功能梯度矩形板的近似理论与解析解

提出了压电功能梯度矩形板在竖向载荷作用下的近似理论与解析解. 引入了板理论的Kirchhoff假设、Reissner-Mindlin假设和提出的补充假设, 并假设材料常数在板厚方向按指数规律变化. 推导了板在周边简支同时又接地情况下中性层法线转角的解和用Fourier级数表示的电势解. 该解在形式上比精确解简单得多, 进行数值计算时也相当方便与快捷. 计算结果与ANSYS软件用三维实体单元的有限元计算结果进行了比较, 证实了该方法即使在厚板情况下仍然具有很高的精度.      

面内功能梯度矩形板的近似理论与解答

提出了面内功能梯度矩形板在竖向载荷作用下的近似 理论与解析解. 假设材料常数在面内x轴方向按指数规 律变化.引入了板理论的Reissner-Mindlin假设, 并考虑了板中面上的剪切变形的影响.推导了板在平行于y轴的两边简支, 平行于x轴方向的两边简支或固支情况下中性层法线转角和挠度用Fourier级数表示的解.讨论了退化为Kirchhoff假设下经典薄板理论的解的情况.提供了经典薄板理论在和Reissner-Mindlin假设下的算例并与三维有限元的计算结果进行了比较, 说明了该方法在厚板情况下也是相当精确的.     

功能梯度与均匀圆板弯曲解的线性转换关系

基于一阶剪切理论, 研究了功能梯度材料圆板与均匀圆板轴对称弯曲解之间的线性转换关系. 通过理论分析和比较 功能梯度材料圆板和均匀圆板在一阶剪切理论下的位移形式的轴对称弯曲控制方程, 发现了功能梯度材料圆板的转角与均匀圆板的转角之间的相似转换关系. 在假设材料性质沿板厚连续变化的情况下, 给出了相似转换系数的解析表达式. 在此基础上, 进一步导出了一阶剪切理论下功能梯度圆板的挠度与经典理论下, 均匀圆板的挠度之间的线性关系. 从而, 可将功能梯度材料圆板在一阶剪切理论下的弯曲问题求解, 转化为相应均匀薄圆板在经典理论下的弯曲问题求解, 以及转换系数的计算问题. 这一方法为功能梯度非均匀中厚度圆板的求解提供了简捷途径, 而且更便于工程应用. 作为例子, 采用上述方法分别求得了周边简支和夹紧条件下, 梯度圆板在均布横向载荷作用下的弯曲解析解, 该解答与Reddy得到的结果完全吻合.      

功能梯度压电圆板轴对称自由振动问题精确解

将功能梯度压电圆板的位移变量和电势变量写为分离变量的形式，由压电动力学平衡方程导出以位移、电势及其一阶导数为状态变量的状态方程，考虑周边固支接地的边界条件，导出了求解功能梯度压电圆板自振频率精确解的方程。将方程退化至一般的非梯度纯弹性圆板的形式，求解其自振频率，得到的结果与相应的理论解完全吻合，从而验证了本文方法的正确性。更进一步地对梯度函数沿板厚以指数形式变化的功能梯度压电圆板的自振频率进行了计算，并得到了梯度化对板自振频率的影响规律。     

基于平板振动精确化方程求解动应力集中问题

基于文献[8]给出的平板弯曲振动精确化方程,对含圆孔平板中弹性波散射与动应力集中问题进行了研究.文中给出了分别基于Mindlin板与精确化板方程在不同参数下圆孔动弯矩集中系数的数值结果,并对结果进行了对比分析和讨论.结果表明:在较低频率和薄板情况下,基于文献[8]的方程与基于Mindlin板理论得到的动弯矩结果是基本一致的;在较高频率和厚板情况下,基于文献[8]的方程与基于Mindlin板理论的动弯矩结果相差较大,最大值超出可达16%.由于文献[8]给出的平板振动精确化方程是在没有任何工程假设条件下得到的,因此其分析计算结果更精确一些.      

一种非经典的近似的叠层板弯曲理论

本文利用Hellinger-Reissner变分原理建立复合材料叠层板小变形弯曲的一般理论。这个理论放弃了经典板理论的Kirchhoff-Love假设,考虑了横向剪切变形的影响。利用这个理论计算了一些叠层板的弯曲问题,并与其它方法得到的结果比较,证明这种理论能够获得与精确的弹性解颇为满意一致的结果。     

四边固定加劲板的非线性自由振动

针对工程中常用的加劲板, 研究了非线性振动的求解方法与振动特性. 将加劲板分为板与加劲肋两个部分考虑, 其中板视为考虑几何非线性的大挠度板, 加劲肋视为Euler梁. 假定加劲板的位移, 利用Lagrange方程结合系统能量和振型叠加推导了加劲板的动力平衡方程. 运用椭圆函数及摄动法计算加劲板非线性振动的单模态解, 多模态解则通过增量迭代法进行求解. 最后, 结合有限元软件ANSYS对一个四边固定且不可移动的加劲板进行分析, 讨论解的收敛性, 并分析两个方向设置不同数量加劲肋的情况下非线性自振频率与振幅的关系, 得到了一些加劲板非线性振动特性.      

