section 8.4) . karEbgEckm:Um:g;eLIgvijminRtUv)anGnuvtcMeBaHGgt;Edl
KNnaedayviFIKNnaedaypal;sRmab;RbBnkRmalxNeT eyagtamemeronTI 17.
Casegb PaBryekIneLIg nigPaKryfycuHnm:Um:g;GviCmanmandUcxageRkam
- enAeBlEdl t < 0.0075 karEbgEckm:Um:g;eLIgvijminRtUv)anGnuBaat / b > 0.476 .
- enAeBlEdl t = 0.0075 PaKrynkarEbgEckm:Um:g;eLIgvijKW 7.5% / b = 0.476 .
- enAeBlEdl t > 0.020 PaKrynkarEbgEckm:Um:g;eLIgvijKW 20% / b = 0.217 .
- enAeBlEdl 0.0075 < t < 0.020 PaKryEbgEckm:Um:g;eLIgvijKW
q '= 1000 t
!^>@$

FwmCab; nigeRKagCab;

495

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

]TahrN_ RbsinebI t = 0.010 enaHPaKryEbgEckm:Um:g;eLIgvijKW 10% . TMnak;TMngrvag

PaKryEdk enAkgmuxkat; nigsac;lUteFobsuT t KWmandUcxageRkam eyagtamEpk 3>9

fy
0.003 +
Es
t =

sRmab;Edk

f y = 400MPa

0.003

#>@$

nig Es = 200000MPa . edaysnt;

0.005

0.003
t =



b

f y / E s = 0.002

enaH
#>@%

sRmab; t = 0.0075 / lImItnPaBsVitKW t / y = 0.0075 / 0.002 = 3.75 . PaKryEbgEckm:Um:g;

eLIgvijERbRbYlGaRsyeTAnwgkarkMNt;TaMgenH nigsRmab; f y = 400MPa RtUv)an[enAkgtaragTI
16>1 nig 16>2.

T.Chhay

496

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

RKb;PaKryEbgEckm:Um:g;eLIgvijEdlRtUv)aneRbI vacaM)ac;RtUvFanafamuxkat;minrgnUvkarxUc
xat bmansameRbHFMedaysarbnkeFVIkar ehIyvaRtUvrkSalTPaBTb;rgVilenARKb;muxkat;eRKaHfak;enA
kgeRKOgbgM. karEbgEckm:Um:g;eLIgvijenAkgeRKOgbgMsaTicminkMNt;eFVI[m:Um:g;GviCmanenAelITRm
nigm:Um:g;viCmanenAkNalElVgfycuH. karfycuHenHmin)aneFVI[suvtiPaBrbs;eRKOgbgMfycuH beFVI
[maneRKaHfak;enaHeT ebIeRbobeFobCamYyeRKOgbgMsaTickMNt;. CakarBit PaBCab;enAkgrcna
sm<npl;nUversIusg;/ sirPaB nigesdkicbEnmeTotkgkarKNna.
emKuNEbgEckm:Um:g;eLIgvij q EdlQrelI (ACI Code 318-99) RtUv)anKNnadUcxageRkam
( ')
!^>@%
q = 20 1

enAkgsmIkarTI !^>@% PaKryEdk b ( ' ) enARtg;muxkat;Edlm:Um:g;RtUv)ankMNt;

RtwmPaKryGtibrma 0.5 b . PaKryEdkGb,brmaenAkgmuxkat;rgkarBt;RtUv)ankMNt;Rtwm
f ' c /(4 f y ) 1.4 / f y . edayeRbIkarkMNt;dac;xatTaMgenHPaKryEbgEckm:Um:g;eLIgvijGtibrma
nigGb,brmaRtUv)anbgajenAkgtarag 16>3.

taragTI 16>1 karpas;brPaKryEbgEckm:Um:g;eLIgvij (q' )

f y = 400MPa

0.0075

0.010

0.0125

0.0150

0.0175

0.020

0.0225

/ b

0.476

0.385

0.323

0.278

0.244

0.217

0.196

q ' (%)

7 .5

10

12.5

15

17.5

20

20

taragTI 16>2 karpas;brPaKryEbgEckm:Um:g;eLIgvij (q' ) sRmab;kar[PaKry /

/ b

0.48

0.45

0.40

0.35

0.30

0.25

0.20

0.0074

0.0081

0.0095

0.0113

0.0137

0.017

0.022

q' (%)

0. 0

8.1

9 .5

11.3

13.7

17.0

20.0

FwmCab; nigeRKagCab;

497

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

taragTI 16>3 karEbgEckm:Um:g;eLIgvijGtibrma nigGb,brma q smIkar !^>@%

f 'c ( MPa)

f y ( MPa)

min

q max %(for min )

q min %(for 0.5 b )

21

400

0.0215

0.0033

16.9

10

28

400

0.0285

0.0033

17.7

10

35

400

0.0339

0.003

17.9

10

]TahrN_16>8

kMNt;m:Um:g;eGLasicGtibrmaenARtg;TRm nigenAkNalElVgnFwmCab;EdlmanbYnElVgdUcbgajenA
kgrUbTI 16>32 a. Fwmmanmuxkat;esI nigRTnUvbnkefrBRgayesI 117kN / m nigbnkGefr
87.5kN . snt;PaKryEbEckm:Um:g;eLIgvijGtibrma 10% nigBicarNanUvkrNIBIrxageRkam
- enAeBlEdlbnkGefrRtUv)andak;enAelIElVgqas;Ka KNnam:Um:g;viCmanGtibrmaenAkgElVg
- enAeBlEdlbnkGefrRtUv)andak;enAelIElVgCab;Ka KNnam:Um:g;GviCmanGtibrmaenAelITRm

dMeNaHRsay
1>

Fwmmanm:Um:g;niclPaB I esIKa nigman E dUcKa. dUcenH EI efr. smIkarbIm:Um:g;

sRmab;viPaKFwmnigsRmab; EI efrKW
M A L1 + 2 M b (L1 + L2 ) + M C L2 =

w1 L13 w2 L32

4
4

edaysarElVgesIKa
L2
(w1 + w2 )
4

!^>@^
enAkg]TahrN_enH M A = M E = 0 . krNInkardak;bnk6EbbepSgKaRtUv)an
BicarNadUcbgajenAkgrUbTI 16>32
krNITI1> bnkefrRtUv)andak;enAelIFwm ABCDE TaMgmUl rUbTI 16>32 b.
krNITI2> bnkGefrRtUv)andak;enAelIElVg AB nig CD sRmab;m:Um:g;GtibrmaenAkg
ElVg AB nig CD rUbTI 16>32 c.
krNITI3> dUckrNITI 2 sRmab;ElVg BC nig DE rUbTI 16>32 d.
krNITI4> bnkGefrRtUv)andak;enAelIElVg AB / BC nig DE sRmab;m:Um:g;GviCmanenA
Rtg; B rUbTI 16>32 e.
M A + 4M B + M C =

T.Chhay

498

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

2>

Department of Civil Engineering

krNITI5> bnkGefrRtUv)andak;enAelIElVg CD nig DE rUbTI 16>32 f.

krNITI6> bnkGefrRtUv)andak;enAelI BC nig CD
sRmab;m:Um:g;GviCmanGtibrmaenARtg; C rUbTI 16>32 g.
krNITI1> GnuvtsmIkar !^>@^ eTAelIFwmkMNat; ABC / BCD nig CDE erogKa
4M B + M C =

62
(117 + 117) = 2106kN .m
4

M B + 4 M C + M D = 2106kN .m
M C + 4 M D = 2106kN .m

KNnasmIkarTaMgeyIgTTYl)an
M B = M D = 451.28kN .m nig M C = 300.9kN .m
sRmab;karkat;bnym:Um:g; 10%
M ' B = M ' D = 0.9 ( 451.28) = 406.2kN .m

M 'C = 0.9 ( 300.9) = 270.81kN .m

m:Um:g;kNalElVgEdlRtUvKanwgm:Um:g;Edl)ankat;bnyehIyKW
2
wD L2 1
ElVg AB = DE = 8 + 2 M ' B = 1178 6 12 406.2 = 323.4kN .m
2

3>

ElVg BC = CD = 1178 6 12 (406.2 + 270.81) = 188kN .m

krNITI2> GnuvtsmIkar !^>@@ eTAelIkMNat;Fwm ABC / BCD nig CDE erogKa
4M B + M C =

62
(87.5) = 787.5kN .m
4

M B + 4 M C + M D = 787.5kN .m
M C + 4 M D = 787.5kN .m

KNnasmIkarTaMgeyIgTTYl)an
M B = M D = 168.75kN .m

nig M C = 112.5kN .m
m:Um:g;eGLasickNalElVgEdlRtUvKanwgm:Um:g;xagelIKW
2
2
Fwm AB = wL8L + 12 M B = 87.58 6 12 168.75 = 309.4kN .m
1
BC = 0 (168.75 + 112.5) = 140.6kN .m
2

FwmCab; nigeRKagCab;

499

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

wL L2 1
87.5 6 2
(168.75 + 112.5) =
140.625 = 253.1kN .m
8
8
2
1
DE = 0 168.75 = 84.4kN .m
2

CD =

edIm,Ikat;bnym:Um:g;viCmankgElVg eKRtUvbegInm:Um:g;Rtg;TRmeday 10% nigKNnam:Um:g;

kgElVgEdlRtUvKa. m:Um:g;viCmanEdlTTYl)anRtUvEtesIy:agtic 90% nm:Um:g;Edl
KNnadMbUg.
M ' B = M ' D = 1.1( 168.75) = 185.6kN .m

M 'C = 1.1(112.5) = 123.75kN .m

m:Um:g;eGLasickNalElVgEdlRtUvKanwgm:Um:g;xagelIKW
2
2
Fwm AB = wL8L + 12 M ' B = 87.58 6 12 185.6 = 300.95kN .m

4>
5>

1
BC = 0 (185.6 + 123.75) = 154.7 kN .m
2
w L2 1
87.5 6 2
CD = L (123.75 + 185.6) =
154.7 = 239.05kN .m
8
2
8
1
DE = 0 185.6 = 92.8kN .m
2

krNITI3> RsedogKanwgkrNITI2> ehIym:Um:g;RtUv)anbgajenAkgrUbTI 16>32 d.

krNITI4> BicarNaElVg AB / BC nig DE EdlRTbnkGefredIm,IKNnam:Um:g;GviCman
GtibrmaenARtg;TM B
4M B + M C =

62
(87.5) = 1575kN .m
2

M B + 4 M C + M D = 787.5kN .m
M C + 4 M D = 787.5kN .m

KNnasmIkarTaMgeyIgTTYl)an
M B = 379.7 kN .m / M C = 56.3kN .m
nig M D = 182.8kN .m
sRmab;karkat;bnym:Um:g; 10% Rtg; B
M ' B = 0.9( 379.7 ) = 341.7 kN .m

m:Um:g;kNalElVgEdlRtUvKaKW

T.Chhay

500

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

1
87.5 6 2 1
MB =
341.7 = 222.9kN .m
2
8
2
2
w L
1
87.5 6 2
BC = L (341.7 + 56.3) =
199 = 194.75kN .m
8
2
8
1
CD = 0 (56.3 + 182.8) = 119.55kN .m
2
w L2 1
87.5 6 2
DE = L 182.8 =
91.4 = 302.35kN .m
8
2
8

Fwm AB = wL8L

6>
7>

krNITI5> RsedogKanwgkrNITI4> elIkElgEtElVgxagcugminRtUv)andak;bnkedIm,I

TTYl)anm:Um:g;viCmanGtibrmarnARtg;TRm B bTRm D sRmab;kardak;bnkRsedogKa.
daRkamm:Um:g;Bt;RtUv)anbgajenAkgrUbTI 16>32 f.
krNITI6> BicarNaElVg BC nig CD EdlRTbnkGefredIm,IKNnam:Um:g;GviCman
GtibrmaenARtg;TM B
62
(87.5) = 787.5kN .m
4
wL L2
M B + 4M C + M D =
= 1575kN .m
2
w L2
M C + 4 M D = L = 787.5kN .m
4

4M B + M C =

KNnasmIkarTaMgeyIgTTYl)an
M B = M D = 112.5kN .m

nig M C = 337.5kN .m
sRmab;karkat;bnym:Um:g; 10% Rtg; C
M 'C = 0.9( 337.5) = 303.75kN .m
M ' B = M ' D = 0.9( 112.5) = 101.25kN .m

m:Um:g;kNalElVgEdlRtUvKaKW
Fwm AB = DE = 12 101.2 = 50.6kN .m
BC = CD =

8>

wL L2 1
87.5 6 2
(101.2 + 303.95) =
202.6 = 191.15kN .m
8
2
8

m:Um:g;Gtibrma nigGb,brmaeRkaykarEbgEckm:Um:g;eLIgvijRtUv)anbgajenAkgtarag
16>4. Moment envelop RtUv)anbgajenAkgrUbTI 16>32 h.

FwmCab; nigeRKagCab;

501

T.Chhay

mhaviTalysMNg;sIuvil

9>

NPIC

enAkg]TahrN_renH muxkat;kNalElVgRtUv)aneRbIsRmab;PaBgayRsYl. m:Um:g;kNal

ElVgmincaM)ac;Cam:Um:g;viCmanGtibrmaeT. kgkrNIElVgcug AB nig DE m:Um:g;Gtibrma
eRkayeBlBIkarEbgEckbnkeLIgvij 10% KWwesInwg wD L2 / 12.2 nigekItmanenAcmay
0.4 L BI BC nig D .

taragTI 16>4 m:Um:g;cugeRkayn]TahrN_TI 16>8 eRkayBIkarEbgEckm:Um:g;eLIgvij

krNI 1
2
3
4 1 + 2
m:Um:;gGviCman
TItaMg m:Um:;g m:Um:;gGviCman m:Um:;gviCman
muxkat; DL Gtibrma LL Gtibrma LL Gtibrma DL + LL
TRm

5 1 + 3
m:Um:;gviCman
Gtibrma DL + LL

406.2

341.7

+ 28.1

747.9 *

378.1

300.9

303.75

604.65 *

300.9

406.2

341.7

+ 28.1

747.9 *

378.1

kNalElVg
AB

+ 323.4

92.8

+ 300.95

+ 230.6

+ 624.35 *

BC

+ 188

154.7

+ 239.05

+ 33.3

+ 427.05 *

CD

+ 188

154.7

+ 239.05

+ 33.3

+ 427.05 *

DE

+ 323.4

92.8

+ 300.95

+ 230.6

+ 624.35 *

(*)

m:Um:g;KNnaGtibrma nigGb,brmacugeRkay.

T.Chhay

502

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

FwmCab; nigeRKagCab;

Department of Civil Engineering

503

T.Chhay

mhaviTalysMNg;sIuvil

T.Chhay

NPIC

504

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

]TahrN_16>9

kMNt;karEbgEckm:Um:g;GviCmaneLIgvijGnuBaatenAelITRm B / C / D nig E nFwmCab; ABCDE

EdlbgajenAkgrUbTI 1633. FwmEdlmanmuxkat;ctuekaNEkg b = 300mm / h = 550mm nig
d = 490mm nigvaRtUv)anBRgwgdUcbgajenAkgtarag f 'c = 28MPa nig f y = 420MPa
- edayeRbI]bsm<n B .
- edayeRbIkarkMNt; ACI Code.

dMeNaHRsay

1> sRmab; f 'c = 28MPa nig

RtUv)an[dUcxageRkam

f y = 420MPa b = 0.0285

'
q = 20 1
b

. emKuNEbgEckm:Um:g;eLIgvij
!^>@%

2> emKuNEbgEckm:Um:g;eLIgvijrbs; ACI Code CaGnuKmn_nbERmbRmYlrageFobEdksuT t

nigERbRbYlcenaHBI 7.5% nig 20% dUcbgajenAkgrUbTI 16>31
q '= 1000 t
!^>@$
0.003 +

t =

FwmCab; nigeRKagCab;

fy
Es

0.003

505

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

nig t = 0.005
0.003

sRmab;

f y = 420MPa

taragxageRkambgajBItm q nig q' EdlminRtUvKa.

TRm

EdkTaj

Edksgt;

'

A' s

As

'
b

q%

q '%

3DB 28

0.0126

0.44

11.2

0.008

3DB32

0.0164

0.58

8 .4

0.0056

3DB 20

0.0064

0.225

15.5

0.0192

19.2

4DB 25

0.0134

0.0064

0.246

15.1

0.0173

17.3

T.Chhay

3DB 20

506

Continuouse Beams and Frames

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

XVII.

karKNnakRmalxNBIrTis

Design of Two-Way Slabs

17>1> esckIepIm (Introduction)

kRmalxNGacRtUv)anBicarNaCaGgt;eRKOgbgMEdlmankRmas; h tUcCagRbEvg L nigTTwg
S . TRmg;dsamBarbs;kRmalxNKWkRmalxNEdlRtUv)anRTedayTRmQmKa Edlvapl;nUvPaB
dabcMbgkgTismYy EdleK[eQaHfa kRmalxNmYyTis (one-way slab). karKNnakRmalxN
mYyTismanniyayenAkgemeronTI 9.
enAeBlkRmalxNRtUv)anRTedayRCugTaMgbYn nigmanpleFobbeNay L elITTwg S tUc
CagBIr ehIykRmalxNdabBIrTis elIsBIenHbnkenAelIkRmalxNRtUv)anbBaneTATRmTaMgbYn
RCug. kRmalxNEbenHRtUv)aneK[eQaHfa kRmalxNBIrTis (two-way slab). m:Um:g;Bt; nigPaB
dabenAkgkRmalxNEbbenHtUcCagenAkgkRmalxNmYyTis kRmalxNdUcKaGacRTbnk)aneRcIn
CagenAeBlEdlvamanTRmTaMgbYnRCug. bnkenAkgkrNIenHRtUv)anRTBIrTis ehIym:Um:g;Bt;kgTis
nImYytUcCagm:Um:g;Bt;enAkgkRmalxNRbsinebIbnkrbs;vaRtUv)anRTkgTisEtmYy. karteRmob
rt-Fwm-kRmalxN (slab-beam-girder) KMrUnkRmalxNmYyTis nigBIrTisRtUv)anbgajenAkgrUbTI
17>1.
17>2> RbePTkRmalxNBIrTis (Types of Two-Way Slabs)
kRmalxNebtugBIrTisGacRtUv)ancat;cMNat;fak;dUcxageRkam
a. kRmalxNBIrTisenAelIFwm (two-way slab on beam) krNIenHekItmanenAeBlEdl
kRmalxNBIrTisRtUv)anRTedayFwmenAelIRCugTaMgGs;rbs;va rUbTI17>1. bnkBIkRmal
xNRtUvbBaneTATRmFwmTaMgbYnrbs;va EdlnwgbBanbnkbneTAssr.
b. Flat slab CakRmalxNBIrTisEdlRtUv)anBRgwgBIrTisedayKanFwmRT ehIybnkRtUv)an
bBanpal;eTAssrTRm. ssrcg;TMluHkRmalxN EdlRtUv)ankarBaredaybIviFIxageRkam
rUbTI 17>2 nig 17>3
- edayeRbI drop panel CamYynwg column capital.
karKNnakRmalxNBIrTis

507

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

- edayeRbI drop panel EdlKan column capital. ebtugEdlBTCMuvij column capital KYr
EtRkas;RKb;RKan;edIm,ITb;Tl;nwgkugRtaMgTajGgt;RTUgEdlekItBIkmaMgkat; punching
shear.
- edayeRbI column capital edayKan drop panel EdlCaviFImYyminFmta.
c. Flat-Plate floor CaRbBnkRmalxNBIrTisEdlmankRmas;kRmalxNesI nigsitenABIelI
ssredaypal;edayKanFwm b column capital rUbTI 17>2 a. kgkrNIenHssrcg;TMluH
kRmalxNedaykugRtaMgTajGgt;RTUg. dUcenH CaTUeTAeKRtUvkarbegInkRmas;kRmalxN
bdak;EdkBiess.
d. Two-way ribbed slabs nig waffle slab system kRmalxNRbePTenHekItBIkRmalxN
EdlmanpleFobbeNayelITTwgtUcCag 2. CaTUeTA kRmas;rbs;kRmalxNsitenAcenaH
5cm eTA 10cm nigRtUv)anRTedayrnUt (rib or joist) TaMgBIrTis. rnUtRtUv)anteRmobkgTis
nImYyCamYyKMlatRbEhlBI 50cm 75cm EdlbegItragkaer bctuekaNEkg rUbTI 17>2
c. rnUtkGacRtUv)anteRmobedaymMu 45 o b 60 o BIGkSrbs;kRmalxN EdlbegInesaPN
PaBsabtkm. sRmab; two-way ribbed slabs RbBnepSgGacRtUv)anTTYlyk
- RbBnrnUtBIrTisCamYynwgRbehagcenaHrnUtEdlTTYledayeRbIBum<Biess EdlCaTUeTA
manragkaer. rnUtRtUv)anRTedayrtTaMgbYnRCugEdlsitenABIelIssr. kRmalxN
RbePTenHRtUv)aneK[eQaHfa two-way ribbed (joist) slab system .

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- RbBnrnUtBIrTisCamYyeRKOgbMeBj (filler) enAcenaHrnUtEdleFVI[BidanerobesI. eRKOg

bMeBj (filler) CagRbehag nigeFVIBIebtugTmn;Rsal bTmn;Fmta bBIsmarTmn;Rsal
epSgeTot. rnUtRtUv)anRTedayrtenARCugTaMgbYnEdlRtUv)anRTbnedayssr. kRmal
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xNRbePTenHkRtUv)aneK[eQaHfa two-way ribbed slab system b hollow-block

two-way ribbed system .
- RbBnrnUtBIrTisCamYyRbehagcenaHrnUt nigKanrt bFwmRTrnUt. vaQrenAelIssreday
pal;CamYynwgbnHebtugtan;. kRmalxNRbePTenHRtUv)anehAfa waffle slab.
17>3> kareRCIserIsRbBnkRmalxNebtugEdlmanlkNesdkic
Economical Choice of Concrete Floor Systems

RbBnkRmalxNCaeRcInRbePTRtUv)aneRbIsRmab;GKarTUeTA dUcCa eKhdan kariyaly nigGKar

rd)alepSg. kareRCIserIsRbBnkRmalxNEdlmanlkNkRmalxN nigRKb;RKan;GaRsyelI
RbePTGKar rUbragsabtkm esaPN nigRbEvgElVgEdlenAcenaHssr. CaTUeTA bnkGefrenAelI
GKarERbRbYlcenaHBI 3.8kN / m 2 7.2kN / m 2 . karENnaMTUeTAsRmab;kareRbIR)as;RbBnkRmal
xNEdlmanlkNesdkicRtUv)ansegbdUcxageRkam
- Flat plate saksmbMputsRmab;ElVgEdlmanRbEvgcenaHBI 6m 7.5m nigbnkGefrERb
RbYlBI 2.9kN / m 2 4.8kN / m 2 . GtRbeyaCn_nkarTTYlyk flat plate rYmmankarcM
NayelIBum<Gs;fefak TTYl)anBidanrabesI nigkarsagsg;qab;. Flat plate manlTPaB
Tb;kmaMgkat;TTwgTab nigPaBrwgRkajtUc EdleFIV[manPaBdabFM;. Flat plate RtUv)an
eRbIy:agTUlMTUlayenAkgGKarCakRmalxNBRgwgedayEdk bkebtugeRbkugRtaMg.
- Flat slab saksmbMputsRmab;ElVgEdlmanRbEvgBI 6m 9m nigsRmab;bnkGefrERb
RbYlBI 3.8kN / m 2 7.2kN / m 2 . vaRtUvkarBum<eRcInCag flat plate CaBiesssRmab;
column capital. kgkrNICaeRcIn eKeRbIEt drop panel edayKan column capital .
- Waffle slab saksmsRmab;ElVgEdlmanRbEvgBI 9m 14.5m nigsRmab;bnkGefrERb
RbYlBI 3.8kN / m 2 7.2kN / m 2 . vaRTbnk)aneRcInCag flat plate nigmanBidanKYr[Tak;
TajEtBum<mantmf.
- kRmalxNelIFwm (slab on beam) salsmbMputsRmab;ElVgcenaH 6m 9m nigbnk
GefrBI 2.9kN / m 2 5.7kN / m 2 . FwmbegInPaBrwgRkajrbs;kRmalxNEdleFVI[man
PaBdabtUc. eKRtUvkarBum<bEnmsRmab;Fwm.
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- kRmalxNmYyTisenAelIFwm (one-way slab on beam)saksmbMputsRmab;ElVgEdlman

RbEvgBI 3m 6m nigbnkGefrcenaHBI 2.9kN / m 2 4.8kN / m 2 . vaGacRtUv)aneRbI
sRmab;ElVgFMCagenHCamYynwgtmfCag elIsBIenHeKnwgTTYl)anPaBdabFM. eKRtUvkarBum<
bEnmsRmab;Fwm.
- One-way joist floor system saksmbMputsRmab;ElVgEdlmanRbEvgBI 6m 9m nigman
bnkGefrcenaH 3.8kN / m 2 5.7kN / m 2 . edaysarEtrnUteRCA brimaNebtug nigEdkKWtic
EtkarcMNayelIBum<Gs;eRcIn. Bidanrbs;kRmalxNGacnwgemIleTAKYr[Tak;Taj.
17>4> eKalKMnitkgkarKNna (Design Concept)
karviPaKdsuRkitsRmab;kmaMg nigbMlas;TIenAkgkRmalxNBIrTisKWsKsaj edaysarEt
PaBminkMNt;x<s;. vaBitCasKsajebIeTaHbICa\TiBl creep nig nonlinear behavior rbs;ebtugRtUv
)anecalkeday. viFI numerical method dUcCa finite element kGacRtUv)aneRbI b:uEnviFIdsamBadUc
EdlGVI)anbgajeday ACI Code saksmbMputsRmab;karKNnasRmab;karGnuvt. ACI Code,
Chapter 13 snt;fakRmalxNeFVIkarCaFwmTUlayEtrak;begItCaeRKagrwg (rigid frame) CamYynwg
ssrEdlenABIeRkam nigBIelIva. karsnt;nkarEckeRKagCaeRKagsmmUlRtUv)anepgpat;eLIgvij
edaykarsikSaRsavRCavCalkNviPaK nigBiesaFn_ (analytical and experimental research). va
)anbgajfa lTPaBRTbnkcugeRkay (ultimate load capacity) nkRmalxNBIrTisCamYynwgkar
Tb;tamRCug restrained boundary) KWesI RbEhlBIrdgnlTPaBRTbnkcugeRkayEdlKNnaeday
karviPaKtamRTwsI edaysarkarEbgEckm:Um:g;eLIgvijdFMEdlekIteLIgenAkgkRmalxNmunnwg)ak;.
enAeBlbnkFMbMlas;TI nigPaBdabFMRtUv)anrMBwgTuk dUcenHeKRtUvkarkRmas;kRmalxNGb,brmaedIm,I
rkSaPaBdab niglkxNeRbHRKb;RKan; eRkambnkeFVIkar.
ACI Code kMNt;viFIsaRsBIrsRmab;KNnakRmalxNBIrTis
- viFIKNnaedaypal; (direct design method DDM, ACI Code, Section 13.6) CaviFIRbhak;
RbEhl (approximate procedure) sRmab;karviPaK nigkarKNnakRmalxNBIrTis. RtUv
)ankMNt;sRmab;RbBnkRmalxNEdlrgnUvbnkBRgayesI nigssrmanKMlatesIKa besIr
esIKa. viFIenHeRbInUvsMnuMemKuNedIm,IkMNt;m:Um:g;KNnaenARtg;muxkat;eRKaHfak;.
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RbBnkRmalxNEdlmin RtUvKanwgkarkMNt;rbs; ACI Code, Section 13.6.1 RtUv)anviPaK

edayviFIsaRsKNnaEdl manlkNsuRkitCag.
- viFIeRKagsmmUl (equivalent frame method EFM, ACI Code, Section 13.7) CaviFImYy
EdlGKarbITMhM (3D) RtUv)anEckecjCaesrIneRKagsmmUlBIrTMhM (2D) edaykat;GKar
tamExSrcenaHssr. lTplrbs;eRKagRtUv)anBicarNadac;edayELkBIKatamTisbeNay
nigTisTTwgrbs;GKar nigRtUv)anKitBImYyCan;eTAmYyCan; dUcEdlbgajenAkgrUbTI 17>4.
viFIsaRsKNnatam ACI Code BIrKWQrelIelIlTplnkarviPaKeGLasic (elastic analysis) n
eRKOgbgMTaMgmUledayeRbIbnkemKuN. viFIEdlEktRmUv (modified approach) viFI direct design
method RtUv)anbgajenAkg commentary n code qaM 1989 CaviFIPaBrwgRkajEktRmUv (modified
stiffness method MSM). vaQrelIkarbBalemKuNEbgEckdkMNt;mYyCaGnuKmn_npleFobPaB
rwgRkaj ec sRmab;KuNnwgm:Um:g;saTicsrubenAkgElVgxagcug. viFIenHRtUv)anBnl;enAeBleRkay.
bEnmBIelIviFIrbs; ACI Code eKenAmanviFIepSgCaeRcIneTotsRmab;KNna nigviPaKkRmal
xN. CalTpl kRmalxNnwgmanbrimaNEdkticCag beRcInCag. viFIviPaK (analytical method)
GacRtUvcat;cMNat;fak;kgRkumnTMnak;TMngeKalrvagbnk nigbMlas;TI CaeGLasic/ )asic nig
nonlinear .
- enAkgkarviPaKeGLasic (elastic analysis) kRmalxNebtugRtUv)anKitCakRmaleGLa
sic. karBt;kmaMgkat;TTwg nigPaBdabRtUv)anKNnaedaysmIkarDIepr:g;EslTI4 (fourth
differential equation) EdlTak;TgbnkeTAnwgPaBdabsRmab;kRmalesIgCamYynwgbMlas;TI
tUc dUcEdl)anbgaj eday Timoshenko. dMeNaHRsay finite difference solution kdUc
CadMeNaHRsay finite element solution RtUv)anesIeLIgedIm,IviPaKkRmalxN. enAkgviFI
finite element method kRmalxNRtUv)an EbgEckCasMNaj;ragRtIekaN bragkaer (mesh
of triangles or quadrilateral). GnuKmn_bMlas;TIncMNuc (node) Edlkat;KaedaycMNuc
sMNaj; (intersecting mesh point) RtUv)anbegIteLIgCaTUeTA ehIym:aRTicnPaBrwgRkaj
(stiffness matrices) RtUv)anbegItsRmab;karviPaKedaykMuBTr.
- sRmab;karviPaK)asic eKmanbIviFI. viFI yield line method RtUv)anbegIteLIgeday
Johansen edIm,IkMNt;sanPaB (limit state) nkRmalxNedayBicarNafa yield line
EdlekItmanenAkgkRmalxNCaemkanicnkar)ak; (collapse mechanism). viFIceRmok
karKNnakRmalxNBIrTis

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RtUv)an begIteday Hillerborg. kRmalxNRtUv)anEckecjCaceRmok

(strip) ehIybnkenAelIkRmal xNRtUv)anEbgEckTisedABIrEkgKa. ceRmokRtUv)anviPaK
CaFwmsamBa. viFITIbICaviFI optimal analysis method sRmab;eFVI[brimaNEdlTTYl)anman
tmGb,brmaedayQrelIkarviPaK)asic. dMeNaHRsay optimal solution KWsKsajkg
karviPaK nigTTYl)ankarBRgaysrsEdlmYydsKsaj.
- karviPaK nonlinear analysis KitlkNbMlas;TIeRkambnkBitnkRmalxNebtugGarem:
enAeBlEdlviFI finite element method KitBicarNaEpk nonlinear nTMnak;TMngkugRtaMgbERmbRmYlrageFob (stress-strain relationship) nGgt;mYydac;edayELkBIKa. kgkrNI
enH dMeNaHRsaykayCasKsaj RbsinebITMnak;TMngEdl)anBIkarBiesaFedayTTYl)ankar
sRmYlminRtUv)ansnt;eTenaH.
viFIEdl)anerobrab;xagelI RtUv)anbgajedIm,IENnaMGksikSanUvviFIepSgnkarviPaKkRmalxN.
kargarBiesaFn_elIkRmalxNminRtUv)anGPivDeTkgb:unanqaMcugeRkayenH b:uEnkarsikSaCaeRcInRbEhlCaRtUvkaredIm,IsRmYldMeNIrkarKNnabcb,nCamYysuvtiPaB karbMerIkargar niglkNesdkic.
(strip method)

17>5> ceRmokelIssr nigceRmokkNal (Column and Middle Strips)

rUbTI 17>5 bgajkRmalxagkgnkRmalxNBIrTisEdlRtUv)anRTenAelIssr A / B / C
nig D . RbsinebIkRmalRTbnkBRgayesI kRmalxNnwgdabBIrTis CamYyPaBdabGtibrmaenAtMbn;
kNal O . cMNucx<s;bMputsitenAelIssr A / B / C nig D dUcenHEpknkRmalxNEdlenACMuvij
ssrnwgmanrage)a:g (convex shape). karpas;brrUbragrbs;kRmalxNbnicmg BIPaBe)a:genA
elIssreTArkPaBptenAkNalkRmal eFVI[ExSkaMnImYykat;Rtg;cMNucrbt;. muxkat;Rtg; O / E /
F / G nig H nwgmanm:Um:g;Bt;viCman b:uEnenAmMbrievNssrnwgmanm:Um:g;Bt;GviCmanGtibrma. eday
BicarNacMeroktambeNay AFB ceRmoknwgekagdUcFwmCab; rUbTI 17>5 b edaymanm:Um:g;GviCman
enARtg; A nig B nigmanm:Um:g;Bt;viCmanRtg; F . ceRmokenHlatsnwgenAcenaHssrBIr A nig B
nigCab;enAelIRCugTaMgsgagnkRmalEdlbegIt)anCaceRmokelIssr (column strip).

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dUcKasRmab;ceRmoktambeNay EOG nwgmanm:Um:g;Bt;GviCmanenARtg; E nig G ehIy

m:Um:g;viCmanenARtg; O EdlbegItCaceRmokkNal. ceRmokTIbItambeNay DHC nwgeFVIkar
RsedogKanwgceRmok AFB . dUcenH bnHkRmalGacnwgRtUv)anEbgEckbIceRmokKW 1enAkNaltam
beNay EOG Edl eK[eQaHfaceRmokkNal nigBIreTotsgagtambeNay AFB nig DHC
EdleK[eQaHfa ceRmokelIssr rUbTI 17>5 a. ceRmoknImYyeFVIkarCaFwmCab;. tamviFIdUcKa
bnHkRmalkRtUv)anEbgEckCabIcMeroksRmab;TisedAmYyeTotKW ceRmokkNalmYytambeNay
FOH nigceRmokelIssrBIreTot tambeNay AED nig BGC erogKa rUbTI 17>5 e.
tamryrUbTI 17>5 a eyIgeXIjfaceRmokkNalRtUv)anRTedayceRmokelIssr EdlbBan
bnkbneTAssr A / B / C nig D enAkgbnHkRmalenH. dUcenHceRmokssrRTbnkeRcInCag
ceRmokkNal. dUcenH m:Um:g;Bt;viCmanenAkgceRmokelIssrnImYy enARtg; E / F / G nig H
mantm FMCagm:Um:g;Bt;viCmanenARtg; O EdlsitenAceRmokkNal. dUcKa m:Um:g;GviCmanenAelI
ssr A / B / C nig D enAkgceRmokelIssrmantmFMCagm:Um:g;GviCmanenARtg; E / F / G
nig H enAkgceRmokkNal. Epknm:Um:g;KNnaEdlRtUv)ankMNt;enAmuxkat;eRKaHfak;nImYyn
ceRmokssr nigceRmokkNalRtUv)anbgajenAkgEpkTI 8.
TMhMnceRmokelIssr nigceRmokkNslnImYyenAkgbnHkRmalRtUv)ankMNt;eday ACI
Code, Section 13.2. ceRmokelIssr x EdlRtUv)ankMNt;edayTTwgkRmalxNenAelIRCugnImYy
nGkSssr esInwgmYyPaKbYnnTMhMbnHkRmal l1 nig l2 mYyNaEdltUcCageK rYmbBalTaMgFwm
RbsinebIman.
l1 = RbEvgElVg KitBIGkSeTAGkS kgTisedAEdlm:Um:g;nwgRtUv)ankMNt;
l 2 = RbEvlElVg KitBIGkSeTAGkS kgTisedAEkgnwg l1
EpknbnHkRmalcenaHceRmokelIssrkMNt;ceRmokkNal.

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17>6> kRmas;kRmalGb,brmaedIm,IkMritPaBdab
(Minimum Slab Thickness to Control Deflection)

kMNt;kRmas;kRmalxNsRmab;kRmalxNBIrTisedIm,IkMrit
PaBdab. TMhMnPaBdabrbs;kRmalxNGaRsynwgGefrCaeRcInEdlrYmbBalTaMgPaBrwgRkaj
Tb;karBt; (flexural stiffness) rbs;kRmalxNEdlbBalCaGnuKmn_nkRmas;kRmalxN h .
enAeBlbegInkRmas; kRmalxN enaHPaBrwgRkajTb;karBt;rbs;kRmalxNkekIneLIg ehIyPaB
dabrbs;kRmalxNnwgRtUvkat;bny. edaysarkarKNnaPaBdabsRmab;kRmalxNBIrTismanPaB
sKsaj nigedIm,IeCosvagPaBdabFM ACI Code kMNt;kRmas;kRmalxNTaMgenHedayTTYlykkar
ACI Code, Section 9.5.3

karKNnakRmalxNBIrTis

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kMNt;Edl)anBIkarBiesaFbI. RbsinebIkarkMNt;rbs;eyIgminsitenAkgEdnkMNt;TaMgbIenHeT eKcaM

)ac;RtUvKNnaPaBdab.
a. sRmab; 0.2 fm 2 /
fy

l n 0.8 +

1400

h=
36 + 5 ( fm 0.2)

b.

!&>!

b:uEnminRtUvtUcCag 125mm
sRmab; fm > 2
fy

l n 0.8 +
1400

h=
36 + 9

c.

T.Chhay

!&>@

b:uEnminRtUvtUcCag 90mm
sRmab; fm < 0.2
h = kRmas;kRmalxNGb,brmaedayKanFwmxagkg tarag 17>1
!&>#
Edl ln = clear span sRmab;TisEvgEdlvas;BIpQmKarbs;ssr
= pleFobn clear span EvgelI clear span xI
fm = tmmFmn f sRmab;RKb;FwmnRCugTaMgGs;rbs;bnHkRmal
f = CapleFobnPaBrwgRkajTb;karBt;nmuxkat;Fwm Ecb I b lIPaBrwgRkajTb;
karBt;nkRmalxN Ecs I s EdlBTCMuvijedayGkSbnHkRmalenABIelIFwmRCug
nImYy.
E I
f = cb b
!&>$
E cs I s
Edl Ecb nig Ecs Cam:UDuleGLasicrbs;ebtugenAkgFwm nigkRmalxN erogKa.
I b = m:Um:g;niclPaBTaMgmUlnmuxkat;FwmeFobGkSTIRbCMuTmn; muxkat;FwmrYmTaMg
beNaykRmalxNenAelIRCugTaMgsgagrbs;FwmEdlesInwgkm<s;FwmBIelI bBI
eRkamkRmalxNykmYyNaEdlFMCageK b:uEnminRtUvFMCagbYndgkRmas;
kRmalxN.
I s = m:Um:g;niclPaBnmuxkat;kRmalxNTaMgmUl.
b:uEn kRmas;kRmalxNminKYrtUcCagtmxageRkam
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- sRmab;kRmalxNEdlman fm < 2.0

- sRmab;kRmalxNEdlman fm > 2

125mm
90mm

tarag 17>1 kRmas;kRmalxNGb,brmaedayKanFwmxagkg

Yield
Stress
fy *
280
420

edayKan Drop Panel**

man Drop Panel***
bnHkRmalxageRkA
bnHkRmalxageRkA
bnHkRmalxagkg
bnHkRmalxagkg
KanFwmxag manFwmxag
KanFwmxag manFwmxag
ln
33
ln
30

ln
36
ln
33

ln
36
ln
33

ln
36
ln
33

ln
40
ln
36

ln
40
ln
36

sRmab;EdkEdlman Yield Stress cenaH 280 nig 420 kRmas;Gb,brmaTTYl)anBI linear interpolation.
** Drop panel RtUv)ankMNt;enAkg ACI Sections 13.3.7.7 nig 13.3.7.2
*** kRmalxNEdlmanFwmcenaHssrtambeNayxagkg. tmn f sRmab;FwmminKYrmantmtUcCag
0 .8 .
*

RbsinebIFwmminRtUv)aneRbI dUckgkrNI flat plate enaH f = 0 nig fm = 0 . smIkar ACI

Code sRmab;KNnakRmas;kRmalxN h )anKit\TiBlrbs;RbEvgElVg/ TRmg;bnHkRmal/ yield
stress rbs; Edk f y nigPaBrwgRkajTb;karBt;rbs;Fwm. enAeBlFwmEdlmanlkNrwgxaMgRtUv)aneRbI
smIkar !&>! Gacpl;nUvkRmas;kRmalxNtUc ehIysmIkar !&>@ Gaclub. sRmab; flat plate nig
flat slab enAeBl EdlFwmxagkgminRtUv)aneRbI kRmas;kRmalxNGb,brmaGacRtUv)ankMNt;eday
pal;BItarag 9>5 c n ACI Code EdlRtUv)anbgajenATIenHKWtarag 17>1.
karkMNt;rbs; ACI Code epSgeTotRtUv)ansegbdUcxageRkam
- sRmab;bnHkRmalEdlmanxagminCab;; FwmxagcugEdlman = 0.8 RtUv)aneRbI ebImindUcenH
eT kRmas;kRmalxNGb,brmaRtUv)anKNnatamsmIkar !&>! nig !&>@ RtUv)anbegIn 10%
y:ag tic ACI Code, Section 9.5.3 .
- enAeBl drop panel RtUv)aneRbIedayKanFwm kRmas;kRmalxNGb,brmaKYrRtUv)anbny
eday 10% . drop panel KYrRtUv)anlatsnwgRKb;TisBIGkSrbs;TRmedaycmayminticCag
karKNnakRmalxNBIrTis

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RbEvgElVgelI 6 RKb;TiscenaHGkSeTAGkSnTRm nigTMlak;cuHeRkamkRmalxNy:agtic

h / 4 . karbnyenH)anrYmbBaleTAkgtaragTI 17>1.
- edayminKittmEdlTTYl)anBIsmIkar !&>! nig !&>@ kRmas;kRmalxNBIrTisminRtUvtUcCag
krNIdUcteTA ! 125mm sRmab;kRmalxNEdlKanFwm b drop panel. @ 100mm
sRmab;kRmalxNKanFwmEtman drop panel. # 90mm sRmab; kRmalxNmanFwmenA
elIRCugTaMgbYnCamYynwg fm 2 nig 125mm sRmab; fm 2 ACI Code, Section
9.5.3.
CMhanxageRkamsegbBIkarKNnaTaMgenH
!> sRmab;kRmalxNEdlKanFwmxagkg flat plate nig flat slab
a. KNnakRmas;kRmalxNedaypal;BItarag 17>1. b:uEnsmIkar !&>! nig !&>@ kGacRtUv)an
eRbI ehIyCaTUeTAsmIkar !&>! lub. kRmas;kRmalxNGb,brmaKYrFMCag besInwg 125mm
sRmab;kRmalxNEdlKan drop panel nigFMCagbesI 100mm sRmab;kRmalxNEdlman
drop panel.
b. enAxagEdlminCab; FwmxagEdlman f 0.8 KYrRtUv)aneRbI. ebImindUecaHeT kRmas;
kRmalxNGb,brmaRtUv)anKNnaedaysmIkar !&>! nig !&>@ KYrRtUv)anbegIneday 10% .
karbegIn 10% RtUv)anbBaleTAkgCYrQrTI 2 kgtaragTI 17>1 rYcehIy.
c. RbsinebI drop panel RtUv)aneRbIenAkg flat slab kRmas;kRmalxNGb,brmaRtUv)anbny
eday 10% enAkgkrNIEdl drop panel latsnwgenARKb;TisBIGkSnTRmCamYycmaymin
tUcCag 1 / 6 RbEvgElVg nigTMlak;eRkamkRmalxNy:agtic h / 4 . karbnyenH)anbBal
eTAkgemKuNntarag 17>1.
@> sRmab;kRmalxNEdlmanFwmenARKb;RCug fm > 0
a. KNna fm nigbnab;mkKNnakRmas;kRmalxNGb,brmaBIsmIkar !&>! nig !&>@. kg
krNICaeRcInsmIkar !&>@ lub.
b. kRmas;kRmalxNKYrFMCag besInwg 125mm sRmab;kRmalxNEdlman fm < 2.0 nigKYr
FMCag besInwg 90mm sRmab;kRmalxNEdlman fm 2.0 .

