Application of a refined plate bending element to buckling problems

The buckling stiffness matrix of a refined plate bending element is derived for various continuous and non-uniform distributions of in-plane forces. Elements of this matrix are expressed in an explicit form and can, therefore, be readily used in a finite element computer program for solving plate buckling problems with various loading and edge conditions. A number of problems are solved and the results obtained are compared with analytical and/or numerical results obtained by using other procedures. It is shown that the refined plate bending element is superior to simpler elements when used for solving plate buckling problems.     

Comparison of boundary element and finite element methods for dynamic analysis of elastoplastic plates

Boundary and finite element methodologies for the determination of the response of inelastic plates are compared and critically discussed. Flexural dynamic plate bending problems are considered and a hardening elastoplastic constitutive model is used to describe material behaviour. The domain/boundary element methodology using linear boundary and quadratic interior elements and the finite element method with quadratic Mindlin plate elements are used in this work. The discretized equations of motion in both methodologies are solved by an efficient step-by-step time integration algorithm. Numerical results obtained are presented and compared in order to access the accuracy and computational efficiency of the two methods. In order to make the comparison as meaningful as possible, boundary and finite element computer codes developed by the author are used in this paper. In general, boundary elements appear to be a better choice than finite elements with respect to computational efficiency for the same level of accuracy.     

Evaluation of some multilayered, shear-deformable plate elements

Recently, the author has developed an improved bidimensional transverse shear deformation theory for multilayered anisotropic plates which accounts for piecewise linear distribution across the thickness of the in-plane displacements u and v, and allows the contact conditions at the interfaces between the layers to be satisfied.

Based on this refined theory and on a Mindlin's-type transverse shear deformation plate theory developed by Whitney and Pagano J. appl. Mech. 37, 1031 (1970), several triangular and quadrilateral multilayered anisotropic plate elements which include extension, bending and transverse shear deformation states have been developed by making use of the displacement formulation in conjunction with the principle of virtual work.

In order to show the accuracy and the relative merits of the developed finite elements, results are presented for the sample problems of the bending and free undamped vibrations of a three-layered, symmetric cross-ply square plate that is simply-supported on all edges.

Excepting the conventional triangular and quadrilateral elements with linear shape functions, fast convergence to the respective analytical solutions for the global response (transverse displacements and fundamental flexural frequencies) is observed for all the elements tested. The rectangular finite element developed on the basis of the refined plate theory proposed by the author is also very efficient to model the warpage of the cross-section and to predict accurate values of the flexural stress at the interfaces. The finite elements developed on the basis of the Whitney and Pagano theory fail in this respect.     

Three- and four-noded planar elements using absolute nodal coordinate formulation

This paper investigates two new types of planar finite elements containing three and four nodes. These elements are the reduced forms of the spatial plate elements employing the absolute nodal coordinate approach. Elements of the first type use translations of nodes and global slopes as nodal coordinates and have 18 and 24 degrees of freedom. The slopes facilitate the prevention of the shear locking effect in bending problems. Furthermore, the slopes accurately describe the deformed shape of the elements. Triangular and quadrilateral elements of the second type use translational degrees of freedom only and, therefore, can be utilized successfully in problems without bending. These simple elements with 6 and 8 degrees of freedom are identical to the elements used in conventional formulation of the finite element method from the kinematical point of view. Similarly to the famous problem called “flying spaghetti” which is used often as a benchmark for beam elements, a kind of “flying lasagna” is simulated for the planar elements. Numerical results of simulations are presented.     

A spline wavelet finite element formulation of thin plate bending

The wavelet scaling functions of spline wavelets are used to construct the displacement interpolation functions of triangular and rectangular thin plate elements. The displacement shape functions are then expressed by spline wavelet functions. A spline wavelet finite element formulation of thin plate bending is developed by using the virtual work principle. Two numerical examples have shown that the bending deflections and moments of thin plates agree well with those obtained by the differential equations and conventional elements. It is demonstrated that the current spline wavelet finite element method (FEM) can achieve a high numerical accuracy and converges fast. The proposed spline wavelet finite element formulation has a wide range of applicability since it is developed in the same way like conventional displacement-based FEM.     

