Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的辛叠加解

研究Winkler地基上正交各向异性矩形薄板弯曲方程所对应的Hamilton正则方程, 计算出其对边滑支条件下相应Hamilton算子的本征值和本征函数系, 证明该本征函数系的辛正交性以及在Cauchy主值意义下的完备性, 进而给出对边滑支边界条件下Hamilton正则方程的通解, 之后利用辛叠加方法求出Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的解析解. 最后通过两个具体算例验证了所得解析解的正确性.     

基于遗传算法的弹性地基加肋板肋梁无网格优化分析

基于遗传算法及一阶剪切理论, 提出一种弹性地基上加肋板肋条位置优化的无网格方法. 首先, 通过一系列点来离散平板及肋条, 并用弹簧模拟弹性地基, 从而得到加肋板的无网格模型; 其次, 基于一阶剪切理论及移动最小二乘近似原理导出位移场, 求出弹性地基加肋板总势能; 再次, 根据哈密顿原理导出结构的弯曲控制方程, 并通过完全转换法处理边界条件; 最后, 引入遗传算法和改进遗传算法, 以肋条的位置为设计变量、弹性地基板的中心点挠度最小值为目标函数, 对肋条位置进行优化达到地基板控制点挠度最小的目的. 以不同参数、载荷布置形式的弹性地基加肋板为例, 与ABAQUS有限元结果及文献解进行比较. 研究表明, 采用所提出的无网格模型可有效求解弹性地基上加肋板弯曲问题, 结果易收敛, 同时基于遗传算法与改进混合遗传算法所提出的无网格优化方法均可有效优化弹性地基加肋板肋条位置, 后者计算效率相对较高, 只进行了三次迭代便可获得稳定的最优解, 此外在优化过程中肋条位置改变时只需要重新计算位移转换矩阵, 又避免了网格重构.      

Analytical solution for bending stress intensity factor in an orthotropic elastic plate containing a crack and subjected to concentrated moments

The problem of estimating the bending stress distribution in the neighborhood of a crack located on a single line in an orthotropic elastic plate of constant thickness subjected to out-of-plane concentrated moments is examined. Using classical plate theory and integral transform techniques, the general formulae for the bending moment and twisting moment in an elastic plate containing cracks located on a single line are derived. The solution is obtained in a closed form for the case in which there is a single crack in an infinite plate subjected to symmetric concentrated moments.     

A non-classical Kirchhoff plate model incorporating microstructure,surface energy and foundation effects

A new non-classical Kirchhoff plate model is developed using a modified couple stress theory, a surface elasticity theory and a two-parameter elastic foundation model. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and the complete boundary conditions and provides a unified treatment of the microstructure, surface energy and foundation effects. The new plate model contains a material length scale parameter to account for the microstructure effect, three surface elastic constants to describe the surface energy effect, and two foundation moduli to represent the foundation effect. The current non-classical plate model reduces to its classical elasticity-based counterpart when the microstructure, surface energy and foundation effects are all suppressed. In addition, the newly developed plate model includes the models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases and recovers the Bernoulli–Euler beam model incorporating the microstructure, surface energy and foundation effects. To illustrate the new model, the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulas derived. For the static bending problem, the numerical results reveal that the deflection of the simply supported plate with or without the elastic foundation predicted by the current model is smaller than that predicted by the classical model. Also, it is observed that the difference in the deflection predicted by the new and classical plate models is very large when the plate thickness is sufficiently small, but it is diminishing with the increase of the plate thickness. For the free vibration problem, it is found that the natural frequency predicted by the new plate model with or without the elastic foundation is higher than that predicted by the classical plate model, and the difference is significant for very thin plates. These predicted trends of the size effect at the micron scale agree with those observed experimentally. In addition, it is shown both analytically and numerically that the presence of the elastic foundation reduces the plate deflection and increases the plate natural frequency, as expected.     

