An efficient computer program for the large deflection analysis of rectangular orthotropic plates

A computer program is presented for the static analysis of rectangular specially orthotropic plates undergoing large deflections. In-plane loads and lateral pressure are both catered for, together with various combinations of boundary conditions. However, a restriction is that there must be symmetry about both plate axes. The plate can have an initial geometrical imperfection, which is represented by a double Fourier series with the same terms as that used for the lateral displacement. By specifying in the input data suitable values for the coefficients in the imperfection series, the effect of relatively complicated initial shapes can be examined. The program is called ELDAROP, a FORTRAN listing of which is given in Appendix A. Appendix B contains a listing of the associated subroutine STRRES, which calculates stress resultants. The analysis is based on equations derived in a recent paper (Comput. Struct. 26, 871–884, 1987), in which it was demonstrated that ELDAROP appears to be both accurate, and extremely rapid in its operation. Appendices C-E give examples of the simple arrangement of input data and output.     

Application of dynamic relaxation to the large deflection elasto-plastic analysis of plates

The application of dynamic relaxation, a finite difference based iterative analysis, to the study of plates to date is reviewed. The extension of the method to include both geometrical and material non-linear effects in plates is then described in detail. Particular attention is paid to aspects of the iteration parameters which control convergence. The advantages of interlacing finite difference meshes is discussed and the mesh refinement necessary for the accurate analysis of plates in compression and in shear is considered. The usual elastic out-of-plane boundary conditions are generalised by the inclusion of terms applicable in the elasto-plastic range, and the role of in-plane boundary restraints is discussed in respect of plating in both bridge and ship structures. The scope of the method is demonstrated by examples of a long plate with patch loading normal to the undeflected plane of the plate and a panel of a box girder web. In both cases, the effects of initial out-of-plane geometric distortions and welding residual strains on behaviour up to and beyond collapse are considered.     

Shear flexible element based on coupled displacement field for large deflection analysis of laminated plates

A shear deformable theory accounting for the transverse-shear (in the sense of Reissner-Mindlin’s thick plate theory) and large deflections (in the sense of von Karman theory) is employed in the construction of variational statement. A four-node, lock-free, shear-flexible rectangular plate element based on the coupled displacement field is developed in this paper to carry out the large deflection analysis. The displacement field of the element is derived by making use of the linearized equations of static equilibrium. A bi-cubic polynomial distribution is assumed for the transverse displacement ‘w’. The field distribution for the in-plane displacements (u,v) and plate normal rotations (θ_x, θ_y) and twist (θ_xy) is derived using equilibrium of composite strips parallel to the plate edges. The displacement fields so derived are coupled through material couplings. The transverse shear strain fields of the proposed element do not contain inconsistent terms, so that the element predicts even shear-rigid bending accurately.The element is validated for a series of numerical problems and results for deflections and stresses are presented for rectangular composite plates with various boundary conditions, loading and lay-ups. The influence of the sign of the loading on the deflection of unsymmetrically laminated plates, in the large deflection regime is also investigated.     

On the elasto-plastic large deflection response of stocky annular steel plates

An incremental full-range analysis for pressure loaded annular plates is presented in outline. Von Karman large deflection plate theory coupled with the Prandtl-Reuss associated flow rule and a layered Von Mises yield criterion form the cornerstones of the analysis. The numerical solution of the governing equations is achieved via a finite-difference implementation of the DR algorithm. The DR analysis is used to carry out a numerical parameter study of the full-range response of uniformly loaded stocky steel annular plates. The study is primarily concerned with the effects of the edge support conditions in controlling the response and dimensionless results for deflections, etc. are presented for a total of nine inner and outer edge condition combinations.     

Field/boundary element approach to the large deflection of thin flat plates

The problem of large deflections of thin flat plates is rederived here using a novel integral equation approach. These plate deformations are governed by the von Karman plate theory. The numerical solution that is implemented combines both boundary and interior elements in the discretization of the continuum. The formulation also illustrates the adaptability of the boundary element technique to nonlinear problems. Included in the examples here are static, dynamic and buckling applications.     

Computer program for large deflection elasto-plastic analysis of semi-rigid steel frameworks

This paper presents an efficient computer method for inelastic and large deflection analysis of flexibly jointed steel frames, based on the most refined type of second-order inelastic analysis, the plastic zone analysis. The combined effects of material, geometric and connection behavior nonlinearity sources are simulated into an object-oriented Turbo-Pascal computer program automatically. This program was used to study the ultimate response of several steel frames which have been studied previously by other researchers. The example computations and the comparisons made have proved the effectiveness of the proposed analysis method.     

