r,,"~-::arillJ1IillillI N. M. BelyoevI I ,

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MIR PUBLlSHERS MOSCOW

H. AA 6Ullle.

C0fV>01l1BnEliHE MATEPIo1AnOB

N. M. BELYAEV

Strength of Materials

frallslafed from [he Russial1byN. K. Me/l/a

MIR PUBLlSHERS MOSCOW

Frrst publlsMR;evised lmm the 1976 Russian edlllon

He ..'......"1..... _Il#

e Ku;~ .H-rll'. \17610 En(llt"" lranslallon. Mi. Pub""".." 1mP,ill/H '" 1M U"t/Pl o/ SOO;ll SedctW RlpubtkJ

Nikolai Mikhailovich Belyaev(1890-1944)

Nikolal "\ikhaHovich Bdyaev occupied a leading rosltlon amongeminenl Soviet selenOsis who worked on Ihe technica applicatlon oIIhecry 01 elasticity and strength al materials and slruclures.

Alter graduating rolO lhe SI. Pelersburg Institute of Railway foglneering in 1916, Nikolai Mikhallovich Belyaev was nvlled lo sta y alIhe Strength 01 Malerials Departmen!. where he worked under S. P. Timoshenko.

Nikolai Mikhailovich Belyaev was associaled wlth thls Instilule(now the Lc!Oingratl Institute 01 RalJway Engineering) lhroughoulhis Iife. Al lhe institule he laught subjeets like engineering slruc!ures.bridges, IhooreUcaJ mechanics, slrength el malerials and thecty 01eJaslicity, and (rolO 1924 to theend 01 his lile was Head of theSlrenglh01 Mlllerials Deparlmcnt.

AH his life Nikollli Mikhllilovich Belyaev was a leadiog engincerand research worker. He was the first lo formulate and wlve the probolem 01 stability 01 prismlllic bus under variable axial loading-aproblem inleresting from lhe throretical llsped and important fromIhe point of view of applicallons. Simullllnoously, Nikolal Mikhailo-vkh Belyaev worked on !he problem 01 local slresses In bodies in contadunder compression. Here he considerably developed the works of Herlz.The work fint published by Nikolai Mikhatlovkh Belyaev in 1924 hascompletel)' relained ils value te this day.

In lhe Soviet Union Belyaev was one 01 the first to undertake Ihestud)' 01 the theory 01 plasUc deformatlon, and he oontributed a 101lowards lhe development 01 his field.

Nlkolai Mikhailovkh Belyaev spenl Ihe lasl yean of his lile infruitlul r~arch on problems 01 creep and relaxalion 01 melals llOdcrhigh temperalures.

Nikolai Mikhailovich Belyaev was 11 rare la len! who successfull)'combined Iheory wilh experimental research. In 1924 he cok o\'(r asHead of the mechanical engineering laboralory 01 lhe Leningrad loslitute 01 Railway Eogineering, and in lhe course 01 16 years 01 adnll-nistralion changed lhe laborator)' into a leading sdentific res~an:hcenlre.

New le

The pedaeocieal actlvily el Nikolal Mikh.lIovlch ~Iy..l"v was no!

restrided lo l~ ltninrld Institut! of RailwlY Engillet'f1ne. He'o\IOfked.1 the Leningr.d Technologicll InsUtute (1919-1926), tening-rad rutltule 01 Civl\ Avlatlon (I93J.J934j, and !rom 1934 onwards

was Head o{ Ihe Slrenath of Material! Department al lhe LeningradPolytechnl

Prefaceto the Fifteenth Russian Edition

The new edltion of Slrmglh o/ MaJeriaJ$ by N. M. Bdy~v hu\)cen publlshed alle!' 11 )'dr5. In 33 years thal lapsed between lIJef,ublication by N. M. Bely.ev 01 lhe first IlIlon In 1932 and Ihe tastourteenlh edition in 1965 a 1(1181 ol 675 000 copies of the book "",eresoldo teslilying \O lb ..... Ide popularity. Outl", th's pedod lile book wasperiodlcally enlarged and revlsed by N. M. Selyaev and, alter hisdeath in 1944, by IVOUP 01 lout or his co-workers. This group, whichprepared [rom the fiflh lo lhe lourteenth edltions lor publlcation, didno! oonslder Jt proper to make subslantial changa in Ihe orIginal...:ork 01 N. M. Belyaev. AddiUON .....ere done al Orte time or anolheronly when lhey beeame absotu!el)' neConsidering the availabllity 01 arge number 01 problern books(S@ot, for instan

corresponding secHons. The ob3otete sted pruliles grading has bernreplaccd by ntw ones.

A" in \he previous edlttons lt ""1$ Out endeavour lo pfl:5efVe Belyaev'sstylt Ind melhod ol presentalkm oi malerll/. Therl'rof1! Ihl' author's161 ha! In general been preserved. 11 Nlkolll Mikhailovich BeJy~\~ allve loday he would pclS:5ibly write man)' thmgs In a dill"erentway. However, slnce the book WOIl wide popularlly as II,'ritlen byN. M. Bel)'aev, we tried 10 preserve tM original lal as {Ir as pos.sible.

The work involved In preparing lhe liH~th edltion loe publicatlon '#ias dlstributed amoog lhe group as follows: Olapler 13, 80of Olapter 14, Qapters 1519, 2425-L. A. Belyavskii; Chaplers 6,812, 27-28-Ya./. Kipnls; Chapters \-5, 26and appendices-N. Yu.Kushelev; Chapter 1, 79 of Chapter 14, Ch8pter.~ 21)23. 2932-A. K. SinHskiL

A. K. Sinitski

Contents

"!toJaI Mikhailovldl Belr~ 5Pre/~ lo !be FiIlftlllb RuUlln EdItlon 7

PART 1. Inlroduction. rendon .nd CompressionOu.plff l. lnlrodoo

Ir::".1

",.J

Coniltnts 11

'",,,.

,"-

,"-

57.

1,,59....61.

In.'73.

,,,.,m.

PART IV. Befldlng. Sb"cngth 01 ReamsCIlIpler lO. I.. Iema! f.,... la "'

"PART V. Ddormation 01 Bums due to Bendloa

a...>ln I$.. """lyt\cal Mtfhod el DIle_mm, Del_tlont 2ftt 81. lA""tion IIIId lohtlon d btam atIC:liOl1l$ mt 82. Dilett>li.l pc\uIIlUll IX tbe IoIect.e

13

llU.n2.113.I"..115.".117.".

PART VII. Rt$lslance Undtr Compound LoadingOoapl(O' to. Unoro-trlc hdllll 37'

~ 119. Func!&lMnt.1 concepb 37.

\'20. Uns)'tl'lrnd.,lt be'll11nc. Dtlermillltlon al lol.reI&a 31912t. Otolermlnlne clSfl1.lrnents In ulII)'TIlll'.flrfc bendln

Oollltl:r 21. Gomblned Ilondln. and T~,l... 1M ~lon :188

1122. OtokUon 01 bum subJec:1td lo ul.1 llld l.ler.l lon:aJZ!. E~QUlttic I~oo or CompresslOll 392124. ~ al !eCUon 396 '"

1,"-".,>l.131.

l"..,35.,35.\

,JI.'39."..141.,..,,,.

Ollljlltr 22. Combl""d bondln, and 10TlIl)/1 401!i 12~. Dtll!l:mln.llon 01 lwlstln: .nel bendln momtnl.s 401f 126. DtltrmlnlUon ol.treM.e$ IlId .lrtntll check In eo:nbined blOndin.

,"d torslon 404

Ch.p'" 13. Goneral Compound l.oIdln, 408f 121. Stressts In I b,r _tlon subJ.dtd

lo.dllW .fOlIDtltmllfllllJon 01 normal w_ 410DeU!,mln.Uon 01 ~.. str_ 413OriermlnaUon of cIIlJ1llOtll'.ents 414DelifOl ol I $Im\!~ c;nnll: rod 411

0lafI1tt 24.~ fb" 423, 132. Gener.l Cdapb ~f 133. ~minallon of bcndlnc _b Ind IlOl"mlJ 1M Ihurj~

kRa 424DtlermlnaUm of str_ due lo IlCIffll.ll ond ~nc ion;e; 42\1DttmnlnatlOll al SlreHS due lo bendln mcmcnl U7QmpullllOll al lhe. t.dJ ... al CW'Y.tllJ1l el tbe neutral ~yer In.rect.r.lcUbc wclJoa '33

1137. OdtnniIIIUon al lile rldlll$ ouv.ulI'e al lile ,,",Iral 11)'8 lardale ancI trlptZOld 434[)etaminfne lile kallon al ne.IIJII laye- ltom bblcs 436Anal",s 01 !be lonnula kr norma.l s~ In 1 OUV" bJ u:;Ad41l1onal remarb on lile JomNla lar IlOl"mal Illesses C9An lIamp~ on delen:tln1nr IIressa In a

149. ISO. ISI.

l '5'- 153.

O!.pl~r 'U. Thlc:k_walled and Thln_wulled Vessel< 446

114'. Analyj 01 Ihlcl

"9/H ''061 t6l "rel

n ~lI",dold II)WP.1pOdUlQ;l pUl ~JI' "1'lIJ.dl.\ .Ropun JP'Iq~ IIJllu~J1S

Lli9 _JI" il1jl"!UU'ld JO 1IOIP"I"SI:l)g '~~In ~n8n'J l0 Imw al".....p pUl :t;ltmlRllI'

1" s:>snO'J :lu1PI0\ ill.(:l '~I

lIS q~IJalllU l0IlIl!tRll< J'n uo ~Jl' "1'lIIB" Je> p;l1l' '41 Bu!uJi'JlIC>J S1l~P! 'Isua "IS1

ILq 8UlPlO1 'I"'[JIA Rlun "111""1'W 10 ~~'O 4IB"'lJIS "I~ 1OId 1'i13 'OS[I999 ~mll'l 101 :tunSilI xdiu '6.1.1

l':9S ~J,dlUJ 00 lU~Ji~' JJiwI~ ~ql 10 S'lIW JO l:.i1l1~ illJ. 1lL1u I 111 _Jli pldw "UlI m lJ9dw I Rpon "'~Ifl

,,Bu"JI' !ulpnPllC>J PUB R&SOIJ1' !uIUIIU.liI13P 10 SllSI~ "lWUCO "9![ i

""IJ~IIJ0:J

PART IInlroduclion.

Tension and Compression

CH......tn'Introduction

1. The Scitl'Kl' of Slrenglh 01 Mal~ialsIn designing struCIUftS and machines, 3n enllinel'l'" has 10 select I~

material ami the cfoss-scclional area 01 t;tl'h tlemen! of the strulementshould fundioo proerl y. Slrenglh 01 malerials provjdes Ihe englneerwilh lundamenlllls or a proper solution 01 his problem.

Stf(!I1gll1 llf moferfals deals wilh lhe behavlour 01 various malcrialsunder lhe aelioo of external lorces aod poiots out ho\\' lo select lheappropri.te material aod the crosS-Stclional area of each ciernen! ofIhe slruclure $O as lo prQvide fully rtliable Functioning and Ihe mosleconomic designo

SomelillK$. slrength o( materials hu jo deal with Ihe problem in amodified lorm-Io check Ihe dimensions of a designed or existingstruclurc.

The conditions for maximum eoonom)' in rlesign and reliabilil) o(lunclioning are COfIlradiclory. T~ lormcr demand minimum consump-tion 01 materials whereas the btter lead lo inttease in consumplOfl.This contradiction form5 the buis o tM Itchnique, which has fatili-(;IIe! the development o slrenglh 01 material$.

orlen lhe uisling methDds 01 checkine: he stren::lh and the available materials a~ unablc lo meet Ihe pradical rtquiremenu for providintanswffS lo nl.'W problems tfor example. allaining high speeds in enei.neering in general and in aerosl.lics in particular. 1ongspan struclures,dynamic slabililr, ele.). This initiales a searcb for new malerialsand sludy 01 Iheir proprties, and lapires r~arch lor improving {heexisltng melhods 01 designing and devising lhe new ones. Slreng1h ofmaterials mus! keep pace wilh lhe llenera1 deve10pmenl 01 eneineerinand techno1eey.1-3uo

.. [Pa,t I

Somelimcs, b{'$ides he chiel Tequirf'rnl.'nls of max;mum reliabllil)'and econom)'. an engine

Ch. 11 /lllrodllrfiM"

the weight of lhe train through the whe~ls and rails and Iransmil ji loIhe slone supporls, aml the latler, in lurn, c.ommunicale Ihis load lolhe foumJation. lhe steam pressure in t1le cylinder of a sleam engine isIransmitled lo a piston rod. lhe pulling forte or Ihe locomolive isIranmitled to Ihe Irain lhrou'h a coupler which conneds Ihe tenderwilh Ihe wagons. Hence, Ihe elemenls of slrl.lctures are subjecl lo eitherlXI/Unte forCl'5 acting on eDeh elem~nt 01 lhe slruclure (dead weighl) or!urces uf inleraclion* be!ween lhe element uoder conslderalion andDdjolning elements or belween lhe ~leOlenl and lhe surrounding medi-UIII (w3ter,1\leam or air).ln fulure.when "",e S3)' lhal an e:dernal forceis being applied lo an eltmen! of the slruclure, Ibis \\'ill imply lhelransmission 01 forceol pressl.lre (molion) lo lhe element under consideraUon Irom adjoining elemenls 'O Ihe slruclure or Ihe surroundingmedum.

lhe rarees may be classifled according lo a number 01 crileria.We dislinguish belween Ihe concenlraled and distributed forees.A C1Jncentrated fura: is delined as lile force al pressure Iransmillll lo

Ihe element or slruclure lhrough an area whieh is very smllll ~s compa-red lo (he size 01 the elemenl, lor e;(ample, Ihe pressure 01 Ihe wher>tsof a moving Irain on Ihe rails.

