Analysis of partly corrugated rectangular diaphragms using theRayleigh-Ritz method

In this paper, the Rayleigh Ritz method is applied to static analyses of partly corrugated four-edges-clamped rectangular diaphragms. We use coupled variational equations (derived from the principle of virtual displacement), and describe some practical considerations needed to analyze multiregional diaphragms by the method. We approximate a sinusoidal corrugation by an equivalent flat single layer, and derive equivalent in-plane and bending stiffnesses using the theory of engineering mechanics. We demonstrate the effectiveness and usefulness of the present method through examples (in one example, we compare the calculated deflection with that obtained by a finite-element program). We examine a partly corrugated rectangular plate with residual stress, and investigate the effects of corrugation parameter variations. The analysis technique presented in this paper is very convenient, efficient, and reliable for analyzing seemingly difficult microelectromechanical system devices built on partly corrugated thin diaphragms with various residual stresses     

Linear and non-linear deflection analysis of thick rectangular plates—II. Numerical applications

Variational methods are widely used for the solution of complex differential equations in mechanics for which exact solutions are not possible. The finite difference method, although well known as an efficient numerical method, was applied in the past only for the analysis of linear and non-linear thin plates. In this paper the suitability of the method for the analysis of non-linear deflection of thick plates is studied for the first time. While there are major differences between small deflection and large deflection plate theories, the former can be treated as a particular case of the latter, when the centre deflection of the plate is less than or equal to 0.2–0.25 of the thickness of the plate. The finite difference method as applied here is a modified finite difference approach to the ordinary finite difference method generally used for the solution of thin plate problems. In this analysis thin plates are treated as a particular case of the corresponding thick plate when the boundary conditions of the plates are taken into account. The method is first applied to investigate the deflection behaviour of clamped and simply supported square isotropic thick plates. After the validity of the method is established, it is then extended to the solution of rectangular thick plates of various aspect ratios and thicknesses. Generally, beginning with the use of a limited number of mesh sizes for a given plate aspect ratio and boundary conditions, a general solution of the problem including the investigation of accuracy and convergence was extended to rectangular thick plates by providing more detailed functions satisfying the rectangular mesh sizes generated automatically by the program. Whenever possible results obtained by the present method are compared with existing solutions in the technical literature obtained by much more laborious methods and close agreements are found. The significant number of results presented here are not currently available in the technical literature. The submatrices involved in the formation of the finite difference equations from the governing differential equations are generated directly by the computer program. The subroutine SOLINV using the change of variable technique illustrated elsewhere takes care of the solution of the general system. Simplicity in formulation and quick convergence are the obvious advantages of the finite difference formulation developed to compute small and large deflection analysis of thick plates in comparison with other numerical methods requiring extensive computer facilities.     

Unilateral plate contact with the elastic-plastic winkler-type foundation

In this work contact problems of a plate with the elastic-plastic Winkler-type foundation has been solved. An unilateral character of bonds between plate and foundation has been taken into consideration. The effect of friction forces in the contact plane has been neglected and an incremental approach has been applied. An incremental problem has been formulated in a variational manner and solved by use of the finite element method. The results of numerical calculations for rectangular plates subjected to the simple and complex load processes have been presented.     

Stability of plates using a mixed finite element formulation

A rectangular plate finite element is developed according to a variational principle due to Prager The element is applied to plate stability analysis. The results obtained compare very favorably with results based on previous formulations.     

An improved numerical process for solution of solid mechanics problems

This paper gives an overview of the development and status of an improved numerical process for the solution of solid mechanics problems. The proposed process uses a mixed formulation with the fundamental unknowns consisting of both stress and displacement parameters. The problem is formulated either by means of first-order partial differential equations or in a variational form by using a Hellinger-Reissner-type mixed variational principle.

For presentation purposes, the components of a numerical process are characterized and the criteria for an ideal process are outlined. Commonly used finite-difference and finite-element procedures arc examined in the light of these criteria and it is shown that they fall short in a number of ways. The proposed numerical process, on the other hand, satisfies most of the optimality criteria and appears to be particularly suited for use with the forthcoming generation computers (e.g. STAR-100 computer).

The paper includes a number of examples showing application of the proposed process to a broad spectrum of solid mechanics problems. These examples demonstrate the versatility and high accuracy of the numerical process obtained by using mixed formulations in conjunction with improved discretization techniques.     

