A BEM approach to static and dynamic analysis of plates with internal supports

A boundary element approach is developed for the static and dynamic analysis of Kirchhoff's plates of arbitrary shape which, in addition to the boundary supports, are also supported inside the domain on isolated points (columns), lines (walls) or regions (patches). All kinds of boundary conditions are treated. The supports inside the domain of the plate may yield elastically. The method uses the Green's function for the static problem without the internal supports to establish an integral representation for the solution which involves the unknown internal reactions and inertia forces within the integrand of the domain integrals. The Green's function is established numerically using BEM. Subsequently, using an effective Gauss integration for the domain integrals and a BEM technique for line integrals a system of simultaneous, in general, nonlinear algebraic equations is obtained which is solved numerically. Several examples for both the static and dynamic problem are presented to illustrate the efficiency and the accuracy of the proposed method.     

The BEM for plates of variable thickness on nonlinear biparametric elastic foundation. An analog equation solution

The BEM is developed for the analysis of plates with variable thickness resting on a nonlinear biparametric elastic foundation. The presented solution is achieved using the Analog Equation Method (AEM). According to the AEM the fourth-order partial differential equation with variable coefficients describing the response of the plate is converted to an equivalent linear problem for a plate with constant stiffness not resting on foundation and subjected only to an `appropriate' fictitious load under the same boundary conditions. The fictitious load is established using a technique based on the BEM and the solution of the actual problem is obtained from the known integral representation of the solution of the substitute problem, which is derived using the static fundamental solution of the biharmonic equation. The method is boundary-only in the sense that the discretization and the integration are performed only on the boundary. To illustrate the method and its efficiency, plates of various shapes are analyzed with linear and quadratic plate thickness variation laws resting on a nonlinear biparametric elastic foundation.     

Fracture of plates with variable thickness

We study the mechanics of fracture of plates with variable thickness weakened by rectilinear through cracks and propose a method for the solution of problems of the theory of elasticity for plates of this kind. We also generalize theoretical results of the investigation of crack propagation in deformable elastoplastic plates to plates with variable thickness and present solutions of new problems in the mechanics of structural arrest of propagating through cracks by local changes in the thickness of platelike structural members near the crack tip. Azerbaijan Technical University, Baku, Azerbaijan. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 3, pp. 46–54, May–June, 1996.     

A BEM solution for plates on elastic half-space with unilateral contact

The method developed to analyze the stress and deflection of plates on unilateral foundation is different from the traditional method. The behavior of plates on unilateral foundation belongs to that of free-boundary problems. It has been proved effective to solve the problems of free-boundary using the linear complementary equation method. In this paper, a boundary element–linear complementary equation is derived according to contact theory. This equation is used to analyze plate-bending on elastic half-space foundation, especially considering the impact on the internal force and displacement of plate, which is caused by the neighboring loads acting on the foundation around the plate. The effectiveness of the method is illustrated by numerical results.     

A semi-analytical solution for free vibration of variable thickness two-directional-functionally graded plates on elastic foundations

The present paper investigates free vibration of variable thickness two-directional-functionally graded circular plates, resting on elastic foundations. The results are obtained for clamped, free, and simply supported edge conditions. Variations of the material and geometrical parameters are monitored by five distinct exponential functions. Therefore, the resulted non-dimensional solution may be used for a wide range of the practical problems. Mindlin’s plate theory and the differential transformation technique are used to obtain the governing equations of the natural frequencies of the circular plates. Effects of variations of the material properties in the radial and thickness directions, geometric parameters (e.g., the thickness-to-radius ratio in the center of the plate), stiffness parameters of the foundation, and various boundary conditions on the natural frequencies are investigated. Results reveal that by choosing a suitable combination of the material properties, the free vibration behavior of the thick plates may be enhanced without the need to change the geometric parameters.     

Vibration analysis of radially FGM sectorial plates of variable thickness on elastic foundations

This paper deals with free vibration analysis of radially functionally graded circular and annular sectorial thin plates of variable thickness, resting on the Pasternak elastic foundation. Differential quadrature method (DQM) is used to yield natural frequencies of the circular/annular sectorial plates under simply-supported and clamped boundary conditions on the basis of the classical plate theory (CPT). The inhomogeneity of the plate is characterized by taking exponential variation of Young’s modulus and mass density of the material along the radial direction whereas Poisson’s ratio is assumed to remain constant. The validity of the present solution is first examined by studying the convergence of the frequency parameters. Then, a comparison of results with those available in literature confirms the excellent accuracy of the present approach. Afterwards, the frequency parameters of the circular/annular sectorial thin plates with uniform, linear, and quadratic variations in thickness are computed for different boundary conditions and various values of the material inhomogeneity constants, sector angles, and inner to outer radius ratios.     

