N\ SUPPLEMENT To : ANALYSIS & BESIGN OF FLIGHT VEHICLE STRUCTURES Bruhn FOR INCREASED SCOPE AND USEFULNESS William F. McCombs >PUBLISHED BY DATATEC DALLAS, TEXAS Current price $15.95 plus $2,00 postage/handling. For insured mailing add $2,00, Pay by postal money order or cash only, please, to avoid delay of weeks, Make Payments To William P, McCombs P.O, Box 763576 Dallas, TX 75376-3576 Tel. 214 337-5506 Other planned books includes Engineering Colum Analysis (Columns, Beam-Columns, Truss Members, Arches and Ben‘ An Introduction To Stress Analysis Of Light Structures Te wisi PaconseA SUPPLEMENT TO ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES Willian PF, McCombs ‘TABLE OF CONTENTS Article and Main Topic Page Beam-Columns Beam-Column Pormulas Beam-Column Deflections Bean-Column Formulas Margins of Safety ‘Truss Analyais Tangent Modulus Multispan Bean-Colunns Approximate Buckling Formula Plastic Torsion Beam Deflections Numerical Analysis Moment Distribution Column Analyt Column Analysis Torsional Buckling Colunn/Bean-Column Date Material Properties XD Crushing Loads Margins of Safety Dealing With Tolerances Combined Stres: Column Curve ¢ Free-Ended Columns Shear Effect on Buckling Multispan Columns Stepped Columns Nunerical Column Analysis Initially Bent Columns Column Design Data Column Elastic Supports Need for Successive Trials Column Elastic End Restraint Tangent and Effective Moduli Buckling Load Data End Friction Effects References for Chapter C2 Flastic Bending Data Plastic Bending Example Plastic Bending Procedure Complex Plastic Bending Shear Stres: ruction nding Modulus Apparent Margins of Safety Flat Plate Shear Buckling Bending Buckling Unequal Angle Leg Thickn Crippling Method Three Torsional Buckling Article and Main Topic C9,13a Frame Stiffness criteria C10,15b Thick-Wed Beam Analysis 611.29a Tension Field Beam Hole: C11, 2he Rivet Design C11,31a Stringer Construction C11,32a Diagonal Tension - Stringe: 611.33a Stringer syst ¢11.34a Stringer Diagonal Longeron Longeron Construction Example Tension Summary for Part 2 for Part 1 Monocoque Shell Buckling Data Shell Axial Comp, Buckling Effect of Internal Pressure Shell Bending Buckling Effect of Internal Pressure External Hydrostatic Pressure Shell Torsion Buckling Effect of Internal Pressure Combined Loads Buckling Conical Shell Buckling Buckling of Spherical Caps Fitting Design Pitting Margins of Safety Factors of Safety Bolte Wate ‘ly Loaded Lugs Obliquely Loaded Lugs Riveted Splice: ‘Tnread Design and Strength Filler (shim) Effects Curved Beam Data Tension Clip Allowable Data Flush Rivet Joints Blind Rivet Joints Bolt Strengths 31|Design Check List 31|Margins of Safety 31) Tension Clips 31|Preload Torque Factors for Bolts 1 ‘in Plastic Range32|Rockwell and Brinell Hardness Data 3| 35| Minimum I for stiffeners 33| stress-strain curves 3 Sheet and Other Buckling Data 36| Material Properti 35| Joint Design No 39|Fastener/Joint Allowable Data 0] Nut/Cotlar Tension Allowable 43] Bolt Allowable Bending Mosente 44] Bean Fornulas us| Additional References Bulkhead, Praze and Arch Analyses wii P hecomePREFACE The widely used and recommended college/industry textbook “analysis and Design of Flight Vehicle Structures" by Dr, E.P, Bruhn has had only one revision since its inception in 1965, That was the 1973 edition in which Chapter A23 was revised and expanded, Chapter C13 was completely rewritten by another author and a few minor changes were made in Chapter 11, Aside from these the book remains in its original form, ‘The purpose of this Supplement is to increase the scope and usefulness of the textbook in numerous specific areas of analysis, These include columns, beam-columns, bending strength, margins of safety, tension field analyses, fastener/ joint data, arches, bulkheads and numerous others, The practical use of the Supplement is discussed in the Introduction, Only one or two applications of the Supplement's contents can be worth much more than its coi The Supplement may be expanded in some future year, so any suggestions for this, or for corrections or changes in its current text will be appreciated, Those readers who wish to be informed of any future revisions or who have suggestions can contact the author at P.O, Bor 763576, Dallas, TX 75376-3576 (Tel.214 337-5506). ‘The author is one of the coauthors of the textbook, His career includes over forty years of experience in structural analysis and design of numerous aircraft and missile projects in the aerospace industry, Also included are technical papers and the preparation and teaching of practical courses in structural design and analysis for engineers working in the aerospace and other industries, William F, McCombs POREWORD I am pleased to have the opportunity to recommend this Supplement to late fathe: widely used college/industry textbook “Analysis and Design of Plight Vehicle Structures", I hope its practical applications will be a benefit to all who have the textbook, The additional data contained in this Supplement can be applied to both the study and the work of structural design and analysis, The Supplement should also be of interest to, and eventually benefit, those who may be considering purchasing the textbook. Patricia Bruhn Beachler aLINTRODUCTION Tne purpose of this Supplement 1s to increase the scope and usefulness of the widely used and reconmended college/industry textbook “analysis and Design of Flight Vehicle structures" by E.F.Bruhn. As auch it is by no means a revision of that book” Rather, it 1s an expansion and clarification of numerous topics and data in the book, along with the introduction of additional topics and data. For best coordination with the textbook the following has been done, Where an existing article such as, for example, Art. Cl.13 has been expanded or otherwise caanged or corrected, the Supplement includes it as art. Cl.13a, the letter indicating @ change or addition. When a new topic is added to a chapter it 1s given an article number which is subsequent to the last article number in that chapter and includes no letter. For example, the jast article in Chapter C3 is art. 03.14. Two additional topic, pyield Stress Bending Modulus" and "Residual stresses Following Plastic Bending" have been added, so they have been given article numbers 03.15 and 3.16 respectively with no letters. The same thing has been done with figure numbers and table numbers, Where a figure has been changed or added to, it retains the same figure number with an added letter. For example, Figure 2.27 has added information go it has the designation Fig.2.27a in the Supplement. Ynen a new figure is added it 19 given a number which is subsequent to the last figure number in the caapter, so no letters are used with its number in the Supplement. ‘This procedure also applies to tables. Although the Supplement provides an Index, for most usefulness the textbook aust be marked in such a manner as to guide the reades directly to revised, corrected and new topics, figures, tables and references in the Supplement. A highly recommended scheme for doing this is provided later. Structural design and analyses are based on theory, empirical methods and data, various assumptions and individual judgement. The assembled structure is a result of various specific manufacturing methods and procedure: Because of these things and also the Possi- bility of inadvertent calculation errors, it 18 alnays necessary to Brove,vhe adequacy and safety of the completed structure by meas of 2 sufficient test program before it 1s put into use. Such teste most denonstrate the structure's adequacy as to ultimate and yield strength, fatigue life, fracture and stiffness. The test results must be properly evaluated since the test article's materials usually gave properties in excess of the miniaun required values. Frocuring agencies such as the military, the airlines and other government and private organizations usually specify the design and test requirements and other criteria which gust be used or met. wutoordis ton of Su nt _and the Textbo. In order to easily guide the reader from the textbook to the Supplement, the following marking of the textbook is recommended, 1 2 5. On the textbook's Table of Contents place an asterisk, in red ink, after "Contents" and add the following footnote at the bottom of the page: * A red asterisk preceding any article or figure number in the textbook means that additional material is available in the same’ article or figure in the Supplement, For each textbook article listed in the Supplement's Table of Contents, place a large asterisk, in red ink, just to the left of each corresponding article number in the textbook (e.g., #A11,2), The 12 articles in the Supplement's Table of Contents not having a letter in the article number are new articles, (e.g. C3,18 Beam-Column Analyses), Their numbers and titles should be written in at the end of their chapters, with a red asterisk just to the left of their article numbers, The following figures in the textbook should have an asterisk, in red ink, placed just to the left of their figure numbers, to indicate that a revision or additional data 1s in the Supplements A5.1_ C2.17 62,26 «C5414 C8415 C8428 = c11, A18.8 2,18 C2,27 C8.8a ¢8.20 8,29 cit, C1.8 2,19 3,27 C811 C8125 c10.15 D145 2,2 2,20 03,28 c8.13 68.26 «011.43 €2.16 2.25 5,118.14 C827 C11 ab 7 8 The additional references shown on p, 30 (for Chapter C2) and on p.A21 (for Chapters C3, C4, C7 and C11) should be written in at the end of the list of references for these chapter: Put 6 black asterisk after "RINGS" on the title of p.A9.1 and at the bottom of the right hand column add the footnote * See Appendix B of the Supplement for alternative analyses of bulkheads, frames, arches and bents, tvA SUPPLEMENT 10 “ANALYSIS AND DESIGN OF PLIGHT VEEICLE STRUCTURES 1 45.230 Introduetion ‘Ma_the following at the end of art. (5.23. For a discussion of all types of beam-columns, including non-unifora bers, with numerous example probleme see “Engineering Coluan Analysis" described in Art.Al6.278, 25.240 Effects of Combined Axial and Lateral Loads Add the following at the end of Art. 5.24, Porgulas and example problems for beams in tension are available in the ook described in Art. Al8.27a. 258 Equations for a compressive axially Loaded 10-050 Serut sit Miugerely Besteibaved Coad Add the following at the end of Art. 45.25. The deflection, y, at any beam station can be calculated as follows. Mon ig + lp # ig + Py y = -(M = Mg)/P = Wig = w/e therefore Te negative sign is introduced since a positive bean bending moment produces « Regative (downward) deflection. a te the goaent due to the lateral ioad and Ute the final aoment. only very slightly to the left of the station and then very slightly to the right, the sane amount, AL. The slope is then slope = (yg ~ y,)/24L 45.264 Formulas for Oter single Bpen Loedings sla presents numerous addi- of single span loadings. ‘The case varying EI or a varying axial loading re- of a beam-column with a quires a numerical analysis, just as do @ column of this nature. ‘The numerical analysis procedure is presented in (3.18. 