STRENGTH 0F MATERIALS

Mechanics : I#
TH the bonanch o ciene which deals the study
with the

ances and ih electa On

bocdy
Mechanica
nechanic, Salida Mechanic Fluld
Hechanic o NechantcA of (Eluid Mechanlca)
Rigid bocls. de<osmabte body.
Eng Nechania)
Intennal HoCe
Staeas:Intenal one per nit. anea at
pen unit qiven point on any planc
Intanal foe : When any ext. oace i agplied AB
in the ba then ponticle AaBstans Ext.
Internal Fose
qoingauay do the intencl moleculu honds Eoce
ing
act'as a spning tohich taies to putl back A
Bback into otginad shage. when chen atoms
in the ame plane exiences duch phengmenon
thein sum is eguals
equas to "p'
P' aas bods îa in the
cquilibzium (it. Ef=0)
Stoay:Suppa ek
ckbody pe. kuch Ext toncea lq ahe hai oun
hme tind kaha" hai plane A8 pe sirck|AB neH
Lbection pe stoes to hm AB plahe se tody cut
Kake Kii ek pat koha leke dekhenqe
dekhengs KiMi. ABpe kya
fcaca clevclap ho onhe hai
How to find the distsibution o Intexnal yaue in the matoial ?
Koi bhi alsi deuice nhi hai jo dinect bta de hi int dunc cbject rne

Lot.e detamalan
Internal hota hai ancn, tuh hm hooks Lau
itna hai q ge
Kajse - Kaçse ho aha hai uko ta kuake iuan akte hai ki int. yazes
 Taansucue) Lateial ii

ka diatuibution Kaisa hoye (ie Panabolic |lincan |cunved ete). Stacsa Anayi Unden Gtnenal laading
Hook'% law Int. force ox defozmation Ouenuie: PA-0 se pass hane wde o planes hongs an sbpe stme
hoga Kim Sin haga to kiai pe Shun Jitne bhi nes
Eqn staes depenca gn the distani buhon of Int g ahe hai unko opp0se Kanc Ke ise intenIal fone alag
Int. foe ane cicnectly pvopaztional to the deyamation innea
elastic limtt) -äkag diaed me dg the hoga so hm 3- mutualy h plans
Unden dierent Ioaditg|Favto the de<oznation of the bocdy ie denge juke acTA hn inko aesolue Kake oestane nikalenge.
about pt.-0
dileneat. hati ahy u delined type ona a thein dehaom js diaent. let us take yiat Zplane :
Heie the ean o stita is ditecni"
X-dinet ke oe ki uyh be
X
Lo¡d
Foce Moment
Axal Tzanvesc. Synholi "Say
Tenaile Compneaie Bending Luiaing in Y-diaection
bimilonHan othezs.

Ao we take a small pont dA the qiuen CnOJS -Jecton lenaana alc the quanities tahich am chaacteaizd
the ate Vaniation in the magnitud, disechon &plan. eg: Staesa, Abain 2
nealy Zeno_Hence,
Mament ot incntia. &o tan Staus e Ned maq, diae plana.
dA m OA -Jecion

|but sually ie take anea in mmi so the stess,. We can Ahoo thEJe tensA jn Mataix amm Qnd thêse mat. iA
G- = N
Mm m

Stzess
E Axial foze
hee is a dhea stess on one plane then taeze must be an equal &
öppaaite shecr sxcss on the penpendiculz plans Knouan as
Shean Stne ss. Conpkaay
Tuisting noment
moment
Axea
intenuieo: lahy Complemenlag Ahcan saes anides ?to ance
Lcs anises to balance the Nament (EN =0).
ar Moment ?
 Nomalstotcse l ) e NODmal Stooh (e aega
She or taiir )

tga eB plant pe to dae ko balane kane ke liye develop hoga Tay on extenal Tensile aus (P) which will inc.
the length
but indo ki ajh de couple on hua to Moment (EN =0) Keliyc ek Cw. lbut dec. the ciamcten. Wheneven Nomal stain come inside the
bods it uwill Come in al 3-dinectiona Unill &tendes body i body ia
Fosa M=0 e alongany axis thcn it maube aiaunt.
(Tyhc)xa =(Tya.abjxc in htial

Hnall
IBi- Axial Staeasplane Stseakol |latenal Staain : a dinec Re aong "P" lqa
SometimeA Sthes along ong one axis is Zeno: Qutiothaee axeA hai ute alawa baki - » ke te Atncins ko -l

