)V1ESI_} RETURNING MATERIALS: P1ace in book drop to LIBRARIES remove this checkout from -_. your record. FINES will be charged if book is returned after the date stamped below. A COMPARATIVE STUDY OF BOUNDARY ELEMENT SHAPE FUNCTIONS FOR LINEAR ELASTOSTATICS BY David Christopher Zwier A THESIS Submitted to Michigan State University in partial fulfillment oi the requirements tor the degree of MASTER OF SCIENCE Department of Metallurgy, Mechanics and Materials Science 1988 ABSTRACT A COMPARATIVE STUDY OF BOUNDARY ELEMENT SHAPE FUNCTIONS FOR UNEAR ELASTOSTATICS BY David Christopher Zwler Many types of boundary elements are available in the boundary element method. With so many available, however, one needs to know what elements are best suited for different problem types. This thesis begins to explore this issue. 'Three simple elements are applied to a select group at problems and compared to exact results. The tirst element utilizes constant dis- placement and constant traction approximations, the second employs linear displacements and constant tractions, and the third employs linear approximations tor both displacements and tractions. it is found that an element which uses linear displacements and constant trac- tions yields the most consistent and accurate results. To the loving memory of my mother, Evelyn iii ACKNOWLEDGEMENTS I wish to thank my academic advisor, Dr. Nicholas Altlero, for his inspiration and invaluable research notes. i also thank my typist, Carrie Coker, for her patience and hard work. Finally, a special thank you to my father, Richard Zwier, for his support and thoughtful guidance. iv TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES CHAPTER 1 INTRODUC'nON CHAPTER 2 THE BOUNDARY ELEMENT METHOD CHAPTER 3 NUMERICAL TREATMENT 3.1 BOUNDARY APPROXIMATION 3.2 NUMERICAL MODELS 3.3 MATRIX EVALUATION 3.4 INTERIOR FIELD POINTS CHAPTER 4 EXAMPLES AND RESULTS 4.1 PROBLEM DESCRIPTIONS 4.. TABLES 4.3 DISCUSSION CHAPTER 5 CONCLUSIONS APPENDIX COMPUTER CODE LIST OF REFERENCES iv vi 12 12 13 16 24 27 27 31 31 V 75 76 117 Table Table Table Table Table Table Table Table 7 LIST OF TABLES Pure Extension - 21 Nodes X-Displacements Pure Extension - 21 Nodes Y-Displacements Pure Extension - 21 Nodes X-Tractions Pure Extension - 21 Nodes Y-Tractions Pure Extension - 21 Nodes interior Values Pure Extension - 21 Nodes Percent Differences Pure Extension - 63 Nodes X-Dlsplacements Pure Extension - 63 Nodes Y-Dlsplacements Table 9 Pure Extension - 63 Nodes X-Tractions Table 10 Pure Extension - 63 Nodes Y-Tractions Table 11 Pure Extension - 63 Nodes interior Values Table 12 Pure Extension - 63 Nodes Percent Differences Table 13 Square - 21 Nodes X-Displacements Table 14 Square - 21 Nodes Y-Displacennnts Table 15 Square - 21 Nodes X-Tractlons Table 16 Square - 21 Nodes Y-Tractions Table 17 Square -21 Nodes interior Values Table 18 Square - 21 Nodes Percent Difference Table 19 Square - 63 Nodes X-Displacements Table 20 Square - 63 Nodes Y-Dispiacements Table 21 Square - 63 Nodes X-Tractions Table 22 Square - B3 Nodes Y-Tractions vi 33 33 34 34 35 36 38 39 4O 42 43 44 47 48 48 49 49 SO 52 54 55 56 Table 23 Square - 63 Nodes interior Values Table 24 Squar. - 63 Nodes Percent Differences Table 25 Table 26 Table 27 Table 26 Table 29 Table 30 Table 31 Table 32 Table 33 Table 34 Table 35 Table 36 Disk Disk Disk Disk Disk Disk Disk Disk Disk Disk Disk Disk - 19 Nodes X-Dispiacements - 19 Nodes Y-Displacements - 19 Nodes X-Tractions - 19 Nodes Y-Tractions - 19 Nodes interior Values - 19 Nodes Percent Difierences - 35 Nodes X-Displacements - 35 Nodes Y-Displacements - 35 Nodes X-Tractions - 35 Nodes Y-Tractions - 35 Nodes interior Values - 35 Nodes Percent Ditferences vii 58 58 62 62 63 63 64 65 67 68 69 7O 7O 72 Figure Figure Figure Figure Figure Figure Figure Figure Figure LIST OF FIGURES 1 Planar Region 2 integration Around Singular Point 3 Boundary Approximation 4 Local Coordinates 5 Free Term Coefficient 6 Local Coordinates On element m= n 7 Local Coordinates On element m=n + 1 6 Problem One - Pure Extension 9 Problem Two - Square With Variable Boundary Conditions Figure 10 Problem Three - Circular Disk Under Uniform Pressure viii 10 12 13 18 19 20 28 29 3O CHAPTER 1 lNTRODUCT ION Engineers and scientists routinely develop mathematical models to describe the systems they wish to study. These models, however, typically involve differential equations and bound- ary conditions which are not easily solved. Therefore, in order to Obtain a solution, some form Of approximate method is employed. Many such methods exist today. However, the three most powerful are finite differences, finite elements and boundary elements. Although this thesis deals with boundary elements, all three will be discussed briefly because, when viewed historically, they show a trend which, in part, prompted this work. Before examining the individual methods, it should be pointed out that they possess com- mon characteristics. First, they each require the region of interest to be discretized using a set of points called nodes. Second, through differing mathematical approaches, each method reduces the system Of differential equations to a system of algebraic equations, which, when solved, give values for the physical variables at the nodes. Of the three subject methods, finite differences evolved first (1). it is probably the most natural way to approximate a set of differential equations because it utilizes the basic defini- tion of the derivative. Applying this definition and a finite nodal spacing to the derivatives in the governing equations results in a system of algebraic difference equations. in theory, finite differences may be applied to any differential equation. in practice, however, it is somewhat limited. This is due to the fact that certain boundary conditions can be cum- bersome to incorporate and it is difficult to work with curved or irregular boundaries. Also, since the solution accuracy is primarily dependent on nodal spacing, many equations are necessary to achieve good results. On the other hand, the method works well on problems involving first and second derivatives such as heat transfer, fluid mechanics and two dimen- sional problems with straight boundaries (2,3). 2 The finite element method is by far the most popular method in use today. it is, in a sense, a progression from finite differences in that it approximates the physical domain itself with small regions, or elements, which simulate the behavior Of the portion Of the domain they rep- resent. The differential equations are then applied to each element in turn and rewritten in terms of variational, weighted residual, or virtual work expressions (2). Finite elements have an advantage over finite differences in that irregular regions pose no significant problems. The method allows for a great deal Of flexibility in terms Of element types and shapes and, therefore, may be applied to many types of problems and regions. This kind of flexibility is partly responsible for the method’s popularity and has prompted the develop- ment Of many types Of elements (2). Another reason for its popularity lies in the fact that it has its roots in the study of differential equations which was well established in the nineteenth cen- tury (1) and, therefore, the underlying mathematics are generally understood today by practicing engineers and scientists. The boundary element method is, in a sense, a progression from the finite element method because it attempts to integrate the differential equations before it discretizes the region. The result is a system of integral equations which involves only the boundary values Of the sys- tem. Thus, only the region boundary needs to be discretized and the problem dimensionality is reduced by one. it should be noted that the boundary element method is similar to the finite element method in that they both can be considered weighted residual methods (4,14). However, the bound- ary element method does have the advantage of only requiring boundary discretization and thus fewer nodes. in addition, numerical integration is more accurate than numerical differentiation and there is little or no numerical differentiation in the boundary element method (1). Perhaps the big- gest disadvantage for the method is that the study Of integral equations is more recent and mathematically more difficult than the study of differential equations. This has, until recently, hampered its development. 3 Integral equations have been studied rigorously since the turn of the century (1,5). However, the contributions which have developed into the boundary element method have come as recently as the 1950’s and 1960’s (5,15). The nature Of the mathematics involved in these works prevented them from being recognized by the majority of engineers and applied scien- tists. However, they were recognized by some, and in the late 1960’s and early 1970's, papers began to be published which applied Integral equations to the solution Of engineering problems (6.7.8.9,13). Throughout the 1970’s, the number of papers increased and the term "boundary elements" emerged from works by Brebbia and others within the department of civil engineering at the University of Southampton (10). By the end of the 1970’s, it became evident that the bound- ary element method was becoming a legitimate analysis tool and international conferences on the topic were established (11). These conferences are designed to solicit work in the field and promote the method. To that end, they have been successful as evidenced by the increas- ing number and diversity of the papers presented at each conference (11). Now in the late 1960's, efforts such as the BETECi-i conferences (12) have begun which are designed to in- crease the popularity of the method and bridge the gap between industry and academia. Efforts such as these are increasing the popularity of the method and already, several com- merciai software packages are available (13). This is not to say that boundary elements are replacing finite elements. it does appear, however, that they will reach equal levels of develop- ment and coexist as complementary methods. in fact, since most problems solved by finite elements may also be solved by boundary elements (1), such a coexistence is desirable to make use of the relative merits of both methods and advance the quality Of solutions. Similarly, within each method it is desirable to use different element types in a complemen- tary manner depending on the problem characteristics. By knowing when to apply certain element types, more efficient and accurate solutions may be obtained. This does not mean that each problem should have its own type of element. It does, however, suggest that efforts should be made to study the applicability of basic element types and indeed such works have begun to appear in the literature (17). The purpose of this thesis then is to examine three types of boundary elements in the direct boundary element method for linear isotropic elasticity. Each element type will be applied to a range of problem types and assessed in terms of accuracy and quality of solution. It is hoped that this work will help establish guidelines for element applicability and show the usefulness of simpler, extrapolated models. CHAPTER2 THE BOUNDARY ELEMENT METHOD There are three natural formulations for the boundary element method. The indirect for- mulation expresses the integral equations in terms of a unit singular solution of the original differential equations. This solution is distributed over the boundary by way of density func- tions. The density functions have no physical significance and once they are obtained from the numerical solution of the integral equations, the values of the physical variables may be found by simple integration at any point within the region. The semi-direct formulation utilizes unknown functions, analogues to airy stress functions, as unknowns in the inteng equations. Once these functions are found, the physical variables may be obtained by differentiation at any point within the region. This is the only numerical differentiation required in the boundary element method. As pointed out in Chapter 1, numeri- cal differentlatlon is less accurate than numerical integration and, therefore, the semi-direct formulation is less useful than the other two formulations. The direct formulation expresses the inteng equations in terms of the actual physical vari- ables. Numerical solution of the integral equations gives the boundary values of the physical variables. interior values are subsequently found by simple numerical integration. This thesis deals specifically with the direct formulation because it is well suited to problems of plane isotropic elasticity. We will now proceed to develop the basic integral equations and in Chap- ter 3, we will develop the numerical solution schemes for these equations. Consider the following planar region: x 2 Figure 1. Planar Region 6 At any point x_ in R, we can write the equilibrium equations as m1.1+m2.2+X1=0 012.1 + m2 + X2 = 0 "I where a isthestresstensorandxisthe bodyiorcevector. Also,the notation of“ indicates differentiation ofoIgwithrespect tocoordlnatedlrectlonl. Hooke’s law for plane stress is given by: 011 = 26(um + u uz,z)l(1 -u) 022 = 26 (112,2 + v 01.1) I (1 ~11) [2] 012 = G (um i» U2.1) whereg is the displacement vector, G is the shear modulus and u is Poisson’s ratio. Substitution of eqs. [2] into eqs. [1] results in Navier’s equations. (u1.11-i-uz.21)(1+u)/(1-v)+V2ur+X1/G=0 [31 (01.12 4- 112.22) (1 + u)/(1-u) + V2112 + leG = 0 Shifting attention now from the region to the boundary B, we have the traction vector t given by: it = (mm -i- 012112 32 = 012m + 02202 [4] for any surface defined by unit normal 31. inserting eq. [2] into eq. [4] gives: it = G [um + 01,1111 + 112.1112 + 2» (01.1 + 112.2)111/(1 -u)] 12 = G [uzn + 111.2111 + 112.2112 -i- 21; (um +U2,2) n2! (1 -u)] Making use of the coordinate ,s, measured along B, we specify the boundary conditions as "i s 91(8) or tI/G = M» [6] for i :- 1,2. We now have the basic forrnuiation of the elasticity prObiem Wlth field equations, eq.