有限圆柱厚壳的一个分析

本文研究了一个弹性有限圆柱厚壳的轴对称弯曲问题.文中对这类壳体给出了一个表达成两个无穷级数之和的三维弹性理论解.这些级数的选择使它们能满足所有给定的边界条件.利用级数的正交性质,其中任一级数中的每项系数都可表达为另一级数的各项系数的线性函数.通过在级数中各取有限个项数进行计算,即可求得问题的近似数值解. 文中处理了一个两端刚性固定的有限圆柱壳体承受内、外均布压力作用的问题作为示范.还给出了计算实例,并将计算结果绘成曲线.这些曲线表明在两端刚性固支的情况下,经典板壳理论即使对薄壳的计算也是难以适用的.此外,这些结果还可作为各种厚壳理论进行比较的一个依据.     

考虑横向拉伸影响的FGM板的静态分析

在三阶剪切变形理论的基础上,添加关于厚度坐标z的幂函数项,并假设板结构的上下表面剪切力为0,提出了一种考虑横向拉伸影响的高阶剪切变形理论。并且研究了简支边界条件下受静态载荷作用的功能梯度材料矩形板的静态弯曲行为。基于虚功原理推导出了功能梯度矩形板的基本方程,利用Navier双三角级数法计算了功能梯度材料矩形板在静态载荷作用下沿厚度方向的位移及应力分布的数值结果。计算结果与三维精确解理论、其他高阶剪切变形理论得到的数值结果进行了比较。对比结果表明,改进的考虑横向拉伸影响的高阶剪切变形理论的正确性和优越性。     

APPROXIMATE THEORY AND ANALYTICAL SOLUTION FOR DYNAMIC CALCULATION OF VISCOELASTIC LAYERED PERIODIC PLATE

由于周期性隔振结构动力计算中较少考虑轨道交通载荷及材料黏弹性,因此,本文以黏弹性层状周期板为研究对象,提出了垂向移动简谐载荷下,可以考虑材料黏弹性及板内横向剪切变形的黏弹性层状周期板动力计算近似理论并给出解析解答.设板中性面的横向剪切变形为横截面的整体剪切变形,利用Reissner-Mindlin假设及提出的剪切变形补充计算条件,得到了中性面法线转角与中性面剪应力的关系.基于平衡方程和应力连续条件,建立了黏弹性层状周期板振动控制方程,推导了对边简支对边自由条件下,板垂向位移的简化Fourier级数形式解.与经典层合板模型和有限元计算结果进行了比较,验证了本文解答的有效性.结果表明：（1）黏弹性层状周期板可以显著降低单一材料板在自振频率处的振动响应,但会引起局部低频频段的振动放大;（2）板的垂向位移随着载荷速度的增大而增大,当载荷速度超过300 km/h后,其对板振动响应的影响减弱;（3）黏弹性层剪切模量存在最佳设计值,可使结构的隔振性能最佳;（4）黏弹性层的阻尼特性在低频范围内对结构振动影响较小;（5）可在满足工程实际的情况下适当增加板长,以提高结构的隔振性能.     

Similarity solution of self-weight consolidation problem for saturated soil

IntroductionA greatdeal of engineeringand environmental projects,such as the tailings reservoir,thehydraulic fill dam,the lake dredging and the estuarine sedimentation,etc.,need to deal withthe problems about the consolidation and sedimentation of recentl…     

Winkler地基上变厚度圆板的轴对称弯曲

本文提出了Winkler地基上变厚度圆板轴对称弯曲的传递矩阵算法。首先,根据贝塞尔函数理论获得了等厚度圆板和环板单元在任意荷载作用下轴对称弯曲的解析解,这些解均由通解和特解两部分组成。基于这些解析解,导出了等厚度圆板和环板单元的传递矩阵。然后沿径向将变厚度圆板划分成一个等厚度圆板单元和一系列等厚度环板单元,应用传递矩阵算法原理获得了变厚度圆板的整体传递矩阵。引入圆板的边界条件,给出了该板每条节线上的挠度、径向转角、径向弯矩和径向剪力。最后,讨论了受均布荷载作用的简支线性变厚度圆板的弯曲,将本文数值解与解析解进行比较,证实了本文方法的有效性,并简要地讨论了地基参数对板挠度和径向弯矩的影响。     

A new exact integration method for the Drucker–Prager elastoplastic model with linear isotropic hardening

This paper presents the exact stress solution of the non-associative Drucker–Prager elastoplastic model governed by linear isotropic hardening rule. The stress integration is performed under constant strain-rate assumption and the derived formulas are valid in the setting of small strain elastoplasticity theory. Based on the time-continuous stress solution, a complete discretized stress updating algorithm is also presented providing the solutions for the special cases when the initial stress state is located in the apex and when the increment produces a stress path through the apex. Explicit expression for the algorithmically consistent tangent tensor is also derived. In addition, a fully analytical strain solution is also derived for the stress-driven case using constant stress-rate assumption. In order to get a deeper understanding of the features of these solutions, two numerical test examples are also presented.     