Free vibration analysis of variable thickness thin plates by two-dimensional differential transform method

This paper aims at extending the application of two-dimensional differential transform method (2D-DTM) to study the free vibration of thin plates with arbitrarily varying thickness. First, the differential equation of motion governing thin plates with varying thickness is derived using Hamilton’s principle. Afterward, the 2D-DTM, a numerical method which is capable of reducing the size of computational work and can be applied to various types of differential equations, has been applied to derive the natural frequencies of variable thickness thin plates with different boundary conditions. Several numerical examples have been carried out to demonstrate the applicability and accuracy of the present method in free vibration analysis of both uniform plates and plates with variable thickness.     

Inelastic transient dynamic analysis of Reissner–Mindlin plates by the D/BEM

A direct domain/boundary element method (D/BEM) for dynamic analysis of elastoplastic Reissner–Mindlin plates in bending is developed. Thus, effects of shear deformation and rotatory inertia are included in the formulation. The method employs the elastostatic fundamental solution of the problem resulting in both boundary and domain integrals due to inertia and inelasticity. Thus, a boundary as well as a domain space discretization by means of quadratic boundary and interior elements is utilized. By using an explicit time‐integration scheme employed on the incremental form of the matrix equation of motion, the history of the plate dynamic response can be obtained. Numerical results for the forced vibration of elastoplastic Reissner–Mindlin plates with smooth boundaries subjected to impulsive loading are presented for illustrating the proposed method and demonstrating its merits. Copyright © 2000 John Wiley & Sons, Ltd.     

Thermal-stress analysis of plates with variable fiber spacing

Plane-stress problems of a square composite plate with variable fiber spacing under a uniform thermal loading are investigated. The problem with two edges perpendicular to the fiber direction of the plate, being itself constrained from normal displacement, is first solved analytically. It is then analyzed by the energy method together with the Rayleigh–Ritz approximation, and the computed results are verified by the analytical solution showing very good agreement. Another problem, with two edges parallel to the fiber direction, the plate again being constrained from normal displacement, which is more complex and cannot be solved analytically, is tackled by the afore-mentioned numerical procedure. To investigate the effect of the variation of fiber spacing on the distributions of the displacement-component fields and the equivalent stress field, two types of variation in fiber spacing are assumed in solving the second problem.     

Temperature fields and displacement of plates of linearly variable thickness

The problem of thermoelastic bending of plates of linearly variable thickness is solved by the method of a small parameter.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 49, No. 5, pp. 833–839, November, 1985.     

A BEM solution to transverse shear loading of beams

In this paper a boundary element method is developed for the solution of the general transverse shear loading problem of beams of arbitrary simply or multiply connected constant cross section. The analysis of the beam is accomplished with respect to a coordinate system that has its origin at the centroid of the cross section, while its axes are not necessarily the principal ones. The transverse shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. Two boundary value problems that take into account the effect of Poisson’s ratio are formulated with respect to stress functions and solved employing a pure BEM approach, that is only boundary discretization is used. The evaluation of the transverse shear stresses at any interior point is accomplished by direct differentiation of these stress functions, while both the coordinates of the shear center and the shear deformation coefficients are obtained from these functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The accuracy of the proposed shear deformation coefficients compared with those obtained from a 3-D FEM solution of the ‘exact’ elastic beam theory is remarkable.     

Thermal stability analysis of circular functionally graded sandwich plates of variable thickness using pseudo-spectral method

In the present study, the thermal stability of laminated functionally graded (FGM) circular plates of variable thickness subjected to uniform temperature rise based on the first-order shear deformation plate theory is presented. Furthermore, two models for FGM plates with variable thickness, corresponding with two manufacturing methods, are proposed. The laminated FGM plate with variable thickness is considered as a sandwich plate constituted of a homogeneous core of variable thickness and two constant thickness FGM face sheets whose material properties are assumed to be graded in the thickness direction according to a simple power law. In order to determine the distribution of the prebuckling thermal load along the radius, the membrane equation is solved using the shooting method. Subsequently, employing the pseudo-spectral method that makes use of Chebyshev polynomials, the stability equations are solved numerically to evaluate the critical temperature rise. The results demonstrate that the thermal stability is significantly influenced by the thickness variation profile, aspect ratio, the volume fraction index, and the core-to-face sheet thickness ratio.     