In practice he eoncenlrllled force is considered lo be acting al a poinlowing fo Ihe small area through whieh Ihe pressure is transmiHed,We must kcep in llIind Ihat Ihis Is an approxlmalion wheh has beeninlrodueed lo simplily Ihe calculalions; actually, no pressure can belransmi!t('d lhrough a poin!. However, lile error due lo lhis appro);i-mation is so smalllhal l may be generally gnored.

A disfrbu/cd furre is defined liS Ihe loree applied conlinually o\'er aeertain lenglb or al'(8 al Ihe slructurl', A la)'l!rol sand 01 uniform !hiek-ness spread OVl!r Ihe sidewalk 01 a bridge repreSl!nts a raree which suniforl11lv dislribuled over a certain area; il lhe tllickness of Ihe sanfllayer is nol unlorm \Ve shall oblain II non.uniformly dislribuled load.lhe dearl weighl ola bellm in Ihe eeiling represents a load dislribull!dover ils lenglh.

lhe conel!ntrated loads are measured in unils of force (lons, kilog-rams, newlons .*); OH.' IOlldsdistribuled over an area are measured Inlerms oi lorce per unit area (II/m'. kgl/cm', NilO', etc.); Ihe loadsdislribuled along lhe lenglh 01 an elemenl are expressed as loree peruni! lengtb (kgllm. N/m. ele.).

The loads ma}' further be classified 3S permanl'nt and lemporar~'.lhe fJCrmanenl loads ael Ihroughout Ihe whole lile 01 lile slruclnrt'.e.g. de3d weighl. lhe (cmporur!J loods ael on Ihe slruelure only lor:1

To he !,'(clS!', lhe ,,(lght

/lIlroductwlI. Ttllsil/l clld CompltSSiOll {Porl /

I.:etlain period of time-Ihe welghl of lhe train moving along lhe bridgemay be ciled as an ex.ample.

ACl.:ording lo Ihe nature of adion, Ihe loads may be c:Iassified asstatic amI dynamic.

Sla!ic ioos act on Ihe struclure gradually, alter being applied lo Iheslructure they eilh~r do nol change al a1\ or ehange nsignificantly;the majorly of loads acling in civil and hydraulic slruclures are 01Ihis nalure. Under the influence of slalic loadin:: al! elel1lents 01 Iheconslrudion r

'" JI " 3. OeformatioJlS and Slr($5(S

In theorclil:al mechanics (slalics) Il'e sludy Ihl:' ~lIilibriulll or aJlffltclly riglll body; Ihis eoncepl ol" material in sllies is sulliclMlt lodell'l'mine (he contlilions in Ili!ich Ihe bodr \l.'iII remain In equihbrlumunde.. Ihe oclion of e.~lemal (orces applied lo il. HO\\1!ver, Ihis roughIlId appro~imale COOC't'Pl 01 Ihe properlics of material, dots 1101 hall!good in slrenglh of malerinls; here .....1' musllake illto acrounl lhe rae!Ih;,! Ihen' dtJel; 1101 exisl a pedectly rigld borly.

lhe elemenls of 11 struclure. as well as lhe slruclure a~ a wh1:>le,chllnge lht.'ir dinumsion.'i and shlpe lo some ('xh.'nt under Ihe clion 01exlernallorces lid are llable lo complete failure In Ihe tud. lhis changein shape and size is called rleformution.

lile magniludeand natureol Ihe deformation deflend upon [hesITuc,ture of lhe material llsetl. AII malerillls may be divilled into 111"0groups: crystalJine anll alllOfphous,

Crl/sfallinE maJuiafs COf1sisl o a ver)' arge number o edrelllely sma!1crysials. Each o th~ is a syslem of a\oms arrang~ very clase lo eachollJer ill regular rows. Thesf: I"O\\"S rorm Ihe so

Pad I

the interalomic lorces Ihal appear as a resull oF the deformation causedby Ihe action 01 external rorces.

In slrenglh of malerjals thi5 i5 achie\'l'd by Ihe mel1wd uf sec/iUf!S,which we shall Iry lo explain by Ihe following example. Lel us imaginea bar (Fig, 1), which is subj~lcd lo the clion or l\Va equal and eppo-sile lorces, P, nd lel us imagine lhat lhe bar is cut in t\Vo Ilarls I

1 m 1P~-----4--------tJ....P

,

and // by a p1ane mil. Under Ihe action 01 forc{'5p both ha1vesoF Ihebar tend lo go aparl, bular" h"ld logelher owing lo lhe forces 01 mleradion betwl.'t'1l Ihe toms localed on bolh sities of plane nUl. Theresultanl 01 Ihe forces of inleraction is Ih" infernal !(}fe.! IransmittedIhrough mil from one hall of Ihr b:lr to the ather, and vice versa. Theinternal force or inleraclion per unit area around any poinl 01 st'elionmil is called lhe slress al lhe poinl of Ihe given seclion. The slressesading rrom parl If on par! I nd from parl f on part If are equa1in magnitude according lo Ihe law 01 arlion aud reaelion.

" Ilumber of planes dividing Ihe bar in t\Va pnrls in t1ifferen! wayscan be dra\\'n thrOllgh a single poinl 01 lhe bar. Thl' rnagnHude anddirection ollhr ~Iresses Iransmilled Ihrough lhe given point [mm onepart ollhe bar lo the olher will clepcnd upon ho\\' lhe plano.' culs lhebar.

ThllS, il is wrong to speak of slrfSSl'S wilholl{ ndiellting {he planethrough which they are being Iransmillecl. Therefore \Ve speak bout"lhe slress on a parliclllar area in

Ch. JI 2J

Thus, to \etermin~ Ihe strcsses. it is nettsSary lo imagine Ihe elemenllo be cut in 1'4"0 parts and write clown Ihe: condilions of e:quilib-rium lor lhe system of Jorct:S aciing on one of t~ cutoll' par\s.: Ihi~s)'sle:m IllCludcs the ulernal {orl'U appHtd to lhe parl of the bar unrk'rronsidn-ation and also Ihe: lorce tro1ll5mitttd through lhe givcn'JL'~ami uprt"SSCd in lerms 01 slresse::s sou!!hl. This is Ihe melhod so.'C.tioM, whlch we shall \,:ollSlanlly apply in lulure.

Let us point out, that, in slrcngth oC materillb. lhe lerm -slrcu~is very oCien uscd nstead oC Ihe: exprcssion -inlernal lorces 01 inlenle-lion Ix!twccn parls 01 the bar~; lherelore in futurc whm \Ve nlcnlion~unilorl1l ami nonunirorm dislribu\ion oC strl':SSCS OVl'f the Sl'Cton~and ~rorce as be SUll1 or slrC$SCS", we must bcar in lIlind tbal l~expressions ar ... lo j cerlain degree convenUonal. For example. lo del('r-nline Ihe force on(' caJlnol 5um up the slresscs al various points; (I,~Il1cnlioned above. it is nccessary lo find al each point 01 sed ion thetlcmelllafY fafre whiclt is transmittetl lhrough an clelllenl"ry arca dt1and Ihen sum up a\1 the~ values. RL"Capilulating what has beL'fl wriLlen above, .....e come to lhe conclusion thal when ao exlernal [orce jsapplit'd lo an elelllt'nt of structure, Ihe hlller gets delorlOed alld lhedefOfmation is lK'ronlpanitd by slresses In Ihe clemen!.

Slrenglh or malerials sludies. on lite one hand, the rdalion bc.'t\l,'eCT\lhe external lorces ando OCI the 01111'1' hand. Ihe deformaUons 2ndslres5l'S due lo hem. This mables the engintc'r to salve the import.1nlproblem 01 :lell'Cling a bar of proper dimeru;ions and approprialt mate-rial lo relicsof resiStallCe 01 malerial, lo faHure tnd plaslic dclormatiom. To e:llSurtthe smooth funelioning or lhe structure wilhoul a risk ollailur('. \1,'(-musl see lo it lhal Ihe element is unly subjedtd lo stresses whicb ar~11":$$ Ihan its ullimatl' strenglh.

"lhe ~Mi~'blr s/rlSS is ck-nol~ bv Ihe ~IDC' letler bul is pul in

square bracbls: il is ~Ialed lo Ihe uit imale ~Iren&th P. b)' Ule fol-klwing expression:

[p]-1'whffe k ~ thC:S(lfl'ly {De/o' whkh 5hOl\'5 ho\\' many times lhe permi~il'l('stress [5 [ess Ihan Ihl" ultima!e (tenslle) strenglh. 1h(' value 0' Ibisfactor vares lrom 1.11.8 lo 810 and depends upon IheoperaHng condilions al the slrurlure. It will be di5cl.'ssed in grealer detall In 16and 17.

Cenoling b)' P... Ihe muimum stress Ihal ap~ars in lhe deslntdeJemenl uoder lhe acllao d external forces. we may ~Tite lhe bnsicoondilion, \\'hlch lhe si~ ami material oi lhe elenK'nl musl satisry .J5(0110\\"'5:

(1.1 )This is lhe sfreflgtl (XmdilioJl, whlch slales Iha! the' adual slrE'SS mllS!

be not grtaler lhan Ihe permi!>Sible.Nov.' we ma)' compile lhe plan rOl" solving lhe problelllS al slrcngth

01 materlals as folkN.'S.(1) Ascerbin lhe magnilude and nalure of all Ihe external forces,

includug lhe reattions, IICting on Ihe element IInder cornideution.(2) Selecl an appropriale material Ihat is mosl suilable in lhe

"..orking conditions 01 lhe eleml!nt (struclure) and lhe natureof loading;dell'fmine lhe permissible stress.

(3) Set the cross-seclional arra oi lhe elenlCnl in numerical ur algeb-raie iorm, and calculale 1he maximum aclual stress p",,, which

Ch. 1I 1tll,odu.~tll)1t"

king dO\1m. Thus, the design based on permissible stresses is replacedby the design based on permissible loatls.

[n Ihis case, il Is necessar~' to:(1) ascerlain the magnltude and nature 01 all Ihe external forces,

acting on Ihe given elemenl;(2) select the approprate malerial that is mest suilable in lhe

WQrking conditions of the structue and al50 lakes into consideration the nature 01 loading; determine the safety factor;

(3) set he crosssectional area of the elemenls of 5lructure in numerical or algebraic formo and calcula!e the maximum permi5Sibleload P",,;

(-t) II'rite down the strmglh tondilion Ps;;,P ptl and wilh ils helpcalculale Ihe cmo;ssectional area of Ihe elemenls 01 slructure orcheck whether lhe set d.imensions are sllllicienl.

[n a numbl'r 01 cases, as wi.'shall see later ( lOO). both methods givesimilar results.

In general, we shall be using the conventional method 01 designbase

IPurl I

coo~iderl1ble aHenlion lo Ihis lIspeci in our book. Expl'rimen!s playan jmporlllo! role in !he understallding 01 a subjecl ami must be ca,r-rieJ oul bv lhe studfnts along wilh lheir theoretical studies. Thesel!xperiments, worked out on lhe basi:; 01 facilities and equipmenlavailabl

Ch. 1I S/rnJ llJId 51r"llI "Ilhill EIIJJllt Llmil"

5. T)-pe5 01 DefonnaliON

Havin~ a~pttd a eenenl method of $Olvini problems c:l strenglhal malenals. \lOe may now go over lo slud)'ing individual problems.These ma)' be dh'ided inlo a numbl'r al group1 depending upon theI"'pe of deforlMtions. The common types 01 ddOfmalions (Fig. 2) are: (1) lernioo oc como

prwion. (tll and (11), as in ehaios. ropes. c:ables, b.no ollrusses \\''OI"kingunt'r lension or cclilpression, columns; (2) shearing Ir), as Jn bolls.nd rivels; (3) torsion (ti), as m shails; (4) bendmg (1'). as in beams 01aH Iypes. These- iour Il'Pes uf dciormatioos art Clllk-d simple ut!vrmaIU)IIs.

The operation 01 el~lIumls in slruclurcs is (enemll)' more tomple;Ihe)' are subjectcd lo 1\\"0 or more Iype$ 01 deformallons simullancousl y.lor exampl.. , Iomsion ar eompression and bending, I)('nding and lorslon.ele. These ar.. cases 01 Ihe SQcalled composilt dl'!Oflllolion. FOI' t'aeh allhe abovemenliont'ulrpes of deformations ....e shall lind oul melhodslor delermining Ihe slresses. selecling the material and cross-setlional aru 01 the elemenls and delermining lhe magnilude 01 deformation.

lo make il easy 101' Ihe reacler ta undenland, iniHall)' ~ shallINicler only Ih~ elernen15 of slNClur~ and mtdunes which are inIhe rorm of prismu/ic btus wllh asifaighfaxlS. A bodr which has a un-lorlO CfO$-wdiooal area aH long ils Irnglh m.lY be considefl'll aprismalic bar. The cenlres 01 gcavil)' 01 alllhe secHons of Ihe body Heon one slraigbt line, whkh is caUro Ihe axu vf IN lxir. Lal("l" on we511.11/ aiso consider bars ..... ith a non-unilorm cross-sectional ,lita andeurved ois.