45,274 Goabinations of Load syatene, uargine of Safety and Acoureey of Caleulations For a bean-coluan the true margin of| safety aust be calculated as discussed in| Art, 0h.23 oF, bending moments are calculated as dis- cussed in art, 03.18. 15.208, Beaay In Example Problem 2 member BD 16 conaidered to be pinned to ABC at Joint This 18 why it does not pick up any of the 36,000 in 1b bending moment at joint B. 45.298 Stresses Above Proportional Limit sree If @ column curve is not available Probiews for the beam material, the axial str P/A, Le caloulated, fo/Fo,7 18 calculated Fig. 62.16 1s entered with this value and Evi fe obtained. Tuon Br S BEy/E). “Et in Art. 45.29 18 actually Ey. 45.318 Bean-Columne in continuous Structures Multispan deam-columns require a mo- ment distribution analysis to determine the end moments acting on each apan. Cnce these are known the applicable single span formulas in Table A5.1 and A5.1a can be used to determine the bending aonents at any station within a span.” Tats te discussed in Art. 03.16, 45.32 Approzinate Formula for Besa-dolumns For preliminary sizing when there are no end moments the following formula can be used to determine the final bend- ing moment at any station Mm Me/(1 = P/Fon) where Mg is the moment due to the tran: verse dads only and Per te the critical load as a column (Chapter 2). Tis le Gost accurate for a uniform lateral load and least accurate for a concentrated load. Por is calculated member hae a stable o: if this is not the case. For a bean in tension the negative sign in the formula is replaced with @ positive eign (ten- Sion makes the bending moment smaller). 46.7a Torsion of Seli4 Hon-Cireular Mapes All of the previous formulas are jd on the shear atré being in the etic range. With ductile materials failure (rupture) does not occur until the shear deformation has gone well in- to the plastic range (similar to the plastic bending case). The toreson: Moment at which rupture occurs can b Predicted ae discussed in Art. C4.20a. For the special case of tubes ha- ving @ circular cross-section the failing torque ean be calculated as discui art. 0é.20 and ite associated Fig 4.17 to 04.30 and in the exanple lems of art. 04,21. AT-1a Introduation For practical purposes the deflec- tion and slopes of beams are calculated 4 discussed in Art. A7.12a, standard formulas being used for unifora bean and for varying section beans using tables such as Table C3.3.and the "ind Fixity" discussion, 41-328 Dethections and Angular cuanges 9 70128 Setas‘by uethod ofStkinerte Sates For beams of uniform section def- Lections and slopes are most easily cal- © Provided tuat 1/369 for the apnSINGLE-SPAN BEAN-COLUMN FORMULAS Table A5.1a Continuation of Table AS.1 M = Cysinx/) + Cocosx/j + f(x) Fain irr) | Pr pleteing 7) Pawnee uy | totais ana | 0 4 mem edt at [Tears se ve Eo sews com —| vm ey Peet « [ravsoen} - a? Seen aeeg ‘anbel there for same reaulte 2 f Or uae Case IIT oy poser + with iegeang wa lave ane v there bmn - for sane reaulte or eateu 2 = Mssine/4_sosb/ -zint/(1cest/1)/atnt/3 > (Occurs under the loads w tis tsa superposition of Cases VMOMENT DISTRIBUTION WITH AXIAL LOAD culated by using beam deflection and slope formulas widely available in the structural literature. The are based on bending atresi are accurate unle or otherwise quite flexible due to hole. eto. only and ‘the wed 1s very thin ieflections and the at any station can be calculated din Art. A5.25% For beams with a varying ET a nun- 1 analysis 1a necessary to deter- mine the deflections and slopes. This of be done as discussed and illustrated in Art. 03.18 where Table C3.3 shows the de-| flections due to the transverse loade on-| ly and Tables 03.4 and 03.5 show the ad- ditional deflections due to the axial loads, the final bending moments being in| Table'c3.5. The deflections at any sta- tion are then calculated per the formula in Art. A5.25a using the final momenta from Table 03.5. Tables for end fixity are discussed in art, 03.18. With elastic end restraint a t distribu- tion analysis 1a required to det the end med to be a "Joint" en though the member may be continuous across the Joint, The sketoh in Fig. 11.92 shows the direction of (+) and (-)| moments as they act upon the spane and also upon the Joint. As seen there (+) oments act clockwise on the span and counterclockwise on the Joint. This is different from conventional beam sign convention where a (+) moment produces compression in the "upper" surface. "Joint" Fig, 411.92 ign Convention 411.508 Example Probleas The stiffness factor, K, ae used in Art. All.5a and subsequently, 1s a "rel- je" factor rather than a true one. It has the value EI/L for the far end fixed and .75 EI/L for the far end pinned. As such 1t applies only when there is no ex- ternal elastic restraint, k (in-lbs per radian)at the ends or at any joint, and when there is no axial load in the spans, In auch cases “correction factors" aust be applied to it, as shown in Art.All.14 for example. In’ general, to avoid inad- Vertent calculation errors it 4. the true value of the » SF, which te in in-1bs per radian ana’is t to SF = 4SCEI/L coefficient ana obtained from Fig. 411.47 (as "o") or calculated per Art. Ail.l3a. Doing this eliminates the need for introducing the correction factors otherwise neededs ‘The carry over factor, COF, can be ob- tained from Fig. A11.46 or calculated per Art. All.13a. In doing moment distribution calcu- lations, unleas the “far end” of a span ie pinned (or free) it 1s assumed to be fixed for determining the values of the etiftn and carry over factor 32.006 Fined Red uomante, seicraees ane Over Fastors for Beams Golldina of Constant Grose-section value: All.5aa) for larger values of L/j than are given in Fig. Al1.46, All.47° and 11.56 (for 28-4). Those values can be culated as follows for compre: of the SC and the COF 8¢ (far end pinne ) = 3/48 far en: : 6 a (fay gra fix Ae ) where & = 6(coseo4- 1)/(L/)) B= 301 - Soot (1/3) jane formulas 1 but the trigonometric functions are replaced with the hyperbo- lic functions, cosech and coth. Exten- sive tables of the 3¢ and COP values (to 6 significant figures) fron L/} = 0 to 2n for compression members and’ from 0 to 50 for tension members are in the book @esoribed in art. A18.27a, 421.150 secondary Beading tmente in FRIED nighe eants Art. A11.25 gives a procedure for deteraining the secondary bending moments in such trusses but no illustrative exam ple is provided. ‘The following illus- trates the procedure except that in step one one finds the relative rotation of each member using the method of virtual, work, step 2 19 omitted and in step 3 "ita relative rotation" 1 used 11 ‘PP ri jointe move, and therefore the members& SECONDARY BENDING MOMENTS IN TRUSSES Fotate relative to each other. These ro-| fixed end moments 1. Por the applied loads the resulting internal loads in the members are de- termined 2. Me Ao arbitrarily the "base member" from ‘tations of all other a oulated. 3. For each other meaber (one at a tine) a olockwise 1 Sn-1b moment in the form of couple loads (2/L) at ite ends ie applied and reacted with a counter clockwise gouple at the ends of the Teaber BG. The resulting loads mombers are then deter Pig. Al1.93 and note that (3 oF 4) meabers are losd- in this procedure, 4, Then for each loaded member the quan- tity SuL/AE is caleulated and the re- aulte eummed to obtain the relative rotation, 0, of the to which ‘the clockwise couple was applied. 5. Steps (3) and (4) are repeated for each of the remaining members to get ‘their relative rotation: ‘Table All.4 presents the basic data and Table All.5 summarizes the calou- ‘Teble Alla Fig. ALL.95 Relative Rotation Losds For each of the above members ti fixed end moments are calculated as FEMS -6E10/L( 28 - ) where 24 -c 18 obtained from Fig. A11,56 and socounts for the effect of the’axtal load, $, in the meaber. A positive (clockwise) relative rotation, &, produces negative (counterclookwi sa PEu's (per Example 2 sketch on p.All.2) hence the minus sign ir the formula. If @ member has one end pinned a fixed nd moment ocours only at the other end and 1s caloulated ai FEM= ~6£10(1 + COF)/L(28- «) where COF is from the pinned end to the fixed end. Table Ai1.5 the calculations for the value The fixed end coments at all truss Joints are nom known and the moment Atetripution procedure oan be carried out as illustrated in Fig. All.43 to obtain the final moments at the ends of each member. Table All.5 Calculation of Relative Rotation: fuem=| SL7AE, Gouple joer 4 2s : BC 0289] 593 BD 0577 | 1183 cD 1155 |-2367 2 TBs sol 8 Gop = -59T| NotSECONDARY BENDING MOMENTS IN TRUSSES je Effecke of Tharensed internal. ‘SNASD Pesan oh dasondery Bending Wonents Although the secondary bending mom- ente in the truss of Art. All.15a were relatively small, they can become quite large as the internal axial loads in the truss weabers increase. As the applied loading on the truss approaches the crit~ eal loading these moments will approach infinity. This 16 why the theoretical critical loading for a truss can n attained; bending failure will pre: The following example, be it] A 1s 8 rigid joint and ends B and ¢ are