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#> sRmab;RKb;kRmalxN kRmas;kRmalxNEdltUcCagkRmas;Gb,brmaEdl[enAkgCMhan !>

nig @> GacRtUv)aneRbI RbsinkarKNnabgajfaPaBdabminFMCagkarkMNt;rbs; ACI Code,
Table 9.5 b EdlBnl;enAkgemeronTI 6.

]TahrN_17>1 RbBnkRmal flat plate EdlmanTMhM 7.5 6m RtUv)anRTenAelIssrkaer

. edayeRbIsmIkar ACI Code kMNt;kRmas;kRmalxNGb,brmacaM)ac;sRmab;bnH

kRmalxagkg nigbnHkRmalkac;RCug dUcbgajenAkgrUbTI 17>6. FwmxagminRtUv)aneRbI. eK[
f 'c = 28MPa nig f y = 420MPa .
500mm

dMeNaHRsay
1>

ln
f y = 420MPa nig
sRmab;bnHkRmalxNkac;RCugelx ! kRmas;Gb,brmaKW 30
KanFwmxagRtUv)aneRbI emIltarag 17>1.

karKNnakRmalxNBIrTis

521

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

l n1 = 7500 500 = 7000mm

7000
250mm
hmin =
= 233mm
30

yk
ma:gvijeTot smIkar !&>! nig !&>@
GacRtUv)aneRbIedIm,IKNnakRmas;Gb,brmaCamYy f = fm = 0 .
2> sRmab;bnHkRmalxagkgelx #> CamYy
f y = 420MPa kRmas;kRmalxNGb,brmaKW
l
hmin = n = 212mm yk 220 mm
33
ma:gvijeTot smIkar !&>! nig !&>@ GacRtUv)aneRbI.
RbsinebIRKb;bnHkRmalxNTaMgGs;eRbIkRmas;dUcKa enaHeKGacyk
hmin = 250mm .

]TahrN_17>2 RbBnkRmalxNdUcbgajenAkgrUbTI 17>7 EdlpSMeLIgedaykRmaltan; nigFwm

enAelITaMgBIrTisEdlRTedayssrkaerEdlmanRCug 500mm . edayeRbIsmIkar ACI Code kMNt;

kRmas;kRmalxNGb,brmacaM)ac;sRmab;bnHkRmalxagkg. eK[ f 'c = 21MPa nig
f y = 420MPa .

dMeNaHRsay

1> edIm,IeRbIsmIkar !&>! fm RtUv)anKNnamun. dUcenH eKcaM)ac;kMNt; I b / I s nig f sRmab;

Fwm nigkRmalxNtamTisEvg nigTisxI.
2> m:Um:g;niclPaBrbs;FwmTaMgmUl I b RtUv)anKNnasRmab;muxkat;dUcbgajenAkgrUbTI 17>7 b
EdlRtUv)anbegIteLIgedayFwm nigEpksgagxHrbs;kRmalxN x = y b:uEnminRtUvFMCag 4
bYndgkRmas;kRmalxN. snt; h = 18cm ehIyvaRtUv)anepgpat;enAeBleRkay enaH
x = y = 56 18 = 38cm < 18 4 = 72cm . dUcenH be = 40 + 38 2 = 116cm nigmuxkat;
GkSr T RtUv)anbgajenAkgrUbTI 17>7 c .
kMNt;TIRbCMuTmn;rbs;muxkat;edayKitm:Um:g;eFobkMBUlrbs;sab
RkLapsab = 18 116 = 2088cm 2
RkLapRTnug = 40 38 = 1520cm 2

T.Chhay

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Department of Civil Engineering

RkLapsrub 3608cm 2
2088 9 + 1520 37 = 3608 y
y = 20.8cm
116
(18)3 + 2088(11.8)2 + 40 383 + 1520(19 2.8)2 = 928924.6cm 4
Ib =
12
12

3> m:Um:g;niclPaBnkRmalxNtamTisedAEvgKW
bh 3
Il =
Edl b = 600cm nig h = 18cm
12
Il =

600 3
18 = 291600cm 4
12
EI
928924.6
= b =
= 3.19
EI s
291600

tamTisedAEvg
4> m:Um:g;niclPaBnkRmalxNtamTisedAxIKW
fl

karKNnakRmalxNBIrTis

523

T.Chhay

mhaviTalysMNg;sIuvil
Is =

NPIC

760 3
18 = 369360cm 4
12
EI
928924.6
= b =
= 2.51
369360
EI s

tamTisedAxI
5> fm CatmmFmn fs nig fl
fs

3.19 + 2.51
= 2.85
2
7.6 0.5
=
= 1.29
6 0.5

fm =

6>
7> kMNt; hmin edayeRbIsmIkar !&>@ ln = 7.1m
hmin

420

7.1 0.8 +

1400

=
= 0.148m
36 + 5 1.29(2.82 0.2 )

b:uEn tmenHminRtUvtUcCag h Edl[edaysmIkar !&>@ fm > 2.0

h=

7.81
= 0.164m
36 + 9 1.29

ma:geTot hmin = 90cm .

dUcenH h = 16.4cm lub.
eKGacTTYlykkRmas;kRmalxNEdl)ansnt; h = 18cm .
cMNaMfa enAkgkrNIGnuvtn_CaeRcIn smIkar !&>@ manlkNlub.
17>7> ersIusg;kmaMgkat;TTwgrbs;kRmalxN (Shear Strength of Slabs)
sRmab;RbBnkRmalxNBIrTis bnHkRmalRtUvEtmankRmas;RKb;RKan;edIm,ITb;nwgm:Um:g;Bt;
TaMgBIr nigkmaMgkat;TTwgenARtg;muxkat;eRKaHfak;. edIm,IGegtlTPaBTb;kmaMgkat;TTwgnkRmal
xNBIrTis krNIxageRkamRtUv)anBicarNa.
17>7>1> kRmalxNBIrTisEdlRTedayFwm (Two-Way Slabs Supported on Beams)
muxkat;eRKaHfak;rbs;kRmalxNBIrTisEdlRTedayFwmKWsitenAcmay d BIpnFwmTRm
ehIy lTPaBTb;kmaMgkat;TTwgnmuxkat;nImYyKW Vc = f 'c bd / 6 . enAeBlEdlFwmman
lkNrwg nigGacbBanbnkkRmaleTAssr vaRtUv)ansnt;[RTbnkEdleFVIGMeBImkelIpkRmal
T.Chhay

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Department of Civil Engineering

xNEdlBT edaybnat; 45o EdlKUsecjBIRCugEkg dUcbgajenAkgrUbTI 17>8. bnkenAelIp

ctuekaNBaynwgRtUv)anRTedayFwmEvg AB nig CD b:uEnbnkenAelIpRtIekaNnwgRtUv)anRTeday
FwmxI AC nig BD .

kmaMgkat;TTwgkgmYyktaTTwgrbs;kRmalmantmx<s;bMputenAcenaH E nig F tam

TisTaMgBIr ehIy Vu = wu (l2 / 2) Edl wu CabnkemKuNBRgayesIkgmYyktap.
RbsinebIEdkTb;kmaMgkat;TTwgminRtUv)andak; kmaMgkat;TTwgenAcmay d BIpnFwm Vud
RtUvEtesInwg
Vud Vc

Edl Vud

f ' c bd

6
l

= wu 2 d
2

17>7>2> kRmalxNBIrTisEdlKanFwm (Two-Way Slabs without Beams)

Flat plate nig flat slab KanFwmeT dUcenHkRmalxNRtUv)anRTedayssredaypal;. sRmab;
kRmalxNEbbenHkugRtaMgkmaMgkat;TTwgBIrRtUv)aneFVIkarGegt TImYyKWkmaMgkat;TTwgmYyTis b
kmaMgkat;TTwgFwm (one-way shear or beam shear). muxkat;eRKaHfak;RtUv)anykenAcmay d BIp
karKNnakRmalxNBIrTis

525

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

nssr ehIykRmalxNRtUv)anBicarNadUcFwmEdlmanTTwgFMsitenAcenaHTRm dUckgkrNIFwmmYy

Tis (one-way beam). lTPaBTb;kmaMgkat;TTwgnmuxkat;ebtugKW Vc = f 'c bd / 6 . RbePTTI
BIrnkmaMgkat;TTwgEdlRtUvsikSaKWkmaMgkat;TTwgBIrTis bkmaMgkat;pug (two-way shear or
punching shear) dUcEdl)anerobrab;enAkgkarKNnaeCIgtag. kar)ak;edaykmaMgkat;ekItmantam
beNaykMNat;ekaN bkMNat;BIra:mIt (truncated cone or pyramid) CMuvijssr. muxkat;eRKaHfak;
sitenAcmay d / 2 BIpssr/ column capital/ b drop panel rUbTI 17>9 a. RbsinebIEdkkmaMg
kat;TTwgminRtUv)andak; ersIusg;kmaMgkat;TTwgrbs;ebtugKWtmEdltUcCageKkgcMeNamsmIkar !&>%
nig !&>^
f ' c bo d
1 1
f ' c bo d
Vc = +
!&>%
3
6 3
Edl bo = brimaRtnmuxkat;eRKaHfak;
= pleFobnRCugEvgrbs;ssrelIRCugxI bRkLapbnk

d
!&>^
Vc = s + 2 f 'c bo d
12 b

Edl s esI 40 sRmab;ssrxagkg/ esI 30 sRmab;;ssrxag nigesI 20 sRmab;ssrkac;RCug.

enAeBlEdlEdkkmaMgkat;TTwgRtUv)andak; ersIusg;kmaMgkat;TTwgminKYrelIs

Vc
f ' c bo d
!&>&
2
17>7>3> EdkkmaMgkat;TTWgenAkgkRmalxNBIrTisEdlKanFwm
Shear Reinforcement in Two-Way Slabs Without Beams

enAkgRbBnkRmalxN flat plate nig flat slab kRmas;kRmalxNEdl)aneRCIserIsGacnwg

minRKb;RKan;edIm,ITb;nwgkugRtaMgkmaMgkat;TTwgEdlGnuvteT. kgkrNIenH eKGacbegInkRmas;
kRmalxN bdak;EdkTb;kmaMgkat;TTwg. ACI Code GnuBaatkareRbIEdkTb;kmaMgkat;TTwgCa
shearhead nig anchored bar b wire.
Shearhead pSMeLIgedayEdkragGkSr I bGkSr C EdlpSarExVgCabYn nigRtUv)andak;enAkg
kRmalxNBIelIssr rUbTI 17>9 c, d . karKNna Shearhead minGnuvtsRmab;ssrxageRkA
Edlm:Um:g;Bt; nigm:Um:g;rmYlmantmFMEdlRtUv)anbMElgcenaHkRmalxN nigssr. ACI Code,
Section 11.12.4 bgajfaenARtg;muxkat;eRKaHfak; ersIusg;kmaMgkat; nominal Vn minKYrelIs
T.Chhay

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Design of Two-Way Slab

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Department of Civil Engineering

b:uEnRbsinebIEdk shearhead RtUv)andak; Vn minKYrelIs 7 f 'c bo d / 12 . edIm,IkMNt;

TMhMrbs; shearhead, ACI Code, Section 11.12.4 pl;nUvkarkMNt;dUcteTA
f ' c bo d / 3

karKNnakRmalxNBIrTis

527

T.Chhay

mhaviTalysMNg;sIuvil

T.Chhay

NPIC

528

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

!> pleFob v rvagPaBrwgRkaj Es I rbs;d shearhead nigPaBrwgRkajnmuxkat;EdleRbH

smasEdlmanTTWg c2 + d minRtUvtUcCag 0.15 .
@> sabrgkarsgt;nEdkragminRtUvmanTItaMgenAmM 0.13d nprgkarsgt;rbs;kRmalxN.
#> km<s;rbs;EdkragminRtUvFMCag 70 nkRmas;RTnug.
$> lTPaBTb;m:Um:g;)asic M P ndnImYyrbs; shearhead RtUv)anKNnaeday
V
c

ACI Code, Eq. 11.37

!&>*
M P = u hv + v lv + 1
2n
2

Edl

= 0.9

kmaMgkat;TTwgemKuNCMuvijbrievNnpssr
n = cMnYnd
hv = km<s;rbs; shearhead
lv = RbEvg shearhead Edlvas;BIGkSssr
%> muxkat;kRmalxNeRKaHfak;sRmab;kmaMgkat;TTWgRtUvEtkat;d shearhead enAcmayesInwg
(3 / 4)(l v c1 / 2) BIpssreTcugndrbs; shearhead dUcbgajenAkgrUbTI 17>9 c. mux
kat;eRKaHfak;RtUvEtmanbrimaRtGb,brma bo b:uEnvaminRtUvenACitCag d / 2 BIprbs;ssr.
^> Shearhead RtUv)anBicarNa[cUlrYmkgkarEbgEckm:Um:g;eLIgvij M v eTAceRmokkRmalxN
elIssrnImYydUcxageRkam

c
Mv =
vVu l v 1
ACI Code, Eq. 11.38
!&>(
2n
2
b:uEnvaminRtUvtUcCagtmtUcCageKkgcMeNam 30% nm:Um:g;emKuNEdlcaM)ac;enAkgceRmok
elIssr karpas;brm:Um:g;ceRmokelIssrelIRbEvg lv b M p Edl[enAkgsmIkar !&>*.
kareRbI anchored bent bar b wire kRtUv)anGnuBaateday ACI Code, Section 11.12.3. Edk
Edldak;enAxagEpkxagelIrbs;ssr niglTPaBkarteRmobEdkRtUv)anbgajenAkgrUbTI 17>9 e.
enAeBlEdl bar b wire RtUv)aneRbICaEdkTb;kmaMgkat;TTwg enaHersIusg;kmaMgkat;TTWg nominal KW
f 'c bo d Av f y d
Vn = Vc + Vs =
+
!&>!0
s
6
Edl Av CaRkLapEdkkgsrub nig bo CaRbEvgnmuxkat;eRKaHfak;nkmaMgkat;BIrTisenA
cmay d / 2 BIpssr. ersIusg;kmaMgkat; nominal Vn minRtUvFMCag f 'c bo d / 2 .
Vu =

karKNnakRmalxNBIrTis

529

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

kareRbIEdkkmaMgkat;enAkg flat plate kat;bnykRmas;kRmalxN nigenAEtrkSaPaBrabesI

rbs;BidanedIm,Ikat;bnyfBum<. TRmg; stirrup cage sRmab;EdkkmaMgkat;TTwgRtUv)anbgajenAkg
rUbTI 17>9 f . RbePTm:ageTotnEdkkmaMgkat;pSMeLIgeday studded steel strip rUbTI 17>9 g.
Steel strip RtUv)andak;CamYy bar chair nigRtUv)anPab;eTAnwgBum< edayCMnYs stirrup gage . ersIusg;
yalrbs;Edk stud RtUv)ankMNt;enAcenaH 280MPa nig 420MPa edIm,ITTYl)an anchorage eBj
eljenAeBlbnkemKuN.
17>8> karviPaKkRmalxNBIrTisedayviFIKNnaedaypal;
Analysis of Two-Way Slabs by the Direct Design Method

CaviFIRbhak;RbEhl (approximate method) RtUv)anbegIteLIgeday

ACI Code edIm,IKNnam:Um:g;KNnaenAkgkRmalxNBIrTisEdlRTbnkBRgayesI. edIm,IeRbIviFIenH
kar kMNt;xHRtUv)anelIkeLIgeday ACI Code, Section 13.6.1.
Direct design method

17>8>1> karkMNt; (Limitations)

!> vaRtUvmankRmalxNCab;Kay:agticbIkgTismYy
@> kRmalxNRtUvEtkaer bctuekaNEkg. pleFobElVgEvgelIElVgxIrbs;kRmalminRtUvFM
CagBIr
#> ElVgEdlenAEk,rkgTisnImYyminRtUvxusKaedayFMCagmYyPaKbInElVgEvgCag.
$> ssrminRtUvlyecjBIGkSssrdTCaeRcIneTotedaytmGtibrma 10% nRbEvgElVg
enAkgTislyecj.
%> bnkTaMgGs;RtUvEtBRgayesI ehIypleFobnbnkGefrelIbnkefrminRtUvFMCag 2 .
^> RbsinebImanFwmenARKb;RCug pleFobnPaBrwgRkajEdlTak;TgkgTisEkgTaMgBI
f 1l 22 / f 2 l12 minRtUvtUcCag 0.2 nigFMCag 5.0 .
17>8>2> m:Um:g;saTicemKuNsrub (Total Factored Static Moment)
RbsinFwmTRmsamBaRTbnkBRgayesI w kN / m enaHm:Um:g;Bt;viCmanGtibrmaekItmanenA
T.Chhay

530

Design of Two-Way Slab

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Department of Civil Engineering

kNalElVgnigesInwg M o = wl12 / 8 Edl l1 CaRbEvgElVg. RbsinebIRtUv)anbgb;cugTaMgsgag bCab;

CamYynwgm:Um:g;GviCmanesIKaenAcugTaMgsgag enaHm:Um:g;srub M o = M p m:Um:g;viCmanenAkNal
ElVg + M n m:Um:g;GviCmanenAelITRm = wl12 / 8 rUbTI 17>10.
\LvRbsinebIFwm AB RTbnk W BIkRmalxNEdlmanTTwg l2 Ekgnwg l1 enaH W = wu l2
ehIym:Um:g;srubKW M o = (wl2 )l12 / 8 Edl wu = GaMgtg;sIuetbnkKitCa kN / m 2 . kgsmIkarenH
m:Um:g;BitR)akdEdlekItmanenAeBl l1 esInwg clear span cenaHTRm A nig B . RbsinebI clear span
RtUv)ankMNt;eday ln enaH
M o = (wu l 2 )

l n2
8

(ACI Code, Eq. 13.3)

RtUv)anvas;BIpeTApTRmkgTisedAEdlm:Um:g;RtUv)anBicarNa b:uEnminRtUvticCag
0.65 dgRbEvgElVgBIGkSeTAGkSTRm. pnTRmEdlmanm:Um:g;GviCmanKYrRtUv)anKNna RtUv)an
bgajenAkgrUbTI 17>11. RbEvg l2 RtUv)anvas;kgTisedAEkgnwg ln ehIyesITisedAcenaHGkSeTA
GkSrbs;TRm TTwgkRmalxN. m:Um:g;srub M o EdlKNnakgTisedAEvgRtUv)anKitCa M ol
nigkgTisedAxIRtUv)anKitCa M os .
enAeBlm:Um:gsrub M o RtUv)anKNnakgTisedAmYy vaRtUvEbgEckCam:Um:g;viCman M p nigm:U
m:g;GviCman M n GBawgehIyeTIb M o = M p + M n rUbTI 17>10. enaHm:Um:g;nImYy M p nig
M n RtUv)anEbgEckqgkat;TTwgkRmalxNcenaHceRmokssr nigceRmokkNaldUcEdl)anBnl;
y:agxI.
Clear span l n

karKNnakRmalxNBIrTis

531

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

17>8>3> karEbgEckm:Um:g;tambeNaykgkRmalxN
(Longitudinal Distribution of Moment in Slabs)

enAkgkRmalxagkg m:Um:g;saTicsrub M o RtUv)anEbgEckenAkgm:Um:g;BIr m:Um:g;viCman M p

enA kNalElVgesInwg 0.35M o nigm:Um:g;GviCman M n enATRmnImYyesInwg 0.65M o dUcbgajenA
kgrUbTI 17>12. tmm:Um:g;TaMgenHQrelIkarsnt;fakRmalxagkgCab;kgTisTaMgBIr ehIyman
RbEvgElVg nigbnkRbhak;RbEhlesIKa dUcenHtMNxagkgKanmMurgVilFMeT. elIsBIenHeTot m:Um:g;man
tmRbEhlnwgm:Um:g;rbs;Fwmbgb;cugTaMgBIrEdlrgbnkBRgayesI Edlm:Um:g;GviCmanenAelITRmesI
BIrdgm:Um:g;GviCmanenAkNalElVg. enAkgrUbTI 17>12 RbsinebI l1 > l2 / enaHkarEbgEckm:Um:g;enA
kgTisedAEvg nigTisedAxIKW

T.Chhay

l n21
8
l2
= (wu l1 ) n 2
8

M ol = (wu l 2 )

M pl = 0.35M ol

M n1 = 0.65M ol

M os

M ps = 0.35M os

M ns = 0.65M os

532

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

RbsinebITMhMnm:Um:g;GviCmanenAelITRmxagkgmantmxusKaedaysarRbEvgElVgminesIKa
ACI Code kMNt;[eRbIm:Um:g;EdlFMCagsRmab;KNnasrsEdk.

enAkgbnHkRmalxageRkA bnkkRmalxNEdlGnuvtelIssrxageRkA)anmkEtBIRCugmag
bNal[ekItmanm:Um:g;minesI (unbalanced moment) nigmMurgVilenAtMNxageRkA. dUcenH m:Um:g;
viCmanenAkNalElVg nigm:Um:g;GviCmanenAelITRmxagkgTImYynwgekIneLIg.TMhMnmMurgVilntMNxag
eRkAkMNt;nUvkarelIneLIgnUvm:Um:g;kNalElVg nigm:Um:g;enAelITRmxagkg. ]TahrN_ RbsinebIRCug
xageRkACaTRmsamBa dUckgkrNIkRmalxNenAelICBaaMg rUbTI 17>13 m:Um:g;kRmalenARtg;p
CBaaMgesI 0 m:Um:g;viCmanenAkNalElVgGacykesInwg M p = 0.63M o nigm:Um:g;GviCmanenATRmxag
kgKW M s = 0.75M o . tmTaMgenHbMeBjlkxNsmIkarsaTic
karKNnakRmalxNBIrTis

533

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

M o = M p + 12 M n = 0.63M o + 12 (0.75M o )

sRmab;RbBnkRmal-ssr (slab-column floor system) tMNxageRkAmankarTb; (restraint) xH

Edlpl;edayPaBrwgRkaJTb;karBt;nkRmalxN nigedayPaBrwgRkajTb;karBt;nssrxageRkA.
eyagtam ACI Code, Section 13.6.3 m:Um:g;saTicsrub M o enAkgElVgcugRtUv)anEbgEckeday
pleFobepSgKaedayeyagtamtarag 17>2 nigrUbTI 17>14. emKuNm:Um:g;enAkgCYrQrTI 1 sRmab;
RCugEdlminmankarTb;KWQrelIkarsnt;fa pleFobnPaBrwgRkajTb;karBt;rbs;ssrelIPaBrwg
RkajTb;karBt;smasrvagkRmalxN nigFwmenARtg;tMN ec KWesIsUn. emKuNnCYrQrTI 2 KWQr
elIkarsnt;fapleFob ec esInwgGnn. emKuNm:Um:g;enAkgCYrQrTI 3/ TI4 nigTI5 RtUv)anbegIt
eLIgedaykarviPaKRbBnkRmalCamYynwglkxNragFrNImaRt niglkxNTRmepSgKa.

taragTI 17>2 karEbgEckm:Um:g;enAkgbnHkRmalxagcug

kRmalxNEdlKan
kRmalxN
RCugxageRkA
FwmenAcenaHTRmxag
EdlmanFwm
kg
enAcenaHRKb;
minRtUv
manFwm KanFwm
TRm
)anTb; Tb;eBj
xageRkA xageRkA
#
!
$
%
@
m:Um:g;emKuNGviCmanxageRkA
m:Um:g;emKuNviCman
m:Um:g;emKuNGviCmanxageRkA

T.Chhay

0.65

0.16

0.30

0.26

0.63

0.35

0.57

0.50

0.52

0.75

0.65

0.70

0.70

0.70

534

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

17>8>4> karEbgEckm:Um:g;tamTTwgkgkRmalxN
(Transverse Distribution of Moment in Slabs)

m:Um:g;tambeNayEdl)anBnl;xagelIKWsRmab;TTwgTaMgmUlrbs;eRKagGKarsmmUl. TTwg
eRKag enHCaplbUknTTwgceRmokelIssrBIr CamYynwgTTwgceRmokkNalBIrnbnHkRmalBIr
Ek,rKa dUcbgaj enAkgrUbTI 17>15. karEbgEcktamTTwgnm:Um:g;tambeNayeTAceRmok
kNal nigceRmokelIssrKW CaGnuKmn_npleFob l2 / l1
E I
beam stiffness
f = cb b =
!&>!@
E I
slab stiffness
cs s

karKNnakRmalxNBIrTis

535

T.Chhay

mhaviTalysMNg;sIuvil
t =

NPIC

E cb C
torsional rigidity of edge beam section
=
2 Ecs I s flexural rigidity of a slab of width equal to beam span length

!&>!#
3
Edl C = torsional constant = 1 0.63y x x3y

!&>!$

Edl x nig y CaTTwg nigbeNayrbs;muxkat;ctuekaN. PaKrynm:Um:g;KNnanImYyEdl

nwgRtUvEbgEckeTAceRmokelIssr nigceRmokkNalsRmab;bnHkRmalxagkg nigbnHkRmal
xageRkA RtUv)an[enAkgtarag 17>3 dl; 17>6. enAkgbnHkRmalKMrUxagkg EpkxHnm:Um:g;
KNna EdlminRtUv)andak;eTAkgceRmokelIssr tarag 17>3 RtUv)anTb;edayceRmokkNal
Bak;kNalEdlRtUvKa. kareFVI linear interpolation sRmab;tm l2 / l1 EdlenAcenaH 0.5 nig
2.0 nigsRmab;tm f 1l 2 / l1 EdlenAcenaH 0 nig 1 RtUv)anGnuBaateday ACI Code. BI
tarag 17>3 eyIgGacemIleXIjfa enAeBlFwmminRtUv)aneRbI dUckgkrNI flat plate nig flat
slab f 1 = 0 . PaKrycugeRkaynm:Um:g;enAkgceRmokelIssr nigceRmokkNalCaGnuKmn_n
M o RtUv)an[enAkgtaragTI 17>4.
sRmab;kRmalxageRkA Epknm:Um:g;KNnaEdlminRtUv)andak;enAkgceRmokelIssr tarag
17>5 RtUv)anTb;edayceRmokkNalBak;kNalEdlRtUvKa. mgeTot kareFVI linear interpolation cenaHtmEdlbgajenAkgtarag 17>5 RtUv)anGnuBaateday ACI Code, Section
13.6.4.2. enAeBlEdl FwmminRtUv)aneRbIenAkRmalxageRkA dUckrNI flat plate nig flat slab
edayKanFwmxag (spandrel beam) f 1 = 0 / C = 0 nig t = 0 . enHmannyfacugssr pl;
nUvkarTb;sRmab;cugkRmalxageRkA. tmGnuvtn_ntarag 17>5 sRmab;krNIBiessenHRtUv)an
bgajenAkgtarag 17>6 nigrUbTI 17>15.

tarag 17>3 PaKrynm:Um:g;tambeNayenAkgceRmokelIssr sRmab;bnHkRmalxagkg

(ACI Code, Section 13.6.4)

pleFob l2 / l1

f 1l 2 / l1

m:Um:g;GviCmanenAelITRmxagkg
m:Um:g;viCmanenAEk,rkNalElVg

T.Chhay

0 .5

1 .0

2 .0

75

75

75

1 .0

90

75

45

60

60

60

1 .0

90

75

45

536

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

tarag 17>4 PaKrynm:Um:g;enAkgkRmalxNxagkgBIrTisEdlKanFwm (

m:Um:g;KNnasrub

= 0)

l n21
n!
M o = (wu l 2 )
8 r!(n r )!

m:Um:g;GviCman

m:Um:g;viCman

m:Um:g;tambeNayenAkgkRmalmYy
ceRmokelIssr

0.65M o

0.35M o

0.75( 0.65M o ) = 0.49 M o

0.60(0.35M o ) = 0.21M o

ceRmokkNal

0.25( 0.65M o ) = 0.16 M o

0.40(0.35M o ) = 0.14 M o

karKNnakRmalxNBIrTis

537

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

tarag 17>5> PaKrynm:Um:g;tambeNayenAkgceRmokelIssr sRmab;bnHkRmalxageRkA

(ACI Code, Section 13.6.4)

f 1l 2 / l1

m:Um:g;GviCmanenAelITRmxag
eRkA
m:Um:g;viCmanenAEk,rkNalElVg
m:Um:g;GviCmanenAelITRmxagkg

pleFob l2 / l1

0 .5
75

1 .0
75

2 .0
75

1 .0

2.5
0

90
60

75
60

45
60

2 .5

90

75

45

60

60

60

1.0

90

75

45

75

75

75

1 .0

90

75

45

tarag 17>6> PaKrynm:Um:g;tambeNayenAkgceRmokelIssr nigceRmokkNal

sRmab;pleFob l / l eday[ = = 0
2

f1

m:Um:g;GviCmanenAelITRmxageRkA
m:Um:g;viCman 0.6 0.52M o
m:Um:g;GviCmanenAelIMTRmxagkg
0.75 0.70M o

m:Um:g;cugeRkayCaGnuKmn_n
ceRmokelIssr ceRmokkNal
M o nig ec
ceRmokelIssr

100

0.26 M o

60

0.312M o

0.208M o

75

0.52 M o

0.175M o

0.65

M o
(1 + 1 ec )

0.28
M
0.63
(1 + 1 ec ) o

0.10
0.75
M o
(1 + 1 ec )

BItarag 17>6 eyIgeXIjfaenAeBlEdlFwmxagminRtUv)aneRbIsRmab;kRmalxageRkA t = 0

nigm:Um:g;KNna 100% RtUv)anTb;edayceRmokelIssr. ceRmokkNalnwgminTb;m:Um:g;NamYy
eT dUcenHbrimaNEdkGb,brmaRtUv)andak;. ACI Code, Section 13.6.4.3 kMNt;faenAeBlTRm
xageRkACassr bCBaaMgEdlRtUv)anBntsRmab;cmayesInwgbIPaKbYnRbEvgElVgTTwg l2 Edl
T.Chhay

538

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

eRbIedIm,I kMNt; M o m:Um:g;GviCmanxageRkAEdlRtUv)anEbgEckesIkat;tam l2 . enAeBlEdlFwm

RtUv)andak;tambeNayGkSssr ACI Code, Section 13.6.5 tRmUvfam:Um:g;RtUvEtsmamaRtedIm,I
karBarm:Um:g; 85% enAkgceRmokelIssr RbsinebI f 1 (l2 / l1 ) 1.0 . sRmab;tm f 1 (l2 / l1 )
enAcenaH 1.0 nig 0 m:Um:g;EdlmankgFwmRtUv)ankMNt;edayeRbI linear interpolation . m:Um:g;k
RtUvEtsmamaRtedIm,IkarBarm:Um:g;bEnmEdlekItedaybnkTaMgGs;EdlGnuvtedaypal;eTAelIFwm
edaybBal TaMgTmn;rbs;tYFwm EdlKitBIeRkamkRmal. Epknm:Um:g;Edlmin)andak;eTAkgFwm
RtUv)anTb;edaykRmalxNenAkgceRmokelIssr.
17>8>5> karpl;rbs; ACI sRmab;\TiBlrbs;KMrUnkardak;bnk
(ACI Provisions for Effects of Pattern Loading)

enAkgrcnasm<nCab; m:Um:g;Bt;Gtibrma nigGb,brmaenARtg;muxkat;eRKaHfak;RtUv)anTTYl

edaykardak;bnkGefrtamKMrUkMNt;mYyedIm,IbegIttmx<s;bMput. kardak;bnkGefrenARKb;ElVg
TaMgGs; nwgminbegItm:Um:g;Bt;viCman bm:Um:g;Bt;viCmanGtibrmaeT.m:Um:g;Gtibrma nigGb,brma
GaRsyCacMbg nwgkrNIxageRkam
!> pleFobnbnkGefrelIbnkefr. pleFobx<s;nwgbegIn\TiBlrbs;KMrUnkardak;bnk
(patter loading).
@> pleFobPaBrwgRkajssrelIFwm. pleFobtUcnwgbegIn\TiBlrbs;KMrUnkardak;bnk.
#> KMrUnkardak;bnk. m:Um:g;viCmanGtibrmaenAkgElVgrg\TiBltictYcBIKMrUnkardak;bnk.
edIm,IkMNt;m:Um:g;emKuNKNnaenAkgrcnasm<nCab; ACI Code, Section 13.7.6 kMNt;dUc
xageRkam
!> enAeBlKMrUnkardak;bnkRtUv)ansal; eRKagsmmUlKYrRtUv)anviPaKsRmab;bnkenaH.
@> enAeBlbnkGefrERbRbYl b:uEnminFMCagbIPaKbYnnbnkefr wL 0.75wD benAeBl
EdlRKb;kRmalTaMgGs;RtUv)andak;bnkGefrkgtMNalKa karviPaKeRKagEdlmandak;
bnkGefremKuNeBjeRKagRtUv)anGnuBaat.
#> sRmab;lkxNnkardak;bnkepSgeTot eKGnuBaat[snt;fa m:Um:g;emKuNviCman
GtibrmaenAEk,rkNalElVgekItmanCamYynwg 0.75 nbnkGefremKuNeBjenAelI
kRmal nigenAelIkRmalqas;.
karKNnakRmalxNBIrTis

539

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

sRmab;m:Um:g;emKuNGviCmanGtibrmaenAkgkRmalxNelITRm RtUv)aneKGnuBaat[
snt;fa 0.75 nbnkGefremKuNGnuvtEtenAelIkRmalEk,r.
$> m:Um:g;emKuNminKYryktUcCagm:Um:g;EdlekIteLIgCamYybnkGefremKuNeBjenAelIkRmal
EdlCab;TaMgGs;eT.
17>8>6> karlMGitsrsEdk (Reinforcement Details)
eRkayeBlPaKryTaMgGs;nm:Um:g;saTicenAkgceRmokelIssr nigceRmokkNalRtUv)an
kMNt;brimaNsrsEdkkGacRtUv)anKNnasRmab;m:Um:g;viCman nigGviCmanenAkgceRmoknImYy
dUcEdl)aneFVIsRmab;FwmenAkgemeronTI4
a

!&>!%
M u = As f y d = Ru bd 2
2

KNna Ru nigkMNt;PaKryEdk edayeRbItarag]bsm<n B beRbIsmIkarxageRkam

f y

Ru = f y 1
!&>!^
1.7 f '

Edl = 0.9 . RkLapmuxkat;EdkKW As = bd . enAeBlEdlkRmas;kRmalxNRtUvnwgkar

kMNt; kRmas;kRmalxNEdl)anerobrab;kgEpkTI 4> enaHeK minRtUvkarEdkrgkarsgt;eT. rUbTI
13>3>8 n ACI Code bgajRbEvgGb,brmanEdk nigkar lMGitsrsEdksRmab;kRmalEdl
KanFwm ehIyvakRtUvbgajenATIenHEdr rUbTI 17>16. KMlatEdkenAkgkRmalxNminRtUvFM
CaglImItGtibrmarbs; ACI EdlmanKMlat 450mm bBIrdgkRmas; kRmalykmYyNaEdltUc
CageK.
17>8>7> viFIPaBrwgRkajEdlRtUv)anEktRmUvsRmab;ElVgcug
(Modified Stiffness Method for End Spans)

enAkgviFIenH PaBrwgRkajrbs;FwmxagcugkRmal nigrbs;ssrxageRkARtUv)anCMnYsedayPaB

rwgRkajnssrsmmUl K ec . PaBrwgRkajTb;karBt;nssrsmmUl K ec GacRtUv)anKNnaBI
smIkarxageRkam
1
1
1
Kc
b
!&>!&
=
+
K ec =
K
1+ K / K
K K
ec

T.Chhay

540

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

karKNnakRmalxNBIrTis

Department of Civil Engineering

541

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

Edl

PaBrwgRkajTb;nwgkarBt;rbs;ssrsmmUl
K c = PaBrwgRkajTb;nwgkarBt;rbs;ssrBitR)akd
K t = PaBrwgRkajTb;karrmYlrbs;Fwmxag
plbUknPaBrwgRkajrbs;ssrxagelI nigxageRkamkRmalxNGacRtUv)anykdUcxag
K ec =

eRkam
I c1 I c 2
+
Lc1 Lc 2

K c = 4 E

!&>!*

Edl I c1 nig Lc1 Cam:Um:g;niclPaB nigRbEvgrbs;ssrxagelInIv:UkRmalxN nig I c2 nig

Lc 2 Cam:Um:g;niclPaB nigRbEvgrbs;ssrxageRkamnIv:UkRmalxN. PaBrwgRkajTb;nwgkarrmYl
rbs;Fwmcug K t GacRtUv)ankMNt;dUcxageRkam
9 E cs C
Kt =
!&>!(
3
c
l 2 1 2
l2

TMhMrbs;ssrctuekaNEkg bctuekaNEkgsmmUl/ capital column b

bracket Edlvas;enAelIElVgTTwgnRCugnImYyrbs;ssr.
E cs = m:UDuleGLasicrbs;ebtugkRmal
C = efrrmYl (torsion constant) EdlkMNt;BIsmIkarxageRkam

x x 3 y
!&>@0
C = 1 0.63
y 3
Edl

c2 =

Edl x CaTMhMTTwgrbs;ctuekaN nig y CabeNayrbs;ctuekaN. kgkarKNna C

vimaRtrbs;muxkat;ctuekaNRtUv)aneRCIserIsy:agNaedIm,IeFVI[)antm C FMCageK.
smIkarxagedImEdl)anENnaMenATIenH nwgRtUv)anykmkeRbIenAkgEpk 12 Equivalent
Frame Method .
RbsinebIkRmalmanFwmRsbKanwgm:Um:g;EdlRtuvKNna enaHPaBrwgRkajTb;karrmYl K t
Edl[kgsmIkar !&>!( RtUv)anCMnYseday K ta EdlmantmFMCag ehIy K ta RtUv)anKNna
dUcxageRkam
K ta = K t

T.Chhay

I sb
Is

542

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

Edl I s = l212h = m:Um:g;niclPaBrbs;kRmalxNEdlmanTTwgesInwgTTwgeBjcenaHGkS

kRmal edayminrYmbBalEpkrbs;tYrFwmEdlbnayeTAelI beTAeRkamkRmal
xN.
I sb = I s / edaybBalTaMgtYrFwmEdlbnayeTAelI beTAeRkamkRmalxN.
muxkat;nGgt;rgkarrmYlxHEdlmanPab;mkCamYyRtUv)anbgajenAkgrUbTI 17>17.
enAeBlEdl K ta RtUv)anKNna enaHpleFobPaBrwgRkaj ec RtUv)anTTYldUcxageRkam
K ec
ec =
!&>@!
(K + K )
s

Edl

4 E cs I s
l1
4 Ecb I b
Kb =
l1

Ks =

=PaBrwgRkajTb;karBt;rbs;kRmalxN

=PaBrwgRkajTb;karBt;rbs;Fwm
I b = m:Um:g;niclPaBTaMgmUlrbs;muxkat;FwmbeNay
karEbgEcknm:Um:g;saTicsrub M o enAkgkRmalxageRkARtUv)an[CaGnuKmn_n ec
dUcxageRkam

0 .1
Interior negative factored moment = 0.75
M
(1 + 1 / ec ) o

0.28
Positive factored moment = 0.63
M
(1 + 1 / ec ) o

0.65
Exterior negative factored moment =
M o
(1 + 1 / ec )

tmTaMgenHRtUv)anbgajenAkgkRmalxageRkAKMrUkgrUbTI 17>18. emKuNTaMgenHRtUv

)anBicarNaBI\TiBlrbs;PaBrwgRkajrbs;ssrxageRkAkdUcCa\TiBlrbs;PaBrwgRkajrbs;Fwm
cugkRmalEdleFVI[karEbgEckm:Um:g;manlkNRKb;RKan;.
17>8>8> segbviFIKNnaedaypal; (Summary of the Direct Design Method (DDM))
krNITI1 kRmalKanFwm
!> RtYtBinittRmUvkarnkarkMNt;Edl)anBnl;enAkgEpk 8>1. RbsinebIvaminRtUvnwg
karkMNt;eT eKminGaceRbIviFI DDM )aneT.
karKNnakRmalxNBIrTis

543

T.Chhay

mhaviTalysMNg;sIuvil

T.Chhay

NPIC

544

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

@> kMNt;kRmalxNGb,brma hmin edIm,IRKb;RKgPaBdabedayeRbItmenAkgtarag

17>1. kRmalxageRkAEdlKanFwmxag[ hmin x<s;bMput ln / 30 sRmab;
f y = 420MPa . vaCa karGnuvtFmtaEdleRbIkRmas;kRmalxNesIKasRmab;RKb;
kRmalxageRkAnigxagkg.
#> KNnabnkemKuN Wu = 1.2WD. + 1.6WL
karKNnakRmalxNBIrTis

545

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

$> epgpat;kRmas;kRmalxN h edIm,IkarBarkmaMgkat;TTwgmYyTis nigkmaMgkat;TTwg

BIrTis. RbsinebIkRmas;kRmalxN h minRKb;RKan; eKRtUvbegInkRmas; h bdak;Edk
Tb;kmaMgkat;TTWg.
%> KNnam:Um:g;saTicsrub M o sRmab;TisedATaMgBIr smIkar !&>!!
^> kMNt;emKuNEbgEcksRmab;m:Um:g;viCman nigm:Um:g;GviCmanenAkgTisedAbeNay nig
TisedATTwgsRmab;ceRmokelIssr nigceRmokkNalnImYyTaMgenAkgkRmalxagkg
nigkRmalxageRkA dUcxageRkam
a. sRmab;kRmalxagkg eRbIemKuNm:Um:g;Edl[enAkgtarag 17>4 brUbTI 17>15
b. sRmab;kRmalxageRkAEdlKanFwmxag emKuNm:Um:g;kRmalRtUv)an[enAkgtarag
17>2 brUbTI 17>14 krNITI5. sRmab;karEbgEckm:Um:g;enAkgTisedATTwg
eRbItarag 17>6 brUbTI 17>15 sRmab;GRtaceRmokelIssr. ceRmokkNalnwg
Tb;Epknm:Um:g; EdlminRtUv)andak;eTAkgceRmokssr.
c. sRmab;kRmalxageRkAEdlmanFwmxag emKuNm:Um:g;kRmalRtUv)an[enAkgtarag
17>2 brUbTI 17>14 krNITI4. sRmab;karEbgEckm:Um:g;enAkgTisedATTwg
eRbItarag 17>5 sRmab;ceRmokelIssr. ceRmokkNalnwgTb;lMnwgnm:Um:g;
kRmal.
&> kMNt;brimaNEdksRmab;RKb;muxkat;eRKaHfak;;nceRmokelIssr nigceRmokkNal
TaMgGs; nigBntsrsEdkeBjkRmalxN rUbTI 17>16
*> KNna unbalanced moment nigRtYtBinitemIlfaetIkarbMElgm:Um:g; unbalanced
moment edaykarBt;RKb;RKan;bGt;. RbsinebIGt;RKb;RKan;eT kMNt;brimaNEdk
bEnmEdlcaM)ac;enAkgTTwgeRKaHfak; eyagtamEpkTI 10.
(> RtYtBinitemIlfaetIkarbMElgm:Um:g; unbalanced moment edaykmaMgkat;TTwgRKb;
RKan; bGt;. RbsinebIGt;eT begIn h bdak;EdkTb;kmaMgkat;TTwg. eyagtamEpkTI
10
krNITI2 kRmalEdlmanFwmxagkg nigFwmxageRkA
!> RtYtBinittRmUvkarnkarkMNt;Edl)anBnl;enAkgEpk 8>1.

