Boundary element analysis of unilateral supported Reissner plates on elastic foundations

 A boundary element–linear complementary equation method (BE–LCEM) is developed for the bending of thick plates with free edges on unilateral elastic foundations with particular emphasis on the non-contact phenomenon between the plates and the subgrades. The theory of thick plate was used, and three boundary conditions on free edge have been adopted. Following numerical discretization by using the boundary integral equation method for this contact problem, an effective linear complementary equation is then established with two complementary variables for each contact node. Complementary variables are taken as the normal contact force and the relative deflection between the plate and the foundation. The solution of which can be obtained using mathematical programming. A number of examples are presented to demonstrate the effectiveness of the features as implemented. Two types of foundations (Winkler and half-space) are examined and the method is shown to provide good agreement with available analytical solutions obtained by other investigators. Received 16 July 2000     

Boundary element formulations for Mindlin plate on an elastic foundation with dynamic load

Boundary element method (BEM) for a shear deformable plate (Reissner/Mindline's theories) resting on an elastic foundation subjected to dynamic load is presented. Formulations for both Winkler and Pasternak foundations are presented. The boundary element formulation in Laplace domain is presented together with complete expressions for the internal point kernels (i.e. fundamental solutions). Quadratic isoparameteric boundary elements are used to discretise the boundary of plate domain. Time domain variables are obtained by the Durbin's inversion method from transform domain. Numerical examples are presented to demonstrate the accuracy of the boundary element method and the comparisons are made with other numerical technique.     

A simple finite element for beams on elastic foundations

A simple "routine" beam on elastic foundation finite element using a polynomial displacement function has been developed which yields acceptably accurate deflection, shear and bending moment values for prismatic or non-prismatic beams of elastic material resting on foundations with varying or nonlinear subgrade reactions. Limited extension of the formulation to an "exact" finite element using the exact displacement function of a beam on elastic foundation has also been carried out. The subgrade is represented by a non-homogeneous solid medium to include nonlinear parameters if required. The iterative solution is extended to cases where the beam may uplift because the foundation is a no tension material. The model is also suitable for calculating the elastic deflections, membrane. and bending stress resultants for axisymmetrically loaded variable thickness shells of revolution. A computer program called FEBEF [finite element: beam on elastic foundation] incorporating the routine finite element has been prepared for the solution of beams on elastic foundations and axi symmetrically loaded shells of revolution.     

Nonlocal plate model for nonlinear analysis of thin films on elastic foundations in thermal environments

Postbuckling, nonlinear bending and nonlinear vibration analyses are presented for a simply supported stiff thin film resting on a two-parameter elastic foundation in thermal environments. The stiff thin film is modeled as a nonlocal orthotropic plate which contains small scale effects. The elastomeric substrate with finite depth is modeled as a two-parameter elastic foundation. The thermal effects are included and the material properties of the substrate are assumed to be temperature-dependent. The governing equation that includes plate-foundation interaction is solved by a two-step perturbation technique. The numerical results reveal that the small scale parameter e₀a reduces the postbuckling equilibrium paths, the static large deflections and natural frequencies, but increases the nonlinear to linear frequency ratios of the thin film slightly. The results also reveal that the effect of the small scale parameter is significant for compressive buckling, but less pronounced for static bending and marginal for free vibration of the thin film resting on an elastic foundation.     

Analysis of thin plates on elastic foundations with boundary element method

An alternative Integral Equation Formulation to deal with thin plates on elastic foundations is developed. The field equation is decomposed into two equations in partial derivatives of second order, which are formulated in an integral form by application of a reciprocity theorem. In this way, no divergent integrals appear in the formulation and the auxiliary function is the fundamental solution of Laplace equation. The domain integrals are transformed into equivalent boundary integrals, although in general it is necessary to introduce some internal points. Two examples will be studied to prove the efficiency of the formulation proposed.     

