Analytical treatment of boundary integrals in direct boundary element analysis of plate bending problems

A direct-type Boundary Element Method (BEM) for the analysis of simply supported and built-in plates is employed. The integral equations due to a combined biharmonic and harmonic governing equations are first established. The boundary integrals developed are then evaluated analytically. The domain integrals due to external body forces are also transformed over the boundary and subsequently evaluated analytically. Thus, it needs only the boundary to be discretized. Without loss of generality, the exact expression for the integrals would enhance the solution accuracy of the BEM. This is due to the fact that at locations where the fundamental solutions approach their singular points the value determined by numerical quadrature may be inconsistent and inaccurate. Also, another major advantage of the exact expressions for integrations is the insensitivity to the geometrical location of the source point on the boundary. The distribution of boundary quantities is approximated either over linear or quadratic boundary elements. General type of plate bending problems, with plates of different geometrical shapes supported simply or fixed can be handled. Loading may be applied point concentrated, uniformly distributed within the domain or over the boundary. Also, hydrostatic pressure can be applied. The results obtained by BEM in comparison with those obtained by analytical or other approximate solutions are found to be very accurate and the solution method is efficient.     

A general procedure transferring domain integrals onto boundary integrals in BEM

This paper is concerned with transferring to the boundary the domain type integrals occurring in the boundary integral approach applied to boundary value problems with nonzero body force terms. The framework used encloses many interesting engineering applications, e.g. elastostatics, heat conduction and magnetostatics. Beside this are pseudo plastic strains are incorporated due to interaction phenomena with relevant quantities. The proposed method is based upon the homogenization of the governing differential equation by using a particular solution of the inhomogeneous one. Various methods deriving such particular solutions are considered.     

Evaluation of boundary integrals for plate bending

The alternative to quadrature, as a procedure for dealing with the integrations required in the direct boundary element method (DBEM), is to carry out the integration analytically and code the results directly. The potential benefits are efficient computer programs; the avoidance of numerical instability; and generally, better accuracy. The technique is developed in this paper. Serious problems arise when Gauss quadrature is employed for the integration of functions which contain, or are close to singularities. A numerical integration approach may fail at the first stage of the analysis, that is, during the assembly of the discrete equations; or it may fail at the subsequent stage of computing domain points near the boundary. The severity of the problem is dependent both on the strength of the singularity, and on geometry. These points are illustrated with examples.     

A Galerkin formulation for shear deformable plate bending dynamics

A Galerkin boundary element formulation for shear deformable plate bending dynamics is developed. The formulation makes use of the static fundamental solutions for the weighted residual integral equations. The domain integrals carrying the inertia terms and generic static loads are considered as body forces and approximated with boundary values using the dual reciprocity method. The load is modelled as a series of impact loads of time varying intensity and moving in space in a predetermined path. The formulation was implemented and tested solving a benchmark problem. The results are compared with finite element solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

The BEM applied to plate bending elastoplastic analysis using Reissner''s theory

An application of the boundary element method (BEM) to plate bending elastoplastic analysis is presented. Reissner's plate bending theory, which caters to thin and thick plates, is considered.

First, the governing equations are shown, in which bending plastic strains are allowed for. Thereafter, the integral equations are presented, including those for moments and shear resultants at internal points. The numerical implementation is carried out using the integral equations discretized in quadratic boundary elements and constant internal cells. An incremental-iterative method is employed to solve the elastoplastic equations.

Numerical examples are presented at the end of the work to illustrate the applicability of the formulation.     

Plane stress and plate bending coupling in BEM analysis of shallow shells

In this paper the shear deformable shallow shells are analysed by boundary element method. New boundary integral equations are derived utilizing the Betti's reciprocity principle and coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Two techniques, direct integral method (DIM) and dual reciprocity method (DRM), are developed to transform domain integrals to boundary integrals. The force term is approximted by a set of radial basis functions. Several examples are presented to demonstrate the accuracy of the two methods. The accuracy of results obtained by using boundary element method are compared with exact solutions and the finite element method. Copyright © 2000 John Wiley & Sons, Ltd.     

