Finit Elmnts from th arly bginning to th vry nd A(x), E(x) g b(x) h x =. x = L An Introduction to Elasticity and Hat Transfr Applications x Prliminary dition LiU-IEI-S--8/535--SE Bo Torstnflt
Contnts 1 Prlud 1 1.1 Background............................... 2 1.2 Th Big Pictur............................. 3 I Linar Static Elasticity 5 2 Introduction 7 3 Bars 9 3.1 Th Bar Displacmnt Assumption.................. 9 3.2 Th Local Equations.......................... 1 3.3 A Strong Formulation......................... 12 3.4 A Wak Formulation.......................... 13 3.5 A Galrkin Formulation........................ 14 3.6 A Matrix Formulation......................... 2 3.7 A 2-Nod Elmnt Stiffnss Matrix.................. 21 3.8 A 3-Nod Elmnt Stiffnss Matrix.................. 25 3.9 An Elmnt Load Vctor........................ 26 3.1 Th Assmbly Opration........................ 28 3.11 Strss and Strain Calculations..................... 28 3.12 Multi-dimnsional Truss Fram Works................ 29 3.13 Numrical Exampls.......................... 32 A 1D bar problm........................... 32 A 2D truss problm.......................... 36 A 3D truss problm.......................... 39 3.14 Common Pitfalls and Mistaks.................... 4 3.15 Summary................................ 42 i
ii CONTENTS 4 Bams 45 4.1 Th Bam Displacmnt Assumption................. 46 4.2 Th Local Equations.......................... 46 4.3 A Strong Formulation......................... 49 4.4 A Wak Formulation.......................... 5 4.5 A Galrkin Formulation........................ 52 4.6 A Matrix Formulation......................... 54 4.7 A 2D 2-nod Bam Elmnt...................... 55 4.8 An Elmnt Load Vctor........................ 59 4.9 A 2D Bam Elmnt with Axial Stiffnss............... 61 4.1 A 3D Spac Fram Elmnt...................... 62 4.11 Strss and Strain Calculations..................... 64 4.12 Numrical Exampls.......................... 66 A 2D consol bam........................... 66 4.13 Summary................................ 69 5 Solids 73 5.1 Displacmnt Assumptions...................... 74 Th 2D Mmbran Displacmnt Assumption............ 74 Th Axisymmtric Displacmnt Assumption............ 75 Th 3D Displacmnt Assumption.................. 77 5.2 Unknown Strss Componnts..................... 77 5.3 Th Local Equations.......................... 8 Th Balanc law............................ 8 Th Constitutiv rlation....................... 81 Th Compatibility rlation...................... 83 A Summary of Local Equations.................... 84 5.4 A Strong Formulation......................... 85 5.5 A Wak Formulation.......................... 86 5.6 Th Galrkin Formulation....................... 9 5.7 Th Matrix Problm.......................... 94 5.8 Th Assmbly Opration........................ 96 5.9 Th Elmnt Load vctor....................... 99 5.1 Th 2D Constant Strain Triangl................... 1 Th Elmnt Stiffnss matrix..................... 1 A Numrical Exampl......................... 16 5.11 Four-nod Rctangular Alignd Elmnts.............. 111 A Numrical Exampl......................... 112 5.12 Isoparamtric 2D Elmnts...................... 115 A 2D 4-nod Quadrilatral Elmnt................. 115 A Numrical Exampl......................... 12 Numrical Intgration......................... 125 Th Numrical Work Flow....................... 129 2D 8- and 9-nod Quadrilatral Elmnts.............. 131
CONTENTS iii Sub- or Hyprparamtric Elmnt Formulations........... 134 5.13 Isoparamtric 3D Elmnts...................... 135 5.14 Isoparamtric Axisymmtric Elmnts................ 137 5.15 Distributd Loads........................... 139 Lin Loads............................... 139 Surfac Loads.............................. 142 Volum Loads.............................. 143 5.16 Raction Forcs............................. 143 5.17 Strss Evaluation............................ 145 5.18 An Elmnt Library.......................... 148 2D and Axisymmtric lmnts.................... 148 3D lmnts............................... 149 5.19 Summary................................ 15 II Linar Hat Transfr 153 6 1D Stady-stat Hat Transfr 155 6.1 Th Local Equations.......................... 156 6.2 A Strong Formulation......................... 157 6.3 A Wak Formulation.......................... 158 6.4 A Galrkin Formulation........................ 159 6.5 A Matrix Formulation......................... 161 7 Multi-dim Transint Hat Transfr 163 7.1 Local Equations............................ 164 Th Balanc Law............................ 165 Constitutiv Rlation.......................... 167 7.2 Strong Formulation........................... 168 7.3 Wak Formulation........................... 169 7.4 Galrkin Formulation.......................... 17 7.5 Matrix Formulation.......................... 172
iv CONTENTS
Prfac Th writing of this book has arisn as a natural nxt stp in my profssion as a tachr, rsarchr, program dvlopr, and usr of th Finit Elmnt Mthod. Of cours on can wondr, why I am writing just anothr book in Finit Elmnts. Th answr is qually obvious as simpl. Aftr many yars in th fild I hav, as hav many othrs, discovrd a larg varity of pitfalls or mistak don by othrs and myslf. I hav now rachd a point whr I would lik to dscrib my viw of th topic. That is, how to undrstand it, how to tach it, how to implmnt it and how to us it; ths will b main goals for th discussion to com! Th discussion to com will b influncd by xprincs from all four of ths branchs. As a tachr I hav taught both basic and advancd courss in Finit Elmnts with focus on both Solid Mchanics and Hat Transfr applications. Ths courss hav bn givn at th Linköping Univrsity in th mchanical nginring programm. Th txt to com is writtn by an nginr for nginrs. On ovrall goal for th dscription is to try to covr vry stp from how a crtain mathmatical modl appars from basic considrations basd on fact from rality, to a classical formulation with its possibl analytical solutions, and finally ovr to a study of numrical solutions usd by a Finit Elmnt program basd on a crtain finit lmnt formulation. That is, discussing th finit lmnt mthod from th arly bginning to th vry nd. Th grat challng is to mak this as short and intrsting as possibl without loosing or braking th mathmatical chain. It is strongly blivd that for succss in larning Finit Elmnts it is an absolut prrquisit to b familiar with th local quations and thir availabl analytical solutions. I think most popl who hav trid to tach Finit Elmnts agr upon this, traditionally howvr, most ducation in Finit Elmnts is givn in sparat courss. Why not try to tach Finit Elmnts in clos connction to whr th basic matrial is taught. That is, intgrat Finit Elmnts togthr with basic matrial in th v
