Geometrically nonlinear analysis of a Reissner type plate by the boundary element method

The boundary element method is used in the geometrically nonlinear analysis of laterally loaded isotropic plates taking into account the effects of transverse shear deformation. This paper presents the general equations for finite deformation of a Reissner type plate, and also gives an integral formulation of the Von Kármán type nonlinear governing equations which involve coupling between in-plane and out-of-plane deformation. The boundary and the domain of the plate are discretized to solve nonlinear plate bending problems. All unknown variables are at the boundary. An iterative procedure is applied to achieve linearization of the nonlinear equations. Some numerical results of the computation are compared with the analytical solutions and other numerical techniques, and good agreement is obtained.     

Elasto-plastic finite element analysis of double chord rectangular hollow section T-joints

Elasto-plastic response of T-joints consisting of double chord, rectangular hollow sections (RHS) has been modelled by treating the chord's mated flanges as thin plates supported by coupled springs that simulate the action of the side walls and bottom flanges. Two loading conditions, namely, branch axial force and branch bending are analyzed. The finite element formulation includes rectangular plate and edge boundary springs in which both in-plane and out-of-plane actions are considered. This paper is an extension of a previous study of the joints' elastic behaviour. Material nonlinearities are incorporated through the Von-Mises yield criterion and its associated flow rule and the Newton-Raphson method is used for the nonlinear analysis. The model is used to determine the ultimate strength and the load-deformation curves for both double and single chord T-joints. The numerical results obtained are in good agreement with some experimental results available in the literature.     

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.     

Comparison of boundary element and finite element methods for dynamic analysis of elastoplastic plates

Boundary and finite element methodologies for the determination of the response of inelastic plates are compared and critically discussed. Flexural dynamic plate bending problems are considered and a hardening elastoplastic constitutive model is used to describe material behaviour. The domain/boundary element methodology using linear boundary and quadratic interior elements and the finite element method with quadratic Mindlin plate elements are used in this work. The discretized equations of motion in both methodologies are solved by an efficient step-by-step time integration algorithm. Numerical results obtained are presented and compared in order to access the accuracy and computational efficiency of the two methods. In order to make the comparison as meaningful as possible, boundary and finite element computer codes developed by the author are used in this paper. In general, boundary elements appear to be a better choice than finite elements with respect to computational efficiency for the same level of accuracy.     

Application of boundary-domain element method to the free vibration problem of plate structures

The boundary-domain element method is applied to the free vibration problem of thin-walled plate structures. The static fundamental solutions are used for the derivation of the integral equations for both in-plane and out-of-plane motions. All the integral equations to be implemented are regularized up to an integrable order and then discretized by means of the boundary-domain element method. The entire system of equations for the plate structures composed of thin elastic plates is obtained by assembling the equations for each plate component satisfying the equilibrium and compatibility conditions on the connected edge as well as the boundary conditions. The algebraic eigenvalue equation is derived from this system of equations and is able to be solved by using the standard solver to obtain eigenfrequencies and eigenmodes. Numerical analysis is carried out for a few example problems and the computational aspects are discussed.     

The performance of Mindlin plate finite strips in geometrically nonlinear analysis

The finite strip method is used in the geometrically nonlinear analysis of laterally loaded isotropic plates within the context of Mindlin plate theory wherein the effects of transverse shear deformation are included. The analysis is of the Lagrangian type with the nonlinearity introduced by the inclusion of certain nonlinear terms in the strain-displacement equations. Following on from a related earlier investigation which dealt with a particular finite strip model, the performance of a range of different models is investigated. Linear, quadratic, cubic, and quartic polynomial interpolation is used in the different models in representing the variation of the five relevant displacement type quantities across a strip: also, both analytical (exact) and numerical (reduced, selective) schemes of integration are used in the crosswise direction in evaluating the stiffness properties of the various models. The ends of the finite strips are simply supported for out-of-plane behaviour and immovable for inplane behaviour. Detailed results are presented of the application of seven types of finite strip model to a range of plate problems, all involving uniformly loaded, square plates but with thin or moderately thick geometry and with simply supported or clamped longitudinal edges.     

On the design and analysis of simply supported and clamped thick rectangular plates

The finite difference method, although well known as an efficient numerical method, was applied in the past, in the case of plate problems, only for the solution of thin plates. In the present study, the suitability of the method for problems involving thick plates is studied. 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. Thin plates are treated as a particular case of the corresponding thick plates. The method is first applied to investigate the behaviour of clamped, square isotropic homogeneous thick plates. After the validity of the method is established, it is then extended to the solution of similar problems for simply supported square plates. Once the solution for a thick plate with a particular plate aspect ratio and boundary condition is obtained using a limited number of mesh sizes, a more refined solution to investigate the accuracy and convergence of the problem is then extended by providing more detailed functions satisfying the mesh sizes generated automatically by a computer program.

