Mixed elements in the optimal design of plates

The theory of design sensitivity analysis of structures, based on mixed finite element models, is developed for static, dynamic and stability constraints. The theory is applied to the optimal design of plates with minimum weight, subject to displacement, stress, natural frequencies and buckling stresses constraints. The finite element model is based on an eight node mixed isoparametric quadratic plate element, whose degrees of freedom are the transversal displacement and three moments per node. The corresponding nonlinear programming problem is solved using the commercially available ADS (Automated Design Synthesis) program. The sensitivities are calculated by analytical, semi-analytical and finite difference techniques. The advantages and disadvantages of mixed elements in design optimization of plates are discussed with reference to applications.     

Flexural failure analysis of anisotropic elliptic plates

An initial flexural failure analysis of laminated anisotropic elliptic clamped plates subjected to a uniform lateral pressure is presented. Two failure modes are distinguished: general lamina failure and interlaminar shear failure. For each of these modes dimensionless initial failure loads have been computed for a range of aspect ratios and fibre orientations. These results are complemented by a set of critical thickness ratios which serve to distinguish which mode of failure governs for a particular case and hence which failure loads apply. It is concluded that general lamina failure is the dominant failure mode for the practical range of thickness ratios, though the likelihood of interlaminar shear failure occurring is greater in CFRP plates than in GFRP plates.     

Elastic-plastic dynamic analysis of anisotropic laminated plates

This paper presents the application of a refined finite element model to the elastic and elastic-plastic dynamic analysis of anisotropic laminated plates. Dynamic analysis is based on Newmark's algorithm used in conjunction with the Hughes and Liu predictor-corrector scheme resulting in an ‘effective static problem’ which is solved using a Newton-Raphson-type process. Flow theory is used in the inelastic range and the Huber-Mises yielding surface extended by Hill for anisotropic materials is adopted. Numerical results obtained for two categories of anisotropic structures, namely cross-ply laminated plates and angle-ply laminated plates, are presented and the effects of anisotropy and bending/ stretching coupling on the dynamic elastic and elastic-plastic responses are discussed. The effects of lamina stack sequences and lamina angle sequences on the dynamic responses are also considered.     

Shape sensitivity analysis of natural frequencies of structures using boundary elements

The boundary element approach to shape sensitivity analysis of eigenvalues of free vibrating elastic structures is presented. The eigenvalue problem is described in terms of the boundary integral equation method. Using the variational approach for variable regions, first-order sensitivities of simple frequencies are derived. Dependence of eigenvalues with respect to the stochastic shape of the boundary is considered. The numerical procedure of discretization of the problem is characterized. Numerical examples for two-dimensional problems are presented.     

Static,vibration and buckling analysis of axisymmetric circular plates using finite elements

The static, vibration, and buckling analysis of axisymmetric circular plates using the finite element method is discussed. For the static analysis, the stiffness matrix of a typical annular plate element is derived from the given displacement function and the appropriate constitutive relations. By assuming that the static displacement function, which is an exact solution of the circular plate equation ?²?²W = 0, closely represents the vibration and buckling modes, the mass and stability coefficient matrices for an annular element are also constructed. In addition to the annular element, the stiffness, mass, and stability coefficient matrices for a closure element are also included for the analysis of complete circular plates (no center hole). As an extension of the analysis, the exact displacement function for the symmetrical bending of circular plates having polar orthotropy is also given.     

The analysis of solids without mesh generation using trimmed patch boundary elements

The tools available to the mechanical engineer—for example, geometric modeling and computer-aided analysis—are individually quite powerful, but they are based on different geometric representations. Hence, they do not always work well together. In this paper an analysis method is presented that operates directly on the geometric modeling representation. Therefore, the time-consuming and error-prone procedure of generating a mesh is skipped. The method is based on boundary integral equations, but unlike previously published methods, the boundary elements aren-sided trimmed patches, the same patches that are used by modern geometric modelers to represent complex solids. The method is made practical by defining shape functions over the trimmed patches in such a way that the number of degrees of freedom can be controlled. This is done by using a concept called virtual nodes. The paper begins by deriving the trimmed patch boundary element. Then its properties are discussed in comparison with existing boundary element and finite element methods, and several examples are given.     