sam cours! Of cours, Finit Elmnts can b taught as a wightd rsidual mthod for approximat solutions of sts of coupld partial diffrntial quations without discussing any physical application and just focusing on xistnc, uniqunss and rror bounds of th solution. This is of cours also important but for most studnts studying diffrnt nginring disciplins Finit Elmnts will b a tool for trying to undrstand and prdict th bhavior of rality. An important focus in studis of finit lmnt formulations of diffrnt nginring disciplins is to b awar what should b xpctd of th quality of th approximat numrical solution. This can only b larnt by knowing th most important dtails of th mathmatical background and thn solving numrical problms having an analytical solution to compar with. Anothr aspct important for nginrs working with th mthod as a daily tool, is how to us th mthod as fficintly as possibl both from a timconsuming and a computr rsourcs point of viw. Ovr a priod of at last 15 yars I hav workd with th graphical finit lmnt nvironmnt TRINITAS. This is a stand-alon tool for optimization, concptual dsign and ducation as wll as for gnral linar lasticity and hat transfr problm both as stady-stat, transint or as ignvalu problms. It is an Objct Orintd program basd on a graphical usr intrfac for manipulation of th databas of th program. This program contains procdurs for gomtry modling, domain proprty and boundary conditions dfinition, msh gnration, finit lmnt analysis and rsult valuation. Th program is usd in duction at diffrnt lvls. It is usd in basic courss in Finit Elmnts at an undrgraduat lvl and also in advancd cours whr th studnts add thir own routins for instanc; lmnt stiffnss matrix, strss calculations in lasticity problms or utilizing rady-to-us routins for crack propagation analysis. This finit lmnt nvironmnt has also bn usd for tsting of diffrnt rsarch idas and for solving of diffrnt industrial nginring applications. This program will b usd throughout th txt in this book as a tool for analysis of all xampls givn during th discussion of diffrnt finit lmnt applications. Th idas and th argumnts givn abov hav bn th main driving forc for doing this work. Hopfully, this book will prov usful as both an introduction of th mthod and also a standard tool or companion to b usd during daily finit lmnt work. Bo Torstnflt Novmbr, 27
Radr s Instruction Radrs that hav nvr studid Finit Elmnts ar rcommndd to first rad th bar chaptr (chap. 3) from th arly bginning to th vry nd vry carfully. It is th author s blif that this chaptr is dtaild nough to srv as a stand-alon bas for slf-studis whr th radr is rcommndd to, during rading th txt, prform a complt rwriting of th basic mathmatical chain. Evry chaptr to com is writtn in a mannr and with an aim to b mor or lss slf-containd for th radr with sufficint pr-qualifications. Typical rquird qualifications ar 2 yars studis at undrgraduat lvl of any of th most common nginring programms. Concrning th layout of th txt; thr ar important kywords which will appar as Margin txt. As a studnt rading th txt for th first tim on should, aftr rading a crtain chaptr or sction, go back and us ths margin txts as rmindrs for having rachd a sufficint lvl of undrstanding of diffrnt important concpts. On major challng whn trying to dscrib Finit Elmnts is to giv sufficint dtail without making it too lngthy. That is th rason why som in-dpth matrial is givn at th nd of th book rathr than whr it appars for th first tim. Margin txt vii
Notation Notation principls usd in this book ar summarizd blow. If a lttr or symbol is usd twic with a diffrnt maning, th lttr or symbol will b givn twic in this list. As such, if a concpt dfind by a lttr or symbol has svral diffrnt usd nams dscribing th sam and quivalnt intrprtation, all will b givn blow. Gnral mathmatical symbols a a a i A C C 1 L S W G M n sd n l n n n n n f n p A scalar valu A column vctor writtn as a bold lowr-cas lttr A cofficint in a vctor A matrix writtn as a bold uppr-cas lttr Th st of all continuous functions Th st of all functions having a continuous first-ordr drivativ Local quations Strong formulation Wak formulation Galrkin formulation Matrix problm Numbr of spatial dimnsions Numbr of lmnts blonging to th msh Numbr of nods blonging to th msh Numbr of nods blonging to an lmnt Numbr of unknown frdoms in th msh Numbr of prscribd frdoms in th msh ix
Latin symbols A An ara or cross-sctional ara a Th global unknown vctor, th global displacmnt vctor, th global dgrs of frdom (d.o.f.) vctor a An lmnt-local unknown vctor, lmnt-local dgrs of frdom (d.o.f.) vctor b, b Load pr unit volum or lngth B B C Global kinmatic matrix Elmnt-local kinmatic matrix Boolan connctivity matrix c, c i An arbitrary vctor usd for th wight function c c p x i, y i, z i D D E E f f d f g f h f r G G Elmnt nodal coordinat vctor Spcific hat cofficint Nodal coordinat componnts Elasticity matrix in flxibility form Thrmal conductivity matrix Young s modulus of lasticity Elasticity matrix in stiffnss form Global load vctor Global load vctor from intrnal distributd forcs Global load vctor from ssntial boundary conditions Global load vctor from natural boundary conditions Global raction forc vctor Shar modulus Global oprator matrix, gradint matrix g, g Essntial boundary conditions (multi-dim or 1D) h, h Natural boundary conditions (multi-dim or 1D) I J K K L M N Ara momnt of inrtia Jacobian matrix Global stiffnss or conductivity matrix Elmnt stiffnss or conductivity matrix Lngth Bnding momnt Global shap function matrix
N i N Ni N Q A global shap function Elmnt-local shap function matrix A lmnt-local shap function Axial forc in bars or bams Hat gnration pr unit lngth q, q Hat flux pr unit surfac (multi-dim or 1D) q n q R r T T T t t t t i S S S S g S h s u Hat flux prpndicular to th surfac Load pr unit lngth Rsidual vctor Unbalancd forc vctor or discrt rsidual vctor Tmpratur Surrounding tmpratur Shar forc Tim Thicknss Traction vctor A tst function Surfac Statical momnt, th first momnt Strss tnsor Th part of th surfac whr ssntial boundary conditions ar known Th part of th surfac whr natural boundary conditions ar known Strss vctor Displacmnt vctor u, v, w Displacmnt vctor componnts w, w Wight function (vctor-valud or scalar-valud) V Volum x, y, z Global coordinats Grk symbols α α δ ij Thrmal xpansion cofficint Thrmal convction cofficint Kronckr dlta
φ i ε ε θ i λ ν ϱ σ ij τ ij A wight function Normal strain Strain componnt vctor Nodal rotation componnt Hat conductivity Poisson s ratio Dnsity Normal strss componnt Shar strss componnt ξ, η, ζ Local coordinats
