Geometrically nonlinear theory of thin-walled composite box beams using shear-deformable beam theory

https://doi.org/10.1016/j.ijmecsci.2009.10.005Get rights and content

Abstract

A general geometrically nonlinear model for thin-walled composite space beams with arbitrary lay-ups under various types of loadings is presented. This model is based on the first-order shear deformable beam theory, and accounts for all the structural coupling coming from both material anisotropy and geometric nonlinearity. The nonlinear governing equations are derived and solved by means of an incremental Newton–Raphson method. A displacement-based one-dimensional finite element model that accounts for the geometric nonlinearity in the von Kármán sense is developed. Numerical results are obtained for thin-walled composite box beams under vertical load to investigate the effects of shear deformation, geometric nonlinearity and fiber orientation on axial–flexural–torsional response.

Introduction

Fiber-reinforced composite materials have been used over the past few decades in a variety of structures. Composites have many desirable characteristics, such as high ratio of stiffness and strength to weight, corrosion resistance and magnetic transparency. Thin-walled structural shapes made up of composite materials, which are usually produced by pultrusion, are being increasingly used in many engineering fields.

The theory of thin-walled closed section members made of isotropic materials was first developed by Vlasov [1] and Gjelsvik [2]. A large number of practical problems require a geometrically nonlinear formulation, such as the postbuckling behavior, load carrying capacity of structures used in aeronautical, aerospace as well as in mechanical and civil engineering. Thin-walled composite structures are often very thin and have complicated material anisotropy. The studies of these structures carried out so far may broadly be divided into two groups. The first and most common approach is based on an analytical technique, while the other approach requires a two-dimensional finite element analysis to obtain the cross-section stiffness matrix. Atilgan and Hodges et al. [3], [4] pioneered the second approach, which was referred to as the so-called “variational asymptotic beam section analysis” (VABS). VABS used the variational asymptotic method (VAM) to split a three-dimensional nonlinear elasticity problem into a two-dimensional linear cross-sectional analysis and a one-dimensional, nonlinear beam problem. Hodges and co-workers (e.g., Cesnik et al. [5], [6], Volovoi et al. [7], Yu et al. [8]) further applied the concept introduced by VAM to two dimensional cross-sectional problem and derived closed-form expressions for the cross-sectional stiffness coefficients of thin-walled beams. By using Atilgan and Hodges's beam model, Jeon et al. [9] developed an analysis model of large deflection for the static and dynamic analysis of composite box beams.

In the present investigation, an analytical approach is adopted for the derivation of the cross-sectional stiffness matrix considering different effects and their coupling to yield a general formulation. Bauld and Tzeng [10] presented nonlinear model for thin-walled composite beams by extending Gjelsvik's formulation to the balanced symmetric laminated composite materials. However, the formulation was somewhat not consistent in the sense that it used coordinate mapping when developing nonlinear stresses instead of variational formulation. A finite element formulation was developed by Stemple and Lee [11], [12] to take into account the warping effect of composite beams undergoing large deflection or finite rotation. Kalfon and Rand [13] derived the nonlinear theoretical modeling of laminated thin-walled composite helicopter rotor blades. The derivation was based on nonlinear geometry with a detailed treatment of the body loads in the axial direction which were induced by the rotation. Pai and Nayfeh [14] developed a fully nonlinear theory of curved and twisted composite rotor blades. The theory accounted for warping due to bending, extensional, shearing and torsional loadings, and three-dimensional stress effects by using the results of a two-dimensional, static, sectional, finite element analysis. Bhaskar and Librescu et al. [15], [16] developed nonlinear theory of thin-walled composite beams, which was employed in a broad field of engineering problems. In this model, the transverse shear deformation was taken into account but the warping torsion component was neglected. Special attention deserved the works of Machado, Cortinez and Piovan [17], [18], [19] who introduced a geometrically nonlinear theory for thin-walled composite beams for both open and closed cross-sections and took into account shear flexibility (bending and warping shear). This nonlinear formulation was developed by using a nonlinear displacement field, whose rotations were based on the rule of semi-tangential transformation. It was used for analyzing the stability of thin-walled composite beams with general cross-section. By using a geometrically higher-order nonlinear beam theory, Machado [19] investigated the influence of large rotations on the buckling and free vibration behavior of thin-walled composite beam. However, it was strictly valid for symmetric balanced laminates and especially orthotropic laminates. Recently, Sapountzakis and Panagos [20] introduced the nonlinear analysis of a composite Timoshenko beam with arbitrary variable cross-section undergoing moderate large deflections by employing the analog equation method (AEM), a BEM-based method.