Accurate evaluation of support reactions in finite element plate-bending analysis

Two simple approaches are presented which allow the distribution of support reactions to be predicted with as high degree of accuracy as the displacements. In the first approach the plate element assembly is completed with special one-dimensional elastic support elements. If their Winkler coefficient is suitably tuned, an accurate prediction of reactions is obtained as a part of the finite element analysis without unduly affecting the displacements and moments of the plate. In the second approach, a standard finite element calculation (without elastic support elements) is performed first and the distribution of reactions is then evaluated based on the known nodal forces at boundary nodes of the plate.

The two approaches are indiscriminately applicable with Kirchhoff and Reissner-Mindlin plate bending elements. Their practical efficiency is illustrated by numerical examples.     

Experience with finite element modeling of thin plate bending

A finite element modeling technique which utilizes different elements for the boundary region and the interior region is tested for plate bending analysis. Numerical results indicate that model types designated as TYPE 1, TYPE 2 and TYPE 3 do not exhibit detrimental locking effect even for very thin plate situations. For extremely weakly constrained problems, TYPE 2A and TYPE 3A can be used for additional stabilization against possible occurrence of kinematic modes.     

A review and catalogue of plate bending finite elements

A brief review of developments in the field of plate finite elements is presented. This review is followed by an extensive tabular listing of plate bending elements.     

A family of rectangular bending elements

By using separate independent transverse and rotational displacement variables in terms of a polynomial it is possible to produce high order conforming elements for plate bending and, at the same time, to include the effect of shear deformation in the analysis. The procedure for constructing a family of conforming rectangular plate bending elements with any number of nodes and the derivation of the stiffness matrix are illustrated. A computer programme is developed to generate the stiffness coefficients of the elements in this family; whereupon the characteristics of elements with as many as 17, 21 or 25 nodes and so on can be investigated. It is demonstrated that accurate results can be obtained for thin and moderately thick plates with various boundary conditions under bending by using just one or a few high order elements in this family. Hence the procedure for solving a problem in plate bending can be much simplified and the total number of nodes in a problem can be much reduced. Highlight in this family is the 17-node element which yields good results without involving too many nodes for many plate bending problems.     

Finite element analysis of static and dynamic stresses around suddenly punched holes in plates and shells

The magnitude and distribution of stresses around suddenly punched holes in initially stressed plates and shells is of interest to insure that cracks will not precipitate from stress concentration. This problem is of practical interest to pressure vessel designers to preclude catastrophic failure when holes are punched in vessels to release gas. This paper presents a finite element analysis of several problems investigating static and dynamic stress fields around suddenly punched circular holes.

The first problem deals with the investigation of the radial and tangential stress fields in the vicinity of a suddenly punched hole in a stretched, elastic, isotropic plate subjected to an initial hydrostatic stress field. The wave propagation from a punched hole in the plate under a hydrostatic state of stress was solved analytically, using transform techniques, by Miklowitz; the finite element analysis of this problem presented in this paper confirms the analytical solution. Two grid meshes were investigated and results are presented to show the effect of grid mesh on solution accuracy and the power of finite element techniques for solving stress unloading problems. A formula for determining integration step size is found to be a function of the minimum element length and the wave propagation velocity. A similar investigation into the stress effects around a suddenly punched hole in the plate subjected to an initial uniaxial state of stress was also carried out as a prerequisite for the final problem studied.

