RECIPROCAL THEOREM METHOD FOR THE FORCED VIBRATIONOF ELASTIC THICK RECTANGULAR PLATES

In this paper, reciprocal theorem method (RTM) is generalized to solve the problems for the forced vibration of thick rectangular plates based on the Reissner's theory. The paper derives the dynamic basic solution of thick rectangular; and the exact analytical solution of the steady-state responses of thick rectangular plates with three clamped edges and one free edge under harmonic uniformly distributed disturbing forces is found by RTM. It is regarded as a simple, convenient and general method for calculating the steady-stare responses of forced vibration of thick rectangular plates.

THE DERIVATION OF EXACT STATIC CONDITIONS AT THE CORNER POINTS FOR THE BENDING OF THICK RECTANGULAR PLATES

In this paper, exact static conditions at the corner points for the bending of thickrectangular ptates are strictly. derived from the theorem of minimum potentialenerg[1].

THE BENDING OF A THICK RECTANGULAR PLATE WITH THREE CLAMPED EDGES AND ONE FREE EDGE

The exact solution of the bending of a thick rectangular plate with three clamped edges and one free edge under a uniform transverse load is obtained by means of the concept of generalized simply-supported boundary[1] in Reissner's theory of thick plates. The effect of the thickness h of a plate on the bending is studied and the applicable range of Kirchhoffs theory for bending of thin plates is considered.

Effect of Non-Homogeneity on Thermally Induced Vibration of Orthotropic Visco-Elastic Rectangular Plate of Linearly Varying Thickness

The analysis presented here is to study the effect of non-homogeneity on thermally induced vibration of orthotropic visco-elastic rectangular plate of linearly varying thickness. Thermal vibrational behavior of non-homogeneous rectangular plates of variable thickness having clamped boundary conditions on all the four edges is studied. For non–homogeneity of the plate material, density is assumed to vary linearly in one direction. Using the method of separation of variables, the governing differential equation is solved. An approximate but quite convenient frequency equation is derived by using Rayleigh-Ritz technique with a two-term deflection function. Time period and deflection at different points for the first two modes of vibration are calculated for various values of temperature gradients, non- homogeneity constant, taper constant and aspect ratio. Comparison studies have been carried out with non-homogeneous visco-elastic rectangular plate to establish the accuracy and versatility.

NONLINEAR VIBRATION FOR MODERATE THICKNESS RECTANGULAR CRACKED PLATES INCLUDING COUPLED EFFECT OF ELASTIC FOUNDATION

Based on Reissner plate theory and Hamilton variational principle, the nonlinear equations of motion were derived for the moderate thickness rectangular plates with transverse surface penetrating crack on the two-parameter foundation. Under the condition of free boundary, a set of trial functions satisfying all boundary conditions and crack＇s continuous conditions were proposed. By employing the Galerkin method and the harmonic balance method, the nonlinear vibration equations were solved and the nonlinear vibration behaviors of the plate were analyzed. In numerical computation, the effects of the different location and depth of crack, the different structural parameters of plates and the different physical parameters of foundation on the nonlinear amplitude frequency response curves of the plate were discussed.

Thermal Effect on Free Vibration of Non-Homogeneous Orthotropic Visco-Elastic Rectangular Plate of Parabolically Varying Thickness

A simple model presented here is to study the thermal effect on vibration of non-homogeneous orthotropic visco-elastic rectangular plate of parabolically varying thickness having clamped boundary conditions on all the four edges. For non-homogeneity of the plate material, density is assumed to vary linearly in one direction. Using the separation of variables method, the governing differential equation has been solved for vibration of non-homogeneous orthotropic viscoelastic rectangular plate. An approximate frequency equation is derived by using Rayleigh-Ritz technique with a two-term deflection function. Results are calculated for time period and deflection at different points, for the first two modes of vibration, for various values of temperature gradients, non-homogeneity constant, taper constant and aspect ratio and shown by graphs.

