Shear deformation effect in non-linear analysis of composite beams of variable cross section

In this paper the non-linear analysis of a composite Timoshenko beam with arbitrary variable cross section undergoing moderate large deflections under general boundary conditions is presented employing the analog equation method (AEM), a BEM-based method. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson's ratio and are firmly bonded together. The beam is subjected in an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, the axial displacement and to two stress functions and solved using the AEM. Application of the boundary element technique yields a system of non-linear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. Numerical examples are worked out to illustrate the efficiency, the accuracy, the range of applications of the developed method and the influence of the shear deformation effect.     

Nonlinear analysis of beams of variable cross section,including shear deformation effect

In this paper the analog equation method (AEM), a BEM-based method, is employed for the nonlinear analysis of a Timoshenko beam with simply or multiply connected variable cross section undergoing large deflections under general boundary conditions. The beam is subjected in an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, the axial displacement and to two stress functions and solved using the AEM. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The influence of the shear deformation effect is remarkable.     

Nonlinear analysis of shear deformable beam-columns partially supported on tensionless three-parameter foundation

In this paper, a boundary element method is developed for the nonlinear analysis of shear deformable beam-columns of arbitrary doubly symmetric simply or multiply connected constant cross-section, partially supported on tensionless three-parameter foundation, undergoing moderate large deflections under general boundary conditions. The beam-column is subjected to the combined action of arbitrarily distributed or concentrated transverse loading and bending moments in both directions as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to two stress functions and solved using the Analog Equation Method, a BEM-based method. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. The proposed model takes into account the coupling effects of bending and shear deformations along the member as well as the shear forces along the span induced by the applied axial loading. Numerical examples are worked out to illustrate the efficiency, wherever possible, the accuracy and the range of applications of the developed method.     

Nonlinear dynamic analysis of Timoshenko beams by BEM. Part I: Theory and numerical implementation

In this two-part contribution, a boundary element method is developed for the nonlinear dynamic analysis of beams of arbitrary doubly symmetric simply or multiply connected constant cross section, undergoing moderate large displacements and small deformations under general boundary conditions, taking into account the effects of shear deformation and rotary inertia. Part I is devoted to the theoretical developments and their numerical implementation and Part II discusses analytical and numerical results obtained from both analytical or numerical research efforts from the literature and the proposed method. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading and bending moments in both directions as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to two stress functions and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique yields a nonlinear coupled system of equations of motion. The solution of this system is accomplished iteratively by employing the average acceleration method in combination with the modified Newton–Raphson method. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. The proposed model takes into account the coupling effects of bending and shear deformations along the member, as well as the shear forces along the span induced by the applied axial loading.

     

Dynamic analysis of 3-D beam elements including warping and shear deformation effects

In this paper the dynamic analysis of 3-D beam elements restrained at their edges by the most general linear torsional, transverse or longitudinal boundary conditions and subjected in arbitrarily distributed dynamic twisting, bending, transverse or longitudinal loading is presented. For the solution of the problem at hand, a boundary element method is developed for the construction of the 14 × 14 stiffness matrix and the corresponding nodal load vector of a member of an arbitrarily shaped simply or multiply connected cross section, taking into account both warping and shear deformation effects, which together with the respective mass and damping matrices lead to the formulation of the equation of motion. To account for shear deformations, the concept of shear deformation coefficients is used, defining these factors using a strain energy approach. Eight boundary value problems with respect to the variable along the bar angle of twist, to the primary warping function, to a fictitious function, to the beam transverse and longitudinal displacements and to two stress functions are formulated and solved employing a pure BEM approach that is only boundary discretization is used. Both free and forced transverse, longitudinal or torsional vibrations are considered, taking also into account effects of transverse, longitudinal, rotatory, torsional and warping inertia and damping resistance. Numerical examples are presented to illustrate the method and demonstrate its efficiency and accuracy. The influence of the warping effect especially in members of open form cross section is analyzed through examples demonstrating the importance of the inclusion of the warping degrees of freedom in the dynamic analysis of a space frame. Moreover, the discrepancy in the dynamic analysis of a member of a spatial structure arising from the ignorance of the shear deformation effect necessitates the inclusion of this additional effect, especially in thick walled cross section members.     