Whenever possible results of the present method are compared with existing solutions in the technical literature obtained by much more laborious numerical techniques, and close agreements are found. The submatrices involved in the formation of the finite difference equations from the governing differential equations are generated directly by the computer program. Simplicity in formulation and quick convergence are the obvious advantages of the method in comparison with other numerical methods requiring extensive computer facilities.     

Large deflection analysis of skew plates by lumped triangular element formulation

A finite difference scheme with triangular mesh is presented for the analysis of skew plate problems with large deflections. The suggested formulation is independent of the boundary condition and uses energy principles to derive a set of nonlinear algebraic equations which are solved by using Newton-Raphson iterative method with incremental loading. The investigation is concerned with the behaviour of constant thickness clamped and simply supported isotropic skew plates with immovable edges and subjected to uniformly distributed transverse load. The effects of skew on plates with large deflections are investigated and comparisons are made with existing results; good agreement is shown.     

Analysis of plate resting on elastic supports and elastic foundation by finite strip method

A procedure incorporating the finite strip method together with spring systems is proposed in this paper for treating plates on elastic supports. The spring systems can simulate different elastic supports, such as elastic foundation, line and point elastic supports, and also mixed boundary conditions. To illustrate the application of this procedure, two numerical examples are presented. A three-span simply supported plate is first considered and the effects of support stiffness on the static and free vibration responses and on the critical buckling stress are discussed. A plate resting on Winkler elastic foundation is next studied and the effects of dimension ratio on the static and free vibration responses are discussed. Numerical results show that the spring system can successfully simulate different kinds of elastic supports.     

Semi-analytical postbuckling analysis of stiffened imperfect plates with a free or stiffened edge

A large deflection, semi-analytical method is developed for pre- and postbuckling analyses of stiffened rectangular plates with one edge free or flexibly supported, and the other three edges laterally supported. The plates can have stiffeners in both directions parallel and perpendicular to the free edge, and the stiffener spacing can be arbitrary. Both global and local bending modes are captured by using a displacement field consisting of displacements representing a simply supported, stiffened plate and an unstiffened plate with a free edge. The out-of-plane and in-plane displacements are represented by trigonometric functions and linearly varying functions, defined over the entire plate. The formulations derived are implemented into a FORTRAN computer programme, and numerical results are compared with results by finite element analyses (FEA) for a variety of plate and stiffener geometries. Relatively high numerical accuracy is achieved with low computational efforts.     

Deflection and stress analysis of orthotropic plates in flexure

A finite difference (FD) method is developed for computing deflections and stresses in rectangular orthotropic plates subjected to transverse flexural loadings. The plate material is elastic with symmetry axes parallel to the plate edges, and plate deflections are assumed to be small. The FD equations are solved iteratively using successive over relaxation techniques. The numerical procedure converges rapidly and is simple enough to be implemented in a minicomputer program for the design analysis of composite plates. As an example the program is applied to the analysis of simply supported and clamped square plates under uniform pressure or point loads, and the effect of material anisotropy on plate behaviour is discussed.     

Thermal stress analysis of skew plates by finite element method

Thermal stresses are induced in general due to nonuniform temperature distribution or due to the boundary restriction. Most of the work reported so far deals with either plates with edges clamped in plane of the plate or plates with stress free edges. While studying buckling or post-buckling problems, one should ideally analyse the plates with mixed in-plane boundary conditions. Hence, in the present analysis, thermal stress analysis of skew plates with mixed in-plane boundary conditions using finite element approach is attempted. In addition, the effect of in-plane boundary conditions on the thermal stresses is also discussed.     

Finite element analysis of crack closure in plate bending

The effect of crack closure in plate bending is studied using the finite element method. Elastic plates containing through-wall-thickness stationary cracks under transverse pressure loading are considered with different plate thicknesses and boundary conditions, respectively. Crack closure on the compression side is modeled two different ways: line closure and surface closure models. A plate bending element degenerated from a three-dimensional solid element is used to model such crack closure. Effects of crack closure are compared using the line or surface closure model for different plate thicknesses and boundary conditions, respectively.     