Chaptr 1 Prlud Nowadays Finit Elmnts ar th standard tools for doing simulations in a larg varity of nginring disciplins. Finit Elmnts ar no mor a tool for just a limitd numbr of nthusiastic xprts; thy ar somthing all of us as nginrs hav to larn. On rason for why this mthod still, to som xtnt, is lookd upon as a tchniqu which you as an nginr can dcid not to larn is probably bcaus it is blivd to b too difficult and tim-consuming. It is now tim to chang this option onc for all. All of us can larn Finit Elmnts. Evry nginr must know at last som basic facts from Finit Elmnts applid to som of th most important filds of application. Whn trying to larn Finit lmnts it is important and usful to hav a solid knowldg of th physical problm, modls of it and thir analytical solutions. That is why Finit Elmnts should b studid in clos connction to ovrall basic studis of a crtain nginring disciplin. Finit Elmnts is just a approximat numrical tool for solving som basic local quations constituting a mathmatical modl of rality. A rason for why this tchniqu is still lookd upon as difficult to larn is probably that most txt books ar writtn by ddicatd rsarchr in diffrnt filds of finit lmnt applications. As an author on probably tnds to dscrib th mthod from a mathmatical point of viw as consistntly as possibl and with a notation prhaps nvr prviously sn by th studnts. Th tchniqu is now so wll-stablishd that from a mathmatical point of viw most faturs and dficincis ar known in a varity diffrnt mathmatical formulations of diffrnt important problms. In this txt w will larn why th mthod works, how th mthod works both analytically and numrically, how to us th mthod in typical daily nginring applications and, probably most important lsson what proprtis on should 1
2 1.1. BACKGROUND xpct form th numrical approximation of th unknown ntitis. It is th author s intntion to writ a txt covring dtails from th arly bginning of a discussion of th modl of rality, which w as nginrs would lik analyz, to th vry nd whr w hav th rsults from th finit lmnt analysis. Gnral Faturs of th txt: Evry finit lmnt application will start from th arly bginning of its application with a discussion concrning which ar th basic quations, why must thy hold and what ar th basic physical assumptions. Evry important concpt and xprssion will b dducd and th mathmatical chain will b unbrokn throughout th txt. Th mathmatical languag will b simpl and concis Th txt will not b wighd down by any rigorous mathmatical proof of important statmnts. Evry finit lmnt application will nd up in on or mor solvd xampls by th finit lmnt program TRINITAS. Th txt will also srv as a thortical dscription of what is implmntd in this program 1.1 Background Finit Elmnts hav bn dscribd ovr th last dcad in svral diffrnt ways. In th arly bginning it was dscribd as a Rayligh-Ritz mthod for lasticity problms and latr on as a gnral tool for solving of partial diffrntial quations of various kinds, always basd on a so calld wak formulation. From a mathmatical point of viw, th first dscription was basd on Calculus of variations and a modrn formulation is now basd on Functional analysis and th thory of linar vctor spacs. Important basic work was don by Courant during th first part of th 194 s and th word finit lmnt was coind in 196 by Clough. Intrst from nginrs working with diffrnt aronautical industrial applications was on of th main driving forcs during th dvlopmnt of th finit lmnt mthod. During th 197 s th first gnral-purpos commrcial finit lmnt packags wr availabl and othr nginring disciplins startd to us th mthod. Th dvlopmnt of diffrnt computr basd support activitis such as prprocssing of finit lmnt input and postprocssing of finit lmnt output, and th ovrall succss of th mthod has bcom possibl du th fast incrasing computr powr which has bn going on in paralll. Today Finit Elmnts ar on important cornrston in th ntir Computr-Aid Enginring (CAE) nvironmnt containing most nginring activitis ndd to b don in most nginring branchs.
CHAPTER 1. PRELUDE 3 1.2 Th Big Pictur Nowadays Finit Elmnts ar usd in a larg varity of nginring disciplins. Typical filds ar lasticity and hat transfr problms in solid bodis and acoustics and fluid flow problms in fluids. A larg numbr of diffrnt linar or non-linar, stady stat or transint problm classs xist. All ths applications ar somtims calld Computational mchanics. If th scop is vn furthr xtndd, us of Finit Elmnts is also possibl and straightforward in magntic fild and diffusion problms tc.. This txt will concntrat on lasticity and hat transfr problms which ar th most important applications of Finit Elmnts among all diffrnt computational mchanics disciplins. A rathr limitd numbr of physical ntitis wll-known by most mchanical nginrs will b usd in ths formulations. In lasticity problms th displacmnt vctor u and in hat transfr problms th tmpratur T is of grat importanc. In fluid flow problms th vlocity vctor v, th prssur p and th dnsity ρ ar basic unknowns. In acoustic problms th prssur p onc again is of grat importanc. Plas obsrv that th vlocity v is just th tim drivativ of th displacmnt u. In svral transint (that is tim-dpndnt) problms w will also hav nd for furthr tim drivativs such th acclration vctor a. In lasticity problms th strss componnts σ ij and th strain componnts ε ij will b important ingrdints. In hat transfr problms w will also hav to put focus on th hat flux vctor q. To b vry dtaild th list can b mad longr but th gnral conclusion so far is that th total numbr of physical ntitis ndd to b familiar with is rathr limitd vn if w ar discussing th ntir fild of computational mchanics. Typical to modls of all thos disciplins is that thy consist of a limitd numbr of quations of diffrnt typs. Th first group of quations to b brought up in this discussion is th Balanc laws motivatd from basic bhaviour of natur. Thr is th Nwton s scond law, f = ma rquiring that all forcs acting on a body most b in quilibrium. This balanc law is th bas for lasticity problms. In hat transfr problms th govrning balanc law is th Consrvation of Enrgy, th first law in Thrmodynamics. This quation only mans that nrgy is undstroyabl. Thr is also a third important balanc law govrning fluid flow problms; this is Consrvation of Mass. Ths thr balanc laws govrn most computational mchanics applications. In mor complx, and probably non-linar applications, somtims svral or all of ths balanc laws hav to b utilizd. A scond group of quations is th Constitutiv rlations. Typical to ths quations ar that thy all ar mpirical quations stablishd through xprimntal studis. Common to ths quations ar also that thy try to dscrib th bhavior of a solid matrial or a fluid in trms of som usful masurs. In lasticity problms a first choic is th gnralizd Hook s law and in hat transfr th Fourir s law is qually common. In fluid flow calculations a Nw- Computational mchanics Balanc law Constitutiv rlation
