Showing papers in "International Journal for Numerical Methods in Engineering in 1970"

Analysis of thick and thin shell structures by curved finite elements

Abstract: A general formulation for the curved, arbitrary shape of thick shell finite elements is presented in this paper along with a simplified form for axisymmetric situations. A number of examples ranging from thin to thick shell applications are given, which include a cooling tower, water tanks, an idealized arch dam and an actual arch dam with deformable foundation. A new process using curved, thick shell finite elements is developed overcoming the previous approximations to the geometry of the structure and the neglect of shear deformation. A general formulation for a curved, arbitrary shape of shell is developed as well as a simplified form suitable for axisymmetric situations. Several illustrated examples ranging from thin to thick shell applications are given to assess the accuracy of solution attainable. These examples include a cooling tower, tanks, and an idealized dam for which many alternative solutions were used. The usefulness of the development in the context of arch dams, where a ‘thick shell’ situation exists, leads in practice to a fuller discussion of problems of foundation deformation, etc., so that practical application becomes possible and economical.

Finite element method for piezoelectric vibration

Abstract: A finite element formulation which includes the piezoelectric or electroelastic effect is given. A strong analogy is exhibited between electric and elastic variables, and a ‘stiffness’ finite element method is deduced. The dynamical matrix equation of electroelasticity is formulated and found to be reducible in form to the well-known equation of structural dynamics, A tetrahedral finite element is presented, implementing the theorem for application to problems of three-dimensional electroelasticity.

A frontal solution program for finite element analysis

Abstract: The program given here assembles and solves symmetric positive–definite equations as met in finite element applications. The technique is more involved than the standard band–matrix algorithms, but it is more efficient in the important case when two-dimensional or three-dimensional elements have other than corner nodes. Artifices are included to improve efficiency when there are many right hand sides, as in automated design. The organization of the program is described with reference to diagrams, full notation, specimen input data and supplementary comments on the ASA FORTRAN print-out.

Thermomechanical analysis of viscoelastic solids

Abstract: SUMMARY This paper is concerned with the development of a computational algorithm for the solution of the uncoupled, quasi-static boundary value problem for a linear viscoelastic solid undergoing thermal and mechanical deformation. The method evolves from a finite element discretization of a stationary value problem, leading to the solution of a system of linear integral equations determining the motion of the solid. An illustrative example is included.

Finite element analysis of torsional and torsional–flexural stability problems

Abstract: Structure of flexural members, analyzing torsional and lateral stability by finite element method and matrix formulation

Transient field problems: Two‐dimensional and three‐dimensional analysis by isoparametric finite elements

Abstract: The transient field problem of the type encountered in heat conduction problems is formulated in terms of the finite element process using the Galerkin approach. Curved two-dimensional and three-dimensional, isoparametric elements are used in a time-stepping solution and their advantages illustrated by means of several examples.

New displacement hybrid finite element models for solid continua

Abstract: The proposed finite element model is based on separate assumptions of interior and interelement displacements and on the assumed boundary tractions of each individual element. The associated variational functional for this model is presented. This method has the same merits of the assumed stress method (References 3 and 4) in that a compatible displacement function at the interelement boundary can be easily constructed, while it can easily be used for shells with distributed loads.

A multidata method of approximate Laplace transform inversion

Abstract: : A newly developed approximate Laplace transform inversion method is described. A derivation of the method, termed the multidata method, is presented along with a derivation of Schapery's collocation method. Similarities and differences between the two methods are described and discussed. Results of parameter studies of both methods are presented which demonstrate the sensitivity to error displayed by the collocation method and the improved accuracy obtainable with the multidata method as compared with the collocation method when errors exist in the function to be inverted. Computer program listings and operating instructions for both methods are presented. (Author)

Finite element analysis of elastic contact problems using differential displacements

Abstract: A technique of differential displacements is presented whereby problems involving elastic contact are solved by the finite element method. The technique is applied to axisymmetric situations in which statically indeterminate conditions occur and is shown to provide a means for resolving these conditions in terms of contact stresses. Three typical engineering problems are analysed to demonstrate the technique in cases where body forces, thermal gradients and external applied forces are acting.