CHAPTeR 2

Stress and Strain in Tensionand Compression Within the Elastic limlt.

Selection 01 Cross-secional Area

6. Oelermloing lite Stresses io Planes Perpendicular lo !he Axis01 Ihe Bar

We shall 51ad Ihe sludy 01 slrength of malerials wilh the simpleslcase al tension or compresskln of a prismalit bar.

Axial hnsion or aJmprnsion of suth a baT is ils deformatioo underthe acHon o 1\\'0 equal and opposile lorces applied al Ih.e end lacesoflIJe ~ar along i.ts axis. lf these lorces are direcled oo(wards, lhe bar is

"tPcrf' 1

saiJ lo be untler lemioo IFig. 311); in th(' opposile C.iI!>e. under comprl.'SIoo (Fig. Jb).

Accol'"ding to 1M g~.1 method of sol\'ing problems of strength elIUlrrials......e rnusllirst determine Ihe magnitulH' of Ihe exlemal 'orenP stretching (compressing) lhe bar. The v.lue 01 force P can usuallybe tlctermined by considering lhe n!encUoo 01 lhe bar wilh Iht' othereleolenls of Ihe slruclu~.

4- nn+" !:+__m_~(IlI rN

FJl[. J

Ih a simple extllllple, consider a round stee) couph.'r Ihreaded al Iheends antl loaded br axial lensile rorces P...25 ti (Fig. 4). Our tMk is losele

(l. 2J 5Irt$.< (md SI,~ill Wi/hill E/usli, mil

The equivalen! forces musl balancl.' lorce P. Therefore Ihey musloompose a rcsultan! force 11' equal in mllgnilude lo lorce P and direcll'rlalong lhe as in Ihe opposile direclion (Fig. 6). This resultanl N

i~ the force acting in thebar.lo lulure Ihe resulta ni of inlernal C'laslic 10rces,lransFerrl.'d irom

one part lo the olher across lhe imaginar}" sed ion, will becalled normalor axial ;orCf'. However, since lhe culolI portian ollhe bar must remainin equi1ibrium under lhe el ion 01 lhe normal rorce and Ihe externalforces acting on tI, Ihe normal forre may also be ca!clllaled Ihroughlhe externa\ for

30 (Parf 1

7. Permissible Strtue5.Stltcling Ihe Crossstdional A,u

To asctrlain lhe perrnissible sIras limil lO!' proper (UJ'M:lioningd. bar of Ihe given malerial, 1Io'e must t'xpcrimel'llally eslablish lilerelallon bel....-een lhe slrqlh el he bu and Ihe stressn Ihal appearin il. Por Ihis. il i.s esscnlial lo prepare a spttirntn (ll.~uall}' of a roulldor rectangular cross seclion) ef lhe iven malerial, clamp lts ends in amachillt' for l('flsile looding and gradually incrt'llse IhE' ltnsile force P.Th~ specimen "",JlI slretch and ullimalely break 00\\"0.

Let p. be th(> maximum load \\'lIieh IhE' spe'Cinllm can sus!ain befareruplure. Tlle normal slrm due lo this load is

p.0'.="and is cal1et11he ultimalr (/ensile) slr~lIglh ef lhe material under lension.It ls USUllJly ell.pressed in l\lt units kgl mm' or kgi cm'.

As poinled oul earlier in 4, lhe maximum permissible normalstress 10'1 is Sl'veral times 1m Ihan lile ultimate slrength (l.; lhe peromissible slress is oblained by dividing Ihe ullimale slrength by lhesarely factor k. The value of k dt'pends upon a lIumb 01 faclors, whic:h$hall ~ dist::ussed in ~I.il laltr on ( 16). Al any rale, lhe value ofthe saftl}' factor must tnsUfl.' nol on1y Ihe normal "''Ol''king of lhe elt-menl, i.t. wOfkmg witlW)lJ1 failufl.'. bul also prevenl Ihe formaHon ofplaslic deformations whkh nlaY affte! lhe ....-orking 01 lhe Illac:hilW orslruc:ture. Tbe salely htctor depends upon lile mattrial al' lhe eleml'l'll.natllrt- 01 lhe lorces ac:ling on the e1emml, economic. l:OOdiHoos ami anumber 01 oiher laclors_

[n \'ie"" of lhe importance of properly st"ltdin Ihe salely faclor andIhe pennissiblc str

Ch. 1} SI.as llN SIra;" lVithi~ ElaJllc '""iI 31Somdimes Ihe cross-secliOllal &r('1I is prest't. Then, rom formula

(2.3) we can find lhe permissible loadPOS; AloJ (2,01)

Retuming lo Ihe design ol lhe w

32 J~I,od.u'I'-O'l. T~n'M and ComprrsJiof1 [P.ar/ I

by A~.l Of A." The crossscc( ional area wthout weakening 15 called (hegr{!s-~ aTea and denoled by Al""" or Arlo Having computerl Ihe ntllareil An> we can obtain Ihe gross area A , ftom design consideralions.

Tlle formulas derived above are val id fOf Icnsion. rhey can be usedior compression as well wi,thoul any changes. The difference wlll bein ,he direction 01 normal slrcSS('S antl Ihe magnitude of the-permissiblestress, 10"1. The compression of bars ls morecomplex in Iha! lhe bar lIlaybecorne unstable, Le. il may suddenly hendo Designing IOt stabilitywill be discu.ssed in Parl VIII.

Figure 1 shows normal dislribution in a secUan perpendiculaF lo liteaxis of Ihe bar fOf lension and compression. ror a number oi'materii1ls(l.'.g. slceJ) the perrnissiblr slress value is thesame in tension ancl com-

r('n~iOll

p-E!3i Ir EEj-PCOl!lprt~iOll

p->I- -I-J~ ~ -1- +PF)G. 7

pression (lor short bar,;, i.c. bars in whkh lhe Jenglh does nol exceedfive lilll~ Ihe diame{er or UQI;S secUon). In other malerials (e.g. castiron) (he permissible slress is different in (ension and compresslon,depending upon Ihe ullimale Slrl'nglh lor the recrdcd dl!formations.

In a number or cases. compressive stresses are IranSlnilted from oneeJement 01 conslrudion lo another through a comparalively smaJl area01 conlad between Ihelll. This type 01 slress is generally caHed the/Jrorillg. or amlart s/,,!ss. Slress dislribution arollnd the nea 01 conladis \'er~' comptex and can be analyzed only by melhods 01 Ihe !heary01 elastkity. Usuallj\ in si mple designing. these stresses are ~'Qnsideredas ..:ompressive s!resses and a special permissible slress limi! is Fixed.Laler on Ihe qlleslion or sele

Ch. 21 SUtil' a/ld. Slrairt Wilhlrt E/a.nc Lim/J 33

These cxperimenls are conducletl in Ihe laboralor)' on special machines which delorlll the specimen lill il breaks down snl! measurc Iheforce required for Ihis purpose.

Simultaneously. Ihe delormation or Ihe specimen is measured-withlhe help 01 sufftcienllr accurale measuring inslrumt'nts - slra!ngauges (Iensolllcters). The testin8. machines are capable 01 appl ,ng asufftciently large load on lhe specimen and accurately measure Ihesorne. Whole parls of struc!ures (columns. portons of lI'all5) can beIl'Stetlfor compression-on presses having a capacHy 01 up lo 5000 ti.Tension fesl can be conducled in- Ihe laboraler~' on machines wheh arecapable of exerUng a lensi1t' load 01 up lo 1500 tr. However, in a ma-jorHy 01 Ihe laboralories machines 01 considerably less capacitl' (from5 to 100 tr for lension test and 200 to 500 tllor compression test) areemployed.

A detailed de5Criptlon er these machines ond measuring instruments,parUcuJ~rly or the well-known Gagarin Press, is available in Ihe bookLabofQtry Experlment in Slrenglh 01 Ma/erials, and also in specialmanua)s on mechanical lesling 01 malerials. With Ihe help or lhesemachines and measuring inslrumentsone can eslablish how Ihe material specimen wlll change its dimensions under ension or cornpression ..

Experimenls enable us lo concJude Ihat up. to a certain limit ofloadlng, lhe elongation is dired.ly propertional lo the lensile force Pand length 01 Ihe specimen 1 bl~t inversel\, proportional lo Ihe cross-seetional art'a A. Denoling by Iillhe t'longation of Ihe specimen rJuelo force P, we mal' write rJown the following relation between Ihesequantilies:

A/=.E..!.EA (2.5)

where E is lhe proporlionalily lactor which depends upon Ihe m

[Pan J

strm 0', '4'e gel anolher expresskm 101" Hooke's Jaw:

12.61

'" 0'=t.E (2.7)Thus, the normal slress under lension or comprmion is diredly

proporlional lo the relaUve elongation or shorttning 01 the bar.ProportionalHy [actor E, which Iinks Ihe normal strl$ ..... ith Ihe

relative ('Iongation, is called the modufus vI tlaslicify 01 the materialunder lcnsion (compression). The grelller the moduJU5 01 elasticily 01 amalerial, the le.ss the bar is slrelched (compressed) provided all olhercondilions (Iength, crosssedional area, lorce P) renlajn unchanged.Thus. in physical interprelalion, the modull1s01 elasticity characleriteS lhe resjstance 01 a material to elastic deformalion under ttnsion(comprl;:SSion).

Since relalive elongalion t is dhnensionless quanlity, it 10Uows fromformula IZ.7) Ihal Ihe! rnodulU5 01 elaslicily has lhe same unHs as slress0', i.e. il is expressed in unils al lorce divid1t should be borne in mind that Hooke's la\\' 1135 been represented bya formula which SIlIllS up the experimental f.!ata only approximalely:ji canllot lhcrelorl' be considered an accurate relalion.

In al! mnlerials Ihe derormation unf.!er knsion or compression moreor lcss deviales (rom Hooke's J:l\\'. In some materials (mast or 1he!melals) this deviatiOl1 IS ncgligible and il may be assumed Ihat {hereis exacl proporlionality between deformation anf.! load; in other materials (casi iron. slOlle, concrete) the devialioll ls conside-rably greattr.

liowever. far practica! purposes we may ignore Ihe small deviallonrrom Imulas (2.5) and (2.6) and use lhem as such in delerminingdefOflnalion of lhe bar.

The mean values of Ihe modulus 01 elaslicily E for a number of1I1ateriais are given in Table 1.

From formula (2.5) it is evideot Ihat the greater ils del1ominator the1m is Ihe elongalion fpliability) or, in other words, lhe grealer is I~ri,(idityof a bar. Therelore, lhe dmominaior oi formula (2.5). Ihe quan-lily EA, is called the tigidi/y "1 l!le bar Uluiu kllslurt or romptnSiOfI.\Ve sec Ihal lhe rigidil)' 01 a bar under tension oc compressi

Ch. 2J S"eu Qlld SI'Pi" IVithi" Ela$li~ Limil

Table 1

Mg~ulus 01 Ela'tlell~ and Lalual Delormallon Coo/llc1enl (Palsson'5 'talla)

lron grey, ",hileCo/bOJ1l 511AIIOJY sleelRol1ecl C'lf'f'erRolJed pho'pltor bronz.CQld.drall'" bussRollesRolied m~nl!"aneSl' blonteRolled aluminiumRolled ,.ineLUdGlassGranil., Iimeslon., marbleSn",lsl0ne

(110m I!ranlle

Ma>(lnry' 110m IlmestoneImm brick

Cnnerele I!aving ultima le (l~ ~:~~~slren::lh 200 kgl/cm

Thnher olool( lhe libresTlmber auoss lhe f,bres

(-,"

"PO'I I

ih~ othn- hand. upon i1s c:rossseclional area A. Somelimes il is moreC'Onvenitrlllo U~ thl' lerro ulatifJt rigidity E/l, Le. Ihe ratio of rigiditylo lhe length al lhe bu.

Formulas (2.5) lmd t2.6) enable us lo tll'l~'fmil'M! lhe elongaUOll or~bortening in liJe bar o( a structure uoder Imsion or rompression.Con\'tI~ly. koowing lhe elongalion, dirncR$ions and lhe material 01the bar onc ma)' cakulale lhe normal stres.ses acling in l. Thus. normalstress can be deierminrd by 1"'0 methods. If the !eosHl' or compressivelortt P is knoWII, o IS calculated from he formula {2.l):

,o"""A

H lhl' externa! force is nol lmown bul Ihe elongation 01 the bar can ben1l'usurctl, a is de-terminl."d frum formula (2.7):

o=tE

The relalive t'longaliOll ma)' be eakulated according lo lhe follo\\'ingformula lhe {olal elona:!Iion M lar a lenglh 1aflhe bar can be measurl'd:

A't=TWe shall show lalt'r lhat lhe sond mdhoofltn lo de~(rmine sllUSotS in a number oi cases.

