AN I N F I N I T E F I N I T E ELEMENT by RONALD FREDERICK DNGLESS B.A.Sc. The University (1971) of B r i t i s h Columbia A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS MASTER in FOR THE DEGREE OF OF APPLIED SCIENCE t h e Department of C I V I L ENGINEERING We a c c e p t to The this thesis the required University as conforming standard of B r i t i s h September 1973 Columbia In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements Columbia, available f o r an advanced degree at the U n i v e r s i t y I agree for that reference the Library and study. permission f o r e x t e n s i v e copying of t h i s purposes may shall I further thesis be granted by the Head of my representatives. I t i s understood make for it freely agree that scholarly Department or by h i s that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed written of B r i t i s h without permission. R. P. Department of C i v i l Engineering The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada September, 1973. Ungless my AS INFINITE FINITE ELEMENT ABSTRACT This element of model o f three extending infinity finite semi-infinite solid or in infinite element numbers element of the i n three in the the i n f i n i t e use of the node involve on a two governed i s devised. having element three i s used has previously finite element computational which has been limited number o f d e g r e e s element semi-infinite in models. time plan form There because The ii one the t o be zero The large of the finite A large i s possible accuracy of loaded infinite by the a r t i f i c i a l whose e x a c t s o l u t i o n s solid, of problems. of freedom are conjunction caused the use dimensional halfspace examples by displacement assumed problem finite analysis to represent the s t i f f n e s s p r e v i o u s models i s e l i m i n a t e d . tested the s t r e s s of t r i a n g u l a r direction infinite which developing a bodies element one elements elements method with to f a c i l i t a t e each The reduction of in regular relieves size e q u a t i o n s . An of freedom. regular concerned semi-infinite nodes per e l e m e n t , degrees with linear to i s infinite dimensional, independent three thesis with boundary the element a r e known. with a i s Eoth surface perpendicular point l o a d . The elasticity point l o a d , and the other with r e s u l t s compare f a v o u r a b l y solutions. iii a surface with the parallel theory of TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS iv LIST OF FIGURES vl ACKNOWLEDGEMENTS vii CHAPTER 1. 2. INTRODUCTION 1 1.1 Background 1 1.2 Purpose and Scope 2 ELEMENT GEOMETRY 4 2.1 Topology. 4 2.2 Transformation o f Axes 9 2.3 Transformation of G l o b a l Side Vectors to L o c a l S i d e V e c t o r s 13 3. AREA COORDINATES 4. ELEMENT DISPLACEMENT FUNCTION 20 5. ELEMENT STIFFNESS MATRIX 26 5.1 » Formulation o f Element S t i f f n e s s Matrix 5.2 15 26 A n a l y t i c I n t e g r a t i o n o f the S t i f f n e s s Coefficients 28 iv CHAPTER 6. Page S E M I - I N F I N I T E NUMERICAL INTEGRATION 38 6.1 Goodwin-Moran Method 38 6.2 Analytical versus Numerical 39 Integration 7. 47 NUMERICAL EXAMPLES 7.1 Example 1 - S e m i - I n f i n i t e Solid Loaded by a S u r f a c e P o i n t Load Perpendicular to the Surface 7.2 Example 2 - S e m i - I n f i n i t e Solid Loaded by a S u r f a c e P o i n t Parallel 8. 49 Load to the Surface DISCUSSION 55 60 BIBLIOGRAPHY 62 APPENDIX A. ELEMENT STIFFNESS MATRIX FOR LOCAL COORDINATE SYSTEM B. 63 GOODWIN-MORAN METHOD OF S E M I - I N F I N I T E INTEGRATION C. D. 66 L I S T I N G OF FORTRAN SUBROUTINE OF THE GOODWIN-MORAN METHOD 70 GENERALIZED LOAD VECTOR 72 v L I S T OF FIGURES Figure Page 2.1 A Three Dimensional Iiifinite Finite 2.2 P l a n View o f the Base P l a n e 2.3 L o c a l and Global Coordinates Infinite Finite Element. 6 f o r an 10 Coordinates 16 D e f i n i t i o n of Area 4.1 Adjacent 7.1 S e m i - I n f i n i t e S o l i d . L o a d e d by L o c a l Coordinate Systems a P o i n t Load P e r p e n d i c u l a r to 24 Surface the Surface Finite 50 E l e m e n t Mesh o f H e m i s p h e r i c a l w i t h One 7.3 7.4 Displacement Quarter of Hemispherical Boundary U n i t Load (X-Z Semi-Infinite Solid L o a d e d by a Displacement Loaded to S u r f a c e of Hemispherical Parallel Plane) to S u r f a c e vi 54 Surface to t h e S u r f a c e of Hemispherical Loaded P a r a l l e l 7.6 53 under a V e r t i c a l Displacement Bowl Symmetry P o i n t Load P a r a l l e l 7.5 5 Element 3.1 7.2 . . . 55 Boundary (X-Z Plane) 58 Boundary (X-Y Plane) 5& ACKNOWLEDGEMENTS I wish advice given preparation and Brian t o e x p r e s s my a p p r e c i a t i o n by my a d v i s o r . of this Council Dr. D. L . A n d e r s o n , thesis. I would o f Canada f o r t h e i r like Vancouver, British Columbia Vll the discussions. t o thank financial S e p t e m b e r , 1973 throughout T h a n k s a l s o a r e due t o I a n M i l l e r Badomske f o r many h e l p f u l Finally, f o r t h e g u i d a n c e and the support. Rational Research 1 CHAPTER 1 IIIHODUCTION }.± Background Much use soil and rock halfspaces. halfspace of the f i n i t e element method has mechanics problems dealing a finite with infinite I t has been normal p r a c t i c e to r e p l a c e the under study by a finite boundary at some a r b i t r a r y d i s t a n c e from (e.g., been made i n body a with an point infinite artificial of interest point of l o a d i n g , l o c a t i o n of s u r f a c e s t r u c t u r e ) . Since number of elements i s a n e c e s s i t y , the reason f o r this boundary i s q u i t e apparent. Because dimensional and of computer storage f i n i t e element analyses t h e r e f o r e , approximating with cannot a f f o r d many be i n normal r e q u i r e d to achieve may elements be q u i t e l a r g e . simultaneous gross error. The ) which may elements -number a certain T h i s w i l l r e s u l t i n very l a r g e equations, three a f i x e d or f r e e boundary near the point of i n t e r e s t may finite limitations, of accuracy numbers place a severe l i m i t a t i o n of on 2 the usefulness band width leading of the method f o r p r a c t i c a l problems. of to element o f the large resulting computer infinite size, equation storage with system becomes requirements. which this Also the A thesis large finite concerns i t s e l f , could minimize some of these problems. 