A BEM-based meshless solution to buckling and vibration problems of orthotropicplates

In this paper a BEM-based meshless solution is presented to buckling and vibration problems of Kirchhoff orthotropic plates with arbitrary shape. The plate is subjected to compressive centrally applied load together with arbitrarily transverse distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting buckling and vibration problems are described by partial differential equations in terms of the deflection. Both problems are solved employing the Analog Equation Method (AEM). According to this method the fourth-order partial differential equation describing the response of the orthotropic plate is converted to an equivalent linear problem for an isotropic plate subjected only to a fictitious load under the same boundary conditions. The AEM is applied to the problem at hand as a boundary-only method by approximating the fictitious load with a radial basis function series. Thus, the method retains all the advantages of the pure BEM using a known simple fundamental solution. Example problems are presented for orthotropic plates, subjected to compressive or vibratory loading, to illustrate the method and demonstrate its efficiency and its accuracy.     

Nonlinear analysis of non-uniform beams on nonlinear elastic foundation

In this paper a boundary integral equation solution to the nonlinear problem of non-uniform beams resting on a nonlinear triparametric elastic foundation is presented, which permits also the treatment of nonlinear boundary conditions. The nonlinear subgrade model which describes the foundation includes the linear and nonlinear Winkler (normal) parameters and the linear Pasternak (shear) foundation parameter. The governing equations are derived in terms of the displacements for nonlinear analysis in the deformed configuration and for linear analysis in the undeformed one. Moreover, as the cross-sectional properties of the beam vary along its axis, the resulting coupled nonlinear differential equations have variable coefficients which complicate the mathematical problem even more. Their solution is achieved using the analog equation method of Katsikadelis. Several beams are analyzed under various boundary conditions and load distributions, which illustrate the method and demonstrate its efficiency and accuracy. Finally, useful conclusions are drawn from the investigation of the nonlinear response of non-uniform beams resting on nonlinear elastic foundation.     

A boundary element solution to the soap bubble problem

The boundary element method (BEM) is applied to the soap bubble problem, that is to the problem of determining the surface that a soap bubble constrained by bounding contours assumes under the action of molecular forces. This is also the shape of a uniformly stretched membrane bounded by one or more non-intersecting curves. As the slopes of the membrane surface are finite, their square can not be neglected and the resulting governing equation is non-linear. The problem is solved using the analogue equation method (AEM). According to this method the non-linear membrane is substituted by a linear one subjected to a fictitious transverse load. The fictitious load is established using the BEM. Numerical examples are presented which illustrate the method and demonstrate its accuracy. This application of the BEM to non-linear problems shows that BEM is a versatile computational method for all-purpose use in engineering analysis. The solution of the problem at hand is very important in engineering, since the soap bubble surface can be used as the best initial form for membrane roofs.     

The ponding problem on elastic membranes: an analog equation solution

 In this paper, the nonlinear response of elastic membranes with arbitrary shape under partial and full ponding loads has been analyzed. Large deflections are considered, which result from nonlinear kinematic relations. The problem is formulated in terms of the displacements components and the three coupled nonlinear governing equations are solved using the analog equation method (AEM). The membrane may be prestressed either by prescribed boundary displacements or tractions. Using the concept of the analog equation the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on the BEM and the displacement components as well as the stress resultants at any point of the membrane are evaluated from their integral representations. In addition to the geometrical nonlinearity, the ponding problem is itself nonlinear, because the ponding load depends on the deflection surface that it produces. Iterative schemes are developed which converge to the equilibrium state of the membrane under the ponding loads. Several membranes are analyzed which illustrate the method and demonstrate its efficiency and accuracy. The method has all the advantages of the pure BEM, since the discretization and integration is limited only to the boundary. Received 28 July 2001     

