theory and anlysis of nonlinear framed structures - y. bin yang.pdf

THEORY & ANALYSIS ---OF---

YEONG-81N YANG Departmenl of Civil EngIneering

National Taiwan UniYerslty

SHYH-RONG KUO DepaItrTIent 01 Harbof and RiYer El~ .. *-~_~o"'ilQ

National Taiwan ooean UrWersitY

PRENTICE HALL New York London Toronlo Sydney Tokyo Singapore

Fint poobI,ohooI I~ by --~ sa. A ~ ( ....... ) .... LId _o..ipwt 11oct 00611 ,..... I'ao\jq Ro.d S_o:Ill

o I~ Simon A S

Contents

fOl'l'word ~r.c:~ ACMO"'IflIC_nlJ Ust or SymbolJ

1 Introduction 1.1 BlI!kground 1.2 Notation and nomeDCLa1Ure 1.3 Ddinition of stnillil

\,). 1 Grtta-ugrangc stnoin tcnsor 1.3.2 Green SInI;a increment tell$Ol" 1.3.3 Updated GrD stnin illCll'menl ttnsor

1.4 Ddlniticla of $lrC$5CS 1.4. 1 Seamd PioIaKirdIhoff' IIU'CSS tensor 1.4.2 Cludly wess ttnsor 1.4.3 Updated Kirchhoff Stn:s5 I(mar 1.4.4 T'ransfonnation rules

I.S Incrcmentl' constitutive laws

.m ..

.. I ulil

I I 4 7 8

JO 13 IS

I' I' 18

I' "

.; ,--, . Prirw;iplc of winual dispiace""'lllS

" '.1 Incre .... ntal L.a&;ran,ian rom1u~lions '" 1.1.1 TouJ Laglln&ian {ormuiltion '" 1.7.2 Updltcd i.Agangian formulation " References "

2 Linur Anllyslf and E"m~nt Quali ty " "' " 2. , Discretization of i\tIUCIures " 2.' Derivltion of clemcnt slirrntSS ~lrices "

'" Planar frame dement ..

21' Spaclc frame dement ,.

'" ~imcll5io""llnlSS clement 62

2.2.4 Planar and 5pk'e trllSS elcmcnlS .. ,., Formation of structure equalions 66 , .. Solution of limulliJleOUS equations

" " . Quality IeSIS (or linear elements " 2..5.1 ConvCf,CO enlcna " 2.5.2 The ~ldJ \ell 80 ~3 Eigenvalue test

" 26 Rigid body Ie$! for nonliMH ck"",nu " 2.6.1 hxnmcnlJl' Jliffness equations " 2.6.2 Rigid body rule for initially Sl/'l:sscd clements 86

2.6.3 Ri,id body lUI 81 ", Gcneralize4 ci,cnvlhle leSI for nonlinear clcmc/lUi

" References "

, Noa.lincar TnlSHS a nd IJl(ftlMnbl CollSlltuUvc L..wl .. ,., lotrodoction

" l2 Formulation of nonlinear planar trusS cle""111 " ,., Pbysical intClpl'Clllion of dement ml~

'OS 3.3.1 Mcm\x. forces duc 10 stretching ' 06 3.3.2 Mcm\xr forces due 10 rigid body rOUlion ' 09

l.6 lIighc:r order stiffness matrices in symmetric form ,,,

" Genertliulion 10 llIree-dimcnsionaJ ca5e

'" , . Fora: _cry pI'OQCdllru based 00 malri~ cqullions ",

,-" vii

3.' Case 51udies by finil~ dement I~h '20 3.7. ] Two.lMmb(r uuss '20 3.72 204-membu shallow dome '22

3.' hlCfcmcnLal CODSIilU!i~ laws ". 3.' formulas for caJcullT;ng dl!ICnl (oras '26

3.9.1 TauJ.form foonulas '26 3.9.2 lnculMnwform formulas '28

3.10 Euci ~lulioM for 11. and UL formulat ions "' 3. 10.1 Uneuly (Iascie maleri.1 (!Mlen.1 A) '" 3.]0 .2 NonlillC~rly elutk: mlterial (mile rial 1/) '" 3.11 !ncrcmcmal (orm vs.. 1OI11form solUlloM '" Rdercoces , ..

, NonlJ llur An. lysll of Planar Fnm.,. ". .. , lnuoduClion " . ' .2 PriDcipk of virtual displKClIICnlJ ".

4.2. 1 Two-dimensional beam '" . 2..2 Statics.nd kinematics '" . 2.3 OeIM:.aI vs. simplif~ theofy ISS

'3 Differential equations and boulldaty conditions '" 4.3.1 Oenenllhcol)' of two-dimensio ... 1 beams '" 4.3.2 Rigid body lest ,.,

4.3.3 SimplifK'd IhWI)' of Iwo-

c....... i~ .19. 1 Symmeuic (rame I- Simply t.uppol:nl case "7 6 . .1 .1 E1aslic stiffness mall;. JJ2 6..s .2 Geometric Sliffnus mau;x lined on generaL

.- J" 6..s .3 Geomeuic stiffness mau;,t based on iimpliliw

.- '" 6.3.' EJElcmal vinUllI work iI!o;nmc:nt J60 6.S..s Element stiffness equatIOn 362

Rigid body ~SI 36J ' .7 JOi/l1 e

, ,-

6.10 Numerical examples '" 6.10.] Symmetric fTJ.me I- Simply supported OUI or ,,~ ",

6.10.2 Symmeu>c frame 2- Fi~cd oul 01 plane m 6.10.3 Symmetric franw. 1 with lip ~n' ", 6.10.4 Fiue! Ingll frame wltlt momcnt kJad '79 6 .10.5 Find angle'" 6.10.1 Find IngJed frame ",;tII rorsional load '" 6.10.8 Single ~am in bo:l!ding 38'

6.10.9 Angkd (",me in bending 38' RcfcmlCeS

'"

7 ThNI')' I nd Analy5i:5 on Ruckllnl or CurYflI ~.ms '" 7.' lmrodUC1ion '" 7.' !It.lics and kinematics of CIIrvod solid bnms 393

1.2. 1 DISpI.uIM!IlS.!IId 'lnoins '96 7.2.2 Slre5KI and ~S'Iion.1 rOf(:~s '99

7.' Equations of equilibrium rOf crossseCliooll fO'a4 '" ,.. Principk or .. inual displ.act;menlJ ."

1.U Strain t"".gy due 10 line .. uial SlrlIin .,. 7 . 2 Sm.;" energy due 10 liMar.\hear sinins

'" 1.4.3 POlen"., ene.gy due 10 uial Wen '" 1.4.4 l'oIenli,1 eMlgy due 10 tninsvcf$C she., SllesKS '" 7..4.5 POItnhal energy dIM: 10 normalsues5 "n '" 7.4.6 PoIenlial energy due 10 dislOl1iom11 shear stress m

1.4.7 PQlenlial enelgy due 10 !IIdial $UC:U '" 1.4.8 External .. il1ual wurlts .,.

1 .. 9 Incftmcntil ";111111 wort equation '" B Differen!ial (qUl!ioas and bounduy cond;UDR!I '" 7.' Rigid body lesl '" 7.7 Buckling Inalysis by analy!ical Il'P"'*'h . .,

1.1.1 CUrvtd bum ",*1 unirorm bendi~g . ., 1.7.2 CUrved beam "'*' nodial loads ..,

7.' Finile demenl fOlmul'I>on ..,

l __ _

,-" ,;

7.8.1 ln1c.poIa,ioo funclions '" 7.8..l ElclMn' $Iif(ness cqUilioo ".

7.' BpcklinS 'nlIlysis by CUl"~-bwn c1CllW'nl approach '" 7.9. ' o.,..,w ~,m under unif

8.8.2 Oispl:ott.mnt cont rol meU,ad 8.8.l Arc length method 8.8.4 Work control method

8.9 Gcnerllliml displacement toIllrOl method 8.10 GcnenoJ YS. Qlrrerli sairr_ plrllmelcn 8.11 Algorithm for geomctrk nonl iM" anaIyt:is 8 .12 /'Juml:ric:al examples

8.12. 1 1'wo-membcr truss 8.12.2 Shallow .rch 8.12.3 Circular arch wilh ~mr"ll load 8.12.4 Circular Ilch under uniform bending

RefefC'1ICU

Apptndl. A Apptndll B Apptad!1 C AuUlor Indu SutUKlladex

I.JsI or In lr,nl Malrls i>nIHdun for Unumfd Buckl inl Analysis Etrtd or T"'lIaI lfd H'eMr Ordtr Terms

m

'" '"

'" '"

'" '" '" '" 55.

'" m

55. m

'" '" '"

Foreword

The seardI fOf ways 10 rcpraenl tile true nonIinurily ofstJUctures I"" back 10 Renl;$S3nCI: times, lnd )mSCn! theories of nonlinur elastic .nd inelastic behavior are the usulc of approximately two hundred yo:an of ~y development. But only r~nlJy has the compulCr mack il possible 10 put mucb of Ihis knowledge to ulle ;n design. In !he mid 1970s,' grOUp.1 Comrn joined the drive to uploil this opponunilY and to advance. If we could, the existing knowledge of IIOnlilitar heh.aviot. By thaI time, small computers powclful enough 10 enable design etlallIfS 10 11K hitherto impnaicablc mell'lo

I

principles of aUuctur.J mechanics 10 practiable oompulalional proce.-du,,~s. ~ work doM by Profe$$Oi Yan, and billludcnts, nocably Dr. ShybRon, Kuo, hall bcerr widely published In leadi", professional journals. II is known for its precision, imaginalion, and lhe insighl il has gi~en inlo complu problems of nonlineJr behavior.

This book is in pan I synthesis of thll ruelrdt and ., sudI, ;1 upruses OM poinl of ~icw in In evol~ing fie ld in whieb there lIe still numerous Ipproac:hu. But it is more thm tha.. In its coverage of undr:rlYiD& principles aDd OIbcr COoDtemporary ruearcb, ir is I lulbook from which IlIe stlllkn. Qn obtain I .borough ulllkrslanding of 1M 51a.e-of.tM111 of nonlinear elutic ..... ,ysis of framed structures. And il shook! be I basic I~roe for IlIe resurche .

This book, theref .. , is limel)'. I. J.hould prove 10 be map contribution to .M cause of advancing the lime when realislic trCalmenl of suuaum nonlineari.), be

Preface

This book has bem wrinea in recognition of tile great advlllCel in ~nl yCIll5 of the geomtlfie Il(Inlinur analysis of framed stnICturu, and tile lact of ~Il-orpnlzcd book dedicated 10 this sub;ect. It bas b'n ~Ioped as result of the resew:h worD conducted by the autbon and co-workus, part of whicb has :appeamI in I scm of ledlllial pa~rs, limrd I' tbe application of tile finite clemod 10 nonHDUr lnJly$i.s of framed Sln>duru. However, this book II DO( inl(lKied IMrely as I collection Of repetition of any previous wom. To meet the pcda,ogital needs. the materills acquired from different sou.rrpniud ia I unified IlWIDI:r, while new romtituC'nlS have been added 10 n:1kcI tile t'Um:nl trend toward I'llionaJlzatiou of _l!Dear aru.l)'5i.s proooiures. We be!it.lc thai with !he pt"C$tllllMthodolo&Y. I~rs sbuuld be ""Ie 10 di$a)vc. the myl.lery

,and beauty of structural nonlinearity. Over !be pW tluee dtadts, tbr. rlllile eknxlll IMthod, stim"ll,ed

by !hi: rapidly growing power of electronic:: digitll o;oml"'lers, has SUl:lI&tllened its po$ition ip Q(lmpulaticNW ElKCbanics from its urly $IagC of pcrfonning linear analyses for simple probLcIllS into an en of conducting the mofC dullicnaing tub, such 1$ DOfllillur, inclutic. dylWllic aoalyses and 10 011, for problems thai _ len, hundred, or Ihou$lnd times more complk.llcd than Lho$e ever aucmpled btCore.

Nowadays, many people I~IICIIO view !be fmile ~Ie_nl "",!bod as well-ulablisMd I0OI WI c:an be readily appIkd 10 !be solution of various oonliocar problema. Expcriwce "- 5bowJI III tbal1his c:an be 100 optimistk in many cues. as !be varioIIs pbuc:s involvc

xvii

or (ora: _Ibod. This book is wriucn for engi ..... '" and ... oenti", woo are """I].

-.:quainled with the tllcory of matrill SUUClUnJ aIIIIlysas and have ~ some uPOSUIl! to the mechanics of dcfOl"llUble bodies. The materials pruentcd bell! nn be: ~1tW as the subjKI maltcr of. thlu-credit. houl, ORe-Kmestcr COUISt on adYanttd 51ructUIII analysis for tbe i5Cnior level undelgraduate or graduale sludenls who have Ilkcn coulStS on finite elemenl method 01 malt i. structulII analysis. 1'01 researd~rs already working on the nonlilltilr and buckling behavior' of slructures, lbe book 0((.1& I Slate-of-Ille-an leview of lbe OXImpulltlional proce_ dules thaI are deeply rooted in ooolinuum mhanics pr1nciplQ.. The crileria pruenled in the tUt CUI be employed 10 ehed: wlwlhcy have adlievtd, while lhc p=uc:nl fnmcwork c;an be relted upon 10 utcnd lUCardl into arus DOl c:ovend by !he lUI, sudlas nonunifarm 10rsion. upcml members, plmk and ykld m1tInisms, eanhquake effeas, dynamic and impacl loadings. nono:ot\5ervllive loadings. elC. Thc IcnSOI noutlions adopted bere sboold plesenl no problems 10 firsl ycar graduate or senior unde rgraduale students. since in most cnuol definit ions. with only. limited number of operalions performed 011 them.