pinned. Assuming A to be a pinned joint only to determine the axial loada in AC and BC, they are as shown in the figure, With the axial loads know this simple truss can be analyzed for stability (buck2ing) ‘tro-span caluan with is Allustrated and di st By oue- ions, when P ‘Pap © 68072 and Pag = 50850. (1/3)4p = 30/-V5-% 1078B0TR =3.9816 S0qp = 2.5428 Pap = 4(01.5428)(5 x 10° )/30 =:1028500 (1/3)ac =17.3/V5 x 10°750850 = 1.7445 S0ag = «89059 SFag 2 4( .89059)(5 x 109/17.3 = 1029600 Hence, at Joint A ESF = 1029600-1028500% 0 1 101700 16 the buckling (critical) load| F If A were Ascumed to be al aa pin= ended coluin, since for pinned ends P; for AB ie 54831 and +866 63915 = S46s0, ffeate of applied| loading on eondary bending moments, assume that P=100000 and find the resulting moments, + Een Far at aay joint $ 0 there 16 ne reaiewanee te rotation TnFiniceml Sonsae sili eauae rotation tod faltanes Let AG be the base member, apply a 1 in- Yb couple on AB and react iv with an op- Posite couple on AC as in Fig. al1.95. Pigs 11.95 virtua York Lost % The relative rotation of member AB is then calculated as follo ‘Ssul/ As] x10 1380 | Brats =BBOS The only FEM ie at A since end 8 is pinned (and AC has no relative rotation). FEMAB= ~6(.5x10* )(-5603x10% )(1+ 2-325)/ 30(1.44) = 13391 in lbs Doing the moment distribution for the final moments at A ayy pe 19 aoe rou 4 gc .0068|-1.398 BF 1027109 |-933670 Bes 8 oF 30.877 (°9.877 ee ees tie vanax Kon 102635] zs Fig. 411,96 aooent ptatribution of since bending fail- ure would occur (if P were 101700 the ao- ments would be infinite). Therefs when the applied loading on a truss is near the critical loading bending failure will ocour (and prevent the eritical loading from being reached), due te the deflection of the truss joints under load, peating the abeve caloulatiens for successively snaller values of the app- lied loading, P, results in the final mo- Mente shown in Table All.6. Nete that as the applied loading decreases from near the critical loading the final moments decrease very rapidly at first. nen the loading decreases to the value which is ‘the critical value assuming pin Joints6 CURVED BEAMS. Table 411.6 Variation of Final Moments mith Applied Loading ‘Applied | % of | Final Loading | Gritical| woments Loading 101700 | 100.0 eo 100000] 96.4 | 145640 96000 | 94.5 50000 g1440 | 90.0 24638 81280} 80.0 12169 63315 | 62.3 6049. 40000 | _39.3 2380 (63325) the final moments are relatively small. This ie hy an assumed pin- Joint analyaie which ignores the secondary mon- ents, @ common precedure, is unlikely te reault in strength failures. Fatigue life might be a concern fer very light structures. If a trusa neaber ts subject te @ lateral loading, that causes an add- Ational type sf secondary moment te occur and additienal bean-column effects. Although this illustrative example uses only the simplest type of rigid jeint truss, the results would bo similar fora conventional type of truss. The analysis would, of cours, be more te- dious. More about trusses 1s in the beok mentioned in Art.A18.27a (and Fig.Al1.43}| 423,118 curved Benge For compact cross-sections the bean 1m net subject te flange instability ef- fects (Chapter 7). Therefore, with duc- tile materials, the ultimate bending strength can be calculated as discussed in Chapter C3, ignoring the curvature. hen the cross-section has relative: ly thin flanges, ae with an I, channel, Z etc. there is another effect of curv It causes the flange to bend a Fig. A13.22a, and therefore be. effective, resulting in higher bending stresses forthe beam. It also generates bending stresses in the flange: in a direction normal to the plane of the web woloh are a maxioun at the flange-to- curved Beam Section Bending’ Fig. At3,220 For syametrical cross-sections the oir- cumferential bending strese at any point on the section can be caloulated as fp = AD 2 GAS] where A= Area of crosa-seotion R=Radius of curvature at the centroidal axis Ms Applied coment, positive for tension in the outer fibera and vice-ver: y=diatance from centroidal ax1, being + outward from thie ax! and = if inwar. z width at distance y Table 413.4 presente formulas for Z for several cross-sections. Where a flange width 1s required, it is not the actual width, b, but rather an effective width, to the deflection shown in Fig. This can be caloulated bores ai Alsv2za. Derr = cyb where 0) 1 obtained from Fig. A13.22b. Derr 18 used for determining A and Z when flanges are pré Being less than the actual b, 1t results in high- er bending stre: Fig.A13.22b curved Beam Bending Coefficients Pint The transverse bending stress in the flange, fpe, aan De caloulated ae fot = Cafy muere Cp 4s obtained fron Fig.A13.220 and fp fa the stress calculated previ- ously uaing C1. Again, thie diecuseion applies only to sysnetrical cros: ations ae in Fig. Al3.22 When weight 18 important and rele thin flanges reault in high oan be reduced by using thin, closely spaced, machined in place "bulkheads" between the flanges, or Beneath a T-member's flange. This reduaeCURVED BEANS, the flange deflection and therefore re reduces the beam bending stresses and the flange's trangverse bending etre: uns fortunately, there 18 apparently no de. sign criteria for this, ao one aust rely on judgement and tei to, both, lim it and ultimate load adequacy. Unaymmetrical oross-sectiona should be avoided in fittings and hooke with e curvature since there are no fora or data for predicting bending atr in such flanged members (teste required ). Finally, much curvature with flanged ‘The: 6 03.28 and stiffeners are needed there to prevent crushing or buckling. ‘The machined bulkheads also do thie in in the case discussed for gachined iteus. Equation 16(b) applies only when the axial stress is in the elastic range and when the cross-section of the column re mains stable (1.¢., no local buckling oo- cure before Por ta attained). hen th conditions do not exist the critical load ‘than predioted by Eq. 16(b), jd in Art. 418.88 428.60 Iapertect columns, tangeat hips Add the following at the end of art. 18,8, Since Ey (in Eq. 16d, 30 and 34) ‘Ae the compressive stress in- ‘above the proportional limit (Ey slope of the stress-strain being th curve), @ successive triale solution for Por 18 indicated. That ie, Ft must be the value corresponding to'Ser.. ‘Tae suo cessive trials oan be avoided if a colum| curve ie available ae in Fig. A16.11, en-| tering with L/r and reading Gor on the ordinate. If a column curve fe not ave ailable the procedure illustrated in Art. 2-10 can be used. Equations 16b, 30 and 34 also do not| apply, nor does the column curve, if part of the cross-section t@ thin enough to have @ local buckling stress which ie eualler than Gor as predicted by the Dove equations. In this cage a special column curve which accounts for this lo- cal instability must be constructed and used ap shown in Art. C7.25 and C216, TORSIONAL BUCKLING 7 Table A13,4 Some Formulas for z “Gy ot Berth fp te donne ee a it T see hey soa ih ?) eae ; : uJ i Se 2 [bimttecdeirns Wes teats 428.00 Torsional Buckling The previous discussions are for the conventional form of general instability involving only a bent (buckled) shape, also referred to as bending buckling. There is another form of general instabi- lity which oan result in amaller values of Por . This form of buckling involves a twisting of the colunn (even tno there is no applied twisting wouent) and ie called “torsional buckling". Depend- upon the cross-sectional enape, the buck- led shape aay be either a pure twisting or @ combination of twisting and bending about one or both axes. It ocours for open crose-seotions having thin elements and 46 usually gore critical than bend- ing buckling in the short to gediun range column range. Toraional buokling is dis: cussed further in Art. 07,31 and in de- tail in the book described in Art.Al6.2788 RIB CRUSHING LOADS. The broad coverage (180 pages) of coluan design referred to in Art. 18.27 ag being in an originally planned “Volume| 2" was not included when 1t was decided to publish only the current single volune| rather than two volumes because of space requirements. However, this material 1s available in the book “Engineering Column| Analysia" by W.F. MoComba, Datatee , P.0. Box 763576, Dallas, TX 75376-3576. ‘Top- ies include columns, beam-columns, truss meabers, Dents, arches, torsional buck- ling, local buckling, crippling, buckling of shells and members on elastic or oth- erwise sagging supports. Both, unifora and varying section menbers are included. 428.260 uchantcal_ and Physical Proper ees*oe "tome Aurorare waterfels ‘The last paragrapbe of Art. A18.26 refer to @ planned Voluze 2 which was finally included as Parte B, ¢ and D of the textbook. Therefore, it 1s Chapters Bl and B2 of the textbook to which the reference is made. 429-23 crushing Loade on Ribs Dus to Wing Bending. chapter a9 rovides methods for de-| termining the stress in the flang shear webs, stringers and skin panels of the wing structure. Chapter 4.21 discus-| sea the shear loada and stresses in wing ribs. Neither of these chapters discus: sea the crushing loade on the rib webs at| a rib-stringer Joint which are caused by the bending of the wing or of any box besa having stringers supported by ribs. Since the ribs provide simple eupport for| the stringers in compression, if a rib fails locally its simple support for the wringer will vanish and tip stringer 111) fail as a column. Mnen the rib web is ‘thin mith no local reinforcement and the crushing loads are large such failure can| occur. The manner in which these crush- ing loads arise 1s as follows. Fig.A19.44 showa a front view of a wing in ite bent form (greatly exaggera- ted), the upper stringers being in com pression, the lower ones being in ten- sion and'the load line froa rib to rib shown by the broken line. Due to the bonding radius of curvature, R, the angle| @ will be @=L/R, @ very small’ angle since R is very large in a practical win, structure. The load, P, in any stringer, located at a rib will have components of, FACTORS OF SAFETY acted by the rib load, Q, in the amount @ = 2Pein(@/2) = Pe or, Q= PLR Since