SDSS Rio 200 STABIITY AND DUCTIITY OF STEE STRUCTURES E. Batista, P. Veasco,. de ima (Eds.) Rio de Janeiro, Brazi, September 8-0, 200 STABIITY ANAYSIS FOR 3D FRAMES USING MIXED COROTATIONA FORMUATION Rabe Asafadie*, Mohammed Hjiaj* and Jean-Marc Battini** * Structura Engineering Research Group/GCGM, INSA de Rennes, 20 avenue des Buttes de Coësmes 35043 Rennes Cede France e-mais: Rabe.Asafadie@insa-rennes.fr, Mohammed.Hjiaj@insa-rennes.fr ** Department of Civi and Architectura Engineering, KTH, Roya Institute of Technoogy, SE-0044 Stockhom, Sweden e-mai: Jean-Marc.Battini@sth.kth.se Keywords: Geometricay noninear, 3D beams, corotationa formuation, mied finite eement anaysis, arbitrary cross-sections, easto-pastic materia behavior, Heinger-Reissner functiona. Abstract. The corotationa technique is adopted for the anaysis of 3D beams. The technique appies to a two-noded eement a coordinate system which continuousy transates and rotates with the eement. In this way, the rigid body motion is separated out from the deformationa motion. Then, a mied formuation is adopted for the derivation of the oca eement tangent stiffness matri and noda forces. The mied finite eement formuation is based on an incrementa form of the two-fied Heinger-Reissner variationa principe to permit easto-pastic materia behavior. The proposed eement can be used to anayze the noninear bucking and postbucking of 3D beams. The mied formuation soution is compared against the resuts obtained from a corotationa dispacement-based formuation having the same beam kinematics. The superiority of the mied formuation is ceary demonstrated. INTRODUCTION In recent iterature, there have been notabe contributions to improve the accuracy and efficiency of dispacement-based finite eements. This approach has the imitation in easto-pasticity since the approimations of the aia strains and curvatures are constrained by the eement's assumed dispacement fieds. Nonetheess, these curvatures can vary in a highy noninear fashion aong the ength of an eastopastic structura member. For eampe, Izzuddin and Smith [] found that a arge number of dispacement-based beam finite eements are typicay required to represent easto-pasticity behavior accuratey. In the mied formuation, both interna forces and dispacements are interpoated independenty. This formuation addresses the fundamenta imitation of conventiona dispacementbased eements: the inabiity of simpe dispacement poynomias to represent the highy noninear distribution of the curvatures aong the member engths due to genera distributed yieding. The corotationa approach has been recenty adopted by severa authors to hande the geometric noninearity in 3D dispacement-based beam modes (Asafadie et a. [2], [3], Battini and Pacoste [4], [5], Crisfied and Moita [6]). This paper etends the works of Battini on corotationa beam eements by appying the two-fied Heinger-Reissner variationa principe for the deveopment of a mied oca formuation. The corotationa approach is empoyed to hande the geometric noninearity, where, in the corotationa frame, the eement rigid body motion has been removed and the formuations focus soey on the eement deformationa degrees of freedom. 547
2 COROTATIONA FRAMEWORK FOR 3D BEAMS The centra idea in the corotationa formuation for a two-noded 3D beam is to introduce a oca coordinate system which continuousy rotates and transates with the eement. Then, oca deformationa dispacements d are defined by etracting the rigid body movements from the goba dispacements d g. The oca dispacements are epressed as functions of the goba ones, i.e. d = d ( d ) () g Then, d is used to compute the interna force vector f and tangent stiffness matri K in the oca frame. The transformation matri B between the oca and goba dispacements is defined by δd = Bδd (2) and is obtained by differentiation of (). The epression of the interna force vector in goba coordinates f g and the tangent stiffness matri K g in goba coordinates can be obtained by equating the interna virtua work in both the goba and oca systems, i.e. g T T T f = B f, K = B K B+ ( B f )/ d (3) g g g f Reations (), (2) and transformations (3) are epained in detais in [4]. 