Solution of De Saint Venant flexure-torsion problem for orthotropic beam via LEM (Line Element-less Method)

In this paper the numerical technique, labelled Line Element-less Method (LEM), is employed to provide approximate solutions of the coupled flexure-torsion De Saint Venant problem for orthotropic beams having simply and multiply-connected cross-section. The analysis is accomplished with a suitable transformation of coordinates which allows to take full advantage of the theory of analytic complex functions as in the isotropic case.A boundary value problem is formulated with respect to a novel complex potential function whose real and imaginary parts are related to the shear stress components, the orthotropic ratio and the Poisson coefficients. This potential function is analytic in all the transformed domain and then expanded in the double-ended Laurent series involving harmonic polynomials.The solution is provided employing an element-free weak form procedure imposing that the squared net flux of the shear stress across the border is minimum with respect to the series coefficients.Numerical implementation of the LEM results in system of linear algebraic equations involving symmetric and positive-definite matrices. All the integrals are transferred into the boundary without requiring any discretization neither in the domain nor in the contour.The technique provides the evaluation of the shear stress field at any interior point as shown by some numerical applications worked out to illustrate the efficiency and the accuracy of the developed method to handle shear stress problems in presence of orthotropic material.     

A new super convergent thin walled composite beam element for analysis of box beam structures

In this paper, a new composite thin wall beam element of arbitrary cross-section with open or closed contour is developed. The formulation incorporates the effect of elastic coupling, restrained warping, transverse shear deformation associated with thin walled composite structures. A first order shear deformation theory is considered with the beam deformation expressed in terms of axial, spanwise and chordwise bending, corresponding shears and twist. The formulated locking free element uses higher order interpolating polynomial obtained by solving static part of the coupled governing differential equations. The formulated element has super convergent properties as it gives the exact elemental stiffness matrix. Static and free vibration analyses are performed for various beam configuration and compared with experimental and numerical results available in current literature. Good correlation is observed in all cases with extremely small system size. The formulated element is used to study the wave propagation behavior in box beams subjected to high frequency loading such as impact. Simultaneous existence of various propagating modes are graphically captured. Here the effect of transverse shear on wave propagation characteristics in axial and transverse directions are investigated for different ply layup sequences.     

Free vibration analyses of axially loaded laminated composite beams based on higher-order shear deformation theory

The dynamic stiffness matrix method is introduced to solve exactly the free vibration and buckling problems of axially loaded laminated composite beams with arbitrary lay-ups. The Poisson effect, axial force, extensional deformation, shear deformation and rotary inertia are included in the mathematical formulation. The exact dynamic stiffness matrix is derived from the analytical solutions of the governing differential equations of the composite beams based on third-order shear deformation beam theory. The application of the present method is illustrated by two numerical examples, in which the effects of axial force and boundary condition on the natural frequencies, mode shapes and buckling loads are examined. Comparison of the current results to the existing solutions in the literature demonstrates the accuracy and effectiveness of the present method.     

Reddy高阶组合梁弯曲的一般解析解法

基于Reddy高阶梁的轴向位移模式,考虑组合梁界面滑移变形,利用最小势能原理建立了Reddy组合梁弯曲问题的控制微分方程和边界条件,,并将控制方程转化为含12个基本未知量的一阶常微分方程组,给出一般求解方法和解表达式。其次,研究了横向均布荷载作用下Reddy简支组合梁的弯曲,所得结果与有限元解吻合良好,说明本文解析解的有效性和可靠性。最后,数值分析了组合梁界面滑移剪切刚度kcs、弹性模量-剪切模量比E/G、梁长-高比L/h和子梁厚度比hs/hc等参数对Reddy简支组合梁弯曲的影响。分析表明：滑移刚度显著影响横截面应力的分布;组合梁长-高比越小、弹性模量-剪切模量比越大或界面滑移刚度越大,组合梁的剪切效应对其挠度影响越显著,此时不宜忽略其剪切变形。     

热载荷作用下梯度多孔材料梁弯曲及过屈曲力学行为的研究

基于Bernoulli-Euler梁理论,引入物理中面解耦了复合材料结构的面内变形与横向弯曲特性,研究了梯度多孔材料矩形截面梁在热载荷作用下的弯曲及过屈曲力学行为．假设沿梁厚度方向材料的性质是连续变化的,利用能量法推导了矩形截面梁的控制微分方程和边界条件,并用打靶法对无量纲化的控制方程进行数值求解．利用计算得到的结果分析了材料的性质、热载荷、边界条件对矩形截面梁非线性力学行为的影响．结果表明,对称材料模型下,固支梁与简支梁均显示出了典型的分支屈曲行为特征,而其临界屈曲热载荷值均会随着孔隙率系数的增加而单调增加．非对称材料模型下,固支梁仍显示出分支屈曲行为特征,但其临界屈曲热载荷不再随着孔隙率系数的变化而单调变化;而对于两端简支梁,发生了弯曲变形,弯曲挠度随载荷的增大而增大．     

