Showing papers on "Finite element method published in 1968"

The Triangular Equilibrium Element in the Solution of Plate Bending Problems

Abstract: Further details are given of a recently developed triangular equilibrium element which is then applied, in conjunction with the complementary energy principle, to the finite element analysis of some plate bending problems. The element is demonstrated to have a straightforward and satisfactory application and to possess advantages over the conventional triangular displacement element.

Finite Element Analysis of Nonlinear Structures

Abstract: The principle of virtual work is employed to derive general relations governing geometrically nonlinear structural behavior. From these basic relations, three general finite-element analysis models, namely, potential energy, direct and incremental, are formulated for nonlinear pre- and post-buckling analyses. In addition, a quadratic eigenvalue model is developed for the prediction of critical load levels. The analysis models are expressed in matrix notation within the framework of the finite-element technology and an inter-consistency is observed among the component matrices. These matrices are given geometrical interpretation and a hierarchy of nonlinearity is developed. Specific representative finite elements are considered and alternative computational procedures are associated with the several levels in the hierarchy of nonlinearity. Recommendations are made concerning the conduct of geometrically nonlinear finite-element analysis.

Matrix and Interface Stresses in a Discontinuous Fiber Composite Model

Abstract: The method of finite element analysis is applied to an axially symmetrical model of a single filament glass-resin composite under tension. Fiber end geometries are varied by considering ellipsoids ...

Finite-Element Analysis of Thin Shells

Abstract: Equations describing small deformations of thin shells of arbitrary shape are written in terms of the displacements of the middle surface and rotation of the normals to the middle surface. The theory accounts for transverse shear deformations. By then using simple bilinear approximations of the displacement and rotation fields within finite elements of the shell, a consistent discrete model is obtained; the model provides complete interelement compatibility, and applies to the analysis of thin shells of arbitrary shape. A discrete equivalent of the Kirchhoff hypothesis is then introduced, which greatly improves convergence rates, and which forces the finite element model to convergence to the continuous Kirchhoff model. Numerical examples are included.

Finite element analyses of pavements

Abstract: AN APPLICATION IS DESCRIBED OF THE FINITE ELEMENT TECHNIQUE TO THE ANALYSIS OF SYSTEMS REPRESENTATIVE OF PAVEMENT STRUCTURES USE IS MADE OF A DIGITAL COMPUTER PROGRAM WHICH GENERATES SUITABLE FINITE ELEMENT CONFIGURATIONS FOR AXISYMMETRIC STRUCTURES AND ACCOMMODATES APPROXIMATIONS OF NONLINEARITY WHICH APPEAR APPROPRIATE TO REPRESENT THE BEHAVIOR OF GRANULAR BASE AND COHESIVE SUBGRADE MATERIALS UNDER CONDITIONS CORRESPONDING TO MOVING TRAFFIC EXAMPLES ARE PRESENTED FOR SYSTEMS WITH LINEAR MATERIAL PROPERTIES SHOWING COMPARISONS BETWEEN DISPLACEMENTS AND STRESSES COMPUTED USING THE FINITE ELEMENT TECHNIQUE AND THOSE COMPUTED USING ELASTIC HALF-SPACE AND LAYERED SYSTEM ANALYSES TO ESTABLISH CRITERIA FOR BOUNDARY CONDITIONS IN THE FINITE ELEMENT PROCEDURE FOR THE ELASTIC HALF-SPACE SUBJECTED TO A UNIFORM CIRCULAR LOAD, DISPLACEMENTS AND STRESSES COMPUTED BY THE FINITE ELEMENT TECHNIQUE COMPARE FAVORABLY WITH THOSE DETERMINED FROM THE BOUSSINESQ SOLUTION WHERE THE NODAL POINTS IN THE FINITE ELEMENT PROCEDURE ARE FIXED AT A DEPTH OF 18 RADII FOR THE BOTTOM BOUNDARY AND CONSTRAINED FROM MOVING RADIALLY ON THE VERTICAL BOUNDARY AT A DISTANCE OF ABOUT 12 RADII FROM THE CENTER TWO ANALYSES ARE PRESENTED FOR DEFLECTION DETERMINATIONS FOR AN INSERVICE PAVEMENT NEAR GONZALES, CALIF, ONE FOR A CONDITION WHERE THE ASPHALT CONCRETE WAS AT A COMPARATIVELY HIGH TEMPERATURE (STIFFNESS MODULUS IN THE RANGE 120,000 TO 280,000 PSI), AND THE OTHER WITH THE MATERIAL AT A LOW TEMPERATURE (STIFFNESS MODULUS APPROXIMATELY 1,500,000 PSI) NONLINEAR MATERIAL PROPERTIES, DETERMINED FROM THE RESULT OF REPEATED LOAD TRIAXIAL COMPRESSION TESTS, WERE USED TO REPRESENT THE BEHAVIOR OF THE UNTREATED GRANULAR BASE AND SUBBASE AND THE FINE-GRAINED SUBGRADE SOIL DEFLECTIONS PREDICTED BY THE FINITE ELEMENT PROCEDURE ARE IN THE SAME RANGE AS THOSE MEASURED WITH THE CALIFORNIA TRAVELING DEFLECTOMETER INDICATING THAT THE METHOD HAS POTENTIAL TO SIMULATE ACTUAL PAVEMENT BEHAVIOR TO A REASONABLE DEGREE THE ANALYSIS ALSO INDICATES THAT WHEN THE EXTENSIONAL STRAINS IN THE ASPHALT CONCRETE ARE LARGE FOR THIS PAVEMENT (IE, WHERE THE STIFFNESS OF THE ASPHALT BOUND MATERIAL IS LOW), THE GRANULAR MATERIAL EXHIBITS A VERY LOW MODULUS UNDER THE LOADED AREA AND A LARGE PROPORTION OF THE SURFACE DEFLECTION CAN BE ATTRIBUTED TO DEFORMATIONS WITHIN THIS MATERIAL ON THE OTHER HAND, WHEN THE ASPHALT LAYER IS STIFF, THE MAJORITY OF THE SURFACE DEFLECTION IS CONTRIBUTED BY THE SUBGRADE AN ANALYSIS IS ALSO PRESENTED FOR THE RESULTS OF PLATE LOAD TESTS ON A TWO- LAYER PROTOTYPE PAVEMENT CONSISTING OF GRANULAR BASE AND A COHESIVE SUBGRADE SOIL USING THE SAME NONLINEAR CHARACTERIZATION FOR MATERIAL PROPERTIES AS FOR THE IN- SERVICE PAVEMENT /AUTHOR/