3 NONINEAR BERNOUI MIXED OCA EEMENT FORMUATION In this section, the interna force vector f and tangent stiffness matri K of a mied oca eement formuation based on the kinematics assumption of the Bernoui beam theory are derived. 3. Kinematics and oca dispacements interpoations a y a r y a z P R u r z I r G II Figure : oca beam configuration. et 0 ( yz P,, ) denote the position vector of an arbitrary point P in the initia configuration and et ( yz,, ) denote the position vector of P in the current configuration (see figure ). P ( yz,, ) = ( ) + yr + zr 0 0 P G y z ( yz,, ) = ( ) + ya ( ) + za ( ) + α( ) ω( yz, ) a ( ) P G y z (4) 0 where G and G denotes the position vectors of the centroid G in the initia and current onfigurations, respectivey. In the case of thin-waed open cross-sections, the normaized warping dispacement is epressed as the product of the warping parameter α ( ) and the warping function ω ( yz, ). To hande in a convenient way nonsymmetric cross-sections with distinct shear center and centroid, the warping function ω is defined according to Saint-Venant torsion theory and refers to the centroid G, [7]: 548
ω( yz, ) = ω yz c + zy c (5) and ω refers to the shear center defined by its coordinates y c, z c. The orthonorma triad ai, i = (, y, z) which specifies the orientation of the current cross-section, is given by ai = Rri, i = (, y, z) (6) The rotation defined by the matri R can be considered as the sum of two bending rotations and a twist rotation, and given by (cf. [8]). v, w, ϑ w, + v, ϑ R = v, ϑ (7) w, ϑ where v, w and ϑ are the transverse dispacements and the twist rotation of the cross-section centroid reative to the oca coordinates system, respectivey. Introducing the oca rotation matri defined in (7) into(4), the dispacement vector can be evauated as U = u y( v + w ϑ ) z( w v ϑ ) + ωα V = v zϑ W = w+ yϑ,,,, (8) To obtain the strain vector the foowing assumptions are adopted: the noninear shear strain components generated by warping are omitted since warping effects are rationay taken into account in a inearized way ony, the warping deformations are proportiona to the variation of the torsiona ange (Vasov assumption), an average vaue of the aia strain is taken in order to avoid membrane ocking and finay the noninear terms in the epressions of the bending curvatures and are negected. With these modifications, the foowing strain epressions are obtained: I ε = ε yκ + zκ + ( r ) κ + ωκ 2 A 2 ε = ( ω z) κ 2 o 2 av z y, y, y 2 ε = ( ω z) κ z, y (9) κ = ϑ κ = w yϑ κ = v zϑ with,, y, c,, z, c, 2 2 2 and r = ( z + y ), ε I = u + v + w + ϑ d, I = r da 2 2 o 2 2 av,,,, o 2 A A Since the strain fied in (9) is obtained from the oca dispacement fied d, therefore, a the components of the strain vector deduced from d wi be designated with a superimposed hat and combined as ˆ = ( ˆ ε 2ˆ 2 ˆ εy εz). Based on the above epression for strain vector ideaization, the strain at any point in the cross-section of the beam eement can be reated to the cross-sectiona generaized strain 2 vector e ˆ = ( εav κy κz κ κ, κ) as 2 ˆ = A( yz, ) e ˆ( ) (0) In the present formuation, the aia rotation ϑ is interpoated with shape functions based on the cosed-form soution of the torsiona equiibrium equation for an eastic prismatic and geometricay inear beam. Cubic Hermitian shape functions are chosen for the transverse dispacement v and w of the 549
