A mixed finite element method for beam and frame problems

Transcription

1 mixed finite eement metod for beam and frame probems R.. Tayor, F. C. Fiippou,. Saritas, F. uriccio Computationa Mecanics 3 (2003) Ó Springer-Verag 2003 DOI 0.007/s y 92 bstract In tis work we consider soutions for te Euer- Bernoui and Timosenko teories of beams in wic materia beavior may be eastic or ineastic. Te formuation reies on te integration of te oca constitutive equation over te beam cross section to deveop te reations for beam resutants. For tis case we incude axia, bending and sear effects. Tis permits consideration in a direct manner of eastic and ineastic beavior wit or witout sear deformation. finite eement soution metod is presented from a tree-fied variationa form based on an extension of te Hu Wasizu principe to permit ineastic materia beavior. Te approximation for beams uses equiibrium satisfying axia force and bending moments in eac eement combined wit discontinuous strain approximations. Sear forces are computed as derivative of bending moment and, tus, aso satisfy equiibrium. For quasi-static appications no interpoation is needed for te dispacement fieds, tese are merey expressed in terms of noda vaues. Te deveopment resuts in a straigt forward, variationay consistent formuation wic sares a te properties of so-caed fexibiity metods. Moreover, te approac eads to a sear deformabe formuation wic is free of ocking effects identica to te beavior of fexibiity based eements. Te advantages of te approac are iustrated wit a few numerica exampes. Keywords Ineastic beam, Finite eements, Mixed metod, Sear deformation Introduction Te deveopment of computationa modes for beam bending probems dates from te eariest days of structura R.. Tayor (&), F. C. Fiippou,. Saritas Department of Civi and Environmenta Engineering, University of Caifornia at Berkeey 727 Davis Ha, Berkeey, Caifornia, US e-mai: rt@ce.berkeey.edu F. uriccio Dip. Meccanica Strutturae Universita di Pavia 2700 Pavia, Itay Dedicated to te memory of Prof. Mike Crisfied, for is ceerfuness and cooperation as a coeague and friend over many years. anaysis and te iterature is too extensive to fuy cite ere. Mike Crisfied considered soution of beam probems from many perspectives as indicated in approaces contained in is books [ 3] and papers wit co-workers [4 8]. Most of is work was for finite dispacement appications many using co-rotationa formuations for wic e was we known. Here we woud ike to remember im for is pioneering work in tis fied of endeavor. In a dispacement formuation, te noninear straindispacement reations are postuated and poynomia interpoation functions are used for te dispacement approximation [9 ]. Because te postuated dispacement interpoation functions are approximate in noninear materia and geometric beavior, eac structura member needs to be discretized into severa eements in order to capture te actua variation of deformations aong its axis. Tis fine discretization resuts in a arge number of degrees of freedom in te fina numerica mode of te structure, tus, reducing te computationa efficiency of tis approac. ternativey, iger order dispacement interpoation functions can be used. Tis approac resuts in severa interna degrees of freedom tat need to be condensed out during te eement state determination. Even so, mes discretization is often required for accuracy. Wit bot of tese approaces numerica instabiities are not uncommon, particuary under cycic oading conditions. Probems wit te dispacement formuation of beam eements encouraged researcers to seek a soution wit force interpoation functions. One of te earier studies in te fied of structura anaysis is by Menegotto and Pinto [2] wo interpoated bot section deformations and section fexibiities. Backund [3] proposed a ybrid beam eement for te anaysis of easto pastic pane frames wit arge dispacements. In tis study te fexibiity matrix is determined from an assumed distribution of forces aong te eement. However, tis metod aso uses dispacement interpoation functions corresponding to inear curvature and constant axia strain distribution for te determination of section deformations from end dispacements. arge dispacement effects are taken into account by updating te eement geometry. Te paper does not provide detais on te numerica impementation of te eement, wic is critica to te approac. ater study by Maasuveracai and Powe [4] proposed fexibiity-dependent sape functions tat are continuousy updated during te anaysis. Tis study is foowed by te fexibiity-based eement of Kaba and Main [5] and its ater improvement by eris and Main [6]. However, te atter studies ack a consistent framework for te

