International Journal of Solids and Structures

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1 Intrnational Journal of Solids and Structurs 46 (29) Contnts lists availabl at ScincDirct Intrnational Journal of Solids and Structurs journal hompag: Nonlinar analysis of lastic high-shar dformabl plan frams by a mixd FEM path-following approach Antonio D. Lanzo * Univrsitá dlla Basilicata, DiSGG (Dip. Struttur, Gotcnica Gol. Appl.), Campus Macchia Romana, 871 Potnza, Italy articl info abstract Articl history: Rcivd 18 Sptmbr 28 Rcivd in rvisd form 16 Dcmbr 28 Availabl onlin 14 January 29 Kywords: Nonlinar lasticity Cossrat bam modl Finit lmnt Path-following analysis Mixd formulation This papr dals with th nonlinar analysis of high-shar dformabl lastic plan fram, frquntly usd to modl bam of composit matrials, lastomric bam-lik barings of bridg and sismic isolation, or to modl DNA strands and polymr chains. Th work uss a FEM approach basd on a Cossrat Timoshnko bam modl with an xact gomtrical dscription, valuating all th axial, shar and bnding contributions. It follows a mixd implmntation of a Riks-path following locking-fr stratgy of analysis, xploring its implmntation dtails and proving its ffctivnss and rliability. At last th papr rports th quantitativ and qualitativ influnc of shar contribution on known classical xampls. Ó 29 Elsvir Ltd. All rights rsrvd. 1. Introduction FEM analysis of gomtrically nonlinar lastic plan fram is a topic widly discussd in litratur, ithr using Riks standard path-following arch-lngth stratgis (Riks, 1972, 1979) or Koitr prturbativ approachs (Salrno and Lanzo, 1997). A hypothsis that is widly accptd in th rlvant litratur is that of rprsnting bam structural lmnts taking into account only thir flxural and, somtims, axial dformation ffcts, whil narly always nglcting thir shar dformation contributions. Nvrthlss, situations can b found that do not fit this hypothsis, whr shar dformation ffcts ar not ngligibl and fundamntal to th analysis. For instanc, th analysis of bam of composit matrials and lastomric bam-lik barings (frquntly usd in bridg and sismic isolation), or th modling of DNA strands and polymr chains, tc. This work aims to tak into account ths situations. Whil a prvious papr (Lanzo, 24) focuss on th bam modl in a prturbation stratgy of analysis, th prsnt papr aims to xamin aspcts connctd to th us of a path-following numrical rconstruction stratgy of th quilibrium path. In this rspct, th work follows a mixd variant of th classical arch-lngth mthod of Riks, initially suggstd in Garca t al. (1998) as a solution of som locking problms, using a stp control dfind both in trms of displacmnts and intrnal strss componnts, in addition to th load paramtr. In th prsnt papr, th intrnal strss componnt rfrrd to ar th axial and th transvrsal tnsion componnts (N; T) of th bam lmnts, and not mrly th axial * Tl.: ; fax: addrss: antonio.lanzo@unibas.it componnt N as considrd in th papr by Garca t al. (1998), whr th shar ffcts ar substantially nglctd. In this study a gomtrically xact Cossrat Timoshnko bam modl (Rubin, 2) and an accurat discrt intrpolation (xact in trms of strss componnts) ar usd. Th mixd path-following stratgy suggstd in th prsnt papr provs to b rliabl and numrically stabl. A carful implmntation also involvs computational costs comparabl to that of a traditional displacmnt-basd path-following stratgy. 