Ortly suucturrs thaI are of \Ix frame Iype. and composed of solid cross sealoas are ronsideml ill lhc lUI, whicta include in pania.>lar the following four calegories o f .IlI\ICtIlrcS: uusscs. pllllll frames. spKe fn.mes, and curved beams. Eadt of these fOUl calcSOries is C/Jvend in "parale chaptel, upcthe Jpace frames, whid! ". coyered by two ehaplers. The lUI hn been organized in a ptogIessive mantler in Ihal il sc.rtS wilh lhe simpiesl theory of lrusses and ends wil h lhe mosl compliated theo)ry of curved bc:ams, followed by chapter on nonlin-til solulion pr-oc:edure$.

In ChapleT I. the "nins, wessa. and oonsIltuli~e laws thlt are 10 be u.sed tlIrougiloul the boot are filS! introduced. The principle of virtual displacements llUillble for incRmenUlI farm"laliom of the lIgrangian type is the:n delived, whid\ l.ys very IIItlural foundal.ion for .11 the theories to be dcli~ed in laltt ehaplCrs.

In the firsl half of Orapler 2, Cl)nvcntionallincar Ilnalysis procedures for framed structUles Ire oullined, followed by review of the qualilY lests for line.r and nonUnell finlle elements. Of Ihese IeSU, Ille rigid body Ies! appua 10 be of pamnount imposance .inoc il provides Ihe guidelincs for alatLalina the element forces in a Slcp-by-SlCp non-linear aIIIIlysis. The planar and space truss clements are derived in

Jviii

OJap'er 3. Of paniClllar inte~ in this chapteT is the formulation of 1M pi'"Odure for obtaining exKl solmions for trusses of . ny complexi-ty loaded into the range of large suains. Two key issues are addressed in this regard: tile updating \If material coRStants and tile aolcuJatioll of bar fOI"5.

In a.lplcr 4, planar frame elemen15, as wen as buckling differential equations and nalUral boundary conditions, Ire derived for lwo-dimen-sional beams based on tbe Bernoulli-Euler hypDthesis of plane sections. This chapler gives us I very good example of how pDwerfultbe rigid body \eSI can be. II can be used nOi only in the tesl of a finile elemenl and ils underlying tbeory, but also for calculating the member fOl"ccs in an incremental nonliDe.af analysis.

One key step in the buckling analysis of space frames is that an phys.ical relalions ~ld be established (or lhe buckling configuration of a wuaure, based on tile Pfinciple$ of continuum mhanics. By Slicking rll1T1 ly t\l Ihis rule, In loalytiao) appro;och based on the COm_ monly used buckling equations is prUenled in OIapicr 5 for Ihe analysis o( the lateral buckling IoIod of some simple (rames, whicb are tlIt:n iranslated into the finite elemenl equations in ClI'plCf 6. The physical link belwun the two a~ helps in Tcsolving some existing conlrOversies on the illH:kling of space: frames. Also presented in a.apler 6 is I general thne-dimensional elemenl suitable for the analysis of space framcs.

In OIaptet 7, a comprehensive treatment o( the bl>CkJing of Cllrved bea./n$ is Pf~nted. One (eature of the curv.d beam equations present-ed in this chapler is that they can be derived either from the principle of virtual displacements or (rom the SlTaight beam equations. By sticking to the rule that IU physical relations should be established for !he buckling configuration of a struclUle, il is demOnSlnlted that the straighl-ileam element can be employed 10 y~ld solutions that are as accurate as lho$e by the QIfVed_beam de"",n!. In Ihis regard. previous arguments conarning \he. inappliCllbility of straightbeam elemeots to modeling the bu~kling of eurved beams have bn shown 10 be invalid. The book concludes with OIapter 8 On the procedure of iT\Cfemenlal nonlinear analysis for structures o( the framed type, with surrlC~Dt details given. Particular emphases are placed on the updating of geometry of structures involving finite rotations and the solulion of nonlinear matrix equations by iterative procedures.

In shOI"1, thi5 book is intended to serve as a bridge that connecl5 the tnld.itional continuum mechanics with modI:m computational pr0ce-dures. "The endeavor to ..... it. this book is result of !be excitement

.nd ellallen", t/W have bo:en uperien'eol"",lIy led 10 1IIc wl;ling or this boolt. Owing the prepantion of 11M: lIWtusalpl. he has .eaiVl:d constant encourage-IMnl and spilitlW support from his colleagues at !he National Taiwan Univ.Il;ty and ocher irmilutes, which Jbould include, in PfJtien Ycn (former Dean, College or Enginrios), ChauSbroung Ych, OIing o.um Olem, Ow:n-Oiana 1

IIolidlya. IiO as 10 rnabk lite, II~ and the!ir dad to wo

Acknowledgments

Parts of the materials presented in this book bay~ been baKd on papers publW!ed by the authors and oo-workcri in I number of journals, especially those listed in the ~rcn:1lCCS below. Efrons hive been undertaken 10 upda.e and rewrite the materials aequi.red from eac:b 1IOIlroC, such IhIl unified and procre5Sivc pre5(lItatioG can be adlic:Ytd throughout tbe book. The IU!bon; would like 10 thank the copyright holders for permission .0 usc 1M Aid materials In the book; in particu tar, the followinl' (I) The materials from KIlO and Yang (1991), KIlO t './' (\993), UU and Yang (l99O). and YanS and 1

l

nil

of ~ frames ... ilh finite fOIltionlo," J . Sln,c,. t:~,. ASCE, 119(1). 1-1$.

leu, L J., Ind Yang, Y. B. (1990), "Effects or rigid body and 5UC\C.h-in, on nonlinear arWysi$ ofu-." J . S"yct. C~, .. ASCE, 11 '(10), 2S82-l>8.

YIII" Y. B., and KIlO, S. R. (19911), "Out-d-piIM ",,"Iina of IllgIed fnmea,"/~I. J . ,1111. Sci., 3)(1). 5s-61.

YIlII" Y. B., and KIlO, S. R. (199]b). "Consi~nl f~ b\K:klina lna]y.is by finite element IlKlhod," J S'rNCt. E~,., ASCE, 117(4). 10$3-69.

YllIIa, Y. B., Ind KIlO, S, R. (199lt), "Buc:klinll of (rimes under ~lrlou5 torsional ]oadings," J. Eft,. MuA., ASCE, I 17(8), 1681-97.

Ylna, Y. B., .nd Kuo. S. R. (1992), 'Frame bu(itlinllanl lysis ... ith full consi!lclltion of joint compatibililies,' J . Eft,. Mull., ASCE, 11 8(5),811-89.

YII\&. Y. B., and leu, L J, (1991), 'ConsIiluti~c I ..... and force IOVcl)' proo;edu~ in nonlinear ..... Iysis of Iruues,' Ctlmp. M~lll. AppL M1t. E~,., n , 121-31.

Yang. Y. B.. and Sbieh, M. S. (1990), 'Solution metllod for nonlinear problenl$ with multiple aitial pointI,. AIM J., 28(12). 2110-16.

List of Symbols

A &eMnl &uidl:line 10 the notation UKd ill the Ie", i$ given in Sec-lion 1.2. The followina; ill lis! of the ,ymbols USI:(\ IlIrougbool e.cll chaple,. ALl the symbol' are defined wilen they fLrSl appeu in the .eKL TIle left ,uperscripl of I symbol dcllOle lhe oonfigunriQn in which the qUlnti,y 0IJrl, while. lite: Loft liub$Cripl di:=noles !he configurlIllon 10 which the quanlily is lIII:asured. The Idl subscriplS may be dropped in lho$e cues wilen: the ,.fnem:. oonfigu'liion con be easily icknlificd, Quantities lilal Ire generaled during cach illaCment.l seep Ire denoted wilh no Ii'll liupcrscriptS. Matrices and .. ectors are c""lO$Cd by brt(:kcll I J aDd bnceI { }, respectively. A rAr [A I " _ , a. G,._ ,I>. C C C, C,

Irea of member aoss 5e(1ions Inlisymmetric paIl of [i,] matrix (Ch. 6) mall;" [AI in global un (Ot. 6) inlcgrlliion ronmnlS (01. 7) intc8J3I11on CO/ISWII$ (01. 7) nltOid of CfOIi KClion bounduy C'IIlVe of CfOIi $lion (Ols. 5, 7) inllial undcrOfmed configuration las! alcut_ted ronfigunltion

uii;

uiv

C, C, C" ,f: ... ,C~ [CJ. [C,I

(C) CSP ,

D, (D( ,

('(

"

" '"

" " , , (,} {e' Ie,}, {e,1 (i,). {l,} F ... F ..

'F.. 'F,. 'F. 'F.. 'F,. 'F. iF. F (F}

f f,

l!~ !t. lI. :" If}

l

CUtRnL dleformed configuIIILion CURcnL dleformed oonfiguIIILioft (01 . 8) bsI alcubted oonfigullliioft (CII. 8) OOIISIilulivc twfflCicn15 refellN 10 Co and C, curved-bcllm twfflCicnt maLrias II %.0 and %./(0..7) COIl$InIint vcaar (OL. 8) cunent lIiITJ!CS5 puametcr (Ot. 8) COl. (OL.7) (nllies in diagonal malli. {D] diagonal matrix diITelential opelaLor vwor for higher order deliul ivt$ of dlsplaecment5 (' ~ Ienglh of in finilcsimal line clement (O!. I) modulllS of elasticity infUlilalmal ~illl {linear components of ~ infllliLe5imal stnins {linear components of ,.~ inftni't$imal smins (linur components of tf.J IImch of ItUA member base vwor (OL. 6) avenae of {pt vIOn .. clement IIOdc:I (O!. 8) nontUIliud vCClOl of {el (O!. 8) vectors defined in Figur~ 8.S(b) (Ot. 8) nonnlliud vectors for Ie,} and (e,) (0.. 8) lineal and nonlinear fOlce inclcmenl$ of truss member mc:mber forces I]oog \he. I ', Y', and .-axes II C, mc:mber forces liong lhe I', r-, and . -axes II C, nial force of \ruSS mc.mber actina II C, and Ie-feRed 10 C, force vector (O!. 6) inlCnW demeot forttS summed II SlrIK1UTll nodes (OL, I) single-valued constitutive function (01. 1) .sin + + 2 00$+ -2 (01, 7) body forces II C, and C" referred 10 C. body forces II C, and C .. referred 10 C, fOlg: inaemenls from C, to C,

"..",5""""

If,}, {I. I Ii.}, (i,), Ii.I Ii.} til {'f}, I'f} fJ} G G

"" , H,

1'1 I,. I, II} ; , " , J ~" '. K 1'1 [1(,1, 11(,[, /K,[

[I(~[ [1(:':1 [k[ [I.:,J, (k,l, IX,I Ik.l (1.:,..,1

''''HI (.1, (I]

fot componenlS generated by [t,llnd Is,l (CIt. 4) .lenw:nt force VIOn ~ing to {wI, I~I, {wI

...

elemenl fot vector (OfrespoDding to Ill,} I!Km~. actions comspondlllj to {I'I (Ap\xn. 8) clement force. at C, .nd C" .derrcd to C, clement forca at C,.OO rd.rred 10 C, .... ar modulus sbear modulus con$Kkring the Wagner dfed (CIt. 7) gellfntlizcd stiffneu ~ter (01. 8) defiMd in (7.11.56) (CIt. 7) COfISIaI>t ~tCl (Ch. 8) height of beam (CIt. 4) Of lever arm or moment (CIt. 7) subnuuUr in Is,l (Ch. J) mOlllCnlS of illfnio about r- . nd l-Ues Kkntity malrD dilrll:nsionleSl coordinak (_ x/L) IOrsitmal romtant JKClbiaru: (Ch. I) stiffllC$$ CO(rrlCicms in IKJ maUD WIgncr efft stNCtUfC stiffJ>Q$ matri~ clutic, geometric, 100 joint moment matri. for ~ruetUfC appli0:4 /IIOIIKnt mwi~ for WlICtUfC integral matri~ stiffness matrix COJrCSfXM>

~I

[i.1 [.t,1 [.t.l. [.t,I" [.,h. [t,!. [.t,] [t}l

l.t~l /.t,]. [t,l. [.tIl '. ' L, 't., 't. {'{ , /, ... ,11

M. M~ 'M,. '/II,. 'M, ' . 'M,. 'M, M (MI 1M) N [N.J. (N.I "

" ", , (" ,), (~.) OS P p ... p ... p ..

P {PI {PI. {fol {'PI, {'PI ,,).{pl ")

cllSlic lIiff_ mlInu geometric stiffness matrix u lo,n'. stiffnus mllr;K submatrica in [Je,l (01. 4) Induoed momt:nt mltr;x joint IIlOl1Knt matrix IWlied mornc:n. lIUIui. &ubmau;oe$ (Ch. 2) entries ia Il.] m.trix

u",,~,

length of Slruclural member .1 C .. C, nd C, lower triangular mauix length of infinitesimal clement (0.. 7) direction cosines applied IJIOIIWnl (filial moment morm:nl$ about the Z 'o JI", and z_ues .1 C, _illS about the r -, y-, and znes II C, moment vectOr (0. 6) _lit yec\O( (a... 6) {M} VCClor IIliMformed 10 g10balucs (Ch. 6) lOIal numller of degrees of frdom of 5tnw;Iurc ooeff.aelll m.1fit:es (Ch. 1) 101.1 number of degrees of freedom of clement (01. 2)

nomber of infinitesimal tkmcnl1i (ClI . 7) difCClion CCllllIIu

IIn~ YeclOf .Ion, OS uis (0.. 8) lilKII .nd cubic interpolation functions His of rOIalion (Oa. 8) .pplkd .:rlal load axla! buckling Ioad$ in lorsioPal.nd fleulal modea (01. S) force vector

"lU

(i) (p) fO.I. [Q,I ,

(f,) , ,

~R. :,R :R, :R I') (R',...) , , ,.

5' ... 5" ..

'S. 's, 's .S" ,5. 's :S" :S, 'So 's

~. IS) [SL. [SL

("]' (',I, 1',1 ",L.. [',I,.. ["1., T, T. T ... 1 ...