Large deflection analysis of clamped skew sandwich plates by parametric differentiation

The differential equations governing the large deflection behavior of skew sandwich plates are highly complex in nature and do not lend themselves for easy solution. In the study reported herein, this problem is solved by a numerical technique known as “parametric differentiation”. The non-linear differential equations in terms of displacements are transformed into a set of linear differential equations with variable coefficients. These are solved to give the gradients of the displacements in the load direction. The subsequent solution of a set of initial value problems yield the displacements proper. The results obtained by this method are compared with available results of other investigators and the agreement is found to be good. Load deflection characteristics have been presented for clamped skew sandwich plates.     

Some variational formulations for large deformation analysis of plates

Four formulations for classical large deformation analysis of thin plates are briefly examined. The relative merits and disadvantages of these formulations for developing finite element models are pointed out. In particular, it is shown that by a priori satisfaction of some of the governing equations the order and degree of the remaining set of differential equations may be altered.     

Large deflection analysis of skew plates by lumped triangular element formulation

A finite difference scheme with triangular mesh is presented for the analysis of skew plate problems with large deflections. The suggested formulation is independent of the boundary condition and uses energy principles to derive a set of nonlinear algebraic equations which are solved by using Newton-Raphson iterative method with incremental loading. The investigation is concerned with the behaviour of constant thickness clamped and simply supported isotropic skew plates with immovable edges and subjected to uniformly distributed transverse load. The effects of skew on plates with large deflections are investigated and comparisons are made with existing results; good agreement is shown.     

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.     

Large deflection analysis of plates and shells by discrete energy method

The discrete energy method—a special form of finite difference energy approach—is presented as a suitable alternative to the finite element method for the large deflection elastic analysis of plates and shallow shells of constant thickness. Strain displacement relations are derived for the calculation of various linear and nonlinear element stiffness matrices for two types of elements into which the structure is discretized for considering separately energy due to extension and bending and energy due to shear and twisting. Large deflection analyses of plates with various edge and loading conditions and of a shallow cylindrical shell are carried out using the proposed method and the results compared with finite element solutions. The computational efforts required are also indicated.     

Large deflection of clamped skew plates

A theoretical analysis is presented for the large deflection elastic behaviour of clamped, uniformly loaded orthotropic skew plates. The governing nonlinear partial differential equations are transformed into a set of nonlinear algebraic equations. For this purpose a network of discrete points over the domain is considered, and the integral equation of beams along the skew directions is used with appropriate boundary conditions. The resulting nonlinear algebraic equations are then solved by a Newton-Raphson procedure.A convergence study of the solution has been made, and numerical results for a few cases of orthotropy and skew angles are presented in graphical form. The results obtained by this method are compared with available results of other investigators.     

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.     

Alternative FEM formulations for the inelastic large deflection analysis of plane frames

The present paper is concerened with the analysis of plane frames of arbitrary geometry undergoing large elasto-plastic deformations. On the basis of so-called mixed-hybrid variational theorems wherein displacements and stress resultants represent independent variables, three alternative finite element schemes are proposed. Their theoretical and computational advantages in comparison with the standard assumed displacement formulation are pointed out and discussed in detail.     

Higher order finite strip analysis for curved bridge decks

The development of a higher order finite strip method for improved accuracy and its application to orthotropic curved bridge decks are discussed. A quintic polynomial in the radial direction is employed along with a basic function series in the angular direction which satisfy a priori the boundary conditions along the radial edges. Thus a two-dimensional plate bending problem is reduced to a one-dimensional one. As a result, both the size and bandwidth of the global stiffness matrix are greatly reduced. The method is easy to program, requires a minimum input data and small computer storage. In order to estimate the reliability of the present formulation, three examples of curved plates are solved and the results are compared with the existing solutions.     

Axisymmetric elasto-plastic large deflection behaviour of circular steel plates with initial deflections—A parametric study

An outline of the incremental equations governing the elasto-plastic large deflection response of circular plates with initial deflections is presented. A few comments on their numerical solution using a finite-difference implementation of the Dynamic Relaxation (DR) method are also included. The numerical analysis is used to undertake a parameter study of the full-range response of imperfect plates and encompasses the effects of edge support condition and plate slenderness as well as imperfection amplitude. The numerical results for deflections, etc. are presented in dimensionless graphical format. Apart from demonstrating the effects of imperfections, which are significant in the elastic and initial elasto-plastic regimes, the results are of potential application for preliminary design.     

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.     

A finite element large displacement analysis of stiffened plates

A finite element analysis of the large deflection behaviour of stiffened plates using the isoparametric quadratic stiffened plate bending element is presented. The evaluation of fundamental equations of the stiffened plates is based on Mindlin's hypothesis. The large deflection equations are based on von Kármán's theory. The solution algmrithm for the assembled nonlinear equilibrium equations is based on the Newton-Raphson iteration technique. Numerical solutions are presented for rectangular plates and skew stiffened plates.