[3], and boundary conditions, eq. [6]. However, we desire eq. [3] in a more convenient form. To accomplish this, we define a vector function 4), such that: 7 111 = v2¢1 - (1 + a) (41.11 + 1:12.12) /2 [7] U2 = V2¢2- (1 + u)(¢1,21 + 132.22) /2 and substitute into eq. [3] to yield the inhomogeneous biharmonic equation 1‘2 (v24) = Xi I G [8] for i = 1,2. The biharmonic operator is well understood and is known to possess a principal solution which satisfies V2(V25)=- 8(x-£)ei IG [9] for i = 1,2.. Here 3 is a unit vector and 601 - g) is a Dirac delta function which represents a point singularity at the source point 5 . For equations such as eq. [9], the solution is a func- tion only of the distance between the source point 5 and the field point x. This distance is given by: , = [In 4:02 + (112-1er 0-5. The solution to eq. [9] is then: ¢1=-pzln(p)eilaec. [101 Substitution Of eq. [10] for 411 in eq. [7] gives the displacement field for the principal solution as: U1 = {(- (3-11) Il‘lp + (1 + u)q12] B1 -I- (1+ U)Q1Q262}/8-.TG [11] U2 = {Ha-1.) Ii‘ip + (1 + 191122] e2 + (1 + u)q1qze1}/8-.:G Here we have defined q1and q2 as: 91=(X1-§1)/P and€lz= (XZ'EZIIP Similarly, substitution Of eq. [11] for U1 into eq. [5] gives the traction field for the principal solution T1 = {[2 (1 + 11) (-n1 (:13 + n2 q23) - (1 -11) n1 qr . (3-11) n2 (:21 e1 + (2(1+ 1.) (1121.13 + 1111123143 + 1,) 112111 -(1 + 31,) n1 c121 e2}I4-I:p T2 = {[2(1 + 1:) (n2 q13 + n1 q23) - (1 + 311) R2 q1 - (3 + u)n1q2] E1 4- [2(1 + 11) (mm3 - 02 ((23) - 3+ U) n1 Q1 - (1 - 1:) n2 Q2102} I417p. 8 if we now define the displacement and traction vectors for the principal solution to be: Ui(X)=Ui1(X1£)Of(£)+Ui2(X.£)82(£) Tim=T11(x.£)e1(£)+Tiz(x.£lez(£). [13] then we can write U11 = [- (3-1.) in p + (1 + 191112] [83:6 U12 = (1 -i- 11) Q1 Q2/8176 [14) U21 = (1 + 11) m q2/seG U22 = [-(3-11) lnp + (i + u)Q22]/8-.:G and T11 = [2(1 + U) (-i11 ([13 + n2q23)-(1-U)n1QI- (3 + 11) I12 QZ] [47.1) T12 = [2(1 + 1,)(1121113 + n1 q23) - (31-11) R2 41 - (1 + 31) n1q21/47: T21 = [2(1 + 11)(n2q13 + R1 423) - (1 + 311) R2 q1 - (3 + 11) m :12] I4: T2 = [2(1 + 1) (R1 1113- nzqz’) - (3 + u)n1Q1-(1-u)i12CI2]/41Tp. The quantities given by eq. [14] and eq. [15] represent the effects caused at the field point x by a unit force at the point tin the infinite plane. Thus U11 , I = 1, 2, are the displacements and fractions respectively at a point 21 due to a unit force in the x1- direction applied at 5. Similarly, U12 and T12 are generated at x by a unit force in the X2 - direction at ,5. We now recall Betti’s reciprocal work theorem which states that, if two elastic equilibrium states (XI,ti,UI) and (XI*,tI*,UI*) exist in a region R bounded by B, then the work done by the forces of the first system on the displacements of the second is equal to the work done by the forces of the second on the displacements of the first. Thus, we write: fii*(x)u1(x)ds(x) +RI)U*(IQUI(x)da(X) B [16] =ftI(x)ui‘(x)ds(x)+fXI(x)ui*(x)da(x) B R for i = 1,2 and where we have employed Einstein's summation convention, l.e. we infer sum- mation over repeated indlces. 9 Now, if we choose the principal solution to be one of the two elastic states, than Ui" = Ui = Uij (11.1) e) (g) V=F=Wkfiqw * = 5 (x - 5) ea (2). Substituting these into eq. [16] gives egwéTIjmflUImdsm + cr(£)|{6(x-.g)ur(x)da(x) [17] =eiliiiéUiIOL£itilxidslxi+éUiI(x.£)Xs(xida(x)i- We can write eiltiéblx-BUleidalxi=0I(£lUI(£)=¢I(£)UI(£) provided 2 is in R. Then, equating coefficients of B], we Obtain U](£)=-éUi(X)Ti](L£)dS(X)+éUmUijmfld8m [13] +IXi(x)Uii(X1£)d3(X)- R For convenience, at this point, we recognize that Uil (ex) = Uii (u) = W (at) T1 (5.4) = 'Tii (at) in eq. [14] and eq. [15] provided that we let u = n (i) in eq. [15]. This allows us to interchange x and g and rewrite eq. [16] as mix) =é"i(£)7ii(&£)dfl£) +BiiI(£)UII(x.£)dsiti [19] + I{XIieiUiIlII-ticltlti for x lnRand l=1,2. Equation [19] also applies for x on B. However, when x is on B and : approaches x, the integrands in the boundary integrals become singular. Specifically, W (11.2) varies as Up and U1) (11.5) varies as in p. Since tip is not integrable, we need to integrate around x_as follows: 10 Figure 2. integration Around Singular Point and take the limit as 1»; goes to 0. Notice that if the boundary has no corner, in = 1: Now on I‘ we have q1 = -cose q2 = -sin6 R1 = case R2 = sine p = E d3 = ado. if we think Of the boundary as being in two parts, i.e., B and B-I‘, then the 1/p type integral becomes IUJTjidS = lim [ijledS + [UjTjids]. e-vo B-l' in the limit as 5 goes to o, the B- I‘ integral becomes a cauchy principal value integral, and we have [u] Tji ds :1; U] Tji ds + lim f U] Tji ds. e-P 0 1' If we define the coefficient (3;; such that lim 1' u] Tji ds = Bji Uj 5-»0 I‘ and let oqi = 511- B]; at x on B, we Obtain (xii u; -f U] Tji ds = [1] Uij ds + [XI U5) da. [20]. B B R Taking X = o for convenience, we have an u; -~f Uj Tji ds = f t; U1] ds [21] B B where U1) and T3; are given by eq. [14] and eq. [15], respectively. 11 Equation [21] is the dashed boundary integral equation. After solving it numerically, we will have the displacement and traction values everywhere on B. Then, to find the displace- ments at any point within R, we simply integrate, l.e. usfuands-i-ftwqu [22] B B Next, employing eq. [2], we have the stresses at any point within R given by mks‘fWSjkldD‘t'fqolkdi [23] B B where Sm and ‘Dikj are given by S111 . G(1 + humid-41112. 1) n1 + 211142 (41112- 1) 1121/21192 8211 = G(1 + 11)[2q1q2(4q12 - 1) m + (411121122- 1) na] [2pr 8122 = 6(1 -i- 1I)[(Bq1:"qz2 - 1) n1 4- 2:11:12 (4:122 - 1) n2] I21rp2 $222 1: on + 1,)(211142 (4422- 1) 111 + (3112‘- «22-1) 112] [211112 S121 = 6(1 + u)[2(l1Q2 («112- 1) m + (8q12q22- 1) 112] [21112 S221 = OH + 11)[(BQ12422- 1) 111 + 24111201422- 1) n2] [211112 [24] D111 =- [-2 (1 + 11):]13-(1 -u)¢]1]/41rp 0121=[2(1 + 11) (123- (3 + 1942] [hp 0221 = (2(1 + 41113— (1 + 31) iii] Map 3 [25] 0112 =[2(1+ 1042 -(1+ 311) 421/4111) 0122 = [2(1 + 11) 1113- (3 + 11):]1] l411p D222 = l-2(1 + 11112" - 11- 4421/4111 CHAPTER 3 NUMERICAL TREATMENT 3.1 BOUNDARY APPROXIMATION TO numerically solve the boundary integral equation, eq [21], we will first approximate the boundary 8 with N straight line segments as shown below. x2 ll N (N) \ (1) / 2 (2) Figure 3. Boundary Approximation 1 Now for any boundary source point, m , and boundary field point , n. , we define the following _x" = Element Field Point _x‘“’ = Nodal Field Point 35'“ = Element Source Point 2““) = Nodal Source Point 113'“: Length Of Element m = [(X1(mi , x,('“"i)2 + (x2011) , x2Inn-11,2] 0.5 _nm = Unit Normal to Element m m = 0‘2"") -X2(m'1)) IAS'“ n2 = (X1‘m"’- X1‘m’) I 43'" p = Distance Between Source Point and Field Point (11 = (X1(")- x1(m))/p Of (X1"- 19me 42 = (x2(“) - x2"”))/ P or 0‘2" - szil 9 12 13 Here the “element points“ are at the midpoints of boundary segments and element quantities . take their midpoint values. Now consider a typical straight segment; m , Figure 4. Local Coordinate with local coordinate, 1) , where -1 s 1) s 1 ds = 0.5 As'" d1) [25] g = 0.5 2"“ “’ (1-11) + 0.5 2"“) (1 + 1,). We can now rewrite the boundary integral equation ajI Ill-£11] Til ds =41; ui) ds. [21] at a given boundary point, x , as N amnion-0.5 2i AsmIUIlniTIIIXmNRI m=1 m [27] N = 0.5 2[ Asmft)(1))UI)(x.n)dn-I m=1 m 3.2 NUMERICAL MODELS Next, consider three cases on segment m (a) "i = 01'" t) = 15'" 14 (b) 1:1 = 0.5 11”" (1 ~11) + 0.5 11"” (1 + 11) ti = ‘1'" (c) “l = 0-5 01"" ‘ " (1 -11) + 0-5 Ui‘m’ (1 + 11) t1 = one" (1 -11) + 0.5111111 + 11) Notice the differences among the cases. Case (a) is the simplest because it assumes both the displacements and fractions to be constant over segment m. Case (b) is slightly more complex in that the displacements are allowed to vary linearly. Case (c) takes the next iogi- . cal progression and allows the fractions to vary linearly also. Clearly, further progressions are possible such as quadratic and cubic variations, however, for our purposes, these three are sufficient. Also, keep in mind that the constant quantities refer to values at the midpoint of segment m. We now rewrite eq. [27] for the field point in each of the three cases. (a) Let x = x“ 241“ uI"-o.5 $31.11)”). (£111)an U)” m: "Is!" [28] n =0.5 2 [AOmeijunr'II’dnl‘Im .m=1 m (b) Let x = xi") " (n) In) N m In) (m -1) “ii "i 43-252 [As iii-ninth: .nidniui m=1 m insert-+1 N -o.25 2 (42m {(1 + 1,)T11u‘"’.11)d111u1‘"" [29l m=1 m . Math N =o.5 2 [AsmIUI)(x(").n)an‘im m=1 m 15 (c) Letx = xi") A N . 21‘"’u1‘“’-o.2s 2 [Asm i (1 ~11) Ti 01‘"). «1)d111 ui‘m'” m=1 m mach -i- 1 N -o.25 2 [Asan [(1 + 11m (x‘n’.11)d11] Uj(m) [30] m=1 m ”1:311 N = 0.25 2 [Asme-n) Uii(1!(n)1‘q)dnltj(m'1) m=1 m + 0.25 : [Asm f (1 + 11) U1) (16"),11) d 11 1 11"") m = 1 m Notice that any integrals multiplying U)" or u,“ were incorporated into the coefficients 01,1" or 041‘") in eqs. [28], [29] and [30]. Also notice that the displacements and fractions are now either nodal or midpoint values and are, therefore, taken outside the integrals. This allows us to express the equations in matrix form as Case (a): [As] {U}e = [Ba] {t}. Case (b): [AB] {U}n = [3b] {1}. [31] C359 (0)3 [Ac] {UIn = [3c] {0n Essentially, we now have our numerical models. A more convenient form, however, would be tO have [F] {X} = {b} for each case where [F] and {b} are known. We can Obtain this desired form by recognizing that {u} and {t} are not known everywhere on the boundary and simply rearrange the equa- tions to have all unknown quantities on the left and known quantities on the right. For convenience, we will make one further adjustment to the models. Since we will be com- paring results for each case on a number of example problems, we desire the input data to be consistent. As it stands, our models use both element and nodal quantities. We may change this if we let 16 Me = Ir)T {Uin me = IIIT {tin so that [A2] {UIe = [32] {Us (At) m" Me = [81:] {tie [32] [Ac] IrI'T Me = [Bci Ir I‘T me where the number of nodes is Odd and Ir)" -T T —I g f i -i I E o o o I7 1 3 I I -I I I (I) 8 9 _ ‘ . . . I . 0 I U = I I I I . [i‘i‘= 0 0 I I 0 [33‘ ' p o o o I] I -1 1 ~I I_ L 3.3 MATRIX EVALUATION The coefficients in matrices [A] and [B] for each case may be calculated using Simple Gauss quadrature. This is a well-known technique and will not be discussed here, however, details Of the procedure may be found in (1). TO apply Gauss quadrature, we must have T): and Us) in proper form. This requires p , q1, (12, R1 and n2 in terms 0111. Employing eq. [26], we have 2‘“) - I: = 12‘“) 12'“) - Ix‘m’ - 2m] 11 = 50"“) - 5mm 11 and 21” - 8 = [21” - Km] - [210“) - am] 11_ _ finm _ 8mn'I where we have defined ., m (34) 51m = 21' ~21 17 Then for case (a) 22 = 11"” -a""‘1-2 14"” - 4"“) 11 + 14"“ - 4mm) 112 (I1 = (Otnm - 81mm 1)) I p Q2 = (82"m - 62mm 1)) I p [35] n1 = 2 52mm” 112 = -2 61”” I As'“ and for cases (b) and (c) 2’ = in”) - a‘"“"l - 21 am" 2 5"“"1 11 + 111mm”! 11" Q1 = (81 (m) - 61mm 1)) / p 112 = (a2‘"'“’-52'""‘ 11m 1361 m = 2 82mm/Asm n2 = -2 51"“ I 43'“ Equations [24), (29), (so), (35] and [33], together 111111 the 1111111111 forms given in (31) and [32], provide the basic numerical. models for each of the three cases. However, further dis- cussion is required because the integrals become singular when the source point and field point coincide. The coefficients given by the singular integrals will be referred to as "special" coefficients and are: case (a): Asm([T)1d11)/2 m=n m Asm(fU1)d1))/2 m=n m case(b): As'“[[(1-1))T)Id1)]/4 m=n,n+1 m Asm[](1 +11)T)1d1)] l4 m=n,n+1 In A8m(fUijd11)/2 m=n,n+1 m 18 case(c):As"‘[fi(‘(1-11)T)Id11]/4 m=n,n+1 MmEfiIU-niTiian/‘i m=n,n+1 A8m[n{(1-11)Uijdnl/4 m=n,n+1 A'mlrfii'fl'tfliuid'fiI/‘I m=n,n+1. Altogether then, sixteen special coefficients need to be dispositioned. Recall, however, that five of these are incorporated into the aji coefficients as shown in eqs. [26], [29] and [30]. Let’s look at these coefficients more closely. Suppose x is a corner point on B and define the following angles: IX Figure 5. Free Term Coefficient Then, proceeding as we did in Chapter 2, we get (1)/211:0+u)[SIn2(a+W)-Sin2a]/Ba |=i ail (1+u)[sin2(a+w)-sin2a]/4w 1.21 so that when m = 11, 1111 = 81/2. it is clear , then, that these coefficients represent only rigid- body displacements and may, therefore, be ignored. Now we are left with eleven special coefficients. However, those involving T1 in cases (b) and (c) are identical so we are actually left with nine. To evaluate these, we need the follow- ing local coordinates: Onm=n: (n) Ix ix 1%”, xfn-I) QTY lffl-l) 5‘") 19 051151 n=n’ dn=dn' p=ASm11’/2 [37] q1=n2 q2=-n1 -1 s 11 s D n=-n’ dn=-dfi p=As"‘-11’Iz (33] 41=-n2 q2=n1 -1 S 11 st 1) = 1-211' 61) =-2d'rj' p-n'As" 1391 (113-02 q2=n1 Figure 6. Local Coordinates On m c n 20 On m=n+1: -15 11 st (n+1) T" n a 211"1 d1] = 2611' p=11'A3m [40] (11) Q1 3 "2 Ix qz = -m Figure 7. Local Coordinates On m = n + 1 Now consider case (a), which has the lollowlng special coelflclent: AsmUUijdnHZ m=1. m Using equations [37] and [38], this becomes 1 n 0 1 {”1 (X m) d “n = '{Uii (11’) d 11’ +6! "ii (11') d 11’- [41] Equations [37] and [38] both yield identical results when substituted Into equations [14] and we have u" = {-(3-u) In (Aam we) + (1 + u)l122]/81tG U12 a - (1 + 1:) n1 02/8116 [42] U21 = - (1 + v)n1n2/81:G U22 = [- (3-u) In (Asm “'12) + (1 + 1:) ma] 181:6. This result allows us to write 0 1 1 '11' ”ii (11’) d 11' + 1‘; ”ii (11') d 11' = 2‘1; "ii (11') d n’- [43] 21 Using equations [42] and [43],wenow writethespeciaicoetlicientsiorcase(a) AO"([U11d1|)/2 B (As"l8c6) [(3-11)-(3-11)in(As"‘/2) + (1 + u) 022] n As"(]U12d-n)/2=-(As"/8w6)(1 +11)n1n2 n n n [44] As(IU21dn)/2=-(As [Second-10111112 n As"([U22d-n)/2 = Min/8116)[(3-u)-(3-u)ln(A8m/2) + (1 +11) n12] l'l Both case (b) and case (c) have the following special coefficients: A8m[[(1-n)T]id'q]/4 m=n m Asm[[(1 +n)T]1d11]I4 m=n-+1 m The singularity in T11, however, ls oi the tom 1 / p, and p goes to zero at the same rate as (1 :1.