I~ [T.). [T. I [TO]. [TO]

projcdions of {PIon'S pl'l\(: (ClI. 8) norm~liw.l ' -eao< of (p', (01. 8) c:oofflCient mauica (0.. 7)

xxvii

patalMt~ fO\' i

XXyiiI

(TI, ITF] [TR]

{T,I, {T,I ~_ !I, , ,

,,~ ,I, U, U, {U} 'O}

" " .. y .. "'.

(II I. (II I.

(11,1, {II.} (MI, (~) . (';'1 (Ii)

V.W.I. v 'v V

(VI)' (V,..,) X, r. Z .r, y, : 1,;, f 'i, 'j, 'f ,. .., " .r, r,

1'1

~ , II, y If , II, V , In y

CO(fflCicn' ml/fices (Ch. 7) transfel mall;" for ~ wm of fini.e Itni''' (00.7) Y'(lq dc.finc4 in (8.7.1 1) and (S.7.12) (Ot. 8) swf_ l1aCIiollII .. C, and C .. ,trelled 10 C. surfacel1aCliolllll' C, and C" ref trIed 10 C, ar,,1 el""gll,on of beam (Ots. 4 , S) SI ln energy (a.. 2) SlruCiUri: displaccmcn. Yee/or SInICIure dis~mcnts Qlfrespondins 10 (~) (Appen. B)

~ntroidaJ dispb

11M,. 11M,. IJ.M, . S IJ.II. IJ. IJ.~ '

.,

." (A,). (11 ,1

(AP) (AU) {AU/I. (IJ.U,) .0 ,

GU, GV

" ~, ,'.

-. ~ .. :e, c c, , ,

". " . 9 .. Ell' 9, o 0, 0" 0,. 0, 0", 6,.. 0"

(01. 7)

loW elongation of beam (0.. 4) "",mbe, defonnations due 10 various dfccts (Ch. .) linear Ind nonlinn! ronlpotlC'nli of deformllion (01 . ) inclllocd Il"IIImCnt inac"",nlS arc length incn"",nt (01. 8) II, - " ". - w and .... - .... for Iru$iI membe, (01. 3) displacemen1 inn.ment (01. 8) work increment (01. 8) in Ind out-ofplane di$pI~menli of curved beam (0. 7) load incrcmmlS (OI. 8) dispbccmenl incremcnl5 (01. 8) displ.amenl vtorl oornspondi", 10 {/Ind (R',..) {OI. 8} tobtion ;natmenl (CII. 8) symbol preding w;nUll quanl;ties (variational Optrator) v ... uuion in strain ene' D .nd in potenti.1 energy K>onccker delll Green strain iocum.nt, updated Gr.en sua;n ineremenlS Euler slrain incremenIJ G.ecn.Lagrange fil , .illl II C, and C1 'M, IGJ -IIR (Ot. 7) embedded axes defined in Figu, .. 5.2-5.3 (01. 5) defined in (7.1l.25) (Ch. 7) nonlinear I;OIIlporKnIJ of tilt nonlinea, comporKnIJ of ,~~ nltural roulion (01 . ) n.lural routions about Z.:/". and .. u .. (0.. 8) rotalion about .. axis (01. 2) rigid body roution rotations (increments) about Z.,.... Ind zue$ rigid routions about Z', y. and l'Ues

... u .. .,,,._

Ill,} rkmcnt component displxemcnt V1OI [A[ eigenvaluc ""uix (0.. 1) (A,), [A,). '" (AI rnalru in Taylor'. up&nsion (Appen. q A load filClo. (ClI. 8) , ci&cnvaluc (Q. 2), Cl'itkal ~ fK:lQl (Appon. B) , defined in (5.9.8) 01' (S.10.33) (0.. 5) , dcfillCd in (8.3.26) (01. 8)

" , load inaemcnl p,arameter (Ch. 8)

" , I/o, v di=lioa cosines (01. 2) , defined in (ljIIIlions (H.8) Of (5.10.33) (01. 5) , "eM, (Cf,GJ) (ClI, 7) unit norm.l to $u.facc o f body

~ , it. C IcfeKflCC lJtC5 fOI clement nod (Ot. 8) , defined in (7.11.90) (Ch. 7) , R + I (radial roordilUlle) (01. 7) .p, 'p. ' p density of material It C. C,. and C, 0 defined in (1.11.97) (01. 7) 'l 'l Q,UI,:hy $UUSU II C, and C, . '. [OJ lnIISformalion rmtrill from Ioa.I 10 &Iobal axes

(01. 6) [OJ modal matrix (01. 1) defined in equltj()n$ (5.9.8) or (S. 10.33) (ClI. 5) z/R (tangential coordinate) (Cli. 7) or ri&id rotation , (01.8) m cicenvtdor (01. 1) [VJ inverse of (41) mal.ix (01. 7)

'" V""" in ('P) (0.. 7)

defined in (7.11.z.) ICh. 1) ' .. r(lUtions (00. I)

Righl .upersa-ip!5 I, _ ,j-I,; number of incremental SIC,. (01. 8)

Righl5Ubscripts

-,' nodes A and B of 5llUdUni member '. J indcJ for coordinalf. Uf.S , lui ilcntiVf. Sttp (0.. 8) ", y, z coordinatt ilia CI , p, T coordinalf. ilia (01. 6)

l"''''st-~ 2 O. I ... 1- 1./ un SUpefKIipu

member numbcr numbcr or il~ral ;ve 51~p$ (0.. 8)

0, 1.2 OC(.'UITing ronfigunuions: C. C,. C,

un su~ripts O. \, 2 rcf~~~ COnfig\lflllions: C. C,. C,

XlIX;

Chapler 1

Introduction

1.1 Background

The pu~ of SIllICNral analysis is to determine tbc: 5trU5eS, $lf1.iM, Ktmg f01'CC$, and displalIImts of given stlUCCu rc undtr given IolIding conditiom. EIa5eiI on the anal)";s results, IllUCCural engineers Ire able to clltck whelher a pcoposl design meets the requirements of ,tlequate lesisWICC 10 a romb;ntlon of loading conditions and, if ntceSSllry, 10 revise a pcoposed design until .11 such requirements are met. At tbc: present time. illlUI' elastic analysis rellllins the staple of 1be 6esign pl'ofessioo, in the Kn5e that the results obtained from ....,h analysis have boen UKd primarily IS 1be basis for the calculation o f forta and Wesse!I &lid for the proponionill& o f lillUdurai rmmbef'l..

One drawbadr. of linear elastic anal)";' hIS boen its inability to reflea the real bclllvior of structures undtr abnormal or ultimate loading ronditions, since almost.ll structures bchJve in some nonlinear manner prior to reaching their limit of resislance. For thlJ reason, IIlO$t modem codes based on the ultimate strength deliign ronpI have inrorponlled certain provisions for s\fu(:lUral engiMers to rons.ider tile nonlinear or $CO(lnd order effccu using cilber UK! or approximate analysis LedutOques (A1SC 1986). The basic: c:onsideBtlon here ;, lhal a more realislie evallLllion of the otrength of suuctures "Pin$I the

,

2 I .. , ' _

ratlun: ooodilions, Or tIM: fa

3

All of !he suuaUrc$ that Ire analyud in thil tut Ire called "o.ttI # rvUMTQ. A framed WUC!II,e eonsisls of members that are loll, ill O(lDlparison witlllbtir cross-sec!ional dimensions. """" as width and depth. Members of this $Oft are called sknder dClmnts and Ire con,-cntionally repr~nted by li~ ek:!DCnlS in the finite cLement .nalysis. Five categories of framed llruClu

w~ hive 10 Iodmit WI ans .... erlng !hi, q~lion can be very

difflOJlt, MalIK a r>Iinear analysis Q)nlJl;ns I number or eoonPUUl-lionaI pIwcs of "'hid! many have I'I()I bRa unifJed or are subjled 10 up .... ots of diffnen! levels. What is more, 1M underlyioglbeoria and IOlulion prooedum of IIlOSI 1IuIic:a1 publicatiofts have no! been

displ.l~ 10 I okg.., lhat is clear enough for I matMmatical Of phYla' judgmenllO be rn*, noI 10 mention !be possibili.y of hidden tm:Jf1 in computer codc:s.

From the poin! of oompulCf programming. we mly 'pLKe Ihe aboye qUe$IioTdin4lru, i.e., the coordinates IS$OclIted with the deformtd body, are employtd as the reference COOIdin.lles, while in Ihe Lagrangian formulation, the "'~'eri4l1 ~oord'''4Ilu, Ihal is. the

1.1 Ii __ ."" __ "' .. ,.. ,

eootdinatell uaoa.led ... ith \be body before it ddonnJ, arc employed instcad. The ~ fonnu\.alion ill pII1icularly suitable for the sup-by.llcp J>OGlin.ear llDlIIyllls of fOlid bodies, in .... 1Iic:h We are interested in \be history of deformalion of cKh poinl of \be body during tile loading pro$S. In oontnst, !be EuleliaA fOfmu\.aUon has been widely .dopc, the laS! kllO .. n deformed configuration (C,), .nd the current deformed configu",lion (C,). It is assumed thai ,lillie slale yariablc:a s\lth as Slresscs. strains. and displa~menlS. tQ8('lhr:r wilh the leNding bistory, are Uowa up \0 the C, ronfiguraOOn. Our problem is Ihr:n 10 formulale III irxnmcntal theory fOf delerminil\& III the SUlle variables of !he body in \be =111 ddorrncd oonr"umion C" ISSlImit:rg!hat the uu:mallnadinp atling on !be body I I C, haye been ina"used by IIIlIIi amount. The step c:h&ractermn, !be deformation prCMSS of tile body fROM !BE C, \0 \be C, (II.f"u"'llon ... iII be rderml 10 typita.lIy as an lIte'tlll~I .. 1 sttp. Wbile 11M: deformllions wilbin Ibe inm:mental JlCp from C, 10 C, are assumed \0 be genen.l ly IIIlIII,Ibe

'"

6 KtCUmulated defomwions of !he body from C. 10 C, 01' C, an be arbitrarily wge..

Dqlc:nding 011 wbich previous wnfiguntion is sdecltd IS Ibc reference lUte for C$Ubli$hina the governing equ.acioM or the body II the current ODIIfiguralion C .. IWO types of t..vanJiaa fonnulllion can funhc. be identified. In the Npd.led L"S,,,,,,I.,, for"'~IDI/Dft, the IQ\ calculated configuration C, is $eleae.:! .... the ,derenoc sule, whereas in the lo,a/ LIIg.D"g/"" formulal/on, the initial undeformed configu-ration C. is usro for 1M. same purpose. BoIh the updlled and 1(11.11 Lagrangian (onnul.lion. !IUIy be reprded 11$ !he spc

7

'A, 'A; kDd the specifIC mass by ' p, ' po ' p . FurtMr. we shall

8 ,.

Cons.ickr I HIIC ekmcnt PQ of Ienl1h '

- J

, - .!(.:I'.>:,.:I'.", 'w); the C, COOfrlllllllU r .. ,. .." -x,> by ('x, 'y, ,): 1M C, displacements ('II " 'II" 'II,,) by (II , 'v, ' ... ); and !he C, coordinates ('x" 'x,. 'x,) by ex, ',. 'Z). Then:for., the compo-nenlS of GI n_ Lagrange "ra;n lellS(lr ~" , C, can be ",riUen explidlly u follows:

,

"'. ,'. 0-,', . ~ [(:::r . (:~r . (:~r1 (l.J.I4I) ~. :~ . ~ [(:~r . (:~J . (:~rl (L3.14b) ~ 0 ,' ! ir "')' (''')' ("'rl (13.1",

.. a~ 2Ra~ aOz aOz

" I a'II ,~ .-. ,~

,--(1 .). I4d)

(1.3.J

/.) ~"'-- .. and tM _IUI~'" c""'f'O"~"'s . 'If are

". (1.3.19)

Ii lhould be noced thai in equations ( 1.3.18) and (\.3.19) tM quantity 'M, denoco:s the tQtaJ d~l_nts of the body from C. to C, and the quantity M, the disp/amcnl inqcmcnts from C, to C,.