The last problem is an anisotropic composite shell of varying thickness under an initial stress field due to internal pressure. The static and dynamic stress fields are computed from an unloading wave that radiates outward from a reinforced circular hole that is cut in the shell in 20 μs. A finite-element model of the shell is developed using quadrilateral and triangular plate elements and both in-plane and bending stiffness is included in the analysis as is nonlinear differential stiffening incorporated into the analysis as a single step approximation. Both bending and in-plane waves radiate outward from the cut hole and the dynamic stresses around the hole edge are computed for both unloading waves. The effects of the unloading waves are temporally spaced due to different wave velocities.

The paper demonstrates that fast response stress problems are readily amenable to finite-element analysis. For holes other than circular, the power of finite-element methods is apparent since these shapes lead to mathematically intractable problems if closed form solutions are attempted.     

Modeling and simulation of asymmetrical three-roll bending process

Asymmetrical three-roll bending is one of the three-roll bending processes widely used in metal forming due to its simple configuration. Modeling and simulation of the process based on finite element analyses has great interests to industrial practices. This paper deals with the prediction of the position of the lateral roll during cylindrical roll bending. In the numerical model, the rolls are assumed to be rigid bodies and the plate is assumed to be made of elasto-plastic material, with a bilinear material model corresponding to the results from tensile tests. Shell elements are applied to the plate and automatic node-to-surface contacts are selected for the interfaces between the plate and the rolls. The nonlinear equations are resolved by fully integrated Belytschko–Tsay shell formulation under the well-known ANSYS/LS-DYNA environment with explicit time integration. The positions of the lateral roll predicted by the numerical simulations agree well with experiments.     

Contact problems for the elastic beam

The paper describes numerical experience on the use of variational inequalities and finite elements to obtain approximate solutions to bending problems for elastic beams in the presence of a rigid barrier. The methods are applicable to plate problems but in order that comparisons could be made with exact solutions, only one dimensional beam problems have been considered in this paper. Even these problems display a variety of contact situations and in all cases the numerical solution represented the exact solution well.Convergence rates were examined and the effect of a free boundary adjustment routine which allowed a more accurate location of the contact region was assessed.     

Finite element analysis of stiffened plates

A finite element method is presented in which the constraint between stiffener and member is imposed by means of Lagrange multipliers. This is performed on the functional level, forming augmented variational principles. In order to simplify the initial development and implementation of the proposed method, two-dimensional stiffened beam finite elements are developed. Several such elements are formulated, each showing monotonic convergence in numerical tests. In the development of stiffened plate finite elements, the bending and membrane behaviors are treated seperately. For each, the stiffness matrix of a standard plate element is modified to account for an added beam element (representing the stiffener) and additional terms imposing the constraint between the two. The resulting stiffened plate element was implemented in the SAPIV finite element code. Exact solutions are not known for rib-reinforced plated structures, but results of numerical tests converge monotonically to a value in the vicinity of an approximate “smeared” series solution.     

Insight into 3-node triangular shell finite elements: the effects of element isotropy and mesh patterns

In this paper, we study the convergence characteristics of some 3-node triangular shell finite elements. We review the formulations of three different isotropic 3-node elements and one non-isotropic 3-node element. We analyze a clamped plate problem and a hyperboloid shell problem using various mesh topologies and present the convergence curves using the s-norm. Considering simple bending tests, we also study the transverse shear strain fields of the shell finite elements. The results and insight given are valuable for the proper use and the further development of triangular shell finite elements.     

A generalised technique for fracture analysis of cracked plates under combined tensile,bending and shear loads

The objective of this paper is to propose a generalized technique called numerically integrated modified virtual crack closure integral (NI-MVCCI) technique for fracture analysis of cracked plates under combined tensile, bending and shear loads. NI-MVCCI technique is used for post-processing the results of finite element analysis (FEA) for computation of strain energy release rate (SERR) components and the corresponding stress intensity factor (SIF) for cracked plates. NI-MVCCI technique has been demonstrated for 4-noded, 8-noded (regular and quarter-point) and 9-noded (regular and quarter-point) isoparametric plate finite elements. These elements are based on Mindlin’s plate theory that considers shear deformation. For all the elements, reduced integration/selective reduced integration techniques have been employed in the studies. In addition, for 9-noded element assumed shear interpolation functions have been used to overcome the shear locking problem. Numerical studies on fracture analysis of plates subjected to tension–moment and tension–shear loads have been conducted employing these elements. It is observed that among these elements, the 9-noded Lagrangian plate element with assumed shear interpolation functions exhibits better performance for fracture analysis of cracked plates.     

