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UBC Theses and Dissertations

Infinite finite element Ungless, Ronald Frederick 1973

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AN INFINITE FINITE ELEMENT by RONALD FREDERICK DNGLESS B.A.Sc. (1971) The U n i v e r s i t y of B r i t i s h Columbia A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of CIVIL ENGINEERING We accept t h i s t h e s i s as conforming to the r e q u i r e d standard The U n i v e r s i t y of B r i t i s h Columbia September 1973 In presenting t h i s thesis in p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. R. P. Ungless Department of C i v i l Engineering The University of B r i t i s h Columbia Vancouver 8, Canada September, 1973. AS I N F I N I T E F I N I T E ELEMENT ABSTRACT T h i s t h e s i s i s c o n c e r n e d w i t h d e v e l o p i n g a f i n i t e e l e m e n t model o f i n f i n i t e s i z e t o f a c i l i t a t e t h e s t r e s s a n a l y s i s o f t h r e e d i m e n s i o n a l , s e m i - i n f i n i t e b o d i e s g o v e r n e d by t i m e i n d e p e n d e n t l i n e a r e q u a t i o n s . An e l e m e n t o f t r i a n g u l a r p l a n f o r m e x t e n d i n g t o i n f i n i t y i n one d i r e c t i o n i s d e v i s e d . T h e r e a r e t h r e e nodes p e r e l e m e n t , e a c h node h a v i n g t h r e e d i s p l a c e m e n t d e g r e e s o f f r e e d o m . The i n f i n i t e e l e m e n t i s u s e d i n c o n j u n c t i o n w i t h r e g u l a r f i n i t e e l e m e n t s t o r e p r e s e n t t h e s t i f f n e s s o f t h e s e m i - i n f i n i t e s o l i d w h i c h has p r e v i o u s l y been assumed t o be z e r o o r i n f i n i t e i n r e g u l a r f i n i t e e l e m e n t m o d e l s . The i n f i n i t e e l e m e n t r e l i e v e s t h e c o m p u t a t i o n a l p r o b l e m c a u s e d by l a r g e n umbers o f e l e m e n t s w h i c h h a s l i m i t e d t h e use o f t h e f i n i t e e l e m e n t method i n t h r e e d i m e n s i o n a l h a l f s p a c e p r o b l e m s . A l a r g e r e d u c t i o n i n t h e number o f d e g r e e s o f f r e e d o m i s p o s s i b l e w i t h t h e u s e o f t h e i n f i n i t e e l e m e n t b e c a u s e t h e a r t i f i c i a l b o u n d a r y o f p r e v i o u s m o d e l s i s e l i m i n a t e d . The a c c u r a c y o f t h e e l e m e n t i s t e s t e d on two e x a m p l e s whose e x a c t s o l u t i o n s a r e known. E o t h i n v o l v e a s e m i - i n f i n i t e s o l i d , one l o a d e d w i t h a s u r f a c e i i p e r p e n d i c u l a r p o i n t l o a d , and the other with a s u r f a c e p a r a l l e l p o i n t l o a d . The r e s u l t s compare f a v o u r a b l y with the theory of e l a s t i c i t y s o l u t i o n s . i i i TABLE OF CONTENTS P a g e ABSTRACT i i TABLE OF CONTENTS i v LIST OF FIGURES v l ACKNOWLEDGEMENTS v i i CHAPTER 1. INTRODUCTION 1 1.1 Background 1 1.2 Purpose and Scope 2 2. ELEMENT GEOMETRY 4 2.1 Topology. 4 2.2 Transformation of Axes 9 2.3 Transformation of Global Side Vectors to Local Side Vectors 13 3. AREA COORDINATES » 15 4. ELEMENT DISPLACEMENT FUNCTION 20 5. ELEMENT STIFFNESS MATRIX 26 5.1 Formulation of Element S t i f f n e s s Matrix 26 5.2 Anal y t i c Integration of the S t i f f n e s s C o e f f i c i e n t s 28 i v CHAPTER Page 6 . SEMI-INFINITE NUMERICAL INTEGRATION 38 6.1 Goodwin-Moran Method 38 6 . 2 A n a l y t i c a l v e r s u s N u m e r i c a l I n t e g r a t i o n 39 7 . NUMERICAL EXAMPLES 47 7 .1 Example 1 - S e m i - I n f i n i t e S o l i d Loaded by a S u r f a c e P o i n t Load P e r p e n d i c u l a r to t h e S u r f a c e 49 7 . 2 Example 2 - S e m i - I n f i n i t e S o l i d Loaded by a S u r f a c e P o i n t Load P a r a l l e l t o t h e S u r f a c e 5 5 8. DISCUSSION 60 BIBLIOGRAPHY 62 APPENDIX A. ELEMENT STIFFNESS MATRIX FOR LOCAL COORDINATE SYSTEM 63 B. GOODWIN-MORAN METHOD OF SEMI-INFINITE INTEGRATION 66 C. LISTING OF FORTRAN SUBROUTINE OF THE GOODWIN-MORAN METHOD 70 D. GENERALIZED LOAD VECTOR 72 v LIST OF FIGURES F i g u r e Page 2.1 A Three Dimensional I i i f i n i t e F i n i t e Element. . . . 5 2.2 P l a n View o f the Base Plane 6 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 F i n i t e Element 10 3.1 D e f i n i t i o n o f Area Coordinates 16 4.1 Adjacent L o c a l Coordinate Systems 24 7.1 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 P e r p e n d i c u l a r to the S u r f a c e 50 7.2 F i n i t e Element Mesh o f H e m i s p h e r i c a l Bowl w i t h One Quarter Symmetry 53 7.3 Displacement of H e m i s p h e r i c a l Boundary under a V e r t i c a l U n i t Load (X-Z Plane) 54 7.4 S e m i - I n f i n i t e S o l i d Loaded by a S u r f a c e P o i n t Load P a r a l l e l to the S u r f a c e 55 7.5 Displacement o f H e m i s p h e r i c a l Boundary Loaded P a r a l l e l to S u r f a c e (X-Z Plane) 58 7.6 Displacement of H e m i s p h e r i c a l Boundary Loaded P a r a l l e l to S u r f a c e (X-Y Plane) 5& v i ACKNOWLEDGEMENTS I wish to express my a p p r e c i a t i o n f o r the guidance and adv i c e given by my a d v i s o r . Dr. D. L. Anderson, throughout the p r e p a r a t i o n of t h i s t h e s i s . Thanks a l s o are due to Ian M i l l e r and B r i a n Badomske f o r many h e l p f u l d i s c u s s i o n s . F i n a l l y , I would l i k e to thank the R a t i o n a l Research C o u n c i l of Canada f o r t h e i r f i n a n c i a l support. September, 1973 Vancouver, B r i t i s h Columbia Vl l 1 CHAPTER 1 IIIHODUCTION }.± Background Much use of the f i n i t e element method has been made in s o i l and rock mechanics problems dealing with i n f i n i t e halfspaces. I t has been normal practice to replace the i n f i n i t e halfspace under study by a f i n i t e body with an a r t i f i c i a l boundary at some ar b i t r a r y distance from a point of interest (e.g., point of loading, location of surface structure). Since a f i n i t e number of elements i s a necessity, the reason for this boundary i s quite apparent. Because of computer storage l i m i t a t i o n s , three dimensional f i n i t e element analyses cannot afford many elements and therefore, approximating with a fixed or free boundary near the point of i n t e r e s t may be i n gross error. The -number of normal f i n i t e elements required to achieve a certain accuracy may be quite large. This w i l l r e s u l t in very large numbers of simultaneous equations, which may place a severe l i m i t a t i o n on ) 2 the usefulness of the method for p r a c t i c a l problems. Also the band width of the r e s u l t i n g equation system becomes large leading to large computer storage requirements. A f i n i t e element of i n f i n i t e s i z e , with which this thesis concerns i t s e l f , could minimize some of these problems. 