9. lateral Ddormalion Codficienl.P(lis$On's R,atio

~\P3rl [rom longitudinal dcformalinn. Ihl" ban workinl umll.'r lensi or eompression an:,' a150 slIbjeded lo Intt'ral (if'krmnlion.!cpt'riml.'nls show Ihal under tension tFig. 8) lhe !,nilJh 01 n hnrn('rellses by 6.1. whereas its wldlh de;creases by o.b_(u_u,). Tlle re-latlve elongalion

.,t=

:md Ihe relative l:lleral derormalion,.

f l -.

In compre:ssion. lhe shorlening 01 lhe bar is illlongiludinal deforma-lion aM lhe incRase in its erosssn:lional area is llit' lalNal IIclorlllll-lion. 11 has been expefimenlally proved Ihal for 3 majorit)' f !he 013-lerials ti Is [rgrn 3 to 4 times less th.m e.

Ch. 21 SIU>5

38 {"lrodudiOtl. TtnJlDIl DIId Cwnp"JJiatt [Parl /

The relative ncrtase in volume (vofume (rain) [5v,-v (-,-_~ 1-2,,)

11 Poisson's ratio }I=0.5. lhere is no changf in lhe \"()Iume duf' lodeformation. Ho\\'ever. si/lC(' JlBy plolling lhe equllibrium triangll,' 1(11" lhese lOlas (Fig. 10), \\~ gelN,= s'?:JIJr:a:02Q=81fN.= Q (QI3O'- QYJ - 6.93 ti

Ch. 21 Stress and Slro;n Wi/hill C/mUe Limit"

lhe requircd crossseclional areas 01 rods AB and AC areN HOOOA, = 1" ~J = 900 = 8.89 cm'

A N, G930 277 '=l"-J=Y==; cmDiameler o[ rod AB is

d=~= y' ~X:.B9 =3.34 CJn:>:$ 3.4 cmSide of Ihe square section of rod AC is

a= 'JI""A: = 11 277 = 16.6 cm ;=:: 17 cm80lh 1he values have belln rounded-Ior Ihe slccl rod lo the nearest

mlll, and for lhe wooden rod lo Ihe neare;,t cm.To determine 1he displacement f 01 point A. lI'e dlsconnecl the rods

and represen! Ihem by lheir new lengths 8A, and CA,. increasing and

,e

(b,Fig. 11

decreasing thcir initiallengths by M,=AA,- and 6.I.=AA,. respective-Iy, wilhout ehanging Iheir diredion (Fig_ 11(a)). lhe new posilion 01point A can be located by bringing logether the delormed rods by rotating thcm aboul paints B and C. Poinls A, and A, will move alongares A,A. and A.A., which duc lo their smaJllel1gth may be conside-ref,-AA.=~I,

JlllroilucUOfI. TClllion Illld CClflprtuifJlt

antl Ihe v('fliu.1 dispraccment {fig. II (bt, =A.A. = A,A.+ ".A~

The segml'llt

Ir.\1,+ 411, lS a: O.507+0.2~xT

(I- SIn" = IU) _1.43 mm

B,'Therdore

A A _(.11 cosox+M I_'__.)/,cvs ' M,ros'"., I IJlI" U"

Consequenlly,

I -A A +A A _l' +~I!rotla:+:1lt

ClI. JI E.pcrimcntal Sludy 01 TmslOll Il1ld Comprfss;on

01 elasticity, limil 01 proporlionality) bul also beyond il up to liscomJllete breakdown (ullimate strenglh). To have a goorl idea abau!the mechanical Jlroperlies of malerials under tcnsion orcoll1prcs~ion, ilis essential to study expel"imenlaJly lhe phenomena Iha! accomJlanylhese processes.

By the differcnce in their mechanical propert ies under simple Icnsionor compression al room temperature, Ihe malerial~ may be classifiedas britlle and durtile. The brillle malerials break down under a verysmall residual deformation. The (ailure in case of ductile malerialsoccurs afler a considerable resiclual delonnalion. Casi iron, slone and

~oncrete, are examptes of britUe malerials. Tlle low...:arbon stcels andcopper belong to (he group of duclilc malt'rials.

Let us examine Ihe behav iour of bolh Iypes of material when subjl'Cled to tension Iill !ailure. A prismatk speclmen of roundor rectangularsection is prepared. the working porlion 01 Ihe spedmen i~ calibraledIn centimelres or [raclions of cenlimelre ,lo be able lo ascerlainIhe change in iis lenglh arter lhe experiment. The specimen is placedon the lesling machine ;Jnd ils ends are clampl.'tl. B)' straining the SPl.'cimen axially il is slretchetl with a load, which incre;Jses graduall~'wilhoul shocks or impacls. A number 01 successive loacl vallles areapplied, antl Ihe correspontling ncreases in the length I marketJ on!he specimen are measuretl.

The experimental resulls can besl be represenled in the form oi alension test tliagram; a majorily 01 Ih~ lesling machines llave an attachmenl wheh aulolllatically plols Ihis tliagram when the specimcn isstretched. In this diagram. 100ld P is plottC'!l along Ihe verlical axis amielongalion M along Ihe horizontal axis.

The lensiOIl lesl tliagram for a specimen !rom ducHle material, e.glow-carbon sleel, is of Ihe pa1tern sholVn in Fig. 12. The first par! of thediagram up lo the point A correspontling lO Ihe timil of propor1ionaliIy is a straighl lineo Ordinate DA, is Ihe value orlhe lensile force lhatcorresponds to Ihe limi( o{ proporliurla/ily

Inlroduclion. Tmsion a/ld ComPltS.

(;h. 3) Eperimenll1l SlUdfl 01 TensiOtl "n~ CompreS$i{m

Iring bctwe

.. tltt,cd~tIiM, TtllsJon rmd ComprtssiOlt [P"d I

Table 2

Mtrlepends lo a large exlenl upon ils shape and chiefly upon the ralio ofi1s l('ngth to its cross-sedional :!Tea. There\ore, in laborafor)" eJlperl-ments the residual elongalion afler breakdown rs not measured overIhe fulllenglh of the specimen, but only over Hs eertain par! ealled Iheredllud lenglh. In round speci mens the reduced length is generally takenequallo IOd; sometimes it is taken as Sd. In rectangular specimens thereduced englh is seleded in such a way Ihat Ihe ratio oi lenglh andcrosssectional area 01 a round spell should be 11.3fji. The speci mens are.]lfepared

CIl. JI,"

SllCh lhallhe Imglh belween the heads somewhal exceeds the estiro..-I~ lenglh.

Tht pliabilily al a material under tension lit oompressiOfl c.m abo bl:3SCt!flained by anolher quanUl)' called lhe ptI71UJl/Lfll (tlarhlt ,rola/ion(01 area). AHer the maximum load i5 reached. a "neck" starls iorroingin a partic::ularsedion 01 lhe bar. andal the place of failure the CC0;S5'SC'C-Ilonal area of lhe sptdmen is gene Prall)' 1m than ils nitia! value (Fig.

I~). Lel us denole the lnitial crosssedlonal uea by Al. and the area oflhe secUon al which the specimenbreaks dawn by A.; Ihe quantily

$"",A.-II,X 100A,

(in per eenls) is called Ihe relalivereduclion aHer breakdown. The 41Iln'ater is this quantil)',lhe more(lliable is the mall"rial.

Finall)'. Ihe ll"nSiOfl test diag- FI,. 15ram $ho\\'f1 in Fig. 12 enables us losludyone more mechanical property 01 malerials relal~ lo their ~.si$tance lo impad loadin. Thegrealer is the amount oS ...."OI"k requiredlo break the specimen. the higher is ils re:sislance lo impacl loadin. Therefore. lhe alllOUnl 01 ......ork done in slretc:hing Ihe specimenup lo Ihe elaslic limit or c:rushing poiot ma)' be {akm as a c:haracteris-tic: oIlhe resistllnc:e of malerial to sudden!y applied loods. This wockis reprcsentl'd by thellrea of he tension tesl diagram (Fig. 12).

Let us considl"r the part or the diagram whic:h Is wilhin Ihe limils "fapptitabililyof Hookc's 13W (Fig. 15). Wheo the spedmen ftxeAn clongation l (scgmenl 08. in Fig. 15) c:orrespoods lo a partkular value of Ihe force P (stg"ment B,B.). JI we ocrease the orce by dP.lhe. elongalion inc:reases by dl and Ihe lensile force havng an averagenlue of P+~dP wi1l perform Ihe ....'()I"k

dIV -(P+tdP )dl =Pd~l++dP~

For &,~~ttr ~IaJl, _ ! 1i9.

.. lP",/ I

Negltcting the $CCOnu-order ferm ydP dtil, we geldW = Ptill

Graphically "'Ofk dW ls exprt;$~ by the area o lhe wded redangleol helghl P and base d~l.

Coosidering lhe gradual incrcllSe of lorce P as 11 number 01 SUl"C1!S.~ivtelementar\' atltlilJOf\S 01 loads dP. we find tha! lhe \llork doJJe by lheexterna! fOrce; in glddually slrelching he spedmen is the sum of he

area~ ol lhe elementarr t\'1:langles (Fig. 15). Whtn he load P is increllsl."d cool inuous]y. l.l!. IIP and df,l are infinilesimal quanlities, lor11Ilrlkular values of P anLl Al, Ihissum ma)' be obtained liS lile areacf (rangle OB,8. tqual lo

, ,28,8,08, ='jPtil

Thll5. lhe work perlorrned in elaslic Jeformation of amay be l."xpres.sed by Ihe formula

111" -'2Pj,/

bar by c.l

(3.1)nd graphic;lIty represenled by lhe corresponding part 01 lhe tensiontest diagram.

TIK' ~lTK'argumenl holdsgood lor Ihe w'hole ollhe lension lest dlagraUl (Fig. 12). The arra of the digram repreents th" to!al v.-ork 11",~xpend~ in breaking /1 ~imen oF lenglh land Cfoss-secUonal are/l A.

To obtan the quanlity \vhich is a characleristic of a material andnol lhe specimen, ~ divide work W by Ih" "olume of lhe specinlen.The ralio a=:;' is called lhe sptcifu; r.wrk "/ e/as/ie di!/urllltlli()1l unuer[,.mslon.

Sirnilarl~, we may del"rmine lhe lo/al sptci~c IJXIfk WI=;; (hisis lhe work. required lo break he specimen The grea!er Ihis quanlily.Ihe more rdillbly lhe material wilhslands ~hock and sudd!'nly appliedloads.

We have.set'n above thallhe specimen malerlal conlinues lo ellperi-cnce elaslic deformalion in acrordanc:e wilh Hooke's law even alter Iheyield slrMS has been pas.sed; in Ihis tase are added residual deforma-lions. ThlS is observed v.-hile unloading the specimen aher loading itbe'OOd tbe yk>ld slrtsS (point lon Ihe temion test diagram in Fig. 12).

f ....'e now slart slrelching Ihe specimen alter unloading it, Ihel(IIding diagram will be r~rf'$('nled by alnJO!>llhc same unloading JineO,l parallel lo DA, and bll'yond poinl l, by the ume curve lDK asprior lo unloadi!li. Hence, il \\"ecomparc tension lesl dilllUam OClDK01 a specimefl nol expcrienting unloatling wilh dillgram O,ZDK 01 a

31

spechnen of lhe !.ame material, whkh has ~erI :rl.'liminaril)' loadetl upto poinl Z ami Ihen loode 11. Strtss-slrain OlagramThe Irnsion lest diagram sho,,"Jl in Fig. 12 iIIuslrales he 1lehaviollr

of a maleri.ll for a spedmen or Ihe g;vM dimensioll5: Iherefore. lo gela Cllrv(' characleristic of the ~avbJr 01 Ihe material itrtSpec:li\'e orIhe dilOO\Siorts 01 Ihe specimt1l, Ihe trnskm tesl diagram is slighlly

modifi~.lhe ordinales orlhe curve in Fig. 12 depicling I08us are ui"ided by

the nitial (helore Ihe slart of uperiment) ctOSBttlional area of thespeci men A.. and lhe absdssas Al are divided by 1M eslimaled ltrlglh l.Then in lhe ne..... dillgram "'01' plol along he vertical axis

p11=-

'.llnd along the horizontal axis

SlKh a diagram, s.hown in F;i!, 16, is called Iill' giresss/rain dill!lramlor {he given malerial lUid" 'lIsiO/l. JI is similar lo Ihe lensiGn lesldiagralll in Fig. 12. In this diagram al! lhe slreSSfS Ihal chatllClerizethe mtthanical properties 01 lhe malerialare markecl: Iimit 01 propartion.lily a,.. yield slress a. and ultimje ~Irenglh 0.

11 \l'e oonsider a portion of lhe diagram .OA. up lo Ihe limil ol prO'porlionalily. Ihen lar a crrlain stress a and lb corrtsponding r('lativeelongalion t. lhe area oflrlangle OAB (Fig. 11) equal lo ":.....ill repre-aent Ihe spedlic: vootk in slrelching lhe malerill lo slress!J. We k~'

..

