An improved rectangular element for plate bending analysis

This paper presents a new, simple, rectangular finite element with twelve degrees of freedom for the bending analysis of thin plates. Three interpolation functions corresponding to the normal deflections and tangential slopes at the nodal points are written in parametric form. Convergence requirements are then used to find relationships among the parameters included in these functions. To identify the optimal values of the still undetermined parameters extensive comparisons are carried out using plate problems with different loading and boundary conditions. Certain values of the unknown parameters are found to produce displacement results with faster rate of convergence than those of other simple elements. When comparisons are based on a measure representing the actual computational effort rather than the mesh size the proposed element is found to excell higher-order elements as well. Stress results are also calculated for the proposed element and found to be fairly close to the exact values.     

Semi-numerical analysis of rectangular plates in bending

Bending analysis of rectangular plates is proposed using a combination of basic functions and finite difference energy technique. The basic function satisfying the boundary conditions along the two opposite edges of the plate is substituted in the integral expression for the total potential energy of the plate thereby reducing a two dimensional functional into an unidirectional one. The discretized form of the total potential energy of the plate expressed as a functional of the displacement field is obtained by replacing the derivatives by the corresponding difference quotient. Using the principle of minimum potential energy a set of algebraic equations is obtained which is subsequently solved for unknown displacements. Examples have been presented for a variety of isotropic and orthotropic plates of rectangular plan-form with different edge conditions and loadings. Results have been compared with other numerical results and available analytical solutions.     

Calculation of the natural frequencies and steady state response of thin plates in bending by an improved rectangular element

This paper presents an application of a new improved rectangular finite element to the problems of free and forced harmonic oscillations of thin plates in bending. The proposed element, called the parametric element, has been presented in a previous paper by the same authors and applied to the problem of static bending of plates. The shape functions corresponding to the various nodal movements are expressed in simple parametric forms which scan the space between the Adini-Clough-Melosh model and the Papenfuss model. The performance of the parametric element is found to be at its best in both the statical and dynamical applications when the parameters included in the shape functions assume certain values. Like what happened in the statical application, the optimal parametric element has shown remarkable superiority over other simple elements when used in the prediction of the natural frequencies and harmonic response of several plates having different boundary conditions.     

An efficient analysis for thin plates of general quadrilateral shape subject to bending stresses

Isotropic quadrilateral plates subject to bending are analysed by the Rayleigh-Ritz method. It is shown that, by defining the plate in a warped coordinate system, suitable global functions for the plate displacements which fit most boundary conditions of practical significance can be readily derived. The performance of the method is assessed by comparing the results with those obtained from different theoretical solutions and/or experiment. Good correlation is obtained between all sets of results, whilst those due to the proposed method indicate considerable advantages over the alternative solutions, both in terms of computational efficiency and the ease with which the requirements for an acceptably accurate solution can be pre-determined.     

Computation of elastic buckling loads of rectangular thin plates using the extended Kantorovich method

In this paper, the extended Kantorovich method proposed by Kerr is further extended to the eigenvalue problem of elastic stability of various rectangular thin plates. By taking advantage of the availability and reliability of state-of-the-art ordinary differential equation solvers, multi-term trial functions have been employed, which is a significant extension to Kerr’s single term approach. As a result, the accuracy is greatly improved and some special problems that a single-term trial function fails to solve are now accommodated. A large number of numerical experiments have been carried out and the computed buckling loads are either exact or more accurate than the best known results in the literature.     

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.     

Vibrations of thin rectangular plates with arbitrarily oriented stiffeners

Free vibration of plates with arbitrarily oriented stiffeners are studied using high precision plate bending and stiffener elements. Good convergence of frequency values for coarse mesh is demonstrated. Natural frequencies of square plates with various arrangement of stiffeners are determined for both simply supported and clamped boundary conditions.     