2 formuation and, tus, suffer from imitations and numerica probems. Te first study to provide a consistent formuation for a force-based eement and its numerica impementation in a genera purpose computer program is te work of Ciampi and Caresimo [7]. n independent attempt in te same direction is reported by Caro and Murcia [8] wo proposed a ybrid frame eement for noninear materia and second-order pane frame anaysis. Second-order effects are accounted for, but te use of a inear strain-dispacement reation imits te formuation to reativey sma deformations. t about te same time Kondo and turi [9] used an assumed-stress approac to derive te tangent stiffness of a pane frame eement under genera oading. Te eement is assumed to undergo arbitrariy arge rigid rotations but sma axia stretc and reative (non-rigid) point-wise rotations. Tey sow tat te tangent stiffness can be derived expicity if a pastic-inge metod is used. Si and turi [20] extended tese ideas to tree-dimensiona frames. Detais of te force-formuation of Ciampi are expounded and refined in severa studies pubised subsequenty [2 27]. Wit te work of youb and Fiippou [28, 29] attempts are undertaken to generaize te formuation to mixed metods wit independent interpoation of force and dispacement variabes for appications wit dispacement dependent equiibrium equations. Finay, te study by Souza [30] formuates a force-based eement under noninear geometry and noninear materia response on te basis of te variationa framework of te Heinger Reissner principe. n independent attempt in tis direction is reported by Hjemstad and Tacirogu [3] and for stee-concrete composite beams by imkatanyu and Spacone [32]. Tese attempts, owever, ack fu variationa consistency. Te purpose of tis study is to formuate a beam eement witin te generaized and variationay consistent framework of te Hu Wasizu principe. 2 Formuation wit section integration We present ere a beam formuation in wic integration of oca constitutive equations is carried out on eac cross section. We start at te oca eve were we assume tat dispacements vary ineary over eac cross section. ccordingy, for te Euer Bernoui teory in two dimensions we ave u ðx; yþ ¼uðxÞ yw ;x u 2 ðx; yþ ¼wðxÞ ðþ were ðþ ;x denotes differentiation wit respect to x. Tis dispacement fied resuts in te axia strain expression ðx; yþ ¼u ;x yw ;xx ¼ ðxþ yvðxþ ð2þ wit a oter strains being zero. If we consider te effects of ony te stress r we can identify te axia force resutant as Nð Þ¼ r ðþd ð3þ and te bending moment resutant as Mð Þ¼ yr ðþd : Equiibrium of te beam requires on ox b x ¼ 0 in te axia direction and ð4þ ð5þ o 2 M ox 2 b y ¼ 0 ð6þ in te transverse direction. Here b x and b y are oadings per unit engt of beam in te x and y directions, respectivey. 2. Constitutive beavior For inear eastic beavior te stress is deduced from r ¼ E : ð7þ If te beam axis is paced at te centroid were d ¼ ; y d ¼ 0; y 2 d ¼ I ð8þ we obtain from (3) and (4) te expressions for resutants as Nð Þ¼Eu ;x ¼ EðxÞ Mð Þ¼EIw ;xx ¼ EIvðxÞ : ð9þ In te seque we sa aso consider an ineastic form in wic an easto pastic mode is given by r ¼ Eð p Þ; _ p ¼ _cf ;r and f ¼ f ðr ; HÞ 0 in wic p is te pastic strain, c is te consistency parameter and f is a yied function in terms of stress and ardening parameter H. Equation (0) may be integrated in time using a backward Euer sceme to obtain an incrementa form for use in numerica cacuations. Te soution may be obtained ocay using a return-map agoritm [33, 34]. Te soution for te stress is ten inserted into te resutant equations to compute te interna force resutants. inearization can provide a means of computing te easto-pastic moduus wic can aso be used to compute section tangent stiffness properties. Te force resutants on eac cross-section are computed from Eqs. (3) and (4). Performing a inearization on te resutant equations gives dnð Þ¼ dr ðþd dmð Þ¼ y dr ðþd : ðþ Te inearization of te stress invoves bot te axia strain and cange in curvature as dr ¼ E T ½dðxÞ ydvðxþš ð2þ were E T is te tangent moduus from te constitutive equation. Insertion of (2) into () ten gives 93

