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     

Ponding on floating membranes

Membranes subjected to ponding loads and floating on a liquid are analyzed. The initially flat membrane, which may be prestressed by edge in-plane tractions or displacements, is subjected to the weight of a liquid (e.g. rain water) filling the space created by the deflection of the membrane. Large deflections of membranes are considered, which result from nonlinear kinematic relations. The three coupled nonlinear equations in terms of the displacements governing the response of the membrane are solved using the analog equation method, which reduces the problem to the solution of three uncoupled Poisson's equations with fictitious sources. The problem is strongly nonlinear. In addition to the geometrical nonlinearity, the ponding problem is itself nonlinear, because the ponding load and the liquid reaction are not a priori known as they depend on the produced deflection surface. Iterative schemes are developed which converge to the equilibrium state of the membrane. Example problems are presented, which illustrate the method and demonstrate its efficiency. The method has all the advantages of the pure BEM.     

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.     

Dynamic analysis of nonlinear membranes by the analog equation method: a boundary-only solution

 A boundary-only solution is presented for dynamic analysis of elastic membranes under large deflections. The solution procedure is based on the analog equation method (AEM). According to this method, the three coupled nonlinear second order hyperbolic partial differential equations in terms of displacements, which govern the response of the membrane, are replaced with three Poisson's quasi-static equations under fictitious time dependent sources. The fictitious sources are established using a BEM-based procedure and the displacements as well as the stress resultants at any point are evaluated from their integrals representations. Numerical examples are presented which illustrate the method. Received 16 December 2000 / Accepted 25 April 2002     

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.     

Geometrically nonlinear analysis of elastic membranes with embedded rigid inclusions

In this paper the deformation of membranes containing rigid inclusions is analyzed. These rigid inclusions can significantly change the entire stress distribution in the membrane and therefore create major difficulties for the design. The initially flat membrane, which may be prestretched by boundary in-plane tractions or displacements, is subjected to externally applied loads and to the weight of the rigid inclusions. The composite system is examined in cases where its deformation reaches a state for which the undeformed and deformed shapes are substantially different. In such cases large deflections of membranes are considered, which result from nonlinear kinematic relations. The three coupled nonlinear equations in terms of the displacements governing the response of the membrane are solved using the analog equation method, which reduces the problem to the solution of three uncoupled Poisson's equations with fictitious domain source densities. The problem is strongly nonlinear [Katsikadelis JT, Nerantzaki MS. The boundary element method for nonlinear problems. Eng Anal Boundary Elements 1999;23:365–73]. In addition to the geometrical nonlinearity, the problem is itself nonlinear, since the membrane's reactions on the boundary of the rigid inclusions are not a priori known as they depend on the produced deflection surface. Iterative schemes are developed for calculation of deformed membrane's configuration, which converge to the final equilibrium state of the membrane with the given external applied loads. Several example problems are presented, which illustrate the method and demonstrate its accuracy and efficiency. The method employed for the solution is boundary only with all the advantages of the pure BEM.     

Nonlinear dynamic analysis of heterogeneous orthotropic membranes by the analog equation method

In this paper the analog equation method, a BEM-based method, is employed to analyze the dynamic response of flat heterogeneous orthotropic membranes of arbitrary shape, undergoing large deflections. The problem is formulated in terms of the three displacement components. Due to the heterogeneity of the membrane, the elastic constants are position dependent and consequently the coefficients of the partial differential equations governing the dynamic equilibrium of the membrane are variable. Using the concept of the analog equation, the three-coupled nonlinear second order hyperbolic partial differential equations are replaced with three uncoupled Poisson's quasi-static equations with fictitious time dependent sources. The fictitious sources are represented by radial basis functions series and are established using a BEM-based procedure. Both free and forced vibrations are considered. Membranes of various shapes are analyzed to illustrate the merits of the method as well as its applicability, efficiency and accuracy. The proposed method is boundary-only in the sense that the discretization and the integration are restricted on the boundary. Therefore, it maintains all the advantages of the pure BEM.     

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     

Large deflection analysis of elastic space membranes

In this paper a solution to the problem of elastic space (initially non‐flat) membranes is presented. A new formulation of the governing differential equations is presented in terms of the displacements in the Cartesian co‐ordinates. The reference surface of the membrane is the minimal surface. The problem is solved by direct integration of the differential equations using the analogue equation method (AEM). According to this method the three coupled non‐linear partial differential equations with variable coefficients are replaced with three uncoupled equivalent linear flat membrane equations (Poisson's equations) subjected to unknown sources under the same boundary conditions. Subsequently, the unknown sources are established using a procedure based on the BEM. The displacements as well as the stress resultants are evaluated at any point of the membrane from their integral representations of the solution of the substitute problems, which are used as mathematical formulae. Several membranes are analysed which illustrate the method and demonstrate its efficiency and accuracy as compared with analytical and existing numerical methods. Copyright © 2005 John Wiley & Sons, Ltd.     