4 1.2. THE BIG PICTURE Scalar-valud Vctor-valud Compatibility rlation Systm of linar algbraic quations Linar Eignvalu problms tonian fluid flow bhavior is th first and simplst choic for domain proprty charactrization. Anothr important classification of a typical finit lmnt formulation is whthr th problm nds up in a Scalar-valud or Vctor-valud problm. In th discussion to com w will find out that th displacmnt vctor u will b th basic unknown and in th hat transfr problm th tmpratur T will b th basic unknown. That is th lasticity problm is a vctor-valud problm and hat transfr problm is scalar-valud problm which b dscribd in dtail latr on. In cass whr w ar studying vctor-valud problms, thr is also a nd for a rlation coming from a third group of quations. Th group rfrrd to hr is th group of Compatibility rlations. Typical to this group of quations is that thy try to prdict how dformations in a mattr will tak plac. Such quations will always put up som rlations for how diffrnt componnts must rlatd to ach othr. In scalar-valud problms thr is nvr nd for any rlation blonging to this group. What has bn discussd so far is what is typical or is in common btwn diffrnt mathmatical formulations of diffrnt filds of application of th finit lmnt mthod. Also, from a numrical point of viw, svral ovrall important commnts can b mad for what is typical or shard btwn diffrnt finit lmnt applications. As a usr of a finit lmnt program it is probably qually important to b awar of what is going on in th computr during diffrnt typs of finit lmnt analysis. In lasticity and hat transfr stady-stat problms w will find out what th computr has to solv of Systm of linar algbraic quations. In cass of studying tim-dpndnt problms our mathmatical discussion will nd up in systms of algbraic coupld ordinary diffrntial quations in tim which hav to b solvd numrically by any of som tim intgration schm. An important aspct of such tim intgration schms is if th schm is implicit or xplicit. Ths schms hav diffrnt mrits and whr filds of application rarly ovrlaps. From a numrical point of viw w will also find anothr typical group of Linar Eignvalu problms. Th most important applications ar dynamic ignvalu problms and linar buckling problms. In non-linar problms on soonr or latr has to introduc a linarization of th quations and from a numrical point of viw an itrativ schm basd on Nwton s mthod has to b mployd. This sction only tris to giv th radr an ovrviw of th topic. Prhaps som of th kywords discussd hav bn touchd upon in som othr courss or contxts. Som of th algorithms and numrical tchniqus ndd hr probably hav bn studid in prvious mathmatical courss. If som of th matrial discussd hr is hard to undrstand it is vry natural bcaus this is an ovrviw and mor dtail will b givn latr on. This sction will probably srv qually wll as a summary and not only as an introduction of th topic.
Part I Linar Static Elasticity 5
Chaptr 2 Introduction In most nginring activitis whr Solid Mchanics considrations hav to b takn into account, a good start is to assum a linar structural rspons and a load that is applid in a quasi-static mannr. This is on of th simplst modls to study and such an analysis can b classifid as a Linar Static Elasticity analysis. A hug majority of all nginring analysis work don, with th purpos of trying to invstigat Solid Mchanics proprtis of a structur, blongs to this class of analysis and in many cass such an analysis will srv as a propr final rsult from which most ovrall nginring dcisions can b takn. In this part of th txt linar strain-displacmnt rlations (small displacmnts) and linar lastic strss-strain displacmnts will b assumd. If also all boundary conditions ar constant and indpndnt of th applid load th structur will show a linar rspons. In this part of th txt th discussion will also b limitd to problms with quasi-static load application; no inrtia forcs will b includd. In Solid Mchanics thr xists a squnc of approximation lvls basd on diffrnt displacmnt assumptions giving a tru 3D dformabl body mor or lss frdom to dform. In th following, svral of th most important of ths basic displacmnt assumption idas will b discussd in trms of th basic local quations, strong and wak formulations, and finally appropriat finit lmnt formulations. In th txt blow th discussion will start with th bar assumption which is th assumption that givs a ral 3D body th last dforming possibilitis. In th following chaptrs thr ar also finit lmnt formulations givn for bams, 2D and 3D solids and finally, Mindlin-Rissnr shll lmnts. In all ths chaptrs, motivatd by diffrnt basic displacmnt assumptions, th ntir chain of quations will b givn. Th xprincd radr will quickly look through this and undrstand that vry much of th structur and ovrall Linar Static Elasticity 7
8 basic natur of th quations ar closly rlatd in-btwn ths diffrnt formulations for bars, bams and solids. That is, basic rlations could hav bn writtn mor gnrally onc and only rfrrd to in th nxt chaptrs. But th txt to com is, as alrady mntiond, writtn with a goal that a chaptr or an application should b slf-containd with minimum rquirmnt for jumping in on dirction or anothr in th txt. Anothr typical fatur for th txt is that vry discussion will start at th arly bginning of th application by a thorough discussion of th local quations constituting th modl of rality. Aftr ths lmnt-spcific topics th txt will continu with gnral discussions concrning how to assmbl and solv th systm of linar algbraic quations. Diffrnt dirct and itrativ algorithms and tchniqus for finding th solution will b givn. In th last chaptrs in this Linar static lasticity part of th txt ar discussions of som furthr important aspcts concrning how to trat and analyz linar static lasticity problms. Somtims thr is nd for transformations of diffrnt kinds. For xampl, on probably would lik to introduc a skw support not paralll to any of th global dirctions; that is thr will b nd for a transformation of on or svral lmnt stiffnss matrics. In a larg typical industrial finit lmnt analysis thr is likly to b a nd for combining diffrnt lmnt typs to ach othr. This can b don by th imposing of constraints on th systm of quations. A larg varity of diffrnt possibilitis xist. Th txt will also covr how to numrically solv systms of quations containing constraints. Th last sctions in th Linar static lasticity part of th txt will actually discuss a problm which is non-linar. That is frictionlss contact problms whr th basic problm is to find th xtnt of th contact surfac. Th contact surfac is th part of th boundary whr two contacting dformabl bodis only transmit comprssiv normal strsss. In a gnral cas th xtnt of this surfac is a rsult of th analysis and it has to b stablishd by itrations. A forc-displacmnt rlation is in th gnral cas non-linar bcaus of chang of contact surfac. An obvious typical ral situation is a ball or rollr baring.