1.2 Purpose and The purpose with which the infinite Scope stress bodies may of this t h e s i s i s to c o n s t r u c t a model analysis be three facilated. which the element i s of i n f i n i t e present a r t i f i c i a l boundary of dimensional, A f i n i t e element model i n s i z e i s proposed to r e p l a c e the conditions. A r e d u c t i o n o f the degrees of freedom and of the an i n f i n i t e h a l f s p a c e problem may infinite finite element. The c a r e f u l placement of an a r t i f i c i a l displacements can be assumed are l e s s e r band needed band width be p o s s i b l e with the use of standard method r e q u i r e s boundary where stresses negligible while the i n f i n i t e element method makes no such requirement. elements semi- As a result, or fewer l e a d i n g to fewer degrees of freedom and a width than the standard method. A t t e n t i o n w i l l be c o n f i n e d linear elastostatic linear equations. to the consideration of problems c h a r a c t e r i z e d by time independent This thesis considers an element of 3 triangular plan form extending Each node of t h i s t r i a n g l e has to i n f i n i t y three degrees of a p p l i c a t i o n of the element developed i s i l l u s t r a t e d by freedom and these nodes form the usual a finite infinite element elements. displacement boundary. s o l u t i o n of the Boussinesg problem using the The r e s u l t s a r e compared s o l u t i o n s of the problem. the i n one d i r e c t i o n . A computer program to exact a n a l y t i c a l was developed f i n i t e element a n a l y s i s employing the i n f i n i t e element. for 4 CHAPTER 2 ELEMENT GEOMETRY 2.J To£olocjy_ The element as an " i n f i n i t e model, shown i n F i g . 2 . 1 , f i n i t e element." i s r e f e r r e d to T h i s p a r a d o x i c a l term has been coined because the element extends to i n f i n i t y i n the direction normal to the f i n i t e plane t r i a n g l e formed by the nodes. Fig. 2 . 1 right hand shows the geometry of the element and the two cartesian l o c a l (x,y,z). c o o r d i n a t e systems - g l o b a l ( X,Y,Z The o r i g i n of being at node I the local coordinate ) and system and the x a x i s p a r a l l e l to the I - 3 shown as side. T h i s c h o i c e of a x i s permits i n t e g r a t i o n over the area the t r i a n g l e without The to element has three nodes l o c a t e d at the c o r n e r s . as the "base plane". i n t o the element. of d i v i d i n g the i n t e g r a l i n t o two p a r t s . These nodes a l l l i e i n the l o c a l z=0 referred is plane. T h i s plane w i l l be The l o c a l z a x i s i s d i r e c t e d 5 Fig. 2.1 A Three Dimensional I n f i n i t e F i n i t e Element 6 F i g . 2.2 Plan View of the Base Plane Three g e n e r a l i z e d displacements (u, v, w) are to each c o r n e r node of the element shown i n F i g . 2.2, in nine degrees of freedom per element. The assigned resulting in-plane 7 displacements and u and v a c t i n the x and the out of plane displacement y direction respectively w a c t s i n the z d i r e c t i o n . The element edges or " r a y s " s t a r t from the travel off to infinity V_ ( L , 3 ) represented by v e c t o r s The i n the z d i r e c t i o n . = , v e c t o r s are shown on F i g . 2 . 1 s i n c e the v e c t o r i s s t r a i g h t . are constants local and I ,J the z=0 flane ,TT z=constant the and can be expressed nodes. as the Q k 1 b % v e c t o r s are u n i t v e c t o r s i n the The superscript denotes a are the nodal c o o r d i n a t e s of the element ) and = Q-L V. element are at be i n the l o c a l c o o r d i n a t e system. (*-_;y- If origins and edges can In the above eguation (x,y,z) c o o r d i n a t e system. g u a n t i t y expressed in the with The nodes if the d i r e c t i o n s of the edges of the + b~j +• k then on any plane given by the nodal c o o r d i n a t e s are y. J «• = y° J 1 + bx 1 I n i t i a l l y the " r a y s " of each element w i l l be i n terms of g l o b a l c o o r d i n a t e s by the v e c t o r s (2.2) specified 8 V* = A I - B,J + Q K t where Ci-1,3) are u n i t v e c t o r s i n the g l o b a l The s u p e r s c r i p t coordinate global side vectors carried out. ( X,Y,Z ) system. <y denotes a q u a n t i t y expressed system. The element input data In Section ( V ) to or may side A[, B;,Cr be in the global 2.3 the t r a n s f o r m a t i o n of the local constants (2.3) vectors may form automatically ( V ) is part of the generated. One method of doing t h i s i s by adding the u n i t v e c t o r s normal to the base planes of a l l elements common to a node. The r e s u l t a n t v e c t o r becomes the " r a y " o r i g i n a t i n g a t t h a t node. The approach written f o r has infinite been adopted i n a computer program latter f i n i t e element problems. Some r e s t r i c t i o n s on the d i r e c t i o n of the rays must be observed i f the method i s to produce realistic results. An element which has a l l t h r e e rays p e r p e n d i c u l a r to the base plane ( i . e . , CX^ b. - O ) t will produce chosen displacement f u n c t i o n the exact erroneous (see Chapter U). the due to the The s t i f f n e s s s o l u t i o n goes to zero i n t h i s case y i e l d i n g displacements whereas t h e f i n i t e At results other element stiffness is of infinite finite. extreme, an element i n which any one of i t s rays lies in the same (i.e. , , )o •-*•<») plane will as the base experience plane, numerical or almost so d i f f i c u l t i e s and 9 w i l l g i v e poor r e s u l t s . A third restriction requires c r o s s s e c t i o n a l area i n c r e a s e with i n c r e a s i n g z. satisfied, likely not the I f t h i s i s not zero c r o s s s e c t i o n a l area a t some value of i a occur producing be that allowed to numerical cross as difficulties. Also will rays must t h i s a l s o l e a d s to a zero c r o s s s e c t i o n a l area. 2.. 2 Transformation To expressed requires system base transform in system, and a of Axes the the local stiffness matrix transformation plane and and element stiffness matrix between the l o c a l the g l o b a l c o o r d i n a t e system the two an element c o o r d i n a t e system to a g l o b a l then assemble i n t o the s t r u c t u r e (x,y,z) of ( X,Y,Z matrix, coordinate ). c o o r d i n a t e systems of an i n f i n i t e element are shown i n F i g . For t h i s t h e s i s the l o c a l c o o r d i n a t e system be chosen as having finite 2.3. There are s e v e r a l c h o i c e s f o r d i r e c t i o n s of the coordinates. The the x a x i s d i r e c t e d along the the t r i a n g l e as shown i n F i g . 