Nonlinear dynamic analysis of Timoshenko beams by BEM. Part II: applications and validation

In this two-part contribution, a boundary element method is developed for the nonlinear dynamic analysis of beams of arbitrary doubly symmetric simply or multiply connected constant cross section, undergoing moderate large displacements and small deformations under general boundary conditions, taking into account the effects of shear deformation and rotary inertia. In Part I the governing equations of the aforementioned problem have been derived, leading to the formulation of five boundary value problems with respect to the transverse displacements, to the axial displacement and to two stress functions. These problems are numerically solved using the Analog Equation Method, a BEM based method. In this Part II, numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. Thus, the results obtained from the proposed method are presented as compared with those from both analytical and numerical research efforts from the literature. More specifically, the shear deformation effect in nonlinear free vibration analysis, the influence of geometric nonlinearities in forced vibration analysis, the shear deformation effect in nonlinear forced vibration analysis, the nonlinear dynamic analysis of Timoshenko beams subjected to arbitrary axial and transverse in both directions loading, the free vibration analysis of Timoshenko beams with very flexible boundary conditions and the stability under axial loading (Mathieu problem) are presented and discussed through examples of practical interest.

     

A new trigonometric shear deformation theory for isotropic,laminated composite and sandwich plates

A new trigonometric shear deformation theory for isotropic and composite laminated and sandwich plates, is developed. The new displacement field depends on a parameter “m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature. The results show that the present model performs as good as the Reddy’s and Touratier’s shear deformation theories for analyzing the static behavior of isotropic and composite laminated and sandwich plates.     

Free vibration analysis of third-order shear deformable composite beams using dynamic stiffness method

The dynamic stiffness method is introduced to investigate the free vibration of laminated composite beams based on a third-order shear deformation theory which accounts for parabolic distribution of the transverse shear strain through the thickness of the beam. The exact dynamic stiffness matrix is found directly from the analytical solutions of the basic governing differential equations of motion. The Poisson effect, shear deformation, rotary inertia, in-plane deformation are considered in the analysis. Application of the derived dynamic stiffness matrix to several particular laminated beams is discussed. The influences of Poisson effect, material anisotropy, slenderness and end condition on the natural frequencies of the beams are investigated. The numerical results are compared with the existing solutions in literature whenever possible to demonstrate and validate the present method.     

A continuum approach to the analysis of the stress field in a fiber reinforced composite with a transverse crack

The stress field in a periodically layered composite with an embedded crack oriented in the normal direction to the layering and subjected to a tensile far-field loading is obtained based on the continuum equations of elasticity. This geometry models the 2D problem of fiber reinforced materials with a transverse crack. The analysis is based on the combination of the representative cell method and the higher-order theory. The representative cell method is employed for the construction of Green’s functions for the displacements jumps along the crack line. The problem of the infinite domain is reduced, in conjunction with the discrete Fourier transform, to a finite domain (representative cell) on which the Born–von Karman type boundary conditions are applied. In the framework of the higher-order theory, the transformed elastic field is determined by a second-order expansion of the displacement vector in terms of local coordinates, in conjunction with the equilibrium equations and these boundary conditions. The accuracy of the proposed approach is verified by a comparison with the analytical solution for a crack embedded in a homogeneous plane.Results show the effects of crack lengths, fiber volume fractions, ratios of fiber to matrix Young’s moduli and matrix Poisson’s ratio on the resulting elastic field at various locations of interest. Comparisons with the predictions obtained from the shear lag theory are presented.     

Shear deformation effect in plates stiffened by parallel beams

In this paper a general solution for the analysis of shear deformable stiffened plates subjected to arbitrary loading is presented. According to the proposed model, the arbitrarily placed parallel stiffening beams of arbitrary doubly symmetric cross section are isolated from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. These tractions are integrated with respect to each half of the interface width resulting two interface lines, along which the loading of the beams as well as the additional loading of the plate is defined. Their unknown distribution is established by applying continuity conditions in all directions at the interfaces. The utilization of two interface lines for each beam enables the nonuniform distribution of the interface transverse shear forces and the nonuniform torsional response of the beams to be taken into account. The analysis of both the plate and the beams is accomplished on their deformed shape taking into account second-order effects. The analysis of the plate is based on Reissner’s theory, which may be considered as the standard thick plate theory with which all others are compared, while the analysis of the beams is performed employing the linearized second order theory taking into account shear deformation effect. Six boundary value problems are formulated and solved using the analog equation method (AEM), a BEM based method. The solution of the aforementioned plate and beam problems, which are nonlinearly coupled, is achieved using iterative numerical methods. The adopted model permits the evaluation of the shear forces at the interfaces in both directions, the knowledge of which is very important in the design of prefabricated ribbed plates. The effectiveness, the range of applications of the proposed method and the influence of shear deformation effect are illustrated by working out numerical examples with great practical interest.     