(3.3)

lbal~ P.11-r-:lAJ-::l

Knowin tba! t.=.z. one may ,.,Tite do\m lhe exprcssion lor sp:ille\\uk 01 dcformation "",ilhln Ihe dastk limi15 as 10110""5:

.. '"f:J= T "'" 'il (3.2)By analogy. lhe tolal area of he diagram shown in Fig. 16 reprt'SCnts

lbe total spific \\"Ork Wl al lhe mOffi(!n{ 01 breakdowTI 01 a specimen oflhe h'en material. fbis quantity may be expresscd lIS lile produd 01

,,

,..

O I Fi(. 16 F1c

"lhe 1eng1.h 6 by he maximum ordinale lJ~ and a roefficie~t '1 whiehrepre5tnls lhe raBo 01 Ihe atea o tllt' diagrllm to thal 'Of a te

o,. JI ..

(3.5)

Howcver, {bis diagrlllll ill only a condilional characlerislk or Ihemhllnical propcrties oi lhe mah:riill. In Ibe initial slages or lhe lesl,Ihe crosssccliona1l1rea of Ihe spec:inlM almosl remams constanl, bulbeginninl: from lhe l'icld slress a noliccable reduchan lakes place,whkh is initilll1y uni orm over the ","Iire Imglh of the specillM'fl., andartl/f c!cming Ihe ultilnale slrength il becomes localjztd. Therefore.tk'yond lhe ultima!e slrenglh Ihe ordinales 01 lhe curve shown in Fig. 16represen! rondilional slresse.s ca1culated lor lh{' iniUal cross-sectionalarea amI nol Ihe real one.

Similarly. unlil Ihe ullimale slrenglh s reached Ihe abscissa.s inFig. 16 depcntl only uponlhe abilily oi rnalerialto ekmgate. Howf.'ver,once lhe neck is formeTo plot Ihe lru~ stress-slrain rliagram it is essenlial lo regisler Ihetensile IOIW al various moments and at Ihe sanie lime mossure lhecross-sectional area of lhe speclIlt'n in Ihe narrO\l.'t'$1 place.

Lel lhe ltu

ro

where8=~

'.

{Par! I

is th(l condilional slrain.Formula (3.6) cannot be used in case of non-uniform deformation

because il is diflicult lo rneasure 0./ lar computing e.It is known lhal the specimen I/alume does nol change uoder non-

uniform deformalion br.ginning ram Ihe momenl of neek formalion.This is known as the lawof constancyof ooLume and may be expressed as

A~l.= Alwhere A. is Ihe original crosssectional area. lt ensues thst

AJ. = (A.-aA) (lo+M)after dividing by Aol.

1=A._AA I.+ll.lA, 1,

wherefrom

I+e=-'-,->

Upon subsliluting thellas! expression in formula (3.6), \Ve finally ob-tain

- ,e=ln--J-'~ (3.7)

It should be noled lha! lf is determincd in the narrov..'es! part 01 l!Jeneek.

In arder lo obtain the relationship belween Irue amI oonditionlllstresscs il should be re

el_ J) Experime~1111 Slud o, TenslO/l Ilfld CompresWm"

Considering lhe relationship between ( and 'P. earlier oblained forconditions of uniform deformaUon, we obtain

ii_o(l+e) (3.8)Under condilions 01 non-uniform deformation, beginning from the

momenl of nKk formation, lrue stress a is found dirKlly from formula(3.4) as il is meaningless lo determine conditional slresses in Ihis slaleof Ihe specimeo because of the large dilference between A and A._

,

Fig. 18

orup

~UP'--F"ig. 19

,

The Ime slress-slrain diagram is shown io Fig. 18. However. !orpractica! use lhis diagram is somewhatsimplilied.lI is considered {hal:~~::;HJI1 and a small portion 01 the curve jusi preceding ruplure isignored. The diagram is then ploUed as shown in Fig. 19.P, -Yield slress 11~=A;' The [rue ultima[e slrengt} 11~ is calcuJaled lromIhe formula (3.8).

The (rue rup/ure stress is lound from formula (3.4), Le.a =~t1.I~ Ji

The troe unifarm elongation is determined from formula (3.6) Le.e~=ln(l+e), where a is lhe conditional strain at lhe mament when lheneck begins lo formo

Finally, Ihe lo/al true rupfure s/rain is found from formula (3.7),where.p is compuled lar the cross scctioo 01 rupture:

- ,erop=ln l_>/>

11 is evidenl from Ihe diagram of Figs. 18 and 19 Ihul the stress ainereases righl up lo Ihe moment of rupture, rapidly al firsl but com-paralively slowly after Ihe maximum (stress a~) is reached. Al lhe,.

"ftllroduetlOll. Ttllsfutl and ComprrDarI IP", J

momen! of ruptUfl: lhe stress corr~pondillg lo the actual ('roMsecliMal area is more han lhe ullimate slrengthoblai~ by thecoo\'en-lional IDf'lhod.

HO\\"t'ver. it ...."ould be ermneous lo use the laller yalueforcalculalilll: lhe maximum lo.ad whieh lhe bar can wilhstand belore breakingckr,1m, \\itich i.s very importanl from the practical poio! oi "iew. Thisis dear {rom Ihe lension test diagram in Fig. 12. lile maximum loadthat the specimenwithslandscorresponds nol lo lhe morncnt of breakdowII bul lo an earlier moment-the magnilude of Ihis load is characterizect by lhc ulttmate strenglh oc lhe Spl'l.:tmen of a given cross-K"Ctional area. Th... aclual stress increase in Ihi~t

Ch. 31

fZOO,o8Ql10

as casi iron, i~ strelched, lnconsitlerable dcformation is observed righlup t Ihe mOlllenl of ruplure. The spedmcn breaks clown suddl'llly.The relalve ('Iongalion !lnd relafive reduclion in area are iountl fo be

~t~k~~ ~./~tdo - 9f55tgl/'ff! t

!~zg,,9'"Ip ..ZZ.5% ".-7f3fxpf/Ut t: d'-5J.8%

I I, ,I I I 800I I I .!I rs,.-JJ78Kf/CIf/! : , "1I ,I 'i I I ':\. '.I ,

'QlII.

1:247gg~ I I11d'- %r IlOi' JO I O O

JO"

50'"

o.!Q() azoo {MUO %

"fig. 21

,

(b)o

O'---~,,;-'-~z:;;o-'--\:;--':llt,;--'=--t-r;,.,

lIIiJO

6000

Fig.

-

OL-_-~,

Fig. 22

,

very sma!!. The slresssirain diagram of casi iron under lension isgiven in Fig. 21. 11 should be noted lhal in Fig. 21 lhe horizontalscale of lhe diegram is appro:dmalely 40 limes more, ami lhe verticalscale ls approxllnately 6 limes moreIhan the corresponding scales inFig. 20.

As a rule, briltle materials havepoor resislance to tension; ther ullimale strength is less than tha! 01the duetlle mall'rials.

The rl'lation belwt'en stress andstrain when slrelehing briltle materials does nol concur well wilhHooka's law; even al low slres>e$we get a slightly curved line insteador the strnitiht Jine on the diag-ram, Le. a slrclly linear preparotionalty bel\\'l'C'n the lorce or slressand lhe eorresponding delormationis absenl.

Therelore, lhe modulus 01 elasticit) r. which is equal lo lheslope or lhc ,liagram (see 11) cannot be eonsidercd a eon~l;nl lJu~n'lily lor britllc l11~terla1s: il changcs dependillg lIpOtl (he strl.'!>S lar

54 IPwt f

Fi:. 23

,- ,

/ \,.-~- ---- --I-~,i ,~__~J--r-"-L_ ...'f

whch lhe deiormalion is to be calculated. As lhe stress iocreases, lhemodulus or e-Iastkly increases Q( decreases clepending upon Ihe direc-lion in which lhe curve 15 bulging-upwards or dO'l\Tlwards.

HO'o\1!:vtf, Ihe devialion from Hooke's law i.s insigni6canl rOl lhestres.s tange in which lhe malerial! groerally function in slruclurn.Thereiore. in procHce. lhe curve 14. Rupluu In Compression 01 Ductlle and 8rlllle Materials.Comprmion Test Diagram

Speci~ in Ihe shape or a cube or a C}'] joder whose heghl is jusi aIilUe lllOn' Ihm ils diameler are uscd in studying Ihe slrtllglh of ma-lerials under coUlprt55ion. In longer specirr.ens 11 is dlfficull lo avoidbending.

The !liizt of the sped~ns varies for different malerialsand i1uctUlltes(lor lhe cube tdge) from 2 cm (....'00

el

WIlft /

The nalure 01 ruplure In a sl()lle spccimeR is shown in Fig. 27; th('cru.;!led specimm represents truncaled p)'umids joined by Iht'irsmaHer bases. This lorm 01 ruplure i5 due lo Ihe friclion lar' bet'A1.'efIthe spccimro and Ihe bcaring plates 01 I~ preu. If we renJOve thisrriclion, rar examplc. by rf:lISing Ihe specimen rllCC5 ""ith paraffin.lhe nalure of ruplure "'.11 be difterent: the stone ",,U break into parlswilh crac.ks running parallel lo t~ dirl'Ctioll 01 t~ compressi\'e force(Fie. 28). The crushing load lor sllcil a cube wlll be less Ihan lar a cubcusted by the col\\Inon llIcthod, wilholll reasing. Tberel"0f('. thc ulllmaLc slrenglh in comprCS5ion is to a considerable exlent a COfldifiol/C1

p

"Fil. 29

characleristic of lhe slrenglh 01 material. TIlis musl be laken inloconsideralion when fixing Ihe salety lactor.

lt has beef observed Ihat ""hen a prismalic specimen madeof sloneor COOctete is compresse

P \ f1 eh 26) ",.

n al 1fte el Intral x ,lrou~h le

In ..taticall\" int!t'tHminatt' bl'an s 1111.' formatlon)f nl' ductlh hlI~ not l.'nou~hfor full ulilization of Iheir bl'ndln!: .. pae 1 11 l nJi I Ihal 1 Il' ..t om more dudJlt' lunge be fo md \\" f4J1l c\p1 111th \\ ith tlil' Iwlp of 4J1l t'\amplt'

-h :;:J. ,O z:,.~~

",, ,

,T,,' I,I I .;1'1 r:'l?'iT PI( L ~'-

J"t'i.: 1101\ hy '!:'olJ 11 fhe p; +,

.,,

r' l ,-i

,

eq 1 Jon I)rlJl hl

rt~ anJ ,n "ore, Ihl.' , ,

~l S('(IIOI1."

6"(~t' Sl't.' tha!

, K

r r."

..1

1\\l:r haln'S1 t:umlJ 11011

(2G 5)prt ,1.:111.:(' of

alio, dt:ll r 11-

Ld U\ rfll1 .. idlr a Iwo-..pan t'onlinllou" !Jt lJl of lllliform l'dlon(f J.: :377111)) 11 .. Ill'nding: 1ll01llt'nt dilgralll ior \\orl\ \\ Ilhin tlll' l'I,1 tll.:lir lit-; fig:. ;~7i/'J) i" !1It' difTt'rt'nn' uf 111l' hl'nlting muuIl'nl dlilcrlm..fOl forn'P and ,upport tllOllll'lll .\1 1 ;~;'l (jr;lphir ,u!Jtradion 01tlw diilJ.:rallls is ,110\\'11 b~ dotle(l lint... lhe r( ..ullan! lwndin.: monwnldLIJ.!fam i" halt'iw(1. TIH' milxilllUIll ... trl''''I'(I ~tdiOlls rt I1w '('1,'1 ion 01application of thl' Iclil"l'lJ. In: "l'" 1I1 Ihl.' tlt'am b~()ml' lqual lo lht )1t'ld In!'i nf1r t uf all m tht top and botlom la~H of Iht' ('(:11011 JI1dt"r load P,n 1 ma~ b(" l'\fm", t'tI b\ Ihl' rC"alion

JI r" (J, \\ lh'rt'frurn P11

"lnlrod..e/iOfl. Terls/an and Campre!slon [Part 1

The amounl ai wark required to crush dutlile materiaJs i5 grealerthan lhal required for brittle maleriaJs. Therdore, ductile materials are moTI" suilable lor struclures designed lo absorb the maximumpossibJe kinelic ellergy al impact wilhout [ilure.

lhe brittle materials {aiJ easily undee impacts jusi because Iheiespecific work af delormation is ver)' smalL Oue lo Ihe!e small deforma-lion up lo stresse5 c10se lo lhe ultimatestrength. thesame belltle mal

CIt. 31"

,lruclure. it is nec~ry lo bcnd er lo ,Iraighlen a bent elemenl.Since Ihe britile material, are eapablt' of willulanding only very malldclormations, such operalions on them l1Mlalty ive rise lo Cfad,s. Theductilc rnalerials. eapable ol taking ronsderable deformations "'-'ilhoulruplurc, elln be bMt and straightened wiihoul an)' duflcullY

Thus. britt1e mal~'fials ha"e poor resislanee lo lemion and impacb.are ver sensitive lO local ircsses and eannol bear ehange in lheshape el t'terr.ents made from tlern.

The doclile malerials are free from lhesl'. drawbacks; therefore ducH-lii)' is one of Ih(' most important and dt:'Sirable preperty in malerials.

