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FINES will be charged if book is returned after the date stamped below. MODIFICATIONS OF THE BOUNDARY- ELEMENT METHOD FOR APPLICATION TO BENDING OF LAYERED COMPOSITES BY David H. Harry A DISSERTATION Submitted to Michigan State University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics, and Materials Science 1982 ABSTRACT MODIFICATIONS OF THE BOUNDARY- ELEMENT METHOD FOR APPLICATION TO BENDING OF LAYERED COMPOSITES BY David H.Harry Modifications of the Boundary-Element Method (BEM), which improve accuracy of bending problem solutions, are presented. The direct and indirect methods are discussed as are the modifications of both methods for the purpose of improving accuracy. Prior to application to layered composites, it is necessary to concentrate on beam bending problems since BEM has not modeled beam bending satisfactorily. The modific- ations are illustrated by modeling beam problems and compar- ing the results with beam.theory and elasticity solutions. The most beneficial modification is obtained, with the indirect-method, by moving the potential sources outside the problem boundary. This procedure is not defined in the direct-method, so further work is limited to the indirect- method. Another modification which resulted in improvement is the addition of rotation and moment boundary conditions along with additional required potential sources. Accuracy is further enhanced by averaging the boundary data rather than specifying the data pointwise as in the initial work. These additions to BEM result in accurate solutions to beam problems with aspect-ratios up to 100:1. Beams of this length, however, are sensitive to the arrangement of the boundary meshes and the type of beam loading. An alternative to BEM is developed for beams com- posed of layers with large aspect-ratios. This alternative is a solution of a curved beam-string with a shape described by a quadratic polynomial. The solution is subjected to beam-theory assumptions, with deflections due to axial loads included, to model the membrane behavior of the thin layers. The layer-coupling problem is presented using both stiffness and compliance techniques. Compliance-coupling is the more accurate technique since additional matrix opera- tions required in the stiffness technique. The straight sandwich beam problem is solved using three layers modeled by BEM as well as two curved beam-string solutions coupled to a BEM modeled core. The latter model yields results more comparable to the sandwich theory solution. The curved beam- string solutions, are also applied to a tapered curved sandwich structure with a BEM modeled core. This structure has curved sandwich faces coupled at the ends. The solution compares favorably with experimental results. ACKNOWLEDGEMENTS I express my thanks to all those persons who gave their assistance in the undertaking of this work as well as those who were able to tolerate my distraction. They include Professor NicholasiL Altiero, major adviser, who made suggestions and supplied the encouragement I needed. I also thank Professors William A. Bradley, Robert W. Little, and Donald J. Montgomery, who seemed to ask the proper challenging questions on the the appropriate occasions. :1 wish also ix) thank Professors Gerald R. Schneberger and Joseph Lestingi, former and present Heads of the General Motors Institute Department CHE Mechanical Engineering, ‘who arranged nnr teaching schedule to accommodate the numerous classes and meetings required of me. My thanks are extended to my other friends who tolerated a change in my level of association. I express a warm gratitude to my wife, Norita, and to my children, Timothy and Tina, for their understanding. ii ——--.- v—v— LIST OF LIST OF LIST OF Chapter Chapter Chapter Chapter TABLE OF CONTENTS TABLES FIGURES APPENDIXES I II III IV INTRODUCTION CURVED BEAM-STRING SOLUTION 11.1 Background 11.2 Influence Function 11.3 Curved-Beam Coupling 11.4 Illustrative Problems BOUNDARY-ELEMENT ANALYSIS 111.1 Background 111.2 Direct-Method 111.3 Indirect-Method 111.4 Boundary-Element Difficulties 111.5 Boundary-Source Shape Functions 111.6 Corner Treatment OUTER-BOUNDARY TECHNIQUES IV.l Background 1V.2 Bending Problem Modeling 1V.3 Boundary-Data Averaging 1V.4 Moment and Rotation Boundary Conditions IV.5 Higher Aspect-Ratios iii Page vii ix 33 62 TABLE OF CONTENTS (CONTINUED) Page Chapter V REGION COUPLING 84 V.l Background v.2 Stiffness-Matrix Solution v.3 Compliance Coupling v.4 Compliance-Coupled Illustrative Problems Chapter VI CLOSURE 106 APPENDIXES 108 BIBLIOGRAPY 172 iv Table 2.1 2.2 2.3 2.4 2.5 2.6 2.7 3.1 3.2 3.3 LIST OF TABLES Straight-Beam Verification Least-Squares Quarter-Circle Response Point—Matched Quarter-Circle Response Half-Quadrant Responses Calculated Twin-Beam Responses with an End Load of Py = 1 Responses of the Stiffness-Coupled Spring-Core Responses of the Compliance-Coupled Spring-Core End Loaded 4:1-Cantilever-Beam Shape- Function Results Constant Shape-Function Sources Applied to a Traction-Specified Cantilever-Beam Quadratic Shape-Function Sources Applied to a Traction-Specified Cantilever-Beam Comparison of Results with and without Corner Treatment of a 36-Mesh Square in Tension "H' Spacing Sensitivity for a Tension- Loaded Strip “B” Spacing Sensitivity for a 10:1 End- Loaded Cantilever Sensitivity for a 48-Mesh Cantilever Mesh Data Averaging for a 10:1-Beam Page 19 21 21 23 27 30 32 48 49 50 60 65 66 67 71 4.5 4.6 5.6 5.7 LIST OF TABLES (CONTINUED) 10:1-Beam with Mesh-Averaging with Distributed Outer-Boundary Sources Cantilever-Beam Utilizing Moment and Rotation Conditions End-Loaded 100:1-Canti1ever End-Moment 100:1-Cantilever Uniformly Distributed Load 100:1-Cantilever Stiffness-Matrix Solution of a Square Boundary in Uniaxial Tension Stiffness-Matrix Solution of a 5x1 Rectangular Strip in Tension Results from Bi-region Tension-Strip Straight-Sandwich Results (66-mesh model) Straight-Sandwich with Beam-String Modeled Faces (44-mesh model) Straight-Sandwich with Beam-String Modeled Faces (52-mesh model) Results of a 32-Mesh Curved-Sandwich vi page 73 79 81 82 83 86 87 94 96 98 101 103 3.1 3.2 3.3 3.4 3.7 3.8 3.9 LIST OF FIGURES Curved Sandwich-Beam Curved Beam-String Definitions Internal Beam Data Coupled Curved-Beams Column Matrix of Known Data Right Side of Matrix Equation Free Surface Response Spring-Core Model Arbitrary Beam Embedded in a Simply Supported Beam Beam L Embedded in Beam L* Showing Sources to Satisfy End Data Stress Field of Two Adjacent Unit-Sources in the Infinite Plane in Plane Stress with v=.35 Quadratic Shape-Function of Boundary- Sources End-Loaded 4:1-Canti1ever Deflection Averaging for a Source- Placed Mesh Source Distribution for the 4:1-Beam with 9 Constant Sources per Mesh Boundary-Traction Singularity Treatment on Corner Indirect-Method Corner Implementation Vii Page 10 10 15 16 16 26 29 37 40 43 45 47 52 54 55 58 5.7 A.l ACZ D.1 E01 LIST OF FIGURES (CONTINUED) Corner-Treatment Mesh Arrangement for the Square-Problem Layout for the Outer—Boundary Technique Loaded End Shear Stresses for the 24 and 48-Mesh Model Cantilever-Beams Rotation on Fixed End and Moment on the Loaded End for 10:1, 24-Mesh Beam Source Additions to Accommodate Moment and Rotation Conditions Region Coupling Definitions Column Matrix Format for Coupled Problems Bi-region Tension-Strip 66-Mesh End-Loaded Straight Sandwich-Beam Sandwich Core and Face Compatability Check Modeling System for the Curved Sandwich- Beam End Deflection of Curved Sandwich-Beam versus Core Modulus for a Unit End-Load Curved Beam-String Definitions Internal Beam Data Three-Ply Beam Structure Experimental Bending Response for the Curved Composite-Beam viii page 59 63 69 74 76 89 92 93 97 100 102 105 109 109 121 124 Appendix A B C D E F LIST OF APPENDIXES Curved Beam-String Solution CBSR (Curved-Beam-String Subroutine) Least-Squares Curve Fitting Layered Composite Stiffness Experimental Results Green's Functions for the Two-Dimensional Problem in the Infinite Plane Program COUPLE and Subroutines Output from Program COUPLE Sandwich-Theory Solution Program SANDWICH and Subroutines Output from Program SANDWICH ix page 108 115 120 121 123 125 128 141 148 150 158 Chapter 1 INTRODUCTION Sandwich structures are extensively used by the aircraft industry to achieve high strength and rigidity while minimizing structure weight. The need for weight savings in ground transportation vehicles and the advent of new materials and fabrication techniques have increased the interest in these structures. As a result, interest in analysis techniques suitable for sandwich structures has also increased. The straight sandwich-beam problem has been solved by several investigators [1,2,3] by developing and solving the sandwich beam differential equation. Finite-Element Methods (FEM) are very widely used for difficult geometries where the boundaries do not coincide with a coordinate systenu hence exact methods are not easily applied” Since the FEM couples geometric elements by the addition of element stiffness matrices into a general stiffness matrix, it is a simple extension to characterize the elements with different material-property constants. Another potential analysis tool is the Boundary-Element Method (BEM), alter- natively called the Boundary-Integral Method. The BEM involves discretizing the boundary into boundary elements instead of defining internal elements as in the FEM. The BEM 1 2 has been used primarily for the solution of elasticity and plasticity problems and has not often been applied success- fully to thin regions of the type encountered in beam theory. Since both the FEM and BEM are numerical tech- niques, they are subject to error in modeling and computation. With FEM, the modeling error involves an approximate displacement function and, in this regard, it is similar to the Rayleigh-Ritz analysis. The boundary con- - ditions and equilibrium are satisfied at the nodes and the compatibility and the constitutive equations are enforced throughout. If the displacement function is incomplete, equilibrium within each element is not satisfied. Equilibrium is satisfied throughout only if the assumed displacement function is the actual displacement function. The BEM modeling satisfies internal equilibrium, compatibility, and the constitutive relationships. The boundary conditions are not satisfied exactly, but only in a point-wise or average sense, and the tractions on the bound- ary may fluctuate with large amplitudes. With some BEM versions the boundary tractions have singular points. The large fluctuations on the boundary result in stress solu- tions which are erratic near the boundary but improve as the distance from the boundary increases. With slender shapes, all internal regions are in the near-boundary region. This characteristic creates severe problems with beam modeling. 