centroid of the cross-section reative to the oca eement aes. And finay, inear interpoation is adopted for the aia eongation u of the oca eement. Thus, the variation in the cross-section deformation ê can be written as δeˆ N δd. Hence, an infinitesima change in strain vector can be written as = e ˆ δˆ AN δd () = eˆ 3.2 Equiibrium and generaized stress interpoation functions The generaized stress resutants vector S, which is work conjugate to the generaized strains ê, may be epressed in vector form as S=( N M y M z B T sv ) where N is conjugate to av, M y and M z are conjugate to y and z, respectivey. The bimoment B, Wagner stress resutant, and the uniform torque T sv are conjugate to,, 2 /2 and, respectivey. Within each eement, the generaized stress resutant interna force vector is approimated as S = N S f S (2) where f S =( N M I y M I z B I T M II y M II z B II ) the corotationa force degrees of freedom of the mied formuated eement (where I: first node, II: second node and T a constant torque) and NS is the force shape functions matri satisfying the equiibrium equations. where f I B 0 0 0 0 0 0 0 0 w / 0 0 0 / 0 0 0 v 0 / 0 0 0 / 0 0 N S = (3) 0 0 0 0 0 0 0 0 I II 0 0 0 fb 0 0 0 fb 0 I II 0 0 0 fb, 0 0 fb, 0 [ ] sinh ( ) II sinh( ) = k, fb = k and k = sinh( k) sinh( k) The resuting eement wi subsequenty be termed as bmw3d eement. It shoud be mentioned that reation (2) incudes P- effects in the interna moment fieds, based on the interpoated transverse dispacements. The variation of the generaized stress resutant interna force vector, may be epressed as 3.3 Heinger-Reissner potentia for beams GJ E I ω δs = N δd + N δf (4) S2 S S In the Heinger-Reissner mied formuation, both the dispacement and the interna forces are approimated by independent shape functions. This principe is appied to a beam eement of ength oaded by end forces ony. This two-fied variationa principe yieds two sets of noninear equations E = N S + N ( e e ) F = 0 (5) T T ˆ et ˆ d 2 d Q e S E = N ( e e ) d = 0 (6) C T ˆ S where E Q and E C are the eement equiibrium and eement strain-dispacement compatibiity equations, respectivey. A third equation, the cross-section equiibrium, may be epressed as S = S S = 0 (7) Q Σ 550
where S is given by the noninear cross-section constitutive reation and represents a genera function that permits the computation of cross-section stress resutants for given cross-section deformations. The inearization of the cross-section constitutive reation S = S (e) is obtained using the cross-section tangent stiffness matri k = SΣ/ e. The cross-section tangent eibiity matri q is obtained by inverting the cross-section tangent stiffness matri: q=k -. Furthermore, S is the interpoated generaized stresses acting over a cross-section and defined by (2). 3.4 inearization of the Heinger-Reissner functiona The noninear system of equations E Q =0, E C =0 and S Q =0 may be soved using various combinations of Newton iteration at the eement, and cross-section eves. Since intereement compatibiity is not enforced for the generaized stress variabes interpoation, the noninear discretized strain-dispacement compatibiity equation E C =0 can be soved iterativey at the eement eve for every goba equiibrium iteration. Simiary, the noninear constitutive equation S Q =0 can be soved iterativey at the cross-section eve for every eement eve iteration. In the foowing Subsections, the consistent inearization of the above noninear equations is presented. In the process of consistent inearization, it is important to recognize the arguments of any given function. 