2. Th lastic bam modl A Cossrat Timoshnko planar bam modl (Rubin, 2) is kinmatically dfind in trms of displacmnts componnts ðu½sš; w½sšþ of th cntroid axis in th problm plan (x; z) p o ½sŠ ¼ðs þ u½sšþi þ w½sšk and in trm of th rigid rotation u½sš of its gnric sction, with rfrnc to an initial configuration with rctilinar axis of lngth l and sctions orthogonal to it (s Fig. 1). Th strain allowd by th kinmatical modl includs axial, shar and bnding dformations. For finit displacmnts, xact strain masurs ð½sš; c½sš; v½sšþ ar dfind by th rlations (s Antman, 1977, 1995; Pignataro t al., 1982; Lanzo, 1994; Salrno and Lanzo, 1997) r; s ¼ð1 þ Þa þ cb; v ¼ h; s whr ðþ; s stand for drivation with rspct to th abscissa s and th unit vctors /$ - s front mattr Ó 29 Elsvir Ltd. All rights rsrvd. doi:1.116/j.ijsolstr

2 A.D. Lanzo / Intrnational Journal of Solids and Structurs 46 (29) " # t ¼ Ni þ Tj; N þ cos u sin u N ¼ T þ sin u þ cos u T ffl{zffl} fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ffl{zffl} ~r R t r u In fact, bcaus of th intrnal quilibrium conditions (2a), ths nw tnsion paramtrs ~r ðn; TÞ ar constant along th gnric bam fn; s ¼ ; T;s ¼ a ¼ cos h i sin h k; b ¼ sin h i þ cos h k ar, rspctivly, normal and tangnt to th plan of th sction in th dformd configuration. Th dvlopmnt of th abov dfinitions lads to th following nonlinar strain displacmnt rlationship 1 þ ¼ð1þu; s Þ cos h w; s sin h ð1aþ c ¼ð1þu; s Þ sin h þ w; s cos h ð1bþ v ¼ h; s ð1cþ Lt (N; T; M) b forc masurs so that intrnal forcs and coupls of contact action ar rprsntd by th vctors t ¼ Na þ Tb; m ¼ Mb a In th absnc of loads along th axis of th bam, th intrnal quilibrium conditions ar th following t; s ¼ ð2aþ m; s þ r; s t ¼ ð2bþ A simpl linar lastic bam modl is obtaind by mans of th following constitutiv rlations N ¼ k n ; T ¼ k t c; M ¼ k m v whr k n, k t d k m ar, rspctivly, th axial, shar and flxural stiffnss moduli, from now on rfrrd to as, GA d, for similarity with th classical linar bam modl, i.. N ¼ ; T ¼ GAc; M ¼ v 3. A mixd formulation of th problm A mixd formulation of th bam modl can b obtaind by stting its normal and shar intrnal strss componnts ðn; TÞ as th primary variabls of th problm, in addition to th displacmnt variabls, and dfining its intrnal strain nrgy by th following functional U½u; rš ¼ r t u 1 2 rt Fr þ 1 2 v2 ds whr u ¼ c Fig. 1. Kinmatic of Cossrat bam modl. ; r ¼ N ; F ¼ T " # 1 1 GA Altrnativly, a mor convnint mixd formulation can b obtaind by rfrring instad to th nw st of static variabls ð N; TÞ dfind by and thn trivially rprsntd in a FEM discrt approach. Th intrnal strain nrgy can b rwrittn in th nw st of variabls r ðu; ~rþ as U½u; ~rš ¼ ~r t ~ u 1 2 ~rt R t u FR u~r þ 1 2 v2 u ds with ~ u ¼ ~ ~c ¼ R t u u () ~ u ¼ 1 þ u cos u ~c u ¼ w sin u 4. Th variations of th strain nrgy In ordr to dvlop a path-following analysis stratgy, th first and scond variations of th strain nrgy of th bam modl ar ndd. Th rlativ xprssions follow n U dr ¼ d~r t ~ u R t u FR u~r þ ~r t ð~ u duþ U _rdr ¼ 1 2 du~rt ðr t u FR u þ R t u FR u Þ~r þ u du ds n d~r t ð _uþþ u _~r t ð u duþþ~rt ð u _uduþ d~rt R t u FR u ~r _ ð_ud~r þ du _~rþ t ðr t u FR u þ R t u FR u Þ~r þ _udu~r t ðr