BASIC EQUATIONS OF THE PROBLEM OF THE NONLINEAR UNSYMMETRICAL BENDING FOR ORTHOTROPIC RECTANGULAR THIN PLATE WITH VARIABLE THICKNESS

Under the case of ignoring the body forces and considering components caused by diversion of membrane in vertical direction (z-direction),the constitutive equations of the problem of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate with variable thickness are given;then introducing the dimensionless variables and three small parameters,the dimensionaless governing equations of the deflection function and stress function are given.

UNIFORMLY VALID ASYMPTOTIC SOLUTIONS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR ORTHOTROPIC RECTANGULAR THIN PLATE OF FOUR CLAMPED EDGES WITH VARIABLE THICKNESS

By using “the method of modified two-variable”,“the method of mixing perturbation” and introducing four small parameters,the problem of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate with linear variable thickness is studied.And the uniformly valid asymptotic solution of Nth-order for ε_1 and Mth-order for ε_2 of the deflection functions and stress function are obtained.

Winkler地基上四边自由正交各向异性矩形中厚板弯曲的有限积分变换解

本文应用有限积分变换法研究Winkler地基上四边自由正交各向异性矩形中厚板的弯曲问题.具体由正交各向异性矩形中厚板弯曲的基本方程组和边界条件出发,结合有限积分变换法及其对应的逆变换法推导出正交各向异性矩形中厚板弯曲问题的解析解.该解析解统一适用于计算各向同性/正交各向异性矩形薄板、中厚板和厚板的弯曲问题,并且通过具体算例验证了所得解析解的正确性.

矩形中厚板屈曲问题Hamilton算子本征函数系的完备性

研究各向同性矩形中厚板的屈曲问题。首先将各向同性矩形中厚板的控制方程组转化为Hamilton系统,然后应用Hamilton体系的分离变量方法得到对边简支条件下对应的Hamilton算子的本征值及本征函数系,并通过符号运算证明了该本征函数系的辛正交性和Cauchy主值意义下的完备性,进而得到各向同性对边简支矩形中厚板屈曲问题的通解。最后通过具体算例,结合通解与另外两侧边的边界条件,得到了四边简支矩形中厚板屈曲问题的屈曲荷载因子。

求解厚矩形板弯曲问题的功的互等定理法

在本文中，功的互等定理法（ＲＴＭ）被推广于求解基于Ｒｅｉｓｓｎｅｒ理论的厚矩形板弯曲问题。首先，本文绘出了厚矩形板弯曲的基本解；其次，给出了三边固定一边自由在均布载荷作用下厚矩形板弯曲的精确解析解；最后，我们分析了本文解的数值结果。

弹性地基上厚矩形板弯曲问题的边界积分法

提出求解弹性地基上厚矩形板弯曲问题的边界积分法，并应用该法首次给出了在均布载荷作用下四边简支弹性基厚矩形板弯曲问题的双曲函数和三角函数混合表示的解的表达式．与双三角级数解相比较，该解具有收敛快的优点．同时用付里叶级数展开法求解了同一问题，所得结果与边界积分法的结果相同，证明了由边界积分法所得结果的正确性．

矩形中厚板自由振动问题的哈密顿体系与辛几何解法

以矩形中厚板的胡海昌方程为基础,将中厚板自由振动问题导入哈密顿体系,然后利用辛几何中的分离变量和本征函数展开的方法求出了对边简支板自由振动的精确解.文中采用的辛方法不必事先人为地引入试函数,而是通过完全理性的推导,从而突破了传统半逆解法的限制,使得问题的求解更加合理,易于推广.计算实例证明了本文推导结果的正确性.

弹性地基上四边自由厚矩形板的弯曲问题解

Reissner厚板理论基础上,利用功的互等定理法和迭加法求解了集中载荷作用下,弹性地基上四边自由厚矩形板的弯曲问题,得到了完全一致的解析解.可见,功的互等定理法更简便易行.