For the special case of linea: anllyail, tlt~rc will be 00 initial dispIJc~m(nt$ '"'' implying that the tWO configurations C, and C. arc icknlic;oJ, i.t, C, C .. aad thai 1M Gtn-ugrange $lrlin te/15Ol" vanisbes, i.e., ~ O. FunMr, it is N5lImed ttgl 1M displament iDcrcments M, are so small t/utt the ~ and products of tlttir tim derivatives an be neglected. For such problems, the diStinction between the two configurations C, and C, disappears, since it is immllerial whether the derivatives arc "'adated II the position of point before Of after deformation . By nc:gkaing 1M ItOfllinc:ar WlllPOOCDts and the initial displattment effects, the Green $trI.in in.cnment lellSOf

" ,- ,

-2,. " .. Ow 2,., (1 .3.2Ie) .. a,

2,. .. .. . 2 ...... (I .J.21f) " - .. .. In rng.inrin, uJage, 11M: lIrIin oompoMfll! ,. (I .. j) doubled, i.e., :z,.. .. ar~ callo:d the ShCII';1I811r1l;'U T .. which repreKnt!he dccruse in lhe right angle inilillily formed by Ihe sidts puali.l to the:lt' and x ( .. ~

However, lhc C()nOI:pI of inrmilesim.aJ Sln.in tensor is IlOl lesuicIW 10 hnclf anal)'$i$ ~;bed above.. UI 11$ 00CISidn IlIc gcDCral c:ase of. body moving from C.IO C, by' number of incumcn \.ll 5'cps and then from C, 10 C, by. single ilKnmcnla] Slep. Though the accumuillcd displacements '~I of the body from C. to C, can be arbitrarily wgc, lhc inacmenlal displaeemenlS~, within the incremental .tcp from C, to C, I,e 5rD.tlJ by definition. In tbis cue, if we ref., lhc suainJ 10 !be C, oonfigumioll, we c:an dcflM the EMler IUIIUt'ttlJl# ~ (or the body'I' C .. wilb .deunce 10 tbe ues "' tIM: $arne oonficuralion C, as follows:

(I.J.22)

Substillllini tqu.tions (1.3.3) Ind (1.).4) into equation (1.3.22) lnd lIIMing thaI

d'x ,

From equalions (1.2. \) through (1.2.3), we have

(1.3.23)

(1 .3.24)

(1.3.2S)

,J~"'- " whidl an be SUbstilUled inlo equalion (1.) ,24) 10 y~ld

(1.3.26)

For 1M pruenl cue. lbo: linear part of lbe Eukr wain ~nsor ,.. is called tM infinitesimal S1~n teMOf ,e,:

(1.3 ,27)

which is identical in form 10 1M inronitesimal SlBin kllSOr t/', gi~n in equation (1.),20). na:pt WI the refeanc:c aMlfipration is du.nge

"

\

and 11K IIOIIlinear rompollCnlS I'I)~ as

.5.) ,'. ,

,.., t _

( 1.3.30)

(1 .3.31)

(1.3.32)

As ""tWo. of flK1. the possibility uisu for lhe rducnce ((lnfigunlion of the .'linin tensors 10 ehange from one 1'1 the DlMr. In _ \2IU, for insunec. We may need 10 uansform the updated Orn stnifl inaeRlen! tensor ,t" which is .... asu,ed II C I' to lhe Grcen lIU1Iin increment IeIl$Ot JI. .. which is meuured I' C. To derive the lr&nSfot-mation rule for the two Itraln tensors ,I, and of!(t> let UI cxlll.'lider their dc finillol11 as giVC'n in eqUilioll$ (1.3.28) and ( 1.3. 16). By lbe use of cqul,iOl1 (1.3.1). we can obtaiJI the following ""Ie of Il1IlI$f(HmllioD for Ie. and ... , from theK definitions:

(\.3.33)

(1.3.34)

1bex relations n:pr~nl ulCtly the tralI$formation rules for ICnson of the seoond order, such as strains and litres:su, [rom one n:fercrw:e ronfigurllion 10 the Othe. . SimiU: transformalion equations CIJI be established fOf other 5IJlIin 1Cn.wn wilen Ihe reference configuration is to be elwtJed. In the analysis of large &U1Iin problems by tile updalCd lagranail.n approach. when: aU quant ities have bun referral 10 the moving configuration C" we nud 10 ",fer all tbe strains . nd st,esses

" \0 fi~o:d common ronfigunllion. such as tile C. ronfilUrilion, befo,~ tbe constitutive oodflCienu can be oonsislO:ntly dcriw:d for tile JliUClural IMmMrJ a' each ~mcnlat step (sec &aion 3.8). RcI.tionl $llCh as ,,- givcn in (IJ.33) and (1.J.34) provide 1M basis (01' performinS the necessary uaMfonnation.

1.4 Definition or stresses

Allhough variou, kinds of Slress tensor! hive been proposed by .Iasticilns and mathematician. in Ihe 11M" only lhe ,neu tensors lhal Ire oonjugll. in 'erms of energy 10 th. Slrain tensors presented in Ih. p'.oedi"g section will be discussed lie . Such .Iress and Slrain tensors I.e Il>O$l dfCClivc 10 the Llgrangian formulations 10 be presented in L.tcr chaplets. In this section, we wI! diSCUII first tM 1011,1 Piol.-KI,cltltol! "'UI IUtHI. For cases "'IIen: 1M "'" of "second fiola-Kirc:hboIT" may .uull in l~i

16

I

""?-p ""-----::---; . ... ,,,. ...

~ "". .... .,.,

p

, , OS, ~ OS, .,;~

1.4.2 c"ucby,llUI lt n_

, .. ' -

(1.4.2)

TIle CawcltJ' '''UI I(~_ i$ chlracterittd by 1M facl Ihal il is Ilw~ys upressed witll rtsjXCI 10 tbe configuration ill whiclllhe 51"_ occur. 11 is alio known ... lhe: EIlIt!:, ,frus 111_. 'The pbyskal mUlling for Cloudy "'t'S$oeS can be usily .ppKC'illtd (rom tile 8llIphicai illustrat ion Jhown in FiaUJc 1.3. Consider .pin 1M motion of I gCMfic: poim P, whicb is clIClosed 11 C, by all infinitesimal rUIlgubr paraLIeWpiped wilh \he following si. surfaces:

I.." '" COIISI., 'J,' d1J, '" COIISI. (I. 1, 2,3) (1.4.3)

Ind at C. by .nother infiniluimal rec:ungular ~'"lIek:piped Wilh the following loUt ~Ifacu:

"

c. ox. 'x. _ 'x.

,k-;:--Ox" 'Xo. ')C, 'x. 'Jr, 'lI.

""Ow: Cauchy ,tr

18 , .. ' -where p add 'p ,cpn:soenl 1M II\.lSS densi,ics of.1Ie: ml.erial ., C. and C, .c::s.pW.ivcly.

1.4.3 Updl ltd KIrchhoff ItrHI ICHOr

II is possible 10 ddi~. third Idod of stl'C:Sl I.MOT called lhe ~pdd'etl Kirchhoff S'reSS WIsor , Consider the infini.csimal .cClangular panJlelcpiped COIIlainilll the poiol l' I' C L endosed by lbe six l'lf1keS Jivcn by fiIII.lion (1 .4.3) (tee also f igure 1.4). The CallCby sUesse$ kti", .. this rectangular pualleLcpiped arc deDOled by ' t .. As 100& as the body a.>ntaining the point P IDO\U from C, 10 C .. Ihis rcaan&\llar panllelepiped .... iII ~ ddonncd uuo an infillilCSimal paraLlelepiped at 1M. C, configunllion, wIIidI is no longer ~lar. In this case, tile eoordinatc$ '''', of the body at C, are employed 1$ the m.atc:n.l roordi-nlte l)'$Um fot the body It tile C, OOlIfigutition. Tbe updated Kirthlloff stresses lS, ate Ikfincd as rhe intcmaJ ro~ pel unit area .cling along the normal and two W1genlial directions of cacb of the .ide Hrfaoes of the parallelepiped 11 the C, wnfigWlllions (Washizu 1982). 10 an i.rI=mcnUlI analysis, tM updated Kirchhoff $lr~:S, ean be decomposed as follOW$:

l~ 'f -. .

p '"T",

o ,,----;::-1:C ' x. .' ~ ....

"Jr, 'X. OX,

(1.4.7)

" .... hcr~ 't, &bouJd be .ecopittd III tIM: Kirchhoff SIrUSU acting I' and re'cncd lO.be C, COfIr"ullIlion. i.e . 't~. :S., and ,s. will be called Ihe ""''''ed KIFdlwff .,U~ i"c'tillc~, W.JDr (\Vashizu 1982).

1.~.4 Tnn5 ronn.uio~ rulu

~ Oouchy Sl.ess tensor 't, .' C, can be- ,el.ted 10 Ihe upd~l.d "',chlloff $IKS.'l I.n_ ;Sf by Ihe following formulas:

I '0 a'A', a'x, I ,s --"-- t (14.8)

, 'p el' ... al ... "

(1.4.9)

... he.e ' p p,,~scms the mass dell$ity of lbe material ., .he C, c:oo.fig-\IllIlion. Sin thc C.uchy suus Iemot' I" is symmetric, c'l~lions (1.4.5) .nd (IA.8) indicate that boIh thc sea,md PioIaKirdlhoff stless ICIIJOf :S, and !be updl.cd Kircllhoff .suess tensor :Sf lie .I!o Iymrmlrie.

lbc ulinsformilion betwffn the Stcond Piol.,Kirdlho/l stress '.1IJOfI wilb different rc:f.n:n ronfigullIliOO$ is abo po55iblc. For iMance, from eqIIl' ioruI (1.4.6) and (1 .4.9). IJ~ can derive Ih. following . elation:

I 0" a.f, aO"j I Of --"-_ S r, ,0 ,0 0" P .1', X.

Similuly, [be following .dation uisu fOf.he body" Co:

, " " 1 -0 .I", _ Ji" OS, --"---~ , 'p a'x a'x '"

(1 .4.10)

(IA.II)

.... Ile'c il should be n::wgnizcd Ihal ' T .:SO' SutH'1clin& cqullion (1.4. LI) from cqtWion (1.4. 10) yields .k ,dation for the ;m-Il'mcnlal $IrcS5e$ .S. and ,s.;

20

Of inversely.

I .. ao.r, a'x/ ~ ......J;. _ S rO'a'a"" p x, x,

, ... ....

(1.4. 12)

(1 .4. \3)

The preceding ''''v .c!alions an: pI"icululy ~rulln the n.lculltio

" By tqUIllons (1.4.14) and (1.4.15). we WI $how thaI the following ,dalion mUSl bold for any INleri.l :

Con$eqllCnlly. equations (1.4.5) and (1.4.6) can be ~wlill(n,

1 I ""x, a.l", 1 -, " ,J- , ~. a'a'

.1", x.

Similarly. by defrning Ihc: Jacobian determinant ~ as

a'. , a'. , a'. , a'. , a'~ a'~

1 101

.1", a'~ a'~ a'~

jJ . a' " a'. a'. a'~ "

, ,

a'. , a'. , a'.., a'. , a'~ a'. ,

llle foUowing relalion rcmaillS valid:

'p - 'p!J

(1.4. 17)

(1.4, [8)

( 1.4.19)

(1.4.20)

(1.4.21)

It follows thai equations (1.4.8) and (1.4.9) c:an be Icwritten as

(1.4.22)

(1.4.23)

I

" ,M ...... ' ...

In 1M form~lal;on of inc.rmenlalrhcories for _Ii,..,., problems ... e DUd 10 Kite! appropriately ronjuptc was and Slnin ",usuru.. By "ronjup\e" ... -co ..... :on tIw che stress and strain ,"eUII," Klecud .... itll I(spec! 10 ox""'in ",fc"'Me c:onfigUllcion ". furm Wfljuptc pair. By the application of Irlndo.malion rules, we shall show in Section 1.7 lhill for tile 100al ugrangian formulation, which is ,dened (0 tbe Co ronfigurllion, we may ,de(! tbe Kl.1,Ind Piola.Ki.chhorr.mss ICIlSO< is, and Qrcen'U&rlnge ~njn lensor~. IS 1M C'OfIjuple pair, and IIIaI for 1M updatc4 laanngl.n fO)f11lulation, of .... hich the reference configuration is Klecctd 10 be C , ..... e may selea 1M uj,dattd Kirchhoff suess ICnsor !Sf aIId IIpdaltd Green linin ilKUlMnl tensor ,e. _ the pouiblc: pair of candidates. 1lM: selc:Clion of approprialel)' C'OfIjuple stJtsIand linin meas\lrta is .150 important in !he der;Vlltion of malUial QOC:ffw;lenlS for inerc .... nt.1 ~lllltive laws.

1.5 Incremenlal consliluth'e laws

Constitutive relations, equilibrium conditions, and compatibility conditions an lhe tluu m(I$! essential ~it\ICnts that muS! be COfISldertd in the analysis of boundary,vllue prob~ms. In the Ii.enture, I geal number of maleriallaws have bun prop

J.J I~~" ' to ..... (:O!ISIitutive law 4;arI be u~. in. terms of the Kirchhoff suess increment teMOl" ~. and Green strain onerermnl tCMOl" ,eo'

(105. 1)

.... hile for the updated Lagnn&W1 formulalion. it can be upresslln terms of the updated Kirdlhoff SItUS inc.ermnIICIl$I)I,s~ and updated

G~en litra;n incrcment l(n50f '-.:

(1 .5.1)

.... bc~ ,.c"" and IC"" denote ~ il>cremtnlal corunil\llivc \e1LSOI"I .... itb raped 10 !be C. and C, confJgUrations, rupectivcly,

ConventionaUy, ~ incremental IIIIleriall ...... (1.5.1) and (I.S.2) .... ith idenlica.l coc:fficienl!., i.e.,,.c,,,. IC .... have been employed in the IOIal .nd updated I..a&rangian fonnlliatiorts In \he derivalion of incremental equations of equilibrium for v.riouI finite clcmcnl$. Since the ptopeny of the m.terial bu been specirocd in an incrcmenll.l or pieawise sense, $Ucb aD assumption of identif;al IIIIterial coc:ffocXnl$ docs nOl imply idenlical material propeniu in ~ lotal or IoCCIImul'ted IiCIlSe for bntlI (onnulalions. As.sud!, the resulting loadfieclion CUNU calculated by botb ~ for !he same problem can be of liiilllirlcant differc!IoOI:. Even with the $III'Ie formulation, the Cllcul.lcd responsa (:;In still be lfl'ectcd by the Slop lilzu UKd in cadi run, if lhe incremcnuJ constitutive law as derIDed in (1.5.1) Of (U2) is ISSIImed to hive oonSllnl malerial

, .. , -

(oJlowinll.rans(onna.ion rule (or .he l:QfIl.i.u.ive 'ensors:

,cO' (1.5.3)

or inver$ely,

(1.5.4)

With .h~ .wo equations, only ooc: Kt of roc:rrK:icnLS, i.c.,.c>' or IC .... need be prescribed. The ocher Je' of codftdenLS can be obtained IlmpJy by uansfomoaUon. In doina so, We an tbe.dore cnso", 'hat lbe material propenies implkd by tbe tOlal and updated ugrantian fonnul.,lons an: physically ldet!1ic:aJ 10 cadi OIl1tr, provided tMI lilt same stcp sius are used in bOIb fomoulations.

Another question in illCfemcnlll formulalions penains 10 lbe way 1M incremental malerial roc:fflcicnts .c"" and IC"" ,.e speciroed. Among tile greal number of a1lcl1lltives uisting In tbe literature, .1Ie simplest method is 10 define tile malerial law in temu of tbc seoond Piola-Kitchboff 6trC$$CS :s~ and OrecnUpnge slr'ins~ .. whim I'" energetially oonjugau:, based 0fI uperi.rnenlll lUll or by postulation.