A quadrilateral finite difference plate element for nonlinear transient analysis of panels

A quadrilateral plate element for the analysis of nonlinear transient response of panels has been developed based on the variational finite difference method for an irregular mesh. Due to the superior computational characteristics of the variational finite difference method with a lesser degree of continuity constraint on the interpolation functions and the use of lower-order polynomials allowing faster numerical integration methods to be implemented, this plate element is quite competitive or perhaps even superior when compared with the conforming finite elements. Three illustrative problems have been solved using this plate element to demonstrate its capability and accuracy in analyzing the large deformation response of panels subject to dynamic loadings.Very favorable correlation was observed between analysis and experiment on large deformations of elasticplastic rectangular plates subject to intensive impulsive loadings. Similar correlation was also observed for circular plates modeled with this quadrilateral plate element under impulsive loadings. Finally, the large dynamic deformations of composite rectangular panels of graphite-epoxy, boron-epoxy, glass-epoxy, and isotropic material were analyzed and found in good agreement with other analytical results.     

Nonsmooth computational mechanics algorithms, quasidifferentiability and related topics

Several extensions of smooth computational mechanics algorithms for the treatment of nonsmooth and possible nonconvex problems are briefly discussed in this paper. A potential or dissipation energy minimization problem approach is used for the structural analysis problem, so as to make the link with mathematical optimization techniques straightforward. Variational inequality problems, hemivariational inequality problems and systems of variational inequalities can be treated by the methods reviewed in this paper. The use of quasidifferentiable and codifferentiable optimization techniques is proposed for the solution of the more general class of nonconvex, possibly nonsmooth problems. Established and new directions in path-following techniques are discussed and are linked with nonsmooth mechanics algorithms.     

Generalized mixed variational principles and solutions of ill-conditioned problems in computational mechanics. Part II: Shear locking

Although the finite element method (FEM) has been extensively applied to various areas of engineering, the ill-conditioned problems occurring in many situations are still thorny to deal with. This study attempts to provide a high-performing and simple approach to the solutions of ill-conditioned problems. The theoretical foundation of it is the parametrized variational principles, called the generalized mixed variational principles (GMVPs) initiated by Rong in 1981. GMVPs can solve many kinds of ill-conditioned problems in computational mechanics. Among them, four cases are investigated in detail: the volumetric locking, the shear locking, the inhomogeneousness and the membrane locking problems, composing four parts of the study, Part I–Part IV, respectively. This paper is Part II, wherein a GMVP specially suited to the Reissner plate theory and Timoshenko beam theory is constructed, providing a mathematical foundation for establishing FEM formulations which can automatically unlock the shear locking and produce no spurious zero-energy modes.     

Constraint variational schemes for analysis of rectangular plates on an elastic half space

The flexural interaction of a rectangular thin elastic plate resting in smooth contact with an isotropic homogeneous elastic half space is analysed by using constraint variational schemes. The deflected shape of the plate is represented by a double power series of spatial variables with a set of generalized coordinates. The contact stresses are expressed in terms of the generalized coordinates by discretizing the contact area into several rectangular regions and solving an appropriate flexibility equation based on generalized Boussinesq's solution. Using the representations adopted for displacement and contact stresses, a constraint energy functional is constructed to determine the generalized coordinates. The constraint term in the variational functional corresponds to plate edge boundary conditions and formulations corresponding to both Lagrange multiplier and penalty types are presented. It is noted that for the present class of problems, penalty type formulations are numerically efficient. The convergence and numerical stability of the solution scheme is confirmed. Selected numerical results are presented to illustrate the dependence of flexural response of plate on the governing parameters of the plate-half space system.     

Panel flutter analysis of plate element based on the absolute nodal coordinate formulation

In this study, aeroelastic analysis of a plate subjected to the external supersonic airflow is carried out. A 3-D rectangular plate element of variable thickness based on absolute nodal coordinate formulation (ANCF) has been developed for the structural model. In the approach to the problem, a continuum mechanics approach for the definition of the elastic forces within the finite element is considered. Both shear strain and transverse normal strain are taken into account. Linearized first-order potential (piston) theory is coupled with the structural model to account for pressure loading. Aeroelastic equations using ANCF are derived and solved numerically. Values of critical dynamic pressure are obtained by a modal approach, in which the mode shapes are obtained by ANCF. All the formulations and the computations are built up in a FORTRAN 90 computer program after it was confirmed by Mathematica^?, ver. 5. The results of free vibration analysis and flutter are compared with the available references and reasonable good agreement has been found. However, some results indicate that the known problem of locking (ANCF with uniform thickness) still persist in the current developed formulation.     