Accuracy estimation in the finite element analysis of transverse bending of thin flat plates

As a basic study for the establishment of an accuracy estimation method in the finite element method, this paper deals with the problems of transverse bending of thin, flat plates. From the numerical experiments for uniform mesh division, the following relation was deduced, ε ∝ (h/a)^k, k 1, where ε is the error of the computed value by the finite element method relative to the exact solution and h/a is the dimensionless mesh size. Using this relation, an accuracy estimation method, which was based on the adaptive determination of local mesh sizes from two preceding analyses by uniform mesh division, was presented.

A computer program using this accuracy estimation method was developed and applied to 28 problems with various shapes and loading conditions. The usefulness of this accuracy estimation method was illustrated by these application results.     

Mixed finite-difference scheme for analysis of simply supported thick plates

A mixed finite-difference scheme is presented for the stress and free vibration analysis of simply supported nonhomogeneous and layered orthotropic thick plates. The analytical formulation is based on the linear, three-dimensional theory of orthotropic elasticity and a Fourier approach is used to reduce the governing equations to six first-order ordinary differential equations in the thickness coordinate. The governing equations possess a symmetric coefficient matrix and are free of derivatives of the elastic characteristics of the plate. In the finite difference discretization two interlacing grids are used for the different fundamental unknowns in such a way as to reduce both the local discretization error and the bandwidth of the resulting finite-difference field equations. Numerical studies are presented for the effects of reducing the interior and boundary discretization errors and of mesh refinement on the accuracy and convergence of solutions. It is shown that the proposed scheme, in addition to a number of other advantages, leads to highly accurate results, even when a small number of finite difference intervals is used.     

An improved three-dimensional full-vectorial finite-difference imaginary-distance beam propagation method

1 Introduction Recently, the importance of the planar lightwave circuits or photonic integrated cir- cuits (PLCs/PICs) has been widely recognized because the monolithic integration of various kinds of photonic or optoelectronic devices can be achieved via…     

Finite element bending analysis of nonuniform circular plates

The finite element method is applied to the small deflection bending analysis of nonuniform thin axisymmetric circular plates made of linear elastic material. Elements with annular and circular geometry with only 4 degrees of freedom are used in the analysis of both symetrically and nonsymmetrically loaded plates. Non-symmetric loads are expanded in Fourier series and elements restricted to deform with specified number of nodal diameters are used for each component of loading. The method is checked with several numerical examples. Although applicable to only axisymmetric plates, the method gives better results compared to other finite element methods besides offering savings in computer storage and time.     

均布载荷下四边固支矩形薄板的挠度

为求解四边固支矩形薄板在均布载荷作用下的挠度表达式,以利维解为基础利用叠加法将复杂问题分解为多个简单问题,然后进行叠加,获得四边固支矩形薄板在均布载荷作用下的挠度表达式.利用有限元模拟验证表达式的正确性,发现有限元结果与公式结果吻合较好.     

On minimum compliance problems of thin elastic plates of varying thickness

The paper deals with two minimum compliance problems of variable thickness plates subject to an in-plane loading or to a transverse loading. The first of this problem (called also the variable thickness sheet problem) is reduced to the locking material problem in its stress-based setting, thus interrelating the stress-based formulation by Allaire (2002) with the kinematic formulation of Golay and Seppecher (Eur J Mech A Solids 20:631–644, 2001). The second problem concerning the Kirchhoff plates of varying thickness is reduced to a non-convex problem in which the integrand of the minimized functional is the square root of the norm of the density energy expressed in terms of the bending moments. This proves that the problem cannot be interpreted as a problem of equilibrium of a locking material. Both formulations discussed need the numerical treatment in which stresses (bending moments) are the main unknowns.     

Axisymmetric bending of circular and annular sandwich plates with nonlinear elastic core material

This paper compares the analytical model of the axisymmetric bending of a circular sandwich plate with the finite element method (FEM) based numerical model. The differential equations of the bending of circular symmetrical sandwich plates with isotropic face sheets and a nonlinear elastic core material are obtained. The perturbation method of a small parameter is used to represent the nonlinear differential equations as a sequence of linear equations specifying each other. The linear differential equations are solved by reducing them to the Bessel equation. The results of the calculations with the use of the analytical and FEM models are compared with the results obtained by other authors by the example of the following problems: (1) axisymmetric transverse bending of a circular sandwich plate; (2) axisymmetric transverse bending of an annular sandwich plate. The effect of the nonlinear elasticity of the core material on the strained state of the sandwich plate is described.     

