Stability Analysis of Laminated Composite Thin-Walled Beam Structures

Transcription

1 Paper 224 Civil-Comp Press, 2012 Proceedings of the Eleventh International Conference on Computational Structures echnolog, B.H.V. opping, (Editor), Civil-Comp Press, Stirlingshire, Scotland Stabilit nalsis of Laminated Composite hin-walled Beam Structures D. Lanc, I. Pešić and G. urkalj Department of Engineering Mechanics, Facult of Engineering Universit of Rijeka, Croatia bstract he paper presents an algorithm for the buckling analsis of thin-walled laminated composite beam-tpe structures. one-dimensional finite element is emploed subject to the assumptions of large displacements, large rotation effects but small strains. he geometric stiffness matri of thin-walled beam finite element is derived. Stabilit analsis is performed in a load deflection manner using the co-rotational formulation. he cross-section mid-line contour is assumed to remain not deformed in its own plane and the shear strains of the middle surface are neglected. he laminates are modelled on the basis of classical lamination theor. he results are validated on test eamples. Kewords: beam, stabilit, buckling, composite, laminate, critical load. 1 Introduction Composite materials are ver suitable for structural applications where high strength to weight and stiffness to weight ratios are required. Fibre reinforced laminates in the form of thin-walled beams have been increasingl used during the past few decades in engineering practice, particularl in aerospace engineering, shipbuilding and in the automobile industr. Such weight-optimised structural components are commonl ver susceptive to buckling failure because of their slenderness so stabilit problems should be considered in their design [1]. finite element model for stabilit analsis of 3D framed structures with thinwalled laminated composite cross section is presented in this paper. he beam crosssection geometr is discretized b quadratic monitoring areas and the structural discretization is performed throughout one-dimensional finite element. Model takes into account effects of large displacements on the response of space frames subjected to conservative and static eternal loads. 1

2 Classical lamination theor for thin fibre-reinforced laminates is emploed in this paper. he model is applicable to an arbitrar laminate stacking sequence, shape of the cross section and boundar conditions. he shear strain of middle surface is assumed to be zero and the cross-section is not distorted in its own plane. he stabilit analsis is performed in a load-deflection manner [2, 3]. he non-linear response of a load-carring structure should be solved using numerical methods, e.g. the finite element method, and some of incremental descriptions like the total and updated Lagrangian ones, respectivel, or the co-rotational description. Each description utilizes a different structural configuration for sstem quantities referring and results in the form of a set of non-linear equilibrium equations of the structure. his set can further be linearized and should be solved using some incrementaliterative scheme. he co-rotational description, used in this work, is linear on element level and all geometricall non-linear effects are introduced through the transformation from the local to the global coordinate sstem. he local co-rotational sstem follows the element chord during the deformation and allows the use of simplified straindisplacement relations on the local element level. [4, 5] Verification eamples utilizing a numerical algorithm developed on the basis of abovementioned procedure are presented to demonstrate accurac of this model [6]. 2 heoretical background 2.1 Displacements In a local Cartesian coordinate sstem in which beam ae, that connects all cross sectional centres of gravit, coincides with z ae while and are principal aes, cross sectional rigid bod displacements are: w0 = w0( z), us = us( z), vs = vs( z), dvs du s dϕ. (1) z ϕz = ϕz( z), ϕ =, ϕ =, θ = dz dz dz In equation above, w 0 is translational displacement in z direction defined for cross-sectional centre of gravit; u s and v s are translational displacements in and directions, defined for shear centre; while ϕz, ϕ and ϕ are rotational displacement around z, and aes. Displacement θ is cross-sectional warping parameter. Cross-sectional displacement field is defined as: { W U V } { w w u u v v } U (2) uk = = where w, u and v are the standard linear displacement field components, while wu, andvare second order components that arise from large rotation effects according to [2, 3]. 2

3 In the contour coordinate sstem (z, n, s) on Figure 1, the part of the contour has been shown. Contour mid-line displacements are wu,, v, while displacements of an arbitrar point of contour are: u u w( zsn,, ) = w n ; v( zsn,, ) = v n ; u( zsn,, ) = u z s (3) where r is contour radius and q is the distance of pole P from normal n. Figure 1: Cross-sectional contour displacement components. Connection between the contour and beam displacements has been established as: (,, ); ( ) β ( ) ( ) β ( ) w= W z s n v= U zsn,, cos + V zsn,, sin β; u= U zsn,, sin V zsn,, cosβ (4) 2.2 Strains Using nonlinear beam displacement field the strain tensor consists of three parts: (,, ),, (,, ) εij = eij + ηij + e ij; eij = 0.5 ui j + uj i, η ij = 0.5 uk i uk j, e ij = 0.5 u i j + u j i. (5) Supposing that the shair strain γ zs of the middle surface is zero, and that the contour of the thin wall does not deform in its own plane, the non-zero strain components are: 2 2 w w u ; w v 2 u e = = = + = ; zz n e 2 zs n z z z s z s z (6) w u v w w u u v v ηzz = + + ; η zs = + + ; 2 z z z z s z s z s (7) 3