Post-buckling analysis of viscoelastic plates with fractional derivative models

The post-buckling response of thin plates made of linear viscoelastic materials is investigated. The employed viscoelastic material is described with fractional order time derivatives. The governing equations, which are derived by considering the equilibrium of the plate element, are three coupled nonlinear fractional partial evolution type differential equations in terms of three displacements. The nonlinearity is due to nonlinear kinematic relations based on the von Kármán assumption. The solution is achieved using the analog equation method (AEM), which transforms the original equations into three uncoupled linear equations, namely a linear plate (biharmonic) equation for the transverse deflection and two linear membrane (Poisson’s) equations for the inplane deformation under fictitious loads. The resulting initial value problem for the fictitious sources is a system of nonlinear fractional ordinary differential equations, which is solved using the numerical method developed recently by Katsikadelis for multi-term nonlinear fractional differential equations. The numerical examples not only demonstrate the efficiency and validate the accuracy of the solution procedure, but also give a better insight into this complicated but very interesting engineering plate problem     

The boundary element method for nonlinear problems

In this paper a boundary-only boundary element method (BEM) is developed for solving nonlinear problems. The presented method is based on the analog equation method (AEM). According to this method the nonlinear governing equation is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. The solution of the substitute equation is obtained as a sum of the homogeneous solution and a particular one of the nonhomogeneous. The nonhomogeneous term, which is an unknown fictitious domain source distribution, is approximated by a truncated series of radial base functions. Then, using BEM the field function and its derivatives involved in the governing equation are expressed in terms of the unknown series coefficients, which are established by collocating the equation at discrete points in the interior of the domain. Thus, the presented method becomes a boundary-only method in the sense that only boundary discretization is required. The additional collocation points inside the domain do not spoil the pure BEM character of the method. Numerical results for certain classical nonlinear problems are presented, which validate the effectiveness and the accuracy of the proposed method.     

Nonlinear bending of FGM plates subjected to combined loading and resting on elastic foundations

A nonlinear bending analysis is presented for a simply supported, functionally graded plate resting on an elastic foundation of Pasternak-type. The plate is exposed to elevated temperature and is subjected to a transverse uniform or sinusoidal load combined with initial compressive edge loads. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The formulations are based on a higher-order shear deformation plate theory and general von Kármán-type equation that includes the plate-foundation interaction and thermal effects. A two step perturbation technique is employed to determine the load–deflection and load–bending moment curves. The numerical illustrations concern nonlinear bending response of functional graded plates with two constituent materials resting on Pasternak elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The results reveal that the characteristics of nonlinear bending are significantly influenced by foundation stiffness, temperature rise, transverse shear deformation, the character of in-plane boundary conditions and the amount of initial compressive load. In contrast, the effect of volume fraction index N becomes weaker when the plate is supported by an elastic foundation.     

The analog equation method for large deflection analysis of membranes. A boundary-only solution

 In this paper the analog equation method (AEM) is applied to nonlinear analysis of elastic membranes with arbitrary shape. In this case the transverse deflections influence the inplane stress resultants and the three partial differential equations governing the response of the membrane are coupled and nonlinear. The present formulation, being in terms of the three displacements components, permits the application of geometrical inplane boundary conditions. The membrane is prestressed either by prescribed boundary displacements or by tractions. Using the concept of the analog equation the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on BEM and the displacement components as well as the stress resultants are evaluated from their integral representations at any point of the membrane. Several membranes are analyzed which illustrate the method and demonstrate its efficiency and accuracy. Moreover, useful conclusions are drawn for the nonlinear response of the membranes. The method has all the advantages of the pure BEM, since the discretization and integration is limited only to the boundary. Received 21 November 2000     

Nonlinear dynamic response of laminated plates resting on nonlinear elastic foundations by the discrete singular convolution-differential quadrature coupled approaches

In this paper, nonlinear dynamic response of rectangular laminated composite plate resting on nonlinear Pasternak type elastic foundations is investigated. First-order shear deformation theory (FSDT) is used for modeling of moderately thick plates. The plate formulation is based on the von Karman nonlinear equation. The resulting nonlinear governing equations for transient analysis of laminated plates on elastic foundation are integrated using the discrete singular convolution-differential quadrature coupled approaches. The nonlinear governing equations of motion of plate are discretized in space and time domains using the discrete singular convolution and the differential quadrature methods, respectively. The validity of the present method is demonstrated by comparing the present results with those available in the open literature. The effects of the foundation parameters, boundary conditions and geometric parameters of plates on nonlinear dynamic response of laminated thick plates are investigated.     