(1 .5.5)

where I is I sinp-value funaion.. A Khc:ml.k dr:iwing of tqIIllion (1.5.5) is given in Figulc 1.5. Auuming lha'lhe lDOlion of.1Ie body (10m C. to C, ean be divided in10 an infinite number of infinitesimal steps and tMt C, is infinitesimally close 10 C .. from equation (1 _505) ..-c can (k. ive tile following diffe"'ntill law for the material a, lhe C, ronfiguralion:

( 1.5.6)

where fGtJ reprcsc:nll lilt tangent modulus of lilt. SU"esS-stnin aJ~ c""Julto:

slop' " /,(,'

,----,..,

(\.6.2)

where 't. dellOlts tbe oomponrnts of body (orca pcr un il volume refmed 10 Il1e C, conl'ipralioa. In unabfid,ed nolalion, " '. have

a', .. ', a', -" -" '1. .. 0 a'. a" a',

(1 .6 .3)

a', a', a', ~ .. 0 '--'-" '--'" -'--'"

a'. a" a', , (1.6.4)

a', a', a', 'I, .. 0 ~ '---"-

a'. a" a" (U.s)

,..,

" . " " " " . " ~ ~ , (1.6.6)

The equations of equilibrium mUll be Sltisfted II aU points lhrou&houl tlw. volume of the body COIISide.td.

In ,eneral, Ihe surface ea S of the body can be divided inlo 110'0 parts: the pan S, over whid! the eXlernal surfMlC II'aClionI arc "'.-Ia'ibed and IlM: part S, ovn whicb !he di$plaamenu are pre$l;l'ibed. The boundaJy conditions associated with the prescribed lractions lie called tbe .... ,~rlll Of .. b~Ic/J1 bowtul .. ry .Mdj,itHu. and those. 'uochlc'II .. ith Ihe prncribed displ.umcnll an: QUed lbe ,"H,wric or . .. ,Id lx>I

" ,_/pit: .,f ""'-I ~I".&...,..""

" " I " m " " "

,~ I " m " "

" (1.6.8)

m ,

" I " m " " " m m ,

on 'S" wbere the dirc:c(ion cosines I. m, and" are defined as

1 - ros

28 1= t _

( 1.6.12)

in which the surf!ll:e I .e. is assumed 10 ~ 's. 'S, + '5, . ~d on ~ fact lha1 't 't~ "I fOf the part 'S, with JlUSCfib(d lractionJ and lila! h , .. 0 for the pan 'S, ""ith presaibed displa

"

( 1.6.18)

MoreOver,;1 can be $hown Ih.1 'f, ~"'f . 0, because " a is symmetric .J>Cl b,"', is skc .... -symmeuie. Thus equation ( 1.6.14) wC$ the f(lf1ll

(1.6.19)

So far we have provw thaI if the "less rleLd is 5UOtically ~miui ble. the following ,el.tion will be ... alid f(lf any admissible vinual dis-placement 6w;

f "f 6,.., JdV .. f 't,6",ldS f '!, liM, 'dY Iy ~ 'V

(1 .6.20)

The converse pmposilion StaleS 1mu if the virtu.ll work equal ion (1 .6.20) is valid for tvery kinem.tica.l.ly admissible virtual displacement rleld, \ben the $IJeSS rICk! is swi

JO ' .... l ,1M

vinuaJ displ.Kl::ments will be: adopted ud.m"~ly in the le.1 as !he basill for deriving the theories for v.rious structural elemenll.

1.7 Incremental Lagrangian rormulaUons

A fund' IMolI I difficulty !p the 'ppliCilion of Ihe principle of vUtuII displaamcn15 pruenlcd iJIlhe foem 01 equalioo (1.6.20) is Ihal it has IKen rdared 10 the c, configuration ofth. body, which is unkllOWn (or IIOnliour problem bc:fore the equation illOl~. This is different from the problem confronting us in tinear an.lysis, in which the displa

" firs1 tM rdation bel"'n ' " and :So and IMt belween 6,... and 6~ Sipoe the . elation belween ' , _ and :S~ is available in fqUl tions (US) and (1.4.6). ln the following we need 10 Cl,)D()entratc only 00 !be leillion between the vinullWainJ b,.., and 6;'r

NOIin& thaI "I'x, 0, from eqIlIl ions (1..3. \) and ( 1..).) we CUI obtain !he followina:

which can be substituted inlo equation (1.7.2) 10 yield

Funh , $;rn;c

from equation (1.3 .27), we oluain

Sy the. dlaio rule,

a6111) ,'. ,

(1.7.2)

(1 .7.3)

( 1.7.4)

(1 .7.5)

(1.7.6)

(1 .7.1)

"

(1.7.8)

(1.7.9)

By I~ use of I~ rcLalions in ( 1.4.6) ,00 (1 .7.9) _nd DOlinl lhat for conserved mass pJV. ' p'oW, we an I~refore pr

,..~-~,~,~,:_~,_.,., ~'"' -hM.' 33 of which the nt~m.1 v;lIu.l WOft on thc right-hand side wiU be derW1ed 1S:,rt througbout lll tut. i.e .

(1.7.15)

~ vil'1ual wmt cquatioP 1$ given in (1.7.14) is by itself I nonlineat cqua1ion of equilibrium for the body unde, eonside .... lion. Though the reference lias been changed (rom the: I;IIfl'Cnt mnfiguralion C, to the initial c:onr""rllion C. the equation um.ins In nan Slate_ol of ~uilibril,lJll fO\' the $IJUCIUI'C under consick .... lion. It thcnforc c:an be USII$' v.lid basis for deriving the incremcnlal nonlinu.r equ.tions In the locall...agmlgian formulation, a5 wm be do:monstn.te

For I body in equilibrium I' C,. !he foilowiQl can be wriuen:

!It f ~,I 6u, OdS f ~ 611, dV , ~

(1.1.20)

(1.7.21)

",hich can be Qbuoined simpl y by _irching ,II !he supcrscriplS in cqultiou ( 1.7.15) from "2" 10 "I ', A pIlpical int~clltion for eqtIIItioa (1 .1.19) can be pnn 1$ fo1low1.; The flfSl !erm 011 !be left-hand $ide represenll 1M dIan&e in suaill eDUIY (in varialional form) durin. !he iDcn:mental Jlep from C, 10 C, ...,;I tbe Krond !erm !he change in poIential cnerIY due: 10 lbo: initl'" 5Uessa ~r The Ienm ~ and iR on 1M right-hand sick rCprcKDI the exlernal vinu.l works done by the surface uactions 11>11 body forces acting on Lbo: bodY'1 C, and C1 rcsplivcly. To puL it IOgctllcr, equation (1.7. 19) StllCS that the dirresellCC bet"'"n 11M: virtual "'orb done by ulemll Igencies II C. and lh.clK I( C, is equal 10 me incr~ in tile. wain energy and potential enuiJ' of !he body duriJI& the motion from C, 10 C .. SilIce no appro~iDwioN have been made in 1M liI:riVllion of equation (1 .7.19), Ibis eqtllltioa ~mailll ... uaa ib.lemcnt of 1M equilibrium of the body II C> iD which aU quantities of 1M body II C,. in(:ludi"l the inili.tJ ~~". ue a$$UJIIed 10 be bIOwI1 .1 the beginning of cac:h ilKUmeotai Step in I nonline'l anllysi$..

For lhe cuts ",here aillhe suess increments..s, ean be lelaled 10 the slllIin ineKmems tl. by tbe incremc:ntal constitutive II'" given in (1.!l . I), equation (1.7.19) can be ~wriucn as follOWJ:

f tC.,,,E.~6 ... ,dY. f ~S, 6.'l,dV ' ~ - ~ (1 .7.22) ~ ~

Either equation (1.7.19) Of (1 .7.22) CU\IlOI be solved directly, since !hey .~ nonlinear in the displac>:ment incn:mcolS"~ For problems of ",hich lhe slllIin increments ean be OOMsilk,ed small within e&Cb incremental "cp. IpproJimate solutions ean be obtained by making lbe following usumplions:

" (1.7.23)

(1.7.24)

Accordinsly. tqIl&lion (1.1.22) red_to

f gC'I'>lrlll"rI,odV . f ~Sf 6o'l ,odV . !oR . ~ (1.7.25) ~ ~

Thil is the li/ICarized version of the incremental equation of equilibrium (Of describing the motion of the body from C, 10 C, with ,derence 10 thI: axes I' 11M: ilIil;a! undoformcd confl&\ln.tion C ..

1.7.1 Updated La,",,,,,",o formulaUolI

In \be IIpd1tcd LIgran&iaD formublion, .u physical qllIDlilic:s should be rdcncd 10 the last akubled oonfigunlioa C, ins~. 'l'bI: ftISI COIIa'm here is 10 prove the validity of the following ,dllion:

f ll, lJ~f ' dV f ~S, 6 1~' 'dV (1.1.26) .~ I y

where il sl>ould be OOItd thaI the tWO symbols ,I, and :., be identical, i.e., Ie, :'''' SiDCe the rdation between the CalK:hy Itresses 'T, and the Jotoond Piol.-Kirdlhoff ~ :S. is a1rndy Ivailable in (1.4.8) and (1.4.9), only the ,dation betwn the infini\eS.ima' stn.iDs 6"., and

G.ccn . l..agran~ ltnIins 6,~ Us 10 be derived. Notillllhal 64'. ,.0, from equatioIII (1.3.18) and (1.3-3). we can

obtain the folkrwm, . elation:

(1.7.21)

Using equations (\.7.3) and (1.7.5). we hi""

36

8y the chain rule,

Hence,

J

6.., d'x d'" r", ~

a'x a'x ',

" With the relalions givtn in (1.7.26), (1.7.34), and (1.7.35), we are able 10 trU5(onn the rcftrcocc ronfigunuion of the vinu.al work equation (1.6.20) from C, IO C,'

f :.,h, 'dS f ~h, 'dV (1.7.36) " 'r

of .... hich the ule"",1 vinual wad on 1he right-hand side wilt be dcr>(lled as:R (or 'R wilh subscrip'"'" dropped) Ihrou&houllhe lext, I.e .

!R .. f :" ~ ~I 'dS f V. 611, 'dV (1.7.37) " 'r

The vinult work tqUalion as givaI in (1.7.36) ill exact and nonJinur in the iI1

" , ... r ....

f IS~6'''f'dV' f 1,.6,'l,'dV .. ~R -:It (1 .7.41) , ~ 'f

:R f I~, 6'~f 'dV "

(1 .7.42)

For I body in ~quil;brjum II C,. we can write

} :R .. f :,,611, 'dS + f ~ 611, 'dV (1.7.43)

'1 I ~

whic;h

" ",,1M:rc: Ie"" UPfe$C:nlll !be _iN.iye cocff..:ifcnu, 1I.s.bould be ('"~ized Ilw !hi, tqu.alioo ,emains valid only fOf eLUlie bodies rOf .... bidl all !be stress c:ompoMOIS ,s, .;an be ~Lated to !be stn.iq COfIIPO"WIS 1~ by the il"lCttmcnlal cons!i!U';Vf law ( 1.5.2). For ca5oe$ wbcrc: some SI!'eSS oompoMnlS o;:an~ be ,dllCd 10 the stnin compo-Milts, modir.aolions have 10 be made 10 ac:coUn! for slJdl facl. One .ypial example of this is on the slud), of two- and Ihlee-dimcnsional bUms employing lbe IkmouUi-Eulcr hypolbesis of pllllc 5CCIions for deJi;rLbing 1M

, .... '-Balhe, K, J" Ramm, E", and Wilson, E" 1- (1975), "fi nite demenl

fwmulalioo for I.", deformalion dY""mic analysis,' 1M, ) , N"",~, M~I., E~", ',35l-86,

Bathe, K, I I; and Bolourchi, S, (1979), "1I1,e d;,,;placemcnl Analysis or three-dimensional bcllll WUdII'CS: I~l, J, II .. ",", MCIA, E~,_ , I", 96 1-86,

CcscoIIO, S" Frey, F" and FOndel, 0, (1979), "'foul and updated Lagrangian dcsaiplions in nonlinur structural analys~: I umned IJlPIO"ch," in Glowinski, R" Rodin, E, Y" Ind Zienkiew~ 0 , C, (cds), E~.rv MCIIuHb ill Fillile EI.",ul A~.Iy.i., John Wiley, New YOO:, N,Y" 283 96.

Fung, Y, C (l96S), Fo"nd./iD~J of $olid MIInlcs, P,cnli Hall, Englewood Oiffs, NJ. I

Malvern, L E. (1969), inlrodMCIIOII lei II. MullnlCI 0/ " Ct>~lin .. 1)JU Mcdi""" Ptenti lIall, EnaJewood Oirr$, NJ,

Wl$hizu, K. (1982), V"""Iit>u1 MclluHls ill EI"./icrlJ nd PI.mid/y, 3rd

Chapler 2

Linear Analysis and Element Quality Test

2.1 Dlscrttl:r.alion or structures

In 1)1IicaI suuaunl analysis, \be crog..seaional behavior of each member of. fnme4 suuaure under ccrtl.in loading conditions can be delcribed by lite so-called ruvu~j~r dilf~r~~rltll eq~"I/(JM (W djlf~r~~ li,,1 tq""ll(Ju of tq"lIibr/"",. A fnmed SlruClule, IS w

" 'The !iniac clement melbod is aimiIN 10 the. nwneric:aJ mcUlods,

such as the finite diffc.cncc method and boundary element method, in that it approKimaleS !be original structural system that has an Infmite nWllber of de~ of frftdom by. Jimplirled INlbematic:al model IbM 1m OIIly flllile numbcl of degreu of freedom. By rcplacinglhe origi. nal differeDlial equatiOllll and continuity conditions by the fmite element malfill: equations, We IlOl only m111CC the nllmber of degrees of freedom for the problem COns.idc:Icd., but also timunVC'nl the insUlll'lOllnlable diffICUlties involved in classical .,Iulian pnx:edura.