The poinls in favour of brHtle malerials are Ihal Ihey a~ usuallycheaper and often havc a high ultimale strength under compresslen:this properly ma~' be ujj]izecllor work under placd loading.

Thus. we sec that ductile Imd britlle materials llave cxceedinglydilferent and contrasling properties as far as their slrength undertemion ami compres.slon ls concern('d. However, this dilJerenee in properlies is only relalive. A brittle material may acquire the properliesof a ductile material. and vice versa. 80th brillleness and ductilil)'depend upon lhe trealllK'nl ollhe material, strM&td slate and lempera-tune:. Slone. ""hich is convcnlionally a brillle malerial under compres-Slon, may be made lo deferm like a ductile malerial; in sorne eJ:periIIJtnls Ihis W2S achicvt

lllfroiJurfiQl'. Tmsiu1I aml COmprtniOfl 1P~FI I

However, Ihe in[ormalion given lill no\\' is nol surticien! to flnd outthe pcrmissible stresses suilable lor differenl types o[ loading. Tlle- va-lues 01 atl the mechanical prpperlies of materials (ultimJlle strenglh,relative elongalion. limil 01 proportionality, elc.) are obtained fromlaboratory experiments under sta tic loading, Le, when he load inuea'" raduall y wHhout impacts, shocks and change 01 sign. Similar1~'.{he ormulas correlating normal stress 1] with the lensile or compressive10fl;e P have becn derived lor stalic looding. lt was assumed lhal theexternal (orces and stresses acting on the cutoil portion 01 Ihe b;lrbalance each other. In pracliee, however. we often come aeross dynamicand s)"slemalically chllnging loads.

As compared to lhe slalic load, he suddenly applied load has a Iwo-lold effect; on Ihe cne hand, Ihe briltle and ductile rnalerjals readdifferently to lhe dynarnicadion of the load and, on lhe other hand, thestresses are also differenl. This problem will be discussed in greaterdelaijs in Ihe chapters on dYllamic loading. Here we shall pay alten-lion only lo Ihe ad hal stresses are generally higher under a dyn~micload Ihan under a static load 01 the sam~ magnitude. This statemenl isconlirmed by experimental resuHs and may also be proved Ihoorell-c.1I1y, as has becn done in Parl IX.

Tb~ ratio or stress 0,) due lo dynamic aetion 01 Ihe load lo str~ (Jdue lo siatic adion or the same load is called the coe!rcienJ of dynamic

reSfJOIl~ and denolcd by Ka:

'!i = KdThe coellicient of dynamic response depfmds upon the typc 01 dyna-

me looding and has a very large value in a number 01 cases.B. The slrcnglh 01 TIlalcrials undc.r loads systemalicaHy changing their

lIla~l1itude or lnagnilude aud sign is mllch dilTerenl hom lheir strengthlInder slalic ami impact loads.

1, for eX

"practlcally an unlimiltil nUIllber o alleraliOfls al Ihe force withoutbrellking down. This slres.s in denoted by Oc anri js laken, the more so bee.1use Iheir strenglhunder Impacl loading is IU""et Ihan lhat 01 duclile materials.

Undcr an allernalLll:: load, \\Ohm we have lo reckon wilh Ihl" possib1 i.Iy of devcloping craek$ due to faligue, il is very essenlial lo lake inloconsideration lhe local stresses, which .seriously alfed Ihe selcclion 01safely laclor el duc!ile malerials. Fur lhe criK'k due lo fatigue to appear,fhe actual slresses in a pulK:ular section musl exCffil !he endura~limil. Sioce Ihe local slrts5t$ are grealer lhan Ihe ~tresses elsewhere(acUne over a larger portian of Ihe secUon), lhe chances of Ihe crackappearlng are due to name.lr the local stresses e,tcee

"{Par' 1

lioos el the maximum gMml\ slresses Irom lhe formula

a.... -~llO;[(fJ=~lhe ~{ely faclor wleded lar lhe general permissible slresses shouldensure that lhe local slresses do nol exceed the endurance limil. Thisrequres considerable increase of lhe salely factor k as comp.red toiu v.lu!' under static loadiog.

In lhe case of ductile materials, when lhe endurance timil tXl:eedslhe yield stress. lhe local stresses mar be Ignored as yielding reducesIhe pOMibilit}' af lhelr spreading. playing Ihe role of a buller.

Far brittle materials, which do no! have a yield platean, Ihe danger01 faligue cracks appearing under variable loadiog la lUore pronounced,and tllls requir(!S lhal Ihe corresponding safel)' musl be inct(!aseu incomparison with Ihal under stalic loading.

Thus, sillCe lhe choice 01 Ihe safe!)' laclor dt'pcnds upon Ihe properlies of Ihe malt'rial aoel lhe method of applying the ulemal lorces, ilsvalue I.S generally grealer lor brittle malerials lhan 101' duclile ones:similarly. a higher \'alue ollhe salely faclor hasto be taken 101' dynamicand varying 100ds as compared lo slalic loads.D. A number DI olher laclan ha\'e lo be takm inlo accounl whenselec:1i1lR Ihe- ~mi5Sible stres.ses. The mllgniludes 01 lorces require-dSor compulatioos are 001 koo'll..n exacll)': the mec:hankal properties ofmaltfials frequently devlale considerably from lheir krlOllm values;lhe methods 01 computlllion and OUt koo'll..led~ al lhe: inletadion be-Iween dilf('ftJlt parls 01 slroclutes ar? usually slmplilied amlllpprolli-mate. The salcly fllClor must COVef all these unavoidable inaccuraciesof compulalion and designo

Tlle les.s lhe hornogeneily al malerial, Ihe poorer Isour knowledge 01Ihe lorces ading on il, Ihe more simplilied is our presenlalion oJ lheinlcracHon between various e!cmCflls of Ihe slrutlure, and the realerhas lo be lhe sarely lactor. In opetlllion, machine parls we-ar oul; therelore, in 11 number or cases a '\\;ear factor" has lo be provided lor.Similarly ('orros/on and ro1ting have lo bl' tllkp.n iolo 3

CI>.31 EXp~,imtlltal Slufg ~! Tt~slan all4 Compnmion

able lo properly apply lhen\. However, in e:'cplienal cases, sal'. lercxample. during war timt:, lhe enginecr has lo abandon tlJ{' slandardvlllues; he mal' Ihen 101l01V Ihe general considerations, 1ald down inthis .secUon aud in Chapler 3l.E. Summing up al! that has been stated above, we may lormulate Ihefollowing main poinls.

The sarely lactar should be seleded in a .....lJy so as lo provh.le adefioite reserve againsl the appearallce 01 Ihe soThest: in general are Ihe basis consideralions essenlial for properlyeva1uatiog Ihe permissible slresses.f. Passing over lo Ihe considerations in assignjng lhe value of the saofety laclor. we shall gjve some very brief inslruction5. lhe nonhorno-geneily of lhe malerial, inaccuracy in force deterrninalion, error ofcomputation, Le. the cornmon factors are ae

.. IPutt 1

:sarely faclor k. For ductile matCl"ials (sleel) it b laken as k~-k._1.41.6, lor brillle matE'fials and \\"000 k.""k,=2.5-3. Olher laclors.suc:h as the d)'namic nalure 01 the rorces, alteralion al their adionand lhe decl of Ial slre!>5e5 are takl!n lK'COtJnled far by addilionalcoeflicienu. by which he main safety lactor is mulliplied.

11 should be borne in mind that lhe perrnissible stress 101 obtaintdaccording lo formula (3.9) should be comparcd lo lhe actual slres.5eS InIhe part of lhe slructure .",.Hhout considering Ihe d)'naU1K:: icUon 01the forre and olher addilional factor!.

JF only lhe general poinls are coru.idered while asslgning the safel)'factor, Le. Ihe overall salel)' factor is laken eqllallo (he maio safel)'faclor, the d~'namic lIelion al ortes and Ihe local stresses are takenaccount or, as lar as possible. in he value of the actual stress, multip.tring Ihe lAajn slress under slaiic loading by Ihe coeflicients of dvnam cloading and stress ooncenlrations. lt is IlOt diHicult lo see tla\ Iheresulls in both Ihe cases will be identical.

Table 4 oonlains approxlmate vaJues 01 Ihe ov('rall salel)' factorwHh respecl lo the ultimale strength ror varlous Iypes of materialsand loads induding the f.elon. accounting lor lhe d)'narnic nalureof loading and local slrtsseS.

Sblic INU

"I(lad IwdV~ryln

Ch. 3)

T~bJ" 5Tent'tiv. Valuu 01 lh. Permlsslbl. Strrs~s fur Son,. Commanly U...d

~lat..ial.

/1101,'1.1

Und.. IO",iOn t'ud,., ''''''_p....,lon

00130

up to 3up lo :2,-,

Gral' tosl lrenL,,"",cMbon 51cclStructural cariJo" slcal u..eJ in machinc buil

PART 11Complicated Cases

01 Tension and Compression

CHAPTER.

Design 01 Statlcally Indeterminate Systentsfor Pennissible Stresses

18. SallelUy Indetl:rminale S)'StemsOur abilily to ca!eulat!! lhe ddormation oi baTS under lension and

compression enablts lIS lo determine tbe changes in Ihe shape and ize01 parts 01 slructures under the ctiao 01 exlernal forces. Usually IhesedeformatJons are ~ snwll tho! lhey seem devoid 01 ;Jny pradical Im-porlance.

However. in a numbel" o trutlures, it s impossible lo check theslrenglh lInd determine Ihe crosssectionai area ollhl;' variolls elemenls""ithalll lhe knowledge of deformalion; lhese struclures are known asstalica/ly jndr:~rmira1e systcms: linding Ih!! fon:es acting in lhe e1e-ments of Ihe5(' slnx:lul't'S in a sUJJical/y illdrft'fmillalt probltm.

In all Ihe uamples \1.ilich \\'E' have ronsidered lill now, Ihe ensileor compreuive fOl't't'S acling on the bar were determined [rom slalicconditions of a salid body.

In case of weight Q suspended (rom hI..o bars (Fig. 31). AB and AC,we find tensile Jorces N, and N. strttclling lhe ban rrom lhe equilibrium condilions 01 poinl A. Three rorces appl ied lo poinl A musl satisfylwo equations 01 t'quilibrium: lhe sum 01 the projections of Iheseforces on lhe two coordinale axes mnst be ~ro. Tllus. we ~ Ihat Ihenumber 01 unknown quanWics (Iwo) is equal to the numbcr 01equaHons (tu'O), thereiore forces N, and N, ma)' be determinedIrom hese equations. This is slatical1y t1elermilUlte probll'tn.

The oonditions will be different il weight Q is suspe-nded rom Ihreeban (Fig. 32). In Ihis case poinl A is in equilibrium un~ Ihl' actiooof fout ion:es: Q, N N,. and N tbree of (he forccs belng unkno\lm.The nllmber al eqllations rem.ins lhe same, Le. 111.'0. Hence, Ihe num-ber- o unknown qu.nlities eJ[c~ by one tbe number al equalions.,lhe strudure is one dtgree indelerminate. and lhe problem cannotbe sol\-ed wiUl !he help al slalic equalions only.

Ch. 4'1 Deslgll 01 Sta/ieal/v fnMlrrmillal~ Syslems G7

The addilional equalion required lor lhe solulioo 01 Ihe problemcan be eompiled using Ihe ideas gained in passing over rrOIn Ihe thoorelieal mechanics to the slrength or materials. We must lake accounllar the ddormobility 01 maleria!. One more equalion eao be loundin studying lhe delormations of Ihe slruclu]e. 11 lurns out Ihal it isalways pessible lo fintl as man)' addHional equalions as is requirerJlo complete Ihe number 01 slatic equlltions so Ihal the number 01equations be equa) to Ihe number or unknown quantilies.

ng. JI Flg. 32

The extra equalions are lormed on Ihe basis oi lhe eommon principIe;they should express Ihe condilions 01 joim deformatioll$ 01 Ihe syslem.

Any slructure dcforms in such a way that IhITe are no rupluresof the bars, Iheir disconnection or ;lOY unrorescen relaUve tlisplaccmentef one part of Ihe struelure with respecl to the olher. This in briells the principie 01 joill! deformalioll of lhe elements 01 a s)'slem.

lhe general melhod 01 solving statically indelerminale systems isas follo\\'s. Firsl 01 al1 we must decide whal are the.larces to be deter-mlned, lhen wrJte down alllhe slatie equalions 01 a salitl body, andfinal1y derive the requirec\ number of exlra equalions to nd Ihe un-known forces.

A course of Ihe solution 01 Ihe problem js shown for Ihe particularI?xample (Fig. 32). Suppose the side bors 01 equal cross-se

axes are vi ....en in Fig. 33. Lel us equate to:ero the SURI or projeclionsof Ihe forces acling on poml A 0fI Ihe eoordinale axes:

N.sina-N, SilH1 "'iOQ -N.-N, cosa-N. cosGt= O

From the nrsl ualion \\"e gel N,=N.; replacing N. b~' N, in lile se-cond cqualloo, we oblain

Ns+2N,coso:_Q (4d)Now we have ene ell~lr loilial ll'flglhs by l, ... AB, and L\l.=AC~. The ncw posilion ofpoinl ti is oblaincd by ro13ling lhe elongated bats CC, and BB. abolltpoinls B and C. Poillls B, arW Cs will coincide al point A" moving310ng Ihe ares CoA, and B.A, which dUe lo tho SllJall dt:-iormalionma)' be considcretl 3S slraight Ines perpendicular lo CC. and BB,.respt.'C ti vel y.