A meshless method for generalized linear or nonlinear Poisson-type problems

This paper presents a meshless method, based on coupling virtual boundary collocation method (VBCM) with the radial basis functions (RBF) and the analog equation method (AEM), to analyze generalized linear or nonlinear Poisson-type problems. In this method, the AEM is used to construct equivalent equations to the original differential equation so that a simpler fundamental solution of the Laplacian operator, instead of other complicated ones which are needed in conventional BEM, can be employed. While global RBF is used to approximate fictitious body force which appears when the analog equation method is introduced, and VBCM are utilized to solve homogeneous solution based on the superposition principle. As a result, a new meshless method is developed for solving nonlinear Poisson-type problems. Finally, some numerical experiments are implemented to verify the efficiency of the proposed method and numerical results are in good agreement with the analytical ones. It appears that the proposed meshless method is very effective for nonlinear Poisson-type problems.     

A BEM based domain decomposition method for nonlinear analysis of elastic space membranes

In this paper the Domain Decomposition Method (DDM) is developed for nonlinear analysis of both flat and space elastic membranes of complicated geometry which may have holes. The domain of the projection of the membrane on the xy plane is decomposed into non-overlapping subdomains and the membrane problem is solved sequentially in each subdomain starting from zero displacements on the virtual boundaries. The procedure is repeated until the traction continuity conditions are also satisfied on the virtual boundaries. The membrane problem in each subdomain is solved using the Analog Equation Method (AEM). According to this method the three coupled strongly nonlinear partial differential equations, governing the response of the membrane, are replaced by three uncoupled linear membrane equations (Poisson's equations) subjected to fictitious sources under the same boundary conditions. The fictitious sources are established using a meshless BEM procedure. Example problems are presented, for both flat and space membranes, which illustrate the method and demonstrate its efficiency and accuracy.     

Dynamic response of finite sized elastic runways subjected to moving loads: A coupled BEM/FEM approach

The transient response of a finite elastic plate, resting on an elastic half-space, and subjected to moving loads is considered here. Both the cases of an elastic foundation alone, as well as a finite sized elastic plate resting on an elastic foundation are considered. The numerical methods employed are: (1) the time-domain boundary element method for the elastic foundation and (2) a combination of the time-domain boundary element method for the soil and the semi-discrete finite element method for the finite sized elastic plate. Both constant as well as linear-time-interpolation schemes are included in the BEM. The integration is carried out analytically in time. The analytical solution for a moving point load on an infinite elastic plate resting on an elastic half-space is derived here. This is used as a benchmark against which the present numerical solution is compared with. The accuracy of the numerical method is also verified by comparing the solutions with some existing numerical results; the comparison with the solutions based on a Winkler foundation model reveals the limitations of the applicability of such a model, especially in the cases of high velocities of the moving load. This is because neither the inertia of the foundation, nor the behaviour of the foundation as a continuum, can be properly accounted for in Winkler's model. A parametric study is conducted, and the influences of velocity of the moving load, load distribution, etc. on the dynamic response of the soil/runway system are investigated. Furthermore, the present computational method is applied to the problem of a transport airplane taxiing on a concrete pavement resting on a typical soil. The responses of pavements are presented for different taxiing velocities.     