R = EI/i Q 5 PLW/EI If Ly and Lg are different rib spacings Q = P(Ly + Lg) M/2er Mnere Q is the crushing load on the rib web at the stringer, Mis the bending mo- Went on the wing at the rib station and EI te that of the wing at the rib station. Therefore, each stringer will need a clip to the rib (or ivs equivalent) «hich can pasa the load Q into the rib web, and if, the load ie large enough to crush the rib the clip will also need to be extended and fastened ao as to serve as a stiffen- for the rib, to prevent crushing of the rib web. C1.A36 Feotore of safety and uargine of Safety von These items are discussed in article 4.2 and in Cl.13 through C1.15, but they do not provide specific assoclated num- bers for factors of safety or aethods for calculating the margin of eafety for nem bers under many combined load system The purpose of the following digcussion is to do that, Eactors of Safety Ultimate loads, also called design loads, are obtained by multiplying the Limit'loads (the actual or expected loads) by a factor of safety. For pilo- Ved aerospace vehicles the factor of sat ty le usually 1.5. For miasiles, which are not piloted, the factor of safety 1s Ae usually 1.25 except for any load con- dition where the safety of people is in- volved where it is 1.5 (e.g., for the ej= ection loads from a carrier airplane). Naval airplanes are designed for a very fast sinking speed, about 26 feet per load parallel to the rib which are re- cond, so no safety factor 1s applied toMARGINS OF SAFETY ‘these loads. However, the landing gear guat continue to function after such a landing and the wajor attach fittings (landing gear to wing or fuselage, wing to fuselage and major fuselage section splices) must chow a aargin of safety of £25 for thie landing condition. In gen- eral, factors of safety are specified by the procuring agency. Margins of Safety The aargin of safety for a structur- al member subject to a single load or stress 1s calculated a Allowable Load MeS= TT ei and gust tw zero or more. However, in some cases it must have a specified posi- tive value as aentioned for Naval air- planes and .15 for fittings. This margin| ie also required if addit{onal fitting or| ing factors have been specified. For| shear Jointe a ainiaum wargin of safety of «15°16 required and for tension jointe| At de .50. Also for these jointa there Bust be no yielding at limit load. For shear Joint attachments if the bearing yield strength ts less than 2/3 of the ultimate bearing strength, the bearing allowable strength 1s taken as 1.5 times the bearing yield strength and the joint Ae said to be "yield critical". For riv- r and tension Joint allowable Chapter Dl. Also see Appendix A. The previous simple formula for the U.S. gives the deciual fraction by which the load may be increased and still have a M.S. of zero. For e: Af 8.2.20) the applied loads may ised by a factor of 1.20. However, the simple for-| aula applies only when ail of the follow conditions exist vo @ single| type of loading, hot to several types such ae shear plus compression etc, The internal loads vary linearly with applied loads. All applied loads are variable, 1.6 none are fixed in magnitude such as a conatant pressurization. ‘the conditions are discussed ae follons| For exaaple a bolt may be subject to shear, bending and tension simultan- gously. In such cases so-called “interaction equations", derived from tests rather than theory, are used to show structural adequacy. A typical in- Yeraction equation os of the form RU + R34 85+ ----- R= 1.0 Were R1=f4/F), Ro=fo/Po, Ry = f5/F5 and Baz fo/Fa, } deing*tne“ultinate atrede (Or load)"and F being the allowable (or load). Ris called the ratio". When the left side of the equation 18 1.0 the M.S. 1e zero. Woen it 1s less than 1.0 the M.S. is po: itive but undefined and wnen it is more than 1.0 the M.S. i negative but unde- fined. Interaction Equatio +333 + 667 = .778 (<1.0) Hence the M.S. 16 positive but undefined. There are two cases for which the K.S. ean be calculated directly using the in- teraction stress ratios, R, as follows. 4) When all exponents (a, b, c, n) have values of 1.0 and/or 2.0: In this c ee MS° SRy 4 VERE aaRe ~ ++? b) When all exponents have the same value, n, in whieh ease MAS. = QR REG seer Ryn w2+0 For all other cases the M.S. 18 found by using plot of tne applicable interaction equation (discussed later) or by succe sive trial calculations. The latter 18 done by finding, by successive trials, by what common factor, A, all of the loads must be multiplied to satisfy the interaction equation. The M.S. te then A= 1.0. Or, when (anf (ang? (ars f+ (argh = 1.0 M8. = A= 10 Example: For tne previous example Ax 333+ (Ax .667P = 1.019 MARGINS OP SAFETY, INTERACTION EQUATIONS After several trials the equation is sat-| uonoc! Linder (Various) Table 08.1 defied when A= 1.271, 90 Gurved_sheet 095 M.8. = A - 1.0 = 1,171-1,0=.173/ Compression or Tension and Shear Since all exponent are 1.0 and/or 2.0 | gtiffened cylinder 9.41 the M.S. oan also be calculated Shear and Compression by formula as . Torsion and Bending 09.13 M.8,2 2 -1,02 a 3334 Vesa ee x OTE Shear and Bending 09613 When using this formula with other more Jengthy interaovion equations the paren- | Stringer Friaary and 011.338 @ around the termER) gust be noted | Secondary stre: tnd useds ER te the aumlof ait strege. ratios having the exponent 1.0 and z! Ring (Frame) Primary and 011.338 the sus of Ulose having the exponent £.0:| Secenuary Sereases (for Stringer systen) hen plots of interaction equations are available the M.3. oan be determined | Longeron Prigary and Secondary 11.368. by @ graphical construction. This is 11-| stre, lustrated in Art. C4.2ka in the textbook using Fig. C4.36. However, graphical so-| Ring (Frame) Primary and 11.360 lutions are not needed when the prior formulas apply, and in many other oa: ‘the successive trials procedure may be easier with calculators or when the inte- raction curves are not available Listing of Interaction Equation: Numerous interaction equations ap- rin various parts of the book per fhe following Iiating. 03.13 Bending and Shi 03.12 Bending and Bending 03.8 ‘Tuping Bending and Compression chee Bending and Tension ch. 238 Bending and Torsion ohm Compression, Bending and Torsion Ch. 24a, Bending and Shear 4.25 Compression, Bending , shear ch. 26 and Torsion on and Torsion ch, 27 Tat, Bending and Compression 5.9 Bending and shear 5.10 Shear and Tension or Compression (5.12 Compression, Bending and shear 5.12 Secondary stro: (tor Longeron system) 2. There are cases where the internal loads in a member do not vary linearly with the applied loads. One example is ‘the beam-column where the bending mom- ent inort plie es Taster than does the sp- loading. Another exanp: in etringera due t id action. For such cai M.S; ae previously caloulated would be an “apparent” M.S., not a true M.S, Whether the apparent M.S. 18 positive a8 follow ive trials find the com gon factor, A, by which the applied Joade (which produce the int in the members) aust all be gultiplied to give a calculated M.S, of zero. The true M.S. ta then A- 1.0, Por a beam golumn the applied loada are the axial load and the transverse loads. Multi- plying these by any factor, F, will generate a bending moment in the beam ooluan which increases faster than the eonmon factor, F. A te the value of F for which the calculated M.S. 1a zero, and then the true M.S. 1a A= 1. ‘Tie procedure applies for any inter- action equation that. may be applicable. Note that this requires a greater ef- fort than that in (1) previously. This ia because increasing the applied loads requires another analysis to determin the internal loads, since they do notDEALING WITH TOLERANCES, vary linearly with the applied loads. Thie ie important because mar- gine of safety are reported for struc- tural gembere and are use!to seo if the gembers can withetand an increase in the applied loads. If the applied Joada increased by a factor of, aay, 1,20'and the report showed a XS. of, say, 1420 1t would be Judged to be ac- ceptable as an incr But if thie happened to be an apparent K.S. the true M.S. would be smaller and the in- crease in applied loads would not acceptable. Hence, the true value ‘the M.S. should be in the report. not they should be “flagged” as being Apparent ones so that a proper eval- uation can be made when needed. The intemal loads in a member are due to the applied loads. Some~ times the applied loads consist of "fixed" loads which do not vary such as, for example, constant pres- urization loads. When such are pres-| ent the internal loads or stresses in the member due to them should not be gultiplied by the common factor, A, in (1) previously or by the factor, F, in (2) since they are constant. If auch were done the calculated M.3. conservative (too loads were "additive" and vice-versa Af they were "subtractive". Tnis alad| applies when the M.S, 16 calculated by| the formulae in (1a) and (1b), and woen the M.S. 18 determined graphical. ly, which makes these calculations no applicable for for such cases (the successive trials procedure is then needed). 3 1,13 Dealing with Tolerances When calculating margins of safety for @ structural member nominal (mean) @imensions are used to determine the rand its allowable Ideally, the M.S. 18 zero, gener- ally, manufacturing tolerance which accompan- ies each dimension such as, for example However, all drawinge specify the +03" and these are considered as follow) In aerospace structures 1t 18 com- mon practice to aoneider only the two worst tolerances affecting any dimension and to compute the reduced margin of safety based on the reduced diuensions. Tois will give a negative M.S. when the nominal M.S. 16 zero. Such negative warging are acceptable if they do not COMBINED STRESS EQUATIONS a or ~.25 for redundant mombers. ‘These are arbitrary liaits, some companies allowing ore negative values such as -.19 and =.39 respectively. Such negative margins esused by tolerances are acceptable since the probability is quite low that, eigul- taneously, the material will have minigun properties, the tolerances will be as large as alloned, the loading condition will be achieved and the internal loads in the members will be as large as pre- dicted. If the M.S. had to be zero or more based on euch @inimum, rather than nominal,dimensions coneiderable weight would be added to the structure. When dimensioning structural members care should be used to prevent the bulld-up of large tolerances affecting the final din- eneion. Unacceptable tolerance effects are most likely to occur when digension- ing small machined or cast protuberances or holes, where email internal corner However, the above should in no way be construed as sanctioning negative aar- gine based on nominal diueneions or en- dorsing the salvage of parte having less strength than required por tne drawing. 