1.2 Purpose and Scope The purpose of th i s thesis i s to construct a model with which the stress analysis of three dimensional, semi-i n f i n i t e bodies may be f a c i l a t e d . A f i n i t e element model i n which the element i s of i n f i n i t e size i s proposed to replace the present a r t i f i c i a l boundary conditions. A reduction of the degrees of freedom and band width of an i n f i n i t e halfspace problem may be possible with the use of the i n f i n i t e f i n i t e element. The standard method requires c a r e f u l placement of an a r t i f i c i a l boundary where stresses or displacements can be assumed negli g i b l e while the i n f i n i t e element method makes no such requirement. As a r e s u l t , fewer elements are needed leading to fewer degrees of freedom and a lesser band width than the standard method. Attention w i l l be confined to the consideration of l i n e a r e l a s t o s t a t i c problems characterized by time independent l i n e a r equations. This thesis considers an element of 3 triangular plan form extending to i n f i n i t y i n one d i r e c t i o n . Each node of t h i s t r i a n g l e has three displacement degrees of freedom and these nodes form the usual boundary. application of the element developed i s i l l u s t r a t e d by a f i n i t e element solution of the Boussinesg problem using the i n f i n i t e elements. The r e s u l t s are compared to exact a n a l y t i c a l solutions of the problem. A computer program was developed for the f i n i t e element analysis employing the i n f i n i t e element. 4 CHAPTER 2 ELEMENT GEOMETRY 2 . J To£olocjy_ The element model, shown i n F i g . 2 . 1 , i s referred to as an " i n f i n i t e f i n i t e element." This paradoxical term has been coined because the element extends to i n f i n i t y i n the d i r e c t i o n normal to the f i n i t e plane tr i a n g l e formed by the nodes. Fig. 2 . 1 shows the geometry of the element and the two right hand cartesian coordinate systems - global( X,Y,Z ) and l o c a l (x,y,z). The o r i g i n of the l o c a l coordinate system i s shown as being at node I and the x axis p a r a l l e l to the I - 3 side. This choice of axis permits integration over the area of the t r i a n g l e without dividing the in t e g r a l into two parts. The element has three nodes located at the corners. These nodes a l l l i e in the l o c a l z=0 plane. This plane w i l l be referred to as the "base plane". The l o c a l z axis i s directed into the element. 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 generalized displacements (u, v, w) are assigned to each corner node of the element shown in F i g . 2.2, r e s u l t i n g i n nine degrees of freedom per element. The in-plane 7 displacements u and v act in the x and y di r e c t i o n respectively and the out of plane displacement w acts in the z d i r e c t i o n . The element edges or "rays" st a r t from the nodes and t r a v e l off to i n f i n i t y in the z d i r e c t i o n . The edges can be represented by vectors V_ ( L = , , 3 ) with origins at the nodes. The vectors are shown on F i g . 2 . 1 and can be expressed as since the vector i s straight. In the above eguation the Qk 1 b % are constants and the I ,J ,TT vectors are unit vectors in the l o c a l (x,y,z) coordinate system. The superscript denotes a guantity expressed in the l o c a l coordinate system. If (*-_;y- ) are the nodal coordinates of the element in the z=0 flane and i f the directions of the edges of the element are V. = Q-L + b~j +• k then on any plane given by z=constant the nodal coordinates are y . = y ° + b x J «• J 1 1 ( 2 . 2 ) I n i t i a l l y the "rays" of each element w i l l be specified i n terms of global coordinates by the vectors 8 V * = A t I - B,J + Q K Ci-1,3) (2.3) where are unit vectors i n the global ( X,Y,Z ) system. The superscript <y denotes a quantity expressed in the global coordinate system. In Section 2.3 the transformation of the global side vectors ( V ) to l o c a l side vectors ( V ) i s carried out. The constants A[,rB;,C- may form part of the element input data or may be automatically generated. One method of doing this i s by adding the unit vectors normal to the base planes of a l l elements common to a node. The resultant vector becomes the "ray" originating at that node. The l a t t e r approach has been adopted in a computer program written for i n f i n i t e f i n i t e element problems. Some r e s t r i c t i o n s on the direction of the rays must be observed i f the method i s to produce r e a l i s t i c r e s u l t s . An element which has a l l three rays perpendicular to the base plane ( i . e . , CX^ t b. - O ) w i l l produce erroneous re s u l t s due to the chosen displacement function (see Chapter U). The s t i f f n e s s of the exact solution goes to zero i n t h i s case yielding i n f i n i t e displacements whereas the f i n i t e element s t i f f n e s s i s f i n i t e . At the other extreme, an element in which any one of i t s rays l i e s i n the same plane as the base plane, or almost so (i . e . , , )o •-*•<») w i l l experience numerical d i f f i c u l t i e s and 9 w i l l give poor r e s u l t s . A t h i r d r e s t r i c t i o n requires that the cross sectional area increase with increasing z. I f this i s not s a t i s f i e d , a zero cross se c t i o n a l area at some value of i w i l l l i k e l y occur producing numerical d i f f i c u l t i e s . Also rays must not be allowed to cross as t h i s also leads to a zero cross sectional area. 2.. 