2.3. I k local shall side of 10 F i g . 2.3 L o c a l and G l o b a l Coordinates f o r an I n f i n i t e Finite Element — fr The vector d e f i n e s the which i n g l o b a l c o o r d i n a t e s i s ik s i d e of the t r i a n g l e 11 r v T.-Y, The components (2.4) direction cosines are found by of t h i s v e c t o r by i t s length g i v i n g dividing the a vector of unit length. r \ x... x X J Y 7\ (2.5) k(. 7 . with 2 + Y. + I* 1 k<. The plane". The local z direction v e c t o r product t h i s normal d i r e c t i o n , thus will of two be normal to the s i d e s of the t r i a n g l e "base defines 12 r Y.Z.. V = V. x V. Y7, JL - •< X..Z J (2.6) kl L X« Y.. the magnitude of Vz i s equal Kt to twice ji Kc the area of the z the triangle. LThe simply direction by d i v i d i n g cosines r > these If directions length 2.A . (2.7) K I Y -• - X . . J J L axis i s perpendicular so i s found by the c r o s s product directions. its - 2A k y by z obtained Y Z - Y- Z . X . The a x i s are vector f *Y V the components of T h i s gives the u n i t k = of vectors of of unit as d e f i n e d by Eqs. l Y ki to the x and vectors in the z axes, x and and z l e n g t h are taken i n each of 2.5 and 2.7, we have 13 r r k J X L - (2.8) A A t r a n s f o r m a t i o n matrix can be w r i t t e n c o n t a i n i n g direction cosines in the v e c t o r s of the l o c a l system. T global directions T h i s matrix will of the be the L iJ , K denoted by and i s given by T = Y \ x \ V V V \* \t \ z \ Z : J (2.9) : The element s t i f f n e s s matrix i n g l o b a l c o o r d i n a t e s is derived from the local element stiffness matrix by the following relation [4] k'= T k T 9 - r T i * (2-10) 2.3__Transformatign_gf_Glcbal To complete the p o s i t i o n i n g of an the element edges element or r a y s must be d e s c r i b e d . in space, Their direction 14 can be d e s c r i b e d by a v e c t o r whose form i s given i n terms of described local by global transformation l o c a l and coordinates. coordinates the thereby to l o c a l c o o r d i n a t e s . rays = global: OL T be necessitating a I f the ray v e c t o r s i n the + the unknown c o e f f i c i e n t s The scalar + 4 lo[ J V. ? = A; I from the known c o e f f i c i e n t s BtJ CL^ C.' ^ (.= 1,3 + C.K , i b i r L = l must C _ product of the u n i t l o c a l vector d i r e c t i o n . a- - V*- t - A ; \ „ c;.V. -IT - < \ . \ T x be '3 determined A:.Bi.C:. the g l o b a l ray v e c t o r with a unit, l o c a l v e c t o r produces the component of the then will in g l o b a l systems are w r i t t e n as local-: then Initially Eg. 2.1 ray vector in Thus + B Ji, + C A ; Y L * B A, *C A ; f o l l o w i n g the form of Eg. 2.1; Y ; > z l Z the constants are r e d e f i n e d as b ^ = b i.7 C l ' c. = c / / , < , I ,2.11, 15 CHAPTER 3 MM_COORDINATES In the development of a basic necessities continuous aid i s to displacements finite establish of normalized relation one of the between the u,'v,w and the nodal displacements. To i n t h i s matter, i t i s convenient a element to i n t r o d u c e a s p e c i a l s e t coordinates for a triangle c a l l e d area or natural coordinates. A, . A , Let and r\ be the areas s u b t r i a n g l e s subtended by P and the c o r n e r s index designating c o o r d i n a t e s 0- the opposite of the p o i n t P corner of the three (see F i g . 3.1), the number. The area a r e d e f i n e d by (3.1) where A / \ = + 1 A 3 L +A 3 = area of the element. c o o r d i n a t e s of the nodes of the element a r e : Thus the area 16 node I node 2. pi-f.-o pt-i,f,=o node 3 = 0, also, a useful ?3 = identity i s proved using the d e f i n i t i o n of 3 (1,0,0) (0,1,0) I (0,0,1) ^ = P • and A w n l c h c a n . (0,0,1) (l,0.0) Fig. 3.1 Definition of Area Coordinates. D e easily 17 The relationship between the x-y c o o r d i n a t e s and the area c o o r d i n a t e s i s l i n e a r and can be shown to be given expression x, x 2 x P3 Pi y Inverting the _ x t h i s equation (3.2) and using the i d e n t i t y + ^3 ' = I ?, ' by >- ?x I 2A (3.3) y J where y. 2A X. y* 18 -(v^/)(y;^ i) 3 = A + + (<^v)(y>bz) B z + Cr 2 (3.4) Rewriting r Pi Eg. 3;3 i n index form g i v e s ( v v)(y;*i° 0 - ( x ; k + a )( ; b ) kZ y + jZ (3.5) 19 Or J ' A + B ^ + C z 1 where i n Eg. 3.5 the i n d i c e s take on the permutation J 1 Z 3 Z 3 I 3 I 1. In a coordinates K plane are parallel functions t r i a n g l e and as we proceed coordinates become to the x,y plane, the area o f the x,y nodal c o o r d i n a t e s of the i n the z d i r e c t i o n these f u n c t i o n s of z. same nodal Thus, the area c o o r d i n a t e s become f u n c t i o n s of z. Area c o o r d i n a t e s of plane a triangle in a p a r a l l e l to top s u r f a c e t r i a n g l e or "base plane" 20 CHAPTER 4 ELEMENT DISPLACEMENT FUNCTION The r o l e of the displacement f u n c t i o n of an element i s to e s t a b l i s h a r e l a t i o n between the nodal displacements and displacements of a l l p o i n t s i n the There are two selection of displacement selected. a element. i n t e r r e l a t e d f a c t o r s which i n f l u e n c e the displacement function. First, the p a r t i c u l a r degrees of freedom t h a t d e s c r i b e the model must Chapter 2 these were chosen as the In of the nodes i n the t h r e e coordinate Second, between adjacent elements providing and ability to d e s c r i b e r i g i d body displacements of the element. fixed boundary at infinity whole to the c o r r e c t s o l u t i o n , the f u n c t i o n should s a t i s f y c e r t a i n c o n d i t i o n s such as displacements be displacements directions. ensure that the numerical r e s u l t s approach compatible the the A will prohibit a rigid body element. However a rigid body displacement of the displacement of each c r o s s s e c t i o n of the element (z=constant) i s a p o s s i b l e c o n d i t i o n which the displacement f u n c t i o n must be 21 able to describe. T h i s d e s c r i b e s a s t a t e i n which only shear s t r e s s e s i n the plane z=constant The displacements shape function u,v,w to exist. which relates the continuous the nine nodal displacements takes the form U FKy) Pfz) = s = A s (4.D where r \ U U = \ v > ocal w V and conlinuous J d i s p l a c e me n l s coorolinale directions in 22 s =< > nodal = displacements