3 94 dn ¼ dm y E T ½ y Šd du ;x dw ;xx ð3þ Note tat te beavior may become couped wen bot axia and bending deformations occur at a cross-section and E T is variabe. 2.2 Tree fied variationa formuation Te approac we now present is based on te use of a tree-fied (dispacement, strain, stress) formuation based on te Hu Wasizu variationa principe. For an eastic materia wit stress r and strain te Hu Wasizu principe may be written as P w ðr ; ; uþ ¼ Wð Þd ou r d P ext ox ð4þ In (4) Wð Þ is te stored energy function from wic stresses are computed as r ¼ ow ð5þ o and P ext is te potentia for te body and boundary oading. Setting te variation of Eq. (4) to zero yieds dp w ¼ d ow o r d ou dr ox d odu ox r d dp ext ¼ 0 ð6þ Wen te term invoving a derivative on du is integrated by parts and combined wit te boundary terms te above functiona incudes a equations for soution of static probems in one-dimensiona easticity. To permit soution of ineastic constitutive forms, we repace te term invoving te variation of te stored energy by ow d d ) d ^r ð Þd ð7þ o in wic ^r ð Þ denotes a stress computed from any constitutive mode in terms of specified strains, strain rates (pasticity), or functiona of strain (viscoeasticity). In tis form we can directy introduce te beam approximations for a of te fied variabes, tus affording a very genera variationa formuation basis Beam formuation et us now appy te Hu Wasizu functiona to te soution of beam probems. We first write Eq. (6) wit te aid of (7) for te beam approximations. To accompis tis we assume tat te dispacements and strain over te cross section are given by () and (2) to obtain 8 0 < dp w ¼ d@ ½^r ð;vþ r Šd : 0 9 = dv@ y½^r ð;vþ r Šd ; dx 8 < dr du ;x 9 = ydr dw ;xx v ; dx : 8 9 < = du ;x r d dw ;xx yr d : ; dx dub x dwb y dx dpbc ð8þ Introducing te definitions given in Eqs. (3) and (4) we may write (8) as dp w ¼ d½ ^Nð; vþ NŠdv½ ^Mð; vþ MŠ dx dn u ;x dm w;xx v dx du ;x N dw ;xx M dx dub x dwb y dx dpbc ð9þ Finite eement approximation Te soution of te tree fied form of te beam probem given by (9) provides considerabe fexibiity in coice of approximating functions. In genera we need to ensure tat te number of terms taken for eac variabe satisfy consistency and stabiity conditions. n essentia requirement is te mixed patc test count condition [9]. Considering a soution in two-dimensions were te dispacement degrees of freedom at eac node are ~a a ¼ð~u a ; ~w a ; ~w a ;xþ; a ¼ ; 2 ð20þ tere are six degrees of freedom for eac eement. For tis case we must ave tree rigid body modes of dispacement and tree straining ones. Considering te functiona form given by Eq. (9) we can sow tat te conditions of approximation for stress and strain for tis case must satisfy te mixed patc test count conditions [9] n n N n v n M 2 ð2þ were n N, n M are te number of unknown eement parameters in N, M and n, n v are te number of unknown eement parameters in, v, respectivey. Te approximations for eac of te variabes are commony taken as

4 continuous poynomias witin eac eement; owever, we sa find tat considerabe advantages arise by using discontinuous or discrete (quadrature) approximation for te strains. For te finite eement approximation we consider a typica eement of engt ¼ x 2 x. We begin by integrating by parts a terms wit derivatives on dispacements. Te terms invoving u and du become dnu ;x du ;x N dx ¼ dn ;x u dun ;x dx fdn u dungj C ð22þ were C is te eft and rigt boundary of te eement. If we assume approximations suc tat N ;x b x ¼ 0 and dn ;x ¼ 0 : ð23þ and add te b x oading term to (22) we obtain dnu ;x du ;x N dx dub x dx ¼ fdnudungj C ð24þ Simiary, we can integrate by parts te terms invoving derivatives on w to obtain dmw ;xx dw ;xx M dx: ¼ dm ;xx w dwm ;xx dx dmw;x dw ;x M jc dm ;x w dwm C ;x ð25þ If we now assume approximations for M and dm tat satisfy M ;xx b y ¼ 0 and dm ;xx ¼ 0 ð26þ ten from (25) we obtain dmw ;xx dw ;xx M dx dwb y dx ¼ dmw ;x dw ;x M C dm ;x w dwm C ;x : ð27þ Introducing Eqs. (24) and (27) into (9) we obtain te reduced variationa functiona dp w ¼ d½ ^Nð; vþ NŠdv½ ^Mð; vþ MŠ dx fdn dmv gdx fdn u dun gj C dmw ;x dw ;x M C dm ;x w dwm C ;x dp bc ¼ 0 ð28þ wic is te form from wic we wi make our approximations. Dispacement approximation We note tat in te form given in (28) no interpoation for u or w is needed in eac eement. We merey use teir noda vaues togeter wit noda vaues of te first derivative of w (i.e., te usua noda vaues for a dispacement eement wit 2 nodes). toug to tis point we do not need an approximation for te dispacement witin te eement, tere are occasions for wic one is needed. One case is merey for grapica dispay of te fina dispaced sape and oters are for beams on eastic foundations, transient anaysis, and non-inear geometric beavior. Tere is no fuy consistent means to recover te dispacements from te variationa formuation presented above. Indeed, any fied wic satisfies te end conditions (20) is sufficient. Consequenty, we wi rey on computing te dispacement by a doube integration of te curvature fied over te eement. Tis approac is discussed in previous work (e.g., see [26]) and is used ere