Shear coefficients for thin-walled composite beams

The shear coefficient in Timoshenko beam theory is obtained for thin-walled beams constructed of laminated panels of composite material using a variation of the method due to Cowper. Formulae are presented for a class of such composite beams. Comparisons are made with Cowper's original formulae for the case of an isotropic beam. The effect of shear deformation under static loading of typical composite beams is investigated. A procedure is outlined for the distribution of plies in the laminated panels to achieve optimal response under static or dynamic loading.     

Instability of laminated composite thin-walled open-section beams

Instability of thin-walled open-section laminated composite beams is studied using the finite element method. A two-noded, 8 df per node thin-walled open-section laminated composite beam finite element has been used. The displacements of the element reference axis are expressed in terms of one-dimensional first order Hermite interpolation polynomials, and line member assumptions are invoked in formulation of the elastic stiffness matrix and geometric stiffness matrix. The nonlinear expressions for the strains occurring in thin-walled open-section beams, when subjected to axial, flexural and torsional loads, are incorporated in a general instability analysis. Several problems for which continuum solutions (exact/approximate) are possible have been solved in order to evaluate the performance of finite element. Next its applicability is demonstrated by predicting the buckling loads for the following problems of laminated composites: (i) two layer (45°/−45°) composite Z section cantilever beam and (ii) three layer (0°/45°/0°) composite Z section cantilever beam.     

单闭室复合材料薄壁梁的结构阻尼

研究单闭室复合材料薄壁梁的结构阻尼特性。基于变分渐进法（VAM) 和Hamilton原理,分别建立薄壁梁的截面力位移关系和运动方程;采用Galerkin法对薄壁梁进行自由振动分析;在获得薄壁梁振动模态矢量的基础上,根据最大应变能理论,对薄壁梁的模态阻尼性能进行预测,并且将阻尼预测的结果与现有的有限元计算结果进行对比,验证了本文阻尼分析模型的有效性。进一步针对周向均匀刚度配置(CUS)和周向反对称刚度配置（CAS）两种构型复合材料薄壁箱形梁以及一个翼型截面梁,进行阻尼计算,揭示了纤维铺层角和截面宽高比等参数的影响     

Buckling failure modes of FRP thin-walled beams

A study on buckling phenomena in pultruded Fiber Reinforced Polymer (FRP) beams, based on two mechanical models recently formulated by the authors with regard to composite thin-walled beams, is presented in this paper. Global buckling behavior is analyzed by means of a one-dimensional model in which cross-section torsional rotation is divided into two parts: the first one, associated with Vlasov’s axial warping, the second one, associated entirely with shear strains. The study of local behavior is based on the individual buckling analysis of the components of FRP profile, assumed as elastically restrained transversely isotropic plates. Both mechanical models take into account, within the field of small strains and moderate rotations, the contribution of shear deformation in the kinematic hypotheses. Design charts suitable to evaluate the buckling load of FRP “I” beams with either narrow or wide flanges are obtained and presented in this paper.     

Shear coefficients for multicelled thin-walled composite beams

The shear coefficient is obtained for multicelled thin-walled composite material beams in terms of the elastic properties of the walls, or panels, which form the beam. The method presented follows one discussed in a previous paper in which shear coefficients were found for open or single cell symmetrical cross-sections by Cowper's method. In order to apply Cowper's method to the multicelled section the shear flow around the cross-section must be modified to account for lateral distortion of the cross-section due to Poisson effects. As an example, a detailed calculation for a double-cell composite box beam is given. Additional results are given for a triple-cell box beam. The effect of varying the properties of the panels around the cross-section is investigated.     

A beam theory for thin-walled composite beams

A beam theory is presented that is formulated in terms of the in-plane elastic properties of the panels of the cross-section of a thin-walled composite beam. Shear deformation is accounted for by using a suitable form of the Timoshenko beam theory together with a modified form of the shear coefficient. The theory gives both the bending deflection and the shear deflection of a beam loaded by an applied transverse load. Numerical and graphical results obtained from a computer code show the effects of using different composite material systems and lay-ups in the panels of typical beams.     