Buckling analysis of shear deformable plates by boundary element method

In this paper, the derivation and numerical implementation of boundary integral equations for the buckling analysis of shear deformable plates are presented. Plate buckling equations are derived as a standard eigenvalue problem. The formulation is formed by coupling boundary element formulations of shear deformable plate and two dimensional plane stress elasticity. The eigenvalue problem of plate buckling yields the critical load factor and buckling modes. The domain integrals which appear in this formulation are treated in two different ways: initially the integrals are evaluated using constant cells, and next, they are transformed into equivalent boundary integrals using the dual reciprocity method (DRM). Several examples with different geometry, loading and boundary conditions are presented to demonstrate the accuracy of the formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

A new boundary element formulation for shear deformable shells analysis

A new domain‐boundary element formulation to solve bending problems of shear deformable shallow shells having quadratic mid‐surface is presented. By regrouping all the terms containing shells curvature and external loads together in equilibrium equation, the formulation can be formed by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. The boundary is discretized into quadratic isoparametric element and the domain is discretized using constant cells. Several examples are presented, and the results shows a good agreement with the finite element method. Copyright © 1999 John Wiley & Sons, Ltd.     

Winkler plate bending problems by a truly boundary-only boundary particle method

This paper makes the first attempt to use the boundary particle method (BPM) to solve the problems of Winkler plate under lateral loading. In this study, we find that the standard fundamental solution does not work well with the BPM. Instead we construct the modified singular fundamental solution, which satisfies the homogeneous governing equation of Winkler plate and is employed in the BPM to calculate the homogeneous solution. Unlike the other boundary discretization methods, the BPM does not require any inner nodes to evaluate the particular solution of inhomogeneous problems, since the method is a truly boundary-only meshfree technique by using the recursive composite multiple reciprocity technique. Our numerical experiments demonstrate efficiency and high accuracy of the BPM in the solution of Winkler plate bending problems.     

Stiffened plate bending analysis by the boundary element method

 In this work, the plate bending formulation of the boundary element method (BEM) based on the Kirchhoff's hypothesis, is extended to the analysis of stiffened elements usually present in building floor structures. Particular integral representations are derived to take directly into account the interactions between the beams forming grid and surface elements. Equilibrium and compatibility conditions are automatically imposed by the integral equations, which treat this composite structure as a single body. Two possible procedures are shown for dealing with plate domain stiffened by beams. In the first, the beam element is considered as a stiffer region requiring therefore the discretization of two internal lines with two unknowns per node. In the second scheme, the number of degrees of freedom along the interface is reduced by two by assuming that the cross-section motion is defined by three independent components only. Received 6 November 2000     

The boundary element method applied to orthotropic shear deformable plates

This work presents a formulation for thick plates following Mindlin theory. The fundamental solution takes into account an assumed displacement distribution on the thickness, and was derived by means of Hormander operator and the Radon transform. To compute the inverse Radon transform of the fundamental solution, some numerical integrals need to be computed. How these integrations are carried out is a key point in the performance of the boundary element code. Two approaches to integrate fundamental solutions are discussed. Integral equations are obtained using Betti's reciprocal theorem. Domain integrals are exactly transformed into boundary integrals by the radial integration technique.     

Buckling analysis of shear deformable shallow shells by the boundary element method

In this work a boundary element (BE) formulation for buckling problem of shear deformable shallow shells is presented. A set of five boundary integral equations are obtained by coupling two-dimensional plane stress elasticity with shear deformable plate bending (Reissner). The domain integrals appearing in the formulation (due to the curvature and due to the domain load) are transferred into equivalent boundary integrals. The BE formulation is presented as an eigenvalue problem, to provide direct evaluation of critical load factors and buckling modes. Several examples are presented. The BE results for a cylindrical shallow shell with different curvatures are compared with other numerical solutions and good agreements are obtained.     

A BEM formulation for inelastic transient dynamic analysis using domain decomposition and particular integrals

This study deals with the domain decomposition method and particular integrals for multi-region inelastic transient dynamic analysis. The particular integral formulation for single-region inelastic transient dynamic analysis is obtained by eliminating the acceleration volume integral and treating the initial stress term by volume cell. The Houbolt time integration scheme is used for the time- marching process. The Newton-Raphson algorithm for plastic multiplier is used to solve the system equation. In order to extend to multi-region problems, the domain decomposition method is examined. The domain of the original problem is subdivided into subregions. The interface boundary conditions are updated by using the iterative coupling employing Schwarz algorithm. Numerical results of two example problems are given to demonstrate the validity and accuracy of the present formulation.     