A generalized Ritz method for partial differential equations in domains of arbitrary geometry using global shape functions

A boundary element method (BEM)-based variational method is presented for the solution of elliptic PDEs describing the mechanical response of general inhomogeneous anisotropic bodies of arbitrary geometry. The equations, which in general have variable coefficients, may be linear or nonlinear. Using the concept of the analog equation of Katsikadelis the original equation is converted into a linear membrane (Poisson) or a linear plate (biharmonic) equation, depending on the order of the PDE under a fictitious load, which is approximated with radial basis function series of multiquadric (MQ) type. The integral representation of the solution of the substitute equation yields shape functions, which are global and satisfy both essential and natural boundary conditions, hence the name generalized Ritz method. The minimization of the functional that produces the PDE as the associated Euler–Lagrange equation yields not only the Ritz coefficients but also permits the evaluation of optimal values for the shape parameters of the MQs as well as optimal position of their centers, minimizing thus the error. If a functional does not exists or cannot be constructed as it is the usual case of nonlinear PDEs, the Galerkin principle can be applied. Since the arising domain integrals are converted into boundary line integrals, the method is boundary-only and, therefore, it maintains all the advantages of the pure BEM. Example problems are studied, which illustrate the method and demonstrate its efficiency and great accuracy.     

Nonlinear analysis of sandwich plates with FGM face sheets resting on elastic foundations

Nonlinear vibration, nonlinear bending and postbuckling analyses are presented for a sandwich plate with FGM face sheets resting on an elastic foundation in thermal environments. The material properties of FGM face sheets are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equation of the plate that includes plate-foundation interaction is solved by a two-step perturbation technique. The thermal effects are also included and the material properties of both FGM face sheets and homogeneous core layer are assumed to be temperature-dependent. The numerical results reveal that the foundation stiffness and temperature rise have a significant effect on the natural frequency, buckling load, postbuckling and nonlinear bending behaviors of sandwich plates. The results also reveal that the core-to-face sheet thickness ratio and the volume fraction distribution of FGM face sheets have a significant effect on the natural frequency, buckling load and postbuckling behavior of the sandwich plate, whereas this effect is less pronounced for the nonlinear bending, and is marginal for the nonlinear to linear frequency ratios of the same sandwich plate.     

The analog equation method for large deflection analysis of heterogeneous orthotropic membranes: a boundary-only solution

In this paper, the analog equation method (AEM) is applied to nonlinear analysis of heterogeneous orthotropic membranes with arbitrary shape. In this case, the transverse deflections influence the in-plane stress resultants and the three partial differential equations governing the response of the membrane are coupled and nonlinear with variable coefficients. The present formulation, being in terms of the three displacement components, permits the application of geometrical in-plane boundary conditions. The membrane may be prestressed either by prescribed boundary displacements or by tractions. Using the concept of the analog equation, the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on the BEM and the displacement components as well as the stress resultants are evaluated from their integral representations at any point of the membrane. Several membranes are analyzed which illustrate the method, and demonstrate its efficiency and accuracy. Moreover, useful conclusions are drawn for the nonlinear response of heterogeneous anisotropic membranes. The method has all the advantages of the pure BEM, since the discretization and integration are limited only to the boundary.     

Fourier series solution for a rectangular thick plate with free edges on an elastic foundation

A Fourier series solution is presented for a system of first-order partial differential equations which describe the linear elastic behaviour of a thick rectangular plate resting on an elastic foundation and carrying an arbitrary transverse load. The lateral edges of the plate are unstressed. A central step in the method for solving the system of equations is to combine a complementary function with a particular solution of the system in order to satisfy the boundary conditions. The complementary function is the sum of two series. The terms of the first series are products of a Fourier term in one space variable with the solution of an eigenvalue problem in the other space variable. The second series is similar and comes from reversing the roles of the space variables.     