Lateral buckling analysis of thin-walled laminated channel-section beams

The lateral buckling of a laminated composite beam with channel section is studied. A general analytical model applicable to the lateral buckling of a channel-section composite beam subjected to various types of loadings is derived. This model is based on the classical lamination theory, and accounts for the material coupling for arbitrary laminate stacking sequence configuration and various boundary conditions. The effects of the location of applied loading on the buckling capacity are also included in the analysis. A displacement-based one-dimensional finite element model is developed to predict critical loads and corresponding buckling modes for a thin-walled composite beam with arbitrary boundary conditions. Numerical results are obtained for thin-walled composites under central point load, uniformly distributed load, and pure bending with angle-ply and laminates. The effects of fiber orientation, location of applied load, and types of loads on the critical buckling loads are parametrically studied.     

Nonlinear free vibrations of thin-walled beams in torsion

Nonlinear torsional vibrations of thin-walled beams exhibiting primary and secondary warpings are investigated. The coupled nonlinear torsional–axial equations of motion are considered. Ignoring the axial inertia term leads to a differential equation of motion in terms of angle of twist. Two sets of torsional boundary conditions, that is, clamped–clamped and clamped-free boundary conditions are considered. The governing partial differential equation of motion is discretized and transformed into a set of ordinary differential equations of motion using Galerkin’s method. Then, the method of multiple scales is used to solve the time domain equations and derive the equations governing the modulation of the amplitudes and phases of the vibration modes. The obtained results are compared with the available results in the literature that are obtained from boundary element and finite element methods, which reveals an excellent agreement between different solution methodologies. Finally, the internal resonance and the stability of coupled and uncoupled nonlinear modes are investigated. This study can be a preliminary step in the understanding of complex dynamics of such systems in internal resonance excited by external resonant excitations.     

Stiffness control in adaptive thin-walled laminate composite beams

The aim of this paper is to verify the control of the stiffness that is feasible to achieve in a thin-walled box-beam made from a laminate by including an adaptive material with variable stiffness. In this work, a material having a strongly varying Young Modulus under minor temperature changes was included in the cross-section. An analytical model was used to estimate the position of shear centre and the axial, bending, torsional, and shear stiffnesses of the cross-section. Two cross-sections were analysed, one with an adaptive wall and another with two adaptive walls. In both sections, the torsional stiffness could be strongly altered with minor temperature variations. In the section with one adaptive wall, the shear centre and thus the bending–twist coupling was also strongly modified. A study was made of the influence on the control of stiffnesses exerted by the overall cross-section thickness and the thickness of the adaptive walls.     

复合材料空间薄壁梁的有限元分析模型

在剪切梁理论的基础上, 采用9 节点平面单元模拟梁任意截面形状; 采用27 节点体单元, 模拟截面出平面外的二次翘曲位移, 从而建立了空间复合材料任意截面薄壁梁考虑二次翘曲的有限元分析模型。根据本文中导出的复合材料有限元模型编制了相应的分析计算程序。算例表明: 本文中建立的复合材料薄壁梁模型正确, 可以用于考虑多种耦合影响因素作用下复杂结构空间薄壁复合材料梁的有限元分析计算。      

Coupled deflection analysis of thin-walled Timoshenko laminated composite beams

For the deflection analyses of thin-walled Timoshenko laminated composite beams with the mono- symmetric I-, channel-, and L-shaped sections, the stiffness matrices are derived based on the solutions of the simultaneous ordinary differential equations. A general thin-walled composite beam theory considering shear deformation effect is developed by introducing Vlasov’s assumptions. The shear stiffnesses of thin-walled composite beams are explicitly derived from the energy equivalence. The equilibrium equations and force-deformation relations are derived from energy principles. By introducing 14 displacement parameters, a generalized eigenvalue problem that has complex eigenvalues and multiple zero eigenvalues is formulated. Polynomial expressions are assumed as trial solutions for displacement parameters and eigenmodes containing undetermined parameters equal to the number of zero eigenvalues are determined by invoking the identity condition to the equilibrium equations. Then the displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to nonzero and zero eigenvalues, respectively. Finally, the stiffness matrices are evaluated by applying the member force-displacement relations to the displacement functions. In addition, the finite beam element formulation based on the classical Lagrangian interpolation polynomial is presented. In order to verify the validity and the accuracy of this study, the numerical solutions are presented and compared with the finite element results using the isoparametric beam elements and the detailed three-dimensional analysis results using the shell elements of ABAQUS. Particularly the effects of shear deformations on the deflection of thin-walled composite beams with the mono-symmetric I-, channel-, and L-shaped sections with various lamination schemes are investigated.     