2 Transformation of Axes To transform the s t i f f n e s s matrix of an element expressed i n the l o c a l element coordinate system to a global system, and then assemble into the structure s t i f f n e s s matrix, requires a transformation matrix between the l o c a l coordinate system (x,y,z) and the global coordinate system ( X,Y,Z ). The base plane and the two coordinate systems of an i n f i n i t e f i n i t e element are shown in Fig. 2.3. There are several choices for directions of the l o c a l coordinates. For t h i s thesis the l o c a l coordinate system s h a l l be chosen as having the x axis directed along the I k side of the t r i a n g l e as shown i n Fig. 2.3. 1 0 Fig. 2.3 Local and Global Coordinates for an I n f i n i t e F i n i t e Element — fr The vector defines the ik side of the triangle which i n global coordinates i s 11 r v T.-Y, (2.4) The d i r e c t i o n cosines are found by dividing the components of t h i s vector by i t s length giving a vector of unit length. r \ 7\ x X J x... Y k(. 7 . (2.5) with 2 + Y.1 + I* k<. The l o c a l z d i r e c t i o n w i l l be normal to the "base plane". The vector product of two sides of the triangle defines t h i s normal d i r e c t i o n , thus 12 r V = V. x V. - •< Y.Z.. X..Z J L k l X« Y.. Y7, JL K t j i Kc ( 2 . 6 ) the magnitude of Vz- i s equal to twice the area of the t r i a n g l e . L-The d i r e c t i o n cosines of the z axis are obtained simply by dividing the components of V z by i t s length 2.A . This gives the unit vector f r k = * Y > - 2A Y Z - Y- Z . X k . Y -• - X . . Y K I J L J l k i (2.7) The y axis i s perpendicular to the x and z axes, and so i s found by the cross product of vectors i n the x and z d i r e c t i o n s . If vectors of unit length are taken in each of these directions as defined by Eqs. 2.5 and 2.7, we have 13 r J r A k X L - (2.8) A transformation matrix can be written containing the di r e c t i o n cosines in the global directions of the L iJ , K vectors of the l o c a l system. This matrix w i l l be denoted by T and i s given by T = \ x \ Y \ Z V V V \* \t \ z : J : (2.9) The element s t i f f n e s s matrix in global coordinates i s derived from the l o c a l element s t i f f n e s s matrix by the following r e l a t i o n [4] 9 - r T i * (2-10) k'= T k T 2.3__Transformatign_gf_Glcbal To complete the positioning of an element in space, the element edges or rays must be described. Their d i r e c t i o n 14 can be described by a vector whose form i s given in Eg. 2.1 in terms of l o c a l coordinates. I n i t i a l l y the rays w i l l be described by global coordinates thereby necessitating a transformation to l o c a l coordinates. If the ray vectors i n the l o c a l and global systems are written as local-: = OL T 4 lo[ J + C . ' ^ (.= 1,3 global: V.? = A; I + B t J + C . K L = l ' 3 then the unknown c o e f f i c i e n t s CL^  , i b i r C _ must be determined from the known c o e f f i c i e n t s A : . B i . C : . The scalar product of the global ray vector with a unit, l o c a l vector produces the component of the ray vector in the unit l o c a l vector d i r e c t i o n . Thus a- - V*- t - A ; \ „ + B ; Ji, Y + C L A > z c;.V . T - IT - < \ . \ x * B ; A , Y * C ; A l Z then following the form of Eg. 2.1; the constants are redefined as b ^ = b i . 7 C l ' ,2.11, c . = c / / , < , I 15 CHAPTER 3 MM_COORDINATES In the development of a f i n i t e element one of the basic necessities i s to establish a r e l a t i o n between the continuous displacements u,'v,w and the nodal displacements. To aid in thi s matter, i t i s convenient to introduce a special set of normalized coordinates for a triangle ca l l e d area or natural coordinates. Let A, . A , and r\ be the areas of the three subtriangles subtended by P and the corners (see F i g . 3.1), the index designating the opposite corner number. The area coordinates 0- of the point P are defined by (3.1) where A = / \ 1 + A 3 L + A 3 = area of the element. Thus the area coordinates of the nodes of the element are: 16 node I node 2. node 3 also, a useful i d e n t i t y i s proved using the d e f i n i t i o n p i - f . - o ( 1 , 0 , 0 ) p t - i , f , = o ( 0 , 1 , 0 ) = 0 , ?3= I ( 0 , 0 , 1 ) = ^ w n l c h c a n D e e a s i l y of P • and A . 3 (0,0,1) (l,0.0) F i g . 3 .1 D e f i n i t i o n of Area C o o r d i n a t e s . 1 7 The relationship between the x-y coordinates and the area coordinates i s l i n e a r and can be shown to be given by the expression _ P-x y x , x 2 x 3 Pi (3.2) Inverting this equation and using the identity + ^3 = ' ?, I ' ? x > - I 2A (3.3) J - y where 2 A X. y. y* 18 - ( v ^ / ) ( y ; ^ 3 i ) + ( < ^ v ) ( y > b z ) = A + B z + Cr2-(3.4) Rewriting Eg. 3;3 i n index form gives r Pi ( v v) (y;* i ° k 0 - ( x ; + a k Z ) ( y ; + b j Z ) (3.5) 19 Or J ' A + B ^ + C z 1 where i n Eg. 3.5 the indices take on the permutation J K 1 Z 3 Z 3 I 3 I 1. In a plane p a r a l l e l to the x,y plane, the area coordinates are functions of the x,y nodal coordinates of the tr i a n g l e and as we proceed i n the z di r e c t i o n these same nodal coordinates become functions of z. Thus, the area coordinates become functions of z. Area coordinates of a tr i a n g l e i n a plane p a r a l l e l to top surface tr i a n g l e or "base plane" 20 CHAPTER 4 ELEMENT DISPLACEMENT FUNCTION The role of the displacement function of an element i s to establish a r e l a t i o n between the nodal displacements and the displacements of a l l points i n the element. There are two i n t e r r e l a t e d factors which influence the se l e c t i o n of a displacement function. F i r s t , the p a r t i c u l a r displacement degrees of freedom that describe the model must be selected. In Chapter 2 these were chosen as the displacements of the nodes i n the three coordinate directions. Second, to ensure that the numerical r e s u l t s approach the correct solution, the function should s a t i s f y certain conditions such as providing compatible displacements between adjacent elements and the a b i l i t y to describe r i g i d body displacements of the element. A fixed boundary at i n f i n i t y w i l l prohibit a r i g i d body displacement of the whole element. However a r i g i d body displacement of each cross section of the element (z=constant) i s a possible condition which the displacement function must be 21 able to describe. This describes a state in which only shear stresses i n the