Resonant effects of local loads on circular sandwich plates on an elastic foundation

The axisymmetric forced vibrations of a circular sandwich plate on an elastic foundation are studied. The plate is subjected to axisymmetric surface and mechanical loads with frequency equal to one of the natural frequencies of the plate. The foundation reaction is described by the Winkler model. To describe the kinematics of an asymmetric sandwich, the hypothesis of broken normal is used. The core is assumed to be light. The analytical solution of the problem is obtained and numerical results are analyzed     

On the Vibrations of a Simply Supported Square Plate on a Weakly Nonlinear Elastic Foundation

In this paper an initial-boundary value problem for a weakly nonlinear plate equation with a quadratic nonlinearity will be studied. This initial-boundary value problem can be regarded as a simple model describing free oscillations of a simply supported square plate on an elastic foundation. It is assumed that the foundation has a different behavior for compression and for expansion. An approximation for the solution of the initial-boundary value problem will be constructed using a two-timescales perturbation method. The existence and uniqueness of the solution of the problem will be proved. Also the asymptotic validity of the constructed approximations will be shown on long timescales. For specific parameter values, it turns out that complicated internal resonances occur.     

CC series solution for bending of rectangular plates on elastic foundations

The analytical solution for the bending problem of the rectangular plates on an elastic foundation is investigated by using the Stockes' transformation of a double variables function. The numerical results for the rectangular plates with free edges on the elastic foundations under a concentrated force are given in the example. First Received Dec. 14 1993     

双参数弹性地基板计算新方法

弹性地基板的弯曲问题,尤其是自由边板,一直是学者和工程师们所十分关切的问题。本文用无单元法研究双参数弹性地基板的弯曲问题,由最小二乘法和变分原理导出了双参数弹性地基板的无单元法刚度短阵,编制相应的无单元法计算程序,并给出计算实例。结果表明本方法精度良好,可求出任意荷载作用下板中任一点的挠度、转角、弯矩和扭矩,且有广泛的工程应用前景。     

弹性地基板的无奇异边界元解法

本文用无奇异边界单元法,分析弹性地基上薄板的弯曲问题,考虑Winkler和双参数地基模型,选择第三类复变量的Bessel函数作为该问题的基本解,由此导出了一组权函数,在数值解法中,对面荷载积分项均统一化为边界积分,避免了在域内划分网格,文中还给出了一种考虑桩支承的边界元分析方法及域内点弯矩和接触应力的计算公式。     

A free rectangular plate on the two-parameter elastic foundation

This paper provides a rigorous solution of a free rectangular plate on the V.Z. Vlazov two-parameter elastic foundation by the method of superposition[1]. In this paper we derive basic solutions under the various boundary conditions. To superpose these basic solutions the most generally rigorous solution of a free rectangular plate on the two-parameter elastic foundation can be obtained. The solution strictly satisfies the differential equation of a plate on the two-parameter elastic model foundation, the boundary conditions of the free edges and the free corner conditions. Some numerical examples are presented The calculated results show that when the plane dimension of plate is given and the ratio between the laver depth and the plate thick is equal to 15, the two-parameter elastic model is near the Winkler’s. It shows that the Winkler model can be applied to the thinner layer.     

Equilibrium of a Prestressed Strip Reinforced with Elastic Plates

The contact problem for a prestressed elastic strip reinforced with equally spaced elastic plates is considered. The Fourier integral transform is used to construct an influence function of a unit concentrated force acting on the infinite elastic strip with one edge constrained. The transmission of forces from the thin elastic plates to the prestressed strip is analyzed. On the assumption that the beam bending model and the uniaxial stress model are valid for an elastic plate subjected to both vertical and horizontal forces, the problem is mathematically formulated as a system of integro-differential equations for unknown contact stresses. This system is reduced to an infinite system of algebraic equations solved by the reduction method. The effect of the initial stresses on the distribution of contact forces in the strip under tension and compression is studied     

Dynamics of a prestressed incompressible layered half-space under moving load

The paper addresses a plane problem: a concentrated force acts on a plate resting on an elastic half-space with homogeneous prestrain. The equations of motion of the plate incorporate shear and rotary inertia. The half-space is assumed to be incompressible and isotropic in the natural state. The elastic potential is given in general form and is only specified for numerical purposes. The dependence of the critical velocity of the load and the stress-strain state on the prestresses is analyzed for different ratios between the stiffnesses of the layer and half-space and different contact conditions. The calculations are carried out for a half-space with Bartenev-Khazanovich potential __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 36–54, March 2008.     

