Large deflection problem of a clamped elliptical plate subjected to uniform pressure

In this paper, the perturbation solution of large deflection problem of clamped elliptical plate subjected to uniform pressure is given on the basis of the perturbation solution of large deflection problem of similar clamped circular plate (1948), (1954). The analytical solution of this problem was obtained in 1957. However, due to social difficulties, these results have never been published. Nash and Cooley (1959) published a brief note of similar nature, in which only the case λ=a/b=2 is given. In this paper, the analytical solution is given in detail up to the 2nd approximation. The numerical solutions are given for various Poisson ratios v=0.25, 0.30, 0.35 and for various eccentricities λ=1, 2, 3, 4, 5, which can be used in the calculation of engineering designs.     

Dynamic response of plates due to moving vehicles using finite strip method

ＩｎｔｒｏｄｕｃｔｉｏｎＤｙｎａｍｉｃｒｅｓｐｏｎｓｅｏｆｂｅａｍ＿ｌｉｋｅｓｔｒｕｃｔｕｒｅｓｔｏｍｏｖｉｎｇｖｅｈｉｃｌｅｓｈａｓｂｅｅｎｅｘｔｅｎｓｉｖｅｌｙｓｔｕｄｉｅｄ .Ｈｏｗｅｖｅｒ,ｔｈｅｓｔｕｄｙｏｎｄｙｎａｍｉｃｒｅｓｐｏｎｓｅｏｆｐｌａｔｅｓｔｏｍｏｖｉｎｇｖｅｈｉｃｌｅｓｈａｓｓｏｆａｒｒｅｃｅｉｖｅｄｂｕｔｓｃａｎｔａｔｔｅｎｔｉｏｎ .Ｗｕｅｔａｌ.[1],ＴａｈｅｒｉａｎｄＴｉｎｇ[2 ],ＹｅｎｅｒａｎｄＣｈｏｍｐｏｏｍｉｎｇ[3],ＨｕｍａｒａｎｄＫａｓｈｉｆ[4 ]ｅｔａｌ.ｓｅｐａ…     

基于有限条带的厚/薄板矩形通用单元

基于两广义位移梁理论，利用解析试函数来建立厚／薄梁单元的横向位移、转角位移、曲率、剪应变等位移模式，构造出厚／薄梁通用单元．应用有限条带，将厚／薄梁单元的位移模式应用于厚／薄板矩形弯曲单元，直接构造出单元的横向位移、转角、曲率、剪应变，导出了单元的刚度矩阵和结点等效力，编制了计算程序，进行了数值计算和比较，结果表明，所研究的单元不出现剪切闭锁且精度较好．     

Mixed finite element and differential quadrature method for free and forced vibration and buckling analysis of rectangular plates

This paper presents a combined application of the finite element method (FEM) and the differential quadrature method (DQM) to vibration and buckling problems of rectangular plates. The proposed scheme combines the geometry flexibility of the FEM and the high accuracy and efficiency of the DQM. The accuracy of the present method is demonstrated by comparing the obtained results with those available in the literature. It is shown that highly accurate results can be obtained by using a small number of finite elements and DQM sample points. The proposed method is suitable for the problems considered due to its simplicity and potential for further development.     

A numerical method for clamped thin rectangular plates carrying a uniformly distributed load

A numerical method for clamped thin rectangular plates carrying a uniformly distributed load and the exact solution of the governing equations are given. The solution is presented in a simple form suitable for direct practical use. The method is a very simple and practical approach. The results are compared with those reported in the previous papers Published in Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 126–143, June 2007.     

Large deflection of a rectangular magnetoelectroelastic thin plate

Based on the von Karman plate theory of large deflection, we derive the nonlinear partial differential equation for a rectangular magnetoelectroelastic thin plate under the action of a transverse static mechanical load. By employing the Bubnov-Galerkin method, the nonlinear partial differential equation is transformed to a third-order nonlinear algebraic equation for the maximum deflection where a coupling factor is introduced for determining the coupling effect on the deflection. Numerical results are carried out for the thin plate made of piezoelectric BaTiO₃ and piezomagnetic CoFe₂O₄ materials. Some interesting results are obtained which could be useful to future analysis and design of multiphase composite plates.     

