Local integral equation method for viscoelastic Reissner–Mindlin plates

A meshless local Petrov-Galerkin (MLPG) method is applied to solve static and dynamic bending problems of linear viscoelastic plates described by the Reissner–Mindlin theory. To this end, the correspondence principle is applied. A weak formulation for the set of governing equations in the Reissner–Mindlin theory with a unit test function is transformed into local integral equations on local subdomains in the mean surface of the plate. Nodal points are randomly spread on the mean surface of the plate and each node is surrounded by a circular subdomain to which local integral equations are applied. A meshless approximation based on the moving least-squares (MLS) method is employed in the numerical implementation.     

BEM formulation for reinforced plates

This article presents a BEM formulation developed to analyse reinforced plate bending. The reinforcements are formulated using a simplified scheme based on applying an initial moment field adopted to locally correct the stiffness of the reinforcement regions. The domain integrals due to the presence of the reinforcements are then transformed to the reinforcement/plate interface. The increase in system stiffness due to the reinforcements can be taken into account independently for each coefficient. Thus, one can conveniently reduce the number of degrees of freedom required in considering the reinforcement. Only one degree-of-freedom is required at each internal node when taking into account only the flexural stiffness of beams. Examples are presented to confirm the accuracy of the formulation.     

Calculating the energy-norm FEM-error for Reissner–Mindlin plates without known reference solution

The validation of (recently introduced conforming) finite element technologies for the numerical treatment of Reissner–Mindlin plate models requires comparisons with the unknown exact solution. Since mathematical results are often provided for the error in energy norms only it is not sufficient to compare a typical displacement or moment at one point of the domain. Instead of computing a reference solution on a very fine mesh (and then providing a lot of data for the public) we propose the storage of one (problem depending) constant C which then allows an error representation which merely involves known quantities. Based on this approach we could verify convergence rates which were theoretically predicted and give experimental evidence that new adaptive automatic mesh-refining algorithms yield superior approximations. Given any reasonable guess of C (computable from known quantities), our error representation yields an approximation for the unknown error. This establishes a method for a posteriori error control to be employed as a termination criterion. Received 9 May 2000     

Analysis of anisotropic Kirchhoff plates using a novel hypersingular BEM

In this article a hypersingular boundary element method (BEM) for bending of thin anisotropic plates is presented. A new complex variable fundamental solution is implemented in the algorithm. For spatial discretization a collocation method with discontinuous quadratic elements is adopted. The domain integrals arising from the transversely applied load are transformed analytically into boundary integrals by means of the radial integration technique. The considered numerical examples prove that the novel BEM formulation presented in this study is much more efficient than previous formulations developed for the analysis of this kind of problems.     

BEM based on discontinuous solutions in the theory of Kirchhoff plates on an elastic foundation

We examine the construction of discontinuous solutions for Kirchhoff plates on a generalized elastic foundation. By discontinuous solutions we mean solutions which, when crossing certain lines, have discontinuities of the first type. In the theory of Kirchhoff plates, there may be jumps of the transverse deflection, slope angle, bending moment and equivalent shear force. Initially we construct the solutions due to concentrated jumps. Using them as Green functions, we express discontinuous solutions that are the base for the indirect method of boundary integral equations.     

Closed and approximate analytical solutions for rectangular Mindlin plates

Summary Basing on the Nádai-Lévy and the Vlasov-Kantorovich methods closed and approximate analytical solutions of Mindlin's plate equations in the case of rectangular plates are discussed. For elastic, homogeneous and isotropic plates three unknowns of the governing two-dimensional boundary value problem are formulated as series of products of functions depending on a single coordinate. Specifying the functions for one of the in-plane coordinate directions the governing partial differential equations for a special type of boundary conditions and the principle of virtual displacements for the general case yield a set of ordinary differential equations. The analytical solution of these equations provides expressions for functions depending on the other in-plane coordinate. For plates with simply supported edges for one of the coordinate directions and for arbitrary homogeneous boundary conditions for the other one the Nádai-Lévy method provides a closed or exact solution in the sense that the infinite series for displacements and stress resultants can be truncated to obtain any desired accuracy. In the general case of nonsimply supported edges the iterative Vlasov-Kantorovich method yields an approximate analytical solution. Both methods are nonsensitive to a reduction of the thickness with respect to accuracy and represent the boundary layer solutions in terms of exponential functions. Applications to rectangular plates with various types of boundary conditions are presented.     