Lateaal stocin bolenge

mataik paopenty Te= Iay =ooill alds be zeno. Case me cincalan stcd hisiye
dono me di dg hai eloe its cd'
Staes componcnis in thi Qne only along tao mutually lat. atain i alaays appoaite of lang sbain( )in natun.
AL the Atsess componenta ane oingle plane
that' why its alo Knawn as plans stes Volometatc Staain: E = Vs -VË
Jplane Vi
Noxmal staaina
btesses devebped
lUni -Axial Stae Sheon Styain.:lts the meaawne.
changein Ahape Ih delincd
When
ony 1staes is thene out o the nang n angle bla tuo mutually h planca.
thee axeLes ike in X-Y plant y= ly =Gy =0. plane Shean Stcin = (ie. Aangle)

Staain ia Tupea
I t the measwnc
Note : due to complementay Ahean ataains the angle o Y-tve
matenial unden extenal Laad. Tts
a unitlees quanity Y= AA' BB'
AC BD

Staain
Nozroal (E) Shean ( ) Stnain Tenas
Longitudinal lateaal Valurmckaic Staes Ki tan hi dame qki cdonotensar qucut hu but yha
T-- hai neplace Kado ti ans
Nonalbtaain isthe meaunc o change in Size. lI3 defincd a chunge
in a imension p uoit calginl diriensi on. Note: isiyc huigkitotal ohean stanis r'a donome eqcualy diuidehua
 4YEyy.
Yxz mateials (due to Jymmetaic slue of pune metas is paopa ane same in
Yyy =Yyx diecion Hence ito hormsg. mat.) to the nsymmetnical Atauctne lueley
hic
Yxz = Yzk Exx Change the pnap. indifeent dinsctionk.
Yyz= Yzy
Lec-2
Onthotaapic matesiala which mat. poops e dienl in mutualy h.
ie in X,Y,7 axea) dlinctions at a point.
pea of Mateiala Woocd
typa matenial. Note:
HomogenaA. botopic ACauthataopic mat. me Shean stes (T) ki wh de sheor stainY)
Non Homogenoua. Anibotaople ayaldeuzlap huga and Nozmul Atau ((s) i tajh se Nonm staain (E) Tska
an Kuch nh hai natunaly uhi behauiaun hai inha.
Orthotaopic Dut e have Anisotopiemat. then T Ki tajh be dono aucga.
Honogenaus mateiak ane thobe muterias o cohich
mateaial pnopettiea ane same at al points in the
Aame disec tion. Knao 2:H oe know V=IR, Staes, = Y. Stnain ibme likhBe bale aua hai but Stes
Note: he calculation prop shoud be tlen in dant cizect" i checking Homog prp voltsge elastt const Rpe depend nhi Kata. Staes depend upon Jaueload
applicd, Sain clatic Conal pe depend Kat hai. Similanls Cuient R
|Non-Homag
pcp. ane
mateni als ane mateniau Hoa which mat.
at al pints in thc same dinection.
pe depend kata hai.
qUA Sane looN ka oe
Steel A Aluminium hi Aame shape &aize ki arcd de
kne to Atoes dono god) me Jame haga but
Atoain will_be diydenentin bothAL hus mone tharn atec
HormcgNlon- lHomcg. telh whethen pop. ne Aam ah ditwn pla Elaatic conatant am matenial paperties.
Taal Ahisal athotopi dinecion v Hm shes thecnit<ally deziue nh ka skte qs uske liye hme pia hona chahiye
ki /nt. yota ka dataibution kitna hai ic.
cauallunequal|linan) paaakal ct" ds
botaopic material : nmatias yn ohich material p10penfies a Aame. in loe cont measue int.
once isiliye stain he dath relation nikalke incd kate ha
dneLtion at o point
3ay Ycungs mod ià same evenydinetatpl-A in the lypea o Elasic constanta
matenlas bke metals glass c.
1.
Modulus
of Elaticity Young Modala (E)
Anisotaropit. matenial which matenlal_pnop. e. Tts the natio Naama stieas,and Nomal atcin.
dent in cery diecion at apoint. I oies
wen mit pate riateias dik mthals weL mlxed oith
ancthen mateiab which chunges. ihe. symnclaie intonal stauutune f th.
 Modulus o Rigidity| Shean Modalas ( CanG): atio o Sheanátas Ashecn staasg Gencnalihed Haakes law
We aill ube discusing thin yan Isotaopi. matenlal We wil haue 9-atbeses 9-stbaln.
due to complementay cond" loe lill haue Totul of 6ean. This
Y Law can be applled in any Cond' 1

3 Bulk Modulua (K): Rato Håcbnoatutic stneas t Volumctic atmin. psincipal o Supenposihan: Pehle 1 load Kujase staan nikal lo. Jia 2 oad k cojh se ksa on.

