Geometrcally exact mult-layer beams wth a rgd nterconnecton Leo Škec, Gordan Jelenć To cte ths verson: Leo Škec, Gordan Jelenć. Geometrcally exact mult-layer beams wth a rgd nterconnecton. 2nd ECCOMAS Young Investgators Conference (YIC 2013), Sep 2013, Bordeaux, France. <hal- 00855911> HAL Id: hal-00855911 https://hal.archves-ouvertes.fr/hal-00855911 Submtted on 30 Aug 2013 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.
YIC2013 Second ECCOMAS Young Investgators Conference 2 6 September 2013, Bordeaux, France Geometrcally exact mult-layer beams wth a rgd nterconnecton L. Škec a,, G. Jelenć a a Unversty of Rjeka, Faculty of Cvl Engneerng Radmle Matejčć 3, 51000 Rjeka, Croata leo.skec@gradr.hr Abstract. In ths work, a fnte-element formulaton for geometrcally exact mult-layer beams wthout consderng the nterlayer slp and uplft s proposed. Numercal examples ndcate that, n comparson wth the exstng geometrcally non-lnear sandwch beam models, the 2D plane-stress elements and the analytcal results from the theory of elastcty, the mult-layer beam model s very effcent for modellng thck beams where warpng of cross-sectons has to be consdered. Keywords: mult-layer beam; geometrcally exact theory; non-lnear analyss; cross-sectonal warpng. 1 INTRODUCTION Research and applcaton of layered composte structures usng beam elements n many areas of engneerng has ncreased consderably over the past couple of decades and contnues to be a topc of undmnshed nterest n the computatonal mechancs communty [1, 2, 3, 4, 5, 6, 7]. Ths work ntoduces a fnte-element formulaton for a geometrcally exact mult-layer beam element. The number of layers (n) s arbtrary and they are assembled n a composte beam wth the nterlayer connecton allowng only for the occurrence of ndependent rotatons of each layer. In other words, nterlayer slp and uplft effects are not consdered. Vu-Quoc et al. [2] also proposed a formulaton for a geometrcally exact mult-layer beam and they used the Galerkn projecton to obtan the computatonal formulaton of the resultng non-lnear equatons of equlbrum n the statc case, whle n the present work the equlbrum equatons are derved form the prncple of vrtual work. Whle the resultng numercal procedure s of necessty equal, here we focus on the actual transformaton of the dsplacement vector for each layer to the dsplacement vector of the beam reference lne and show that t may be wrtten n a remarkably elegant form allowng for smple numercal mplementaton. Furthermore, we specfcally analyse the problems wth large number of layers and on the thck beam problems wth pronounced cross-sectonal warpng compare the performance of the elements derved to the analytcal results and the fnte-element results obtaned usng 2D plane-stress elements. Detaled analyss of ths problem s gven n [8]. 2 PROBLEM DESCRIPTION Consder an ntally straght composte beam of length L and a cross-secton composed of n parts wth heghts h and areas A, where [1, n] s an arbtrary layer. Layers are made of lnear elastc materal wth E and G actng as Young s and shear modul of each layer s materal. The reference axes of all layers n the ntal undeformed state are defned by the unt vector t 01 whch closes an angle ψ wth respect to the axs defned by the base vector e 1 of the spatal co-ordnate system. Durng the deformaton the cross-sectons of the layers reman planar but not necesarly orthogonal to ther reference axes (Tmoshenko beam theory wth the Bernoull hypothess). 3 GOVERNING EQUATIONS In the assemlby equatons, the dsplacements of each layer (u ) are expressed n terms of the basc unknown functons u and θ. For each layer, the knematc and consttutve equatons are derved. The equlbrum equatons are
2 L. Škec and G. Jelenć Young Investgators Conference 2013 derved from the prncple of vrtual work. The dsplacement of each layer can be, accordng to Fg. 1, expressed n terms of the dsplacement of an arbtrarly chosen man layer α (denoted by u) and correspondng rotatons θ as [8] u = u + d,ζ (t ζ,2 t 0,2 ) + d,ξ (t ξ,2 t 0,2 ) + ξ 1 s=ζ+1 d,s (t s,2 t 0,2 ), (1) where ζ represents the bottom and ξ the upper layer between layers and α, whle d,j, (j = ζ, ξ, s), are the dstances dependng on the mutual poston between layers and α. The reference axs of the layer α then becomes the reference axs of the composte beam and u, θ, [1, n], become the basc unknown functons of the problem whch are assembled n a vector as p T f = u θ 1... θ n T. Fgure 1: Undeformed and deformed state of the multlayer