coordinate The variation directions of the continuous displacements u,v,w i n the x-y plane i s given by F(x^) in loca F(X , which w i l l s a t i s f y y) • T h e simplest function f o r compatibility of displacements when the base planes of adjacent elements l i e i n the same plane, is a linear relation Since a linear between nodal and continuous r e l a t i o n between the (x,y) c o o r d i n a t e s and the area c o o r d i n a t e s e x i s t s , a l i n e a r f u n c t i o n of can be used f o r i n t e r p o l a t i o n plane (i.e., displacements, z=constant). area over the t r i a n g l e Therefore, coordinates l y i n g i n the x-y F ( ,y0 will x be expressed as f o l l o w s o f, O f, o O ?, Rx,y)- o o O f, O o Of, o o o p. 0 f t 0 ° o (<*.2) ° p. In order f o r the displacements to vanish at infinity 23 u,v and w must approach zero as z tends t o i n f i n i t y , be a that decay function d e c r e a s i n g them vanish at function yet f o r POO. remains m o d i f i e s the displacements u,v,w monotonically infinity. must with Also increasing must be until unity which s a t i s f i e s the above simple enough to c a r r y z by they at z=0. requirements A and out a n a l y t i c i n t e g r a t i o n i n some cases, i s P(z)= 0 + Z)" (4.3) and w i l l be used throughout The chosen continuity plane. the t h e s i s . displacement f u n c t i o n will satisfy between elements whose base t r i a n g l e s l i e i n the same However this i s , in general, d i s c o n t i n u i t i e s at the i n t e r f a c e can occur. this discontinuity « jf(*,_y) not To the case and illustrate how may occur, an example i n F i g . 4.1 i s g i v e n . 24 Fig. 4.1 Adjacent l o c a l Coordinate Systems A d i s c o n t i n u i t y a r i s e s because between adjacent elements local coordinate produce the z, would have as a p o i n t on the boundary different shown i n F i g . 4.1. values T h i s i n turn d i f f e r e n t values of displacement f o r the same boundary. of T h i s d i s c o n t i n u i t y can be minimized the f i n i t e non-conforming on i f the angle same. element d e r i v e d i n t h i s t h e s i s i s based on a displacement In summary a very function. elementary form function has element. T h i s y i e l d s a simple f i n i t e element will will point of the ray v e c t o r to each base plane i s approximately the However the of displacement been chosen t o r e p r e s e n t the displacements of the but one which, i t be shown, g i v e s good r e s u l t s f o r the deformations o f semi- infinite solids. 26 CHAPTER 15 ELEMENT STIFFNESS 5^1^ Formulation The homogeneous, theory MATRIX of Elejment_Stiffness element linear stiffness elastic matrix material. is developed Classical for a elasticity i s used i n the development with only the l i n e a r p o r t i o n s of the s t r a i n - d i s p l a c e m e n t relations l i m i t i n g the a n a l y s i s t o s m a l l The being retained, thereby displacements. s t r a i n - d i s p l a c e m e n t equations are given by: (5.1) 27 or ( where 6 , £^, £ z X L : U are x X 0 the normal strain are the components of shear compoments and strain. I f the assumed form of the displacement f u n c t i o n given i n Eg. A.3 i s i n s e r t e d (Eq. 5.1), the into strains the strain-displacement may be expressed equation i n matrix n o t a t i o n i n terms of the unknown nodal displacements as ( L u : = L A s = B s (5.2) I t can be shown [ 1 ] that the element s t i f f n e s s in l o c a l coordinates, K B _ T E_ matrix , i s given by B dV _ (5.3) V where the element. integration E is carried i s the c o n s t i t u t i v e s t r a i n , and i s given by out over the volume of the matrix relating stress to 28 I /|-v/-v O v O V I /-v O O v O - O O O ,-2v (lrVXl"2v) O 20-v) 5YM O o l-2v (5.4) O 2(l-v) 1-2V where E. V and are Y,oung s f respectively. Terms of the given i c f u l l i n appendix A. 5_.2 ftPgl£tic_ The The stiffness the to of z. stiffness coefficients A must be i n t e g r a t e d first two The t h i r d as matrix area of volume are of the c a r r i e d out i n the x-y the element for any i n t e g r a t i o n i n the z - d i r e c t i o n completes volume i n t e g r a l by ranging from the base plane infinity. k given by Eg. 5.3 and over the integrations, p l a n e , cover the c r o s s - s e c t i o n a l value and P o i s s o n ' s r a t i o Integration,,,of_the_Stiffness_Coefficients shown i n Appendix element. local modulus (i.e., z=0) 29 As more and the more desirable. the f i r s t was laborious, proceeds thus making two (x,y) i n t e g r a t i o n s in integration a l l cases to evaluation are examined will, orientations. local integration reduce computing time. i n the z d i r e c t i o n proved in to be too i n a l l but a s i n g l e case. If c r o s s - s e c t i o n s triangles numerical with reasonable ease, and was made to numerical i n t e g r a t i o n plane the c a l c u l a t i o n s become However a n a l y t i c i n t e g r a t i o n can be c a r r i e d out f o r done analytic integration at general, be The t h i r d difficult for Therefore r e s o r t i n every case. of the various this element lying values of found to z, in the x-y the r e s u l t i n g have different For t h i s reason a r o t a t i o n and t r a n s l a t i o n of the coordinate system analytic integration. of z are shown below. is convenient Two c r o s s - s e c t i o n s in carrying for different out the values Consider x'-y* two rectangular c o o r d i n a t e systems x-y (see f i g u r e below) with axes r o t a t e d and r e s p e c t to each translated and with other. 01 a point relative to transformation P these i n space has c o o r d i n a t e s coordinate systems. between c o o r d i n a t e s are given The by (x,y) cr (x' y*) eguations # of 31 x '= (x-a) c o s c x - (y-b) sm^ y' = (x-a)sm^ +(y-b)cos^ (5-5) or x r x ' c o s ^ y - -x'smoc where the angle j' and + y'sin©< + + a y' coS°< + b ( c*- i s the c o u n t e r c l o c k w i s e the x'-y' c o o r d i n a t e system i s l o c a t e d a t to the x-y c o o r d i n a t e system. The x*-y» coordinate c r o s s - s e c t i o n a l area a therefore section. will be distance orientated system z from 1-3 shown ) relative be attached to a the base plane and f o r each new c r o s s The x-y c o o r d i n a t e system w i l l be attached to the base plane thereby f i x i n g i t s o r i e n t a t i o n . direction 6 Also the o r i g i n (o^b) will differently ' r o t a t i o n of the x* axes with r e s p e c t t o the x and y axes. of 5 of each particular, the x c o o r d i n a t e system w i l l be a l i g n e d along the edge with the o r i g i n a t the below. In 1 node. These axes are 32 O Zl= Using the above diagrams, the s i n e and angle ^ Sin where and y.