Finite element analysis of eccentrically stiffened plates in free vibration

A compound finite element model is developed to investigate eccentrically stiffened plates in free vibration. The plate elements and beam elements are treated as integral parts of a compound section, and not as independent bending components. The derivation is based on the assumptions of small deflection theory. In the orthogonally stiffened directions of the compound section, the neutral surfaces may not coincide. They lie between the middle surface of the plate and the centroidal axes of the stiffeners. The results of this study are compared with existing ones and with those of the orthotropic plate approximation. Modifications to the existing equivalent orthotropic rigidities are proposed.     

Implementation of a C1 triangular element based on the P-version of the finite element method

The implementation of a computer code CONE (for C¹ continuity) based on the p-version of the finite element method is described. A hierarchic family of triangular finite elements of degree p ≥ 5 is used. This family enforces C¹-continuity across inter-element boundaries, and the code is applicable to fourth order partial differential equations in two independent variables, in particular to the biharmonic equation. Applications to several benchmark problems in plate bending are presented. Sample results are examined and compared both with theoretical predictions and with the computations of other programs. Significant improvements are shown for the results obtained using CONE.     

Axisymmetric bending of circular and annular sandwich plates with nonlinear elastic core material

This paper compares the analytical model of the axisymmetric bending of a circular sandwich plate with the finite element method (FEM) based numerical model. The differential equations of the bending of circular symmetrical sandwich plates with isotropic face sheets and a nonlinear elastic core material are obtained. The perturbation method of a small parameter is used to represent the nonlinear differential equations as a sequence of linear equations specifying each other. The linear differential equations are solved by reducing them to the Bessel equation. The results of the calculations with the use of the analytical and FEM models are compared with the results obtained by other authors by the example of the following problems: (1) axisymmetric transverse bending of a circular sandwich plate; (2) axisymmetric transverse bending of an annular sandwich plate. The effect of the nonlinear elasticity of the core material on the strained state of the sandwich plate is described.     

Development of special fastener elements

A special finite element (FASNEL) is developed for the analysis of a neat or misfit fastener in a two-dimensional metallic/composite (orthotropic) plate subjected to biaxial loading. The misfit fasteners could be of interference or clearance type. These fasteners, which are common in engineering structures, cause stress concentrations and are potential sources of failure. Such cases of stress concentration present considerable numerical problems for analysis with conventional finite elements. In FASNEL the shape functions for displacements are derived from series stress function solutions satisfying the governing difffferential equation of the plate and some of the boundary conditions on the hole boundary. The region of the plate outside FASNEL is filled with CST or quadrilateral elements. When a plate with a fastener is gradually loaded the fastener-plate interface exhibits a state of partial contact/separation above a certain load level. In misfit fastener, the extent of contact/separation changes with applied load, leading to a nonlinear moving boundary problem and this is handled by FASNEL using an inverse formulation. The analysis is developed at present for a filled hole in a finite elastic plate providing two axes of symmetry. Numerical studies are conducted on a smooth rigid fastener in a finite elastic plate subjected to uniaxial loading to demonstrate the capability of FASNEL.     

A finite element procedure for plates with curved boundaries

The difficulties associated with the use of the finite element method on regions with curved boundaries are discussed. An approach based on the Lagrangian multiplier technique is presented which overcomes all of the difficulties mentioned and which is of particular value for higher order problems such as plate bending. The method is tested numerically using a refined triangular plate element and is found to be a significant improvement over more conventional finite element methods, for this class of problems.