Dynamic stability of a rectangular plate on non-homogeneous winkler foundation

The dynamic stability of a rectangular plate on non-homogeneous foundation, subjected to uniform compressive in-plane bi-axial dynamic loads and supported on completely elastically restrained boundaries is studied. The non-homogeneous foundation consists of two regions having different stiffnesses but symmetric about the centre lines of the plate. The equation governing the small amplitude motion of the system is derived by a variational method. The use of Galerkin method with reduced beam eigenfunctions transforms the system equations in matrix form. The system of coupled Mathieu-Hill equations thus obtained, are analysed by the method of multiple scales which yields the stability boundaries for different combinations of the excitation amplitude and frequency. The effects of stiffness and geometry of the foundation, boundary conditions, static load factor, in-plane load ratio and aspect ratio on the stability boundaries of the plate for first- and second-order simple and combination resonances are studied.     

Discrete energy method for the analysis of cylindrical shells

This paper presents an investigation into the use of the closely associated finite difference technique for the analysis of shell structures as a feasible alternative to the finite element method. The method discretises the total energy of the structure into energy due to extension and bending and that due to shear and twisting, contributed by two separate sets of rectangular elements formed by a suitable finite difference network. The derivatives in the corresponding energy expressions are replaced by their finite difference forms and the nodal displacements then constitute the undetermined parameters in the variational formulation. The formulation is also extended to a cylindrical shell element of rectangular planform. The results obtained by DEM are compared with existing results and they show excellent agreement.     

基于修正变分法开孔板和加强板结构的自由振动分析

为研究船舶开孔板和加强板结构的振动特性,用1阶剪切变形板理论描述各向同性板的位移场,并采用修正变分原理和区域分解方法建立板的离散动力学模型.每一块子域板的位移和转角分量通过第一类切比雪夫正交多项式展开.针对加强板模型,将该方法获得的结果与已经发表的文献和有限元商用软件计算结果进行对比,验证该方法的收敛性和正确性.基于修正变分法探讨多种开孔和加强板模型的自由振动特性,充分说明该数理模型和半解析方法是一种适合处理复杂板结构问题的数值工具.     

压电弹性厚板动力学的简化Gurtin型变分原理

根据古典阴阳互补和现代对偶互补的基本思想,通过罗恩提出的一条简单而统一的新途径,系统地建立了压电弹性厚板动力学的各类简化Gurtin型变分原理.这种简化Gurtin型变分原理能反映动力学初值-边值问题的全部特征.文中首先给出压电厚板动力学的广义虚功原理的表式,然后从该式出发,不仅能得到虚功原理,而且通过所给出的一系列广义Legendre变换,还能系统地成对导出压电弹性厚板动力学的8类、6类、4类变量简化Gurtin型变分原理的互补泛函以及3类和2类变量简化Gurtin型势能形式的泛函.同时,通过这条新途径还能清楚地阐明这些原理的内在联系.     

Shear flexible element based on coupled displacement field for large deflection analysis of laminated plates

A shear deformable theory accounting for the transverse-shear (in the sense of Reissner-Mindlin’s thick plate theory) and large deflections (in the sense of von Karman theory) is employed in the construction of variational statement. A four-node, lock-free, shear-flexible rectangular plate element based on the coupled displacement field is developed in this paper to carry out the large deflection analysis. The displacement field of the element is derived by making use of the linearized equations of static equilibrium. A bi-cubic polynomial distribution is assumed for the transverse displacement ‘w’. The field distribution for the in-plane displacements (u,v) and plate normal rotations (θ_x, θ_y) and twist (θ_xy) is derived using equilibrium of composite strips parallel to the plate edges. The displacement fields so derived are coupled through material couplings. The transverse shear strain fields of the proposed element do not contain inconsistent terms, so that the element predicts even shear-rigid bending accurately.The element is validated for a series of numerical problems and results for deflections and stresses are presented for rectangular composite plates with various boundary conditions, loading and lay-ups. The influence of the sign of the loading on the deflection of unsymmetrically laminated plates, in the large deflection regime is also investigated.     

Bifurcation, pre- and post-buckling analysis of frame structures

A procedure is developed for investigating the stability of complex structures that consist of an assembly of stiffened rectangular panels and three-dimensional beam elements. Such panels often form one of the basic structural components of an aircraft or ship structure. In the present study, the stiffeners are treated as beam elements, and the panels between them as thin rectangular plate elements, which may be subject to membrane and/or bending and twisting actions.