-y oc, = f o r example X 3 i s the shifting of The X i s the 3 coordinate takes , ^ step = <x = x, f of of place, the to be = X 3 - K ! c o o r d i n a t e of the point the such point that 3 the . 3 Also t a translation jf • is to evaluate the limits of the upper l i m i t on the i n t e g r a t i o n with r e s p e c t to (see above diagram on the r i g h t ) i s x and COS Q - *, next integration. 1 X axes The 5 . 6 are found 3 X parameters are x 5 . 5 and of Eqs. cosine (* 3 ~* | y the lower l i m i t i s x 33 X The • = range of y integration with r e s p e c t to y 1 The i n d i v i d u a l terms of the s t i f f n e s s i s from zero to matrix K have the form r I UL O (5.7) LL where UL and L L r e f e r to the upper and lower l i m i t s of the x' variable. The term i n s i d e the brackets w i l l be i n t e g r a t e d On examination of the i n t e g r a n d of Eg. 5 . 7 f o r a l l of analytically. the terms of the s t i f f n e s s matrix, i t i s found that there are s i x d i s t i n c t types which manner. Type 1 : Type 2 : Type 3 : Type H : Type 5 : f( .y, ) x z ' can be expressed in the following 34 ^(x,y Type 6 : z) Since = ^ ( z ) Oj (z.^ i n each case y* we can remove t h i s term from within i s not a f u n c t i o n of x* or the bracket c f Eg. 5.7. Let I. - dx,' d y KCT) The e v a l u a t i o n of X, triangular section area cross has a l r e a d y reduces to f i n d i n g the area l y i n g i n the x-y plane, been derived i n Eg. 3 . 4 . I = (A+ B Z + C Z Thus X. i n t e g r a l of a however this becomes z (5.8) Let dx'd I y J Using Eg. 5.6 r e s u l t s i n I. = The as X, a x ' c o S <* + y ( Jx' oly' term may be i n t e g r a t e d by i n s p e c t i o n by r e c o g n i z i n g type Carrying Sinc< + X 1 integral since out the i n t e g r a t i o n with X, is respect it a f u n c t i o n of z only. to x* and y' and 35 s u b s t i t u t i n g the l i m i t s y i e l d s (x '+x ')co5o< 2 With COS the «< 3 substitution there of the + y ' s m < * ) + x, (A + Bz + Cz*) I expressions for 5"in << and results , + X.CA+BZ + CZ -) 7 (5.9) In an i d e n t i c a l manner the may be evaluated to give the f o l l o w i n g y I four remaining results. cJx'dy' ^ J dx'dy' I 4 - 1 integrals (x -x.) 3 x' ( x / + X,') + X , t (5. 10) 36 y* x, + 2x, z (A + BZ + C z ) z (5.11) \ y 1 l ^ y ' >i 12 x ; 1 - ( y , - y * ) ( 3 - * . ) [ y * Y » ' 2 x ; ) j + (x -x,) x + 2y, [ I , ] x + - y ' (A+BZ 3 + C 2 Z |_y 'j z ) (5.12) x y d x' ol y ' 37 + i + 2 ~J r ( V O -(y.-Ja) >i (x,'+ X ') 2 I. (vO^-y^Ty; - vy, + X. tl.] (A + B Z + C Z ^ ) 2 (5.13) The primed Eg.- 5.5 and so the i n t e g r a l s that the coordinates out. functions integrand Chapter 6. of z which be eliminated by using identity contain The i n t e g r a t i o n carried may only with unprimed respect The X values only are multiplied will coordinates. to 2 o f E q s . 5.8 be n u m e r i c a l l y by ^"^0 remains t o 5.13 t o to be which a r e f° r n i the i n t e g r a t e d as d e s c r i b e d i n 38 CHAPTER 6 SEMI-INFINITE NUMERICAL 6.1 INTEGRATION Goodwin-Moxan^Method The Goodwin-Moran w h i c h i s a form for the of the f a m i l i a r semi-infinite trapezoidal method rule of numerical trapezoidal integration. rule integration,* method, i s used I t has been n o t e d g i v e n by t h e e x p r e s s i o n fa)dt = h 2 ^ i M (6.1) J often gives surprisingly doubly infinite transformation the doubly in this is used. range rule range. that infinite thesis, accurate results transforms i n t e g r a l s range namely i s expressed E g . 6.1 for integrals To t a k e a d v a n t a g e i s made. \ (-(x)cJx T h i s t r a n s f o r m s the whereupon that the o f t h i s phenomenon, over other For the i n t e g r a l ranges a a to of i n t e r e s t # the t r a n s f o r m a t i o n * = £ integral c a n be u s e d . as f o l l o w s over to a doubly The r e s u l t i n g infinite quadrature 39 r J= N =i 4- The c o m p l e t e d e r i v a t i o n o f E q . 6.2 a p p e a r s i n A p p e n d i x S parameter area which regions The makes parameter do significant not P formulation spaced a serves as the i n t e r v a l progression contribution a node h nodes f o r an i n f i n i t e The while avoiding to the i n t e g r a l . positioner or interval i n the t r a p e z o i d a l r u l e . spacing which B. t h e summation on t h e contribute appreciably parameter analogous to geometric 6.2 i s a scale f a c t o r to center which spacing this (6.2) pJ ' J is mere between logical nodes In forms a than uniformly the numerical integral. Anajivtical_yersus_Numerical Integration n In integration, order a to test comparison the with accuracy the exact of integral c a n be made I/* for This some o f t h e e n t r i e s i n comparison does n o t c o n t a i n integrated is made o n l y local comparison with will stiffness f o r one i n t e g r a l a d e r i v a t i v e with analytically The the some respect matrix whose j\ integrand t o z and which c a n be ease. be made f o r e l e m e n t kj-, of k^ 40 which i s given i n Appendix A as E (i-v)(o< /3) + k (6.3) 0+vXl-2v) xn 6.2^1 A n a l y t i c a l . Integratign_.of The area c o o r d i n a t e At+B z+C z*+ l Pi From Chapter A + 4, ^ (DI l k tl i s given i n Chapter 3 by + E L Z ) X (F^Q.z)y + Bz + Cz' (3.6) i s g i v e n as (4-3) Differentiating E(l-v)(* + ;fl) (l+v)0"2v) _ Eg. 3.6 f as indicated in Eg. 6.3 yields ^ (D.+E.ZXFJ + S.Z) Jxdyolz J (\*zy (A+Bz-Cz X A and letting 41 (D, + E z ) ( F + G , z ) [ V e ( z ) ] o l z i (l.z) o 1 l a ( A + B z + Cz. )* 1 Since the i n t e g r a n d i s not a f u n c t i o n integrations function yield of z. the area of of x or y, the the triangular T h i s expression f o r the area obtained i n Chapter 3 as ZA = A+Bz+Cz. has first prism as a already been Therefore A.+ B z + C z ' Area(z) = Z The integration K " - ^ z -1 2 with r e s p e c t to z can now be c a r r i e d out (D,+E,zYF>G,z)d; \ ( l z ) V A + Bz + + Cz ) r «o Oz)* (A + Bz + Cz*) J two 42 Let (| +z) ~ "t 1 z J, Z t ( A * + B*t+C*t* ) 2 9 olt + t T + Jt T I 43 and edge d i r e c t i o n parameters (see E q s . 