t u FR u R t u FR u Þ~r þ _u du ods In a FEM approach, th valus of th rotation u½sš can b mad small nough using an adquat discrtization msh. This allows th simplifications sin u u; cos u 1 ð3þ By stting k ¼ 1 1, w thrfor obtain th following simplifid GA xprssion of th strain nrgy variations U dr fdn u 1 N kut þ dt w u 1 T GA kun þ du N þ dw T Þþu du gds U _rdr þ du Nu T þ kuð N 2 T 2 Þ k N T fd N _u þ d T _w þ _ Ndu þ _ Tdw 1 d N _ 1 N GA d T _ T kuðdn T _ þ dt NÞ _ þð_ud N þ du NÞðu _ þ 2kuN ktþ þð_ud T þ du _ TÞð 1 kn 2kuTÞ þ _uduð N ut þ kðn 2 T 2 Þþ4kuN TÞþ _u du gds 5. Th finit lmnt Th intrnal strss variabls ~r ð N; TÞ, constant on ach bam lmnt, hav a natural discrt rprsntation. Howvr by dvl-

3 1766 A.D. Lanzo / Intrnational Journal of Solids and Structurs 46 (29) oping th intgral xprssions of th strain nrgy variations, it can b dmonstratd that th displacmnt filds ðu½sš; w½sšþ contribut only through th discrt valus (u j u i ; w j w i ) whr (u i ; w i ; u j ; w j ) ar th rlativ nodal displacmnts. Thrfor, to obtain a discrt rprsntation of th mixd problm, only th discrt rprsntation of th rotation fild u½sš is ndd. For this purpos w assum a classical quadratic polynomial as intrpolation function, which is th solution of th Timoshnko bam linar problm that taks into account th shar ffct. In trms of natural variabls (Argyris t al., 1977, 1979) dfind by :¼ u j u i l ; / r :¼ w j w i l ; / s :¼ u u i j ; / 2 :¼ 1 1 þ b u i þu j / 2 r whr b ¼ 12 GAl 2, th function u½sš thn has th following xprssion u½sš ¼/ r þð1 2 s l Þ/ s þð1 þ b 6 s l þ 6s l2 Þ/ By using a co-rotational approach, th bam is rfrrd to a rfrnc systm ðx; zþ which is rigidly connctd to th dformd actual configuration, with th x axis along th straight lin joining th nodal bam positions (s Fig. 2). This allows us to brak down th total displacmnt filds of th bam into th sum of two componnts, th first a rigid rotation from th initial undformd configuration to th co-rotational on, th scond a pur dformation dscribd and rprsntd in th co-rotational rfrnc systm: u½sš ¼u r ½sŠþu ½sŠ; con u r ½sŠ ðu r ½sŠ; w r ½sŠ; u r Þ; w u r j w i ¼ arctan l n þ u j u i As th first displacmnt componnt u r ½sŠ is a rigid rotation, th corotational approach filtrs its ffcts in th xprssion of th strain nrgy and of its variations, now valuatd only in th scond componnt displacmnt u ½sŠ. As th gomtrical nonlinar ffct is almost narly totally connctd to th rigid componnt, it mans that th gomtric nonlinar trms only partially affct th strain nrgy and its variations. This nabls us to simplify som gomtric nonlinar trms in thir xprssions as shown in Eq. (3) in th prvious sction. In addition, th rotation fild u½sš is now rfrrd to th corotational rfrnc and can b minimizd simply by a finr FEM discrtization grid. Discrt xprssions of th strain nrgy variations wr finally obtaind by obsrving that in th th co-rotational rfrnc systm th following rlations ar valid w i ¼ w j ¼ ; / r ¼ 5.1. Th first strain nrgy variation (th lastic intrnal forc vctor) Shown blow is th discrt xprssion of th first strain nrgy variation: U dr ¼ dnl þ d/ r ððn þ nþlb/ kn TlÞ 1 þ d/ s 3 ð N þ nþl/ s þ 4 l / s þ d/ ðn þ nþ 1 5 þ b2 l/ ð1þknþb Tlþ 12 l / þ dn Nl l! k Tlb/ þ dt Tl! GA ð1þk NÞlb/ with n ¼ kð N 2 T 2 Þ This dfins th lastic intrnal forc vctor