T.Chhay

546

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

@> kMNt;kRmalxNGb,brma hmin edIm,IRKb;RKgPaBdabedayeRbItmenAkgsmIkar TI

!&>! dl; !&>#. kgkrNICaeRcIn smIkarTI !&>@ lub. smIkarTI !&>! KYrRtUv)an
KNnadMbUgdUcbgajenAkg]TahrN_TI 17>1.
#> KNnabnkemKuN Wu = 1.2WD. + 1.6WL
$> epgpat;kRmas;kRmalxN h tamrykmaMgkat;TTwgmYyTis nigkmaMgkat;TTwgBIr
Tis. CaTUeTA kmaMgkat;TTWgminmanlkNeRKaHfak;sRmab;kRmalxNEdlRTeday
FwmeT.
%> KNnam:Um:g;saTicsrub M o sRmab;TisedATaMgBIr smIkar !&>!!
^> kMNt;emKuNEbgEcksRmab;m:Um:g;viCman nigm:Um:g;GviCmanenAkgTisedAbeNay nig
TisedA TTwgsRmab;ceRmokelIssr nigceRmokkNalnImYyTaMgenAkgkRmalxag
kg nigkRmalxageRkA dUcxageRkam
a. sRmab;kRmalxagkg eRbIemKuNm:Um:g;kgrUbTI 17>14 krNITI 3 brUbTI
17>12. sRmab;karEbgEckm:Um:g;kgTisedATTwg eRbItaragTI 17>3 sRmab;
ceRmokelIssr. cM erokkNalnwgTb;Epknm:Um:g;Edlmin)andak;eTAkgceRmok
elIssr. KNna 1 BI smIkar !&>!@.
b. sRmab;kRmalxageRkA eRbIemKuNm:Um:g;kRmalenAkgtarag 17>2 brUbTI 17>14
krNI TI3. sRmab;karEbgEckm:Um:g;enAkgTisedATTwg eRbItarag 17>5
sRmab;ceRmokelIssr. ceRmokkNalnwgTb;lMnwgnm:Um:g;kRmal.
c. kgkrNITaMgBIr (a) nig (b) FwmRtUvTb; 85% nm:Um:g;enAkgceRmokssr enAeBl
Edl f 1 (l2 / l1 ) 1.0 b:uEnGRtaERbRbYlcenaH 85% nig 0% enAeBl f 1 (l2 / l1 )
ERbRbYlcemaHBI 1.0 nig 0 .
&> kMNt;brimaNEdksRmab;RKb;muxkat;eRKaHfak;;nceRmokelIssr/ Fwm nigceRmok
kNalTaMgGs; bnab;mkBntsrsEdkeBjkRmalxN rUbTI 17>16
*> KNna unbalanced moment nigbnab;mkRtYtBinitemIlkarbMElgnm:Um:g; edaykar
Bt; nigkmaMgkat;TTwg eyagtamEpkTI 10.

karKNnakRmalxNBIrTis

547

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

]TahrN_TI17>3 edayeRbIvIFI direct design method KNnakRmal flat plate xagkgKMrUdUc

Edl)anbgajenAkgrUb TI 17>6 nig 17>19. RbBnkRmalpSMeLIgeday kRmalbYnenARKb;Tis

EdlkRmalmYy manTMhM 7.5 6m . kRmalTaMgGs;RtUv)anRTedayssrTMhM 50 50cm man
RbEvg 3.6m . kRmalxNRTbnkGefreFVIkar BRgayesI 4.8kN / m 2 nigbnkefreFVIkarEdlrYm
mankRmalkargarbegIy (floor finish) 1.5kN / m 2 rYmTaMgbnkpal;rbs;kRmal. eK[
f 'c = 28MPa nig f y = 420MPa .

T.Chhay

548

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

dMeNaHRsay

1> kMNt;kRmas;kRmalxNGb,brmaedayeRbItarag 17>1 sRmab; flat plate. BI]TahrN_

TI 17>1 kRmas;kRmalxNKW 25cm .
2> KNnabnkemKuN
wD = 1.5 + weight of slab = 1.5 + 0.25 25 = 7.75kN / m 2

wu = 1.2 7.75 + 1.6 4.8 = 17kN / m 2

3> RtYtBinitkmaMgkat;TTwgmYyTis nigkmaMgkat;TTwgBIrTis

a. RtYtBinitkmaMgkat;pugenAcmay d / 2 BIpssr GMeBIBIrTis.
Edaysnt;kRmas;ebtugkarBarEdk 2cm nigeRbIEdk DB16 . enaH d mFmKW
25 2 1.6 = 21.4cm nig bo = 4(50 + 21.4 ) = 285.6cm emIlrUbTI 17>19 c
Vu = [l1l 2 (71.4 71.4)] wu = (750 600 5098) 17 10 4 = 756.3kN
0.75

Vc =
f ' c bo d =
28 2.856 0.214 10 3 = 808.5kN
3
3

EdlFMCag Vu
b. KNnakmaMgTTwgFwmenAcmay d BIpssr. d mFmKW 21.4cm . BicarNaceRmok
1m rUbTI 17>19 d CamYyRbEvgceRmokKW
x = 3.75 0.25 0.214 = 3.286m
Vu = wu (1 3.286) = 17 3.286 = 55.862kN
0.75

Vc =
f 'c bd =
28 1 0.214 10 3 = 141.5kN
6
6

EdlFMCag Vu . Kugkardak;bnkFmta kmaMgkat;TTwgmYyTisGt;lub.

4> KNnam:Um:g;saTicsrubenAkgTisedAEvg nigTisedAxI
2
kgTisedAEvg M ol = wu l82ln1 = 178 6 7 2 = 624.75kN .m
wu l1l n22 17
=
= 7.5 5.5 2 = 482.11kN .m
8
8

kgTisedAxI M os
edaysarEt l2 < l1 TTwgnBak;kNalceRmokelIssrenAkgTisedAEvgKW
0.25 6m = 1.5m ehIyTTwgnceRmokkNalKW 6 2 1.5 = 3m . TTwgnBak;kNal
ceRmokelIssrkgTisedAxI KW 1.5m ehIyTTwgnceRmokkNalKW
7.5 2 1.5 = 4.5m . edIm,IKNnakm<s;RbsiTPaB d kgTisedAnImYy snt;faEdkenA
karKNnakRmalxNBIrTis

549

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

kgTisedAxIRtUvBIelIEdk enAkgTisedAEvg. dUcenH

d (long direction ) = 25 2 0.8 = 22.2cm nig
. sRmab;karGnuvtn_ d (average) =
25 3.5 = 21.5cm GacRtUv)aneRbIsRmab;TisedATaMgBIr.
dMeNIrkarKNnaGacRtUv)anerobcMCaTRmg;tarag dUcbgajenAkgtarag 17>7 nig 17>8.
karlMGitsRmab;kareRCIserIssrsEdkRtUv)anbgajenAkgrUbTI 17>20 edayeRbIRbBn
EdkRtg;. eKRtUveKarBkardak;RbEvgGb,brmarbs;EdkdUcEdl)anbgajenAkgrUbTI
17>16.
Gksagsg;cUlciteRbIEdkRtg; nigEdkEdlmanersIusg; f = 420MPa .
of panel 3000
=
= 375mm
KMlatGtibrma = width
8
no. of bars

d (short direction) = 25 2 1.6 0.8 = 20.6cm

taragTI17>7 karKNnabnHkRmal flat platexagkg kgTisEvg

M o = 624.75kN .m
M n = 0.65M o = 406.1kN .m

TisEvg
karEbgEckm:Um:g; %

M p = +0.35M o = 218.66kN .m

ceRmokelIssr
GviCman
viCman
75

60

0.75M n = 304.6 0.6 M p = 131.2

M u (kN .m)

TTwgceRmok b(mm)

ceRmokkNal
GviCman
viCman
25

40

0.25 M n = 101.5

0.6 M p = 87.5

3000

3000

3000

3000

222

222

222

222

2.06

0.89

0.69

0.59

PaKryEdk (%)

0.57

0.24

0.19

0.16

As = bd (mm 2 )

3796.2

1598.4

1265.4

1065.6

1350

1350

1350

1350

20DB16

8DB16

12DB12

12DB12

150

375

250

250

km<s;RbsiTPaB d (mm)
Ru =

Mu
bd 2

( MPa)

As (min) = 0.0018bhs (mm 2 )

EdkEdleRCIserIs Rtg;
KMlat 2h = 500mm
s

T.Chhay

550

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

taragTI17>8 karKNnabnHkRmal flat platexagkg kgTisxI

M o = 482.11kN .m
M n = 0.65M o = 313.4kN .m

TisEvg
karEbgEckm:Um:g; %

M p = +0.35M o = 168.7kN .m

ceRmokelIssr
GviCman
viCman
75

60

0.75M n = 235.05 0.6 M p = 101.2

M u (kN .m)

TTwgceRmok b(mm)

ceRmokkNal
GviCman
viCman
25

40

0.25 M n = 78.35

0.6 M p = 67.5

3000

3000

4500

4500

206

206

206

206

1.85

0.79

0.41

0.35

PaKryEdk (%)

0.51

0.21

0.11

0.09

As = bd (mm 2 )

3151.8

1297.8

1019.7

834.3

1350

1350

2025

2025

16DB16

8DB16

18DB12

18DB12

187.5

375

250

250

km<s;RbsiTPaB d (mm)
Ru =

Mu
bd 2

( MPa)

As (min) = 0.0018bhs (mm 2 )

EdkEdleRCIserIs Rtg;
KMlat 2h = 500mm
s

KMlatEdkenAkgceRmokelIssrkgTisxIKW 250mm . vamanlkNRKb;RKan; edaysar

vatUcCag 2hs = 500mm nigtUcCag 450mm EdlkMNt;eday ACI Code.
cMNaMfa PaKryEdkTaMgGs;KWticCag max = 0.0182 . dUcenH = 0.9 .

]TahrN_TI17>4

edayeRbIviFI direct design method KNnakRmal flat plate xageRkAEdlmanTMhM bnk ersIusg;ebtug
nigersIusg;EdkdUcKanwgGVIEdl)an[enAkg]TahrN_TI 17>3. FwmminRtUv)aneRbI rUbTI 17>21.

dMeNaHRsay

1> kMNt;kRmas;kRmalxNGb,bramedayeRbItarag 17>1 sRmab; flat plate.

BI]TahrN_TI 17>1 kRmas;kRmalxNKW 25cm .

karKNnakRmalxNBIrTis

551

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

2> KNnabnkemKuN wu = 17kN / m 2

3> RtYtBinitkmaMgkat;TTwgmYyTis nigkmaMgkat;TTwgBIrTis eyagtam]TahrN_TI 17>3
nigrUbTI 17>9.

a.
b.
c.

RtYtBinitkmaMgkat;pugenAssrxagkg Vu = 756.3kN < Vc = 808.5kN

RtYtBinitkmaMgkat;TTwgmYyTis Vu = 55.862kN < Vc = 141.5kN
RtYtBinitkmaMgkat;pugenAssrxageRkA d = 21.4cm
x = 50 +

T.Chhay

21.4
= 60.7cm
2

552

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

y = 50 + 21.4 = 71.4cm

karKNnakRmalxNBIrTis

553

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

bo = 2 x + y = 192.8cm
750

Vu = 600
+ 25 60.7 71.410 4 17 = 400.6kN

Vc =
d.

f ' c bo d = 545.8kN > 400.6kN

RtYtBinitkmaMgkat;pugenAssrkac;RCug d = 21.4cm
x = y = 50 +

21.4
= 60.7cm
2

bo = x + y = 121.4cm
600

750
Vu =
+ 25 60.7 60.7 10 4 17 = 214.7kN
+ 25

2
2

Vc =

f ' c bo d = 343.7kN > 214.7 kN

4> KNnam:Um:g;saTicsrub BI]TahrN_TI 17>3

M ol (long direction ) = 624.7 kN .m

d = 22.2cm

M os (short direction) = 482.11kN .m

d = 20.6cm

TTwgrbs;ceRmokelIssrKW 300cm nigTTwgceRmokkNalKW 450cm

5> KNnam:Um:g;KNnaenAkgTisedAEvg l1 = 7.5m eyagtamtarag 17>5 brUb
17>15. karEbgEckm:Um:g;srub M ol enAkgceRmokelIssr nigceRmokkNalKWRtUv
)anKNnadUcxag eRkam
a. ceRmokelIssr
m:Um:g;GviCmanxagkg = 0.525M o = 0.525(624.75) = 328kN .m
m:Um:g;viCmanenAkgElVg = 0.312M o = 0.312(624.75) = 195kN.m
m:Um:g;GviCmanxageRkA = 0.26M o = 0.26(624.75) = 162.4kN .m
b. ceRmokkNal
m:Um:g;GviCmanxagkg = 0.175M o = 0.175(624.75) = 109.3kN .m
m:Um:g;viCmanenAkgElVg = 0.208M o = 0.208(624.75) = 129.9kN .m
m:Um:g;GviCmanxageRkA = 0

T.Chhay

554

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

6> KNnam:Um:g;KNnaenAkgTisedAxI l s = 6m . vaRtUv)anKitdUckRmalxagkgEdr BIeRBaH

vaCab;TaMgsgag. eyagtamtarag 17>4 brUbTI 17>15 karEbgEckm:Um:g;srub M os
enAkgceRmokelIssr nigceRmokkNalRtUv)anKNnadUcxageRkam
a. ceRmokelIssr
m:Um:g;GviCman = 0.49M o = 0.49(482.11) = 236.2kN .m
m:Um:g;viCman = 0.21M o = 0.21(482.11) = 101.2kN.m
b. ceRmokkNal
m:Um:g;GviCman = 0.16M o = 0.16(482.11) = 77.1kN .m
m:Um:g;viCman = 0.14M o = 0.14(482.11) = 67.5kN.m
dMeNIrkarKNnaRtUv)anteRmoby:aggayRsYlenAkgtarag 17>9. karlMGitsRmab;kar
eRCIserIssrsEdkRtUv)anbgajenAkgrUbTI 17>22 edayeRbIRbBnEdkRtg;;enAkgTis
Evg. karlMGitsrsEdkenAkgTisxImanlkNRsedogKanwgkarBRgaysrsEdkenA
kgrUbTI 17>20 edayeRbIkareRCIserIssrsEdkenAkgtarag 17>9.
cMNaMfa RKb;PaKryEdkTaMgGs;tUcCag max = 0.0182 . dUcenH = 0.9 .

karKNnakRmalxNBIrTis

555

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

tarag 17>9 karKNnakRmal flat platexageRkAsRmab;]TahrN_TI 17>4 d = 22.2cm

ceRmokelIssr
ceRmokkNal
TisEvg
xageRkA viCman xagkg xageRkA viCman xagkg
162.4

195

328

129.9

109.3

3000

3000

3000

3000

3000

3000

1.10

1.32

2.22

0.88

0.74

PaKryEdk (%)

0.30

0.36

0.62

0.24

0.20

As = bd (mm 2 )

1998

2398

4129

1598

1332

As (min) = 0.0018bhs (mm 2 )

1350

1350

1350

1350

1350

1350

22DB16

12DB12

18DB12

12DB12

136

250

167

250

M u (kN .m)

TTwgceRmok b(mm)
Ru =

Mu
bd 2

( MPa)

EdkEdleRCIserIs Rtg;
KMlat 2h = 500mm
TisxI
s

10DB16 12DB16
300

250

ceRmokelIssr

ceRmokkNal

236.2

101.2

77.1

67.5

TTwgceRmok b(mm)

3000

3000

4500

4500

km<s;RbsiTPaB d (mm)

206

206

206

206

1.86

0.79

0.40

0.35

PaKryEdk (%)

0.52

0.21

0.11

0.09

As = bd (mm 2 )

3214

1298

1020

834.3

As (min) = 0.0018bhs (mm 2 )

1350

1350

2025

2025

16DB16

8DB16

18DB12

18DB12

187.5

375

250

250

M u (kN .m)

Ru =

Mu
bd 2

( MPa)

EdkEdleRCIserIs Rtg;
KMlat 2h = 500mm
s

]TahrN_TI17>5

eFVI]TahrN_TI 17>4 eLIgvij edayeRbIviFI modified stiffness method. eKRtUvkarKNnaRsedogKa

sRmab;viFI equivalent frame method, EpkTI 12.
T.Chhay

556

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

dMeNaHRsay

1> GnuvtRsedogKasRmab;CMhanTI 1 dl; 4 dUckg]TahrN_TI 17>4

2> KNnaPaBrwgRkajssrsmmUl/ K ec
1
1
1
=
+
K ec K c K t

eyIgGacsnt;faEpknceRmokkRmalEdlenAcenaHssrxageRkAeFVIkarCassrTb;nwgkar
rmYl. muxkat;rbs;kRmalxN-ssrKW 50cm TTWgrbs;ssr 25cm kRmas;kRmal
xN dUcEdlbgajkgrUb.
a. kMNt;PaBrwgRkajTb;karrmYl K t BIsmIkar !&>@0

x x3 y
x = 250mm
y = 500mm
C = 1 0.63
y 3

250 250 3 500

C = 1 0.63
= 17.84 10 8 mm 4

500
3

Kt =

9 Ec C
c
l 2 1 2
l2

9 E c 17.84 10 8
500

60001

6000

= 3.47 E c 10 6

sRmab;kRmalxNEk,rKaBIr enAelIRCugTaMgsgagrbs;ssr EdleFVIkarCaFwmTTwg

K t = 2 3.47 E c 10 6 = 6.94 E c 10 6

b.

KNnaPaBrwgRkajrbs;ssr K c / km<s;ssr Lc = 3.6m

Kc =

4 Ec I c
4 E c 500 4
=

= 5.79 E c 10 6
Lc
3600 12

sRmab;ssrBIrenABIelI nigBIeRkamkRmalxN
K c = 2 5.79 E c 10 6 = 11.58E c 10 6

c.

KNna K ec
1
1
1
=
+
K ec 11.58 E c 10 6 6.94 E c 10 6

K ec = 4.34 E c 10 6

3> KNnaPaBrwgRkajrbs;kRmalxN nigemKuN ec

Ks =

4Ec I s
l1

karKNnakRmalxNBIrTis

l 2 = 6000mm

hs = 250mm

557

Is =

l 2 hs3
12

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

4 E c 6000 250 3

= 4.17 E c 10 6
7500
12
K ec
=
(K s + K b )

Ks =

ec

Kb = 0

dUcenH

edaysarKanFwm
ec =

4.34 E c 10 6

4.17 E c 10 6
1
Q = 1+
= 1.96

= 1.04

yk
ec
4> KNnam:Um:g;KNnaenAkgTisedAEvg ll = 7.5m .
karEbgEckm:Um:g;enAkgkRmalmYyRtUv)anbgajenAkgrUbTI 17>18.
m:Um:g;GviCmanxagkgKW

0.10
0.10

M ni = 0.75
M ol = 0.75
(624.7) = 436.6kN .m

Q
1.96

m:Um:g;viCmanKW

0.28
0.28

M p = 0.63
M ol = 0.63
(624.7) = 304.3kN .m

Q
1.96

m:Um:g;GviCmanKW
M ne =

0.65
0.65
(624.7 ) = 207.2kN .m
( M ol ) =
Q
1.96

5> KNnakarEbgEckm:Um:g;kRmalenAkgTisxIeTAceRmokelIssr nigceRmokkNal. m:Um:g;

M ni / M p nig M ne RtUv)anEbgEckdUcxageRkam eyagtamtarag 17>6
a. m:Um:g;xagkg M nl = 436.6kN .m RtUv)anEbgEck 75% sRmab;ceRmokelIssr nig
25% sRmab;ceRmokkNal
column strip = 0.75( 436.6 ) = 327.5kN.m
Middle strip = 0.25( 436.6 ) = 109.1kN.m

b.

m:Um:g;viCman M p = 304.3kN .m RtUv)anEbgEck 60% sRmab;ceRmokelIssr nig

40% sRmab;ceRmokkNal
column strip = 0.60(304.3) = 182.6kN.m
Middle strip = 0.40(304.3) = 121.7 kN.m

c.

T.Chhay

m:Um:g;GviCmanxageRkA M ne = 207.2kN .m RtUv)anEbgEckGaRsytamtarag 17>5

558

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca
t =

E cb C
C
=
2 E cs I s 2 I s

Department of Civil Engineering

ebtugkRmalxN nigebtugssrmanm:UDuleGLasicdUcKa

250 3
= 78.125 10 8 mm 4
12
17.84 10 8
=
= 0.114
2 78.125 10 8
E I
l
f 1 = cb b = 0
f1 2 = 0
l1
Ecs I s
I s = 6000

l2
= 0 .8
l1

BItarag 17>5 nigedayeFVIviFanmUlvacar (interpolation) cenaH t = 0 PaKry

=100% nig t = 2.5 PaKry = 75% sRmab; t = 0.114 PaKryKW 98.9% .
m:Um:g;GviCmanxageRkAenAkgceRmokelIssrKW 0.989 ( 207.2) = 204.92kN.m
nigenAkgceRmokkNalKW 2.28kN.m . kgkrNIenHeKGacKitfaceRmokelIssr
RTm:Um:g; M ne 100% KWesInwg 207.2kN.m
6> kMNt;srsEdkEdlcaM)ac;enAkgTisedAEvgkgtaragEdlmanlkNRsedogKanwg]TahrN_
TI 17>4. lTplEdlTTYl)anmanlkNERbRbYlticbMputxusBItarag 17>9.
7> eRbobeFoblTplrvag]TahrN_TI 17>4 nig 17>5 eyIgeXijfam:Um:g;xageRkAenAkg
ceRmokelIssr 207.2kN.m FMCagcMelIyEdlTTYl)ankg]TahrN_TI 17>4
162.4kN.m eday 27.6% b:uEnm:Um:g;viCman 182.6kN.m RtUv)ankat;bnyeday
6.8% eFobnwg 195kN.m tmdTeTotesIrEtRtUvKa.

]TahrN_TI17>6

KNnakRmalxagkgnRbBnkRmalBIrTisEdl)anbgajenAkgrUbTI 17>7. kRmalpSMeLIgeday

kRmal EdlmanTMhM 7.6 6m cMnYn 6 kgTisnImYy. kRmalTaMgGs;RtUv)anRTedayssrEdlman
TMhM 50 50cm RbEvg 3.6m . kRmalRtUv)anRTedayFwmtambeNayGkSssrEdlmanmuxkat;dUc
bgajkgrUb. bnkGefreFVIkarRtUv)anyk 4.8kN / m 2 nigbnkefreFVIkarpSMeLIgeday 1kN / m 2
sRmab;kargarbegIybEnmBIelITmn;pal;rbs;kRmal. cUreRbI f 'c = 21MPa / f y = 420MPa nigviFI
direct design method.

karKNnakRmalxNBIrTis

559

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

dMeNaHRsay

1> eKRtUveFVItamkarkMNt;rbs; ACI Code. kMNt;kRmas;kRmalxNGb,brmaedayeRbIsmIkar

17>1 nig 17>2. kRmas;kRmalxNRtUv)anKNnarYcCaeRscenAkg]TahrN_TI 17>2
ehIy eyIgTTYlykkRmas; 18cm . CaTUeTA kRmas;kRmalxNenAkgRbBnkRmalRtUv)an
RKb;RKgedaykRmalkac;RCugdUcCakarKNna hmin kRmalxageRkApl;nUvkRmas;kRmalFM
CagsRmab;kRmalxagkg.
2> KNnabnkemKuN
wD = 1 + 0.18 25 = 5.5kN / m 2

wu = 1.2 5.5 + 1.6 4.8 = 14.28kN / m 2

3> kugRtaMgkmaMgenAkgkRmalxNminmanlkNeRKaHfak;eT. muxkat;eRKaHfak;mancmay d

T.Chhay

560

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

BIpFwm. sRmab;TTwg 1m
1
0.4

0.15 = 37.84kN
Vu = wu 3 beam width d = 14.28 3
2
2

0.75 21

f ' c bd =
1000 150 10 3 = 85.9kN
Vc =
6
6

4> KNnam:Um:g;saTicsrubenAkgTisEvg nigTisxI

wu
14.28
l 2 (l n1 )2 =
6(7.1)2 = 539.9kN .m
8
8
w
14.28
= u l1 (l n 2 )2 =
7.6(5.5)2 = 410.4kN .m
8
8

M ol =

M os

5> KNnam:Um:g;KNnaenAkgTisEvg ll = 7.6m

a. karEbgEckm:Um:g;enAkgkRmal
m:Um:g;GviCman M n = 0.65M ol = 0.65 539.9 = 350.9kN .m
m:Um:g;viCman M p = 0.35M ol = 0.35 539.9 = 189kN .m
b. karEbgEckm:Um:g;kRmalkgTisTTwgeTAFwm/ ceRmokelIssr nigceRmokkNal
EI
l2
6
=
= 0.79
f 1 = s = b = 3.19 BI]TahrN_TI 17>2
7 .6
l
EI
1

l
f 1 2 = 3.19 0.79 = 2.52 > 1
l1

karEbgEckm:Um:g;GviCman M n . Epknm:Um:g;GviCmanxagkgedIm,IkarBaredayceRmok
elIssrRtUv)anTTYlBItarag 17>3 edayeFVI interpolation nigesInwg 81.3% sRmab;
l 2 / l1 = 0.79 nig f 1 (l 2 / l1 ) > 1.0 .
ceRmokelIssr = 0.813M n = 0.813 350.9 = 285.3kN .m
ceRmokkNal = 0.187M n = 0.187 350.9 = 65.6kN .m
edaysarEt f 1 (l2 / l1 ) > 1/ ACI Code, Section 13.6.5 bgajfa 85% nm:Um:g;
kgceRmokelIssrRtUv)anEck[eTAFwm nigenAsl; 15% RtUv)anEck[eTAkRmal
ceRmokelIssr.
Fwm = 0.85 285.3 = 242.5kN.m
ceRmokelIssr = 0.15 285.3 = 42.8kN.m
ceRmokkNal = 65.6kN.m
d. karEbgEckm:Um:g;viCman M p . Epknm:Um:g;viCmanxagkgEdlRtUv)anTb;edayceRmok
c.

karKNnakRmalxNBIrTis

561

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mhaviTalysMNg;sIuvil

NPIC

elIssrRtUv)anTTYlBItarag 17>3 edayeFVI interpolation nigesInwg 81.3% sRmab;

l 2 / l1 = 0.79 nig f 1 (l 2 / l1 ) > 1.0 .
ceRmokelIssr = 0.813M n = 0.813 189 = 153.7kN .m
ceRmokkNal = 0.187M n = 0.187 189 = 35.3kN .m
edaysarEt f 1 (l2 / l1 ) > 1/ ACI Code, Section 13.6.5 bgajfa 85% nm:Um:g;
kgceRmokelIssrRtUv)anEck[eTAFwm nigenAsl; 15% RtUv)anEck[eTAkRmal
ceRmokelIssr.
Fwm = 0.85 153.7 = 130.6kN.m
ceRmokelIssr = 0.15 153.7 = 23.1kN.m
ceRmokkNal = 35.3kN.m
karlMGitm:Um:g;RtUv)anbgajenAkgrUbTI 17>23.
6> KNnam:Um:g;KNnaenAkgTisxI ElVg = 6m . viFIKNnaRsedogKanwgCMhanTI5>
m:Um:g;GviCman M n = 0.65M os = 0.65 410.4 = 266.8kN .m
m:Um:g;viCman M p = 0.35M os = 0.35 410.4 = 143.6kN .m
EbgEck M n / M p eTAFwm/ ceRmokelIssr nigceRmokkNal
EI
l 2 7.6
=
= 1.27
f 1 = s = b = 2.51 BI]TahrN_TI 17>2
l
6
EI
1

f1

l2
= 2.51 1.27 = 3.19 > 1
l1

PaKrynm:Um:g;GviCman nigGviCmanenAkgceRmokelIssrRtUv)anTTYlBItarag 17>3

edaykareFVI interpolation. sRmab; l2 / l1 = 1.27 nig f 1 (l2 / l1 ) > 1.0 PaKryEbgKW
67% .
m:Um:g;GviCmanceRmokelIssr = 0.67M n = 0.69 266.8 = 178.8kN .m
m:Um:g;GviCmanceRmokkNal = 0.33M n = 0.33 266.8 = 88kN .m
eday f 1 (l2 / l1 ) > 1.0 / 85% n 178.8kN.m RtUv)andak;eTAkgFwm. dUcenH
m:Um:g;GviCmanelIFwm = 0.85 178.8 = 152kN.m
m:Um:g;GviCmanceRmokelIssr = 0.15 178.8 = 26.8kN.m
m:Um:g;viCmanelIFwm = 0.85 0.67 143.6 = 81.8kN.m
T.Chhay

562

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

m:Um:g;viCmanceRmokelIssr = 0.15 0.67 143.6 = 14.4kN.m

m:Um:g;viCmanceRmokkNal = 0.33 143.6 = 47.4kN.m
7> brimaNEdkcaM)ac; nigcMnYnEdkRtUv)anbgajenAkgtarag 17>10.
cMNaMfaPaKryEdkTaMgGs;tUcCag max = 0.00137 . dUcenH = 0.9 .

tarag 17>10 karKNnankRmalxNmanFwmBIrTisxagkg

TisEvg
ceRmokelIssr

ceRmokkNal

42.8

23.1

65.6

35.3

3000

3000

3000

3000

150

150

150

150

( MPa)

0.63

0.34

0.97

0.52

PaKryEdk (%)

0.17

0.09

0.26

0.14

As = bd (mm 2 )

765

405

1170

630

As (min) = 0.0018bhs (mm 2 )

972

972

972

972

9DB12

9DB12

11DB12

9DB12

M u (kN .m)

TTwgceRmok b(mm)
km<s;RbsiTPaB d (mm)
Ru =

Mu
bd 2

EdkEdleRCIserIs

TisxI
ceRmokelIssr

ceRmokkNal

M u (kN .m)

26.8

14.4

88

47.4

TTwgceRmok b(mm)

3000

3000

4600

4600

140

140

140

140

( MPa)

0.46

0.24

0.98

0.53

PaKryEdk (%)

0.12

0.06

0.27

0.14

As = bd (mm 2 )

504

252

1739

902

As (min) = 0.0018bhs (mm 2 )

972

972

1490

1490

9DB12

9DB12

16DB12

14DB12

km<s;RbsiTPaB d (mm)
Ru =

Mu
bd 2

EdkEdleRCIserIs Rtg;
karKNnakRmalxNBIrTis

563

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

]TahrN_TI17>7 edayeRbIviFI direct design method kMNt;m:Um:g;viCman nigGviCmancaM)ac; sRmab;

karKNnankRmalxageRkA elx @ nRbBnkRmalBIrTisEdlmanFwmdUcEdlbgajenAkgrUbTI
17>7. eRbIbnk nig TinnyEdl[enAkg]TahrN_TI 17>6.

dMeNaHRsay

1> karkMNt;EdlTamTarbs; ACI Code RtUv)anbMeBjenAkg]TahrN_enH. kMNt;kRmas;

kRmalGb,brma hs edayeRbIsmIkar !&>! nig !&>@ nigCMhanbnbnab;.
snt; hs = 18cm . muxkat;rbs;Fwmxagkg nigxageRkARtUv)anbgajenAkgrUbTI 17>7.
cMNaMfa karbnykRmalxNenARKb;RCugrbs;FwmKW x = y = 38cm
2> a. m:Um:g;niclPaBrbs;Fwm nigkRmalxagkgEdlRtUv)anKNnaenAkg]TahrN_TI 17>2 KW
TaMgBIrTis I b = 928924.6cm 4
TisEvg I s = 291600cm 4
I s = 369360cm 4
TisxI
b. KNna I b nig I s sRmab;Fwmxag nigkRmalxNxag
30 38 3

183
+ (68 18)(13.5)2 +
+ (30 38)(14.5)2
Fwmxag I b = 68 12
12

= 632987cm 4
500
KNna I s sRmab;ceRmokxagcugEdlRsbeTAnwgFwmxag EdlmanTTwg = 7600
+
2
2
= 4050mm

kRmalxagcug I s = 40512 18 = 196830cm 4

3> a. KNna f f = EI b / EI s
.6
TisEvg l = 928924
= 3.19
291600
.6
= 2.51
TisxI s = 928924
369360
632987
Fwmxag = 196830
= 3.22
3.19 + 2.51 2 + 3.22
= 2.86
mFm = fm =
4
b. = pleFob clear span EvgelI clear span xI
3

T.Chhay

564

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca
=
c.

Department of Civil Engineering

7.1
= 1.29
5.5

KNna hs
hs min

420

7100 0.8 +

1400

=
= 150mm
36 + 5 1.29(2.86 0.2 )

b:uEntmenHminRtUvtUcCag
420

7100 0.8 +

1400

=
= 164mm
36 + 9 1.29

lub
eRbI hs = 18cm > 9cm tmGb,brmaEdlkMNt;eday code
4> KNnabnemKuN wu = 14.28kN / m 2 BI]TahrN_TI 17>6
5> KNnam:Um:g;saTicsrub M ol = 539.9kN .m M os = 410.4kN .m BI]TahrN_mun
6> KNnam:Um:g;KNnaenAkgTisxI ElVg = 6m edaysarkRmalCab;kgTisenH m:Um:g;mantm
dUcnwgm:Um:g;Edl)anKNnaenAkg]TahrN_TI 17>6 ehIyRtUv)anbgajenAkgrUbTI 17>23
sRmab;kRmalxagkg.
7> KNnam:Um:g;enAkgmYykRmaledayeRbIemKuNEdl[enAkgtarag 17>2 brUbTI 17>14
krNITI 3
m:Um:g;GviCmanxagkg M ni = 0.7M o = 0.7 539.9 = 378kN .m
m:Um:g;viCmankgElVg M p = 0.57M o = 0.57 539.9 = 307.7kN .m
m:Um:g;GviCmanxageRkA M ne = 0.16M o = 0.16 539.9 = 86.4kN .m
cMNaM RbsinebIviFI modified stiffness method RtUv)aneRbI enaH C = 3857.27 106 mm 4 /
k ta = 39.13E c 10 6 / K c = 5.79 E c 10 6 / K s = 1.56 E c 10 6 / k b = 4.95E c 10 6 /
k b = 8.94 Ec 10 6 nig ec = 1.37 . m:Um:g;GviCmanxagkgkayCa 373.7 kN.m Rbhak;
RbEhlKa m:Um:g;viCmankayCa 252.75kN.m fycuH 18% nigm:Um:g;GviCmanxageRkAkay
Ca 202.86 ekIneLIg 235%
8> karEbgEckm:Um:g;kRmaleTAFwm ceRmokelIssr nigceRmokkNal
hs min

l2
6
=
= 0.79
f 1 = l = 3.19
l1 7.6
l
f 1 2 = 3.19 0.79 = 2.52 > 1.0
l1

karKNnakRmalxNBIrTis

565

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

KNna C

x x 3 y

C = 1 0 . 63
y 3

Eckmuxkat;rbs;FwmxagCactuekaNBIrEdleFVIy:agNaedIm,ITTYl)an C Gtibrma.
ykmuxkat;Fwm 300 560mm x = 300mm / y = 560mm nigmuxkat;kRmal
180 380mm x = 180mm / y = 380mm
300 300 3 560
180 180 3 380

+ 1 0.63
C = 1 0.63
= 3857.27 10 6 mm 4

560
3
380
3

t =
a.

b.

T.Chhay

E cb C
3857 10 6
=
= 0.66
2 Ecs I s 2 291600 10 4

EbgEckm:Um:g;GviCmanxagkg M ni eyagtamtarag 17>3 nigedaykareFVI

interpolation PaKrym:Um:g;Edldak;enAkgceRmokelIssr sRmab; l 2 / l1 = 0.79
nig f 1l2 / l1 > 1.0 KW 81.3%
ceRmokelIssr = 0.813 378 = 307.3kN.m
ceRmokkNal = 0.187 378 = 70.7kN.m
edaysar f 1l2 / l1 > 1.0 / 85% nm:Um:g;enAkgceRmokssrKWRtUv)andak;eTAkgFwm
dUcenH
Fwm = 0.85 307.3 = 261.2kN.m
ceRmokelIssr = 0.15 307.3 = 46.1kN.m
ceRmokkNal = 70.7kN.m
EbgEckm:Um:g;viCman M p eyagtamtarag 17>3 nigedayeFVI interpolation PaKry
m:Um:g;EdlRtUvdak;kgceRmokelIssrKW 81.3% 85% ntmenHRtUv)andak;eTAFwm.
dUcenH
Fwm = 0.85(0.813 307.7) = 212.6kN.m
ceRmokelIssr = 0.15(0.813 307.7) = 37.5kN.m
ceRmokkNal = 0.187 307.7 = 57.5kN.m

566

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca
c.

Department of Civil Engineering

EbgEckm:Um:g;GviCmanxageRkA M ne eyagtamtarag 17>5 nigedaykareFVI

interpolation PaKrym:Um:g;Edldak;eTAkgceRmokelIssr sRmab; l 2 / l1 = 0.79
nig f 1l2 / l1 > 1.0 KW 92.5% nig 85%
Fwm = 0.85(0.925 86.4) = 67.9kN.m
ceRmokelIssr = 0.15(0.925 86.4) = 12kN.m
ceRmokkNal = 0.075 86.4 = 6.48kN.m

17>9> m:Um:g;KNnaenAkgssr (Design Moments in Column)

enAeBlEdlkarviPaKeRKagsmmUl (equivalent frame) RtUv)aneFVIeLIgedayviFI direct
design method m:Um:g;enAkgssrEdlbNalBIbnkminesI (unbalanced load) enAelIFwmEk,rRtUv)an
TTYlBIsmIkarxageRkam EdlRtUv)ankMNt;eday ACI Code, Section 13.6.9:
M u = 0.07[(wd + 0.5wl )l 2 l n2 w' d l ' 2 (l ' n )2 ]
!&>@@ a
RbsinebIviFI modified stiffness method EdleRbI K ec nig ec enaHm:Um:g; M u RtUv)ankMNt;
dUcxageRkam
0.08[(wd + 0.5wl )l 2 l n2 w' d l ' 2 (l ' n )2 ]
!&>@@ b
Mu =

1 +
ec

Edl

nig wl = bnkefr nigGefremKuNenAelIElVgEdlEvgCag

w' d = bnkGefremKuNenAelIElVgEdlxI
l n nig l' n = RbEvgnElVgEdlEvgCag nigxICag erogKa
wd

!&>@!
m:Um:g;enAkgsmIkarTI !&>@@ KYrRtUv)anEbgEckrvagssrxagelI nigxageRkamkRmalxNRtg;
tMNreTAtamsmamaRtPaBrwgRkajrgkarBt;rbs;va rUbTI 17>24. sRmab;ElVgEdlesIKa l2 = l '2
nig ln = l 'n
M u = 0.07(0.5wl l 2 l n2 )
!&>@# a
ec =

karKNnakRmalxNBIrTis

K ec
(K s + K b )

567

T.Chhay

mhaviTalysMNg;sIuvil

Mu =

NPIC

0.08 0.5wl l 2 l n2

1 +

ec

!&>@@ b

karbegItsmIkarTaMgenHedayQrenAelIkarsnt;fabnkGefrBak;kNaleFVIGMeBIenAelIElVg
EvgCag b:uEnbnkGefreFVIGMeBIenAelIElVgTaMgBIr. smIkar !&>@@ kGacRtUv)anGnuvtsRmab;ssrxag
eRkA edaysnt;faRbEvgElVgEdlxICagKW 0 rUbTI 17>25.

T.Chhay

568

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

17>10> karbMElgm:Um:g;minesIeTAkgssr (Transfer of Unbalanced Moments to Columns)

17>10>1> karbMElgm:Um:g; (Transfer of Moment)
kgkarviPaKeRKagsmmUl (equivalent frame) enAkgGKar m:Um:g;ekItmanenARtg;tMNkRmal
xN-ssr EdlekItBIkmaMgxagdUcCa kmaMgxl; kmaMgrBaydI bkmaMgTMnajminesI (unbalanced
gravity load) edayeFVI[manm:Um:g;minesIenAkgkRmalenAelIRCugEdlpynwgssr. mYycMENkn
m:Um:g;minesIRtUv)anbMElgeTAssredaykarBt; ehIycMENkEdleFVI[m:Um:g;manlMnwgRtUv)anbBan
edaykmaMgkat;bBarEdlmanGMeBIenAelImuxkat;eRKaHfak;sRmab;kmaMgkat;pug. RbEhl 60% nm:U
m:g;EdlRtUv)anbMElgeATcugssrTaMgsgagRtg;tMNRtUv)anbMElgedaykarBt; ehIy 40% EdlenA
sl;RtUv)anbMElgedaykmaMgkat;cakpit bkmaMgrmYl enARtg;muxkat;EdlmanTItaMg d / 2 BIp
ssr. ACI Code, Section 13.5.3 bgajfacMENkn unbalanced moment EdlRtUv)anbMElg
edaykarBt; M f enARtg;tMNkRmalxN-ssrEdlRtUv)anKNnadUcxageRkam
M f = f Mu
!&>@$
1
1
!&>@%
=
f =

2 b
2 c1 + d
1 +

3 c2 + d

1+

3 b2

ehIym:Um:g;EdlbMElgedaykmaMgkat;TTwgKW
(

M v = 1

)M u = M u M f

!&>@^

Edl c1 nig c2 CaRbEvgrbs;RCugBIrnssrctuekaN bssrctuekaNEkgsmmUl/ b1 = (c1 + d ) nig

b2 = (c 2 + d ) . enAeBlEdl c1 = c 2 / M f = 0.6M u ehIy M v = 0.4 M u .
17>10>2> karRbmUlpMsrsEdkenAelIssr
(Concentration of Reinforcement Over the column)

sRmab;karbMElgm:Um:g;edaypal;eTAssr vaCakarcaM)ac;kgkarRbmUlpMsrsEdkdak;kg
ceRmokssrkgTTwgkMNt;mYyenAelIssr Epknm:Um:g;EdlbMElgedaykarBt; M f RtUv)anBicarNa
[ eFVIGMeBItamryTTwgkRmalxNesInwgTTwgssr c2 bUknwg 1.15hs enARKb;RCugTaMgGs;rbs;ssr
b tamryTTwgkRmalxNesInwg (c2 + 3hs ) (ACI Code, Section 13.5.3). srsEdkGacRtUv)an
RbmUlpMenAelIssredayKMlatCitKa bedayeRbIsrsEdkbEnm.
karKNnakRmalxNBIrTis

569

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

17>10>3> kugRtaMgkmaMgkat;EdlbNalBI M f
(Shear Stress Due to) M f
kugRtaMgkmaMgkat;EdlbegItedaycMENkn unbalanced moment M v RtUv)anpSMeLIgCa
mYynwgkugRtaMgkmaMgkat;EdlbegItedaykmaMgkat; Vu Edl)anBIkmaMgbBar. kugRtaMgkmaMgkat;
TaMgBIrRtUv)ansnt;[eFVIGMeBIenACMubrievNbg;EdlmanTItaMgenAcmay d / 2 BIpssrdUcbgajenA
kgrUb 17>26. smIkarsRmab;KNnakugRtaMgkmaMgkat;KW
V
M C
!&>@&
v1, 2 = u v
Ac
Jc
Edl Ac = RkLapmuxkat;eRKaHfak;enACMuvijssr
J c = m:Um:g;niclPaBb:UElnRkLapEdlRsbeTAnwgm:Um:g;EdlmanGMeBIbEnmBIelIm:Um:g;nicl
PaBnmuxkat;xagcugEdleFobnwgGkSTIRbCMuTmn;nmuxkat;eRKaHfak;.

T.Chhay

570

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

sRmab;ssrxagkg
nig

!&>@*
!&>@(

Ac = 2d (x + y )
Jc =

xd 3
d x 3
+ x2 y +

2 3
6

sRmab;ssrxageRkA Ac = d (2 x + y )
!&>#0
2dx 3
xd 3
nig
!&>#!
(2 x + y )dx12 +
Jc =
3
6
Edl x1 / x2 nig y Rtuv)anbgajenAkgrUbTI 17>26. kugRtaMgkmaMgkat;Gtibrma
v1 = Vu / Ac + M v C J c RtUvEttUcCag f 'c 3 ebImindUecaHeT EdkTb;kmaMgkat;RtUv)andak;.

]TahrN_TI17>8 kMNt;m:Um:g;enAelIssrxagkg nigxageRkAkgTisedAEvgrbs; flat plate kg]Ta-

hrN_TI 17>4.

dMeNaHRsay

1> kMNt;m:Um:g;ssrxageRkA. BI]TahrN_TI 17>4 nig 17>5

wd = 1.2 7.75 = 9.3kN / m 2
0.5wl = 0.5 1.6 4.8 = 3.84kN / m 2

Unbalanced moment
Mu =

1
1 +
ec

l n = l ' n = 7m

l 2 = l ' 2 = 6m

= 1.96

EdlRtUvbMElgeTAssrxageRkAedayeRbIsmIkar !&>@@ b KW

0.08
(9.3 + 3.84)6 7 2 0 = 157.7kN .m
1.96

RbsinebIsmIkar !&>@@ a RtUv)aneRbI/ M u = 270.4kN .m EdlCatmEdlsnSMsMc.

2> enATRmxagkg PaBrwgRkajrbs;kRmalenAsgagssrRtUv)aneRbIedIm,IKNna ec
K ec
ec =
!&>@!
(K s + K b )
BI]TahrN_TI 17>5/ K ec = 4.34Ec 106 / K s = 4.17 Ec 106 nig K b = 0 dUcenH
ec =

4.34 Ec 10 6
2 4.17 Ec 10 6

1
1 +
ec

= 0.52

= 2.92

BIsmIkar 17>22 b m:Um:g; unbalance enAssrxagkgKW

karKNnakRmalxNBIrTis

571

T.Chhay

mhaviTalysMNg;sIuvil
Mu =

NPIC

0.08
(9.3 + 3.84)6 7 2 9.3 6 7 2 = 30.93kN .m
2.92

RbsinebIsmIkar !&>@@ a RtUv)aneRbI/ M u = 79kN .m EdlCatmEdlsnSMsMc.