62.60 Combined streas Equations For practical calculation purposes the following summary and exaaple problen are helpful. Fig.Cl.8a@) shows the posit direction of known or given applied Fig. (b) shows the resulting % endo , on any plane @ (posi- tive directions shown), which can be calculated as follons (Gy. + dy 1# HC, ~dy 00828 + Tysin2e [= Hoy,-dy Jain 26 - Fy coa2e 1. The principle stresses, culated as follows. on = HOyt dy 18 VRS, dy Prag The plane for the largest principle stress, Op, 1s meaaured froa the plane of the larger of dy ordy and te calculated as follows. » are cal- Op = tarctan(2%y/(d, -dy )) The plane of the aller principle stress 1s 90° amay from this plane. 3+ The maxioum ahear etr exceed -.15 for single load path meabers + Toaxe 18 calculated ag folewes t” TSX Taax = Vile - oy + Tae12 COMBINED STRESSES, Tote plane on entch Tyq te located Le farctan( (oy -o5 /-eay) ies 4 % ty eI, Fig. Chee Exaople Problem (>) 500 Fz000 For the stresses shown in the sketch above (no! ty 18 compressive, hence nogative) what areg and on a plane at @ = 60°? What are cq, Gy Tnax and @| G, =4 (10000 - 2000)+ }(10000+ 2000 )cosi26| + 4500sinlat = 4897 (10000 + 2000)sinl2d"=4500c08 120° = Tas #(10000 - 2000) + $(10000+2000FFu5 00" | = 11509 and 3500. = tarctan(2 x 4500/(10000 + 2000)) Taax = #110000 + 2000}"+ A50G = 7500 @y = tarctan( 10000 + 2000/-2( 4500 % " on Gy 1a always 45° away from ep If any of the above calculated values had deen negative they would be acting or lo cated in a direction opposite to that shown in Fig.c-1Ba. Usually one is interested only in deter- FREE-ENDED COLUMNS mining the value of the largest a and of the largest Tmax: G2sla Methods of Colum Fatlure. ua Tauatione ‘The last paragraph 19 extended to include the following. Predicting fail- ure due to local instability requires that tne column curve be reconstructed in the short to intermediate ranges. Tals Feconstruction is discusaed and illus~ trated in Art. C7.25 through C7.27 62.20 Free-Mded colume Fig.G2.2 shows only a special case of the free-ended column, one end being fully fixed and the load remaining paral- el to the axis of the member in it straight (unbuckled) form. In general, however, the load, P, may be directed either to or from’a given point, "o", as shown in Fig.2.2¢ a= aflasd) (Ls Le)/ty Fig, 02:28 Pree-Rded Goluma, Fixed tn As snown in Ref.4 (Art.Al8.27a), the buckling load 18 defined by the tran cendental equation (L/j)etnl/y = 2 = (1/n)" of m (Fig.C2.28), Por ye trials. ‘That is, @ value for P, calculates thie’ in the equation, Pop the equation will be satin However, thia effort can be great- ly reduced by using Fig.c2.2b, entering the figure with mand obtaining (1/4 )on. Pop 18 then” calculated Por = (1/3 )ap( EI/1? ) jote that shen a orl,zoo, a= 1.0 and cen, te Caryie) mona and Por = q?(EI/L?); whe and Per = 0. Toat ie, ae a°decteasca Pop increase, but aa Ly decreases Por ) tgeRtermate fore te pant) = =e/3FABE-ENDED COLUMNS WITH ELASTIC END RESTRAINT . — fe an as -— ght te Ss lf | Figs 62.2 Wi)ge ¥ dof being fully fixea, the ond may be elastically restrained as shown by the “torsion spring”, K, in Fig. 2.20, which has a value in in-1be/radien Tole results in a smaller buckling load than when fully fixed (k=e), (a) Load advected eee k) EIS k aoe wey 7" tb) Lond directed from "9" Be (Lt lol/ty Pree-inded colvan Having a9 Hisetiosldy Restrained Bnd Fig. 2. For this case, as suown in Ref.4, the Duckling load’1s given by tne equation Pep t8 found by successive trial calcu- 13 lations as follows. For a given k and a one assumes a value of P, calculates J and uses these values in the equation. When P= Por the equation te satisfied. For the spectal case whore a or Ly to infinity, 20 that a= 1.0, Pop can be e- termined directly by entering Fig. C2.44 witn KL/El, obtaining the value of Cand caloulating Pop 88 Pop = Om EI/I Note tnat when k= (fully fixed) ¢=.25 as given in Fig.c2.2. It must be remembered that in all coluan calculations E is Et, which de- creases as the compressive stress ox- ceeds the proportional limit. Also, when the coluan has an unstable (thin) erose- section E is an “effective” modulus* To determine it one finds the buckling stress from the aodified column curve (Art.7.26) and then calculates 1t aa ef = Fon L/ph /ar? For free-ended columns Fig.02+17 18 ap plicable only when a = 1.0 If the column has lateral Lo: initially bent eam-col! For @ column as in Fig.C2-2a(a) assuge that_a = 30", L = 30", E = 10.5 x 10¢ (1075-76 Extrusion, p.Bl.11), I = «191, A > 150 sq in and & stable cross-section Wnat is Per? & = 30/(30 + 30) for m= 45, (L/J)or: Por = 2,05°(10-5210° }( 192)/30° = 9183 For = 7183/.50 = 18366 150. Per Fig.c2.2b 03. Therefore, Since Fer< Prop. Limit, Ey=E as assused. Example Problem 2 Repeat Example 1 assuming the fixed end ie replaced with an elastic restraint of & = 50000, ae in Fig.c2.2c. This must be solved by’ succeseive trial: Trial 1: Aseume Pop = .Sr® 51/417 = 4 Then) = VERS Oe Oe SEES = 21.35 and L/j = 1,405 i ‘Then 50000- 0, 0: =1.405 ctaTy (#0) =i 56849 ‘ee art, 2.1614 EFFECT OF SHEAR ON COLUMN BI After several gore trials it 1s found ‘hat when P 19 2622 the equation to st Leried, so thie 1a the critical load, follows. 3 = Vi0.5 x10"(. 191) 72688 L/} = 30/27.656 = 1.0848 25)(30. 50000 = =e HORAN oBnBT 2-47 27.656 which 1 essentially sero. So thie particular elastic restraint has reduced the critical load from 9148 for a fully fixed end to only 2622 lbs. 2.38 Te ert ot of shear on Dusxling stress Equations (1) and (2) of Art.c2.2 consider only the bending stiffness of columns. Wien a bean bends because of applied transverse loads and the reoult- ing bending moments, the usual def tion formulas consider only the bending moment. There 1s, nowever, an addition al deflection due'to shear. For example, a simply supported bean of uniform ET having @ load, Q, at mid-span will have a| maxinum deflection given by y= QU /sBEI + ngl/saa ‘Tne first term 1s due to bending and the second term is due to shear, where n 1 a| form factor which depends upon th of the cross-section. Thia te us negligible, but ae n increases due to @ thin or perforated or otherwise more flexible web it can become significant. For @ column there te no applied transverse load, but a shear load 18 gen-| erated by the axial load as the column takes on a bent (buckled) shape, aa shown in Pig,C2,24 heart Fig. 02.24 Generation of Sear in « column At.any station the bending aoment te M= Py and the shear in the aember te V = al/ax = Pay/ax where dy/dx 1a the slope of the bent enape. Hence, for the uniform simply upported colucn the shear varies from a waxinun at the ends to zero at the load used in UCKLING.. MULTISPAN COLUMNS middle where dy/dx, the elope, 18 zero. Consequently there is arin the aen- ber which, in effect, aakea the ooluan more flexible and tnereby reduces the buckling load or stresi As discussed in Ref.5, for unifora coluans the buckling load, conaidering shear, 1 EA TITIAN 2n7AC Fer © where Por is the buckling load ignoring sh ry n 1s the form factor for the ction, A ta the cross-sectional and @ ie’tne shear modulus of ela theity (@ = £/2(1+ p)) where yp te Pois- son's ratio. The values of n for several etions are shown in Table C2.2- Table 02.2 Cross-Section Form Factors Cross-Section Rectangle T.200 solid Circle 1100 thin Round Tube 2.000 [i-Beam or Rectangular Tube| Area/Web Areal n can be calculated for other cros: sections as ne (Mt) {a ano @ A 18 cross-sectional area, Q is the etatic moment of area beyond dA about the neutral axis and b is the width at the neutral axis. For columns having lat- ‘iced etruts (trussed columne or "batten" plati Ref. 5 r stiffne: that the column buckling load te signifi- cantly reduced.* 62.30 Multtepan columns Unfortunately, this and aost other textbooks do not disouss coluans having Bore than one span. Such columns are e ily checked for stability by applying a 1 in-1b couple at any “Joint™ (support) not having full fixity and carrying out the moment distribution procedure as di cussed and illustrated 1n Art. Alls13 ~ All,14 and its example problem: The larger the axial load slower" the convergence. If they diverge at any Joint the column is un- stable. Te critical load 19 that for whioh they do neither, found only by su0- cessive trial analyses, varying the axial ch Moment distribution.MULTISPAN COLUMNS: There are some helpful techniques for doing this in Ref. 4, including some “quick checks" for detecting instability sometimes without the ubove procedure. Ref. 4 also contains numerous example problems for aultispan columna including those on elastic or otherwise deflecting supports, those having free-ended meabers and those having varying loads and El Values within the apang The alternative to doing the moment @istribution analysis 18 to conservative- Ly assume that each epan 1s simply sup- ported and check each span individually. This 1s accurate only when all spans are identical, the two ends are simply sup- ported and the axial load is constant. "Quick Check" Ingtability Criteria 1) The following can be done when each span is uniform and the axial load does not vary along the span (j is constant). If any span has a value of L/} greater than shown in Table C2.3 the column is known to be unstable. If less than these values it aay or may not be unstable, but a mouent die-| tribution analysis is required. Table C2.3 Instability criteria’ * Ty 3 Suport Tapa tne end elastically aw restrained, one end free Both ends eimply aupported One end simply supported, one elastically restrained| Both ends elastically restrained AND an Elastic restraint 1s provided by an adjacent span in a multiepan column or by @ torsion spring as in Fig.C2.2¢ If at any joint the aum of the stiff- ness factors, ESF, 12 negative the column 1s unstable (see the footnote for art. All.15b). ‘Tmo-Span Column The following applies not only to a ‘wo-span column but to any number of spans meeting at a common Joint (support}| If the outer ends of the members aro ei- ther siaply supported or fully fixed erthee| (no elastic supports) the coluan is known| to be stable if at the eommon joint ESF is positive. If it is negative the col- umn is known to be unstable. No moment distribution 1s,therefore, required go the analyeis is quite simple. The criti-| 2) 45 cal load 1 that for which ESF = 0, found by sugcesaive trial values of ‘the axial load.