1be flniae dement procedure tOl' an.alyzing suuo;:!ura of the fnme\l type bl.'led on Ihe linear theory can be lummarized as foliollo'S. Fint, the cnlif1l framed 'U"t!\l,e is broken down inlo I number of line dements which ~ coru>eaed 10 ~ other II Ihe IIOdaI points or nodes.. Each dement is given an clemenl number and nch 1IOCk. node number. Nexl, the stiffness equaTionS lie derived for each clemenl in tenn. of !he nodIl deguu of frelom. This s.ltp CIlSUI'CS tile equilib-rium of iDdividual ckmcllU 10 be satisfoell in ..... uk or .""nae __ By tn.ns!onnllll \be: clcment stifIncM equations from Ioc:al aJOfdinale$ for each elemoent 10 I common gklbal coordinllc l)'$Icm, we then bave ,II dIe element equationl usembkd 10 yield the struculre ItiffJlC$S equations. This 5Iep, in eoaln$l, e!I$U~ tbt uti$fldion of compIIt-ibllity and equilibrium ooaditions for !be Hltire _un 11 eadI oodal point. Further, by impo&illl appropriale geometric: bounduy conditions, !he 'trvetu~ will ~hieve ill kinematic 'lability, in the sense thlt rigid body motioDl are ~moved, IS indated by the po&itive definlltlltSi of the Stiffness lIlIIriJI. For. Jiven ICI of applied loadings, the oodal displactmtnts can then be 1OIvec1 from tbt ItJudItn: atiffntSS equations. 1bc: final 5Itp in finite element 1.1111)'$;' is the processing nf elements 10 obtain qlWllitits such IS forctJ and $UC$$CS, for the p"rpoM: of desianin& mtlIIbcr aou sealoos.

As .... Stlled ~, the disaruzltion or ideIolizMion of $U\lClUreI constitutes !be first step in the (mite element analysis

proccdu~. In this step, usumplions hive to be made regalding the ,eGmell)', auu seaions, 0DIIfIeCti0ns, mIICIiab, ~ ODIIditions, and Ioadiaa conditions.. It is wilb .fuc:h assumptions that SlnIdU.e originally 1ppear1Dg as I continuum .. iIb eomplel Ioadi.., and bounduy conditions can be approxlmlted by I mathematical model wilb a finite numbcr of del'ees of fR>Cdom to such an exlcnt \hat ac:auate anaI)'fis can be carritd out. All tbt iDformatioo ptrtainiD& to the l1nK:ture model should be prepared IS !Ix illpul data before. finite clelnt.nt analysis program can be excl;Utcd

Conventlnnally, the data prcparllion -.C of I finite clement analysis was referred to III tbt prcprou"" phuc. ill contrU! with the

" of'-~-" 4) J.1 Do" ilIt p/IaSc: for uution of analysis program', and the !'OJlp'O ~ p/I'" fOI im~rpI'ct'lioo of analysis results, Although n"ous ~ have been developed in the puc to enhance the dfici~ncy

dlta prcpMUion, for instance, using !he tcchniqu~ of ince~ivc of cer p1Ip/lit:I (puqucno el .. I. 1983). dIolI pupuatJ(ln rCIDa'1I$ !be ,:oIII~ilicJ,1 ""It of. fini1e demmc 1UI,lysis, in telms of lhe liIM. and n;i" spcnc by (ngiMe" in design offl&. One reUOfl fOf chi, is 'hac :;"'O~dUle for de'elmining.nd checking che gcornel/klll'yOOt of $lIVdU1U is icenltive in na,ure, and thai for large, complex SlructUru, hu""" em .... S all: ,~I)' likd)' '0 oo;ur.

In Ihit.nl, il is usumcd lhal 11M: leaders arc "R~y Kqll.inlcd ".i!h 1M proccdUTC for liM.r lUIalysis and 11M: preparalion of SllII(1ure dill required by such. proccdulc. Therefore, no .l1empl wi]l be: made 10 conside. 11M: ekmcnlll)' programming aspccu of finile elcmcm .... lysis. Th.oughout tIM: Inl, $l/U(tuntl rncmbcl1l lhal art n rl)' only axial forca, IlUCh as the bncing membe:rs commonly used in Sleel rramcWOl'kl, will be: referred 10 as lhe bD. o. "yn ./.IPleMJ, In UJIII/ISI, wuaural mc.mbe:", lhal can resist nisI, flexural, Ind eVen 1(11. J.ionallClions. such .. bums and eoIumns., will be referred 10 15 lhe be .. ", 01" fT ... e duo ... u . In lhe KClion 10 follow, 1he stiffness millitn for ...... ioIIs fn.me and truss elemcnts based on the linear elaslic ISSUmpr ionJ will be: derived fil'$l.

2.2 Deriva t ion or clement stiffneSS matrices

A liMIi e1l5lic .nllYli, distinguishes it~lf from the nonlinear Inalysis in thaI .he cqII.lions of equilibrium for the body or stnw;:lurc under eonsidcllllion a ... CSI~blishcd at the iniliat undcrorrncd eonfiguralion C,. wilh 11M: effect of change in gWIIICl1)' of the $11\lC!ure assumed 10 be IIC'gligibte. For problenu of this IYPC, there will be no ,n'liat displau IIlCnl$ or in ilillloadinp, i.e., '~ 0 and~".:".!t, . :". O. Th is means lhal confiXuralions C. and C, an idenlical, i.e., C C" Ind lhal Ibc Ulemll vi"ual works : R and lit IS defined in equa1ions (1.7.21) aDd (1.7.43) an be laken as u:ro. Moreover, lbe displacemenl incre.

~~ts It , from C, 10 C, arc $0 sman lhal lhe 5quarC5 and produC!s o f 1lM:" derivatives CIIn be I>C'gIeCI~. Thus, no nonliMlr components o f tIM: stllliltS need be considered. Both lbe IOIIl and Updaled Grn.

~~nge stra;n ICII$OIS "'. and ,t, reduce 10 the same infinllesimal SlIIIln ten50. , i.e., p(~ It, rI' ,~, t~ in liMar analys is.

..

With 11M: foregoing assumptioas, I linuJ analysis I;U be C(HIsidercd I 'pecial cue of. nonlinear analy,js that requires '1.dy I""I~ iIIC'_~'" 0/ /o

'l. +-1,---=+--. I- L I

(.J

- yv'

(bJ

.,

of tIN: beam dinal uis of 1M beam remains planar and IIOrm:;lIlo the Arne uis afier do:forml lion. As. ronsrqUCIlC'C , the u ial and (r:oRSverse displacements w, llId ", of I geMric p

" II - ",, ' (2.2.5)

In which I prime denotes differentiation wi1.b ,upe

" IIIMTe It is IS!iwned l~l A A and l~ ",rf_ Ilk1ions I, and" an \.lk(n posiljv~ wilen ~d along the positive)t . and y.ues, lespeC-

u~ly. BasnI on !be oonditions of equilibrium, rile uial force F ... shear

foroo F,.. and belldina; moment M. 1\ Stion 8 am be related 10 the surfKt: lJaClions', as

F". L ',ItA ,

F,. f l, dA "

(2.2.12)

(2.2. 13)

(2.2. 14)

Substilurina equations (2.2.5) and (2.2.6) for tile disp!attllM:ntlll. Ind ", inlO tqullioo (2.2.1 1), and mating UK of tile preceding n preaions for thc nodal action.!., we an o;aJcuta\e the n leml' v,"val work R. as the following produa;

(2.2. 15)

wlw:u ( ... )Ihoold be recogniml as the displacement veaor auocialcd with tile centroid of Stion 8 of the beam Ind V.l the corre5p01ldin, fora: vtaor,

(2..2.16)

(2.2.17)

W~IC e v; for mill! rOOItions. In I limilar minner, the external vinual work R done by the

"".face lrlctions I, II tDd A can be wrineR as

..

(2.2.18)

FoJ\Qwing the AIIIC proedure, we Cian fll.nMr Upre&I the ulenuol .. inuaJ work R. in !emu of 11M: DOdaI quantities II end A ,

(2.2.19) in which

(2.2.20)

(2.2.21)

with 6 ~: fOf smlll rotations. Consequently, the 100ai utonw .. inual wo,k R \:all ~ obtained

1.$ the sum of R. in (2.2.15) Ind R. in (2.2. 19),

when (M) npr$ol$ !he ek:menl d~mcnt "CdOr and {f} the associated foroe .. tor (Re Figun 2.2).

(2.2.23)

(2.2.24)

1bu.&, we hI .. e shown that !be .. if1ual won: equAli.oIl of cquilibrituD (2.2.3)

" Dtri>'o- ." ,I ... "" "\If- .... ,,""'

"

'I "'

L "' ~ , ~.(;:- , .*. -~ ( . ,

,

" "~I t..: .. , r {:- A

.7;: - -~ ( .,

r" .... 1.1 PlaIW f"-- de .... ft': (.) Nodol dtgrea of r.eecIDm: (b) Nodil fotI [=~, =::::11 I' ,.(1'----'-, _-'1 )~

so LI ...... ..... 1)0 ....... ,/non, 0 ... 1..,. Tu' As ,. side ,emark, we like [0 emphasiu lhat a fini te demom

formull!ion based on the preceding vinual wru\( uprusion ;$ just as raliOflaJ as (hal bucd on the ,~.~iJJg diff~r.nrigl cquD.i()M and

tw,,~daf)' ctHIdiljo~s fOllhe problem considered. Though such a finite clemenl formulalion can be carried 0111 wilhoul any knowledge of Ih. aswcia,ed governing differential equal ions, when starl ing from the principle of vin"ai displKC'mcnLs. as will be prescnlllalcr on in ' his section, a preview of these equations provides valuable clues for KI _ , ing Ille inlcrpolalion funccions and nodal degrees of freedom in I finite clement formulalWn. f or Ihe presclll problem, Ihe governing equalions will be derived as the Eliler.l.llgrilng. equations of Ihe funct;onal Nscd on Ihe varial;oruoJ prQUdufIIIl.

First, lei us integral. equalion (j2.2S) by pans 10 oolain Ihe vin,,_ al quantities h and /Iv,

1 v"ll/lv)dr ,

(2.2.26) Rewriting,

- (M - h '~/I ,, ~L - (F 1 "/I~/I,,IL 0 (2.2.27) "101'10

wh~re the follQWing relations have bc:tn utilized for the nodal displace-ments,

.. . "., " . ".,

.. -....

,,, e

,,, e

fo r :c - L

and tbc: follQWing for \he forces a1 \he two tDd$,

(2.2.29)

F - F F - F AI . -M "'rx - 0 (2.2.30) -"''''.'~'' F .. - Fz- F,. - Fyo M,. - M. [or x - L (2.2.31)

" Admitting lhal !be vinual displamenlS h .1Id lw art arbitrvy

in nalu,e, from equation (l.2.27) we can obtain lhe differential equations of equilibrium as !he EulcrLagranac eqUllions of Ibc (unctionaJ.

.t" 0

EI ~ "" 0

(2.2.32)

(2.2.33)

along "jlh the boundary oondiliom for the 1WO ends of !he lKam,

6 .. . 0 ~

6v' 0 ~

,!;v - 0 ~

F .t,, '

M _ E/ v"

F - EJ ylft , .

(2.2.34)

(2..2.35)

(2.>3"

"'M't lbe: rJJ$l Sotl of conditions, i.e., b 0, tw' 0, and lw z; 0, is Down as the 8ftH11C"''' boouui.a'1 cONiifio,u, and \he KWIId SCI as the

~8,",.1 """"da,., cOMtiitiolu. The diffe~ntial equations of equilibrium as derived above in (2.2.32) and (2.2.33) I'c encely "Ilid fOf planar rome clemenll! free of l.IIy dislributed loads. f or such problems, the uaer ,01,,';011 for the uial displacemrnl " can be ,epc_oleIn the above deriv"ion, we have shown that the vinull "'Ofl: equation as presented in (2.2.25) iii equivalent 10 the ~vcming diffeltntial equalioos and boundary Wlldilio ... oommonly used in cllSIli

interpolate the nil.! di$pl.""ment w and transverse displac

53

(2.2.44)

"' Io(re 1M Jub$c:npu ',. aDd - denote the ckgru of tile inlerpolalion fWlcuo!lS, tM SIlptnaipls s and 'r' denote 1M (Ink. of differen'QI_ ,ion .nd .~. represclWi 1M upoMOI of llie multiplying fac10f i. V.riou$ (nrep matrio::! bae4 on Ibis definition have bn g;"'cn in

Appcnd;~ A. Using (he.rove _I;on rOl' the (nlegnls, cqualion (2.2.43) CIIn

be: ,ewriucn ;n a lIIQ.e compaC1 fonn,

1&;,T !2![K,"I~ liil 16;I T EI'IK::'lI;;j [6uI T[f1 (2.2.45) L L'

"'01111, thallbc virtual dispba::mcolll (6ii) and (6~) ... 0: rorrelaled IQ (6u) [_ cqualioas (ll41). (2.2.42). and (2.2.2J)J. by applying tile arbinary natUAi of yin""l displacements, .... c can derive 1M equ.tions of equilibrium fOf the clenc:nt from equ.ation (2.2.43):

(2.2.41)

wlle.c the nodal foru Vedol$1i.1 and if,) c:orrespond 10 the displace ment VIOn (oi) and (i') respectively.

_ M VI -IF -!!

, " L

(2.248)

(2.2.49)

With tbe IIIbrtulrioes (K::'I and [K:"J Jiven in AJIPC'ndi" A, the cqllliions of equilibrium as given in (2.2.46) Ind (2.2.47) rot the axial Ind nuuol .aions can be C(lmbiMd ;0 m.tr;~ form,

(2.2.50)

" ...-Mte Ik] .epruenlS Ibc d.uric IriJf"UII Ma"h, I") 1M displaccmenl vector, and {f} the fOf()l: YCCIOr of tht. planar frune "LcIMnt. For the pruent ~ both 1M 1M} and {/} wecton luiv" ~n slWII in ~iooru (1.2.23) and (2.2.2-4) respectively. The stiffness mwi~ [ll can be given IS folJ(nII'S;

< 0 0 < 0 0 L L

121, 6EI, 0 12E1, 6EI, L' L' LI L'

.U, 0

_ 6EJ, lU, L L' L (2.2.5]) It)

< , 0 0 L 12/, _ W,

" L'

"...'""

