Special elements for point singularities

In this paper a simple method for developing special elements for modelling point singularities is suggested. The proposed elements are a direct extension of Williams eigenvalues for different vertex angles. Three numerical examples for cracked plates are used in the calculation of stress intensity factors for linear fracture mechanics problems. The solutions compare favourably with those obtained by using the quarter-point crack-tip elements. The use of the proposed elements preserves all the advantages of the finite element method.     

A refined higher-order C° plate bending element

A general finite element formulation for plate bending problem based on a higher-order displacement model and a three-dimensional state of stress and strain is attempted. The theory incorporates linear and quadratic variations of transverse normal strain and transverse shearing strains and stresses respectively through the thickness of the plate. The 9-noded quadrilateral from the family of two dimensional C° continuous isoparametric elements is then introduced and its performance is evaluated for a wide range of plates under uniformly distributed load and with different support conditions and ranging from very thick to extremely thin situations. The effect of full, reduced and selective integration schemes on the final numerical result is examined. The behaviour of this element with the present formulation is seen to be excellent under all the three integration schemes.     

Study of elastic behaviour of plates containing cracks by finite element analysis

This work applies finite element analysis very simply to cracked plates. An infinite plate and a finite plate, both with a central crack, are considered to study their elastic behaviour and some fracture mechanics concepts, such as the geometry factor and the fracture toughness. These magnitudes are calculated by means of finite element methods and the results are in very good agreement with the established theory, which proves that the finite element approach is very appropriate. The fracture toughness fraction is defined and calculated for a finite plate to predict its failure.     

A quadrilateral finite difference plate element for nonlinear transient analysis of panels

A quadrilateral plate element for the analysis of nonlinear transient response of panels has been developed based on the variational finite difference method for an irregular mesh. Due to the superior computational characteristics of the variational finite difference method with a lesser degree of continuity constraint on the interpolation functions and the use of lower-order polynomials allowing faster numerical integration methods to be implemented, this plate element is quite competitive or perhaps even superior when compared with the conforming finite elements. Three illustrative problems have been solved using this plate element to demonstrate its capability and accuracy in analyzing the large deformation response of panels subject to dynamic loadings.Very favorable correlation was observed between analysis and experiment on large deformations of elasticplastic rectangular plates subject to intensive impulsive loadings. Similar correlation was also observed for circular plates modeled with this quadrilateral plate element under impulsive loadings. Finally, the large dynamic deformations of composite rectangular panels of graphite-epoxy, boron-epoxy, glass-epoxy, and isotropic material were analyzed and found in good agreement with other analytical results.     

Nonlinear finite element analysis of reinforced concrete plates and shells under monotonic loading

Plane stress constitutive models are proposed for the nonlinear finite element analysis of reinforced concrete structures under monotonic loading. An elastic strain hardening plastic stress-strain relationship with a nonassociated flow rule is used to model concrete in the compression dominating region and an elastic brittle fracture behavior is assumed for concrete in the tension dominating area. After cracking takes place, the smeared cracked approach together with the rotating crack concept is employed. The steel is modeled by an idealized bilinear curve identical in tension and compressions. Via a layered approach, these material models are further extended to model the flexural behavior of reinforced concrete plates and shells. These material models have been tested against experimental data and good agreement has been obtained.     

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.     

Finite element free vibration analysis of eccentrically stiffened plates

A new finite element model is proposed for free vibration analysis of eccentrically stiffened plates. The formulation allows the placement of any number of arbitrarily oriented stiffeners within a plate element without disturbing their individual properties. A plate-bending element consistent with the Reissner-Mindlin thick plate theory is employed to model the behaviour of the plating. A stiffener element, consistent with the plate element, is introduced to model the contributions of the stiffeners. The applied plate-bending and stiffener elements are based on mixed interpolation of tensorial components (MITC), to avoid spurious shear locking and to guarantee good convergence behaviour. Several numerical examples using both uniform and distorted meshes are given to demonstrate the excellent predictive capability of this approach.     

A three-dimensional nonlinear analysis of cross-ply rectangular composite plates

The results of a three-dimensional, geometrically nonlinear, finite-element analysis of the bending of cross-ply laminated anisotropie composite plates are presented. Individual laminae are assumed to be homogeneous, orthotropic and linearly elastic. A fully three-dimensional isoparametric finite element with eight nodes (i.e. linear element) and 24 degrees of freedom (three displacement components per node) is used to model the laminated plate. The finite element results of the linear analysis are found to agree very well with the exact solutions of cross-ply laminated rectangular plates under sinusiodal loading. The finite element results of the three-dimensional, geometrically nonlinear analysis are compared with those obtained by using a shear deformable, geometrically nonlinear, plate theory. It is found that the deflections predicted by the shear deformable plate theory are in fair agreement with those predicted by three-dimensional elasticity theory; however stresses were found to be not in good agreement     

Linear free flexural vibration of cracked functionally graded plates in thermal environment

In this paper, the linear free flexural vibrations of functionally graded material plates with a through center crack is studied using an 8-noded shear flexible element. The material properties are assumed to be temperature dependent and graded in the thickness direction. The effective material properties are estimated using the Mori–Tanaka homogenization scheme. The formulation is developed based on first-order shear deformation theory. The shear correction factors are evaluated employing the energy equivalence principle. The variation of the plates natural frequency is studied considering various parameters such as the crack length, plate aspect ratio, skew angle, temperature, thickness and boundary conditions. The results obtained here reveal that the natural frequency of the plate decreases with increase in temperature gradient, crack length and gradient index.     

