A Displacement Solution to Transverse Shear Loading of Composite Beams by BEM

In this paper the boundary element method is employed to develop a displacement solution for the general transverse shear loading problem of composite beams of arbitrary constant cross section. The composite beam (thin or thick walled) consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli and are firmly bonded together. The analysis of the beam is accomplished with respect to a coordinate system that has its origin at the centroid of the cross section, while its axes are not necessarily the principal bending ones. The transverse shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. The evaluation of the transverse shear stresses at any interior point is accomplished by direct differentiation of a warping function. The shear deformation coefficients are obtained from the solution of two boundary value problems with respect to warping functions appropriately arising from the aforementioned one using only boundary integration, while the coordinates of the shear center are obtained from these functions using again only boundary integration. Three boundary value problems are formulated with respect to corresponding warping functions and solved employing a pure BEM approach. Numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The accuracy of the obtained values of the resultant transverse shear stresses compared with those obtained from an exact solution is remarkable.     

Shear deformation effect in second-order analysis of frames subjected to variable axial loading

In this paper a boundary element method is developed for the second-order analysis of frames consisting of beams of arbitrary simply or multiply connected constant cross section, taking into account shear deformation effect. Each beam is subjected to 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. Three boundary value problems are formulated with respect to the beam deflection, the axial displacement and to a stress function and solved employing a BEM approach. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress function 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 both the shear deformation effect and the variableness of the axial loading are remarkable.     

Shear deformation effect in nonlinear analysis of spatial composite beams subjected to variable axial loading by bem

Summary In this paper a boundary element method is developed for the nonlinear analysis of composite beams of arbitrary doubly symmetric constant cross section, taking into account the shear deformation effect. 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 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. Numerical examples with great practical interest are worked out to illustrate the efficiency and the range of applications of the developed method. The influence of both the shear deformation effect and the variableness of the axial loading are remarkable.     

Flexural - Torsional Nonlinear Analysis of Timoshenko Beam-Column of Arbitrary Cross Section by BEM

In this paper a boundary element method is developed for the nonlinear flexural - torsional analysis of Timoshenko beam-columns of arbitrary simply or multiply connected constant cross section, undergoing moderate large deflections under general boundary conditions. The beam-column is subjected to the combined action of an arbitrarily distributed or concentrated axial and transverse loading as well as to bending and twisting moments. To account for shear deformations, the concept of shear deformation coefficients is used. Seven boundary value problems are formulated with respect to the transverse displacements, to the axial displacement, to the angle of twist (which is assumed to be small), to the primary warping function 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 as well as the angle of twist 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 both the shear deformation effect and the variableness of the axial loading are remarkable.     

Shear deformation effect in flexural–torsional vibrations of beams by BEM

In this paper, a boundary element method is developed for the general flexural–torsional vibration problem of Timoshenko beams of arbitrarily shaped cross section taking into account the effects of warping stiffness, warping and rotary inertia and shear deformation. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Both free and forced vibrations are examined. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy.     

Higher order refined computational model with 12 degrees of freedom for the stress analysis of antisymmetric angle-ply plates – analytical solutions

Analytical formulations and solutions for the stress analysis of simply supported antisymmetric angle-ply composite and sandwich plates hitherto not reported in the literature based on a higher order refined computational model with twelve degrees of freedom already reported in the literature are presented. The theoretical model presented herein incorporates laminate deformations which account for the effects of transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-plane displacements with respect to the thickness coordinate thus modelling the warping of transverse cross sections more accurately and eliminating the need for shear correction coefficients. In addition, two higher order computational models, one with nine and the other with five degrees of freedom already available in the literature are also considered for comparison. The equations of equilibrium are obtained using Principle of Minimum Potential Energy (PMPE). Solutions are obtained in closed form using Navier’s technique by solving the boundary value problem. Accuracy of the theoretical formulations and the solution method is first ascertained by comparing the results with that already available in the literature. After establishing the accuracy of the solutions, numerical results with real properties using all the computational models are presented for the stress analysis of multilayer antisymmetric angle-ply composite and sandwich plates, which will serve as a benchmark for future investigations.     