S of th bam lmnt on th bas of th following quivalnc rlation: U dr :¼ S t dr whr th vctor dr rprsnts variations of all th discrt kinmatical ð; / r ; / s ; / Þ and statical ( N; T) paramtrs of th gnric bam lmnt, i.. dr :¼½d; d/ r ; d/ s ; d/ ; d N; d TŠ t ¼½d t ; d~r t Š t Th lastic intrnal forc vctor S :¼½S ; S /r ; S /s ; S / ; S N ; S T Š t ¼½S t u ; St r Št consists of th following componnts: S ¼ Nl S /r S /s S / ¼ðN þ nþlb/ kn Tl ¼ 1 3 ð N þ nþl/ s þ 4 l / s ¼ðN þ nþ 1 5 þ b2 l/ ð1þknþb Tlþ 12 l / S N ¼ l Nl k Tlb/ S T ¼ Tl GA ð1 þ k NÞlb/ ð4þ ð5þ ð6þ 5.2. Th scond strain nrgy variation (th mixd lastic tangnt stiffnss matrix) Th discrt xprssion of th scond strain nrgy variation dfins th mixd local stiffnss matrix K m on th bas of th following quivalnc rlation U _rdr :¼ _r t Km dr Th componnts ar: Fig. 2. Th bam lmnt in a corotational dscription. ð7þ

4 A.D. Lanzo / Intrnational Journal of Solids and Structurs 46 (29) with K N N ¼ l K N T ¼ kbl/ K N ¼ l K N/s ¼ 1 3 ð1 þ 2k NÞl/ s K N/ ¼ kbtlþ 1 5 þ b2 ð1 þ 2kNÞl/ K N/r ¼ Tklþð1 þ 2k NÞbl/ K T T ¼ l GA K T ¼ lbð1þknþ 1 / 5 þ b2 K T /s K T /r K /s / s ¼ 2 3 k Tl/ s 2k Tl/ ¼ Nkl 2kTlb/ ¼ 4 l þ 1 3 ð N þ nþl þ 2 15 þ b ð 1 þ 4kNÞ 3 Tl/ K /s / ¼ 2 15 þ b ð 1 þ 4kNÞ 3 Tl/ s K /s / r ¼ 1 3 ð 1 þ 4k NÞ Tl/ s K / / ¼ 12 l þ 1 5 þ b2 ðn þ nþl þ 2 35 þ 3 5 b þ b3 ð 1 þ 4kNÞ Tl/ K /r / ¼ðN þ nþlb þð 1þ4kNÞ 1 5 þ b2 Tl/ K /r / r ¼ð N þ nþl þð 1 þ 4kNÞb Tl/ 6. Th path-following stratgy In a mixd FEM approach, th stat variabls q of th structur ar xprssd by th nodal displacmnts d and by strss lmnt paramtrs s. For loads that incras by a k paramtr p½kš ¼k^p, th bhavior of th structur dscribs a curv in th spac (d; s; k) rprsntd in implicit form using a gnric scalar paramtr n (th curvilinar abscissa) d½nš; s½nš; k½nš ð8þ Rfrring to S d ½d; sš and S s ½d; sš as th intrnal lastic rspons in trms of nodal displacmnts and strss componnts rspctivly, assmbld on th basis of th rlativ componnts (6) at th local lvl of ach bam, th points of th curv (8) ar solutions of th nonlinar quations systm k^p S d ½d; sš ¼ S s ½d; sš ¼ g½d; s; kš ¼n ð9aþ ð9bþ ð9cþ whr Eq. (9a) xprsss th nodal quilibrium conditions of th problm, Eq. (9b) th intrnal kinmatic compatibility conditions insid th lmnts and Eq. (9c) th scalar rlation that dfins th curvilinar abscissa n. Th path-following numrical algorithm for th rconstruction of th curv (8) is basd on a succssion of stps ach dfind by a prdictor phas, i.. an xtrapolation starting from a known point d 1 ¼ d o þ Dd; s 1 ¼ s o þ Ds; k 1 ¼ k o þ Dk and a corrctor phas, i.. an itrativ squnc convrging to a nw point of th curv d jþ1 ¼ d j þ _ d; s jþ1 ¼ s j þ _s; k jþ1 ¼ k j þ _ k It should b notd that th itrativ squnc is rgulatd by th jacobian matrix J of th systm (9). Th problm can b xprssd in th following form K " zfflfflfflfflfflfflffl} fflfflfflfflfflfflffl{ m #" # " d _ ¼ ðk j þ kþ^p _ # S j d d s d Du t M _u þ ldk _ k ¼ S j s ð1aþ ð1bþ in trms of tangntial matrix of th mixd problm K m assmbld on th basis of th local matrics K m of th bam lmnts givn by (7), and suitably transforming th Eq. (9c) into th condition (1b), which constraints th corrctor componnts ( d; _ k) _ to blong to th normal plan of th prdictor componnts (Dd; Dk), making us of a suitabl mtric oprator dfind in trms of th