'U, L

From this equ ion, il can be seen Iba' tlw: uial and bendina actions of the planar fnme demeo! .re 001 coupled. The eluti

"

".hc:~ G is \be sbw" modulus. 1M f.cwr .". hal been iacluded in tejUllion (2.2.52) 10 III:Q)UlII fOi the diffucnce bctwn:n tbe nulllcmlti-o;:al shew stnins (~.,. ~.,) and coginUTing s.but Stnm. (Y..,. T.J. 1bt.y have I difference of ty,~ lilnC$, Le., c" ~T" and c ~T ...

The di5pllttmcnlS of. generic point of the sp&oc rnome .,kmenl no .... contain ch,e

wM,e J i$ !he lO ... ional COIl$l&DI, and I, .ncI I, afe !he moIMnl$ of i""nia of tM cross section .bout the y- and . -aril ' espectively.

I . f.,'" , . (2..2..60)

(2.2.61)

In addition 10 tllultion (2..2.10). lhe following OI}hogonalilY conditions for prineipal centroidal eoordinll1es havt bun uK('2.2.64)

whc~ I, dtlllMes tbe tractiollll on the .section of node B Ilong the :_axis. The same modiftcation should alJo M made fOllht vinu.al work R. in (2..2.18). Based OIIlhe COIIdiliolll of equilibrium., tbret more eompn-IICnlS o f wess resultaots CUI be defined fOl lht space frune clement.. in addition 10 those of (2.2.12)-{1.2.14). They aft tIIo sbear force F .. along tile zuis, ModiDg momen. 101., boultbe ,...uis, ancl torsion 1.1 .. about the z -axis,

(2.2.65)

(2.2.66)

(2.2.67)

"

, 1

I'" ---.!:....':. fc:'-____ !'fr;. ~ ~

Iu.. 1>

" interpolation fulKtiom and element dcgrus of freedom , I n overview o f the usoc:iatcd diffcrcnl ill e

.,

(2.2.83)

By inlerpotali1l& the displacements M, ~, ... , and &, by their nodal !!epees of freedom , i.e., through $I,Ib$Utution of (2.2.37). (2..2.38). (2.2.80). and (2.2.81). from equation (2.2.72) we can obtain the followillJ:

It/w IT 5:[K~ Iw[ [661T GJIK:I~ tiil _ [6I1JT If! (2.2.84) L' L

Sin the virtual displacements (6ii). {6~1 , and (66)lrc COIl'claCW IO {hi [see (2.2.41). (2.2.42), (2.2.82). (2.2.83), and (2.2.73)1. we caJt take advantage of their IrbiUUy nature to obtain the following matri~ equation. .. from equ..tion (2.2.&4):

(2.2.85)

(2.286)

ill addition to tbo:/$e prucnted in cqu.tions (2.2.46) Ind (2.2.47). 1lM: force vectors {{,lind If.} are ddinod as

_ M if ,T . IF _.:...:l! F.

L (2.2.87)

(2.2.88)

CombiDin& equations (2.2.8S) aDd (22.86) wiIb c'{\Ialions (2.2M).Dd (2.2.47) yieLds the Slitrneu equations f()! lhe space (!'lillie element, [k]{l/) {fl , in whkh the ckllKnl dispba:mcnt \'lOr (III and Coree vcc;\OI'ill are cUell)' those: defined in equations (2.2.73) and (2.2.74) respectively. 1M Sliffness matriz [I:) for the space frune ci(menl, which has a dimtnsiOll of 12 x 12, can be given as follows:

~-"'-..."...-- " ~ [t' l (.l::IJ

It 1 1.1::1' (kJ (2.2.89) .. ~re 1M submatriccs an

" 0 0 L 0 0 0

121, 0 0 0 ' I, L' L'

121, 0 _ 61, 0

It,1 L' L' (2.2.!IOa) GJ 0 0 L

41, 0 L

". . '1, L

A 0 0 0 0 0 --L

0 _ 121, 0 0 0 6EJ, L' L' 121

'I 0 0 ----' 0 ----' 0

1.1::1 L' L' (2.~) 0 0 0 GJ 0 0 --L

0 0 6EJ, 0 21, 0 L' L

0 _ 6/,

0 0 0 21, L' L

" t_"...,. ... _11"_ QNIi(y T#n

.. L 0 0 0 0 0

121, L'

0 0 0 ~ 61, L'

12/, 0 tiEl, 0

Ik,I L' L' (2.2.9Oc) OJ 0 0 L ,

"I, L 0

-.

~~",.-~..nu. 63

"

1'11 .... 2,5

~ , , -

- - -,.) , .. , , , ..

-- - --,

O!Ie4imermanal "'" ok_ftt: (a) NodIoI do,,", of frtedom: (b) Nooal forteS.

(2.2.93)

Jince only uial deforrrurions and loClions are involved. Adopll"8 the PIM linear interpol'rioJa .. 1M OM in tqUllion

(2.2.37) fOf \he uial di$plaa:fikm . , ..., carr write

(2.2.94)

where: the clement displacement vcctOf {"I hal !e" defined in (2.2.92). SubstilUtln8 Ihe prtcedin8 expression for tM ui.l di5p1ace ment w inlo cq~liOll (2.2.91), and usin8 1M notarion of (2.2.44). we wllin

(2.2.~)

By the Ifburary lIIlure of the virtual displacement {II..}. it is possible: 10 derive

(2.2.96)

Dr in 'ymbolic fOm! 1.$ [k){. } _ {f} . For tbe one-ilimtll5ion.lllrUSII elellH:m, the 511ffllU.\: maUiE Ik] is simply

..

It) .. L .. L

(2.2.97)

wlIicll is Ihc simpksl ~iff_ matriJ: llIal can be found for singk clemenl in SlnIaliral mhanicl.. Again, il should be Idded llIal ;n the nonlinear analysis (If \J1IS5eI in t.lel chaplel'S, the claslic Iliffnus m.lfill: [t] will "50 be rcremd 10 ill tbe (k.! m.trix.

,

One basic clwKtuistic of the truss or bar element is thai ;t Qln .e:sist only uiaJ forces, but DIM War forces or belldi", momenlS. Such. chlneteristic remaim InIe ,cptdle:ss or whetMr the demenl is one, two-, or thn:e-dimelt!lionl'. To ldlieve I .ySlcmatk: trUI_III for the assembly of dement Sliffncu malri

,., ~.,_ "'11->-~ "

\' ~ r I ~ ~ ,

- - -, B ,.,

I' 1'" I'r , .. ,. ,

-- -- -, B

", fl, .... 1.' PI ...... INSl dtmml: (I) Nodal dc11ft' of freedom:

(b) Nodal fortes.

On ~ OIlIer hand, for IIJMC~ n.u tI .... ~III, !he diJp!acemrnl vectOr 1M} and fora VKUlf V} c:&II be au&IMnled 10 consist of tIm:e ... "ees of f.~m I' cadi DOCIe of the ell:ment:

(2.2.101)

(2.2.102)

TIv:se are plotte

..

I' ... 1\10 "", ... 7 T,---;-7,,- , -;{~ ~

( .)

( .)

from the above upressioll$ of the stiffness mllrix It). 11 is obvious !hal I truss clement cannot resist .ny traosvcrse sheu foroes.

Before elating 11111 Jediol!, one commu\ should be given for !be clemenl loads In !be fongoi"l derivations, we hive IU$IImW lhatllli tbc applied loads an lXNICI:otnolcd aI the DOdal poilus .imply for the We of c:oavcnicnc:e. Sud! all _mptioa should IlOl be regarded ... res:lricIion or Ibe theory. if it is ruliZ1Cd IbM &lI1

lJ , ........ .,-- - 67

LO ing tbc stiffness method. the f("mullli~. of SlruCiurc .equ.~ioll5 (PIP Y rall)' scparltro in10 IW(J pam pI',UJnlll& 10 1M IndIvidual IS ."';.nd the uscmbl.,e. Each of lhe twO paru is formed in terms dc:;:ng>nlctry, JUlies, and ronstiMive cqualioll5. In the preding of . --.hods bave ban prdtnted for deriving the dement stiffness - .". fqU.ljo1!S,

[tllwl [/1 (2.3.1 )

bued on the plillCiplc of vinual di5pla~mCnls. In Ihis section, We ~I procd 10 demoPSlrJ.1C 110 ... 11le: stiffnc:ss eqlllltions for fIlCh clement ~ be assembled 10 fonn the SUUC1u,e stiffness equations.

IXlIUJ [PI (2.3.2)

follow;n, tile prooedure of dir'l stiffness method. Here 1,1,7) dcIl(lLC$ the iliffr>e$l m.ui:c, {U} 1M d.ispboo:mcnl ~ctor. 100 [p} 1M load VK'IOf of the SlIUdIin. 11 is 110( the pIIrp

68

' I

, -- .

-

-{' Ft,un 1.' Element o;:oordinalel.

of I thin! node 011 1M uis or I dirKtion ve

" F~"'-"''''I'''-' "

IJ '. '. '. I::! " " " '. '. '. (2.3.3) or. in shor1 f(>l1ll,

IPI - [yll,llJ (2.3.4)

in ",hleb (P)lnd lPllknole the components of the ro,~ yeao' {>'n tbe loW and glob,l./ coordinate sySCCRlS, respo:c1ivdy. With the prucot OOIluon, In clernen1 manu Of "'WO!' with ,deft,," 10 the global coooIilW is ck5ignaleG b~ I 1ui1 .... The rotation mlltix [y I is knoW" 10 be an 0ftII0&0nlI m.lrm-, because of the faa Iba, its inverse i, equal 10 ilS Uampo5C', i.e., [y]"' [r]' (McGuire and Gallagher \9'79). ROgniringlhal the arne rule of tnonsIormation applic:s to the dired foroes. momrnlS, translationl, and roulions. we can usc equation (2.3.") dire

Oearly, tile transronN.lioo matrix In is onhoaonal, as !he submatrices h I contained in il an OfIho&onal.

Now, by subslitutiD& 1IIe ~lalioDl (2.3.5) and (2.3.6) into tbe elemenl Sliff_ eljlWioo (2.3.1), _ p:I

(2.3.8)

Using 1M propc:ny of onhoaonalily for the WI mallix, l.e., (n'ln (rl'(r] /I),

,

rr)T[k)[rHII I til (2.3.9)

Thus, we have shown lhal 1M elemenl IIiffllUl equations an be tn.nsformcd 10 the global QOOrdWIC$"

(2.3.10)

whe~ the matrix (I) in tile global QOOrdinala is ~1aIed 10 the IIIMrb It) in the local c:oonlinalU.,

(2.3.11)

and the load ~eclor {}} is defined as

(2.3.12)

The po:ccdilll IWO eljWllioos defillC .... haI ill ealkfI, ...... frll1lS/f>r ... n-. The IraDSformed matrix 11) will be sym .. nellic, 10 Ioa& as IIIe. matrix It) is symmetric.

The InInsfonnalioa equalions for oIltcr typc:s of ekmenlS, SlId! lIS tIIc. planar frame clement, and p/lIlI.r Ind spII;e tnass clenw:nlS, an be conside~ .. tpe

,

~~.-l-_ X "I.~r< :. 10 for in .1emtnllnd WUCIUI\! coordinates- pl."" Q$

"f1u!od ItpOU :H.jl II ~lddl ~I~!P :lI1111(l speD[ P~I"'U~ ~I glOU:Ip "(J) J()p;I" ~41 ~J:>II""

,.,

(L1"("t) "IJ) IJI 3: - IJ )

0 1 PQlJ!potu ~ Plnosp (s ITU tIO!llnb~ '~Jn I:MUIS;oI,j1 JO Slu!od ItpOU ~41 II 051" 1/"1 J1UOKU~I~ ~ \10 ~IUO IOU P:I!ldob:ll1 ~ [1I1U~n;ol,j1 ~~"""XCI J1l:lll~ ~JO\U

~ JO.:I ":KI~ 8u!41~WOII JO SPO:O( [!pO-~:lIJ JO S~2J!;,p P""!"1S21 241 01 8u!pUOOsmoo r~fllu~ '"'I1l1U!121:>P ~q pili p!IIU~ ~WWnp;oI,j1 VI f()J:n; IU!P"IJIi! ~II '~l~A!]:l;K!RJ " "N pili N " N '"2"1 ';un]:)nJl! ~41 JO (J) JO);nA ptOI pili LvI rflllw -JJ!II '"'fl JO UO!su;>W!P ;oI,j1 01 jlu;ruJInl ~1I'nidtJO.) 1>

1,1 ~." , ,. -_-" It should be IIOI~ I~I bodl equalions (2.3.1S) and (2.3.16)

.emain VlIlid even if lhe auemblaae conlains diffe.enl Lypn of elemenlS. In (his case, LlIe number of element degrees of fludc)m. II. may e!wlge (rom .lemenl to ekmenL. II $hOu1d be ado:.kd (hal, in practice. lbe upansion of lhe clemen( mmiccs [.fW] and (l 1iJ) from lheir original dimensions of ~ ,,~ and ~ " I to malcb LlIc: dimensions of lhe SUUC!u", maujcu IKI Ind (PI, i.e., N " Nand N )( I , is 1101 necessary. One simple way ill to fi/"5l identify lbe IoCIlions of (lie cnlties of !be clemc:nl mllrK:e$ [l"ll and (ii'l) in the SU\ICIurc malrices [KJ Ind (PI, rcsplivdy. Ind (hen add (hem 10 (he ui$1ing vlllIC$ in tlIose Ioo;ations as we loop /)VC' all IlIe elc:mc:nlS, i.c., w,lh IlIe indeJr I

,~ngini: from 1 10 E. In Ihe prcJWalion of Ille fini(e clement mesh, ,,"c alLxb 10 exh node of IlIe at6C111blagc a node number, and ach dement an clemen( numbel. The connccdon of each elemenl 10 (he assemblage.nd therefore, IlIe MUons of lhe enuiQ of 1iV').nd awl in IKI and (PI rcspec1i~ly. can be dttennilltd from the tkgrtts of fI.cdom US(lciatcd with the labels of tbe nodal poin~ II Lhc IWO ends of each clement.