plane z=constant e x i s t . The shape function which relates the continuous displacements u,v,w to the nine nodal displacements takes the form U = FKy) Pfz) s = A s (4.D where r \ U U = \ v > w V J c o n l i n u o u s dis pla ce me n l s in ocal coorolinale d i r e c t i o n s and 22 s = < > = nodal d i s p l a c e m e n t s in loca c o o r d i n a t e d i r e c t i o n s The variation of the continuous displacements u,v,w in the x-y plane i s given by F(X y) • T h e simplest function for F ( x ^ ) , which w i l l s a t i s f y compatibility of displacements when the base planes of adjacent elements l i e i n the same plane, i s a lin e a r r e l a t i o n between nodal and continuous displacements, Since a li n e a r r e l a t i o n between the (x,y) coordinates and the area coordinates e x i s t s , a lin e a r function of area coordinates can be used for interpolation over the triangle lying in the x-y plane ( i . e . , z=constant). Therefore, F (x,y0 w i l l be expressed as follows ?, o o f , o O f , o o Rx,y) - O f, o O o o p. ° (<*.2) o O f, O 0 f t 0 ° p. In order for the displacements to vanish at i n f i n i t y 23 u,v and w must approach zero as z tends to i n f i n i t y , must be a decay function that modifies the displacements u,v,w by decreasing them monotonically with increasing z u n t i l they vanish at i n f i n i t y . Also must be unity at z=0. A function for POO. which s a t i s f i e s the above requirements and yet remains simple enough to carry out analytic integration in some cases, i s P ( z ) = 0 + Z ) " (4.3) and w i l l be used throughout the thesis. The chosen displacement function jf(*,_y) w i l l s a t i s f y continuity between elements whose base triangles l i e i n the same plane. However this i s , i n general, not the case and d i s c o n t i n u i t i e s at the interface can occur. To i l l u s t r a t e how th i s discontinuity may occur, an example i n Fig. 4.1 i s given. « 24 Fig. 4.1 Adjacent l o c a l Coordinate Systems A discontinuity arises because a point on the boundary between adjacent elements would have di f f e r e n t values of the l o c a l coordinate z, as shown i n Fig. 4.1. This i n turn w i l l produce d i f f e r e n t values of displacement for the same point on the boundary. This discontinuity can be minimized i f the angle of the ray vector to each base plane i s approximately the same. However the f i n i t e element derived i n t h i s thesis i s based on a non-conforming displacement function. In summary a very elementary form of displacement function has been chosen to represent the displacements of the element. This yields a simple f i n i t e element but one which, i t w i l l be shown, gives good r e s u l t s for the deformations of semi-i n f i n i t e s o l i d s . 2 6 CHAPTER 1 5 ELEMENT STIFFNESS MATRIX 5^ 1^  Formulation of Elejment_Stiffness The element s t i f f n e s s matrix i s developed for a homogeneous, lin e a r e l a s t i c material. C l a s s i c a l e l a s t i c i t y theory i s used in the development with only the lin e a r portions of the strain-displacement r e l a t i o n s being retained, thereby l i m i t i n g the analysis to small displacements. The strain-displacement equations are given by: (5.1) 27 or ( : L U where 6 x , £^, £z are the normal s t r a i n compoments and X X 0 are the components of shear s t r a i n . If the assumed form of the displacement function given in Eg. A.3 i s inserted into the strain-displacement equation (Eq. 5.1), the strains may be expressed in matrix notation i n terms of the unknown nodal displacements as ( : L u = L A s = B s (5.2) It can be shown [ 1 ] that the element s t i f f n e s s matrix in l o c a l coordinates, K , i s given by B T E B dV _ _ _ (5.3) V where the integration i s carried out over the volume of the element. E i s the constitutive matrix r e l a t i n g stress to s t r a i n , and i s given by 28 - (lrVXl"2v) I v/|-v V/-v O O O I v/-v O O O 5 Y M O ,-2v 20-v) O O l-2v 2 ( l -v ) O o O 1-2V (5.4) E. and V are Y,oungfs modulus and Poisson's r a t i o where respec t i v e l y . Terms of the l o c a l s t i f f n e s s matrix k given i c f u l l in appendix A. are 5_.2 ftPgl£tic_ Integration,,,of_the_Stiffness_Coefficients The s t i f f n e s s c o e f f i c i e n t s as given by Eg. 5.3 and shown i n Appendix A must be integrated over the volume of the element. The f i r s t two integrations, carried out in the x-y plane, cover the cross-sectional area of the element for any value of z. The th i r d integration i n the z-direction completes the volume i n t e g r a l by ranging from the base plane ( i . e . , z=0) to i n f i n i t y . 29 As the integration proceeds the calculations become more and more laborious, thus making numerical integration desirable. However analytic integration can be carried out for the f i r s t two (x,y) integrations with reasonable ease, and t h i s was done in a l l cases to reduce computing time. The t h i r d integration in the z dir e c t i o n proved to be too d i f f i c u l t for analytic evaluation in a l l but a single case. Therefore resort was made to numerical integration i n every case. If cross-sections of the element lying in the x-y plane are examined at various values of z, the resulting t r i a n g l e s w i l l , in general, be found to have di f f e r e n t orientations. For t h i s reason a rotation and tra n s l a t i o n of the l o c a l coordinate system i s convenient i n carrying out the analytic integration. Two cross-sections for d i f f e r e n t values of z are shown below. Consider two rectangular coordinate systems x-y and x'-y* (see figure below) with axes rotated and translated with respect to each other. 01 a point P in space has coordinates (x,y) cr (x' #y*) r e l a t i v e to these coordinate systems. The eguations of transformation between coordinates are given by 3 1 or x '= ( x - a ) c o s c x - (y-b) s m ^ y' = (x-a)sm^ +(y-b)cos^ (5-5) x r x ' c o s ^ + y ' s i n © < + a y - -x'smoc + y' coS°< + b ( 5 ' 6 ) where the angle c*- i s the counterclockwise rotation of the x* and j' axes with respect to the x and y axes. Also the o r i g i n of the x'-y' coordinate system i s located at (o^ b) r e l a t i v e to the x-y coordinate system. The x*-y» coordinate system w i l l be attached to a cross-sectional area a distance z from the base plane and therefore w i l l be orientated d i f f e r e n t l y for each new cross section. The x-y coordinate system w i l l be attached to the base plane thereby f i x i n g i t s orientation. In pa r t i c u l a r , the x dir e c t i o n of each coordinate system w i l l be aligned along the 1 - 3 edge with the origi n at the 1 node. These axes are shown below. 