]TahrN_TI17>9 sRmab; flat plate kg]TahrN_TI 17>4 KNnakugRtaMgkmaMgkat;enAkgkRmal

xNRtg;muxkat;eRKaHfak;EdlbNalBI unbalanced moment nigkmaMgkat;TTWgenAelIssrxagkg

nig ssrxageRkA. RtYtBinitkarRbmUlpMsrsEdk nigkmaMgrmYlkgrbs;ssrxageRkA. eK[
f 'c = 28MPa nig f y = 420MPa

dMeNaHRsay

1> m:Um:g;EdleFVI[KanlMnwg (unbalanced moment) enAssrxagkgKW M u = 30.93kN .m

]TahrN_ TI 17>8 Edl f = 0.6 edaysar c1 = c2 = 50cm .
m:Um:g;EdlRtUvbMElgedaykarBt;KW M f = f M u = 0.6 30.93 = 18.56kN .m
m:Um:g;EdlbMElgedaykmaMgkat;KW M v = M u M f = 30.93 18.56 = 12.37kN .m
ma:gvijeTot m:Um:g;E;dlKNnaBIsmIkar !&>@@ a GacRtUv)aneRbIedIm,IbegItkugRtaMgkmaMgkat;
FMCag.
edayeRbI d = 22.2cm ]TahrN_TI 17>4
Vu = 17(6 7.5 0.722 2 ) = 756.1kN
BIrUbTI 17>27
Ac = 4 72.2 22.2 = 6411.4cm 2
Jc =
=

xd 3
d x 3
+ x2 y +

2 3
6

72.2
22.2 72.2 3
+ (72.2 )2 72.2 +
22.2 3 = 5.7 10 6 cm 4

2 3
6

v max =

756.1
6411.4 10 4

0.722
2 = 1258kN / m 2
2
5.7 10

12.37

v min = 1179.3 78.34 = 1100kN / m 2

vc = f 'c / 3 = 0.75 28 / 3 = 1.323MPa = 1323kN / m 2 > 1258kN / m 2

T.Chhay

572

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

2> sRmab;ssrxageRkA unbalanced moment EdlRtUvbMElgedaykarBt; M f enARtg;tMN

kRmalxN-ssrKWesInwg f M u Edl M u = 157.7kN .m . cMNaMfa c1 = c2 = 50cm /
d = 22.2cm enAkgTisedAEvg nig f = 0.6 sRmab;ssrkaer.
M f = 0.6 157.7 = 94.6kN .m

m:Um:g;EdlRtUvbMElgedaykmaMgkat;TTwgKW
M v = M u M f = 157.7 94.6 = 63.1kN .m

3> sRmab;karbMElgedaykmaMgkat;enAssrxageRkA muxkat;eRKaHfak;sitenAcmay d / 2

BIpssr rUbTI 17>28.
wu = 17kN / m 2
Vu = 17(6 3.9 0.611 0.722) = 390kN

kMNt;TItaMgTIRbCMuTmn;rbs;muxkat;eRKaHfak;edayKitm:Um:g;eFob AB
61.1

2 61.1
= (2 61.1 + 72.2 )xl
2

dUcenH xl = 19.2cm . RkLapmuxkat;eRKaHfak; Ac KW

22.2(2 61.1 + 72.2 ) = 4315.7cm 2 . KNna J c = I x + I y sRmab;RkLapesIKaBIr
22.2 61.1 EdlmanRCugRsbeTAnwgTisedAn m:Um:g; CamYynwgRkLap 22.2 72.2
EkgeTAnwgTisedArbs;m:Um:g;. m:Um:g;niclPaBnRkLapTaMgBIrRtUvKiteFobnwgGkS CD .
bh 3

Jc = I x + I y =
+ Ax 2
12

karKNnakRmalxNBIrTis

573

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

61.13
61.1
61.1
22.2 3 + 72.2 22.2 19.2 2
= 2 22.2
+ 22.2 61.1
19.2 +
12
2
12

= 1.84 10 6 cm 4

bedayeRbIsmIkar !&>#! sRmab;ssrxageRkA. KNNakugRtaMgkmaMgkat; nominal Gtibrma

nigGb,brmaedayeRbIsmIkar !&>@&
Vu M v C
390
63.1 0.192
+
=
+
= 1562.1kN / m 2
4
2

Ac
Jc
4315.7 10
1.84 10
V
M C
= u v = 903.7 658.4 = 245.3kN / m 2
Ac
Jc

v max =
v min

vc = f 'c / 3 = 0.75 28 / 3 = 1.323MPa = 1323kN / m 2 < 1562.1kN / m 2

T.Chhay

574

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

kugRtaMgkmaMgkat;TTwgFMCagkugRtaMgkmaMgkat;GnuBaat dUcenHeKRtUvbegInkRmas;kRmalxN
beRbIEdkTb;kmaMgkat;.
4> RtYtBinitkarRbmUlpMsrsEdkenAssrxageRkA KWRtYtBinitPaBRKb;RKan;rbs;lTPaBTb;
karBt;rbs;muxkat;edIm,IbMElgm:Um:g;GviCmaneTAkgssrxageRkA. muxkat;eRKaHfak;rbs;kM
ralxNRtUv)anbnay 1.5hs enAelIRCugnImYyrbs;ssr Edl[TTwg (50 + 3 25)
= 125cm nigkRmas; 25cm . m:Um:g;srubenAkgTTwgceRmokelIssr 3m KW 162.4kN.m
Edl)anKNnaenAkg]TahrN_TI 17>4 CMhan %. m:Um:g;enAkgTTwg 125cm KWesInwg
125
= 67.67 kN.m .
162.4
300
RbsinebIeKeRbIKMlatEdkesIKaenAkgceRmokelIssr enaHeKRtUvkarbrimaNEdkbEnmenAkg
TTwg 125cm sRmab;m:Um:g;EdlesInwg M f 67.67 = 94.6 67.67 = 26.93kN .m . brimaN
EdkcaM)ac; As = 340mm 2 dUcenH 4DB12 452mm 2 GacRtUv)aneRbI.
dMeNaHRsayepSgeTotKWedIm,IteRmobEdkenAkgceRmokelIssredIm,IbegInbrimaNEdkenA
kgTTwkceRmok 125cm .
brimaNEdkcaM)ac;enAkgTTwgceRmokKYrRKb;RKan;edIm,ITb;nwgm:Um:g;EdlesInwg 0.6 dgnm:Um:g;
GviCmanenAkgceRmokelIssr b 0.6 162.4 = 97.44kN.m .
Mu
snt; a = 25mm
As =
a

f y d

As =

97.44 10 6
= 1230mm 2
25

0.9 420 222

2

As f y
1230 420
a=
= 17.4mm
=
0.85 f 'c b 0.85 28 1250

RtYtBinit
eRbI 11DB12 enAkgTTwg 125cm edayEckesIKasgagssredayKitBIGkSssr rUbTI
17>29. kardak;EdkbEnm 4DB12 Edl)anbgajxagelICadMeNaHRsaykan;EtRbesI.
5> kmaMgrmYlenAkgkRmalxN kmaMgrmYlBIRCugTaMgsgagrbs;ssrxageRkAesInwg 40%
nm:Um:g;elIceRmokTTwg.
Tu = 0.4 162.4 = 64.96kN .m
64.96
= 32.48kN.m
2

kmaMgrmYlenAelIRCugmag
karKNnakRmalxNBIrTis

575

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

TTwgnmuxkat;kRmalxNEdlesInwgTTwgssrRtUv)ansnt;[Tb;nwgkugRtaMgrmYl
1
Tu = vtu x 3 y
3

Edl x = 25cm nig y = 50cm . muxkat;eRKaHfak;KWenAcmay d BIpssr rUbTI 17>30.

edaysnt;fakmaMgrmYlERbRbYlmanragCaExSekag)a:ra:bUleTAGkSkRmalxN enaHkmaMg
rmYlenAcmay d KW
2

3.5 0.222
Tu = 32.48
= 28.42kN .m
3 .5

sRmab;ersIusg;Tb;karrmYlrbs;ebtug Acp = 25 50 = 1250cm 2 / Pcp = 2(25 + 50)

2
= 150cm tamsmIkar !%>!( Tcp = 0.75 28 (1250 10 2 ) 10 6 / (3 1500 )
= 13.78kN.m / Ta = 13.78 / 4 = 3.445kN .m < 28.42kN .m .

T.Chhay

576

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

eKRtUvkarEdkTb;kmaMgrmYl. EdkkgbiTCitcaM)ac; nigEdkbeNaybEnmRtUv)anKNnadUc

Bnl;enAkgemronTI 15. muxkat;cugeRkayRtUv)anbgajenAkgrUbTI 17>30. vaCakar
RbesIrRbsinebImanFwmxagenAssrxageRkAedIm,IbegInPaBrwgRkajTb;karrmYlrbs;kRmal
xN.

]TahrN_TI17>10 kMNt;brimaNEdkTb;kmaMgkat;TTwgcaM)ac;sRmab;kRmal flat plate xagkg

edayBicarNadUcteTA kmaMgkat;put Vu = 870kN / kRmas;kRmalxN = 23cm / d = 19cm /
f 'c = 28MPa / ehIyTMhMssrKW 50 50cm .

dMeNaHRsay

1> kMNt; Vc =

f 'c bo d / 3

sRmab;kmaMgkat;BIrTis

bo = 4(50 + d ) = 4(50 + 19 ) = 276cm

Vc = 0.75 28 2760 190 10 3 / 3 = 693.7kN

edaysar Vu = 870kN > Vc EdkTb;kmaMgkat;TTwgRtUvkarcaM)ac;.

2> Vc GnuBaatGtibrma EdleRbIEdkkmaMgkat;TTwgesInwg
f ' c bo d / 2 = 1.5(Vc ) = 1040.6kN > Vu EdkTb;kmaMgkat;TTwgGacRtUv)aneRbI.
3> EdkTb;kmaMgkat;TTWgGacpSMeLIgedayEdksrs Edkrag dUcCaFwmGkSr I b special
large-head studs welded to a steel strip emIlrUbTI 17>9. kg]TahrN_enH dMeNaHEdl
minfedayeRbIEdkTb;kmaMgkat;FmtaRtUv)anTTYlykmkeRbI rUbTI 17>9 f . EdkTb;kM
laMgkat;TTwgRtUv)andak;enAelIRCugTaMgbYnrbs;ssrxagkg bbIRCugsRmab;ssrxageRkA
sRmab;RbEvg d + a rUbTI 17>31. cmay a RtUv)ankMNt;eday[ Vc = Vu enAmux
kat; bo EdlbgajedayExSdac; nigsnt;fa Vc = f 'c bo d / 6 .
bo = 4(c + a 2 ) = 4(500 + 2a )
0.75 28 4(500 + a 2 )190 / 6 = 870000
enATIenH a = 87cm ehIy (a + d ) = 106cm dUcenHyk 110cm .

karKNnakRmalxNBIrTis

577

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

4> KNnaEdkkmaMgkat;TTwg
Vs = (Vu Vc ) = 870 693.7 = 176.3kN
Vs = 235kN

sRmab;pmYyrbs;muxkat;eRKaHfak; V4s = 58.75kN

eRbI DB10 EdkkgGkSr U / Av = 157mm 2 sRmab;eCIgBIr. KMlatKW s = Av f y d / Vs
= 157 420 190 / 58750 = 213mm . KMlatGtibrmKW d / 2 = 190 / 2 = 95mm / yk
s = 90mm .
5> karEbgEckEdkkg cMnYnEdkkgkgmYyRCugrbs;ssrKW 110 / 9 = 12.2 b 13 kg.
cmaysrubKW 13 90 = 1170mm rUbTI 17>31
Vs

]TahrN_TI17>11 Flat-Slab Floor System

edayeRbIviFI direct design method KNnaRbePTkRmalxN flate slab xagkgEdlmanEt drop

panel ehIyvamanTMhM 7.5 6m rUbTI 17>32. RKb;kRmalTaMgGs;RtUv)anRTedayssrTMhM
50 50cm RbEvg 3.6m . kRmalxNRTnUvbnkBRgayesIGefrKanemKuN 4.8kN / m 2 nigbnkefr
KanemKuN 1.15kN / m 2 edayKanrYmbBalTmn;pal;. eK[ f 'c = 28MPa nig f y = 420MPa
dMeNaHRsayKWRsedognwg]TahrN_TI 17>3.
T.Chhay

578

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

karKNnakRmalxNBIrTis

Department of Civil Engineering

579

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

dMeNaHRsay

1> kMNt;kRmas;kRmal nigkRmal; drop panel

a. RbEvg clear span KW 7.5 0.5 = 7m . sRmab;kRmalxageRkA tmGb,brmarbs;
h = l n / 33 = 21.2cm b:uEnkRmalxagkg tmGb,brmarbs; h = l n / 36 = 19.4cm .
eRbIkRmalxNEdlmankRmal 20cm kRmas;TMlak;cuHBIeRkamkRmalxNKW h / 4 = 5cm
dUcenHkRmas; drop panel KW 25cm .
b. TMhMrbs; drop panel KW L / 6 = 7.5 / 6 = 1.25m sRmab;TisnImYyBIGkSrbs;TRmkg
TisedAEvg nig 6 / 6 = 1m enAkgTisedAxI. dUcenHTMhMsrubrbs; drop panel KW
2.5 2m rUbTI 17>32.
2> KNnabnkemKuN
slab load = 1.15 + 0.2 25 = 6.15kN / m 2
Wu = 1.2 6.15 + 1.6 4.8 = 15.06kN / m 2
drop panel load = 1.15 + 0.25 25 = 7.4kN / m 2

Wu = 1.2 7.4 + 1.6 4.8 = 16.56kN / m 2

edaysarEt drop panel manRbEvg L / 3 kgTisnImYy tmmFm

1
2
Wu = 15.06 + 16.56 = 15.56kN / m 2
3
3

3> RtYtBinitkmaMgkat;BIrTis enAcmay d / 2 BIpssr

a. enAkg drop panel
d = 25 2 0.8 = 22.2cm
bo = 4(50 + 22.2 ) = 288.8cm

Vu = 15.56 7.5 6 (0.722 )2 = 692kN

0.75
Vc =
f 'c bo d =
28 (2888 222 ) 10 3 = 848.1kN > Vu
3
3

b.

enAkgkRmal d = 20 2 0.8 = 17.2cm ehIy bo RtUv)anvas;ecjBI drop panel

enAcmay 17.2 / 2 = 8.6cm .
bo = 2(250 + 17.2 ) + 2(200 + 17.2 ) = 968.8cm
Vu = 15.56[7.5 6 2.672 2.172] = 609.9kN

0.75
Vc =
f 'c bo d =
28 (9688 172 ) 10 3 = 2204kN > Vu
3
3

T.Chhay

580

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

tarag 17>11> karKNnaRbBnkRmal flat slab xagkg

M ol = 571.8kN .m

emKuNm:Um:g;

TisEvg
ceRmokelIssr

ceRmokkNal

0.49 M o

0.21M o

0.16 M o

0.14 M o

M u (kN .m)

-280.2

120.0

-91.5

80.0

TTwgceRmok b(mm)

3000

3000

3000

3000

210

160

160

160

( MPa)

2.12

1.56

1.19

1.04

PaKryEdk (%)

0.59

0.43

0.32

0.28

As = bd (mm 2 )

3717

2064

1536

1344

As (min) = 0.0018bhs (mm 2 )

1350

1080

1080

1080

19DB16

11DB16

14DB12

12DB

km<s;RbsiTPaB d (mm)
Ru =

Mu
bd 2

EdkEdleRCIserIs Rtg;
M os = 441.3kN .m

emKuNm:Um:g;

TisxI
ceRmokelIssr

ceRmokkNal

0.49 M o

0.21M o

0.16 M o

0.14 M o

M u (kN .m)

-216.2

92.7

-70.6

61.8

TTwgceRmok b(mm)

3000

3000

4500

4500

km<s;RbsiTPaB d (mm)

210

160

160

160

( MPa)

1.63

1.21

0.61

0.54

PaKryEdk (%)

0.45

0.33

0.16

0.14

As = bd (mm 2 )

2835

1584

1152

1008

As (min) = 0.0018bhs (mm 2 )

1350

1080

1620

1620

15DB16

15DB12

15DB12

15DB12

Ru =

Mu
bd 2

EdkEdleRCIserIs Rtg;
karKNnakRmalxNBIrTis

581

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

kmaMgkat;mYyTisminmaneRKaHfak;.
4> KNnam:Um:g;saTicsrubenAkgTisEvg nigTisxI
c.

6 72
= 571.8kN / m 2
8
7.5 5.5 2
= 15.56
= 441.3kN / m 2
8

M ol = 15.56
M os

TTwgrbs;ceRmokelIssrkgTisnImYyKW 600
= 300cm
2
b:uEnTTwgrbs;ceRmokkNal KW 300cm kgTisedAEvg nig 450cm kgTisedAxI.
5> KNnam:Um:g; nigsrsEdkRtUv)anbgajtarag 17>11. eRbIkm<s;RbsiTPaBmFm
d = 250 40 = 210mm sRmab;ceRmokelIssr nig d = 200 40 = 160mm sRmab;
ceRmokkNal. karBRgaysrsEdklMGitmanlkNRsedogKakg]TahrN_ flat plate.

T.Chhay

582

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

17>11> kRmalxN Waffle (Waffle Slabs)

RbBnkRmalxN waffle slab BIrTispSMeLIgedayrnUtebtugEdlFmtaRbsBVKaCamMuEkg.
kRmalxNenHGacsagsg;edayKanFwm EdlkgkrNIenHk,alssrtan;RtUv)aneFVIenAelIssredIm,I
karBarkarpugEdlbNalBIkmaMgkat;. FwmTUlaykGacRtUv)aneRbIenAeLIGkSssrsRmab;km<s;esI.
Bum<canEdkragkaer b fiberglass pan RtUv)aneRbICaTUeTAedIm,IbegItRTnug. kRmalxNesIgkRmas;
75mm eTA 125mm RtUv)ancak;CamYyRTnugedIm,IbegItCa waffle slab.
kRmalnImYyRtUv)anEbgEckCaceRmokelIssr nigceRmokkNal. ceRmokelIssrrYm
bBalRTnugTaMgGs;EdlCab;eTAnwgk,alssrtan; ehIyceRmokkNalsitenAcenaHceRmokelI
ssr. EdkRtg; bEdkBt;GacRtUv)aneRbICaEdkBRgwgenAkg waffle slab. karKNna waffle slab
BIrTismanlkN RsedogKanwgkarKNna flat slabEdr edayKitk,alssrtan;Ca drop panel.
edIm,IkarBarkmaMgTaj FMenAkgk,alssr eKRtUveRbITMhMssrFMRKb;RKan; bdak; shear cap.
kgkarKNna waffle slab kRmalxagelI nigrnUtnImYybegItCamuxkat;GkSret EdlesIkm<s;
kg flat plate. dUcenH ElVgEvgEdlRTbnkFn;GacRtUv)anKNnaCamYynwgkarsnSMsMcebtug)an
eRcIn. kRmal waffle slab k)anpl;nUvBidandKYr[Tak;Taj Edl)anBIkarelcecjnUvRTnug beday
karlMGePIg. kRmalsg;darnBum<canEdlRtUv)aneRbICaTUeTAenAkg waffle slab
GacsitenAkgCeRmIsBIrxageRkam
1> Bum<cankaerTMhM 75cm 75cm CamYynwgkRmas;kRmal 75mm nigTTwgrnUt 150mm
manKM lat 90cm KitBIGkS. sg;darRbePTenH mankm<s;ERbRbYlBI 200mm eTA
500 mm eday ekIneLIgmg 50mm . eyagtam]TahrN_TI17>12 nigrUbTI 17>33.
2> Bum<cankaerTMhM 48cm 48cm CamYynwgkRmas;kRmal 75mm nigTTwgrnUt 125mm
manKM lat 60cm KitBIGkS. sg;RbePTenH mankm<s;ERbRbYlBI 150mm eTA 300mm
edayekIneLIgmg 50mm . BtmanepSgBIkRmalRtUv)anbgajenAkgtarag 17>12.

karKNnakRmalxNBIrTis

583

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

tarag 17>12 lkNmuxkat;

sRmab;rnUt Bum<canTMhM 75cm 75cm
maDxl;
RkLap
kRmalxag km<s;rnUt
elI (mm ) (mm) ( 10 6 mm 3 / pan ) ( 10 4 mm 2 )

Ycg

Ig

(mm )

( 10 mm )
8

75

200

107.4

10.09

81.91

5.45

75

250

132.7

11.04

98.58

9.02

75

300

157.4

12.02

116.54

13.85

75

350

181.4

13.05

135.4

20.08

75

400

204.9

14.12

155

27.88

75

500

250

16.39

195.7

48.8

115

200

107.4

13.69

95.12

8.25

115

250

132.7

14.64

109.4

12.89

115

300

157.4

15.62

141.8

19.06

115

350

181.4

16.65

141.8

26.95

115

400

204.9

17.72

159.45

36.72

115

500

250

19.99

196.88

62.59

sRmab;rnUt Bum<canTMhM

48cm 48cm

75

150

32

6.56

72.14

2.31

75

250

42.01

7.33

89.11

4.25

75

250

51.57

8.15

107.57

7.03

75

300

60.75

90

127.08

10.78

115

150

32

8.96

87.47

3.84

115

250

42.01

9.73

102.21

6.44

115

250

51.57

10.55

118.53

10.08

115

300

60.75

11.4

136.12

14.91

T.Chhay

584

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

karKNnakRmalxNBIrTis

Department of Civil Engineering

585

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

]TahrN_TI17>12 Waffle Slab

KNnaRbBnkRmalxN waffle floor EdlpSMeLIgedaykRmalkaerKanFwmedayKitnUvTinnyxag

Rkam rUbTI 17>33
RbEvgElVgedayKitBIGkSssreTAGkSssr= 10m
TTwgrnUt = 150mm manKMlat 900mm BIGkSeTAGkS
km<s;rnUt = 350mm
kRmas;kRmalxN = 75mm
TMhMssr = 500 500mm
bnkefr minrYmbBalbnkpal;= 2.4kN / m 2
f ' c = 35MPa
f y = 420 MPa
bnkGefr= 4.8kN / m 2
T.Chhay

586

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

dMeNaHRsay

1> KNnakRmas;Gb,brmarbs;kRmalxNedayeRbItarag 17>1 hmin = ln / 30 /

l n = 10 0.5 = 9.5m / hmin = 9.5 / 30 = 0.317 m sRmab;kRmalxagkg ehIy
hmin = l n / 33 = 0.288m sRmab;kRmalxagkg. smIkar !&>! nig !&>@ GacRtUv)aneRbI.
edaysnt;km<s;srub 425mm EdlrYmmankRmalkRmas; 75mm nigrnUtkm<s; 350mm .
2> KNnabnkEdlmanGMeBIenAelI waffle floor
a. bnkemKuNnEpktan;enAelIssr = 1.2 0.425 25 = 12.75kN / m 2
b. maDxl;rbs;rnUtkm<s; 350mm KW 181.4 10 6 mm 3 elIRkLap 900 900mm 2 .
Tmn;srubrbs;p 810000mm 2 KW
1.2 25(81 10 4 425 181.4 10 6 ) 10 9 = 4.89kN
bTmn;kg 1m 2 KW 4.89kN / m 2
c. bnkefrbEnmemKuNnigbnkGefremKuNKW 1.2 2.4 + 1.6 4.8 = 10.56kN / m 2 .
bnkBRgayesIemKuNenAelIptan; wu = 10.56 + 12.75 = 23.31 23.5kN / m 2
bnkBRgayenAemKuNenAelIpelIrnUt wu = 10.56 + 4.89 = 15.45 15.5kN / m 2
d. bnkenAelImYykRmal eyagtamrUbTI 17>34 enAelIptan;
W = 23.5 3.6 + 15.5 6.4 = 183.8kN / m

enAelIpelIrnUt W = 15.5 10 = 155kN / m

3> KNnakmaMgkat; nigm:Um:g;saTicsrub
Vu (at face of column) = 183.8 1.55 + 155 3.2 = 780.89kN
M o (at midspan ) = 780.89 4.75 183.8 1.55 3.975 155

3.2 2
= 1783.2kN .m
2

4> RtYtBinitkmaMgkat;pug eyagtamrUbTI 17>35

a. enAkgptan;enAcmay d / 2 BIpssr h = 425mm / d = 425 30 = 395mm /
c(column) = 500mm / bo = 4(500 + 395) = 3580mm /
Vu = 155 6.4 + 183.8 3.6 23.5 0.895 2 = 1634.9kN nig
Vc = f 'c bo d / 3 = 0.75 35 (3580 395) / 3 = 1952kN > Vu
b.

enAkgpelIrnUtenAcmay d / 2 BIRCugrbs;ptan;. kRmas;kRmalKW 75mm . yk

d = 60mm enaH

karKNnakRmalxNBIrTis

587

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

bo = 4(3800 + 60 ) = 15440mm
Vu = 155 6.4 + 183.8 3.6 23.5 3.860 2 = 1304kN

Vc = f 'c bo d / 3 = 0.75 35 (15440 60) / 3 = 1370kN > Vu

5> KNnam:Um:g; nigsrsEdk

a. kRmalxageRkA M o = 1783.2kN .m
m:Um:g;GviCmanxageRkA = 0.26M o = 463.6kN.m
m:Um:g;viCman = 0.52M o = 927.3kN.m
T.Chhay

588

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

m:Um:g;GviCmanxagkg = 0.7M o = 1248.2kN.m

b. kRmalxagkg M o = 1783.2kN .m
m:Um:g;GviCman = 0.65M o = 1159.1kN.m
m:Um:g;viCman = 0.35M o = 624.1kN.m
karlMGItBIkarKNnaRtUv)anbgajenAkgtaragTI 17>13 nigrUbTI 17>36. cMNaMfa
PaKryEdkTaMgGs;tUc ehIy = 0.9
6> KNna unbalanced moment enAkgssr nigRtYtBinitkmaMgkat;sRmab; Vu nig M v
dUckg]TahrN_TI 17>8 nig 17>9.

karKNnakRmalxNBIrTis

589

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

tarag 17>13> karKNnakRmal waffle slab xagkg nigxageRkA rnUt 5enAkgceRmokelI

ssr nigrnUt 6enAkgceRmokkNal
kRmalxageRkA

ceRmokelIssr
xageRkA

emKuNm:Um:g;

ceRmokkNal
xagkg
M

100

60

75

25

40

M u (kN .m)

463.6

556.4

936.2

312

370.9

TTwgceRmok b(mm)

3800

5000

3800

900 (6 ribs)

5000

395

395

395

395

395

( MPa)

0.78

0.71

1.58

2.22

0.48

PaKryEdk (%)

0.21

0.19

0.43

0.61

0.13

As = bd (mm 2 )

3152

3753

6454

2169

2567

As (min) = 0.0018bhs (mm 2 )

2907

765/rib

2907

688.5

637.5/rib

11DB20

2DB25/rib

21DB20

7DB20

2DB22/rib

km<s;RbsiTPaB d (mm)
Ru =

Mu
bd 2

EdkEdleRCIserIs Rtg;

kRmalxagkg

ceRmokelIssr
xageRkA

emKuNm:Um:g;

ceRmokkNal
xagkg
M

60

75

25

40

M u (kN .m)

374.5

869.3

289.8

249.6

TTwgceRmok b(mm)

5000

3800

900 (6 ribs)

5000

395

395

395

395

( MPa)

0.48

1.47

2.06

0.32

PaKryEdk (%)

0.13

0.40

0.57

0.09

As = bd (mm 2 )

2567

6004

2026

1777.5

As (min) = 0.0018bhs (mm 2 )

765/rib

2907

688.5

637.5/rib

EdkEdleRCIserIs Rtg;

2DB25/rib

20DB20

7DB20

2DB22/rib

km<s;RbsiTPaB d (mm)
Ru =

T.Chhay

Mu
bd 2

590

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

karKNnakRmalxNBIrTis

Department of Civil Engineering

591

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

17>12> viFIeRKagsmmUl (Equivalent Frame Method)

enAeBlEdlRbBnkRmalxNBIrTismineKarBtamkarkMNt;rbs; direct design method m:Um:g;
KNnaRtUv)ankMNt;edayviFI equivalent method. enAkgviFIenH GKarRtUv)anEckCaeRKagsmmUltam
BIrTis nigbnab;mkviPaKeRKagkgsanPaBeGLasicsRmab;RKb;lkxNnkardak;bnk. PaBxusKa
rvagviFITaMgBIrsitenAelIviFInkarKNnam:Um:g;tambeNayrbs;ElVg. tRmUvkarsRmab;karKNnaRtUv
)anBnl;dUcxageRkam
k> karBiBNnaBIeRKagsmmUl eRKagsmmUlCaeRKagGKarkgbg; BIrTMhM EdlTTYl)an
edaykarkat;GKarkglMhM bITMhMtambeNayExSkNalElVgcenaHssr rUbTI 17>4.
eRKagsmmUlEdlTTYl)anRtUv)anBicarNadac;edayELkenAkgTisbeNay nigTisTTwg
rbs;GKar. sRmab;bBar kRmalnImYyRtUv)anviPaKdac;edayELk edaysnt;cugsagelI
nig cugxageRkamrbs;ssrRtUv)ansnt;fabgb;. kRmal-Fwm (slab-beam) RtUv)ansnt;
fabgb; enARtg;TRmkRmalBIrEdlQmnwgTRmEdleYigRtUvBicarNa BieRBaHbnkbBarcUlrYm
CamYym:Um:g; tictYcenARtg;tMNrenaH. sRmab;bnkedk eRKagsmmUlpSMeLIgedaykRmal
TaMgGs;sRmab; km<s;TaMgmUlrbs;GKar edaysarkmaMgenAkRmalnImYyCaGnuKmn_n
bnkedkenARKb;kRmalBIelInIv:UEdlRtUvBicarNa. karviPaKeRKagkGaceFVIeLIgedaykmviFI
kMuBTr.
x> karsnt;bnk enAeBlpleFob service live load nig service dead load tUcCagbesInwg
0.75 KMrUnkardak;bnk pattern loading RtUv)aneRbIedayBicarNanUvlkxNxageRkam
- manEt 75% nbnkefremKuNTaMgmUlGacRtUv)aneRbIsRmab;karviPaK.
- m:Um:g;Bt;GviCmanGtibrmaenAkgkRmalRtg;TRmRtUv)anTTYledaykardak;bnkEtelI
kRmalxNBIrEdlCab;Ka.
- m:Um:g;viCmanGtibrmaenAEk,rkNalElVgRtUv)anTTYledaykardak;bnkEtenAelIElVg
qas;.
- m:Um:g;KNnaminRtUvtUcCagm:Um:g;KNnaEdlekIteLIgedaykardak;bnkemKuNeBjelIk
Rmal (ACI Code, Sectoion 13.7.6).
- m:Um:g;GviCmaneRKaHfak;RtUv)anBicarNa[manGMeBIenARtg;pssrctuekaN benA
Rtg;ssrkaersmmUlEdlmanRkLapdUcKasRmab;muxkat;minEmnctuekaNEkg.
T.Chhay

592

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

K> m:Um:g;niclPaBkRmal-Fwm ACI CodekMNt;fakarERbRbYlm:Um:g;niclPaBtamGkS

beNay ssr nigFwmkRmalRtUv)anykmkKitkgkarviPaKeRKag. tMbn;eRKaHfak;sitenA
cenaHExSGkSssr nigpssr/ bracket b capital. tMbn;enHRtUv)anBicarNaCamuxkat;
Rkas;nkRmalxN. edIm,IBicarNakm<s;dFMrbs;ssr nigkarkat;bnyTTwg\TiBlrbs;va
EdlCab;nwg FwmkRmal ACI Code, Section 13.7.3.3 kMNt;fam:Um:g;niclPaBrbs;Fwm
kRmalrvagGkSssr nigpTRmRtUv)ansnt;esInwgFwmkRmalenARtg;pssredayEcknwg
tm (1 c2 l2 )2 Edl c2 CaTTwgssrkgTisedATTwg nig l2 CaTTwgrbs;FwmkRmal.
RkLaprbs;muxkat;TaMgmUlGacRtUv)aneRbIedIm,IKNnam:Um:g;niclPaBrbs;FwmkRmal.
X> m:Um:g;niclPaBrbs;ssr ACI Code, Section 13.7.4 bgajfam:Um:g;niclPaBrbs;ssr
RtUv)ansnt;faesIGnnBIpxagelIrbs;kRmaleTA)atrbs; column capital bFwmkRmal
rUbTI 17>37.
g> PaBrwgRkajrbs;ssr K ec RtUv)ankMNt;eday
1
1
1
=
+
!&>!&
K ec K c K t
Edl K c CaplbUkPaBrwgRkajrbs;ssrxagelI nigssrxageRkamenARtg;cugrbs;va
9 E es C
Kt =
!&>!(
3
c
l 2 1 2
l2

x x 3 y
C = 1 0.63
y 3

!&>@0

c> m:Um:g;ssr enAkgkarviPaKeRKag m:Um:g;EdlkMNt;sRmab;ssrsmmUl enAcugxagelIssr

eRkamkRmal nigenA)atssrelIkRmalRtUv)aneRbIkgkarKNnassr.
q> m:Um:g;GviCmanenAelITRm ACI Code, Section 13.7.7 bgajfasRmab;ssrxagkgm:Um:g;
GviCmanemKuNRtUv)anykenARtg;pssr bp capital b:uEnenAcmayminFMCag 0.1175l1
BI GkSssr. sRmab;ssrxageRkA m:Um:g;GviCmanemKuNRtUv)anykmkKitenARtg;muxkat;
EdlsitenABak;kNalcmayrvagpssr nigEKmrbs;TRm. ssrmuxkat;mUlRtUv)anKit
CassrkaerEdlmanmuxkat;esIKa.
C> plbUkm:Um:g; RbBnkRmalxNBIrTisEdleKarBtamkarkMNt;rbs;viFI direct design
method kGacRtUv)anviPaKedayviFI equivalent frame method . edIm,IFanafaviFITaMgBIr
karKNnakRmalxNBIrTis

593

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

pl;nUvlTpldUcKa ACI Code, Section 13.7.7 bgajfam:Um:g;EdlkMNt;edayviFI

equivalent frame method GacRtUv)ankat;bnyedaysmamaRtEdlplbUknm:Um:g;viCman
nigm:Um:g;GviCmanenAkgkarKNnaminRtUvFMCagm:Um:g;saTicsrub M o .

]TahrN_TI17>13

edayeRbIviFI equivalent frame method viPaKeRKagKMrUxagkgnRbBnkRmal flate plate Edl[enA

kg]TahrN_TI 17>3 EtkgTisedAEvg. RbBnkRmalpSMeLIgedaykRmalbYnkgTisedAnImYy Edl
kRmalnImYymanTMhM 7.5 6m . kRmalnImYyRtUv)anRTedayssrTMhM 50 50cm RbEvg 3.6m .
Service live load KW 3.8kN / m 2 Service dead load KW 6kN / m 2 edayrYmbBalTaMgTmn;pal;
rbs;kRmal. eK[ f 'c = 21MPa / f y = 420MPa . FwmxagminRtUv)aneRbI. rUbTI 17>38

dMeNaHRsay

1> kRmas;kRmalxN 25cm RtUv)aneRCIserIs dUcEdlBnl;kg]TahrN_TI 17>3.

2> bnkemKuNKW wu = 1.2 6 + 1.6 3.8 = 13.28kN / m 2
pleFob Service dead load elI Service live load KW 3.8 / 6 = 0.63 < 0.75 . dUcenHeRKag
GacRtUvKNnaCamYynwgkarbnkemKuN wu [manGMeBIRKb;kRmalCMnYs[karBRgaybnk
tamKMrUnkardak;bnk.
3> kMNt;PaBrwgRkajrbs;kRmal K s

T.Chhay

594

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca
Ks = k

Edl

Department of Civil Engineering

EI s
ls

CaemKuNPaBrwgRkaj nig

Is =

l 2 hs3 6 0.25 3
=
= 7.81 10 3 m 4
12
12

emKuNPaBrwgRkajGacRtUv)ankMNt;edayviFI column analogy method EdlRtUv)anBnl;

enAkgesovePAviPaKrcnasm<n (structural analysis). edayKitfam:Um:g;niclPaBsRmab;kRmal
I s esInwg 1.0 dUckgrUb m:Um:g;niclPaBrvagGkSssr nigpssrKW
1.0
c2
1
l2

1.0
500

1
6000

= 1.19

TTwgn analogous column ERbRbYlCamYy 1 / I dUcbgajenAkgrUbTI 17>38 b 1 / 1.19

= 0.84

emKuNPaBrwgRkajkRmal k = l1 A1

Edl

Mc

I a

!&>#@

RkLapnmuxkat; analogous column

I a = m:Um:g;niclPaBrbs; analogous column
M = m:Um:g;EdlekItBIbnkktaenAsrsxageRkAbMputn analogous column
EdlenAGkSrbs;kRmal.
Aa =

M = 1.0

l1
2

Aa = 7 + 2 0.25 0.84 = 7 + 0.42 = 7.42

I a = I (for slab portion of 7m) + I (of end portion) about the centerline
2

Ia =

73
0.250

+ 0.42 3.75
= 34.1
12
2

edayecalm:Um:g;niclPaBnGgt;xIxagcugeFobnwgTIRbCMuTmn;rbs;va.
75 3.75
emKuNPaBrwgRkaj k = 7.5 7.142 + 1 3.34
= 1.01 + 3.09 = 4.1
.1
emKuN carryover = 3.094.11.01 = 0.507
10
dUcenH PaBrwgRkajrbs;kRmalxNKW K s = 4.1E 77..81
5

karKNnakRmalxNBIrTis

595

= 4.27 10 3 E

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

4> kMNt;PaBrwgRkajrbs;ssr/ K c
EI
K c = k ' c
lc

T.Chhay

596

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

sRmab;ssrenABIelI nigBIeRkamkRmal.
k ' = emKuNPaBrwgRkajrbs;ssr
l c = 3.6m

Ic =

0.5 4
= 5.2 10 3 m 4
12

emKuNPaBrwgRkaj k ' GacRtUv)ankMNt;dUcxageRkam

1
Mc

+
k ' = l c
A
I
a
a

sRmab;ssr c = lc / 2 nig M = 1.0(lc

2) = lc 2

Aa = l c hs = 3.6 0.25 = 3.35

(lc hs )3

3.35 3
=
= 3.133
12
12
1.8 1.8
1
k ' = 3.6
+
= 4.8
3.133
3.35
Ia =

K c = 4.8 E

5.2 10 3
2 = 0.0139 E
3.6

enAkg flat-plate floor system PaBrwgRkajrbs;ssr K c GacRtUv)anKNnaedaypal;

dUcxageRkam
Kc
Ic
3I c l c2
=
+
!&>##
3
(l h )
E
c

(lc hc )

5> KNnaPaBrwgRkajTb;karrmYl K t rbs;kRmalenARtg;prbs;ssr

x x3 y
9 Rcs C

Kt =
C
=
1

0
.
63
ehI
y

3
y 3

c
l 2 1 2
lc

enAkg]TahrN_enH x = 250mm kRmas;kRmalxN nig y = 500mm TTWgssr

emIlrUbTI 17>17.
0.25 0.253 0.5

= 3.42 10 3 m 4
C = 1 + 0.63

0.5
3

Kt =

9 E cs 3.42 10 3
0.5
61

= 6.66 10 3 E

sRmab;kRmalxNBIrCab;Ka K t = 2 6.66 10 3 E = 13.32 10 3 E

6> KNnaPaBrwgRkajssrsmmUl/ K ec
karKNnakRmalxNBIrTis

597

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

1
1
1
1
1
=
+
=
+
K ec K c K t 0.0139 E 13.32 10 3 E
K ec = 6.8 10 3 E

7> emKuNEbgEckm:Um:g; (D.F)

- sRmab;tMNxageRkA
Ks
4.27 10 3
=
= 0.386
K s + K ec 4.27 10 3 + 6.8 10 3
K
D.F (columns) = ec = 0.614
K
D.F (slab) =

ssrxagelI nigssrxageRkamkRmalxNmanPaBrwgRkajdUcKa dUcenHemKuNEbgEck

0.614 RtUv)anEckesIKarvagssrTaMgBIr EdlmYyTTYl)an D.F= 0.614 / 2 = 0.307 .
- sRmab;tMNxagkg
Ks
4.27 10 3
=
= 0.278
K s + K ec 2 4.27 10 3 + 6.8 10 3
K
6.8 10 3
D.F (columns) = ec =
= 0.444
K 2 4.27 10 3 + 6.8 10 3
D.F (slab) =

ssrmYyman D.F= 0.444 / 2 = 0.222 .