k = Iaiaxial:
ly
2

|PoiAson's RaioVon m: Ratio o magnitude o Latera dtran longitucin

Stoain.
X

Elongit

alsays bcaz i Elan te then Eat =-ve and vicc-v0

The vae o |Azsi. calculting only Er ond making thnee Companenu a;
0.25 to 0.33

Relaion F=3K{1-2) E 9kG

3k+ G. Total Stain in I =_Stacún due t Stucin due to + Sban due to
akase is a Tactacpi matenids. Gyin X~i. zin X-ditect

Hooke's law:. Staes,= Elastic Conbt. X Staain Note: i Oy i the longitucinal stnain o Y- dinecian due to 6y.
Fo bimple Atass candi ficn (ie twhens single stzcss act on.
-y S is the Latenal stbrain in X- dinection due to Sy and '-ye' sign Ahous
E
that we apply tenile stes Gy in Y-axis then Stuain in X-axis
wil be -ve ic. shinkagc abg X-axid Qr Compacasiaa.
i) y is Comaneasiue JanLE then we take fhen lwe get an expasis
Shea Shesn in X- diecton amgunt
Stnain i 6 ke case me waa nhi hota ki ch TA ejh se 3-stacin in X,l,Z me ata bo.
 Univenaal testing Machine (UTM) me hm pehle dedanmahon dete hai say Imin
E Intenuiea:- Kiya bpeci men lon stucs calulate kale hai iolye UTN e /o cuave mita stretch
hi wme
Ey = Gy AL la on X-QN But engincenlng paoblems me hm Eng. stnes -&tsain CLNC we
E whlch ib dimilar to UTM CUe:
konge
A

E E

Stnoin
G AL
G G Gncph by LTN Eng. Staca -atJain
Note:. 4 Ductile mateniala hove Small plastie elasti. de<aamion and Lange plastic cefa
taa above egn. lwenecd at least -Rlastic conata anc the gau one can be abtüned Hente hcue Vey lanqe total demm beytone bäeaking
by thiea Ielation
Buittle. matenials haue amal elasti delanm. Gnd Keno plastic dedam. Hence hax
Biaxial : lule dor G Tulu nhi hog then E
haue wale oel" hi zanoaat nhi hai Uens Amal tota
bcor Oa.G. Txy inse aajayeng to itne cqn Kaf hai
Loading Stae
(G)

A
E
Staain
Elastic Plasic hancdning
Rementhen: Minimum na. Tndependent Elosic Constonta Stuala(E)
ig-l (lcw Canben mild steel)
Matenial Tani axial StbeA Bioxial Stacs
Tsotmepic
Oathotaupic. L

Anischaopie 6

CuTUC ) We tuke oaiginad A. lo bcoz instantanioua Ancu i length ko hm

CIaUe Ka hm uc haie hi las calcalafons. lame instuntaniou. Llbhoot
hegigidle hata hai.
-