The l1ew posHion or the side bllrs BA, and CA, is shown by dottedIines. Sincl' tho entl 01 Ihe middle bar s also laslened lo lhe hinge,il will also come to poiot A and elongalion bl. will be equal ta AA,.

Aceording lo Hooke's latw. ihe elongallons .dI" 61" amI 61. of al!the thrce bars will be directly proporlionaI lo Lhe {ensile lorces slrel

Ch. 41 Ddg~ "1 Sla/icul/'l I"d.>tmll;nille SI/51rms

ching them. \fler linding Ihe relalion between Ih

"Complimlcd CD$t$ (JI Ttnsion and Comp",ssion [Parl 11

In thl' statically indelerminate systl!m (Fig. 32), Ihe localion ofpoio! A alter dcformation combines Ihe elong:nions 01 all Ihe lheeebars. For lhe condilon 01 jOlol deformalon to be satisfied it is es!enlial Ihn! lhe elangalions should be in a definite ratio. This coodiliongives U~ lhe extra cqualion (4.4) requlr('] for determining lhe unknownforce.

v,HIlinuing lhe sollltion el Ihe problem \Ve pul Ihe value 01 Njrom (4.4) ulo Eq. (4.1) and oblato

N +2N EA, Q "r;:::-;coso;=

whencelrcm

and rom (olA)

N, Q (4.5)

(4.6)

(4.7)

II is evideo! (ram Ihe formulas obtained Ihat Ihe value al N dependsno! upon Ihe absolute values uf lhe cros,s-sediona] arcas A and moduli01 l.'lnsticily E, bul upoo tlleir ratio. By selting difieren! vallles alIhe ratio 11- t we ohlain variou~ combinallons 01 the forces N"N" and N,.

Knowing lhe force" and Ihe pe-nnissible stress we can find Al andA, hum Ihe CQndilions

Nt";;:[I1];,calculaling Al from Ih(' lirs1 conditon and knowing lhe seJecled

ratio II=AA....!' we can lind A."'~, This value ran be checked up, "by seeing II"helher il salisfles Ihe second conrlition o[ (4.7); jf not,

the value 01 A, is lound {rom this condtion, and A, IS delermined byIhe formula

A, =nA, (4.8)Thus, in a slatirally ndeterminate slrurlure witl! a given load we

may obtain a number 01 dilfl.'rent modifirations 01 force dislributionbelween Ihe bars by changing Ihe ratio 01 Iher cross-sec\ional areas.Le! us lake a oumerical example [or greater clari6calion.

Let Q=4 U; a-3O'; (cr,l=lllCXl kgl/cm'; E,=2xI0' kgl:cm',Icr

Ch. 41 Dcsign 01 S((Jlical/!I fndfltrmlMlc S!ls(rms

For preliminary cakulation let us assume sn arbitrary value ofn=~=lA.

Then4X 2X 10' '30-i'X'i'Oi cosN,=N,= 1.67tf

1+' 2x 10" ~~MxfXi(iicos"VN.= 2)(10' =1.lItf

1+2x i"XiOi (OS' 30'

From strength condilion we oblsinA ~-!E!! 167 cm'

,"" la.1 -1000'" .

As we have assumed A,=A., lhen A,=-1.67 cm'Let us check whelher these lIimcnsions will satisfy the strength

condition lor Ihe middle bar:

~;= :~~~ = 667 kgf/cm'> 600kgfcm'The assumed value 01 A~ is nol enough; It should be

A N. lUlO L 85 ',-=~= 600 =. cm

To maintain Ihe conclition A,=A. which lormed the basls 01 ourcalculation, we musi lake A,=A.= 1.&5 cm' instead 01 the rrquiredvalue 01 1.67 cm' oblained fram the first l:ondilion. In this way weshall have an additional reserve in Iho:! side ban.

lf we wish to avoid lhis extra reserve and lakeA,=A,=1.67 cm', A.=I.83 cm' (4.9)

Ihen forces N" N,. and N, will change immellialcly; Ihe ratio A,/A,will no longer equal 1, as assumed earlier. bu! wil1 beO.9. In formu13(4.5) the denominator becomes less and N, ncreases; in formula (4.6)the decrease in Ihe value o[ the denomnalor will be less as compare

12 Compliwrtd Cast:s 01 TensiaR "nd Comprtssion [P(4.12)

i.e. in arder lo select the cross-sectional arca wilhoul excessive reservefor anr va[ue 01 {he ratio 11 it s cssential that cos el. should satisfy con-ditioo (4.12).

Tabl~ 6

Rtlulls 01 Calcul.lloM for VuJous V.luts 01 n

.4, .. .4, m',

ft~ ~; !'I,-N, N N.lUl Uf)r.q"I ..~ ...uroed (,ro')

o., (.5(; LOO LOO 1.74 2.170' l.(jO 1.20 0." LOO 2.00LO 1.67 1.11 1.()7 I.lI.~ 1.85L2 1.75 0.97 1.75 1.94 LO'0., 0." 0.82 0." 2.06 1.37

eh. .fJ"

In our numerical nample ,"-e oolllin l000XI)(IO" h~ ct_ llOOX2XIO" E'nce

Since in {he given slrudure ct-JO', then for an atbilrary n we !halleilher hllve lo gi\'e uce:;.s reserve in one group of bars, or overload lheother ifOUP. The valut 01 /1 ilseU should bE' seleded frolll ecooomic consjdefalions. Table 6 conlail1$ lhe values 01 dilterenl quanlflies rarvarious values of /l. KnoI\ring Ihe crosssedional areM, lenglhs nnd]lkllerials of Ihe bars. we can select a combinalion whkh is economically mos! etieclive as [ar as Ihe cmt oS material is ooncerned.

19. The Eftect of Manulatluring Inact.uradeson the Fort.l'S Acll ng in Ihe flemenls o/ Slallt.8lly IndeterminateSlruclures

fll. 35

6

In Ihe pr~'l;CtIing secliom wt' cstllblished Ihe main ealures oS tlleI\'Ol'king ami dcsign 01 slalically indeterminale systems.

1. Th shart of force Ihal ilwill tah.

in Ihis seclion we shall studyanollltr propl'rly of slalically inde-terminate slrudures which is 01greal Ilractkal imporlanee.

11 is impcssible lo manuractureparls of struc1ures with absoluteaccuracy: small manulacturing er-rors and inaCCllracies musl alwaysbe laken i"lo acCOU"1. In a sla!ieally dtlerminale struclure. Iheseinaccuracies cannol give rise 10 slrl'SSl'S in Ihe syslem. Thus,lor eJamplt, if bar AB (Fig. 31) is madea liUle shorter lhan ilshould be accord-ing lo the drawing. alllhat will happen is a slight diSlorlion al tri-angle CAB. In Ihea~ of force Q the forces in bus AB and ACwill be equal lO tero irrespedive or Ihe manufaduring accuracy ofthe bar lenglh.

C side bal'5 by 4l,-A8._AC,_ Orawing Ihrough poinlsC. and 8, ~diculrs lo lhe lnilial posHions oi Ihe sde bars asexplalntd in 18, we get lhe point 01 junction A, of the ends of al]lhe lhree bars. From (he figure we may wrile dmrm the c:ondilion ofjoint defOlmation:

"(4.13)

Sin' lhere are no external forces and N, is a rompressive forcewhertas N. is tensile, the equilibrium condition (4.1) takes Ihe form

N.-2N,cosa=O (4.14)Replacing in (4.13) tr.l, and 6f. by Ihelr values

M, ~.!0,'AI. and M _ N.t. N,I, cosaE,A, '-l';A,"'~

and solving equations (4.13) and (4.14), we have(4.15)N_2L

, 2COSllN OC.A,.- [ O~'

l. 1+2E.A, fcJ7iJThe plus sign befare Ihe values 01 N, and N. signifies that our assump

lions abaot Ihe dlf('CIIOI1s 01 these lorces are ccrrt'C1.II should be poinledoot lha! in formula (4.3) ttK-length of Ihe middle

b;r can be replllced b)I. and not b)' l. -, bause 6 is an infinitesimalas fomp

CA 41"

Ihe nitia! stresses i"mase the non-unilormily In the ..,rorking 01lhl' bars and are therelort harmful.

Had the middle bar been !onger by ti, Ihe initia! slres.ses ....,ould haveh.d opposile signs .nd \\'Ould have levelled 011 to sorne ulefll lhe nonuniforrnil)' in force distribulion beN:een tlle middle and the sitie bars

"1FIJ, 36

un&!r Ihe adion or weight Q. In Ihis case, the particular properly 01Ihe statically indelerminale s}'stems discussed .bove lIo'Olild havehelped in belter \\:orking o lhe slructure.

Anothef eumple 01 Ihe expedienl use 01 iniHal slresses is putlingon the tyre on the ""heels 01 a roIling slock. The wh~l consists elt....,o parts: the central cast porlion and lhe st~l tyre ....hich is pul on