3 The computational error in both the FEM and BEM arises from the solution of a large system of equations. In general, the FEM matrix is larger than that for the BEM for the same problem. Since the FEM matrix is symmetric and banded, the actual storage space for the coefficients is less than half the total matrix size. The storage is generally arranged so that the zeros outside the bandwidth do not require storage space. The matrices for BEM are not banded and are not always symmetric so that space is not conserved by special storage arrangement. The computational accuracy difficulty with BEM is caused by the matrix ill-conditioning associated with modeling of some problems. Modeling features which contribute to BEM ill-conditioning are displacement boundary conditions and source locations distant from the boundary. Both of these features weaken the matrix diagonal coefficients. The equation system is solved in this research by Gaussian E1imination.'The error normally result- ing from variations in matrix conditioning is expected. Interactive refinement methods as reported by Martin and associates [4] are applicable, but require a doubling of storage since the original matrix must be retained. The primary objective of this research is to develop a BEM technique, which will successfully model the deflection responses of two-dimensional layered-composites. 4 Of special interest are the structure geometries and loading conditions which produce bending, such as those for the sandwich beam shown in figure 1.1. Prior to addressing the issue of coupling structural components with different material properties, it is necessary to develop a modific- ation of BEM which will adequately model bending problems. Modifications of BEM to enhance modeling accuracy have been reported by several investigators [5-10]. These modifications, as well as others developed in this research, will be tested for solution accuracy in beam-bending applic- ations. The combination of modifications which provide the most improvement will be utilized in the sandwich applic- ations. As an alternative method, a curved beam-string solution.isldeveloped for thin uniform thickness sandwich faces. This method will provide a solution for the structure shape (high aspect ratio) which has been difficult to model-using BEM. The question of coupling the layers in a composite will be approached by arranging the numerical deflection solutions of the individual layers into stiffness matrices. This is a format which will permit addition of the stiffness coefficients at the common positions to assemble a general stiffness matrix for the entire structure. Since the BEM and curved beam-string solutions are more directly expressed as compliance equations, the coupling problem will also be expressed in a compliance format. The compliance format rigid skin. compliant core rigid skin Figure 1.1 Curved Sandwich-Beam 6 will eliminate the need to apply matrix-inversion operations prior to coupling the equations. Chapter 11 presents the deflection solution of a curved beam-string. This solution is shown for straight and circular beams. A curved sandwich-beam is simulated by coup- ling two curved beam-string solutions to a simple spring element core model. The spring-core model is to provide resistance to the faces moving together or separating. The beam-string solutions are also shown, in Chapter V, coupled to a BEM solution of the sandwich core. Chapter III presents the general basis of the BEM (direct and indirect) by illustrating a trivial, but instructive one-dimensional beam example. Several modific- ations of the two-dimensional elasticity BEM are also shown which are designed to improve the bending problem accuracy. Chapter IV presents the outer boundary source tech- nique as applied to beam-bending problems. A system for the addition of moment and rotation conditions is presented which also shows improved solution accuracy. Chapter V presents techniques for coupling regions of different properties by both stiffness and quasi- compliance methods. The solutions for several sandwich structures are obtained using the quasi-compliance technique. Chapter VI presents a summary of the techniques and results from this research. Chapter 11 CU RVED BEAM-STRING SOLUTION 11.1 W Beams, due to their narrow geometries, present solution difficulties when using numerical elasticity tech- niques. Large stress gradients across the narrow dimension and insufficient St Venant decay diminish modeling accuracy. However, when subjected to the beam theory assumptions, the deflection solutions of straight beams are quite simple. The FEM employs an element utilizing beam theory, which may be coupled to elasticity elements to overcome the bending deficiencies of the two-dimensional FEM. The beam-element yields the exact beam data correspondence at the nodal points and can simulate an arbitrarily shaped beam with a number of straight or constant curvature elements. The BEM techniques, though effective for elasticity problems for boundary shapes having a large area-to-boundary length ratio, have not been successfully applied to bending of slender shapes. In general, increasing the number and decreasing the size of the divisions improves accuracy. This tendency is true only to the point where matrix size becomes so large that error accumulation is unacceptable. Slender- beam solutions follow the same pattern, but the interior 8 stress values improve at a faster rate than the deflection values. The deflection magnitudes are generally less than the exact values. As shown in Chapter III, the deflection solutions for aspect-ratios greater than four display large error when using the simpler BEM versions. Curved beams have been the subject of studies by investigators such as Odenlll], TimoshenkollZ], and Love [13]. These studies range from circular beams loaded both in and out-of-plane with various boundary conditions, to the development of differential equations for arbitrary beam shapes. Morris [14] uses a strain energy format to develop curved beam finite elements capable of handling out-of—plane loads and torsion. This approach is direct since the deflection solutions depend upon integral evaluation rather than solution of a set of differential equations. The strain energy method, to be detailed here, is similar to the Morris method in the initial approach. The Morris solution was applied to beam elements of constant radius, whereas the solution presented here uses a quadratic polynomial beam contour. The purpose of this solution is to represent the curved faces in a sandwich model. 11.2 Influence Functinn This development is subject to the conventional beam-theory assumptions which limit the strain energy to the contribution of strain in the direction of the beam axis. 9 The contribution of the net internal axial stress is included, since the solution is later applied to sandwich beams requiring membrane model characteristics. The beam is modeled as a cantilever, but may be subjected to other constraints by superposition. The beam shown in Figure 2.1 is subjected to real loads of Px' Py, and M at the horizontal coordinatett and the virtual load counterparts Px*, py*, and u* at x. The beam shape is specified as y = fin). (2.1) At any position where n< x and x < t- , the internal moment and the net axial force illustrated in Figure 2.2 may be represented as M(n) = Py(§-n) + Px(f(n)-f(§))+M + p *(x-n) + Px*(f(n)- f(x))+M* (2.2) Y and F(n) = (P+ P*)'n . where the boldface type indicates a vector. 10 \\\\ Figure 2.1 Curved Beam-String Definitions shear force (neglected) \\\\ Figure 2.2 Internal Beam Data 11 The deflections, using Castigliano's theorem are represented as f f aUe 1 6M0!) 1 3F('l) 11x 3 an* 3 E MUN—5;;— dS + E FUH—BFxT ds, 0 O K I aUe 1 aM(n) l aF(n) 75;: 8—1 ““717“ as + fif“"’"3'§T 68' O O and V (2.3) f O aUe l aM(n) am ' ET ”(v—air- ds ' where Px*,P *, and M* I O. Y The upper limit f isx for x<§ and t for x>§, since this represents the entire non-zero region of the integrand. 12 The beam shape may be represented by any function f(n). A simple function which represents curvature and curv- ature variation is a quadratic polynomial. The quadratic representation y = a + bn + on2 (2.4) combined with equations 2.2 and 2.3 yields integrals of the type [1:1 T(n)dn for i = 0 *4, and i -1 A . II T(n) dn for l = 0 +2, (2.5) where T(fl) = (An2 + an + C)1/2, A = 4C I B = 4bC ' and C = 1 + oz. The details for the integrals and final compliance equations are contained in Appendix A. The equations for the deflection responses have been structured in a computational subroutine CBSR to be used in various programs discussed later. The subroutine CBSR is in Appendix B. 13 11.3 Cnrxedzneam Sampling The deflection responses are obtained from concen- trated loads representing the static equivalents of the real loading along the beam length. The result of evaluating equations 2.3, are compliance equations in the form of ux(X) .zcxx(x'§j)Px(§j)+ny(x'§j)Py(§j)+me(x'§j)M(§j) I j j and (2.6) 6(x) IZGQX(X,¢j)Px(§j)+G9y(x,¢j)Py(§j)+Ggm(x,§j)M(§j) . J The summation is over J loads at J different positions represented by the position variable {j- Single-beam statically determinate problems are easily handled in the aforementioned format. Statically indeterminate problems with mixed boundary conditions are most conveniently structured in a compliance matrix form as {u} = [G] {P} , (2.7) where {u} is the column of deflections and rotations, [P] represents the column of loads and moments, and [G] is the square matrix of influence function coefficients. 14 The influence function terms are obtained at all deflection-specified and non-zero loading points. Since the solution is ”exact” within beam theory definitions, there is no need to evaluate coeffients at the zero loading points unless the responses at these points are of interest. The compliance matrix may be inverted to yield a stiffness equation as {P} = [S] {u} (2.8) and coupled to other structural components as employed in FEM. Certain compliance matrices, which will be discus- sed in Chapter V, present serious inversion error due to ill-conditioning. For this reason, a coupling scheme is also presented in the compliance form. This procedure is shown as an example with two coupled cdrved beams in Figure 2.3. To uncouple the beam solutions, the column matrix of the specified boundary conditions is constructed as shown in Figure 2.4, where entries 7 through 9 represent equilibrium of the end loads and entries 10 through 12 equate the end deflection and rotation conditions to satisfy compatibility. The associated square matrix is a combination of compliance entries, identity terms, and end compatibility. This matrix takes the form shown in Figure 2.5. In the example problem, four of the rows are redundant allowing a \\\\ \\\\ 15 M ye xe e Figure 2.3 Coupled Curved-Beams {/T'le Pyl e1 ux2 PYZ .M2 M3+M4 93-94 K Px3+Px4 ux3‘ux4 uy3’uy4 J 16 represents end load Pxe (represents end load Pye represents end moment Me end compatibility of ux end compatibility of uy end compatibility of 9 Figure 2.4 Column Matrix of Known Data OH 17 0 0 0 0 0 0 0 0 0 0 ('le‘w 0 0 0 0 0 0 0 0 0 0 Pyl x 0 0 0 x x x 0 0 0 M1 0 x x x 0 0 0 x x x sz 0 0 l 0 0 0 0 0 0 0 Pyz 0 0 0 l 0 0 0 0 0 0 M2 0 0 0 0 l 0 0 l 0 0 Px3 0 0 0 0 0 l 0 0 1 0 Py3 0 0 0 0 0 0 1 0 0 1 M3 x x x x x x x x x x Px4 x x x x x x x x x x Py4 x x x x x x x x x x \ M4 ‘) Figure 2.5 Right Side of Matrix Equation 18 collapse of the 12 x 12 system into an 8 x 8 matrix prior to solving for the unknown loading data. The first row for example is the trivial statement, le=le- The two regions are now uncoupled and the unknown variables may be determined. 