* Example Problem 2 Ie the three-span coluan shown in the sketch below stable or unstable? ee The analysis 8 carried out as snown, ap- plying a 1 in=lb couple to the Joint at B. Note tnat the SC, SF and CoF are on- itted at the ends, A’and D, since they are simply supported and no initial mom ents (FEM's) are present there. As the Moment distribution process shows, the successive carry over momenta (COM's) at all Joints are decreasing. Therefore the coluan 1s known to be stable. 85 Actually, one should aleays calculate the compressive stross, P/A, for each span and use the corresponding value of f% (or Bete) in calculating the value of J. Al- s0, Ref. 4 contains extensive tables giv- tae values of SC and COF to aix.sianifi- cant figures (better than Fig.All.%-47). Example Problea 2 Ie the two-span coluan shown in the sketch below stable or unstable? bw 4 —__ before for the span column, it is seen that SSF at the common Joint, B, 1s negative. There fore, the column'ts inown to be unstable. By successive trials 1t 1s found that when the axial load 1s 93040 1baZisr = 0, so that 1s the buckling load. Tan AF ertterta (1) steve is aes + The Listed 1/3 vatea fare tor Po For = ontaritt If the axial load were 168750 to 219600 © Por 4 tree-ended column when a = 1 (Pig.c2, 2a) the S.P. ie =Pitant/316 STEPPED COLUMNS ZF would be positive, indicating stabi1-| For (= Per/A) should be calculated for ity, but this would not be applicable bi used cause L/J for span BC would exceed the waximun (4.49+) allowed in Table ¢2.3. Taie 1s why the Table C2.3 criteria snould always be checked first, before proceeding with any other analy. In auamary, the moment distribution procedure can be used to check any multi- span column for stability. But before doing thie instability can be detected quickly per the criteria of Table C2.3 and if at any Joint ESF 19 negative. Toon, for any two-span colunn if TSF 1s positive at the common joint the column ia known to be stable. Uniform Two-Span Coluan Foraula For the special case of a uniform member on three sinple supports an ap- proximate formula for the buckling load ie Per = of81(2 ~ b/a)/at whore a is the longer apan and b is the shorter span. When a= b (=L/2) the for- aula is exact. When D becomes zero (1: L) the error ia only 2.44% id is conservative (one end becomes The formula can be verified numerically by caloulations for a two-span column previously illustrated, or theoretically ae in the book "Theory of Limit Design" by J.A. Van Den Broek (J. Wiley and Sone)| €2.60 Stepped Colum are a epecial case for which formulas (buckling equations) are avail- din Ref. 7, Formulas are presented for the cages of simply supported members having one, two and thr Por more than three steps the formulas become too lengthy for| practical useage, so a simple tabular calculation form’is presented with an ex- ample problea show. re transcenden-| tal in form, a successive trials proce dure 1s necessary to determine the criti- eal load. This consiate of assuming a value for P (and Ey) ,oaleulating any Associated parameters’ and using these v: lues in the equations. When the assumed P te Pop the equation will be satisfied. for Ey (Fig.¢2.16) or Egee (art.c2,16) correspond to Fer, which they auat.® Sin 4 19 possible for more than of P to satisfy the equations (quite different values), 1t 19 best to Use a longth-weignted avérage EI, calou- lgte a, corresponding value for Por as WEI/I? and use this as the initially as- sumed value for P. For the special cases of one-step and symmetrical two-step col- uma graphical plote are available in Fig. 02.21 and 02.22 for a direct deter- mination of Per. L -eant VB © wnere 9 = VEE Syametricel Coluan ‘Two-Ste) Tr ‘gyaaetrieal Coluan x tan(Le V2/EaTa/2) Veena ‘Three-step Column petrect sto 71g.62.20b Taree-step column Tapes, sasattons are eastiy programed for rep! SOLIEAOG ty computer or etitanle salsulators®STEPPED COLUMNS, NUMERICAL COLUMN ANALYSIS woere Bn=VP¥iala and f= Gain The equations are easily programmed for rapid evaluation using # suitable calculator (or computer). The derivation| of the equations is available in Ref. 7. Goluans Having More than Three Steps For the: 8 (and also for 2 or 3 ateps) the following tabular numerical procedure can be used, as illustrated in Table (2.4, It, too, 18 a auccesaly trials procedure (aucceasive table; umes 4 value for P (and E) as us: ed later and carries out the tabular calculations as shown. When the assumed P 1a Pop the value in Col. 6 for the last sognent, W, in| gnent N-1 multiplied by of segments can be used. The exaaple below shows the last of seve- ral auccesetve trials with different va lues for P, the last being 79200 lbs. As discussed previously, to start with a reasonable value for # a length- weighted average EI should be used and P calculated as sEI/L?. When Col. 6y Ae less than ~Col. Gy.) a larger value for P should be assumed for the next trial and vice-versa for 6y > -8y-1- petospee py Fig.c2.200 bata for Table c2.4 Exeaple Probiet Table 62.4 Stepped Column Analyeis oI {tat ore P= 79200 lbs, ‘As discussed for the formulas in Art, the proper values of E must be used for each segment 62.60 Mumerteal Column Analyeee casee discussed in 1 Fig.C2-21 through Except for tho Art.c2.1 through G2: heck for atability under a given load, Py age cnet EUG Sic be eek ashe) a thee 17 02.27 or other similar type data, a nun- erical analysis is required to determine the oritical load or stress. Art.c2.6a and G2.6b presented a numerical solution for the special case of etepped coluans on aimple supports. The following pre- sents @ procedure for any variation in shape and for simple and fixed supports. These procedures are in tabular form, ra- ther than as computer programa, since this gives a better understanding of what 1a being done. ‘The procedures are, of Course easily programmable for solution by computer. As before, the critical load is determined byauccessive trial: high means successive tables of caleu- lationa. Example problems illustrate the procedures.* The basic procedure uses the method gf discrete elastic weights (alao called “Mobr's Method" and the "Conjugate Bean Method") to replace the M/EI diagram and calculate deflections and 1s due to New= mark, Ref. 6. ‘The formulas for the el- Qetic melghta are discussed in Ref. 4, Simply Supported column, Referring to Fig.C2.20d and Table (2.5, the procedure 1s aa follows. 1. Divide the column into several equal length segments, S, at least five or six ate, but ten or more will Give @ more accurate value of Por+ Assume (sketch) an initial buckled (deflected) shape. Any initial shape will do (even two straignt linea), but the gore realistic it 13 the goon- er the effort mill be completed. At the station in (2) above which hae the largest deflection, y, let ite lection be taken as 1", and let the deflections at the other stations be Proportional to thie (per the aketah). Lot P be one 1b. Then at any station, By the bending goment will be Mn = Py, Yns 0 the values in (3) above are ef- Vered in Col. 2. Positive deflections are upward and poaltive bending som- nte produce compression in the upper surface, hence the ainua sign used in ‘ne headings for Col. 2 and 4, At each station enter the value of EI in Col. 3. If there 1a a “atep" in EI an adjustaent 1a made to EI at the vation nearest to the step, discussed Fy and ite EI is "flagged" with an risk to indicate thie sd justaont, * Baee (and ovter) tantes are saatty programmed rapid solucion ty conputer or multatls talents18. VARYING SECTION COLUMNS 6. The “equivalent concentrated elastic loads" ("elastic weights") are then calculated per the foraulas below the table for each atation and are entered| in Gol. 5. These formulas are dis- cussed in Ref. 4, 7. The tabular operations are then car- ried out as shown, and a range of Pep Values is obtained in Col. 10. If the axial load is below this range the oo-| lum ia stable, and if above it the column is unstable. An average value for Pop 19 calculated as shown, U4, 150 ibs, “Note that several checks are Gade to detect any errors made after the Col. 5 data are calculated. Co: lumn 9 (and 8) defines a new shape which will be different from that as- sumed in Col. 2. 8. Using the deflections tn col. 9 or in Gol. 8, let the larges.of these be one| inch and get the others by dividing thetr deflections by the largest de- flection. Hnter these in a second table's Gol. 2 and complete the table. 9. Repeat (8) as needed until the range of Pop Values is quite small oan then be taken ae the average of the values in Gol. 10. 