8> m:Um:g;cugbgb; edaysar L.L / D.L tUcCag 0.75 m:Um:g;emKuNTaMgmUlRtUv)ansnt;faman
GMeBIelIElVgTaMgGs;.
Fixed end moment = k"wu l 2(L1 ) 2

emKuN k" GacRtUv)ankMNt;eday column analogy method sRmab;bnkkta

w = 1kN / m enAelIbeNayElVg 7.5m daRkamm:Um:g;Bt;samBaRtUv)anbgajenAkgrUbTI
17>38 b . RkLapnm:Um:g;Bt; edayBicarNabERmbRmYlm:Um:g;niclPaBtambeNayElVgKW
RkLapsrub ( Am ) = A1 + A2 + 2A3
=

emKuNm:Um:g;bgb;cug =

2
1

7(7 0.91) + 7 0.91 + 2 0.25 0.910.84 = 34.98

3
2

Am
Aa l12

Edl Aa sRmab;kRmalxNKW 7.42 dUcEdl)anKNnaenAkgCMhanTI 3.

k" =

T.Chhay

34.98
7.42 7.5 2

= 0.0838

598

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

eyIgeXIjfaemKuNm:Um:g;bgb;cug k"= 0.0838 mantmEk,rnwgemKuN 1 / 12 = 0.833 Edl

eRbICaTUeTAedIm,IKNnam:Um:g;bgb;rbs;Fwm. eK)anrMBwgTuknUvlTplenH edaysarEpknElVg
manm:Um:g;niclPaBERbRbYltUcenAkg flat plate EdlKan column capital b drop panel.
enAkg flat plate EdlmanpleFobElVgelITTwgssrFMCag 20 enaHeKGacykemKuNm:Um:g;
bgb;cug 0.0833 mkeRbIedIm,IKNnam:Um:g;cugRbhak;RbEhl. m:Um:g;bgb;cug EdlbNalBI
wu = 13.28kN / m 2 = 0.0838 13.28 6 7.5 2 = 375.6kN.m
9> karEbgEckm:Um:g;GaceFVIeTA)ansRmab;Bak;kNaleRKag edaysarvamanlkNsIuemRTI. enA
eBlEdlm:Um:g;cugGviCmanRtUv)ankMNt; m:Um:g;viCmanenAkNalElVgGacTTYl)anedaydk
tmmFmnm:Um:g;cugGviCmanBIm:Um:g;viCmanFwmTRmsamBa. karEbgEckm:Um:g;RtUv)anbgaj
enAkgrUbTI 17>39. m:Um:g;Bt; nigkmaMg kat;TTwgcugeRkayRtUv)anbgajenAkgrUbTI
17>40.
10> kRmalxNCagRtUv)anKNnasRmab;m:Um:g;GviCm anenARtg;pssrdUcbgajenAkgrUbTI
17>40.

karKNnakRmalxNBIrTis

599

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

]TahrN_TI17>14

edayeRbIviFI direct design method edIm,IKNnakRmal flate slab xagkgEdlman drop panel edIm,I
RTbnkefr 8.6kN / m 2 nigbnkGefr11kN / m 2 . RbBnkRmalpSMeLIgedaykRmalcMnYn ^ kgTis
nImYy EdlkRmalmYymanTMhM 6 5.4m . RKb;kRmalTaMgGs;RtUv)anRTeday ssrEdlmanGgt;
pit 0.4m ehIy column capital manGgt;pit 1.0m . km<s;Can;KW 3.0m . eK[ f 'c = 28MPa nig
f y = 400 MPa

T.Chhay

600

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

dMeNaHRsay

1> RKb;karkMNt;rbs; ACI edIm,IeRbI direct design method RtUv)anbMeBj. kMNt;kRmas;

kRmalGb,brma hs edayeRbIsmIkar !&>! nig !&>@. Ggt;pitrbs; column capital esInwg
1.0m . muxkat;ssrkaersmmUlEdlmanRkLapesIKamanRCugesInwg r 2 = (500 )2

karKNnakRmalxNBIrTis

601

T.Chhay

mhaviTalysMNg;sIuvil
= 885mm

NPIC

b 900mm .

clear span (long direction) = 6.0 0.9 = 5.1m

clear span (short direction) = 5.4 0.9 = 4.5m

edaysarFwmminRtUv)aneRbI fm = 0 / s = 1.0 nig = 6 / 5.4 = 1.11 BItarag !&>!/

kRmas;Gb,brma hs = ln / 33 = 5.1 / 33 = 155mm b:uEnedaysarman drop panel hs GacRtUv
)ankat;bny 10% RbsinebI drop panel latsnwgcmayy:agtic l / 6 kgTisnImYyBIGkS
rbs;TRm nigTMlak;cuHBIeRkamkRmaly:agtic hs / 4 . dUcenH eRbIkRmas;kRmal
hs = 0.9 155 = 140mm ehIyRbEvgbeNay nigTTwgrbs; drop panel KW
l1 6.0
=
= 2m
3
3
l
5 .4
Short direction = 2 =
= 1 .8 m
3
3
Long direction =

kRmas;rbs; drop panel KW 1.25hs = 1.25 140 = 175mm . begInkRmas;rbs; drop panel
dl; 220mm edIm,Ipl;nUvkRmas;RKb;RKan;sRmab;kmaMgkat;pug nigedIm,IecosvagnUvkareRbInUv
PaKryEdkFM. TMhMTaMgGs;RtUv)anbgajenAkgrUbTI 17>41.
2> KNnabnkemKuN
wu = 1.2 8.6 + 1.6 11 = 28kN / m 2

3> RtYtBinitkmaMgkat;BIrTis. dMbUgenAkg drop panel muxkat;eRKaHfak;enAcmay d / 2 CMuvij

column capital. yk d = 220 30 = 190mm . Ggt;pitrbs;muxkat;rgkmaMgkat;KW
1 + 0.19 = 1.19m

Vu = 286 5.4 1.19 2 = 876kN

4

1.19
bo = 2
= 3.74m
2
0.75 0.33
Vc = 0.33 f ' c bo d =
28 3740 190 = 930kN > Vu
1000

bnab;mkeTotepgpat;kmaMgkat;BIrTisenAkgkRmalxN. muxkat;eRKaHfak;KWsitenAcmay
d / 2 BIeRkA drop panel
d (slab ) = 140 30 = 110mm

RkLapeRKaHfak; = (2 + 0.11)(1.8 + 0.11) = 4.03m 2

T.Chhay

602

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

bo = 2(2.11 + 1.91) = 8.04m

Vu = 28(6 5.4 4.03) = 794kN
0.75 0.33
Vc =
28 8040 110 = 1158kN > Vu
1000

kmaMgkat;mYyTismineRKaHfak;eT.
4> KNnam:Um:g;saTicsrubenAkgTisEvg nigTisxI

wu
28
l 2 l n21 =
5.4 5.12 = 491.6kN .m
8
8
w
28
= u l1l n22 =
6 4.5 2 = 425.2kN .m
8
8

M ol =
M os

edaysar l2 < l1 TTwgrbs;ceRmokelIssrkgTisedAEvgKW 2(0.25 5.4) = 2.7m . TTwgcM

erokelIssrkgTisedAxIKW 2.7m . edaysnt;Ggt;pitEdk 12mm ehIyEdkkgTisedAxI
sitenABIelIEdkkgTisedAEvg enaHkm<s;RbsiTPaBkgTisedAxIKWtUcCagkm<s;RbsiTPaBkg
TisedAEvgRbEhl 10mm . tmrbs; d nigdMeNIrkarKNnaRtUv)anbgajenAkgtarag
17>14. RbEvgGb,brmarbs;EdkEdl)aneRCIserIsRtUveKarBtamRbEvgtRmUvkarrbs; ACI
Code Edl)anbgajenAkgrUbTI 17>16. cMNaMfaPaKryEdkTaMgGs;tUcCag max .
dUcenH = 0.9 .
5> PaBrwgRkajrbs;ssrKW
l1
D.L 8.6
ratio
=
= 0.782
ehI
y
= 1.11
L.L 11
l
2

kMNt; min
viFIRbhak;RbEhlRtUv)aneRbIenATIenHedIm,IkMNt;PaBrwg Rkajrbs;ssrCamYynwg capital
rbs;va.
140 3
I s m:Um:g;niclPaBrbs;kRmalkgTisxI = 6000
= 1372 10 6 mm 4
12
1
=
= 1.15
0.782 1.11

Ks =

4 Ec I s 4 E c 1372 10 6
=
= 1016 10 3 E c
l2
5400

sRmab;ssrmUlEdlmanGgt;pit
D 4

400 mm =
(400) 4 = 1257 10 6 mm 4
=
64
64
Ic

Kc =

4 Ec I c 4 E c 1257 10 6
=
= 1676 10 3 E c
3000
lc

karKNnakRmalxNBIrTis

603

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

tarag 17>14 karKNnankRmal flat plate xagkgEdlman drop panel

M o = 491.6kN .m
M n = 0.65M o = 319.5kN .m
M p = +0.35M o = +172.1kN .m

TisEvg
ceRmokelIssr

ceRmokkNal

emKuNm:Um:g;

0.75M n

0.60 M p

0.25M n

0.40M p

M u (kN .m)

-239.6

103.3

-79.9

68.8

190

110

110

110

2.7

2.7

2.7

2.7

( MPa)

2.46

3.16

2.44

2.10

PaKryEdk (%)

0.71

0.93

0.7

0.6

As = bd ( mm 2 )

3642

2762

2079

1782

As (min) = 0.0018bhs ( mm 2 )

1070

680

680

680

18DB16

14DB16

20DB12

16DB12

150

193

135

170

TTwgceRmok b(mm)

km<s;RbsiTPaB d (mm)
Ru =

Mu
bd 2

EdkEdleRCIserIs Rtg;
KMlat (mm)

TisxI

M o = 425.2kN .m
M n = 0.65M o = 276.4kN .m
M p = 0.35M o = +148.8kN .m

ceRmokelIssr

ceRmokkNal

emKuNm:Um:g;

0.75M n

0.60 M p

0.25M n

0.40 M p

M u (kN .m)

-207.3

89.3

-69.1

59.5

TTwgceRmok b(mm)

180

100

100

100

km<s;RbsiTPaB d (mm)

2.7

2.7

3.3

3.3

( MPa)

2.37

3.30

2.10

1.80

PaKryEdk (%)

0.69

1.00

0.60

0.50

As = bd (mm 2 )

3353

2700

1980

1650

As (min) = 0.0018bhs ( mm 2 )

1070

680

680

680

18DB16

14DB16

18DB12

16DB12

150

195

185

205

Ru =

Mu
bd 2

EdkEdleRCIserIs Rtg;
KMlat (mm)
T.Chhay

604

Design of Two-Way Slab

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

pleFobrvagPaBrwgRkajrbs;ssrnigPaBrwgRkajrbs;kRmal
=

K c 1676 10 3
=
= 1.65
K s 1016 10 3

EdlFMCag min = 1.15 . RbsinebI I s kgTisEvgRtUv)aneRbI pleFobnPaBrwgRkajssr

elIPaBrwgRkajkRmalEdl)anKNnanwgFMCag 1.65 . dUcenHssrKWRKb;RKan;.
6> kMNt; balanced moment enAkgssr nigRtYtBinitkugRtaMgkmaMgkat;enAkgkRmal dUc
Edl)anBnl;enAkg]TahrN_TI 17>8 nig 17>9.

karKNnakRmalxNBIrTis

605

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

XVIII.

CeNIr

Stairs

18>1> esckIepIm (Introduction)

RKb;GKarTaMgGs; TaMgTab nigx<s;EtgEtmanCeNIr eTaHbICamanCeNIreyagRKb;RKan;keday.
CeNIrpSMeLIgeday km<s;kaM (rise), CMhankaM (run or tread) nigfasCeNIr (landing)rUbTI 18>1.

T.Chhay

606

Stairs

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

TMhMFmtarbs;km<s;kaM nigCMhankaMenAkgGKarEdlTak;Tgnwg empirical rule xH.

Rise + run = 430mm
2Rise + run = 635mm

Rise run = 0.05m 2

km<s;kaMGaRsynwgRbePTGKar. ]TahrN_ sRmab;GKarsaFarN km<s;kaMKWRbEhl 15cm

b:uEnsRmab;GKarsak;enAvaERbRbYlcenaH 15cm eTA 19cm . CMhankaMKWRbEhl 25cm enAkgGKar
saFarN nigERbRbYlcenaH 23cm eTA 32cm sRmab;GKarsak;enA. CaTUeTA km<s;kaMminRtUvFMCag
M an;eTACan;rbs;GKarnwgkm<s;kaM
20cm nigminRtUvtUcCag 10cm ehIycMnYnkm<s;kaM)anBIkarEckTMhC
snt;. kargarbegIyrbs;kaMCeNIrman fdus (troweling Alundum grit) kar:U asphalt kar:U terrazzo
kar:U marble bRBM. bEnmBIelIbnkefr CeNIrRtUv)anKNnasRmab;bnkGefrGb,brma 4.8kN / m 2 .
18>2> RbePTCeNIr (Types of Stairs)
eKmanCeNIrCaeRcInRbePTEdlGaRsyCacMbgeTAnwgRbePT nigtYnaTIrbs;GKar nigGaRsy
eTAelItRmUvkarsabtkm. RbePTCeNIrFmtabMputmandUcxageRkam
!> Single-flight stair kareFVIkarrbs;rcnasm<nntYCeNIrRsedogeTAnwgkRmalxNmYyTisEdl
RtUv)anRTenAcugsgag. kRmalrbs;kRmalRtUv)aneKehAfa waist rUbTI 181. enAeBltY
CeNIrmanfas eKRtUvdak;FwmenARtg; B nig C edIm,IlkNesdkic rUbTI 18>2. RbsinebI
FwmTaMgBIrenHminRtUv)andak; enaHRbEvgElVgCeNIrnwgekIneLIgedaysarTTwgrbs;fasBIrEdl
latsnwgcenaH A nig D . enAkgGKarsak;enATTwgfasKWsitenAcenaH1.2m eTA1.8m ehIy
cmaysrubcenaH A nig D KWRbEhl 6m .
viFIepSgeTotkgkarRT Single-flight stair KWRtUveRbI stringer bFwmxag (edge beam)
enARCugBIrtambeNayCeNIr. kaMRtUv)anRTedayFwm rUbTI 18>3.
@> Double-flight stair GkeRbInwgmanlkNgayRsYlkgkareRbIR)as;GKarEdlman doubleflight stair. RbePTEdlRtUv)aneRbIR)as;CaTUeTAKWRbePT quarter-turn rUbTI 18>4nig
closed-well stair b open-well stair dUcbgajenAkgrUbTI 18>5. sRmab;karviPaKeRKOgbgM
CeNIr tYmYyRtUv)anKitdUc single-flight ehIyRtUv)anBicarNaedayRTenAelIFwmBIr beRcIn
CeNIr

607

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

dUcbgajenAkgrUbTI 18>2. fasCeNIrlatsnwgkugTisedATTwgcenaHTRmBIr ehIyRtUv)an

KNnadUc one-way slab. kgkrNI open-well stair EpkkNalrbs;fasCeNIrRTbnkeBj
cMENksgagBIrRTbnkEtBak;kNal dUcbgajenAkgrUbTI 18>5 d. bnkBak;kNaleTot
RtUv)anRTenAkgTisbeNayedaytYCeNIr muxkat; A-A nig B-B.

T.Chhay

608

Stairs

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

#> Three or more flight of stair kgkrNIxH EdlTMhMTaMgmUlrbs;CeNIrRtUv)ankMNt; three or

more flight of stair GacRtUv)anTTYlyk rUbTI 18>6. tYCeNIrmYyRtUv)anKitdac;eday
ELkBIKadUckgkrNI double-flight staircase.

$> Cantilever stair RtUv)aneRbICaTUeTAsRmab;CeNIr fire-escape stair ehIyvaRtUv)anRTeday

CBaaMg bFwmebtug. CeNIrRbePTenHGac full-flight EdltYCeNIrQrelIEtCBaaMgmag nigGac
half-flight EdltYCeNIrQrelICBaaMgsgag bRbePT semispiral dUcbgajenAkgrUbTI 18>7.
enAkgCeNIrRbePTenH kaMnImYyeFVIkarCaFwm cantilever ehIyEdkemRtUv)andak;enAkgtMbn;
rgkarTajnCMhankaM nigEdkRtUv)anf<k;eTAkgCBaaMgebtug. EdkTb;sItuNPaB nigkarrYmmaD
RtUv)andak;kgTisedATTwg.
CeNIr

609

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

TRmg;epSgeTotrbs;CeNIr cantilever stair KWeRbI open-riser step EdlRTedayFwm

kNaldUcbgajenAkgrUbTI 18>8. FwmmanCRmaldUctYCeNIr ehIyEpkxagelIrbs;Fwm
GnuBaat[kaMCeNIrKgelI. kgkrNICaeRbIkaMebtugGarem:cak;eRscGacRtUv)aneRbICamYynwg
karpl;BiesssRmab;rwtbULgedIm,IPab;kaMeTAnwgFwm.

T.Chhay

610

Stairs

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

%> Precast flights of stair el,nkgkarsagsg;enAkgKMeragxHtRmUv[eRbI precast flights of

stair rUbTI 18>8. tYCeNIrGacRtUv)ancak;edayELk ehIybnab;mkRtUv)anPab;eTAnwgfas
CeNIrEdlcak;enAnwgkEng. kgkrNIepSgeTot tYCeNIrrYmTaMgfasCeNIr RtUv)ancak;nig
bnab;mkdak;enAelICBaaMg bFwmTMr. vaRtUv)anKNnaCakRmalxNmYyTisEdlmanTRm
samBaCamYynwgEdkemenA)at waist rbs;CeNIr. brimaNEdkRKb;RKan;RtUv)andak;enAelI
tMN dUcbgajenAkgrUbTI 18>9.
TMBk;RtUv)aneKeFVIedIm,IkargarelIkdak; nigdMeLIg. EdkBiessRtUvdak;enARtg;mux
kat;eRKaHfak; edIm,ITb;nwgkugRtaMgTajEdlekItmanenAkgCeNIrkgdMNak;karelIkdak;.

CeNIr

611

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

^> Free standing stair sRmab;CeNIrRbePTenH fasCeNIrsitenAkgxl;edayKanTRmenA

xagcug rUbTI 18>10. CeNIreFVIkarkglkNCakaelat (springboard) EdlbegItCa
kugRtaMgrmYlenAkgkRmalxN.

RbBnnkardak;bnkbIRbePT RtUv)anykmkBicarNakgkarKNnaCeNIrRbePTenH
edayKitBicarNafam:Um:g;rmYlnwgekItmanenAkgkRmalRKb;lkxN
- enAeBlbnkGefreFVIGMeBIEtenAelItYCeNIrxagelI nigBak;kNalfas rUbTI 18>12
tYCeNIrxagelInwgrgnUvkmaMgTajbEnmBIelIm:Um:g;Bt; EttYCeNIrxageRkamnwgrgnUv
kmaMgsgt; EdlGacbegItCakmaMgPat; (buckling) enAkgkRmal .

T.Chhay

612

Stairs

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

- enAeBlbnkGefreFVIGMeBIEtenAelItYCeNIrxageRkam nigBak;kNalfas tYCeNIrxag

elInwgrgkmaMgTaj EttYCeNIrxageRkamnwgrgm:Um:g;Bt; nigkmaMgsgt;.
- enAeBlbnkGefreFVIGMeBIenATaMgelItYCeNIrxagelI nigxageRkam kardak;bnkenAelI
tYCeNIrmYynwgbNal[tYrCeNIrmYyeTotrmYl. kugRtaMgrmYlekItenAkgCeNIr
TamTarsrsEdkRKb;RKan;enAkgpTaMgsgagrbs;CeNIr nigfas. srsEdkTTwg
enAkgkRmal nigfasRtUv)andak;enAkgpTaMgsgagrbs;ebtugkgTRmg;CaEdlGkSr
U biTCit enABak;kNalTTwgrbs;CeNIr. srsEdklMGitRtUv)anbgajenAkgrUbTI
18>13.
sabtkrcUlcitCeNIrRbePTenH ehIyCeNIrRbePTenHeBlxHRtUv)aneKehAfa pliersshaped staircase b jackknife staircase.
&> Run-riser stair CaRbePTCeNIrEdlpSMeLIgedaycMnYn km<s;kaM nigCMhankaMCaeRcInEdl
Pab;Kaedaypal;y:agrwgedayKankRmal waist rUbTI 18>14 a. CeNIrRbePTenHTTYl
karcab;GarmN_edaysabtkredaysarrUbragxageRkArbs;va. karviPaKeRKOgbgMrbs;
run-riser stair GacRtUv)ansRmYledaysnt;fa\TiBlrbs;kmaMgtamGkSRtUv)anecal nig
snt;fabnkenAelICMhankaMnImYyRtUv)anRbmUldak;enAxagcugrbs;CMhankaM rUbTI 18>
14 b. sRmab;karviPaKntYCeNIrEdlRTedayTMrsamBa edayBicarNatYCeNIrsamBan
CMhankaMBIr ABC EdlrgnUvbnkcMcMNuc P enARtg;cMNuc B rUbTI 18>14 b. edaysar
tMN B nig B CatMNrwg m:Um:g;enARtg;tMN B esIm:Um:g;enARtg;tMN B b M B = M B'
= PS / 2 Edl S CaTTwgrbs;CMhankaM. m:Um:g;enAkgkm<s;kaM BB' mantmefr ehIyesI
nwg PS / 2 .
enAeBlEdlKankm<s;kaM CeNIr ABC eFVIkarCaFwmTMrsamBa ehIym:Um:g;Bt;Gtibrma
EdlekItmanenAkNalElVgCamYytm M B = PL / 4 = PS / 2 . sRmab;tYCeNIrEdlpSM
eLIgedaykm<s;kaM nigCMhankaMeRcIn viFIdUcKaRtUv)aneRbIdaRkamm:Um:g;Bt;RtUv)anbgaj
enAkgrUbTI 18> 15 a. m:Um:g;enAkg BB' mantmefrsInwgm:Um:g;enARtg;tMN B b 2PS .
dUcKa M C = M 'C = 3PS / M D = M 'D = 3PS nig M E = M 'E = 2PS .

CeNIr

613

T.Chhay

mhaviTalysMNg;sIuvil

T.Chhay

NPIC

614

Stairs

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

RbsinebImanvtmanfasCeNIrenAcugmag bcugsgag bnkenAelIfaskRtUv)anRbmUl

CabnkcMcMNucdUckrNICMhankaMEdr. karviPaKrcnasm<nkGaceFVIedayBicarNabnkBRgay
esIenAelItYCeNIr. m:Um:g;enAkgkm<s;kaMTaMgGs;mantmefr nigTTYl)anBIrdaRkamm:Um:g;
Bt;nFwmTMrsamBaEdlrgbnkesI rUbTI 18>15 b. ]TahrN_TI 18>3 bgajBIkar
KNnaCeNIredayeRbIkarsnt;BIr KWbnkcMcMNuc nigbnkBRgayesI.
RbsinebItYCeNIrRtUv)anbgb; bCab;enAcugmag bcugTaMgBIr m:Um:g;GacRtUv)anTTYl
edayeRbIviFIviPaKeRKOgbgMNamYyk)an. edIm,IBnl;krNIenH cat;TuktYCeNIrEdlpSMeLIg
edayCMhankaMBIr nigRtUvbgb;enAcugTaMgBIr rUbTI 18>16 a. m:Um:g;enAcugbgb; A nig B
EdlbNalBIbnkenARtg; B esInwg PL / 8 = PS / 4 . lTplenHRtUv)anTTYledaysnt;
faCeNIrminmankm<s;kaM ABC eFVIkarCaFwmbgb;cugEdlrgnUvbnkcMcMNucenAkNalElVg
rUbTI 18>16 b. m:Um:g;enAkNalElVg muxkat; B esInwg
PL
PS PS PS
MA =

=
4
2
4
4

m:Um:g;Bt;ntYCeNIrEdlmankm<s;kaMmYyRtUv)anbgajenAkgrUbTI 18>16 a. cMNaMfam:Um:g;

enAkgkm<s;kaM BB' mantmefr ehIy M B = M 'B = PS / 4 .
sRmab;tYCeNIrEdlmanlkNsIuemRTI cugbgb;TaMgsgag ehIyrgnUvbnkcMcMNucenA
Rtg;cMNucCMhankaM m:Um:g;enAcugbgb;GacRtUvKNnadUcxageRkam
M (fixed end ) =

Edl

PS 2
(n 1)
12

bnkcMcMNucenARtg;cMNucCMhankaM
S = TTwgCMhankaM
n = cMnYnCMhankaM
(4 1) = PS
enAeBl n = 2 enaH M (fixed end) = PS
12
4
EdllTpldUcnwglTplTTYl)anBIxagedIm.
RbsinebImanfasCeNIrenAcugmag bcugsgag bnkenAelIfasCeNIrGacRtUv)an
dak;CabnkcMcMNucenAKMlat S .

CeNIr

P=

615

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

*> Helical stair (open-spiral stair) CaRbePTeRKOgbgMbITMhM EdlCaTUeTAmanragCargVg;enA

kgbg; rUbTI 18>17. vaCaRbePTCeNIrEdleKeRbIenAkg entrance hall, theater foyer
nigGKarkariyalyEdlmankm<s;Tab. tmrbs;vafCagCeNIrRbePTFmta.
CeNIrGacRtUv)anRTenARCugxHEdlCab;nwgCBaaMg bGacKNnaCa free-standing
helical staircase EdlTTYlkareBjniymCaeK. eRkamGMeBIrbs;bnk tYCeNIrGacrgnUvkug
RtaMgrmYl. tYCeNIrxagelInwgrgnUvkugRtaMgTaj cMENkkugRtaMgsgt;ekItmanenAEpk
xageRkamrbs;CeNIr. kmaMgEdleFVIGMeBIenARtg;muxkat;NamYyGacpSMeLIgedaym:Um:g;
bBar m:Um:g;edk m:Um:g;rmYl kmaMgtamGkS kmaMgkat;TTwgEdlkat; waist rbs;CeNIr nigkM
laMg radial horizontal shearing force. EdkbeNayempSMeLIgedayEdk helical bar
Edldak;enAkg waist rbs;CeNIr nigCMhankaMBIfasxagelIeTATMrxageRkam. EdkTTwgRtUv
EtmanTMrgCaEdkkgbiTCitedIm,ITb;kugRtaMgrmYl bCaTRmg;EdkGkSr U pbKadak;enA
kNalTTwgrbs;CeNIr.
viFIepSgeTotsRmab;pl;[ helical stair KWeRbIFwm helical girder enAkNalTTwg
rbs;CeNIr. kaMnImYyRtUv)anKitCaFwm cantilever ehIyEdkRtUv)andak;enAEpkxagelI
rbs;kaM.

T.Chhay

616

Stairs

viTasanCatiBhubeckeTskm<Ca

CeNIr

Department of Civil Engineering

617

T.Chhay

mhaviTalysMNg;sIuvil

T.Chhay

NPIC

618

Stairs

viTasanCatiBhubeckeTskm<Ca

CeNIr

Department of Civil Engineering

619

T.Chhay

mhaviTalysMNg;sIuvil

T.Chhay

NPIC

620

Stairs

viTasanCatiBhubeckeTskm<Ca

CeNIr

Department of Civil Engineering

621

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

]TahrN_TI 18>1 KNnaCeNIrdUcbgajenAkgrUbTI 18>18 edIm,IRTbnkGefrBRgayesI

. snt;km<s;kaMesInwg 15cm ehIyCMhankaMesInwg 30cm . eK[
f y = 400MPa .

4.8kN / m 2

f 'c = 21MPa

nig

dMeNaHRsay

1> bnk snt;kRmas;kRmal waist esInwg 10cm .

Tmn;rbs;kaMmYy = trapezoidal area 25
0.1118 + 0.2618
=
0.3 25 = 1.4kN / m
2

eyagtamrUbTI 18>18 b snt;Tmn;kargarbegIyEdlRKbBIelIkaMCeNIrKW 0.3kN / m .

dUcenHbnkefrsrub D.L. = 1.7kN / m
Wu = 1.2 D + 1.6 L = 1.2 1.7 + 1.6 4.8 0.3 = 4.35kN / m

T.Chhay

622

Stairs

viTasanCatiBhubeckeTskm<Ca

CeNIr

Department of Civil Engineering

623

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

2> m:Um:g;Bt;GtibrmaenAkgmYykaMKW Wu l 2 / 2
Mu =

4.35 2
1.8 = 7.05kN .m
2
11.18 + 26.18

= 18.68cm
2

kRmas;mFmrbs;kaMCeNIrKW
yk d = 18.68 2 0.6 = 16.1cm
a

snt; a = 12.5mm
M u = As f y d
2

As =

7.05 10 6
= 127 mm 2
12.5

0.9 400161

As f y
127 400
a=
= 9.5mm
=
0.85 f 'c b 0.85 21 300

mantmRbEhlkarsnt;
RtYtBinit
brimaNEdkGb,brma As = 0.00333(300)(161) = 161mm 2
eRbI 2DB12 sRmab;mYykaM. kRmas;kRmalGacyktUcCagkarsnt;enH b:uEnedIm,IeCosvag
PaBdabFM nigPaBrMjrrbs;CeNIr eKRtUveRCIserIskRmas;kRmalsmrm.
3> KNnakmaMgkat;enAcmay d BIpTRm
Vu = 4.35(1.8 0.161) = 7.13kN
0.75
0.75
Vc =
f 'c bd =
21(300 161) 10 3 = 27.7 kN
6
6

edaysarEt Vu < Vc / 2 dUcenHeKminRtUvkarEdkkgeT. b:uEneKRtUvdak;Edkkg DB10 @100

edIm,IcgCamYyEdkembeNay.
4> CeNIrRtUvrkSalMnwgedayTmn;rbs;CBaaMg bedayFwmebtugGarem:enAkgCBaaMg. kgkrNIenH
FwmrgnUvm:Um:g;rmYl 7.05kN .m / m .
5> srsEdklMGitRtUv)anbgajenAkgrUbTI 18>18.

]TahrN_TI 18>2 KNnaCeNIrdUcbgajenAkgrUbTI 18>19 EdlRTbnkGefrBRgayesI

. snt;km<s;kaMesInwg 18cm ehIyCMhankaMesInwg 27.5cm . eK[
f y = 400 MPa .

5.7kN / m 2

T.Chhay

624

f 'c = 21MPa

nig

Stairs

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

dMeNaHRsay

1> RbBneRKOgbgM RbsinebIeKmineRbIFwm stringer enaH eKGacTTYlykdMeNaHRsaymYykgcM

eNam dMeNaHRsaybYnEdl)anbgajenAkgrUbTI 18>2. enAeBlTRmkNalminRtUv)aneRbI
tYCeNIrRtuv)anRTenAxagcugnfasCeNIrxagelI nigxageRkam. RbBnrcnasm<nenHRtUv)an
ykmkeRbIkg]TahrN_enH.
2> bnk snt;kRmas;kRmal (waist) KW 20cm
Tmn;rbs;kaMmYy = trapezoidal area 25
0.239 2 + 0.18
=
0.275 25 = 2.26kN / m
2

2.26
=
= 8.2kN / m 2
0.275

Tmn;rbs;tYCeNIr
Tmn;fasCeNIr = 0.2 25 = 5kN / m 2
edaysnt;kargarbegIyRKbelICeNIrKW 1kN / m 2 .

Wu (on stair ) = 1.2(8.2 + 1) + 1.6 5.7 = 20.16kN / m 2

Wu (on landing ) = 1.2(5 + 1) + 1.6 5.7 = 16.32kN / m 2

edayKNnaCeNIrkg 1m TTwgrbs;CeNIrdUcenH
Wu (on stair ) = 20.16kN / m
Wu (on landing ) = 16.32kN / m

edaysarbnkenAelIfasCeNIrRtUv)anRTBIrTis dUcenHbnkBak;kNalRtUv)anKitenAkgTis
mYy.
3> KNnam:Um:g;Bt;Gtibrma nigEdkBRgwg rUbTI 18>19 d
a. m:Um:g;enAkNalElVgKW
M u = 34.42 2.6 8.16 1.5 1.85 20.16

yk d = 200 20 6 = 174mm

CeNIr

625

1.12
= 54.65kN .m
2

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627

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NPIC

b. M u = As f y (d a / 2);
As =

edaysnt; a = 20mm

54.65 10 6
= 926mm 2
0.9 400 (174 10)
As f y
926 400
a=
=
= 20.75mm c = 24.4mm
0.85 f 'c b 0.85 21 1000

RtYtBinit
brimaNEdkGb,brma As = 0.0033 1000 174 = 574mm 2 < 926mm 2
eRbIEdk DB12 @120 As = 942mm 2 . sRmab;CeNIrTTwg 1.5m eRbI 13DB12 .
d t = 174mm

c.

c = 24.4mm

bERmbRmYlrageFobTajsuT (net tensile strain)

d c
dUcenH = 0.9
t = t
0.003 = 0.0184 > 0.005
c
EdkTTwgRtUv)andak;edIm,ITb;karrYmmaD
As = 0.0018 1000 200 = 360mm 2

eRbIEdk DB12 EdlmanKMlat 300mm As = 377mm 2

d. RbsinebICeNIr nigFwmEdlRTvaRtUv)ancak;CamYyKa EdkbEnmRtUv)andak;enAEpkxagelI
nigEpkxageRkamrbs;fasCeNIr. srsEdklMGitRtUv)anbgajenAkgrUbTI 18>19.
4> kRmas;kRmalxNGb,brmasRmab;karBarPaBdabKW
L 5.2
=
= 0.208m sRmab;kRmalxNTRmsamBa
25 25
kgkrNIEdlbgajenATIenH edaysarcugkRmaltYCeNIrRtUv)ancak;CamYynwgFwmEdlRTva
ehIymandak;EdkbEnmsRmab;m:Um:g;GviCman enaHkRmas;kRmalGb,brmaGacRtUv)ansnt;
esInwg
L
= 0.186m < 0.2m Edl)aneRbI
28
5> KNnafasCeNIr edayBicarNafasCeNIrkgRbEvgTTwg 1m bnkenAelIfasCeNIRtUv)an
bgajenAkgrUbTI 18>20. EpkkNal 0.6m RTbnkeBj EtEpksgag 1.5m RTbnkEtBak;
kNal.
2
m:Um:g;Bt; = 17.15 1.8 8.16 1.5 1.05 16.32 0.26 = 15.08kN .m
edaysarEdkenAkgfasCeNIrRtUv)andak;enABIelIEdkCeNIrem enaH
d = 200 20 12 0.6 = 167.4mm

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edaysnt; a = 10mm
As =

15.08 10 6
= 258mm 2 < As (min) = 574mm 2
0.9 400 (167.4 5)

eRbIEdk DB12 KMlat 180mm As = 628mm 2

6> FwmTTwgenAcugfasCeNIrRtUv)anKNnaeday[RTbnkBICeNIr 34.42kN / m bEnmBIelI
bnkpal;rbs;va nigbnkrbs;CBaaMgEdlenABIelIva.
7> RtYtBinitkmaMgkat;dUcFmta.

]TahrN_TI 18>3 KNnaCeNIr run-riser stair EdlRTedayTRmsamBadUcbgajenAkgrUbTI

18>21 sRmab;RTbnkGefrBRgayesI 5.7kN / m 2 . eK[

f 'c = 21MPa

dMeNaHRsay

nig

f y = 400MPa

1> bnk snt;kRmas;rbs;CMhankaM nigkm<s;kaMesInwg 15cm .

bnkcMcMNucenAelICMhankaMnImYyRtUv)anKNnadUcxageRkam eyagtamrUbTI 18>21 a .
edayKitkg 1m TTwgrbs;CeNIr
PD = (0.15 0.4 + 0.025 0.15) 25 = 1.6kN

cMNaMfa bnkefrcMcMNucenAelIfasCeNIrmantmtUcCag 1.6kN buEnvaGacRtUv)ansnt;esI

nwg PD edaykarKNnay:agsRmYl. KitbnkGefrenAelITTwgCeNIr 1m ehIyRbmUlvaCa
bnkcMcMNuc enaHeK)an
PL = 5.7 0.25 = 1.425kN

bnkemKuN Pu = 1.2PD + 1.6PL = 1.2 1.6 + 1.6 1.425 = 4.2kN

CeNIr

629

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630

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2> KNnam:Um:g;Bt;enAkNalElVg bnkenAkg]TahrN_enHmanlkNsIuemRTIeFobnwgkNal

ElVgRtg;muxkat; B . RbtikmenARtg;cMNuc A
RA =

15
4.2 = 31.5kN
2

m:Um:g;enARtg;cMNuc B = R A (8S ) 7 Pu (4S )

= 31.5(8 0.25) 7 4.2(4 0.25) = 33.6kN.m

3> KNnasrsEdkcaM)ac;enARtg;muxkat;kNalElVg sRmab; h = 15cm / d = 150 25 = 125

Ru =

Mu
bd 2

sRmab;

33.6 10 6
1000 125 2

nig Ru = 2.15 enaHPaKryEdkKW

= 0.0064 < max = 0.0135 / = 0.9
f 'c = 21MPa

= 2.15MPa

f y = 400MPa

As = 0.0064 1000 125 = 800mm 2

eRbIEdk DB12 nigmanKMlat 14cm As = 807mm 2 tamTisedk nigTisQrkgTRmg;Edk

kgbiTCit.
sRmab;Edkrg eRbIPaKryEdkGb,brmasRmab;Tb;nwgsItuNPaB nigkarrYmmaD = 0.0018
As = 0.0018 400 150 = 108mm 2

eRbIEdk DB10 manKMlat 15cm As = 209.3mm 2 . dak;Edk DB10 enARCugmYyrbs;

kaM As = 235.5mm 2 dUcbgajenAkgrUbTI 18>21 c.
4> m:Um:g; nigsrsEdkEdlcaM)ac;sRmab;muxkat;muxkat;epSgeTotRtUv)aneFVICataragdUcxag
eRkam
TItaMg
B.M .(kN .m)
Ru ( MPa)

(% )

As mm 2

CeNIr

7.88

14.7

20.48

25.2

28.88

31.5

33.08

33.6

0.5

0.94

1.31

1.61

1.85

2.02

2.12

2.15

0.14

0.27

0.38

0.47

0.55

0.60

0.63

0.64

175

338

475

588

688

750

788

800

631

T.Chhay

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eRbIEdk DB12 @ 200 sRmab;fas nig DB12 @150 sRmab;kaMCeNIr. sRmab;

As = 588mm 2 eRbIEdk DB12 @ 200 enAkgfasCeNIr. karlMGitsrsEdkRtUv)anbgaj
enAkgrUbTI 18>21 c.
5> RtYtBinitbrimaNEdkcaM)ac;enAkgTisTTwgrbs;fas bnkenAelIfaskgmYyktaRbEvgKW
4.2
= 16.8kN / m dUcenH
0.25
16.8
3.6 2 = 27.2kN .m
8
27.2 10 6
Ru =
= 1.74 MPa
1000 125 2

Mu =

= 0.0051

As = 638mm 2

eRbIEdk DB12 @175 As = 646mm 2

6> RbsinebIbnkBRgayesIRtUv)ansnt;famanGMeBIelItYCeNIr eKnwgTTYl)anlTplRsedog
Ka.
]TahrN_ bnkemKuNcMcMNucEdl)anKNnaKW 4.2kN manGMeBIelITTwgCMhankaM 0.25m . dUc
enHbnkkgmYyktaRbEvgKW 4.2 / 0.25 = 16.8kN / m . m:Um:g;GtibrmaenAkNalElVgRtg;
muxkat; B
Mu =

16.8 2
4 = 33.6kN .m
8

m:Um:g;enAmuxkat;dTeTotGacRtUv)anKNnay:agRsYl ehIykarKNnaGacRtUv)anerobcMCa
taragdUcBnl;kgCMhan 4.

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633

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XIX.

esckIENnaMBIebtugkugRtaMg

Introduction to Prestressed Concrete

19>1> ebtugeRbkugRtaMg (Prestressed Concrete)

19>1>1> eKalkarN_nkareFVIeRbkugRtaMg (Principles of Prestressing)
karGnuvteRbkugRtaMgeTAelIGgt;eRKOgbgMKWCakarbegItkugRtaMgGciRny_xagkgEdlmanGMeBI
RbqaMgnwgkugRtaMgTajenAkgebtugEdl)anbnkxageRkA. karGnuvteRbkugRtaMgenHbegItCaEdnkug
RtaMgEdlGgt;GacTb;Tl;)any:agmansuvtiPaB. eKGacGnuvtkmaMgeRbkugRtaMgmun bkgeBldMNal
KankarGnuvtbnkxageRkA. kugRtaMgenAkgGgt;eRKOgbgMRtUvEtenAsl; RKb;TIkEng nigsRmab;
RKb;sanPaBnkardak;bnk enAkgEdnkMNt;rbs;kugRtaMgEdlsmarGacRTRTg;)anKanTIkMNt;. Ca
TUeTA karGnuvtkugRtaMg PaKeRcInCakugRtaMgsgt; RtUv)anbegIteLIgeday high-strength steel
tendon EdlrgkarTaj nig anchor eTAnwgGgt;ebtug. kugRtaMgRtUv)anepreTAebtugeday bond
tamprbs; tendon beday anchorage enAxagcugrbs; tendon.
edIm,IgayRsYlkgkarBnl; cUrBicarNaFwmmYyEdleFVIBIebtugsuT ehIyFwmenaHRtUvRTnUv
bnkTMnajxageRkA (external gravity load) dUcbgajkgrUbTI 19>1 a. muxkat;FwmRtUv)aneRCIs
erIsCamYynwg tensile flexural stress EdlCalkxNeRKaHfak;sRmab;karKNna dUcenHeKTTYl)an
muxkat;EdlminmanlkNesdkic. mUlehtuKWedaysarebtugxaMgkgkarsgt;CagkarTaj. flexural
tensile strength bm:UDuldac; (module of rupture) rbs;ebtug f r esInwg 0.62 f 'c rUbTI 19>1a.
kgkarKNnaebtugGarem:Fmta eKminKit tensile strength rbs;ebtugeT ehIyEdksrs
RtUv)andak;enAkgtMbn;Tajrbs;ebtugedIm,ITb;Tl;nwgkugRtaMgTaj b:uEnebtugTb;Tl;nwgkugRtaMgsgt;
rUbTI 19>1 b.
kgkarKNnaebtugeRbkugRtaMg kugRtaMgsgt;dMbUgRtUv)anGnuvteTAkgFwmedIm,I[manGMeBITb;
nwgkugRtaMgTajEdlekIteLIgedaysarbnkxageRkA rUbTI 19>1 c. RbsinebIkugRtaMgEdldak;
dMbUgenHesInwgkugRtaMgTajenAsrseRkambMput enaHkugRtaMgTaMgBIrRtUv)anlubecal b:uEnkugRtaMg
sgt;enAsrsEpkxagelIbMputnwgmantmDub. kgkrNIenH muxkat;Ggt;TaMgmUlrgkarsgt;. Rb
sinebIkugRtaMgsgt;EdlGnuvtdMbUgtUcCagkugRtaMgTajenAsrsEpkxageRkambMput enaHsrsenA
EpkxageRkamenHrgkarTaj srsEpkxagelIbMputrgkarsgt;.
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kgkarGnuvt Ggt;ebtugGacrgeRbkugRtaMgtamviFImYykgcMeNamviFIxageRkam
esckIENnaMBIebtugeRbkugRtaMg

635

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!> karTajeRkay (Posttensioning): kgkareFVI posttensioning, eKTaj steel tendon

eRkayeBlebtugRtUv)ancak; nigkkrwg. kareFVI posttensioning RtUv)aneFVIeLIgtamviFI
saRsdUcteTA dMbUgeKeRbI hydraulic jack Taj steel wire b strand [lUt bnab;mk
CMnYs jack eday anchorage EdlGacrkSa[ steel strand enAEtrgkarTaj. CaTUeTA
tendon RtUv)aneFVIeLIgBI wire, strand b bar. eKGacTaj wire nig strand CaRkum)an Et
eKTaj bar mg)anEtmYy. kgdMeNIrkareFVI posttensioning, eKdak; steel tendon eTAkg
Bum<muneBlcak;ebtug ehIy tendon RtUv)ankarBarkarsitCab;eTAnwgebtugeday waterproof paper wrapping b metal duct (sheath). tendon EdlsitCab;eTAnwgebtugRtUv)an
eKehAfa boded tendon. Unbonded tendon/ RtUv)andak;edayKan grout bRtUv)anlab
eRbg.
@> karTajmun (pretensioning): kgkareFVI prettensioning eKTaj steel tendon muneBlcak;
ebtug. eKTb; tendon CabeNaHGasneday abutment ehIyeKkat;va bnab;ebtugRtUv)an
cak; nigkkrwg. kmaMg prestessing RtUv)anepreTAebtugeday PaBsitenAtamRbEvgrbs;
tendon. CaTUeTA eKeRcIneFVI prettensioning enAkgkardan beragcRkpliteRKOgbgMebtug
eRbkugRtaMgcak;eRsc EdlmankRmalrwgmaMCaGciRny_.
#> kareFVIeRbkugRtaMgxageRkA (external prestressing): kgkareFVI external pretessing, eK
GnuvtkmaMgeRbkugRtaMgeday flat jack EdlRtUv)andak;enAcenaHcugGgt;ebtug nig
permanent rigid abutments. Ggt;minman prestressing tendon dUcviFITaMgBIrxagelI
EdleKGacehAfakareFVIeRbkugRtaMgxagkgeT. External prestressing mingayRsYlkg
karGnuvteT edaysar shrinkage nig creep enAkgebtugEdlnaM[mankarkat;bnykug
RtaMgsgt;EdlGnuvtdMbUg.
Profile rbs; tenden GacRtg; ekag bragrgVg;GaRsyeTAelIkarKNnaGgt;eRKOgbgM. CaTU
eTAeKeRbI straight tendon enAkg solid slab nig hollow-cored slab b:uEneKeRbI bent tendon enA
kgFwm nigGgt;eRKOgbgMPaKeRcIn. eKeRbI circular tendon enAkgeRKOgbgMEdlmanTRmg;mUldUcCa
tank, silo nig pipe. eKGacGnuvtkmaMgeRbkugRtaMgEtkgmYydMNakkal beRcIndMNakkaledIm,Ikar
BarebtugkMu[rgkugRtaMgelIs.
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Department of Civil Engineering

eK)anbegItnUvRbBneRbkugRtaMgCaeRcIn EdlkgcMeNamenaHman Freyssinet, Magnel Blaton,

B.B.R.V., Dywidag, CCL, Morandi, VSL, Western Concrete, Prescon, nig INRYCO. eBlxH
eKCYbnUvbBaakgkareRCIserIsRbBneRbkugRtaMgsRmab;kargarBiess.
visVkrKYrBicarNanUvktacMbgbIEdlnaMdl;kareRCIserIsRbBnenH
!> GaMgtg;sIuetnkmaMgeRbkugRtaMgEdlRtUvkar
@> ragFrNImaRtrbs;muxkat; nigKMlatEdlGacmansRmab; tendon
#> tmnRbBneRbkugRtaMg smar nigkmaMgBlkm
]TahrN_xageRkambgajBIlkNBiessrbs;ebtugeRbkugRtaMg.