11.4 Illustratine Emblems 11.4.1 Straight Beam The solution of the straight beam is trivial, but was used as a partial verification of the analysis. The straight beam is a degenerate case of the quadratic shape beam, so another purpose of the example is to determine the minimum values permitted for the c coefficient in equation 24%. 1f the minimum value is too large to simulate a straight beam, a separate computation may be required for a straight beam. The geometry data were a=0, b=0, and c = .001, .0001, and .00001, where a,b and c are the shape coefficents. The comparisons are shown in Table 2.1 for an end loaded straight beam. The three loading cases are a transverse load, an axial load, and a moment. As expected, the most troublesome loading was the axial case which, with a slightly curved beam, results in some bending. The coefficent, c = .00001, which represents a beam deviation of .001 in the total length of 10, did not cause significant degradation of the straight beam response. 19 Table 2.1 Straight-Beam Verification Calculated Response x 10'2 Beam Theory Load c ux uy 6 ux uy e .001 .131 -6.25 1.00 .00001 .052 -.07 -.01 0001 -0063 .5000 .75 Py=l .0001 -.006 5.00 .75 0.00 5.00 0.75 .00001 -.001 5.00 .75 .001 -.010 .75 .15 Mal .0001 -.001 .75 .15 0.00 0.75 0.15 .00001 -.000 .75 .15 Notes: The symbol ui is the i-direction displacement and 8 is the rotation in radians. 20 1134.2 Qua;tgr;§irglel The other shape extreme for a quadratic curve is the quarter circle. This case poses the problem that a vertical end slope cannot be modeled. This case can also provide a comparison between the shape matching techniques of point-matching three points and a least-squares fit. The least-squares fitting technique minimizes the integral of the square of the difference between the real and simulated curves represented as Y actual = (R2 -x2).5 y estimate = a + bi! + CX2 (2.9) 2 =[(ya-ye)2dx 32 OZ . . . Z _ _ _ Wlth the condition that -33 ab 6c 0. This method yields a simulated quarter-circle with a = .9586R, b = .3856, and c = -1.098/R. Additional details are included in Appendix C. The point-match method uses the points at each end- plus the arc center point yielding y = R + x - 2x2/R. (2.10) The results of the above models are given in Tables 2.2 and 2.3. Since both models yielded unsatisfactory deflection 21 Table 2.2 Least-Squares Quarter-Circle Response Load Responses Exact Solution “x uy e ux uy 9 szl 588 548 974 1178 750 1500 py-l 548 548 869 750 2034 856 Mal 974 869 2012 1500 856 2356 Table 2.3 Point-Match Quarter-Circle Response Load ux uy e ux uy 9 Px-l 1604 954 1804 1178 750 1500 pysl 954 622 1013 750 2034 856 M=l 1804 1013 2550 1500 856 2356 22 responses, it is obvious that the quarter circle is too severe for a quadratic shape. 11.4.3 W .m The 45° arc is point- matched with coefficients of a = R, b = .055, and c=.663/R. The deflection solutions are shown in Table 2.4. The most severe error was 1.3% in the uy response from a Px load. II.4-4 W Curxedzfieams. The next example, in the progression to more complex problems, consists of two slightly curved beams with the "free" ends clamped together. The end response equations for an end loaded single beam are expressed as “1k ZGijk ij (2.11) 1 with i,j = 1,2,3 and k 1,2. The symbol uik is the 1th response of the kth beam, and ij is the jth loading of the kth beam. 23 Table 2.4 Half-Quadrant Responses Loads Responses Exact Solution ux uy e ux uy e Px=l 56.8 96.2 227.8 56.6 97.5 227.6 py=1 96.2 182.9 395.1 97.5 182.2 393.4 M=l 227.8 395.1 1178.1 227.6 393.4 1178.0 24 The compatibility and equilibrium of the coupled ends demand that and (2.12) The solution of this set gives the loading pattern for the beams, which is then combined with the original deflection equations to yield the responses of the composite structure. The complete double beam data includes a1 = 0, bl = .0346, c1 = .02054, a2 = .43, b2 = .00382, c2 = .00897 width = .82, thickness as .050 (both beams), EEl :- EE2 = 7.8(105) , and E81 = 13132 = 14.5(105). The symbols EE and EB refer to the extension and bending modulus characteristic of layered composites. Properties of layered composites are discussed in Appendix D. The resulting end responses are Ux=-.000685, uy=.00915, and 9(rotation)=.000843 for an end load of py=l. Since no other solution for this problem is available, it is compared with experimental data from a structure fabricated to the same specifications. The measured “y response ranged from .0081 to .0098 in several tests. Additional experimental results are in Appendix E. 25 11.4.5 W The free-surface response of the twin-beam structure consisted of the two beams moving closer or further apart, depending on the load- ing direction, as shown in Figure 2.6. This phenomenon was observed experimentally, and is shown in the twin-beam calc- ulations. Table 2.5 tabulates these responses from the computations. Since the space between the beams seems to contract or expand more than it distorts, core material inserted between the beams will be subjected principally to tension or compression normal to the core-skin interface. This behavior is counter to that of the conventional sandwich core, which is strongly shear-coupled. Provided that the core compliance is large compared with that of the sandwich faces, the internal shear load should be supported by the coupling point at the beam ends. To check the validity of these assumptions, calculations are made for spring elements coupled to the twin-beam model to simulate the core of a sandwich-beam. The beam-deflection equations 2.6 are written for numerous points (nodes) along the beam length. The resulting compliance matrix is then inverted to form a global stiffness-matrix. The matrices for the spring elements are constructed as 26 -uyl I Beams Closing \\\\\ _T__ +uy1 * Beams Spreading Figure 2.6 Free Surface Response 27 Table 2.5 Calculated Twin-Beam Responses with an End Load of P =1 “x1 uyl ux2 “yz Ax AY x position top bottom difference 10'4 10‘3 10‘4 10'3 10‘4 10'3 048 -02 08 -01 -00 01 -08 1.44 -2.9 4.8 -.5 .4 2.4 -4.4 2.40 -607 8.4 -105 2.2 5.2 -602 3.36 -8.2 9.6 -3.5 5.3 4.7 -4.3 4032 -609 901 -600 8.3 .9 -08 4.80 -609 9.1 -609 901 000 000 Notes: Beams move together for P = l ([§y<1 ) and Y apart for FY = -1. All magnitudes are identical for P with all signs reversed. y=’1 28 1-1 0 . s = Ki -1 1 o (2.13) o o 0 which simulates the arrangement in Figure 2.7. 'The zeros in the local spring matrix indicates no resistance to node rotation or shear. The individual spring stiffnesses are established by determining the stiffness contributions of a slice of core cut midway between the nodes. The stiffness is Ki= Li/AiE (2.14) where Li is the vertical spaCe between the opposite beams at the ith node, A1 is the horizontal area projection (the beams are nearly horizontal), and E is the core material modulus. The beam nodes line up vertically and are evenly spaced.'The core modulus is the experimentally'determined modulus of "rigid"1 polyurethane foam. Table 2.6 presents the simulated response using the stiffness-matrix method. The experimental results for the same load condition ranged from .0028 to .0033 inches for a unit loadz. Repeatable experimental responses were not ob- tained for “x and rotation due to the small magnitudes 1The term ”rigid" is to indicate a material of wood-like consistency as compared with flexible elastomeric type foam materials. 2The remaining experimental data is contained in Appendix E. 29 Node 2,i Upper Beam Node 2,i+1 K1+1 Lower Beam Node 1,1 Node l,i+l Figure 2.7 Spring Core Model 30 Table 2.6 Responses of the Stiffness-Coupled Spring-Core Number End Responses of Springs ux “y ' 9 10‘4_ 10'3 10" 1 -3.46 5.14 -4.83 5 -3.28 4.91 -5.57 10 -3.27 4.90 -S.59 15 -3.27 4.90 -5.59 20 -3.27 4.90 -5.58 Notes: The spring-core has a modulus of 15000 and the structure is loaded with Py=1, 31 involved. The computations converge at .0049 which, as expected, show more flexiblity than the actual core structure. The motivation for running a higher number of nodes and springs is to check for convergence and any i114 conditioning difficulties as the matrix size increases. The system may also be compliance-coupled, and since the curved beam-string solution yields compliance equations directly, a matrix inversion is not required. The results of the compliance solution are contained in Table 25L. The same initial equations are used for both methods and the difference in the results may be explained by the additional error accumulation from the matrix inversion step. 11.5 BEM Represented £913; The spring-core model has accurate deflections considering the crude nature of the model. It would, however, be unsatisfactory for sandwich structures where the core offers appreciable resistance to shear or bending. The objective, as stated in Chapter 1, was to solve sandwich-beam type problems involving regions modeled with BEM. The initial attempts to model the core with BEM were completely unsatisfactory. Several versions of BEM were then examined and found deficient for solving simple beam problems. Chapter 111 will discuss the initial difficulties with BEM as well as describe modifications designed to improve modeling accuracy. 32 Table 2.7 Responses Using the Compliance-Coupled Spring-Core Number End Responses Spgings ux uy 8 10-4 10-3 10'4 1 -3.08 4.91 -5.56 5 -2.84 4.63 -6.48 10 -2.83 4.63 -6.49 15 -2.83 4.63 -6.49 20 -2.83 4.63 -6.48 Chapter 111 BoundarybElement Methods 111.1 mm The Boundary-Element Methods (BEM) are numerical techniques for solving differential equations with specified boundary conditions. The solution is accomplished by selecting a solution of the differential equation with a "convenient" boundary, placing the real boundary inside the convenient boundary, and selecting the solution coefficients which satisfy the real boundary conditions. As previously noted, the real boundary conditions are run: satisfied everywhere, but satisfied at discrete points or in a piecewise-average- sense. The numerical solution of the problem yields either the unknown boundary data in the ”direct-method”, or the potential source functions in the "indirect-method". The indirect-method sources are in turn used to determine data within the real problem boundary. 1f the source functions are concentrated loads, the actual boundary data are often unavailable in the indirect-method due to the resulting boundary singularities. This chapter summarizes the basis for the two general boundary-element methods and presents discussion of the solution difficulties. Several modifications of the methods, many a result of this research, are discussed and evaluated. 