- 11). This means that the integrals are well-behaved and may be evaluated numerically. We now turn our attention to the liq - type coefficients 01 cases (1:) and (c). From equations [39] and [40], we have 1 1 (b) A8"i1f U11 11"“. n) a «11/2 = “"5 U1 (1') d «1' 1 1 (C) A8"[1I (1- 11) U1. 01‘"). 11) d n l / 4 = “"0! 11' "ii (11') d 11’ [45] 1 1 Mi; 11 + 1101 W. n) d «11 I4 - As" g (1 - 11') U» («n d «1' 22 m=n+1: 1 1 (b) 48"“1‘! "101(th n) a 111 I2 = w“: 01m 6 n' 1 1 (c) AI"“[{(1-II)UII(X‘"+".II)¢ «11 I4 = 111"“ g (1 «n on (n'idn' 1461 1 1 A8"+'i.1i(1+ 11) ”ii “(n+1)1"|)d 111/4 = A'M'gn’ ”ii (11') d 11’ Just as equations [45] and [46] are identical in form, substitution of equations [39] and [40] into equations [15] shows that U1] is identical in form on m + n and m = n + 1. U11 = {-(a - u) in (mm 11’) + (1 + u) 112’] I816 U12 = -(1+ u)n1112/81rG i471 U21 = -(1 + 11)n1n2/8nG 022 =- [-(3-11)ln (Aamn') + (1 + u) 11121/811’6 Alter making the substitutions and carrying out the integrations, we get the following results: case (b): 1 Aamend-n’ = Asml-(3-v)lnAsm + (3-u) + (1 +11) “221/816 1 Asme12dn’ = -As"‘ (1 +11) n1 "2/8116 0 [48] 1 AsmgUmd-n's-Asmfl +u)fl1fl2/81rG 1 Asmezzd'q’ = As'“ [-(3-11)lnAsm + (3-11) + (1 + 1.011121/81'6 o 23 cese (c): 1 AOmg‘n1U11d'n’5' As'" [~0.5(3-11)|n mm + 0.25 (3-u) + 0.5 (1 + u)1122]/81rG 1 As'“ I n' U12 (1 11’ = A8.“ (0.5) (1 + u) 111 112/ MG 0 [49] 1 Aim 1' n' U21 d 11' = -A8m (0.5) (1 + u) 111 112/ 816 0 1 _ Asmfn’Uzzd-n' = As'“ {-0.5(3-u) mm” + 0.25 (3-u) + 0.5 (1 + u)1112]/81rG o 1 2 Asm [(1-11’)U11dn' = As'" [(3-11) (0.75- 0.5 In As'“) + 0.5(1 + 11) n2 1 I816 o 1 As'“ f (1 un’) U12 dn’ = - As'" (0.5) (1 + u)111112/81rG o [50] 1 :1st (1 -11’)U21d11’ = - As'" (0.5) (1 + 10111112th 1 Asmg (1 - 11') U2 dn’ = M” [(3-11) (0.75- 0.5 In As'“) + 0.5 (1 + 11) 1112] I 8116 wherem = n,n +1. it is important to note that in case (c), the (1 - 11') - type integrals add together for m = n and m = n + 1. Recall that the coeflicients in [A] , which multiplied U]", for all three cases, were incorporated into the a]: coefficients. These were then found to represent rigid - body displacements and are therefore ignored. In doing this, however, we have created a diagonal void in the [A] matrix. We may fill this void by determining the diagonal entries In the following manner. Consider a rigid - body displacement applied to the system. This will produce no stress and, therefore, [A] {U} 3 [3] {0} = {0}- in other words, 24 1A11u1+A12U2+A13ua+... +A12nUzn=o. So we have that the diagonal entries of [A] are the negative sums of the off-diagonal entries. 01 course, the diagonal entries referred to here are actually 2 x 2 block matrices and are given by N A12n-1)(2n°1) = - 2 A(2n1)(zm.1) m at n m=1 N A(2n-1)(2n) = - }: A(m.1)(2m) mean m=1 [51] N A(2n)(2n-1) = - 2 Amman—1) 1118111 m=1 N . A(2n)(2n) = - 21A(2n)(2m) mean I“: In summary, the following equations are used to calculate the matrices: [A] Matrix: case (a) case 0») case (c) [8] Matrix: case (a) case (b) case (c) 3.4 INTERIOR FIELD POINTS [1511 [35]. [51] [15]. [35]. [51] [151113511 [51] [14]. [3511 [44] [14]. [33]. [48] [14]. [33]. [49]. [50] Following the same reasoning used in the previous section, we may rewrite equations [22] and [23] tor the three cases. 25 N “80(8): u1 = 211(A8m/zim {Tim-11) 111,111,” + m IIMZ 1[( 48m/ 2M ”II 01m) 1'11 l 1'" [52] on: 8111:: 1“ Atm/ 2) I 81111 (3.11)an "J” + 2 [(AIm/zimfbiki (midniiim m=1 _N “3° M m II AsmIai I (1 -nI Tn (m) an I 11"" " N + 2 i( AsmIai I I1 + 107101.11)an uI‘m’ m=1 111 3M2 II AsmIzi .{1 Un (me an I 1'“ [531 M2 aII= II As ”/41 I I1 - «1) $qu (511)an "1"" " 21liA'm/4InI I1 + 11) SN 01.11)an 11"“) + 111 IIMz [( Asm/z) g. Dun (11m) 611 1 i1'" + m 1 N cm (Cr 01 = 2 II tum/4) I I1 -n) In (11.11)an uI‘m'” m=1 m N _2 II Asm/4) .1. I1 + «In» (11.11)an 11"” + 111 1 + 2 [(Asm/II) [(1.19 u" “milieu“ 1) m=1 . ”In: I (mm/4) I I1 + «1) ”ii (am) an i I‘m’ 26 N 154] I = 2 I Mam/4) I (1 - 11)si|d (3.111511 I 11“" m=1 m ilMZ +m§IAsmI4InI (1 + n) slid (8.11)an 11"“) 11.15“ Mm“) "I I1 - n) 011 (m) 4n I 4""'” +m2 1IIAsmIII) I I1 + «1) an (m) an I 4"" ’ Applying equations [24] and [25] and Gauss quadrature, these equations may be used to obtain displacements and stresses at any point in the body CHAPTER 4 EXAMPLES AND RESULTS 4.1 PROBLEM DESCRIPTIONS Our objective is to assess the performance of each of the three numerical models in solv- ing linear isotropic elasticity problems. Three example problems were chosen tor this purpose. They represent a range at geometries and boundary conditions intended to highlight the rela- tive strengths and weaknesses of each model. The example, problems are described in the following figures along with boundary condi- tions, exact solutions and boundary meshes. These figures are followed by tables containing the output data and percent differences tor all three models in each problem. 27 28 Y TX='O. TY=1. 2 EXACT SOLUTION UX = -O.1x UY = 0.4y SXX 8 O. UX = O. TX 8 0. TX=O. ‘U=O.25 TY=0. SXY=O. SYY 1. $ x O 1 TX = O TY = -1 15 12 4. .34 1811 52 0 1 4 1 10 21 W8 * 63 ELEMENTS * * Between each node number, elements are of equal size. Figure 8. Problem. One - Pure Extension 29 y 0 TX = -O.5x TY = 0.75 1 UX = O. TX = 3.75 + 0.5 y1 TY = O. 1} = 0.25 TY’= 0.5y - 1.0 —a- x O 1 IX - x UY = -( + 2) X’ / (1 + ) EXACT SOLUTION (18) ux = 0.5[4x - x’/12 + y’/2 - x(y - y'/4)] / (1 + ) UY - 0.5[yz/2 - y’/1§ - (4y - x’y/4 + y’/6) - (1 + )x2 + x /2]/(1 + SXX = 4 - x’/4 + y’/2 SXY = xy/2 - x SYY = y - y’/4 16 11 46 31 i 1 6 1 16 21 ELEMENTS * 63 ELEMENTS * * Between each node number, elements are of equal size Figure 9. Problem T140 - Square With Variable Boundary Conditions UX 8 O TY = -1 TX 8 1 TX = -1 UY - O UY = O UX = O TY = 1 Otherwise: TX = - nx TY - - my 17 1 19 ELEMENTS * EXACT SOLUTION UX = -O.3x UY - -O.3y SXX - -1 SXY - O SYY = -1 29 800 35 ELEMENTS * * Between node numbers shown, elements are of equal size. Figure 10. Problem'Three - Circular Disk Under Uniform Pressure 31 4.2 TABLES OF RESULTS The following tables summarize the numerical solutions of the three example problems. Each problem was solved twice using an increasing number of nodes. The tables should be self-explanatory, however, two small clarifications are necessary. First, since percent difference is meaningless for elements where the quantity is either specified or equal to zero, these elements are omitted from the percent difference table. Second, all displacement values are multiplied by the shear modulus. 4.3 DISCUSSION The problems described in the previous section represent a range of geometries and boundary conditions. These problems were chosen to include smooth boundaries and boun- daries with corners as well as constant and variable boundary conditions. The performance of each model in these basic problems will then provide information on their applicability in general. Problem one has a straight boundary with corners and linear displacements with piecewise constant fractions. The straight boundaries allow an exact boundary representation and the variable behavior is identical to that assumed in case (b). Therefore, one would expect case (b) to solve this problem exactly and, indeed, it does. The other cases differ in their assumed variable behavior and, therefore, yield approximate results. in case (a), the assumed traction behavior is consistent with the problem, however, the displacements are piecewise constant. Conversely, in case (c), the assumed displacement behavior is consistent, but the tractions are piecewise continuous. Therefore, in cases (a) and (c), we see increasing inaccuracy near comer points where there is a change in variable behavior. These inaccuracies in the bound- ary solution carry over into the interior solution as well. However, interior solutions are, in general, more accurate except for points near the boundary. Problem two also has straight boundaries with corners, however, the boundary conditions are more complex. One would still expect to see lnaccuracy near corners, however, it Is not 32 clear what effect the assumed variable behavior would have. Since the piecewise constant behavior is more forgiving at corners, it would appear cases (a) and (b) might prevail. Problem three has a circular boundary which introduces boundary approximation errors in all three cases. The variable behavior does not lrnmediately favor any case so we expect, and find, similar results in each. Table 1. XEDISPIACEMENTS N0 1X Y 1 0.00000 .12500 2 .16667 0.00000 3 .50000 0.00000 4 .83333 0.00000 5 1.00000 .12500 6 1.00000 .37500 7 1.00000 0.62500 8 1.00000 .87500 9 1.00000 1.12500 10 1.00000 1.37500 11 1.00000 1.62500 12 1.00000 1.87500 13 .83333 2.00000 14 .50000 2.00000 15 .16667 2.00000 16 0.00000 1.83333 17 0.00000 1.50000 18 0.00000 1.16667 19 0.00000 .87500 20 0.00000 .62500 21 0.00000 .37500 Table 2. Y‘DISPIACEMENTS Nihih'h'h'hf hihf PIC>E>G1~10151hnhi x - 0.00000 .1666? .50000 .83333 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 .83333 .50000 .1666? 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 ' Y .12500 0.00000 0.00000 0.00000 .12500 .37500 .62500 .87500 1.12500 1.37500 1.62500 1.87500 2.00000 2.00000 2.00000 1.83333 1.50000 1.16667 .87500 .62500 .37500 UXA 0.00000 -.02301 -.05480 -.08671 -.11030 -.11029 -.10990 -.10980 -.11000 -.11060 -.11162 -.11216 -.08878 -.05730 -.02706 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Pure Extension - 21 Nbdes: 023 .05323 0.00000 0.00000 0.00000 .05362 .15182 .25156 .35152 .45156 .55155 .65119 .74901 .80226 .80312 .79939 .73029 .59941 .46688 .3506? .25100 .15141 33 UXB 0.00000 -.01667 -.05000 -.08333 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.08333 -.05000 -.01667 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 UYB .05000 0.00000 0.00000 0.00000 .05000 .15000 .25000 .35000 .45000 .55000 .65000 .75000 .80000 .80000 .80000 .73333 .60000 .4666? .35000 .25000 .15000 UXC 0.00000 -.0224? -.04920 .00321 -.03224 -.11801 -.10915 -.1059? -.10615 -.10578 -.09857 -.02072 .01120 -.01054 .0279? 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 UYC .1291? 0.00000 0.00000 0.00000 .1608? .14766 .22970 .33063 .42866 .51974 .59108 .61680 .73813 .76959 .87782 .7807? .63285 .49113 .35124 .20905 .15842 Pure.Extension.- 21.nodes: XI- Displacements .EXACT 0.00000 -.01667 -.05000 -.08333 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.08333 -.05000 -.0166? 0.00000 0.00000 0.00000 ‘0.00000 0.00000 0.00000 Y - Displacements EXACT .05000 0.00000 0.00000 0.00000 .05000 .15000 .25000 .35000 .45000 .55000 .65000 .75000 .80000 .80000 .80000 .73333 .60000 .4666? .35000 .25000 .15000 Table 3. XFTRACIICNS NO .x 1 0.00000 2 .16667 3 .50000 4 .83333 5 1.00000 6 1.00000 7 1.00000 8 1.00000 9 1.00000 10 1.00000 11 1.00000 12 1.00000 13 .83333 14 .50000 15 .16667 16 0.00000 17 0.00000 18 0.00000 19 0.00000 20 0.00000 21 0.00000 Table 4. YJTRACTIGNS NO X 1 0.00000 2 .16667 3 .50000 4 .83333 5 1.00000 6 1.00000 7 1.00000 8 1.00000 9 1.00000 10 1.00000 11 1.00000 12 1.00000 13 .83333 14 .50000 15 .16667 16 0.00000 17 0.00000 18 0.00000 19 0.00000 20 0.00000 21 0.00000 Y .12500 0.00000 0.00000 0.00000 .12500 .37500 .62500 .87500 1.12500 1.37500 1.62500 1.87500 2.00000 2.00000 2.00000 1.83333 1.50000 1.16667 .87500 .62500 .37500 Y .12500 0.00000 0.00000 0.00000 .12500 .37500 .62500 .87500 1.12500 1.37500 1.62500 1.87500 2.00000 2.00000 2.00000 1.83333 1.50000 1.16667 .87500 .62500 .37500 g? §§§§§ E §§§§§ . 00000000000000! ggo 3 § -.004?3 -.02091 .02341 .00012 -.00363 TEA -1.00351 -.99128 -1.00616 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 34 TXB .00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 .00000 .00000 .00000 .00000 .00000 .00000 TYB 1.00000 §§§§§§ Pure Extension.- 21 Nbdes: x1-1Tracticns TXC -.11642 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -.32744 -.01711 -.00401 .05992 .04106 -.12700 Pure Extension - 21 Nbdes: Y - Tracticns ‘TYC 0.00000 -.86657 -.9?328 -1.45664 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 1.00000 1.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 §§§§§ é? E 000000000000000000000 Exmcr 0.00000 -1.00000 -1.00000 -1.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 1.00000 1.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Table 5. INTERIGRJX+DISP @GQGUI-bUNi-‘a XI .25000 .25000 .25000 .50000 .50000 .50000 .75000 .75000 .75000 Y .50000 1.00000 1.50000 .50000 1.00000 1.50000 .50000 1.00000 1.50000 INTERICRLY‘DISP omqmmauun-Ia IX .25000 .25000 .25000 .50000 .50000 .50000 .75000 .75000 .75000 INTERIOR.SXX ‘DQQO‘UI-thI-‘a INTERIOR.SXNT mooqosmauNI-Ia IX .25000 .25000 .25000 .50000 .50000 .50000 .75000 .75000 .75000 IX .25000 .25000 .25000 .50000 .50000 .50000 .75000 .75000 .75000 1.00000 1.50000 .50000 1.00000 1.50000 .50000 1.00000 1.50000 1.00000 1.50000 .50000 1.00000 1.50000 1.00000 1.50000 ‘Y .50000 1.00000 1.50000 .50000 1.00000 1.50000 .50000 1.00000 1.50000 UXA -.02964 -.03032 -.02984 -.05507 -.05523 -.05591 -.08047 -.08036 -.08145 UXA .20142 .40116 .59992 .20143 .40114 .6007? .20165 .40132 .60101 A. .00035 .0096? '-.02427 -.00587 -.00348 -.00754 .00081 .00281 -.00018 -.00061 .00162 .00385 .00010 .00056 .00015 .00186 .00019 -.00320 35 UXB -.02500 -.02500 -.02500 -.05000 -.05000 -.05000 -.07500 -.07500 -.07500 .20000 .40000 .60000 .20000 '.40000 .60000 .20000 .40000 .60000 Pure Extension - 21 Nodes: Interior Values UXC -.02002 -.03363 -.00236 -.04395 -.05881 -.04264 -.08055 -.08084 -.07538 UXC .20889 .40849 .61530 .19088 .39594 .59726 .18026 .38623 .58238 4C .05022 -.00209 1.31848 -.04520 .02039 -.06053 -.06660 .06332 -.02828 1C -.24064 -.23927 -.13439 -.15109 -.08109 .00618 .04726 -.01461 -.03663 EXNCT -.02500 -.02500 -.02500 -.05000 -.05000 -.05000 -.07500 -.07500 -.07500 .20000 .40000 .60000 .20000 .40000 .60000 .20000 .40000 .60000 .EXACT 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.0000 EXACT 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Table 5 (Contld.) DNTERIOR.SRY mmqmmhunwa Table 6. IX .25000 .25000 .25000 .50000 .50000 .50000 .75000 .75000 .75000 Y .A .50000 .99402 1.00000 .98745 1.50000 1.00367 .50000 .99949 1.00000 .99635 1.50000 .99900 .50000 .99370 1.00000 .99351 1.50000 .99346 XFDISPIACEMENES gusfisomqmmbunai ._.; In .A -38.07 -9.61 -4.06 -10.30 -10.29 -9.90 -9.80 -10.00 -10.60 -11.62 -12.16 -6.53 -14.61 -62.34 .B .