ㄇ形梁剪力滞效应的解析解

精确分析ㄇ形梁在纵横向荷载共同作用下,其横断面上正应力分布规律对于计算其有效宽度有重要意义,应用力法原理,先将ㄇ形梁和翼板截开成矩形截面梁和平面应力板,在截面上代之以赘余的分布剪力,对于平面应力板,通过利用板变形的对称性来简化其边界条件,然后假设一个满足板的控制方程的Airy应力函数求得板的应力和位移,再利用Timoshenko梁理论求得梁的挠度和转角,根据截面上梁与板的纵向位移相等的变形协调条件便可最终确定截面上的分布剪力,给出的数值算例验证了方法的有效性,并与铜陵长江公路大桥主梁的模型有限元结果和试验结果作了对比,解析解法还可用来检验其他各种数值计算方法的精度,并可推广到其他多跨薄壁结构梁桥的膜应力分析中。     

Advanced 3D beam element of arbitrary composite cross section including generalized warping effects

In this paper, an advanced 20 × 20 stiffness matrix and the corresponding nodal load vector of a member of arbitrary composite cross section is developed taking into account shear lag effects due to both flexure and torsion (the case of the three‐dimensional beam element of arbitrary homogeneous cross section is treated as a special one). The composite member consists of materials in contact each of which can surround a finite number of inclusions. Nonuniform warping distributions are taken into account by employing four independent warping parameters multiplying a shear warping function in each direction and two torsional warping functions. Ten boundary value problems with respect to the kinematical components are formulated and solved employing the analog equation method, a BEM‐based technique. The aforementioned boundary value problems are formulated employing either an improved stress field arising from the correction of shear stress components or the stress field arising directly from displacement considerations. The warping functions and the geometric constants including the additional ones due to warping are evaluated employing a pure BEM approach, that is, only boundary discretization of the cross section is used. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The deviations arising from the use of the advanced 20 × 20 stiffness matrix and the classical 12 × 12 or 14 × 14 ones employed in commercial software packages are illustrated through examples of great practical interest. Moreover, the influence of nonuniform warping effects necessitating the use of the aforementioned ‘advanced’ stiffness matrix is also demonstrated. Copyright © 2015 John Wiley & Sons, Ltd.     

考虑滑移、剪力滞后和剪切变形的钢-混凝土组合梁解析解

通过在Newmark 模型中引入(1)描述横向非均匀分布的纵向位移的翘曲形函数和(2)描述钢梁腹板剪切变形的Timoshenko 梁假定,建立了一个能考虑滑移、剪力滞后和剪切变形的钢-混凝土组合梁模型,并推导了均布荷载作用下的解析解。最后通过4 个算例验证了模型和解析解的正确性和适用性,并显示了考虑组合梁剪切变形的必要性。另外,算例还表明,在组合梁的三维有限元建模中采用Timoshenko 梁单元来考虑钢梁的剪切变形会导致显著的误差。     

Some closed-form solutions in random vibration of Bresse-Timoshenko beams

Random vibration of simply supported uniform Bresse-Timoshenko beams is considered under ‘rain-on-the-roof’ (stationary space- and time -wise ideal white noise) excitation. An approximate differential equation is used with both shear distortion and rotary inertia included, but with the term which covers the simultaneous action of these effects omitted. A closed-form solution is derived for the displacement and velocity space-time correlation function of the Bresse-Timoshenko beam with transverse damping, generalizing the corresponding result by Eringen for the classical Bernoulli-Euler beam. Closed-form solutions are also derived for beams with structural or Voigt damping mechanisms. The mean-square value of the stress diverges for both the classical Bernoulli-Euler and Bresse-Timoshenko beams with transverse damping, but converges for the beam possessing structural damping. The main finding of this study is identity of the space-time correlation functions of displacement according to the refined Bresse-Timoshenko and classical Bernoulli-Euler theories, when the joint action of rotary inertia and shear deformation is neglected for the beam under the ‘rain-on-the-roof’ excitation. This remarkable coincidence takes place for beams possessing (a) transverse viscous damping, (b) Voigt damping, and (c) combined rotary and transverse viscous damping.     