Three‐dimensional vibration analysis of rectangular thick plates on Pasternak foundation

The free‐vibration characteristics of rectangular thick plates resting on elastic foundations have been studied, based on the three‐dimensional, linear and small strain elasticity theory. The foundation is described by the Pasternak (two‐parameter) model. The Ritz method is used to derive the eigenvalue equation of the rectangular plate by augmenting the strain energy of the plate with the potential energy of the elastic foundation. The Chebyshev polynomials multiplied by a boundary function are selected as the admissible functions of the displacement functions in each direction. The approach is suitable for rectangular plates with arbitrary boundary conditions. Convergence and comparison studies have been performed on square plates on elastic foundations with different boundary conditions. It is shown that the present method has a rapid convergent rate, stable numerical operation and very high accuracy. Parametric investigations on the dynamic behaviour of clamped square thick plates on elastic foundations have been carried out in detail, with respect to different thickness–span ratios and foundation parameters. Some results found for the first time have been given and some important conclusions have been drawn. Copyright © 2004 John Wiley & Sons, Ltd.     

Free vibration analysis of functionally graded plates resting on Winkler–Pasternak elastic foundations using a new shear deformation theory

Free vibration analysis of simply supported functionally graded plates (FGP) resting on a Winkler–Pasternak elastic foundation are examined by a new higher shear deformation theory in this paper. Present theory exactly satisfies stress boundary conditions on the top and the bottom of the plate. The material properties change continuously through the thickness of the plate, which can vary according to power law, exponentially or any other formulations in this direction. The equation of motion for FG rectangular plates resting on elastic foundation is obtained through Hamilton’s principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. The numerical results obtained through the present analysis for free vibration of functionally graded plates on elastic foundation are presented, and compared with the ones available in the literature.     

Buckling Analysis of Plates Stiffened by Parallel Beams

In this paper a general solution for the elastic buckling analysis of plates stiffened by arbitrarily placed parallel beams of arbitrary doubly symmetric cross section subjected to an arbitrary inplane loading is presented. According to the proposed model, the stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. These tractions are integrated with respect to each half of the interface width resulting two interface lines, along which the loading of the beams as well as the additional loading of the plate is defined. The unknown distribution of the aforementioned integrated tractions is established by applying continuity conditions in all directions at the two interface lines, while the analysis of both the plate and the beams is accomplished on their deformed shape. The method of analysis is based on the capability to establish the elastic and the corresponding geometric stiffness matrices of the stiffened plate with respect to a set of nodal points. Thus, the original eigenvalue problem for the differential equation of buckling is converted into a typical linear eigenvalue problem, from which the buckling loads are established numerically. For the calculation of the elastic and geometric stiffness matrices six boundary value problems are formulated and solved using the Analog Equation Method (AEM), a BEM-based method. Numerical examples with practical interest are presented. The accuracy of the results of the proposed model compared with those obtained from a 3-D FEM solution is remarkable.     

Analysis of plates stiffened by parallel beams

In this paper a general solution for the analysis of plates stiffened by parallel beams subjected to an arbitrary loading is presented. According to the proposed model, the stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. The aforementioned integrated tractions result in the loading of the beams as well as the additional loading of the plate. Their distribution is established by applying continuity conditions in all directions at the interfaces. The analysis of both the plate and the beams is accomplished on their deformed shape taking into account second‐order effects. Six boundary value problems with respect to the plate transverse deflection, to the plate inplane displacement components, to the beam transverse deflections, to the beam axial deformation and to the beam non‐uniform angle of twist are formulated and solved using the analog equation method (AEM), a boundary element method (BEM)‐based method employing a boundary integral equation approach. The solution of the aforementioned plate and beam problems, which are non‐linearly coupled, is achieved using iterative numerical methods. The adopted model describes better the actual response of the plate beams system and permits the evaluation of the shear forces at the interfaces in both directions, the knowledge of which is very important in the design of prefabricated ribbed plates. The evaluated lateral deflections of the plate–beams system are found to exhibit considerable discrepancy from those of other models, which neglect inplane and axial forces and deformations. Copyright © 2006 John Wiley & Sons, Ltd.