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 Y-DESEIJKIiflflIES B» .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 C -34.83 1.60 103.86 67.76 -18.01 -9.15 -5.97 -6.15 -5.78 1.43 79.28 113.44 78.92 267.81 36 ‘C .86865 1.06814 1.03900 1.03050 1.00854 1.03846 1.07657 .95636 .98414 §§§§§ ”é‘éé’éw § Ema Ebctensim - 21 Nodes: Parcent Differences Table 6 (oa'tt'd.) 19 -019 20 “.40 21 -094 Y‘TRHCTICNS NO .A 2 -.35 3 .87 4 -062 .00 .00 .00 .00 .00 INTERIORLXPDESP .A -18.57 -21.28 -19.38 -10.14 -10.46 -11.83 -7.29 -7.15 -8.60 UGQQUIQUNHS .B .00 .00 .00 .00 .00 .00 .00 .00 .00 INTERIOR.Y‘DESP -4.44 -2.12 -2.55 4.56 1.02 .46 9.87 3.44 2.94 NO A. B 1 -.71 .00 2 -.29 .00 3 .01 .00 4 -.72 .00 5 -.28 .00 6 -.13 .00 7 -.82 .00 8 -.33 .00 9 -.17 .00 INTERIOR.SYY NC .A. B 1 .60 .00 2 1.25 .00 3 -.37 ' .00 4 .05 .00 5 .37 .00 6 .10 .00 7 .63 .00 8 .65 .00 9 .65 .00 13.13 -6.81 -3.90 -3.05 -.85 -3.85 -?.66 4.36 1.59 37 Table 7. X#DISP1ACEMENTS omqmmbuuwoomqmmeuuwommumwauwwo 0‘10 b owmqmmauuw ‘X 0.00000 .05556 .16667 .27778 .38889 .50000 .61111 .72222 .83333 .94444 §§ O 2 5 q 0000.00000 1.54167 1.62500 1.70833 1.79167 1.87500 1.95833 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 1.94444 1.83333 1.72222 1.61111 1.50000 1.38889 UXA 0.00000 -.00785 -.01849 -.02950 -.04055 -.05161 -.06268 -.07374 -.08478 -.09545 -.10337 -.10349 -.10343 -.10338 -.10335 -.10332 -.10330 -.10329 -.103297 -.10330 -.10331 -.10334 -.10338 -.10342 -.10348 -.10355 -.10362 -.10370 -.10379 -.10388 -.10395 -.10400 -.10398 -.10367 -.09569 -.08509 -.O7421 -.06326 -.05230 -.04134 -.03040 -.01952 -.00927 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 38 UXB 0.00000 -.00556 -.01667 -.02778 -.03889 -.05000 -.06111 -.07222 -.08333 -.09444 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 UXC 0.00000 -.01587 -.02705 -.03015 “.03953 “.04983 -.06018 “.06975 -.07539 -.05542 “.07145 -.10864 -.10299 “.09963 -.09837 -.09787 -.09769 “.09767 -.09773 -.09783 -.09809 “.09821 “.09831 -.09841 “.09849 -.09858 -.09869 “.09884 “.09911 -.09971 -.10104 -.10075 “.07178 -.06085 '.07232 -.06603 -.05711 -.04746 -.03783 -.02868 -.01652 .00876 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Examsim-fluodes: x-Displaoalmrts EXACT 0.00000 -.00556 -.01667 -.02778 -.03889 -.05000 -.06111 -.07222 -.08333 -.09444 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 --.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.10000 -.09444 -.08333 -.07222 -.06111 -.05000 -.03889 -.02778 -.01667 -.00556 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Table 7 waned.) 50 0.00000 51 0.00000 52 0.00000 53 0.00000 54 0.00000 55 0.00000 56 0.00000 57 0.00000 58 0.00000 59 0.00000 60 0.00000 61 0.00000 62 0.00000 63 0.00000 Table 8. Y‘DISPIACEMENIS Etgomqmmeuwwa IX 0.00000 .05556 .1666? .27778 .38889 .50000 .61111 .72222 .83333 .94444 §§§§§§§§§§ Eééé’ééééééww i 1.27778 1.1666? 1.05556 .95833 .87500 .7916? .70833 .62500 .5416? .45833 .37500 .2916? .20833 .12500 .12500 .20833 .2916? .37500 .45833 .5416? .62500 .70833 .7916? .87500 .95833 1.04167 1.12500 1.20833 1.2916? 1.37500 1.45833 1.5416? 1.62500 1.70833 éééééééééééééé 0.00000 .0179? .05074 .08398 .1172? .15058 .18390 .21723 .25056 .28389 .31723 .3505? .38390 .41724 .4505? .48390 .51722 .55053 .58383 .61710 .65035 .68355 39 3???? §§§§§§§ UYB .0166? 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 .0166? .05000 .08333 .1166? .15000 .18333 .2166? .25000 .28333 .3166? .35000 .38333 .4166? .45000 .48333 .5166? .55000 .58333 .6166? .65000 .68333 §§§§§§ §§§§§ 00000000000000 00000000000000 § UYC .06515 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 .07223 .03992 .06209 .0957? .13026 .16454 .19858 .23244 .26615 .29973 .33320 .3665? .39985 .43305 .46618 .49923 .53220 .58505 .59??? .63026 .6622? Pure Extension.- 63 ches: Y - Displacements .0166? .05000 .08333 .1166? .15000 .18333 .2166? .25000 .28333 .3166? .35000 .38333 .4166? .45000 .48333 .5166? .55000 .58333 .6166? .65000 .68333 Table 8 mama.) 32 1.00000 33 1.00000 34 1.00000 35 .94444 36 .83333 37 .72222 38 .61111 39 .50000 40 .38889 41 .27778 42 .16667 43 .05556 44 0.00000 45 0.00000 46 0.00000 47 0.00000 48 0.00000 49 0.00000 50 0.00000 51 0.00000 52 0.00000 53 0.00000 54 0.00000 55 0.00000 56 0.00000 57 0.00000 58 0.00000 59 0.00000 60 0.00000 61 0.00000 62 0.00000 63 0.00000 Table 9. XHTRACTICNS NO X 1 0.00000 2 .05556 3 .16667 4 .27778 5 .38889 6 .50000 7 .61111 8 .72222 9 .83333 10 .94444 11 1.00000 12 1.00000 13 1.00000 1.7916? 1.87500 1.95833 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 1.94444 1.83333 1.72222 1.61111 1.50000 1.38889 1.27778 1.1666? 1.05556 .95833 .87500 .7916? .70833 .62500 .5416? .45833 .37500 .2916? .20833 .12500 Y .0416? 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 .0416? .12500 .20833 .71669 .74970 .78223 .80016 .80096 .80115 .80120 .80116 .80106 .80088 .80054 .79916 .7760? .73259 .68852 .64429 .59998 .55563 .51126 .4668? .4224? .38362 .35031 .31701 .28370 .25040 .21709 .18379 .15049 .11719 .08391 .0506? TXA -.00253 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 . 40 .7166? .75000 .78333 .80000 .80000 .80000 .80000 .80000 .80000 .80000 .80000 .80000 .77778 .73334 .68889 .64445 .60000 .55556 .51111 .4666? .42222 .38333 .35000 .3166? .28333 .25000 .2166? .18333 .15000 .1166? .08333 .05000 TXB .00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 .69246 .71451 .71602 .76016 .76540 .76775 .7683? .76879 .76899 .76822 .76479 .81089 .77810 .72948 .67976 .63330 .58801 .54319 .49872 .45473 .41294. .36846 .34443 .30231 .26954 .23668 .20383 .17105 .13840 .10594 .04068 .04905 Pure Extension - 63 Nbdes: 1X - Tractions TXC -.09?36 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 .7166? .75000 .78333 .80000 .80000 .80000 .80000 .80000 .80000 .80000 .80000 .80000 .77778 .73333 .68889 .64444 .60000 .55556 .51111 .4666? .42222 .38333 .35000 .3166? .28333 .25000 .2166? .18333 .15000 .1166? .08333 .05000 0.00000 0.00000 0.00000 0.00000 Table 9 (ant'dd 14 15 16 1? 18 19 20 21 22 23 24 25 26 2? 28 29 30 31 32 33 34 35 36 3? 38 39 40 41 42 43 44 45 46 4? 48 49 50 51 52 53 54 55 56 5? 58 59 60 61 62 63 1.00000 1.00000 1.00000 1.00000 EEEEEEEEEEEEEEEEE .94444 .83333 .72222 .61111 .38889 .27778 .1666? .05556 E EEEEEEEEEEEEEEEEEEEE .2916? .37500 .45833 .5416? .62500 .70833 .7916? .87500 .95833 1.0416? 1.12500 1.20833 1.2916? 1.37500 1.45833 1.5416? 1.62500 1.70833 1.7916? 1.87500 1.95833 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 1.94444 1.83333 1.72222 1.61111 1.50000 1.38889 1.27778 1.1666? 1.05556 .95833 .87500 .7916? .70833 .62500 .5416? .45833 .37500 .2916? .20833 .12500 EEE EE EEEEEEEEEEEEE EE 0000000000000?0000000000000 41 E E E E E E E E E E E E EEEEEEEEEEEEEEEEEEEEEEEEEEEEEE 0000000000000000000000000.0000000000000000000000000 E E E E E E E E E E E E E mg EEEEEEEEEEEEE 00002 00002 EEE 00001 00001 00001 E I 0 U N 3 N -.00?38 -.02542 .10233 .09651 -.22024 EEEEEEEEEEEEEE EEEEEEEE‘EEEEEEEEEEEEEEEEEEEEEEEEEEEE Table 10. YHTRACEIDNS NO .X 1 0.00000 2 .05556 3 .16667 4 .27778 5 .38889 6 .50000 7 .61111 8 .72222 9 .83333 10 .94444 11 1.00000 12 1.00000 13 1.00000 14 1.00000 15 1.00000 16 1.00000 17 1.00000 18 1.00000 19 1.00000 20 1.00000 21 1.00000 22 1.00000 23 1.00000 24 1.00000 25 1.00000 26 1.00000 27 1.00000 28 1.00000 29 1.00000 30 1.00000 31 1.00000 32 1.00000 33 1.00000 34 1.00000 35 .94444 36 .83333 37 .72222 38 .61111 39 .50000 40 .38889 41 .27778 42 .16667 43 .05556 44 0.00000 45 0.00000 46 0.00000 47 0.00000 48 0.00000 49 0.00000 .95833 1.0416? 1.12500 1.20833 1.2916? 1.37500 1.45833 1.5416? 1.62500 1.70833 1.7916? 1.87500 1.95833 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 2.00000 1.94444 1.83333 1.72222 1.61111 1.50000 1.38889 TIA 0.00000 -1.00852 -.99576 -.99?84 -.99804 -.99820 -.99823 -.99818 -.99608 -1.01001 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 .1.00000 1.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 42 TYB 1.00000 Pure Extension.- 63 ches: Y'-»Tractions TYC 0.00000 -.92313 -1.05889 -1.051?9 -1.02182 -1.00505 -.99108 -.98?59 *.?2985 -1.51632 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0400000 0.00000 0.00000 0.00000 0.00000 0.00000 ERICT 0.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Table 10 (cam-Jo.) 50 0.00000 51 0.00000 52 0.00000 53 0.00000 54 0.00000 55 0.00000 56 0.00000 57 0.00000 58 0.00000 59 0.00000 60 0.00000 61 0.00000. 62 0.00000 63 0.00000 Table 11. 1.27778 1.1666? 1.05556 .95833 .87500 .7916? .70833 .62500 .5416? .45833 .37500 .2916? .20833 .12500 INTERIOR.X+DISP UQQO‘UI-wal-‘a DmflmmfiUNI-‘a IX .25000 .25000 .25000 .50000 .50000 .50000 .75000 .75000 .75000 >4 .25000 .25000 .25000 .50000 .50000 .75000 .75000 .75000. INTERIOR.SXX manner-o5 IX .25000 .25000 .25000 .50000 .50000 1.00000 1.50000 1.00000 EEEEEE °EEEEEEE°°°°°° E UXA -.02651 -.02668 -.02686 -.05167 -.051?9 -.05204 -.0?682 -.07689 -.0?726 .20044 .4004]. ' .60018 .20046 .40043 .60041 .20052 .40049 .60043 -.00166 -.00128 -.00140 -.00161 -.0012? 43 EE E 00000000000000 0.0000000000 UXB -.02500 -.02500 -.02500 -.05000 -.05000 -.05000 -.07500 -.0?500 -.0?500 .20000 .40000 .60000 .20000 .40000 .60000 .20000 .40000 EEEEEEEEEEEEEE Pure Extension.- 63 Nodes: Interior“va1ues UYC .18762 .38628 .58542 .18645 .3847? .58321 .18472 .38359 -.58149 .00296 .01982 .0159? .00258 .00792 EEEEEEEEEEEEEE EXACT .20000 .40000 .60000 .20000 .40000 .60000 .20000 .40000 .60000 E 00000 Table 11 (curt-Jo.) UQQO‘ .50000 .75000 .75000 .75000 INTERIORLSXY mmqmmbuupa IX .25000 .25000 .25000 .50000 .50000 .50000 .75000 .75000 .75000 INTERIOR.SYY ‘DQQQUI-hUNI-‘S muen. IX .25000 .25000 .25000 .50000 .50000 .50000 .75000 .75000 .75000 1.50000 .50000 1.00000 1.50000 XFDISPIACEMENES 5:E:::t;K;t:E;MDGD~JO\via-03832; IA -41.21 -10.93 -6.20 -4.26 -3.22 -2.56 -2.11 -1.?4 -1.06 -3.3? -3.49 -3.43 -3.38 -3.35 -3.32 -.00214 -.00145 -.0010? -.00193 -.00011 -.00044 .00114 -.00025 .00001 .00039 -.00017 -.00110 A. .99946 .99902 .99873 .99978 .99928 1.00025 .99960 .99942 .99949 B C .00 -185.61 .00 -62.30 .00 -8.53 .00 -1.64 .00 .34 .00 1.53 .00 3.42 .00 9.54 .00 41.32 .00 28.55 .00 -8.64 .00 -2.99 .00 .3? .00 1.63 .00 2.13 EEEEEEEEE .00068 .0073? .00631 .00033 -.00569 -.01226 - 00895 00481 00663 00678 00607 00480 00648 ‘C .98993 .99072 .9851? .99343 .99285 .99240 .99394 .99454 .98585 EEEEE EEEEE°°°° E? E HyHHrHHHH mmnnmfim163m&$ mmthfiamms Table 12 (ca'tt'd.) '1? 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 “3.30 .3029 “3.29 “3.30 “3.31 “3.34 “3.38 “3.42 “3.48 “3.55 “3.62 “3.70 “3.79 “3.88 “3.95 “4.00 “3.98 “3.67 “1.32 “2.11 “2.75 “3.52 “4.60 “6.30 “9.43 -17012 “66.85 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 YBDISPIACEMENES N0 1 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 2? 28 29 30 A. “7.37 “7.80 “1.48 “.77 “.51 “.38 “.31 “.26 “.22 “.20 -0 18 “.16 “.15 “.14 “.13 “.12 “.11 “.10 “.08 “.07 “.05 B .00 .oo .oo .oo .oo .oo .oo .oo .oo .oo .oo .oo .oo .oo .oo .oo .oo .oo .oo .oo .oo 0.00 \IQUH h‘k‘h‘h‘h‘h‘f‘h’k’h’h’h) h)huUIUIO\\lu) P‘k’h’h’ raoora\0\0\o+asz 1.16 .89 .29 -1.04 -.75 28.22 35.57 13.22 8.58 6.55 5.08 2.73 -3.23 .26 257.77 ‘C -290.90 -333.39 20.16 25.49 17.91 13.16 10.25 8.35 7.02 6.06 5.35 4.80 4.3? hibih’h’h’h’hlbu 285838212 45 Table 12 (Contfd.) 31 -.03 32 .00 33 .04 34 . .14 35 -.02 36 -.12 3? -.14 38 -.15 39 -.15 40 -.13 41 -.11 42 -.0? 43 .10 44 .22 45 .10 46 .05 47 .02 48 .00 49 -.01 50 -.03 51 -.04 52 -.06 53 -.07 54 -.09 55 -.11 56 -.13 57 -.16 58 -.20 59 -.25 60 -.32 61 -.45 62 -.69 63 -1.34 YJTRACTIONB NO .A 2 -.85 3 .42 4 .22 5 .20 6 .18 7 .18 8 .18 9 .39 10 -1.00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 Om .00 .00 -2.02 46 Table 12 (ccntfid.) UflflO‘t’I-fiu .7046 .3033 -3058 “4.08 -2042 “2.53 “3.01 Om Om Om .00 .00 .00 .00 INTERIGRLY‘DESP N0 .A. B 1 -.22 .00 2 -.10 .00 3 -.03 .00 4 -.23 .00 5 -.11 .00 6 -.07 .00 7 -.26 .00 8 -.12 .00 9 -.07 .00 INTERIGRLSYY N0 IA. B 1 .05 .00 2 .10 .00 3 .13 .00 4 .02 .00 5 .07 .00 6 -.03 .00 7 .04 .00 8 .06 .00 9 .05 .00 Table 13. XEDISPIACEMENTS NO ‘x Y 1 0.00000 .08333 2 .10000 0.00000 3 .30000 0.00000 4 .50000 0.00000 5 .70000 0.00000 6 .90000 0.00000 7 1.00000 .10000 8 1.00000 .30000 9 1.00000 .50000 10 1.00000 .70000 11 1.00000 .90000 12 .90000 1.00000 HHHyHH 288383332 1.01 .93 1.48 .66 .72 .76 .61 .55 1.42 UXA 0.00000 .16661 .48854 .81024 1.12711 1.43617 1.57028 1.56760 1.57974 1.60466 1.62674 1.47094 47 UXB 0.00000 .15998 .47872 .79528 1.10769 1.41354 1.55964 1.55867 1.57500 1.60855 1.65723 1.52260 Square.- 21.Nbdes: Xi- Displacements UXC 0.00000 .12491 .42453 .75824 1.10029 1.47046 1.65303 1.42869 1.44284 1.43724 1.37350 1.0519? EXACT 0.00000 .1599? .47910 .79583 1.10857 1.41570 1.55892 1.55692 1.57292 1.60692 1.65892 1.52820 Table 13 (cont!d.) 48 13 .70000 1.00000 1.16171 1.19346 1.02668 14 .50000 1.00000 .8368? .85689 .77929 15 .30000 1.00000 .5045? .51575 .49220 16 .10000 1.00000 .17078 .17233 .16281 17 0.00000 .9166? 0.00000 0.00000 0.00000 18 0.00000 .75000 0.00000 0.00000 0.00000 19 0.00000 .58333 0.00000 0.00000 0.00000 20 0.00000 .4166? 