Analytical solution of rectangular plate with in-plane variable stiffness

The bending problem of a thin rectangular plate with in-plane variable stiffness is studied. The basic equation is formulated for the two-opposite-edge simply supported rectangular plate under the distributed loads. The formulation is based on the assumption that the flexural rigidity of the plate varies in the plane following a power form, and Poisson’s ratio is constant. A fourth-order partial differential equation with variable coefficients is derived by assuming a Levy-type form for the transverse displacement. The governing equation can be transformed into a Whittaker equation, and an analytical solution is obtained for a thin rectangular plate subjected to the distributed loads. The validity of the present solution is shown by comparing the present results with those of the classical solution. The influence of in-plane variable stiffness on the deflection and bending moment is studied by numerical examples. The analytical solution presented here is useful in the design of rectangular plates with in-plane variable stiffness.     

横观各向同性弹性半空间地基上四边自由各向异性矩形薄板弯曲解析解

选用更具广泛性的横观各向同性弹性半空间地基模型,来分析四边自由各向异性矩形地基板的弯曲解析解．将异性薄板的弯曲控制方程,与基于横观各向同性弹性半空间地基位移解建立的板与地基变形协调方程相结合,先按对称性分解,然后用三角级数法,得出横观各向同性弹性半空间地基上四边自由各向异性矩形薄板的弯曲解析解,包括地基反力、板的挠度及内力的解析表达式．该解析解克服了数值法的弊端,取消了对地基反力的假设,板的内力及地基反力求解更切实际．算例结果与文献结果吻合良好,证明本文方法的可行性．     

ANALYTICAL RELATIONS BETWEEN EIGENVALUES OF CIRCULAR PLATE BASED ON VARIOUS PLATE THEORIES

Based on the mathematical similarity of the axisymmetric eigenvalue problems of a circular plate between the classical plate theory(CPT), the first-order shear deformation plate theory(FPT) and the Reddy's third-order shear deformation plate theory (RPT), analytical relations between the eigenvalues of circular plate based on various plate theories are investigated. In the present paper, the eigenvalue problem is transformed to solve an algebra equation. Analytical relationships that are expressed explicitly between various theories are presented. Therefore, from these relationships one can easily obtain the exact RPT and FPT solutions of critical buckling load and natural frequency for a circular plate with CPT solutions. The relationships are useful for engineering application, and can be used to check the validity, convergence and accuracy of numerical results for the eigenvalue problem of plates.     

Fibre-reinforced composites with fibre-bending stiffness under azimuthal shear – Comparison of simulation results with analytical solutions

In Ref. [1], Spencer and Soldatos proposed an enhanced modelling approach for fibre-reinforced composites which accounts for the fibre-bending stiffness in addition to the directional dependency induced by the fibres. Although analytical solutions for simple geometries have been derived over the past years, often subject to specific assumptions such as small deformation kinematics, the application to more general and non-academic boundary value problems is desirable. Motivated by the latter, the numerical solution of the general system of partial differential equations by means of a multi-field finite element approach is proposed in Ref. [2] and the principal model properties are studied for a specific form of the elastic energy potential. In the present contribution a comparison of the numerical solution by means of the multi-field finite element approach against the analytical solution is presented for the azimuthal shear deformation of a tube-like structure. To this end, the general deformation pattern and especially the distribution of the stress and couple stress tensor are taken into account. We find that, although the analytical solution is derived subject to the assumption of small deformations, whereas the numerical solution is based on the finite strain counterpart of the theory, the simulation results are quasi identical, which verifies the numerical framework proposed.     

Exact three-dimensional analysis for static torsion of piezoelectric rods

Based on the theory of elasticity, exact analytical and numerical solutions of piezoelectric rods under static torsion are studied. In this paper, direct solution method is used. The main scope is to check the extension of validity of assumptions in previous papers that had been made based on linear distribution of electric potential through the cross section and their influences on deflection and the angle of rotation. Stress and electric induction functions are employed to obtain the exact solution of the static and electrostatic equilibrium equations under torsional loading. It is shown that previous assumptions are valid only in some types of piezoelectric materials, while in other types these assumptions lead to considerable deviations from accurate modeling. The present analytical solutions are compared with three-dimensional finite element analysis results and absolute agreements are found. At the end of this article, torsional rigidity, shape-effects on induced piezoelectric deformation and the range of valid region for linear distribution of electric potential assumption have been studied.     

Note on the Unsteady Mixed Boundary Layer in a Porous Medium with Temperature Slip: Exact Solutions

In this note, the mixed unsteady stagnation-point boundary layer over a vertical plate with mass transfer in a fluid-saturated porous medium is revisited. Closed-form analytical solutions are found and presented for a special value of the flow unsteadiness parameter. Multiple solution branches are obtained for certain controlling parameters. These solutions might offer more insights into the mixed convection flow characteristics compared with the numerical solutions.