composte beam Non-lnear knematc equatons are defned accordng to Ressner s beam theory [9] as { } ɛ γ = = Λ T r e 1 = Λ T (t 01 + u ) e 1, κ = θ, (2) γ where ɛ, γ, κ are the axal stran, shear stran and curvature, respectvely, wth respect to the reference axs of the -th layer. The consttuve law s gven as N { } N T = = C M = E A 0 E S ε { } 0 G k A 0 γ M = C γ, (3) κ E S 0 E I where N, T, M are the axal force, shear force and bendng moment wth respect to the reference axs of layer, respectvely, whle k s the shear correcton coeffcent. S and I are the frst and the second moment of area of the cross-secton of layer, respectvely, and ε, γ and κ are the axal stran, shear stran and the curvature. Equlbrum equatons are derved usng the prncple of vrtual work for a statc problem (G G G e 0), whch for a multlayer beam composed of n layers after a seres of transformatons becomes [8] κ G = n L =1 0 p T f B T ( { } { }) { } { } D T L T γ f C dx κ w 1 p T f,0b T F,0,0 p T W,0 f,lb T F,L,L 0. (4) W,L Matrx B transforms the vector of dsplacements and rotaton of the layer to the basc the unknown functons, whle the matrces L and D transform the vector of vrtual strans and curvature to the vector of vrtual dsplacmenents and rotaton of the layer. Indces 0 and L represent the beam ends where the boundary pont forces F j,0, F j,l and bendng moments W j,0, W j,l are appled. The dstrbuted force and moment loads are denoted by f j and w j. Fnally, p f s the vector of the vrtual basc unknown functons.
L. Škec and G. Jelenć Young Investgators Conference 2013 3 4 SOLUTION PROCEDURE The governng equatons of the problem are hghly non-lnear and cannot be solved n a closed form. Thus, t s necessary to choose n advance the shape of test functons (u, θ ), and later also the shape of tral functons (u, θ ), where [1,..., n] as p f = Ψ j (X 1 )p j, p f = Ψ k (X 1 ) p k, (5) j=1 where Ψ j are the matrces of nterpolaton functons, p j s the vector contanng nodal dsplacements and rotatons of node j, whle p f and p j are the vectors of the ncrements of the basc unknown functons and nodal ncrements of the dsplacements and rotatons of the node j, respectvely. From expresson (4) we can easly obtan the vector of resdual forces for the node j as G p T j g j = 0. (6) j=1 After the lnearsaton of the nodal vector of resdual forces g j + g j = g j + the nodal tangent stffness matrces are obtaned as k=1 K j,k p k = 0, j = 1,..., N. (7) k=1 K j,k = n =1 ( L 0 Ψ T j J,1 Ψ k dx 1 δ j,1 δ k,1 J,0 δ j,n δ k,n J,L ), (8) whch are of dmenson (2 + n) (2 + n) and are assembled nto an element tangent stffness matrx of dmenson N(2 + n) N(2 + n). Matrces J,l, (l = 1, 0, L) are produced n the process of lnearsaton of the vector of resdual forces (see [8] for detals). For ntegraton n (8) we use the Gaussan quadrature wth N 1 ntegraton ponts n order to avod shear-lockng [10]. From (7) we fnally obtan p = K 1 g, (9) where vectors p k and g j are assembled nto vectors p and g, respectvely. The soluton s obtaned teratvely usng Newton-Raphson method untl a satsfyng accuracy s acheved. 5 NUMERICAL EXAMPLES 5.1 Roll-up Maneuver A comparson of the presented formulaton wth [1] and [2] s gven for the roll-up maneuver (a moment M = 2EIπ/L at the beam tp whch bends the cantlever beam nto an exact crcle). Both for a sngle layer beam and for a sandwch beam wth three dentcal layers usng the so-called normal moment dstrbuton over the layers (M 1 : M 2 : M 3 = 7 : 13 : 7) the present formulaton shows excellent accordance wth the results from [1, 2] and the analytcal results [8]. 5.2 Thck beam tests The presented mult-layer beam model s further compared to a homogeneous beam dvded nto a fnte number of equal lamnae (layers) wth dentcal geometrcal and materal propertes and no nterlyer slp and uplft. The ndependent cross-sectonal rotatons of each layer allow the cross-secton to deform n a pecewse lnear form. Ths s compared to the results from the theory of elastcty [11] where the cross-sectonal warpng occurs n the deformed state. For a one-pont-clamped thck cantlever beam wth a narrow rectangular cross-secton of unt wdth subjected to a transverse force F at the free end the soluton, for a geometrcally lnear case, the multlayer formulaton s compared to a two-dmensonal fnte element mesh. The results (see Fg.2) show that the multlayer formulaton gves consderably better results wth less degrees of freedom n comparson wth the two-dmensonal fnte element meshes. A more detaled analyss s provded n [8].