32 Zl= O Using the above diagrams, the sine and cosine of the angle ^ of Eqs. 5 . 5 and 5 . 6 are found to be S i n oc, = y . - y 3 COS <x = X 3 - K ! where for example X 3 i s the X coordinate of the point 3 t and X 3 i s the X coordinate of the point 3 . Also a s h i f t i n g of axes takes place, such that the translation parameters are Q - * , , ^ = jf • The next step i s to evaluate the l i m i t s of the integration. The upper l i m i t on the integration with respect to x 1 (see above diagram on the right) i s x = x , f - ( *3 ~ x* | y and the lower l i m i t i s 3 3 X = • y The range of integration with respect to y 1 i s from zero to The i n d i v i d u a l terms of the s t i f f n e s s matrix K have the form r I -UL O LL ( 5 . 7 ) where UL and L L refer to the upper and lower l i m i t s of the x' variable. The term inside the brackets w i l l be integrated a n a l y t i c a l l y . On examination of the integrand of Eg. 5 . 7 for a l l of the terms of the s t i f f n e s s matrix, i t i s found that there are six d i s t i n c t types which can be expressed i n the following manner. Type 1 : Type 2 : Type 3 : Type H : Type 5 : f ( x . y , z ) ' 34 Type 6 : ^ (x ,y z ) = ^ ( z ) Since in each case Oj (z.^ i s not a function of x* or y* we can remove th i s term from within the bracket cf Eg. 5.7. Let I. - dx,' d y K C T ) The evaluation of X, reduces to finding the area i n t e g r a l of a triangular cross section lying i n the x-y plane, however this area has already been derived in Eg. 3 . 4 . Thus X . becomes I = (A+ BZ + C Z z (5.8) Let I J dx'd y Using Eg. 5.6 re s u l t s i n I. = Jx' oly' x ' c o S <* + y S i n c < + X ( The X , term may be integrated by inspection by recognizing i t as a type 1 i n t e g r a l since X, i s a function of z only. Carrying out the integration with respect to x* and y' and 35 substituting the l i m i t s yields (x 2'+x 3')co5o< + y I 'sm<*)+ x, (A + Bz + Cz*) With the substitution of the expressions for 5"in << and COS «< there r e s u l t s , + X.CA+BZ + CZ7-) (5.9) In an i d e n t i c a l manner the four remaining inte g r a l s may be evaluated to give the following r e s u l t s . I y c J x ' d y ' ^ J (5. 10) I 4 -d x ' d y ' 1 (x3-x.) xt' (x / + X,') + X , 36 y* + 2x, x, z (A + BZ + C z z ) \ y l ^ y ' (5.11) 1 >i 12 x ; 1 - ( y , - y * ) ( x 3 - * . ) [ y * Y x » ' + 2 x ; ) j + (x 3 - x , ) |_yz'j + 2y, [ I , ] - y ' (A+BZ + C Z 2 ) (5.12) x y d x' ol y' 37 + i 2 ~J r (VO -(y.-Ja) >i (x,'+ X2') + (vO^-y^Ty; I. + X. t l . ] - vy, (A + B Z + C Z ^ ) 2 (5.13) The primed c o o r d i n a t e s may be e l i m i n a t e d by using Eg.- 5.5 and the i d e n t i t y so t h a t the i n t e g r a l s c o n t a i n only unprimed c o o r d i n a t e s . The i n t e g r a t i o n with r e s p e c t to 2 remains to be c a r r i e d out. The X values of Eqs. 5.8 to 5.13 which are f u n c t i o n s of z only are m u l t i p l i e d by ^"^0 t o f ° r n i the i n t e g r a n d which w i l l be n u m e r i c a l l y i n t e g r a t e d as d e s c r i b e d i n Chapter 6. 38 CHAPTER 6 SEMI-INFINITE NUMERICAL INTEGRATION 6.1 Goodwin-Moxan^Method The Goodwin-Moran method of numerical integration,* which i s a form of the f a m i l i a r t r a p e z o i d a l r u l e method, i s used f o r the s e m i - i n f i n i t e i n t e g r a t i o n . I t has been noted t h a t the t r a p e z o i d a l r u l e given by the e x p r e s s i o n fa)dt = h 2 ^ i M J (6.1) o f t e n g i v e s s u r p r i s i n g l y a c curate r e s u l t s f o r i n t e g r a l s over a doubly i n f i n i t e range. To take advantage of t h i s phenomenon, a t r a n s f o r m a t i o n t h a t transforms i n t e g r a l s over other ranges to the doubly i n f i n i t e range i s made. For the i n t e g r a l of i n t e r e s t i n t h i s t h e s i s , namely \ (-(x)cJx # the t r a n s f o r m a t i o n * = £ i s used. T h i s transforms the i n t e g r a l to a doubly i n f i n i t e range whereupon Eg. 6.1 can be used. The r e s u l t i n g quadrature r u l e i s expressed as f o l l o w s 39 r J = N = i 4- p J ' J (6.2) The complete d e r i v a t i o n of Eq. 6.2 appears i n Appendix B. The parameter S i s a s c a l e f a c t o r to cente r the summation on the area which makes a s i g n i f i c a n t c o n t r i b u t i o n while a v o i d i n g r e g i o n s which do not c o n t r i b u t e a p p r e c i a b l y to the i n t e g r a l . The parameter P s e r v e s as a node p o s i t i o n e r or i n t e r v a l s pacing parameter analogous to h i n the t r a p e z o i d a l r u l e . In t h i s f o r m u l a t i o n the i n t e r v a l spacing between nodes forms a geometric p r o g r e s s i o n which i s mere l o g i c a l than uniformly spaced nodes f o r an i n f i n i t e i n t e g r a l . 6.2 An a j i v t i c a l _ y e r s u s _ N u m e r i c a l n I n t e g r a t i o n In order to t e s t the accuracy of the numerical i n t e g r a t i o n , a comparison with the exact i n t e g r a l can be made I/* f o r some of the e n t r i e s i n the l o c a l s t i f f n e s s matrix j\ T h i s comparison i s made only f o r one i n t e g r a l whose i n t e g r a n d does not c o n t a i n a d e r i v a t i v e with r e s p e c t to z and which can be i n t e g r a t e d a n a l y t i c a l l y with some ease. The comparison w i l l be made f o r element kj-, of k ^  40 which i s given i n Appendix A as E (i-v)(o<+/3) k 0+vXl - 2 v ) (6.3) xn 6.2^1 An a l y t i c a l . Integratign_.of k t l Pi The area coordinate ^ i s given i n Chapter 3 by At+Blz+Clz*+ (DI + E L Z ) X + (F^Q.z)y A + B z + Cz ' (3.6) From Chapter 4, i s given as (4-3) D i f f e r e n t i a t i n g Eg. 3.6 as indicated i n Eg. 6.3 and l e t t i n g E(l-v)(* + ;fl) _ f yields (l+v)0"2v) ^ (D.