Contribution to free boundary problems using boundary elements: Trefftz approach

There are two main approaches to the formulation of boundary methods, these are boundary integral equations and approximations by complete systems of solutions (Trefftz method). The latter has been the subject of extensive studies by one of the authors oriented to clarifying the foundations of the method and increasing its versatility. The present paper is devoted to explain the application of this procedure to free boundary problems such as Signorini and friction problems in elasticity.     

Treatment of singular integral caused by employing linear boundary elements for two-dimensional elastostatic problems

The boundary element method employing linear elements has the advantage of high precision in calculations. However, there exists a problem of singularity when the diagonal terms of the boundary influence coefficients are calculated. A method by which the singular terms are cancelled is proposed in this paper. A computer program for the solution of two-dimensional elastostatic problems using linear boundary elements is developed and verified through two examples. The technique dealing with the nodes at corners is discussed. The results show that the present method of solving singular integrals is credible.     

An asymptotic variational formulation for dynamic analysis of multilayered anisotropic plates

A multilevel variational formulation for dynamic analysis of multilayered anisotropic plates is developed within the framework of three-dimensional elasticity. By means of asymptotic expansions the Hellinger-Reissner functional for the elastodynamic problem is decomposed into a series of functionals with which a computational model can be constructed. In the formulation multiple time scales are introduced so that the secular terms can be eliminated systematically in obtaining a uniform expansion leading to valid asymptotic solution. Modifications to the approximation of various orders are determined by considering the solvability conditions of the higher-order equations. The model is adaptive, when combined with the finite element method, it has many appealing features, including that the displacements and transverse stresses may be interpolated independently, that the nodal degree-of-freedom (DOF) at each level is less than that of Kirchhoff plates, and that the mass and stiffness matrices generated at the leading-order level are always used at subsequent levels. Above all, the solution is three-dimensional in effect yet requires only two-dimensional interpolation. The through-thickness variations of the field variables are determined analytically with no need of interpolating in the thickness direction.     

Stability analysis of anisotropic laminated composite plates by finite strip method

The finite strip method has been applied to the stability analysis of rectangular shear-deformable composite laminates. However, for the plates with two opposite simply supported sides, the existing analysis was restricted to the symmetrical cross-ply laminates under compression loading.In the present study, by selecting proper displacement functions and including the coupling between different series terms, the finite strip method is extended to the stability analysis of any anisotropic laminated plates under arbitrary in-plane loading. Furthermore, a number of numerical results are presented to show the effects of thickness, fibre orientation and stacking sequence on the buckling loads.     

Mixed finite-difference scheme for analysis of simply supported thick plates

A mixed finite-difference scheme is presented for the stress and free vibration analysis of simply supported nonhomogeneous and layered orthotropic thick plates. The analytical formulation is based on the linear, three-dimensional theory of orthotropic elasticity and a Fourier approach is used to reduce the governing equations to six first-order ordinary differential equations in the thickness coordinate. The governing equations possess a symmetric coefficient matrix and are free of derivatives of the elastic characteristics of the plate. In the finite difference discretization two interlacing grids are used for the different fundamental unknowns in such a way as to reduce both the local discretization error and the bandwidth of the resulting finite-difference field equations. Numerical studies are presented for the effects of reducing the interior and boundary discretization errors and of mesh refinement on the accuracy and convergence of solutions. It is shown that the proposed scheme, in addition to a number of other advantages, leads to highly accurate results, even when a small number of finite difference intervals is used.     

Comparison of three-dimensional solid elements in the analysis of plates

This paper is concerned with the comparison and testing of 20- and 27-node hexahedral and ten-node tetrahedral three-dimensional solid elements. Their performance characteristics are examined in the context of the analysis of plate structures and their strengths and weaknesses are considered.     