Developments in structural analysis by direct energy minimization.

Abstract: The equivalence of problems in structural analysis to certain variational statements has traditionally been used as a means of formulating field equations and natural boundary conditions. In recent years developments in computers and minimization algorithms have made it possible to determine the displacement state that satisfies the variational principle without direct recourse to the solution of a system of equations. In this paper an unconstrained minimization approach is applied to a problem in finite-element structural analysis and is shown numerically to be competitive with conventional methods. Adopting an energyminimization viewpoint, important storage advantages of the conjugate-gradient method are extended by eliminating the need for an assembled stiffness matrix. The convergence of conjugate direction methods in quadratic problems is shown to be greatly influenced by scale effects. This problem is studied using the eigenvalues of the stiffness matrix and a practical solution is proposed which by a special scaling transformation can improve the ratio of the maximum to the minimum eigenvalue by several orders of magnitude. This transformation is applied with considerable success to a plate-bending problem.

Elastic Shear Analysis of General Prismatic Beams

Abstract: A procedure is developed for the determination of the transverse shear stress distribution, the shear center location and the shear deformation coefficients for prismatic beams with arbitrarily shaped cross sections. Examples are presented to demonstrate the accuracy and applicability of the procedure. The solution is based upon the elasticity analysis of an end-loaded cantilever beam. The solution to the elasticity problem is developed by utilizing the finite element concept in conjunction with the theorem of minimum potential energy and the Ritz technique.

Application of finite elements to the solution of Helmholtz's equation

Abstract: A novel method, that of finite elements, for the solution of Helmholtz's equation is suggested. Various 2- and 3-dimensional problems are solved using this method, and the results are compared with more conventional techniques, particularly the finite-difference method, which it may be regarded to supersede. The ease with which various boundary conditions may be handled is discussed and illustrated. Nonhomogeneous configurations present no difficulty, nor do they require any special formulation. There is considerable scope for the further development of the technique, which has, until now, been applied mainly to the solution of Laplace or Poisson equations.

Undrained Stress Distribution by Numerical Methods

Abstract: Stress distribution in incompressible or undrained soil is important to the engineer, but the usual finite difference and finite element methods break down for incompressible materials. A revised procedure is described in which the pore pressure is treated as another unknown. Specific application is made to linearly elastic materials; however, the technique can be used for other stress-strain relations. The paper shows the form of the procedure used for finite elements, and that used for a lumped parameter model equivalent to a finite difference method. Results for a sample problem illustrate the insensitivity of vertical stress to compressibility and show the larger effects on the other stress components. Comparison of undrained and drained analysis shows the motion of soil towards the load during consolidation.

Matrix bandwidth minimization

Abstract: In the analysis of a structural problem by the finite element method, a large order stiffness matrix is created which describes mathematically the inter-connectivity of the system The structure is defined in three dimensional space by discrete points called nodes Each node is represented by its coordinates in the space The nodes are then connected by the various finite elements that the particular computer program may utilize (ie, bar members, rectangular or triangular panels, three dimensional tetrahedrons, etc)

Accuracy and Convergence of Finite Element Approximations

Abstract: : The paper reports on a theoretical investigation of the convergence properties of several finite element approximations in current use and assesses the magnitude of the principal errors resulting from their use for certain classes of structural problems. The method is based on classical order of error analyses commonly used to evaluate finite difference methods. Through the use of the Taylor series differential or partial differential equations are found which represent the convergence and principal error characteristics of the finite element equations. These resulting equations are then compared with known equations governing the continuum, and the error terms are evaluated for selected problems. Finite elements for bar, beam, plane stress, and plate bending problems are studied as well as the use of Straight or curved elements to approximate curved beams. The results of the study provide basic information on the effect of interelement compatibility, unequal size elements, discrepancies in triangular element approximations, flat element approximations to curved structures, and the number of elements required for a desired degree of accuracy.

A high precision triangular plate-bending element,

Abstract: A fully conforming, displacement-type, finite element of triangular shape is developed and applied to the solution of several problems in the bending of plates The deflection is represented by a fifth degree polynomial in the Cartesian co-ordinates of the mid-plane Six deflection parameters, namely the displacement and its first and second derivatives, are specified at each vertex, yielding a total of 18 degrees of freedom for the element Examples treated include static and dynamic analyses of a square plate with edges simply supported or clamped; statics of a triangular, simply-supported plate; and vibrations of cantilevered triangular plates Rates of convergence of the finite element approximations are investigated both theoretically and numerically Excellent accuracy is achieved in all cases, and the rates of error convergence agree closely with predicted asymptotic values