to pot deformed sapes. Resutant approximations To satisfy (23) and (26) we wi write te approximation for force resutants as N ¼ ~N N p ðnþ dn ¼ d ~N and bending moment resutants as M ¼ 2 ð nþ ~M 2 ð nþ ~M 2 M p ðnþ dm ¼ 2 ð nþd ~M 2 ð nþd ~M 2 ð29þ ð30þ were n, ~N, ~M and ~M 2 are eement parameters and N p and M p are particuar soutions for specified nonzero b x or b y. For exampe, if b x and b y are constant witin eac eement suitabe forms are N p ¼ 2 b xn and M p ¼ 8 b y 2 ð n 2 Þ : ð3þ For simpicity, we use (3) for N p and M p in te remaining deveopment. Te sape functions for unknown parameters in te axia force and bending moment are sown in Fig.. Expanding te boundary terms in (28) we obtain fdnu dungj C ¼ d ~Nð~u 2 ~u Þðd~u 2 d~u Þ ~N ðd~u 2 d~u Þ 2 b x for axia oading terms and dmw ;x dw ;x M C ¼ d ~M 2 ~ 2 d ~M ~ d ~ 2 ~M 2 d ~ ~M Fig.. Beam force and moment sape functions ð32þ ð33þ 95

5 96 dm ;x w dwm C ;x ¼ d ~M d ~M 2 ~w ~w 2 d ~w d ~w 2 M ~ ~M 2 d ~w d ~w 2 2 b y ð34þ for bending moment terms, were ¼ ow=ox. Te product terms between te stress and strains are considered next. For interpoation of te strains and v we can use discontinuous piecewise constant functions were we take ¼ a v ¼ a N e a ~a N e a ~va ð35þ wit typica Na e as sown in Fig. 2. Te integration may be convenienty carried out by defining te Na e as agrange poynomias wit reference to quadrature points and approximating te integras wit a singe point evauation. In tis case te ~ a and ~v a are merey ampitudes at te quadrature points. Wen mutipied by te strain parameters for eac part and superimposed a typica strain distribution in an eement is sown in Fig. 3. Te ine integras in Eq. (28) are now approximated as d½ ^Nð; vþ NŠdx d~ ½ ^Nð~ ; ~v Þ ~N N p ŠW dv½ ^Mð; vþ MŠdx dn dx dmv dx d ~N~ W d~v ^Mð~ ; ~v Þ 2 ð n Þ ~M 2 ð n Þ ~M 2 M p W ½ 2 ð n Þd ~M 2 ð n Þd ~M 2 Š~v W ð36þ were n denotes one quadrature point for eac discontinuous function and W denotes a quadrature weigt and engt. s an aternative to (35), we can use continuous sape functions wit teir definition point coinciding wit Gauss obbato (or oter quadrature type) points as sown for a 4-point case in Fig. 4. Combining te function for strain as ¼ a N e a ~a we obtain a continuous function in eac eement as sown in Fig. 5. Matrix expression for equations ssembing te above approximations, Eq. (9) may be written in matrix form as: 8 9 >< d~a >= dp ¼ d~q >: >; d~e T H T 0 >< ~a >= B6 4 H 0 b T 5 ~q >: >; 0 b 0 ~e 8 9 >< F >= 0 C ¼ 0 >: s p >; ^s were denotes evauation at a quadrature point, Fig. 3. Typica discontinuous strain distribution in a beam eement ð37þ Fig. 4. Beam continuous strain sape functions Fig. 2. Beam discontinuous strain sape functions Fig. 5. Typica continuous strain distribution in a beam eement

6 H ¼ 4 0 = 0 = 0 5 ð38þ 0 = 0 0 = and b ¼ ð n Þ 2 ð n : ð39þ Þ Te particuar soution is given as s p ¼ N pw M p W ð40þ and te constitutive equation evauation by ^s ¼ ^NW : ð4þ ^MW ppying a inearization to (37) gives te incrementa form for a Newton soution process as 8 9T02 38 < d~a = 0 H T 0 d~q 4 H 0 b T 5 d~a < = < R a : d~e ; d~q : 0 b k d~e ; ¼ R : q ; R e ð42þ were d is an increment, te residua expression is given by < R a = < F HT ~q = R : q ; ¼ b T ~e H~a ð43þ : R e b ~q s p ; ^s and te tangent matrix by k ¼ E y T ðn ; yþ½ y Šd ð44þ ðn Þ Te forces F are noda vaues computed from te particuar soutions N p and M p and any appied concentrated forces at te nodes. Soution strategy It soud be noted in te above formuation tat continuity between eements is enforced ony for dispacement degrees of freedom. Forces and strains may be discontinuous between eements. Tus, te parameters for forces and strains may be eiminated at te eement eve resuting in a stiffness matrix for dispacement parameter determination. Te eimination may be performed in two steps:. Eiminate section strain components separatey, resuting in d~e ¼ k ½b d~q R e Š were from (43) we ave R e ¼ b ~q s p ^s : Substitute into te remaining equations to obtain Concentrated forces appied at an eement interior are used to compute soutions for N p and/or M p. T d~a 0 H T d~a ¼ R a d~q H f d~q R q were f ¼ b T k b is te eement fexibiity and R q ¼ R q b T k R e is a modified stress residua. Given an increment d~a te second of te above equations may be soved for increments in ~q Eiminate te stress parameters for eac eement giving 3 d~q ¼ f Hd~a R q Wen te resut of te above two steps is substituted into te remaining equation set we obtain Kd~a ¼ R a were an eement stiffness is given as K ¼ H T f H and a modified eement residua by R a ¼ R a H T f R q Te resuting stiffness and residua now may be assembed into te goba equations in an identica manner to any dispacement formuation. We note tat at convergence te R q and R e residuas are zero for eac eement and, tus, te R a residua becomes te usua eement residua on equiibrium. During iteration steps, owever, te strain residuas R e in genera wi not be zero (except for inear probems). Te update strategy for te parameters may be carried out as foows:. For soution at time t n assume te state at te previous step t n is known. 