Vibrations of a pipe on elastic foundations

A cantilevered pipe subjected to external transverse (or lateral) force is investigated. Two cases of elastic foundations are considered: rotational and both linear and rotational. The major findings are the variations in frequency with flow velocity and displacements at different points and times     

A boundary element method for elastic buckling analysis of assembled plate structures

In this paper, we first discuss the integral equation formulation for the buckling problem of a single plate, using the biharmonic fundamental solution for the plate bending problems. The so called boundary-element method previously proposed by the senior author is applied to the numerical solution of the resulting set of integral equations. The total set of simultaneous equations are derived for nodal unknowns taken out of the whole domain, and reduced to an algebraic set of eigenvalue equations. The proposed method is method to the solution of elastic buckling of assembled plate structures. A few examples are computed and results obtained are compared with other solutions to demonstrate the potential usefulness of the proposed method.     

Frequency analysis of thick orthotropic plates on elastic foundation using a high precision triangular plate bending element

A high precision triangular thick orthotropic plate bending element on an elastic foundation is developed for the free vibration analysis of thick plates on elastic foundation. The element has three nodes with twelve degrees-of-freedom per node, and takes into account the shear deformation and rotatory inertia. The accuracy of the element is established by comparison of the natural frequencies of certain thick and thin plates, determined from a consistent mass matrix formulation, with available results.     

Vibration analysis of orthotropic rectangular plates on elastic foundations

The vibration responses of orthotropic plates on nonlinear elastic foundations are numerically modeled using the differential quadrature method. The differential quadrature technique is utilized to transform partial differential equations into a discrete eigenvalue problem. Numerical results and those from literature closely correspond to each other. Numerical results demonstrate that elastically restrained stiffness, plate aspect ratio and foundation stiffness significantly impact the dynamic behavior of orthotropic plates.     

Boundary element analysis of actual residual stresses in elastic plastic bodies under cyclic loading

The BEM was applied to the evaluation of an actual residual stress field in an elastic — perfectly plastic body subjected to any given cyclic loading. A new mechanical model describing this problem was used.

A corresponding discrete model based on both the direct and indirect versions of the BEM was proposed. The BEM approximation of stresses applied was similar to that one used in the elasto-plastic analysis in so called modified tractions and modified body forces algorithm. Some numerical tests have been carried out where BE solutions were compared with those obtained by means of the FE and FD methods. These comparisons show that the discrete model based on the BEM is efficient in the residual stress analysis.     

Boundary element formulation of axisymmetric problems for an elastic halfspace

Axisymmetric problems for an elastic halfspace are commonly analyzed by the boundary element (BE) method by employing the axisymmetric fundamental solution for the fullspace. In such cases, the discretization of the free surface is required, with its truncation at an appropriate location from the axis of symmetry. This paper presents the BE implementation of the axisymmetric fundamental solution for an elastic halfspace, given in terms of integrals of the Lipschitz–Hankel type, that satisfies in advance the boundary condition of zero traction on the free surface and the decay of displacements in the far field. Explicit equations for post-processing the results at internal points are provided, as well as adequate numerical schemes to evaluate the boundary integrals arising in the method. This formulation can be easily implemented in existing BE computational codes for axisymmetric fullspace problems, requiring only a few modifications. Numerical results are provided to validate the proposed formulation.     

DQEM for free vibration analysis of Timoshenko beams on elastic foundations

 A differential quadrature element method (DQEM) based on first order shear deformation theory is developed for free vibration analysis of non-uniform beams on elastic foundations. By decomposing the system into a series of sub-domains or elements, any discontinuity in loading, geometry, material properties, and even elastic foundations can be considered conveniently. Using this method, the vibration analysis of general beam-like structures is to be studied. The governing equations of each element, natural compatibility conditions at the interface of two adjacent elements and the external boundary conditions are developed in a systematic manner, using Hamilton's principle. The present DQEM is to be implemented to Timoshenko beams resting on partially supported elastic foundations with various types of boundary conditions under the action of axial loading. The general versality, accuracy, and efficiency of the presented DQEM are demonstrated having solved different examples and compared to the exact or other numerical procedure solutions. Received: 11 October 2002/Accepted: 26 November 2002     

Analysis of plates on elastic foundations

The modification required to provide plate on elastic foundation analysis capability with existing finite element algorithms is presented. As shown by an example, the modification is applicable to configurations with varying ratios of plate to foundation stiffness.     

