Nonlinear vibration of laminated plates on an elastic foundation

This paper studies the effects of initial stresses on the nonlinear vibrations of laminated plates on elastic foundations. The nonlinear partial differential equations based on Mindlin plate theory are derived for the nonlinear vibration of laminated plates in a general state of nonuniform initial stress. Both rotary inertia and transverse stress are considered. Using the derived governing equations, the nonlinear vibration behavior of an initially stressed cross-ply plate on a Pasternak or Winkler elastic foundation is studied. The Galerkin's approximate method is applied to the governing partial differential equations to yield ordinary differential equations. The ordinary differential equations are solved by Runge–Kutta method to obtain the ratio of nonlinear frequency to linear frequency. The initial stress is taken to be a combination of a pure bending stress and an extensional stress in the plane of the plate. It is found that the foundation stiffness and initial stresses may result in a drastic change of the nonlinear vibration behavior. The frequency responses of nonlinear vibration are sensitive to the initial stress, vibration amplitude, modulus ratio and foundation stiffness. The effects of various parameters on the nonlinear vibration are discussed.     

On the contact pressure of circular plates on elastic subgrade

A methematical model is presented based on combined methods for determining the contact pressure of circular plates on elastic subgrades. The proposed numerical approach considers two parameters of the subgrade simultaneously, the modulus of elasticity $\text{E}{}_{\mathrm{s}}$ and the modulus of subgrade reaction $\text{K}$. This approach can be utilized in the design of circular foundations such as oil and water towers, silos, chimneys and other structures of similar nature.     

高耸构筑物厚地基圆板的混合变分解

根据应力变分和位移变分原理,导出圆板的混合变分方程,并将其转换成状态方程,使状态空间理论和变分解相结合。将地基反力作为非独立状态变量直接引人到状态方程表达式中,得出任意厚地基圆板的变分解,同时给出了数值结果。     

弹性半空间上圆板的弯曲分析

对各向同性弹性半空间地基上薄圆板一般弯曲问题进行了系统的分析.板的挠度、荷载、地基反力及板下地基表面的沉降均被展开为二重Fourier-Bessel级数,这些级数中的待定系数,由板的边界条件、板的控制方程及板-地基的相容条件加以确定,从而将弹性半空间地基与薄圆板的相互作用问题转化为数值积分和代数方程组的求解问题.数值计算表明,该级数解答具有较快的收敛速度.     

双参数弹性地基上周边固支叠层厚板的解析解

基于三维弹性力学的基本方程，引入Dirac函数，导出了周边固支叠层板的状态方程。采用Cayley-Hamilton定理，得出了双参数地基上叠层厚板在任意荷载作用下的封闭解。此解满足弹性力学全部方程和层间连性条件，适用于任意厚跨比，可求出全部位移的应力分量，计算结果吻合较好。     

A parameter study of the full-range axisymmetric response of uniformly loaded circular plates

Incremental elasto-plastic large deflection equations and their solution via a finite-difference version of the DR-algorithm are outlined as background to a parametric study of the axisymmetric full-range response of uniformly loaded circular plates. Stocky and slender plates with four combinations of flexural and extensional edge conditions are studied. The parametric results for deflections etc. illustrate the limits on the response variation in the full-range regime due to changes in the slenderness as well as the flexural and extensional edge conditions. The results of the study are presented in dimensionless graphical format and, therefore, may be useful in the preliminary design of circular plate structures.     

Solution for large amplitude vibrations of circular plates via modified Berger's approximation

Complex mathematical formulations for the geometrically nonlinear analysis of structural members have been presented by many researchers using the von-Karman strain-displacement relations. Berger proposed a simplifying approximation to take care of the geometric nonlinearity by neglecting the second strain invariant. Although it is a reasonable approximation, it poses problems when applied to circular plates. This is mainly due to the coupling between the radial and circumferential inplane strains, by the radial displacement. In this article, a modification to the Berger's approximation, applicable to the circular plates, is proposed. The main motivation of the present study is to examine how the modified Berger's approximation works in the case of the large amplitude vibrations of circular plates. It is shown that the modified Berger's approximation gives consistently accurate results, for both the simply supported and clamped circular plates, when compared to the classical results.     