Dynamic analysis of inelastic plates by the D/BEM

A direct boundary element method is developed for the dynamic analysis of thin inelastic flexural plates of arbitrary planform and boundary conditions. It employs the static fundamental solution of the associated elastic problem and involves not only boundary integrals but domain integrals as well. Thus boundary as well as interior elements are employed in the numerical solution. Time integration is accomplished by the explicit algorithm of the central difference predictor method. A viscoplastic constitutive theory with state variables is employed to model the material behaviour. Numerical results are also presented to illustrate quantitatively the proposed method of solution.     

Numerical and experimental studies on aluminum sandwich plates of variable thickness

Abstract

The elastic flexural behavior of static deformation and free vibration of sandwich plates of variable thickness is investigated numerically and experimentally. In the analysis, the face plates are treated as Marguerre shells, and the core is assumed to be an antiplane core and to provide resistance to transverse shear and normal stresses only. Displacement continuity conditions are used at the interfaces between face plates and the core to derive the displacement field. Energy formulations are obtained and solved by the isoparametric finite element method. The numerical results are obtained to compare with the results in the existing literature and to show the effects of taper constant and face plate thickness on deflections and natural frequencies. Finally, experimental works based on the method of holographic interferometry are conducted to confirm the theoretical findings. Experimental and numerical data agree quite well in this work.     

Levy-type solution for buckling analysis of orthotropic plates based on two variable refined plate theory

This paper presents closed-form solution for buckling analysis of orthotropic plates using two variable refined plate theory. The theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Governing equations are derived from the principle of minimum total potential energy. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, loading condition, and variations of modulus ratio, aspect ratio, and thickness ratio on the critical buckling load of orthotropic plates are investigated and discussed in detail.     

Vibrations of orthotropic square plates having variable thickness (linear variation)

The effect of fibre orientation on the natural frequencies of thin square orthotropic plates having variable thickness is studied using the Rayleigh-Ritz method for various boundary conditions. In the present study plates having linear variation in thickness in one direction (either X or Y) are studied.     

A BEM formulation for linear bending analysis of plates reinforced by beams considering different materials

In this work, a boundary element method (BEM) formulation to perform linear bending analysis of building floor structures where slabs and beams can be defined with different materials is presented. The proposed formulation is based on Kirchhoff's hypothesis, the building floor being modelled by a zoned plate, where the beams are treated as thin sub-regions with larger rigidities. This composed structure is treated as a single body, the equilibrium and compatibility conditions being automatically taken into account. In the final integral equation, the tractions are eliminated along the interfaces, therefore reducing the number of degrees of freedom. The displacements are approximated along the beam cross-section, leading to a model where the values remain defined on the beam skeleton line instead of their boundaries. The accuracy of the proposed model is shown by comparing the numerical results with a well-known finite element code.     

A time-domain collocation-Galerkin BEM for transient dynamic crack analysis in anisotropic solids

This paper presents a time-domain boundary element method (BEM) for transient elastodynamic crack analysis in homogeneous and linear elastic solids of general anisotropy. A finite crack subjected to a transient loading is investigated. Two-dimensional (2D) generalized plane-strain or plane-stress condition is considered. The initial-boundary value problem is described by a set of hypersingular time-dependent traction boundary integral equations (BIEs), in which the crack-opening displacements (CODs) are unknown quantities. The hypersingular time-domain BIEs are first regularized to weakly singular ones by using spatial Galerkin method, which transfers the derivatives of the fundamental solutions to the unknown CODs and the weight functions. To solve the time-domain BIEs numerically, a time-stepping scheme is developed. The scheme applies the collocation method for temporal discretization of the time-domain BIEs. As spatial shape-functions, two different functions are implemented. For elements away from crack-tips, linear spatial shape-function is used, while for elements near the crack-tips a special ‘crack-tip shape-function’ is applied to describe the local ‘square-root’ behavior of the CODs at the crack-tips properly. Special attention of the analysis is devoted to the numerical computation of the transient elastodynamic stress intensity factors for cracks in general anisotropic and linear elastic solids. Numerical examples are presented to verify the accuracy of the present time-domain BEM.