2.1 and 3.4) a = O b, = -I a» =' O a - | b = | b = 0 3 2 3 A= *'Xy;-y;) + *:(y;- ;) y o o + I B -^(b,-b )^-(b,-b ) 1 + 1 = 0 + 0 (i-o) 1 , + 1 [ i - ( - o ; ; - y : ) y x;(b,-b ) + + y,'(a,-a,) + y (q,-a ) + 1 < ( 1 y;(a,-a ) + o +i - o ] = 3 C = ,(b -k) a 3 0 = 2 + + flt(b,-b,) O + a.fk-b,) 2 +o U5 A* = A - B + C = O B * =' B- 2 C = -I C* = C - 2 D, = yl-y: F, = -i E, = b - x;- ; x = -i (i.'VO.- -i 3 = -i 9, - D,F, = ( - 0 ( - l ) = I <J = D,G, +E,F, = (HY-I) + (-O(-i) = z t = (-O(-i) - | Sa = E.Gt, g ; » 3,-293 = 2 - 2 ( i ) = o ^ = B * - 4 A * C * -(-i)(-0-(4)(o)(2)= ! l L g,' p 00 J, it T <- — = J z ^ 1 > 46 = 1 2 2(2)+(-l)+ | I 2 ( 2 ) + ( - | ) - = 1 | - \ Lz - ? L r r - y Using with progression output, gave a value of is only 2 (.34457) the Goodwin-Moran numerical i n t e g r a t i o n scheme N = number of nodes = 15,; S geometric 2 factor = scale factor = 2 = '/^ and using double K^, = (. 34£>5 5 ^ ^ • This and P - precision guadrature approximate of course but the accuracy a t t a i n e d i n the p r e v i o u s example shows t h a t the numerical r e s u l t i s an e x c e l l e n t replacement f o r the exact r e s u l t . 47 CHAPTER 7 NUMERICAL EXAMPLES Two examples were chosen a g a i n s t which the r e s u l t s of an a n a l y s i s using the i n f i n i t e f i n i t e elements could be compared t o exact analytical solutions. The two examples b a s i c a l l y d e r i v e from the s o l u t i o n s f o r a p o i n t load on an e l a s t i c h a l f s p a c e , perpendicular halfspace. in one The a c t u a l examples considered r e g i o n removed from the top the h a l f s p a c e surface, loads a p p l i e d to the s u r f a c e o f t h i s h e m i s p h e r i c a l stresses that e n t i r e h a l f space. examples are displacements applied and i n the other t a n g e n t i a l , to the s u r f a c e of the hemispherical the case and with a with bowl egual to would a r i s e from a concentrated lead on the The s t r e s s e s and displacements i n the a c t u a l therefore identical to the stresses and i n the h a l f s p a c e problem i n the region o u t s i d e of the h e m i s p h e r i c a l cutout. U8 Point L o a d on a rlal£space. Finife There were s e v e r a l reasons First f o r choosing was the d e s i r e to t e s t the element i n an similar to a situation finite elements regular three dimensional least primarily a cutout. reasonable applied in such examples. roughly to be u s e f u l , Second was the d e s i r e to t e s t infinite hemispherical Example example where i t was c o n s i d e r e d thus the h a l f space problem. at Element only and not finite in combination elements, thus the with the T h i r d the d e s i r e to t e s t the element, or combination the i n f i n i t e a p p l i e d p e r p e n d i c u l a r t o the i n f i n i t e cf (2) elements, for loads d i r e c t i o n and f o r loads direction, thus the examples with loads normal and t a n g e n t i a l to the s u r f a c e . two 1»9 11... -, fi ? § ,B P1 §,., 3 _ Z „,. ? § J tz I *? I i B .1 f .§,_ ,j? ,9 1 I ,£L £ °. § £? 3 _ I? Y. 3. Surface P o i n t e Load Perpendiculat m The equations f o r the d e f l e c t i o n s of p o i n t s i n a semiinfinite solid loaded s u r f a c e can be found with a p o i n t load p e r p e n d i c u l a r to the i n [ 1 ] as P U V 1 IT + E _p lit P ZTT i n which r •f r (7.1) 50 x,u X \R Z ZW F i g . 7.1 S e m i - I n f i n i t e S o l i d Loaded by a Surface Perpendicular to the Surface The s t r e s s e s due to a p e r p e n d i c u l a r s u r f a c e of a s e m i - i n f i n i t e s o l i d are given 0~ = — ** 2TT CT XY cr xz cr YY — R2 P =7 — R lir _ 3P 2TT P 2ir I — z R ^ 3 3XYZ XZ y \R (l-2v) p o i n t load on the i n [ 1 ] as R +Z R(R+Z)' (2R+Z)XY R (R + Z ) 2 Z (7.2) R J_ 3 Z Y R* P o i n t Load R 3 -(l-Zv)/Y^ZR+ZU V \ R R+Z R(R+Z)7 51 - 3P XZ 2.TT R cj- 5 YZ _ _3P YZ_* D 5 o ZZ Z7T R These s t r e s s e s are used i n the g e n e r a t i o n of j? vector to axial symmetry, any r e p r e s e n t the s t i f f n e s s of the h a l f s p a c e . hemispherical elements stress section However the is considered. The quarter and of bowl forms the base plane f o r the i n f i n i t e X the load are V The Y the axis. allowed W only. Appendix D describes the load vector used approximate the s t r e s s e s d i s t r i b u t e d over the base plane. s t r e s s e s , given by Eqs. 7.2, point. finite a x i s and are assigned to be zero along the T h e r e f o r e p o i n t s d i r e c t l y beneath the fifty-four t h i r t y - s e v e n nodes as shown i n F i g . 7.2. displacements displacements boundary surface are set equal to zero along the may of the semi- T h i s s u r f a c e i s subdivided i n t o a mesh of displacements U radial are most e a s i l y i n s e r t e d i f one-quarter infinite solid elements. nodal (see Appendix D) . Due conditions the However to vary simplify ncn-linearly from to These point to the c a l c u l a t i o n s to determine the 52 load vector, the each node and then Fig. finite Fig. with The infinite on the load, unit load of 360-67 was numerical 96 example a r e as solution 1 The in a a stiffness central seconds for given the shown by to vary problem matrix the entire stiffness integration with with in Eqs..7,2. 4°/o by the are f o r a N| S = = problem band time on including width a IBM the matrix. parameters I 5 I.O progression factor ninety-seven a half (CPU) follows geometric mesh A l l tests processing unit of the numerical factor from D i r e c t l y beneath i s achieved. results and solution. nodes. obtained the at ). number o f nodes scale using calculated between results elasticity 2. / o within integration The the the exact (i.e., P = I twenty-five. linearly is element r e s u l t s are found from o f freedom vector analysis exact finite Example degrees to vary element average stress compares the agreement of the assumed 7.3 infinite 7.2 value Y" = 0.5 used in this 53 Fig. 7.2 Finite E l e m e n t Mesh o f H e m i s p h e r i c a l Quarter Symmetry Bowl w i t h One FIG. 7.3 DEFLECTIONS OF PERPENDICULAR HEMISPHERICAL TO SURFACE (X-Z BOUNDARY PLANE) LOADED 55 7.2 Example 2 - _ Seffli-Ipf i n i t e S o l i d Load P a r a l l e l The equations Loaded by., a_Surf acg_Point to the Surface f o r the d e f l e c t i o n s of p o i n t s i n a semi- i n f i n i t e s o l i d loaded by a point load p a r a l l e l to the s u r f a c e i n the d i r e c t i o n as shown i n F i g . 7 . 4 X u = ZQTT (uv) E Y = i R R Q (W) XY_ ZTT E R 3 w _Q ZTT (l+v) X E R* Z (R+Z) z I- are [ 1 ] X R(R+Z) (l-2v) R (R + Z ) , (7.3) L (l-2v)R R * x.u Fig. 7 . 