quantitis (l; M). It is known that th dirct factorization of th mixd matrix K m is not th most fficint computational stratgy to solv th problm (1a). It can instad b solvd in a partitiond form, through a block Gaussian limination of th strss variabls _r at ach bam finit lmnt lvl, whr th systm (1) has th following localization K zfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflffl{ m K uu K ru K ur _u K rr _r ¼ " ðk j þ kþ^p _ # S j u S j r ð11þ ^p bing th (unknown) part of th load ^p associatd with th lmnt and using th dfinitions of Eqs. (6) and (7). Th Gauss limination of th strss variabl _r _r ¼ K 1 rr K ru _ u K 1 rr Sj r transforms th problm (11) into ðk uu K ur K 1 rr K ruþ _u ¼ðk j þ kþ^p fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} _ S j u þ K 1 rr Sj r K c now controlld by th local tangntial stiffnss matrix K c of th compatibl formulation. Using standard procdurs to assmbl th trms at th global lvl (A is th matrix of kinmatical compatibility of th lmnt) K c ¼ X A t Kc A S j ¼ X d A t Sj u DSj ds ¼ X A t K 1 rr Sj r transforms th problms (1a) to th condnsd form K c d _ ¼ðkj þ kþ^p _ S j þ d DSj ds which is almost idntical to that of th compatibl formulation (K c is th usual tangntial stiffnss matrix of th structur), nrichd howvr by an additional contribut DS j ds which xprsss th influnc of th condnsd strss variabls. From a computational point of viw, th problm now has a much simplr solution which can b obtaind by a standard factorization procdur of th matrix K c. 7. Numrical rsults Th proposd finit bam lmnt in a mixd path-following formulation was thn chckd in th light of th numrical rsults. Th aim was to vrify th accuracy and th prformanc of th stratgy, comparing th rsults with othr availabl solutions, and at th sam tim to analys th influnc of shar dformation

5 1768 A.D. Lanzo / Intrnational Journal of Solids and Structurs 46 (29) ffcts. Th analysis was prformd varying th finit lmnt discrtization and for svral valus of th stiffnss ratios k ¼ l2 and s ¼ GA Cantilvr bam In th first xampl, a cantilvr bam was studid undr two diffrnt load conditions. Th first is a condition of pur bnding with a concntratd momnt M applid to its fr nd. This is a tst to vrify th accuracy of th suggstd FEM stratgy in rproducing a vry larg rotation of th bam. In fact th bam dforms, in accordanc to Eulr formula, into involvd circular shaps and an xact analytical solution of th quilibrium path can b asily obtaind (s Ibrahimbgovic, 1997). As can b sn in Fig. 3 whr th load paramtr Ml is plottd vs. th latral nd displacmnt paramtr w, l accurat rsults ar obtaind with a discrtization of just two finit lmnts. Th scond condition dals with a concntratd forc P applid to th fr nd of th bam. In this cas, to xplor th influnc of th shar dformability, svral tsts ar prformd varying th ratio s ¼ GA. Th rsults ar plottd in Fig. 4 and show again that only fw finit lmnts ar ndd to rproduc accurat valus (with no xact analytical solutions availabl, th comparison is mad with a discrtization of 1 finit lmnts). This is ssntially tru for rlativly larg valus of th ratio s, whil som discrpancy can b obsrvd for low valus of s and larg valus of th displacmnt paramtr w. l Fig. 3. Cantilvr bam in pur bnding (xampl 1). Fig. 4. Cantilvr bam in shar and bnding condition (xampl 2).