6

~~~'--- "If 91 11Fil", :I1

11 (Flg. 36 (a) and (b. The tyre is fastened lo the central porlion bymean5 al special flxtures; besides, iis interna! diameter d~ is made alillle less han d1 . Usually this dilfereflce is 01 the order 01 ~ dr-approximatel)' ~ d" Befare slipping the tyre on the cefllral portion,It is hule

Compliooted Ca.... o( T~,l5ion and CO"'I',,~5iO/l [Part 11

If we cut Ihe Iyre across Ihe diameler (fig. 37), lhl.! 1\\"0 forcl!sN lllusl blance Ihe lotal pressure on the inlerna! surlace 01 the("U 10fT portion of Ihe Iyre. Le! us wrilc down the equilibriumcondition by proj~'{:ling all Ihe forces on Ihe y-axis (Fig. 37). A presoslIn.' J ds atls upon Ihe elemenl or lenglh ds or Ihe lrrr; i1s projl'Clionon Ihe yaxis Is equal lo -pds sin a--/l; sin a da., brcause ds=

={ da.. The cquilibrium condilion !ak

eh 41 Dtsign of StutlC"l/y lnr/tluminalt Sy&ltm&"

-2x 10' kgf/cm'; A =7.5x 13=97.5 cm". lhen we have0= 2~~' =2000kgFclll'N = 2XlO"x975 = 195000 k,f -. lD5 tf

10002X2XIO"x97,!i 4330k l'p= 9Ox\l)OO = g,lm

20. ICl\liion 8nd Compression in Bars Madeof HeterogeneOIlS Materials

,Jlttl,

This lype of bars belongs lo the group oF stalically ndeterminale.As n example. we shall discuss how lo determine tlle dimension~ of acomposite column (Fig. 38) under Ihe aclon 01 l"ompressive forcesP. The column conslsls ul a rouml steel bl!r 01 diameter d, and 15

l +-d~ ......

'@},:, ', t

, ,_ d, '"-

located nside a bronze jacket 01 external diameter db anu wll Ihickness t.

Lel liS introduce Ihe lollowing notations:Ab - cr055wetion1 arel! 01 Ihe bronze pipe;A.- crosssectlonal arca oF Ihe steel bar;

10",,1, 10,1. Eb E,- permissible stresses and moduli of ('laslicity ofbronz(' and steel, rcspcclively.

lhe reqllir~1 dimensions 01 Ihe bar should be such thal ('nableil to wilhstand load P.

Lel us find slresses Oj. and o,due to load P over arcas A~ and A.,respectively, und write oown the slrenglh condilion.

lhe bar is axially compressed by rorces P applied al lhe centre orgravHy 01 lhe seclion llirough rigid slabs S whose delormations areoonsidered negligible (Fig. 38). lhe parl P~ 01 (he comprcssive [urcesis transmiltt! Ihrough lhe brom.e jacket,

"(4.18)

which bolh lbe bronu: jacket and sleel bar al !.he column (Fil!. 39)must shor1l?'n by Ihe same length M. since the top and lhe boltomplanes el both roinc::ide. From Hooke's law we have

t1l_.!Y.. ... P,IctA. l;r.

This is the second equalioo correlating PI> and p._ From (4.18) \lie'o,

P P A,E.,= tA;!;

Subslituling this value 01 p. in (4.17), \Ve gelP (1 +~!L)_p

1> Ab El>

'"'(4.19)

('.20)

The dislribulion al rClrC'tS betll.~ the demenls 01 slalically indettrminale struclUre5 depends upon the ralios or their crosssectiona[

~",-----~"P'~~' ----- -----------W+\-._._._._._._._.

,~----------------- -Fil_ 39

areas and moduli 01 elaslicily. From ~ullion (4.18), taking intoconsideration (hat

'"'

p.A, =0,

,,e find that the ralio of Ihe stresses in brome and steel depends onlyupon the ralio ol their moduli el elasticity:

~= E,If. E,

o. 41"

(4.21)

and Ihe sl~ are directly proporlional lo Ihe moduJi 01 elaslkily.Aswming Ihal under compres~ion E.-2XI0' kgl!cm' aod E.=lxX 10' ktf/cm" (Table 1), it is obvious that slresst'S in steel bar willalwl)" be 1\00 limes higher Ihan lhe slresses in brome jacket. rhe per-missible slresscs for sl("el are U!ually Ihft.'(' llmes grealer than permis-sibIl" stresscs lar brome. TherefoR', if !he stres.ses in lhe brome jacket

a~ ;qual lo lhe pcrmissible slress Inr bronU', Ihe stre:sses in lhe sltelIJar will be smaller Ihan Ihe permissible stress lor sleel. Hence lhedimcnsions of Ihe colurnn are oblained (rom slrenglh condition ofbronte jacket under compresslon:

p po~=o E - ( A t.' ) '-;.[0.1A.+A.'t A. 1+ A: i'Let P=2S If. rhe ratio A,IAb of Ihe cross-secllonal areas is usually

seleded rom designconsiderations. Let AJA.-=2, and lhe permisslblestress [ObJ-SOO kgf/cm'. Equalion (4.21) will lhe" be wrilten as

2>000A.(l+2X2)';;;;SOO.

\loMe/mm

''''''A~~~_IOcm" and A.=2xlO_20emt

TIte diameter of tbe sleel bar is calculaled from lhe coodition

~ ;> A., wherelrom d. -~= -V ~~I~"CO S.05an z 51 mmTIIe dimensions of thl" bronu jacket scetion can be round ir we assume

a particular value 01 waH Ihiekn~ 1 lrom design coosidentions. LelI-S mm-O.S cm. No\\', applying lhe approximale formula lar a nng.Wl" have

A.:ro;; 11 d.t, wherelrom db~~= 3,,10ti' _ 6.48 cm ~ 65 mmni _ x,~

lhe ddormalions of such structures are calculllled according to thegl"neral principies. Sinct Ihe sleel and bronze porlions 01 Iht' colurnn$horlen by the same amount, it is imm31erilll ....hich al the formula.sin equation (4.18) is employed for calculaling al.

21. Slresscs Duc to Temperalure ChangeIn $talically indtltrminale $yslems. s1r~ withoul any external

looding occur nol only due lo the inaccuracy oC manulacluring andassembling, but lllso due lo a change: in lernperalure.

Considerable siresses 01 Ihis lype may arise: in rails weldl"d into aamlinoou.~ llne. Tht rails are subjecled lo tensile or cornpressht slres-

.. Ip"r 11

S('IS whef1 Ihe lempenture changes wilh respect lo Ihal al whl.:h lIK'Y...."re "",,[ded. lile problem may be chemalicaJly exprC'Ssed as 1011(N:s:\\'e ha ...e a reslran~ bar Ilo'ho.sl! both ends have bten rigidl'l Tixedat atempcrlllure t,; ftn lhe stresses rising \\'ntn thetempeulurechangeslo t. lFig. 40). Tite length al he biU is 1, cross-setUonal area is Aand modulus o elaslkity is E.

Lel us ilSCertain Ihe forces which \\'m ad on the bar 9.'heo 1M lemperl'ltureri.ses 'rom tI lo ' . The bar will tl'nd lo eJangate and push apart

/A--- - .--fl ~, , ti

Fil(. 40

lhe supporls A and B. lhe supporls will resist Ihis wilh rt'adions,Iirecled asshown in Ihe figure. These l()("ces will cause lhe bar lo berompressed.

These lorces cannol be round from static condiUons, becausc 1111Iha! Il~ come lo know Irom Ihe single equilibrium condmoo is lhalthe reactions al points A Ind B 3rt equal in magniludt and opposilelO each olher. Th~ value 01 the reactiol'l P rernllins unklJO\l,T1, and ht':fIQlhe! slrudur~ may be oonsidered slalically indeterminale.

The! addiliooal equalion can be writ\en 100m lhE' oonsideration lh81lenglh f 01 Ihe reslrained bar remains unchanged in spiteol !'le changein ltmperalure. This implres that shortening lJ.{p due lo fo.n P istqual in ils absoluh.' \'alue lo {he temperalure elongation t'!.l, whkhIhe bar \lo'ould have experienced had lhe end A been fixed and end 8Iree lo loove. Hence

(4.22)This is Ihe oondilion 01 joinl deTorrnation; il shows thal Ihe lenglh

01 [h ... bar remains constan! despite lhe tcrnpcrllture change, since JI~ nol tear away frOll1 (he fixOO supports.

Sioc\!

"/p-n. and lJ.1,_4/(I,_I,)

where ~ is fhe linear th

el, 41 D~"grl o{ 510.1ko./I'I frldtle""irlille 5gsltm&"

Leo. lhe stress due to lemperture change in a reslrined bar 01 uniformcrossseclional area depends only upon the modulus of elasticily or lhemlerial, lis linenr expansion coerftcieont and Ihe lemperalure differencc!ami nol upon ils lenglh or lhe cross-secllonal areoa.

force P may be calculated from Ihe expression

P_aEA(l._I,)

Fi. 41

PI,

, 1IfO~Z II

'1IIIII~, I

".II jI1

II~ __JT

In lhis example. if 1;>1" slress a lIIill be compressive, beca use lhediredion 01 reacUon P illSitle lhe bar has been considered posilivc.1f \\'e follow Ihe generally accepled oonvention of writing Ihecompressive slresses with a minus sign, and the tensile slresse5 wilh a plussign, lben formula (4.23) should be..",rilten in lhe following manner loautomalicO.3 mm (Fig. 41). Find lhe slresses in bolh the pars il lhe tem-perature increases by 50C, a.=I25XIO-'.

locrease in lemperature caU5:CS elongalion 01 the bar by 61 10 andIhe compression from the support reacUon P resulls in ils shorteningbyM". The dilf~rence 01 thestl two deformations (in absolule value)4_1110

"Par! fI

is 4. (see Fig. 41):AI,-fJl'=,

This is lhe condition 01 joinl defonnation. The respecti\'~ valul'S offJ, IInd &1,. are

6.I, ""'cd{'.+IJ. l,\ll'=~+ PI. _ PI, [1+"A'JEA, "I'A;"lA; l,iI.Thererore

al (ti +l,)-:!t [1 + ~:~:] _A.whencefromp [a/(/,+'.) a.IEA. 1125xtO-1xlOOx50 O.03]X2X10'XlO- 1.['+.:6] - ,,[,+"lXlO] -/, A. 4O'X!ii3

= 9300 kgf

Stress in the upper porlion Is0'_ :, -~=930kgfcml

Stress in {!le lower portion is0"_ Lc:~=465kgI/cm1A, 20

(bolh lhe st~ are compressive).Had lhere been no gap A Ihe force as well as lhe slresses ,,"ould

ha.e increased 1.92 times.

22. Simullaneous Aceount for Varlous factorsSometimcs, in slalically ndeterminate syslerns, we have lo consiller

simullaneously lhe elfecl of external rorces, change of ternperatllTeand manuFaduring inaccuracies. lhe problem can be solved in twoways: fint method-simultaneous account for alJ the fadors. In lhiscase Ihe eGuation of jolnt deformation must conlain terms reOtclingthe elfed or alt the faclors (load, temperalure and manufacturing inaccurades). The orces and strwes oblairll!d as a rtsult o soch a COmI"U-tatian an' linal.

Jn he sa-ond melhod, ft compule separately lhe forres and slres-ses due to lhe load. lemperature and manufacturing inaccuracies. Inolher Il:ords, a number of separate problems are sol'led. each problrrlltaking mIo account only one factor. The final forces and slresses areobtained as the algebrak: sum of lhe values oblaintd Imm each 01 lhesolulions. The setOl1d method is often slmpler and more convenlenl.

CIl. 41

, ,

1 J

iJl,-o- ... 1._iJz_1 ,----~-~,

FJg. 43

8

Fig. 42

" ", ,

) , 1 J..-, .~L ,er-A

pi

aHhough it calls for more clI!Cullltion. 1t is known as the metlwd alCWllulative action ollorces. This method is valid beeause 01 Ihe appli-cability 01 Ihe principIe 01 superpositon 01 loras. When dE.'formaliomaresmall in magnilud". Ihe ddormalion eallsed by a force or a group offorees either does no! affed Ihe dciormalion due lo another lorct' orgroup of rarees, or lhe effed is so small (less by an arder) Ihal it may

"

be neglC'Cl('d. This principie is nol applieable for extremely l1exibleor highly deformable slructures like long thin bars, membranes,rubber parts, ele.

\Ve shall solve Ihe rollowing example lo illustrale Ihe lechnique01 simultaneous aceounling 01 various [adors.

Three parallel verlical rods 01 equai length 1=2 m support a rigillbeam AB lo which a force P=4 tl (Fig. 42) is applied. lhe dislancebetwt'en lhe rods and rrom Ihe middle rod lo Ihe poinl of applicalion01 force P art' a= 1.5m, b=l In and c=O.25 m, respectively. lhe middlebar is shorter than its design lenglh by 05=0.2 mm. Data aboul thebars are given in Table 7.

T~be 7

No 01 r tensile.

Tht' re.adions at Ihe supporling points, equal to Ihem, are shown in,.

.. (Patt 11

Flg. 42. As lhe rorces 3re paullel, IVe can "'rite down only \Va cqullib-rium equalions. Tlle flrst is Ihe SUIIl 01 Ihe projcclions 01 lorcl'S onlhe vertical axis and has lhe lollO\.1",,2~S8""129kgf/cm'

CIt. 41"

lile probl..m cnuld also have~ solvf'd by considering separaltlylhe etet 01 load. It;'lnjll.."falure llnd mnulaclurin m,ll'Curac~ anJsub5loquently adding Iht;' slresse:s. lile resull would ob"iousl) 1111\'" betnthe HUle.

t 23. Mace Compllcaled~ of SfaliullyInilelenninale Slr~turt'S

In alllhe stalicalll' ndeterminat... s)stems 111111 we llave C0rt5Idf:red(111 oo\\', lhe number 01 unknov.'n lorces execNW by one Ihe numm01 stalie equalions. lhey \\,t;'r~ aH firstorder Slatically indeterminaleproblems: ene of lhe unknown orces mar be eonsiderell as o redundanlunknown, \\-hich eannot b~ llelerminell IrOlll Ihe s11l1ic equalions.

lhtre may be case! when lhe number 01 these rNundanl unk~'mis grealer; in SIJI:h cases il beromes n~arr lo wdle down an equalnUlnbtr 01 exlra equalions Irom lhe condillons of oinl deformaliOI1of the S)'slem. lile slructure shown in Fig, 44 lila)' bt laken as anl'xaluple: a \'ecy rigid bar is hinged to a f.>;etJ supporl suspendl'tlfrom thre(' rods nd lcacleWe may wrile down Ihr~ sfatie equations for bar AB. lhe numb~r01 unknowfls is however five: forces ifl Ihe three rods and lhe horiton!aland v('rtical eomponenls of lhe reaetion af hinge A,

lhe extra equaiions can be wrilten by conslderlng Ihe delorm&lionor lhe syslem. s~ v'e &reeonsidering bar AB lo be very rigid. ilsdeiormation may be i~ored. Remainng slraighf, ji v.'ill oo.:upy po:silion AB,. From lhe slmi1aril)' 01 {rangles we may lind the relalionbelll'l!'ell M" i., and 111.: Ihis will ve us 11.110 ulra equalions. name

I~';~_.!L and.11,

86 CQmJUcaiM. Cases 01 TtflSiI)lI amI Compre~oll [Parl 11CHAPTER S

Account for Dead Weight In Tensionand Compression. Design oi Flexible Strings

24. Selecting the CroS$-~ctional Area with fhe Account for IheDead Weight (in Tension and Compression)

Tl1l no\\', in delermining tlle externa! lorces stretching or compressing Ihe elements al slruclures, we gnoTed lhe dead weighl of Ihesedements. rhe question arises: Ooes nol Ihis simplificalion introducea considerable error in lhe oornfutations? Le! us therefore determinelbe slresses and deformations o a stretched or compressed bar ae-counling for lts dead weigh.

Lel a vertical bar {Fig. 45 (o)) he lixed al its upper end and load psuspended from ils lower end. The Jength 01 Ihe bar is l. Hs cross-sec-

"

I ~ I N

L- 1 M-,I Jp

"!Fig. ~5

lional area is A, modulus of elastieily E and specilic weight y. Lelus calculale Ihe stresses in secUon MN located at a distance x frolnIh' Ir'e end of the bar.

Cut Ihe bar Ihrough section MN and se-parate Ihe lower parl 01Jength x loaded by force P and Hs own dead weight yAx (Fig. 45 (b).These lwo lorces are balancl!d by stressl!S acting on face MN lram cut-oll portion. Thes!' stre.sses will be normal, unirormly dislributl'd lIntltlirected outwards 01 Ihe portion 01 bar under consideration, Le. theywill be lensile. The magnilUdl! of these strcsses wJI be

P+,\,Ax P )a (x) =-A--"A+Yx (5.1

Thus, when Ihe tirad weight is acrOLlnled lar, Ih(' normal slre5S('sarE' loLlnd lo be nol constan along the length of the bar. The moslstressed and lIenee the crilieal section will be the uppennosl section

Ch. 5] Dtlld W~gh(. Dcsign 01