3.4. inearization of the Cross-section Constitutive Equation By epanding S Q =0 about the current cross-section state whie hoding S constant, we can write j+ j SQ j j j j SQ SQ + Δe Δ e = qsq = q( S SΣ) e (8) 3.4.2 inearization of the Eement Compatibiity Equation The incrementa form of the eement compatibiity condition E C =0 may aso be derived by taking a Tayor series epansion of E C =0 about the current state variabes f S and d i+ i EC i EC i EC EC + Δ d + Δ fs = 0 (9) d fs Then, soving for f i S, we obtain i i i Δ f ( κ 2) S = H M + G H Δ d + E C (20) 3.4.3 inearization of the Eement Equiibrium Equation Athough the inearization of the discrete weak-form of the eement equiibrium equation E Q =0 foows standard procedure [35, 36], this inearization is compicated by the presence of dispacementdependent noninear interpoation functions for the generaized stresses. For the case at hand, the consistent oca tangent stiffness equations are obtained by epanding (5) for each of the state variabes d, f S and F et about the current state. Again, a Tayor series epansion of E Q =0 is written as foows + E E E E + Δ d + Δf Δ F = 0 n n Q n Q n et, n Q Q S d fs (2) This equation can be rewritten by substituting where n Δf S from (20), and soving for n et, n+ K is the oca consistent tangent stiffness matri given by n Δd. Hence, K Δ d = F f (22) K = K + G + G H + ( M + G H ) H ( M + G H ) (23) and f is the oca interna force vector T T g 2 2 22 κ 2 κ 2 55
f = G f + N ( eˆ e ) d + ( M + G H ) H E (24) T n T n n T n S S2 κ 2 C 3.5 Noninear state determination agorithm The noninear system of equations is iterativey soved using Newton's method using three imbricated oops at different eves (structura eve, eement eve and cross-section eve). The oca eement straindispacement compatibiity equation E C =0 is soved iterativey for every goba equiibrium iteration. Simiary, the oca cross-section constitutive equation S Q =0 is soved iterativey at the cross-section eve for every eement compatibiity eve iteration. Therefore, residuas at the cross-section equiibrium and eement compatibiity eves are eiminated through iterations at each of these eves. In this subsection, the superscripts n, i and j denote the iteration indices for the goba structura equiibrium, the eement compatibiity and the cross-section equiibrium eves, respectivey. Before aunching the computer program (initiaization), the oca dispacement vector for each finite eement and the generaized strains e at each Gauss point aong the eement ength, need to be stored as zero vectors. Once the oca dispacements are obtained, the state determination procedure, estabished to obtain the interna force vector and tangent stiffness matri in the oca frame, starts as foows:. Evauate the generaized strains ê compatibe with the interpoated dispacements d n n eˆ = eˆ( d ) i 2. Evauate the noda force degrees of freedom f S. (iterate on Δf S ) i i, i+ : i i i T ( n i Δ f = H E f = f +Δ f where E = N e e ) d ˆ S C S S S C S 3. Evauate the generaized stress resutant interna force vector S i + ( i + ) i + S fs = NS fs j i 4. Cross-section equiibrium eve : Consider for the first iteration at this eve that e = : e, then, evauate the generaized strains e derived from the interpoated stress-resutant force fieds T j i+ i+ j j j+ j j j j j Δ e = q( S ( fs ) SΣ( e )), e = : e +Δ e where SΣ( e ) = A ( Ae ) da A i+ j+ i j 5. Repeat the above step unti S S Σ toerance, then consider e + = : e + upon convergence. i+ 6. Repeat the above steps from 2 to 5 unti E C toerance, then consider n : i +, n : i + fs = fs e = e and n i+ EC = : EC upon convergence. 7. Cacuate oca eement forces f and tangent stiffness matri K 4 EXAMPES 4. Cantiever with channe-section Figure 2 contains the probem description. This eampe was first introduced by Gruttmann et a. [7]. The beam is modeed using 4 bmw3d eements and the resuts are compared against 4 and 20 pbw3d dispacement-based beam eements. These meshes used 2 Gauss points per eement ength and 80 integration points within the cross-section. In Figure 2, the oad versus the vertica dispacement v of point O at the cantiever tip is depicted, where the noninear response has been computed up to v = 200. The resuts obtained with 4 bmw3d and 20 pbw3d eements are in very good agreement with those presented by [7] based on she eements. Furthermore, the resuts obtained with 4 pbw3d eements do not agree we over a arge etend of the computed oad defection curve. It can be observed that, in eastopasticity, the number of mied-based beam eements used to discretize the structure is consideraby reduced compared to the number of dispacement-based beam eements needed to obtain the oaddispacement curve with the same accuracy. This probem demonstrates the capabiity of the mied formuation to satisfactoriy predict the noninear behavior of beams with nonsymmetric cross-sections. 552