33 34 111.2 The Direstflethed The direct method [15-18] utilizes the Somigliana form [13] of the Betti Reciprocal Theorem [11,19], which relates solutions of two force SYStEHfiL The equation in two dimensions is = j j ujo [(Px Ux+Py uy)da A j 'j 'IJfltxs uxs+tys uys)ds S - j j J/(txs uxs+tys uys)ds. (3.1) S The symbol ujo is the j displacement at some internal point 0, pi is the i-direction body force, tis is the i-direction boundary traction, and “is is the boundary displacementJ'These variables all represent boundary data for the problem to be solved. The tractions and displace- ments represented with a superscript j are due to a_concen- trated load at some internal point 0 in the j-direction. These functions are the Greenfls Functions for a concentrated j-direction load in the infinite xy plane. If the unit j- direction load at o is multiplied by the left side of equation 3.1, it can be more clearly viewed as a reciprocal equation. A complete set of these Green's Functions is shown in Appendix F. 35 If the point 0 is moved to the boundary, txs' tysr ust and uys represent the problem boundary conditions. To obtain sufficient equations to determine the unknown boundary data numerically, the equation set is rewritten with point 0 at N-number of boundary points where the known conditions exist. The influence of each load is combined in each equation by superposition. For a discretized problem in two dimensions, a set of 2N equations result from the N- points analyzed on the boundary. When point 0 is brought to the boundary, the unit load results in a singular integrand from the Green's Function terms for both of the line integrals. The integration through the point is easily handled, but the form of the result requires a separate computation. These results are also shown in Appendix F. The discretized problem may be represented by [T] {u} [U] {t} (3.2) for a problem without body forces. The [T] and [U] matrices represent the traction and displacement Greenls functions SUCh that Tij is the traction at i due to a unit load at j. These are partitioned properly to identify traction and unit-load directions, and the diagonals of the matrices are the singular integrals. 36 To illustrate the technique, consider a beam with arbitrary boundary constraints, loaded as in Figure 3.1a. The "convenient" solution is a simply supported beam as in Figure 3.1b with length L* larger than L. This condition is to allow the shorter beam to be embedded as shown in Figure 3.1c. If v(x) is a displacement.solution of the beam of Figure 3.1 loaded with 6(x), and G(x,§) is the solution of the convenient problem with a unit load at C , then ‘1 = 61V. (3.3) Placing the solutions in the Galerkin form yields the orthogonal condition [(viv(§)- O(§))G(x,§)d§ = 0. (3.4) Integrating by parts four times inverts the problem to v(x) = J[D(§)G(x,§)dt - G(x,§)v"'(§) + G'(x,§)v"(§) R X1 - G"(x,§)v'(¢) + G"'(x,§)v(§) (3.5) X2 where G' = aG/ar. 37 D /\ X1 1 X2 .L l — x-Xl —> I L >| Figure 3.1a Arbitrary Beam 1 l . l—x» I t L A * r I Figure 3.1b Convenient Beam Problem X1 ) * r L l Figure 3.1c Arbitrary Beam Embedded in Convenient Beam Region 38 Equation 3.5 differentiated in x will yield an equation for rotation. Each of these two equations is solved with x equal to x1 and x2, yielding four equations to determine the set of four unknown end conditions. Although trivial, equation 3.5 is the one dimen- sional form of the reciprocal statement used in the direct BEM. The distributed-load integral is equivalent to the body-force integral, and the limit points are similar to the two-dimensional boundary-integral. 111.3 The Indireetznethed The indirect-method [5,6,20-25] uses concentrated load sources placed in the region of the "convenient" problem in a pattern along the embedded real-problem boundary. The usual ”convenient" larger-region solution is the infinite plane, as in the direct-method. The magnitudes of the sources are simply chosen to satisfy the problem boundary conditions. The Green's Function set is the same as the direct-method set, since each source influence on the boundary is computed at each location where boundary data are specified. The matrix form of the discretized boundary yields {t} = [T] {9*} and ' (3.6) {u} = [U] [P*}. 39 The [T] and [U] matrices are identical to the direct method matrices but are defined as the tractions and displacements as influenced by the P* sources. In the indirect-method, both matrices need not be evaluated and stored, but only those rows which represent the known boundary conditions. The general matrix constructed for a specific problem is a mixture of [T] and [U] entries written as {t,u} = [T,U] {P*}, (3.7) where {t,u} represents the column of known data. The solution of {P*] does not give direct information but must be used in conjunction with the appropriate Greenfls Function in the form N 2 uk(XlY) =2 Z Umk(x,y,§j,nj)Pm*(§j,flj), (3.8) J=l m=l where uk(x,y) is the k-direction displacement at x,y, umk(x,y,§,n) is the k displacement at x,y due to a unit-load in the m-direction at Lu, and Pm*(:,n) is an m-direction source at ¢,n. A similar format is used for determination of the stresses at x,y with Umk replaced by the stress Green's Function. As before, a simple beam problem is shown to illustrate the technique. A beam of length L is embedded in a larger beam of length L*. Each end has four boundary 40 8 V1 /\ V2 81 l V 92 P1 P2 I x-Xl --—-> (a) Arbitrary Beam P1* 0 P2* $Ml* ( M2* I ' l x1 x2 I L e g ——-> L* *= (b) Beam Embedded in Larger Beam of Length L* Figure 3.2 Beam L Embedded in Beam L* Showing Sources Required to Satisfy End Data 41 data “it 91, Pi, and M1 with two of the four magnitudes known at each end. To satisfy the known end point loads and/or constraints, load and moment sources are placed at each end point of the L beam as shown in Figure 3.2. The deflection at any point x in L is u(x)= “/IG(x,§)B(§)d§ + G(x,Xl)P1*+G(x,X2)P2* (3.9) +G'(x,x1)M1* + G'(x,x2)M2* with the pmoperly differentiated forms yielding rotations, moments, and shear loads. The equations which are equal to known end data are used to compute the source loads and moments. The resulting 4 x 4 matrix is then assembled as (P,M,u,9} = [T'U] {P*’M*} (3.10) which is the beam form of equation 3.7. In general, if all of the unknown boundary data is the only information needed, and if sufficient matrix storage space is available for the [T] and [U] arrays, the direct-method is more appropriate. If the interior stress and deflection values are needed, and storage space is limited, the indirect-method is the proper choice, since only half the storage space is needed. The unknown boundary data at the point of source application is singular and not directly obtainable in the indirect-method. Although the methods have been shown to be equivalent by Brebbia [26] , the indirect-method with several modifications is used 42 exclusively in this research: the direct-method has been shown only for completeness. III-4 Beundalxzfllement Difficulties A weakness of both techniques is the degradation of the solution as the boundary shape deviates from a circular shape and as the loading condition results in bending. The boundary stresses are singular at the source points and smooth out as the point of interest moves from the boundary into the interior of the body. As the boundary shape becomes less circular, the disturbance of the stress field does not moderate sufficiently at the intersection of the opposite-side boundary. The smoothing is greatly improved with finer boundary divisions as shown in Figure 3.3, but the computational error increases as the matrix size increases. In situations involving bending stresses, differ- ences in the magnitude of the neighboring sources are large, a condition which further amplifies the distrubances. In all cases, the most elementary forms of either the direct or indirect technique yield excellent results for problems such as uniaxial loading of square shapes. 111.5 Benndarx Senree Shape Eunstiens Shape functions have been used in the direct [7,8,27] and indirect [9] methods by several investi- gators. To use shape-functions, the boundary meshes are subdivided further into mesh sets with the number of sub- meshes in each set determined by the order of the shape function. For example, with polynomial shape functions a 43 .1 4 + .1 v + + + % y=S/4 .359 .347 (Ty/S ‘ I" P 4)- Y’s Figure 3.3 Stress Field of TwolAdjacent Unit-Sources in the Infinite Plane in Plane Stress with v=.35 44 two-submesh set is used for a linear polynomial, a three- submesh set for a quadratic, and so forth. A shape-function presents an opportunity to put more sources on the boundary (closer together) with a smaller magnitude for each source, and provides a more effective smoothing of the disturbances without increasing the matrix size. To illustrate the format, a three-mesh quadratic- set is shown in Figure 3.4. Each of the three meshes is further divided into N-smaller divisions. A partial line of the general matrix may then be constructed with 3 3N k=1 j=l 3N = :5: (c(i,zj)A1(Aj)pl* + G(§,Ej)A2(Aj)P2* J: ' 1 + G(E,Ej)A3(Aj)P3*) (3.11) where A (A ) _ ()j - A2)(Aj - A3) lj'()(-A)().-) 1 2 1 A3 . 3( j - .5) s A]: N , (3.12) {j = 42 ' Aj DY , and .3 [I j "2 + Aj nx . Figure 3.4 Quadratic Shape-Function of Boundary- Sources on a Three-Mesh Set 46 The variable x symbolically represents point x, y and E the point (§,n). The variablez. is the local coordinate position and S is measured in the 1 direction. The results for the problem illustrated in Figure 3.5 are presented in Table 3.1. The results improved slightly as the sources per mesh increased but degraded further as the higher-order shape function was utilized. The problem was reformulated as an all-traction problem by equilibrating the clamped end with traction boundary condi- tions. The results are presented in Tables 3.2 and 3.3. The solution contains rigid-body translation and rotation, but these are subsequently removed resulting in improvement in the end-deflection values and neutral-axis shear stress as well. As initially expected, increasing the number of dis- tributed sources and using a higher-order shape function improved the results. Using only traction-specified prob- lems is not feasible for statically indeterminate systems, so the method should be modified to properly handle mixed- boundary-condition problems. The difference in the mixed problem and the all- traction problem is that the all-traction matrix contains only T(x,t) terms, and-matrix elements with the singular traction integral form the diagonal terms. The displacement Green's Functions are used only to determine displacements after the sources are determined. Neither solution step involves the singular displacement integral when solving an all-traction problem. Since its absence improves the 47 uw>wafiucmunauv coccochcm on» How anewuwccou aumccsom can mosmw: Lou scannesmwmcoo m.m musmfim o u m20wuom»u Ham _ . _ _ _ _ on e.m~.u» _ NH HH ea a a A e m e m m H _ _ _ ma appease name on. _ a + + a I“ s c I. — — O" a I u I o m H _ «H mm.. x + + O“ 3 OHNHsMFofl _ mH QN_ _ _ ma ha ma ma om an mm mm «N mm mm mm _ _ _ _ _ _ _ . o u mcofiuomLu Ham 47 Ho>mHHucm0IHHv cmcmoqucm one How meoHuHccou humccsom can mmnmw: mom coHumHsmfimcou m.m muamflm o u meoHuomHu HHm mcofiuomHu Ham 0 fl _ _ _ _ _ _ Ix . .I» m NH. HH I oH I m I m I A I e I m I e . m . m . H m _ o u m5 u _ NH muwneac nme cm. _ + + a Ix . . Ih _ _ oI u m I e H _ «H a~_. x a + + Ix .I _ mH mm. _ I u. I u o me _ GH AH mH mH om Hm mm mm ea mm em AN _ _ _ . _ _ _ _ 48 Table 3.1 End-Loaded 4:1 Cantilever-Beam Shape-Function Results Shape Sources function per mesh 7§y(2L/3, 0) uy(L,0) -- 1.50a 269b 1C 1.12 105 constant 3 1.15 128 5 1.19 135 9 1.23 141 3 .95 115 quadratic 5 .92 113 7 .91 111 gBeam Theory Stress Beam Theory End Displacement including Shear Displacement cOne source per mesh is equivalent to a non-shape-function solution. 