A triangular plate element for thermo‐elastic analysis of sandwich panels with a functionally graded core

A sandwich construction is commonly composed of a single soft isotropic core with relatively stiff orthotropic face sheets. The stiffness of the core may be functionally graded through the thickness in order to reduce the interfacial shear stresses. In analysing sandwich panels with a functionally gradient core, the three‐dimensional conventional finite elements or elements based on the layerwise (zig‐zag) theory can be used. Although these elements accurately model a sandwich panel, they are computationally costly when the core is modelled as composed of several layers due to its grading material properties. An alternative to these elements is an element based on a single‐layer plate theory in which the weighted‐average field variablescapture the panel deformation in the thickness direction. This study presents a new triangular finite element based on {3,2}‐order single‐layer theory for modelling thick sandwich panels with or without a functionally graded core subjected to thermo‐mechanical loading. A hybrid energy functional is employed in the derivation of the element because of a C¹ interelement continuity requirement. The variations of temperature and distributed loading acting on the top and bottom surfaces are non‐uniform. The temperature also varies arbitrarily through the thickness. Copyright © 2006 John Wiley & Sons, Ltd.     

Boundary element discretization of plane elasticity and plate bending problems

The paper deals with the discretization of the integral equations arising in the boundary formulation of plane elasticity and plate bending problems. Particular attention is paid to the efficiency of the interpolation used in approximating the boundary quantities and to the precision and computational convenience in evaluating the boundary integrals. The proposed discretization model is based on the use of a quadratic B-spline approximation to represent the boundary variables and on the results from the analytical integration to compute the boundary coefficients. The advantages are those of accuracy and the saving of computer time. Some numerical results allow an analysis of the performance of the model.     

Boundary element modelling of flat plate floors under vertical loading

The present paper develops a boundary element model for flat plate floors. The floor slab is modelled using the shear‐deformable plate bending theory. Internal columns or walls are treated using internal collocation technique, where three interaction generalized forces are considered at the slab–column connection: two bending moments in two directions and shear force in the vertical direction. Such forces are considered to be constant over the column cross‐sections. The present technique takes into account the realistic geometric modelling of the column cross‐sections. The effect of considering such geometry in the analysis of the bending moments transferred from slab to columns is studied. Several examples, including solution of practical building slab, are presented. The results are compared to those obtained from other numerical methods to demonstrate the accuracy and the reliability of the present formulation. The present formulation can be considered as an accurate tool to predict moment transferred from the slab to the column. Copyright © 2004 John Wiley & Sons, Ltd.     

Boundary element analysis of Navier-Stokes equations

As arrangements, the fundamental solutions of anisotropic convective diffusion equations of transient incompressible viscous fluid flow and boundary elements analysis of the diffusion equation are presented. Secondly, by considering that convective diffusion equations and Navier-Stokes equations are mathematical formulations of mass and momentum conservation law respectively, and that consequently, both physical contents and equation styles are analogous, boundary integral formulations for Navier-Stokes equations are proposed on the basis of formulation of diffusion equations.     

Boundary element analysis of dielectric waveguides

We apply the boundary element method to the analysis of optical waveguides. After summarizing constant and linear element algorithms for both two- and three-dimensional simulations, we introduce a new recursive series procedure for constructing the diagonal matrix elements. We then demonstrate that our method can be employed to minimize the reflectivity of optical waveguide antireflection coatings with both straight and angled facets.