剪切可变形梁热过屈曲解析解

该文导出了面内热载荷作用下,梁过屈曲问题的精确解。首先基于非线性一阶剪切变形梁理论,推导了控制轴向和横向变形的基本方程。然后,将3 个非线性方程化简为一个关于横向挠度的四阶非线性积分-微分方程。该方程与相应的边界条件构成了微分特征值问题。直接求解该问题,得到了热过屈曲构形的闭合解,这个解是外加热载荷的函数。利用精确解,得到了临界屈曲载荷的一阶结果与经典结果的解析关系。为考察热载荷、横向剪切变形以及边界条件的影响,根据得到的精确解给出了两端固定、两端简支以及一端固定一端简支边界条件下的具体数值算例,讨论了梁在面内热载荷作用下的过屈曲行为,并与经典结果进行了比较。该文得到的精确解可以用于验证或改进各类近似理论和数值方法。     

Bending of sandwich plates with anti-symmetric angle-ply face sheets – Analytical evaluation of higher order refined computational models

The aim of the present study is to assess the accuracy of the few computational models based on various shear deformation theories in predicting the bending behaviour of sandwich plates with anti-symmetric angle-ply face sheets under static loading. Five two-dimensional models available in the literature are used for the present evaluation. The performance of the various models is evaluated on a simply supported laminated plate under sinusoidal loading. The equations of equilibrium are derived using the principle of minimum potential energy (PMPE). Analytical solution method using double Fourier series approach is used in conjunction with the admissible boundary conditions. The accuracy of each model is established by comparing the results of composite plates with the exact solutions already available in the literature. After establishing the correctness of the theoretical formulations and the solution method, benchmark results for transverse displacement, in-plane stresses, moment and shear stress resultants are presented for the multilayer sandwich plates.     

Warping shear stresses in nonuniform torsion by BEM

 In this paper a boundary element method is developed for the nonuniform torsion of simply or multiply connected prismatic bars of arbitrary cross section. The bar is subjected to an arbitrarily distributed twisting moment, while its edges are restrained by the most general linear torsional boundary conditions. Since warping is prevented, beside the Saint–Venant torsional shear stresses, the warping normal and shear stresses are also computed. Three boundary value problems with respect to the variable along the beam angle of twist and to the primary and secondary warping functions are formulated and solved employing a BEM approach. Both the warping and the torsion constants using only boundary discretization together with the torsional shear stresses and the warping normal and shear stresses are computed. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The magnitude of the warping shear stresses due to restrained warping is investigated by numerical examples with great practical interest. Received: 13 November 2001 / Accepted: 2 October 2002     

Analysis of plates stiffened by parallel beams

In this paper a general solution for the analysis of plates stiffened by parallel beams subjected to an arbitrary loading is presented. According to the proposed model, the stiffening beams 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. The aforementioned integrated tractions result in the loading of the beams as well as the additional loading of the plate. Their distribution is established by applying continuity conditions in all directions at the interfaces. The analysis of both the plate and the beams is accomplished on their deformed shape taking into account second‐order effects. Six boundary value problems with respect to the plate transverse deflection, to the plate inplane displacement components, to the beam transverse deflections, to the beam axial deformation and to the beam non‐uniform angle of twist are formulated and solved using the analog equation method (AEM), a boundary element method (BEM)‐based method employing a boundary integral equation approach. The solution of the aforementioned plate and beam problems, which are non‐linearly coupled, is achieved using iterative numerical methods. The adopted model describes better the actual response of the plate beams system and 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 evaluated lateral deflections of the plate–beams system are found to exhibit considerable discrepancy from those of other models, which neglect inplane and axial forces and deformations. Copyright © 2006 John Wiley & Sons, Ltd.     