+E.ZXFJ + S.Z) Jxdyolz J (\*zy ( A + B z - C z A X 41 (D, + E i z ) ( F l + G ,z ) [Ve a ( z ) ]o l z o ( l . z ) 1 ( A + B z + Cz. 1 )* Since the integrand i s not a function of x or y, the f i r s t two integrations y i e l d the area of the triangular prism as a function of z. This expression for the area has already been obtained i n Chapter 3 as ZA = A+Bz+Cz. Therefore A.+ B z + C z ' Area(z) = Z The integration with respect to z can now be carried out ( D , + E , z Y F > G , z ) d ; K - ^ " z \ ( l + z ) V A + Bz + C z J ) - 1 2 r «o Oz)* (A + Bz + Cz*) 42 L e t (| +z) ~ "t 1 z J , t 2 ( A * + B*t+C*t* ) Z 9 olt + t T + J t T I 43 and edge d i r e c t i o n parameters (see Eqs. 2.1 and 3.4) a = O b, = -I a» =' O b 2 = | a 3 - | b 3 = 0 A= *'Xy;-y;) + *:(y;-y;) + < ( y ; - y : ) o o + I ( i - o ) B - ^ ( b , - b 1 ) ^ - ( b , - b 1 ) + x;(b,-b 1) + y,'(a,-a,) + y 1 ,(q,-a 1) + y ; ( a , - a 2 ) = 0 + 0 + 1 = 3 [ i - ( - o ; + o + i -o ] + o C = a , ( b 3 - k ) + flt(b,-b,) + a . f k - b , ) 0 + O = 2 U5 A* = A - B + C = O B* =' B- 2C = - I C * = C - 2 D, = yl-y: = - i F, = x ; - x ; = - i E, = b 3 - - i ( i . ' V O . - - i 9, - D,F, = ( - 0 ( - l ) = I <Jt = D,G, +E,F, = (HY-I) + (-O(-i) = z Sa = E.Gt, = ( - O ( - i ) - | g ; » 3 , - 2 9 3 = 2 - 2 ( i ) = o ^ = B * l - 4 A * C * - ( - i ) ( -0 - ( 4 ) (o ) ( 2 ) = ! p 00 — g,' i t = ^ 1 <-> L J , T J z 46 = 1 2 - \ Lz - ? L r r - y ( . 3 4 4 5 7 ) Using the Goodwin-Moran numerical integration scheme with N = number of nodes = 15,; S = scale factor = 2 and P -geometric progression factor = '/^  and using double precision output, gave a value of K^, = (. 34£>5 5 ^ ^  • This guadrature i s only approximate of course but the accuracy attained i n the previous example shows that the numerical res u l t i s an excellent replacement for the exact r e s u l t . I 2 ( 2 ) + ( - l ) + | = 1 2 ( 2 ) + ( - | ) - | 2 2 47 CHAPTER 7 NUMERICAL EXAMPLES Two examples were chosen against which the res u l t s of an analysis using the i n f i n i t e f i n i t e elements could be compared to exact a n a l y t i c a l solutions. The two examples b a s i c a l l y derive from the solutions for a point load on an e l a s t i c halfspace, in one case applied perpendicular and in the other tangential, to the surface of the halfspace. The actual examples considered the halfspace with a hemispherical region removed from the top surface, and with loads applied to the surface of t h i s hemispherical bowl egual to the stresses that would arise from a concentrated lead on the entire half space. The stresses and displacements in the actual examples are therefore i d e n t i c a l to the stresses and displacements i n the halfspace problem in the region outside of the hemispherical cutout. U8 Point L o a d on a rlal£space. F i n i f e E l e m e n t E x a m p l e There were several reasons for choosing such examples. F i r s t was the desire to test the element in an example roughly s i m i l a r to a situ a t i o n where i t was considered to be useful, thus the half space problem. Second was the desire to test the i n f i n i t e f i n i t e elements only and not in combination with regular three dimensional f i n i t e elements, thus the hemispherical cutout. Third the desire to test the element, or at least a reasonable combination cf elements, for loads primarily applied i n the i n f i n i t e (2) d i r e c t i o n and for loads applied perpendicular to the i n f i n i t e d i r e c t i o n , thus the two examples with loads normal and tangential to the surface. 1»9 11... -, fi ? § ,B P1 §,., 3 _ Z „,. ? § J tz I *? I i B .1 f .§,_ ,j? ,9 1 I ,£L £ °. § £? e 3 _ I? Y. 3. Surface Point Load mPerpendiculat The equations for the deflections of points in a semi-i n f i n i t e s o l i d loaded with a point load perpendicular to the surface can be found in [1] as U 1 IT P E + r V _ p lit r •f (7.1) P ZTT i n which 50 Z W X \ R Z x,u Fig. 7.1 Semi-Infinite Solid Loaded by a Surface Point Load Perpendicular to the Surface The stresses due to a perpendicular point load on the surface of a s e m i - i n f i n i t e s o l i d are given in [1] as 0~ = — — ** 2TT R 2 P I CT =7 — — XY lir R z cr x z c r YY R 3 ^ y \ R R + Z R(R+Z)' 3XYZ _ 3P X Z Z 2TT R P J_ 2ir R* 3ZY R 3 ( l - 2 v ) ( 2 R + Z ) X Y R (R + Z ) 2 (7.2) - ( l - Z v ) / Y ^ Z R + Z U  V \ R R+Z R(R+Z)7 5 1 cj- - 3P XZ YZ 2.TT R 5 _ _3P YZ_* ZZ o D 5 Z7T R These stresses are used in the generation of the nodal stress vector j? (see Appendix D) . Due to a x i a l symmetry, any r a d i a l section may represent the s t i f f n e s s of the halfspace. However the boundary conditions are most e a s i l y inserted i f one-quarter of the semi-i n f i n i t e s o l i d i s considered. The quarter surface of the hemispherical bowl forms the base plane for the i n f i n i t e f i n i t e elements. This surface i s subdivided into a mesh of f i f t y - f o u r elements and thirty-seven nodes as shown in F i g . 7.2. The V displacements are set equal to zero along the X axis and the U displacements are assigned to be zero along the Y axis. Therefore points d i r e c t l y beneath the load are allowed W displacements only. Appendix D describes the load vector used to approximate the stresses distributed over the base plane. These stresses, given by Eqs. 7.2, vary ncn-linearly from point to point. However to simplify the calculations to determine the 52 l o a d v e c t o r , the value of the s t r e s s vector i s c a l c u l a t e d at each node and then assumed to vary l i n e a r l y between nodes. F i g . 7.3 compares the r e s u l t s obtained from the i n f i n i t e f i n i t e element a n a l y s i s using the mesh shown i n F i g . 7.2 with the exact e l a s t i c i t y s o l u t i o n given by Eqs..7,2. The i n f i n i t e f i n i t e element r e s u l t s are found to vary by 4 ° / o on the average from the exact s o l u t i o n . D i r e c t l y beneath the l o a d , agreement wit h i n 2. / o i s achieved. A l l t e s t s are f o r a u n i t load ( i . e . , P = I ). Example 1 r e s u l t s i n a problem with ninety-seven degrees of freedom and a s t i f f n e s s matrix with a h a l f band width of t w e n t y - f i v e . The c e n t r a l p r o c e s s i n g u n i t (CPU) time on a IBM 360-67 was 96 seconds f o r the e n t i r e problem i n c l u d i n g the numerical i n t e g r a t i o n of the s t i f f n e s s matrix. The numerical i n t e g r a t i o n parameters used i n t h i s example are as f o l l o w s number of nodes N| = I 5 s c a l e f a c t o r S = I.O geometric p r o g r e s s i o n f a c t o r Y" = 0 . 