Vibration problems of flexible circular plates with initial deflection

In this paper,the differential equations of flexible circular plates with initialdeflection are derived.The stability of motion is investigated in phase plane.Theperiodical solutions of nonlinear vibration for circular plates with initial deflection areobtained by use of Galerkin method and Lindstedt-Poincare perturbation method.Theeffect of initial deflection on the dynamic behavior of the flexible plates are alsodiscussed.     

计入膜力塑性耗散效应的矩形板塑性动力响应

从能量的观点在小挠度理论中引入表征膜力塑性耗散效应的修正因子,基于刚性板块的总体平衡给出矩形板大挠度塑性动力响应的完全运动方程组,分析了理想刚塑性简支和固支矩形板在矩形脉冲和冲击载荷下包括移行塑性铰相的完全大挠度响应过程。解决了当矩形板的挠度达到厚度量级时弯矩、膜力的联合作用问题,理论预报的结果在板的挠度为10倍板厚的量级与实验结果符合良好,改进了只考虑弯矩作用的小挠度理论结果和模态近似估计。     

BIPARAMETRIC PERTURBATION SOLUTIONS OF LARGE DEFLECTION PROBLEM OF CANTILEVER BEAMS

The large deflection problem of cantilever beams was studied by means of the biparametric perturbation method and the first order derivative substitution from pseudolinear analysis approach. This kind of substitution can transform the basic equation, an integral differential equation into nonlinear algebraic ones, thus simplify computational process. Compared with present results, it indicates that the large deflection problem solved by using pseudolinear analysis can lead to simple and precise results.     

金属板条在圆柱形模中弯曲成形的大变形有限元分析

本文采用大变形弹塑性有限元法对金属板条在柱形模中的压弯成形过程进行了数值模拟,并与实验进行了比较。首先给出了纠正的拉格朗日有限元公式和基于弹塑性乘法分解的超弹性塑性本构关系。对接触摩擦问题的处理采用了罚函数法。通过对数值结果的分析得出了一些对弯曲工艺的设计有指导价值的结论。     

用能量法研究双模量大挠度圆板的轴对称弯曲

双模量圆板在外载荷作用下发生轴对称弯曲变形时,会形成各向同性的拉伸区和压缩区.此种情况下,可把双模量圆板看成两种各向同性材料组成的层合板,采用弹性理论建立了双模量圆板在外载荷作用下的静力平衡方程,利用静力平衡方程确定了双模量圆板的中性面位置.在此基础上,采用能量法研究了双模量大挠度圆板轴对称弯曲变形问题,将该方法计算结...     

Large deflection analysis of unsymmetrically laminated composite plates: analytical-numerical type approach

Large deflection analysis of laminated composite plates is considered. The Galerkin method along with Newton-Raphson method is applied to large deflection analysis of laminated composite plates with various edge conditions. The von Kármán plate theory is utilized and the governing differential equations are solved by choosing suitable polynomials as trial functions to approximate the plate displacement functions. The solutions are compared to that of Dynamic Relaxation and finite elements. A very close agreement has been observed with these approximating methods. In the solution process, analytical computation has been done wherever it is possible, and analytical-numerical type approach has been made for all problems.     

Convergence and exact solutions of spline finite strip method using unitary transformation approach

The spline finite strip method(PSM) is one of the most popular numerical methods for analyzing prismatic structures.Efficacy and convergence of the method have been demonstrated in previous studies by comparing only numerical results with analytical results of some benchmark problems.To date,no exact solutions of the method or its explicit forms of error terms have been derived to show its convergence analytically. As such,in this paper,the mathematical exact solutions of spline finite strips in the plat...     

Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method

This paper describes a method for free vibration analysis of rectangular plates with any thicknesses, which range from thin, moderately thick to very thick plates. It utilises admissible functions comprising the Chebyshev polynomials multiplied by a boundary function. The analysis is based on a linear, small-strain, three-dimensional elasticity theory. The proposed technique yields very accurate natural frequencies and mode shapes of rectangular plates with arbitrary boundary conditions. A very simple and general programme has been compiled for the purpose. For a plate with geometric symmetry, the vibration modes can be classified into symmetric and antisymmetric ones in that direction. In such a case, the computational cost can be greatly reduced while maintaining the same level of accuracy. Convergence studies and comparison have been carried out taking square plates with four simply-supported edges as examples. It is shown that the present method enables rapid convergence, stable numerical operation and very high computational accuracy. Parametric investigations on the vibration behaviour of rectangular plates with four clamped edges have also been performed in detail, with respect to different thickness-side ratios, aspect ratios and Poisson’s ratios. These results may serve as benchmark solutions for validating approximate two-dimensional theories and new computational techniques in future.     