0.00000 0.00000 0.00000 21 0.00000 .25000 0.00000 0.00000 0.00000 Table14. Square-21Nodes:Y-Disp1acanmts YBDISPIACEHENTS NO X Y UYA 'UYB. ‘UYC 1 0.00000 .08333 -.06238 -.02709 -.03212 2 .10000 0.00000 -.00450 -.00450 -.00450 3 .30000 0.00000 -.04050 -.04050 -.04050 4 .50000 0.00000 -.11250 -.11250 -.11250 5 .70000 0.00000 -.22050 -.22050 -.22050 6 .90000 0.00000 -.36450 -.36450 -.36450 7 1.00000 .10000 -.49160 -.48032 -.43473 8 1.00000 .30000 -.55198 -.54201 -.46370 9 1.00000 .50000 -.59429 -.59006 -.50192 10 1.00000 .70000 -.62192 -.62775 -.48306 11 1.00000 .90000 -.6279? -.65726 -.40117 12 .90000 1.00000 -.61124 -.59490 -.40115 13 .70000 1.00000 -.49212 -.46004 -.50421 14 .50000 1.00000 -.39729 -.35799 -.41121 15 .30000 1.00000 -.33432 -.29000 -.34075 16 .10000 1.00000 -.30638 -.25592 -.36235 17 0.00000 .9166? -.26109 -.23466 -.36142 18 0.00000 .75000 -.23952 -.20654 -.24584 19 0.00000 .58333 -.20439 -.17272 -.21656 20 0.00000 .4166? -.16231. -.13256 -.169?1 21 0.00000 .25000 -.11245 -.084?1 -.13878 Tab1e15. Square-ZlNodes: x-Tractims XJTRACTICNS N0 1X Y TXA. TXB 'TXC 1 0.00000 .08333 -4.13240 -4.01886 -3.30061 2 .10000 0.00000 .10000 .10000 .10000 3 .30000 0.00000 .30000 .30000 .30000 4 .50000 0.00000 .50000 .50000 .50000 5 .70000 0.00000 .70000 .70000 .70000 6 .90000 0.00000 .90000 .90000 .90000 7 1.00000 .10000 3.75500 3.75500 3.75500 8 1.00000 .30000 3.79500 3.79500 3.79500 1.1960? .85833 .51660 .1724? near -.03197 -.00450 -.04050 -.11250 -.22050 -.36450 -.48555 -.54585 -.59375 -.63165 -.66195 -.59425 -.45825 -.35625 -.28825 -.25425 -.23712 1.20859 -.17520 -.13556 -.08828 EXACT -4.00347 .10000 .30000 .50000 .70000 .90000 3.75500 3.79500 Table 15 (contid.) 9 1.00000 10 1.00000 11 1.00000 12 .90000 13 .70000 14 .50000 15 .30000 16 .10000 17 0.00000 18 0.00000 19 0.00000 20 0.00000 21 0.00000 Table 16. YHTRACTIONS NO X 1 0.00000 2 .10000 3 .30000 4 .50000 5 .70000 6 .90000 7 1.00000 8 1.00000 9 1.00000 10 1.00000 11 1.00000 12 .90000 13 .70000 14 .50000 15 .30000 16 .10000 17 0.00000 18 0.00000 19 0.00000 20 0.00000 21 0.00000 Table 17. .50000 .70000 .90000 1.00000 1. 1.00000 1.00000 1.00000 .9166? .75000 .58333 .4166? .25000 Y .08333 0.00000 0.00000 0.00000 0.00000 0.00000 .10000 .30000 .50000 .70000 .90000 1.00000 1.00000 1.00000 1.00000 1.00000 .9166? .75000 .58333 .4166? .25000 INTERIOR.XPDESP NO 1 2 3 IX .25000 .50000 .75000 3.87500 3.99500 4.15500 “.45000 -.35000 -.25000 “.15000 “.05000 -4.47629 “4.18303 “4.13571 “4.07330 “3.99275 Square - 21 Nbdes: Y TEA 0.00000 .00965 .03890 .02188 -.00711 -.12226 -.95000 “.85000 “.75000 “.65000 -.55000 .75000 .75000 .75000 .75000 .75000 0.00000 0.00000 0.00000 0.00000 0.00000 UXA .40295 .80191 1.19548 49 3.87500 3.87500 3.99500 3.99500 4.15500 4.15500 -.45000 -.45000 -.35000 -.35000 -.25000 -.25000 -.15000 -.15000 -.05000 -.05000 -4.44??9 -1.86205 -4.24155 -5.08293 -4.18825 -4.504?2 -4.07308 -3.96936 -4.028?7 -5.90831 - Tractions TYB 'TWC 0.00000 0.00000 .0125? -.40149 .00244 .28765 .01382 .27392 -.00094 .11283 .00752 -.43576 -.95000 -.95000 -.85000 -.85000 -.75000 -.75000 -. -.65000 -.55000 -.55000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 UXB .40096 .79890 1.19084 Square - 21.ncdes: Interior'values UXC .4633? .74203 1.02113 3.87500 3.99500 4.15500 “.45000 “.35000 “.25000 -015” “.05000 “4.42014 “4.28125 “4.17014 “4.08681 “4.03125 .40104 .79896 1.19062 Table 17 (cmt'd.) U'l-bUND-‘B Table 18 . .25000 .50000 .75000 .50000 .50000 .25000 .50000 .75000 .50000 .50000 .75000 .25000 .50000 .50000 .50000 .75000 .25000 .50000 .50000 .50000 .75000 .25000 XFDESPIACEMENTS mummbuua IA “4.15 -1.99 -1.81 “1.67 “1.45 “.73 -.69 .81498 .79826 UYA “.21033 “.28775 “.41491 “.34319 “.21685 IA 4.1022? 4.06440 4.00304 4.17205 3.98660 A. “.19181 “.37550 “.54214 “.3324? “.43381 .A .37973 .38910 .42416 .55615 .19261 B. “.01 21.92 .08 11.39 .0? 4.72 .08 .75 .15 -3.8? “.05 -6.04 “.11 8.24 50 .82104 .79032 UYB -.18155 “.26358 “.3999? “.31498 “.19660 8 4.10770 4.06207 3.98706 4.21496 3.96764 ID “.18875 “.37543 -.56115 -.31565 “.43622 B .43102 .43100 .43216 .60311 .22925 .76488 .72923 . UYC “.23262 “.30050 -.39314 “.35766 “.22326 C 3.91982 4.04759 4.05478 3.72216 4.00292 (3 “.15594 -.34196 -.44944 -.28413 “.5919? C: .41555 .26430 .34010 .68714 .31365 Square - 21 Nodes: Pe'ccent Differences .82161 .79036 EXACT “.18359 -.26563 -.40234 “.31641 -.19922 EXACT 4.10938 4.06250 3.98438 4.21875 3.96875 -.18750 “.37500 -.56250 -.31250 -.43750‘ EXACT .43750 .43750 .43750 .60938 .23438 Table 18 (contld.) 9 “.43 10 .14 11 1.94 12 7 3.75 13 2.87 14 2.50 15 2.33 16 .98 Y“DISFIACEMENTS N0 .A 1 “95.10 7 -1.25 8 “1.12 9 -.09 10 1.54 11 5.13 12 “2.86 13 “7.39 14 “11.52 15 -15.98 16 “20.50 17 “10.11 18 -14.83 19 “16.66 20 -19.?3 21 -27.38 XHTRACTICNS N0 IA 1 -3.22 1? -1.27 18 2.29 19 .83 20 .33 21 .95 “.13 -.10 .10 .3? .22 .1? .16 .08 15.2? 1.08 .70 .62 .62 .71 -.11 “.39 “.49 -.61 -.66 1.04 .98 1.42 2.21 4.04 “.38 “.63 .93 -.43 .34 .06 INTERIOR.X#DESP’ ammopfi .A -048 “.37 “.41 .81 “1.00 B> .02 .01 -.02 .0? .01 INTERIGRHY-DESP 1 IA -14 056 1.11 8.27 10.56 17.21 31.16 14.16 9.21 4.72 5.60 Table 18 (”It'd.) 2 -8.33 .77 3 -3.12 .59 4 -8.47 .45 5 -8.85 1.31 INTERIORHSXX no A. B 1 .17 .04 2 -.05 .01 3 -.47 -.07 4 1.11 .09 5 -.45 .03 INTERIOR.SXY N0 .8 B 1 -2.30 -.67 2 -.13 -.11 3 3.62 .24 4 -6.39 -1.01 5 .84 .29 INTERICRLSYY N0 .A. B 1 13.21 1.48 2 11.06 1.49 3 3.05 1.22 4 8.73 1.03 5 17.82 2.19 Table 19. XFDISPIACEMENTS NO ‘X Y 1 0.00000 . 2778 2 .03333 0.00000 3 .10000 0.00000 4 .16667 0.00000 5 .23333 0.00000 6 .30000 0.00000 7 .36667 0.00000 8 .43333 0.00000 9 .50000 0.00000 10 .56667 0.00000 11 .63333 0.00000 12 .70000 0.00000 13 .76667 0.00000 14 .83333 0.00000 11.7? (3 16.83 20.10 9.08 “35.31 (I 5.02 39.59 22.26 “12.76 “33.82 UXA 0.00000 .05511 .16156 .26859 .37552 .48225 .58869 .69480 .80049 .90570 1.01036 1.11439 1.21772 1.32029 52 UXB 0.00000 .05332 .15993 .26648 .3728? .47906 .5849? .69056 .7957? .90052 1.00477 1.10846 1.21151 1.3138? Square:- 63 Nodes: IX — Displacements UXC 0.00000 .00493 .10622 .21864 0 32572' 3 .43269 .53930 .64551 .75136 .85691 .96229 1.06778 1.1740? 1.28300 EXACT 0.00000 .05333 .1599? .26651 .37291 .47910 .58502 .69062 .79583 .90060 1.0048? 1.10857 1.21165 1.31404 Table 19 (cart'dJ 15 16 1? 18 19 20 21 22 23 24 25 26 2? 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 5? 58 59 60 61 62 63 .90000 EEEEEEE .00000 EEE HyHHHHHyHHHHHH ESE E E 1.00000 .9666? .90000 .83333 .76667 .70000 .63333 .5666? .50000 .43333 .36667 .30000 .23333 .16667 .10000 .03333 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000' 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 .03333 .10000 .1666? .23333 .30000 .3666? .43333 1.42203 1.52258 1.56699 1.56300 1.56058 1.55991 1.56106 1.56409 1.56901 1.57580 1.58445 1.59491 1.60711 1.62093 1.63609 1.65191 1.66326 1.61185 1.50966 1.40348 1.29534 1.18590 1.07546 .96419 .85224 .73971 .62670 .51328 .39953 .28553 .17138 .05782 0.00000 0.00000 0.00000 0.00000 ' 0.00000 EEEEEEEEEEEE B 1.41548 1.51619 1.56354 1.55899 1.55640 1.55577 1.55713 1.56048 1.56582 1.57316 1.58248 1.59381 1.60712 1.62243 1.63971 1.65893 1.67983 1.63651 1.52768 1.41785 1.30720 1.19585 1.08386 .97129 .85821 .74469 .63078 .51653 .40202 .28730 .17243 .05749 EEEEEEE °EEEEEEEE°°°°°°°° E 1.40058 1.53791 1.59452 1.47722 1.47243 1.47058 1.47129 1.47415 1.47882 1.48506 1.49268 1.50141 1.51085 1.52022 1.52695 1.51862 1.45536 1.32756 1.38275 1.31287 1.21982 1.12010 1.0168? .91144 .80462 1.41570 1.51656 1.56358 1.55892 1.55625 1.55558 1.55692 1.56025 1.56558 1.57292 1.58225 1.59358 1.60692 1.62225 1.63958 1.65892 1.68025 1.63739 1.52820 1.41821 1.30748 1.19607 1.08403 .97143 .85833 .74479 .63086 .51660 .40208 EEEEEEEEEE Table 20. Y“DISPLACEHENTS Egugtsomqmmeuuwa IX 0.00000 .03333 .10000 .16667 .23333 .30000 .3666? .43333 .50000 .5666? .63333 .70000 .7666? .83333 .90000 .9666? 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 .9666? .90000 .83333 .76667 .70000 .63333 .5666? .50000 .43333 .36667 .30000 .23333 .1666? .10000 .03333 0.00000 0.00000 0.00000 'Y .02778 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 .03333 .10000 .1666? .23333 .30000 .36667 .43333 .50000 .5666? .63333 .70000 ' .76667 .83333 .90000 .9666? 1.00000 1.00000 1.00000 1.00000 1.00000 .97222 .9166? .86111 on -.02059 -.00050 -.00450 -.o1250 -.02450 -.04050 -.06050 -.oe4so -.11250 -.14450 -.18050 -.22050 -.26450 -.31250 -.36450 -.42050 -.46505 -.48979 -.51133 -.53101 -.54904 -.56563 -.53085 -.59479 -.60751 -.51906 -.62947 -.63871 -.64665 -.65275 -.65355 -.64727 -.60051 -.55417 -.51049 -.47006 -.43311 -.39975 -.37006 -.34407 -.32181 -.30332 -.28861 -.27774 -.27085 -.26840 -.25372 -.24767 -.23845 % UYB -.01039 -.00050 “.00450 -.01250 “.02450 -.04050 “.06050 -.08450 “.11250 -.14450 -.18050 -.22050 -.26450 -.31250 “.36450 -.42050 -.46169 “.48510 “.50674 -.52683 “.54544 -.56268 “.57863 “.5933? “.60700 -.61961 -.63128 -.64210 -.65214 “.66148 -.6?016 -.64714 “.59444 -.54536 -.50003 “.4584? -.42068 -.38668 “.35645 -.33000 -.30733 “.28844 -.2?333 “.26200 “.25446 “.25068 “.24555 “.23695 -.22788 Sqaxe-63Nodes: Y-Displacanents UYC .00194 -.00050 “.00450 -.01250 -.02450 “.04050 “.06050 “.08450 “.11250 “.14450 -.18050 “.22050 -.26450 -.31250 -.36450 “.42050 -.43345 “.44758 “.47872 “.50141 “.52122 -.53864 -.55390 -.56710 -.57820 -.58699 “.59291 “.59453 -.58?83 “.56221 -.50541 -.52250 “.5979? -.5529? -.51040 “.47120 -.43535 “.40309 -.3?460 “.34998 -.32931 “.31259 -.29958 “.28882 “.27733 -.31549 -.32345 “.25774 -.24586 EXACT -.01096 -.00050 “.00450 -.01250 “.02450 -.04050 “.06050 “.08450 “.11250 -.14450 “.18050 -.22050 -.26450 -.31250 -.36450 “.42050 -.46228 -.48555 ' -.50718 -.52?25 “.54585 -.56308 -.57901 “.59375 -.60738 -.61998 -.63165 -.64248 “.65255 -.66195 -.67078 -.64?14 -.59425 -.54514 -.49981 “.45825 -.42047 “.3864? -.35625 -.32981 -.30714 “.28825 -.27314 “.26181 -.25425 -.25047 -.24579 “.23712 -.22807 Table 20 (cart'd.) 50 0.00000 .80556 51 0.00000 .75000 52 0.00000 .69444 53 0.00000 .63889 54 0.00000 .58333 55 0.00000 .52778 56 0.00000 .47222 57 0.00000 .41667 58 0.00000 .36111 59 0.00000 .30556 60 0.00000 .25000 61 0.00000 .19444 62 0.00000 .13889 63 0.00000 .08333 Table 21. Square - 63 XHTRACTIONS NO X Y 1 0.00000 .02778 2 .03333 0.00000 3 .10000 0.00000 4 .16667 0.00000 5 .23333 0.00000 6 .30000 0.00000 7 .36667 0.00000 8 .43333 0.00000 9 .50000 0.00000 10 .56667 0.00000 11 .63333 0.00000 12 .70000 0.00000 13 .76667 0.00000 14 .83333 0.00000 15 .90000 0.00000 16 .96667 0.00000 17 1.00000 .03333 18 1.00000 .10000 19 1.00000 .16667 20. 1.00000 .23333 21 1.00000 .30000 22 1.00000 .36667 23 1.00000 .43333 24 1.00000 .50000 25 1.00000 .56667 26 1.00000 .63333 27 1.00000 .70000 28 1.00000 .76667 29 1.00000 .83333 30 1.00000 .90000 31 1.00000 .96667 “.22876 “.21858 “.20785 -0 19650 “.18450 “.17178 “.15820 “.14401 “.12887 “.11282 “.09583 “.07785 “.05885 “.03886 TXA “4.11647 .03333 .10000 .16667 .23333 .30000 .36667 .43333 .50000 .56667 .63333 .70000 .7666? .83333 .90000 .96667 3.75056 3.75500 , 3.76389 3.77722 3.79500 3.81722 3.84389 3.87500 3.91056 3.95056 3.99500 4.04389 4.09722 4.15500 4.21722 55 “.2183? “.2083? “.19783 “.18670 -.17492 -.16245 -.14924 “.13522 “.12036 “.10460 -.08788 “.0701? -.05139 “.03151 Nbdes: X’- Tractions TXB -4.006?8 .03333 .10000 .16667 .23333 .30000 .36667 .43333 .50000 .5666? .63333 .70000 .7666? .83333 .90000 .9666? 3.75056 3.75500 3.76389 3.77722 3.79500 3.81722 3.84389 3.87500 3.91056 3.95056 3.99500 4.04389 4.09722 4.15500 4.21722 “.23541 “.22473 -.2135? -.20180 -.18935 -.1761? “.16222 -.1474? “.13192 “.11562 -.0986? -.08134 “.06268 “.0476? TXC -3.24222 .03333 .10000 .1666? .23333 .30000 .3666? .43333 .50000 .56667 .63333 .70000 .7666? .83333 .90000 .9666? 3.75056 3.75500 3.76389 3.77722 3.79500 3.81722 3.84389 3.87500 3.91056 3.95056 3.99500 4.04389 4.09722 4.15500 4.21722 “.21858 -.20859 -.19807 -.18696 “.17520 -.16275 -.14956 -.13556 -.12072 “.10498 -.08828 “.07058 “.05183 “.0319? EXACT “4.00039 .03333 .10000 .1666? .23333 .30000 .3666? .43333 .50000 .5666? .63333 .70000 .7666? .83333 .90000 .9666? 3.75056 3.75500 3.76389 3.77722 3.79500 3.81722 3.84389 3.87500 3.91056 3.95056 3.99500 4.04389 4.09722 4.15500 4.21722 Table 21 (omt'd.) 32 .96667 1.00000 33 .90000 1.00000 34 .83333 1.00000 35- .76667 1.00000 36 .70000 1.00000 37 .63333 1.00000 38 .56667 1.00000 39 .50000 1.00000 40 .43333 1.00000 41 .36667 1.00000 42 .30000 1.00000 43 .23333 1.00000 44 .16667 1.00000 45 .10000 1.00000 46 .03333 1.00000 47 0.00000 .97222 48 0.00000 .91667 49 0.00000 .86111 50 0.00000 .80556 51 0.00000 .75000 52 0.00000 .69444 53 0.00000 .63889 54 0.00000 .58333 55 0.00000 .52778 56 0.00000 .47222 57 0.00000 .4166? 58 0.00000 .36111 59 0.00000 .30556 60 0.00000 .25000 61 0.00000 .19444 62 0.00000 .13889 63 0.00000 .08333 Table 22. Square - 63 YJTRACTICNS N0 2X Y 1 0.00000 .02778 2 .03333 0.00000 3 .10000 0.00000 4 .16667 0.00000 5 .23333 0.00000 6 .30000 0.00000 7 .36667 0.00000 8 .43333 0.00000 9 .50000 0.00000 10 .56667 0.00000 11 .63333 0.00000 12 .70000 0.00000 13 .76667 0.00000 56 “.48333 “.48333 “.48333 “.45000 “.45000 “.45000 “.4166? “.4166? “.4166? “.38333 “.38333 “.38333 “.35000 “.35000 “.35000 , “.3166? “.3166? “.3166? “.28333 “.28333 “.28333 “.25000 “.25000 “.25000 “.2166? “.2166? “.2166? “.18333 “.18333 “.18333 “.15000 “.15000 “.15000 “.1166? -.11667 “.1166? “.08333 “.08333 “.08333 “.05000 “.05000 “.05000 “.0166? “.0166? “.0166? -4.58134 “4.48431 “.98029 -4.35249 “4.40426 “5.11678 “4.34655 “4.38360 “5.53915 “4.30382 -4.31159 “4.94881 -4.26452 “4.29182 -4.42764 “4.22762 “4.23081 -4.18993 “4.19348 -4.21241 “4.12636 -4.16213 “4.16225 -4.11612 “4.13353 “4.14524 “4.10435 “4.10769 “4.10613 “4.08172 “4.08454 “4.09044 “4.05309 “4.06408 “4.06224 “4.02181 “4.04622 “4.04791 “3.98148 “4.03071 -4.0307? “3.93725 “4.01726 “4.01768 “3.62106 “4.00656 “4.01160 “3.54151 “3.96093 -3.99992 “6.35215 Nodes: Y - Tractions TIA. TYB. 'TMC 0.00000 0.00000 0.00000 “.0119? .00116 “.87809 .01260 .00062 .02971 .01300 .00181 .30095 .01249 .00016 .24658 .01155 .00194 .15212 .01020 “.0001? .09823 .00839 .00206 .07835 .00611 “.00044 .07126 .00339 .00214 .06529 .00029 “.00072 .05820 “.00308 .00226 .05251 -.000650 -.00107 .05223 “.48333 “.45000 “.41667 “.38333 “.35000 “.28333 “.25000 “.21667 “.18333 “.15000 “.11667 “.08333 “.05000 “.01667 “4.47261 “4.42014 “4.37076 “4.32446 “4.28125 “4.24113 “4.20409 “4.17014 “4.13927 “4.11150 “4.08681 “4.06520 “4.04668 “4.03125 “4.01890 “4.00964 “4.00347 Table 22 (cmt'd.) 14 15 16 1? 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 5? 58 59 60 61 62 63 .83333 .90000 EEEEEEEEEEEEEEEE .1666? 00 0" U U U U EEEEEEEEEEEEEEEEE 0.00000 0.00000 0.00000 .03333 .10000 .1666? .23333 .30000 .3666? .43333 .50000 .56667 .63333 .70000 .76667 .83333 .90000 .96667 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 .97222 .9166? .86111 .80556 .75000 .69444 .63889 .58333 .52778 .47222 .4166? .36111 .30556 .25000 .19444 .13889 .08333 “.0096? “.00986 “.07304 “.98333 “.95000 “.91667 “.88333 “.85000 “.8166? “.78333 “.75000 “.7166? “.68333 “.65000 “.6166? “.58333 “.55000 “.5166? .