4 Semi-Infinite Solid Loaded by a Surface Point Load P a r a l l e l to the Surface 56 The paralled cr = s t r e s s e s due t o a p o i n t t o the s u r f a c e o f semi-,inf i n i t e o — 27T R Z7T R _ R R 3 Q X Z — T Again Example 2 X (3R 1 Z R 2 Z +Z) ( R + Z ) + (l-2v)R (R + Z ) : I - Y (3R+zV 2 R (R+Z) 2 : 5 R symmetry solid need to zero argument along that reguires only be c o n s i d e r e d . 1 , F i g . 7 . 2 , may equal symmetry -(l-2v) R' 3 2TI infinite R (R+Z) (7.4) X Y Z a X ^ R + Z ) ' 5 3Q z direction X*Z R (T, = zz ( R +Z ) 2 Q X = — — 11 YZ set R 3 (l-Zv)R + X are [1] 2 3 2TT YY solid 2 3Q cr i n the 3 X _(l. 2 v Wl-2v)RY3R (R+Z) \ X 3X* XY load be u s e d . the the V X one g u a r t e r Therefore The axis. and w \/ of the t h e same semi- mesh a s displacements are I t c a n be shown f r o m displacements along a the 57 I a x i s are a l s o zero. In F i g s . 7.5 and 7.6 obtained from with The the a test comparing exact e l a s t i c i t y 3 A s o l u t i o n given i n Egs. 7.3 i s shown. from the exact Example 2 results CPD time on problem i n c l u d i n g a IBM to vary i n the solution. in a problem with 91 degrees of freedom and a s t i f f n e s s matrix with a h a l f The results the i n f i n i t e f i n i t e element mesh shown i n F i g . 7^2 i n f i n i t e f i n i t e element r e s u l t s are found order of the band width of 24. 360-67 was 99 seconds f o r the e n t i r e the numerical integration of the stiffness matrix. The numerical those used i n Example 1. integration parameters are the same as 58 FIG. 7.5 DEFLECTIONS LOADED OF PARALLEL HEMISPHERICAL TO SURFACE BOUNDARY (X-Z PLANE) FIG. 7.6 DEFLECTIONS OF HEMISPHERICAL BOUNDARY LOADED PARALLEL TO SURFACE (X-Y PLANE) 60 CHAPTER 8 DISCUSSION Instead boundary, using as finite infinite of d e a l i n g has finite some represent great a flexible elements imposed rigid boundary assumed semi-infinite solid t o be zero or i n f i n i t e . be used to e v a l u a t e the s t r e s s e s and solution onl:y or displacements be reduced with since they by the stresses of a to reach halfspace has Close to the elements would problem using r e g u l a r the a r t i f i c i a l boundary can be assumed n e g l i g i b l e . This the i n t r o d u c t i o n of the i n f i n i t e represent which displacements. elements r e q u i r e s numerous elements, many necessary problems The purpose of f i n i t e elements i s not to e v a l u a t e s u r f a c e and near p o i n t s of load, r e g u l a r f i n i t e finite free d i s t a n c e from the s u r f a c e , but to r e a l i s t i c a l l y been The or formed has been i n t r o d u c e d . the s t i f f n e s s of the previously an commonplace i n s o i l mechanics elements, using such i n f i n i t e at been with of which are where s t r e s s e s boundary finite the s t i f f n e s s of the surrounding can elements region. 61 Thus, with the s h r i n k i n g of the boundary, a l a r g e the number of degrees In displacement reduction in of freedom i s p o s s i b l e . l i g h t of the accuracy a t t a i n e d using a very f u n c t i o n , a higher order element i s f o r h a l f s p a c e problems. not simple warranted 62 BIBLIOGRAPHY 1. S c o t t , R. F., F u n d a m e n t a l s o f S o i l W e s l e y , R e a d i n g , Mass., 1963, Mechanics, Addisonpp. 498-500. 2. Squire, William, Integration f o r Engineers A m e r i c a n E l s e v i e r , Hew Y o r k , 1970. 3. Squire, William, "Numerical E v a l u a t i o n of I n t e g r a l s using Moran T r a n s f o r m a t i o n , " West V i r g i n i a U n i v e r s i t y , D e p a r t m e n t o f A e r o s p a c e E n g i n e e r i n g , TR-14, 1969. 4. Z i e n k i e w i c z , 0. C., The F i n i t e E l e m e n t Method i n E n g i n e e r i n g S c i e n c e , M c G r a w - H i l l , London, 1971. and Scientists, 63 APPENDIX A ELEMENT STIFFNESS MATRIX FOB LOCAL COORDINATE SYSTEM The element s t i f f n e s s matrix f\ may c o n v e n i e n t l y be p a r t i t i o n e d as f o l l o w s : SYM : <A.1) : The submatrices are a l l (3x3) matrix i s a (9x9) symmetric matrix. i s shown i n f u l l i n Eq. A.2 i n which Zv Z(\-v) arrays and The p a r t i t i o n e d the matrix 64 0~ p E(i-v) s r where (l+v)(i-2v) E and 'V are Young*s modulus and Poisson's r a t i o . Each term of the submatrix volume of the element. k j L must be integrated over the —> < X +>* cn . _» II • —> 1 + > • —> K "T5 r X N M • *J 4 1^ • ** _i_ Q—> % >^ a - . + —» o + X X • .> LU II + 1 -r Kl X • —> X • -» + 66 APPENDIX GOODWIN—MOBAN The -<?o to + oo METHOD OF S E M I - I N F I N I T E application of Goodwin-Moran t o t h e range method. The j = +N f(x)dx = h X f(j^) (B.1) J = "N - oo vanish the rule rule + oo gives INTEGRATION of the trapezoidal i sthe basis simple trapezoid.al B s u r p r i s i n g l y good quickly a s x —> 0 results f o r integrands 0 whose derivatives . Since + 00 r +00 (B.2) - 00 Eq. B.1 c a n b e r e w r i t t e n - 00 as 67 + 00 J J= - N to center the summation and a v o i d adding n e g l i g i b l e terms f o r e i t h e r l a r g e p o s i t i v e or negative arguments. When i n t e g r a l s over ranges other than -oo to +oo are encountered, a t r a n s f o r m a t i o n to the doubly i n f i n i t e range can be made t o make use of the t r a p e z o i d a l r u l e ' s accuracy. i n t e g r a l to be transformed i s taken as I = food then a t r a n s f o r m a t i o n o f I - •f ( e 1 X = 6 ^ + Ol + a) e 1 will olt - oo S h i f t i n g the v a r i a b l e as i n Eg. B.2 g i v e s give I f the 68 f ( e ^ a)e I = \ t t S + <Jt oo and then approximating the integral in the same manner as Eg. B.1 g i v e s ja+N I K = e > j h * a) J J Now r s e ^ let substitutions - Then Also l e t and f(x)Jx - S | L r | ^ rif(a^SrJ) . j--N . approximation becomes The a b s o l u t e value of the can be l e s s than -J These + The r . c J=* o P —*• = produce the r e s u l t poo A O term r e c i p r o c a l of are T I.O and exact as [S]-> oo term i s taken so the r e s u l t i s unchanged as the and that +J simply interchanged from the case where the i s used. The quadrature r u l e then becomes 69 +N (B.3) 70 APPENDIX L I S T I N G OF FORTRAN SUBROUTINE OF THE GOODWIN-KORAN The f o l l o w i n g t which a the i sa listing u s e s t h e Goodwin-Moran method semi-infinite C range. of a Fortran to evaluate METHOD subprogram an i n t e g r a l over E g . B.3 o f A p p e n d i x E i s t h e b a s i s o f subprogram. The input to listing while returned to the c a l l i n g the subprogram i s explained i n the t h e o u t p u t , QGEOM, t h e v a l u e o f t h e i n t e g r a l , i s program. 