Figure 2: Cantiever with channe-section: data and resuts. 4.2 Right-ange frame The right-ange frame, shown in Figure 3 is subjected to a concentrated out-of-pane oad P acting at the midde of the span of the horizonta member. In the first mode, each member is modeed using 4 bmw3d eements with 3 Gauss integration points aong the eement ength. The square beam cross-section is meshed into a grid of 64 integration points. Four and 20 pbw3d eements per member are aso used for the second and third modes, respectivey, with the same eemnt and cross-section discretization. Noninear anaysis is aso performed with Fineg [9], using 20 corotationa two-noded spatia beam eements. The oad versus the out-of-pane dispacement of point O curves are depicted in Figure 3 for a modes. The comparison between the modes shows a very good agreement between the resuts obtained with the mied mode and those obtained with Fineg and with 20 pbw3d dispacement-based eements per member. 5 CONCUSION Figure 3: Right-ange frame: data and resuts. This paper proposed an efficient oca mied finite eement formuation for the anaysis of 3D Bernoui beams with sma strains and arge dispacements and rotations. The corotationa technique proposed in [4] is empoyed here. The oca strains are derived based on a consistent second-order inearization of the fuy geometricay noninear Bernoui beam theory. A 3D, geometric-noninear, easto-pastic oca beam eement based on the incrementa form of the two-fied Heinger-Reissner 553
functiona has been presented. This eement is targeted particuary for the anaysis of thin-waed beams with generic open cross-section where the centroid and shear center of the cross-section are not necessariy coincident. Severa numerica eampes have demonstrated the superiority of the mied formuation over dispacement-based one: the use of mied formuation eads to a considerabe reduction in the number of eements needed to perform the anaysis with the same accuracy. REFERENCES [] Izzuddin B.A., Smith D.. arge-dispacement anaysis of easto-pastic thin-waed frames. I: Formuation and impementation. Journa of Structura Engineering (ASCE), 22(8), 905-94, 996. [2] Asafadie R., Battini J.-M., Somja H., Hjiaj M. oca formuation for eastopastic corotationa thin-waed beams based on higher-order curvature terms. Finite Eements in Anaysis and Design, submitted. [3] Asafadie R., Battini J.-M., Hjiaj M. Efficient oca formuation for easto-pastic corotationa thin-waed beams. Communications in Numerica Methods in Engineering, in press. [4] Battini J.-M., Pacoste C. Co-rotationa beam eements with warping effects in instabiity probems. Computer Methods in Appied Mechanics and Engineering, 9(7), 755-789, 2002. [5] Battini J.-M., Pacoste C. Pastic instabiity of beam structures using co-rotationa eements. Computer Methods in Appied Mechanics and Engineering, 9(5), 58-583, 2002. [6] Crisfied M.A., Moita G.F. A unified corotationa framework for soids, shes and beams. Internationa journa of Soids and Structures, 33(20-22), 2969-2992, 996. [7] Gruttmann F., Sauer R., Wagner W. Theory and numerics of three-dimensiona beams with eastopastic materia behavior. Internationa Journa for Numerica Methods in Engineering, 48(2), 675-702, 2000. [8] Van Erp G.M., Menken C.M., Vedpaus F.E. The noninear feura-torsiona behavior of straight sender eastic beams with arbitrary cross-sections. Thin-Waed Structures, 6(5), 385-404, 988. [9] Fineg User's Manua. V9.0. Greisch Info S.A. - Department ArGEnCo - iege University (Ug), 2005. 554