Impantant OA is a St. lie wth Alope =E. Let pt-B pe tempnc G=02
E=05
Staenath : Max. magnituce q Staeas that the mat can bustain without
B
elastic deam () =0.2 ha jo k pl-& se lad seleae
Ksone pe de<amation pezmanent nhi ahega. and aay i lield stoncngth Max Stnesthat mat. Can bustain witthout
pl=Cpetotal dfaum (E) =0.5 lieE elatc acft i) Ultimate Aength without Facuie.
plastic de) hai aun is p. de aelcawe Kiya to speinen
ne penmanent plaatc defnm = 0.3 Ka ah jayeqa au Ne haue 3etyga f staengh due to Terailk, Compmcsion Shea Hanes.
Heine ki lope &ean CKacly sane aheqi ine A8 ke tuith lage =E. Sut -ulimate (u) Stacngth (8) in tenaion(t)
Suc/Sye aSus Sys kinilanty tan comasion
to Bse shuna tha
|but pC me smal yiclding thi. Yielding mlts jha de lange Ductile matoial
Iplastic det shunu ho aha hai ie dnon Cjieldng bging Ductile mat - Syt =Syc >Sys
Note: thencs 02% o plat siain ia obsexued at the Buittle mat. Suc >Sus > Syt
point whene yielding Jeqins We should dhas a ot. line g EE (ysmx) awith
Same sloge as of OA line exp: Ifoa duchile mat. Vield stoength unden tenaile &comp laads ae equa &qreatn than
"ùnder shcan Load. Similaaly on baitte mat. ultimat streng undez C2$2T
Sain Hondring: let oe haue a mat wha|e yeildingie
egins at yiel stength looa Duchlity: bility a mat h deaam plasically ithaut cchne k pua
the auia oDAapond t yield pl) and lWe have a Note: Rubber has Lanqs.cde. but its elas tic de wich lcgains ita aigina
shit its .S yaon l00 e 120 NAa then lise aoill' load
bt to 20 MPa and onemoue loacl. ond this will
shape. Ductility- can be measnecl by to loas whethen ung 2clang. oxLAA
the line Yo on aemeval. 4 oe again y h Load the mat hen cu qaph Elongation -liio0 Reducien jn Aszea= Ai-AA y 100
Atots yoIom CrE Hene ou7 Mew A;
Note: èShain' Handing 1 the Yitld Atacagh.
loada ane o tco typea Static and dynamic
Impsct. Stutic
Stutic Loaads means without mauement load lq nhu Load
-F - Faachuc p./1
Staus - Staain diag1am foa Baittk mateai cla. tltmat pt. ho like 4 lke bau Qn thetable in Cst.
Baitte matuuial oill dedeam, Jaialy aJaucta. A-fkopcihnal Limia
It has no plast de<aina tian. I tnocku~cs uilhin. beating Ahset wih hamnen
the elati limit, We ngligil. staan. Note: foa static load Cascs ecl docs o2 the Gtangth o me mat we. dort coxe
ba oactae. egt alus, ctz. abaut cicsilcnce. But i Ingct Load hen wcll yocw on 'Resilicnc'
 Initial : P Lo, A
Hesiliente : Ability of mat. to abonb Stuain encigqy withaut pemancnt Final:
detoinaion
eglike Spoing shauld haue lange meiliene to Abadonb_ Ahocka (impact loadh) ibaat
o Cas oithout penmanent
ie withaut Lange plastic dela (yelaing kng Staeas: Tue StoeA: P

Modulus Resilicne = Staain engH Eng Staain: Ep= AL Ihue stain :

La
Volume, upto Yield point
Le gild pt.
E Anea unden S-S = Syt eann Enom E, = L -Lo = lt-1
Vol-yeld SE/val.
Lo Lo Lo

N-m o:J wheac gypE Sye at yicld pB.

m Ao Ag
T,= G(1+ Ez)
Sometimes loell te givcn P-Al diag. ifs ne Ap
uncden the Cuue gue only stmcin enengy Stoain eng
paoot aesiliencs Staaln encgqy uph Yieldpt. "Note: aue &Eng. Stacss -Stain ka koi biqnidisance nhi hai
upto Yicld point' bcoz Amall cf then. hut Vield UlRmate (Eng)
ve
Note:
ashudy e taae klute imit, aopsationdl linit and Yicld pi yan auag but.
in pialy they ane Ves close so ue take itu a
-singlk pt.So. defooam. ater YR At the Ultimate pt. the necking ttazts
we AsuME upto cld point e oill have a Stacight linc. L Intenviesa:whch Meduces the Int Resistunce yoe P) andd anea (a) Anea oill decacasc
study stacight line is upto propcntionud limit) Huned Haak òlaa valid ugta drastically than P" cn this a Can say that the anaph coill qo
Yield paint. in cun studies. doan in Eng. G-S and go Upuaad ton Tuc S-S digg ata Uimate pt.
Touganca Hility mat. to absozb staain enens without actua. Ao (Corst)
p )
Af (1)
Modulus Sbmain Enengy Powex Lau: Yicld pt.tk Eny S-S yazmulas se kate hai hm wme koi problem
Volumc upto jnuctuze pl. GATE nhi hi cbutt YP Ke haad actuae qzaph mat. Taue Sas' ka dollow kata hi.
Exp: [Erg stasini calcalaBed y dividing the denah. changed jaom lota and The ean oq CUUe JaomYicld pt to Ultiral pt. is,
ond odd1ny bhauin eey smal k,n-Constunts dep on mat.
pat (d/) adn L, ia l,. K-stacngh co
N- Staain hazdaing exponcat
(bls Oto 1)
 |AA Ultimate pt. o matoial.,
Ho to find k?hon ultimate bt. Ka staalie Tun), E, mil jaye to n bhi mi gya
b put Ko1ke

U(1=)
Pemyectls platic maBcaial Gsk Slapt =k

(ns0)
U.P

Linean Staain Honckoing matcaial Aslupe =E

n=1 slope =k

StmeAs-Stmain dioq tn Tdeal matciala.

Rigid mat. Haciy plastik mt.

Higd Haeiy platic madl Shain Hamding mut.

FighCobon
Eng. denvica: |on mild stee, as Canbon %incneases
Hediom Coto
Low Canern
Ductility
Ecomt. ax all thae. upto Y.P

mild teel