Chaptr 3 Bars Considr a straight slndr body with a smoothly changing cross sction A(x) and with a lngth L. Lt us now assum that all loads applid to th body act in th dirction of th xtnsion of th body, which is th local horizontal x-dirction, s figur 3.1. Thr is a distributd load, h pr unit surfac [N/m 2 ] at th right nd and a distributd load, b(x) pr unit lngth [N/m], acting in th intrior of th body. That is, th body will not b xposd to any bnding loads and th body will only b strtchd in it s own dirction. If it is ncssary to includ bnding of such a slndr structur w hav to mov to th bam displacmnt assumption discussd in th nxt sction. In figur 3.1 th lft nd of th bar has bn givn a known prscribd displacmnt g, whr g << L, and E(x) is th Young s modulus of th matrial. A(x), E(x) g b(x) h x =. x = L x Figur 3.1: A typical bar structur 3.1 Th Bar Displacmnt Assumption Undr th circumstancs dscribd abov th Bar displacmnt assumption is 9 Bar displacmnt assumption
1 3.2. THE LOCAL EQUATIONS applicabl. That is, vry plan prpndicular to th x-axis is assumd to undrgo just a constant translation in th x-dirction and th initial plan will rmain flat in it s dformd configuration. By introducing this assumption th displacmnt u will b a function of x only as illustratd in figur 3.2. Undformd body u(x) Figur 3.2: A typical bar dformation Dformd body This mans that only on strss componnt σ(x) and on strain componnt ε(x) will b non-zro at vry cut x through th bar. That is, from a mathmatical point of viw this problm is locally on-dimnsional. This modl of rality will only involv thr diffrnt unknown functions, th displacmnt u(x), th strain ε(x) and th strss σ(x) in th intrior of th body which has to b calculatd undr considration of influnc from th boundary conditions g and h. 3.2 Th Local Equations To b abl to analyz this modl thr is nd for at last thr diffrnt quations. As mntiond in th introduction, all modls proposd for studying diffrnt physical phnomna always hav to fulfill at last on balanc law. In this cas a statical quilibrium rlation will srv as th balanc law. Equilibrium for a x N(x) b(x) x N(x+ x) Figur 3.3: Forcs acting on a slic x of a 1D bar modl short slic of lngth x of th bar rquirs N(x + x) N(x) + b(x) x = (3.1)
CHAPTER 3. BARS 11 whr Taylor s formula givs N(x + x) N(x) + dn(x) x (3.2) dx and th axial forc N(x) can b xprssd in th strss σ(x) and th crosssctional ara A(x) as follows N(x) = A(x)σ(x). (3.3) Ths thr quations (3.1) to (3.3) dfins a Balanc Law in trms of th strss σ(x) and aftr division by x w hav d (A(x)σ(x)) + b(x) =. (3.4) dx Typically, this quation always must hold indpndnt from what strss-strain or strain-displacmnt rlations will b assumd latr on. In this contxt, as w alrady hav indicatd, a linar lastic Constitutiv Rlation (th 1D Hook s law) σ(x) = E(x)ε(x) (3.5) Balanc Law Constitutiv Rlation will b usd and a linar compatibility rlation (small displacmnts) can b dducd by using th displacmnt u at th two positions x and x + x in figur 3.4. x u(x+ x) Undformd body Dformd body u(x) Figur 3.4: Typical bar dformation Th linar strain masur ε(x) is dfind as th chang in lngth ovr th initial lngth x as follows ε(x) = u(x + x) u(x) x u(x) + du(x) x u(x) = dx x = du(x) dx (3.6)
12 3.3. A STRONG FORMULATION Compatibility Rlation and this will srv as a Compatibility Rlation for a linar 1D bar structur. Ths thr basic local quations 3.4 to 3.6 can b summarizd in th box L as follows Box: L Local Equations in 1D Linar Static Elasticity d (A(x)σ(x)) + b(x) = dx σ(x) = E(x)ε(x) ε(x) = du(x) dx and ths quations hav to b fulfilld at any position insid th opn domain Ω =]; L[. On obvious rmark is of cours that thr is no influnc form th boundary conditions so far. 3.3 A Strong Formulation On of svral possibl ways to start th analytical work for solving this systm of quations is to liminat th strss σ(x) and th strain ε(x) by putting th constitutiv rlation and th compatibility rlation into th balanc law. Aftr introducing th boundary conditions from figur 3.1 th following wllposd boundary valu problm S can b stablishd. Box: S Strong form of 1D Linar Static Elasticity Givn b(x), h and g. Findu(x) such that ( d A(x)E(x) du(x) ) + b(x) = x Ω dx dx u() = g on S g E(L) du(l) dx = h on S h Rmarks: 1D scond-ordr mixd Boundary-Valu Problm This formulation S constituts a Strong formulation of a linar static 1D Bar problm and from a mathmatical point of viw this is a 1D scondordr mixd Boundary-Valu Problm
CHAPTER 3. BARS 13 u() = g is a non-homognous Essntial boundary condition. If g = th Essntial boundary condition is homognous. Th total surfac S consists in this 1D cas only of th two nd cross sctions S h and S g. Th diffrntial quation is an xampl of a scond ordr ordinary on. E(L)du(L)/dx = h is a Natural boundary condition. Natural Th boundary valu problm is mixd bcaus thr ar both ssntial and natural boundary conditions. Latr on w will b awar of that som ssntial boundary conditions always hav to xist to b abl to guarant th uniqunss of th solution of th matrix problm M. 3.4 A Wak Formulation A Strong formulation can always b transfrrd into an quivalnt Wak formulation by multiplication of an arbitrary Wight function w(x) and an intgration ovr th domain. L ( ) d du L w (AE dx dx ) + b dx = w d du L (AE dx dx ) dx + wb dx = (3.7) Wight function Aftr partial intgration of th first trm th following is obtaind. [ wae du ] L L dx dw du AE dx dx dx + L wb dx = (3.8) Th first trm in th quation abov can b rwrittn as th natural boundary condition h can b idntifid form box S as [ wae du ] L dx = w(l)a(l) E(L) du(l) dx } {{ } = h w() }{{} = A()E() du() dx (3.9) By putting on spcific rstriction on th wight function w(x) and no longr ltting th function w(x) b compltly arbitrary an infinit st V of functions can b dfind whr vry choic of wight function w(x) must b qual to zro on th part of th boundary whr ssntial boundary conditions (S g ) ar dfind. V = {w(x) w(x) = on S g } (3.1) An appropriat Wak formulation W of this 1D Bar problm can b summarizd as follows.