The procedure [H.-nLed in Lbill5ection ferves only 10 tkmansl/ale the bask concepI involved in the formation 0( the stl1>C1ure stiffness equations. Variant!: of !be prooc:dure Lbal Lake into accoonl lhe .rr.dency of IlIe o;OmpuLer mellIOf)' and OIher flClon., such as (he symmelry and Mlllkdntsli of tbe stiffllt5ll mat';ces.. Irt aVlilable elsewhere. Readers intelfSkd In such Lopes sIIould refn to finil. elemenl ICltbooks, such as BalM and W ilson (1976) Ind Tong Ind ROSKIIO$ ( 1977). for more informlLion.

2.4 Sol ution or simultaneous equations

Assume Ihat a 5HUC1Ufe is p.-operly resLnLinoed against rigid body motions and Ihat III rcwained degrees of freedom of Ibc $trUctU~ hlve been rel/lOVed in lhe fonnalion of Lhe $LruC1UrC stiffness cq~aLions:

IKllUI (PI (2.4. I)

"'here [K] den(lles Ibc Sl,ffllt5ll malri~, (Ul tM dis.placemcnl vector. Ind (I') (he load vectOi" of the 1I11>C1UIe. If the $UUCfUr. has I lou! of .. activc desrees of freedom, Ih(cn Ihe "reding matrix equatiofo will

" rone.in " simullancous C

" where ILl iii lower lI'iangular ml1rix and [0] ;s diagonal malriJ,

'" D,

r." L" IDJ D, (2.4.3) (LI , ,

'. '. '. D,

By expansion,

,

"' E .. , LWD,L,. (j d: / _ 1,2, __ , 11) (2.4.4) Sinc:c !be stiffness matrix [XI is symmetric, namdy, K, K,. the l"ecediDS exprc"ions rontain II most n(1I .. 1)12 ilMkpcndc:ntequations, bIn !here arc "(II + 1}'2 cnlriel of L, and" mira of D, 10 be evaluat-ed. We tbercforc hIve .t least II free chokes ill usigning tile val\1C5 of L, IlDd Dr

With the Clio/est, IfWbd, all of the II entries D, of !be diJ.&oDal matrix [DJ an Jel equal 10 unity. In aocordan.ce, we have the following rttUrrcDC formulas fOf Li-

...

L~. K,- Ei.! (2.4.!i) "'

(/ > j) (2.4.6)

Which may be applied by columns. Onr. drawbKlt wilh the a.olky IMthod is Ihal il works only for positivc.dertnite S)'Jlcms. For l)'SIems IlIaI He DOl pCIIIitiv~llnitc. we may encounter nUlm'rkaI diff"teuUies luch as the ~~1ICe of IM:galive numbers wilhin the IIqIIUC roo! of tqualioQ (2.4.5) in Qk:u~liDg!he diagoll&l enlr1e$ L,.. Beause of Ihis 'eslriClion, the mctllod is DOl recommended for 11K beyond the to/ulion of pl)Sitjyeflllite systems..

" A1IUnltively, IlIC may cl>c>oK 10 oel the ~ d;agolLll entries L.

equal EO unily. In this f;aSC, the 'J

,. ~""".I_--' 77 ~ displamcnt veaor lUI caf1 be IIOlved in three 5t~ps. First, by kUma [DJ[L)'{Ut .. {G}, equation (2.4.10) 7CdUl:S to

(LIlG) .. [PI (24.n)

SInce [LI Is lowcr triangular matrix, we may detcrmine (G) by f(IfWud SUbs1itution Slaning from !he first equllion of the system. This will yield the following for ~ach element of the (G) V~ClOf:

(2.4.12)

... bere fOf I I ,,e have G I "F ,fL". The second stcp Is to divide both side:l of [OJ(L]'{U) .. {O} by [01, ladin& to

ILtlUI .. [HI (2.4.13)

wbere IH) .. [OI ' {G) or, equivalenlJy, H,. GIO~ The rmal stcp is 10 1101"" !he preceding equalion fOf lUI. Th;'..." be KCOmpli5hed in. Ilmple proce$I by ~kward substirution suning from !he 11$1 equation of the 'yllem. The following is the ,eame~ formula for each clement of {UI:

1 U, -(H, - L LflU)

l.. J .... , (2.4.14)

"'heu fori SlaI1 w~ iet U," HJl._. Thul, we have sbownllow the SUU

" the clcmcll~ fOKell (J} in the same Q)(Irdi...;,u. The proo;edu'CI de-ac:ribcd in this Ition and tho: prcccdina IItioII Nt lypK::al for the analysis of $IlUdullIl (I1Imu and lruMeS COfIttrning lhe linear behav_

""

2.S Quality I~ ror linear dtmeDl.!i Before wt uolt Ibou. the qualil), of finite clenxnl analysil, kt us brieny review KIfllC bu~ f~l\Ires of Ole mO$I notable Ipproximllc prIXlCdurc, lhe R,ylclgA.R'" itmAod, for $(lIving the boundary-value problcml. Willi this nxlhod, rJIe rIM Slep is 10 approJ.imate !he UI(:I ~isplvemenl (!tId 0(' structure by fulldions which contain I finite number of indqlende.nt eoefflcicn". By subIIituti", !he Ilial displacement rtcld into Ibe poIenlial of !be JUIJCIIIre, and by rt:

P ~-I-'-'- " mbel of nodal degrees of ~dom, TarMr Ih.n [brOllgh addilion of ~ fulldlonllO rhe ,ri.1 series. To en.lure.~vergtna: of the. rmil~ ekcmcnl solutions 10 Ih;c u-.:t one, a:rtIoln crn~ bve to be Ja1:":~' All rhese crirer;> flll 1010 three groups conCC'.nlng the compatibIlity, """,plc:tellCSS, and stability, of the finite clement II5oI:d. ~~ elements tbat fulfil .11 tbeK aitenl, the convergena: characteristics of the: apptOJ[im.atc solutions cu be lbown 10 be monotonic or DCUly asycnplOl!l;.

1.5.1 Con.-trgcllre criteria

Willi ,espccIlO Ihe rompdibilily of fUlilc clements, it is TUlllimi WI ~ displlccmenlS within the elements and acroll the dement bound-orin: be c:ontinuou$.. For elcmentJ WI employ polynomial funaions tOl' ialcrpolaring rhe demeot di$pllamenlS, wbich is Lhe case for tile frame and IJUSS elements to be discussed Ihro\J.ghoIat tbe Itlll, Lbc:rc is no problem in IIU.lnll.inina' OOIItiollOllS dispbcenxnl field wilbin the fUl;!e clemenl.

Eklmnts thai satisfy both the IlIIpatibilily and COOIpletcneu ClDllditions arc called COlIforM;'" eluoG'ltJ, while !hose that saiUry only the condition of complelcnt$S but violate the oondition of inICrclcmcnl compuibility an Cllled IIDf1CtHlfqr",iII, or incoMplJfible ~I_t"ts. A

~nformin, el~ment Qn still be u.Kd in practice to yield ,ood mailS, if 1M dC'ment CUI apprOIiCb !hi: state of \XIIISUnt stn.iJI as \be fmite dement mesh is relined. One difference I~ that the manJ>C1 of cortverscnce uhibited by fioite element models with IIODQ)IIformiog elements 1liiy not be monotonic. II sboItld be mentioned thai the elements fQ be diseuuc:d in this tU t, includin, the trusS demcnt, frame clemeOt, &od curved beam tlemenr. eilOO two- or thne-dimensiollll, all ~I in the QtelOry of conform in, elcmenl$, as they all satisfy the

Icrclemcnt compatibility condirioos. f.. __ n.e uquiremcnt of compietcllCSl Implies thlt the displacement '""""Ions of the elements must be ~Ic to reIRKot both the ,/,Id bod)' =q ~ II1c COIlS'"", Jrr"", JUllu. From the point of rmile clement I ull"on, we woold e~pea tbe assumed di$plarmnt rleld of I finite

;,:'.ent fQ be able to reptUeol the cua displaamc:nt field IS closoe as dlspJlbIe. II ... ill be found that this Is not lUIinly $0) if the c:boscn it . o.ee~ents .110 such thaI siraining of !be clement Is possible wben olc!e lU~ed 10 rigid body motions. Thus, the ability of I fInile Ial nl ~ lepreso.ntthe rigid body ItIOde$ bea>mes the mosl fundamen

requirement. In I r",ile elclllt'nl formUlation thaI employs 1M

pol)'llOrDiai f\mctions .. IlIc inlcrpobtion functions. !be ri&id body modes can be easily COftSideKd through inclusion of IlIc COOSWIt and lillCar terms in !be po!yllOmial_ics.

Fr(lm , physical point of .. ie .... !be necessity for finite dements 10 represent tile OIJnstlnt 6ln.in $Utes can be ulWkr$lood if ... e try to U$C more and more elemenll to model, structure. In tile limit u the finite e]emenll get smiller 100 smaller, tile 6ln.in in each dcment I.hould Ipproach I consllnt "Ilue. Sudt. capability ill uscntial 10 tile modeling of SllUCtIIra dIowin& complex varillion in 1Uaia. In beams and plates. ... lIcre bending deformations arc involved, IlIc 6ln.illJ IIbould be intclpR'lCd IS !be gencmi=! SU1Iins. Obviously,!be $tate of oonsUnt smillS iDOOi PClIItes !be rigid bod)' mocks IS special case ... ith UfO strains.

By ,"bilit)' . ... e mun thlt tile solution of tile Itructure Sliffl>CSl equltions (2.4.1) mu" remain bounded and unique. This implies that the stiffness matrix [K] must be lIOII5ingl'llr and the ItlUClun: must be

~]e, given jUSl cnough rCSlaint OIJIIditiOtl$ to prevcnt rigid body moiions. For, Sllble stnIeIure, !be displaoemenll (U) sol .. ed ffom equation (2.4.1) OIJntlUlllO zero eocrgy modes. 1M stability condition bas ududed !be potSibility of 1liiy di .. CfSCnce or numcrical overfJow ill !be solution process.. It is therefore I IUfficient condition of coovcrgeDCe. In COOItaSl, !be oomplctenes:l and compatibility conditions mentiCIIIIXI above rcpresent men:ly I _I)' condition of CO

JJ a-~""'Jo

" widdy used pro:4u~ for checkill& \be ekmeot quality. Considtr \be following eigo:nYlllue rqu&lion for an IIIIrtStrained finite eJemem:

((t] - ~ llllllIJ .. [01

whe~ [t) lepresents the oomplc\e clement matrix. (I] is unit matrix, .I. [be eigo:llvalue, and (It) the \XII1'esponding ci,en~ector. TM." arc as many ci&eovalues l , as Ibcre arc degrees of [rudora in \be cigo:nvottof {II} . Lellhc eigenvtof (II), be nmn.alizcd sudllhaI

(2.5.3)

We un premuhiply

OJ (eS1$ p.csenlcd in chis scClion. In lhe sections to follow. we shall prottcd 10 cklllOftSlr.llle bow the ri&id body (\lies p

..

As was shown in Figure 1.1. in an incremental formulation lhe history of mOl ion of a body can be described by IhRe \ypial configu .... lions: IlIe initial undeformed CQIlf"Ul'1Ilion C. the la.st calculated configuration C" and the CUlTCnL deformed configun.lion C,. In an updated ugnongian fonnulation, all physical parameters are referred \0 the lasl QlculalCd configuration C,. ahhough I~ cqualions llf equilibri-um are established for lhe body II tilt: cuneol oonfiguniltion C,_ Based on the principle of vinull worl

8S

where [1:.1 is tile duci

..

Wi,h regard 10 lhe rigid body motion, the following comments can be made. Firsl, I finite element thaI satisfIeS the ~tch res! Of ilS eq",w.lcnt in !be Iinell' acllSe does no! proessorily imply that it will.1so satisfy 1M rigid body aitcriI in 1M _linea. ICIISC, because 1M elU1ie stiffness matrix [i,1 &lid geometric Slillness matrix [,1,,1 "'n been

deri~ed from leoos of diffeRnt natUle in Ihe v;nllil WQ.I< uprusion. Second. jUSlI$ a linear clemen! ml, rlil In pass tile qulli.), lesl, 10 100, I _liMar fmill: element may r,illa mt the rigid body requucmeuts. Third, !lie quality of. nonlineN clement cannol be oonside~ uliable WM'-U it violalcs rhe rigid body rule. FinaUy. Iltbougb IlM:.rove discussions have been restricted In problems wilh sman Sllllins, lhe same rigid body aileli. Cln be oonskkred as lbe limit ~ for lesting clements ,nvolving large wains Of indastie RUIICrial propenict..

2.6.2 RIgid body rule for In ltla Uy llresseci dcrrwnl$

Consider I bar sittiJl& on the surflOl: of tIw: earth and subjcc:lro 10 II1IIv;tationa! force P at tbe top. For equHibriwn ruclion of the same magnitude P will be fonned I' the bottom [Figure 2.11(1). Sucb. loading condition can be regarded AI the C, c:onriguralion of !he bar. Now, suppose WI the ea"h TOIales IS a rigid body by an angle 8,. In 1OCOrdanoe, the bar will move 10 the C, oonfJgU.1lII1on in the rDannef s.hown in Figure 2..1l(b). From Ibis figure, we observe WIllie line of ac:Iion of fora: P rOUtes wilh lhe movement, while il5 mignillllk remains unelwlgeli. An overall result is lhe malntcnAnCcof equilibrium of the bar in Ibe C, confi~ion .

Whal we like 10 &trc$$ ~re is thai if the bar is rcpraclllN by

,

1 'oJ

"J

Vi l a ... 1.11 Inillilly IoocIed bar. (I ) Before riSid body roIallOft;

theory and anlysis of nonlinear framed structures - y. bin yang.pdf

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