]TahrN_ 19>1 sRmab;FwmTMrsamBaEdlbgajenAkgrUbTI 19>2 cUrkMNt;kugRtaMgRtg;muxkat;

kNalElVgEdlbNalmkBITmn;pal;xnva nigkrNInkardak;bnk nigeRbkugRtaMg
!> bnkGefrBRgayesI 13.15kN / m
@> bnkGefrBRgayesI 13.15kN / m nigkmaMgsgt;tambeNaycMpit P = 1132kN
#> bnkGefrBRgayesI 30.61kN / m nigkmaMgsgt;tambeNaycakpit P = 1132kN Edl
manGMeBIRtg;cMNakpit e = 10cm
$> bnkGefrBRgayesI 39.28kN / m nigkmaMgsgt;tambeNaycakpit P = 1132kN Edl
manGMeBIRtg;cMNakpitGtibrmasRmab;muxkat; e = 15cm
%> bnkGefrGtibrmaenAeBl P = 1132kN EdlmanGMeBIRtg; e = 15cm
eRbI b = 30cm / h = 60cm / f 'c = 31MPa nigkugRtaMgGnuBaat f 'c = 14.14MPa

dMeNaHRsay
!> kugRtaMgEdlbNalmkEtBIbnkefr nigbnkGefr

bnkpal;rbs;Fwm = (0.3 0.6)24 = 4.32kN / m

2
2
m:Um:g;bnkefr M D.L. = wL8 = 4.32(87.2) = 28kN .m
kugRtaMgenARtg;srsEpkxageRkAbMputEdlbNalBIbnkefrKW
=

esckIENnaMBIebtugeRbkugRtaMg

Mc M (h / 2) 6M
= 3
=
I
bh / 12 bh 2

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D =

6 28
0.3 0.6

10 3 = 1.56MPa

m:Um:g;EdlbNalBIbnkGefr L1 = 13.15kN / m KW
M L.L. =

13.15 7.2 2
= 85.2kN .m
8

kugRtaMgEdlbNalBIbnkGefrKW
L1 =

6M
bh

6 85.2

0.3 0.6

10 3 = 4.73MPa

edayeFVIkarbUkbBakkugRtaMgEdl)anBIbnkefr nigbnkGefr rUbTI 19>2 a eyIg)an

kugRtaMgxagelI = 1.56 4.73 = 6.29MPa rgkarsgt;
kugRtaMgxageRkam = +1.56 + 4.73 = 6.29MPa rgkarTaj
edaysarkugRtaMgTajFMCagm:UDuldac;rbs;ebtug f r = 0.62 31 = 3.45MPa dUcenHFwmnwg)ak;.
@> kgkrNIEdlkugRtaMgbNalBIeRbkugRtaMgesI RbsinebIeKGnuvtkmaMgsgt; P = 1132KN
Rtg;TIRbCMuTmn;rbs;muxkat; enaHmuxkat;tambeNyFwmnwgrgkugRtaMgesI
p =

P
1132
=
10 3 = 6.29MPa
area 0.3 0.6

kugRtaMgcugeRkayEdlbNalBIbnkGefr nigbnkefrbUknwgbnkeRbkugRtaMgenARtg;srs
xagelI nigxageRkambMputKW 12.58MPa nig 0 erogKa rUbTI 19>2 b. kgkrNIenH kmaMg
eRbkugRtaMg)anebgInkugRtaMgsgt;enARtg;srsEpkxagelIbMput[mantmBIrdg nig)ankat;
bnykugRtaMgTajenARtg;srsEpkxageRkam[esInwg 0 . kugRtaMgsgt;Gtibrma12.58MPa
mantmtUcCagkugRtaMgGnuBaat f 'c = 14.14MPa .
#> sRmab;kugRtaMgEdlbNalBIeRbkugRtaMgcMNakpit e = 100mm
RbsinebIeKGnuvtkmaMgeRbkugRtaMg P = 1132KN enARtg;cMNakpit e = 100mm BI eRkamTI
RbCMuTmn;rbs;muxkat; kugRtaMgenARtg;srsEpkxagelI nigEpkxageRkambMputRtUv)ankMNt;
dUcxageRkam. m:Um:g;EdlbNalBIeRbkugRtaMgcMNakpitKW Pe
P (Pe )c
P 6(Pe )

=
A
I
A bh 2
1132
6(1132 0.1) 3
=
10 3
10
0.3 0.6
0.3 0.6 2

p =

= 6.29 6.29

enAsrsEpkxageRkam p = 12.58MPa nig p = 0 enAsrsEpkxageRkambMput.

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638

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BicarNaGMBIkardak;bnkGefr L2 = 30.61kN / m
30.61 7.2 2
= 198.36kN .m
8
6(198.36) 3
=
10 = 11.02MPa
0.3 0.6 2

M L. L. =

L2

kugRtaMgcugeRkayEdlbNalBIbnkefr/ bnkGefr nigkmaMgeRbkugRtaMgenAsrsEpkxagelI

nigEpkxageRkambMputKW 12.58MPa nig 0 erogKa rUbTI 19>2 c. cMNaMfa kugRtaMgcug
eRkaydUcKanwgkrNIelIkmunenAeBlEdlbnkGefresInwg 13.15kN / m . edayGnuvtkmaMg
eRbkugRtaMgenARtg;cMNalpit 10cm FwmenHGacRTbnkGefrEfmeTot 17.46kN / m .

esckIENnaMBIebtugeRbkugRtaMg

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$> sRmab;kugRtaMgEdlbNalBIeRbkugRtaMgcakpitRtg;cMNakpitGtibrma
snt;facMNakpitGtibrmasRmab;muxkat;enHKW e = 15cm .
m:Um:g;Bt;EdlekItedaysarkmaMgeRbkugRtaMgcakpitKW Pe = 1132 0.15 = 169.8kN .m
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kugRtaMgEdlbNalBIkmaMgeRbkugRtaMgKW
p =

6(169.8) 3
1132
10 3
10
0.3 0.6
0.3 0.6 2

= 6.29 9.41
= 15.7 MPa

nig

+ 3.12

begInbnkGefrdl; L3 = 39.28kN / m . m:Um:g;Edl)anBIbnkenHKW

M L. L. =

39.28 7.2 2
= 254.5kN .m
8

kugRtaMgEdlbNalmkBIbnkGefrKW
L3 =

6(254.5)

0.3 0.6

10 3 = 14.14MPa

kugRtaMgcugeRkayenAsrsEpkxagelI nigEpkxageRkambMputEdlbNalBIbnkefr nigbnk

GefrKW 12.58MPa nig 0 erogKa rUbTI 19>2 d. cMNaMfa kugRtaMgcugeRkaydUcKanwgkrNI
munEdr b:uEnbnkGefr)anekIneLIgdl; 39.28kN / m . kugRtaMgTaj 1.56MPa RtUv)an
begIteLIgenAsrsEpkxagelIbMput enAeBlEdleKGnuvtkmaMgeRbkugRtaMg. kugRtaMgenH
mantmtUcCagm:UDuldac;rbs;ebtug f r = 3.45MPa dUcenHvaminekItmansameRbHenAelIFwm
eT.
%> eKkMNt;bnkGefrGtibrmaenAeBlkmaMgeRbkugRtaMgcakpiteFVIGMeBIenARtg; e = 15cm dUct
eTA. kgkrNImun kugRtaMgsgt;cugeRkayesInwg 12.58MPa EdltUcCagkugRtaMgGnuBaat
f 'c = 14.14MPa . dUcenH bnkGefrGacekIneLIgdl; L4 = 43.6kN / m .
43.6 7.2 2
= 282.5kN .m
8
6 282.5 3
=
10 = 15.7 MPa
0.3 0.6 2

M L.L. =

L4

kugRtaMgcugeRkayEdlbNalBIbnkefr nigbnkGefr L4 ehIynigkmaMgeRbkugRtaMg KW

14.14 MPa nig + 1.56 MPa rUbTI 19>2 e. kugRtaMgsgt;KWesInwgkugRtaMgGnuBaat
14.14 MPa ehIykugRtaMgTajKWtUcCag modulus of rupture rbs;ebtug 3.45MPa . kg
krNIenH eKGacKNnabnkGefrBRgayEdlesInwg 43.6kN / m dUcteTA bUkkugRtaMgsgt;
GnuBaatGtibrma 14.14MPa CamYynwgkugRtaMgTajdMbUgenARtg;srsEpkxagelIbMput
1.56 MPa edIm,ITTYl)an 15.7 MPa . m:Um:g;EdlnwgbegItkugRtaMgenAsrsEpkxagelIbMput
esckIENnaMBIebtugeRbkugRtaMg

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mhaviTalysMNg;sIuvil
15.7 MPa

NPIC

esInwg
bh 2

M =
6

15.7
(0.3)(0.6)2 103 = 282.6kN.m
=
6
W L2
M= L
8
8 282.6
WL =
= 43.6kN / m
7.2 2

eyIgTTYl)an

cMNaMfa
!> muxkat;ebtugTaMgmUlKWskmkgkarTb;Tl;CamYykmaMgxageRkA
@> kugRtaMgTajcugRkayenAkgmuxkat;tUcCag modulus of rupture rbs;ebtug Edlbgaj
famuxkat;ebtugminmansameRbHeRkamGMeBIrbs;bnkGtibrma
#> bnkGnuBaatenAelIFwmekIneLIgeRcInKYrsmedaysarkarGnuvtrbs;kmaMgeRbkugRtaMg
$> karekIneLIgnUvcMNakpitrbs;kmaMgeRbkugRtaMgkbegInkmaMgGnuvtn_GnuBaat EdleFVI[kugRtaMg
enAelImuxkat;minFMCagkugRtaMgGnuBaat.
19>1>2> karGnuvteRbkugRtaMgedayEpk (Partial Prestressing)
eKkMNt;Ggt;ebtugeRbkugRtaMgedayEpk (partially prestressed concrete member) CaGgt;
Edl
- kugRtaMgxagkgEdlmanGMeBITb;EpknkugRtaMgEdlekItBIbnkxageRkA
- kugRtaMgTajekItmanenAkgebtugeRkamGMeBIbnkeFVIkar (service load)
- EdkBRgwgminEmnCaEdkeRbkugRtaMgRtUv)andak;bEnmedIm,IbegInlTPaBrbs;Ggt; edIm,ITb;
nwgm:Um:g;
eKGacBicarNa partially prestressed concrete tamBIrkrNI
!> eKeRbIEdkeRbkugRtaMg nigEdkminEmneRbkugRtaMgenAkgmuxkat;EtmYy. ExSkabeRbkugRtaMg
begItkugRtaMgxagkgRtUv)anKNnaedIm,ITTYl)an ultimate capacity rbs;muxkat;ebtugEt
mYyEpkb:ueNaH. cMENk capacity EdlenAsl;RtUv)anTTYlBIEdkminEmneRbkugRtaMg
Edldak;tamTisdUcKanwgkabeRbkugRtaMg. EdkEdleRbICaEdkminEmneRbkugRtaMgGacCaRb
T.Chhay

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ePTEdkFmta dUcCaEdk carbon steel bCaEdk high-tensile-strength. kabeRbkugRtaMgk

CaRbePTEdkFmtadUcEdkminEmneRbkugRtaMgEdr Etvaman ultimate strength esInwg
1725MPa 250ksi . kareRCIserIsGaRsynwgktacMbgBIrKW PaBdabGnuBaat nigTMhM
sameRbHGnuBaat. dUcKa ACI Code kMNt;nUvpleFobGtibrmanRbEvgElVgelIkkm<s;n
Ggt;ebtugGarem: sRmab;PaBdab. eKminGnuBaat[manPaBdabFMelIslubCamYynwgkm<s;
rbs;muxkat;ebtugeRbkugRtaMgtUc nigedaysarkareRbIPaKryEdktic. sameRbHekItmanenA
tMbn;rgkarTajrbs;muxkat;ebtug benARtg;nIv:UEdkedaysareKGnuBaat[kugRtaMgTajekIt
maneRkamGMeBI working load. eKGnuBaatsameRbHGtibrmaRtwm 0.016in. (0.41mm)
sRmab;Ggt;xagkg nig 0.013in. (0.33mm) sRmab;Ggt;xagkg.
@> kugRtaMgxagkgEdleFVIGMeBIelIGgt;)anEtBI prestrssed steel b:ueNaH b:uEnvaRtUv)anTajCa
mYynwgEdnkMNt;TabCag. kgkrNIenHsameRbHekItmanelOnCagGgt;rgeRbkugRtaMgeBj
eljeRkambnkdUcCaKa.
eKGacBicarNa partially prestresssed concrete kgTRmg;kNalrvagebtugGarem: nigebtug
eRbkugRtaMgeBj (fully prestressed concrete). enAkgGgt;ebtugGarem: sameRbHekItmaneRkamGMeBI
bnkeFVIkar dUcenHeKdak;EdkBRgwgenAkgtMbn;Taj. CaTUeTAenAkgGgt;ebtugeRbkugRtaMg sameRbH
minekItmaneRkamGMeBIbnkeFVIkareT. kugRtaMgsgt;EdlbNalBIkmaMgeRbkugRtaMgGacesI belIsBI
kugRtaMgTajEdlbNalBIbnkxageRkA. dUcenHeKGacBicarNaGgt; partially prestressed concrete
CaGgt;ebtugGarem:EdlkugRtaMgxagkgrbs;vamanGMeBITb;nwgEpkxH rbs;kugRtaMgEdl)anBIbnkxag
eRkA dUcenHkugRtaMgTajenAkgebtugminRtUvFMCagtmkMNt;eRkambnkeFVIkareT. eKKitvaCaebtug
Garem:enAeBlNaEdlminmankugRtaMgxagkgeFVIGMeBIelIGgt;. ebtugeRbkugRtaMgeBjCakRmitx<s;bMput
rbs;ebtugeRbkugRtaMgedayEpk EdlenAkgenaHEdkminEmneRbkugRtaMgRtUv)ankat;bnydl;sUn.
enAcenaHGgt;ebtugGarem:EdlmaneRbH nigGgt;ebtugeRbkugRtaMgeBjEdlminmaneRbH eK
manEdndFMsRmab;KNnaebtugeRbkugRtaMgedayEpk rUbTI 19>3. kareRCIserIskRmitnkareFVIeRb
kugRtaMgdl nwgbegItnUveRKOgbgMEdlmansuvtiPaB nigmanlkNesdkic.
rUbTI 19>3 bgajBIExSekagPaBdab-bnkrbs;FwmebtugGarem:EdlmanbrimaNEdk nigRbePT
EdkxusKa. ExSekag a bgajBIFwmebtugGarem: EdlmansameRbHFmtaeRkambnktUc Wcr . eKGackM
Nt;m:Um:g;EdleFVI[eRbH (cracking moment) dUcxageRkam
esckIENnaMBIebtugeRbkugRtaMg

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M cr =

Edl

NPIC

fr I
c

m:UDuldac;rbs;ebtug = 0.62 f 'c

I = m:Um:g;niclPaBn gross concrete section
c = cmayBIGkSNWteTAsrsrgkarTajxageRkAbMput
eKGackMNt;ebtugEdleFVI[eRbH (cracking load) BI cracking moment enAeBlRbEvg ElVg
nigRbePTnkardak;bnkRtUv)ankMNt;. sRmab;FwmTRmsamBaEdlrgbnkcMcMnugenAkNalElVg
Wcr = (4 M cr ) / L .
ExSekag e nig f bgajBIFwmebtugeRbkugRtaMgeBjEdlmanEdktic nigEdkeRcIn erogKa. Fwm
ebtugGarem:EdlmanbrimaNEdkeRcIn)ak;edaysarkarEbkebtugmunnwgEdkeTAdl; yield strength b
T.Chhay

fr =

644

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Department of Civil Engineering

rbs;va. FwmmanPaBdabtUc nwgrgkar)ak;edayPaBRsYy (brittle failure). FwmEdlman

brimaNEdktic)ak;edaysarEdkeFVIkardl; yield nig ultimate strength rbs;va. vabgajnUv)abdab
nigsameRbHEdlbNalBIkarlUtsac;rbs;EdkmuneBlebtugEbkCabnbnab; ehIyFwm)ak;rlM.
enAcenaHExSekag a nig e CaEdndFMrbs;FwmebtugCamYynwgbrimaNERbRbYlrbs;Edk nigrg
nUvbrimaNERbRbYlrbs;kmaMgeRbkugRtaMg. FwmEdlrgkmaMgeRbkugRtaMgtUcenAEk,rExSekag a ehIy
FwmEdlmaneRbkugRtaMgFMenAEk,rExSekag e . eKeRCIserIsbnSMEdkeRbkugRtaMg nigEdkminEmneRbkug
RtaMgsRmab;karKNnaKWGaRsyelIkugRtaMgebtugGnuBaat PaBdab nigTMhMsameRbHGtibrma.
ExSekag b tMNag[FwmEdlnwgeRbHeRkamGMeBInbnkeFVIkareBjelj. RbsinebIEtEpkxH
rbs;bnkGefrekItmanenAelIeRKOgbgMCaerOy enaH W1 tMNag[bnkefrsrub nigEpkxHrbs;bnk
Gefr L1 .
ExSekag c tMNag[Fwmcab;epImeRbHeRkamGMeBI working load. kugRtaMgTajGtibrmaenA
kgebtug = f r = 0.62 f 'c .
ExSekag d tMNag[FwmEdlmankmaMgeRbkugRtaMgkMNt;. muxkat;eRKaHfak;rbs;Fwmnwgmin
eRbHeRkambnkeFVIkareBjeljeT b:uEnvanwgmankugRtaMgTajGtibrma 0 < f r < 0.62 f 'c . ACI
Code GnuBaatkugRtaMgTajGtibrmaenAkgebtugRtwm 0.5 f 'c .
ExSekag e nig e' tMNag[FwmebtugeRbkugRtaMgeBjeljEdlminmankugRtaMgTajeRkam
bnkeFVIkar emIlrUbTI 19>4.
sarRbeyaCn_dsMxan;bMputrbs;kmaMgeRbkugRtaMgedayEpkKWlTPaBkgkarRKb;RKgkMeNag
(camber). edaykat;bnykmaMgeRbkugRtaMg camber nwgRtUv)ankat;bny ehIysnSMnUvbrimaNEdk
eRbkugRtaMg brimaNkargarkgkarTaj nigcMnYn end anchorage.
GaRsynwgGaMgtg;sIuetnkmaMgeRbkugRtaMg sameRbHenAkg partially prestressed member
ekIteLIgelOnCagenAkg fully prestressed concrete member eRkamGMeBIrbs; service load. enA
eBlEdlsameRbHekItman m:Um:g;niclPaBRbsiTPaBrbs;muxkat;eRKaHfak;RtUv)ankat;bny ehIyeK
nwgTTYl)anPaBdabFMCagmun. b:uEn kareRbIkmagM eRbkugRtaMgedayEpkKWeKTTYl)anlTplKYrCaTI
eBjcit ehIyvaTTYl)ankareBjniym.
proof stress

esckIENnaMBIebtugeRbkugRtaMg

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NPIC

19>1>3> karcat;cMNat;fak;Ggt;rgkarBt;ebtugeRbkugRtaMg
(Classification of Prestressed Concrete Flexural Members)

)anEckGgt;ebtugeRbkugRtaMgCabIfak;edayQrelIkugRtaMgTaj
enAelIsrsxageRkAbMput f t enAkgtMbn;TajeRkamGMeBIbnkeFVIkardUcxageRkam
ACI Code, Section 18.3

!> fak; U (uncracked section) Edlman f t 0.62 f 'c . enAkgmuxkat;ebtugEdlKansam

eRbHenH eKeRbIlkNn gross section edIm,IRtYtBinitPaBdabeRkamGMeBIbnkeFVIkar. Kan
sameRbHekItmanenAkgmuxkat; nigeKminRtUvkar skin reinforcement eT.
@> fak; T (muxkat;enAkg transition zone) Edlman 0.62 f 'c < f t f 'c . muxkat;
RbePTenHmankugRtaMgTajenAkgebtugFMCagm:UDuldac; (modulus of rupture) rbs;ebtug
f r = 0.62 f 'c EdlbegItnUvkrNIcenaHmuxkat;eRbH nigmuxkat;Gt;eRbH. enAkgkrNIenH
eKeRbIlkNn gross section edIm,IRtYtBinitkugRtaMg ehIyeKeRbI bilinear section rbs;
muxkat;eRbHedIm,IKNnaPaBdab. eKmincaM)ac;eRbI skin reinforcement enAkgtMbn;TajeT.
#> fak; C (cracked section) Edlman f t > f 'c . kugRtaMgTajenAkgmuxkat;FMCag
modulus of rupture rbs;ebtug 1.6 dg. dUcenH sameRbHnwgekItmandUckgkrNIGgt;eb
tugeRbkugRtaMgedayEpk. enAkgkrNIenH eKeRbIlkNnmuxkat;eRbHedIm,IRtYtBinitkug
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RtaMg sameRbH nigPaBdab. eKKYreRbIkarpl;[edIm,IRKb;RKgsameRbH nigeRbI skin

reinforcement dUckarBnl;enAkgEpk 6>7 sRmab;Ggt;ebtugGarem:Edlmankm<s;
RbsiTPaB d > 915mm .
19>2> smar nigtRmUvkarsRmab;beRmIbRmas; (Material and Serviceability Requirement)
19>2>1> ebtug (Concrete)
lkNrbs;ebtugRtUv)anbgajenAkgCMBUk 2. eTaHbICaerOy Ggt;ebtugGarem:RtUv)anplit
BIebtugEdlmanersIusg;sgt; 21MPa eTA 35MPa keday kGgt;ebtugeRbkugRtaMgRtUv)anplitBI
smarEdlmanersIusg;x<s;Cag CaTUeTAsitenAcenaH 28MPa eTA 56MPa . eKeRbIebtugersIusg;x<s;
sRmab;Ggt;ebtugcak;eRsc nigGgt;ebtugeRbHkugRtaMg Edlkarlay karcak; karbgab; nigkarEfTaM
ebtugsiteRkamkarRtYtBinity:agm:t;ct;.
kugRtaMgGnuBaatenAkgebtugEdleyagtam ACI Code, Section 18.4 mandUcxageRkam
!> kugRtaMgeRkayeBlepreRbkugRtaMg (prestress transfer) nigmuneBl)at;bg;eRbkugRtaMg
(prestress losses):

kugRtaMgsgt;GtibrmaesInwg 0.6 f ci
b. kugRtaMgTajGtibrma elIkElgGVIEdl)anGnuBaatdUcxageRkam esInwg 0.25 f ci
c. kugRtaMgTajGtibrmaenARtg;cugnGgt;TRmsamBaesInwg 0.5 f ci
Edl f ci CaersIusg;rbs;ebtugenAeBlepr
RbsinebIkugRtaMgTajmantmFMCagenH eKRtUvdak;EdkenAtMbn;sgt;edIm,ITb;Tl;kmaMgTaj
srubenAkgebtug edayQrelI uncracked gross section.
@> kugRtaMgeRkamGMeBIbnkeFVIkareRkayeBlkMhatbg; (loss) TaMgGs; sRmab;fak; U nigfak;
T mandUcteTA kugRtaMgsgt;Gtibrma 0.45 f 'c EdlbNalBIkmaMgeRbkugRtaMgbUknwgbnk
efr nigkugRtaMg 0.05 f 'c EdlbNalBIkmaMgeRbkugRtaMgbUknwgbnksrub.
#> kugRtaMgTaMgenHGacmantmFMCagenH RbsinebIkarBiesaF nigkarviPaKbgajfakaRbRBwteTA
rbs;vaRKb;RKan;.
a.

esckIENnaMBIebtugeRbkugRtaMg

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NPIC

19>2>2> EdkeRbkugRtaMg (Prestressing Steel)

Edk tendon EdleKniymeRbICageKenAkgebtugeRbkugRtaMgCa strands bkab Edlplit
eLIgCamYynwglYssrsr (wire) CaeRcIn CaTUeTAmancMnYn 7 b 19 . Wire nig bar kRtUv)aneKeRbI
R)as;pgEdr. Stand nig wire RtUv)anpliteLIgedayeKarBtam ASTM Standard A421 sRmab;
uncoated stress-relieved wire nig A416 sRmab; uncoated seven-wire stress-relieved strand.
lkNrbs;EdkeRbkugRtaMgRtUv)an[enAkgtarag 19>1.
tarag 19>1
Type
Seven-wire strand (grade 250)

Seven-wire strand (grade 270)

Prestressing wire grades (250)

(250)
(240)
(235)
Prestressing bars (smooth)
(grade 145 or 160)

Prestressing bars (deformed)

(grade 150-160)

Diameter
(mm)
6.350
7.950
9.525
11.125
12.700
15.240
9.525
11.125
12.700
15.250
4.877
4.978
6.350
7.010
19.050
22.225
25.400
28.575
31.750
34.925
15.875
19.050
25.400
31.750
34.925

Area
(mm2)
23.2
37.4
51.6
69.7
92.9
139.4
54.8
74.2
98.7
138.7
18.7
19.4
31.6
38.7
283.9
387.1
503.2
638.7
793.5
954.8
180.6
271.0
548.4
806.5
1006

Mass
(kg/m)
0.179
0.298
0.402
0.551
0.729
1.101
0.432
0.595
0.789
1.101
0.146
0.149
0.253
0.298
2.232
3.036
3.973
5.030
6.206
7.515
1.458
2.218
4.480
6.535
8.274

EdkeRbkugRtaMgEdleRbIenAkgebtugeRbkugRtaMgRtUvEtmanersIusg;x<s; CaTUeTAman
enAcenaH 1730MPa eTA 1860MPa . eKcaM)ac;GnuBaat[EdkersIusg;x<s;
mansac;lUtFM nigrkSakugRtaMgenAkgebtug[RKb;RKan; nigGciRny_bnab;BI inelastic shortening rbs;
ultimate strength f pu

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ebtug.
kugRtaMgGnuBaatenAkgEdkeRbkugRtaMgeyagtam ACI Code, Section 18.5 mandUcxageRkam
!> kugRtaMgGtibrmaEdlbNalBI tendon jacking force minRtUvFMCagtmEdltUcCageKk
gcMeNam 0.8 f pu nig 0.94 f py . tmEdltUcCagminRtUvFMCagkugRtaMgEdlENnaMedayGk
plit tendon b anchorage eT.
@> kugRtaMgGtibrmaenAkg pretensioned tendon PameRkayeBleprminRtUvFMCagtmtUc
CageKkgcMeNam 0.74 f pu nig 0.82 f py .
#> kugRtaMgGtibrmaenAkg postensioned tendon eRkayeBl tendon RtUv)an anchor KW
0.70 f pu .
19>2>3> EdkBRgwg

(Reinforcing Steel)

CaTUeTA eKeRbIEdkBRgwgminEmneRbkugRtaMgenAkgGgt;eRKOgbgMebtugeRbkugRtaMg Ca
BiessenAkgsMNg;ebtugeRbkugRtaMgcak;eRsc. eKeRbIEdkBRgwgCaEdkkmaMgkat;TTwg Ca
EdkbEnmsRmab;kardwkCBan nigkarelIkdak;Ggt;cak;eRsc ehIynigeRbIenAkgGgt;ebtugeRb
kugRtaMgedayEpkEdlcUlrYmCamYynwgEdkeRbkugRtaMg. RbePT nigkugRtaMgGnuBaatrbs;Edk
RtUvENnaMenAkgCMBUk 2 nigCMBUk 5 rYcehIy.
19>3> kMhatbg;eRbkugRtaMg
19>3>1> Lump-sum losses

(Loss of Prestress)

kMhatbg;nkmaMgeRbkugRtaMgCabnbnab;ekItmaneRkayeBlkmaMgeRbkugRtaMgRtUv)an
eprBI jack eTaGgt;ebtug. kMhatbg;eRbkugRtaMgCakarkat;bnykmaMgeRbkugRtaMgkgmYy
CIvitneRKOgbgM. brimaNkMhatbg;enAkg tendon ERbRbYlcenaHBI 15% eTA 30% nkugRtaMg
edIm edayGaRsynwgktaCaeRcIn. sRmab;eRKOgbgMebtugGarem:Tmn;FmtaPaKeRcInEdlsagsg;
eday standard method, tendon stress loss bNalmkBI elastic shortening, shrinkage, creep nig
relaxation rbs;EdkKWmantmRbEhlnwg 35ksi(241MPa ) sRmab; pretensioned member
esckIENnaMBIebtugeRbkugRtaMg

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NPIC

nigsRmab; posttensioned member KWRbEhlbwg 25ksi(172MPa) . kmaMgkkit nig anchorage slip

minRtUv)an rab;bBaleT.
karENnaMBIrsRmab;kar)a:n;sankMhatbg;srubenAkgGgt;ebtugeRbkugRtaMgRtUv)anbgajeday
AASTHO nig Posttensioning Institute (PTI). AASTHO ENnaM[ykkMhatbg;srub
(edayminKitkmaMgkkit) 45ksi(310MPa ) sRmab; pretensioned strand nig 33ksi(228MPa )
sRmab; postentioned strand nig wire enAeBlEdleKeRbIersIusg;sgt;ebtug f 'c = 35MPa . PTI
ENnaM lump-sum prestress loss sRmab; posttensioned member 35ksi(241MPa) sRmab;Fwm nig
30ksi(207 MPa ) sRmab;kRmalxN edayminKitkmaMgkkit. eKGaceRbItmTaMgGs;enH)anluH
Rta EteK)aneFVIkar)a:n;RbmaNkMhatbg;eRbkugRtaMgedayRbPBnkMhatbg;nImYydac;edayELkBI
Ka)anl dUcEdl)anENnaMy:agsegb.
CaTUeTA RbPBnkMhatbg;eRbkugRtaMgKW
- Elastic shortening rbs;ebtug
- Shrinkage rbs;ebtug
- Creep rbs;ebtug
- Relaxation rbs;Edk tendon
- kmaMgkkit

Anchorage set

19>3>2> kMhatbg;edaysar (Elastic Shortening of Concrete)

kar)a:n;RbmaNkMhatbg; elastic shortening rbs;ebtugenAkg pretensioned member
RtUv)aneFVIeLIgdUcteTA. BicarNa pretensioned concrete member nmuxkat;efr nigkugRtaMgBRgay
esItambeNayGkSTIRbCMuTmn;rbs;vaedaysarkmaMg Fo . eRkayBIkareprkmaMgeRbkugRtaMgFwmebtug
nig prestressing tendon rYjxIedaybrimaNesIKa edaysarPaBsitrvagsmarTaMgBIr. dUcenH kmaMg
eRbkugRtaMgEdlcab;epIm Fo Fak;mkRtwm Fi ehIykMhatbg;kmaMgeRbkugRtaMgKW Fo Fi . dUcKa
strain enAkgebtug c RtUvEtesInwgbERmbRmYlrageFob (strain) rbs; tendon s . dUcenH
c = s b ( f c / Ec ) = (f s / E s ) ehIykMhatbg;kugRtaMgEdlbNalBI elastic shortening KW

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f s =

Edl

Department of Civil Engineering

nF nF
Es
f c = nf c = i o
Ac
Ac
Ec

(19.1)

RkLaprbs;muxkat;ebtug
n = E s / Ec = pleFobm:UDul (modular ratio)
f c = kugRtaMgrbs;ebtugenARtg;TIRbCMuTmn;rbs;EdkeRbkugRtaMg
KuNkugRtaMgnwgRkLaprbs;EdkeRbkugRtaMg Asp edIm,ITTYl)ankmaMgsrub . Elastic loss KW
Ac =

nF
ES = Fo Fi = f s Asp = (nf c )Asp o Asp
Ac
Fi = Fo (nf c )Asp

(19.2)
(19.3)

sRmab;karKNnaGnuvtn_ kMhatbg;kugRtaMgnkmaMgeRbkugRtaMg f s kgmYyktap Asp mantm

Rbhak;RbEhlnwg nFo / Ac . RbsinebI kmaMg Fo manGMeBIRtg;cMNakpit e enaH elastic loss Edl
bNalBIvtmann Fo nigbnkefrGnuvtn_enAeBleprKW
ES = (nf c )Asp EdlbNalBIeRbkugRtaMg + (nf c )Asp bnkefr
F F e2
M e
ES = Fo Fi = i + i nAsp + D nAsp
A
I
I

(19.4)

eKGaceRbItmRbhak;RbEhln Fi = (0.63 f pu )Asp enAkgsmIkarxagelI.

1 e 2
Fo + f c (D.L.)nAsp = Fi 1 + nAsp +
A I

Fo + nAsp f c (D.L.)
Fi =
1 e2
1 + nAsp +
A I

(19.5)

sRmab; posttensioned member Edl tendon nig individual strand minrgkugRtaMgdMNalKa enaHeK
GackMhatbg;eRbkugRtaMgesInwgBak;kNalntm ES sRmab; prestensioned member.
dUcKa eKGacyk elastic shortening loss enAkgkRmalxNesInwgmYyPaKbYnntm ES
sRmab; prestensioned member edaysarkarlUtrbs; tendon
mYyman\TiBltictYceTAelIkugRtaMgn tendon dTeTot.
19>3>3> kMhatbg;edaysarkarrYmmaD
(Loss Due to Shrinkage)
kMhatbg;eRbkugRtaMgEdlbNalBIkarrYmmaDKWGaRsynwgeBl. eKGac)a:n;RbmaNvadUcxag
eRkam
esckIENnaMBIebtugeRbkugRtaMg

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mhaviTalysMNg;sIuvil

NPIC

SH = f s (shrinkage) = sh E s

(19.6)

Edl Es = 2 105 MPa nig sh = shrinkage strain enAkgebtug

eKGacsnt; Strain mFmEdlbNalBIkarrYmmaDmantmdUcxageRkam
- sh1 = 0.0003 sRmab; pretensioned member
- sh2 = 0.0002 sRmab; posttentioned member
RbsinebIeKGnuvt posttensioning kgcenaH 5 eTA 7 feRkayBIcak;ebtug/ eKGacyk shrinkage strain esInwg 0.8 sh1 . RbsinebIeKGnuvt posttensioning kgcenaH 1 s)ah_eTA 2 s)ah_ eK
GaceRbI sh = 0.7 sh1 nigRbsinebIeKGnuvt posttensioning eRkayeBlcak;ebtugeRcInCag 2 s)ah_
enaHeKGacyk sh = sh2 . eKkGac)a:n;RbmaNkMhatbg;edaykarrYmmaD SH dUcxageRkam
0.06V
SH = 8.2 10 6 K sh E s 1
S

(100 RH )

Edl V / S = pleFobmaDelIp nig RH = average relative humidity. K sh = 1.0 sRmab;

pretensioned member nigesInwg 0.8 / 0.73 / 0.64 nig 0.58 sRmab; posttensioned member
RbsinebIeKGnuvt posttensioning eRkayeBlcak;ebtug 5 / 10 / 20 nig 30 f erogKa.
19>3>4> kMhatbg;edaysar creep rbs;ebtug
Creep CakMhUcRTg;RTayGaRsynwgeBlEdlekIteLIgenAkgebtugeRkamGMeBIbnkefr. kMhUc
RTg;RTayEdlekIteLIgedaysar creep eFVI[)at;bg;kmaMgeRbkugRtaMgBI 5% eTA 7% .
Creep strain ERbRbYlCamYynwgGaMgtg;sIuetnkugRtaMgedImenAkgebtug relative humidity
nigeBl. eKGackMNt;kMhatbg;kugRtaMgEdlbNalBI creep dUcxageRkam
CR = f s (creep) = Cc (nf c ) = Cc ( cr E s )
creepstrain, cp
Cc =
creep =
initial elastic strain, i

(19>7)

emKuN
Edl
eKGacyktm Cc dUcxageRkam
ersIusg;ebtug

f 'c 28MPa

f 'c > 28MPa

Relative humidity

100%

50%

100%

50%

Cc

1 2

24

0 .7 1 .5

1 .5 3

T.Chhay

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Department of Civil Engineering

eKGaceFVI interpolation sRmab;tmEdlenAcenaHtmEdlenAkgtaragxagenH. edayKitfa

creep EtBak;kNalekIteLIgkgGMLg 134 ExdMbUgn 6 ExdMbUgbnab;BIkarepreRbkugRtaMgeTAebtug
nigeRkamlkxNsMeNImFmta enaHeKGacsnt; creep strain sRmab;karKNnaGnuvtn_dUcxageRkam
!> sRmab; pretensioned members, cr = 7 10 5 kugRtaMgenAkgebtug
@> sRmab; postensioned members, cr = 5.2 105 kugRtaMgenAkgebtug. eKeRbItmenH
enAeBlEdleKGnuvt posttensionning kgGMLg 2 eTA 3 s)ah_. sRmab;karGnuvt
postensioning elOnCagenH eKGaceRbItmkNal.
eKeRbItmTaMgenH enAeBlEdlersIusg;rbs;ebtugenAeBleprKW f 'c 28MPa . enAeBlEdl
f 'c < 28MPa creep strain KYrekIneLIgkgGRta 4 / ersIusg;Cak;Esg.
(19.8)
kMhatbg;eRbkugRtaMgsrubEdlbNalBI creep = cr Es
19>3>5> kMhatbg;edaysar Relaxation rbs;Edk
Relaxation rbs;EdkbNal[mankMhatbg;enAkgEdkeRbkugRtaMgGaRsynwgeBl Edl
RsedogKanwg creep enAkgebtugEdr. kMhatbg;edaysar relaxation ERbRbYleTAtamRbePTEdk.
CaTUeTA tmrbs;vaRtUv)anpl;[edayGkplitEdk. CaFmta eKsnt;kMbaatbg;enHesInwg 3% n
kugRtaMgedImrbs;EdksRmab; posttensioned member nig 2% eTA 3% sRmab; pretensioned
members. RbsinebIeKminmanBtmanBIkarBiesaFeT eKGacPaKrykMhatbg;sRmab; relaxation enA
1000h dUcxageRkam
!> enAkg low-relaxation strands, enAeBlEdleRbkugRtaMgedImesInwg 0.7 f pu nig 0.8 f pu /
relaxation (RE) KW 2.5% nig 3.5% erogKa.
@> enAkg stress-relieved strand b wire, enAeBlEdleRbkugRtaMgedImesInwg 0.7 f pu nig
0.8 f pu / relaxation (RE) KW 8% nig 12% erogKa.
19>3>6> kMhatbg;edaysarkmaMgkkit (Loss Due to Friction)
CamYynwgEdkrg pretensioning kMhatbg;edaysarkmaMgkkitekItmanenAeBlEdl wires b
strand dabtam diaphragm. CaFmtakMhatbg;enHmantmtUc ehIyeKGacecalva)an.

esckIENnaMBIebtugeRbkugRtaMg

653

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

enAeBlEdl strand dabtam concordant profile enaHkMhatbg;edaysarkmaMgkkitGacmantmFM.

enAkrNIEbbenH CaFmtaeKeRbI]brkrN_Edlvas;bnkCak;EsgedIm,IkMNt;kmaMgenAkg tendon.
CamYynwgEdkrg posttensioning, \TiBlnkmaMgkkitmantmFMedaysarktacMbgBIrKW
kMeNagrbs; tendon nigkar)at;bg;PaBRtg;rbs;bMBg; (wobble). RbsinebIeKTb;cugbgb;magrbs;
tendon edaykmaMg P2 nigeKTajcugTMenrmageTotrbs; tendon edaykmaMg P1 edIm,I[ tendon
enaHrGiltamTisrbs;kmaMg P1 )anluHRtaEt
P1 = P2 e

px

(19.9)

Edl = emKuNmMukkitsaTic nig px = mMurvag P1 nig P2 . CaTUeTAeKKit wobble effect tamviFI

RsedogKa
Px = Ps e ( + Klx )

Ppj = Ppx e
Ppx = Ppj e

Edl

+ Kl px + p px
Kl px + p px

(19.10)

kmaMgeRbkugRtaMgenARtg;cMNuc x
Ppx = kmaMgeRbkugRtaMgenARtg; jacking end
p = emKuNkmaMgkkitedaysarkMeNag
px = bMErbMrYlmMusrubnragtambeNayrbs;EdkeRbkugRtaMgBI jacking end eTAdl;cMNuc x
KitCara:dg;
RbEvgkMeNag
=
kaMkMeNag
K = emKuNkmaMgkkit wobble kgmYyktaRbEvgrbs; tendon CakarsRmYl ACI Code
[nUvsmIkarxageRkamsRmab;krNI ( p px + Kl x ) 0.30 . lTplEdlTTYl)anBI
smIkarCatmRbEhl
Ppj =

Ppx = Ppj 1 + Kl px + p px

)1

(AIC Code, eq. 18.2)

(19.11)

emKuNkmaMgkkit nig K GaRsynwgRbePTn strand b wire, RbePTbMBg; niglkxNp

b:H. ACI Commentaru, Sectin 18.6 nigenAkgtarag 19.2 [nUvtmRbhak;RbEhlrbs; nig K .
kMhatbg;edaysarkmaMgkkitenAkg jack ERbRbYl nigGaRsynwgktaCaeRcIn edayrab;bBal
TaMgRbEvgrbs; tendon. eKENnaM[eRbI accurate load ceel edIm,Ivas;kmaMgedaypal;. kareRbI
pressure gauge pl;nUvlTplminsuRkit luHRtaEteKeFVIkarEktRmUveTAtamkmaMgEdleKsal; enAkg
T.Chhay

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Introduction to Prestressed Concrete

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Department of Civil Engineering

.
kMhatbg;edaysarkmaMgkkitenAkg cnchorage KWGaRsynwgRbePT anchorage nigbrimaN
n deviation rbs; tendon Edlqgkat; anchorage. CaFmtakMhatbg;enHmantmtUcEdlGac
ecal)an. karENnaMkgkrNIBiessKYrTTYl)anBIplitkr.

tendon

tarag 19>2 emKuNkmaMgkkitsRmab; post-tensioned tendon

RbePT tendon

emKuNkmaMgkkit wobble K emKuNkmaMgkmaMgkkit

kgmYyktaRbEvg (10 3 ) edaysarkMeNag

Tendon in flexible metal sheathing

(grouted)
Wire tendon
Seven-wire strand
High-strength bars

3.33 5.0

0.15 0.25

1.67 6.67

0.15 0.25

0.33 2

0.08 0.30

1 6.67

0.05 0.15

0.33 0.67

0.05 0.15

Pregreased unbonded tendon

Wire tendon and seven-wire strand
Mastic-coated unbonded tendon
Wire tendon and seven-wire strand

19>3>7> kMhatbg;edaysar Anchor set

enAeBlkmaMgenAkg tendon RtUv)aneprBI jack eTA anchorage unit, clnart;cUlkgbnic
rbs; tendon ekIteLIgedaysarkardak; gripping device b wedge. karrGilenHbNal[man
tendon rYjxI EdleFVI[)at;bg;kmaMgeRbkugRtaMg. RbEvgrGilERbRbYlBI 2.5mm eTA 6mm ehIy
CaTUeTARtUv)ankMNt;edayplitkr. eKGacKNnakMhtabg; anchor set edayrUbmnxageRkam
f s = E s =

Edl

L
Es
L

(19.12)

GaMgtg;sIuetnkarrGil anchor

E s = 2 10 5 MPa

L=

RbEvgrbs; tendon

esckIENnaMBIebtugeRbkugRtaMg

655

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

edaysarkMhatbg;eRbkugRtaMgCacMras;smamaRteTAnwgRbEvgrbs; tendon bRbEhlCaBak;kNal

nRbEvgrbs; tendon RbsinebIvargkugRtaMgBIcugsgxagkgeBlEtmYyPaKrykMhatbg;enAkgkug
RtaMgEdknwgRtUv)ankat;bnyenAeBlEdlRbEvgrbs; tendon ekIneLIg. RbsinebI tendon lUteday
enAeBlepr enaHeKecalkMhatbg;eRbkugRtaMgedaysarkarrGil.

]TahrN_ 19>2 FwmTRmsamBarg pretensionning RbEvg 11m manmuxkat;ctuekaNEkgCamYynwg

nig h = 80cm . KNnakMhatbg;eGLasic nigkMhatbg;EdlGaRsynwgeBlTaMgGs;. eK

[ kmaMgeRbkugRtaMgenAeBleprKW Fi = 1935kN / RkLaprbs;EdkeRbkugRtaMgKW Aps =
1935mm 2 / f 'c = 35MPa / Ec = 34500MPa / E s = 2 105 MPa / profile rbs; tendon manrag
Ca)a:ra:bUl/ cMNakpitenAkNalElVg = 15cm nigcMNakpitenAcug = 0 .
b = 45cm

dMeNaHRsay

!> kMhatbg;edaysar elastic shortening: kugRtaMgEdl)anBIkmaMgeRbkugRtaMgenAeBleprKW

Fi
1935 3
=
10 = 1000MPa
A ps 1935

rbs;EdkeRbkugRtaMg = Efs = 21000

= 5 10 3
5
10
s
edayeRbIsmIkar 19>1
E
2 105
n= s =
= 5.8 yk 6
E
34500

strain

f s =

nFi
6 1935 3
=
10 = 32.25MPa
Ac 450 800

edayKitbERmbRmYlcMNakpittambeNayFwm
Fi
1935
strain enAmuxkat;xagcug =
=
10 3 = 1.56 10 4
450 800 34500
AE
c c

strain

enAkNalElVg = AFEi

c c
3

Fi e 2
IEc

bh
450(800 )3
=
= 1.92 1010 mm 4
12
12
1935 150 2
strain = 1.56 10 4 +
103 = 2.22 10 4
10
1.92 10 34500
1
strain
= (1.56 + 2.22)10 4 = 1.89 10 4
2
I=

mFm

T.Chhay

656

Introduction to Prestressed Concrete

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Department of Civil Engineering

kMhatbg;eRbkugRtaMg = strain Es = 1.89 10 4 2 105 = 37.8MPa

37.8
PaKrykMhatbg; = 1000
= 3.78%
@> kMhatbg;edaysar shrinkage:
shrinkage strain = 0.0003

f s = sh E s = 0.0003 200000 = 60MPa

60
=
= 6%
1000

PaKrykMhatbg;
#> kMhatbg;edaysar creep rbs;ebtug edaysnt; Cc = 2.0 enaH f s = Cc ( cr Es )
Elastic strain =

Fi
= 1.56 10 4
Ac Ec

f s = 2 1.56 10 4 200000 = 62.4 MPa

62.4
=
= 6.24%
1000

PaKrykMhatbg;
bedaytmRbhak;RbEhl eyIgyk cr = 7 10 5 kugRtaMgenAkgebtug
1935

103 = 3.76 10 4
450
800

cr = 7 10 5

f s = cr E s = 3.76 10 4 200000 = 75.2MPa

75.2
=
= 7.52%
1000

PaKrykMhatbg;
vaCatmEdlmansuvtiPaB ehIyeKnwgTTYl)anGRtadUcKasRmab;karKNnaxagelIRbsinebIeKyk
Cc = 2.41 .
$> kMhatbg;edaysar relaxation rbs;Edk sRmab; low-relaxation strand eKsnt;ykkMhatbg;
esInwg 2.5%
f s = 1000 2.5% = 25MPa

%> snt;kMhatbg;edaysarkarBt; kmaMgkkitrbs; cable spacer nigbkxagcugrbs;RbBn

pretensioning KW 2% .
f s = 0.02 1000 = 20MPa

^> kMhatbg;edaysarkmaMgkkitenAkg tendon KWsUn.