14 3.5. A GALERKIN FORMULATION Box: W Wak form of 1D Linar Static Elasticity Givn b(x), h and g. Findu(x) such that L dw(x) dx A(x)E(x)du(x) dx dx = u() = g on S g L w(x)b(x) dx + w(l)a(l)h for all choics of wight functions w(x) which blongs to th st V Rmarks: This wak formulation W srvs as an fficint platform for applying numrical tchniqus such as wightd rsidual mthods for solving this bar problm approximatly. Th partial intgration stp is prformd bcaus it opns th possibility to nd up in a symmtric systm of linar algbraic quations that is mor fficintly solvd in th computr compard to a non-symmtric systm. Th natural boundary condition is now implicitly containd in th intgral quation. It is possibl to show that th Strong and Wak formulations ar quivalnt. 3.5 A Galrkin Formulation Wightd Rsidual Mthods Th basic rason for first turning th local quations into a Strong formulation and aftr that transfr th problm into an quivalnt Wak formulation is that th wak form can b utilizd as a bas for a varity of diffrnt Wightd Rsidual Mthods that all ar capabl of solving our basic bar problm, at last approximatly. Gnral to ths mthods ar that both th unknown function u(x) and th wight function w(x) ar built up from finit sums of n functions. n u(x) u h (x) = t 1 (x)a 1 +t 2 (x)a 2 +...+t n (x)a n +t (x) = t i a i +t (x) (3.11) i=1 and n w(x) = φ 1 (x)c 1 + φ 2 (x)c 2 +... + φ n (x)c n = φ i c i (3.12) i=1
CHAPTER 3. BARS 15 Rmarks: Th functions t i (x) ar calld Tst functions. Latr on furthr dtails Tst functions and ruls will b givn concrning how to slct ths functions and what proprtis thy must fulfill. Th function t (x) must b thr to scur that th non-homognous ssntial boundary condition u() = g will b fulfilld. Furthr dtails will b givn blow. All a i ar unknown scalar constants. In th cas whn all tst functions has bn stablishd th only unknowns ar all ths a i. Th arbitrarinss of th wight function slction w(x) is by this tchniqu furthr limitd to th choic of th n functions φ i (x) and th valu of ach of th scalars constants c i. By this introduction of finit sris consisting of n functions our problm turns ovr from a Continuous on with infinit numbr unknowns to a Discrt on with a limitd numbr of unknowns On of th most popular wightd rsidual mthods is th Galrkin mthod. On rason for this is that this mthod always will gnrat symmtric systms of linar algbraic quation which is mor fficintly solvd in th computr compard to non-symmtrical ons. Hr th basic ida is Continuous Discrt Galrkin mthod t i (x) = φ i (x) = N i (x) i = 1, 2,,, n (3.13) that if a slction is don of th tst functions t i vry function φ i also is dfind and vic vrsa. Plas obsrv, that from hr ths functions (th tst and th wight functions) most oftn will b calld Shap functions and th notation N i (x) will b usd. By moving ovr from sums to a matrix notation th approximation u h (x) and th wight function w(x) can b rwrittn as follow Shap functions u h (x) = N(x)a + t (3.14) w(x) = N(x)c w(x) = c T N T (x) (3.15) whr a 1 c 1 N(x) = [ N 1 (x) N 2 (x)... N n (x) ] a 2 c 2.., a =., c =. (3.16). Concrning quation (3.15) th two altrnativs ar qual and actually th latr will b mostly usd. a n c n
16 3.5. A GALERKIN FORMULATION Bfor som gnral and mathmatically mor prcis ruls will b givn concrning what proprtis a crtain choic of shap functions N i hav to fulfill, on possibl choic among many othrs, will b givn and discussd from an intuitiv point of viw. In this particular bar problm w hav now accptd an ida whr an approximation is introducd for th displacmnt u(x) in th bar. Latr on w will find out that this will of cours also gnrat approximat solutions for th strss and strain in th bar. Th simplst possibl assumption is to think of th displacmnt approximation u h (x) as a pic-wis linar polygon chain. Such a function is continuous in its slf but th first drivativ is discontinuous. Th qustion is now how to u(x) Exact solution g Pic-wis linar approximation x =. x = L x Figur 3.5: A 1D bar displacmnt approximation assumption Nods Finit lmnt xprss such a pic-wis linar function as convnint and fficint as possibl with n linar indpndnt paramtrs typically stord in th column vctor a. Lt a numbr of n + 1 so calld Nods x i b dfind insid and at th nds of th domain Ω from to L. Th intrval btwn two nods is calld a Finit lmnt. As a first choic of functions N i a st of pic-wis linar functions 1. N 1 (x) N 1 (x) N i-1 (x) N i (x) N i+1 (x) N n (x) x =. x 1 x 2 x i-2 x i-1 x i x i+1 x i+2 x n-1 x n Figur 3.6: On possibl choic of shap functions N i for a 1D bar problm will srv as a bas for furthr discussions and so far thy ar only dfind from intuitiv rason and from figur 3.6 as follows
CHAPTER 3. BARS 17 N 1 (x) = { (x1 x)/(x 1 x ). x x 1. x 1 x x n (3.17). x x x i 1 (x x N i (x) = i 1 )/(x i x i 1 ) x i 1 x x i (3.18) (x i+1 x)/(x i+1 x i ) x i x x i+1. x i+1 x x n { N n (x) =. x x x n 1 (x x n 1 )/(x n x n 1 ) x n 1 x x n (3.19) Rmarks: In a typical intrval th unknown function will b approximatd by a linar function as follows u(x) u h (x) = N i 1 (x)a i 1 + N i (x)a i x i 1 x x i (3.2) whr only two shap functions at th tim will b non-zro and influnc th approximation at an arbitrary point insid th intrval. All ths functions N i hav a unit valu at on nod and ar zro at all othr nods. That is, th following holds N i (x j ) = δ ij = { 1. i = j. i j (3.21) which mans that th shap functions ar linar indpndnt at th nods. That is, th vctor a will rprsnt th displacmnt in th nods. It is also possibl to show that ths shap functions N i (x) ar linarly indpndnt at an arbitrary position insid th intrvals. From a gnral mathmatical point it is possibl to show that such a linarly indpndnt choic of shap functions N i (x) will span a n-dimnsional subspac from which th approximation will b rcivd. Anothr important proprty that has to b fulfilld by a crtain choic of a shap function N i (x) is that th function must blong to th st C which consists of all continuous function N i (x) which fulfills ( ) 2 dni (x) dx <. (3.22) Ω dx