A conforming solution to curved plate bending problems

A curved triangular element for plate bending is developed, using the displacement method of formulation. The element has one outwardly curved side and conforms on an arbitrary quadratic boundary. To the user, the element appears as a single curved triangle. Internally, however, the element is subdivided into a straight-sided triangle and a ‘curved segment’. The straight-sided triangle is extensively documented in the literature^2-5 and is referred to in the paper as the Cowper triangle. The curved segment is a two-noded element with 12 degrees-of-freedom. It has one straight side and one curved side. Independent expansions are assumed in each domain and explicit shape functions derived. The theoretical solution to a few practical problems is contained in each of the two expansions. In such cases, the curved triangle produces perfect results. Otherwise, the accuracy is of the same order as that obtained in similar problems with straight boundaries. The method is easily extended to elements of higher order.     

Post buckling analysis of shear deformable shallow shells by the boundary element method

This paper presents four boundary element formulations for post buckling analysis of shear deformable shallow shells. The main differences between the formulations rely on the way non‐linear terms are treated and on the number of degrees of freedom in the domain. Boundary integral equations are obtained by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Four different sets of non‐linear integral equations are presented. Some domain integrals are treated directly with domain discretization whereas others are dealt indirectly with the dual reciprocity method. Each set of non‐linear boundary integral equations are solved using an incremental approach, where loads and prescribed boundary conditions are applied in small but finite increments. The resulting systems of equations are solved using a purely incremental technique and the Newton–Raphson technique with the Arc length method. Finally, the effect of imperfections (obtained from a linear buckling analysis) on the post‐buckling behaviour of axially compressed shallow shells is investigated. Results of several benchmark examples are compared with the published work and good agreement is obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

Solution of one mixed problem of plate bending for a domain with partially unknown boundary

The paper addresses the problem of finding a full-strength contour in the problem of plate bending for a cycle-symmetric doubly connected domain. An isotropic elastic plate, bounded by a regular polygon, is weakened by a required full-strength hole whose symmetry axes are the regular polygon diagonals. Rigid bars are attached to each component of the broken line of the outer boundary of the plate. The plate bends under the action of concentrated moments applied to the middle points of the bars. An unknown part of the boundary is free from external forces. Using the methods of complex analysis, the analytical image of Kolosov?CMuskhelishvili??s complex potentials (characterising an elastic equilibrium of the body) and of an unknown full-strength contour are determined. A numerical analysis is performed and the corresponding plots are obtained by means of the Mathcad system.     

The direct boundary element method in plate bending

The direct boundary element method based on the Rayleigh-Green identity is employed for the static analysis of Kirchhoff plates. The starting point is a slightly modified version of Stern's equations. The focus is on the implementation of the method for linear elements and a Hermitian interpolation for the deflection w. The concept of element matrices is developed and the Cauchy principal values of the singular integrals are given in detail. The treatment of domain integrals, the handling of internal supports, the properties of the solution and the effect of singularities are discused. Numerical examples illustrate the various techniques. In the appendix the influence functions for the second and third derivatives of the deflection w are given.     

Vibro-acoustic formulation of elastically restrained shear deformable stiffened rectangular plate

A method constructed on the basis of the Rayleigh–Ritz method and the first Rayleigh integral is presented for the vibro-acoustic analysis of elastically restrained shear deformable stiffened rectangular orthotropic plates. In the proposed method, the displacement fields of the plate and stiffeners are formulated on the basis of the first-order shear deformation theory. The theoretical sound pressure level (SPL) curve of the plate is constructed using the responses at different excitation frequencies and the first Rayleigh integral. The experimental SPL curve of an elastically restrained stiffened orthotropic plate was measured to verify the accuracy of the theoretical SPL curve of the plate. The effects of Young’s modulus ratio E₁/E₂ on the sound radiation characteristics of elastically restrained stiffened orthotropic plate with different aspect ratios are studied using the proposed method. It has been shown that the effects of Young’s modulus ratio become more prominent as the plate aspect ratio gets larger.