4 e zz = w ; e = w zs + v z s z. (8) 2.3 Internal forces he constitutive equations for one lamina are: * * σ z Q 11 Q16 ε z = * * τ zs Q16 Q66 γ zs (9) * where Q ii are so called transformed reduced stiffnesses [7]. Integrating over the laminate thickness n and the contour direction s, and transforming into the beam coordinate sstem follows the cross-sectional internal force components: F = σ dn ds; M = τ n dnds; M = σ ( ω nq) dnds; z z z zs ω z M = σ ( ncos β) dnds; M = σ ( + nsin β) dnds; z z (10) 2.4 Beam finite element In Figure 2 two-nodded beam finite element with eight degrees of freedom is presented. ϕ ϕ ϕ B ϕb l B B zb w B z M M M l M B B M B M B M zb F zb z Figure 2: wo-nodded spatial beam element in local coordinate sstem. he nodal displacement vector of the beam element is: ( u e ) = { B, ϕzb, ϕ, ϕb, ϕ, ϕb, θ,, θb} w (11) 4

5 and an appropriate nodal force vector is: ( ) = { zb, zb,, B,, B,M ω,mωb} f e F M M M M M. (12) Incremental analsis supposes that a load-deflection path is subdivided into a number of steps or increments. his path is usuall described using three configurations: the initial or undeformed configuration C 0 ; the last calculated equilibrium configuration C 1 and current unknown configuration C 2. dopting corotational formulation, all sstem quantities should be referred to configuration C 2, [3]. ppling the virtual work principle and neglecting the bod forces, the equilibrium of a finite element can be epressed as, [2, 3]: δ U= δ W (13) in which U is potential energ of internal forces, W is the virtual work of eternal forces, while δ denotes virtual quantities. fter making the first variation of eq. (10), the following incremental equations can be obtained: e e e e e δw = δu f ; δu= δ u k u (14) ( ) ( ) e In Equation (14) k denotes the local tangent stiffness matri of the e-th beam element which can be evaluated according to procedure eplained in [4, 5]. Now the incremental equilibrium equation can be written in the following form: k e Δ u e =Δf e (15) he element global tangent stiffness matri k e can be obtained as follows: e e e e e k = t k t + t f e. (16) Matrices t e 1 and t e 2 are standard transformation matrices from local co-rotational to global coordinate sstem eplained in [4]. Matri t e 1 is of dimension 14 8 and contains first derivations of local with respect to global displacements while t e 2 is matri containing second derivations. Matri t e 2 presents geometric stiffness contribution because it contains effects on global forces caused with change in geometr. Element force vector transformed from local to global coordinate sstem is: e e f t f e. (17) = 1 fter the standard assembling procedure, the overall incremental equilibrium 5

6 equations can be obtained as: KU = P, K =, k e e 2 1 P= P P, (18) where K is tangential stiffness matri of a structure, while U and P are the incremental displacement vector and the incremental eternal loads of the structure. 2 P and 1 P are the vectors of eternal loads applied to the structure at C 2 and C 1 configurations, respectivel. 3 Numerical eamples cantilever and simpl supported beam subjected to an aial force are considered, Figure 3. he beam and cantilever length is L = 2 m and thin-walled cross section is of I-tpe with dimensions 10 cm 10 cm 1 cm. he web and the flanges are considered to be four laered smmetric angle-pl laminates with stacking sequence [0, 0, 0, 0], [0, 90, 90, 0] and [45, -45, -45, 45]. he all plies are of the same thickness of 2.5 mm. he analsed material is graphite-epo (S4/3501) whose properties are: E 1 =144 GPa, E 2 =9.65 GPa, G 12 =4.14 GPa and ν 12 =0.3. o initiate the fleural instabilit, the horizontal perturbation force ΔF of value 10-3 F is added in both cases at positions shown on Figure 3. L 100 F F 10 F F z 10 Figure 3: Simpl supported beam, cantilever and cross-section geometr. he Figures 4 and 5 illustrate the load-deflection curves for different laminate configuration for both cases of constraints, simpl supported beam and cantilever. ll the obtained results correspond to the fleural buckling modes in the z-plane and used mesh consist of eight beam finite elements. 6

7 Figure 4: Load vs. displacement curves for simpl supported beam. Figure 5: Load vs. displacement curves for cantilever. he buckling response of this beam was considered previousl b Cortinez and Piovan [6] where the critical buckling loads were obtained b eigenvalue analsis. Buckling predictions in this paper depicted on Figures 4 and 5 show that ver good accurac is achieved in comparison with their results, marked b dashed line. 7

8 4 Conclusion he co-rotational formulation for the non-linear analsis of beam columns with laminated composite cross section is proposed. he governing incremental equilibrium equations of a two-node space beam element are developed using the linearised virtual work principle. he presented test eamples suggest that the numerical model developed is an accurate tool for modelling composite cross section beam structure nonlinear behaviour. cknowledgement he research presented in this paper was made possible b the financial support of the Ministr of Science, Education and Sports of the Republic of Croatia, under the project No , and also b the grant of Zaklada Sveucilista u Rijeci. References [1] N.. lfutov, Stabilit of Elastic Structures. Springer-Verlag, Berlin, [2] G. urkalj, J. Brnic and J. Prpic Orsic, Large rotation analsis of elastic thinwalled beam-tpe structures using ES approach, Computers & Structures 81: , [3] Y.B. Yang and S.R. Kuo, heor & nalsis of Nonlinear Framed Structures, Prentice Hall, New York, [4] B.. Izzuddin, Conceptual issues in geometricall nonlinear analsis of 3D framed structures, Computer Methods in pplied Mechanics and Engineering 191: , [5] D. Lanc, G. urkalj and J. Brnic, Finite-element model for creep buckling analsis of beam-tpe structures, Communications in Numerical Methods in Engineering 24: , [6] V.H. Cortinez, M.. Piovan, Stabilit of composite thin-walled beams with shear deformabilit, Computers & Structures, 84, , [7] J. Lee and S.E. Kim, Lateral buckling analsis of thin-walled laminated channel-section beams, Composite Structures 56: ,