Two‐dimensional automatic mesh generation for structural analysis

Abstract: This paper presents a labour-saving method of discretizing irregular and inhomogeneous two-dimensional continua into triangular elements. The method uses a magnetic pen to record node point data and a computer program to generate element data. This technique eliminates the tedium in the manual generation of data and the delay due to mistakes which would otherwise arise frequently for a complex mesh.

A gradient computational procedure for the solution of large problems arising from the finite element discretization method

Abstract: A computational procedure based on gradient iterative techniques is proposed for the solution of large problems to which the finite element method is applicable. In linear problems the procedure can be used either for solving the set of algebraic equations or for the complete inversion of the matrix of coefficients. Special attention is focused on the practical aspects of the procedure concerning its realization on the digital computer.

Finite element solution to an elastica problem of beams

Abstract: In this study, finite element solution procedures are developed for an elastica problem of inextensible beams. The element stiffness matrices are obtained by using Galerkin's method. Results of a numerical example compare reasonably well with those obtained by using the elliptical integral. Extensions of this research are currently being made to include membrane effects of beams and to develop plate element stiffness matrices for elastica problems of plate structures.

Discrete element methods for the plastic analysis of structures subjected to cyclic loading

Abstract: Discrete element method for plastic analysis of complex built-up structures subjected to cyclic loading causing membrane stress and stress reversal

Convergence studies of eigenvalue solutions using two finite plate bending elements

Abstract: The convergence rates of eigenvalue solutions using two finite plate bending elements are studied. The elements considered are the well-known 12 degree of freedom, non-conforming rectangular element and the 16 degree of freedom, conforming rectangular element. Three problems are analysed, a square plate simply supported on two opposite sides with the other two sides clamped, simply supported, or free. Closed form, finite element solutions for these problems are obtained by using shifting E-operators. With few exceptions, eigenvalue solutions found with the non-conforming element converge from below the exact answers at an asymptotic rate of n−2, where n is the number of elements on a side. However, since the array size needed for such convergence is very large, little can be said about the convergence rates for practical arrays. The conforming element solutions converge from above at an asymptotic rate of n−4. A comparison of the errors involved in using these two elements shows that the conforming element is far superior to the non-conforming element.

A conforming triangular finite element plate bending solution

Abstract: A conforming plate bending solution using simple polynomial deflection functions of third-degree inside each triangular element is presented In order to avoid normal slope discontinuities along the sides of the elements, the plate displacement parameters are subjected to ‘slope continuity conditions’ acting as constraints to the minimum potential energy problem This is then solved by the classical method of Lagrange introducing multipliers as new auxiliary variables If a special variational formulation of the problem is used, it can be shown that the Lagrangean multipliers are generalized stress parameters The suggested solution is therefore basically a ‘mixed’ solution, the unknown variables of the problem being both displacement and stress parameters Several numerical results are presented

Behaviour and design of pin connected structures

Abstract: The feasibility of achieving a preassigned stress state by changing the stiffness of determinate and indeterminate pin connected structures is presented. A new force method with all element forces (or stresses or areas) as independent variables is developed. The possibility but futility of getting the element areas by inverting and pre-multiplying the coefficient matrix with the external load vector when areas are chosen as variables is shown. The relationship existing between the compatibility equations and the plastic strength of pin connected structures is illustrated. Possible design methods for trusses under single or multiplicity of load conditions, the design of truss geometry and the relations between fully stressed and minimum weight structures are examined and illustrated through several examples.

Kinematically consistent unsteady aerodynamic coefficients in supersonic flow

Abstract: A finite element method to determine unsteady aerodynamic influence coefficients, consistent with the stiffness and inertia properties of a lifting surface in supersonic flow, is described. This is basically a kinematic method, which reduces the dynamical equations of a non-conservative system to a simple and elegant form. It is illustrated by application to a delta wing using triangular elements to calculate steady and unsteady lift and moment coefficients. Throughout the calculations only a coarse grid system has been employed and the answers have been compared with available results.