Dynamic stiffness elements and their applications for plates using first order shear deformation theory

Dynamic stiffness elements for plates are developed using first order shear deformation theory to carry out exact free vibration analysis of plate assemblies. The analysis has been facilitated by the application of Hamiltonian mechanics and symbolic computation. The Wittrick–Williams algorithm has been used as the solution technique. Results have been extensively validated using published literature for both uniform and non-uniform plates. Some finite element results are also provided. The accuracy and computational efficiency of the method are demonstrated. In the final part of the investigation, significant plate parameters are varied and their subsequent effects on the free vibration characteristics are studied.     

On optimal stabilized MITC4 plate bending elements for accurate frequency response analysis

The mixed interpolation technique of the well-established MITC4 quadrilateral plate finite element is combined with shear and generalized least-squares stabilization methods for accurate frequency response analysis. Dispersion analysis is used to determine optimal combinations of stabilization parameters, which, for a given mesh, provide for a three-fold increase in the frequency range over which accurate solutions are obtained, thus allowing for accurate solutions at significantly lower cost. Numerical results for the forced vibration of Reissner–Mindlin plates validate the observations made from the dispersion analysis.     

Elastic buckling of rectangular mindlin plate with mixed boundary conditions

An approximate method for analyzing the elastic buckling of Mindlin plates is proposed. The solutions of the differential equations of buckling of Mindlin plate are obtained in discrete forms by applying numerical integration. The discrete solutions give the transverse shear forces, twisting moment, bending moments, rotations and deflection at the all discrete points which are the intersections of the vertical and horizontal equally dividing lines on the plate. In order to confirm the convergency and accuracy of numerical solutions, the buckling loads of simply supported square Mindlin plates are calculated, and the results are compared with other published finite element solutions. As an application, the buckling loads and buckling modes of Mindlin plates with various types of mixed boundaries are calculated.     

Three-dimensional finite element analysis of thick laminated plates

A displacement-based, three-dimensional finite element scheme is proposed for analyzing thick laminated plates. In the present formulation, a thick laminated plate is treated as a three-dimensional inhomogeneous anisotropic elastic body. Particular attention is focused on the prediction of transverse shear stresses. The plane of a laminated plate is first discretized into conventional eight-node elements. Various through-thickness interpolation is then denned for different regions of the plate; layerwise local shape functions are used in the regions where transverse shear stresses are of interest, while an ad hoc global-local interpolation is used in the region where only the general deformation pattern is concerned. For satisfying the displacement compatibility between these two regions, a transition zone is introduced. The model incorporates the advantages of the layerwise plate theory and the single-layer plate theory. Details of formulation will be presented together with several numerical examples for demonstrating the proposed scheme.     

An elastoplastic cracked-beam finite element for structural analysis

A finite element model has been developed in this paper to analyse statically indeterminate skeletal cracked structures. The model is based on elastic-plastic fracture mechanics techniques in order to consider the crack tip plasticity in the analysis. Stiffness matrices for single-edge and double-edge cracked structural elements have been derived using transfer matrix theory. These matrices take into account the effects of axial, flexural and shear deformations due to crack presence. The present model has been applied to investigate the effects of crack size, structure cross-section depth and crack tip plasticity on the redistribution of internal forces in structures. Hence, this analysis can be employed to identify the overstressed regions in cracked structures.     

Application of the spline finite strip to the analysis of shear-deformable plates

A spline finite strip is proposed to analyse thick isotropic or laminated composite plates. The formulation is based upon the principle of virtual work and the third-order plate theory developed by Reddy. The variational functional requires the satisfaction of C₁,-continuity of the assumed vertical deflection variable which can be easily fulfilled by the present method. The proposed spline finite strip is a conforming element with a smaller number of unknowns at each node compared to other existing elements based on the third-order theory. For the analysis of thin isotropic or laminated plates, the present element shows no sign of shear locking. A number of computational examples are given to demonstrate the efficiency and the accuracy of the present method.     