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 EEEEEEEE °EEEEEEEE°°°°°°°° E 57 .0024? “.0012? .00171 “.98333 “.95000 “.9166? “.88333 “.85000 “.8166? “.78333 “.75000 “.7166? “.68333 “.65000 “.6166? “.58333 “.55000 “.5166? .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 0.00000 .0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 .0841? .06669 “.57466 “.98333 “.95000 “.9166? “.88333 “.85000 “.8166? “.78333 “.75000 “.7166? “.68333 “.65000 “.6166? “.58333 “.55000 “.5166? .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 0.00000 0.00000 0.00000 0.00000 EEEEEEEEEEEEE 0.00000 0.00000 0.00000 “.98333 “.95000 “.9166? “.88333 “.85000 “.8166? “.78333 “.75000 “.7166? “.68333 “.65000 “.6166? “.58333 “.55000 “.5166? .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 .75000 EEEEEEEE °EEEEEEEE°°°°°°°° E Tibia 23. INTERIORlXFDESP N0 1X Y 1 .25000 .50000 2 .50000 .50000 3 .75000 .50000 4 .50000 .75000 5 .50000 .25000 INTERICRLY“DISP N0 1X ~ Y 1 .25000 .50000 2 .50000 .50000 3 .75000 .50000 4 .50000 .75000 5 .50000 .25000 INTERIORLSXX N0 1X Y 1 .25000 .50000 2 .50000 .50000 3 .75000 .50000 4 .50000 .75000 5 .50000 .25000 INTERIOR.SXY NO X Y 1 .25000 .50000 2 .50000 .50000 3 .75000 .50000 4 .50000 .75000 5 .50000 .25000 INTERIORLSYY N0 1X Y 1 .25000 .50000 2 .50000 .50000 3 .75000. .50000 4 .50000 .75000 5 .50000 .25000 Table 24. XbDESPIACEMENTS N0 .3. B 2 -3.34 .02 UXA .40186 .80031 1.1927? .82009 .79309 UXA “.1920? “.27281 “.40690 “.32509 “.20494 .A 4.10915 4.06554 3.99143 4.20651 3.97444 A. “.18875 “.37560 “.55765 “.31989 “.4654 A. .41995 .42242 .4347? .5941? .22260 90.75 58 UXB .40104 .79896 1.19065 .8215? .79036 UYB “.1833? “.26540 “.40210 “.31625 “.19893 8’ 4.10917 4.0624? 3.98465 4.21840 3.96861 “.18758 “.37504 “.56239 “.31281 “.4735 ‘8 .43683 .43678 .43686 .60871 .23379 Square - 63 nodes: Interior“values UXC .65922 .76110 .86015 .82222 .70008 UYC “.19715 “.27561 “.40024 “.33046 “.20476 ‘C 4.04201 4.00402 3.9333? 4.07419 3.9530? (3 “.19350 “.37101 “.51642 “.32148 “.43464 (2 .37092 .37659 .4177? ‘.55431 .18115 Square - 63 Nodes: Percent.Diffierences IEXACT .40104 .79896 1.19062 .82161 .79036 EXACT “.18359 “.26563 “.40234 “.31641 “.19922 IEXACT 4.10938 4.06250 3.98438 4.21875 3.96875 “.18750 “.37500 “.56250 “.31250 “.43750 EXACT .43750 .43750 .43750 .60938 .23438 Table 24 (cart'd.) 3 -1.00 .02 4 “.78 .01 5 -.7o .01 6 “.66 .01 7 “.63 .01 8 “.60 .01 9 -.59 .01 1o -.57 .01 11 -.55 .01 12 -.53 .01 13 -.so .01 14 “.48 .01 15 -.45 .02 16 -.4o .02 17 -.22 .oo 18 “.26 .oo 19 “.28 -.01 20 “.28 -.01 21 -.27 -.01 22 -.25 -.01 23 -.22 -.02 24 -.18 -.02 25 -.14 -.01 26 “.08 -.01 27 -.01 -.01 28 .08 -.01 29 .21 -.01 30 .42 .oo 31 1.01 .02 32 1.56 ..05 33 1.21 .03 34 1.04 .03 35 .93 .024 36 .85 .02 37 .79 .02 38 .75 .01 39 .71 .01 4o .68 .01 41 .66 .01 42 .64 .01 43 .63 .01 44 .63 .02 45 .63 .02 46 —.55 .02 Y“DISPLACEMENTS N0 A. :8 1 “87.88 5.15 17 —.6o .13 18 -.87 .09 19 “.83 .09 Pm HNUUbhmg‘QDBQh-D O O O 0 0 3888888888838 11? . 67 6.24 7.82 5.61 59 Table 24 (m'dd 20 “.71 21 “.59 22 “.45 23 “.32 24 “.18 25 “.02 26 .15 27 .35 28 .59 29 .90 30 1.39 31 2.5? 32 “.02 33 “1.05 34 “1.66 35 “2.14 36 “2.58 3? “3.01 38 “3.44 39 “3.88 40 “4.32 41 “4.78 42 “5.23 43 “5.66 44 “6.09 45 “6.53 46 “7.16 4? “3.22 48 “4.45 49 “4.55 50 “4.66 51 “4.79 52 “4.93 53 “5.10 54 “5.30 55 “5.55 56 “5.85 5? “6.23 58 “6.75 59 “7.47 60 “8.55 61 “10.30 62 “13.55 63 “21.55 XHTRACTIONS NO IA 1 “2.90 4? “2.43 48 1.53 49 .55 .08 .0? .07 .07 .06 .06 .06 .06 .06 .06 .0? .09 .00 “.03 “.04 “.04 “.05 “.05 “.05 “.06 “.06 “.06 “.07 “.07 “.08 “.08 “.08 .10 .07 .08 .09 .11 .12 .14 .16 .18 .21 .25 .29 .36 .45 .59 .85 1.44 B» -016 “.26 .36 “.29 4.90 4.51 4.34 4.34 4.49 4.80 5.32 6.13 7.46 9.92 15.07 24.65 19.26 “.63 “1.44 “2.12 “2.83 “3.54 “4.30 “5.15 “6.12 “7.22 “8.45 “9.68 “10.32 “9.08 “25.96 “31.59 “8.70 “7.80 “7.70 “7.74 “7.82 “7.94 “8.08 “8.25 “8.47 “8.78 “9.28 “10.14 “11.77 “15.24 “20.93 “49.08 18.95 78.08 “15.76 “26.73 60 Table 24 (cmt'd.) 50 51 52 53 54 55 56 57 58 59 60 61 62 63 .48 .39 .32 .25 .19 .14 .09 .06 .03 .01 .01 .04 .08 1.06 .30 -025 INTERIOR.X“DISP UI-bUNO-‘S .A “.20 “.1? “.18 .19 “.35 'B .00 .00 .00 .01 .00 INTERIORIY“DESP mthl-‘a INTERIOR.SXX UI-bUNl-‘a INTERIOR.SXY 01.51.710.45 .A ‘4062 “2.70 “1.13 -2074 “2.87 IA .01 “.07 “.18 .29 “.14 .A “.66 “.16 .86 “2.3? .22 .12 .08 .06 . .05 .15 .01 .00 “.01 .01 .00 IE “.04 “.01 .02 “.10 .03 “14.44 “3.42 1.21 1.85 1.30 .84. .72 .83 1.07 1.61 2.33 9.90 11.68 “58.6? “64.38 4.74 27.76 “.07 11.42 “7.39 “3.76 .52 “4.44 “2.78 61 Table 24 (cart'd.) INTERIOR.SYY UlubUNl-‘a Table 25. XHDISPIACEMENTS 55585555886m46m80~H6 Table 26. Y“DISP1ACEMENES flSDGQGUbUNI-‘a A. 4.01 3.45 .62 2050 5.02 XI “.41620 .22404 .52101 .76269 .92341 .98612 .94416 .80199 .57470 .28641 “.03228 “.34754 “.62592 “.83786 “.9608? “.98189 “.89254 “.69636 IX “.41620 “.09671 .22404 .52101 .76269 .92341 .98612 .94416 .80199 .57470 .28641 ‘8 .15 .17 .15 .11 .25 Y “.89254 “.98189 “.9608? “.83786 “.62592 “.34755 “.03228 .28641 .57470 .80199 .94416 .98612 .92341 .76269 .52102 .22404 “.09671 “.41620 “.69636 Y' “.89254 “.98189 “.9608? “.83786 “.62592 “.34755 “.03228 .28641 .57470 .80199 .94416 050 N IUD-b O Q 010” ._.2 FROM UXA .09604 0.00000 “.08836 “.17181 “.24019 “.28573 “.30439 “.29276 “.25069 “.1833? “.09732 0.00000 .08575 .16330 .22274 .25702 .26296 .23594 .17842 Disk-19Nodes: Y- UYA .23594 .26296 .25702 .22274 .16330 .08575 0.00000 “.09732 “.1833? “.25069 “.29276 62 Disk-19Hodes: X-Dimlacalaxts UXB UXC .0974? .09219 0.00000 0.00000 “.09208 “.10161 “.17603 “.17864 “.24563 “.24145 “.29242 “.28136 “.31165 “.29958 “.29985 “.28646 “.25682 “.24143 “.18732 “.17379 “.09753 “.08950 0.00000 0.00000 .09300 .08594 .16836 .15901 .22891 .21336 .26424 .2424? .27063 .24123 .2441? .20644 .18455 .15161 Displacanents UYB’ UYC .2441? .22012 .27063 .2581? .26424 .23543 .22891 .19908 .16836 .14050 .09030 .07150 0.00000 0.00000 “.09753 “.11315 “.18732 “.19350 “.25682 “.25599 “.29985 “.29518 EXACT .12486 0.00000 “.06721 “.15630 “.22881 “.27702 “.29583 “.28325 “.24060 “.17241 “.08592 0.00000 .10426 .18778 .25136 .28826 .2945? .26776 .20891 EXACT .26776 .2945? .28826 .25136 .18778 .10426 0.00000 “.08592 “.17241 “.24060 “.28325 63 Table 26 (ca'tt'd.) 12 “.03228 .98612 “.30439 “.31165 13 “.34754 .92341 “.28573 “.29242 14 “.62592 .76269 “.24019 “.24563 15, “.83786 .52102 “.17181 “.17603 16 “.9608? .22404 “.0883? “.09208 1? “.98189 “.09671 0.00000 0.00000 18 “.89254 “.41620 .09604 .0974? 19 “.69636 “.69636 .17842 .18455 Table27. Disk-DNodes: X-Tractims XHTRACTICNS NO X Y TXA. TXB 1 “.41620 “.89254 .41620 .41620 2 “.09671 “.98189 .05623 .04803 3 .22404 “.9608? “.22404 “.22404 4 .52101 “.83786 “.52101 “.52101 5 .76269 “.62592 “.76269 “.76269 6 .92341 “.34755 “.92341 “.92341 7 .98612 “.03228 “.98612 “.98612 8 .94416 .28641 “.94416 “.94416 9 .80199 .57470 “.80199 “.80199 10 .57470 .80199 “.57470 “.57470 11 .28641 .94416 “.28641 “.28641 12 “.03228 .98612 .08018 .08478 13 “.34754 .92341 .34754 .34754 14 “.62592 .76269 .62592 .62592 15 “.83786 .52102 .83786 .83786 16 “.9608? .22404 .9608? .9608? 17 “.98189 “.09671 .98189 .98189 18 “.89254 “.41620 .89254 .89254 19 “.69636 “.69636 .69636 .69636 Table28. Disk-19m: Y-Tractias Y4TRACTTONB NO IX Y TYA. TYB 1 “.41620 “.89254 .89254 .89254 2 “.09671 “.98189 .98189 .98189 3 .22404 “.9608? .9608? .9608? 4 .52101 “.83786 .83786 .83786 5 .76269 “.62592 .62592 .62592 6 .92341 “.34755 .34755 .34755 ? .98612 “.03228 .08019 .08475 8 .94416 .28641 “.28641 “.28641 9 .80199 .57470 “.57470 “.57470 10 .57470 .80199 “.80199 “.80199 11 .28641 .94416 “.94416 “.94416 “.30243 “.28109 “.16457 “.08488 0.00000 .07830 .13606 TXC .41620 .09908 “.22404 “.52101 “.76269 “.92341 “.98612 “.94416 “.80199 “.57470 .28641 .04518 .34754 .62592 .83786 .9608? .98189 .89254 .69636 'TYC .89254 .98189 .9608? .83786 .62592 .34755 .18549 “.28641 “.57470 “.80199 “.94416 “.29583 “.27702 “.22881 -0 15630 “.06721 0.00000 .12486 .20891 EXACT .41620 .09671 “.22404 “.52101 “.76269 “.92341 “.98612 “.94416 “.80199 “.57470 .28641 .03228 .34754 .62592 .83786 .9608? .98189 .89254 .69636 EXACT .89254 .98189 .9608? .83786 .62592 .34755 .03228 “.28641 “.57470 “.80199 “.94166 Table 28 (cont'd.) 12 -.03228 13 -.34754 14 -.62592 15 -.83786 16 “.9608? 17 “.98189 18 -.89254 19 -.69636 Table 29. .98612 .92341 .76269 .52102 .22404 “.09671 “.41620 “.69636 INTERIOR.X“DISP F'F' Hommqmmbunwg XI “.75000 “.50000 0.00000 .50000 .75000 0.00000 0.00000 0.00000 0.00000 .50000 Y 0.00000 0.00000 0.00000 0.00000 0.00000 .75000 .50000 “.50000 “.75000 .50000 .50000 INTERIOR.Y“DISP XI “.75000 “.50000 0.00000 .50000 .75000 0.00000 0.00000 0.00000 0.00000 “.50000 .50000 E E \lOfiUI-bUNF-‘B XI “.75000 “.50000 0.00000 .50000 .75000 0.00000 0.00000 Y 0.00000 0.00000 0.00000 0.00000 0.00000 .75000 .50000 “.50000 “.75000 .50000 .50000 Y 0.00000 0.00000 0.00000 0.00000 0.00000 .75000 .50000 “.98612 “.92341 “.76269 “.52102 “.22404 .05623 .41620 .69636 UXA .19790 .12514 “.02036 “.16586 “.23854 “.01614 “.01862 “.02141 “.02309 .12698 “.16492 UYA “.02309 “.02141 “.02036 “.01862 “.01614 “.23854 -. 16586 \ .12514 .19790 “.16602 “.16492 .A “.9702? “.96936 “.96990 “.97086 “.9649? “.97838 “.97191 64 “.98612 “.92341 “.76269 “.52102 “.22404 .04803 .41620 .69636 DiSK - 19 ches: Interior'values UXB .20154 .12773 “.02006 “.16801 “.24203 “.01534 “.01835 “.02120 “.02359 .12932 “.16710 UYB “.02359 “.02120 “.02006 “.01835 “.01534 “.24203 “.16801 .12773 .20154 “.16821 “.16710 IB “.98360 “.98436 “.98602 “.98694 “.98748 “.98944 “.9875? “.98612 “.92341 “.76269 “.52102 “.22404 .04756 .41620 .69636 UXC .17640 .10661 “.02803 “.16305 “.23132 “.01275 “.02011 “.03115 “.03178 .11774 “.15513 UYC “.03124 “.03198 “.02984 “.02799 “.02288 “.24701 “.17536 .11918 .19559 “.17369 “.17396 C! -.99003' “.96725 “.95970 “.97295 “.97296 “.98242 “.97390 “.98612 “.92341 “.76269 “.52102 “.22404 .09671 .41620 .69636 EXACT .22500 .15000 0.00000 “.15000 “.22500 0.00000 0.00000 0.00000 0.00000 .15000 “.15000 EXACT 0.00000 0.00000 0.00000 0.00000 0.00000 “.22500 “.15000 .15000 .22500 “.15000 “.15000 EXACT “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 Table 29 (curt'd.) 8 9 10 11 0.00000 0.00000 .50000 E E a E E H000~IO§UIIFUNH P'h‘ “.75000 “.50000 0.00000 .50000 .75000 0.00000 0.00000 0.00000 0.00000 “.50000 .50000 “.50000 “.75000 .50000 0.00000 0.00000 0.00000 0.00000 0.00000 .75000 .50000 “.50000 “.75000 .50000 .50000 “.96750 “.96429 “.96944 “.97850 .01565 .00521 .00293 .00575 .00570 .00570 .00575 .00521 .01565 .01015 .00738 A. “.96429 “.96750 “.96990 “.97191 “.97838 “.9649? “.97086 “.96936 “.9702? “.95983 “.97850 65 “.98468 “.98053 “.97989 “.99558 IB .01243 .00561 .00179 .00640 .01358 .01358 .00640 .00561 .01243 .01084 .01195 ‘8 “.98053 “.98468 “.98602 “.9875? “.98944 “.98748 “.98694 “.98436 “.98360 “.97670 “.99558 “.95030 “.97728 “.99931 “.95242 “.01252 .01081 .00686 “.0015? .02050 .05688 .01720 “.00616 “.01394 .02086 .04680 (3 “.93358 “.94655 “.97341 “.9729? “.9866? “.95095 “.96699 “.99498 “1.01433 “.92942 “.99591 Table 30. Disk - 19 Nodes: Peleent Differences XFDESPTACEMENTS NO IA. B 1 23.08 . 21.94 3 “31.4? “37.00 4 “9.92 “12.62 5 “.498 “.735 6 “3.14 “5.56 7 “2.89 “5.35 8 “3.36 “5.86 9 “4.19 “6.74 10 “6.36 “8.65 11 “13.26 “13.51 13 17.76 13.40 26.1? “51.17 “14.29 “5.52 “1.56 “1.27 “1.13 “.35 “.80 “4.16 17.57 “1.00000 “1.00000 “1.00000 “1.00000 EEEEEE °EEEEE°°°°° E. “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 Table 30 (catt'dd 14 13.03 10.34 15 11.39 8.93 16 10.84 8.33 1? 10.73 8.13 18 11.88 8.81 19 14.59 11.66 Y“DESP1ACEMENTS NO .A B 1 11.88 8.81 2 10.73 8.13 3 10.84 8.33 4 11.39 8.93 5 13.03 10.34 6 17.76 13.40 ‘8 “13.26 “13.51 9 “6.36 “8.65 10 “4.19 “6.74 11 “3.36 “5.86 12 “2.89 “5.35 13 “3.14 “5.56 14 “4.98 “7.35 15 “9.92 “12.62 16 “31.4? “37.00 1? 100.00 100.00 18 23.08 21.94 19 14.59 11.66 INTERIOR.X“DISP NO .A 1 12.04 2 16.57 4 “10.57 5 “6.02 10 15.35 11 “9.95 B 10.43 14.85 “12.01 “7.5? 13.79 “11.40 INTERIOR'Y-DISP ND IA 6 “6.02 7 “10.5? 8 16.5? 9 12.04 10 “10.68 11 “9.95 B’ “7.5? “12.01 14.85 10.43 “12.14 “11.40 15.32 15.12 15.89 18.11 22.90 27.43 17.79 12.36 18.33 20.80 25.18 31.42 “31.69 “12.23 “6.40 “4.21 “2.23 “1.4? “2.58 “5.29 “26.28 100.00 37.29 34.87 21.60 28.93 “8.70 “2.81 21.51 “3.42 “9.78 “16.90 20.55 13.07 “15.80 “15.97 66 Table 30 (ca-nt'd.) INTERIORLSXX tBOGQGMhUNHa 3' nuuunyuuuuu 58383883383 E 3 H toomqmm¢uwwa Table 31. 3' waununuuyu 0‘0 \0 H0030“ GNQRHgmt-‘HUIQ XFDISPIACEMENTS KGKUSfiSoaqmm.u~98 IX “.18499 .00000 .17299 .34072 .49810 .64034 .76313 .86273 .93612 .98106 .99619 .98106 .93612 .86273 .76313 .64034 NHHHHHHHHH abblnb' bb'.’ ' meuogmHogg m NHHHrHHHHH aw up mamm .628pm8.ouu Y .97768 .99619 .98106 .93612 .86273 .76313 .64034 .49810 .34072 .17299 .00000 .17299 .34072 .49810 .64034 .76313 NHNN‘UH 0 UXA .05388 0.00000 “.05166 “.10143 “.14803 “.19011 “.22642 “.2558? “.27756 “.29086 “.29550 “.29119 “.27791 “.25618 “.22670 “.19034 67 Disk - 35 Nodes: X - Displacanents UXB .05511 0.00000 “.0517? “.10203 “.14912 “.1916? “.22840 “.25819 “.2801 “.29359 “.29816 “.29369 “.28025 “.25829 “.22849 “.19175 UXC .05228 0.00000 “.05979 “.10351 “.14748 “.18669 “.21993 “.24673 “.26555 “.27728 “.28271 “.27870 “.2643? “.24259 “.2138? “.17840 EXACT .05550 0.00000 “.05190 “.10222 “.14943 “.19210 “.22894 “.25882 “.28084 “.29432 “.29886 “.29432 “.28084 “.25882 “.22894 “.19210 68 Table 31 (ant'dd 17 .49810 .86273 “.14820 “.14918 “.13809 18 .34072 .93612 “.10154 “.1020? “.09378 19 .17299 .98106 “.05169 “.05181 “.04715 20 0.00000 .99619 0.00000 0.00000 0.00000 21 “.17299 .98106 .05051 .05164 .05022 22 “.34072 .93612 .09975 .1016? .09623 23 “.49810 .86273 .1460? .14866 .13998 24 “.64034 .76313 .18796 .19115 .17875 25 “.76313 .64034 .22412 .22782 .21213 26 “.86273 .49810 .25345 .25755 .23865 2? “.93612 .34072 .27503 .27944 .25771 28 “.98106 .17299 .28820 .29280 .26888 29 “.99619 .00000 .2924? .29716 .27142 30 “.97768 “.18499 .28548 .29144 .26604 31 “.92164 “.37504 .26901 .27461 .24728 32 “.82906 “.55023 .2418? .24696 .21789 33 “.70359 “.70359 .20515 .20955 .17904 34 “.55023 “.82906 .16030 .16384 .13310 35 “.37504 “.92164 .10914 .11166 .0924? Table 32. Disk - 35 Nodes: Y - Displacanents Y“DISPLACEMENES NO IX Y UYA. UYB 1 “.18499 “.97768 .28548 .29144 2 .00000 “.99619 .2924? .29716 3 .17299 “.98106 .28820 .29280 4 .34072 “.93612 .27503 .27945 5 .49810 “.86273 .25345 .25755 6 .64034 “.76313 .22412 .22782 7 .76313 “.64034 .18796 .19115 8 .86273 “.49810 .1460? .14866 9 .93612 “.34072 .09975 .1016? 