5 53 F i g . 7.2 F i n i t e Element Mesh of Hemispherical Bowl with One Quarter Symmetry F I G . 7 .3 D E F L E C T I O N S OF H E M I S P H E R I C A L BOUNDARY L O A D E D P E R P E N D I C U L A R TO S U R F A C E ( X - Z P L A N E ) 55 7.2 Example 2 - _ Seffli-Ipf i n i t e Solid Loaded by., a_Surf acg_Point Load P a r a l l e l to the Surface The equations for the deflections of points in 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 surface i n the X d i r e c t i o n as shown in F i g . 7 . 4 are [ 1 ] u = Q (uv) i ZTT E R R z (R+Z) Y = Q (W) X Y _ ZTT E R 3 I -(l-2v) R (R + Z ) L w _ Q (l+v) X ZTT E R* Z , (l-2v)R R X R(R+Z) ( 7 . 3 ) * x.u F i g . 7 . 4 Semi-Infinite Solid Loaded by a Surface Point Load P a r a l l e l to the Surface 56 The s t r e s s e s due t o a p o i n t l o a d i n t h e X d i r e c t i o n p a r a l l e d t o t h e s u r f a c e o f s e m i - , i n f i n i t e s o l i d a r e [ 1 ] o X cr = — 27T R 3 XY Z7T R 3 3 X R2 _(l. 2 vWl-2v)RY3- X ^ R + Z ) ' ( R + Z ) 2 \ R 2 ( R + Z ) 3 X * + ( l - Z v ) R 1 R 2 ( R + Z ) 2 3 Q X*Z 2TT R 5 X Z ( 3 R + Z ) R Z ( R + Z ) ( 7 . 4 ) Q X cr = — — YY R 3 Y Z 1 1 R ' 3 Q X Y Z 2TI R 5 -(l - 2 v ) + ( l - 2 v ) R : ( R + Z ) : I - Y2 ( 3 R + z V R 2 ( R + Z ) _ 3 Q X Z ( T , = — -zz z a T R A g a i n s y m m e t r y r e g u i r e s o n l y one g u a r t e r o f t h e s e m i -i n f i n i t e s o l i d need be c o n s i d e r e d . T h e r e f o r e t h e same mesh a s E x a m p l e 1, F i g . 7.2, may be u s e d . The \/ d i s p l a c e m e n t s a r e s e t e q u a l t o z e r o a l o n g t h e X a x i s . I t c a n be shown f r o m a symmetry a r g u m e n t t h a t t h e V and w d i s p l a c e m e n t s a l o n g t h e 57 I axis are also zero. In Figs. 7.5 and 7.6 a test comparing the results obtained from the i n f i n i t e f i n i t e element mesh shown in Fi g . 7^2 with the exact e l a s t i c i t y solution given i n Egs. 7.3 i s shown. The i n f i n i t e f i n i t e element r e s u l t s are found to vary in the order of 3 A from the exact solution. Example 2 re s u l t s i n a problem with 91 degrees of freedom and a s t i f f n e s s matrix with a half band width of 24. The CPD time on a IBM 360-67 was 99 seconds for the entire problem including the numerical integration of the s t i f f n e s s matrix. The numerical integration parameters are the same as those used i n Example 1. 58 F I G . 7.5 D E F L E C T I O N S O F H E M I S P H E R I C A L B O UNDARY L O A D E D P A R A L L E L TO S U R F A C E ( X - Z P L A N E ) FIG. 7.6 DEFLECTIONS OF HEMISPHERICAL BOUNDARY LOADED PARALLEL TO SURFACE (X-Y PLANE) 6 0 C H A P T E R 8 D I S C U S S I O N Instead of dealing with an imposed r i g i d or free boundary, as has been commonplace in s o i l mechanics problems using f i n i t e elements, a f l e x i b l e boundary formed by the i n f i n i t e f i n i t e elements has been introduced. The purpose of using such i n f i n i t e f i n i t e elements i s not to evaluate stresses at some great distance from the surface, but to r e a l i s t i c a l l y represent the s t i f f n e s s of the sem i - i n f i n i t e s o l i d which has previously been assumed to be zero or i n f i n i t e . Close to the surface and near points of load, regular f i n i t e elements would be used to evaluate the stresses and displacements. The solution of a halfspace problem using regular f i n i t e elements requires numerous elements, many of which are necessary onl:y to reach the a r t i f i c i a l boundary where stresses or displacements can be assumed n e g l i g i b l e . This boundary can be reduced with the introduction of the i n f i n i t e f i n i t e elements since they represent the s t i f f n e s s of the surrounding region. 61 Thus, with the shrinking of the boundary, a large reduction i n the number of degrees of freedom i s possible. In l i g h t of the accuracy attained using a very simple displacement function, a higher order element i s not warranted for halfspace problems. 62 BIBLIOGRAPHY 1. S c o t t , R. F., Fundamentals of S o i l Mechanics, Addison-Wesley, Reading, Mass., 1963, pp. 498-500. 2. Squir e , W i l l i a m , I n t e g r a t i o n f o r Engineers and S c i e n t i s t s , American E l s e v i e r , Hew York, 1970. 3. Squire, W i l l i a m , "Numerical E v a l u a t i o n of I n t e g r a l s using Moran Tran 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 , Department of Aerospace 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 Element Method i n E n g i n e e r i n g Science, McGraw-Hill, London, 1971. 63 APPENDIX A ELEMENT STIFFNESS MATRIX FOB LOCAL COORDINATE SYSTEM The element s t i f f n e s s matrix f\ may conveniently be partitioned as follows: SYM : : <A.1) The submatrices are a l l (3x3) arrays and the matrix i s a (9x9) symmetric matrix. The partitioned matrix i s shown in f u l l i n Eq. A.2 in which Zv Z(\-v) 6 4 0~ p s E ( i - v ) r ( l+v ) ( i - 2 v ) where E and 'V are Young*s modulus and Poisson's r a t i o . Each term of the submatrix k Lj must be integrated over the volume of the element. X —> < cn + . _» II 1 >* • —> + • —> K r > " T 5 X M • *J _ i _ • ** Q — > 1 % - r Kl >^ + + a - . X — » • —> • -» o + X • .