In this paper, which is an extension of the authors’ previous works [21], [22], [23], [24], [25], a geometrically nonlinear model for thin-walled composite beams with arbitrary lay-ups under various types of loadings is presented. This model is based on the first-order shear deformable beam theory, and accounts for all the structural coupling coming from both material anisotropy and geometric nonlinearity. The general nonlinear governing equations are derived and solved by means of an incremental Newton–Raphson method. A displacement-based one-dimensional finite element model that accounts for the geometric nonlinearity in the von Kármán sense is developed. Numerical results are obtained for thin-walled composite box beam under vertical load to investigate the effects of shear deformation, geometric nonlinearity, fiber orientation, laminate stacking sequence and load parameter on axial–flexural–torsional response.

Section snippets

Kinematics

The theoretical developments presented in this paper require two sets of coordinate systems which are mutually interrelated. The first coordinate system is the orthogonal Cartesian coordinate system (x,y,z), for which the x- and y-axes lie in the plane of the cross-section and the z-axis parallel to the longitudinal axis of the beam. The second coordinate system is the local plate coordinate (n,s,z) as shown in Fig. 1, wherein the n-axis is normal to the middle surface of a plate element, the s

Variational formulation

Total potential energy of the system is calculated by sum of strain energy and potential energyΠ=U+Vwhere U is the strain energyU=12v(σzεz+σszγsz+σnzγsz)dv

The strain energy is calculated by substituting Eq. (18) into Eq. (20)U=12v{σz[εz+(x+nsinθ)κy+(y-ncosθ)κx+(ω-nq)κω+(2rn+n2)χz]+σszγxzcosθ+γyzsinθ+γωr-F2t+κszn+F2t+σnz[γxzsinθ-γyzcosθ-γωq]}dvThe variation of the strain energy, Eq. (21), can be stated asδU=0l(Nzδεz+Myδκy+Mxδκx+Mωδκω+Vxδγxz+Vyδγyz+Tδγω+Mtδκsz+Rzδχz)dswhere Nz,Mx,My,M

Constitutive equations

The constitutive equations of a k th orthotropic lamina in the laminate co-ordinate system of section are given byσzσszk=Q¯11*Q¯16*Q¯16*Q¯66*kεzγszwhere Q¯ij* are transformed reduced stiffnesses. The transformed reduced stiffnesses can be calculated from the transformed stiffnesses based on the plane stress (σs=0) and plane strain (εs=0) assumption. More detailed explanation can be found in Ref. [26].

The constitutive relation for out-of-plane stress and strain is given byσnz=Q¯55γnz

The

Governing equations

The nonlinear equilibrium equations of the present study can be obtained by integrating the derivatives of the varied quantities by parts and collecting the coefficients of δW,δU,δV,δΦ,δΨy,δΨx and δΨωNz+Pz=0Vx+[Nz(U+ypΦ)]-[MxΦ]+Vx=0Vy+[Nz(V-xpΦ)]+[MyΦ]+Vy=0Mt+T+[Nz(rp2Φ+ypU-xpV)]+[MyV]-[MxU]+[RzΦ]+T=0My-Vx+My=0Mx-Vy+Mx=0Mω+Mt-T+Mω=0

The natural boundary conditions are of the formδW:NzδU:Vx+Nz(U+ypΦ)-MxΦδV:Vy+Nz(V-xpΦ)+MyΦδΦ:T+Mt+Nz(rp2Φ+ypU-xpV)+MyV-MxU+RzΦ

Finite element formulation

The present theory for thin-walled composite beams described in the previous section was implemented via a one-dimensional displacement-based finite element method. The linear, quadratic and cubic elements with seven degrees of freedom at each node as shown in Fig. 2 are derived. The same interpolation function is used for all the translational and rotational displacements. Reduced integration of shear terms, that is, stiffness coefficients involving laminate stiffnesses (Ei,j, i=68,j=68) is

Numerical examples

Throughout numerical examples, a tolerance of ε=10-3 and maximum allowable iterations of 20 (per load step) are used to check for convergence of nodal displacements in the Newton–Raphson iteration scheme. Ten quadratic elements with three nodes are used in the numerical computation. The initial solution vector is chosen to be the zero vector, so that the first iteration solution corresponds to the linear solution. The results of the present analysis are given for both the linear and nonlinear

Concluding remarks

A geometrically nonlinear analysis for thin-walled composite beams with arbitrary lay-ups under various types of loadings has been presented. This model is based on the first-order shear deformable beam theory, and accounts for all the structural coupling coming from both material anisotropy and geometric nonlinearity. General nonlinear governing equations are derived and solved by means of an incremental Newton–Raphson method. A displacement-based one-dimensional finite element model that

Acknowledgments

The support of the research reported here by Seoul R&BD Program through Grant GR070033 and by Korea Ministry of Construction and Transportation through Grant 2006-C106A1030001-06A050300220 is gratefully acknowledged. The authors also would like to thank the anonymous reviewers for their suggestions in improving the standard of the manuscript.

References (27)

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