# Showing papers in "International Journal for Numerical Methods in Engineering in 1971"

••

TL;DR: In this article, a simple extension is made which allows the element to be economically used in all situations by reducing the order of numerical integration applied to certain terms without sacrificing convergence properties.

Abstract: The solution of plate and shell problems by an independent specification of slopes and middle surface displacements is attractive due to its simplicity and ability of reproducing shear deformation. Unfortunately elements of this type are much too stiff when thickness is reduced.
In an earlier paper a derivation of such an element was presented1 which proved very successful in ‘thick’ situations. Here a very simple extension is made which allows the element to be economically used in all situations.
The improved flexibility is achieved simply by reducing the order of numerical integration applied to certain terms without sacrificing convergence properties. The process is of very wide applicability in improvement of element properties.

1,336 citations

••

TL;DR: In this article, a selective integration scheme for evaluating the stiffness matrix of a quadratic thick shell element was proposed, in which each component of the strain energy was evaluated separately using a different Gaussian integration grid for each contribution.

Abstract: A quadratic thick shell element derived from a three-dimensional isoparametric element was first introduced by Ahmad and co-workers in 1968. This element was noted, however, to be relatively inefficient in representing bending deformations in thin shell or thin plate applications. The present paper outlines a selective integration scheme for evaluating the stiffness matrix of the element, in which each component of the strain energy is evaluated separately using a different Gaussian integration grid for each contribution. By this procedure, the excessive bending stiffness of the element, which results from the use of me quadratic interpolation functions, is avoided.
The improved performance of this element, as compared with the original thick shell element, is demonstrated by analyses of a variety of thin and thick shell problems. 1

320 citations

••

281 citations

••

Laval University

^{1}TL;DR: In this paper, a detailed study of the deformations and stresses produced in an elastic-perfectly plastic half-space indented by a rigid sphere was done using the finite element method, which covers the transition region from the maximum elastic contact load to a state where this load has been increased one hundredfold.

Abstract: Using the finite element method, a detailed study of the deformations and stresses produced in an elastic-perfectly plastic half-space indented by a rigid sphere was done. The analysis covers the transition region from the maximum elastic contact load to a state where this load has been increased one hundredfold. Experimental results available in the literature are in good agreement with the analysis. In solving repeatedly the large number of linear equations involved in the solution of the problem, it was found profitable, in order to save computer time, to modify the direct elimination method. This technique is described in some details in the paper.

236 citations

••

TL;DR: In this paper, the theory of conjugate approximations is used to obtain persistent approximation of stress fields in finite element approximation based on displacement and assumptions, and these consistent stresses are continuous across interelement boundaries and involve less mean error than those computed by the conventional approach.

Abstract: The theory of conjugate approximations is used to obtain persistent approximations of stress fields in finite element approximations based on displacement and assumptions. These consistent stresses are continuous across interelement boundaries and involve less mean error than those computed by the conventional approach. (Author)

195 citations

••

TL;DR: In this article, a combined global and local dependent variable representation which couples the conventional and finite element Ritz methods is presented. But the method is not suitable for the case of a beam and a plate vibration problem.

Abstract: Finite element procedures usually require more degrees of freedom for a specified accuracy than does a classical Ritz procedure if suitable coordinate functions are available. This paper develops a combined global and local dependent variable representation which couples the conventional and finite element Ritz methods. This hybrid method preserves much of the flexibility of the finite element method while increasing the solution accuracy for a specified system order. The method is illustrated by examination of a beam and a plate vibration problem.

164 citations

••

TL;DR: In this article, a finite element technique is used to determine the natural frequencies and the mode shapes of a cantilever plate mounted on the periphery of a rotating disc, where the plate is assumed to make any arbitrary angle with the plane of rotation of the disc.

Abstract: A finite element technique is used to determine the natural frequencies and the mode shapes of a cantilever plate mounted on the periphery of a rotating disc. The plane of the plate is assumed to make any arbitrary angle with the plane of rotation of the disc. The distributed centrifugal force is resolved into two components—one acting in the plane of the plate and the other normal to the plate. The stresses produced in the middle surface of the plate due to the in-plane forces are first determined. The increase in the bending stiffness of the plate elements due to these in-plane stresses is obtained in a manner similar to that used in the stability analysis of plates. The component of the distributed centrifugal force normal to the plate surface is added to the inertia force.
From the results of computations carried out for various values of the aspect ratio, the speed of rotation, the disc radius and the setting angle, empirical formulae are derived giving the effect of these parameters on the natural frequencies. These empirical formulae are observed to be in agreement with the corresponding known formulae for rotating cantilever beams, when the aspect ratio is high.