作大范围运动矩形板的动力学建模理论研究

研究了初应力法的作大范围运动矩形板的建模理论。根据连续介质理论,考虑应变-位移中的非线性项,用一致质量有限元法对柔性板进行离散,基于Jourdain速度变分原理导出定轴转动下大范围运动为自由的柔性板刚-柔耦合动力学方程。从其刚柔耦合动力学方程出发,考虑在大范围运动已知情况下的结构动力学方程。通过引入准静态概念,把其结构动力学方程转化为准静态方程。对纵向和横向变形节点坐标进行坐标分离,解出与纵向变形相关的准静态方程,得到准静态时的纵向应力表达式,从而获得附加刚度项;并对此非惯性系下作大范围运动柔性板的结构动力学方程进行数值仿真,验证了采用初应力法柔性板的动力学建模方法来计算经历大范围运动的不规则柔性板的动力学响应是可行的,体现了初应力法对柔性板建模的优越性。     

THE METHOD OF THE RECIPROCAL THEOREM OF FORCED VIBRATION FOR THE ELASTIC THIN RECTANGULAR PLATES (III)—CANTILEVER RECTANGULAR PLATES

In this paper, applying the method of the reciprocal theorem, we give the stationary solutions of the forced vibration of cantilever rectangular plates under uniformly distributed harmonic load and concentrated harmonic load acting at any point of the plates, the figures and tables of number value of bending moment and the deflection amplitudes as well.     

ON THE METHOD OF ORTHOGONALITY CONDITIONS FOR SOLVING THE PROBLEM OF LARGE DEFLECTION OF CIRCULAR PLATE

In this paper,we reexamine the method of successive approximation presented byProf.Chien Wei-zang for solving the problem of large deflection of a circular plate,and findthat the method could be regarded as the method of strained parameters in the singularperturbation theory.In terms of the parameter representing the ratio of the centerdeflection to the thickness of the plate,we make the asymptotic expansions of thedeflection,membrane stress and the parameter of load as in Ref.[1],and then give theorthogonality conditions(i.e.the solvability conditions)for the resulting equations,bywhich the stiffness characteristics of the plate could be determined.It is pointed out thatwith the solutions for the small deflection problem of the circular plate and theorthogonality conditions,we can derive the third order approximate relations between theparameter of load and the center deflection and the first-term approximation of membranestresses at the center and edge of the plate without solving the differential equ     

An approximate solution with high accuracy of transverse bending of thin rectangular plates under arbitrarily distributed loads

Ritz method is an effective way widely used to analyze the transverse bending ofthin rectangular plates.Its accuracy depends completely on the basis functions selected.This paper selects the superposition of sine series with polynomials as the basisfunctions of thin rectangular plates.The calculating formulae are not only simple andeasily programmed,but also have high accuracy.Finally,two numerical results aregiven and compared with those obtained by the classical method.     

高层筒体结构整体稳定及二阶位移分析的改进条元法

根据柱壳理论,构造了一种柱壳曲条,本文结合柱壳曲条和平壳条元求解高层筒体结构的整体稳定及二阶位移。采用三次H erm ite插值函数模拟条元横截面的翘曲位移变化,能较好地反映筒体受力“剪切滞后”效应;采用一族能较好地逼近弯剪型变形曲线的正交多项式作基函数来描述位移沿竖向变化。用最小势能原理建立稳定及二阶位移分析方程。该方法适用于任意平面形状的高层建筑筒体结构及剪力墙结构的稳定及二阶位移分析。与其它方法相比,该方法具有精度高、通用性强、计算量小等优点。     

结构几何非线性分析的一种修正的有限元迁移矩阵法

本文提出一种修正的有限元法与迁移矩阵法相结合的方法用于结构几何非线性劳力分析。该法可克服一般的ＦＥ－ＴＭ法只适用于规则边界结构的缺点，同时可避免由于传递矩阵连续相乘而产生的误差传递。采用修正的步长增量法与修正的牛顿。拉斐逊方法相结合的算法于非线性问题的求解，编制了在ＩＢＭ－ＰＣ／ＡＴ上实现该方法的计算机程序ＴＮＯＮＤＬＷ１，数值计算结果表明了本文方法及所编制程序的有效性。