&> kMhatbg;srubmandUcxageRkam

esckIENnaMBIebtugeRbkugRtaMg

657

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mhaviTalysMNg;sIuvil

NPIC

Elastic Shortening
Shrinkage loss
Creep of concrete loss
Relaxation of steel loss
Other loss
Total loss

37.8MPa
60.0MPa
62.4MPa
25.0MPa
20.0MPa
205.2MPa

3.78%
6.00%
6.24%
2.50%
2.00%
20.52%

eRbkugRtaMgRbsiTPaB = 1000 167.4 = 832.6MPa

kmaMgeRbkugRtaMgRbsiTPaB F = 832.6 1935 103 = 1611kN
F = (1 0.167 )Fi = 0.833Fi

sRmab; F = Fi
dUcenH = 0.833

]TahrN_ 19>3 KNnakMhatbg;TaMgGs;n post-tensioned beam EdlmanRbEvg 36m . RkLap

rbs;muxkat;ebtug ( Ac ) = 49 10 4 mm 2 / m:Um:g;niclPaB (I g ) = 6.83 1010 mm 4 / kmaMgeRbkugRtaMg

enAeBlepr (Fi ) = 4950kN / RkLapEdkeRbkugRtaMg (Aps ) = 4840mm 2 / f 'c = 35MPa /
Ec = 34500MPa / nig E s = 2 10 5 MPa . Profile rbs; tendon manragCa)a:ra:bUl/ cMNakpitenA
kNalElVg = 50cm nigcMNakpitenAxagcug = 0 .

dMeNaHRsay

!> kMhatbg;eday elastic shortening:

kugRtaMgEdkenAeBlepr = AFi

ps

4950 3
10 = 1022.7 MPa
4840

kugRtaMgenAkgebtugRtg;muxkat;xagcug = 494950
10 3 = 10.1MPa
4
10
2

kugRtaMgenAkgebtugRtg;muxkat;kNalElVg = AFi + FiIe MID e

c
Tmn;rbs;Fwm = 49 10 2 25 = 12.25kN / m
36 2
= 1984.5kN .m
8
2
4950
1982.5 500 6
3 4950 500
=
+
10
10 3
10
4
10
49 10
6.83 10
6.83 1010

M D = 12.25

kugRtaMgenAkNalElVg
kugRtaMgmFm
T.Chhay

= 10.1 + 18.12 14.5 = 13.72 MPa

10.1 + 13.72
=
= 11.9 MPa
2

658

Introduction to Prestressed Concrete

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Department of Civil Engineering

11.9
mFm = 11E.9 = 34500
= 3.45 10 4
c
kMhatbg;eGLasicKW f s = c Es = 3.45 10 4 2 105 = 69MPa edaysnt;faeKTaj tendon
mgBIrkgeBlEtmYy. KUTImYynwgmankMhatbg;FMCageK b:uEnKUcugeRkaynwgmankMhatbg;esIsUn.
dUcenH kMhatbg;eGLasicmFm f s = 69 / 2 = 34.5MPa .
34.5
PaKrykMhatbg; = 1022
= 3.37%
.7
@> kMhatbg;edaysarkarrYmmaDrbs;ebtug
strain

f s (shrinkage) = 0.0002 E s = 0.0002 200000 = 40 MPa

40
=
= 3.91%
1022.7

PaKrykMhatbg;
#> kMhatbg;;edaysar creep rbs;ebtug snt; Cc = 1.5
elastic strain =

Fi
4950
=
10 3 = 2.93 10 4
4
Ac Ec 49 10 34500

f s (creep) = Cc ( cr E s ) = 1.5 2.93 10 4 200000 = 87.9 MPa

87.9
=
= 8.59%
1022.7

PaKrykMhatbg;
$> kMhatbg;edaysar relaxation rbs;Edk sRmab; low-relaxation strand, kMhatbg;KW 2.5%
f s = 0.025 1022.7 = 25.6 MPa

%> karrGilrbs; anchorage: sRmab;karTajEtBIcugmag snt;RbEvgrGil 3.8mm

f s =

L
3.8
Es =
200000 = 21.1MPa
L
36000

edIm,IGnuBaat[mankarrGilrbs; anchorage eKRtUvkMNt;kugRtaMgkgkarTaj 1022.7 + 21.1

= 1043.8MPa enAelI pressure gauge edIm,ITTYl net stress 1022.7 MPa enAkg tendon.
^> kMhatbg;EdlbNalBIkmaMgkit smIkar parabolic profile KW
4e
e x = 2 (Lx x 2 )
L
Edl ex = cMNakpitenARtg;cmay x Edlvas;BITRm nig e = cMNakpitenAkNalElVg
d (e x ) 4e
= 2 (L 2 x )
dx
L

CaCRmal (slope) rbs; tendon enARKb;cMNucTaMgGs;. enARtg;TRm e = 0 eyIgTTYl)an slop

d (e x ) 4e 4 500
=
=
= 0.056
dx
L 36000

esckIENnaMBIebtugeRbkugRtaMg

659

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

enAkNalElVgesIsUn. dUcenH px = 0.056 . edayeRbI flexible metallic sheath,

p = 0.5 nig K = 0.00333 . enAkNalElVg x = 18m . RtYtBinitfaetI ( p px + Kl x ) 0.30

slope

p px + Kl x = 0.5 0.056 + 0.00333 18 = 0.088 < 0.3

Ppx = Ppj 1 + Kl px + p px

= Px (1 + 0.088) = 1.088 Px
= 1.088 1022.7 = 1112.7 MPa

kmaMgenAcug jacking

f s = 1112.7 1022.7 = 90 MPa

90
=
= 8.8%
1022.7

PaKrykMhatbg;
&> kMhatbg;srub

Elastic Shortening
Shrinkage loss
Creep of concrete loss
Relaxation of steel loss
Friction loss
Total loss

34.5MPa
40.0MPa
87.9MPa
25.6MPa
90.0MPa
278.0MPa

3.37%
3.91%
8.59%
2.50%
8.80%
27.17%

eRbkugRtaMgRbsiTPaB = 1022.9 243.5 = 779.2MPa

kmaMgeRbkugRtaMgRbsiTPaB (F ) = (1 0.238)Fi = 0.762Fi
F = 0.762 4950 = 3772kN

sRmab; F = Fi
dUcenH = 0.762
(Analysis of Flexural Members)
19>4> viPaKGgt;rgkarBt;begag
19>4>1> kugRtaMgEdlbNalBIlkxNmanbnk niglkxNKanbnk
Stresses Due to Loaded and Unloaded condition

enAkgkarviPaKFwmebtugeRbkugRtaMg CaTUeTAkardak;bnkBIrmaneRKaHfak;CageK. TImYyKWekIt

manenAeBlepr KWenAeBlFwmrgkmaMgeRbkugRtaMg Fi ehIyTmn;rbs;Fwm bbnkefrGnuvtn_enAxNn
kareprkmaMgkugRtaMg. eKminKitlkxNbnkefrbEnm bbnkGefreT. kglkxNKanbnkenH kug
RtaMgenAsrsEpkxagelIbMput nigEpkxageRkambMputnmuxkat;eRKaHfak;minRtUvFMCagkugRtaMgenA
eBlepr f ci nig f ti sRmab;kugRtaMgrgkarsgt; nigkugRtaMgrgkarTajrbs;ebtugerogKa.
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Department of Civil Engineering

krNITIBIrnkardak;bnkekIteLIgenAeBlEdlFwmrgkmaMgeRbkugRtaMgeRkayBIekItmankMhat
bg;TaMgGs; nigrgnUvbnkefr nigGefr. enAkglkxNmanbnkenH kugRtaMgenAsrsEpkxagelIbM
put nigEpkxageRkambMputnmuxkat;eRKaHfak;dac;xatminRtUvFMCagkugRtaMgGnuBaat f c nig f t
sRmab;kugRtaMgrgkarsgt; nigkugRtaMgrgkarTajrbs;ebtugerogKa.
lkxNTaMgenHGacsresrCaTRmg;KNitviTadUcxageRkam
!> sRmab;lkxNKanbnk enAeBlepr
- enAsrsEpkxagelIbMput
ti =

Fi (Fi e ) yt M D yt
+

f ti
A
I
I

(19.14)

Fi (Fi e ) yb M D yb

f ci
A
I
I

(19.15)

- enAsrsEpkxageRkambMput
bi =

@> sRmab;lkxNmanbnk bnkTaMgGs;RtUv)andak;eRkayBIkMhatbg;eRbkugRtaMg

- enAsrsEpkxagelIbMput
(19.16)
- t = FA + (FeI)yt M DI yt M LI yt f c
- enasrsEpkxageRkambMput
(19.17)
- b = FA (FeI)yb M DI yb M LI yb f t
Edl Fi nig F = kmaMgeRbkugRtaMgenAeBlepr nigeRkayBIkMhatbg;
f ti nig f t = kugRtaMgrgkarTajenAkgebtugenAeBlepr nigeRkayBIkMhatbg;
f ci nig f c = kugRtaMgrgkarsgt;enAkgebtugenAeBlepr nigeRkayBIkMhatbg;
M D nig M L = m:Um:g;EdlbNalBIbnkefr nigbnkGefr
yt nig yb = cmayBIGkSNWteTAsrsEpkxagelIbMput nigEpkxageRkambMput
enAkgkarviPaK eKsnt;fasmarmanlkNeGLasicenAkgEdneFVIkarnkugRtaMgEdlGnuvt.
19>4>2> EdnkMNt;sl
(Kern Limits)
RbsinebIeKGnuvtkmaMgeRbkugRtaMgenARtg;TIRbCMuTmn;rbs;muxkat; vanwgekItmankugRtaMg
BRgayesI. RbsinebIGnuvtkmaMgeRbkugRtaMgenARtg;cMNakpit e BIeRkamTIRbCMuTmn; EdleFVIy:agNa
[kugRtaMgenAsrsEpkxagelIbMputesIsUn enaHeKcat;TukkmaMgeRbkugRtaMgenHmanGMeBIRtg;cMNuc
esckIENnaMBIebtugeRbkugRtaMg

661

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

rUbTI 19>5. enAkgkrNIenH e RtUv)ansMKal;eday K b ehIIykarBRgaykugRtaMgman

ragRtIekaN EdlmankugRtaMgsgt;GtibrmaenAsrseRkameKbMput. kugRtaMgenAsrsxagelIbMputKW

lower kern

t =

Fi (Fi e ) yt
+
=0
A
I

e = K b = lower kern =

I
Ayt

(19.17)

dUcKa RbsinebIeKGnuvtkmaMgeRbkugRtaMgenARtg;cMNapit e' BIelITIRbCMumn; EdleFVIy:agNa[kug

RtaMgenAsrsEpkxageRkambMputesIsUn enaHkmaMgeRbkugRtaMgRtUv)aneKcat;TukfamanGMeBIRtg;cMNuc
upper lower rUbTI 19>5. enAkgkrNIenHcMNapit e' RtUv)ansMKal;eday K t ehIykarBRgaykug
RtaMgmanragRtIekaN EdlmankugRtaMgsgt;GtibrimaenAsrsEpkxagelIbMput. kugRtaMgenAsrs
EpkxageRkambMputKW
b =

Fi (Fi e ) yb
+
=0
A
I

e' = K t = upper kern =

Kern limits

I
Ayb

(19.18)

nmuxkat;RtIekaNRtUv)anbgajenArUbTI 19>5.

19>4>3> karkMNt;tmncMNakpit
(Limiting Values of Eccentricity)
eKGacsresrsmIkarkugRtaMgTaMgbYn BIsmIkar 19.13 dl; 19.16CaGnuKmn_ncMNakpit e
sRmab;lkxNnkardak;bnkepSg. Ca]TahrN_ eKGacsresrsmIkar 19.13 dUcxageRkam
T.Chhay

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Fi (Fi e ) yt M D yt
+

f ti
A
I
I
(Fi e )yt f + Fi + M D yt
ti
I
A
I
I Fi M D yt

e
+ f ti
+
Fi yt A
I

ti =

(19.19)

RbsinebIeKeRbI lower kern limit K b = I / Ayt / enaH

e Kb +

M D f ti AK b
+
Fi
Fi

(19.20)

tm e CacMNakpitGtibrmaEdlQrelIsrsEpkxagelIbMputsRmab;lkxNKanbnk.
dUcKa BIsmIkar 19.14
I Fi M D yb

+ f ci
+
Fi yb A
I

f
AK
M
t
e K t + D + ci
Fi
Fi
e

(19.21)
(19.22)

tm e CacMNakpitGtibrmaEdlQrelIsrsEpkxageRkambMputsRmab;lkxNKanbnk.
eKKNna tmGtibrma e BIsmIkarelITaMgBIredayyktmEdltUcCagmkeRbI.
BIsmIkar 19.15
I F M T yt

fc
+
Fyt A
I

f AK
M
e Kb + T c b
F
F
e

(19.23)
(19.24)

Edl M T = m:Um:g;EdlbNalBIbnkefr nigbnkGefr = (M D + M L ) . tmenHCacMNakpitGb,brma EdlQrelIsrsEpkxagelIbMputsRmab;lkxNRTbnk. BIsmIkar 19.16

I F M T yb

ft
+
Fyb A
I

f
AK
M
t
e Kt + T t
F
F

(19.25)
(19.26)

tmenHCacMNakpitGb,brmaEdlQrelIsrsEpkxageRkambMput sRmab;lkxNmanbnk. eKKYr

KNnatmGb,brma e TaMgBIrenHBIsmIkarTaMgBIrxagelI ehIyeKykcMNakpitGb,brmaNaEdl
mantmFMCageKmkeRbI.

esckIENnaMBIebtugeRbkugRtaMg

663

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

19>4>4> tmkMNt;mkmaMgeRbkugRtaMgenAeBlepr
(Limiting Values of the Prestessing Force at Transfer)

edayKitfa F = Fi Edl CapleFobn net prestressing force eRkayBIkMhatbg; nig

sRmab;krNIepSgnkardak;bnk eKGacsresrsmIkar 19.20, 19.22, 19.24 nig 19.26 eLIgvijdUc
xageRkam
(e K b )Fi M D + f ti AK b
(e + K t )Fi M D + f ci AK t
(e K b )Fi M D

ML

(19.27)
(19.28)
1

( f c AK b )

(19.29)

(e + K t )Fi M D + M L 1 ( f t AK t )

(19.30)

CMnYsmIkar 19.27 eTAkgsmIkar 19.30 eKTTYl)an

b

f AK t
1 M
Fi (K b + K t ) M D 1 + L t
f ti AK t

M L f t AK t
1
( f ti AK b )

Fi
1 M D +
(K b + K t )

(19.31)

tm Fi CatmGb,brmankmaMgeRbkugRtaMgenAeBlepredaymin[FMCagkugRtaMgGnuBaateRkam
lkxNmanbnk nigKanbnk. CMnYssmIkar 19.29 eTAkgsmIkar 19.28 edIm,ITTYl)an
Fi

1
M L f c AK b
1
+
1 M D
(K b + K t )

+ ( f ci AK t )

(19.32)

tm Fi CatmGtibrmankmaMgeRbkugRtaMgenAeBlepredaymin[elIskugRtaMgGnuBaateRkam
lkxNmanbnk nigKanbnk. edayCMnYssmIkar 19.31 eTAkgsmIkar 19.32
1

f
f
2M L
+ fti + c AKb + f ci + t AKt 0
1 2 M D

(19.33)

smIkarenHbgajfa Fi max Fi min 0 . eKeRbIsmIkarenHsRmab;bgajfamuxkat;NamYymanlkN

RKb;RKan;.

]TahrN_ 19>4 FwmTRmsamBaEdlrgeRbkugRtaMgmunEdlmanRbEvg 14.4m manmuxkat;dUcbgaj

enAkgrUbTI 19>6 a. FwmenHRTnUvbnkefr 13.15kN / m edayminrYmbBalTmn;pal; nigrgnUvbnk

Gefr 16kN / m . edaysnt;faEdkeRbkugRtaMgpSMeLIgeday tendon 20 Edl tendon mYymanGgt;

T.Chhay

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pit 11.125mm CamYynwg Es = 2 105 MPa / Fo = 1200MPa nig ultimate strength

f pu = 1725MPa .

!> kMNt;TItaMgEdnkMNt;xagelI nigEdnkMNt;xageRkamrbs; tendon profile TIRbCMuTmn;rbs;

EdkeRbkugRtaMg sRmab;muxkat;enAkNalElVg nigsRmab;muxkat;bIepSgeTotenAcenaH
muxkat;kNalElVg nigcugFwm.
@> dak; tendon cMTItaMgedIm,IbMeBjEdnkMNt;TaMgenH eday[ tendon xHegIbeLIgcab;BIcMNuc
mYyPaKbInRbEvgElVg. RtYtBinittmkMNt;nkmaMgeRbkugRtaMgenAeBlepr.
#> RtYtBiniteLIgvijnUvkMhatbg; edayKit tendon profile Edl)aneRCIserIs nigbERmbRmYl
cMNakpit e .
esckIENnaMBIebtugeRbkugRtaMg

665

T.Chhay

mhaviTalysMNg;sIuvil

eRbI fci enAeBlepr = 28MPa /

NPIC

f 'c = 35MPa Ec = 27600MPa

nig Eci = 24800MPa .

dMeNaHRsay

!> kMNt;lkNrbs;muxkat;
RkLap = 600 150 + 450 150 + 300 250 = 23.25 104 mm2
kMNt;TIRbCMuTmn;rbs;muxkat;edayKitm:Um:g;eFob)atrbs;muxkat;
1
(
yb =
75 103 125 + 90 103 550 + 67.5 103 925) = 522mm
4
23.25 10
yt = 1000 522 = 478mm

KNna gross moment of inertia I g

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450(150)3
150(600 )3
Ig =
+ (150)(600)(28)2
+ (450)(150)(403)2 +
12
12

300(250 )3

+
+ (300)(250 )(397 )2
12

= 2.607 1010 mm 4
I
2.607 1010
Kb =
=
= 235.6mm
Ayt 23.25 10 4 478
Kt =

I
2.607 1010
=
= 214.8mm
Ayb 23.25 10 4 522

@> )a:n;RbmaNkMhatbg;eRbkugRtaMg eday Fo = 1200MPa

a. snt; elastic loss esI 4% b 0.04 1200 = 48MPa
b. kMhatbg;edaysarkarrYmmaDKW 0.0003Es = 0.0003 2 105 = 60 MPa
c. kMhat;bg;edaysar creep rbs;ebtug kar)a:n;RbmaNdMbUgdlbMputnkMhatbg;eday
sar creep KW 1.67 dgn elastic loss
1.67 48 80 MPa

d.

e.

kMhatbg;edaysar relaxation nEdkKW 4% 0.04 1200 = 48MPa

kMhatbg;GaRsynwgeBlKW 60 + 80 + 48 = 188MPa
188
PaKrykMhatbg; = 1200
= 15.7%
kMhatbg;srubKW 188 + 48 = 236MPa
PaKrynkMhatbg;srubKW
236
= 19.7%
1200

f.

kugRtaMgkmaMgeRbkugRtaMg
Fi = 1200 48 = 1152 MPa

enAeBlepr

F = 1200 236 = 964 MPa

F = Fi

= 1
=

pleFobkMhatbg;GaRsynwgeBl

964
= 0.837
1152

esckIENnaMBIebtugeRbkugRtaMg

667

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

#> EdnkMNt;ncMNakpit e Rtg;kNalElVg kMNt;kugRtaMgGnuBaat nigm:Um:g;. enAeBlepr

f 'ci = 28MPa / f ci = 0.6 28 = 16.8MPa nig f ti = 0.25 f 'ci = 1.32 MPa . enAeBlrgbnk
eFVIkar f 'c = 35MPa / fc = 0.45 f 'c = 15.75MPa nig ft = 0.5 f 'c = 2.96MPa .
bnkpal;rbs;Fwm = 23.25 102 25 = 5.81kN / m
5.81(14.4 )2
M D bnkpal; =
= 150.6kN.m
8
2

Ma

bnkbEnm nigbnkGefr = wa8L

=

(13.15 + 16)14.4 2
8

= 755.6kN.m

m:Um:g;srub (M T ) = M D + M L = 906.2kN .m
Fi = kugRtaMgenAeBlepr RkLapEdkeRbkugRtaMg
RkLaprbs; tendon 20 Edl tendon nImYymanGgt;pit 11.125mm KW
20 69.7 = 1394mm 2
Fi = 1394 1152 10 3 = 1606kN
F = 1394 964 10 3 = 1344kN
a.

BicarNamuxkat;enAkNalElVg
srsxagelIbMput kglkxNminrgbnk
e Kb +

M D f ti AK b
+
Fi
Fi

235.6 +

150.6 3 1.32 23.25 10 4 (235.6) 3

10 +
10 374.4mm
1606
1606

srsxageRkambMput kglkxNKanbnk
e Kt +

M D f ci AK t
+
Fi
Fi

214.8 +

150.6 3 16.8 23.25 10 4 214.8 3

10 +
10 401.4mm
1606
1606

yktm e EdltUcCageKkgcMeNamlTplTaMgBIrxagelICatmGtibrma.
dUcenH tmGtibrmarbs; e = 374mm
srsxagelIbMput kglkxNrgbnk
e Kb +

T.Chhay

M T f c AK b

F
F

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Introduction to Prestressed Concrete

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Department of Civil Engineering

235.6 +

906.2 3 15.75 23.25 10 4 235.6 3

10
10 268mm
1344
1344

srsxageRkambMput kglkxNmanbnk

M T f t AK t

F
F
906.2 3 2.96 23.25 10 4 214.8 3
10
214.8 +
10 349.5mm
1344
1344

e Kt +

b.

tmGb,brmarbs; e CatmtUcCageKkgcMeNamlTplTaMgBIrxagelI.
dUcenH tmGb,brmarbs; e = 350mm
BicarNamuxkat;EdlenAcmay 2.4m BIkNalElVg muxkat;elx @ kgrUbTI 19>6 a
w
M D bnkpal; = R A (4.8) D (4.8)2
2
= (5.81)(7.2)(4.8)

5.81
(4.8)2 = 133.9kN.m
2
29.15
(4.8)2 = 671.6kN .m
M a = (29.15)(7.2)(4.8)
2
M T = 133.9 + 671.6 = 805.5kN .m

srsxagelIbMput sRmab;lkxNminrgbnk
133.9 3 1.32(23.25 10 4 )(235.6) 3
10 +
e 235.6 +
10 364mm
1606

1606

srsxageRkambMput sRmab;lkxNminrgbnk
133.9 3 16.8(23.25 10 4 )214.8 3
10 +
e 214.8 +
10 391mm
1606

1606

tmGtibrmarbs; e = 364mm
srsxagelIbMput sRmab;lkxNRTbnk
805.5 3 15.75(23.25 10 4 )235.6 3
10
e 235.6 +
10 193mm
1344

1344

ssrxageRkambMput sRmab;lkxNRTbnk
805.5 3 2.96(23.25 10 4 )214.8 3
10
e 214.8 +
10 274.5mm
1344

c.

1344

tmGb,brmarbs; e = 274.5mm
BicarNamuxkat;Rtg;cmay 4.8m BIkNalElVg muxkat;elx # kgrUbTI 19>6 a
M D bnkpal; = 83.7kN.m

esckIENnaMBIebtugeRbkugRtaMg

669

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

M a = 419.8kN .m
M T = 503.5kN .m

- srsxagelIbMput sRmab;lkxNKanbnk e 333mm Gtibrma

- srsxageRkambMput sRmab;lkxNKanbnk e 360mm
- srsxagelIbMput sRmab;lkxNrgbnk e 32mm
- srsxageRkambMput sRmab;lkxNrgbnk e 50mm Gb,brma
d. BicarNamuxkat;Rtg;cmay 0.9m BIcugFwm RbEvg anchorage
M D bnkpal; = 35.3kN.m / M a = 177.1kN .m nig M T = 212.4kN .m
- srsxagelIbMput sRmab;lkxNKanbnk e 303mm Gtibrma
- srsxageRkambMput sRmab;lkxNKanbnk e 330mm
- srsxagelIbMput sRmab;lkxNrgbnk e 248mm
- srsxageRkambMput sRmab;lkxNrgbnk e 167mm Gb,brma
$> Tendon profile RtUv)anbgajenAkgrUbTI 19>6 b. cMNakpitEdl)aneRCIserIsenAkNalElVgKW
e = 364mm EdlvaRKb;RKan;sRmab;muxkat; B enAcmay 2.4m BIkNalElVg. TIRbCMuTmn;nEdk
eRbkugRtaMgmanlkNedkcenaH A nig B nigbnab;mkeTreLIgEdlmanlkNCabnat;cenaHBI
B eTA E . cMNakpitenARtg;muxkat; C nig D RtUv)anKNnaedayeRbIbnat;eRTt BE Edlman
CRmal 364 / 4.8 = 75.83mm / m . cMNakpitRtg; C KW 182mm nigRtg; D KW 68mm . Tendon
profile Edl)aneRCIserIsbMeBjlkxNEdnkMNt;xagelI nigEdnkMNt;xageRkamrbs;cMNakpit
enARKb;muxkat;TaMgGs;.
karelIk tendon eLIgRtUv)aneFVIdUcxageRkam
a. dak; tendon TaMg 20 Ggt;pit 11.125mm
enAmYyPaKbInkNalElVgrbs;FwmedaymanKMlat 50mm BIKadUcbgajenAkgrUbTI 19>6 a.
edIm,IKNnacMNakpitCak;EsgenARtg;muxkat;kNalElVg Kitm:Um:g;sRmab; tendon eFob
nwg)at rbs;muxkat;
cmayBI)at = 201 (16 125 + 4 275) = 155mm
e kNalElVg = yb 155 = 522 155 = 367mm

T.Chhay

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Department of Civil Engineering

EdlvaEk,rnwg 364mm Edl)ansnt;. RbsinebIebIeKdak; tendon BIrenAcmay 75mm BI

tendon EdlenAxageRkam enaHcmayBI)atkayeTACa
1
(16 125 + 2 250 + 2 325) = 158mm
20

b.

enaHcMNakpitnwgkayeTACa 522 158 = 364mm EdlesIweTAnwgcMNakpitEdl)ansnt;. Ca

karGnuvt eKdak; tendon TaMgGs;edaymanKMlatBIKa 50mm .
elIkEt tendon EdlenAkNalcMnYn 12 [egIbeLIg. karBRgay tendon enAmuxkat;xagcug
RtUv)anbgajenAkgrUbTI 19>6 a. RtYtBinitcMNakpitrbs; tendon edayKitm:Um:g;eFobTI
RbCMuTmn;rbs;muxkat;ebtugsRmab; tendon 12 enAxagelI nig tendon 8 eTotEdlRtUv)andak;
enAxageRkam
e=

1
(8 364 12 226) = 10mm
20

tm e enHtUc nigRKb;RKan;. cMNakpitCak;EsgenAcmay 0.9m BImuxkat;xagcug

e=

0.9
(364 10) + 10 = 76mm
4.8

cMNakpitCak;EsgenAcmay 2.4m BImuxkat;xagcugKW

e=

1
(364 10) + 10 = 187mm
2

%> tmkMNt;rbs; Fi tmrbs; Fi EdleRbIsRmab;karKNnaBIxagelIKW Fi = 1606kN .

RtYtBinit Fi Gb,brmaedayeRbIsmIkar 19.31:
Fi min =

1
M L ( f t AK t )

( f ti AK b )
1 M D +
(K b + K t )

1
755.6 106 2.96 23.25 10 4 214.8

1150.6 106 +

10 3
0.837
0.837
=
0.837

(235.6 + 214.8)
4

1.32 23.25 10 235.6

= 1516.8kN

vamantmtUcCag Fi EdleRbI. RtYtBinit Fi GtibrmaedayeRbIsmIkar 19.32:

Fi max =

esckIENnaMBIebtugeRbkugRtaMg

1
M L f c AK b
+
+ f ci AK t
1 M D
(K b + K t )

671

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

1
15.75 23.25 10 4 235.6
6 755.6 10
1
150
.
6
10

10
0.837
0.837
=
0.837

(235.6 + 214.8)
4

16
.
8
23
.
25
10
214
.
8
+

= 2081.9kN

vamantmFMCag Fi EdleRbI. dUcenHmuxkat;eRKaHfak;enAkNalElVgKWRKb;RKan;.

^> RtYtBinitkMhatbg;eRbkugRtaMg edayeyIgman Fo = 1200MPa nig Aps = 1394mm2
kmaMg Fo srub = 1200 1394 103 = 1672.8kN
Ec = 27600MPa
E
200000
n= s =
= 7.25
Ec
27600
MD

enAkNalElVg = 150.6kN .m
Fi =

Fo + nAps f c (D.L.)

1 + nA ps

a.

yk n = 7

2
3
1 e2
+
A I

tmrbs; f c Edl)anBIkarBRgaybnkefrRtUvKuNnwg 2 / 3 edIm,IbgajBIbERmbRmYlrag)a:ra:

bUl rbs;kugRtaMgbnkefrtambeNayFwm Edlpl;[nUvtmRbhak;RbEhlrbs; Fi )an
RbesIrCag.
kMNt;tmmFmrbs; e2 EdlTTYlenAkgFwm. ExSekagtMNag[ e2 RtUv)anbgajenAkgrUb
TI 19>6 c
1

e2

mFm = 71.2 3 (5776 0.9) + (5776 3.9) + 3 (126720 3.9)

1

+ (2.4 132496 )

= 70414.7 mm 2
e = 265mm

b.

RkLaprbs;)a:ra:bUlesInwgRkLaprbs;ctuekaNEkg.
kugRtaMgEdlbNalBIbnkefrenARtg;nIv:Urbs; tendon KW
f c (D.L.) =

T.Chhay

150.6 265
2.607 1010

10 6 = 1.53MPa

672

Introduction to Prestressed Concrete

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dUcenH

Department of Civil Engineering

1672.8 103 + 7(1394 ) 1.53

2
3
Fi =
10 3 = 1575.1kN
1
70414.7

1 + 7(1394 )
+

4
2.607 1010
23.25 10

elastic loss

KW 1672.8 1575.1 = 97.7kN = 5.8% . tmenHKWFMCag elastic loss Edl)an

snt; 4% .
97.7 3
kgmYyktaRkLapEdk = 1394
10 = 70 MPa
1575.1 3
Fi kgmYyktaRkLap =
10 = 1130 MPa
1394
kMhatbg;GaRsynwgeBl
kMhat;bg;edaysarkarrYmmaDebtug = 60MPa dUcelIkmun
kMhatbg;edaysar creep
elastic loss

c.

elastic strain =

Fi
1575.1
=
103 = 2.45 10 4
4
Ac Ec
23.25 10 27600

f s = Cc ( cr Es )

yk Cc = 1.5 enaH

f s = 1.5 2.45 10 4 200000 = 73.5MPa

73.5
=
= 6.5%
1130

PaKrykMhatbg;
kMhatbg;edaysar relaxation rbs;EdkKW 48MPa dUcelIkmun.
kMhatbg;GaRsynwgeBl esInwg 60 + 73.5 + 48 = 181.5MPa ehIyPaKrykMhatbg;KW
181.5 / 1130 = 16% Edlman tmEk,rnwgtmEdl)ansnt; 15.7% .
F = Fi = (1 0.16 )Fi = 0.84 Fi

= 0.84

19>5> KNnaGgt;rgkarBt;begag
(Design of Flexural Members)
19>5>1> sBaaNTUeTA (General)
EpkBIedIm)anbBaak;fakugRtaMgenAsrsxagelIbMput nigsrsxageRkambMputnmuxkat;eRKaH
fak;rbs;Ggt;ebtugeRbkugRtaMgminRtUvFMCagkugRtaMgGnuBaatsRmab;RKb;krNITaMgGs; bdMNak;kaln
kardak;bnk. bEnmBIelIlkxNTaMgenH eKRtUvKNnaGgt;ebtugeRbkugRtaMgCamYynwgemKuNsuvtiesckIENnaMBIebtugeRbkugRtaMg

673

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

PaBRKb;RKan;edIm,IRbqaMgnwgkar)ak;. ACI Code tRmUv[m:Um:g;Edl)anBIbnkemKuN M u minRtUvFM

CagersIusg;rgkarBt; M n nmuxkat;Edl)anKNna.
sRmab;krNI tension-controlled section, FwmebtugeRbkugRtaMgcab;epIm)ak;enAeBlEdlkug
RtaMgEdkFMCag yield strength rbs;EdkEdleRbIenAkgmuxkat;ebtug. EdkeRbkugRtaMgersIusg;nwgmin
bgajcMNuc yield c,as;las;dUcEdkFmtaEdleRbIenAkgebtugGarem:eT. b:uEneRkamkarbEnmbnk
strain enAkgEdkekIneLIgedayGRtay:agelOn ehIykar)ak;ekIteLIgenAeBl compressice strain
Gtibrmarbs;ebtugmantmesInwg 0.003 rUbTI 19>7.

EdnkMNt;sRmab;EdkBRgwgrbs;Ggt;rgkarBt;ebtugeRbkugRtaMgEdlGaRsyeTAtam ACI
Code, Section 18.8 KWQrelI net tensile strain sRmab; tension-controlled, transition b
compression-controlled section edayeKarBtam ACI Code, Section 10.3
T.Chhay

674

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Department of Civil Engineering

dUcEdl)anBnl;enAkgCMBUk 3. em KuNkat;bnyersIusg; RtUv)an[enAkgCMBUkTI 3 edayQrelI

ACI Code, Section 9.3.
19>5>2> muxkat;ctuekaN (Rectangular Sections)
eKGackMNt; Nominal moment capacity rbs;muxkat;ctuekaNdUcxageRkam eyagtamrUb
TI 19>7
a
a

M n = C d = T d
2
2

(19.34)

Edl T = Aps f s nig C = 0.85 f 'c ab . sRmab; C = T

a=

A ps f ps
0.85 f 'c b

p f ps

0.85 f 'c

(19.35|

EdlpleFobEdkeRbkugRtaMgKW p = Aps / bd ehIy Aps nig

EdkeRbkugRtaMg. yk

f ps

CaRkLap nigkugRtaMgTajrbs;

f ps
0.32 1
f 'c

p = p

p
(19.36)
bnab;mk
a=
d
0.85
tm p CakmaMgenAkg tendon EdlRtUv)anvas;edaypal;. edIm,IFananUv tesion-controlled
behavior, ACI Code, Section 18.8.1 kMNt;fa p minRtUvFMCag 0.32 1 EdkRtUvKanwg net tensile
strain t = 0.005 . cMNaMfa 1 = 0.85 sRmab; f 'c 28MPa nwgkat;bnyeday 0.05 sRmab;ral;
7 MPa sRmab; 28MPa < f 'c < 56 MPa ehIyesInwg 0.65 sRmab; f 'c > 56 MPa .
eKkGacsresr
a

M n = A ps f ps d
2

p f ps

M n = Aps f ps d 1
1 .7 f ' c

(19.37)

M n = A ps f ps d 1
1 .7

(19.38)

nig M u = M n

esckIENnaMBIebtugeRbkugRtaMg

675

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mhaviTalysMNg;sIuvil

NPIC

enAkgsmIkarBImun f ps CakugRtaMgenAkgEdkeRbkugRtaMgenAeBl)ak;. eKminGackMNt;tm

Cak;Esgrbs; f ps edaygayRsYleT. dUcenH ACI Code, Section 18.7.2 GnuBaat[)a:n;RbmaN
tm f ps dUcxageRkam.
sRmab; bonded tendons
p
f ps = f pu 1
1

p pu
f 'c

(19.39)

sRmab; unbonded tendon enAkgGgt;EdlmanpleFobElVgelIkm<s;tUcCag besI 35

f 'c
f ps = f se + 69 +
f py

100

(19.40)

RbsinebI f se 0.5 f pu nigRbsinebI f ps sRmab; unbonded tendon minFMCag f py b f se + 415MPa .

sRmab; unbonded tendon enAkgGgt;EdlmanpleFobElVgelIkm<s;FMCag 35

f 'c
f ps = f se + 69 +

300 p

(19.41)

b:uEnminRtUvFMCag f py b f se + 207MPa Edl

p = emKuNsRmab;RbePTrbs; tendon eRbkugRtaMg
= 0.55 sRmab; f py / f pu EdlmintUcCag 0.8
= 0.4 sRmab; f py / f pu EdlmintUcCag 0.85
= 0.28 sRmab; f py / f pu EdlmintUcCag 0.9
f pu = ersIusg;TajEdlkMNt;rbs;EdkeRbkugRtaMg
f se = kugRtaMgRbsiTPaBenAkgEdkeRbkugRtaMgeRkayeBlkMhatbg;TaMgGs;
f py = specified yield strength rbs;EdkeRbkugRtaMg
enAkgkrNIEdl p > 0.321 FwmebtugeRbkugRtaMgCa compression-controlled section.
edIm,IFananUv ductile failure eKkMNt; p RtwmtmGtibrma 0.321 . sRmab; = 0.321 nig
a = 0.377 1d eyIgTTYl)an
0.32 1
M n = Aps f ps d 1

1.7

= p bd f ps d (1 0.1881 )
= p f 'c (1 0.1881 )bd 2

= 0.32 1 0.0612 f 'c bd 2

T.Chhay

(19.42)

676

Introduction to Prestressed Concrete

viTasanCatiBhubeckeTskm<Ca

sRmab;

Department of Civil Engineering

. enaH

f 'c = 35MPa 1 = 0.8

M n = 0.22 f 'c bd 2 = 1.09bd 2

dUcKa/ sRmab;

f 'c = 28MPa M n = 0.915bd 2

nigsRmab;

f 'c = 42 MPa M n = 1.238bd 2

19>5>3> muxkat;Edlmansab (Flanged Sections)

sRmab;muxkat;mansab (T- or I-section) RbsinebIkm<s;bk a sitenAkgsab eKnwgKitvaCa
muxkat;ctuekaNEkg. RbsinebI a sitenAkgRTnug enaHeKKitRTnugCamuxkat;ctuekaNEkgedayeRbI
TTwgRTnug nigTTwgsabEdlelIs (b bw ) RtUv)anKitdUcKanwgebtugGarem: T-section Edl)anBnl;
enAkgCMBUk 3 nig4. eKGacKNna design moment strength rbs; flanged section dUcxageRkam
emIlrUbTI 19>7.
M n = M n1 ersIusg;m:Um:g;rbs;RTnug + M n2 ersIusg;m:Um:g;rbs;sabEdlelIs
hf

M n = Apw f ps d p + A pf f ps d p
2
2

A pw f ps
a=
M u = M n
0.85 f 'c bw

(19.43)

nig

Edl

A pw = Aps Apf

A pf = 0.85 f 'c (b bw )h f / f ps

kRmas;rbs;sab
cMNaMfaRkLapEdkeRbkugRtaMgsrub Aps EckecjCaBIrEpk Apw nig Apf EdlbegIt web
moment capacity nig flange moment capacity. sRmab;muxkat;Edlmansab snsSn_EdkBRgwg
(reinforcement index) pw minRtUvFMCag 0.32 1 sRmab; tension-controlled section Edl
Apw f ps
f

= GRtaEdkRTnugeRbkugRtaMg ps
pw =
b d
f'
f'
hf =

(Nonprestressed Reinforcement)
19>5>4> EdkBRgwgrgeRbkugRtaMg
kgkrNIxH eKdak;EdkminrgeRbkugRtaMg As enAkgtMbn;Tajrbs;Ggt;rgkarBt;ebtugeRbkug
RtaMgCamYynwgEdkeRbkugRtaMg Aps edIm,IbegInersIusg;m:Um:g; (moment strength) rbs;Fwm. enAkg
krNIenH Edksrub Aps nig As RtUv)anBicarNaenAkgkarviPaKm:Um:g;.

esckIENnaMBIebtugeRbkugRtaMg

677

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

sRmab;muxkat;ctuekaNEdl manEdkrgeRbkugRtaMg nigEdkminrgeRbkugRtaMg eKKNna design

moment strength M n dUcxag eRkam
a
a

M n = A ps f ps d p + As f y d
2
2

A ps f ps + As f y
a=
dp
d
0.85 f 'c b

(19.44)

/ ehIy nig CacmayBIsrsrgkarsgt;xageRkAbMputeTATIRbCMuTmn;

Edl
rbs;EdkrgeRbkugRtaMg nigEdkminrgeRbkugRtaMg erogKa. sRmab;muxkat;Edlmansab
hf

a
a

M n = Apw f ps d p + As f s d + Apf f ps d p
2
2
2

Edl

(19.45)

A pw = A ps Apf
a=

Aps f ps + As f y
0.85 f 'c bw

sRmab;muxkat;ctuekaNEkgEdlmanEdkrgkarsgt; ehIym:Um:g;RtUv)anKiteFobnwgkmaMg C
a
a

M n = Aps f ps d p + As f y d + A's f y d '

2
2

Aps f ps + As f y A's f y
a=
0.85 f 'c b

(19.46)

Edl
smIkarenHmannyEtenAeBlEdlEdkrgkarsgt; yield. lkxNsRmab;Edkrgkarsgt; yield KW
Aps f ps + As f y A' s f y
f ' d ' 600

0.851 c
bd
d 600 f y

RbsinebIeKminCYblkxNenHeT enaHEdkrgkarsgt;min yield eT. enAkgkrNIenH eKGacecal A's

yk A's = 0 bm:ageTot eKGackMNt;kugRtaMgenAkg A's edaykarviPaKTUeTA dUcEdlBnl;enAkg
CMBUk 3.
enAeBleKeRbIEdkrgeRbkugRtaMg nigEdkminrgeRbkugRtaMgenAkgmuxkat;dUcKa eKKYrGansmIkar
19.39 dUcxageRkam
p
f ps = f pu 1
1

f
p pu + d ( ') (ACI Code, Eq. 18.3)

f 'c d p

RbsinebIeKKitEdkrgkarsgt;TaMgGs; enAeBlKNna
p

nig
T.Chhay

f pu
f 'c

f ps

(19.47)

enaHtY

d
( ') 0.17
dp

d ' 0.15d p

678

Introduction to Prestressed Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

Edl d / d ' nig d p CacmayBIsrsrgkarsgt;xageRkAbMputeTATIRbCMuTmn;rbs;EdkTajEdlminrg

eRbkugRtaMg Edkrgkarsgt; nigEdkrgeRbkugRtaMg erogKa.
p = emKuNsRmab;RbePTrbs; tendon eRbkugRtaMg
= 0.55 sRmab; f py / f pu EdlmintUcCag 0.80
= 0.4 sRmab; f py / f pu EdlmintUcCag 0.85
= 0.28 sRmab; f py / f pu EdlmintUcCag 0.9
1 = 0.85 sRmab; f 'c 28MPa nwgkat;bnyeday 0.05 sRmab;ekIneLIg 7 MPa rbs; f 'c
Et 1 0.65 .
!> sRmab;muxkat;ctuekaNEkg ACI Cose,Section 18.8 kMNt;pleFobEdkdUcxageRkam
t 0.005 sRmab; tension controlled section
wp +

Edl

d
0.321
dp
f ps

f 'c

p = p

fy

f 'c

nig

p =

nig

Aps
bd

As
bd

EdkrgeRbkugRtaMg
EdkminrgeRbkugRtaMg

@> RbsinebIeKeRbIEdkFmta A's enAkgtMbn;sgt; enaHlkxNkayeTACa

p +

d
( ') 0.321
dp

Edl ' = ' ( f y / f 'c ) nig ' = A's / bd . EdnkMNt;Edk (reinforcement limitation)
mansarsMxan;edIm,I Fana plastic failure rbs;FwmebtugGarem:EdkmanEdktic.
#> sRmab;muxkat;mansab RkLapEdlcaM)ac;edIm,IbegItersIusg;rbs;RTnug Apw RtUv)aneRbI
edIm,IRtYtBinit reinforcement index.
f ps
0.32 1
pw RTnug = pw
f'

A pw

pw =
Edl
bw d d
RbsinebIeKeRbIEdkminrgeRbkugRtaMg enaH reinforcement limitation KW
pw +

esckIENnaMBIebtugeRbkugRtaMg

d
d pw

(w 'w ) 0.321

679

T.Chhay

mhaviTalysMNg;sIuvil

Edl

NPIC

w =

As f y

bw d f 'c

nig

'w =

A's f y

bw d f 'c

enAeBleKmineRbIEdkrgkarsgt; A's enaH 'w = 0 . kgkarKNna nigkarviPaKGgt;

ebtugeRbkugRtaMgedayEpk (partially prestressed concrete member) eKRtUvEtCYb
lkxNEdkxagelI.
sRmab;fak; C nGgt;ebtugeRbkugRtaMgrgkarBt; Edl ft > f 'c muxkat;Edl
eRbH eKKYreRbI crack control provision EdlBnl;enAkgEpk 6>7 . enAeBleRbIsmIkar
6.18 sRmab;KMlatGtibrma s / ACI Code, Section 18.4 kMNt;dUcxageRkam
a. sRmab; tendon eRbIKMlat s = 17 mm .
b. sRmab;bnSMnEdkminrgeRbkugRtaMg nig tonden eRbIKMlat s = 20mm .
c. sRmab; tendon eRbI f ps CMnYs[ f s Edl f ps CaPaBxusKarvagkugRtaMgEdl
KNnaenAkg tendon eRbkugRtaMgeRkambnkeFVIkaredayQrelImuxkat;eRbH nig
decompression stress f dc enAkg tendon eRbkugRtaMg EdlRtUv)anKit[esInwgeRb
kugRtaMgRbsiTPaB (effective prestress) f se . cMNaMfa f ps minKYrelIsBI 250MPa .
RbsinebIvaticCag besInwg 140MPa eKminGacGnuvtKMlatEdlTamTareT.
eKGacsresrsmIkar 8.18 dUcxageRkam

2 3700
2.5Cc
s =
3 f ps

19>6> m:Um:g;eRbH (Cracking Moment)

sameRbHekItmanenAkgFwmebtugeRbkugRtaMgenAeBlEdlkugRtaMgTajenAsrsEpkxageRkA
bMputrbs;muxkat;eRKaHfak;esI belIs modulus of rupture rbs;ebtug f r . eKGacsnt;tmrbs;
f r sRmab;ebtugTmn;Fmta[esInwg 0.62 f 'c . kugRtaMgenAsrsxageRkambMputrbs;FwmTRm
samBaEdlbegItedaykmaMgeRbkugRtaMg nig cracking moment KW
b =

F (Fe ) yb M cr yb

+
A
I
I

enAeBlEdl b = f r = 0.62

T.Chhay

f 'c

enaH cracking moment KW

680

Introduction to Prestressed Concrete

viTasanCatiBhubeckeTskm<Ca
M cr =

Department of Civil Engineering

I
F (Fe ) yb
0.62 f 'c + +

yb
A
I

(19.48)

kugRtaMgTajGtibrmaeRkayBIkMhatbg;KW 0.62 f 'c EdltMNag[ f r . enAkgkrNIenH Fwmebtug

eRbkugRtaMgenArkSamuxkat;minmansameRbHeRkamGMeBIbnkefr. edIm,IFanaPaBRKb;RKan;ersIusg;Rb
qaMgnwgsameRbH ACI Code, Section 18.8.3 TamTarfa ultimate moment strength rbs;Ggt; M n
y:agticRtUvesInwg moment cracking 1.2 dg.