复合材料层板层间缺陷分析——剪切滑移

根据三维弹性平衡方程和层间剪切滑移条件,导出了一个复合材料层板层间剪切滑移模型。本文模型具有一般形式的二维板壳理论的位移场及其平衡方程,但因引入了能反映层板界面粘贴情况及板面条件的剪切变形函数,模型因而简单又精确。层板的弯曲问题和屈曲问题被考虑,层间弱粘贴的影响被讨论。数值结果与精确解比较,表明了本文模型的高精度。      

反对称正交铺层剪切圆柱壳在外压下的屈曲和后屈曲

给出了反对称正交铺层剪切圆柱壳广义大挠度Donnell 型方程, 并运用位移型摄动技术构造出该圆柱壳在均匀外压作用下的后屈曲渐近级数解。考虑到边界效应对中短圆柱壳的影响及边值问题摄动解的一致性, 详细研究了该圆柱壳端部边界层方程和奇异摄动解, 以便与中部正则摄动解相匹配。文中同时给出一些典型例子并讨论了横向剪切变形、Batdo rf 数、弹性模量比和初始几何缺陷对圆柱壳屈曲与后屈曲性态的影响。比较显示, 横向剪切变形对圆柱壳的屈曲与后屈曲有重要影响。      

大范围运动柔性梁动力学分析

针对有偏心、考虑剪切的变截面大范围运动柔性梁，使用有限元方法离散柔性梁，结合非线性变形场得到耦合形函数；在此基础上得到动力学方程，仿真算例验证了有限元离散的有效性和耦合形函数的正确性。     

复合材料梁的几何精确非线性建模及非经典效应

基于Hodges的广义Timoshenko梁理论对具有任意剖面形状、任意材料分布及大变形的复合材料梁进行几何精确非线性建模,采用旋转张量分解法计算梁内任意一点的应变,采用变分渐近法确定梁剖面的任意翘曲,采用平衡方程由二次渐近精确的应变能导出广义Timoshenko应变能,采用广义Hamilton原理建立梁的几何精确非线性运动方程。将所建模型用于复合材料梁的静动力分析,通过与实验数据的对比,验证了建模方法的准确性,并进一步研究了剖面翘曲及横向剪切变形非经典效应对复合材料梁的影响。研究表明,剖面翘曲对复合材料梁的静变形和固有频率有显著影响,横向剪切变形对复合材料梁的静变形和固有频率的影响与梁的长度/剖面高度比有关。     

Stress analysis of the double cantilever beam specimen

A higher-order plate theory which includes transverse shear deformation and a thickness-stretch mode is utilized to analyze a complete double cantilever beam specimen. Homogeneous, orthotropic materials are considered. The beam is divided into a section along the crack and a second section along the uncracked region. Complete continuity of inplane force resultant, transverse shear force resultant, bending moment, and displacements are satisfied across the boundary between the two sections. This analysis allows one to obtain an approximate distribution of the interlaminar normal stress ahead of the crack. The effect of specimen geometry on energy release rate is investigated numerically. Consideration is also given to the average stress criterion as an alternative to a fracture mechanics approach for characterizing interlaminar peel strength.     

Analytical and numerical investigation of the stiffness matrix for edge‐cracked circular shafts

This paper examines two engineering methods of evaluating the stress intensity factors for cracked beams and bars subjected to a combined loading and proposes innovative formulations, as far as the circular cross section is concerned. Based on the definition of the stress intensity factors, the compliance matrix is determined as the inverse of the stiffness matrix, modelling the cracked section of a beam through a line‐spring approximation with interactive forces computed within fracture mechanics. A comparative evaluation of numerical predictions based on the proposed methods is also performed with methods available from the literature. Results for free vibration analyses of beams with transverse non‐propagating open cracks are presented and compared in order to estimate the accuracy and efficiency of the proposed methods, where a good agreement is generally found. More specifically, two different coupling effects are herein analysed for circular beams subjected to a combined bending, axial and shear loading, first, and a combined bending, shear and torsion loading, subsequently.