49 .ccw coxfim on» Scum :oHuMDOH can :oflumHmcmuu mcH>OEmu an coauommcmHu ucosoomammfinm a .mmm u ucmswomammwn cam coHuooufinI» uomxm one . mp.I.m.H.me.I I e nee mHI.e.eH I an .em .eee .mm .mm magmas eo ”mmuoz m.mmH H.bl H.h .wm .Hm MH.H m m.va o.hl 0.5 .hm .Nm hm.H m m.mNH v.mI ¢.m .cml .mwl No.H H «195»: G\£I.ovx= $\n.ovxs 8.9»: 8.5»: HO.M\ANC. amoe— mom monusom Emomluo>mHHucmo coHMHoommIcoHuocua m on noHHaQ< moousom newuocsmemmnm ucmumcou ~.m wanna 50 new coxfim on» on o>Huchu :oHumu0u can :oHumamcmLu mcfi>oewu an coauommcmuu ucmaoomammwam .mwm u ucoewomHmmwn new coHuooHHcI> uoexo was "ouoz «.mom mH.hI mH.b . b.¢h . m.oa~ mN.H m H.mcm ho.hl bo.h «.mh m.HHH mN.H m b h s x § x s a s x s Q m8 5 s G)? 9 a G\£ 9 a 8 9 a 8 5 s 8 mEuNCI smma um moousom Emoquo>oHHucmu cmwmwowmmIcoHuomua m on ccHHmmc moousom coHuocsmIommnm cwumucmso m.m wHema 51 solution the singular displacement integral is further ex- amined. The displacement Greenfis Function for the x-direc- tion source is Uxx(0,i) = (A4log r + A5xy) 4nr2 e (3.13) U (0.2) = (A5y2) yx 4nr2 where x and y are the respective distances from the source to the point of interest. The displacement is expressed in the average sense as S/2 - - -l _ _ Uxx(x,x) = -§— Uxx(X,x+S)dS (3.14) -S/2 yielding _ - ‘1 2 Uxx(x,x) = Z;— (A4(loge(S/2)-l)+A5nx ) (3.15) U (‘ ‘) - '1 (A5 ) Y3 XIX - ‘1',"— nxny for the mesh illustrated in Figure 3JL The boundary load term, on the other hand, is exact at the point of source application although the stresses are singular. For a source on a straight mesh the singular integral value is .5, ‘ * which relates a Fx,y boundary force to the Fx,y source. XI 52 XY Plane Problem Region ( ----- —> Fx* Figure 3.6 Displacement Averaging for the Source-Placed Mesh 53 111.6 CQLDEL Treatment Examination of the source magnitudes in Figure 3.7 indicates a tendency for the values to become larger near the corners. This characteristic suggests a need to place the end-sources on the corner with special treatment as first used by Ricardella [10] in the direct-method. To illustrate the corner treatment, consider the mesh in Figure 3.8, where it is of interest to evaluate the singular traction integral on a corner. The boundary source is shown in the center of a small circular boundary depression. Writing the traction Green's Function in polar coordinates and integrating along the are from 91 to 62 yields e F = -l— ((Al-A2)sin49-(A2+A3)c0849) F * 2 1 (3.16) 1 , 92 F = -—— (2(A1+A2)6+(A2-Al)sin26) Fy* . y 16 91 The functions for Fx* are an image of equations 3.16. The results are not dependent upon the radius of the depression, allowing the source to be taken on a boundary corner for r80. The definitions of the A1 coefficients give A1+A2=4, and A1-A2=2+2v for plane stress and Al-A2=2/(1-V) for plane strain. 54 it r—f—r—‘T _,,I—.'-. y l l l L l l I I l r i j T T T T T T T T T T T T 3 2. I J 1 0 28 29 30 1 2 4 6 8 10 12 13 14 15 -2. Mesh Numbers Clamped End Loaded End _'Corner-——-’w ‘¢--Corner - . .II. 4 ,_I'-I__. I“ I. r Figure 3.7 Source Distribution for the 4:1-Beam with 9 Constant Sources'per Mesh 55 XY Plane Problem Region Figure 3.8 Boundary-Traction Singularity Treatment on Corner 56 The results for a straight mesh, 62-91:=n,Iare C11=C22=.5 and C12=C21=0 in the form Fx=C11Fx*+C12Fy* and (3.17) Fy=C12Fx*+022Fy*- A 90° corner gives c11=c22=ezs and C12=C21=(A2-Al)/8n. The influence of the sources on the other meshes on the boundary data on a corner depends upon the direction cosines of the point where the boundary data are analyzed. This condition creates a problem since each corner has two surface normals.‘This problem has been handled by specifying double corner-points by Morjaria and Mukherjee [8], Besuner and Snow [27], and Chaudonneret [28]. A boundary condition is assigned to each of the nodal points on the corner. Since the two points at the same corner have the same coordinate values, the stresses are identical and equilibrium requires the boundary tractions be related by .n'xtx-nyt'y+n'ytY-nxt'x=0. (3.18) A particular situation exists if both of the corner nodes are specified with displacement boundary conditions such as when two adjacent surfaces are clamped. IA corner of this type has only two independent conditions ux and “y- 57 Since each corner adds four sources to the entire source set. the source-set is over specified by two for each such corner. In the indirect-method two sources per corner may be assigned arbitrary magnitudes. In the direct-method, the influence tractions are actual boundary tractions and cannot be made arbitrary. Chaudonneret augments the corner conditions by using shape-functions for theIdisplacements and tractions in the meshes leading to the corner in question. The displacements and traction shape-functions on these meshes are split into normal and tangential components, and these eight quantities are placed into the two identities 01+ 02 0.1+U'2 and (3.19) 611‘52 5'1”r 5'2- The case of only one of the corner nodes being clamped,Ias in the cantilever-beam, does not present the same situation, since one node is traction specified. Figure 3.9 shows the arrangement of sources and resulting boundary forces (”1 the corners. Figure 3.10 describes an example problem, and the results with corner- treatment are compared with those of the conventional method in Table 3.4. For this problem the results are degraded by use of the corner treatment. In the case of the ux deflec- tions at (2.9,l.5), the expected result of 2.9 compares 58_ (a) Representation of a Corner Source in the Infinite Plane * ’ FxgcllFx t (b) Resulting Force Contribution at the Corner Boundary (C11=C22=u25 and C12=C21=--107 for the 90 degree corner with v=.35) Figure 3.9 Indirect-Method Corner Implementation 59 (0,3) / / C / 9 equally L / spaced A / meshes M / on each P / side E / D / / (0,0) (a) Problem / mesh center 422. (3,0) (7% = 1. Definition (b) Conventional Corner Mesh Arrangement mesh center.\\\ I I (c) Special Corner Meshes with Overlap Corners Figure 3.10 Corner-Treatment Mesh Arrangement for the Square Problem 60 Table 3.4 Comparison of Results with and without Corner Treatment of a 36-Mesh Square in Tension (a) Corner Treatment Results Location ux uy 03‘ 03, Txy .1,.1 .04 .04 .96 .41 .82 1.5,.1 1.34 .44 .88 .01 -.03 2.9,.1 2.66 .55 .90 -.07 -.17 .l,1.5 .01 0.00 1.53 -.03 0.00 1.5,1.5 1.27 0.00 .94 .02 0.00 2.9,1.5 2.65 0.00 1.36 -.19 0.00 (b) Conventional Mesh Handling Location ux “y 0x 037 Txy .1, l .11 .23 1.02 -.36 -.53 .5, l 1.49 .50 .95 -.00 -.04 l 2.89 .55 1.03 -.04 .17 5 -.01 0.00 1.74 .06 0.00 5 1.42 0.00 1.03 .00 0.00 5 2.87 0.00 1.32 -.18 0.00 Cll-C22-.5 and C12=C21=0 on straight boundaries. Notes: C11=C22-.25 and c12.c21=+.107 on the corners. 61 favorably with 2.89 for the conventional BEM shown in Table 3.4b. The result of 2.65 for the corner-treatment case shows no improvement for this particular problem. It is appropriate to note that Riccardella reported results from the direct-method for a 24-mesh 5:1-cantilever which yielded an end deflection of 50% of the beam-theory deflection. These results were presented as an improvement over the conventional direct-method beam solution. The results for the constant shape-function displayed in Table 3.1 show an end deflection of equal accuracy to the Riccardella solution. For these reasons, the corner treat- ment was discarded in favor of more promising modifications. Chapter IV OUTER-BOUNDARY TECHNIQUES Iv.l Baeksmnnd. A method using sources outside of the problem geometry was first used by Kupradzel29] to evaluate Fredholm Integrals, and the numerical pr0perties orouter-boundary sources were studied by Heise[30]. Wu[l9,20] applied outer- boundary sources to avoid a 1/r2-strength singularity obtained from the Green's Function for a boundary moment source in plate bending problems. The method has been suggested for in-plane loading in two-dimensional problems by Keshavarzi[31] and applied to problems involving material nonlinearity by Burgess[32] . This technique avoids the problems caused by the large stress gradients near the sources. It also avoids the displacement-singular boundary integral discussed in Section 111.4 since neither the traction nor displacement singular integral is necessary. The question arises concerning the best location of the sources. The usual choice is to establish an outer boundary of the same general shape»as the real boundary as shown in Figure 4.1. The ”real" boundary is divided into N meshes as before and the known boundary data satisfied at the boundary-mesh center-point. The sources are located on 62 63 XY Plane Real Boundary Outer Boundary Figure 4.1 Layout for the Outer-Boundary Technique 64 lines normal to the boundary through the mesh center-point at some distance (H) from the boundary measured in mesh- length units. The choice of H is not trivial, since a large value will result in an ill-conditioned matrix, and a small value will dilute the smoothing benefits of the technique. One difficulty with the outer boundary placement of sources arises with a geometry involving re-entrant corners. The sources may be erroneously placed inside the problem boundary, but this difficulty may be avoided by separating the body through the corner and using region-coupling as discussed in chapter V. 1V.2 Benflnuzmblem Madeline The method is in general dependent on loading con- ditions as well as geometry. A boundary loading condition which results in a uniform stress field will tolerate an extremely large space between the boundary and the source placement. A problem with large gradients, such as bending, is more sensitive to ill-conditioning resulting from the more distant source locations. These points are demonstrated in Tables 4.1 and 4.2 with a tension strip and cantilever- beam. The tension-problem results improve with the larger outer-boundary distance, and the bending-problem seems to optimize near a spacing of 10. Table 4.3 illustrates the same problem modeled with 48 meshes.'The meshes are arranged by halving each mesh length in the 24-mesh model. The results were low with the 24-mesh model, but high in the 48- 6S Table 4.1 ”H" Sensitivity for a Tension Loaded Strip . = l /| Px=1 /I 10 X 1 -————" /| I y =- o x . 