4 L. Škec and G. Jelenć Young Investgators Conference 2013 Fgure 2: A comparson between the warped cross-secton of the left-hand end of the beam accordng to the theory of elastcty and the lnear-pecewse cross-sectons obtaned by the mult-layer beam model and the two-dmensonal fnte element models for dfferent meshes 6 CONCLUSIONS In ths work we have presented a geometrcally exact mult-layer beam element wth rgd connecton between the layers and arbtrary poston of the layers and the composte beam s reference axes, thus allowng for arbtrary poston of the appled loadng. We have shown that the knematc constrant relatng the dsplacement vector of an arbtrary layer and the dsplacement vector of the beam reference lne may be wrtten n a unque way regardless of the postons of the layer and the beam reference axes. The element has been verfed aganst the results n [1, 2] and ts capabltes tested on a thck beam example aganst analytcal and numercal results comng form 2D elastcty. Whle the beam theory utlsed obvously cannot recognse the exstence of the transverse normal stresses and strans, t shows remarkable ablty to capture the cross-sectonal warpng effect and gve good approxmaton of 2D elastcty results usng far less degrees of freedom. ACKNOWLEDGEMENT The results shown here have been obtaned wthn the scentfc project No 114-0000000-3025: Improved accuracy n non-lnear beam elements wth fnte 3D rotatons fnancally supported by the Mnstry of Scence, Educaton and Sports of the Republc of Croata. REFERENCES [1] Vu-Quoc, L., Deng, H.. Galerkn Projecton for Geometrcally Exact Sandwch Beams Allowng for Ply Drop-off. J. Appl. Mech. 62: 479-488, 1995. [2] Vu-Quoc, L., Deng, H., Ebcoğlu I. K.. Multlayer Beams: A Geometrcally Exact Formulaton. J. Nonlnear Sc. 6: 239-270, 1996. [3] Grhammar, U. A., Pan D. H.. Exact statc analyss of partally composte beams and beam-columns. Int. J. Mech. Sc. 49: 239-255, 2007. [4] Schnabl, S., Saje, M., Turk, G., Plannc, I.. Analytcal Soluton of Two-Layer Beam Takng nto account Interlayer Slp and Shear Deformaton. J. Struct. Eng., ASCE 133(6): 886-894, 2007. [5] Kroflč, A., Plannc, I., Saje, M., Čas, B.. Analytcal soluton of two-layer beam ncludng nterlayer slp and uplft. Struct. Eng. Mech. 34(6): 667-683, 2010. [6] Kroflč, A., Saje, M., Plannc, I.. Non-lnear analyss of two-layer beams wth nterlayer slp and uplft. Comput. Struct. 89(23-24): 2414-2424, 2011. [7] Škec, L., Schnabl, S., Plannc, I., Jelenć, G.. Analytcal modellng of multlayer beams wth complant nterfaces. Struct. Eng. Mech. 44(4): 465-485, 2012. [8] Škec, L., Jelenć, G.. Analyss of a geometrcally exact mult-layer beam wth a rgd nterlayer connecton. Submtted to Acta Mechanca, 2013. [9] Ressner, E.. On one-dmensonal fnte-stran beam theory; the plane problem. J. Appl. Math. Phys. (ZAMP) 23(5): 795-804, 1972. [10] Smo, J.C., Vu-Quoc, L.. On the Dynamcs of Flexble Beams Under Large Overall Motons - The Plane Case: Part I and II. J. Appl. Mech. 53(4): 849-863, 1986. [11] Tmoshenko, S.P., Gooder, J.N.. Theory of Elastcty. McGraw-Hll, 1951.