98 citations

••

TL;DR: In this article, the authors compare different Gaussian-type rules, some of them new and all of them designed to integrate complete polynomials, with the corresponding product-Gauss rules which are normally used, and demonstrate how to further halve the cost by using simpler integration formulae having the same order of truncation error.

Abstract: Already isoparametric hexahedral (brick) finite elements with 20 or 32 nodes1 are highly com petitive in practice, despite the observation that 50 per cent of the total computation is often absorbed in numerically integrating the coefficients of the equations.2 This cost is approximately halved3 by a method based, essentially, on using a 9 x 9 [D] matrix which operates on iu/ox, ou/oy, ou/cz, ot•/c1x, ... ow/oz-a technique which, moreover, is more general than the classical S BT DB x constant algorithm.4 The purpose of this note is to demonstrate how one may further halve the cost by using simpler integration formulae having the same order of truncation error. We compare certain Gaussian type rules, some of them new and all of them designed to integrate complete polynomials, with the corresponding product-Gauss rules which are normally used.5 The former integrate correctly

95 citations

••

TL;DR: Dynamic relaxation, an iterative method for use with digital computers, is described and is shown to be suitable for the solution of a system of linear equations and in particular for such problems derived from structural frame analysis.

Abstract: Dynamic relaxation, an iterative method for use with digital computers, is described and is shown to be suitable for the solution of a system of linear equations and in particular for such problems derived from structural frame analysis. It is further shown that the method may be modified to include non-linear equations relating to these problems.
Some specific examples of linear and non-linear solutions are given and comparisons are made with another computer method which performs the same tasks.

86 citations

••

TL;DR: In this paper, the finite element method is applied to the stability analysis of structural systems subject to non-conservative forces, and the specific application considered here is the stability of thin-walled members subject to follower forces.

Abstract: The finite element method is applied to the stability analysis of structural systems subject to non-conservative forces. The development of the method is general, but the specific application considered here is the stability of thin-walled members subject to follower forces. The method predicts the type of instability, whether it be buckling or flutter. Example problems, for which exact solutions are known, illustrate the accuracy and convergence characteristic of the finite element formulation.

75 citations

••

••

TL;DR: In this paper, the simultaneous iteration method of obtaining eigenvalues and eigenvectors is employed for the solution of undamped vibration problems, and a method of allowing for body freedom is given and some numerical tests are discussed.

Abstract: The simultaneous iteration method of obtaining eigenvalues and eigenvectors is employed for the solution of undamped vibration problems. This method is of significance when a few of the dominant eigenvalues and eigenvectors are required from a large matrix, and hence is particularly suitable for vibration problems involving a large number of degrees of freedom. It is shown that advantage may be taken of both the symmetry and the band form of the mass and stiffness matrices, thus making it feasible to process on a computer larger order vibration problems than can be processed using transformation methods. A method of allowing for body freedom is given and some numerical tests are discussed.

••

TL;DR: In this article, a hybrid technique combining continuum and finite element concepts is proposed for finite element analysis with stress singularities, where each region of stress concentration is covered by one large primary element whose description includes term(s) identifying the type and order of concentration, while the remaining structure is split into a few secondary elements.

Abstract: An important limitation of finite element analysis, namely, the need for a large number of small elements in regions of finite or infinite stress concentrations and the difficulties of convergence in such cases, is well known. Rao1 suggested a possibility of overcoming this by developing hybrid techniques combining continuum and finite element concepts. In such techniques, each region of stress concentration is covered by one large primary element whose description includes term(s) identifying the type and order of concentration, while the remaining structure is split into a few secondary elements which are conventional finite elements. In this paper a procedure incorporating this concept is developed and its effectiveness is clearly demonstrated by successful application to two important examples, one of them with stress singularities. The concept, in fact, can be applied equally well to other two- and three-dimensional problems of continua with discontinuities and concentrations.

••

TL;DR: In this article, a very simple equation solver of very large capacity is developed, which does not require a very large in-core memory; however, a fast random access device like a magnetic disk drive is required.

Abstract: A very simple equation solver of very large capacity is developed. The method described here does not require a very large in-core memory; however, a fast random access device like a magnetic disk drive is required.

••

TL;DR: In this article, the authors used the discrete element displacement method to analyze the finite-deflection behavior of shallow arches and obtained numerical solutions of the geometrically non-linear problem by directly minimizing the total potential energy of the system.