71 FUNCTION QGEOM (GRAND) C C C C C C C C C C C C C C C 30 EVALUATION OF INTEGRAL OVER SEMI-INFINITE RANGE BY GOODWTNMORAN METHOD USING POINTS IN A GEOMETRIC PROGRESSION A=LOWER LIMIT OF INTEGRAL,IT CAN BE POSITIVE,NEGATIVE OR ZERO GHAND=EXTERNAL FUNCTION SPECIFYING INTEGRAND S=POSITIVE NUMBER SERVING AS SCALE FACTOR RAT=POSITIVE NUMBER (NOT 1.) USED AS FACTOR IN GEOMETRIC PROGRESSION N=NUMBER OF NODES (ACTUALLY 2*N+1 NODES ARE USED) THE FUNCTION GRAND IS EVALUATED AT POINTS BETWEEN A+S*RAT**N AND A+S/RAT**N. SELECT VALUES TO COVER SIGNIFICANT RANGE. DEFINE GRAND TO GUARD AGAINST UNDERFLOWS. COMMON/INTPAR/A,S,RAT,N RP=S RM=S QGEOM=S*GRAND (A+S) DO 30 J=1,N RP=RP*RAT RM=RM/RAT QGEOM=QGEOM+RP*GRAND(A + RP)+RM*GRAND (A*RM) QGEOM= QGEOM*ABS (ALOG (RAT)') RETURN END 72 APPENDIX D GEOHAL1ZED_LOAD_VECTOB In the finite forces acting at taken to produced the be by the the acting of this stresses to nodal The point and of elasticity for base p l a n e global coordinates virtual coordinates, must is. be are equal to of on that the these would surface surface. are It i s given be of the boundary p l a n e s t r e s s e s C|_(^ ) a c t i n g on the the the The Bcussinesq area (i.e.,z=0). are done by of (3 is R the node forces to . statically C) P work done by the solution coordinate Let displacements p r o d u c e d , the to that convert boundary forces load nodal B from that node a point the the loads. [1], only to this hemispherical out obtained theory When on C h a p t e r 7, hemispherical equivalent appendix in base p l a n e are the s t r e s s e s , from purpose in nodes of statically halfspace, the element examples of surface be B equivalent t also these stresses defined in global node forces • That 73 dA (D. 1) where ^.(^&) system and tl— = A distributed stresses point to point vary linearly. from each end along the nodal interpolate linear relate points the but With the two the will linearly C|_ Eg. 4.2. non-linearly from be restricted stresses 0^ node v a l u e s o f the nodal are s p e c i f i e d i t w i l l be function displacements element. This stresses has to been the used 0|_ , one at . If 0|_ necessary CJ_ . i n Chapter displacements same f u n c t i o n c a n be to the s t r e s s e s a p p l i e d to varying between n o d e s i n o r d e r t o d e f i n e interpolation global by the b o u n d a r y , a r e n e c e s s a r y t o s p e c i f y used on of to A 4 to a l l to r e l a t e the base Then, •4 in they in defined ^(^g)vary distributed node t o node, o n l y nodal a p p l i e d plane. f o r convenience values of i n an displacements i s an i n t e r p o l a t i n g f u n c t i o n The linearly continuous = which A ^ i s a linear U A o interpolation function and therefore 74 is identical values to Eg. 4.2. The vector i s 9*1 o f s t r e s s e s C|_(^ )and B _P . contains Rewriting the Eg. nodal D. 1 g i v e s That i s . (D.3) Uo confusion are interpolating while the M should arise over the functions defined pertaining different for to the A' S . base i s the area The A'S plane only of the base plane. The contain yield integration c r o s s product the can terms. be carried I t can be out explicitly shown t h a t t h e following results ' 1.3 and will integrals 75 Finally, are given upon integration forces by 2 IX + r r IX + 2.X +r R IY IY 2 R IZ A + r IZ + r + 2Z >"3I r + IY + 3Y r 2Y r iz R2Z 3 X r + Zr + R 2X < o f E q . D.3, t h e node 2. r 3X P (0.») 3V 2z R 3X R 3Y R 3Z where the X the terms # Y and nodal force and The components given R^ x Z , R^y and components a t node the d i s t r i b u t e d s t r e s s vector ^ CTT^. a c t i n g roust at a point. as T; p and be •ix ' PiV of the i a n a PiZ are generalized respectively. found from the s i x s t r e s s In t e n s o r n o t a t i o n this is 76 where the hj j lh terms in evaluated C~ i s a component o f a u n i t n o r m a l t o t h e b a s e p l a n e i n direction. the _|0 , when e v a l u a t e d matrix a t node 2. i . e. P zt = TY a t the nodes, = become « C T n, + C r f i + CC ( x The s i x u n i q u e t e r m s o f t h e s t r e s s a r e g i v e n f o r each problem i n S e c t i o n s 7.1 and by summing the 7.2. g e n e r a l i z e d f o r c e s i n any d i r e c t i o n and c o m p a r i n g i t t o t h e p r o d u c t same direction equlibrium, these and the area over two r e s u l t s s h o u l d be e g u a l . be particular of the s t r e s s which n matrix A v a l u a b l e c h e c k cn t h e g e n e r a l i z e d l e a d v e c t o r may obtained 3 i t acts. in the From 3
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Infinite finite element Ungless, Ronald Frederick 1973
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Title | Infinite finite element |
Creator |
Ungless, Ronald Frederick |
Publisher | University of British Columbia |
Date Issued | 1973 |
Description | This thesis is concerned with developing a finite element model of infinite size to facilitate the stress analysis of three dimensional, semi-infinite bodies governed by time independent linear equations. An element of triangular plan form extending to infinity in one direction is devised. There are three nodes per element, each node having three displacement degrees of freedom. The infinite element is used in conjunction with regular finite elements to represent the stiffness of the semi-infinite solid which has previously been assumed to be zero or infinite in regular finite element models. The infinite element relieves the computational problem caused by large numbers of elements which has limited the use of the finite element method in three dimensional halfspace problems. A large reduction in the number of degrees of freedom is possible with the use of the infinite element because the artificial boundary of previous models is eliminated. The accuracy of the element is tested on two examples whose exact solutions are known. Both involve a semi-infinite solid, one loaded with a surface perpendicular point load, and the other with a surface parallel point load. The results compare favourably with the theory of elasticity solutions. |
Subject |
Structural analysis (Engineering) |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-03-18 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0050530 |
URI | http://hdl.handle.net/2429/32603 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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