10 .98106 “.17299 .05051 .05164 11 .99619 .00000 0.00000 0.00000 12 .98106 .17299 “.05169 “.05181 13 .93612 .34072 “.10154 “.1020? 14 .86273 .49810 “.14820 “.14918 15 .76313 .64034 “.19034 “.19175 16 .64034 .76313 “.22670 “.22849 1? .49810 .86273 “.25618 “.25829 18 .34072 .93612 “.27791 . “.28025 19 .17299 .98106 “.29119 “.29368 20 .00000 .99619 “.29550 “.29816 21 “.17299 .98106 “.29086 “.29359 22 “.34072 .93612 “.27756 “.28013 23 “.49810 .86273 “.2558? “.25819 24 “.64034 .76313 “.22642 “.22840 25 0.76313 .64034 “.19011 “.1916? 26 “.86273 .49810 “.14803 “.14912 UYC .26595 .28391 .26343 .25045 .29910 .20108 .16642 .12742 .08480 .04168 0.00000 “.05944 “.10503 “.15006 “.19019 “.22400 “.2513? “.27059 “.28230 “.28550 “.28035 “.2663? “.24460 “.21581 “.18035 “.14011 “.14943 “.10222 0.00000 .05190 .10222 .14943 .19210 .22894 .25882 .28084 .29432 .29886 .29331 .27649 .24872 .21108 .1650? .11251 EXACT .29331 .29886 .29432 .28084 .25882 .22894 .19210 .14943 .10222 .05190 0.00000 -.05190 -.10222 -.14943 -.19210 -.22894 -.25882 -.28084 -.29432 “.29886 -.29432 “.28084 “.25882 -.22894 -.19210 -.14943 mm. 32 (cant'd.) 27 “.93612 28 “.98106 29 “.99619 30 “.97768 31 “.92164 32 “.82906 33 “.70359 34 “.5502 35 “.37504 Table 33. XHTRACTICNS NO X 1 “.18499 2 .00000 3 .17299 4 .34072 5 .49810 6 .64034 7 .76313 8 .86273 9 .93612 10 .98106 11 .99619 12 .98106 13 .93612 14 .86273 15 .76313 16 .64034 17 .49810 18 .34072 19 .17299 20 .00000 21 “.17299 22 “.34072 23 “.49810 24 “.64034 25 “.76313 26 “.86273 2? “.93612 28 “.98106 29 “.99619 30 “.97768 31 “.92164 32 “.82906 33 “.70359 34 “.55023 35 “.37504 .34072 .17299 .00000 “.18499 “.37504 “.55023 “.70359 “.82906 “.92164 Y “.97768 “.99619 “.98106 “.93612 “.86273 “.76313 “.64034 “.49810 “.34072 “.17299 .00000 .17299 .34072 .49810 .64034 .7631 .86273 .93612 .98106 .99619 .98106 .93612 .86273 .76313 .64034 .49810 .34072 .17299 .00000 “.18499 “.37504 “.55023 “.70359 “.82906 “.92164 “.10143 -a 05166 0.00000 .05388 .10914 .16030 .20515 .2418? .26901 TXA .18499 .00552 “.17299 “.34072 “.49810 “.64034 “.76313 “.86273 “.93612 “.98106 “.99619 “.98106 “.93612 “.86273 “.76313 “.64034 “.49810 “.34072 .17299 .00960 .17299 .34072 .49810 .64034 .76313 .86273 .93612 .98106 .99619 .97768 .92164 .82906 .70359 .55023 .37504 69 “.10203 “.0517? 0.00000 .05511 .11166 .16384 .20955 .2469? .27461 Disk-35Nodes: X-Tractia'a TXB .18499 .0031? “.17299 “.34072 “.49810 “.64034 “.76313 “.86273 “.93612 “.98106 “.99619 “.98106 “.93612 “.86273 “.76313 “.64034 “.49810 “.34072 -“.17299 .00276 .17299 .34072 .49810 .64034 .76313 .86273 .93612 .98106 .99619 .97768 .92164 .82906 .70359 .55023 .37504 “.09540 “.0484? 0.00000 .05124 .10496 .15235 .19281 .20781 .21729 TXC .18499 .06044 “.17299 “.34072 “.49810 .-.64034 “.76313 “.86273 “.93612 “.98106 “.99619 “.98106 “.93612 “.86273 “.76313 “.64034 “.49810 “.34072 “.17299 “.01894 .17299 .34072 .49810 .64034 .76313 .86273 .93612 .98106 .99619 .97768 .92614 .82906 .70359 .55023 .37504 “.10222 0.00000 .05550 .11251 .1650? .21108 .24872 .27649 EXACT .18499 .00000 “.17299 “.34072 “.49810 “.64034 “.76313 “.86273 “.93612 “.98106 “.99619 “.98106 “.93612 “.86273 “.76313 “.64034 “.49810 “.34072 .17299 .00000 .17299 .34072 .49810 .64034 .76313 .86273 .93612 .98106 .99619 .97768 .92614 .82906 .70359 .55023 .37504 70 Tab1e34. Disk-35Nodes: Y-Tractiau Y“TRACT10NB NO IX Y TYA. TYB» TYC 1 “.18499 “.97768 .97768 .97768 .97768 2 .00000 “.99619 .99619 .99619 .99619 3 .17299 “.98106 .98106 .98106 .98106 4 .34072 “.93612 .93612 .93612 .93612 5 .49810 “.86273 .86273 .86273 .86273 6 .64034 “.76313 .76313 .76313 .76313 7 .76313 “.64034 .64034 .64034 .64034 8 .86273 “.49810 .49810 .49810 .49810 9 .93612 “.34072 .34072 .34072 .34072 10 .98106. “.17299 .17299 .17299 .17299 11 .99619 .00000 .00960 .00275 .11302 12 .98106 .17299 “.17299 “.17299 “.17299 13 .93612 .34072 “.34072 “.34072 “.34072 14' .86273 .49810 “.49810 “.49810 “.49810 15 .76313 .64034 “.64034 “.64034 “.64034 16 .64034 .76313 “.76313 “.76313 '“.76313 1? .49810 .86273 “.86273 “.86273 “.86273 18 .34072 .93612 “.93612 “.93612 “.93612 19 .17299 .98106 “.98106 “.98106 “.98106 20 .00000 .99619 “.99619 “.99619 “.99619 21 “.17299 .98106 “.98106 “.98106 “.98106 22 “.34072 .93612 “.93612 “.93612 “.93612 23 “.49810 .86273 “.86273 “.86273 “.86273 24 “.64034 .76313 “.76313 “.76313 “.76313 25 “.76313 .64034 “.64034 “.64034 “.64034 26 “.86273 .49810 “.49810 “.49810 “.49810 2? “.98612 .34072 “.34072 “.34072 “.34072 28 “.98106 .17299 .17299 .17299 .17299 29 “.99619 .00000 .00551 .00316 .0045? 30 “.97768 “.18499 .18499 .18499 .18499 31 “.92164 “.37504 .37504 .37504 .37504 32 “.82906 “.55023 .55023 .55023 .55023 33 “.70359 “.70359 .70359, .70359 .70359 34 “.55023 “.82906 .82906 .82906 .82906 35 “.37504 “.92164 .92164 .92164 .92164 Table 35. Disk - 35 Nodes: Interior Values INTERldkiXFDTSP N0 IX Y IUXA IUXB 'UXC 1 “.75000 0.00000 .22094 .2259 .13353 2 “.50000 0.00000 .14664' .14888 .08700 3 0.00000 0.00000 “.00179 ‘“.00053 “.00619 4 .50000 0.00000 “.15009 “.14991 “.09844 5 .75000 0.00000 “.22426 “.22460 “.14488 6 0.00000 .75000 “.00105 “.0002? “.00856 7 0.00000 .50000 “.00139 “.00039 “.00798 EXACT .97768 .99619 .98106 .93612 .86273 .76313 .64034 .49810 .34072 .17299 .00000 “.17299 “.34072 “.49810 “.64034 “.76313 “.86273 “.93612 “.98106 “.99619 “.98106 “.93612 “.86273 “.76313 “.64034 “.49810 “.34072 .17299 .00000 .18499 .37504 .55023 .70359 .82906 .92164 .22500 .15000 0.00000 “.22500 0.00000 0.00000 Table 35 (cart'd.) 8 0.00000 -. 9 0.00000 “.75000 10 “.50000 .50000 11 .50000 .50000 INTERIOR.Y“DESP NO IX Y 1 “.75000 0.00000 2 “.50000 0.00000 3 0.00000 0.00000 4 .50000 0.00000 5 .75000 0.00000 6 0.00000 .75000 7 0.00000 .50000 8 0.00000 “. 9 0.00000 “.75000 10 “. .50000 11 .50000 .50000 INTERIOR.SXX N0 IX Y 1 “.75000 0.00000 2 “. 0.00000 3 0.00000 0.00000 4 .50000 0.00000 5 .75000 0.00000 6 0.00000 .75000 7 0.00000 .50000 8 0.00000 “.50000 9 0.00000 “.75000 10 “. .50000 11 .50000 .50000 INTERIOR.SXY N0 XI Y 1 “.75000 0.00000 2 “. 0.00000 3 0.00000 0.00000 4 .50000 0.00000 5 .75000 0.00000 6 0.00000 .75000 7 0.00000 .50000 8 0.00000 “. 9 0.00000 “.75000 10 “.50000 .50000 11 .50000 .50000 “.00178 “.0013? .14728 “.14999 UYA “.0013? “.00178 “.00179 “.00139 “.00105 “.22426 “.15009 .14664 .22094 “.14983 “.14999 IA “.99078 “.98928 “.98899 “.9891? “.98932 “.99070 “.99044 “.98722 “.98676 “.99036 “.99073 71 “.00052 “.00040 .14912 “.14986 “.14991 .14888 .22359 “.14982 “.14986 ID “.99593 “.99593 “.99594 “.99601 “.99610 “.99651 “.99644 “.99539 “.99531 “.99644 “.99652 “.00396 “.00044 “.08554 “.10125 IUYC “.00522 “.00773 “.00943 “.00890 “.0076? “.22755 “.1546? .13703 .21340 “.15251 “.15515 C: “.97552 “.97486 “.97109 “.9716? “.97701 “.97460 “.97591 “.95302 “.94871 “.97754 “.97310 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 “.22500 .15000 .22500 EEEEEEEEEEE Table 35 (cart'd.) INTERIOR.SYY NO IX 1 “.75000 2 “. 3 0.00000 4 .50000 5 .75000 6 0.00000 7 0.00000 8 0.00000 9 0.00000 10 “.50000 11 .50000 Table 36. )QTEEEEAIIAENHS mmqmubuwa .A 2.92 838 P‘F‘F‘F‘f‘h‘k‘f‘f‘ 888555838 0.. O 0000 GNNG uwuuywyyy 383388383 .8 .70 .24 .19 .21 .22 .24 .24 .25 .25 .23 .22 .21 .20 .19 .18 .16 .14 .18 .54 .51 .50 .49 .49 .50 .52 .5? .64 .68 .70 .72 F‘ P’ 01K;C>E>K>didl~I:IO\O\UIhlHDGD~I\JO\O\UIUI A. “.98676 “.98722 “.98899 “.99044 “.99070 “.98932 “.9891? “.98928 “.99078 “.98864 “.99073 ‘10:.“ uwguquoumhpuuuguau N BUSOQ O 0 0 UOuhUINUIN-FUI‘D n B» “.99531 “.99539 “.99594 “.99644 “.99651 “.99610 “.99601 “.99593 “.99593 “.99578 “.99652 C “.96770 “.9639? “.96913 “.97358 “.97341 “.97538 “.97121 “.9839? “1.02409 “.96878 “.98002 Disk - 35 Nodes: Parcent Differences “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 “1.00000 Table 36 (cmt'd.) 34 2.89 .74 35 3.00 .76 Y“DESP1ACEMENTS NO IA. 8 1 2.67 .64 2 2.14 .57 3 2.08 .52 4 2.07 .49 5 2.08 .49 6 2.10 .49 7 2.16 .50 8 2.25 .51 9 2.41 .54 10 2.68 .50 12 .39 .18 13 .66 .14 14 .82 .16 15 .92 .18 16 .98 .20 17 1.02 .20 18 1.04 .21 19 1.06 .22 20 1.12 .23 21 1.1? .25 22 1.17 .25 23 1.14 .24 24 1.10 .24 25 1.04 .22 26 .94 .21 27 .77 .19 28 .45 .25 30 2.92 .70 31 3.00 .76 32 2.89 .74 33 2.81 .72 34 2.75 .70 35 2.71 .68 INTERIOR.X“DESP N0 A. ‘8 1 1.81 .63 2 2.24 .75 4 “.06 .06 5 .33 .18 10 1.82 .59 11 .01 .09 NH Hmmqarxmmmmmuuno 3.65 4.47 58838888258858 40.65 42.00 34.37 35.61 42.97 32.50 73 Table 36 (cont'd.) INTERIOR.Y“DESP .A .33 “.06 2.24 1.81 .11 .01 E E 8 IA .92 1.0? 1.10 1.08 1.0? .93 .96 1.28 1.32 .96 .93 HOOQQOUIubuNH P'P' INTERIOR.SYY 6 .A 1.32 1.28 1.10 .96 .93 1.0? 1.08 1.0? .92 1.14 .93 HO‘OmdeIwaH P'P' I8 .18 .06 .75 .63 .12 .09 .41 .41 .41 .40 .39 .35 .36 .46 .47 .36 .35 .47 .46 .41 .36 .35 .39 .40 .41 .41 .42 .35 “1.13 “3.11 8.65 5.16 “1.67 “3.43 O 0U\OI-'U'l NNUIbNNNNNNN 0 O‘NHQ-fi-RUQGUIh $001000.“ 0 A)h)h)h*h)§)h)k)hlhlhfi 85388888888 74 CHAPTER 5 CONCLUSIONS The results from each problem favor case (b) as the best model. It is a closer approxima- tion for linear elasticity since displacements are linear, but tractions are not, in general, piecewise continuous. Case (a), with its simple assumptions, is the next best approximation. It appears that an inaccurate assumption in displacement behavior is less damaging than an inaccurate traction assumption. The corner effect in case (c) is too pronounced to favor it over the other two cases. It is possible to devise schemes for avoiding or reducing the corner effect. However, this will only add to the complexity of the input data or the model itseli and, beiore taking that step, one should examine the benefits. This brings us tothe ideas bought out in Chapter 1. Since the time oi the first approximation methods, we have progressed to ever-increasing levels of sophistication. As we keep progressing, however, we should learn to use what we have more efficiently. That is what prompted this work and the results indicate that much can be learned about the applicability of our models through simple comparative studies. 75 APPENDIX C C C C C moors dimension xtm) .yim) .xmfmn) .ymim) dinensim rmr(m),n1y(m),s(m) dimension r(10) ,w(10) ,w1(10) ,m(10) dimensim ua(1mn),ub(1mn) ,uc(mm),ue(mm) dimensicn ta(nnn) ,tb(mnn) ,tc(mnn) ,te(nnn) dimersim mra(nm) ,mrbamm) ,uxc(nm) dimersim uya(mn) .uybimn) .uyc(mn) dimensim uxeamm) ,uye(1man) dimensim mum) ,sxxb(nmn) ,sacac(mm) dimerBim main-m) .scymml .swumn) dimmer! syyamm) .syyb(mn) .syyclmn) dimension mama) ,syye(nmn) ,sxye(nmn) dimension ibc(mn,2) ,bc(mnn) ,bcm(nnn) ,rhs(1mn) dimension a(na1n,mm) ,b(1mm,mm) ,ucc(1mn,mnn) dinensim xfamln) ,yf(nmn) character-no title m(tmit=3,file-'sol') OBIAININIUTDA‘IA write(*,*) "nus m ms ma: om mm mm' write(*,*) 'MEDDD 'IO SOLVE PLANE meme or LINEAR mums) merry' “ii-'80.") ' ' write(*,*) '11) YO] WANT '10 FIND VALUES AT INTERICR TOMS? wr-i-I‘T‘e(*r*) ' ' write(*,*) 1 1 - YES' write(*,*) ' 2 I 10' write(*.*)' ' write(*,*) 'ENTER THE W OF YQIR GDICE' read(*,*)ifie1d 76 77 write(*,*) ' ' write(*,*) 1mm: A mm or m: mom on our: 1313' write(*~,*) 'cmmns 1-70. (moss IN @1713) ' read(*,*)title _ write(*,*) ' ' write(*,*) 'ENI‘ER THE mm NIMBER' write(*,*) ' 1 - PLANE ems' write(*,*) ' 2 8 PLANE SIRAJN' read(*,*)iplane writ-30.“) ' ' write(*,*) 'ENI‘ER 'IHE NIMEER or may read(*,*)n write(*,*) ' ' write(*,*) 'm missm's mo' 193603)!!! if(ip1ane.ne.1)then MAL-Pr) else endif write(*,*) ' 1 can wiry (n.X.y.m.ym.M.rny.s) write(*,*) 'mw MANY mon POMS?‘ ”(*fflm write(*,*)' ' write(*,*) 'ENI'ER XF,YF (11E SET PER LINE' do 5 i=1,m reaIr1(*.“')>.eq.2)ther do 20 i-l,n 1122*(1-1)+1 12-2*(i-1)+2 m=n(i)*)nn(i) WHWmm WWM) Wm“) it(ym(i) .eq.0. )then bc(il)=1nn(i) bc(12)=-0.45*m2 ibc(i,2)=l boar(il)=hc(il) boar(i2)=bc(12) ue(il)=(4.*m(i)-m3/12.)/2.5 m(12)=(-1.125*m2)/2.5 te(il)-aun(i) te(iZ)-0. else it(xm(i) .eq.1.)the1 bo(i1)=3.75+ym2/2. bc(12)=0.5*ym(i) -l. boar(il)=-bc(il) boar(iZ)-bc(12) 93 mun-(47. /12. +0. 5625*ym2-ym(i)/4. )/2. 5 m(12)=(ym2/2.-ym3/8.-15. *ym(i)/16.-1.125)/2.5 te(il)-yrn2/2. +3. 75 te(12)‘YM(l)/2- -1 else it(ym(i) .eq.1.)t.hen bc(il)-O.5*aan(i) bc(12)=0.75 boar(il)=bc(il) bal'l(12)=bc(12) ue(11)-(69.m(1)/16.-m3/12.V2.5 11e(iZ)=(-5./8.-17.*3an2/16.)/2.5 te(i1)-0.5*m(i) te(12)-O.75 else bc(i1)-0. bc(12)=0. ibc(i,1)=l boar(il)-o. boon(12)=0. ue(il)=0. a(12)=(ym2/2--ym3/8o-ym(i))/2-5 te(il)--ym2/2.-4. te(12)-o. edit cartinJe do 25 i=1,mn xt2=xt(i)*xt(i) yf2=yf(i)*yf(i) mum) WWW snore(i)=4.-xt2/4.+yf2/2. syye(l)-yf(i)-yf2/4- sxye(i)=xt(i)*yt(i)/2.-xf(i) uxe(i)=(xt(1)*(4.-xt2/12.tyt2/2.-(yt(i)-yt2/4.)/4.))/2.5 uye(1)-(yt2/2.-yt3/12.-yt(i)*(4.-xt2/4.+yt2/6.)/4.-1.125* xt2)/2.5 oartirue else it(iptcb.eq.3)ther write(*,*) 'ENIER mamas mm uxao' reed(*,*)bcl,ix2 write(*.*) ' ' write(*, *) 'EJTER mm MERE UY=0' reed(*, *)iy1, iyz write(*, *) ' do 30 i-l,n 11-2*(1-1) +1 94 12-2*(I-1)+2 it(i.eq.ix1.,or.i.eq.ix2)ther - bc(il)=0. bc(12)=-my(i) ibc(i,1)-1 boar(il)-bc(il) bam(12)=bc(12) else it(i.eq.iy1.or.i.eq.iy2)then be(il)-mx(i) bc(12)-=0. ibc(i,2)=1 boa1(il)=bc(il) boa1(12)=bc(i2) else bc(il)=-mx(i) bcuzzdmyq) boar(11)=bc(1l) boa1(12)-bc(12) edit 11e(il)-—O.3*rnx(i) m(iZ)=-O.3*rny(i) te(il)-zmc(i) te(12)°-mY(i) 30 oartirue do 35 i-l,nln uxe(i)=-O.3*xt(i) uye(i)-0-3*yf(i) eone(i)=-1. syye(i)=-1- sxyem=0- oartinue else goto 100 edit write(*,*) 'W CINDITIm VECICFS' write(*.*)' ' write(*,115) 115 format( my ,4x, 'Bcan ' ,4x,"3<:(12) ' ,4x,‘Im(I,1)',4x, '13:: 8 (IR-P) do 40 i=1,n 11=2*(1-1)+1 12-2*(1-1)+2 write(*.120)i.b<=(11) .bC(12) .ibC(i.l) .ibC(i.2) 120 tormet(12.43:,t6.3,4x,t6.3,7x,il,11x,il) 40 cartinre C C C 95 write(*.*) ' ' write(*,*) 'm m mm m cmrmna' mnd(*.*)1qo writa(*.