The main objective of the study is the determination of the critical buckling loads and the generation of the complete force-deformation behavior of such structures within a specified load range, based on the use of a computer program developed for this purpose. The present formulation can trace through the postbuckling or post limit behavior whether it is of an ascending or descending type. A limit load extrapolation technique is automatically initiated within the computer program, when the stability analysis of an imperfect or laterally loaded structure is being carried out.

The general approach to the solution of the problem is based on the finite element method and incremental numerical solution techniques. Initially, nonlinear strain-displacement relations together with the assumed displacement functions are utilized to generate the geometric stiffness matrices for the beam and plate elements. Based on energy methods and variational principles, the basic expressions governing the behavior of the structure are then obtained. In the incremental solution process, the stiffness properties of the structure are continuously updated in order to properly account for large changes in the geometry of the structure.

The computer program developed during the course of this study is referred at as GWU-SAP, or the George Washington University Stability Analysis Program.     

A stabilized MITC element for accurate wave response in Reissner–Mindlin plates

Residual based finite element methods are developed for accurate time-harmonic wave response of the Reissner–Mindlin plate model. The methods are obtained by appending a generalized least-squares term to the mixed variational form for the finite element approximation. Through judicious selection of the design parameters inherent in the least-squares modification, this formulation provides a consistent and general framework for enhancing the wave accuracy of mixed plate elements. In this paper, the mixed interpolation technique of the well-established MITC4 element is used to develop a new mixed least-squares (MLS4) four-node quadrilateral plate element with improved wave accuracy. Complex wave number dispersion analysis is used to design optimal mesh parameters, which for a given wave angle, match both propagating and evanescent analytical wave numbers for Reissner–Mindlin plates. Numerical results demonstrates the significantly improved accuracy of the new MLS4 plate element compared to the underlying MITC4 element.     

Numerical analysis on snapping induced by electromechanical interaction of shuffling actuator with nonlinear plate

When a voltage is applied between two electrodes consisting of a rigid ground and a deformable rectangular plate that has two parallel free edges, and one movable edge and one fixed edge, the plate bends under the electrostatic force generated in it. Accompanying to the bending deformation, the movable edge of the plate results in some displacement along the movable direction, which is the main clue of the MEMS actuator of shuffling movement. As the control voltage is applied up to a critical value, the phenomenon of snapping occurs at the plate electrode under the interaction of electromechanical coupling such that the shuffling displacement is obtained as possibly as large. Based on the nonlinear theory of plates with the von Karman’s type deformation and the electrostatic theory, a theoretical analysis for the snapping behavior is quantitatively displayed in this paper. For this purpose, a numerical code is proposed by associating the increment finite element methods for deformation with the moment method for electrostatic fields as well as the arc-length control approach in the quantitative calculations. The numerical results for some case studies show that the path of bending deformation from stability into instability passing through the snapping criterion can be tracked well by this numerical code, while the characteristic curves of the maximum deflection in the plate and the shuffling displacement versus the control voltage, which are mainly concerned in the design of the MEMS shuffling actuator, are obtained additionally.     

考虑刚-柔-热耦合的板结构多体系统的动力学建模

对热载荷作用下中心刚体与大变形薄板多体系统的动力学建模问题进行研究.基于Kirchhoff假设,从格林应变和曲率与绝对位移的非线性关系式出发,推导了非线性广义弹性力阵,用绝对节点坐标法建立了大变形矩形薄板的有限元离散的动力学变分方程.为了考虑刚体姿态运动、弹性变形和温度变化的相互耦合作用,推导了热流密度与绝对节点坐标之间的关系式.引入系统的运动学约束方程,建立了中心刚体-矩形板多体系统的考虑刚-柔-热耦合的热传导方程和带拉格朗日乘子的第一类拉格朗日动力学方程.为了有效地提高计算效率,将改进的中心差分法和广义-α法相结合,求解热传导方程和动力学方程,差分后的方程通过牛顿迭代法耦合求解.对刚-柔耦合和刚-柔-热三者耦合两种模型的仿真结果进行比较表明,刚体运动对温度梯度和热变形的影响显著.此外,本文建模方法考虑了几何非线性项,因此也考虑了热膨胀引起的轴向变形对横向变形的影响.     

An edge problem having no singularity at the corner

The problem of thermal stresses induced by a uniform temperature rise in a rectangular plate clamped along an edge is formulated as a variational problem with a subsidiary condition. It is demonstrated that the solution in the clamped corner is known a priori and is non-singular. An approximate state of stress through the plate is obtained by the method of Kantorovich. It is shown that the finite element method fails to solve this problem. It is also shown that a previous investigation of this problem suffers from serious drawbacks.