4 For te first iteration step j ¼ 0, set ~a ð0þ n ¼ ~a n, ~q ð0þ n ¼ ~q n and ~e ð0þ n ¼ ~e n. 2. Form eement matrix and residua for state at iteration j. 3. Condense arrays as described above and assembe goba stiffness and residua for te noda parameters ~a. 4. Sove equation system Kd~a ¼ R and update soution ~a j n ¼ ~aj n d~a 5. For eac eement determine d~q and d~e and update te stress and strain parameters 2 In soution of igy non-inear probems we sometimes find it necessary to subincrement te dispacement increments d~a in order to converge te soution for te ~q. 3 For non-inear materias it may be desirabe to use a singuar vaued decomposition and construct an inverse or pseudo-inverse to avoid numerica precision probems [35]. 4 In some definitions we deete te individua cross-section identifier wen defining te ~e to avoid cumbersome notation. 97

7 98 ~q j n ¼ ~qj n d~q ~e j n ¼ ~ej n d~e 6. Ceck convergence on goba residua for te dispacements R and eac eement residua R q and R e. (a) If converged: set n ¼ n and go to Step. (b) If not converged: set j ¼ j and go to Step 2. Generay, it is more efficient to compute and perform te update step for stress and strain just before te next eement matrix and residua are computed for iteration j. Tis requires one to eiter recompute te eement matrix and residua at te vaues of te j iteration or store te matrices for ater use. In eiter case, owever, it is necessary to save te parameters for stress, ~q, and strain ~e for eac eement. 2.3 Beam formuation wit sear deformation et us now appy te Hu Wasizu functiona to te soution of beam probems wic incude searing deformation. Te dispacement approximation for a beam wic incudes te primary effect of sear deformation is given by u ðx; yþ ¼uðxÞ yðxþ and u 2 ¼ wðxþ ð45þ were is te rotation of te beam cross-section and c is te average cross-section searing strain. In tis case we ave two strain components at eac point in te beam wic are given by ¼ ou o y ox ox ¼ ðxþ yvðxþ c 2 ¼ ow ð46þ ¼ cðxþ : ox We first modify Eq. (4) to incude r 2 and c 2 for te sear effects. We ten assume strain distributions given by (46), define te sear resutant by V ¼ s d ð47þ and integrate over te cross section to obtain dp w ¼ d½ ^Nð;c;vÞ NŠdN ou ox odu ox N dx dv½ ^Mð;c;vÞ MŠdM o ox v od ox M dx dc½ ^Vð;c;vÞ VŠdV ow c dx ox odw d V dx ox dub x dwb y dx dpbc ð48þ Finite eement approximation We again use te mixed patc test count condition as a guide to construct finite eement approximations. Considering a 2-node eement in wic te dispacement degrees of freedom at eac node are a a ¼ð~u a ; ~w a ; ~ a Þ; a ¼ ; 2 ð49þ tere are again six degrees of freedom for eac eement (Note tat now te ~ a is a parameter describing te rotation of te beam cross section and, in genera, is not equa to w a ;x). For te case wit sear deformation we must ave tree rigid body modes of dispacement and tree straining ones. Considering te functiona form given by Eq. (48), te conditions to approximate stress resutants and strain functions must satisfy te mixed patc test count conditions n n N n c n V n v n M 2 ð50þ were n N, n V, n M are te number of unknown eement parameters in N, V, M and n, n c, n v are te number of unknown eement parameters in, c, v, respectivey. By using force soutions wic satisfy equiibrium we wi be abe to satisfy tese conditions easiy. For te finite eement approximation we again consider a typica eement of engt ¼ x 2 x and integrate by parts a terms wit derivatives on dispacements. Te terms invoving u and du again yied (22) and we assume approximations wic satisfy (23). Incuding te eement force b x we obtain (24) again. Simiary, we can integrate by parts te terms invoving derivatives on to obtain dm o ox od ox M dx ¼ odm om d dx ox ox fdm dmgj C ð5þ and tose invoving derivatives on w to obtain dv ow ox odw ox V odv ov dx ¼ w dw dx ox ox fdvw dwvgj C ð52þ We assume tat te approximations for M; V and dm; dv satisfy te equiibrium reations ov ox b odv y ¼ 0 and ox ¼ 0 om ox V ¼ 0 and odm ox dv ¼ 0 : ð53þ We ten combine (52) and (53) wit te remaining terms for w and in (48) to obtain