10 x = 0 H Location 03, 03, Txy “x. (.1y 0 0,.5 -1.19 -.55 0.00 ..23 0.00 0 5,05 1.01 -004 0.00 4098 0000 0 10,05 .55 -006 0.00 9083 0000 5 0,05 2059 ‘066 0000 -044 0000 5 5,.5 1.03 -.02 0.00 5.22 0.00 5 10,.5 .78 -1.22 0.00 10.25 0.00 10 0,0 .83 .29 .10 .08 -.00 10 0,.5 1.05 .37 0.00 -.03 0.00 10 5,0 1.00 0.00 -.00 5.02 .18 10 5,.5 1.00 0.00 0.00' 5.02 0.00 10 10,0 1.25 -.35 .08 10.17 .37 10 10,.5 .91 -.39 0.00 10.01 '0.00 15 0,.5 .80 .32 0.00 .04 0.00 15 5,.5 1.00 0.00 0.00 5.01 -.03 15 10,.5 .84 -.57 0.00 9.98 -.10 Notes: There are 10 equal meshes on each long side and 2 equal meshes on each end (24 total). The material properties are E81 and v8.35. Table 4.2 66 ”H" Sensitivity for a 10:1 End Loaded Cantilever H Location (Ty uy 0 10,.5 0.0 39. 5 10,.5 0.0 -1052. 10 0,.5 0.0 1.4 0.0 -1. 10 5,.5 0.0 1.4 0.0 1165. 10 10,0 -0.1 0.1 6.7 3714. 10 10,.5 0.0 1.4 0.0 3714. 15 0,.5 0.0 1.3 0.0 -l. 15 5,.5 0.0 1.4 0.0 1137. 15 10,00 -000 0.0 906 3622. 15 10,.5 0.0 1.3 0.0 3622. Notes: This 24-mesh model is in the form (10,2,10,2) with E=l and v=.35. The elasticity solution for u is 4032.2. y(x=10) 67 Sensitivity for a 48 Mesh Cantilever Table 4.3 Pyll 10x1 /// Location O'x (TY Txy 555555 Notes: The elasticity solution uy(x=10) is 4032-2 68 mesh model. The 24-mesh model with H=10 gives the best results, but it should be noted that H=5 for the 48-mesh model yields better results than H=5 for the 24-mesh model. In summary, the BEM modification of placing the sources outside the problem boundary is shown to produce significant improvement. The tension-strip results were accurate with boundary-sources but were improved with outer-boundary sources. The end-loaded cantilever-beam, however, yields meaningless results for boundary-sources, but is much improved using soures outside the boundary. 1V.3 Bennderxznata W The loads applied to the beam are such that the stress at a particular mesh center point corresponds to this load as if it were uniform across the mesh. For example, the 24 mesh cantilever model has two meshes on the loaded end with center points at (10,.25) and (10,.75), and the shear stresses at these points are set at unity by applying a load of .5 on each. The four end meshes on the 48 mesh model were loaded by setting the shear stresses at .6,l.4,l.4, and .6 at points (10,.125), (10,.375), (10,.625) and (10,.875) respect- ively. Figure 4.2 shows a plot of the stresses at several points across the loaded end for both models.‘The load of a .5 on a mesh of .5 length sets the stress at 1. A more realistic system would require an average mesh stress of l, and permit the actual mesh center stress to find its own specific magnitude. With the present situation, the average stress across the loaded end is different for each model and 69 .75 .5 .25 .875 .75 .625 .375 .25 .125 Figure 4.2 Loaded-End Shear Stresses for Model Cantilever-Beams 48-mesh, (10,.5)=3714 24-mesh, 8810 (10,.5)=4387 H=10 the 24 and 48-Mesh 70 the averages are not equivalent to the desired load in either model. The non-averaging method constructs the boundary condition equations as N BC(§) = Gk(§r zj)Pk(-§-j) (4.1) 2 k=1 j=l where R is the mesh center-point and Ej is the coordinate position of the two sources. The averaging equation, how- ever, can be represented as u. M 2 ‘ 1 _ _ _ where the integer M is the number of locations within each mesh used for averaging, and the floating mesh point xm is defined as _ 3 gm = x + .75 (1+m 2). (4.3) The variable 8 represents the length of the respective mesh in the x direction. 71 Table 4.4 Mesh Data Averaging for a 10:1-Beam Model Location 03, (Ty Txy ux uy 0,0 59.9 .1 -.1 .3 3. 0,.5 0.0 0.0 1.5 0.0 -2. 24 5,0 30.0 -.0 0.0 224.9 1265. mesh 5,.5 0.0 0.0 1.5 0.0 1263. 10,0 0.0 -.1 .-.0 299.9 4026. 10,.5 0.0 0.0 1.5 0.0 4026. 0,0 64.3 3.4 6.3 0.5 1. 0,.5 0.0 0.0 -5.2 0.0 0. 48 5,0 30.5 0.0 -0.0 228.4 1287. mesh 5,.5 0.0 0.0 1.5 0.0 1284.. 10,0 0.0 -1.2 .0 305.3 4093. 10,.5 0.0 0.0 1.6 0.0 4093. Notes: The outer-boundary distance "H" = 10 with 5 boundary data averaging points per mesh. The elasticity solution is 4032.2. 72 Table 4.4 shows responses from both models with good results for the 24-mesh model, and improved results for the 48-mesh model, compared with the point-matching tech- nique. It seems strange, however, that the course model is better than the more finely divided 48-mesh model. Since some improvement was experienced with source shape-functions in Section 111.5, this notion was tried for distributing sources on the outer-boundary. The results in TableI4.5 show results for both models using five sources, distributed uniformly, for each boundary mesh. The results were different than those of the one-source-per-mesh method, with some of the results showing slight improvement and the others slight degradation. The use of shape-function sources in later problems, in general, yields equivalent results with no clear improvement or degradation. In later work, the results will involve one source set for each boundary mesh. In search of other factors which affect solution accuracy, it is noted that ux deflections on the clamped end and 0x stresses on the loaded end possess unwanted distor- tional rotations and moments as illustrated in Figure 4g3. Although not displayed here, some of the unloaded meshes on the long beam sides show the same tendencies. This situ- ation means that the problem was solved with unwanted moments which undoubtedly have an influence on long beams. This difficulty suggests a need to specify moment and 73 Table 4.5 10:1-Beam with Mesh Averaging and Distributed Outer-Boundary Sources Model Location 03, (7y Txy ux uy 0,0 59.8 .1 -.0 .3 4. 0,.5 0.0 ' 0.0 1.6 0.0 -2. 24 5,0 30.0 -.0 -0.0 224.9 1266. mesh 5,.5 0.0 0.0 1.5 0.0 1263. 10,0 0.0 -.l 0.0 300.0 4026. 10,.5 0.0 0.0 1.5 0.0 4026. 0,0 64.3 3.2 6.3 .5 l. 0,.5 0.0 0.0 -5.3 0.0 0. 48 5,0 30.5 0.0 0.0 228.6 1287. mesh 5,.5 0.0 0.0 1.5 0.0 1284. 10,0 0.0 -l.3 0.0 305.5 4096. 10,.5 0.0 0.0 1.5 0.0 4096. Notes: Five constant sources are used for each mesh at a distance 'H' - 10. Five points on each mesh are used for boundary-data-averaging. 74 Figure 4.3 Rotation on Fixed End and Moment on Loaded End for 10:1, 24-Mesh Beam 75 rotation boundary conditions so that the correctly loaded structure is being modeled. IV.4 Memenh and RQLaLiQn.QQndiLiQnS The addition of the moment and rotation condition on the boundary meshes requires an additional source specified for each mesh, increasing the degrees-of—freedom by 50%. One choice is to specify a set of sources-moments located at the same point as the line-load sources as shown in Figure 4.4a. This choice requires additional Green's Functions relating the deflections and rotations to the moments and functions relating the boundary rotations and moments to the original force sources. These may be generated from the existing functions as auiy(i,2) auix(§,f) “1m(x") ' a: - an ' auyj(i,t) auxj(i,E) 91(2’E) = 3x - ay ' (4.4) timeIE) z a: ' an ' and "1(3'5) = 3877 ' The term uim is the i-direction displacement due to a moment source, Gj is the rotation due to a j-source (x,y,or M), and so forth. The tn is the normal boundary traction defined as tn=nxtx+nytyp (4.5) 76 T —’ j)» b . 7., _:p ix)“ (“1 (I; (a) Source Moments *> (b) Source Couples Figure 4.4 Source Additions to Accommodate Moment and Rotation Conditions 77 and the variable S is the boundary tangent coordinate. The alternative to the aforementioned functions is the use of source couples. These couples are obtained by specifying additional load-sources in the boundary tangent- direction, displaced from the other source pair on a line normal to the boundary as illustrated in Figure 4.4(b). This arrangement permits the use of the original Green's Function equations where c: A N I ‘ "II [I uix(§,f)ny-uiy(§,t)nx and _ _ (4.6) tic(ilf) tix(§’z)ny'tiy(§'z)nx 0 These equations are displacement and traction influences in the i- direction at 2 due to a boundary tangent direction load at f. The position i is displaced from E in the boundary normal direction.at aIdistance specified as some multiple of the mesh length denoted by BC. An alternative to using the 9j and Mj functions is a numerical format which is more compatable with the mesh-averaging routine. For a mesh with M averaging points, M ( - M ‘ ' M R) -' —M—_-l—- tn(xm) (3(X)-S(rm)) m=l and (4.7) M M - 9(2) 3 1T :Un(x)/(S(X)'S(xm)), m=l where Rm # R. 78 The tn and un indicate the normal direction data and x1“ the intermesh coordinate, as defined in equations 4.3 and 4.5. Table 4.6 displays the results of the 24 and 48- mesh beam with moments and rotations specified as zero at the appropriate meshes. The solutions shown are with a load source spacing of 5 mesh lengths (H=5) and a couple source spacing of 6 mesh lengths (HC=6). For the first time, the 48 mesh model yields better results than the 24-mesh model. The end deflection of 4027.9 compares favorably with the 4032.2 exact solution. The solutions with 8:10 and HC=12 broke down completely. The breakdown was noted by a loss of symmetry of the problem in addition to totally absurd results for stress and deflection. As noted previously, the increase in H contributes to matrix ill-conditioning, which coupled with a 125% increase in the number of matrix elements (50% increase in the degrees of freedom), accumulated unreasonable error. These factors also may be reasons that the 24-mesh model has, in some cases, had better results than the 48-mesh model. For unsymmetric problems where the solution is unknown, a breakdown may easily be identified by comparing calculated boundary data with the specified boundary conditions. 79 Table 4.6 Cantilever-Beam Utilizing Moment and Rotation Conditions “y Model Location ux 0,0 0. 3. 0,.5 0. -2. 24 5,0 222. 1254. mesh 5,.5 0. 1251. 10,0 296. 3982. 10,.5 0. 3982. 0,0 g 0.0 0. 2. 0,.5 0.0 0. 0. 48 5,0 0.0 225. 1267. mesh 5,.5 0.0 0. 1268. 10,0 0.0 300. 4028. 10,.5 0.0 0. 4028. Notes: The load sources are at Has, couple sources at HC=6, and 5 boundary-data-averaging points are used per mesh. 80 1V.6 HigherAsneetznaties To establish more confidence in the technique, a 100:1-cantilever is shown in Table 4.7 with an end load, an end moment in Table 44% and a uniformly distributed load in Table 4.9. The 70-mesh model was better for both the end- loaded and moment-loaded cases. The 70-mesh uniform—load model results are meaningless, since the solution broke down. An example of the solution break-down is the 11x mismatch at (100,0) and (100.1) which are 5.7(104) and -5.4C105) respectively. These should have identical absolute values. . The 100:1-beam results show that the problem is clearly more severe than that for the 10:1-beam. The results also confirm an earlier suggestion that the solution accuracy is sensitive not only to geometry, but loading patterns as well. The applicability of the method is questionable with aspect ratios approaching 100, but will be examined further with region-coupling in Chapter V. 81 .Hooavv mH ucwEoomHmmwc 0cm uomxw one .mm.u> 0cm Hum wee mmHuemsoea HeHewees mes .HH.em.H.eme emeHsHe mH Hmeos ewes as see see AH.e~.H.o~c eeeHsHe mH Hmeos emesIue mes .mmuoz A00H00.¢ .Nvl H.H m.Nl .H m..00H A00H00.¢ .hhflom H.h H.l .Gl 0.00H IeeHcm.H .aI m.H ~.oI .HI m..em ewes A00HVM.H .mmmmm N.0l N.0I .N0m 0.0m 0F AH0H05.HI .N m.N m.0l .0l m..0 AH0H00.M .v N.ml 0.0l .N00 0.0 A00H00.v .0 m.H 0.0 .0 m.~00H A00H00.v .hN0vm H.0l 5.0! .0 0.00H A00HVQ.H .0 m.0 0.0 .0 m.