A three-dimensional nonlinear analysis of cross-ply rectangular composite plates

The results of a three-dimensional, geometrically nonlinear, finite-element analysis of the bending of cross-ply laminated anisotropie composite plates are presented. Individual laminae are assumed to be homogeneous, orthotropic and linearly elastic. A fully three-dimensional isoparametric finite element with eight nodes (i.e. linear element) and 24 degrees of freedom (three displacement components per node) is used to model the laminated plate. The finite element results of the linear analysis are found to agree very well with the exact solutions of cross-ply laminated rectangular plates under sinusiodal loading. The finite element results of the three-dimensional, geometrically nonlinear analysis are compared with those obtained by using a shear deformable, geometrically nonlinear, plate theory. It is found that the deflections predicted by the shear deformable plate theory are in fair agreement with those predicted by three-dimensional elasticity theory; however stresses were found to be not in good agreement     

An improved boundary-integral equation method for three dimensional elastic stress analysis

An improved numerical implementation of the boundary-integral equation method for three dimensional stress analysis is reported. The new implementation models the boundary data as piecewise-linear variations over the boundary segments. As with all boundary-integral equation models, a system of equations relating unknown boundary data to known boundary data is obtained. The new implementation is described mathematically and verified on several simple test problems. In addition the method is used to study a finite fracture specimen used in material testing. The numerical results and computer run times are compared to an earlier version of the boundary-integral equation method. The results show significant improvement in accuracy for comparable run times for most problems.     

A finite element formulation for large amplitude flexural vibrations of thin rectangular plates

Large amplitude flexural vibrations of rectangular plates are studied in this paper using a direct finite element formulation. The formulation is based on an appropriate linearisation of strain displacement relations and uses an iterative method of solution. Results are presented for rectangular plates with various boundary conditions using a conforming rectangular element. Whenever possible the present solutions are compared with those of earlier work. This comparison brings out the superiority of the proposed formulation over the earlier finite element formulation.     

Buckling of rectangular mindlin plates

The elastic buckling of rectangular Mindlin plates is considered using two related methods of analysis. These methods are the Rayleight-Ritz method and one of its piece-wise forms, the finite strip method. Arbitrary combinations of the standard boundary conditions of clamped, simply-supported and free edges are accommodated by the use in the assumed displacement fields of the normal modes of vibration of Timoshenko beams. The applied membrane stress field leading to buckling can comprise biaxial direct stress plus shear stress. A range of numerical applications is described for isotropic and transversely isotropic plates of thin and moderately thick geometry. The results obtained using the two methods compare closely to one another and to other published results where these are available. A direct relationship between unidirectional buckling stress and frequency of vibration is demonstrated for a category of plates having one pair of opposite edges simply supported.     

A family of rectangular bending elements

By using separate independent transverse and rotational displacement variables in terms of a polynomial it is possible to produce high order conforming elements for plate bending and, at the same time, to include the effect of shear deformation in the analysis. The procedure for constructing a family of conforming rectangular plate bending elements with any number of nodes and the derivation of the stiffness matrix are illustrated. A computer programme is developed to generate the stiffness coefficients of the elements in this family; whereupon the characteristics of elements with as many as 17, 21 or 25 nodes and so on can be investigated. It is demonstrated that accurate results can be obtained for thin and moderately thick plates with various boundary conditions under bending by using just one or a few high order elements in this family. Hence the procedure for solving a problem in plate bending can be much simplified and the total number of nodes in a problem can be much reduced. Highlight in this family is the 17-node element which yields good results without involving too many nodes for many plate bending problems.