Thermoelastic stress analysis of anisotropic composite sandwich plates by finite element method

In the present work a finite element approach is established for the analysis of sandwich plates with different anisotropic composite facings due to aerodynamic and thermal fields. The main contribution of this study is the consideration of the inherent coupling phenomenon of stretching and bending in such non-symmetric composite sandwich plates. Consequently, the stiffness matrix includes now sub-matrices which relate the “generalized” transverse thermal (or aerodynamic) loads to the in-plane displacements, and the in-plane forces to the “generalized” transverse displacements. These sub-matrices do not exist in symmetrically layered sandwich plates and their inclusion in the analysis is shown to be of primary importance. Three examples are included, indicating the sensibility of the stress and displacement fields to the class of heterogeneity and anisotropy of the considered sandwich plates.     

Numerical solution of singular boundary value problems using advanced Adomian decomposition method

Engineering with Computers - In this work, we present a powerful method for the numerical solution of non-linear singular boundary value problems, namely the advanced Adomian decomposition method...     

Quadrilateral finite elements for analysis of thick and thin plates

We discuss two quadrilateral plate elements applicable in the analysis of both thick and thin plates. The elements are based on Reissner-Mindlin plate theory and an enhanced displacement interpolation, which enables the consistent loading vector to be constructed. The constraint on the constant shear strain is enforced explicitly thus eliminating the shear locking phenomena in the analysis of thin plates. As a by-product of this work, we take a new look at a well-known discrete Kirchhoff plate element.     

A variable power singular element for analysis of fracture mechanics problems

A variable power displacement formulated singular element is developed. The stress and displacement fields within the element are expressed in terms of singular and regular components. The singular components of these fields are shown to provide a more accurate representation of the singularity than the total field solutions. The problem of a crack terminating at a bi-material interface is solved for two extreme ratios of singular element radius to crack length. The solution shows that accurate results can be achieved for either case.     

Structural design sensitivity using boundary elements and polynomial response function

This main issue of this paper is a conjunction of the structural design sensitivity analysis using the Boundary Element Method with the polynomial response function determination. The procedure is so general that it enables sensitivity analysis for potential and elasticity problems within both homogeneous and heterogeneous plane and 3D problems. The essential difference with respect to the previous approaches like the Direct Differentiation Method or the Adjoint Variable Method is in discrete evaluation of the structural response using the response polynomials of some state parameters and design variable as the independent parameter. Such a determination is carried out via the several solutions of the given boundary value problem, where design parameter mean value is regularly perturbed in each of the solutions to cover the closest neighborhood of this mean value. Those few solutions make it possible to recover the polynomial response function from node-to node within the boundary elements, so that further symbolic differentiation using MAPLE returns the sensitivity gradients particular values. The entire procedure is tested here twice—first example deals with the homogeneous cantilever beam, where comparison against pure analytical differentiation is done and, separately, for two-component composite cantilever, where such a comparison is made against the central difference method linked with the same BEM solution.     

Stability of partially supported square plates using high precision triangular finite elements

The stability of square plates with partial supports under uniaxial or biaxial uniformly distributed compressive loads is studied using high precision triangular finite elements. Stability parameters are evaluated for both partially simply supported and partially clamped plates for various values of support ratios: limiting values of the support ratios being either fully simply supported/clamped or point supported at the corners/mid-points of the edges.     

Topology optimization of structures with anisotropic plastic materials using enhanced assumed strain elements

The focus of this paper is on consistent and accurate adjoint sensitivity analyses for structural topology optimization with anisotropic plastic materials under plane strain conditions. In order to avoid the locking issue, the Enhanced Assumed Strain (EAS) elements are adopted in the finite element discretization, and the anisotropic Hoffman plasticity model, which can simulate the strength differences in tension and compression, is incorporated within the framework of density-based topology optimization. The path-dependent sensitivity analysis is presented wherein the enhanced element parameters are consistently incorporated in the constraints. The objective of topology optimization is to maximize the plastic work. Several numerical examples are presented to show the effectiveness of the proposed framework. The results illustrate that the optimized topologies are highly dependent on the plastic anisotropic material properties.