A boundary element method solution for anisotropic nonhomogeneous elasticity

A new BEM approach is presented for the plane elastostatic problem for nonhomogeneous anisotropic bodies. In this case the response of the body is described by two coupled linear second order partial differential equations in terms of displacement with variable coefficient. The incapability of establishing the fundamental solution of the governing equations is overcome by uncoupling them using the concept of analog equation, which converts them to two Poisson’s equations, whose fundamental solution is known and the necessary boundary integral equations are readily obtained. This formulation introduces two additional unknown field functions, which physically represent the two components of a fictitious source. Subsequently, they are determined by approximating them globally with radial basis functions series. The displacements and the stresses are evaluated from the integral representation of the solution of the substitutes equations. The presented method maintains the pure boundary character of the BEM. The obtained numerical results demonstrate the effectiveness and accuracy of the method.     

Nonlinear analysis of elastic space cable-supported membranes

In this paper a solution method is presented for the coupled problem of elastic flat or space membranes supported by elastic flexible cables. Both membrane and cable undergo large deflections. Starting from the minimal surface the membrane is prestressed by imposed boundary displacements under the self-weight. Then an iterative procedure is employed, which consists in solving the membrane and the cable large deflection problems separately in each iteration step and checking the continuity of displacements and forces between membrane and cable. The procedure is repeated until convergence is achieved. Both membrane and cable problems are solved using the analog equation method (AEM). The displacements as well as the stress resultants are evaluated at any point of the membrane and the cable from the integral representations of the solution of the analog equations, which are used as mathematical formulae. Example problems are presented, for both flat and space membranes, which illustrate the method and demonstrate its efficiency and accuracy.     

Analysis of cylindrical shell panels. A meshless solution

The Meshless Analog Equation Method, a purely meshless method, is applied to the static analysis of cylindrical shell panels. The method is based on the concept of the analog equation of Katsikadelis, which converts the three governing partial differential equations in terms of displacements into three substitute equations, two of second order and one fourth order, under fictitious sources. The fictitious sources are represented by series of radial basis functions of multiquadric type. Thus the substitute equations can be directly integrated. This integration allows the representation of the sought solution by new radial basis functions, which approximate accurately not only the displacements but also their derivatives involved in the governing equations. This permits a strong formulation of the problem. Thus, inserting the approximate solution in the differential equations and in the associated boundary conditions and collocating at a predefined set of mesh-free nodal points, a system of linear equations is obtained, which gives the expansion coefficients of radial basis functions series that represent the solution. The minimization of the total potential of the shell results in the optimal choice of the shape parameter of the radial basis functions. The method is illustrated by analyzing several shell panels. The studied examples demonstrate the efficiency and the accuracy of the presented method.     

Large deflection analysis of beams with variable stiffness

Summary. In this paper, the Analog Equation Method (AEM), a BEM-based method, is employed to the nonlinear analysis of a Bernoulli-Euler beam with variable stiffness undergoing large deflections, under general boundary conditions which maybe nonlinear. As the cross-sectional properties of the beam vary along its axis, the coefficients of the differential equations governing the equilibrium of the beam are variable. The formulation is in terms of the displacements. The governing equations are derived in both deformed and undeformed configuration and the deviations of the two approaches are studied. Using the concept of the analog equation, the two coupled nonlinear differential equations with variable coefficients are replaced by two uncoupled linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under fictitious load distributions. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Several beams are analyzed under various boundary conditions and loadings to illustrate the merits of the method as well as its applicability, efficiency and 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.     

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.     

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.     

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 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.     

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.     

Flexural-torsional Vibrations of Beams by BEM

In this paper a boundary element method is developed for the general flexural-torsional vibrations of Euler–Bernoulli beams of arbitrarily shaped constant cross-section. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved using the analog equation method, a BEM based method. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. The general character of the proposed method is verified from the fact that all basic equations are formulated with respect to an arbitrary coordinate system, which is not restricted to the principal one. Both free and forced vibrations are examined. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The range of applicability of the thin-tube theory is also investigated through examples with great practical interest.