弹性地基上自由层合圆板弯曲问题的解析解

从三维弹性力学基本方程出发 ,建立了弹性地基上周边自由的横观各向同性层合圆板轴对称弯曲问题的状态方程 ,并将板面的法向载荷展成傅里叶 -贝塞尔级数 ,从而给出问题的解析解。此解满足弹性力学全部方程 ,计及了所有独立的弹性常数 ,并满足层间连续性条件。     

Effect of transverse shear strains on plates resting on elastic foundation using modified Vlasov model

A four-noded quadrilateral (PBQ4) and an eight-noded quadrilateral (PBQ8) plate bending element based on Mindlin plate theory are adopted for the analysis of thin and thick plates resting on elastic foundation using modified Vlasov model. The terms of vertical deflection stiffness matrix and shear deformation stiffness matrix of the subsoil are evaluated using finite element method, and presented in explicit forms. Selective reduced integration technique is used in addition to full integration technique for both the types of the elements to avoid shear-locking problem that occurs under the thin plate limit. It has been demonstrated that the performance of the eight-noded quadrilateral element is excellent for thin and thick plates on elastic foundation when selective-reduced integration technique is used. General conclusion can be drawn from the results that the effect of the shear strain on the behavior of the plate resting on subsoil is always quite small for free plates compare to the supported ones.     

Solution to the problem of axisymmetric and asymmetric dynamic instability of three-layered annular plates

This paper presents the solution to the dynamic stability problem of three-layered, annular plate loaded by compressive stress increasing in time. The solution enables the evaluation of the critical, dynamic loads corresponding to the various modes of plate buckling. The symmetrical cross-section structure of plate is described by the classical theory of sandwich plate with the broken line hypothesis, the nonlinear Kármán’s plate equations and linear physical relations. The solution is based on Bubnov–Galerkin method and finite difference method. The values of critical dynamic loads have been calculated by means of the stability criterion presented by (Volmir (1972) [1]). The comparison of values of critical dynamic and static loads is presented, using the dynamic ratio. The obtained results have been compared with those obtained for plate model built by means of finite element method. The calculations were carried out using the ABAQS system. The dynamic response of plate models examined by two methods is consistent.     

求解弹性地基上四边自由板的功的互等定理法

1　引　　言 弹性地基上的板是土木工程中较常见的一种结构形式 ,如机场跑道、公路路面等。其分析方法得到了广大理论工作者的重视。但由于问题的复杂性 ,至今还很难得到令人满意的解析解。目前 ,处理这类问题的方法可分为两大类 :数值法和解析法。数值法主要有有限元法和边界法[1,2 ] ,此类方法缺乏通用性。解析法是事先选定位移函数 ,再利用边界条件决定出待定参数[3～ 5] ,但这类方法所得到的解非常复杂 ,不便于工程应用。文献 [6]把功的互等原理用于板的分析中 ,并且证明了功的互等原理与叠加原理的等价性 ,但并没有解决弹性地基上自由板的问题。本文利用功的互等原理推导出了弹性地基上自由板的解析解     

弹性地基板的等参有限元法计算

提出了弹性半空间地基板的四节点和八节点等参元计算方法。该方法适用于弹性地基上不同形状的薄板或厚板的计算。算例表明，本文方法适应性强、精度较高，用较少的单元就能获得较好的结果。     

岩层上非均质弹性地基轴对称问题求解

假设地基土体为弹性,不可压缩,剪切模量随深度线性变化,采用Hankel 变换,得到了下卧为岩层,表面分布一般轴对称荷载的地基土体应力位移积分形式解,应用数值方法,就地基的非均质性、岩层深度对位移的影响进行了讨论。获得的解可以退化到半空间情况。     

弹性地基上的框架计算

重点介绍在力法的基础上所建立的“弯矩分配法”和“角变分配法”的计算方法,以烟台市城市主干道新建工程为例进行了计算,并将计算值同有限元法计算所得值作了比较。