Point loading on hyperbolic paraboloidal shells

A combination of simple theory and experiment is used to investigated a complex structural problem. The problem is the point loading of hyperbolic paraboloidal shells. The solution is based on an analogy between the response of these shells to a point load and the localized behavior of a flat plate that rests on an elastic foundation. The subgrade modulus of this analogous foundation is derived from the measured deflection of a point load on shell models. The value of the modulus was found to be about one-fifth that of a spherical shell with the same magnitude of principal curvature.     

Tensionless contact of a finite circular plate

A general formulation is developed for the contact behavior of a finite circular plate with a tensionless elastic foundation. The gap distance between the plate and elastic foundation is incorporated as an important parameter. Unlike the previous models with zero gap distance and large/infinite plate radius, which assumes the lift-off/separation of a flexural plate from its supporting elastic foundation, this study shows that lift-off may not occur. The results show how the contact area varies with the plate radius, boundary conditions and gap distance. When the plate radius becomes large enough and the gap distance is reduced to zero, the converged contact radius close to the previous ones is obtained.     

中厚板弯曲问题的自然单元法

自然单元法是一种新兴的无网格数值计算方法,基于Reissner-Mindlin板弯曲理论,将自然单元法应用于平板弯曲问题的计算中,给出了相关的公式,推导了总体刚度矩阵和荷载列阵的计算列式.算例分析表明,自然单元法应用于中厚板的弯曲问题具有较高的计算精度,并可用于Winkler地基上基础板的计算.同时指出,对于厚跨比较小的薄板,由于对挠度和中面法线转角采用相同的插值形式,当板厚变薄时夸大了虚假的剪切变形影响,因而表现出剪切自锁现象.对进一步开发厚薄板通用的计算程序作了初步探讨.     

Large deplection of corrugated circular plate with a plane central region under the action of concentrated loads at the center

This paper solves the problem of large deflections for corrugated circular plates with a plane central region under the action of concentrated loads at the center by means of the non-linear bending theory for anisotropic circular plates. Using the modified iteration method, the characteristic relation of the plate is obtained. This formula may be applied directly to design of elastic elements of measuring instruments.     

层状横观各向同性地基上异性矩形薄板的弯曲解析解

选用更具广泛性的层状横观各向同性弹性地基模型,来分析四边自由各向异性矩形地基板的弯曲解析解。先基于直角坐标下横观各向同性体的静力胡海昌通解,借助双重傅里叶变换及矩阵传递法,获得层状横观各向同性地基的静力位移场和应力场;然后将异性薄板的弯曲控制方程,与基于层状横观各向同性弹性地基的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,再用三角级数法,得出层状横观各向同性弹性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、板的挠度及板的内力的解析表达式。克服了数值法的弊端,取消了对地基反力的假设,且避免了矩阵指数函数的计算;同时考虑了地基的层状性及板和地基的各向异性,从而得到板的内力及地基反力更切实际的分布规律。算例结果与文献的有限元结果吻合良好,证明本文方法是切实可行的。     

椭圆孔口外受集中力和力偶时的新解法

文中给出了无限平板带椭圆孔又受集中力和力偶时的弹性分析。所讨论问题归结为复变函数的一个特殊问题，即令一个复变函数（单位圆外有奇点）的值和另一个复变函数（单位圆外为解析）的共轭值沿着单位圆相等。最后，得出了闭合形式的解。     

Contact problem for a ring plate on an elastic half-space. Variational approach

A contact problem of an axisymmetrically loaded flexible ring plate lying frictionlessly on an elastic half-space is considered. The plate subsidences are represented as a power series with unknown coefficients, which are determined by the Rayleigh-Ritz method using the minimum condition for the total strain energy of the plate and the elastic foundation. The method of orthogonal polynomials is used implicitly. Belarussian State Polytechnical Academy, Minsk 220027. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 193–198, January–February, 1999.