8 dm o ox M dx ¼ dv ow ox odw ox V dx fdv dvgdx dwb y dx fdm dmgj C fdvw dwvgj C ð54þ Introducing te above into (48) we obtain dp w ¼ d½ ^Nð; c; vþ NŠ dn dx dv½ ^Mð; c; vþ MŠdMv dx dc½ ^Vð; c; vþ VŠdVc dx fdnu dun gj C fdm dmgj C fdvw dwvgj C dp bc ð55þ wic is te form from wic we wi make te approximations. We note tat in tis form, subject to te conditions imposed on N, V and M, no interpoation for u, w or is needed in eac eement. We merey use teir noda vaues (i.e., te usua noda vaues for a sear deformabe dispacement formuation eement wit 2 nodes). We can again use te approximation for force and bending moment resutants given by (29) and (30). We aso note tat te sear in eac eement may be computed from moment equiibrium as V ¼ om ox ¼ ðm M 2 ÞV p wit V p ¼ om p ox ð56þ and, tus, it is not necessary to add additiona force parameters to te eement. From Eq. (55) we again obtain te axia oad terms given by Eq. (32) and for bending moments and sears fdm dmgj C ¼ d ~M 2 ~ 2 d ~M ~ d ~ 2 ~M 2 d ~ ~M fdvw dwvgj C ¼ d ~M d ~M 2 ~w 2 ~w d ~w2 d ~w M ~ ~M 2 d ~w 2 V 2 p d ~w V p were Vp and V2 p are vaues of te particuar soution for sear at te and 2 ends, respectivey. Tese boundary terms may be written in matrix form (38) and we note tat (37) resuts in te eement expression no differences arise by incuding sear deformation. 2 3T02 32 Te product terms between te stress and strains are d~a 0 H T 0 considered next. For interpoation of te strain parts, c 4 d~q 5 4 H 0 b T 5 d~a 3 4 and v we can use discontinuous piecewise constant d~e d~q 5 ¼ 4 0 b functions given by Eq. (35) and evauated at a singe k d~e quadrature point or continuous sape functions wit teir definition point coinciding wit te Gauss obbato ð57þ ð58þ (or oter quadrature type) point as sown for te 4-point case in Fig. 4. Te ine integras in Eq. (55) are again approximated as d½ ^Nð; c; vþ NŠdx d~ ½ ^Nð~ ; ~c ; ~v Þ ~N N p ŠW dv½ ^Mð; c; vþ MŠd d~v ^Mð~ ; ~c ; ~v Þ 2 ð n Þ ~M 2 ð n Þ ~M 2 M p W dc½ ^Vð; c; vþ VŠdx dn dx d~c ½ ^Vð~ ; ~c ; ~v Þ ~V V p ŠW dmv dx dvc dx d ~N~ W ½ 2 ð n Þd ~M 2 ð n Þd ~M 2 Š~v W d ~V~c W ð59þ were n denotes one quadrature point for eac function and W denotes a quadrature weigt and engt. terms except tose invoving ^N, ^V and ^M may be written in matrix form as d~ ~N d~v ½ 2 ð n Þ ~M 2 ð n Þ ~M 2 Š d~c ~V ¼ðd~e Þ T b ~q and d ~N~ ½ 2 ð n Þd ~M 2 ð n Þd ~M 2 Š~v d ~V~c ¼ d~q T b T ~e ð60þ ð6þ were ~e ¼ð~ ; ~v ; ~c Þ T, ~q ¼ð~N; ~M ; ~M 2 Þ T and b ¼ ð n Þ 2 ð n Þ 5 : ð62þ 0 Here a difference wit te formuation witout sear deformation arises from te addition of a tird row in te b array and te incusion of ~c in te definition of ~e. ppying a inearization to te equations equivaent to R a R q R e 3 5 ð63þ were d is an increment and te residua expression is given by 99

9 R a F H T ~q 4 R q 5 ¼ 4 b T ~e H~a 5 : ð64þ R e b ~q s p ^s Te constitutive equation terms are < ^NW = < N p W = ^s ¼ ^MW : ; and sp ¼ M p W : ^VW V p ; W ð65þ and te tangent matrix for a decouped bending-sear beavior becomes k ¼ 4 y 05 E TðyÞ 0 y 0 d ð66þ 0 G 0 T ðyþ 0 0 were E T and G T are tangent Young s and sear moduus, respectivey. Couping between te beavior merey adds off diagonas to te moduus array. Te modifications to incude sear beavior are minima and te soution strategy for te mode is identica to tat aready presented for te case witout transverse searing strains. 3 Numerica exampes 3. Simpy supported beam wit uniform oading Consider a simpy supported beam under uniformy distributed oad of intensity q wit engt, eastic properties E and G, cross sectiona area and moment of inertia I as sown in Fig. 6. Te exact dispacement at mid-span for te Euer Bernoui teory is w E max ¼ 5 q4 ð67þ 384 EI and incuding te effects of sear deformation (Timosenko beam teory) is w T max ¼ 5 q4 384 EI q2 ð68þ 8 j G were j is te sear correction factor. Simiary te crosssection rotation at te eft support for te two teories is te same and is given by E q3 max ¼ 24 EI ¼ T max : ð69þ For te comparison wit te eement presented above we consider a rectanguar cross section wit inear eastic materia. Te properties are: E ¼ 0 6, m ¼ 0:25, q ¼, j ¼ 5=6, ¼ b ¼. Using symmetry, one af of te beam is modeed wit one eement based on te teory given above. Tis probem as been anayzed by Reddy [36] were it is sown tat severa eements are required aong te engt to get satisfactory answers using standard dispacement approaces. We note tat our approac uses sape functions wic invove no materia parameters, contrary to te dispacement sape functions proposed by Reddy to avoid sear ocking. Tabe sows tat te soution at te nodes is exact for te present deveopment. 