Abstract: The discrete element displacement method is used to analyse the finite-deflection behaviour of shallow arches. The arches are idealized as assemblies of shallow curved elements and the necessary properties of the displacement patterns in these elements are discussed. Numerical solutions of the geometrically non-linear problem are obtained by directly minimizing the total potential energy of the system.

••

TL;DR: In this paper, a stiffness matrix for a finite element having the planform of an annular segment is derived using the displacement approach, and rapid convergence to exact solutions is demonstrated on three sample problems.

Abstract: A stiffness matrix for a finite element having the planform of an annular segment is derived using the displacement approach. Numerical problems involved in the derivation are discussed and rapid convergence to exact solutions is demonstrated on three sample problems. It is concluded that the new element will be of great value to engineers concerned with the analysis of slabs of bridge decks curved in plan.

••

TL;DR: In this paper, an extension to the method is suggested which allows the approximate eigenvalue to be improved and bounds are obtained on the eigenvalues of the full problem before reduction.

Abstract: The general eigenvalue problem Ax = λBx as arising from vibration problems tackled by the finite element method is often solved by the economization method in which the two matrices are reduced to a more manageable size and approximate answers are obtained.
This paper analyses the method in a more mathematical way than previous accounts and leads to a definition of the optimum set of variables to be retained during the reduction. An extension to the method is suggested which allows the approximate eigenvalue to be improved and bounds are obtained on the eigenvalues of the full problem before reduction.
An estimate is made of the calculation involved in the method and it is concluded, by reference to examples, that the extended method leads to a more efficient algorithm.

••

TL;DR: In this article, the development of consistent discrete models, via the concept of finite elements, of linear and non-linear electrothermomechanical behaviour of continuous bodies is concerned.

Abstract: This paper is concerned with the development of consistent discrete models, via the concept of finite elements, of linear and non-linear electrothermomechanical behaviour of continuous bodies. In the development, general energy balances are utilized to derive equations governing electromagnetic fields over an element and coupled equations of motion and heat conduction of a typical element of the continuum. These equations make it possible to study a general class of field problems involving arbitrary geometries and boundary conditions. Sample problems are included.

••

••

TL;DR: A refined axisymmetric curved finite element for the analysis of thin elastic-plastic shells of revolution is described in this paper, which is obtained by employing cubic polynomials in terms of local Cartesian co-ordinates for the assumed in-plane and out-of-plane displacements.

Abstract: A refined axisymmetric curved finite element for the analysis of thin elastic-plastic shells of revolution is described in the paper. The improved element is obtained by employing cubic polynomials in terms of local Cartesian co-ordinates for the assumed in-plane and out-of-plane displacements. This introduces into the solution two internal degrees of freedom in the cord direction of each element. These internal degrees of freedom are removed by static condensation before assembling the individual element stiffness matrices, and are subsequently recovered after the nodal displacements are obtained. On comparison with the previous formulation, this procedure greatly improves the accuracy of the solution especially with regards to in-plane stress-resultants at discontinuities in the meridional curvature and interelement equilibrium of forces. The latter fact makes it possible to analyse shells with a discontinuous meridional slope. In using this element, improvement in the convergence of the elastic-plastic solutions has also been observed.
Several examples illustrate the quality of solutions. The reported study is limited to axisymmetric loadings cum boundary conditions.

••

TL;DR: The Dynamic Relaxation (DR) method as discussed by the authors is based on the spectral radius of the matrix, and it is shown that the DR method gives a faster asymptotic rate of convergence than the degenerate Chebyshev method.

Abstract: The Dynamic Relaxation (DR) method of solving a set of simultaneous linear equations requires an estimate of the spectral radius of the matrix. Dividing each equation by the corresponding row sum of moduli of the elements of the matrix gives a convenient upper bound of unity to this. This note shows that the DR method then gives a faster asymptotic rate of the convergence than the degenerate Chebyshev method which it closely resembles.

••

••

TL;DR: In this article, a unified network-topological approach to the formulation and solution of the analysis and minimum weight design of rigid-plastic structural systems is presented, where both analysis and design can be formulated using network concepts, but in contrast to elastic networks, the formulation results in a mathematical programming problem.

Abstract: A unified network-topological approach to the formulation and solution of the analysis and minimum weight design of rigid-plastic structural systems is presented. It is shown that both analysis and design can be formulated using network concepts, but that, in contrast to elastic networks, the formulation results in a mathematical programming problem. Formulations based both on the static and kinematic theorems of plasticity are presented, and it is shown that duality guarantees that the two types of formulation yield identical results. The problem is first formulated on the basis of a simple definition of yielding, and is then generalized to the case where yielding is defined in terms of a yield surface.