6 A.D. Lanzo / Intrnational Journal of Solids and Structurs 46 (29) Fig. 5. Eulr Bam (xampl 3). Fig. 6. Load vs. displacmnt of Eulr bam l2 ¼ 4p. 2 Fig. 7. Load vs. displacmnt of Eulr bam l2 ¼ 1p Eulr bam Th xampl shown in Fig. 5 is a Eulr bam, analyzd with a transvrsal load :1k, whr k is th axial comprssion load. Th tsts wr prformd for two diffrnt valus of th ratio k ¼ l2 and for svral valus of th ratio s ¼ GA, whil at th sam tim varying th numbr n of discrtization lmnts of th bam. Th rsults ar rprsntd in Figs. 6 and 7 in trms of load paramtr kl 2 vs. cntral transvrsal displacmnt paramtr w of th quilibrium paths. l For all th tsts of th cas l2 ¼ 4p 2 (s Fig. 6), th solution of th prfct initial post-buckling bhavior of th prturbation approach obtaind analytically in Salrno and Lanzo (1997) and Lanzo (24) is rportd in dashd lins. In this cas and for s ¼ 1 th solutions corrspond xactly to thos obtaind using th modl by Garca t al. (1998). Furthrmor, only a fw finit lmnts (n = 2 4) wr ndd to obtain accurat rsults. Th prturbativ approximatd curv prdicts this solution quit wll, both in qualitativ (with a stabl post-buckling path) and in quantitativ trms. With a lowr valu of th ratio paramtr s, i.. whn shar stiffnss is noticably lowr than th axial stiffnss, th load valu at which latral buckling phnomna occur ar gratly rducd. Again, this can b accuratly prdictd with only a fw lmnts. This cas xhibits a dcrasing post-buckling bhavior, which is far from th stabl post-buckling bhavior prdictd by an approximatd prturbation approach. Highr valus of th ratio paramtr s incras th buckling load. Mor spcifically, s ¼ 1 and s ¼ 1 tsts initially xhibit an unstabl post-buckling path (corrctly prdictd by th prturbativ approach) but subsquntly show an incrasd load carrying capacity (ascnding path) which cannot b rproducd by th approximatd prturbation approach. In this cas, howvr, accurat solutions rquir a larg numbr of discrtization lmnts (n ¼ 5 for s ¼ 1 and n ¼ 15 for s ¼ 1). For all th tsts of th cas l2 ¼ 1p 2 (s Fig. 7), th solutions ar accuratly obtaind with only a fw lmnts. Th shar ffcts ar ngligibl for highr valus of th ratio paramtr s (for s > 1 th solution substantially coincids with that for s ¼ 1). Th lvl of influnc is snsitiv for lowr valus of s, as can b obsrvd in th quilibrium paths tracd in Fig Roorda s fram This xampl is th classical Roorda s fram (Roorda, 1965) with a load k of ccntricity ¼ l=1 (s Fig. 8). Tsts wr

7 177 A.D. Lanzo / Intrnational Journal of Solids and Structurs 46 (29) prformd for two diffrnt valus of th ratio k ¼ l2 and for svral valus of th ratio s ¼ GA. Accurat solutions in trms of quilibrium paths wr obtaind with a discrtization of just 4 þ 2 finit lmnts. Th rsults ar rprsntd in Fig. 9 plotting th ratio kl2 vs. th rotation u of th cntral nod. For k ¼ 1: 1 3 an influnc of th ratio paramtr s can b obsrvd for valus s < 1, whil th curvs ar practically idntical for s > 1. This influnc is ngligibl throughout for k ¼ 1: 1 7. Kouhia and Mikkola (1989), hr obtaind for th spcific cas s ¼ A rigidly jointd truss Th rigidly jointd truss of Fig. 11 was xamind in Salrno and Lanzo (1997); Garca t al., 1998 (th rsults thr rportd ar affctd by rrors in th graphical rprsntation). Th solution in 7.4. A dp arch Th clampd-hingd dp arch of Fig. 1 has alrady bn th subjct of xtnsiv study. Th solution rprsntd hr was obtaind using 2 finit lmnts and varying th ratio s ¼ GA. Th shar influnc can b obsrvd only for small valus of th paramtr s ðs ¼ 1=1Þ, whil for highr valus of s th rsults largly coincid with thos producd in Garca t al. (1998) and Fig. 8. Roorda s fram (xampl 4). Fig. 9. Load vs. displacmnt of Roorda s fram. Fig. 1. Dp arch (xampl 5).