A BEM formulation based on Reissner’s theory to perform simple bending analysis of plates reinforced by rectangular beams

In this work, the plate bending formulation of the boundary element method (BEM), based on the Reissner’s hypothesis, is extended to the analysis of plates reinforced by rectangular beams. This composed structure is modelled by a zoned plate, being the beams represented by narrow sub-regions with larger thickness. The integral equations are derived by applying the weighted residual method to each sub-region, and summing them to get the equation for the whole plate. Equilibrium and compatibility conditions are automatically imposed by the integral equations, which treat this composed structure as a single body. In order to decrease the number of degrees of freedom, some approximations are considered for both displacements and tractions along the beam width. The accuracy of the proposed model is illustrated by simple examples whose exact solution are known as well as by more complex examples whose numerical results are compared with a well-known finite element code.     

Boundary element frequency domain formulation for dynamic analysis of Mindlin plates

A new boundary element formulation for analysis of shear deformable plates subjected to dynamic loading is presented. Fundamental solutions for the Mindlin plate theory are derived in the Laplace transform domain. The characteristics of the three flextural waves are studied in the time domain. It is shown that the new fundamental solutions exhibit the same strong singularity as in the static case. Two numerical examples are presented to demonstrate the accuracy of the boundary element method and comparisons are made with the finite element method. Copyright © 2006 John Wiley & Sons, Ltd.     

An extended BEM formulation for plates reinforced by rectangular beams

This article presents a BEM formulation developed particularly for analysis of plates reinforced by rectangular beams. This is an extended version of a previous paper that only took into account bending effects. The problem is now re-formulated to consider bending and membrane force effects. The effects of the reinforcements are taken into account by using a simplified scheme that requires application of an initial stress field to locally correct the bending and stretching stiffness of the reinforcement regions. The domain integrals due to the presence of the reinforcements are then transformed to the reinforcement/plate interface. To reduce the number of degrees of freedom related to the presence of the reinforcement, the proposed model was simplified to consider only bending and stretching rigidities in the direction of the beams. The complete model can be recovered by applying all six internal force correctors, corresponding to six degrees of freedom per node. Examples are presented to confirm the accuracy of the formulation and to illustrate the level of simplification introduced by this strong reduction in the number of degrees of freedom.     

An improved three‐node hybrid‐mixed element for Mindlin/Reissner plates

Based on the mixed shear projected (MiSP) approach [6], an enhanced bending approximation for homogeneous isotropic plates is presented. Some hard benchmark tests, as skew plate (30°) problem for instance, have often shown poor convergence when low order elements (3‐ or 4‐node element) are developed using linear approximations for kinematic variables. To put right this weakness, we propose a high‐order interpolation for rotational dofs which results in more rich bending curvatures. The mid‐side rotational nodes are eliminated using a combination of local discrete kinematic and constitutive Mindlin hypothesises. The derived 3‐node triangular element, called MiSP3+, is free of shear locking and passes all patch‐tests for thick and thin plates in an arbitrary mesh. Copyright © 2001 John Wiley & Sons, Ltd.     

Exact solutions for free vibrations of orthotropic rectangular Mindlin plates

Exact closed-form solutions are obtained for free vibrations of orthotropic rectangular Mindlin plates by using the separation of variables method although it is difficult to solve them. The plates have two opposite edges simply supported and all possible combinations of classical boundary conditions at the other two edges. The exact solutions of orthotropic rectangular Mindlin plates are compared with those of isotropic ones and their differences are discussed. The exact solutions are validated through both mathematical proof and numerical comparisons with available p-Ritz solutions and the differential quadrature finite element method solutions calculated by the authors.     

A general degree semihybrid triangular compatible finite element formulation for Kirchhoff plates

We revisit compatible finite element formulations for Kirchhoff plates and propose a new general degree hybridized approach that strictly imposes C¹ continuity. These new elements are triangular and based on nodal polynomial approximation functions that only use displacement and rotation degrees of freedom for assembly, and thereby “nearly” impose C¹ continuity. This condition is then strictly enforced by adding appropriately chosen hybrid constraints and the corresponding Lagrange multipliers. Unlike all other existing approaches, this formulation allows for the definition of elements of arbitrary degree considering a single polynomial basis for each element, without using degrees of freedom associated with second-order derivatives. The convergence is compared with that of alternative approaches in terms of numbers of elements and degrees of freedom.     

