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• Alexandre R. Schuler

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Stv1fX:G'N TABB A SIMPLE SUIDE TO FINITE ELEMENTS bv D. R. J, OWEN and E. HINTON, Department of Civil Engineering, University Col/ege, Swensee, U.K. This book provides the absolute beginner with a brief introduction to the finite element rnethod. Steady stats heat flow in a eylinder is eonsidered using linear one and two dimensional elements. Ali eomputational details are ineluded and a useful FORTRAN program is provided. The further applications of the torsion of prismatic bars and groundwater f,1?ware also considered.

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Approx. 150 pages, , 980, Q-906674-.

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by E. HINTON end D. R. J. OWEN, õepertment of Civil Engineering, University Col/ege,Swensee, UX. This book oflers a basic introduction to the finite element method. Alter a detailed Introduction into th~ numerical analysis of diserete systems, such as frameworks, consideration is given to the solution of some one-dimensional problems using variational and weighted residual finite element methods. Two dimensional finite elements are Introdueed and used to solve problems associated with heat and fluid flow. the torsion of prismatie bars and other stress analysis applieations. Other to pies eonsidered Include : numerieally integrated isoparametrie elsrnents. the flnite strip method, advaneed equation solvers and mesh generation schemes. Computer programs, written in FORTRAN, and worked examples are included for ali applieations. This book, which treats a potentially difficult subjeet in a straightforward and readable manner, should be of interest in the elessroom, for private study or as a professional refaranea. Engineering graduates and under-graduates as well as Qu.II~.d .ngln •• ,. wlll flnd lhe Itlf·contaln.d IIXt I uaeful inlioductlon to finit. elernents espeelally as tha mathematies is kept to a falrly elementary leveI.

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Approx. 400 pages, 1980, Q-906674-, dE2 (dEZ = O). Stress

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FINITE

2.1

ELEMENT

2 BASICS

Introduction

The finite element method now has a very wide range of applications. These in addition to structural engineering include the fields of fluid flow, electricity and magnetismo It started with structural applications, and it is ~ith the formulations for relating loads to the displacement of a structure that we are concerned here. A different formulation is required for seepage analysis. This is covered in Chapter 9. The reader familiar with matrix methods for anaIysing structures such as a pin-jDinted frame is half-way to understanding the f.e.m. Perhaps more than half-way, because the members Df a pin-jointed frame are simple finite elements. The procedure for assembling the stiffness contributions from the individual members to obtain the coefficients Df the overalI stiffness matrix [~) is in principIe the same as when the elements represent an arbitrary subdivision Df a continuum. The only difference is that in the latter case (which is the case Df interest to geotechnical engineers) the stiffness matrix cannot be set up by inspection. More formal procedures are required.

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This chapter sets out to explain these procedures. The conventional 'stiffness' approach is used. (Alternative derivations based on variational caIculus or, e.g., the Galerkin method, are not given both because of the mathematics involved and because they are better suited to formulations where the starting point is a governing differential equation. This is not the case with load-deformation problems, although it is with seepage.) Geometric considerations - elements. shape (or interpolation) functions and certain geometric co-ordinate transformations - are treated first. The virtual work method is then used to set up the stiffness equations, and to determine nodal forces equivalent to body forces and surface tractions. For a more complete treatment of finite element reader is referred in the first instance to two books same stable as the principal authors of this volume: text by Zienkiewicz (1977) now in its third edition,

basics the from the the classic and the

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12 text by Hinton and Owen (1979). The notation used herein is much the same as fn these texts. The second of them devotes a chapter to a review Df the literature. In it there is a use~ fuI list Df 31 other books on the subject, with short comments on each. It is noteworthy that not one in this list is aimed primarily at geotechnical engineers. An earlier text by Hinton and Owen (1977) deals specifically with programming aspects of the f.e.m. .

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2.2

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The starting point of an analysis is the division Df the structure into elements. Eight basic element types are illustrated in figure (2-1), Some of these shapes represent a number of different types. Thus a 2 or 3 noded line element can represent a bar having no bending sti~fness in 1, 2, or 3 dimensions. For these applications it will have respectively 1, 2, or 3 degrees of freedom per node. Alternatively it may be a bending element in which case an extra rotational degree of freedom per node is added for 2-0 applications and three extra rotational degrees Df freedom (making a total Df 6) are added for 3-0 applications. Similarly, the triangular and quadrilateral elements may represent membranes or plates in 3-0. In the former case their bending stiffness is neglected and they have 3 degrees of freedom per nade. Plate bending elements may have 6 d ,of f , per node.

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Figure 2-1

Some basie finite elements

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For geotechnical work the triangular and quadrilateral elements applied in a plane strain analysis are most eommonly used. (They then represent a solid bloekof material with an out of plane thickness which is usually one.) Sometimes the plane elements will represent an axisymmetric geometry. Line elements can represent ties or props, or flexible linings to tunnels. Bending elements are seldom used, perhaps beeause there are not many programs available which mix them with con-

13 ventional elements (there is a compatibility problem at common nodes). They have, however, a role in soil:structure interaction applications. 3-D elements are used when the cost (which lies as much or more in data preparation and output handling as in the actual computing) is justified. Higher order elements with more than one midside node are available but although they give better accuracy per element it is doubtful if they offer any advantage on a 'per nade' basis. It seems to be widely accepted that the so-called 'parabolic' elements which have one midside node offer the best value per nade. The bottom row of elements illustrated in figure (2-1) are therefore widely used, especially the 8 noded quadrilateral. Recent work has drawn attention to the 'Lagrangian' quadrilateral and brick elements which have, respectively, 9 and 27 nades. The ninth node is at the centre of the quadrilaterial, and the seven extra nodes in the 3-D element are at the element centre and the centres of the six faces. These elements may give even better value than the more commonly used type of parabolic elemento With the exception Df the beam and plate bending elements, the elements considered here are 'isoparametric'. That is, the equations describing the shape of their boundaries are of the same arder as those describing the variation of the nodal unknown (e.g. displacement) across the elemento Thus both are quadratic in the case of 'parabolic' elements. Selection of the size and shape of elements is a matter of experience and intuition. Generally, elements ~hould ·be smaller where the 'action' is coneentrated, i.e. wherethere are rapid changes in stress and strain. Figure (2-2 a) illustrates this for a footing. The smallest elements are at the corner of the footing. Figure (2-2 b) illustrates a 'spider web' type of mesh. This efficiently increases the element size with distance from the highly stressed region. Remarkably high accuraey can be achieved with a coarse .mesh when parabolie elements are used. It is good praetiee to carry out a preliminary analysis using a very coarse mesh, e.g. 6 soil and one footing element would eonstitute a suitable coarse mesh for a preliminary analysis of the footing of figure (2-2 a). Substantial agreement of displacements with those obtained from a fine mesh analysis would be expected (say 20% difference in the maximum valuesl. .

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lhese define the'variation of quantities across elements. lhe quantities comprise the nodal unknowns in the first instance, but include any quantity which is rEquired to vary smoothly across the element between fixed (known, or to be determined) values at the nodes. Let O stand for the value of the quantity at some point x,y (~e restrict to 2-D for simplicity), and suffix i indicate the value at node i, then n (2-1 ) O 0i i=1

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which proves the lemma.

Yield surfoce

Lemma 2 has many other implications. Consider a stress situation on the yield surface (figure 6-4) represented by stress ~. Let ~a be another stress situation on the yield surface. Now (6-10) (R:-~a)T kP = I~-~al IkPI cosa where

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