8 A.D. Lanzo / Intrnational Journal of Solids and Structurs 46 (29) Fig. 11. Rigidly jointd truss (xampl 6). Th finit lmnt modl is basd on a Cossrat bam and an accurat discrt intrpolation (xact in trms of strss componnts). At any stp of th stratgy, a partitiond numrical solution is implmntd, using a block Gaussian limination of th strss variabls at ach bam finit lmnt lvl, which lads at a global lvl to a condnsd problm almost idntical to that of th usual displacmnt-basd compatibl formulation. Som numrical tsts ar prsntd in ordr to prov th ffctivnss of th stratgy and th influnc of shar contribution on known classical xampls. Rfrncs th prsnt papr uss a discrtization of four finit lmnts for ach bam of th truss. As with th prvious xampl, th rsults rprsntd in Fig. 12 also xhibit a shar factor influnc only for small valus of th paramtr s ¼ GA, whil for valus s P 1 this influnc is ngligibl. 8. Conclusions Fig. 12. Load vs. displacmnt of xampl 6. This papr prsntd a mixd implmntation of th Riks pathfollowing stratgy for th analysis of lastic plan frams, which ffctivly taks into account thir flxural, axial and shar dformation contributions. Th mixd stratgy uss a stp control dfind both in trms of displacmnts and intrnal strss (axial N and transvrsal T) componnts, in addition to th load paramtr. Antman, S.S., Bifurcation problms for nonlinarly lastic structurs. In: Rabinowitz, P.H. (Ed.), Applications of Bifurcation Thory. Acadmic Prss, Nw York. Antman, S.S., Nonlinar Problms of Elasticity. Springr-Vrlag, Nw York. Argyris, J.H., Dunn, P.C., Scharpf, D.W., On larg displacmnt small strain analysis of structurs with rotational dgrs of frdom. Comput. Mthods Appl. Mch. Eng. 12, Argyris, J.H. t al., Finit lmnt mthod th natural approach. Comput. Mthods Appl. Mch. Eng. (17/18), Garca, G., Trunfio, G.A., Casciaro, R., Mixd formulation and locking in pathfollowing nonlinar analysis. Comput. Mthods Appl. Mch. Eng. 165, Ibrahimbgovic, A., On th choic of finit rotation paramtrs. Comput. Mthods Appl. Mch. Eng. 149, Kouhia, R., Mikkola, M., Tracing th quilibrium path byond simpl critical points. Int. J. Numr. Mthods Eng. 28, Lanzo, A.D La trav com continuo monodimnsional dotato di struttura uclida: studio di un modllo (in Italian), In: Rport 158, Dipartimnto di Struttur, Univrsitá dlla Calabria, Cosnza, Italy. Lanzo, A.D., 24. On lastic bam modls for stability analysis of multilayrd rubbr barings. Int. J. Solids Struct. 41 (2), Pignataro, M., Di Carlo, A., Casciaro, R., On nonlinar bam modl from th point of viw of computational post-buckling analysis. Int. J. Solids Struct. 18 (4), Riks, E., Th application of Nwton s mthod to th problm of lastic stability. J. Appl. Mch. 39, Riks, E., An incrmntal approach to th solution of snapping and buckling problms. Int. J. Solids Struct. 15, Roorda, J., Stability of structurs with small imprfctions. J. Eng. Mch. Div. ASCE (EMI) 91, Rubin, M.B., 2. Cossrat Thoris: Shlls, Rods and Points. Kluwr Acadmic Publishrs, London. Salrno, G., Lanzo, A.D., A nonlinar bam finit lmnt for th post-buckling analysis of plan frams by Koitr s prturbation approach. Comput. Mthods Appl. Mch. Eng. 146,