18 3.5. A GALERKIN FORMULATION Th function t (x) can now b constructd from th N 1 (x) function as follows t (x) = N 1 (x)g t (x x 1 ) =. (3.23) By putting th quations (3.14), (3.15) and (3.23) into th wak formulation W th following discrt Galrkin formulation will b achivd. L d dx (ct N T )AE d dx (Na + N 1 g) dx = Th vctor c T can b brought out as follows c T { L d dx N T AE d dx (Na + N 1 g) dx L L c T N T b dx + c T N T (L)A(L)h N T b dx N T (L)A(L)h } = and a matrix B(x) can b dfind as B(x) = dn(x) [ dn1 (x) dn = 2 (x) dx dx dx dn n (x) dx ] (3.24) which thn can b insrtd into th quation abov and th following is obtaind Global stiffnss matrix K Global load vctor f { L c T B T AEBdx a ( L N T b dx + N T (L)A(L)h L )} B T AE d dx N 1 g dx =. Th Global stiffnss matrix K and th Global load vctor f can b idntifid from this quation as K = L B T (x)a(x)e(x)b(x)dx (3.25) f = L N T (x)b(x) dx + N T (L)A(L)h L B T (x)a(x)e(x) d dx N 1 (x)g dx. (3.26) whr th matrix K is a symmtric matrix with n rows and columns and th vctor f is a column vctor containing on load cas. A discrt Galrkin formulation for this 1D problm now rads
CHAPTER 3. BARS 19 Box: G Galrkin form of 1D Linar Static Elasticity Find a such that c T (Ka f) = c T r = for all choics of th vctor c (th wight function) Not ncssary hr, but convnint in th discussions to com is to introduc th following gnral split of th global load vctor f into thr diffrnt load vctor contributions. f = f d + f h f g (3.27) Th first part f d coms from intrnal distributd forcs and in this 1D cas it is qual to f d = L N T (x)b(x) dx (3.28) and two othr parts ar from ssntial and natural boundary conditions on S h and S g. f h = N T (L)A(L)h (3.29) f g = L B T (x)a(x)e(x) d dx N 1 (x)g dx. (3.3) Th vctor f h can always b valuatd, indpndnt from th xplicit choic of shap functions, as follows. f h = N T (L)A(L)h = A(L)h. (3.31) 1 and vctor f g is shap function dpndnt. In th cas with linar shap functions, as discussd so far, and a constant cross sction A and Young s modulus E, w hav f g = L B T AE d dx N 1 g dx = AE L g 1 1.. (3.32) whr L 1 is th lngth of th first lmnt. From ths xprssions it is asy to conclud that th product A(L)h and th product AEg/L 1 ar both forcs.
2 3.6. A MATRIX FORMULATION 3.6 A Matrix Formulation Unbalancd rsidual forcs It is obvious from abov that th Galrkin formulation mans a scalar product btwn th column vctor c and anothr column vctor r and it is still on singl quation. Th vctor r is calld th rsidual and it can b intrprtd as Unbalancd rsidual forcs. From th basic ida of involving an arbitrary wight function w(x) in th wak formulation now only rmains a vctor c. This vctor still must b possibl to slct compltly arbitrary. From this rquirmnt it is obvious that th vctor r must b qual to a zro vctor r = (3.33) which mans that th structur is in quilibrium. Plas obsrv that this fulfillmnt of quilibrium is hr said to b in a wak sns which mans that w hav quilibrium masurd in nodal forcs! A matrix problm consisting of n linar algbraic quations can now b idntifid. Box: M Matrix form of 1D Linar Static Elasticity Find a such that Ka = f whr K and f ar known quantitis By solving this systm of quations th vctor a will rprsnt th displacmnts at th nods at quilibrium. Th vry last stp in th analysis is to calculat th, in th strong formulation liminatd strsss and strains, by making us of th compatibility and constitutiv rlations from th local quations (S box L). ε(x) = d dx (N(x)a + N 1 (x)g) = B(x)a + d N 1 (x) dx g (3.34) Finit Elmnt Program σ(x) = E(x) d dx (N(x)a + N 1 (x)g) = E(x)(B(x)a + d N 1 (x) g) (3.35) dx This is always don in an lmnt-by-lmnt fashion. What now lacks is a numrical procdur for stablishing of th matrics K and f and solving of th matrix problm M for th vctor a. Such a numrical procdur is typically implmntd as a computr program which can b calld a Finit Elmnt Program.
CHAPTER 3. BARS 21 Rmarks: In th bginning of this discussion thr ar thr unknown functions of x. Ths ar th strss σ(x), th strain ε(x) and th displacmnt u(x) which all now can b calculatd at last in an approximativ mannr. Most of th mathmatical work don so far is of an analytical natur and nds only to b don onc (whn trying to larn and undrstand why th finit lmnt mthod works bfor it coms to us of a computr program) Both th matrix K and vctor f ar compltly dfind by givn data in figur 3.1, th numbr of shap functions N i (lmnts) and th bhavior of ths shap functions (th lmnt typ) From a mathmatical point of viw this discussion can b summarizd as L S W G M and sourcs for rrors in this mathmatical modl of rality ar dviations from rality in th constitutiv and th compatibility rlations, dviations in th slctd boundary conditions and numrical rrors du to us of a limitd numbr of Finit Elmnts with a spcific bhavior in ach lmnt. On can show that th solution to th matrix problm M always xists and has a uniqu solution if th global stiffnss matrix K is non-singular. If thr xist at last on ssntial boundary condition which prvnts rigid body motion th stiffnss matrix will b non-singular. That is th global stiffnss matrix K is positiv dfinit and th following holds a T Ka > a dt(k) > In this 1D cas th matrix K is symmtric with a thr-diagonal population. 3.7 A 2-Nod Elmnt Stiffnss Matrix This 1D finit lmnt analysis discussion is now approaching th nd of th analytical part of th analysis and w ar clos to a position whr w hav to put in numbrs and start th numrical part of th analysis. This is normally prformd by a computr program basd on this analytical discussion. What still has to b discussd is how to valuat th intgrals in box M. Aftr that th numbr and th bhavior of shap functions N i is dcidd, ths intgrals only contain known givn quantitis and th basic qustion is how to valuatd ths as fficint as possibl! Plas obsrv, whn it coms to practical us of a finit lmnt program on always has to slct a crtain numbr of finit lmnts of a crtain lmnt typ which mans xactly th sam as slcting th numbr and th bhavior of th shap functions N i.