*) ' ' return ed mature iJ'gptMrmflLVZ) dinersim r(10,w(10) ,w1(10) ,w2(10) CPOINISANDWEIQIISFUIWQLMW C C r(1)-.97390652851717 r(2)-.86506336668898 r(3)-—.67940956829902 r(4)=-.43339539412925 r(5) -. 14887433898163 r(6)-r(5) r(7)-r(4) r(8)=-r(3) r(9)-r(2) r(10--r(1) w(1)-.066671344308688 w(2)-.14945134915058 w(3)-.21908636251598 w(4)=.26926671931000 w(5)=.29552422471475 w(6)=w(5) w(7)=w(4) w(8)=w(3) W(9)!W(2) - w(10=w(1) do 10 l-l,10 wl(l)=1.-r(l) va(l)=1.r(l) oartirue return ed subrmtim casea(n,pr,x,y,m,ym,mx,my,s,r,w,e,b) parameter (nu-75) pet-emu: (me-QM) dimrsiar x(nn),y(‘1m) dimension mm) .wtm) .mM) .my(m) .somn) dimesim e(nrm,nnn) ,b(1mn,nur) ,r(10) ,w(10) cmmmmmcss 96 mm pi-4.*eten(l.) (1141313!) C2'(1~‘Pfl'-') C3'(3-+Pt) ell-(3.1:?) c5=(l.+3.*pr) do 2 j-1,n 332-2*j 331-332-1 it (j .eq. 1)ther jml-n ijmz-nn else jm1=j-1 332192-3324 edit jj1m2=ijm2-1 it(j .eq.n)the1 jpl=1 33292-2 else jp1=rj+1 jj2p2:jjz+2 edit 33192‘332132'1 rnl=nnc(j) mz-mym dead) m12=ml*m1 m22=rn2*mz b(jj1,jj1)-(c1*rn22-c4* (1og(o. 5*ds) -1. ) ) *ds/Z . b(jj1,jj2)=1c1*m1*rnz*ds/2. b(332.jjl)=b(jjl.jjz) b(jj2,jj2)-(o1*m12-o4*(log(ds*o.5)-1. ) ) ids/2. drnnl-x(j)—m(j) medrwm o-dllnl’tdnml-l-dmz do 2 i-1,n it (i.eq.j)gotoz iiz=2*i iil-iiz-l drnleun(i)-)an(j) Mummy mmww ONUO 97 BID-2 . *. (mmwm) do 3 1-1,10 duo-eqrt(ee+tb*r(l)+c*r(l) *r(1)) qw(dm1-dm1*r(l))/rho qr=(dm2-dun2*r(l) )/rho t11-(c12* (dm2*¢3+dmn1*qy3 ) +c3*dnln2*qx-c2*dm1*qy) /rho u11= (c1*qx2-c4*log(rho) ) *ds/4 . ulz-c1*q:c*qy*ds/4 . 1122- (c1*qy2-c4*log(rho) ) *ds/4 . e(iil,jjl)-a(iil,jj1)+w(l)*t11 ' e(iil,jj2)-a(iil,jj2)+w(l)*t21 3(112.331)-a(112.fil)+¥l(l)*t12 8(1127332)'a(1127332)w(1)*t22 b(iil,jj1)-b(iil,jj1)+w(l)*ull b(ii1,jj2)=b(iil,j32)+w(l)*u12 b(112.33'1)=b(112.fil)+W(l)*'-112 b(112ojjz)'b(112:2532)+w(1)*‘-I22 oartinue cartimxe do 4 i=1,n :Li.2=-2*i iil-iiz-l do 4 k=1,n it(k.eq.i)goto 4 Ida-2*]: mean-1 e(ii1,iil)-e(iil,iil)-a(ii1,ldt1) e(ii1,iiZ)-e(iil,iiZ)-e(iil,ld<2) a(iiz,iil)=a(iiz,iil)-e(iiz,kk1) e(iiz,iiZ)=e(iiz,iiZ)-e(iiz,kk2) oartime write(*,*) 'CnB'EN'I' MMRI‘ W' write(*,*) ' ' retanm ed C 98 W oeseb(n,pr,x,y,s,r,w,w1,w2,e,b) peremter (rm-75) permter (mu-2m) dimensiar x(m) ,y(m) ,s(m) dimesiar r(10) ,w(10) ,w1(10) .102 (10) dimensiar e(nnn,nlm) ,b(mm,nrm) cmmcmermc C m-zm ' pi-4.*atan(1.) CPR-+9!) c12=c1*2. CZ‘U-‘Pfi C3'(3-+P1‘) (3443-1317 c5-(1.+3.*pr) do 2 j-1,n fill-2‘9 311-3324 it (:1 .eq. 1)t.ber jml-n jj2m2=m else jm1aj-1 332m2'j32-2 edit jjm‘jjm'l it(j .eq.n)ther jpl=1 33292=2 else jp1=j+1 2132133132” edit jjlptéijPZ-l dx-xkaunl) dy-y(j)-y(jm1) ds=sqrt( needy/ch max/<18 10031 jjl)=(o4*(1 -log(ds) )+c1*rny(my)*ds/2- 1:031 332)=1<=1 bdjzdjlhbdjl 312) b(jj2,jj2)-(o4*(1.-log(ds))+c1*mx*mx) *ds/Z. 99 due-x(jpl)-x(j) momma) des-eqfi.( nerdy/ass was b(331.331P2)'(°4*(1--1°9(€188) )WIW) “188/2- b(jj1,jj2p2)-c1*mc*n1y*dss/2. b(332.jjlp2)=b(jjl.ijz) b(jj2,jj2p2)-(o4*(1.-log(dss))+c1*nnc*rm¢)*dss/2. ) nun-(x(j)+x(jnl) )/2- Yul-(YOHYdml) )/2- duml-x(j)-m ammo-m mum-1mm do 2 i-l,n ii2-2*i iil-iiZ-l dual-a“ i) -m mama ae-dml*dml+dnn2*dm2 tb-2.*(dm1*dm1+dnn2*dm2) do 3 1-1,1o dw-sqrt(ee+tb*r(l)+c*r(l) *r(l)) qx-(dml-dmnru) )/rho qy~(dm2-dm2*r(1) mm War We! WW! WW tll-(c12*(chm2*q>a+drml*qy3)+c2*dnm2*@(-c3*dml*qy)/(2.*rho) t12-(c12*(ch1m1*qx3-dnn2*qy3)-c3*drun1*qxt-c5*dnm2*qy)/(2.*rho) t21-(c12*(dun1*q:o-dm2*qp) :c5*dm1*qx-l-c3*¢mn2*qy)/(2.*d10) t22-(-c12* (dum2*qx3+dm1*qy3)+c3*dm2*qx-a*dm1*qy) / (2.*rho) it(i.eq.j)then e(ii1,jjm2)-e(iil,jj1m2)+w(l)*w1(l)*tll e(iil,jj2m2)=e(iil,jj2m2)+w(l)*w1(1)*t21 a(112.jjlm2)-a(112.iim2)+w(l)*wl(l)*tn‘ a(112.332n2)-a(112.j:)2m2)+w(l)*w1(1) *t22 else it(i.eq.jml)ther 3(1117331)'=a(iil.fil)w(1)“0(1)*tll e(iil,jj2)-e(iil,jj2)+w(l)mu)*t21 a<112.331)-a(112.331)+w<1)muvm e(112,jj2)-e(112,jj2)+w(l)M(l)*t22 100 else e(iil,jj1n2)q(ii1,jj1m2)+w(l)*w1(l)*t11 e(iil,jj2m2)-1(iil,jj2m2)+w(l)M(1)*t21 e(i:12,jj1m2)=e(iiz,jj1m2)+w(l)*w1(1)*t12 e(122,ijm2)-e(iiz,332m2)+w(l)*wl(l)*t22 a(iil.ijl)=a(iil.jjl)+w(l)*w2(l)*t11 e(iil,ij)=-e(iil,jj2)+w(1)M(l)*t21 a(112.jjl)-a(112.ijl)+W(l)M(l)*t12 6(112.332)a(112.332)w(1)*w2(1) *t22 edit it(i.eq.j.or.1.eq.jm1)goto 3 1- (c1*cpa-o4*log(rho) ) *ds/4. 1112-01 u22- (c1*qy2-04*log(rho) ) *ds/4. b(iil,jj1)-b(iil,jj1)+w(l)*u11 b(iil,jj2)=b(iil,jj2)+w(l)*u12 b(112.jjl)=b(112.jjl)+wl)*u12 b(112.jjz)=b(112.jjz)+wl) *1122 cartime oartirue mmmorm ()0an0 do 4 181,11 iiz=2*i . iil-iiz-l do 4 k=1,n it(i.eq.i)goto4 kk2=2*k kk1=kk2-1 a(iil,iil)=e(iil,iil)-e(iil,ldt1) e(iil,iiZ)-e(iil,iiZ)-a(ii1,ldc2) e(ii2,iil)-e(i12,iil)-e(iiz,kk1) e(iiz,iiZ)-e(iiz,iiZ)-e(iiz,ldc2) oartirue write(*,*) 'Ncmsommmmxc mmx W' write(*,*) ' ' return ed suwatiJ're casec(n,pr,x,y,s,r,w,w1,w2,a,b) parameter (nu-75) 101 pewter (mu-2mm) dinesiar x(m) ,yom) ,s(nn) dimaia: r(10) ,w(10) ,w1(10) ,w2 (10) dine-aim e(nlm,nnn) ,b(nrm,mrrn) C C comm: CIEFFICIENI‘ ”ICES C Ill-2m pi-4,*etan(l.) CPU-JPN!) c12-c1*2. c2-(1.-pr:) C3‘(3-+Pl') 0443-13?) c5-(l.+3.*pr) do 2 j-1,n fill-2‘9 331'532-1 it (3 .eq.1)ther jml-n ijInz-nn else jmlsj—l 332m2-332-2 edit jjm-jjzmz-1 it C) .eq.n)ther jpl-l 11W else jp1=j+1 332p2=332+2 edit jilp2=332P2-1 dx-x(j)-x(jm1) WWW-Him) ds-sqrt(dx*dx+dy*dy) needy/d5 my—dx/ds b(jj1,jj1m2)-(o4* (O.5-log(ds) )+c1*my*my) ids/4. b(jj1,jj2m2)=1c1*mc*nry(de/4. b(332.jilm2)=b(jjl.332) b(jj2,jjzm2)-(c4*(o.s-1og(ds) )+c1*nnc*nm)*ds/4. b(331.3jl)-(a* (1 5-l°9(ds) )lemY) *ds/4 10031 332)--01 #- bdjzdjlkbdjldjz) b(jjz,jj2)-(o4*(1.5-log(ds) )+c1*rmc*rnx)*ds/4. 102 die-X09040) moon-m) deg-sqru . dey/dss was 130 b6 3 jl)-b(jjl,jj1)+(o4*(1.5-1og(o+dml*qy3)+c2*drm2*qx-c3*eml*qy)/(2.*rho) t12-(c12*(dml*qx3-dnln2*qy3)-c3*dnln1*qx-I-c5*dnln2*qy)/(2.*d1o) t21-(c12*(dnm1*qx3-enn2*qy3)-c5*dml*qafic3*dmn2*qy)/(2.*rho) t22-(-012* (drun2*qao+dm1*qy3)+c3*chun2*qx-c2*dml*qy) / (2.1mm) it(i.eq.j)then e(iil,jj1m2)-e(iil,jj1m2)+w(l)*w1(1)*t11 e(iil,jj2m2)=e(iil,jj2m2)+w(l)*w1(1)*t21 a(iiz,jjlm2)=e(iiz,jj1m2)+w(l)*w1(l)*t12 a(iiz,jj2m2)-a(iiz,jj2m2)+«v(l)*w1(1)*t22 OOONUO 103 else it(i.eq.j1n1)ther e(iil,jj1)-e(iil,jjl)+w(l)*w2(l) *tll e(ii1,ij)-e(ii1,jj2)+w(l) “'2 (1) *t21 3(112.fil)-a(112.ijl)+wu)m(l)*t12 e(iiz,j32)-e(iiz,jj2)+w(l) “'2 (1) "=22 else a(iil.jilm2)-a(iil.ijln2)+wl)M(l)*tll e(iil,jj2m2)-e(iil,jj2m2)+w(l)*w1(l)*t21 a(112.ijlm2)-a(112.jjlm2)+wl)*w1(1)*t12 a(112.332n2)-a(112.332n2)+wl)M(l)*t22 8(111o311)'a(111.311)+w(1)m(l)*t11 e(iil,jj2)-e(ii1,jj2)+w(l)*wz(l)*t21 a<112.331)=a<112.331)+w(1)mu) *t12 e(iiz,jj2)-e(iiz,j32)+w(l) W2 (1) “=22 edit it(i.eq.j.or.i.eq.jm1)goto 3 “11'(¢1*@¢2'°4*1°9(fib) ) *ds/ 8 - U12-aways. u22-(c1*qy2-o4*log(rho) ) ids/8 . b(iil.331n2)-b(1il.jjm2)+wl)*w1(1)*ull b(ii1,jj2m2)=b(iil,jj2m2)+w(l)*w1(l)*u12 b(iiz,jj1m2)=b(iiz,jj1m2)+w(l)*w1(l)*u12 b(112.312m2)-b(112.332m2)W(l)*w1(1)*1122 b(iil,jjl)-b(ii1,jj1)+w(l)M(l)*u11 b(111,jjz)=b(111,jj2)+w(1)mmamz b(iiz,jj1)=b(iiz,jj1)+w(l)*w2(l)*u12 b(112:112)‘b(1127312)+w(l)m(1)*‘l22 cartime oartixue mmac’mumorm do 4 i-l,n 112-2*1 111:112-1 do 4 le-lm it (k.eq. i)goto 4 kk2=2*k kklska-l e(iil,ii1)-e(iil,ii1)-e(iil,ldc1) e(iil,iiZ)=e(iil,iiZ)-e(iil,kk2) e(iiz,iil)-e(iiz,iil)-e(iiz,ldcl) a(112,112)q(112,112)-a(112,m) 104 4 cartirue write(*,*) 'IWC mm W' write(*,*) ' ' return ed Mm game(1nc,dm,m) perennter (nu-75) perennter (mu-21mm) permter (mwn dimim uoc(1mn,urm) ,rhsm'm) dinersiar Mariam) ml-nn-l ml-nrH-l C C SET UP 'IHE Hm MA'IRIX HR AX=B C do 2 i-1,m do 1 j-1,rrn mudrwcud) cartinie madden-rad) oartinae mmmmmmmxmmm mmmmm. do 8 i=1,m1 SMFUI'IHEWENIRYINGJHMLMBIWN. IPIWPIS'B‘IEWMOF'BIEWW. GOOD OOOON H pivot-o. do 3 j-lmn WMQUAH it(pivot.ge.telp) go to 3 ipiwb-d oartime it(pivot.eq.0.) goto 13 it(ipivot,eq.i) goto 5 c C W now I AND new mm. e do 4 k-l,rp1 tap-mad» m(1.k)-aug(1piwt.k) «(mimkktap cartinae 105 O C (I+1,I),(I+2,I),..., (N,I) IN 'IHE mm MIX. ipl-i+l do 7 lc-ip1,m mkdflmd.“ eug(k,i)-o. do 6 jBilepl qug(k.i)-a*me(1.i)+m(k.j) cartinie earthen oaatirue it(eu;(m,m).eq.o.) goto 13 W '10 08mm SOIUI'ICN '10 AX=B. UIO GOO couch 1318031)“ (“mum/M (1111.!!!) ® 10 killfilnl $0. do 9 j-1,k ma-l.)*chrm2-(8,*qx2*qy2-1.)idnml) 108 /d'ro2 s221-4.*((8.*qx2*qy2-1.)*dnln2-2.*qx*qy((4.*qy2-1.)*dml) xmoz 8222-4.*(2.*qx*qy()4.*qy2-1.)*drmZ-(8,*qy4-4.*qy2-1.)*dm1/ /rhoz 0mm:- ul(3,jj1)-ul(3,jj1)+v(l) *3111 01(37132)'!11(3:132)+W(1)*8211 u1(r,jjl)=ul(4,jj1)+w(1)*s211 “1(47332)='\11(4:532)+W(1)*8221 ul(5,jj1)-a.rl(5,jj1)+w(1)*3221 91(5.332)-!11(5.332)+W(l)*8222 d111-(-2 . tango-cam) /rho d112- (8 . *c1*qY3-<5*qy) /rho d121= (2 . *c1*qy3-c3*qy) /rho d122= (2 . *c1*qx3-c3*qx) /rho d221=(2 . *c1*q>c3-c5*qx) /rho d222-(-2.*c1*qy3-c2*qy)/rho m(3,jj1)-=ru(3,jj1)+w(l)*d111 m(3.332)-ur(3.332)+w1)*6112 m'(4.iil)-ur(4.jjl)+w(l)*d121 Im(4.332)=11r(4.332)+w1) *d122 m‘(5.331)=m'(5.331)+wu) *6221 ur(5.iiZ)-ur(5.ij)+w(l)*d222 137 cartixue mac-0. uuy-o. seat-O. sety=0. SSYY'O- do 142 i-lmn unmet-ul(1,i)*ua(i)+ur(1,i)*ta(i) many-ul(2,1)*m(i)+ur(2,i)*ta(i) W1*ul(3,i)*ua(i)+ur(3,i)*te(i) Wl’tul(4,i)*ue(i)fiar(4,i)*te(i) Wolm(5,i)*\1e(i)+\m(5,i)*te(i) 142 oartime m(ii)-n.nnc/(8.*pi) “Ya(11)'UUY/(3-*Pi) eoa(ii)=ss)og/(8.*pi) ma(ii)=say/(8-*pi) C syya(ii)=esyy/(8-*pi) 135 cartixue C retina ed 109 Mm tl®(n,m,pr,x,y,r,w,w1,w2,ub,tb,st,yt,m:b,uyb, & eatb,e¢yb,syyb) peremter-(WS) peremter(mn-2*m) permeate-(WWI) pereneter(nlm-50 dimersiar x(m) ,y(nn) dimension r(10) ,w(10) ,w1(10,vf2(10) dimiar ub(mm) ,tbau'rn) ,tbamn) dinesiar xt(nmn) ,yt(1mm) dimiar u1(5,mm) ,ur(5,um) dimensiar mtb(nm) ,uybmun) dimensiar eotbmnn) ,sxyb(nm) ,syyb(mn) Mic“ t331916) $61926) #6936) #8945) I'm-2*!) pi-4.*aten(1.) cl-l.+pr c2-1.-pr c3=3.+pr c4-3.-pr c5-l.+3.*pr do 135 ii-1,nln do 136 i=1,nn do 136 khl,5 ul(k,i)-O. ur(k,i)-0. 136 cartime do 137 j=1,n it(j .eq.1)then jmlnn else jm1=j-1 edit 332-25 331-332-1 it(j.eq.1)ther jjm-am else 33219-3324 edit , ijm-diznz-l die-X6) -X(jml) army-mm) mu ) 110 fb(331)'tb(331) *8 5033413632) *8 mi-(X(j)+X(inl) )/2- Yllli'(Y(j)+Y(jm1))/2- dml-x(j)-xmi dam-y(i)-ymi mmww drn1=a1t(ii)-xmi man-3m mmww bb-2.*(drm1*dm1+dm2*chm2) do 137 l-1,10 dronsqz't(ee+tb*r(l)+c*r(l) *r(l)) mum gt=(dnnl-dnlnl*r(1) )/rho qy=(dnn2-dnln2*r(1) )/rho t11-(2.*c1*(dnln2*q)0+dml*qy3)+c2*dnln2*qx-c3*dnm1*qy)/rho t21-(2.*c1*(chln1*qx3-dm2*qy3)-c5*drun1*qx+c3*dnm2*qy)/dro t12-(2.*c1*(dml*qx3-dnm2*qy3)-c3*dnn1*w*dmn2*qy)/rho t22-(2.*cl*(-dmn2*qx3-dun1*qy3)+c3*dm2*qx-c2*dnm1*qy)/mo ul(1,jj11n2)-ul(1,jj1m2)+w(l)*w1(l)*tll ul(1,jj2m2.)=ml(1,jj2m2)+w(l)*w1(l)*t21 ul(2.jjlm2)-ul(2.ijlm2)+wm*w1(l)*t12 ul(2.332m2)-ul(2.jjzm2)+w(l)*w1(l)*t22 “1(17331)'01(1o311)+w(1)m(l)*t11 u1(1,jj2)+ul(1,jj2)+w(1)m(1)*t21 “1(2.311)W1(2.111)W(1)m(1)"=12 91(2:132)+ul(2o132)+"(1)m(l)*t22 u11-(-o4*log(mo)+c1*qx*qx)/2. warmly/2. u22-(-o4*log(rho)+c1*qy(qy)/2. tn'(l.331)-ur(1.3jl)+W(1)*ull m‘(l.332)-ur(1.332)+w(1)*912 \n'(2,jjl)-m'(2,jj1)+2 (1)*u12 ur(2.332)-ur(2.332)+W(1) *‘122 8111-2.*((8.*qx4-4.*q:0-1.)W-2.*qx*qy*(4.*qx2-2.)*dnlnl) rhoz / 3211-2.*(2.W(4.*qx2-1.)*dm2-(8.*qx2*qy2-1.)*dmn1) /rhoz 111 8221-2.*((8.*ga*qy2-l.)*chln2-2.W(4.*qy2-1.)*dnml) & /mo2 8222-2.*(2.W(4.*qy2-1.)*dnnz-(8.*qy4-4.*qy2-1.)*dmnl) a /rho2 u1(3,jj1n2)-u1(3,jjlm2)+w(l)*w1(l)*3111 ul(3,jj2m2)-ul(3,jj2m2)+w(1)*w1(1)*3211 ul(4,jj1m2)=ul(4,jj1m2)+w(1)M(1)*5211 ul(4,jj2m2)-ul(4,j32m2)+w(1)W1(1)*3221 ul(5,jj1m2)-ul(5,jj1m2)+w(1)*w1(1)*3221 u1(5,jj2m2)-ul(5,j32m2)+w(1)*w1(1) *3222 “1(3:111)+01(3.111)W(1) M (1) *3111 “1(3.332)+01 (3.312)+W(1) *W2 (1) *8211 “1(4le)+\11 (4.131)+V(1) M (1) *8211 ul(4,jj2)+ul(4,jj2)+w(l) *W2 (1) *8221 ul(5,jj1)+ul(5,jj1)+w(1) m (1) *5221 ul(5,j32)+ul (5.312)+W(1) M (1) *8222 d111-(-2 . *c1*qx3-c2*qx) /rho d112-( 2 . *c1*qy3-c5*qy) /rho d121= (2 . *c1*qy3-c3*qy) /rho d122- (2 . *c1*qx3-c3*qx) /rho d221= (2 . *cl*gc3-c5*qx) /rho d222= (-2 . *cl*qy3-c2*qy) /rho ur(3,jj1)-ur(3,jj1)+w(l)*dlll ur(3,jj2)=mr(3,jj2)+w(l)*d112 ur(4,jj1)=m'(4,jjl)+w(l)*d121 m(4,jj2)-ur(4,jj2)+w(1) *d122 ur(5,jj1)-m'(5,jjl)W(1)*d221 ur(5,ij)-1m(5,jj2)+w(l)*d222 C 137 oartixue C C W UL 8! GM!“ mm m C ads-1 do 138 131,5 tepl(l)=ul(l,1) mun-mum 138 oartime do 139 le-2,n Ida-2*]: kklskkz-l do 140 181,5 n1(1,l)a.1l(l,l)+ul(l,kkl)*ee ul(l,2)-u1(l,2)+u1(l,ldc2)*ee 140 oartime, ee-ee 139 oartirue C 141 142 112 do 141 1-1,5 tep3(l)-ul(1,ldc1) W(l)=ul(l.kk2) ul(l,kk1)=-ul(l,ldc1-2)+2.*terp1(l) u1(1,kk2)=-ul(1,)dQ-2)+2.*‘telp2(l) tapl(1)=tap3(l) tapflh'tapfll) cartirue aux-0 uuy-o. seen-o. sexy-0 SSW-0 do 142 i-l,nn mum-ul(1,i)*ub(i)+ur(l, i) *fb(i) W0,” *ub(i)+ur(2, i) *tb(i) Welml (3, i) *ub(i)+ur(3,i) *tb(i) ssxy=esxy+c1*ul (4, i) *ub(i)+ur(4, i) *tb(i) ssyy=ssyy4c1ml (5, i) *ub(i)+ur(5, i) *tb(i) oartin1e md)(ii)nnnq/(8.*pi) uyb(ii)=m1y/(8.*pi) eow mmwwdmz dm1=xt( ii) -xmi wan-mi ee-drml*drm1+dmr2*drm2 m—2.*(drm*dm1+dm2*dm2) do 137 131,10 114 dro-sqzt(ee+tb*r(l)+c*r(l)*r(l)) we» qr(dnn1-dm1*r(l))/d1o qy-(dmZ-dmz*r(l) )/rho M W W“? WW Q‘WW WWW t11-(2.*c1*(mn2*qx34dm1*ey3)+c2*¢nn2*qx-c3*dnm1*qy)/dio t21-(2.*c1*(chm1*qaa-dm2*qy3)-cs*¢m1*qc+c3*dnm2*qy)/dro t12-(2.*c1*(dm1*uc5*dm2*qy)/mo t22=(2.*c1*(-chln2*qao-dm1*qy3)+c3mz*qx-c2*dnn1*qy)/mo u1(1,jjlm2)-ul(1,jj1m2)+w(1)*w1(1)*t11 u1(1,j32m2)a31(1,jj2m2)+w(l)*w1(1)*t21 u1(2,jjlm2)=ul(2,jj11n2)+w(l)*w1(1)*tlz “ladimktllflofimfiflnm(l)*t22 ul(1,jj1)-ul(1,jj1)+w(l) m (10*1111 “1(1o332)‘11(11332)w(1) “’3 (1) ":21 “1(2o331)‘11(21331)w(1) "2(1) *t12 “1‘2 ,jj2)'!11(2,jj2)+W(l) m (1) *t22 ull- (-c4*log (rho) +c1*qx*qx) /4. WW”- 022=(-04*109(fib)+cl*W)/4 m'(1.331n2)-mr(1.331n2)+~v(l)*w1(1)*ull m(1,:!jZm2)-tm(1,jj2m2)+w(l)*w1(l)*u12 m(2.331n2)er(2.331n2)+w(1)*w1(1)*u12 m'(2.jjzm2)-ur(2.jjm)+w(l)*w1(1)*uzz m'(1.jjl)-ur(1.3jl)+w(l)*w2(i)*ul1 ur(1.332)-ur(1.332)+w(l)M(l)*un ur(2.jjl)n1r(2.jjl)+W(l)*w2(l)*qu m‘(2.332)-n'(2.332)+W(1)M(1) *u22 slll-Z.*((8.*qx4-4.*q:a-1.)*drnn2-2.*a-1.)*dnml) éznlfiz.*(2.W(4.*qaa-1.)*dm2-(8.*q>a*qy2-1.)*dnln1) £32232.”(8.*q:a*qy2-1.)*dm2-2.*qx*+w(1)mm *szn ul(4,jj1m2)-ul(4,jj1m2)+w(l)*w1(l) *3211 ul(4,jj2m2)-ul(4,jj2m2)w(l)*w1(1)*3221 ul(5,jj1n2)-ul(5,jj1m2)+w(l)*w1(l)*3221 115 “1(5o332m2)‘ll(57132m2)+w(1)”1(1)sz u1(3,jji)an(3,jj1)+w(1)ma)*3111 u1(3,jj2)-ul(3,jj2)+w(1)*w2(1)*3211 “1(4o111)‘11(4.331)+w(1)m(1) *8211 111(4ofiz)“11(4.3°3°2)w(1) *W2 (1) *8221 “1(5o131)‘01 (sojj1)+w(1) ‘10 (1) *3221 “1(57312)'01(57132)+V(1)*W2(1) *8222 d111-(-2 . *cl*qc3-c2*qx)/ (rho*2 .) d112-(2.*c1*qy3-c5*qy)/(dro*2.) d121-(2 . *c1*qy3-cn*qy) / (rho*2 .) d122-(2.*c1*qx3-c3*cp¢)/ (1110*2.) d221=(2. *cl*