10.If at any station For (=Por/A) 18 sige nificantly above the propertional lis it stress (or above any local buckling stress) the value of Et (or ferr) used in the term EI gust correspond to For (or to the local buckling stress), which can require gore successive trial tabular caloulatione (art. C2.16), Tables ¢2.5 through 2.7 illustrate the procedure. ‘Tree or four tables are Usually sufficient. For the example shown Por = 45000 Ibe per Table 02.7, (actually between 44200 and 45900 lbs). If the applied load were less than 39300 the column would be knom to be atable after only the first table. or, if wore than 47600 it would be known to be unstable, without further effort. Using ten segnents instead of five results in a more accurate value of Por = 4200 Lbs. values of Et and of Eorr at any strees level are obtained as discussed in Art.C2.16+ Ady t for a Step in EI Referring to Fig.C2.20e, this 1s done to keep the tabular operations sinple. a longer table te n NUMERICAL COLUMN ALALYSIS Figsc2.200 Adjuataent for « step decaetry Let the station nearest to the step be ‘n" and that on the other elde of the step be "a", In x Ealy eine fale) R= Eley - Enln (Enla)err = EnIn + (1-2a/3)R (Entn)ore in Column 3 for station Note thal if the etep is aiteay between n and @, (EnIn)ert = Enin so no adjustment 18 necessary. I? a =°0 then (Enln Jerr = Elay- Examp: For the column in Fig.¢2.20d station 3 18 nearest to the atep, which 1s be- ‘tween stations 3 and 4, a = 3.8" and S= 9.4". Therefore, EI at station 3 is adjusted as follows where n=3 and a in'the above formulas. £= 10.5 x 10". Calculate Elay Calculate Calculate Use len =3ande 2 ery 275 x10" 3 Ae R = .75(10”) = 1.5(10") E3I Serr 145(107) + (1-2x3.8/9.4)(~.75)(107) 26 x 10” 5+ Therefore 1.36 x 107 18 entered in Table C2.5 for EI at station 3. Golumn Fixed at tt Left End Tae following procedure is used for @ column fully fixed at it left end. if the right end te fixed, rotate the colunn 80 that it becomes the left end. The pro- cedure is the aame as before except that ded to account for te the left end. @ shown as Table c2.8. Une fixed end moment The table The data in the table are for the aber shown in Fig.c2,.20d except that the Lert end ie fully fixed. ‘Therefore,NUMERICAL COLUMN ANALYSIS 19 seas Det eoged su BE ee oe APN [>-—20. 1s—4 Fig.d2,204 Staply Supported Varying-Section Goluan eee pees Wen 5 es, res lan Hastie Wotghte and Reactions N= No. of Segments 5 Lengtn of Segment Table 2.5 Determination of Por (First Trial) She n36 M Mom. ot | Unit | unt | tre su. a | ae ‘aie | Sdpe | at | bat Fon a | me | re | -O/@ @@O | %-2@s | 7D. | Oxsv2 | -Q/@ Pe [x | aw zie zi | xi [a = z = z cr z z z Tar Pe la-eiatoy as rar Tar | ory CO Tae z eaiesleoe te ei er | ar Eo Tz z re] 6 73a ma ar EBD TE 7 ao ie a ee] Te TB TE z z ed Ta az z “See formulas below . a 46, oo : = 4150 **Adjusted for step cray™ “4 136.27 a ng -2@.. 1627. as Rye t@= Rye Os mas6 -Qy.1+@x= Fr EQUIV. CONC. EL. LOADS OF coL.@ On -0 2 At Ends, Sta. O48, Qo = 3.5x@o +3.9%Q@ - .5xQ; On + 35% Oy + 3.0% Oy - 52 Ox True elope =Unit slope x s/12 1) Av omer Stations, 8 Slope at left ond = Ry x 5/12 @a= @e1*19%@+ Opa Slope at right end = Ry ¥ S//2 Beam Sign Convention (+) Loads act upward (+) Reactions act downward (+) Deflections are upward (+) Slope ts up to the right (+) M puts top in compression (+) Axial load 1s compression The development of the calculation table ie shown tn Ref. 420 NUMERICAL COLUMN ANALYSIS Table C2.6 Determination of For (Second Trial) SOLO lo © |o], oO 16 iene linear ee cntae Poa Sa, | Me EL El, Loads Loads Slope Det. 2 | me | pe Out. | OO | L-2O, | 20, | Oxs72 | -O/@ tote te tet ts = 7 np oee me : at ore wef as pare et ee ee on * Use applicable formula z= 44.80 137.04 Bhat os ae Bia 2Q ae te = 2Q. s31.06 Re t= ge = 2. se necks mhen sketching the assumed initial buck- ied shape its slope at the left end should| Same as before =| i280 be sero, to be realistic. The unit va- lues (due to the left end fixity) in oo ae ed pa — 12 »—| a. got “tere oa Pend Yaying-oection Golan p pot ts Atm ee Elastic Weights and Reactions Fig. Table 2.8 Determination of Por (First Trial) ®|O@lTOToOoToOTel[o/e[sle, ole swat | aye] Co |e, Cone [ temo item ot] oy | Toad | ome | var Kade [vena |G) NG) | ace | donee [stone [be | BT | Pom + | pe | ome} buts |-O/G}-O/YOu tw |Ou ee |OK- DlOH-DIO- nfO- G|-2@|20 |@x5-/O a be ae be peter oee chor ar mess tar aor ore re er Sane a ae os ere or te ae = Te eat =r | aa or fs ear A ststr tt} a} oar or 7 z Te aust ae ganes Ea same ther. os Ponay, = ae mS HID +1O+1@: Ons One: ert f 7 (and 8) te omteulated aitrerentiy for this case (about the right end).NUMERICAL COLUMN ANALYSIS, INITIALLY BENT COLUMNS aa Table 02.9 Determination of For (Secona Trial) {fo efotetTeotToto @lelel ele sia. | ae» [orgy] at | aver | aspen] 4; Sone] Ee; Cone = Sie mes meer peer niet Lei Teor a is Sor ss ae Sareea ao Sar rae ma rer ae pe =H San ne ote ore re arf et ae aoe oa pe ee a mene} —} a ao 5 rere BH apm aw ne a Make eame checks as before Peay, * $6.025¢ 3,6, 8, and 10 do not eaange in successive tables. For any station, n, the entry in coluan 3 ie 1 - n/N where N is tne number of segments used. Hence, these decrease uniferaly from i.0 for station 0 to zero for station N. Columns 1 through 10 are completed, then My is, culated as shown and the reat of the table 1s coapleted. For the meaber in Fig. C2.20f using five segnents Por le found to be 66025 after two tables are completed. A third table would reduce the spread in the Col. 16 values for Pop. Coluan Ft: Bach md The procedure is the same as for the| previous left end fixed case except that additional columns are needed to account for the fixed right end. The initial as- sumed buckled shape should be aketched in| with zero slope at each end to be realie- tic. Table C2.10 shows the procedure, ‘The values entered in Col,3,4,7,8,10 11,134 14 do not change in successive tablée. For any station, n, tne entry in| Golumn 4 is n/N, where Nis’ the number of segments used. Hence, the uniforaly from tero at station 0} t station N. Columns 1 through "9 completed; then My and ig are cal- culated and the reat of the table is com- pleted. Using five segmenta the buckling| load, Por, 18 found to be 172250 lbs after tno tables are completed. lore tablea would give more accuracy. The euggeeted checks should be made to detect any errors made after Column 11. These tables, with a ainor change, are also letermine the bending moments in bean-columns with a varying EI| 38 discussed in Art. C3.18. The value of| for the aeaber aa a column can also be| calculated as shown in Table C3.6%. Qther Uses of the Table. For a complete discussion of the tabular method development, the "equiv- concentrated ic loads" and addition- al examples seo Ref. 4 (described in Art. A18.27a) which also diacusses the follow ing application 1. Axial loads between the ends 2. A column conatating of two pieces ®onnected by a torsion spring (which could also be a splice). Multiepan columns having one or gore spans of @ varying EL Free-ended columns having a varying EI The numerical determination of carry- over factors and stiffness factors Multispan columns on elastic supports which may be present as either dis- crete isolated supports or another beam or colusn. 2.60 Columns Having an Initial Beat saps When a column has an initially bent shape as in Fig. C2.40 the axial load will generate a bending moment along th colum in the amount M=Py, where y is the deflection of the bent shape at any station. The coluan may be uniform or it may have @ varying £1. 0S ve.c2,00 Gstemn Vien Initial bent ape Therefore, the column ia actually a Deam-column, as discussed in Art.C3.18, Beam-Columns Having an Initially Bent Shape" except that there are no trane-22 NUMERICAL COLUMN ANALYSIS Anavaed Dertecged fess no = bP fy F1g.02.205 e Bade FLaed Yarying-Section Column, t nt a 3 Hee R,=0. rat lan=o Elastic eignts and Ranctions Table 2.10 Determination of Por (Firat Trial) Stet Hemche Do lololoT oToToTolT@[6le[6[e/e][e]ele|e, a fs ur, “F l@.mlQx shin] | 3 ia | om [on fon fone ona -B amjom>Phoho Ee a a wa 1810.4 19-2028 988.19 2" 79.8 4) ** See Table 62.5 ahi at im ee 2Q) oy 2 2@ omg x2 G) 2 Pemans WD 10: =O: 10-90 7 Table C2-1i Determination of Por (Second Trial) © |o[olo]/ OTOT@ To [eee |e ele @|elelala “|-8|-8] SRS Boole: 5 & [8+ 40x Bl 6|:@ |O+27u1.0/9 Mer ge verse loads and their initial banding mo-| gin 1s based on ueing Pop. Rents, Mg. Thus, the column would be and alyzed a8 discussed there using an equiv. Therefore, an alternative approuch Pegs yeaeanse loading such as that in| 1s to assume an initial Dent shape having Fig.C3.33 or thereabouts, a "bow" of L/800 inches or thereabout, QF of what the drawing specifies such aa Actually, no colum 1s perfectly traight within X inches" where X is us- straight although this te assumed in us- | ually on the order of L/600 or aos YS ing the numerous buckling load formulas actually a tolerance for deviation from and data for Por. Because of this gany | straightness of tX inches, so the moss analyats and designers maintain a smail | would be *X/2 ani this would be used to Positive margin of safety when tne aar- | Show a cero or positive margin of dafety.COLUMNS WITH ELASTIC SUPPORTS Then a check 18 nade using X to be sure that there ts not a negative margin of safety greater than ~.15 or -.25 (per| ‘rt,Ci.13b). Sonetines X, rather than X/2,18 used to show a positive aargin, which 13 a conservative procedure. In any case @ beaz-column analysis 13 aade. 2.74 Design Coluan Curves for Columns with Rone 778 alters Crose-Sectione For columns with other steps the procedures in Art.C2.6a are used. For columns whose taper does not geet th requirements of Fig. 02.23 or 02.24 ei- ther a conservative assumption (adJust- ment) for the taper aust be used or a numerical analysis as in art. C2.6b is required. 