On the statical behaviour of fibre-reinforced polymer thin-walled beams

The work deals with the formulation of a one-dimensional kinematical model that is able to study the static behaviour of fibre-reinforced polymer thin-walled beams. The proposed model allows us to take into account the effects of shear deformability. Some numerical results obtained via the finite element method are provided and comparisons with the results obtained by Vlasov's classical theory are also presented.     

On sixfold coupled vibrations of thin-walled composite box beams

This paper presents a general analytical model for free vibration of thin-walled composite beams with arbitrary laminate stacking sequences and studies the effects of shear deformation over the natural frequencies. This model is based on the first-order shear-deformable beam theory and accounts for all the structural coupling coming from the material anisotropy. The seven governing differential equations for coupled flexural–torsional–shearing vibration are derived from the Hamilton’s principle. The resulting coupling is referred to as sixfold coupled vibration. Numerical results are obtained to investigate the effects of fiber angle, span-to-height ratio, modulus ratio, and boundary conditions on the natural frequencies as well as corresponding mode shapes of thin-walled composite box beams.     

Shear deformability of thin-walled beams with arbitrary cross sections

Formulas for the computation of the shear deformability of thin-walled prismatic beams can be found in the technical literature only in the special case of symmetric cross sections. In order to fill this gap a formulation of the flexural behaviour of thin-walled beams taking into account transverse shear deflections is developed in the present paper. On this basis, the general expression of the shear centre location and the shear deformability tensor for open and closed sections of arbitrary shape are given and their properties discussed. In the case of polygonal, circular and arc-shaped cross sections explicit formulas, which can be suitably implemented for automatic computations, are provided. For the sake of completeness, the expression of the stiffness tensor for prismatic beams, previously obtained by the first two authors in a co-ordinate-free version, is reported. Finally, a numerical example is carried out and comparisons with the results given by Cowper¹ for symmetric cross sections are presented.     

Vibration analysis of thin-walled composite beams with I-shaped cross-sections

A general analytical model applicable to the vibration analysis of thin-walled composite I-beams with arbitrary lay-ups is developed. Based on the classical lamination theory, this model has been applied to the investigation of load–frequency interaction curves of thin-walled composite beams under various loads. The governing differential equations are derived from the Hamilton’s principle. A finite element model with seven degrees of freedoms per node is developed to solve the problem. Numerical results are obtained for thin-walled composite I-beams under uniformly distributed load, combined axial force and bending loads. The effects of fiber orientation, location of applied load, and types of loads on the natural frequencies and load–frequency interaction curves as well as vibration mode shapes are parametrically studied.     

On a moderate rotation theory of thin-walled composite beams

A small strain and moderate rotation theory of laminated composite thin-walled beams is formulated by generalizing the classical Vlasov theory of sectorial areas. The proposed beam model accounts for axial, bending, torsion and warping deformations and allows one to predict critical loads and initial post-buckling behaviour. A finite element approximation of the theory is also carried out and several numerical applications are developed with reference to lateral buckling of composite thin-walled members. The sensitivity of critical load to second-order effects in the pre-buckling range is pointed out.     

Locking-free finite elements for shear deformable orthotropic thin-walled beams

Numerical models for finite element analyses of assemblages of thin-walled open-section profiles are presented. The assumed kinematical model is based on Timoshenko–Reissner theory so as to take shear strain effects of non-uniform bending and torsion into account. Hence, strain elastic-energy coupling terms arise between bending in the two principal planes and between bending and torsion. The adopted model holds for both isotropic and orthotropic beams. Several displacement interpolation fields are compared with the available numerical examples. In particular, some shape functions are obtained from ‘modified’ Hermitian polynomials that produce a locking-free Timoshenko beam element. Analogously, numerical interpolation for torsional rotation and cross-section warping are proposed resorting to one Hermitian and six Lagrangian formulation. Analyses of beams with mono-symmetric and non-symmetric cross-sections are performed to verify convergence rate and accuracy of the proposed formulations, especially in the presence of coupling terms due to shear deformations, pointing out the decay length of end effects. Profiles made of both isotropic and fibre-reinforced plastic materials are considered. The presented beam models are compared with results given by plate-shell models. Copyright © 2007 John Wiley & Sons, Ltd.