22 3.7. A 2-NODE ELEMENT STIFFNESS MATRIX On of th cornrstons in a finit lmnt formulation is that th ntir domain is split into a finit numbr of sub-domains, so calld finit lmnts. Du to th natur of th shap functions as linar indpndnt and only nonzro ovr vry limitd parts of th ntir domain it is convnint to prform th intgral ovr on lmnt (on sub-domain) at a tim and w hav n l K = i=1 xi+1 x i B T (x)a(x)e(x)b(x)dx (3.36) whr this is a sum of n l matrics whr n l is th numbr of finit lmnts. Each of ths sub-matrics will only contain 4 non-zro cofficints symmtrically positiond around th main-diagonal of th sub-matrix. This is bcaus th vctor B valuatd for x-valus insid th intrval x i and x i+1 will only contain 2 non-zro positions and th non-zro part of th product B T B is a symmtric 2 row and 2 column matrix. Lt us now study such a sub-intrval in mor dtail. W will now mov ovr to an lmnt-local notation, s figur 3.7. Th two linar parts of th global 1. N i (x) N 1 N 2 N i+1 (x) x i-1 x i x i+1 x i+2 x Figur 3.7: Th rlation btwn lmnt-local and global shap functions shap function N i and N i+1 ovr th intrval from x i to x i+1 has bn givn th closly rlatd notations N1 and N2 which is an lmnt-local numbring from 1 to 2 ovr th numbr of nods associatd with this lmnt. Th following nw lmnt-local vctors can now b dfind { } N (x) = [ N1 (x) N2 (x) ] and a a = 1 a (3.37) 2 and b usd for an lmnt-local xprssion of th displacmnt approximation u (x) as follows u (x) = N 1 (x)a 1 + N 2 (x)a 2 = N (x)a. (3.38) Th lmnt-local strain approximation ε (x) can now b writtn as [ ] ε (x) = du (x) dn = 1 (x) dn2 (x) a = B (x)a (3.39) dx dx dx } {{ } =B (x)
CHAPTER 3. BARS 23 and th global matrix B will hr appar in an lmnt-local vrsion B. A rlation btwn th global unknown vctor a and th lmnt-local unknown vctor a is asily stablishd as a 1. { } [ ]. a a = 1 1 a a = i 1 = C a (3.4) 2 1 a i } {{ } =C. a n whr th matrix C is a Boolan Matrix populatd by only unity or zro valus. lmnt-local Boolan Matrix Rmarks: Th lmnt-local vctor a is always a subst of th global vctor a. Thr is always on unity valu in ach row of th matrix C as long as non of th nods in th lmnt blongs to th boundary S g, whr w hav known valus of th displacmnts. In cass whr on or svral nods ar associatd to th boundary S g w can so far think of a matrix C prsrving its numbr of rows and whr a zro row without any unit valu is introducd corrsponding to th givn valu g. A mor thorough and dpnd discussion of this topic can b found in chaptr 5 undr sction 5.8. It can now b shown that th global stiffnss matrix K can b built from a sum of small 2x2 matrics which ar xpandd by a pr- and post-multiplication of th boolan matrix C. n l K = i=1 C T i xi+1 B T (x)a(x)e(x)b (x) dx x } i {{ } =K i C i (3.41) Such a small matrix is an important and oftn discussd topic calld th Elmnt Stiffnss Matrix K. Th subscript i will only b usd whn a spcific lmnt i is discussd. In this analytical discussion it is now tim to prform th vry last analytical stps. Lt us xprss th two local shap functions N1 and N2 in an lmnt-local coordinat axis ξ and in accordanc to th figur 3.8 and w typically hav Elmnt Stiffnss Matrix N1 (ξ) = 1 2 (1 ξ); N 2 (ξ) = 1 (1 + ξ). (3.42) 2
24 3.7. A 2-NODE ELEMENT STIFFNESS MATRIX 1. N 1 (ξ) N 2 (ξ) x i x i+1-1.. 1. ξ x Figur 3.8: Th lmnt-local coordinat systm ξ Th mapping btwn th two coordinat systms can b writtn as x(ξ) = 1 2 (x i+1 x i )ξ + 1 2 (x i+1 + x i ). (3.43) whr th lngth of th lmnt L = (x i+1 x i ). Diffrntiation and th chain rul thn givs dx = 1 2 L dξ and dn i (ξ(x)) dx = N i (ξ) dξ dξ dx = 2 Ni (ξ) L dξ (3.44) and it is asy to valuat th B matrix as follows [ ] B dn = 1 dn2 = 2 [ dn 1 dn ] 2 dx dx L = 1 [ ] 1 1 dξ dξ L (3.45) whr th B matrix in this simpl cas is indpndnt from th local coordinat systm. Th lmnt stiffnss matrix K is thn K = 1 [ 1 L 2 1 ] [ ] 1 1 1 A(x)E(x) L dξ (3.46) 1 2 and if th th cross sction A(x) and th Young s modulus E(x) ar constants with rspct to x and ξ w hav K = EA L [ 1 1 1 1 ] 1 1 dξ 2 1 } {{ } =2 which finally can b summarizd in th box blow. Box: 1D 2-nod bar lmnt stiffnss matrix K = EA [ ] 1 1 L 1 1 (3.47)
CHAPTER 3. BARS 25 This is xactly what has to b implmntd and valuatd numrically in th computr program and th nd of th analytical discussion is rachd at last for this lmnt typ. Evn if th cross sction A(x) changs ovr th domain on typically us th valu of th cross sction at th mid-point of th lmnt. That is, th th cross sction is modld as a stp-wis constant function. 3.8 A 3-Nod Elmnt Stiffnss Matrix Th slction of shap functions discussd so far is actually th simplst possibl with its pic-wis linar natur with a discontinuous first-ordr drivativ. Lt us now introduc a scond choic of shap functions, still with a discontinuous first-ordr drivativ, rquiring a nod at th mid-point of ach lmnt. By doing so our approximation of th displacmnt u(x) will b nhancd by a scond-ordr trm and th approximation will b a pic-wis parabolic polynomial chain. 1. N 1 (ξ) N 2 (ξ) N 3 (ξ) N i (x) N i+2 (x) x i x i+1 x i+2-1.. 1. ξ x Figur 3.9: Elmnt local shap functions for 3-nod lmnt In a typical lmnt of lngth L = x i+2 x i w hav now dfind thr lmnt-local shap functions in accordanc to figur 3.9 and lmnt-local displacmnt approximation can b writtn as u(ξ) = N 1 (ξ)a 1 + N 2 (ξ)a 2 + N 3 (ξ)a 3 = N a. (3.48) whr N = [ N1 (ξ) N2 (ξ) [ ξ N3 (ξ) ] = 2 (ξ 1) 1 ξ2 ξ (ξ + 1) 2 ]. (3.49) It is now possibl to valuat th B matrix as follows [ ] B dn = 1 dn2 dn3 = 2 [ dn 1 dn2 dx dx dx L dξ dξ dn 3 dξ ] (3.5) whr th B matrix in this parabolic cas will b dpndnt on th local coordinat systm. Aftr introducing drivativs of th shap functions with rspct