~0m Sme A00H0v.H .HHmmN N.Hl H.0l .va 0.0m NV AHOHVH.HI .0 0.H 0.0 .0 m..0 AHoHcm.e .eI m.o e.eI .eme e.o a: x: EC. ab xb coHumooq Hence. _\ a u m _\ _\ \ \ uo>mdfiuchIHu00H cocoon 0cm h.v manna 82 .coo.om mH ucoewomammfic cam uomxm one ”ouoz .m00w0 .N H0.0 «HH.0 H.0l m..00H .00000 .0m0 0N.0l m00.0 H.0 0.00H .00NOH .H 00.0 000.0 0.0 m..0m Sme .50NOH .va 00.0 000.0 «.0 0.0m 05 .0 .0 m0n01 000.0] 0.0 m..0 .0] .0! 00.0 000.0 0.0 0.0 .00Nm5 .0 00.0l 000.0 0.0 m..00H .00Nm5 .005 00.0] 5H0.0l m.0 0.00H .0m00N .0 No.0] 000.0 0.0 m..0m zme .Qm00N .00v No.0] H00.0I 0.5 0.00 N0 .0 .0 H0.0 000 .0 0 .0 020 .H .0l No.0 NH0.0I 5.0 0.0 as x: 5.5 mb xb cofiumooa Hope... _\ an: H. _\ _\ eeeaHHueeoIHueeH ceases geese: e.e «Hess 83 Hmoavm.H mH DawsoomHmmH© pew uomxo one "muoz. A50Hv5.m Amova.Nl .5mH .MNvVHI AM0H00.0 m..00H A50H05.m A00H05.m: .m50Hm .mle A00H00.Hl 0~00H H0005 A50va.m A00H0H.Hl .50! .vm5l Am0H00.NI m..0m Smwfi A50H0¢.N A00H0v.N .05] .0m5l Am0HV0.HI 0.00 05 Am0H00.H Am0H00.ml .QNH .550 AM0H00.NI m..0 Am0dvm.m AMOHVH.0I .0HOH .mm A¢0H00.H 0.0 AeeHce.H .HoHce.HI .mHI .H e.e m..eoH ImOHcm.H .eoH.~.H .HI .emI AHOHcm.e e.eeH Hence A50H0N.0 A00H00.Hl .mv .H 0.0 m..00 SmmE A50H0N.0 A00H00.H .Hvl .5l Am0H05.m 0.00 Nv AN0H00.0| AH0H00.0 .00H .0 0.0 m..0 LHOHem.~ .HeHce.H .mm .mH .eoHee.m o.o a: x: 5C. ab xb c0333 Home... _ .\ _ _\ _ « h .\ H I s eeeeHHueeoIHueeH neon emeseHeumeo sHseouHes a.e «Hens Chapter V REGION COUPLING v. 1 Beekemnnd The techniques of FEM involve developing element stiffness matrices and adding the coefficients at the common nodes to form a global stiffness-matrix. The form of the indirect BEM equations, as previously noted, is {u} = [U] {P*], and (5.1) {t} = [T] {P*}. These may be considered quasi-compliant, since the unknowns are the source loads. Equations 5.1 may be combined to yield {t} = [T] [01-1 {u}, (5.2) with the resulting stiffness matrix [S] = [T] [U]"1. (5.3) The matrices may be added to form a global-matrix in the conventional FEM-manner. Zienkiewicz [33] and coworkers have attempted coupling of the direct BEM with FEM without great success. They show that the stiffness-matrix from the direct-method is 84 85 [S] = [01-1 [T]. . (5.4) This matrix is not symmetrical as demanded by the reciprocal theorem. The stiffness matrix for the direct-method is the transpose of the indirect-method when the sources are on the boundary. In either case a matrix-inversion step is required, which increases cost and computation error. Although not as simple to program for computation, the indirect-method may be coupled to other indirect- solution regions by a compliance technique similar to that shown in Chapter II. This solution will yield the source magnitudes for each region which uncouple the problem. These magnitudes are then used to solve for unknown problem data. v.2 Stiffnesszuatrix Selutien The stiffness matrix is-generated from the outer- boundary method discussed in Chapter IV. The direct-method is not defined on the outer-boundary, so the technique used here is indirect. A stiffness-matrix solution for a square is shown in Table 5.1 for various outer-boundary distances (H). The solution does not seem to be H-sensitive, and the results for H=100 are nearly exact. Table 5.2 is a similar problem with a l4-mesh rectangular strip. The results quickly become erratic for the l4-mesh problem as the H-spacing becomes larger. This difficulty is attributed to a 86 .muasmmu cmuommxm may mumHH 30L macaw .mm.u> can Hum mum moHuquOHQ Hewuwume one .Humhm use .ouvmmuqu umxmumamumxm .ouH>suHxs mum mcofiufiocoo >u~©¢30h one ”mouoz cam. msH. oee.H .0 com. emeH.I ee.H .o eeH com. msH. mas. .o com. msH.I ee.H .e eH wee. esH. has. .0 mac. esH.I eo.H .e m men. emu. mes. .o «em. em~.I me. .e e com. msH. eoe.H .o eem. msH.I oe.H .0 II as: ex: mas mes «as we: Ham Has a cofimcwa Hmwxmfica :H >umocaom mueawm e no eoHnsHom xHeuezImmoemLHem H.m oHeea 87 Table 5.2 Stiffness-Matrix Solution of a 5 x 1 Rectanglar Strip in Tension meshes 12—>8 /| /| 13 7 ———> Px7 = .5 /l l x 5 /l 14 6 ——-.' PKG = .5 /| . meshes 1—>5 H Px14 Py14 ux3 - uy3 ux6 uy6 0 .5 -016 2.48 .19 4093 013 5 1.0 -.95 2.83 .20 5.51 .71 10 .6 -.15 2.58 .18 5.12 .24 100 .5 0.00 2.59 .46 5.16 .30 Notes: The expected results are Px14=.5, ux3=2.5, uy3-.175, ux5-5., and uy5-.l75 . The material properties are E-l and v-.35 . 88 combination of matrix size and ill-conditioning. The stiff- ness-matrix approach was not pursued further for the above reasons. Also, since the identity of the sources is lost, the internal problem data are not obtainable with the stiffness-method. The stiffness-matrix approaches symmetry as H becomes large for those matrices which were successfully inverted. For the four mesh problem in Table 5.1, two of the values are S(l,3)= -.58972 and S(3,l)= -.53902 for H=0. For H=100 the same two are S(l,3)= -.56375 and S(3,1)= -.56379. v.3 Compliance Coupling The problem of coupling multiple regions by a compliance-technique with outer-boundary sources is similar to the double-curved-beam problem of Chapter II. The system is defined as having a region R1 with N1 meshes and a region R2 with N2 meshes with N12 meshes common or coupled together as in Figure 5.1. The Nl-le meshes of region 1 and Nz-le meshes of region 2 are then subject to known load or displacement conditions. 'The resulting equations for the uncoupled meshes are 3”“1“1‘112> Fr(I) = Tr(I,J)Pr*(J) j=1 or (5.5) 3”‘h-Mz’ ur(I) = Ur(I,J)Pr*(J) i=1 89 Nl-Niz Region 1 N12 ‘ \¥ / Region 2 N2-N12 Figure 5.1 Region Coupling Definitions 90 where Fr(I) represents the region r forces and moments at mesh I, and so forth. For the coupled meshes, F(I)=F1(I)+F2(I)=z (T1(I,J)P1*(J)+T2(I.J)P2*(J)) i=1 and (5.6) 3N12 0=ul(I)-u2(l)=_ :2: (U1(I,J)P1*(J)- U2(I,J)P2*(J)) j: To increase the versatility of expressing the interface conditions, the boundary-data are converted to tangential and normal components instead of components in x and y. This procedure will permit modeling problems where the interface is not adhered. Tangential loads may be separately uncoupled and set equal to a frictional force. Meshes, where the normal force would be in tension, may be uncoupled and set to zero. 91 The uncoupled meshes have column matrix entries tti or “ti! tni or “ni' and Mi or 91 with the coupled meshes rewiring the entries tti'tt(i+l)' uti+ut(i+l)' tni'tn(i+l)' uni+un(i+1)r M1+M(1+1)rand 91’9(i+1)- The tangential direc‘ tion is taken clockwise around the boundary and the normal direction is outward. The choice of traction (t) or dis- placement (u) depends on which of the two is known data at the particular boundary position. The entries for the coupled meshes are used to satisfy traction and moment equi- librium, and continuity of displacements and rotations. These column matrix entries are set to zero. The column matrices relating to the global matrix are illustrated in Figure 5.2. After the sources are determined for each region, the problem uncouples, and the results for each region are determined independently by the Greenfls Function approach as in single-region problems. The program and subroutines developed for the computation are presented in Appendix G. v.5 Compliancezcounledmustratinmblems \L5.l B1;maje;ial Strip, To illustrate the techni- que, a rectangular composite of two different materials is coupled as shown in Figure 5m3. The problem definitions and results are presented in Table 5.3. Although the exact solution to this problem was not available, it is to be noted that the end displaces upward as expected. It is also 92 Left Column Matrix Right Column Matrix (known data) (unknown data) I I I I I tt or u I I Px* I I t I Region 1 I I Region 1 : tn or un : Meshes I Py* : Sources I M or 0 I I C* I I I I I I I I | I I I ----------------------- I I I I I I I I l t or u I I l t t I - I Region 2 I I : tn or un I Meshes I I I M or 9 I I I I I | I I I I l I I I I ------------------------ I I I I I I I I I -------------------- I P * I Region 2 x | I I I Sources I I I Py* I I 0 I Coupled I I I I Meshes I C* I I 0 I I I I I I I l I 0 I I I I I I I I I I I I I I I I I I I I I I I I Boundary Conditions Outer-Boundary Sources Figure 5.2 Column Matrix Format for Coupled Problems @669 I 93 ® ® 13 I 14 13 14 e . _-©'.; 12 HQ}- 12"11@' 1 T n T 10 Q 11 9 local mesh numbers (uncircled) 3.”. 10®T9 global mesh numbers (circled) ® Figure 5.3 Bi-Region Tension Strip 8 @o o a 5 @18 069 94' Table 5.3 Results From Bi-region Tension Strip \\\\\ region 2 (R2) 82-2 v2=.35 -————a>P2 = 1 region 1 (R1) 8181 v1=.35 —————+>P1 a 1 Interface location x,y O'x (Ty Txy ux uy R1 0.1 1.02 .122 .459 .03 -.07 R2 0']. 1019 0506 .332 -001 -003 R1 .5,1 .60 .158 -.024 .32 .06 R2 1.57 p.141 .004 .34 .07 R1 105,1 080 -0030 .040 1009 055 R2 1.49 -.028 .028 1.09 .56 R1 205']. .76 -0003 -0017 1082 1046 R2 1.44 -.003 -.009 1.84 1.46 R1 3.5.1 .60 .023 .056 2.57 2.75 R2 1.53 .020 .044 2.55 2.74 R1 4.5.1 1.03 -.041 -.202 3.23 4.45 R2 ' 1.20 -.027 -.160 3.30 4.44 R1 5. ,1 1.05 .207 .341 3.83 5.46 R2 .95 -.127 .133 3.55 5.40 Note: More complete results are in Appendix H 95 to be noted that the interface data is to be identical for both regions except for 0}. This condition matches well except for the interface stresses at (0,10) and (5,1). The points (0,10) and (5,1) are at the end of the extreme inter- face meshes and the stresses are equated on the mesh- average-sense whereas the other points shown are in the centers of the respective meshes. v.5.2 Wynn Table 5.4 displays the results of the sandwich-beam problem of Figure 5.4. The loaded ends of the faces deflect 332 with the core center point deflecting 319. The sandwich-theory [2] result is 276. Of the 276, 151 is from face membrane extension and 125 from the core shear deformation. The sandwich theory solution is shown in Appendix I. Besides the noted deflec- tion deviation, there is some mismatch in the interface stresses. The faces have aspect ratios of 100, and as noted in Chapter IV, the method is suspect for ratios of this magnitude. An alternative technique is the coupling of the beam-string solution of Chapter II to the outer-boundary core model. The program "Sandwich" developed for this coup- ' ling is shown in AppendixIL. The solution shown in Table 5.5 utilizes 10 meshes on the coupled surfaces as before. but the total number of model meshes is 44 versus 66 from the previous example. The 10 face meshes are on the neutral axis rather than 10 on each long side as required with the face regions modeled with BEM. Although the core and face 96 Table 5.4 Straight Sandwich Results (66 mesh model) (a) Top Interface Location (Tx (Ty Txy ux I3y Face 0 216. “22.17 “49.8 0.0 0.0 Core 0 19. 2.52 -4.0 -3.7 1.1 F 2 499. 0008 -400 8.6 -3809 C 2 5. 0.13 -509 8.4 -3804 F 4 369. “0.03 “4.2 15.9 “95.2 C 4 3. “0.08 “5.7 15.8 “95.7 F 6 363. “0.12 “8.8 21.1 “164.2 C 6 2. “0.15 “5.2 21.3 “164.9 F 8 156. 0.21 “6.2 24.3 “241.7 C 8 20 -0008 -406 24.7 -24105 F 10 “8. 10.77 “22.2 24.6 “332.4 (b) Core Center X‘O 000 000 -005 0.0 -006 5 0.0 0.0 “5.8 0.0 “128.6 10 0.0 0.0 0.6 0.0 “318.6 (c) Face Center X30 53909 307 -3107 0.0 -00 5 276.0 “0.3 “3.4 20.5 “128. 