纵横载荷作用下功能梯度梁的弯曲和过屈曲

基于一阶非线性梁理论和物理中面概念,导出了纵横向载荷作用下功能梯度材料(FGM)梁非线性弯曲和过屈曲问题的控制方程,并获得了该问题的精确解;据此解研究了梯度材料性质、外载荷、横向剪切变形以及边界条件等因素对功能梯度材料梁非线性力学行为的影响,分析中假设功能梯度材料性质只沿梁厚度方向,并按成分含量的幂指数函数形式变化。结果表明:纵横载荷共同作用下,功能梯度梁的弯曲构形将有无限多个;随着梯度指数的增大,梁的变形减小,临界载荷升高;随着长高比的增大,横向剪切变形的影响减小。     

采用有限单元法计算梁任意形状截面特性

采用将梁截面离散化的方式,用数值积分计算截面的几何特性,并根据梁剪切变形和扭转理论,利用变分原理建立截面的有限元法方程,求解任意形状截面的扭转常数、剪切中心以及剪切面积修正系数等特性.本方法适用于各种形式的截面,具有计算精度高及适应性强的特点.根据上述理论编制了相应程序,按照不同的单元划分方式,分别计算出矩形截面截面特性,与理论解进行比较;又对舟山市定海长峙至岙山预应力混凝土连续箱梁截面进行了计算,并与Ansys结果进行比较,均证明采用本文的计算方法能得到满意的结果,且该方法适用于各种形状的截面形式.     

建立基于力的变截面梁单元质量矩阵的新方法

基于位移的有限梁单元中三次Hermite插值函数不能有效地描述变截面梁单元内部位移变化,只能通过加密网格增加单元数解决,会造成计算量增大。基于力的有限梁单元由于使用的力插值函数不受截面形状变化的影响,在处理变截面梁时有很大优势,可以得到精确的位移插值函数,利用较少的单元可以达到很高的精度,解决了基于位移的有限梁单元在处理变截面梁时的不足。本文得到了考虑剪切变形的位移插值函数和考虑转动惯量的一致质量矩阵。利用算例验证了本文理论的正确性和高效性。     

压电复合材料层合梁的分岔、混沌动力学与控制

研究了简支压电复合材料层合梁在轴向、横向载荷共同作用下的非线性动力学、分岔和混沌动力学响应. 基于vonKarman理论和Reddy高阶剪切变形理论,推导出了压电复合层合梁的动力学方程. 利用Galerkin法离散偏微分方程,得到两个自由度非线性控制方程,并且利用多尺度法得到了平均方程. 基于平均方程,研究了压电层合梁系统的动态分岔,分析了系统各种参数对倍周期分岔的影响及变化规律. 结果表明,压电复合材料层合梁周期运动的稳定性和混沌运动对外激励的变化非常敏感,通过控制压电激励,可以控制压电复合材料层合梁的振动,保持系统的稳定性,即控制系统产生倍周期分岔解,从而阻止系统通过倍周期分岔进入混沌运动,并给出了控制分岔图.      

Dynamic stability analysis of shear-flexible composite beams

Dynamic stability behavior of the shear-flexible composite beams subjected to the nonconservative force is intensively investigated based on the finite element model using the Hermitian beam elements. For this, a formal engineering approach of the mechanics of the laminated composite beam is presented based on kinematic assumptions consistent with the Timoshenko beam theory, and the shear stiffness of the thin-walled composite beam is explicitly derived from the energy equivalence. An extended Hamilton’s principle is employed to evaluate the mass-, elastic stiffness-, geometric stiffness-, damping-, and load correction stiffness matrices. Evaluation procedures for the critical values of divergence and flutter loads of the nonconservative system with and without damping effects are then briefly introduced. In order to verify the validity and the accuracy of this study, the divergence and flutter loads are presented and compared with the results from other references, and the influence of various parameters on the divergence and flutter behavior of the laminated composite beams is newly addressed: (1) variation of the divergence and flutter loads with or without the effects of shear deformation and rotary inertia with respect to the nonconservativeness parameter and the fiber angle change, (2) influence of the internal and external damping on flutter loads whether to consider the shear deformation or not.