3.2 Frame structure Te frame structure considered by Reddy [36] is anayzed using te beam eement deveoped above. Te ony modification from tat presented previousy is te need to transform te member from oca coordinates (were te teory is deveoped) to goba coordinates. Tis standard operation is described in any text on structura anaysis. Te geometry for te frame is sown in Fig. 7. Crosssection properties are: ¼ 0 in 2, I ¼ 0 in 4, E ¼ 0 6 psi, m ¼ 0:3, and j ¼ 5=6. Our mode for te frame consists of tree eements: one for te vertica coumn and two for te incined beam. Te resuts for te dispacement at point B are compared wit te exact soution (given in [36]) in Tabe 2. We note tat te resuts are exact at tis point. Moreover, te force distribution obtained is aso exact. Te abiity of te mixed formuation given ere to produce correct resuts wit and witout sear and no sear ocking is ceary evident. 3.3 Simpy supported beam wit point oad s a fina probem we consider a simpy supported beam wit a centra point oad as sown in Fig. 8. To sow te Tabe. Simpy supported beam: w max ¼ wð=2þ and 0 ¼ð0Þ Exact (no sear) Exact (wit sear) Present (no sear) Present (sear) = ¼ 0 = ¼ 00 w max w max ) ) ) ) ) ) ) ) Fig. 6. Uniformy oaded, simpy supported beam Fig. 7. Frame structure

10 Tabe 2. Dispacement at B for frame structure Dispacement/P at B 0 4 u B w B B Exact (no sear) ) ) Exact (wit sear) ) ) Present (no sear) ) ) Present (sear) ) ) Fig. 8. Simpy supported beam wit centra point oad Tabe 3. Properties for ineastic beam engt ¼ 80 Dept ¼ 0 Widt b ¼ 0 Eastic moduus E ¼ Yied stress r y ¼ 50 Hardening moduus H iso ¼ 290 (%) advantages of te formuation presented ere we aow te beam to ave easto pastic beavior. Te entire beam is modeed wit two eements (one eement for eac symmetric af); five Gauss-obatto curvature stations per eement are used aong te beam axis. rectanguar cross-section is considered wit 0 Gauss obatto points troug te dept to permit modeing of te spread of te pastic zone. No sear deformation is incuded. Te properties for te anaysis are: Te centra oad is aowed to vary using a oad contro strategy [3]. For te comparison we aso consider te soution using a standard dispacement mode wit cubic Hermite poynomia sape functions. Soutions for two, four, and eigt eements for te engt are used (one, two and four on eac af engt). In Fig. 9 we sow te forcedispacement reation, deformed sape and distribution of moment and curvature aong te engt of te beam at te ast computed oad state for eac anaysis. Te dispacement mode permits ony inear cange, wereas te mixed mode presented ere aows for arbitrary cange at eac axia station used (5 in te present case). Te superiority of te mixed form is evident in bot te force-dispacement, te computed deformed sape and te moment and te curvature distribution. 4 Cosure In tis work we ave presented a tree-fied variationa formuation for beams. Te presentation is restricted to two-dimensiona, sma-dispacement teory and incudes 20 Fig. 9. Simpy supported, point oaded beam ineastic soution. DF = Dispacement formuation; MF = Mixed formuation

11 202 te effects of sear deformation. Sear deformation can be readiy incuded witout danger of sear ocking and, tus, beavior is independent of te number of integration points aong te axis of eac eement. Non-inear materia beavior is incuded by integrating te resutants for axia force, sear force, and bending moment over te member cross-section. Te extension to tree dimensions is straigtforward. Geometric noninearity may be incuded wit te approac presented by Sousa [30] for inear and noninear materia response. In tat study fu geometric-noninearity for arge dispacements is incuded wit te co-rotationa formuation tat Crisfied was so instrumenta in deveoping and refining over te years. Te numerica exampes demonstrate te advantages of te mixed approac over resuts from traditiona dispacement based formuations especiay for very coarse mes discretizations. In cosing, we again wis to remember our ate coeague Mike Crisfied and te motivation e as instied in us to pursue our deveopment. Mike s wry wit and insigts into computationa mecanics wi be sorey missed! References. Crisfied M (986) Finite eements and soution procedures for structura anaysis, vo., inear naysis. Pineridge Press, Swansea, UK 2. Crisfied M (99) Non-inear finite eement anaysis of soids and structures, vo., Jon Wiey & Sons, Cicester, UK 3. Crisfied M (997) Non-inear finite eement anaysis of soids and structures, vo. 2, Jon Wiey & Sons, Cicester, UK 4. Crisfied M, Coe G (990) Co-rotationa beam eements for two- and tree-dimensiona structures. In: Kun G, and H. Mang (eds) Discretisation Metods