Post-buckling behavior of thin steel plates using computational models

Thin plates loaded in plane will buckle at very small loads, and due to unavoidable out-of-plane imperfections, the theoretical buckling load cannot be observed experimentally. If the plate is adequately supported along its boundaries, it will be able to carry a much higher load than the theoretical buckling load.Computational models can be used to study the post buckling behaviour of thin plate structures up to failure. Failure of such structures is usually due to large out-of-plane deflections, yielding, and rupture. Therefore, the computational model should include the effects of geometric and material nonlinearities. In this paper, the nonlinear finite element analysis program NONSAP and ANSR-III were modified and used in the analysis. Since these programs did not include the suitable elements and material properties to conduct the subject study, new elements and new material properties were added to the programs. In particular, a thin shell element was added and the solution routines were modified to improve its accuracy and efficiency.The modified programs were used on a Super Computer to calculate the post buckling strength of stiffened and unstiffened plates subjected to uniaxial compression, and plates subjected to in plane bending or shear. Crippling of plates subjected to in-plane or eccentric edge compressive loads was also examined. The results from the computational models were compared with test results and reasonable agreements were obtained. A computational model was developed for a multi-story thin steel plate shear wall subjected to cyclic loading and the results from the model were compared with experimental results, and again agreement was achieved.     

A simple displacement-based 3-node triangular element for linear and geometrically nonlinear analysis of laminated composite plates

A simple displacement-based 3-node, 18-degree-of-freedom flat triangular plate/shell element LDT18 is proposed in this paper for linear and geometrically nonlinear finite element analysis of thin and thick laminated composite plates. The presented element is based on the first-order shear deformation theory (FSDT), and the total Lagrangian approach is employed to formulate the element for geometrically nonlinear analysis. The deflection and rotation functions of the element boundary are obtained from the Timoshenko’s laminated composite beam functions, hence convergence to the thin plate solution can be achieved theoretically and shear-locking problem is avoided naturally. The plane displacement interpolation functions of the Airman’s triangular membrane element with drilling degrees of freedom are taken as the in-plane displacements of the element. Numerical examples demonstrate that the present element is accurate and efficient for linear and geometrically nonlinear analysis of thin to moderately thick laminated composite plates.     

Accurate force evaluation with a simple,bi-linear,plate bending element

A great deal of attention has been given to the development of simple C ° continuous plate and shell elements based on the shear flexible theories for application to thick plates, sandwich or cellular plates and transversely isotropic or laminated plates. After considerable experimentation using unconventional approaches such as reduced integration, selective integration, mixed methods using discontinuous force fields, etc., it has been possible to develop simple displacement-type elements which can be reliably used. The stress recovery at nodes from such elements is often unreliable as the nodes are usually the points where strains or stresses are least accurate in the element domain. Further, nodal values can reflect severe oscillations at some difficult corner or edge conditions. In this paper, we focus attention on the optimal stress recovery from such an element. This is done after an interpretation of the displacement method as a procedure that obtains strains over the finite-element domain in a least-squares accurate fashion. If a shear flexible element is field-consistent, there are optimal locations at which bending moments and shear forces are accurate in a least-squares sense. These points are identified for the present element and used to study stresses in typical plate problems. Another difficulty faced is the rapid variation of twisting moments at free edges and corners of shear flexible plates and its influence on the shear forces at that edge. A related source of difficulty is the distinction made in Kirchhoff theory between shear forces and the effective shear reactions of that theory. The present study is seen to give accurate enough shear force and twisting moment predictions to allow one to draw the severe conclusion that the use of the Kirchhoff shear reaction at edges in classical plate theory is an ambiguous and unnecessary one and can be avoided. The findings confirm a recent suggestion that it may be more appropriate to have three (as introduced originally by Poisson) instead of the two boundary conditions (as modified by Kirchhoff) usually applied on the edge of a thin plate, especially if that edge is unsupported.     

Finite element analysis of shell structures

Summary A survey of effective finite element formulations for the analysis of shell structures is presented. First, the basic requirements for shell elements are discussed, in which it is emphasized that generality and reliability are most important items. A general displacement-based formulation is then briefly reviewed. This formulation is not effective, but it is used as a starting point for developing a general and effective approach using the mixed interpolation of the tensorial components. The formulation of various MITC elements (that is, elements based on Mixed Interpolation of Tensorial Components) are presented. Theoretical results (applicable to plate analysis) and various numerical results of analyses of plates and shells are summarized. These illustrate some current capabilities and the potential for further finite element developments.     

Viscoelastic analysis of nearly incompressible solids

The finite element method is well established for the analysis of structures and other field problems. However, its straightforward application for the analysis of nearly incompressible solids yields erratic results. In the present work, an efficient special purpose code for the Viscoelastic Analysis of Nearly Incompressible Solids (VANIS) is developed using isoparametric elements with selective integration procedure, which is a third order Gauss rule for deviatoric response and second order Gauss rule for volumetric response. The software can be effectively employed for the structures with lower Poisson's ratios. VANIS is based on the direct formulation using linear uncoupled thermoviscoelastic theory for the thermorheologically simple materials. The element library consists of 8-noded plane strain, 8-noded axisymmetric solid and 20-noded three dimensional quadratic isoparametric elements. These elements meet all the possible structural idealisation requirements of the solid continua. Experimentally obtained rigidity modulus can be used directly or expressing it in Prony series. The software is tested on a number of problems and gives very accurate results for all the permissible values of the Poisson's ratio.