Combined use of equivalent center mass and stiffness factors to better estimate frequencies of mass loaded plates

This paper is concerned with the frequency analysis of vibrating plates carrying multiple masses at various positions. By virtue of an equivalent center method, the frequencies of the plates carrying multiple masses can be predicted by using merely the data obtained earlier for the same plate with each corresponding mass component. Further accounting the change of the strain energy into the model, by introducing respective stiffness ratios, enables one to obtain a quick and better estimation of the loaded plates with various boundary conditions. The proposed model is validated through experimentation of a rectangular clamped plate carrying concentrated masses. Analytical and experimental results for the plate system are compared and discussed. It is found that the change in the strain energy should be incorporated in the model, especially for cases of large masses placed on a thin plate, in order to predict well the natural frequency of the amalgamated system from those of the component systems. The excellent performance of the proposed model is due to the fact that the effects of both the different masses and their locations on the frequency have been accounted for implicitly in the respective equivalent center mass and stiffness factors. Although the model considered is rather simple, the proposed methodology can be extended to plates with other geometry and configurations.     

Study of elastic behaviour of plates containing cracks by finite element analysis

This work applies finite element analysis very simply to cracked plates. An infinite plate and a finite plate, both with a central crack, are considered to study their elastic behaviour and some fracture mechanics concepts, such as the geometry factor and the fracture toughness. These magnitudes are calculated by means of finite element methods and the results are in very good agreement with the established theory, which proves that the finite element approach is very appropriate. The fracture toughness fraction is defined and calculated for a finite plate to predict its failure.     

Piezoelectric control of composite plate vibration: Effect of electric potential distribution

The paper deals with the vibration analysis of active rectangular plates. The plates considered are composites containing piezoelectric sensor/actuator layers, which operate in a velocity feedback control to achieve transverse vibration suppression. The piezoelectric layers are poled through the thickness and equipped with traditional surface electrodes. In order to satisfy the Maxwell electrostatics equation the widely used simplification of the electric potential distribution in the actuator layer (linear across the thickness) is replaced by a combination of a half-cosine and linear distribution in the transverse direction. The in-plane spatial variation of the potential instead of applying uniform distribution is determined by the solution of the coupled electromechanical governing equations with the natural boundary conditions corresponding to both the flexural and electric potential fields. The analysis is performed for simply supported plates. Two models of the plate are considered. In the first case the displacement field is based on the Kirchhoff hypothesis. For the second the Mindlin plate model is applied. The governing coupled equations describing the active plate behaviour are derived. The influence of the electric potential distribution and also the thickness of piezoelectric layers on the plate dynamics including the natural frequencies modification is numerically investigated and discussed.     

Application of isogeometric method to free vibration of Reissner–Mindlin plates with non-conforming multi-patch

The isogeometric method is used to study the free vibration of thick plates based on Mindlin theory. The Non-uniform Rational B-Spline (NURBS) basis functions are employed to build the thick plate’s geometry models and serve as the shape functions for solution field approximation in finite element analysis. The Reissner–Mindlin plates built with multiple NURBS patches are investigated, in which several patches of the model have multi-interface and different patches may share a common point. In order to solve the non-conforming interface problems, Nitsche method is employed to glue different NURBS patches and only refers to the coupling conditions in this work. Various plate shapes, different boundary conditions and several kinds of thickness-span ratios are considered to verify the validity of the presented method. The dimensionless frequencies for different cases are obtained by solving the eigenvalue equation problems and compared with the existing reference solutions or the results calculated by ABAQUS software. Several numerical examples exhibit the effectiveness of the isogeometric approach. It shows that the natural frequencies of the Reissner–Mindlin plate can be successfully predicted by the combination of isogeometric analysis and Nitsche method.     

Dynamic response of plate systems by combining finite differences, finite elements and laplace transform

A numerical method for the determination of the dynamic response of large rectangular plates or plate systems to lateral loads is proposed. The method is a combination of the finite difference method, the finite element method and the Laplace transform with respect to time. The plate system is considered as an assemblage of a small number of big rectangular superelements whose stiffness matrices are derived with the aid of the finite difference method in the Laplace transform domain. These superelements are then used to formulate and solve the problem by the finite element method in the transformed domain. The dynamic response is finally obtained by a numerical inversion of the transformed solution. External viscous or internal viscoelastic damping as well as the elastic foundation interaction effect can easily be taken into account. The method is illustrated and its merits demonstrated by means of numerical examples.     

Application of the extended Kantorovich method to the bending of variable thickness plates

The governing equations of the classical plate theory for a uniform or a unidirectional variable thickness rectangular plate under transverse applied loading are solved by means of the extended Kantorovich method. The plate may be either simply supported or clamped along the edges. The solution procedure is iterative and must be carried out numerically. This necessitates the calculation of the two missing pieces of boundary data along the edges of the plate. The missing boundary data are determined utilizing the method of adjoints of the shooting method. The numerical values of the deflection and bending moments for uniform and variable thickness plates are compared with those from the exact solutions and finite element analysis, respectively.