An application of the finite element technique to boundary value problems of potential flow

Abstract: This report summarizes an experimental application of the finite element approach to two-dimensional inviscid fluid flow. The method results in a matrix equation relating the vector of velocities at the nodal points with the vector of singularities. The singularities, which are concentrated at the nodes, consist of a source and a vortex. On solution of this equation (defining the flow in the region), boundary conditions must te stipulated. This usually involves setting the internal singularities to zero and making certain modifications ro those lying on the boundary of the region. Some results of the application to a particular problem are included in the paper.

The effectiveness of excessive nodal continuities in the finite element analysis of thin rectangular and skew plates in bending

Abstract: Some typical results of an extensive series of evaluations of parallelogram-shaped plate bending elements with increasing levels of sophistication are presented. It is shown that, contrary to experience in other fields of stress analysis, this sophistication does not lead to striking improvements in accuracy in the solution of simple rectangular and skew plate problems. A computational procedure is described whereby stiffness matrices may be automatically generated from the minimum of input data specifying the displacement modes prescribed for the element.

Calculations of plate frequencies from complementary energy formulation

Abstract: The complementary variational principle has been used to derive the differential equations and the associated boundary conditions of the vibrating plate in terms of bending moments. It is shown that in this formulation, the plate possesses an infinite number of zero frequency modes in which the plate remains in a state of constant strain under a set of self-equilibrating bending moments. In applying the Rayleigh Ritz procedure for the non-zero frequency modes of the plate, it is shown that it the assumed functions are orthogonal to only a finite number of zero frequency modes, then one may obtain frequencies which are lower than the true frequencies of the plate. An iliustrative example is given in the paper.

A curved element approximation in the analysis of axi-symmetric thin shells

Abstract: The stiffness equation is derived for curved elements of orthotropic axi-symmetric thin shells, and equivalent applied loads are found for shells subjected to initial strains, applied surface loads and body forces. The Lure approximation of thin shells and displacement field approximation by polynomials of arbitrary degree are included in the formulae derived.

A finite element application of the modified Rayleigh-Ritz method

Abstract: Considerable attention has been devoted in the literature on numerical methods towards securing energy convergence of solutions for, say, linearly elastic plate bending problems. Although energy convergence is necessary it by no means follows that the derived bending moments and shearing forces converge uniformly at a given point and it is this kind of feature which the engineer is really seeking. This question is examined in the context of a problem which is of particular interest to the civil engineering field and concerns the bending of a square plate under uniformly distributed load; the plate has simply supported edges and contains a central square hole with free edges. The solution to this multiply connected and mixed boundary value problem is obtained through a recently developed modification to the Rayleigh–Ritz method which has very general application and renders the solution mathematically valid up to the internal corner points where the bending moments are singular. Use is made of triangular equilibrium finite elements in conjunction with continuous eigenfunctions. Although it is already known that the order (i.e. the eigenvalue) of the singularity at the internal corners is available by inspection, it is an interesting feature of the present solution that a good approximation to the amplitude is also obtained by an inspection of the finite element results.

Finite element with high degree of regularity

Abstract: This paper deals with solving two-dimensional variational problems of second- and third-order by the finite element method. To each meshpoint are associated three or six basic functions of class C1 or C2. The expression of the admissible functions on a triangular and rectangular element are given here in a general form which is specially suitable for computation.

On the numerical solution of a class of problems in a linear first strain-gradient theory of elasticity

Abstract: Discrete models for boundary value problems analysis in first strain-gradient elasticity theory, using extended finite element method

The application of three‐dimensional finite elements to a cylinder‐cylinder intersection

Abstract: Recent advances in finite element techniques have enabled full three-dimensional stress analyses to be undertaken Economy and accuracy can only be achieved simultaneously if the characteristics of the element are understood Preparatory work is described in which suitable pitching and disposition of brick-type elements are established for the analysis of cylinders The information derived is put to use in the analysis of a cylinder–cylinder intersection A favourable comparison is made with an independently computed solution