••

TL;DR: For a hierarchy of polynomials on the triangle there is derived an algorithm for computing the stiffness matrix of the plate bending element that is easy to program and means a considerable saving of the computing time.

Abstract: For a hierarchy of polynomials on the triangle there is derived an algorithm for computing the stiffness matrix of the plate bending element. The algorithm is easy to program and means a considerable saving of the computing time. The same approach can be used for any elliptic equation with constant coefficients.

••

TL;DR: In this paper, the point matching numerical method and its generalization, the method of boundary point least squares, have been successfully applied to numerous boundary value and eigenvalue problems.

Abstract: The point matching numerical method and its generalization, the method of boundary point least squares, have been successfully applied to numerous boundary value and eigenvalue problems. The present paper demonstrates the application of these techniques to problems in the micromechanics of fibrous composite materials, i.e. determination of elastic moduli and stress concentrations for parallel-fibre materials which are loaded transversely with respect to the fibres. The solution technique utilizes exact solutions of the governing equations of plane elasticity for each component fibre and its surrounding matrix material in a typical repeating section of the composite material. The continuity conditions for stresses and displacements between fibre and matrix and the repeatability conditions at the boundary of the repeating section are satisfied approximately in a pointwise manner. Some special numerical techniques which were found to be particularly useful in applying the point matching method to these problems are delineated. The method is demonstrated for composite materials having circular, elliptical and square fibres in regular, staggered arrays. Numerical results are given which show the accuracy of the method as well as stress concentration and composite elastic moduli data.

••

TL;DR: In this article, a method of analysis is presented for two structures which have connections with clearance, where the behaviour of the structures is described, by the movement of discrete nodes and that as the load on the structure changes, clearances open or close between corresponding nodes on the structures; each time this occurs the stiffness of the combined structure changes because its configuration has changed.

Abstract: A method of analysis is presented for two structures which have connections with clearance. It is assumed that the behaviour of the structures is described, by the movement of discrete nodes and that as the load on the structure changes, clearances open or close between corresponding nodes on the structures; each time this occurs the stiffness of the combined structure changes because its configuration has changed. Between these changes, the structure will behave linearly and the overall behaviour can be described as piecewise linear. Applications include diesel engine crankshafts with bearing clearances and the wrap-around problem in pin-and-eye connections.

••

TL;DR: In this paper, the central finite difference equations for the plane stress extension of flat plates are derived as a localized Ritz process, and a dual differential-variational discretization of this type enables common classification of the finite difference and finite element methods.

Abstract: The conventional central finite difference equations for the plane stress extension of flat plates are derived as a localized Ritz process. A dual differential-variational discretization of this type enables common classification of the finite difference and finite element methods. Also, it provides alternative methods of establishing sufficiency conditions and relative rates of convergence for discrete systems derived from a localized Ritz process, and the existence of solution bounds for discrete systems derived using difference procedures.

••

TL;DR: In this paper, the numerical solution of the biharmonic equation in a rectangular domain is presented in the context of continuous dynamic programming techniques and the equations are specialized to the solution of elastic rectangular plates.

Abstract: The numerical solution of the biharmonic equation in a rectangular domain is presented in the context of continuous dynamic programming techniques. The equations are specialized to the solution of elastic rectangular plates. A suitable approximate expression of a certain functional equation containing derivatives only in one direction is used to derive equations for the stiffness and flexibility matrices of the plate. It is shown that those matrices satisfy matrix Riccati equations subject to suitable initial conditions. It is also shown that the condition of optimality in the Hamilton-Jacobi-Bellman equation directly expresses a classical variational principle, i.e. the principle of complementary energy. Some numerical examples are finally presented.

••

TL;DR: In this paper, a new class of displacement functions has been developed for rectangular bending elements, in which trigonometric expressions that had apparently failed to yield satisfactory results in earlier attempts can be used along with polynomial terms.

Abstract: A study of bending of plates for small deformations by applying the finite element method is presented in this paper.
A new class of displacement functions has been developed for rectangular bending elements. For the first time a method is proposed for the derivation of displacement functions in which trigonometric expressions that had apparently failed to yield satisfactory results in earlier attempts can be used along with polynomial terms.
The derivation leads to conforming type of displacement functions which satisfy the convergence criterion. The approach also demonstrates how many more new displacement functions can be found.