> X X 1 ^ + N 4 L U II 66 APPENDIX B GOODWIN—MOBAN METHOD OF S E M I - I N F I N I T E INTEGRATION The a p p l i c a t i o n o f t h e t r a p e z o i d a l r u l e t o t h e r a n g e -<?o to + oo i s t h e b a s i s o f t h e Goodwin-Moran method. The s i m p l e t r a p e z o i d . a l r u l e + oo j = +N f(x)dx = h X f ( j ^ ) (B.1) - oo J = "N g i v e s s u r p r i s i n g l y good r e s u l t s f o r i n t e g r a n d s whose d e r i v a t i v e s v a n i s h q u i c k l y a s x —> 0 0 . S i n c e + 00 r +00 (B.2) - 00 - 00 Eq. B.1 c a n be r e w r i t t e n a s 67 + 00 J J = - N to center the summation and avoid adding negligible terms for either large positive or negative arguments. When integrals over ranges other than -oo to +oo are encountered, a transformation to the doubly i n f i n i t e range can be made to make use of the trapezoidal rule's accuracy. I f the in 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 of X = 6 ^ + Ol w i l l g i ve I - •f ( e 1 + a ) e 1 o l t - oo S h i f t i n g the variable as in Eg. B.2 gives 6 8 I = \ f ( e ^ + a ) e t t S < J t oo and then approximating the i n t e g r a l in the same manner as Eg. B.1 gives j a + N I = K > e j h * J a ) J Now l e t r s e ^ - Then Also l e t O = c . These substitutions produce the result poo J=+* A f ( x ) J x - S | L r | ^ r i f ( a ^ S r J ) . o j - - N The approximation becomes exact as [S]-> oo and P —*• . The absolute value of the term i s taken so that r can be less than I.O and the res u l t i s unchanged as the + J and - J term are simply interchanged from the case where the r e c i p r o c a l of T i s used. The quadrature rule then becomes 69 + N (B.3) 70 APPENDIX C LISTING OF FORTRAN SUBROUTINE OF THE GOODWIN-KORAN METHOD The t f o l l o w i n g i s a l i s t i n g of a F o r t r a n subprogram which uses the Goodwin-Moran method to evaluate an i n t e g r a l over a s e m i - i n f i n i t e range. Eg. B.3 of Appendix E i s the b a s i s of the subprogram. The input to the subprogram i s e x p l a i n e d i n the l i s t i n g while the output, QGEOM, the value of the i n t e g r a l , i s ret u r n e d to the c a l l i n g program. 71 FUNCTION QGEOM (GRAND) C C EVALUATION OF INTEGRAL OVER SEMI-INFINITE RANGE BY GOODWTN-C MORAN METHOD USING POINTS IN A GEOMETRIC PROGRESSION C A=LOWER LIMIT OF INTEGRAL,IT CAN BE POSITIVE,NEGATIVE OR C ZERO C GHAND=EXTERNAL FUNCTION SPECIFYING INTEGRAND C S=POSITIVE NUMBER SERVING AS SCALE FACTOR C RAT=POSITIVE NUMBER (NOT 1.) USED AS FACTOR IN GEOMETRIC C PROGRESSION C N=NUMBER OF NODES (ACTUALLY 2*N+1 NODES ARE USED) C THE FUNCTION GRAND IS EVALUATED AT POINTS BETWEEN C A+S*RAT**N AND A+S/RAT**N. SELECT VALUES TO COVER C SIGNIFICANT RANGE. DEFINE GRAND TO GUARD AGAINST C UNDERFLOWS. C 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 30 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 f i n i t e element examples of Chapter 7, the nodal f o r c e s a c t i n g a t the nodes of the h e m i s p h e r i c a l boundary are taken t o be s t a t i c a l l y e q u i v a l e n t to the f o r c e s that would be produced by the s t r e s s e s , from a p o i n t load on the s u r f a c e of the h a l f s p a c e , a c t i n g on t h i s h e m i s p h e r i c a l s u r f a c e . I t i s the purpose of t h i s appendix to convert these given boundary s t r e s s e s to nodal p o i n t l o a d s . The i n and out of plane s t r e s s e s C|_(^B) a c t i n g on the base plane are o b t a i n e d from the Bcussinesq s o l u t i o n of the theory of e l a s t i c i t y [ 1 ] , The area c o o r d i n a t e (3 B i s d e f i n e d f o r the base plane only (i.e.,z=0). Let R be the node f o r c e s i n g l o b a l c o o r d i n a t e s that are s t a t i c a l l y e q u i v a l e n t to . When v i r t u a l node displacements C) P t a l s o i n g l o b a l c o o r d i n a t e s , are produced, the work done by these node f o r c e s must be equal to t h a t done by the s u r f a c e s t r e s s e s • That i s . 73 dA (D. 1) where ^ . ( ^ & ) = tl— = continuous displacements i n the g l o b a l system and A i s an i n t e r p o l a t i n g f u n c t i o n d e f i n e d by Eg. 4.2. The d i s t r i b u t e d s t r e s s e s ^ ( ^ g ) v a r y n o n - l i n e a r l y from p o i n t t o p o i n t but f o r convenience they w i l l be r e s t r i c t e d to vary l i n e a r l y . With the d i s t r i b u t e d s t r e s s e s 0^ v a r y i n g l i n e a r l y from node to node, only two node values of 0|_ , one at each end along the boundary, are necessary to s p e c i f y 0|_ . I f the nodal v a l u e s of C|_ are s p e c i f i e d i t w i l l be necessary to i n t e r p o l a t e l i n e a r l y between nodes i n order to d e f i n e CJ_ . A l i n e a r i n t e r p o l a t i o n f u n c t i o n has been used i n Chapter 4 to r e l a t e the nodal displacements to the displacements of a l l p o i n t s i n an element. T h i s same f u n c t i o n can be used to r e l a t e the nodal a p p l i e d s t r e s s e s to the s t r e s s e s a p p l i e d on the base plane. Then, •4 U A o i n which A ^ i s a l i n e a r i n t e r p o l a t i o n f u n c t i o n and t h e r e f o r e 74 i s i d e n t i c a l to Eg. 4.2. The v e c t o r _P c o n t a i n s the nodal values of s t r e s s e s C|_(^ B)and i s 9 * 1 . Rewriting Eg. D. 1 g i v e s That i s . (D.3) Uo c o n f u s i o n should a r i s e over the d i f f e r e n t A' S . The A'S are i n t e r p o l a t i n g f u n c t i o n s d e f i n e d f o r the base plane only while the M p e r t a i n i n g to i s the area of the base plane. The i n t e g r a t i o n can be c a r r i e d out e x p l i c i t l y and w i l l c o n t a i n c r o s s product terms. I t can be shown t h a t the i n t e g r a l s y i e l d the f o l l o w i n g r e s u l t s ' 1.3 75 F i n a l l y , upon i n t e g r a t i o n of Eq. D.3, the node f o r c e s are given by IX R R R IY IZ 2X < R 2Z R R R 3X 3Y 3Z A iz 2 r + r IX 2.X + r 3 X + r IY 2Y + r 3 Y 2 r + r IZ 2Z + >"3I + Z r + r 3 X r IY + P 3 V + 2. r 2 z (0.») where the terms R^x , R^y and and p are •ix ' P iV a n a P i Z the X # Y and Z components at node i of the g e n e r a l i z e d nodal f o r c e and the d i s t r i b u t e d s t r e s s r e s p e c t i v e l y . The v e c t o r ^ roust be found from the s i x s t r e s s components CTT^ . a c t i n g a t a p o i n t . In tensor n o t a t i o n t h i s i s given as T; 76 where h j i s a component of a u n i t normal t o t h e base p l a n e i n the j lh d i r e c t i o n . , when e v a l u a t e d a t the nodes, become terms i n the _|0 m a t r i x i . e. P z t = TY = « C T ( n, + C r f i x + CC 3 n3 e v a l u a t e d at node 2. The s i x unique terms o f t h e s t r e s s m a t r i x C~ 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 7.2. A v a l u a b l e check cn the g e n e r a l i z e d l e a d v e c t o r may be o b t a i n e d by summing the g e n e r a l i z e d f o r c e s i n any p a r t i c u l a r d i r e c t i o n and comparing i t t o the pr o d u c t of the s t r e s s i n the same d i r e c t i o n and the area over which i t a c t s . From e q u l i b r i u m , t h e s e two r e s u l t s s h o u l d be e g u a l . 

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