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.     

Exact solutions for the in-plane vibrations of rectangular Mindlin plates using Helmholtz decomposition

Deriving frequency equations for in-plane vibration of a rectangular plate with different boundary conditions and uniform thickness in the elastic range is the goal of this research. To derive frequency equations, the kinetic and potential energy for in-plane behavior initially are obtained by using the stress–strain–displacement expressions according to the theory of Mindlin plates in Cartesian coordinates by applying the Hamilton’s principle, which leads to five sets of highly coupled differential equations for the equations of motion. Replacement of Helmholtz decomposition for the coupled differential equations creates uncoupled equations of motion. The hypothesis of a harmonic solution for the uncoupled equations lead to wave equations. The general solutions for the wave equations are obtained by using the separation of variables. Finally, the application of boundary conditions yields the frequency equations for the rectangular plate. The natural frequencies are compared and validated by finite element analysis and previously reported results.     

On the formulation of the fundamental solutions for orthotropic plane elasticity

Summary A method is given, in a compact, schematized form, of representing the fundamental solutions for the infinite orthotropic plane. Attention is particularly focused on the tensorial and quasi-tensorial structure of the orthotropic-elastic factors which occur in the influence functions. It is shown that the proposed formulation leads directly to the limiting case of isotropic material.With 1 Figure     

On the Laplace transform of a matrix of fundamental solutions for thermoelastic plates

The Laplace transform of a matrix D(x,t) of fundamental solutions for the partial differential operator describing the time-dependent bending of thermoelastic plates with transverse shear deformation is constructed, and its asymptotic behavior near the origin is investigated. The differential system is reduced to an algebraic one through the application of the Laplace and then Fourier transformations, and all possible cases of roots of the determinant of the latter system are considered. It is shown that in every case, the asymptotic expansion of near the origin has the same dominant term. This is an important step in the construction of boundary-element methods for the above time-dependent model because it determines the nature of the singularity of the kernel of the boundary-integral-equations associated with various initial-boundary-value problems for the governing system.     

A Galerkin symmetric and direct BIE method for Kirchhoff elastic plates: formulation and implementation

A variational Boundary Element formulation is proposed for the solution of the elastic Kirchhoff plate bending problem. The stationarity conditions of an augmented potential energy functional are first discussed. After addressing the topic of the choice of the test functions, a regularization process based on integrations by parts is developed, which allows to express the formulation in terms of double integrals, the inner being at most weakly singular and the outer regular. Standard integration procedures may then be applied for their numerical evaluation in the presence of both straight and curved boundaries. The normal slope and the vertical displacement must be C⁰ and C¹ continuous, respectively. Numerical examples show, through comparisons with analytical solutions, that a high accuracy is achieved. © 1998 John Wiley & Sons, Ltd.     

A time domain BEM for 3-D elastodynamic analysis using the B-spline fundamental solutions

A Boundary Element for 3-D elastodynamic analysis is introduced in detail. The method uses a new generation of the Stokes fundamental solutions that utilize the B-spline family of polynomials. The integration techniques of the boundary element kernels are also discussed for both the singular and non-singular cases. A number of numerical examples are presented for the validation of the proposed methodology.     

Dynamic analysis of polygonal Mindlin plates on two-parameter foundations using classical plate theory and an advanced BEM

Forced vibrations of moderately thick plates on two-parameter, Pasternak-type foundations are considered. Influence of plate shear and rotatory inertia are taken into account according to Mindlin. Excitations are of the force as well as of the support motion type. Formulation is in the frequency domain. An analogy to thin plates without foundations is given. This analogy to classical plate theory is complete in the case of polygonal plan-forms and hinged support conditions. In that case the higher order Mindlin-problem is reduced to two (second order) Helmholtz-Klein- Gordon boundary value problems. An advanced BEM using Green's functions of rectangular domains is applied to the latter, thereby satisfying boundary conditions exactly as far as possible. This problem oriented strategy provides the frequency response functions for the deflection of the undamped Mindlin plate with high numerical accuracy. Structural damping is built in subsequently, and Fast Fourier Transform is applied for calculation of the transient response.Part of the paper has been presented at the IUTAM-Symposium Advanced BEM, San Antonio, Texas, 1987. Another part has been presented at the 6th Int. Conf. Numerical Methods for Geomechanics, Innsbruck, Austria 1988