变厚度矩形板自由振动的广义微分求积法分析

基于广义微分求积法,对变厚度矩形板横向自由振动的控制微分方程及其不同边界条件进行离散,研究其自由振动的频率特性.数值计算得到不同长宽比,不同厚度变化参数和简支或固定边界条件下变厚度矩形板的无量纲振动基频率,并与其它求解方法的数值进行比较.结果表明,运用广义微分求积法对变厚度矩形板的频率求解结果与其它方法的求解结果相差很小,且在条件相同的情况下用广义微分求积法仅需较少的节点就能达到满意的求解精度,从而为求解此类问题提供一种可供选择的简便有效的数值方法.

功的互等定理在四边固定厚矩形板弯曲问题中的应用

以Reissuer理论为基础，利用功的互等定理得到了四边固定厚矩形板在均布载荷作用下弯曲问题的解析解．从所得结果来看，该法简单实用，精度较高．

弹性厚矩形板受迫振动的功的互等定理法

本文将功的互等定理法（ＲＴＭ）推广应用于求解基于Ｒｅｉｓｓｎｅｒ理论的厚矩形板受迫振动问题·本文导出了厚矩形板动力基本解；给出了三边固定一边自由厚矩形板在均布简谐干挠力作用下稳态响应的精确解析解·这是计算厚矩形板受振动稳态响应的一个简便通用的方法·

中厚矩形板的非线性动力稳定性分析

本文以Ｔｉｍｏｓｈｅｎｋｏ－Ｍｉｎｄｌｉｎ假设及Ｈａｍｉｌｔｏｎ原理为基础，建立了中厚板的非线性运动控制方程。应用参数增量法求解了四边简支矩形板在纵横谐载共同作用下的非线性动力稳定性问题，并对影响其动力稳定性的若干因素进行了讨论。

任意边界条件下变厚度中厚矩形板的静动态特性分析

文章基于改进傅立叶级数方法,在Mindlin一阶剪切变形板理论的基础上,对任意边界条件下变厚度中厚板矩形板的静动态特性进行了研究。矩形板的横向位移函数与旋转位移函数被表示为包含正弦三角级数的改进傅立叶级数,正弦三角级数的引入,能够有效地解决在边界处存在的不连续或者跳跃现象。在此基础上,将位移容许函数的未知傅立叶展开系数看作广义变量,采用能量原理建立结构的能量泛函,结合Rayleigh-Ritz法对未知傅立叶展开系数求极值,将结构的静动态特性问题转换为一个求解标准特征值问题。通过大量的数值算例,并与现有文献解及有限元方法计算结果进行对比,验证了文中方法的合理性,并且具有良好的收敛速度与计算精度。

桩筏结构复合地基中筏板受力分析的理论计算模型与试验研究

由于缺乏刚性桩复合地基之上筏板受力的合理计算公式,目前中国高速铁路桩筏结构复合地基中筏板的设计计算仍采用简化算法,其计算结果的准确合理性尚不满足要求,为此拟对桩筏基础之上的筏板进行理论求解。选取桩筏结构中单桩作用的筏板区域作为求解单元,将该单元结构视为位于单桩和Winkler弹性地基共同支撑的四边滑支Reissner矩形中厚板,将Reissner矩形中厚板的基本方程导入Hamilton体系,利用辛几何中的分离变量及本征函数展开等方法,求出矩形中厚板弯曲问题的解析解,进而可求得复合地基上筏板的内力、挠度、桩顶反力和桩间土压力等设计值。最后将现场试验实测值与理论模型解析解的计算结果进行对比,结果采用文中理论模型计算得到的筏板弯矩、桩顶轴力及桩间土压应力等与现场实测结果吻合较好,表明在桩筏结构复合地基的筏板设计中,提出的理论模型是一种准确可靠的计算方法。