2,80 Colman Pirity coefrielents ¢ for U Ceolasns with Elastic Side Beatraint Known Eod Beoding bestreints Fig.02.25a and ¢2,26a provide the envelopes (for q=eo) for Fig.C2.25 and 02.26. © 8 0a 03 4 Pig.c2, 26a 3 08 O71 Of O08 10 ML 23 €2,10a Solution Without Using Coluan Curves The inside scales for Fig. C2.17 are not shown in the textbook. Flg.02.17a shows the inside scales for the ordinate and for the absciasa. It must be understood that, like a coluan curve, Fig.C2.17 (and 2:17) can- Rot be used when deteraining the critical load requires using a paraneter which is also a function of E. Tis occurs, for example, with Fig.C2.25, 02.26, 2.27 (and 2-27a) where the parameter C 1s a function of E. This also occurs for other such data. In these cases a suc- cessive trials procedure aust be used (as in art.c2.12, Case 2 Inelastic Fail- ure, Portion 2). That is, the value of E used in determining Pop por the figures muet also be the valus of E corresponding Lo For= Por/A when the stress te in the plastic range (£ 18 By) or when local in- stability is present (z ta bere). Ey and eff are discussed in Art.c2ci6 C2198 gel serengen vith Known End Bestraining ents Fig.02.27a 18 auch more useful than Fig.02.27 since 1t provides curves for numerous values of k and kj, wnere the symbol "k" 16 “uY in Fig.C2-27 and in Art 2.12. The curves can be obtained by us- ‘he wonent distribution procedure. Referring to the example shown for Fig.c2.34, for each of the restraining members, AC, AE and AF the value of k is, calculated as k= 4SC(EI)/L where SC is obtained as C in Fig.all.47, conserva- tively using the "far end pinned" curves. for tension or compression in the aeaber. The total restraint at end A of the uea- ber AB will then be ka = kag + Kar + kag The same thing would be done ut end B for members BF, BG and BD to obtain kg. Then having the values of k Por for nem- ber AB ts obtained by using Fig.c2.27a. Note in Fig.all.87 when L/} exeseds a for compression members C becones neg- tive for the pinned end case. This procedure ts more accurate than that in Art.c2.13, but it is still an approximate one. ‘For more exactness the procedure discussed in art.All.15b should be used, Many trusses are designed accuy Bing pinned joints which ts generatiy conservative discussed in All.15b.NON-DINENSIONAL DATA FOR COLUMNS ¢ E sol 0 3 a © e Fa ‘0 | © ® za * «J © 20 Tel RACK TRV 10 -10 Ola ope 20 Sa ag War coe) Pig.C2,l6a Dimensionless Tangent Modulus Curves ay 0: 230 32 14 Fg. c2.17 Fig.C2.17a Non-Dimensional Column Curves Bis also\VF0,7/For wnere For = "*E/(t/p#, E veing Youngs Modulus (not Ey)TANGENT AND EFFECTIVE MODULI 25 Example Problem Proceding as discussed, recalculate For for member AB of Fig.c2.3 asauming that the loads in the restraining members| at ends A and B are due to an applied ups] ward load of 3,000 1bs at joint D, react ed at A and B. The following table showsl the calculations for k at ends A and B. 7 ml? =) aye] ed sles Hes B580h uaa, S83 2 Bono Using Pig,C2,27a, ky = kp (the larger i) and k= eae kyL/EL = 246300(30)/1,121,000 = 6.59 so Wy = 180 and Cia obtained tro Sa” 2,278 as 2,2, For this member D/t = 1.25/.058 = 21.6 and per Fig.Cl.9 its crushing (crippling) strength 45 quite high, 67500 psi,’ There. fore, the coluan curve of Pig.c2,3 for RT| can be used without modification, For member AB Ls LT = 59/V2 = 20,2 Up = 20,2/,422 2 47,9 Then, per the column curve, for member Aa| For = 53,000, This ts slightly less then obtained per the Art,c2,13 procedure, If pinned ends were assumed for all Joints then C = 1.0, L/p = 30/,422 = 71,1 and For = 43,000 (too small), but worde about bending failure in Art-All-isb o applies here. If the truss has meabecs subject to local instability see art, 62.16, Also if the members’ b/t were darEe. say 9 or gore, and of open sestion| torsional buckling might be critical (are,c7,31), 62.16 The Use of By ant Bere ngent Modulus, Ey, As the axial stress in a column be= comes increasingly greater than the prow portional limit stress, the tangent bodu~ jus which ts the slope of the stresas Tats Teduction in Ee ts shown in FigsBles Tight being Et) and also in the following sketeh, has shown and experimental data has veri- fied (Fig.A18.11 and A18.12) that the use of Et is justified for determining the buckling stress of columns, It ts also used for beam-columns having significant axial stresses. Hence, the use of Ey in Art.C2.1 and subsequent articles is tn- derstood, However it must be understood that Et 1s applicable only for columns of stable cross-sections (reasonably thick flanges ete.). That 1s, there must be no flange having a local buckling stress smaller than the column's buckling stress If there ts then Et does not apply and an "effective" aodilus, Eerr, suet oo used instead, Te 4s snaliey than By. The Effective Modulus, Fore When any flange has a local buck- ling stress smaller than Fp for the col- wsnyassuming a stable cross-section, an "effective" value for E, Bere, aust’ be used, Eerr is obtained as follows, A coluan curve is constructed as discussed in Are.C?,25 and iilusteatsd in Art.C?.26, Method i there is the most frequently used, followed by vothou 3: tiaving this adjusted coluan curve chesh accounts for the reducing effort offen cal instability, For can be obtained for any value of Lp, However, when the buckling stress is deterained for other than the bere Simple case it is necessary to use Eery which can be gotten as follows. For’ sty Yalue of Foy the value of Lip ia avualned froa the coluan curve and Eerr is catee, lated a Eert = For(i/pl? /n* 4 ~ Proportional tints stresa It 4s for cases where For 1s determined using a “constant” which ts a function of E that Eerr aust be determined and used: Examples of such cases are Pig.¢2,25 C226, €2.27, C2,41 ete, In these cases As discussed in art.Al8-8, Shanley a successive trials procedure 1s needed26 COLUMNS WITH ELASTIC SUPPORTS, COLUMNS WITH INTERMEDIATE Loap aes Podeety 8 dee, That is, one assuues a value for Eerr. de-| 5 =. termines For and then determines the vs lue| 2 Aytanky by 2 Ayiands 4 of err corresponding to this For per the 1 ‘ above formula, If it ts not the sane (es-| eh 2 fe Pity Pe sentially) as the assured Eger another va-| were 12 arg amg te lue for Eere 45 assumed and’ the procedure 2 > 1 1s repeated. This 1s repeated as neccore ‘The critical load combination, The value ‘cortesmenaaratee 28 essentially | .,.,42%9, ruitcel load combination, | Staata'aS SOUTS2O eg fogs Bia’ ro-) cabled ti and Ya guste, the sate Solng Et ag diarissed ret qreeazegmnen | paete es the applied Toads Pyand Fo 3 using Et as discussed in art.C2.10a. er; oads calculate a = » Assume Pp is 1p noe aed Sith tean-coutans Condy geis] toads calculate a= Py/Fo" Assuse PS Po.and per the foraulas find by sucsesive Since the beas-coluan allowable steect ais the value of My which sotseres considers any local buckling effect when | tials th e fethod 2 theveln ts ween ge eteet whe af Method 1 Eerrwould be used when loca’ tee stability 1s present, see art.c3 18. lo 2.47 sition Dutting ond ate th 8 Tis article provides additional ICT e8 8 buckling load data’ for various trons or 5 goluans, It 1s generally self-explana~ : i‘ ° tory., 3 S a ° { : a : st = . 3 ® Askin taorm pee on ee ~ 5 sal sala self nd : a |e oe nec ae i i} dss Ftg.c2.41 Column On Elastic Foundation | +, s é : 8 be ‘ $ alls y i ® 3 Bole tn pe toe 3 i eddie 3 + [+] sfesfeafon Jordon fo [loo zy. & aes Eeteiepe are ee fev * pas B “3 F1g.c2.62 Column On Elastic Foeieiis | «| sau dl] FE With a Distributed axial Los 27: ze ps] = a ae & or lo Pie. C2.43 column With Intermediate Load Note that the intermediate load, Py, ke in Fig,C2.43 must be located at the step, * Af one exists, The buckling equation ¢> eCOLUMNS WITH ELASTIC END BENDING RESTRAINT the equation and then calculate «= Pi/P3] portion then the value If a’ # @ assume @ different calling 1t Pb, and repeat, Repeat as necessary until a PL and P2 are then the critical load codbination: margin of safety is then Yalue of Po, The MS. = PL/PL - 1.0 (or ¥2/P2 ~ 1.0) If For ts 1n the plastic range for either| 22 of Et aust be used, and the value of Ey corresponding to Fee Bust be the sane , essentially, as the Yelue used in calculating it, (successive trials). If local instability is present Eerr would be used instead of Ey. Pane procedure is, of course, auch siapler for the special case where there ts no step (Ely = Elz), Por = CntEI/L? Bhs ote cot EE Lo = 7 i 3) 35 p- }——T {| | | _—__ [«| | ——J | | 4 3.0 | BX fe, | 09 | _—_—_—7 25 =| c }—1_5, | ce ee el Leh ae Eas, i 0%, _e a Ls >(@ Q)< 4 ie .—s 1.0 1 i i 14 Ls ce 30 100, 150 200, Pig. Note that interpolation for ¢ fhe Lines* is non-linear, For an exact Yalue here, or for KiL/EI > 200, Pee can be found by successive trials, as f5lisus 1.Use the figure to get an approximate value of Cand calculate Per. "between 2, Get the corresponding values of J, L/y COF and SF (= 4SCEI/L). 3. Enter the values in the following buckling equation (see Ref, & “for derivation of the equation), SF (COPF Atky + SP)(k + SP) = 1,0 4 If the lert side of the equation ky, te the larger of the txo eprings ©2+27a Constant Section Colurn Having Elastically Restrained Ends is < 1,0 assume a slightly larger va- lue for Pop and repeat steps (2)=(u); if > 1,0 aSsume a smaller value, 5-Repeat steps (2)-(4) until the equation As satisfied and Por is as assumed. The value used for £ respond to Er is in the plastic range, Bust cor for Fer when Pop or to Farr, To check a column's adequacy under a given load enter its values of SF and COF in the equation, Ir the left side 1s £1,0 tt ts stable; if>1.0 tt is une gtable. Seo All.3a for SC and COF values. ‘The k/ky curves in the above figure were, generated using the buckling equation,COLUMNS WITH DISTRIBUTED AXIAL LOADS Table 2,13 Unifora Colunn with Distributed Load and end Load antes cummcat(suexung voaos =e | ae | om Tem [am G2 cenmcar Gent) ionns = | ame] a ot larity | rowel vot The above data are used as follows. For a Given Py the orit- teal value of g 13 found. Or, for a given value of q the criieci value of P is found. Positive values for the loads Sct as snercs Regative Values actin the opposite direction. if F >i— then ger will be negative. If q>aep (for P= 0) then Zp will be negative. For 2 given q Pep 1s found as follows. 2+ Compute Py using the equation in Table 1 (20th row) 2. Compute gf/P— observing tne proper sign for s 3; Obtain 2/P» from either Table 1 (interpolating) or Fig. 2 42 compute Por = Pal?/Fe) For a given P dcr (the auximum allowable value of q) is found as above, but using P/Pe in step (2), qL/Pg in atep (3) and couputing Ger = Pe/L x (qL/Pe) in step (4). Example 1 (For all cases 1=50, El=10)| Example (UfEI/IE) =80773, P/Pe=.619. Per Fig.2| FoxI9478, aL/Pent.6t, Per Ble. QE/Fe=1.05. So qormt.05(80773)/50=1696| P/Pe=.34. So Bor=,30(39478 )= 13423, 2 Example & FESSBQ50;, case 1, nat 4s P's 133000) Case 3, what 49 aor? eg? Pent! EL/If =39478, P/Penn.76e ber tadie| 2, 80773, B/Peni.65. Per Table L ab/Pen}.17. So ger=3.17 (39478)/50=2503| qh/Pax-2:02, Gex==2,02(80773)/5 263