]TahrN_ 19>5 sRmab;Fwmn]TahrN_ 19>4 cUrRtYtBinit design strength nigtRmUvkarrbs; ACI

Code

EdlRbqaMgnwg cracking moment.

dMeNaHRsay

!> RtYtBinitemIlfakm<s;bkkugRtaMg a sitenAkgsabbGt;.

a=

A ps f ps

(19.35)

0.85 f 'c b

n 20 tendon Edl tendon mYymanGgt;pit 11.125mm = 1394mm2

yk f py / fu = 0.85 / p = 0.4 nig p / 1 = 0.4 / 0.8 = 0.5 . sRmab; bonded tension
A ps

f pd
p

f ps = f pu 1
p
f 'c
1

(19.39)

d = 1000 158 = 842mm

A ps
1394
p =
=
= 3.68 10 3
bd
450 842

edayeK[

f pu = 1725MPa

1725

f ps = 17251 0.5 3.68 10 3

= 1568.6 MPa
35

1394(1568.6)
a=
= 163.3mm
0.85(35)450

edayvamantmFMCakRmas;sab 150mm . dUcenH muxkat;enHeFVIkarCamuxkat;mansab.

@> sRmab;muxkat;mansab
hf

M n = A pw f ps d + A pf f ps d
2
2

Edl Apw = Aps Apf

A pf =

1
0.85 f 'c (b bw )h f
f ps

esckIENnaMBIebtugeRbkugRtaMg

]
681

T.Chhay

mhaviTalysMNg;sIuvil
=

NPIC

1
[0.85(35)(450 150)150] = 853.5mm 2
1568.6

A pw = 1394 853.5 = 540.5mm 2

a=

Apw f ps

0.85 f 'c bw

540.5(1568.6 )
= 190mm
0.85(35)150

190 6
150 6

+ 853.5(1568.6 ) 842
M n = 540.5(1568.6 ) 842
= 1660.2kN .m
10
10
2
2

M n = 0.9(1660.2) = 1494.2kN .m

RtYtBinit reinforcement index sRmab;muxkat;mansab

pw =

A pw
bw d

pw = pw

540.5
= 4.28 10 3
150 842

f ps
f 'c

0.32 1 = 0.32 0.8 = 0.256

= 0.9

#> KNnam:Um:g;emKuNxageRkAEdl)anBIbnkefr nigGefr

bnkefr = Tmn;pal; + bnkefrbEnm
= 5.81 + 13.15 = 18.96kN .m

bnkGefr = 16kN / m
U = 1 .2 D + 1 .6 L
14.4 2
[1.2 18.96 + 1.6 16] = 1253.3kN .m
Mu =
8

m:Um:g;emKuNxageRkAenHmantmtUcCag ultimate moment capacity rbs;muxkat; 1494.2kN .m

dUcenHmuxkat;enHRKb;RKan;.
$> cracking moment smIkar !(>$* KW
M cr =

y
I
F
0.62 f 'c + + (Fe ) b
yb
A
I

BI]TahrN_ 19>4 F = 1344kN / A = 23.25 104 mm2 / e = 364mm / yb = 522mm /

I = 2.607 1010 mm 4 / f 'c = 35MPa nig 0.62 f 'c = 3.67 MPa
6
2.607 1010
1344 103
522
10
+ (1344 103 364 )
M cr =
3.67 +
4
10

522
23.25 10
2.607 10

= 961.2kN.m

RtYtBinitemIlfa 1.2M cr M n
1.2 M cr = 1.2(961.2 ) = 1153.4kN .m
T.Chhay

682

Introduction to Prestressed Concrete

viTasanCatiBhubeckeTskm<Ca

Department of Civil Engineering

tmenHKWtUcCag M n = 1494.2kN .m . dUcenH FwmenHRKb;RKan;kgkarTb;nwgkareRbH.

(Deflection)
19>7> PaBdab
PaBdabrbs;cMNucmYyenAelIFwmCabMlas;TIsrubrbs;cMNucenaH eTaHCaeLIgelI bcuHeRkam
EdlbNalBIkarGnuvtrbs;bnkenAelIFwmenaH. enAkgFwmebtugeRbkugRtaMgEdlRTedayTRmsamBa
CaTU eTAkmaMgeRbkugRtaMgRtUv)aneKGnuvtenABIeRkamTIRbCMuTmn;rbs;muxkat; EdlbegItCaPaBdabeLIg
elI EdleK[eQaHfa camber. Tmn;pal;rbs;Fwm nigbnkTMnajxageRkAEdlmanGMeBIenAelIFwmnwg
eFVI[manPaBdabcuHeRkam. Net deflection CaplbUkBiCKNitnPaBdabTaMgBIr.
enAkgkarKNnaPaBdab eKcaM)ac;RtUvBicarNaTaMgPaBdabryeBlxI(short-term deflection)
bPaBdabPam (immediate deflection) nigPaBdabryeBlyUr (long-term deflection). edIm,IFa
na[eRKOgbgMGaceFVIkareTA)an PaBdabryeBlxIGtibrma nigPaBdabryeBlEvgGtibrmaenARKb;
dMNak;kalnkardak;bnkEdlmanlkNeRKaHfak;TaMgGs;minRtUvFMCagtMnkMNt;Edl)ankMNt;
eday ACI Code emIlEpk 6>3.
eKGacKNnaPaBdabrbs;Ggt;ebtugeRbkugRtaMgedaysmIkarPaBdabsg;dar beday conventional method EdlmanenAkgesovePAviPaKeRKOgbgM. ]TahrN_ PaBdabenAkNalElVgrbs;
FwmTRmsamBaEdlrgbnkTMnajBRgayesI w esInwg 5wL4 / 384EI . m:UDuleGLasicrbs;ebtugKW
Ec = 0.043w1.5 f 'c = 4780 f 'c sRmab;ebtugTmn;Fmta.
eKKNnam:Um:g;niclPaBrbs;muxkat;ebtug I edayQrelIlkNrbs; gross section sRmab;
FwmEdlKansameRbH. eKGaceRbIkrNIenH)anenAeBlEdlkugRtaMgTajGtibrmaenAelIsrsxageRkA
bMputrbs;ebtugminelIs modulus of rupture rbs;ebtug f r = 0.62 f 'c Fwmfak; U . enAeBl
EdlkugRtaMgTajGtibrmaenAelIsrsxagelIrbs; gross section FMCag 0.62 f 'c eKRtUveRbIm:Um:g;
niclPaBRbsiTPaB (effective moment of inertia) I e EdlQrelImuxkat;EdlmansameRbH bGt;
mansameRbH dUcEdl)anBnl;enAkgCMBUk 6 Fwmfak; T nig C . PaBdabkNalE;lVgsRmab;
FwmTRmsamBaEdlbNalBIbnkTMnaj nigkmaMgeRbkugRtaMgRtUv)anbgajenAkgtarag 19>3.

esckIENnaMBIebtugeRbkugRtaMg

683

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

]TahrN_ 19>6 sRmab;FwmenAkg]TahrN_ 19>4 cUrKNna camber enAeBlepr nigbnab;mk

KNnaPaBdabPamEdlrMBwgTukcugeRkayEdlrgbnkeFVIkar.

dMeNaHRsay

!> PaBdabenAeBlepr
a. KNnaPaBdabcuHeRkamEdlbNalBIbnkefrenAeBlepr kgkrNIenHbnkefrCabnkpal;
sRmab;FwmTRmsamBaEdlrgbnkBRgayesI
5wL4
D bnkefr =
384 EI
BI]TahrN_ 19>4/ wD = 5.81kN / m / L = 14.4m / Eci = 24800MPa nig
I = 2.607 1010 mm 4
5(5.81)14400 4
D =
= 5mm
384(24800 )2.607 1010
b.

cuHeRkam

KNna camber EdlbNalBIkmaMgeRbkugRtaMg

sRmab;FwmTRmsamBaEdleKelIk tendon xHenAcMNucmYyPaKbI
CamYynwgcMNakpitenAcMENkmYyPaKbIkNalFwm e1 = 364mm nigenAcugFwm e2 = 0 .
p =
=

23(Fi e1 )L2
216 Eci I

23 1606 103 364 14400 2

216(24800 )2.607 1010

= 20mm

eLIgelI

cugeRkayenAeBleprKW 20 + 5 = 15mm eLIgelI

c. PaBdabeRkambnkeFVIkar
bnkeFVIkarBRgayesIsrubKW WT = 5.81 + 13.15 + 16 = 35kN / m
nig Ec = 27600MPa . PaBdabcuHeRkamEdlbNalBI WT KW
5WT L4
5(35)14400 4
cuHeRkam
w =
=
= 27 mm
384 E I 384(27600 )2.607 1010
Camber

EdlekItBIkmaMgeRbkugRtaMg F = 1344kN nig Ec = 27600MPa KW

23(1344 103 364 )14400 2
eLIgelI
p =
= 15mm

Camber

216(27600 )2.607 1010

PaBdabPamcugeRkayeRkambnkeFVIkarKW
= w p = 27 15 = 12mm

T.Chhay

684

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Introduction to Prestressed Concrete

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Department of Civil Engineering

19>8> KNnasRmab;kmaMgkat;TTwg (Design for Shear)

viFIKNnaedIm,IkMNt;EdkkmaMgTTwg (shear reinforcement) enAkgFwmebtugeRbkugRtaMgesIr
EtdUcKanwgviFIKNnasRmab;FwmebtugGarem:. eKsnt;sameRbHedaysarkmaMgkat;TTwg (shear
crack) ekIteLIgtammMu 45o edayvas;BIGkSrbs;Fwm. CaTUeTA TRmg;nsameRbHEdlTak;TgnwgkmaMg
kat;TTwgcMnYnBIrRbePT. mYyRbePTKWbNalBI\TiBlEdlrYmpSMnkarBt; nigkmaMgkat;TTwg
esckIENnaMBIebtugeRbkugRtaMg

685

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

sameRbHcab; epImeLIgedaysameRbHedaykarBt; ehIybnab;mksameRbHenHcab;epImgak ehIy

raltamTiseRTt EdlbNalBI\TiBlnkmaMgTajGgt;RTUg. RbePTTIBIrCa web-shear cracking
ekItmanenAkgFwm EdlkugRtaMgTajem bkugRtaMgTajGgt;RTUg (principal tensile stress) enAkg
RTnugdtUcrbs;vamantmFMebIeRbobeFobnwgkugRtaMgBt;. eKRtUveRbIEdkkgedIm,IFana principal
tensile stress enAkgkrNI TaMgBIr. eyIgnwgeRbIlkNvinicykgkarKNnarbs; ACI sRmab;kmaMg
kat;TTwg.
19>8>1> viFIcMbg

(Basic Approach)

QrelItRmUvkarersIusg;cugeRkay (ultimate strength requirement)

edayeRbIemKuNbnkEdl)anbgajenAkgCMBUk 3. enAeBlEdlkmaMgkat;TTwgemKuN Vu FMCagBak;
kNaln nominal shear strength ( Vc / 2 ) eKmindak;EdkkgeT. . kmaMgkat;TTwgKNnaEdlRtUv
kar Vu enARtg;muxkat;nImYyminRtUvFMCag nominal design strength Vn rbs;muxkat;EdlQrelI
nominal shear capacity Edl)anBIkarbUkpSMnebtug nigEdkRTnug.
ACI design approach

Vu Vn (Vc + Vs )

(19.49)

rbs;ebtug
Vs = nominal shear capacity rbs;Edk
= emKuNkat;bnyersIusg; = 0.75
enAeBlEdlkmaMgkat;emKuN Vu tUcCag Vu / 2 eKRtUvkardak;EdkkmaMgkat;Gb,brma.
Edl

Vc = nominal shear strength

19>8>2> ersIusg;kmaMgkat;Edlpl;edayebtug
Shear Strength Provided by Concrete

bgajsmIkary:agsamBaEdl)anBIkarBiesaFedIm,I)a:n;RbmaN
nominal shear capacity rbs;Ggt;ebtugeRbkugRtaMg EdlenAkgGgt;enaH tendon mankmaMgeRb
kugRtaMg f se y:agtic 40% ersIusg;Taj f pu
ACI Code, Section 11.4

V d
Vc = 0.05 f 'c + 4.8 u bw d
Mu

Edl

T.Chhay

Vu

(ACI Code, Eq. 11.9)

(19.50)

nig M u = kmaMgkat;emKuN nigm:Um:g;emKuNenARtg;muxkat;EdlBicarNa

bw = TTwgrbs;RTnug
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Department of Civil Engineering

kgtY VMu d = cmayBIsrsrgkarsgt;eTATIRbCMuTmn;rbs;EdkeRbkugRtaMg

u
d enAkgsmIkar Vci b Vcw = tmEdlFMCageKkgcMeNam d xagelI nig 0.8h (ACI Code,
d

Section 11.4.2)

kareRbIsmIkar 19.50 RtUv)ankMNt;tamlkxNxageRkam

!> Vu d / M u 1.0 edIm,IKittmtUcn Vu nig M u
Vc Gb,brma
@> Vc (0.17 f 'c )bwd
#> Vc (0.42 f 'c )bwd
Vc Gtibrma
karERbRbYln shear capacity rbs;ebtugsRmab;FwmebtugeRbkugRtaMgEdlRTedayTRmsamBargnUv
bnk BRgayesIRtUv)anbgajenAkgrUbTI 19>8. cMNaMfa eKcaM)ac;dak;EdkkmaMgkat;GtibrmaEk,rTRm
nig Ek,rcMNucmYyPaKbYnnElVgEdl Vs manxiteTArktmGtibrma. pymkvij FwmebtugGarem:
RtUvkar EdkkmaMgkat;TTwg bKMlatGb,brma EtEk,rTRmEdl Vs mantmGtibrma.

esckIENnaMBIebtugeRbkugRtaMg

687

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

eBlxH tm Vc EdlKNnaedaysmIkar 19.50 GacmantmtUc dUcenH ACI Code, Section

11.4.2 [nUvviFIm:ageTotsRmab;KNna Vc EdlKitBicarNaersIusg;bEnmrbs;ebtugenAkgmuxkat;.
enAkgviFIenH Vc CatmEdltUcCageKkgcMeNamersIusg;kmaMgkat;rbs;ebtug Vci nig Vcw rUbTI
19>8.
ersIusg;kmaMgkat; Vci RtUv)anQrelIkarsnt;fa flexural-shear cracking ekIteLIgEk,rEpk
qaybMputrbs; flexural cracking EdlenAcmayRbEhl d / 2 BIcMNucrgbnknkarfycuHm:Um:g;.
ACI Code kMNt;fa Vci RtUv)anKNnadUcxageRkam

VM
(19.51)
Vci = (0.05 f 'c )bw d + Vd + i cr
M

max

b:uEnvaminRtUvtUcCag (0.14 f 'c )bwd Edl

Vd = kmaMgkat;TTwgEdlbNalBIbnkefrKanemKuN
Vi = kmaMgkat;TTwgemKuNEdlbNalBIbnkEdlGnuvtBIxageRkA EdlekItmandMNalKa
CamYynwg M max
M max = m:Um:g;emKuNGtibrmabNalBIbnkEdlGnuvtxageRkA
M cr = m:Um:g;eRbH (cracking moment)
eKGackMNt; cracking moment BIsmIkarxageRkam
I
M cr = (0.5 f 'c + f pe f d )
(ACI Code, Eq. 11.11)
(19.52)
yt
Edl I = m:Um:g;niclPaBEdlTb;Tl;nwgbnkemKuNxageRkA
yt = cmayBIGkSTIRbCMuTmn;n gross section EdlminKitsrsEdkeTAsrsrgkarTajeRkA
bMput
f pe = ersIusg;rgkarsgt;enAsrsxageRkAbMputrbs;muxkat;ebtugedaysarkmaMgeRbkug
RtaMgeRkaykMhatbg;
f d = kugRtaMgEdlbNalBIbnkefrKanemKuNenAsrsxageRkAbMput EdlkugRtaMgTaj
bNalBIbnkxageRkA
Web-shear strength Vcw KWQrelIsameRbHedaykmaMgkat;TTwgenAkgFwmEdlminEmneRbH
edaysarkarBt;eT. sameRbHEbbenHekItmanenAEk,rTRmEdlmanRTnugtUc. ACI Code, Section
11.4.2 bBaak;fa Vcw RtUv)anKNnatamsmIkarxageRkam
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(
)
(19.53)
Edl V p = bgMkM;laMgeRbkugRtaMgbBarenARtg;muxkat;EdlBicarNa
f pc = kugRtaMgsgt;enAkgebtug eRkaykarGnuBaatsRmab;kMhatbg;eRbkugRtaMg enARtg;TI
RbCMuTmn;rbs;muxkat;EdlTb;Tl;nwgbnkEdlGnuvt benARtg;TIRbsBVrvagRTnug nig
sabenAeBlEdlTIRbCMuTmn;sitenAkgsab
ma:gvijeTot eKGackMNt; Vcw CakmaMgkat;EdlbegItkugRtaMgTajem (principle tensile
stress) 0.33 f 'c enARtg;GkSTIRbCMuTmn;rbs;Ggt; bRtg;RbsBVnsab nigRTnugenAeBlEdlTIRbCMu
TMgn;sitenAkgsab. eKGacsresrsmIkarGkSdUcxageRkam
Vcw = 0.29 f 'c + 0.3 f pc bw d + V p

f
f pc
2
pc
f t = 0.33 f 'c = vcw
+
2
2

f pc
b d
Vcw = f t 1 +

w
f
t

(19.54)

Edl ft = 0.33 f 'c . enAeBlGnuvtsmIkar 19.51 nig 19.53 b 19.54 tmrbs; d RtUv)anKitCa
cmaycenaHsrsrgkarsgt; nigTIRbCMuTmn;rbs; tendon eRbkugRtaMg b:uEnvaminRtUvtUcCag 0.8h .
muxkat;eRKaHfak;sRmab;kmaMgkat;GtibrmaRtUv)anykRtg; h / 2 BIprbs;TRm. eKRtUveRbI
EdkkmaMgkat;dUcKaRtg;muxkat;cenaHTRm nigmuxkat;Rtg; h / 2 .
(Shear Reinforcement)
19>8>3> EdkkmaMgkat;
eKRtUvKNnatm Vs edIm,IkMNt;RkLapcaM)ac;rbs;EdkkmaMgkat;

Vu = (Vc + Vs )
1
Vs = (Vu Vc )

(19.49)
(19.55)

sRmab;EdkkgbBar
Vs =

nig

Av f y d

s
Vs
Av = s
f yd

(19.56)

s=

Av f y d
Vs

(19.57)

Edl Av = RkLaprbs;EdkkgbBar nig s = KMlatrbs;Edkkg. smIkarsRmab;Edkkg

eRTt dUcKanwgsmIkarEdl)anbgajenAkgCMBUk 8.
esckIENnaMBIebtugeRbkugRtaMg

689

T.Chhay

mhaviTalysMNg;sIuvil

19>8>4> EdnkMNt;

NPIC

(Limitation)

!> KMlatGtibrma s rbs;EdkkgminRtUvFMCag 3h / 4 b 60cm . RbsinebI V FMCag

s

max

enaHKMlatGtibrmaxagelIRtUv)ankat;bnymkRtwmBak;kNal (ACI Code,

Section 11.5.4).
@> kmaMgkat;Gtibrma Vs minRtUvFMCag (8 f 'c )bwd . RbsinebImindUcenaHeT eKRtUvbegInTMhM
rbs;muxkat; (ACI Code, Section 11.5.6).
#> EdkkmaMgkat;Gb,brma Av EdlTamTareday ACI Code KW
(4 f 'c )bw d

Av min =

b s
0.35bw s
0.062 f 'c w
fy
fy

enAeBlEdlkmaMgeRbkugRtaMgRbsiTPaB
Av =

Aps f pu s
80 f y d

(19.58)

f pe 0.4 f pu

EdkkmaMgkat;Gb,brma Av Kw

d
bw

(19.59)

eKmincaM)ac;ykkm<s;RbsiTPaB d < 0.8h eT. CaTUeTA smIkar 19.59 pl;[nUvEdkkmaMg

kat;Gb,brmaFMCagkarpl;[edaysmIkar 19.58.

]TahrN_ 19>7 sRmab;FwmenAkg]TahrN_ 19>4 cUrkMNt; nominal shear strength nigEdkkmaMg

kat;caM)ac;. RtYtBinitmuxkat;Rtg; h / 2 nig 3m BIcugrbs;Fwm. eRbI

kmaMgkat; nigbnkGefr = 19.4kN / m .

f y = 400MPa

sRmab;Edk

dMeNaHRsay

!> sRmab;muxkat;enARtg; h / 2
h 1000
=
= 500mm
2
2

BIxagcug
@> bnkBRgayesIemKuNenAelIFwm
Wu = 1.2(5.81 + 13.15) + 1.6 19.4 = 53.8kN / m
h
Vu
= 53.8(7.2 0.5) = 360.5kN
2

enAcmay
kMNt; M u enARtg;muxkat; h / 2

M u = 53.8(7.2 )0.5 53.8

0.52
= 187kN .m
2

tmrbs; d Rtg;muxkat; h / 2 BIxagcug rUbTI 19>6 b KW

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d = (1000 158)

(4.8 0.5) 364 = 516mm

4.8
Vu d 360.5 0.516
=
= 0.995 1.0
187
Mu

dUckarTamTarrbs; ACI Code

V d
Vc = 0.05 f 'c + 4.8 u bw d
Mu

(
)
tmGb,brmarbs; Vc = 0.17 f 'c bwd = 0.17 35 (150)516 103 = 77.8kN
tmGtibrmarbs; Vc = 0.42 f 'c bwd = 0.42 35 (150)516 103 = 192.3kN
tmGtibrmarbs; Vc = 192.3kN manlkNlub.
#> viFIepSgeTotEdlbgajeday ACI Code KWfaeKGacyktm Vc CatmtUcCageKkgcMeNam Vci
nig Vcw
a. edayQrelI flexural-shear cracking strength

VM
Vci = (0.05 f 'c )bw d + Vd + i cr
M
= 0.05 35 + 4.8 0.995 150 516 10 3 = 392.6kN

max

KNnatYnImYydac;edayELkBIKa
(0.05 f 'c )bwd = 0.05 35 (150)516 103 = 22.9kN
Vd = kmaMgkat;efrKanemKuN = (5.81 + 13.15)(7.2 0.5) = 129.4kN
M max = m:Um:g;emKuNGtibrma elIkElgTmn;rbs;Fwm
bnkemKuN = 1.2 13.15 + 1.6 19.4 = 46.8kN / m

0.52
M max = 46.87.2 0.5
= 162.6kN .m
2

Vi = 46.8(7.2 0.5) = 313.6kN

I
M cr =
0.5 f 'c + f pe + f d
yt

/
f pe = kugRtaMgrgkarsgt;EdlbNalBIkmaMgeRbkugRtaMg

I = 2.607 1010 mm 4 yb = 522mm

=
=

F Feyb
+
A
I
1344 103
23.25 10 4

esckIENnaMBIebtugeRbkugRtaMg

1344 103 38 522

2.607 1010
691

= 6.8MPa

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

fd =

kugRtaMgbnkefr = M DI yb

0.52
M D = 18.967.2 0.5
= 65.9kN .m
2

65.9(522 ) 10 6
fd =
= 1.32 MPa
2.607 1010
2.607 1010
M cr =
0.5 35 + 6.8 1.32 10 6 = 412.4kN .m
522
412.4
= 947.7kN
Vci = 22.9 + 129.4 + 313.6
162.6

b.

dUcenH
Vci minRtUvtUcCag (0.14 f 'c )bw d = 0.14 35 (150)516 10 3 = 64.1kN
ersIusg;kmaMgkat;EdlQrelI web-shear cracking KW
Vcw = (0.29 f 'c + 0.3 f pc )bw d + V p
f pc =

1344 103
23.25 10 4

d = 516mm

yk

= 5.78MPa

b 0.8h = 800mm

d = 800mm

V p = 1344

1
= 101.8kN
13.2

Edl 1 / 13.2 = slop rbs; tendon profile = 364 / 4800

0.29 35 = 1.72 MPa

dUcenH Vcw = (1.72 + 0.3 5.78) 150 800 103 + 101.8 = 516.3kN
c. edaysar Vcw < Vci dUcenH Vcw = 516.3kN Ca nominal shear strength enARtg;muxkat; h / 2
BIcugrbs;Fwm. kgkrNICaeRcIn Vcw lubRtg; h / 2 BITRm.
$> EdkRTnug (web reinforcement)
Vu = 360.5kN

Vcw = 0.75 516.3 = 387.2kN

edaysar Vu < Vcw / Vs = 0 dUcenHeRbIEdkkgGb,brma. eRbIEdkkg DB10 .

Av = 2 10 2 / 4 = 157mm 2

KMlatGtibrmaCatmEdltUcCageKkgcMeNam
s1 =

3
h = 750mm
4

s2 = 600mm

KNna s3 BIsmIkarEdkRTnugGb,brma
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Av min =
157 =

Aps
80

f pu

fy

s
d

d
bw

1394 1725 s3 516

80
400 516 150

s3 = 581mm

yk s3 = 500mm

b s
0.35bw s
0.062 f 'c w
fy
fy

Av f y
157 400
=
s4 =
= 1140mm
0.367bw 0.367 150

ma:gvijeTot

0.062 f 'c = 0.367

Av min =

dUcenHeyIgeRbI DB10 @ 500 .

%> sRmab;muxkat;enAcmay 3m BIxagcug viFIsaRskgkarKNnaKWRsedogKasRmab;muxkat;enARtg;
h / 2 . edayeRbIviFIEdlsRmYlrbs; ACI
Vu = 53.8(7.2 3) = 226kN

32
M u = 53.87.2 3 = 920kN .m
2

(4.8 3) 364 = 705mm

d = (1000 158)
4.8
Vu d 226 0.705
=
= 0.173 < 1.0
920
Mu

V d
Vc = 0.05 f 'c + 4.8 u bw d = 0.05 35 + 4.8 0.173 150 705 10 3 = 119.1kN
Mu

nig Vc max = 192.3kN

dUcenH Vc = 119.1kN
lub
^> eRbIsmIkar ACI Code edIm,IKNna Vci nig Vcw . dMbUgKNna Vci EdllubsRmab;muxkat;enH
(0.05 f 'c )bwd = 0.05 35 (150)705 103 = 31.3kN
Vc min = 77.8kN

Vd = (5.81 + 13.15)(7.2 3) = 79.6kN

32
M max = 46.87.2 3 = 800.3kN .m
2

Vi = 46.8(7.2 3) = 196.6kN
f pe =

1344 103
23.25 10 4

esckIENnaMBIebtugeRbkugRtaMg

1344 103 227.5 522

2.607 1010

693

= 11.9 MPa

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

32
M D = 18.967.2 3 = 324.2kN .m
2

fd =

324.2(522 ) 106

= 6.5MPa
2.607 1010
2.607 1010
M cr =
0.5 35 + 11.9 6.5 10 6 = 417.4kN .m
522
417.4
= 213.4kN
Vci = 31.3 + 79.6 + 196.6
800.3

dUcenH

Vci min = 0.14 f 'c bw d = 0.14 35 (150)705 10 3 = 87.6kN

dUcnH
Vci = 213.4kN
bnab;mkKNna Vcw
f pc = 5.78MPa
d = 705mm

dUcelIkmun

V p = 101.8kN

b 0.8h = 800mm

yk d = 800mm
dUcenH Vcw = (1.72 + 0.3 5.78) 150 800 103 + 101.8 = 516.3kN
tmrbs; Vcw minmaneRKaHfak;eT. enARtg;mYyPaKbYnnRbEvgElVg tmkmaMgkat;eRKaHfak;KW
Vci rUbTI 19>8.
&> KNnaEdkRTnug
Vu = 226kN

Vci = 0.75 213.4 = 160kN

Vu = (Vc + Vs )
1
(226 160) = 88kN
Vs =
0.75

Edkkg DB10 / Av = 157mm2 . RtYtBinitKMlatGtibrma smax = 500mm dUcelIkmun

103 500
muxkat;EdlEdlRtUvkar Av = Vf sds = 88400
= 156
705
y

Av

T.Chhay

EdleRbIKW 157mm2 > 156mm2 . dUcenH yk DB10 @ 400 .

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19>9> KNnaCMhandMbUgnGgt;ebtugeRbmugRtaMgrgkarBt;
Preliminary Design of Prestressed Concrete Flexural Members

19>9>1> rUbrag nigTMhM

(Shapes and Dimensions)
kar)a:n;sanTMhMrbs;muxkat;dRtwmRtUveFVI[karKNnacMNayeBltic nigminsKsaj. dUc enH
karKNnaCMhandMbUgmansarsMxan;Nas; edIm,IFanafaTMhMmuxkat;manlkNsmrmmunnwgcab;epIm
KNnalMGit.
enAkgdMNak;kalKNnaCMhandMbUg CaFmtaeKmanTinnyxHEdlGacCYykgkareRCIserIsTMhM
dsmRsb. ]TahrN_ m:Um:g;Bt;Edl)anBIbnkGnuvtn_xageRkA kugRtaMgGnuBaat nigTinnysRmab;
kMNt;kMhatbg;RtUv)andwg bRtUv)ankMNt;.
rUbragrbs;muxkat;Ggt;ebtugeRbkugRtaMgGacCactuekaNEkg GkSr T GkSr I bRbGb;. km<s;
srubrbs;muxkat; h RtUv)ankMNt;edaykarBicarNakm<s;rbs;lMh bkminRtUv)ankMNt;. kareRCIs
erIsTMhMrbs;muxkat;sRmab;karKNnaCMhandMbUgmandUcxageRkam rUbTI 19>9

!> km<s;srubnmuxkat;KW h = 1 / 20 eTA 1 / 30 nRbEvgElVg. sRmab;karrgbnkFn; h = L / 20

nigsRmab;karrgbnkRsal h = L / 30 b h = 43.6 M D + M L Edl h KitCa mm nig
M KitCa kN.m .
@> kRmas;rbs;sabxagelIKW h f = h / 8 eTA h / 6 .
#> TTwgrbs;sabxagelIKW b 2h / 5 .
$> kRmas;rbs;RTnugKW bw 100mm . CaTUeTA eKyk bw = h / 30 + 100 .
esckIENnaMBIebtugeRbkugRtaMg

695

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

%> kareRCIserIs bw nig t KWERbRbYleTAtamkarBRgay tendon eRbkugRtaMg edayeFVIy:agNa

rkSakRmas;ebtugkarBarEdk.
^> RkLapEdlsmrmrbs;muxkat;ebtugEdlRtUvkarKW
M + ML
Ac (m 2 ) = D
Edl M D + M L KitCa kN.m nig h KitCa m
1450h
sRmab;karKNnakgkarGnuvt nigkarKNnaEdlmanlkNesdkic ]sSahkmpliteRKOg
sMNg;BIebtug)anbegItnUvrUbrag nigTMhMbTdanCaeRcIn EdlGkKNnaGaceRCIserIsnUvGgt;NaEdl
RtUvkar. taragnmuxkat;sg;darmanenAkg PCI Design Handbook. AASTHO k)anENnaMnUv
girder sg;daredIm,IeRbIR)as;enAkgsMNg;s<an tarag 19>4.

tarag 19>4 AASTHO Girders, ebtugTmn;Fmta

RbePT
Type II

yb

Zb

Zt

Tmn;

(in. )
2

(in. )

(in.)

(in. )

(in. )

(lb / ft )

369

50979

15.83

3220

2527

384

Type III

560

125390

20.27

6186

5070

593

Type IV

789

260741

24.73

10544

8908

822

19>9>2> kmaMgeRbkugRtaMg nigRkLapEdk

(Shapes and Dimensions)
enAeBlEdleKeRCIserIsrUbrag km<s; nigTMhMepSgeTotrbs;muxkat;rYcehIy eKGac)a:n;santMlRbhak;RbEhlrbs;kmaMgeRbkugRtaMg nigRkLaprbs;EdkeRbkugRtaMg Aps . BIm:Um:g; couple

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xagkg m:Um:g;srub M T EdlekItBIbnkefr nigbnkGefreFVIkaresInwgkmaMgTaj T KuNnwgdXas;

jd .
M T = T ( jd ) = C ( jd )

M T = Aps f se ( jd )

A ps =

MT
f se ( jd )

Edl Aps CaRkLapEdkeRbkugRtaMg nig f se CakugRtaMgeRbkugRtaMgRbsiTPaBeRkaykMhatbg;.

tmrbs;dXas; jd ERbRbYlBI 0.4h eTA 0.8h EdlkgkarGnuvteKykvaenAkgcenaH 0.6h eTA
0.7 h . eKGaceRbItmmFm 0.65 )an. dUcenH
A ps =

MT
(0.65h ) f se

(19.60)

ehIykmaMgeRbkugRtaMgKW
F = T = A ps f se =

MT
0.65h

(19.61)

kmaMgeRbkugRtaMgenAeBleprKW Fi = F / Edl CaemKuNnkMhatbg;GaRsynwgeBl.

kmaMgsgt; C enAelImuxkat;esInwgkmaMgTaj T
C = T = Aps f se

KitCakugRtaMg AC = ApsA f se = f c1
c
c
Edl fc1 CakugRtaMgBRgayesIEdlsnt;enAelImuxkat;
sRmab;karKNnaCMhandMbUg
eKsnt;srsxageRkArbs;karEbgEckkugRtaMgragRtIekaNesInwg kugRtaMgsgt;GnuBaatGtibrma f ca .
dUcenHkugRtaMgmFmKW 0.5 fca = fc1 . kugRtaMgsgt;GnuBaat enAkgebtugKW fca = 0.45 f 'c . dUcenH
eKGacsnt;RkLapebtugEdlRtUvkar Ac BIkmaMgTaj T dUc xageRkam
A ps f se A ps f se
Aps f se
T
=
=
=
0.5 f ca 0.225 f 'c
f c1
f c1
T
MT
MT
MT
Ac =
=
=
=
0.5 f ca (0.65h )(0.5 f ca ) 0.33 f ca 0.15 f 'c

Ac =

(19.62)
(19.63)

karviPaKenHKWQrelIkarKNnasRmab;bnkeFVIkar minEmnsRmab;bnkemKuNeT. cMNakpit e RtUv)an

vas;BITIRbCMuTmn;rbs;muxkat;eTATIRbCMuTmn;rbs;EdkeRbkugRtaMg ehIyeKGacKNnaCatmRbhak;
RbEhl dUcxageRkam
esckIENnaMBIebtugeRbkugRtaMg

697

T.Chhay

mhaviTalysMNg;sIuvil
e = Kb +

NPIC

MD
Fi

(19.64)

Edl Kb Ca lower kern limit ehIy M D Cam:Um:g;Edl)anBIbnkefrKanemKuN.

19>10> kugRtaMgbkxagcug (End-Block Stresses)
19>10>1> Ggt;rgkugRtaMgTajmun
(Pretensioned Members)
dUcKanwgEdkenAkgGgt;ebtugGarem:EdlRtUvkarRbEvgbgb;Cak;lak; enaHkmaMgeRbkugRtaMg
enAkgGgt;ebtugrgkugRtaMgmunRtUv)anepreTAebtugedaykarbgb; bedaykarf<k;cug bedaybnSMTaMgBIr.
enAkgGgt;rgkugRtaMgmun eKehARbEvgrbs;EdkeRbkugRtaMgEdlkmaMgeRbkugRtaMgRtUv)anepreTAebtugfa RbEvgepr (transfer length) lt . eRkayeBlepr kugRtaMgenAkg tendon Rtg;cugbMputrbs;Ggt;
esInwgsUn b:uEnkugRtaMgenARtg;cmay lt BIcugGgt;esInwgkmaMgeRbkugRtaMgRbsiTPaB f pe . transfer
length lt GaRsynwgTMhM nigRbePTrbs; tendon, lkxNpb:H ersIusg;ebtug f 'c kugRtaMg nigviFIn
kareprkmaMg. kar)a:n;RbmaN lt EdlGacTTYlyk)anKWesInwg 50 dgnGgt;pitrbs; tendon b:uEn
sRmab; single wires eKyk lt esInwg 100 dgnGgt;pitrbs; wire.
edIm,I[kmaMgTajenAkgEdkeRbkugRtaMgbegIt ultimate flexural strength eBjelj vaRtUv
karnUvRbEvgcMNg (bond length). eKalbMNgKWedIm,IkarBarkarrGilTUeTAmunnwgkar)ak;rbs;Fwm
eRkam design strength eBjelj. RbEvgbgb; ld (development length) esInwgplbUkrvag bond
length nig transfer length. edayEpkelIkarBiesaF ACI Code, Section 12.9.1 [nUvsmIkarxag
eRkamsRmab;KNnaRbEvgbgb;n three- or seven-wire pretensioning strand:
2

ld = f ps f se d e
3

Edl

(19.65)

kugRtaMgenAkgEdkeRbkugRtaMgeRkam nominal strength

f se = kugRtaMgRbsiTPaBenAkgEdkeRbkugRtaMgeRkaykMhatbg;
d b = nominal diameter rbs; wire b strand
enAkg pretensioned member kugRtaMgTajx<s;manenAtMbn;xagcug EdleKRtUvkardak;EdkBiess.
EdkenHmanTRmg;CaEdkkgbBarehIyRtUv)anBRgayesIkgKMlat h / 5 Edlvas;BIxagcugrbs;Fwm.
CaTUeTA eKdak;EdkkgTImYyenAcmay 25mm eTA 75mm BIxagcugFwm. vaCaKarGnuvtFmtakgkar

T.Chhay

f ps =

698

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Department of Civil Engineering

bEnm nominal reinforcement sRmab;cmay d Edlvas;BIcugrbs;Fwm. RkLaprbs;EdkbBar Av

EdlRtUv)aneRbIenAtMbn;xagcugGacRtUv)anKNnaedaytmRbhak;RbEhlBIsmIkarxageRkam
Av = 0.021

Edl

f sa =

Fi h
f sa lt

(19.66)

kugRtaMgGnuBaatenAkgEdkkg CaTUeTA 140MPa nig ld = 50 dgnGgt; tendon

]TahrN_ 19>8 kMNt;EdkkgcaM)ac;EdlRtUvkarenAtMbn;xagcugrbs;FwmEdl[enAkg]TahrN_

19>4.

dMeNaHRsay

Fi = 1606kN h = 1000mm

dUcenH

Av = 0.021

f sa = 140 MPa lt = 50 11.125 = 556mm

1606 103 1000

= 433mm 2
140 556

h 1000
=
= 200mm
5
5

eRbIEdkkgbiTCit DB10 cMnYnbYnkg EdlEdkkgTImYysitenAcmay 200mm BITRm

Av Edldak; = 157 4 = 628mm 2
(Post-tensioned Members)
19>10>2> Ggt;rgkugRtaMgTajeRkay
enAkg post-tensioned concrete member kmaMgeRbkugRtaMgRtUv)aneprBI tendon sRmab;
bonded nig unbonded tendon eTAebtugRtg;cugrbs;Ggt;eday special anchorage device. enA
kg anchorage zone Rtg;cugrbs;Ggt; kugRtaMgsgt;mantmx<s;Nas; ehIy transverse tensile
stress ekItmandUcbgajenAkgrUbTI 19>10. enAkgkarGnuvt eKeXIjfaRbEvgn anchorage zone
minFMCagkm<s;rbs;cugGgt;eT RbsinebImindUcenaHeT sanPaBnkugRtaMgenAkgtMbn;enaHnwgman
lkNsKsaj.
karBRgaykugRtaMgEdl)anBI tendon mYyenAkg anchorage zone RtUv)anbgajenAkgrUbTI
19>11. enARtg;cmay h BImuxkat;xagcug eKsnt;kugRtaMgBRgayesIelImuxkat;TaMgmUl. edayKit
ExSkmaMg (trajectories) CaFatumYyEdlmanGMeBICa curved strut enaHExSkmaMgmanGMeBItamTisedk
eTArkExSGkSrbs;FwmenAkgtMbn; A edaybegItCakugRtaMgsgt;. enAkgtMbn; B ExSkmaMg)anbEgVr

esckIENnaMBIebtugeRbkugRtaMg

699

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

Tis ehIy strut manGMeBIecjeRkAEdlbegItCakugRtaMgTaj. enAkgtMbn; C strut esIrEtRtg;eday

begItkarBRgaykugRtaMgesI.

T.Chhay

700

Introduction to Prestressed Concrete

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Department of Civil Engineering

EdkBRgwgEdlRtUvkarsRmab; anchorage zone xagcugrbs; posttentioned member CaTU

eTApSM eLIgedayRkLanEdkbBar nigEdkedkEdlmanKMlatCitKaeBjRbEvgnbkxagcugedIm,I
Tb;Tl;nwgkugRtaMgpH (bursting stress) nigkugRtaMgTaj. CakarGnuvtTUeTAeKmin[KMlatEdkFMCag
75mm enAkgTisnImYyeT ehIyminRtUvdak;EdkenAcmayFMCag 50mm BIpxagkgrbs; bearing
plate eT.

esckIENnaMBIebtugeRbkugRtaMg

701

T.Chhay

mhaviTalysMNg;sIuvil

NPIC

]bsm<n

T.Chhay

702

Appendix

viTasanCatiBhubeckeTskm<Ca

]bsm<n

Department of Civil Engineering

703

T.Chhay

mhaviTalysMNg;sIuvil

T.Chhay

NPIC

704

Appendix

viTasanCatiBhubeckeTskm<Ca

]bsm<n

Department of Civil Engineering

705

T.Chhay

mhaviTalysMNg;sIuvil

T.Chhay

NPIC

706

Appendix

viTasanCatiBhubeckeTskm<Ca

]bsm<n

Department of Civil Engineering

707

T.Chhay

mhaviTalysMNg;sIuvil

T.Chhay

NPIC

708

Appendix

viTasanCatiBhubeckeTskm<Ca

]bsm<n

Department of Civil Engineering

709

T.Chhay