in Structura mecanics. Springer-verag, Berin 5. Crisfied M (990) consistent co-rotationa formuation for non-inear, tree-dimensiona, beam-eements. Comput. Met. pp. Mec. Eng. 8: Crisfied M, Moita GF (996) co-rotationa formuation for 2-d continua incuding incompatibe modes. Int. J. Numer. Met. Eng. 39: Gavanetto U, Crisfied M (996) n energy-conserving co-rotationa procedure for te dynamics of panar beam structures. Int. J. Numer. Met. Eng. 39: Jeenic G, Crisfied M (999) Geometricay exact 3d beam teory: impementation of a strain-invariant finite eement for statics and dynamics. Comput. Met. pp. Mec. Eng. 7: ienkiewicz OC, Tayor R (2000) Te finite eement metod: te basis, vo.. 5t edn., Butterwort-Heinemann, Oxford, UK 0. ienkiewicz OC, Tayor R (2000) Te finite eement metod: soid mecanics vo. 2. 5t edn., Butterwort-Heinemann, Oxford, UK. Bate K-J (996) Finite eement procedures. Prentice Ha, Engewood Ciffs, NJ, US 2. Menegotto M, Pinto PE (973) Metod of anaysis for cycicay oaded reinforced concrete pane frames incuding canges in geometry and non-eastic beavior of eements under combined corma force and bending. In: Symposium on Resistance and Utimate Deformabiity of Structures cted on by We Defined Repeated oads, isbon, IBSE 3. Backund J (976) arge defection anaysis of easto-pastic beams and frames. Int. J. Mec. Sci. 8: Maasuveracai M, Powe GH (982) Ineastic anaysis of piping and tubuar structures. Tecnica Report UCB-EERC 82/27, Eartquake Engineering Researc Center, University of Caifornia, Berkeey, C, US 5. Kaba M, Main S (984) Refined modeing of reinforced concrete coumns for seismic anaysis. Tecnica Report UCB-EERC 84/03, Eartquake Engineering Researc Centre, University of Caifornia, Berkeey, C, US 6. eris C, Main S (988) naysis of reinforced concrete beam-coumns under uniaxia excitation. J. Struct. Eng. SCE 4(ST4): Ciampi V, Caresimo (986) noninear beam eement for seismic anaysis of structures. In: Proc. European Conference on Eartquake Engineering. pp , isbon, Portuga 8. Caro I, Murcia J (989) Noninear time-dependent anaysis of panar frames using an exact formuation I: Teory. Comput. Struct. 33: Kondo K, turi SN (987) arge-deformation, easto-pastic anaysis of rames under nonconservative oading, using expicity derived tangent stiffness based on assumed stresses. Comput. Mec. 2: Si G, turi SN (988) Easto-pastic arge deformation anaysis of space-frames: pastic-inge and stress-based expicit derivation of tangent stiffnesses. Int. J. Numer. Met. Eng. 26: Spacone E, Ciampi V, Fiippou FC (995) Mixed formuation of noninear beam finite eement. Comput. Struct. 58: Spacone E, Fiippou FC, Taucer FF (996) Fiber beam-coumn mode for non-inear anaysis of R/C frames.. Formuation. Eartquake Eng. Struct. Dynamics 25: Spacone E, Fiippou FC, Taucer FF (996) Fiber beam-coumn mode for non-inear anaysis of R/C frames. 2. ppications. Eartquake. Eng. Struct. Dynamics 25: Petrangei M, Ciampi V (997) Equiibrium based iterative soution for te non-inear beam probem. Int. J. Numer. Met. Eng. 40: Neuenofer, Fiippou FC (997) Evauation of noninear frame finite eement modes. J. Struct. Eng. SCE 23: Neuenofer, Fiippou FC (998) Geometricay noninear fexibiity-based frame finite eement. J. Struct. Eng. SCE 24: Petrangei M, Pinto PE, Ciampi V (999) Fiber eement for cycic bending and sear of RC structures. I. Teory. J. Eng. Mec. SCE 25: youb, Fiippou FC (999) Mixed formuation of bond-sip probems under cycic oads. J. Struct. Eng. SCE 25(ST6): youb, Fiippou FC (2000) Mixed formuation of noninear stee-concrete composite beam eement. J. Struct. Eng. SCE 26: Magaães de Souza R (2000) Force-based finite eement for arge dispacement ineastic anaysis of frames. P.D dissertation, Department of Civi and Environmenta Engineering, University of Caifornia, Berkeey (ttp://wwwib.umi.com/cr/ berkeey/fucit? p30086) 3. Hjemstad KD, Tacirogu E (2002) Mixed metods and fexibiity approaces for non-inear frame anaysis. J. Contruct. Stee Res. 58: imkatanyu S, Spacone E (2002) Reinforced concrete frame eement wit bond interfaces I: Dispacement-based, forcebased, and mixed formuations. J. Struct. Eng. SCE 28(ST3): Simo JC, Huges TJR (998) Computationa Ineasticity, voume 7 of Interdiscipinary ppied Matematics. Springer- Verag, Berin Germany

12 34. Simo JC (999) Topics on te numerica anaysis and simuation of pasticity. In: Ciaret PG, ions J (eds) Handbook on Numerica naysis, vo. III, pp Esevier Science Pubiser, BV 35. Goub GH, Van oan CF (996) Matrix Computations. 3rd edn., Te Jons Hopkins University Press, Batimore MD 36. Reddy JN (997) On ocking-free sear deformabe beam finite eements. Comput. Met. pp. Mec. Eng. 49: