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I ' '3 3-“ "nu',.,;:',¢.. ,fi,,-........w iii-1"r ‘- ' WWWWWMW %%180q75 1293 00590 3988 , - Door-O... 0* ~ LIBRARY Michigan State University This is to certify that the thesis entitled On the Application of the Boundary Element Method to Plane Orthotropic Elasticity presented by Javad Katibai has been accepted towards fulfillment of the requirements for Master's Mechanics degree in "flit hdflfyk (l [tit-"la Major éj’essor Date May 12, 1989 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE MSU Is An Affirmative ActioNEquel Opportunity lndltution ON THE APPLICATION OF BOUNDARY ELEMENT METHOD TO PLANE ORTHOTROPIC ELASTICITY BY Javad Katibai AN ABSTRACT OF A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Metallurgy, Mechanics, and Material Science 1989 (9040,15 ABSTRACT ON THE APPLICATION OF THE BOUNDARY ELEMENT METHOD TO PLANE ORTHOTRDPIC ELASTICITY 3? Javad Katibai Application of the boundary element method to elastostatic, orthotropic, plane stress problems is presented. The direct boundary element method is utilized. The boundary is approximated as piecewise straight and, on each straight boundary element. constant traction and linear displacement variations are assumed. This model allows for traction discontinuities at the boundary element interfaces. Three forms of the influence functions are discussed for the orthotropic materials depending upon a relation among the material constants. However, only two practical cases are solved. Results are obtained for four example problems and are compared to finite element solutions(NASTRAN). ii ACKNOWLEDGEMENTS The author ‘wishes to express his gratitude and deep appreciation to his advisor. Professor Nicholas Altiero, for his able guidance and patience during the course of this research , and for his spending many hours with the author in the preparation of this thesis. The author also wishes to thank, his parents for their continuous encouragement and moral support in the past several years. The financial support of' Garrett. Turbine Engine Company, Phoenix, Arizona, is also gratefully acknowledged. iii LIST OF TABLES LIST OF FIGURES CHAPTER I CHAPTER II CHAPTER III TABLE OF CONTENTS INTRODUCTION APPLICATION OF THE BOUNDARY ELEMENT METHOD TO ISOTROPIC MATERIALS II.1 11.2 DESCRIPTION OF THE BOUNDARY ELEMENT METHOD PIECEVISE LINEAR ELEMENTS APPLICATION OF THE BOUNDARY ELEMENT METHOD TO ORTHOTROPIC MATERIALS III. III. III. III. III. III. III. III. INTRODUCTION THE INFLUENCE FUNCTIONS FOR ORTHOTROPIC MATERIALS PROBLEM FORMULATION COMPARISON OF THE ISOTROPIC AND ORTHOTROPIC INFLUENCE FUNCTIONS PIECEWISE LINEAR ELEMENTS SINGULAR TERMS NUMERICAL INTEGRATION PRELUDE TO COMPUTER PROGRAMMING iv PAGE vi vii 1-4 5-14 5-7 8-11 15-53 15-16 16-21 22 22-23 23-25 25 26-30 31-33 CHAPTER IV APPENDIX A APPENDIX B APPENDIX C BIBLIOGRAPHY III. III. III. III. III. III. III. PLANE STRESS, ORTHOTROPIC BOUNDARY ELEMENT PROGRAM INPUTTING PROBLEM DATA CALCULATION OF MATERIAL PROPERTIES E1.1 AND C13 CALCULATION OF COEFFICIENT MATRICES TRANSFORMATION OF THE COEFFICIENT MATRIX uR CONSOLIDATION OF KNOWN AND UNKNOWN BOUNDARY VALUES CALCULATIONS OF THE STRESSES AND DISPLACEMENTS EXAMPLES AND DISCUSSION IV.1 IV.2 EXAMPLES DISCUSSION OF RESULTS COMPUTER LISTING OF THE PROGRAM ORTHO.CASE1 AND ORTHO.CASE2 COMPUTER LISTING OF THE PROGRAM TEST FEM AND BEM INPUT DATA FOR EXAMPLE PROBLEM TWO PAGE 33-34 34-35 35 35-36 37433 38 38 54-66 54-56 56-66 67-87 88-93 94-98 99 TABLES 111.1 111.2 111.3 111.4 111.5 111.6 111.7 IV.1 IV.2 IV.3 IV.4 LIST OF TABLES Some composite materials widely used in industry The values of the influence function uR for the source point x-0.00 y-5.00 The values of the influence function uR for the source point x-5.00 y-0.00 The values of the influence function uR for the source point x-3.55 y-3.55 The values of the influence function uc for the source point x-0.00 y-5.00 The values of the influence function uc for the source point x-5.00 y-0.00 The values of the influence function uc for the source point x-3.55 y-3.55 Displacements calculated by NASTRAN and BEM selected points in example problem one Displacements calculated by NASTRAN and BEM selected points in example problem two Displacements calculated by NASTRAN and BEM selected points in example problem three Displacements calculated by NASTRAN and BEM selected points in example problem four vi for for for for PAGE 47 48 49 50 51 52 53 63 64 65 66 FIGURES 11.1 11.2 11.3 111.1 111.2 111.3 111.4 111.5 111.6 111.7 111.8 IV.l 1v.2 _ 1v.3 IV.4 IV.5 LIST OF FIGURES Mixed boundary value problem Plane boundary value problem Boundary discretization Definition of a1. b1. x(m) Flowchart of the program TEST Field and source points selected for the example problem Flowcharts of the programs ORTHO.CASE1 and ORTHO.CASEZ Flowchart for the calculations of the coefficient matrix Singular contributions in [uc] [uR] Non-singular contributions in [uc] [uR] Flowchart for modification of [uR] Flowchart for rearranging equations into final form FEM model of example problem one BEM model of example problem one FEM model of example problem two BEM model of example problem two FEM model of example problem three BEM model of example problem three vii PAGE 12 13 14 39 4O 41 42 43 44 45 46 57 58 59 6O 61 62 CHAPTERI INTRODUCTION Boundary solution techniques are becoming increasingly popular with engineers and have been applied for the solution of a wide range of problems including two and three dimensional elasticity. The most frequently used method employs the fundamental solution of the governing equations as an influence function and constructs the solution to the problem of interest by superposition. This method is presented under different names such as ”Boundary Integral Equation Method", "Boundary Integral Method”,etc. In its most general form this technique consists of subdividing the boundary of the region under consideration into a collection of elements; hence the name “Boundary Element Method" [1,2]. Consider the classical mixed boundary value problem of linear elastostatics, shown in Figure 1-1, consisting of an elastic body, R, loaded by specified tractions, ti, on portion Bt of the boundary and specified displacements, ui, on the remainder of the boundary Bu' Body forces are neglected here. The stress fields and displacements everywhere in R, subject to the given boundary conditions, are sought. Numerical solutions are sought for this problem which allow for orthotropic material properties and arbitrary body shape and loading conditions. The Boundary Element Method derives from the statement of the problem in the form of an integral equation. The integrands consist of known influence functions and both known and unknown boundary conditions. The influence functions satisfy the differential equations exactly. Hence, the solution of displacements and stresses also satisfies the differential equations exactly. Thus, the resolution of high stress gradients by the Boundary Element Method is very good. The integration path of the integral equations of the Boundary Element Method is around the boundary of the body. Thus, for numerical purposes, the discretization needs to be done only on the boundary . This is in contrast to the Finite Element Method in which discretization is done over the. entire domain. The net result of this difference is that the Boundary Element Method requires less data preparation effort to solve a problem. The Boundary Element Method, however, is not without its own share of problems. Though the Boundary Element Method has been used extensively for isotropic problems, literature on its application to anisotropic material problems is relatively sparse[6,ll,l2,l3]. The sparsity of existing literature on anisotropic materials is not the only problem. Many inaccuracies present in these research papers and technical reports make application of the method difficult. It is one of the objectives of this dissertation to clearly define the influence functions and to offer a systematic solution that is easy to understand and implement. The application of the Boundary Element Method to the isotropic case is discussed in chapter 11 of this dissertation. The general problem is formulated and all of the assumptions are outlined. In chapter 111, the application of the Boundary Element Method is extended to orthotropic materials. It is demonstrated that the mathematical formulation is the same except that the influence functions are different from the isotropic case. For plane problems involving orthotropic elastic materials, there are three forms of the influence functions depending upon the relationship among the four material constants [6]. The Boundary Element programs, ORTHO.CASE1 and ORTHO.CASE2, described in chapter 111, are based on the direct approach. The boundary discretization consists of straight segments in which displacements are assumed to vary linearly and tractions are assumed to be constant. In the final chapter, results of the program are presented and it is compared to the isotropic case. Some conclusions are drawn from these results and recommendations are made. Figure 1.1 Mixed boundary value problem CHAPTER II APPLICATION OF BOUNDARY ELEMENT METHOD TO ISOTROPIC MATERIALS 11.1 DESCRIPTION OF THE BOUNDARY ELEMENT METHOD For the plane boundary-value problem of linear elasticity illustrated in Figure 11.1, the displacement at a point x on B is related to the displacements and tractions at all other points on B by Somigliana's identity, [1] i.e. oij(x)uj(x)+J(uc)1.J(x,x)uj(x)ds-J(uR)1.j(x,x)tJ(x)ds (11.1) B B where the integral on the left hand side is interpreted in the Cauchy principal-value sense. The function (uc)1.J(x,x) is the displacement, u1(x), due to a unit displacement discontinuity, cJ(x), applied in the infinite elastic plane and (uR)1.J(x,x) is the displacement, u1(x) due to a unit force, R (i), applied in J the infinite elastic plane. The coefficients oij(x) are equal to 0.56 if the boundary 11 is smooth at x, where 6 is the Kronecker delta. Otherwise 1] oij(x) depends on the corner angle at x. At a point x in R, the displacements and stresses can be calculated from the equations u1(x)-J (uR)1.J(x,x)tJ(x)ds-J(uc)1.J(x,x)uj(x)ds (11.2) B B aij(x)-J (oR)1J.k(x,x)tk(x)di-J(oc)1j.k(x,x)uk(x)ds (11.3) s s where the influence function (0R)ij k(x,x) and (oc)1j k(x,x) are the stress component 01 (x) due to a unit force, nk(i), J and a unit displacement discontinuity, ck(x), ' respectively, applied in the infinite plane. At each point x on B and in each direction, either uj(x) or tJ(x) is known. Therefore, equation (11.1) can be used to solve for the unknown values of uj(x) and tJ(x), thus giving complete boundary information. The displacements and stresses at any internal point can then be determined by simple integration using equations (11.2) and (11.3). It can be shown that, for plane stress and material isotropy the influence functions or the Green's functions (in this dissertation the terms influence function and Green's function are used interchangably) of equations (II.1),(II.2) and (11.3) are given by [3] (uR)1.kf[-(3-v)losp+(1+V)qiqkl/(8«G) (11.4) (uc)1.1-[2<1+u) (filq13-fizq23)+<1-u>filq1+<3+v)62q21mm (uc)1.2-[2(l+u)(-fizq13-n1q23)+(1+3u)n2q1+(3+v)nlq2]/(4sp) (uc)2.1-[2(l+y)(-n2q13-n1q23)+(3+u)n2q1+(l+3u)fi1q2]/(4sp) 7 (uc)2.2-[2(l+v)(-n1q13+nzq23)+(1-v)n2q2+(3+u)n1q1]/(4xp) (11.5) (an)11.1-1-2<1+u>q13-<1-u>q11/<4«p) 12.1-12(1+u)q23-<3+u>q21/(4«p> q13-(1+3v)q11/(4«p> (an)11.2-12<1+u>q23-(1+3v>q21/<4«p> (an)12.2-12<1+v>q13-<3+u>q11/(4«p) 22.2-1-2<1+v>q23-<1-v)q21/<4«p) (11.6) 61+2q1q2<1-4q12)521/<2«p2> (ac)12.1-c<1+u>I<1-8q12q22>62+2q1q2(1-4q12)fi11/(2xp2) 2 (ac)22.1-G(1+v)[(1-891 q221fi1+2q1q2<1-4q22)621/(2«p2> (°°>11.2‘(°°)12.1 (°°)12.2'(°°)22.1 1<1+aq22-8q2“>62+2q192<1-4q22>611/<2«p2> (11.7) where - 2 - 2 1/2 p - [(xl-xl) +(x2-x2) l ql-(xl'i1)/p qZ-(xz'i2)/P 51 and 62 are the components of the outward-directed normal unit vector to the boundary at i. 11.2 PIECEWISE LINEAR ELEMENTS Equation (11.1) can be solved numerically if the boundary B is approximated by N straight segments, as shown in Figure 11.2. For this model, equation (11.1) can.be written as N (n) (n) (n) - . - aij(x )uj(x )+n§1 [(uc)1 J(x ,x)uj(x)ds m N (>- - - - 2 (us)1 J(x “ ,x)tJ(x)ds (11.3) m-l ' m (n) where x is the location of boundary node n. The displacements and tractions in each segment m can be approximated using shape functions so that (m-l) (m) uJ(R)-uj N1(€)+uj N2(€) (11.9) - m tj(x) tJ (11.10) where N1(€)-0.5(1-£) N2(E)-O.5(l+£) i-N1(e)x(“‘1)+N2(e)x(m) ds-O.5(sm-sm_1)dE (11.11) ds-O.5Asmd£ (11.12) where £ is a local coordinate for the segment m with value -1 at node m-l, value 0 at the center of the segment, and, value 1 at node m. Note that u(m) is the displacement at node m whereas t m J J is the traction on the element m. Note that the order of differentiation of t1(x) in the interval is less than that of u3(x). This model allows for discontinuities of t on the boundary. J If equations (11.9-12) are substitued into equation (11.8) the following is obtanied N (n) (n) (n) 2013 U5 +nflAsm[J (uc)1 J(x ,€)N1(£)d€uj (m-l) (n) (m) N (n) m + (ac)1 Jo: .mzameuj 1- 2 (um, Jo: .meAsmtJ C 1 O m m (11.13) or N (n) (n) (m n) (m-l) (m n) (m) Zaij Nuj +§31IA1'J u.J + B1.) uj ] (m.n) ‘ - 2 c F m-l 1'3 j m (11.14) where (n) (m.n) 1.3 "sm (“°)1.J(x .€)N1(£)d£ (11.15) B (m.n)_As (n) 1.3 m (“¢)1.J(x .€)N2(£>d5 (11.16) th--sa'---. 10 (m.n) (n) (31.1 -J (11115.1(): .E)d£ (11.17) III §“'- Asntju (11.18) Note that the functions (uc)1ojand (uR)1 j are singular when the point of coordinate e approaches the node n. This is the case when the element m ends or begins with the node n, i.e. when m-n or m-n+l (n-m or nFm-l). In this presentation, the terms involving logarithmic functions are evaluated analytically and the Cauchy principal-value integrals involving l/p terms are evaluated numerically. One can also evaluate the logarithmic terms numerically provided that a proper table of integrals for numerical solution is used. Note that it is not necessary to mathematically calculate aij since it only contributes to the diagonal of (uc)1.J. The value of this diagonal term can be determined quite easily using rigid-body considerations. This will be shown later. All nonsingular integrals can be evaluated by numerical integration using Guess-Legendre quadrature. If it is defined that( see Figure 11.3): ;(“)- 0.5 [x(")+x("'1)] b1" x1 'xi 2 Rpaiai-2a1bi£+bib1£ ll then substitution into equations (11.15-18) gives expressions that are easy to program. 12 Figure 11.1 Plane boundary value problem 13 Figure 11.2 Boundary discretization 14 .' by ‘In-II A (m) Figure 11.3 Definition of a1, b;, x CHAPTER III APPLICATION OF THE BOUNDARY ELEMENT METHOD TO ORTHOTROPIC MATERIALS 111.1 INTRODUCTION For an orthotropic material for which the xl-x2 axes are alligned with the principal material directions the strain-stress laws for plane stress reduce to 1 u x e - -—- a -‘-- a 11 E 11 (E\ 22 x \UM u l l x c - - -- o + -- o 22 E 11 E 22 x Y 1 e - -- a (111.1) 12 2E 12 where ”x is Poisson's ratio and the constants Ex(Young's modulus in the x1 direction), Ey(Young's modulus in the x2 direction), and Es are the longitudinal, transverse, and axial shear moduli of elasticity, respectively. We shall select the x1 and x2 axes such that Ex > Ey' In general, orthotropic materials should satisfy a fourth order polynomial equation in which the coefficients are in terms of the aforementioned elastic constants, [4]: 15 16 1 a 1 ”x 2 1 ——u+2(—-—)u+—-O (111.2) E 2E E E x s x y The roots of this equation can be expressed as a E Ex 2 " + " 2 n-—-vx_ (—-vx)- 215 21-: E s s y (111.3) where the nature of the roots depends on the quantity under the radical sign. Note that p2 is real if (111.4) For isotropic materials, the equality of (111.4) is satisfied. Furthermore, it appears that the inequality (111.4) is satisfied for materials of practical interest. Table 111.1 lists many materials that are widely used in industry, all of which satisfy inequality (111.4). 111.2 THE INFLUENCE FUNCTIONS FOR ORTHOTROPIC MATERIALS To find the influence functions or the Green's functions, a two-dimensional infinite orthotropic plane is considered [6]. The equilibrium equation, the compatibility equation, and the boundary conditions at infinity are satisfied using the technique 17 of Fourier Transforms. The boundary conditions at infinity are that the stresses and their first derivatives go to zero. Three forms of the Green's function have been found [6]. These three forms of the Green's function correspond to the nature of the roots of equation (111.2). However, for practical purposes, the materials for which the material constants satisfy inequality (111.4) will be considered here. Let us define the distance between the field and the source points in terms of Cartesian coordinate as: where x1 and x2 are the coordinate of the field point and i and 1 i2 are the coordinate of the source point. Then, the influence functions (uR)i j(x51) and (uc) (x,x) have the form [6]: 1.3 (monoch- 1.2- D11 7 12 a D (uR)2.1(X.x)- D 0 22 6 32 T5 41 T7 * D42 8 (“°)1.1"311 T1 n1’312 T2 n1'351 T3 n2'552 Ta n2 ‘“°)1.2"Es1 Ta n1'352 Ta n1'331 T1 n2’532 T2 n2 (“°’2.1"Ez1 T3 n1‘322 Ta n1'361 T1 n2'362 T2 n2 (“°)2.2"561 T1 n1'362 72 n1'341 T3 n2'542 Ta n2 (111'6) (uR)2.2(x.i)- 1 (111.5) 18 and “l and "2 are the roots of equation (111.3), pl 2 p2. Also {'1 11 ['1 m 22 31 32 in m 41 m 42 51 52 61 62 11 12 22 32 41 - (a1 + d3 - a) / (a x as) - (d1 - d3 - a) / (a s d6) d4 / (4 I) - (2d2 - d1 d4) / (a s as d6) (1 + (an / d3)) / (4 w d5) (1 - (64 / d3)) / (4 w 65) -1 / (4 I) (a1 - 2d“) / (a 8 d5 d6) Ea1 'Eaz '63 E31 d3 232 (111.7) E11 ”x E31 s s x x E12 ”x E32 2 s x 31 E22 ” E42 --——-+ s s X X D22 d n 19 D42 - -d3 D12 (111.8) where E x d - 2 ( - u ) 1 2E‘ x E d Y 2 E x d3 - d4 - -vx d5"‘1*“2 ul-pz case 1 d6 - (111.9) 1 case 11 the functions T1 through T8 depend on the relationship among the elastic constants. Two cases are considered. In case 1: Ex Ex - vx )2 > 2E E 3 Y ( and the roots of equation (111.2) can be expressed as E Ex 2 E I‘ - x " U + - u -——-x— 1,2 x - 23 x E 2Es s y then 0-1 I v-l I 20 log (r1) + log (r2) log (r1) - log (r2) where and 21 2 r x T - l r 2 1 2 -2p1rxry T - 2 r 4 l Zulry T - 3 r 2 l 3 rx ry pl ry T - 4 r 4 l r r x Y T - 6 r 2 1 T7 - 2 log ( r1) 2 r T Y 8 2 22 III . 3 PROBLEM FORMUIATION To determine the complete set of boundary information , i.e., the displacements and the tracions at all chosen boundary points, one ‘must solve the numerical form of Somigliana's identity as formulated in Chapter 11. The formulation of the problem for orthotropic materials is exactly the same as in the isotropic case except that the influence functions (uc)1.j(x,x) and (uR)1 3(x,i) are different. 111.4 COMPARISON OF THE ISOTROPIC AND ORTHOTROPIC INFLUENCE FUNCTIONS The isotropic influence functions listed in chapter 11 were compared to the orthotropic influence functions presented in this chapter. This comparison was made by numerically calculating the values of those equations for different field and source points. The computer program TEST in Appendix B performs these calculations. See Figure 111.1 for the flow chart of the the progrm TEST. To better understand the differences and characteristics of these influence functions, the following examples were examined. Field points and source points were selected as shown in Figure 111.2. The values of the influence functions in chapter I for the isotropic case were calculated based upon the Poisson's ratio and shear modulus of 0.25 and 1.25, respectively. Tables 111.2-7 exhibit the results for each of the source points. These 23 tables also list the results for the influence functions of cases 1 and 2 assuming near-isotropic and isotropic conditions, respectively. The *values calculated using the isotropic influence functions of chapter I are identical to the case 2 influence functions. This is expected since isotropic materials belong to the case 2 of the orthotropic influence functions. It appears that the case 1 is also suitable for isotropic materials provided that the material properties selected are slightly anisotropic to satisfy the inequality of equation 111.4. Note that the (uR)1.1 of orthotropic influence functions are off by a constant. This constant represents rigid body motion and has no effect on the boundary element formulation. 111.5 PIECEWISE LINEAR ELEMENTS Somigliana's identity can be solved numerically for the case of orthotropic materials just as in the isotropic case by approximating the boundary B with N straight segments, as shown in Figure 11.2. The displacemnts and tractions in each segment m are approximated using shape functions so that - (In-1) (In) 11301) - uj 511(6) + “3 N2“) - m tJ(x) - tJ where N1(e> - 0.5 <1 - e) 24 N2(£) - 0.5 (l + f) i - N1(e)x("1) + s2(e)x(”) ds -0.5(sn - sn_1)d£ d3- 0.5 As nae (111.10) and E is a local coordinate for the segment m with value -1 at node m-l, value 0 at the center of the segment, and ‘value 1 at node m. By substituting equations (111.10) into Somigliana's identity the following is obtained N (n) (n) x(n) (m~1) 2&1] u.J +n§1 Asm [J (uc)1.J(x ,E) N 1(£)d£uj +J (uc>1.3+1°s<1 i e) d: I. ‘1 +£52b2 2(a1-b15) +3 b ( )(1 i f) 66 31 1 ( >u$e>a l. I 2pl(a2-bze>2 l 1 (111.11e) and A2 1 ° and B2 1(m.n) have the same expressions as . m.n 1,2 and 31.2( > where 551.552.E31and E32 are replaced by E21,E22,E61 and E62 respectivly. 2 (‘2'b2£) 11210g(r1)+D12( r 2 )ide 1 (m.n) c1.1 ' {D .-————.»‘ l (‘1’b1£)(‘2’b2€) ' ”22‘ 2 ’d5 ‘1 l (‘1'b1£)(.2°b2€) D I ldf 32 2 ‘1 (‘2'b2€) 2log(r1)+Da2( r 2 )Idf l (111.11f) C (m.n)_ 2.2 ‘D 41 31 111.8) PREWDE TO COMPUTER PROGRAMMING Equations (111.11) can be written in matrix form as [uc]{u)-[uR]IF) (111.12) This is a system of equations relating nodal displacements to resultant segment forces. In order to solve a well-posed elasticity problem, it is necessary to re-pose this system of equations in terms of nodal forces. Therefore, a transformation (Fl-[rllFI relating the vector of nodal forces (F) to the vector of segment forces {F}, is introduced into the system (111.12). The simplest physical interpretation of the transformation is to replace the segment forces by nodal forces equal to the average of the segment forces adjacent to each node, or (n) ‘ (n) ‘ (n+1) F1 - 0.5 [F1 +F1 ] The form of [r] for this transformation is I I 0 0 0 0 ..... 0 0 I 1 0 0 0 ..... 0 0 0 1 1 0 O ..... 0 [r] - 0.5 0 0 0 1 1 O ..... 0 32 where 1 is a 2x2 identity matrix. For an odd number of nodes, the inverse of [r] is 1 -1 1 -1 ...... I I 1 -I 1 ..... -1 .1 -I I 1 -1 ...... 1 [r] - I -I I 1 ..... -1 L-I I -1 1 ...... 1‘ and (111.12) becomes lucitui-[unllrl'ltri (111.13) It should be noted that for an even number of nodes, [r] has no inverse. A perturbation is introduced into the system of equations (111.13) in order to enforce the equilibrium conditions, i.e. N 2 11(1)-0 1-1 N 1-1 N (1) (1) Z (‘1 F2 -x (1)F1(1))-0 (111.14) 1-1 2 or in matrix form [QllFi-O (111.15) where 33 1 1 0 1 0 ' [0] _ 0 1 . 1 _ _x2(1) x1(1) ,x2(2) x1(2) ,x2(N) x1(N) Coupling equations (111.14) and (111.15): uc uni 0‘ “’ where [uR*] - [uR] [r1'1. 111.9 PLANE STRESS, ORTHOTROPIC BOUNDARY ELEMENT PROGRAM The boundary element program described in this section can be used to solve plane stress, orthotropic elastostatics problems. Linear interpolation is used for the displacements and constant segment tractions are assumed. The logic of the program for calculations for orthotropic materials belonging to case 1 and case 2 is the same. The only difference is that different equations are used to solve for material constants E13 and D11. The flow chart for the program is shown in Figure (111.3). In the first part, the input data is read. Then the weights and points for the numerical integration are assigned to the array W(L) and R(L). Next, E13 and Dij are calculated. In the fourth part of the program, the entries of the matrices [uc] and [uR] are calculated based upon the relationship among the material 34 constants. In part five, the diagonal of the matrix [uc] is calculated. In the sixth part of the program, the operation [uR] [Fl-1 is performed, and in part seven, the system of equations are rearranged in order to have all the unknowns on the same side of the system of equations. Once the system of equations is solved by the Gauss elimination method, the unknowns are printed in part nine of the program. Finally, the stresses and displacements are calculated for the field points. 111.9.1 INPUTTING THE PROBLEM DATA The order and format in which the variables and arrays must be submitted to the program are as follows: TITLE -The title of the problem. This must be written in column 1 through 80. N -The number of nodes on the boundary. It must be an odd number. Enter in free format. EX, EY, ES, PRX -Material properties. Enter in free format. X(I), Y(1) -Coordinates of nodes 1 through N, entered counter-clockwise in free format. J, K -Nodes, J, and corresponding directions, K, at which displacements are specified (zero and non-zero). Enter in free format and end by inputting 0, 0. 35 J,K,F(2*J+R+2) -Nodes, J, and corresponding directions, R, at which non-zero forces are specified, and specified values F. Enter in free format and end by inputting 0,0,0. 111.9.2 CALCUIATION OF MATERIAL PROPERTIES Eij and C1.1 The program calculates E and C based on the criterion ij ij of part three and equations (111.7-8). 111.9.3 CALCULATION OF COEFFICIENT MATRICES A more detail flowchart for part four of the program is shown in Figure 111.4. The order of the loops was chosen in order to avoid repeating operations. For each value of loop J, associated with the element J, parts of the (2x2) singular submatrices, shown in Figure 111.5, are calculated. Note that only singular, logarithmic terms are evaluted in this loop. This type of function appears only in the (uR)1 J influence functions. In the other loop, I, all of the nonsingular integrals and non-logarithmic singular integrals are evaluated numerically. The positions in the matrices are illustrated in Figure 111.6. The functions for the numerical integration are calculated using equations (III.lla-f). The diagonal terms of the matrix (uc)1 J are calculated after completion of the loops J and 1. Each diagonal term is 36 calculated by algebraically summing row elements in the same row as the desired diagonal term. To show this, consider the equation (111.12) [no] {uI-[uR] (F) If rigid body displacement is applied to the body so that (1) (2) <3) (4).. (n) ‘11 WI .111 '11 (1>_u (2>_u (3)_u (4).. (n) ‘12 2 2 2 then there are no stresses. Therefore, (F) is a null vector that N uc(2n-1)(2n-1)' ‘2 “°(2n-1)(2m-1) m-l min N “°(2n-1)(2n)' '2 uc(2n-1)(2m) u°(2n)(2n-1)' '2 u°(2n)(2m-1) “°<2n><2n)' '2 “°<2n>(2m)' m-l mun 37 111.9.4 TRANSFORMATION OF COEFFICIENT MATRIX uR This part of the program performs the operation * -1 [an -[uR] I r] a The most efficient way to obtain the matrix [uR] is not to create [r]'1 and postmultiply by [uR], but to modify [uR] itself. The alogrithm for this modification is better understood by examining the example of a 6 x 6 matrix: - s * a . 1 . [“311] [“312] [“313] l “‘11 “‘12 “‘13 I ’1 I l uR * uR * uR * uR uR uR 1 1 1 I 21] I 22] l 23] ' 12 22 23 ' uR * uR * uR * uR uR uR 1 1 1 - I 31] I 32] l 33] - L 31 32 33 --' - in which the entries [uR21]* are * [“Rzil ' (“R211 * [“R22] ' [“323] [“R22]*' ’[“321] I [“R22] I [“323] 111112213 -11uR211 + 11111221 - [uR23li+2[uR22] * 22] In general it can be shown that [uRk1]*-[uRk1]+[uRk2]-[uR]k3]+[uR]ka-...[uRkn] (111.17) or [uR22]*- -[uR21] +2[uR and [uRk1]*--[uRk(1-1)]*+2[uRk1] , i-2,...,N. (111.18) 38 The flowchart for this modification is illustrated in Figure 111.7. In the first loop, the first two columns of [uR]* are obtained using equation (111.17). In the second loop, the other entries of [uR]* are generated using equation (111.18). 111.9.5 CONSOLIDATION OF RNOUN AND UNRNOUNS BOUNDARY VALUES This part of the program consists of a reorganization of the system of equations (111.16) so that all unknowns go to the left hand side and all knowns go to the right hand side of the equations. This procedure is illustrated with the flow chart of Figure 111.8. The array NTBC(K,I) which specifies the boundary condition at node R, direction 1, is used to decide when to interchange columns, i.e. if NTBC(K,I)-0, there is a specified force, and no interchange is necessary. The final vector of knowns is multiplied by the matrix of coefficients to produce the right-hand side of the final system of equations. The final system of equations is then solved by Guess elimination procedure. 111.9.6 CALCULATIONS OF STRESSES AND DISPLACEMENTS AT THE ELEMENTS AND NODES In this part of the program, the stresses at each element and displacements at each nodes are evaluated. 39 lculation of uh influence tune. of isotropy 1 , Calculation of ac influence (has. of isotropy Cal. latersal constants for the first case of the orthotropic Cal. o! uh influ. tune. of the 1st case of the orth. Cal of us in lu. tune. of the 2nd case of the orth. k Cal. asterial constants for the second case of the orthotropic Cal. of uh influ. tune. of the 1st case of the orth Cal. of us influ. tune. of the and case of the orth Figure 111.1 Flowchart of the program TEST 40 (0e0'3e0) . 0 (1.4.1.4) 3.0 O O)‘' fie (3.55.3.55) (5.0.0.0) (I . flit-'31 0 field point O * source point Figure 111.2 Field and source points selected for the example problem Points for numerical Integration Calculate Materia l‘EI’ Constants a“ and D. _ .1 0 Calculate tntries of uh and us 1 Calculate the Diagonal of us 3 1_ L I Rearrange 0 Equation : .J ’ Solve the System of Equation Output Forces, Displacements, and Stresses Figure 111.3 Flowchart of the program ORTHO.CAS£1 and ORTHO.CASEZ 42 Loop on columns 3 I 1, NM Calculate singular LDOD OD TONS 1 m 1,NN j/I I c 3, 3-1 Loop on points of Integration l. 8 :1, LL l Calculate nonsingular (:EONTINUEj:> .1. Figure 111.4 Flowchart for the calculations of the coefficient matrix ‘13 L J j -_|\ - \ \ J \\\ [0d \\ \A ll [I 1 I: I a . \ \ .. b. . .1 -—k 1 \ \ \ J \ \ [0a] \ \N'Tél \ \ L. 5 Figure 111.5 Singular contribution in [uc]: [uR] [06] [U a] ‘11 /’|| J-I JJ J'I l. l7/JI l V/J//// Figure 111.6 Non-singular contribution in [uc],[uR] 45 fr Loop on 1‘\\ 1.1g" ] l Calculate [UR]: l LDDD DD K Ks2,N l Calculate [UR]; l CONTINUE E‘\— C f .L Figure 111.7 Flowchart for modification of uR 46 Loop on nodes “.1." Loop on degrees of freedom 1.1.2 [KK.NTBC(K,1) | KKsO KKr} Interchange columns [2'(K-1)¢IJ of [0:3 and [us] with signs changed ’1 W CONTINUE (:: j .411 Multiply matrix at RHS bl vector of DOWNS a ‘DVC in SAV(1) ,L Figure 111.8 Flowchart for rearranging equations into final form 47 TABLE 111.1 ”I! ”O5"! MATH!” RIDELY USED IN 1mm MATERIAL D! CY 05 POISSON'S RATIO (0P4) (094) (0P4) IN x-omtcnou Mina/Don 230.0 21.0 7.0 0.20 ”OI/EPOXY 210.0 10.0 4.0 0.25 sonar/Dom! TYPE 0(0/5505 204.0 10.5 5.0 0.23 ULTRA-LIGHT abouws 200.0 0.2 4.0 0.20 GRAPHITE/EPOXY mos-1.000100 GRAPHITE/EPOXY 220.0 0.0 4.0 0.20 GRAPHITE/EPOXY mt 7300/0200 101.0 10.3 7.17 0.20 GRAPHITE/EPOXY mt 40/3001 130.0 0.00 7.17 0.30 HIGH-STRENGTH GRAPHITE/EPOXY 140.0 10.0 4.0 0.20 KELVAR 40 MID/EPOXY 70.0 0.0 2.1 0.34 S-GLASS/EPOXY 00.0 10.0 7.0 0.20 E-GLASS/EPOXY 40.0 12.0 0.0 0.20 GLASS/EPOXY m: 0001va 30.0 0.3 4.0 0.20 1002 oousn'ruon PLIES 000 30.0 10.3 4.0 0.30 005 10.5 10.5 1.2 0.20 070 33.0 17 7 0.7 0.30 070 42.1 1 10. 0.0 0.33 ,3. a 118 TAILI "3.2 1'11! V41.” 0' THE ”I'LL”! MT!“ W M 71‘ “300 001111 1.0.00 YI0.00 :5 £01 f,’.“:'01!0:.0“....:-U 11...-.. §§§§§§§§§§§§§§§2 r.,.;‘ r999§§§§§§§999r9 2230052005505050 I”. O”. 0 O”. O C x ifixixifiiiisiiiii auuuuu-o-uuuuud. £§§§§§§§§§§§§§§i ruuyyuaoryuuuuro §8§§§§§§£!§!!!§!‘ O I I I I 300.004.4090 ‘Odsflyflflgfld. 5120235301122113 W IWIC "tum: WIN W11-0.44040 W12-0.00400 W21-0.00400 urn—0.30000 WHO-0.30332 W12-0.04200 W21-0.04200 W22-0.20100 W11-0.10010 W12—0.02301 1321—0323" W22-0.07042 WHO-0.10074 1312—0.01231 W21-0.01231 Luz—0.00700 WHO-0.10001 W12- 0.00000 W21- 0.00000 W11-0.10074 W12- 0.01231 W21- 0.01231 W11-0.10010 W12- 0.02301 W21- 0.02301 W1 1—0.30332 W12- 0.04200 W21- 0.04200 WHO-0.44040 W12- 0.00400 W21- 0.00400 1311-0.00000 W12- 0.02000 1321- 0. 1.31 1—0.00000 1312- 0.00020 1321- 0. WHO-0.00003 1312- 0.00311 W21- 0. WHO-0.00003 1312- 0.00000 1321- 0. W11-0.00003 1312-0.00311 1321—0. W1 10-0.00000 1312-0.00020 1321—0 W 040! 1 ”rune: 71.067101 W11-0.07000 W11—0.40032 1311—0.31000 1311-0.31021 W11-0.31000 W11—0.31021 W11-0.31050 W11-0.40032 W11—0.07000 1311—0.03003 W11—0.00000 W11-0.00007 W12-0.00402 WWII-0.04200 W12-0.02302 W12-0.01232 1312- 0.00000 W12- 0.01232 1312- 0.02302 1312- 0.04200 1312- 0.00402 1312- 0.02007 131h 0.00021 1312- 0.00311 WHO-0.00000 W12- 0.00000 W11—0.00007 1312-0.00311 W11-0.00000 W12-0.00021 W1 1-0.03003 1312—0.02007 1321-4. 1321—0. 1321—0. 1321—0. W21!- 0. .01232 1321- 0. W21- 0. W21- 0. W21- 0. W21- 0. 1321- 0. W21- 0. W21o-0. W21-0. 1321—0. 1321- 0 W CASE 2 IWUDCE WIN W11-0.07374 1312-0.00400 W210-0. W11—0.47700 1312—0.04200 W21—0. W11-0.31403 1312—0.02301 W21—0. W11-0.31400 W12-0.01231 W21—0. WHO-0.31300 W12- 0.00000 W21- 0. 1311-0.31400 W1b 0.01231 1321- 0. WHO-0.31403 1312- 0.02301 1321- 0. 02000 00020 0031 1 00000 0031 1 .00020 W11-0.00000 W12-0.02000 W21“. 02000 00402 04200 02302 01232 00000 02302 04200 00402 02007 00021 0031 1 0031 1 00021 02007 00400 04200 02301 01231 00000 01231 02301 W11-0.47700 W12- 0.04200 1321- 0.04200 W11-0.07374 1312- 0.00400 W21- 0.00400 W22I-0.30000 W11-0.03204 1312- 0.02000 W21- 0.02000 W22-0.30002 W11-0.00320 W12- 0.00020 W21- 0.00020 W22-0.44014 W11—0.00317 W1fi 0.00311 W21- 0.00311 W22-0.44400 W11-0.00317 W12- 0.00000 W21- 0.00000 W22—0.444-40 WHO-0.00317 W12-0.00311 13210-030311 W22-0.44400 W1 10-0.00320 W12-0.00020 W21—0.00020 W22—0.44014 W1 1-0.03204 W12-0.02000 W21-0.02000 W22-0.30002 W22-0.00027 W22-0.00700 W22-0.07042 W22-0.20100 W22—0.30000 W22-0.30002 W22-0.44014 W22-0.44400 W22-0.44440 W22-0.44400 W22-0.44014 W22-0.30002 NH . 30170 1322—0 . 20170 WZH . 07400 1322—0 . 00703 1322—0 . 00440 W .00703 W22-0 .07400 WZH . 20170 1322—0 . 30170 1322—0. 30000 WZH . 44000 W2“ . 44000 W22-0 . 44-400 WZH . 44000 1322—0 . 44000 W2“ .00000 11122—037042 Luz—0.20100 119 7401.! ”1.3 THE VALUES 0' 1'11! 115m! WIN W 'W 3110 m0 001117 X-0.00 Y-0.00 : E ,3, 3 ”0111070 r§r§§999r9 r999§§§§§“-99‘~ §§§§§§§§§§§ §§§§§§§§§§§0§§§!” 0000000 11 ......“ fiiiiii‘ auuuuuao-ouuuuud 1 a... . -3.000 .200 -3.000 0.000 -3.000 0.200 -3.000 0.400 -3.000 1.400 -1. r999§§§§§10999r9 §§§§§§§§§§§§§§§§ .0“ 13 100100010 31171100! 01067101 WHO-0.00027 W12- 0.00000 W21- 0.00000 ”H.100” WHO-0.20100 W12-0.04200 W21-0.04200 W22-0.30332 W1 1—0.37000 W12-0.00000 W21-0.00000 WZH.42002 W11-0.30470 W12-0.00000 W21-0.00000 W22-0.43027 W1 1-0.30000 W12-0.004-00 W21-0.00400 W22-0.44040 W11-0.30000 W12-0.00302 W21-0.00302 W22—0.40023 W11-0.40300 W1H.00270 W21-0.00270 W2H.40070 W11-0.30002 W12-0.02000 W21—0.02000 1322-0.00000 W11-0.44440 W12- 0.00000 W21- 0.00000 W22-0.00003 W11-0.30002 W12- 0.02000 W21- 0.02000 131 1-0.40300 W12- 0.00270 1321- 0.00270 1311—0.30000 W12- 0.00302 W21- 0.00302 W11-0.30000 W12- 0.00400 W21- 0.00400 W11—0.30470 W12- 0.00000 W21- 0.00000 W11-0.37000 W12- 0.00000 W21- 0.00000 W11-0.20100 131% 0.04200 W21- 0.04200 W 050 1 "I'm mum U311-0. 1311-0. 1311—0. 1311—0. UR11-0. 13110—0. WHO-0. WHO-O. 1311—0. W11-0. W11-0. WHO-0. 1311—0. 1311—0. 1311-0 1311—0 WHO-0. W1 10-0. 13110-0. 1311—0. 13110-0. 1311—0. W11-0. 1311—. 1.31 1-0. 1311—. W11—0. 1311—0. 1311—0 W11-0. W11-0. W11-0 30702 00030 0114-4 01704 02304 02073 02172 07073 02172 02073 02304 01704 01144 .00030 W12-0. W12-0. W12-0. W12-0. W12-0. W12-0. W12—0. W12- 0. W1b 0. W12- 0. W12- 0. W12- 0. W12- 0. W12- 0. 10024 1312- 0.00000 W21- 0.00000 04200 1321-0.04200 00000 13210—0.00000 00000 W21-0.00000 00402 W21-0.00402 00370 W21-0.00370 00270 W21-0.00270 02007 W21-0.02007 00000 1321- 0.00000 02007 W21- 0.02007 00270 1321- 0.00270 00370 1321- 0.00370 00402 1321- 0.00402 00000 1321- 0.00000 00000 1321- 0.00000 .30702 W12- 0.04200 W21- 0.04200 W CASE 2 "WW! 01087101 10001 W12- 0.00000 W21- 0.00000 30000 1312-0.04200 W21-0.04200 00303 W12|-0.00000 W21-0.00000 00012 W12-0.00000 01023 W12-0.00400 02133 W12—0.00302 02743 W12-0.00270 .01000 W12-0.02000 00003 W12- 0.00000 .01000 W12- 0.02000 02743 W12- 0.00270 02133 W1? 0.00302 .01023 W12- 0.00400 00012 W12- 0.00000 00303 W12- 0.00000 W21-0.00000 W21-0.00400 W21I-0.00302 W21-0.00270 W21-0.02000 13.21- 0.00000 W21- 0.02000 W21- 0.00270 1321- 0.00302 1321- 0.00400 1321- 0.00000 W22-0 W22-0 W22I—0 W22-0. .43027 W22-0. W22-0. 1322-0 W22-0 . . 30440 W22-0 . . 44007 1322—0 . W22-0 . . 47020 W22-0 . . 07073 W22-0 . W22-0 . 1322—0 . W22-0 . . 44007 W22-0 . . 30440 W22-0 W22-0 1322-4 1322—0 1322—0 00000 40070 40023 44040 42002 30332 10024 43010 40004 40072 0101 0 0101 0 47020 40072 40004 43010 .10001 .30332 .42002 .43027 .44040 .40023 .40070 .00000 .00003 .00000 UR22-0. .40023 .44040 40070 43027 1.021- 0.00000 1322—0242002 .30000 W12- 0.04200 1321- 0.04200 W22-0.30332 II ...“... 50 140a 111.4 7110 VALUES 0' THE IWUJDCE mum W 3 n 5 § 3 H r9990‘1&§‘§999r9 0250355335305000" ...00999r? §§§§§§§§§§§§§§§§ r999§§ r999§§éél§§999r9 3553520030505033 QL-LOdUUUUH-fi. I I 110100410400 1 d 0 I I I I I I I 0 0 I 0 O O O rOuOuOuouOu.‘ 8§88888 gzxgxxs 0 00000 00 000 I .0 § 0333301053333053‘ P???9?9?P ss§xssx§§ CO 0000 . UUUUU noduuuuu-oo §§§§§§§§§ M 711‘ M! 001111 N.“ Y-3.00 W 100100010 [WWI WIN W11—0.34000 W12- 0.01001 1321- 0.01001 W22—0.22030 W11—0.24203 W12- 0.00217 W21- 0.00217 W22—0.24203 W11—0.10732 W12- 0.02107 1321- 0.02107 W22-0.31430 W11—0.21327 W12- 0.01000 W21- 0.01000 1322-0.33100 W11—0.22030 W12- 0.01001 W21- 0.01001 W22-0.34000 W11—0.24270 W12- 0.01700 W21- 0.01700 W22—0.30100 W11—0.20043 W12- 0.01000 W21- 0.01000 W2H.37004 W11—0.30004 W12- 0.04043 W21- 0.04043 W22-0.44141 W11—0.40324 W12- 0.00200 W21- 0.00200 W22-0.02112 W11-0.47014 W12- 0.00217 1321- 0.00217 1322-0.47014 W11—0.02330 W12- 0.00400 1321- 0.00400 W22—0.40037 W11—0.02220 W12- 0.00301 W21- 0.00301 W22-0.40020 W11—0.02112 W12- 0.00200 W21- 0.00200 W22-0.40324 W11—0.02012 W12- 0.00041 13.21- 0.00041 W22-0.44730 W11-0.01022 W12- 0.04000 W21- 0.04000 W22-0.44100 W11—-0.44141 W12- 0.04043 W21- 0.04043 W22-0.30004 W 04001 11171.“! WHO! W11—0 W11—0. 1311—0. W11—0. W11—0. W11—0. 1311—0. W11—0 1311—0. W11—0. . 47300 .40200 UR12- 0.01002 UR12- 0.00211 UR12- 0.02100 UR12- 0.01000 UR12- 0.01002 UR12- 0.01700 UR12- 0.01000 UR12- 0.04043 UR12- 0.00200 UR12- 0.00211 30040 32201 33001 30307 30030 30200 sum 50730 1321- W21- W21- W21- W21- 1321- W21- W21- W21- 1321- .01002 W22-0.22013 .00211 W22-0.24204 .02100 W22-0.31034 .01000 W22-0.33210 .01002 W22—-0.34000 .01700 W22-0.30300 .01000 W22-0.37720 .04043 W22-0.4-4200 .00200 1322-0.02201 .00211 W22—0.47102 W11—0. W11-0. W11—0. W11—0.04700 13.21- 0.00400 W22—0.40044 W21- 0.00300 W22-0.40027 1321- 0.00200 W22-0.40420 W12- 0.00030 W21- 0.00030 W22-0.44020 W11-0.04070 W12- 0.04000 1321- 0.04000 W22-0.44244 W11—-0.00070 W12- 0.04043 W21- 0.04043 W22-0.30704 W CASE 2 INFLUDCE 010$le W11—0.47124 W12- 0.01001 W21- 0 W11—0.30037 W12- 0.00217 W21- 0 0 0 00002 04074 04000 W12- 0.00400 W12- 0.00300 W12- 0.00200 .01001 W22-0.22030 .00217 W22—0.24203 W11-0.32100 W12- 0.02107 1321- .02107 W22-0.31430 W11-0.33701 W12- 0.01000 W21- .01000 W22-0.33100 WHO-0.30273 W12- 0.01001 1321- 0.01001 W22-0.34000 W11-0.30700 W12- 0.01700 W21- .01700 1322—0.30100 W11—0.30077 W12- 0.01000 W21- .01000 W22-0.37004 W11—-0.40000 W12- 0.04043 1321- .04043 W22—0.44141 W11-0.07700 W12- 0.00200 00200 W22—0.02112 W11-0. 1311—0. W11—0. 1311—I0. W11—0. 1311—0. 00440 W12- 0.00217 04773 W12- 0.00400 04004 W1b 0.00301 04040 W12- 0.00200 044-40 W12- 0.00041 04300 W12- 0.04000 W21- W21- W21- um- 02 W21- 0. 0 0 0 1321- 0. 0 0 0 .00217 1322-0.47014 .00400 W22-0.40037 00301 WZH.40020 00200 W22-0.40324 00041 W22—0.44730 W21- 0.04000 W22-0.44100 W11—0.00070 W1? 0.04043 W21- 0.04043 W22-0.30004 51 TAILI 111.0 THI VALUE OF THI INFLUDCI WHO! W 'W “III MI 001111 1.0.00 Yd.” 03m ”"170 ”"1410 X Y 3.000 0.000 1.400 1.400 0.400 3.000 0.200 3.000 0.000 3.000 -0.200 3.000 -0.400 3.000 -1.400 1.400 -3.000 0.000 -1.400 -1.400 -0.400 -3.000 -0.200 -3.000 0.000 -3.000 0.200 -3.000 0.400 -3.000 1.400 -1.400 3.000 0.000 1.400 1.400 0.400 3.000 0.200 3.000 0.000 3.000 -0.200 3.000 -0.400 3.000 -1.400 1.400 -3.000 0.000 -1.400 -1.400 -0.400 -3.000 -0.200 -3.000 0.000 -3.000 0.200 -3.000 0.400 -3.000 1.400 -1.400 3.000 0.000 1.400 1.400 0.400 3.000 0.200 3.000 0.000 3.000 -0.200 3.000 -0.400 3.000 -1.400 1.400 -3.000 0.000 01.400 -1.400 -0.400 -3.000 -0.200 -3.000 0.000 -3.000 0.200 -3.000 0.400 ~3.000 1.400 -1.400 W 1mm "rum: W10! W11. 0.03702 W12—0.03320 W21—0.02000 1322- 0.00002 W1 1.4.02000 W12-0.01120 W21- 0.04300 W22-0.00070 W11—0.02333 W12-0.04140 W21. 0.00001 W11—0.00104 W12- 0.01300 W21. 0.02041 W11—0.00000 W1h 0.00000 W21. 0.00000 W11—0.00104 W12-0.01300 W210-0.02041 W1 1.4.00474 W12—0.02031 W21-434020 W11—0.00020 W1fi 0.02307 W21-0.03000 W11—0.01101 W1h 0.01020 W21—0.02071 W11. 0.01030 W12. 0.01000 W21. 0.00231 W11. 0.00000 W1b 0.01304 W21—0.01001 W11. 0.01404 W12- 0.00007 W21- 0.00101 W11- 0.01402 W12- 0.00000 W21- 0.00000 W11. 0.01404 W12--0.00007 W21—0.00101 W11. 0.01001 W12-0.00173 W21II-0.00322 W110 0.01407 W1H.02100 W21. 0.00417 W OASI 1 tarmac: 01057104 W11. 0.03702 W11—0.02001 W11—0.02320 W1 1-0.00000 W11—0.00003 W1 1-0.00000 W11—0.00400 WHO-0.00027 W11D-0.01170 W11. 0.01030 W11- 0.00007 W11. 0.01401 W110 0.01400 W11. 0.01401 W11. 0.01407 W11. 0.01404 W12-0.03317 W12-0.01120 W12D-0.04137 W12- 0.01300 W12- 0.00000 W1H.01300 W12-0.02030 W12. 0.02300 W12- 0.01023 W12- 0.01000 W12- 0.01301 W12- 0.00007 W12- 0.00000 W12—0.00007 W12-0.00173 W12-0.02100 W21—0.02000 W210 0.04347 W21. 0.00703 W21. 0.02000 W21. 0.00000 W21—0.02000 W21—0.04040 W21—0.03070 W21—0.02000 W210 0.00230 W21-0.01077 W210 0.00102 W21. 0.00000 W21—0.00102 W21o-0.00323 W210 0.00410 W CASE 2 [WWI mum W11. 0.03702 W12-0.03320 W21—0.02000 W1 1-0.02000 W11—0.02333 W11—0.00104 W11—0.00000 W11-0.00104 W11M.00474 W11—0.00020 W11—0.01 101 W11. 0.01030 W11. 0.00000 W11. 0.01404 W11- 0.01402 W11. 0.01404 W11. 0.01001 W11- 0.01407 W1H.01120 W12-0.04140 W12. 0.01300 W12- 0.00000 W12-0.01300 W12-0.02031 W12- 0.02307 W12- 0.01020 W12- 0.01000 W1b 0.01304 W12. 0.00007 W12- 0.00000 W12—0.00007 W12—0.00173 W1H.02100 W21- 0.04300 W21. 0.00001 W21!- 0.02041 W21- 0.00000 W210-0.02041 W21—-0.04020 W21—0.03000 W21—0.02071 W21. 0.00231 W21—0.01001 W21. 0.00101 W22-0.00007 W22—0.20412 W22U-0.20003 W22-0.20412 W22I-0.24132 W22-0.02240 WZH.02100 W22- 0.00073 W22- 0.03000 W22- 0.00400 W22. 0.00400 W22- 0.00400 W22- 0.00437 W22. 0.00320 W22- 0.00047 W22-0.00077 W22—0.00722 W22—-0.20000 W22-0.20000 W22-0.20000 W22-0.24202 W22-0.02240 W22-0.02104 W22- 0.00001 W22- 0.03000 W22- 0.00402 W22II 0.00400 W22II 0.00402 W22- 0.00400 W22. 0.00330 W22II 0.00002 W22I-0.00070 W22-0.00007 W22-0.20412 W22-0.20003 W22—0.20412 W22-0.24132 W22-0.02240 W22-0.02100 W22- 0.00073 W2fi 0 . 03000 W22- 0 . 00400 W210 0.00000 W22- 0.00400 W21—0.00101 W22- 0.00400 W21-0.00322 W22. 0.00437 W21. 0.00417 W22- 0.00320 11 .20.... -...1111111...-. 0‘OCO‘U 70L! HELD 0011170 MINA?! t Y .000 0.000 .400 1.400 .400 3.000 .200 3.000 .000 3.000 .200 3.000 .400 3.000 .400 1.400 .000 0.000 .400 -1.400 .400 -3.000 200 -3.000 000 -3.000 .200 -3.000 .400 -3.000 .400 -1.400 .000 0.000 .400 1.400 .400 3.000 .200 3.000 .000 3.000 .200 3.000 .400 3.000 .400 1.400 .000 0.000 .400 -1.400 .400 -3.000 .200 -3.000 .000 -3.000 .200 -3.000 .400 -3.000 400 -1.400 :0...“ r999‘§1§}§§999r9 §§§§§§§§§§§§§§§§ 11111 HUUUUJILCdUUBUU-fi. §§§§§§§§§2§§§§§§ 52 "1.0 THE VALUE 0' THE INPLUDCE 01057101 W M 1110 ME 00111? “.00 VI0.00 W 1mm 11171.1”! W101 WHO-0.17031 W12'-0.04402 W21- 0.04402 W22-0.03030 WHO-0.04104 W12. 0.04002 W21—0.01710 W22-0.01010 W1 1-0.03003 W1fi 0.03307 W21-0.00077 W22-0.01024 W12- 0.00114 W21-0.03403 W22. 0.02104 W1$I 0.00207 W21—0.03304 W22- 0.01003 W12. 0.00200 W21—0.03100 W22- 0.01021 W12. 0.00301 W21O-0.03017 W22- 0.01070 lama—0.00m W21—0.01007 W22. 0.01077 W12- 0.01123 W21—-0.01123 W22- 0.00003 W12b-0.00004 W21. 0.02103 W22- 0.01000 W12. 0.02031 W21. 0.02030 W22- 0.03447 W12—0.00200 W21. 0.03100 W22- 0.01021 W12-0.00207 W210 0.03304 W22- 0.01003 W12-0.00114 W21. 0.03403 W22- 0.02104 W12-0.00010 W21. 0.03031 W22- 0.02300 W12D-0.04000 W21. 0.00300 W22-0.02000 W CASE 1 INVLUDCE 01001101 W11. 0.03707 W11. 0.03030 W110 0.03470 W11. 0.03320 W11. 0.07174 W11. 0.04250 W11. 0.00120 W11. 0.00040 W11- 0.03470 W11- 0.03030 W110 0.03707 W11. 0.03004 W11—0.07202 WHO-0.17003 W1 1-0.04100 W11-0.03000 W11. 0.03703 W11. 0.03032 W11. 0.03477 W11. 0.03327 W11. 0.07103 W110 0.04273 W11. 0.00143 W11. 0.00043 W11. 0.03477 W11. 0.03032 W11. 0.03703 W11. 0.03000 WHO-0.07210 W1H.04400 W12-I 0.04007 W12- 0.03301 W12- 0.00110 W12- 0.00202 W12- 0.00204 W12I 0.00300 W12-0.00000 W12- 0.01120 W12—I0.00000 W12. 0.02033 W12-0.00204 W12—0.00202 W12—-0.00110 W1H.00000 W12-0.04007 W21. 0.04400 W21-0.01714 W21—0.00070 W21—0.03403 W21-0.03200 W21-0.03140 W21O-0.03000 W21—0.01004 W21—0.01120 W21. 0.02000 W21- 0.02023 W21. 0.03140 W21. 0.03200 W21. 0.03403 W21. 0.03022 W21. 0.00303 W CASE 2 INFLUDCE WNW W12-0.04402 W21. 0.04402 W12. 0.04002 W21-0.01710 W12- 0.03307 W21—0.00077 W12. 0.00114 W21—-0.03403 W12- 0.00207 W21-0.03304 W12- 0.00200 W21—0.03100 W12. 0.00301 W21-0.03017 W12-0.W001 W21—0.01007 W1? 0.01123 W210-0.01123 W1H.00004 W21. 0.02103 W1 1—0.17031 WHO-0.04104 WHO-0.03003 W11. 0.03707 W11. 0.03030 W110 0.03470 W11. 0.03320 W11. 0.07174 W110 0.04200 W110 0.00120 W11- 0.00040 W11- 0.03470 W11. 0.03030 W11. 0.03707 W11. 0.03004 W1 1.4.07202 W12- 0.02031 W12-0.00200 W12-0.00207 W12-0.00114 W12h-0.00010 W12-0.04000 W210 0.02030 W210 0.03100 W21- 0.03304 W21. 0.03403 W21- 0.03031 W21. 0.00300 W22-0.03020 W22—-0.01011 W22—0.01010 W22- 0.02100 W22- 0.01077 W22- 0.01010 W22I 0.01072 W22- 0.01072 W22- 0.00000 W22- 0.01004 W22- 0.03430 W22- 0.01010 W22- 0.01077 W22- 0.02100 W22- 0.02302 W22-0.02002 W22h-0.03030 W22-0.01010 W22-0.01024 W22- 0.02104 W22I- 0.01003 W22- 0.01021 W22. 0.01070 W22- 0.01077 W22- 0.00003 W22- 0.01000 W22. 0.03447 W22- 0.01021 W22- 0.01003 W22- 0.02104 W22- 0.02300 W22-0.02000 2.2.3. 53 TABLE ”1.7 THE VAU‘S OF THE INFLWCE WIN W M m ME POINT N.” M.” FIELD FCINTS enamx I“? W nature IWUDCE 0W1 101 3.000 0.000 W11. 0.02303 W1H.01400 W21- 0.03022‘W2b 0.00000 1.400 1.400 WHO-0.10440 Wat-0.00207 W21—0.00700 W22-0.10440 0.400 3.000 WHO-0.14040 W12—-0.00040 W21—0.03300 W22-0.03000 0.200 3.000 W11—0.02410 W12- 0.03100 W21-0.03773 W22-0.00010 0.000 3.000 WHO-0.02100 W12. 0.03027 W21—0.03040 W22-0.00040 -0.200 3.000 WHO-0.01040 W12- 0.02007 W21—0.03330 W22—0.00400 -0.400 3.000 WHO-0.01703 W1h 0.02777 W21I-0.03102 W22-0.00430 -1.400 1.400 W11. 0.04700 W1b 0.03340 W21O-0.00200 W22. 0.01010 .3.000 0.000 W11. 0.01203 W12- 0.02000 W21—0.01070 W22. 0.00020 -1.400 -1.400 W11. 0.04037 W12- 0.02722 W21. 0.02040 W22- 0.04037 -0.400 -3.000 W11. 0.02020 W12- 0.02700 W21. 0.01340 W22- 0.04704 -0.200 -3.000 W11. 0.02002 W12- 0.01107 W21. 0.02700 W22- 0.04010 0.000 -3.000 W11. 0.02470 W12. 0.01204 W21. 0.02730 W220 0.00030 0.200 -3.000 W11. 0.02443 W12- 0.01213 W21. 0.02001 W22. 0.00201 0.400 -3.000 W110 0.02407 W12- 0.01210 W21. 0.02030 W22- 0.00407 1.400 -1.400 W11. 0.00002 W12b-0.01700 W21. 0.02007 W22- 0.01240 W CASE 1 IHFLUDCE mum W11. 0.02200 W12-0.01402 W21. 0.03021 W22- 0.00110 W11-0.10410 W12—-0.00202 W21II-0.00773 W22-0.10410 WHO-0.14001 W12—0.00002 W21—0.03303 W22-0.03701 WHO-0.02420 W12- 0.03100 W21—0.03704 W22-0.00010 WHO-0.02172 W12- 0.03010 W21-0.03032 W22-0.00047 WHO-0.01000 W12- 0.02000 W21—0.03327 W22-0.00400 WHO-0.01700 W12- 0.02700 W21-0.03140 W22—0.00430 W11- 0.04771 W12- 0.03347 W21—0.00200 W22- 0.01012 W11. 0.01202 W12- 0.02002 W21—0.01007 W22- 0.00010 W11. 0.04020 W12- 0.02710 W21. 0.02042 W22- 0.04020 W110 0.02022 W12. 0.02700 W21. 0.01347 W22. 0.04700 W11. 0.02400 W12- 0.01103 W21. 0.02707 W22- 0.04010 W11. 0.02400 W12- 0.01200 W21. 0.02730 W22- 0.00037 0. 0. §§99¢r9 §§§§§§§2§§§§§§§§ §§§§§§§§§§§§§§§§ I U“ W11. 0.02437 W12. 0.01210 W21- 02001 W22I 0.00201 W11. 0.02401 W12. 0.01212 W21. 02040 W22- 0.00400 W11. 0.00000 W12-0.01703 W21. 0.02002 W22- 0.01240 W CASE 2 INFUDCE 01007101 W11. 0.02303 W12-0.01400 W21- 0.03022 W22- 0.00000 WHO-0.10440 W12-0.00207 W21—0.00700 WHO-0.10440 W1 1—0.14040 W12-0.00040 W21-0.03300 WZH.03000 WHO-0.02410 W12. 0.03100 W210-0.03773 W22I—0.00010 WHO-0.02100 W12. 0.03027 W210-0.03040 W22—0.00040 W1 1—0.01040 W12- 0.02007 W21—0.03330 W22-0.00400 WHO-0.01703 W12. 0.02777 W21—0.03102 W22-0.00430 W11- 0.04700 W12- 0.03340 W21I—0.00200 W22- 0.01010 W110 0.01203 W12. 0.02000 W21—0.01070 W22- 0.00020 W110 0.04037 W12- 0.02722 W21- 0.02040 W22- 0.04037 W11. 0.02020 W12- 0.02700 W21. 0.01340 W22- 0.04704 W11. 0.02002 W12- 0.01107 W21. 0.02700 W22- 0.04010 W11- 0.02470 W12. 0.01204 W21. 0.02730 W22- 0.00030 W11. 0.02443 W12. 0.01213 W21. 0.02001 W22. 0.00201 W11. 0.02407 W12. 0.01210 W210 0.02030 W22- 0.00407 W11. 0.00002 W12U-0.01700 W21- 0.02007 W22- 0.01240 ‘9-9§§ 0111 uuuu §§§§§§§§§§§3§§§§ . FUFHC‘MUUUUd. §.!!!!§!§§§§§§§§ ...,Ltzazzs...-u 111 ‘00 CHAPTER IV EXAMPLES AND DISCUSSION IV.1 Examples Four example problems are solved utilizing the Boundary Integral Method and the results are compared to NASTRAN. In the first example problem, a triangular shaped geometry is analyzed. The FEM and BEM model of this problem are shown in Figure IV.l and IV.2, respectively. The structure is subjected to the following non-constant boundary conditions. on the side where x-O ux--4.76 y2/2 uy--l.h3 y2/2 on the side where y-O ux--l.43 x2/2 uy--4.76 x2/2 - 167 x and on the side where x + y - l the following tractions are applied tx- y cos a - sin a ty- -cos a + x sin a The orthotropic material properties of Graphite/Epoxy, AS/3501 was used and those properties are as follows: Ex - 0.210084 Ey - 0.014006 Es - 0.002994 5h 55 v‘ - 0.300400 This material belongs to first case of orthotropic formulated in chapter III. The problem was solved utilizing 00H by discretizing the boundary into 13 elements as shown in Figure IV.2. The NASTRAN model shown in Figure IV.l contains six rectangular elements and four triangular elements. Table IV.l lists the displacements calculated by these two codes for three points on the side where x + y -1. In the second example, the same problem is solved, however, the number of nodes used on the boundary was increased to 19 in the BEM model as shown in Figure IV.4. Figure IV.3 exhibits the model used to solved the problem using NASTRAN. Table IV.2 shows the displacements calculated by both codes on the side where x + y - 1. In example problem three, a quarter of a plate with a hole in the middle subjected to a uniform distributed load of magnitude unity is solved. The FEM and BEM model of this problem are shown in Figures IV.5 and IV.6. respectively. Table IV.3 lists the displacements calculated with both codes. The material properties selected for this problem are as follows: Ex - 2.50000 0y - 2.50000 0‘ - 0.50000 vx - 0.25000 In the example problem four, the same problem was solved with 47 nodes on the boundary. The coordinates of the points selected and displacements calculated by NASTRAN and BEM are 56 listed in Table IV.4. Note that this material belongs to the second case of the orthotropic formulated in chapter III. IV.2 DISCUSSION Some conclusions can be drawn from the results of the four example problems. The BEM results are in agreement with FEM results generated from the NASTRAN code. As expected, it is much easier to prepare the data for BEM than for FEM. This factor can be quantified by comparing the input decks of NASTRAN and BEM listed in appendix C. This advantage of the BEM is even more profound when solving three dimensional problems. For the same number of the boundary nodes. fewer equations are solved in the BEM program. However, since the BEM matrix of coefficients is neither symmetric nor banded, a greater effort is necessary for solving the equations. Conversely, the high number of equations in the FEM method are solved quite efficiently due to the symmetric and banded form of the global stiffness matrix. The numerical method used to solve the singular terms in BEM seems to be adequate to obtain accurate results. 57 Figure IV.1 FEM model of example problem one 58 Figure IV.2 BEM model of example problem one 59 IL IIL ...L IIIIL. III-IL Figure IV.3 FEM model of example problem two A I’/ Figure IV.4 60 BEN model of example problem two 61 :5 zr—J Figure IV.5 FEM model of example problem three 62 Figure IV.6 4 a .._.. : TAT r BEM model of example problem three 63 Table IV.l .Displacements calculated by NASTRAN and BEM for ’ selected points in example problem one 0 A 0 T I A 11 0 I M x-eoord. y-coord. 1 s-displ. y-displ. x-displ. y-displ. 0.75000 0.25000 -3.41030 ~120.05080 -3,gs4so -121,34739 0.50000 0.50000 -0.01401 o01.34050 -g,gosoa -31,37oso 0.25000 0.75000 -l9.03734 -44.36070 -19.62609 .43.95847 64 Table IV.2 Displacements calculated by NASTRAN and BEM for selected points in example problem two N A 8 T RNA n 0.0.! x-coord. y-coord. x-displ. y-displ. x-displ. y—displ. 0.66670 0.33333 -4.37904 ~107.70980 -4.69846 -108.1867l 0.03330 0.33333 -0.74310 ~136.10740 -4.09990 -l35.93440 0.00000 0.50000 -0.05747 -01.l4751 -0.72976 -01.05050 0.33333 0.66670 -l§.44l65 -06.306lB o15.06597 -56.63600 0.16670 0.03330 -24.334l4 ~31.12919 -24.l6266 -3l.13294 65 Table 117.3 Displacements calculated by mm and IBM for selected points in example problem three I A 0 0 A A 11 0 2 M record. record. s-displ. y-displ. x-displ. y-displ. 1.00000 1.73210 1.26701 -1.02510 1.30060 -1.06707 1.73210 1.00000 2.44562 -4.05746 2.55095 -5.50470 0.00000 2.00000 3.05404 0.25506 3.01277 0.17675 0.00000 4.00000 2.01640 0.20166 2.05003 0.14799 0.00000 0.00000 2.02305 0.10245 2.13055 -0.00495 4.00000 6.00000 1.24023 -0.65750 1.40710 -0.60455 2.00000 0.00000 0.05251 -1.20541 0.07330 -1.22614 66 Table IV.4 Displacements calculated by NASTRAN and BEM for selected points in example problem four N A S T R A N B E H x-coord. y-coord. x-displ. y-displ. x-displ. y-displ. 3.”.33 3.” 3.”. .1 .0000: 3.” .1 .“414 3.40133 1.30030 3.73133 .1.32200 3.72373 .1 .32343 1 .37330 1 .22030 2 . 03040 .3 .33013 2 .33400 .3 .32374 1 .73330 3.30000 2 .33300 .3 .74102 2 .30730 .3 .73734 1 .30000 3.40000 3 .20137 '3 . 33320 3. 20307 .3 .33042 2.”000 3.“ 3.33423 3.”300 3.30012 3.“330 2.33000 3.30000 3.30007 3.33000 3.30077 3.~300 3.33000 3.”.00 3.42132 3.33000 3.42737 3.”000 3.33000 3.“300 3.40017 3.33030 3.33313 '3.”330 4.30000 3.~300 3.33343 3.”000 3.33043 3.” 4.33000 3.”030 3.72024 3.33000 3.73011 3.” 3.30000 3.33000 3.30000 3.33000 3.30340 3.“000 3.30000 3.30000 4.04202 3.30000 4.30342 3.”000 3.30000 3.33000 4.20270 3.”300 4.23300 3.”330 3.33000 3.33000 4.13003 3.12001 4.17307 3.11213 3.30000 1 .33000 4.37303 3.22000 4.30034 3.22701 3.30000 1.30030 3.32020 3.31440 3.33470 3.31010 3.33000 2.30000 3.73717 3.37120 3.74340 3.37300 3.30000 2.33030 3.32420 3.30022 3.02012 3.30070 3.30000 3.30000 3.20003 3.30010 3.30230 3.30303 3 . 30000 3 . 33000 3 . 37200 3 . 37332 3 .37407 3 . 37333 3.30000 4.30000 2.35027 3.34002 2.30230 3.33400 3.30000 4.33000 2 .33300 3.20447 2.33730 3.20707 3 . 30000 3 .30000 2 . 42300 3 . 24-473 2 . 42330 3 .23373 3.30000 3.33030 2.21743 3.13433 2.22334 3.13013 3.30000 3.33030 2.31203 3.14377 2.34473 3.11330 3.30000 3.30000 1 .31321 -3.33004 1 .32731 -3.30023 3.30000 3.30000 1 .31300 -3.20302 1 .32430 -3.20772 4.33000 3.30000 1 .41473 4.43000 1 .42333 -3.40730 4.30000 3.30000 1 .21077 4.30420 1 .22324 4.30000 3.30000 3.”000 1 .31000 4.30233 1 .32011 -3.30340 3.30300 3.” 3.32000 -1 .30020 3.33200 -1.30030 2.33000 3.“000 3.34000 -1.24003 3.30173 -1.20004 2.33000 3.“ 3.43404 -1 .42334 3.43300 -1.43334 1.33000 3.30000 3.33000 14.30200 3.34137 01.33000 1.“000 3.”000 3.21317 -1.70042 3.21744 -1.71307 3.33000 3.”000 3.13223 -1 .73420 3.11337 -1.73000 3.”000 3.”300 3.33330 -1.31143 3.33333 -1.33021 3.33330 3.” 3.”000 -1.73243 3.”.30 -1 .73032 3.33000 3.“030 3.“000 -1.74007 3.33300 -1.73330 3.33000 4.9030 3.33300 -1 .72041 3.33030 -1.72001 3.”030 4.33000 3.33300 -1.30013 3.”000 -1.73474 3.33000 3.33300 3.30000 -1.30543 3.33300 -1 .33002 3.30000 2.” 3.“.30 -1.37734 3.” -1 .33400 Appendix A) Computer listing of the program ORTHO.CASE1 and ORTHO.CASEZ 67 LISTING or THE PROGRAM ORTHO.CASE1 O......OOOOOOOOOO......OO'OOOOO00......0....00............DOOOOOOOOFFO C C C C C C C C C C C C C C C C C C C C C C C C C C C m MW.CASE1 THIS W D'LOYS THE DIRECT MARY ELDIDIT «moo usmc STRAIGHT MARY ELUENTS CHARACTERIZED CY LINEAR DISPLACDIEN‘TS MD ”GIANT TRACTIOIS TO SOLVE PLANE STRESS “TmTMIC MTERIAL ”BT13 ”L05 0' LINEAR ELASTICITY. WIT THIOOIESS IS ASSLMED. THE POLLUIW IS THE LBER'S DEFINE new PM”. VARIABLES AID AMAYS: N 15 THEMEROFTHEWESNTKW.”TIEM “ER. DUB IN FREE EMT. Ex 15 THE LOJGITLDINAL MULUS OE ELASTICITY. E7 15 THE TRANSVERSE ”ULUS OE ELASTICITY. ES 15 THE AXIAL SHEAR ”ULUS OE ELASTICITY. 'RX 15 THE POISSM' S RATIO IN TflE t-OIRECTIGJ. X(1)AND Y(I) ARE Tt'lE MIMTES DE ”ES 1. R ENTERED IN MTG-CLMISE IN FREE EMT. J. K ARE WES. J. A10 CMREMIIC DIRECTIOG. K. AT MIC" DISPLACDAENTS ARE SPECIFIEDUW A10 non-2m). DITER IN FREE PUD-LAT AID DO 0‘! IWUTTIPKS O. O. J.K.CO~ID ARE WES. J. MD CMRE'SPWIW DIRECTIN. K. AT WICH lat-ZERO FMCES ARE SPECIFIED. AND SPECIFIED VALUE coo. DITER IN FREE EMT MD DO WITH 0Y IIPUTTIDG 0. 0. 0. oocoocowccccc PARAHETER(WN-101) Pma(ww2.m.mws) IPLICIT REALOB (A-H.O-2) some X(wu).r(m) REAL-0 no).w(a).m(a).w2(e).u1(2) REAL-8 RL(e).wLe(e).wL1(e).wL2(a) DIMENSION mac(wu.2).x(wm).W(uxN).swv(wn) REAL-l w(IAerJ.W3) .m(ws.wm) .Rt-6(Mxvl>3) PI¢ACOS(-1 .00000) 0.0...0......UOOOOOOOOOOOO...ODO.......O.......OOOOOOOOOOOOOOOOO... C READ INTHEIIPUTDATA'RUEILES ......O'. .F. ......O...0......OI.0.0000,.........OOD'O...O.O....D'O READ(5.0 READ($.- DK.EY.ES.!RX READ(5.- (X(I).Y(I).I-1.N) C W 0 I-1.N no 0 K01.2 : NIEUJ)‘ 11 READ(5.0)J.K IE(J.E0.0)COTO 12 NTBC(J.K)-1 com 11 68 MIKE NN-ZON no 15 1.1.0“ IC(I)-O. READ(5.0) J.K.OOND IF(J.EO.C)COT 17 I-20(J-‘)+K C11-1/EX C22-1/EY C66-1/(20ES) CIZI-PRX/EX RITEEGJDO) IRITE 6.150) 00 19 1-1.u Il-Zol 11u1-11-1 “ s -, 1r NTBC(1.2 .zo:1 GOTO 20 It NTBC(I.1 .zo.1 0010 21 unxt:(s.2oo I.X(I).Y(I).IC(IINI).BC(II) 0010 as IF(NTBC(I.1 .£0.1) core 22 IRITE(6.201 I.X(I).Y(I).BC(IIH1).OC(II) 0010 19 IRITE(6.202)I.X(I).Y(I).IC(IIH1).BC(II) core 19 NRITE(6.203) I.x(l).Y(I).BC(IIu1).IC(II) CONTINUE AUX-CWT“ CIZ+C66)/C11)o((C12'OCCC)/CH)-C22/C11) an-OSMT C12+C66 /C11+AUX “IQ-05$“ 0121666 [CI I-AUX 69 M1-(c124-cccvcn m—czz/cn WSORNDLNZ) ma-owuowa-m WSORNABSRAD» «nag-050m “HIM UAW: IOSORT nun-nun Miz/CH Duds-MT 1 1(2 MT 1 1(2 ulna-2.00“" E‘ i— “81ng Dub-“PI E12— DLN1-OuD-OLIM / m.n?! E21-OW4/(4-PI) E22D-(2JOlM2-OW10m)/(MOMO4OPI) E31- 1.+ow4/oo.m)/(mw1.4 E32- I.-OUM/DLIL3)/(MOP104 E41—1./(40P1) 542-(ow1-2 . oDlAH)/(ul§onul6040PI ) E51—1./(40PI) E52D-E42 END-(WG/(MO4OPI) E62-(DUD-DWO/(wuofl) 01 1‘1 1 0‘1 14C120E3I 012-(C110E12+C120E32) DZHI IOEZZ-CIZOE42 032-022 MSWT(CZZ/C11) NIMODII “MODIZ R 130-.96828985649753 R 2 -.79666647741362 R 3§O-.52553248991632 R 4 -.18343464249565 R 5 #51; R 6go—R 2 RE? H23; R 8)-R 4 R8 1)-.18122833629838 N 2%..22238183445337 N 3 . . 31 378664587789 '8 4 II. 36268378337836 ”(3 £322§§§§ no 10 1,-1.3 mpg-1.4!“ N2 L .1 .mu comma: 000000 00000‘ 70 uc J.I)-8. IR J.I)-8. JJ-ZOJ IF(J.EO.1) THEN JN1-N JJNQ-NN ELSE JN1-J-1 JJN2-JJ-2 END IF IF(J.EO.N) THEN JP1-1 JJP1-1 ELSE JP1-J+1 JJP1-JJ+1 END IF m-(xU +XEJP1 g/z. w-(Yu «w .m /2. 31-! JP1g-XH 82-Y JP1 oYH 8081-81OBI+ROOT10ROOT1082082 WBZ-IBIOBHROOTZOROOTZOBbBZ TERu1-OLOG(BDB1)+DLOG(BDBZ)+4.0LDC(2.)-4. TERH2-OLOG(BOB1)-DLOC(BDBZ) UR JJ-1.JJP1)-UREJJ-1.JJP1;+O110TERH1+D120TERN2 UR JJ.JJP1+1)-UR JJ.JJP1+1 +041-TERH1+D420TERH2 MEXEJg-tXEJMD/Z. m- Y J +Y Jinn/2. B1-X(J)-XN 82-Y(J)-YH 8081-81081+ROOTIOROOT1082082 8082-81oB1+ROOT20R0012082082 TERM1-DLOG(BDB1g+OLOG(8082)+4.oLDC(2.)-4. TERHR-DLOG(BDB1 -DLOC(BDBZ; UR(JJ-1.JJ-1)-UR(JJ-1.JJ-1 +0110TERM1+O120TERM2 m(JJ.JJ)-UR(JJ.JJ)+041-Taw+o42-‘rmt2 71 0 00000 DO 2 1-1 .N 11.201 A1-X(I)-ni A3-Y(I)-YN ADA1-A10A14RwT1ORwT1oA20A2 ADA2-A10A1¢R00120ROOT20A20A2 ADB1-2.oéA1081+ROOT1oROOT10A2082; Abaz-2.- A1o31+R0012oRoorzoA2oaz DO 3 L-1.8 R1-ADA1-ADBI.R(L +aoa1oR(L)oR(L R2-ADA2-ADB20R(L +80820R(L)0R(L A1a1-A1-a1-au A282-A2-820R(L T1-A1B1/R1+A1B1/R2 T2-A1B1/R1-A181/R2 TJ-ROOT 1 0A282/R1+Rw1’20A282/R2 14-Roor1oA232/R1-R0012-A282/R2 TM 1-E1 1.820T1-E120820T2+E510810T3+E52081IT4 TA22U-E61082-T1-E620820T2+E410810T3+E42081014 TA12—E51 OBZOTJ-ESZOBZoTfiEM 0810T1+E32081 012 TA21—E210821T3-E220820THE61cB1oT1+E620810T2 UC(II-1.JJH2-1)-UC(II-1.JJN2-1)+N1(L)0N6(L)0TA11 UC(11.JJH2)-UC(1 I .JJH2)+VI1 (L)om(L)-TA22 veal-1.JJM2)-UC(lI-1.JJM2)+w1(L)ow0(L)-TA12 UC(lI.JJu2-1)-Lc(ll.JJH2-1)W1(L)owo(L)oTA21 ucm-1 .44-1 )-UC(11-1 .44-1 )+w2(L)owo(1.)on11 [£0 I .JJ)-UC(II.JJ)+W2(L)0WO(L)OTA22 vow-1.JJ)-UC(11-1.JJ)+w2(L)-wo(L).1A12 0cm .44-1)-u¢(11 .JJ-1)+112(L)owo(L)oTA21 mounaoonmzaz ARGUZ-ROOTZ-AZBZ- IF(ABS(A1B1 ) . LT. 1 .oE-3)THEN 1r(mcu1.ct.a.) nun-9V2 1r mou1.u.o.) TANT1—PI/2 1F ARGU2.CT.8.) TANT2-PI/2 Ir momma.) mnz—Px/z ELSE mum-noonuzaz/Mm TANT1-OATAN(RATIO1) unoz-Roonomz/Mm ONUO 72 TANT2-DATAN(RAT102) ENDIF F11-8.50(D110(0Lm(R1 +DLw R2) +0120(DLW(R1)-DLW(R2))) F22-8.50 0410(DLW(R1 +DLW R2) +042-(oLoc(R1)-0Loc(nz))) ”2.0220 TANT1-TANT2 F21-D32o TANT1-TANT2 EgéJQLMJIQUUTHEN mu 1-1 ..14-1 )«(11-1 .JJ-1)+r11owo(1.) gigmmuuuymomm ungu-nuruam-nu #12013“; at 11.44-1 «m.n-1 421-110“ WINE WINE 000000 59 51 0000 455 rom1(1o(1x.rs.3)) ccocccccocccooooocooccc CALCULATION OF THE DIAGOML ENTRIES OF Lb MATRIX. OCCCCCCCCCCCCCOCCCCOCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC DO 59 1-1.HN swx(I)-e. SM(l)-o. DO 59 J-1.N JJ-ZOJ SM(I)-SWX(I)+UC(I.JJ-1) SWY(I)-SWY(I)+UC(I.JJ) WTINUE “—1 J-1 DO 51 I-1JN 1.1-n+1 m(l.J)-SWX(I) ucu.a+1)-suw(1) IF(H.E0.0)GOTO 51 .1-4-12 u—1 CCNTINUE NR11E(5.455) ((LR(N.L).L-1.m).M-1.M~l) IRITE(6.26) NRITE(5.455) ((uc(N.L).L-1.m).u-1.m) W C C POST-WLTIPLY BY CAIN“ INVERSE III-201 IIN1-II-1 0 0.10 0000 9 000 0“ 000 0 73 A0-1. DO 4 102." 1R11.1)“(11.T)+1R(11.20K-I)0A0 W IIN1.2)-IR(IIN1.2)¢LR(IIN1.2010.“ “(II .2)-LR(11 .2)+LR(II.20K)OA6 H no 5 K-2.N 1M 1 [H1 .2-N-1 )—LR(IIN1 .2.1<-3)+2 . 111101111 .2oK-1) IRE” .2-K-1go-LRUI .20K-3)+2.0LR(II .20K—1) LR 111111 .2-K W(IIN1.20K-2)+2. OLR(IIN1.20K) LRUI .20K)-LR(II .20K-2)+2. OLR(II.20K) mgnm.1)-UR(11N1.1)M(11N1.2-K-1)-Ao w 7 1.1.111 LC(201-1.H~P1)-1. W(201.W2)-1. LR(NNP1.201-1)-1. LR(ND~P2.2-I)-1. no a 1-2.N m(2o1-1.NNP3)-Y(x)-v(1) U:(201.NNP3)-x(1)-X(I) m(NNps.2. 1-1 )«(1 )-.-1r(1) m(NNp3.2.1)-x(1)-x(1) cccccc RE-mCDER SYSTDI CBASED GI «mom MARCY CODITIWCS WWCCC DO 13 l-1.N Do 13 K-1.2 II-20(I-1)+K KK-NTBC(I.K) 1F(KK.Eo.o) com 13 Do 11 LIIJN’S TEu-UR(L.11) m(L.11-UC(L.II) uc(1..11 ~19: OONTINUE C DETDNINE mom RIGHT-HAM SIDEN C 711 WWWWWCC DO 18 I-1.M«P3 RHS(I)-O. DO 18 LIL.“ 18 RHS(I)-RHS(I)+LR(I.L)0m(L) DO 23 1.1.088’2 PIVOT-0. DO 24 J-IJOP3 Tan-OABS(UC(J.I)) IF(PIVOT.CE.TDA) COTO 24 PIVOT-TD! IPIVOT-J 24 COJTIMJE C C IE(IPIVOT.EO.I) GOTO 45 oo 27 numb: ransom) UC(1.x)-ucuvaor.x) ucuvaononm 27 00111le C Tat-R1150) RHS(I)-RHS(IPIVOT) RHS(IPIVOT)-TEN C 45 IP1-I+1, . DO 28 K-IPI. NNP3 O—WW. ”MU. I) UC(K.1)-8. RHS(K)-O~RHS(I)+RHS(K) DO 29 Jon-HM UC(K. J)-OOUC(I. J)+LK:(K. J) 29 CWTINUE 28 CWTINUE 23 CWTIWE RHS(HP3)-RHS(N~P3)/LC(HPS.MP3) DO 36 K-1.HP2 0-0. 00 31 .1-1.1< m(w3-K.H~P4-J)0RHS(HP4-J) 31 CONTINUE RHS(NHP3-K)-(RHS(MP3—K)-O)M(NPH.MP3—K) 30 cmmw: WAL DISPLACEHEHTS INTO KRHS VECTM C C PU'T C PUT MAL FMCES INTO 8C VECTG? C ”PUT VECTMS RHTS AID ac C 75 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC DO 32 I-1.N OO 32 KI1.2 KK-NTBC(I.K) II-2-(I-1)+K IF(KK.EO.1) THEN TDD-8C0” 8C(II)-RHS(II) RHS(II)-TEHP ELSE END IF CONTINUE RRITE 6.300) IRITE 6.325) DO 33 I-1.N II-ZOI IIH1-II-1 WRITE(6. 350) I .RHS(IIH1). RHS(II). 8C(IIN1). BC(II) "$f SAV1-BC(1) SAV2-8C(2) A8-1. DO 34 1-2.N II-2-I IIN1-II-1 8C(1)-8C(1)+A008C(IIM1) 8C(2)-8C(2)+AOOBC(II) A8-AO no 35 I-2.N 11-201 11N1-II-1 SAVJ-BC(IIH1) SAV4-8C(II) BC(IIN1)-2.-SAv1-BC(IIu1-2) BC(II)-2.-5Av2-BC(ll-2) SAV1-SAV3 SAVZISAV4 CONTINUE C COMPUTE BOUNDARY STRESSES AND OUTPUT C C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC NR11E(6.4ao) NR11E(6.425) no 36 1-1.N 11-2-1 11u1-11-1 IF(I.EO.1) THEN 1N1-N IIH2-2-N 76 ELSE 1611-31-1 11612-1 1-2 EDD 11" IIN3-I IH2-1 mun-110111 Yu-Y(1)-Y( 1111 SFIIOSORT ( mo XII-ONION) CTF-YN/SF STF—XH/SF SIGN-(CTEOR IIN1)+STFIE(II )ISF SIGNT-(CTFOBC I I )-STFON(I 1N1 )/SF C SIGN-PROSIMU APR). 0(CTFO(RHS(11)-RHS(11|Q))-STFO (RHS(IIN1) C 4-RHS(IIU.3)))O2. [SE SIGTT-O. RRITE(6.450) 1.51W.SIG¢T.SICTT 36 COTTIWE C FMTS ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc 100 EORNAT(//.° BOUNDARY NODEs AND PRESCRIBED BOUNDARY CONDITIONS') 150 EORNAT(/.° NOOE°.sx.'X°.Ox.'Y'.Bx.'cONDITIONS') 209 FORMAT(/.I3.F16.5.F10.5.5X.'FX -°.r1e.5.3x.'rY -'.r10.5) 201 rORuAT(/.15.r1e.5.r1o.5.5x.'rx -’.F18.5.3X.’UY -'.r1O.5) 202 FORMAT(/.IS.F18.S.F10.5.5X.’UX o’.F16.5.3X.’FY -'.r1o.5) 203 FORMAT(/.IS.F18.S.F18.S.5X.'UX -'.F18.5.3X.'UY -'.E1B.5) C see FORMAT(//.' DISPLACEMENTS AND FORCES AT ALL BOUNDARY NOOES') 325 FORMAT(/.' NOOE'.6X.'UX'.11X.'UY'.11X.'FX'.11X.'FY') 459 EORNAT(/.17.2x.r1o.s.3x.r1e.s.3x.r1o.s) 359 FORMAT(/.15.2X.F16.5.3X.F18.5.3X.F10.5.3X.F10.$) 400 FORMAT(//.' STRESSES ON ALL BOUNDARY ELEMENTS') 425 FORNA'r(/.' ELaAENT'.3x.'SIONA NN' .sx.°SIONA NT°.sx.'SIONA TT') 450 FORMAT§2X.'S11- '.E12.B.2x.'512- '.E12.E.2x.'51e- ' +.E12.6 465 FORMAT§2X.'SE11- '.E12.6.2X.'SE12- '.E12.6.2X.'SE1G— ' +.E12.6 470 FORNAT(24x.'S(2.2)- ’.£12.6.2X.'S(2.6)- '.E12.6.2x) 475 EORNA1(24x.'SE22- '.E12.6.2X.'$E26- ’.E12.6.ZX) 45¢ FORMAT(46X.'566- '.E12.6) 485 FORMAT(46X.'SE66- ’.E12.6) see FORNAT(1x.° NATERIAL PROPERTY S(I.J) AFTER TRANSFORMATION ') 403 FORMAT(1X.' THROUGH ANGLE PHI- ’.F5.2.1X.'DEGREE’) 51D FORMAT(1X.///) 50 FORMAT(1X.’POISSON RATIO -'.rs.3) 25 FORMAT(1X.///) 40$ FORNA1(1x./) 505 FORMAT(10(1X.F6.3)) STOP DD 77 LISTING OF THE PROGRAM ORTHO.CASEZ C C C C C C C C C C C C C C C C C C C C C C C C C C C m "TmfiQEZ THIS PMRAH DIPLOYS THE DIRECT ”CARY ELO‘ENT NET” USMC STRAIGHT MARY ELENENTS MACTERIZD DY LINEAR DISPLACDAENTS APO CWSTANT TRACTIOJS TO SOLVE PLANE STRESS "TNTRWIC MATERIAL WGTIES ML” OF LINEAR ELASTICITY. WIT THICKNESS 1S ASSIHD. THE FOLLNIIG IS THE USER'S DEFINE IOPUT PARAMETERS. VARIABLES ADD ARRAYS: N IS THE ”ER OF THE WES 01 THE WY. HIST DE wO, KAISER. ENTER IN FREE FWT. EX IS THE LMITLDINAL IDOULUS OF ELASTICITY. E'Y IS THE TRANSVERSE IDOULUS OF ELASTICITY. ES IS THE AXIAL SHEAR WULUS OF ELASTICITY. PRX IS THE POISSON'S RATIO IN THE K-OIRECTIOJ. X(I) AND Y(I) ARE THE WINATES OF WES 1.N DITUED IN CQJNTER-CLxK'RISE IN FREE FWT. J.K ARE WOES. J. AND CMRESPONDIW DIRECTIN. K. AT WICH DISPLACDAENTS ARE SPECIFIED(ZERO Am 006-2910). ENTER IN FREE FWT AND DID DY INPUTTINC O. O. J.K.CO~ID ARE WOES. J. AM) CORRESPODIm DIRECTIW. K. AT MUCH ”ZERO FMCES ARE SPECIFIED. AND SPECIFIED VALUE ”0. ENTER IN FREE FWT AND END IITH DY IPPU‘TTIW 8. 8. 8. CCCCCCCCCCCCCCCCCQCCCCCCCWCCCWCCCCCCCCCCCCCCCCCCC PARAMETER (Wit 1 81) PARAMETER (Whit-20W .mmfli-fl) lumen REALDB (A-H.O-Z) REAL-B 11(1LAXN).Y(1AAXN) REAL-B R(B).RO(B).N1(B).1Y2(B).NAT(2) REAL-B RL(B).RLO(B).NL1(B).NL2(B) DIUENSION NTBC(11AXN.2).Bc(uAXNN).wa(uAXN).SM(quN) REAL-8 w(wms.ww3).m(ms.wm) .RHS(HAX1~P3) PI-OACOS(-1 .OODOO) READ(5.0 N READ(5.0 EX.EY.ES.FRX READ(5.0 (X(I).Y(I).I-1.N) DO 9 III-1.11 m D K-1.2 NTBC(I.K)-8 READ(5.0)J .K IF(J.EO.8)GOTO 12 NTBC(J.K)-1 ”TO 11 78 CONTINUE ”2011] DO 15 I-1.NN DC(I)-O. READ(5.0) J.K.COND 1F(J.E0.0)GOTO 17 1.29(J-1)+K DC(I)-COND O11-1/Ex C22-1/EY css-1/(2.ES) O12-PRx/Ex 00000 9 00 NRITE(6.1BO) NR1TE(6.150) Do 19 I-1.N 11-2-1 IIN1-II-1 IF(NTBC(I.2).EO.1 GOTO 29 Ir(NTBC(I.1).Eo.1 GOTO 21 NRITE(6.206) 1.x(l).Y(1).BC(11N1).Bc(II) GOTO 19 IF(NTBC(I.1).EQ.1) GOTO 22 NRITE(6.201) I.x(1).Y(I).BC(11N1).BC(II) GOTO 19 NRITE(6.292)I.X(1).Y(I).BC(11U1).BC(II) GOTO 19 lRITE(6.203) I.X(1).Y(I).BC(Ilu1).BC(Il) CONTINUE PARAMETER(NAXN-1B1) PARAMETER(HAXPN-29HAXN.MAXDP3-HAXIN+3) OUTPUT PRESCRIBED INFORMATION f 79 RRITE(6.SO)PR NRITE(6.189) RRITE(6.158) DO 19 I-1.N II-ZOI IIH1-I I-1 IFENTBC(I.2 .EO.1 GOTO 29 IF NTBC(I.1 .EO.1 OOTO 21 IRITE(6.2OO 1.X(I).Y(I).8C(IIN1).DC(II) ”TO 19 IF(NTBC(I.1 .EO.1) COTO 22 NRITE(6.281 I.X(I).Y(I).DC(IIN1).DC(II) GOTO 19 IRITE(6.282)I.X(I).Y(I).DC(IIH1).DC(II) GOTO 19 WRITE(6.283) I.X(I).Y(I).DC(IIN1).8C(II) CONTINUE WWW ARO1-(c12+css)/c11 ROOT-OSORT(ARG1) DW1-(C12+C66)/C11 DM-CZZ/CH DWJ-DSORHDIMZ) DUU4-C12/C11 DLMS-ZJDSORUDWU Dun-24019.41 E11o-(DW1'PDtM3-DLM4)/(DMO4OPI) E12n-(OUH1-OUH3—DUH4)/(40PI) EZTIOW4/(49P1) E22—(2.IDM-OW1OOW4)/(DW5049PI) 531-(1.+DW4/DW3)/(DW50P104) E32-(1.-DUU4/DUN3)/(PIO4) E41I—1 ./(40P1) E42-(DW1-2.ODW4)/(DW5O4OPI) E51—1./(40PI) E52D-E42 E61~(OW3+OW4)/(DU6040PI) E62-(DW3-DW4)/(4OPI) D11-C119E11-I-C120E31 D12-(C110E12+C120E32) D22O-C110E22-C12-E42 032.022 D41-DW30011 D42-DW30012 R(1 )II- . 96028985649753 0000000-4 00000-4 80 R(2)-—.79555547741352 R(5)-.52553245991532 R(4 )--. 15545454249555 R(5)-R(1 R(5)-—R(2 R(7)-R(3) INN-RU) ND(1;-.18122853629838 N0(2 -.22238103445337 l0(3)-.31370664587789 119(4)-.55255575537555 N9(5)dND(1) ID(5)dwO(2) ”(n-1M3) “(M-119(4) Do 19 L-1.B w1(L)-1.-R(L) 112(L)-1.+R(L) CONTINUE 3 INITIALIZE MATRICES CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC NNPa-NN-a Do 1 I-1.NNP3 DO 1 J-1.NNP3 UC(J.l)-O. IF(I.GT.NN) OOTO 1 UR(J.I)-e. CONTINUE CCCCCCCCCCCCCCCCCCCCCCOCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC LOOP ON COLUMN CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC DO 2 J-1.N JJ-2-J IF(J.EO.1) THEN JN1-N JJM2-NN ELSE JH1-J-1 JJM2-JJ-2 END IF IF(J.EO.N) THEN JP1-1 JJP1-1 ELSE JP1-J+1 JJP1-JJ+1 END IF 81 m-(x(.1)+x(JP1))/2. Yu-(Y(J)+Y(JP1))/2. B1-x(JP1)-xu 82-Y(JP1)-YN 8081-81oB1+ROOToROOToBZOBZ TERN1-2.ODLOG(BDB1)+4.4LOC(2.)-4. TERNO-2.-52-32/BOO1 URéJJ-1.JJP1;-UREJJ-1.JJP1 +0110TERN1+D120TERN2 UR JJ.JJP1+1 IUR JJ.JJP1+1 +0410TERN1+D420TERH2 fiat-(X(J;+X(JH1) /2. YH-(Y(J +Y(JH1) /2. D1-X(J)-XH 82-Y(J)-YH 8081-81081+ROOT9ROOT982OB2 TERM1-2.0DLOC(BDB1)+4.0LOC(2.)-4. TERN2-2.082082/BDB1 UR(JJ-1.JJ-1)-UR(JJ-1.JJ-1)+D119TERH1+O129TERH2 UR(JJ.JJ)-UR(JJ.JJ)+041-TERH1+D42-TERM2 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C LOOP ON ROW C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC DO 2 I-1.N II-ZOI C A1-x(I)-xu A2-Y(I)-YH C ADA1-A10A1+ROOTOROOTOA20A2 ADS1-2.O(A1OB1+ROOT9ROOTOA2082) C C C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C LOOP ON POINTS OF INTEGRATION C C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC DO 3 L01.8 C R1-ADA1-ADB10R(L)+BDB1-R(L)0R(L) A181-A1-B19R(L) A282-A2-820R(L) T1-2.0A1B1/R1 T2-2.CROOT-A1B10A2820A282/(R1-R1) T3-2.-ROOT0A282/R1 T4-(A1B10020A282-ROOT0020A282003)/(R19R1) TA11-E110820T1-E120820T2+E510819T3+E520810T4 TA22O-E61082-T1-E620820T2+E410819T3+E42OB19T4 TA12O-E510820T3-E520820T4+E31081IT1+E32OB19T2 TA21-E210820T3-E22082-T4+E610810T1+E62OB10T2 59 51 82 UC(II-1.JJH2-1)-UC(II-1.JJH2-1)+N1(L)0NO(L)0TA11 UC(II.JJH2)-UC(II.JJH2)+N1(L)0ND(L)oTA22 UCEII-1.JJN2)-UC 11-1.JJN2)+R1(L)oNO(L)-TA12 uc II.JJN2-1)-Uc II.JJN2-1)+w1(L)-N9(L)oTA21 UC(II-1.JJ-1)-UC(II-1.JJ-1)+R2(L)-ND(L).TA11 UC(II.JJ)-UC(II.JJ)+R2(L)-NO(L)oTA22 UC(II-1.JJ -UC(II-1.JJ)+N2(L)9NO(L)9TA12 UC(II.JJ-1 -UC(II.JJ-1)+N2(L)0ND(L)9TA21 T8-A1819A232/R1 T7-OLOG(R1) TB-AzaZoAZBZ/RI F11-O119T7+D120T8 F22-D410T7+O420T8 F12-0220T6 F21-O320T6 EF(I.EO.J.OR.I.ED.JN1)THEN LSE UR(II-1.JJ-1)-UR(II-1.JJ-1)+r11-RC(L) UR(II.JJ)-UR(II.JJ)+P22-NO(L) ENOIF UR(II-1.JJ)-UR(II-1.JJ)+E12owD(L) UR(II.JJ-1)-UR(II.JJ—1)+F219N9(L) CONTINUE CONTINUE CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CALCULATION OF THE DIACONAL ENTRIES OF UC MATRIX. CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC DO 59 I-1.NN SUMX(I)-o. SUHY(I)-O. DO 59 J-1.N JJ-2-J SUHX(I)-SUHX(I)+UC(I.JJ-1) SUHY(I)-SUMY(I)+UC(I.JJ) CONTINUE N-—1 J-1 DO 51 1-1.NN ltfifld UC(I.J)-SLNX(I) UC(I.J+1)-—SUHY(I) IF(M.E0.0)GOTO 51 J-J+2 Nn-1 CONTINUE 83 NR1TE(6.26) NRITE(5.505) ((UR(N.L).L-1.m).u-1.NN) WRITE(6.26) RRITE(5.595) ((UC(H.J).J-1.m).u-1.m) 0 0‘0 00000000100! DO 6 I-1.N 11.201 IIN1-II-1 Ah1. DO 4 K-2.N UR(IIH1.1)-UR(IIN1.1)+UR(IIH1.2-K-1)0AO UR(II.1)-UR(II.1)+UR(II.2-K-1)-Ao UR(IIN1.2)-UR(IIU1.2)M(IIN1.2.K).A9 IR(II.2)-LR(II .2)M(II.20K)9AO ADD-AD DO 5 K-2.N UR(IIH1.2-K-1)-—UR(IIH1.2-K-3)+2.0UR(IIM1.29K-1) WU) .24K-1 )-—LR(I I .20K-3)+2.0UR(II.20K-1) UR(IIH1.2-K)-UR(IIH1.2-K-2)+2.4UR(IIH1,29K) UR(II.20K)-UR(II.20K-2)+2.0UR(II.20K) CONTINUE CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCOCCCCCCOCCCCCCCCCCCC AWENT Fm EOUI LIBRIW CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCOCCCCCCCCCCCCCCCCCC C 8 C mp1m1 ”192sz Do 7 131.111 LC(201-1.NNPI).—1. W<2°1.WP2)-—1. LR(NNP1 .291-1)-1. W(W2.201)-1. Do 5 I-2.N UC(2.I-1,NNP3).Y(I)-Y(1) UC(2-I.NNP3)-x(1)-x(1) UR(NNP3.2-I-1)-Y(1)-Y(I) UR(NNP5.2.I)-x(I)-x(1) CCCCCOCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCWCCCCCOCC C 81-1 RE-GIDER SYSTDA BASED 01 Km MARY CGIDITIWS DO 13 I-1.N DO 13 K-1.2 II-2-(I—1)+K KK-NTBC(I.K) IF(KK.E0.0) 0010 13 DO 14 L-1.MP3 Tea-UR(L.II) LR(L.II)-L£(L.II) LC(L.II)-TDI 14 CONTINUE 13 CONTINUE DO 18 1.1.1.1533 RHS(I)-O. DO 18 L-1.I~N 18 RHS(I)-RHS(I)+LR(I.L)OBC(L) C CCCCCCCCCCCCCCCWCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C SOLVE FOR WNW BOUNDARY CWITIWS C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCmCCCCCCCCCCCCCCCCCCCCC C MW DO 23 I-1.NNP2 PIVOT-O. DO 24 J-IJRP3 TU-OABS(L£(J.I)) IF(PIVOT.CE.TDA) COTO 24 PIVOT-TEN IPIVOT-J 24 CONTINUE C IF(IPIVOT.EO.I) GOTO 45 C DO 27 K-IJRP3 TBA-UC(I.K) WU .K)-UC(IPIVOT.K) LE(IPIVOT.K)-TDI 27 CONTINUE c TDAIRHSU) RHS(I )-RHS(IPIVOT) RHS(IPIVOT)-TEM c 45 IP1-I+1 DO 28 K-IP1.NNP3 85 SE?UC(§. .I)/UO(I. I) RHS(N)-OoRHS(I)-1RNS(I<) DO 29 J-IP1. NNP3 UC.(K .I)-O-UC(I. .I)4-UC(1<. J) 29 CONTIw 28 CONTINUE 23 CONTINUE RHS(mP3)-RHS(1AP3)/UC(w3.NP3) OO 30 x-1.NNPz 0-9. 00 31 4.1.x O-O+UC(N1«P3-K.II~P4-J)¢RHS(MP4—J) 31 CONTINUE RHS(MJP3-K)-(RHS(MP3-K)-O)M(H~P3-K.HP3-K) 39 CONTINUE C coccccocwCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCOCCCCOCCCCCCCCCCCCCC C C PUT NODAL OISPLACENENTS INTO KRHS VECTOR C PUT NOOAL FORCES INTO BC VECTOR C OUTPUT VECTORS RNTs AND BC C ccccccccccocccmccccccccccocccccocccccocoocccoocccccmcccccccccccc C no 32 I-1.N Do 32 K-1.2 KK-NTBC(I.K) lI-2-(I-1)+K IF(KK.EO.1) THEN TEMP-Bc(ll) BC(II)-RH$(II) RHS( I 1 )-TM ELSE END IF 32 CONTINUE NRITE(5.3OO) NRITE(5.325) Do 33 I-1.N II-2-I IIN1-II-1 33 HRITE(5.359) I.RHS(IIN1).RHS(II).BC(IIN1).BC(II) c CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCcc c C PRE IAILTIPLY BCI BY cm INVERSE c CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCWCCCwOCCCCCCCCccccccmCC C SAV1-BC(1) SAV2-BC(2) ADD-1 . DO 34 I-2.N II-2-I IIN1-II-1 86 BC(1)-BC(1)+A5.BC(IIN1) BC(2)-BC(2)+A5oBC(II) A5-A5 OO 35 1-2.N 11-2-I IIM1-II-1 SAVS-BC(IIU1) SAV4-BC(II) Bc(IIN1)-2.oSAv1-BC(IIN1-2) BC(II)-2.oSAV2-BC(II-2) SAV1-SAV3 SAv2-SAV4 gs CONTINUE c CI. IOOO‘OOFOOOOO. ODIOOOOOIDI......-0.0....‘OOOQIOOOOOOOOOOOOO'. U 01: C C COMPUTE BOUNDARY STRESSES AND OUTPUT C C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C HRITE(5.455) NRITE(5.425) DO 35 I-1.N II-2-I 11N1-II-1 IF(I.EO.1) THEN IBM-N IIH2-20N ELSE 1H1-1-1 IIH2-II-2 END IF IIN3-IIN2-1 mun-110911 ) YN-Yu )-Y( 11.11) SF-OSORT(XMCNM+YH9YH) CTF-YH/SF STFo-xu/SF SIONN-(CTF.ac(IIN1)+STF.ac(11))/sF SICNT-(CTFoBC(II)-STF-BC(IIN1))/SF C SICTT-PRoSICNN+(1.+PR).(CTF.(RHS(II)-RHS(IIN2))-STF.(RHS(I1N1) c +-RHS(IIN3)))o2./SF SICTT-O. NRITE(5.455) I.SICNN.SICNT.SICTT 35 CONTINUE cccccccccccccccccccccCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCc C c FORMATS c CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCOCCCCCCCCCCCOCCCCCCCCCCCCCCCCC 155 FORNAT(//.' BOUNDARY NOOEs AND PRESCRIBED BOUNDARY CONDITIONS') 155 FORMAT(/.' NODE'.6X.'X'.9x.'Y'.8X.'CONDITIONS') 255 FORMAT(/.IS.F1O.5.F18.5.5X.'FX --.F15.5.3x.-FY -'.F15.5) 251 FORMAT(/.15.F18.5.F10.5.5X.'FX --.F15.5.3x.'UY -°.F15.5) 252 FORMAT(/.I5.F10.5.F10.5.5X.'UX -'.F1C.5.3X.'FY -°.F15.5) 253 FORNAT(/.15.F15.5.F15.5.5x.'Ux -'.F10.5.3X.'UY -’.F10.5) 388 325 458 358 425 468 465 478 475 485 580 483 518 26 485 595 87 FORMAT(//.' DISPLACEMENTS AND FORCES AT ALL BOUNDARY NODES') FORMAT(/.' NODE'.BX.'UX'.11x.'UY'.11x.'Fx'.11x.'FY') FORMAT(/.I7.2X.F18.6.3X.F18.6.3X.F18.6) FORNAT(/.15.2x.F15.5.3x.F15.5.3x.F15.5.3x.F15.5) FORMAT(//.’ STRESSES ON ALL BOUNDARY ELENENTS') FWTU.’ ELENENT°.3x.'SICNA 191351135151“ NT'.5X,'SIGM T’T') FORNAT(2x.'S11- ‘.E12.6.2X.'$12- °.E12.5.2x.°515- ° +.E12.5) FORMAT(2X.'SE11- '.E12.6.2X.'SE12- '.E12.6.2X.'SE18- ' +.E12.5) FORMAT(24X.'S(2.2)- '.E12.6.2X.'S(2.6)- '.E12.C.2x) FORMAT(24X.'$E22- '.E12.6.2X.'SE26- '.E12.6.2x) FORUAT(45X.'555- '.E12 5) FORNAT(45x.°SE55- °.E12.5) FORMAT(1X.' MITERIAL PROPERTY S(I.J) AFTER TRANSFORMATION ’) FORMAT(1X.’ THROUGH ANGLE PHI- '.F5.2.1x.'DEGREE') FORMAT(IX.///) FORMAT(1X.'POISSON RATIO -'.F6.3) FORMAT(1X.///) FORNAT(1x./) FORNAT(15(1x.F5.2)) STOP END Appendix B) Computer listing of the program TEST 88 E 00000000000000000000000 PMRAN TEST THIS PRwRAH CALCULATES THE VALUES OF THE ISOTROPIC AID NTH CASES OF NTI'DTWIC INFLUEICE FLICTIWS. INPUT DATA N IS THE NIGER OF THE FIELD POINTS. PR IS THE POISSON'S RATIO. ISOTROPIC INFLUENCE FUNCTION. C IS THE AXIAL SHEAR NDDULUS OF ELASTICITY. ISOTROPIC INFLUENCE FUNCTION. EX IS THE LOOITUDINAL NODULUS OF ELASTICITY. ORTHOTROPIC INFLUENCE FUNCTION. E'Y IS THE TRANSVERSE WULUS OF ELASTICITY. “TMTRNIC INFLUENCE FUNCTION. EST IS THE AXIAL SHEAR ”ULUS. MTI'DTRWIC INFLUDCE FUNCTION. PRX1 IS THE POISSON'S RATIO IN THE X-OIRECTION. XS AND YS ARE THE X AND Y COORDINATE OF THE SOURCE POINT. X(I) AND Y(I) ARE THE COORDINATE OF THE FIELD POINTS POINT. CCCCCCCCCCCCCCCCC ILPLICIT REAL08 (A-H.O-Z) REAL08 B(5).X(O:25).Y(0:25;.LAAOA1.LALDA2.HAT(5) COHPLEXo16 Z(2).ZE(2).ZD(2 PI-OACOS(-1.88086) READ(5.0 N READ(5.- PR.C READ(5.- EX1.EY1.ES1.PRX1 READ(5.' EX2.EY2.E52.PRX2 READ(5.9 XS.YS READ(5.- (X(I).Y(I).I-1.6) READ(5.0 EX(I).Y I).I-7.12) READ(5.0 X(I).Y I).I-13.16) X(6)-1.4 Y(5)--1.4 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C C CALCULATIONS OF UR INFLUENCE FUNCTIONS FOR THE ISOTROPIC CASE CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C WRITE(5.5501) RRITE(6.20) DO 457 KILN RWSUIT((X(K)-XS)992+(Y(K)-YS)992) OI-(X(K)-XS)/RO 02-(Y(K)-YS)/RO UR11-((1.+PR)/(5.PI.G ).O1oO1-((3.-PR)/(5.PI.O)).OLOG(RO) UR12-((1.+PR)/(5.PI-C g-Ouoz 1821- (1.4-PR)/(80PICC) 901902 111122- (1 .+PR)/(B-PI-G))-02o02-((3.-PR)/(BoPIoC))oDLOG(RO) mfgézuu X(K).Y(K).LR11.LR12.LRZ1.LR22 459 89 CALCULATIONS OF LC INFLUEICE FLICTIWS Fm THE ISOWIC CASE NRITE(5.25) WITE(6.5582) 11RITE(5.25) DO 459 K-1.N ROIDSORT((X(K)-XS)992+(Y(K)-YS)992) OI-(x(K)—XS)/RO 02-(Y(K)-YS)/RO VB-OSORT((X(K)-X(K-1))992+(Y(K)-Y(K-1”992) VNI-2.9(Y(K)-Y(K-1))/VB “20—2,. x(K)-x(K-1)) UCI1-(1/ 49P19RO))9(29(1.4-PR)9(VN1901993-VN2902993)+(1.-PR)9VN1 +9O1+(3.+PR)9VN29OZ) UC12-(1/(49P19RO))9(29(-1.-PR)9(VN2901993+VN1902993)+(1.43. 9PR) +9019VN2+(3.+PR)9VN1902) 1.1521-(-2-(1.+PR)-(VN2-O1uS+VN1o52--3)+(3.+PR)-VN2oO1 4+(1.+3. 9PR)9VN1902)/(49PI9RO) LC22-(29 (1 APR) 9 (W2902-93-VN1901993)+(3.4PR)9VN1901 4+(1.-PR)9029VN2)/(49PIOR0) NRITE(5.2599) x(N).Y(N).UC11.UC12.UC21.Uczz CONTINUE WWWCC 0000 THE MATERIAL COISTANTS Fm THE INFLUENCE mums OF THE CASE 1. CCCCCCCCCCCCCCCCCWCCCCCWCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 67 68 C11-1/EX1 C22-1/EYI C66-1/(29ESI) C12—PRx1/EX1 M1-(C124C66)/C11 M2-C22/C11 Okla-OSORNDM) RAD-01.911.01.411“ 1M1(1)-1. IF(ABS(RAD).LT.1.0E-O3) GO TO 67 IF(RAD.CT.9.) GO TO 55 HAT(1)-1 IAAT(2)-DSORT(5.5-(Dw1+ow3)) NAT(3)-NAT(2) MAT(4)-OSORT(5.5-(Dw3-OLM1)) NAT(5)-IAAT(4) GO TO 59 HAT(1)-8. 01914-5 IF(HAT(1).GT.5.) m-OSORT(ABS(RAD)) u1(2)-OSORT(Dw1-1-Dw4) 1AAT(3)-DSORT(Dw1-Ow4) MAT(4)-O. 1AAT(5)-5. CONTINUE Dun-C12/C11 Dub-NAT(2)+NAT(3) 90 DW6-MAT(2)-MAT(3) IN-MATU) IF(IN.EO.8) DUHGIDUHS IF(IN.EO.-1) Owe-2.91m“) DUN1-2.0DUN1 E11-(OUM1+DUH3—OUH4)/EDUH5949PI E12-(DW1-DW3-Dl~4)/ M949P1 E21-OUN4/(40P1) E22-(2.90UN2-OUH190UH4)/(DUNSODUN6949PI) E31-(1.+OUH4/DUH3)/(DUH59P194 E32-(1.-Ow4/Dw3)/(Dw5-Ph4 E41-—1./(49P1) E42-(OUN1-2..DUN4)/(DUN5.OUN5.4.PI) E51-1./(49P1) E52-E42 E51-(DUH3+DUNK)/(DUU504OPI) E62-(OUN3-DUI4)/(DUN6949PI) 011-C110E11+C120E31 012-(C110E12+C120E32) DZH119E22-CI20E42 D32-022 MSWT(C22/C11) D41-DUNODTI NW9DIZ ARG-OSORT(((C12+C66)/C11)992-C22/C11) LNDAI-DSORT((C12+C66)/C11+ARC) LAADAZ-DSORT((C12+C66)/C11-ARC) WCCWWWWCCCCCCCCCC C C C C CALCULATIWS OF In INFLUDCE “ACTIONS Fm THE MTI'DTRPOIC HATERIALS OF CASE 1 WWWWWWWCC 875 NRITE(6.20) NRITE(6.6544) NRITE(6.20) DO 875 K-1.N R1-OSORT((X(K)-XS§992+LAHDA19929(Y(K)-YS)992) R2-OSORT((X(K)-XS 992+LAAOA29929(Y(K)-Y$)992) T1-(X(K)-XS)/R1992+(X(K-XS))/R2992 T2-(X(K)-XS)/R1992-(X(K)-XS)/R2992 T3-LAIIDA19(Y(K)-YS)/R1992+LAIDA29(Y(K)-YS)/R2092 T4-LAAOA19(Y(K)-YS)/R1992-LAADA29(Y(K)-YS)/R2992 VB-OSORT((X(K)-X(K-1))992+(Y(K)-Y(K-1))992) VN1-2.9(Y(K)-Y(K-1))/VB VN2-2.9(X(K)-X(K-1))/VB m11-(-E119T1-E129T2)9VN1+(-E519T3-E529T4)9VN2 UC22-(-E61911-E629T2)9VN1+(-E41913-E429T4)9VN2 UC12-(-E519T3—E529T4)9VN1+(-E319T1-E32912)9VN2 13214-5219T3-E229T4)9VN1+(-E619T1-E629T2)9VN2 CRITE(6.2609) X(K).Y(K).L¢11.LC12.|X§21.L£22 CWTINUE WWW C C C CALCULATIONS OF THE LR INFLUEKE MCTIW Fm THE WTI'DTROPIC MATERIALS OF CASE 1 91 CCCCCCCCCCCCCCCCCCWCCCCCCWCCCWCCCCCWCCCCCCCCCCCCCCCCCCCCCCCCCCCC C WRITE(6.20) RRITE(6.6545) “ITE(6.28) Do 989 K-1.N RIIOSORT((X(K)-XS)002+LALOA1992C(Y(K)-YS)992) R2-OSORT((X(K)-XS)992+LAADA29929(Y(K)-YS)992) ARCLAbX(K)-XS IF(ARGW.NE.O.)TNEN RATIO1-LALOA19(YEK;-YS /EX(K;-XS) RATIOZ-LAIDA29(Y K -YS / X(K -XS) T6-OATAN(RATIO1)-OATAN(RATI02) TANT1-OATAN(RATIOI; TANTz-DATAMRATIOZ ELSE S1-LANDA19(Y(K)-YS) S2-LAIDA2-(Y(K)-YS) IF(S1.CT.8.)THEN TANT1-PI/2 ELSE TANT1-PI/2 EDDIE IF(SZ.CT.O.)THEN TANT2-PI/2 ELSE TANT2—PI/2 ENDIF T6-TANT1-TANT2 EDDIE T7-OLOC(R1)+OLOC(R2) T8-OL®(R1)-DLm(R2) UR11-O119T74-D120T8 W12-0229T6 11221-53205 LIR22-O41 9T7+O42 9 T8 NR1TE(6.2000) X(K).Y(K).LR11.LR12.IR21.LR22 CWTINUE CCCCCCWCCWWCCCCCCCCCCCCCCCCCCCCCCCC C C C C THE MATERIAL COISTANTS Fm THE INFLUENCE FWCTIWS OF THE CASE 2. WWWCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CI 1-1/EX2 C2281/EY2 C66-1/(29E52) C12D-PRX2/EX2 WWSW‘N (C12+C55)/C1 I) DWI-(C124C66)/CI1 DM-CZZ/CII OW3-OSORT(DLIA2) DUU4-C12/C11 01145-2. 9DSORT(OW1 ) Dun-2.901.811 92 E11-(0UH1+OUM3-OUH4)/(0UH5949PI) E12D-(DLN1-DLM3-DLN4)/(49PI ) E21-OUN4/(49PI) E220-(2 . 90M-0W190W4)/(01¥5949PI) E31-(1 .90W4/0Lu3)/(0LI159P194) E32-(1.-0LM4/OLN3)/(P194) E41-1./(49P1) E42-(0W1-2 . ODLM)/(DLM5949PI) E51—1./(49PI) E52—E42 E51—(Ow3-1-ow4)/(Dw5-49PI) E52.(W4)/(49P1) D11-C119E114-C120E31 DI2-(C119E124-C129E32) 022—C119E22-C12-E42 032-022 D41-0W39011 D42—DW39012 C C CALCULATIWS OF LC INFLUEIICE FWTIWS Fm THE MTI-DTRPOIC C MATERIALS OF CASEZ C CCCCCOCCCCCCCCCCCWCOOCWCCCCCCCCCOCOCCCCCCCCCCCCCCCC FRITE(5.28) RRITE(5.5546) NRITE(6.20) DO 885 K-1.N R1-OSORT((X(K)-XS)992+LAMDA9929(Y(K)-YS)992) T1-29(X(K)-XS)/R1992 T2—29LAADA9(X(K)-XS)9(Y(K)-YS)902/R1994 T3-29LAAOA9(Y(K)-YS)/R1992 T4-((X(K)-XS)9929(Y(K)-YS)-LALOA9929(Y(K)-YS)993)/R1994 VB-OSORT((X(K)-X(K-1))992+(Y(K)-Y(K-1))992) VN1-2.9(Y(K)-Y(K-1))/VB VN2-2.9(x(K)-X(K-1))/VB UC11-(-E119T1-E129T2)9W1+(-E519T3-E529T4)9VN2 UC22-(-E619T1-E62912)9VN1+(-E419T3-E429T4)9VN2 W12-é-E519T3-E529T4)9VN1+(-E319T1-E320T2)9VN2 UC21- -E219T3-E22914)9VN1+(-E619TI-E629T2)9VN2 NRITE(6.2669) X(K).Y(K).LE11.LC12.L£21.UC22 885 CWTINUE COCCCOCCCCWCCOCCWOCCOCCCCCCOWCCCCCCOCCCCOCCCOCCCWOCCCCCCCCC CALCULATIGIS OF THE LR INFLUENCE FLICTIW Fm THE MTI-DTRG’IC MATERIALS OF CASE2 CCCCCCCC RRITE(6.28) NRITE(5.5547) RITE(6.26) DO 999 K-1.N R1-OSORT((X(K)-XS)992+LAADA9929(Y(K)-YS)992) T6-(X(K)-XS)9(Y(K)-YS)/R1992 T7-290LW(R1) T8-(Y(K)-YS)992/R1992 00050000 93 UR11-0119T7+O129T8 UR12-0229T6 UR21-O32916 UR22-O419T7+O429T8 NRITE(6.2888) X(K).Y(K).UR11.UR12.UR21.UR22 CONTINUE FORMAT(18X.'UC INFLUNCE FUNCTION FOR CASE 1- FORMAT(18X.'UC INFLUNCE FUNCTION FOR CASE 2' FORNAT(1ax.'UR INFLUNCE FUNCTION FOR CASE 1- FORNAT(1Bx.'UR INFLUNCE FUNCTION FOR CASE 2' FORNAT(2x.-x-.7x.-Y°.5x.'UR INFLUENCE FUNCTION OF ISOTROPY'; FORMAT(2X.'X'.7X.'Y'.5X.'UC INFLUENCE FUNCTION OF ISOTROPY' FORMAT(F6.3.1X.F6.3.2X.'UR11-‘.F8.5.1X.'UR12-'.F8.5.1X.'UR21-' +.F8.5.1X.'UR22-'.F8.5) FORNAT(1x./) FORNAT(F5.3.1x.F5.3.2x.'UC11-'.FB.5.1x.'UC12-°.F5.5.1x.'Ucz1-' +.F8.5.1X.'UC22-'.F8.5) STOP END Appendix C) FEM and BEM input data for example problem two 911 NASTRAN INPUT DATA FOR EXAMPLE PROBLEM Two [MIN JOB (8282-16).'J. KATIIAI'JGOLML-I. // NSCCLASS-T.TINE-99.PRTY.7.NOTIFY-NNIVON. // USER-xxxxxx.PAsm-xxxxxxxx //9uAIN SYSTMSA //oFORw.T PR.OONANE-.OEST-LwAL jybuAIN LINES-300.CAROS-SOO // EXEC NASTRANJORKSP-nu. /. ALTa-RF24555. // VSAPOSNI’KNIVDINSAPJHV NASTRAN BUFFSIZE-SCSO IO TRIMCULAR ”LEI IITN 1O IDES SOL 24 OIAO 15 TIHE 205 W SERGE S 8 CEND TITLE- TRIAICULAR ”LO! IITN 19 ”ES SLBTITLEII STATIC ANALYSIS T0 OIED‘ DISPLACWTS :CHOISORT SUBCASE 1 LABEL-STATIC ANALYSIS SPC-180 LOAD-2 OLOAD-ALL STRESS(VOUISES .PLOT)-ALL :ISP(PRINT)-ALL DECIN BULK PARAH AUTOSPC YES COUAD4 12 1 246 268 273 247 8. COUAD4 17 1 263 264 269 268 8. COUAD4 18 1 245 263 268 246 8. COUAD4 22 1 259 268 265 264 8. COUAD4 23 1 258 259 264 263 8. COUAD4 24 1 244 258 263 245 8. COUAD4 27 1 255 256 261 268 8. COUAD4 28 1 254 255 268 259 8. COUAD4 29 1 253 254 259 258 8. COUAD4 38 1 243 253 258 244 8. COUAD4 32 1 239 238 257 256 8. COUAD4 33 1 248 239 256 255 8. COUAD4 34 1 241 248 255 254 8. COUAD4 35 1 242 241 254 253 8. COUAD4 36 1 232 242 253 243 8 CTRIA3 37 1 256 257 261 8. CTRIA3 38 1 268 261 265 8. CTRIA3 39 1 264 265 269 8. CTRIA3 48 1 268 269 273 8. CTRIA3 41 1 247 273 231 8. CTRIA3 42 1 238 229 257 8. S : LCS.NANE - LOAD1 LOAD SET HM IS 2 95 FORCEO 2 257 1.89FOE881 9FOE881 -1.389999986E-81-2.779999748E-82 8.888888888E488 LOAD1 FORCEo 2 261 1.89FOE882 9FOE882 -1.111999758E-81-5.559999888E-82 8.888888888E488 LOAD1 FORCE- 2 265 1.89FOE883 9FOE883 -8.339995146E-82-8.339995146E-82 8.888888888E488 LOAD1 FORCE9 2 289 1.89FOE884 9FOE884 -5.559999868E-82-1.111999758E-81 8.888888888E488 LOAD1 FORCEO 2 273 1.89FOE885 9FOE885 -2.779999748E-82-1.389999986E-81 8.888888888E488 LOAD1 GRID 229 1.888 8.888 8.888 GRID 231 8.888 1.888 8.888 GRID 232 8.888 8.888 8.888 GRID 238 8.833 8.888 8.888 GRID 239 8.867 8.888 8.888 GRID 248 8.588 8.888 8.888 GRID 241 8.333 8.888 8.888 GRID 242 8.167 8.888 8.888 GRID 243 8.888 8.167 8.888 GRID 244 8.888 8.333 8.888 GRID 245 8.888 8.588 8.888 GRID 246 8.888 8.667 8.888 GRID 247 8.888 8.833 8.888 GRID 253 8.167 8.167 8.888 GRID 254 8.333 8.167 8.888 GRID 255 8.588 8.167 8.888 GRID 256 8.867 8.167 8.888 GRID 257 8.833 8.167 8.888 GRID 258 8.167 8.333 8.888 GRID 259 8.333 8.333 8.888 GRID 268 8.588 8.333 8.888 GRID 261 8.867 8.333 8.888 GRID 263 8.167 8.588 8.888 GRID 264 8.333 8.588 8.888 GRID 265 8.588 8.588 8.888 GRID 268 8.167 8.667 8.888 GRID 289 8.333 8.867 8.888 GRID 273 8.167 8.833 8.888 HATBO 1 2.1888888888-81 1.488568888E-82 8.388488888E4889HA2881 9HA2881 2.994818888E-83 3 GENERAL RIGID ELUENTS :SHELLO 1 1 1.888888888E488 : SPC2 SPCO 2 229 3456 8.888 X1 SPC9 2 231 3456 8.888 X1 SPCO 2 232 3456 8.888 X1 SPC9 2 238 3456 8.888 X1 SPCO 2 239 3456 8.888 X1 SPCO 2 248 3456 8.888 X1 SPCO 2 241 3456 8.888 X1 SPC9 2 242 3456 8.888 X1 SPCO 2 243 3456 8.888 X1 SPC9 2 244 3456 8.888 X1 SPC9 2 245 3456 8.888 X1 .SPCO 2 246 3456 8.888 X1 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPCo SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 : LCS.NNNE IS X2 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPco SPC9 SPC9 SPC9 SPC9 SPC9 SPC9 SPCADD.188.1.2 ”NNNNNNNNNNNNNNN ““““-“‘dd‘dd““d‘d“““d“‘d“‘ 96 247 3456 253 3458 254 3456 255 3458 258 3456 257 3456 258 3458 259 3456 288 3458 281 3458 263 3458 264 3458 285 3458 268 3458 269 3458 273 3456 SET 10 18.0 IS 232 123 242 242 242 3 241 241 241 3 248 248 248 3 239 239 239 3 238 238 238 3 229 229 229 3 231 231 231 3 247 247 247 3 246 246 246 3 245 245 245 3 244 244 244 3 243 243 243 3 X1 X1 X1 X1 X1 X1 X1 X1 X1 X1 X1 X1 X1 X1 X1 X1 0 O O I C O O 8 O I Q 0 I 8 O O 8.888 X2 1-1.989999786E-82X2 2-2.798499878E+81X2 8.888 ° X2 1-7.939994335E-82X2 2-5.592498779E+81X2 8.888 X2 1-1.788999438E-81X2 2-8.489498596E+81X2 8.888 X2 1-3.177999854E-81X2 2-1.123959961E+82X2 8.888 X2 1-4.964999557E-81X2 2-1.488139954E+82X2 8.888 X2 1-7.149999738E-81X2 2-8.587998962E+81X2 8.888 X2 1-3.569999695E+81X2 2-7.149999738E-81X2 8.888 X2 1-2.478999329E+81X2 2-4.963999987E-81X2 8.888 X2 1-1.586799988E+81X2 2-3.177999854E-81X2 8.888 X2 1-8.924999237E488X2 2-1.779999733E-81X2 8.888 X2 1-3.965869984E+88X2 2 7.942855358E-82X2 8.888 X2 1-9.928599461E-81X2 2-1.986899972E-82X2 8.888 X2 97 BEM INPUT DATA FOR EXAMPLE PROBLEM Two 18 8.21888483 8.814885682 8.88289481 8.3884 8 8 .1667 8 .3333 8 .5 8 .8687 8 .8333 8 1. 8. .8333 .1867 .8887 .3333 ‘5 .25.”33 .8887 .1867 .8333 8 1 8 .8333 8 .8867 8 .5 8 .375 8 .25 1 #9000000905989-9-9 “ 00.000990000009989”. N I ‘ p 0 0 J . can‘N‘N-oN-‘Nd LIIIFIIIAII ...-9.9 “N” 13 14 14 15 15 18 18 17 17 18 18 19 19 8 2 -8.7158 1 -24.798 2 -8.4964 1 -15.868 2 -8.3178 1 -8.925 2 -8.1789 1 -5.82 2 -8.1885 1 -2.2313 2 .8447 1 .5578 .8118 '111 2 8 98 99 BIBLIOGRAPHY 1) Brebbia. C.A. and Walker 8.. Boundary Element Techniques in Engineering. Newnes-Butterworths, London, (1980). 2) Banerjee. P. K.. Boundary Element 1 Methods in Engineering Science. NCGraw Hill Book CO.. London. (1981). 3) Heise, U.. Application of the Singularity Nethod for the Formulation of the Plane Elastostatical Boundary Value Problems as Integral Equations". 1983. h) Lekhnitskii, 8.. Theory of Elasticity of an Anisotropic Elastic Body, San Francisco, Holden-Day. 1963. 5) Jones, R. H.. Mechanics of Composite Materials, Hemisphere Publishing Corporation, New York. 6) Cloud, 6., Vable. M., Experimental an Theoretical investigation of Mechanically Pastened Composites, Technical Report 128h4, U.S.A. Army Tank-Automotive Command. 7) Schwartz, M.M.. Composite Materials Handbook, McCraw Hill Book Co.. New York. 8) Tsai, S. W., Introduction to Composite Materials, Technomic Publishing CO., Wesport Connecticut. 9) Zweben C.. and Hahn H.T.. Mechanical Behavior. McGraw Hill Book Co.. New York. 10) Morely, J. 6.. High-Performance Fiber Composites, Academic Press, London (1987). 11) Benjumea, R.. Sikarskie, D.L.. On the Solution of Plane, Orthotropic. Elasticity Problems by an Integral Method. 12) Zastrow, 0.. Solution of Plane Anisotropic Elastostatical Boundary Value Problems by Singular Integral Equations, 1981. 13) Rizzo, F. 3., Shippy. D. J.. A Method for Stress Determination in Plane Anisotropic Elastic Bodies, 1969.