10 167 17.8 “54.3 27.4 “332. Note: More complete results are in Appendix H 97 10 meshes on free sides / / - _ ,1 / t 10 meshes on interfaces / tlgol / c=2 / 1 mesh E=1 v=.4 l mesh / . / t =01 / 2 II I E=100, v=.35 on faces l P2=5 / . = Py(§-n)+Px(f(n)-f(§))+M +Py*(x-fl)+Px*(f(fl)-f(x))+M* (A.1) FM) = (P+P*) 'n x where n< { l . The symbol n is the axis direction vector such that nx = (An2 +Bn +C)"°5 and n = (b+2cn) n where y x A = 4c , B = 4bc, and c = 1 + b2. The b and c terms represent the beam shape coefficients from the polynomial y = a +bx +cx2. (A.3) 108 \\\\ 109 shear force (neglected) Mm /F(1I) n =J \\\‘K I \f Figure A.2 Internal Beam Data 110 The deflection equations are constructed from Castigliano's theorem in the form 6 6 308 1 8M(fl) 1 3F(fl) ux = 3Px* = 31 MI“) an* ds + AE FIn) an* ds, 0 O aUe 1 g aM(n) 1 g ar uy 3 GP * =3 E MUD—3F“;— (38 + E FUN-SET (38' (11.4) Y x Y O O and aUe 1 g M(n) a BM" 3 E-I-JI— MOUT— d8 1 0 where Px*,Py*, and M* = 0. The upper limit is x for x<§ and §.for x>§, since this represents the entire non-zero region of the integrand. The deflections may be rewritten as “xx =fg7dn jggdn (ux due to a unit x load at f. ), where M(n) aMIn) 97 = EI nx an* ana' . ' (A.5) FIn) as 99 3 AE nx apx* The ux displacement due to pY at c is written as uxy = j[ g6dn +Jfg8dn (A.6) 111 In the same format, the complete list is constructed as uxx ajr(g7+g9)dn = G7+Gg, (A.7) uxy = 66+G8' (A.8) uxm = 610' (A.9) uyx = GZ+G4. (A.10) “yy 8 G1+G3, (A.1l) uym = Gsr (A.12) 8x = G12, (A.13) and 9y a 311' (A.14) 9m = G13. (A.15) The symbol uxm is the x-direction displacement due to a unit moment at t and 9x is the rotation due to an x direction force at t and so forth. In all cases G1 is defined as Jféidn , and the gi terms are represented by EIgl =§xf6-(x+§)f7+f8, (A.16) Elgz = (b§x+c§2x)f6+(b§+bx+c§2)+ (cx-b)f8-cf11, (A.17) AEg3 = b2f3+4bcf14+4c2f15, - (A.18) A894 = bf3+2cf14, (A.19) 8195 = xf6-f7, (A.20) 2196 = -€(bx+cx2)f6+(bx+cx2+b§)f7+ (C§“b)f8“Cfllp (A.21) 112 E197 = 20f6+C4f7+C5f8+ C6f11+2bcf13r AEgg = bf3+2cf14: A399 = f3. Elglo = -(cx2+bx)f5+bf7+cf3. BIg11= fifs'f7r 31912 = -(b§+c§2)f5+bf7+cf3. and E1913 = £5. The variables b and c are the shape coefficients. The fi functions of n are defined as f6 = (An2+Bn+C)-5, f7 = “f6, f8 = n2f5. f11 = “3f6' £13 3 fl4f5r and f14 = "/fs and the Ci functions are defined as C4 = x§(b+cx), c5 = -b2(x+§)-bc(x2+§2), and C6 = -c2(x2+§2)-bc(x+§)+b2 A set of Pi integrals is defined as F1 =‘jrfidn (A.22) (A.23) (A.24) (A.25) (A.26) (A.27) (A.28) (A.29) (A.30) (A.31) (A.32) (A.33) (A.34) (A.35) (A.36) (A.37) (A.38) 113 and are evaluated as F6 = F1F2+F3/2, F7 = (F13/12 -B(2F1F2+F3)/16)/Czp F8 = F13 F4+F6(SBZ-4AC)/16A2. F11 = .4n3F6-.BBF8/A-.3F10/c, ("4F6“BF11-F12/C)/31 F14 = (Fl‘BF3/2)/A. The additional Fi terms are Fl = f6, F2 = (ZAfl +B)/4AI dn F = = 3 1 2E_ ln(4cF1+8c2 +B), - (6An-5B)/24A2, a] .5 I F5 d/pR dfl = 4F2F3C'F1/2C, R = sin-1((2An+B)(B2-4C)-°5). F10 :/;2Rdfl F12 f/;3Rdn n3F5/4-BBFlo/32c2 +3F8/8c, nst'BF9/12C3+F7/3C, (A.39) (A.40) (A.41) (A.42) (A.43) (A.44) (A.45) (A.47) (A.48) (A.49) (A.50) (A.51) (A.52) 114 F15 = (2An-3B)/4A2, and F1617]r :26“ 1 The final list of G1 terms is G1 = (ngG-(x+:)F7+F8)/EI, oz = ((b(x+¢)+c¢2)F7-b§x+ cx§2)F6+(cx-b)F8-CF11)/EI, G3 = (b2F3+4ch14+4c2F16)/AE: G4 = (bF3+cF14)/AE, c5 = (xF6-F7)/EI. G6 = ((x(b+cx)+b§)F7-x§(b+cx)F6+ (c:-b)F8-cF11)/EI. G7 = (2cF6+C4F7+C5F8+ C6F11+2ch13)/EI, GB = (bF3+2cF14)/AE, cg = F3/AE, G10 = (-(cx2+bx)F6+bF7+cF8)/EI, G11 = (th-F7)/EI, 612 = (-(b +c§2)F6+bF7+cF8)/EI, and G13 = F6/EI, (A.53) (A.54) (A.55) (A.56) (A.57) (A.58) (A.59) (A.60) (A.61) (A.62) (A.63) (A.64) (A.66) (A.67) The analysis is further presented in the Fortran subroutine CBSR in Appendix B. APPENDIX B CBSR (Curved-Beam Subroutine) elements: Rarametar XX BT, GM EI, AE CBSR(Curved-Beam Subroutine) The parameter list for CBSR contains the following D . l' A one dimensional array of the x coordinate locations where loads are placed and/or responses are of interest. Location in XXII) where a particular response is required. The location in XX(J) where a load Px(J), PyIJ) and/or MIJ) is located. The b and c coefficients in the shape function. The flexural and axial stiffnesses. The influence coefficients ranging from R(l) to R(9) are defined as UXX, UXY, UXM, UYX, UYY, UYM, OY, and 8M1. The subroutine listing follows on page 116. 1mm is the x-direction displacement at x(I) location due to an x direction load at the location x(J) , GM is the rotation due to a moment, and etc. 115 15 20 25 30 116 SUBROUTINE CBSR(XX,I,J,BT,GM,EI,AE,R) IMPLICIT REAL*8(A-H,O-Z) REAL*8 F(16),FT(16),XX(50),R(9) A=4.*GM*GM B=4.*BT*GM C=1+BT**2 X=XX(I) Z=XX(J) IF(X.GT.Z) GO TO 5 =x GO TO 10 Y=Z CONTINUE C2=(5.*B*B-4.*A*C)/(16.*A*A) C3=DSQRT(A) FT(1)=DSQRT(A*Y*Y+B*Y+C) FT(2)=(2.*A*Y+B)/(4.*A) FT(3)=DLOG(2.*C3*FT(1)+2.*A*Y+B)/C3 FT(4)=(6.*A*Y-5.*B)/(24.*A*A) FT(5)=2.*FT(2)*FT(3)*C3-FT(1)/C3 FT(6)=FT(1)*FT(2)+FT(3)/2. FT(7)=((FT(1)**3)/(3.*A))-(B*FT(1)*FT(2)/(2.*A)) -B*FT(3)/(4.*A) FT(8)=(FT(1)**3)*FT(4)+C2*FT(6) FT(9)=((Y-B/(2.*A))/2.)*FT(5)+FT(6)/(2.*C3) FT(10)=(Y*Y*FT(5)/3.)-(B*FT(9)/(3.*A))+2.*FT(7)/(3.*C3) FT(11)=(.4*(Y**3))*FT(6)-(.3*B*FT(8)/A)-.6*FT(10)/C3 FT(12)=((Y**3)*FT(S)/4.)-(3.*B*FT(lO)/(8.*A))+ 3.*FT(8)/(4.*C3) FT(13)=(((Y**4)*FT(6))-(B*FT(11)/A)-(2.*FT(12)/C3))/3. FT(14)=(FT(1)/A)-B*FT(3)/(2.*A) FT(15)=(2.*A*Y-3.*B)/(4.*A*A) FT(l6)=FT(15)*FT(1)+(3.*B*B-4.*A*C)*FT(3)/(8.*A*A) IF(Y.EQ.0) GO TO 25 D0 20 1131 r 16 F(II)=FT(II) CONTINUE Y=0 GO TO 15 CONTINUE DO 30 II=1,16 F(II)=F(II)-FT(II) CONTINUE 117 Gl=(x*z*F(6)-(X+ZI*F(7)+F(8))lEI G2=(-(BT*Z*X+GM*Z*Z*X)*F(6)+(BT*(Z+X)+GM*Z*Z) 1 *FI7)+(GM*x-BT)*F(8)-GM*F(11))/EI GB=(BT*BT*F(3)+4.*BT*GM*F(14)+4.*(GM**2)*F(16))/AE G4=(BT*F(3)+2.*GM*F(14)I/AE GS=(X*F(6)-F(7))/EI GG=(-Z*X*(BT+GM*X)*F(6)+(X*(BT+GM*X)+BT*Z)*F(7) 1 +(GM*z-BT)*F(8)-GM*F(11))/EI C4=X*Z*(BT+GM*X)*(BT+GM*Z) C5=-BT*(BT*(X+Z)+GM*(X*X+Z*Z)) C6=-GM*GM*(X*X+Z*Z)-BT*GM*(X+z)+BT*BT C7=2.*BT*GM C8=(GM**2) G7=(C4*F(6)+C5*F(7)+C6*F(8)+C7*F(ll)+C8*F(13))/EI GB=(BT*F(3)+2.*GM*F(14))/AE G9=F(3)/AE GlO=((-GM*X*x-BT*X)*F(6)+BT*F(7)+GM*F(8))lEI G11=(z*F(6)-F(7))/EI GlZ=(-(BT*Z+GM*Z*Z)*F(6)+BT*F(7)+GM*F(8)I/EI Gl3=F(6)/EI R(l)=G7+GQ R(2)=GG+G8 R(3)=Glo R(4)=G2+G4 R(5)=Gl+G3 R(6)=GS R(7)=Glz R(8)=Gll R(9)=Gl3 CONTINUE RETURN END 118 The CBSR subroutine is applied in the program CRVBM shown on the following page. This program is designed to solve curved beam problems. The input data for CRVBM is: Data EB EB AL,BT,GM TK,W TL FX,FY,AM Winn beam extension modulus beam bending modulus a, b and c of the shape function beam thickness and width number of beam divisions to generate x(I). (XX(I) in sasn) total beam x-direction size one dimensional arrays specifying loads and moments'at the beam division points. FX(J) is the x-direction load at X(J) and etc. 119 /SYS REG=MAX C ********* CRVBM ********* C IMPLICIT REAL*8(A“H,O-Z) DIMENSION FX(50),FYISO),AM(50),X(50),R(9) NAMELIST/DATAl/EE,EB,AL,BT,GM,TK,W,N,TL NAMELIST/DATAZ/FX,FY,AM READ(5,DATA1) READ(5,DATA2) 100 FORMAT(//) 150 FORMAT(' DIVISIONS',13X,'X LENGTH',18X,'WIDTH',18X, 1 'THICKNESS') 200 FORMAT(IlO,3F25.5) 250 FORMATI'O EXTENSION MODULUS',20X,'BENDING MODULUS') 300 FORMAT(ZEZS.5) 350 FORMAT('O ALPHA',10X,'BETA',1OX,'GAMMA') 400 FORMAT(3P15.5) 500 FORMAT('O X',13X,'UX',13X,'UY',10X,'ROTATION', 1 10X,'FX',13X,'FY',13X,'M') 550 FORMAT(7F15.5) WRITE (6 p 150) WRITE(6,200)N,TL,W,TK WRITE (6 p 250) WRITE(6,300) EE,EB WRITE(6.350) WRITE(6.400) AL,BT,GM WRITE(6,500) EI=EB*W*(TK**3)/12. AE=EE*W*TK DO 5 I=1,N 5 X(I)=I*TL/N DO 10 I=1,N UX=O. UY=0. ROT=0. DO 9 J=1,N CALL CBSR(X,I,J,BT,GM,EI,AE,R) UX=R(1) *FXIJ) +R(2) *FYIJ) +R(3) *AM(J) UY=R(4)*FX(J)+R(5)*FY(J)+R(6)*AM(J) 9 ROT=R(7)*FXIJ)+R(8)*FY(J)+R(9)*AM(J) 10 WRITE(6,550) X(I),UX,UY,ROT,FX(I),FY(I),AM(I) STOP END /DATA &DATA1 EE=1.D7,EB=1.D7,AL=“1.,BT=.2,GM=“.01,TK=.2,W=1.,N=2, TL=10. &END 8DATA2 FX=0.: FY80 o ' AM=0.,1. &END APPENDIX C LEAST-SQUARES CURVE FITTING LEAST-SQUARES CURVE FITTING To fit a quarter circle ( ya = (R2 - x2)°5 ) to a quadratic polynomial ( ye s a + bx +cx2 I. the integral 2 =J[I(R2-x2)'5-a-bx-cx2)2dx (C.1) is minimized with respect to the polynomial coefficients by az _ aa - -2 -(ya-ye)dx - 0 R Oz 3b = -2 [flx(ya-ye)dx = 0 (C.2) R 82 _ 2 _ ac - -2‘/Px (ya-ye)dx -0 . R This yields the three-equation set n/4R2 -aR -bR2/2 -cR3/3 = o. R3/3 -aR2/2 -bR3/2 -cR4 = o. (c.3) and "/16R4 -aR3/3 -bR4/4 -cR5 = 0. An evaluation of the coefficients gives a=.958R, b=.3856, and c=l.908/R. 120 APPENDIX D LAYERED COMPOSITE STIFFNESS LAYERED COMPOSITE STIFFNESS Composite structures are often fabricated using layers of materials with different properties. One common practice is to layer strips of fiber-reinforced polymerics. The fiber direction of the outer layers is in the direction of the beam length, and interlayers are at off-angles to reinforce against splitting. Since the modulus is highest in the principal fiber direction, the beam will have an axial stiffness less than the flexural stiffness. Jones [34], Popov [35], and others have used the terms ”equivalent" and/or "apparent“ to describe the cross- sections and the respective stiffnesses. As an example of this treatment, consider a three- ply beam structure with unequal thicknesses, outer-layer moduli of El and E3, and a core modulus of E2 as shown in Figure D.l. 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(1 _ sinh(ax)+ta::(aL)(l-cosh(ax)) ) (I 1) 148 149 The additional coefficients are defined as a = ( EIf(IfIf/I) '5 (1'2) A = (c+t)2/c (1.3) I = t3/6 + t(c+t)2/2 (1.4) If = t3/6 (1.5) E = face extension modulus (1.6) G = core shear modulus (1.7) APPENDIX J PROGRAM SANDWICH AND SUBROUTINES PROGRAM SANDWICH AND SUBROUTINES The SANDWICH program input data are Data Symbol Dessrintion N Number of meshes in the sandwich core E,PR Modulus and Poisson's ratio for the core H,MP,KT Same as couple program XBJI) The x-coordinate representing the Ith node location ALl,BT1,GMAl AL2,BT2,GMA2 The a.b, and c coefficients in the shape polynomial for faces (beam-string) l and 2 EIl,AEl EI2,AE2 The bending and extension stiffnesses of faces 1 and 2 NBD The number of nodes per sandwich face X(I),Y(I) End points of the core meshes KI,KJ,SGN Same as COUPLE program XF(I),YF(I) The ith core coordinates where displacements and stresses are computed NF The number of XF(I),YF(I) coordinates The SANDWICH program listing follows on page 151. 150 151 /SYS REG=MAX /SYS TIME=3 C C ******* SANDWICH ****** C IMPLICIT REAL*8(A-H,O-Z) REAL*8 X(30),Y(30),GM(200,200),SGN(70,4),R(9), 2 PX(50),PY(50),AM(50),FR(70,4),BC(200),P(200),RM(8), 3 XF(35,4),YF(35,4) DIMENSION IT(70),IN(70),IM(70),KI(70,4),KJ(70,4),NF(4) NAMELIST/DATAl/N,E,PR,H,MP,KT NAMELIST/DATAZ/XBpALl,BTl,GMA1,AL2,BT2,GMA2,EIl,EIZ, 1 AE1,AE2,NBD NAMELIST/DATA3/X,Y,KI,KJ,SGN NAMELIST/DATA4/BC,IT,INpIM NAMELIST/DATAS/XF,YF,NF 100 FORMAT(//) 101 FORMAT(/) 110 FORMAT( ' 0 ************************************** |) 150 FORMAT('0 MESH END X COORDINATE') 151 FORMAT('0 MESH END Y COORDINATE') 200 FORMAT(6F20.10) 205 FORMAT<7F15.S) 206 FORMAT(60X,2F15.10) 210 FORMAT('0 BEAM DATA',10X,'ALPHA',10X,'BETA', 1 10X,'GAMMA') 220 FORMAT(18X,3F15.10) 230 FORMAT('O STIFPNESS',10X,'EI',10X,'AE') 240 FORMAT(15X,2E15.5) 245 FORMAT(10X.'BEAM X COORDINATES') 250 FORMAT('0FIELD',2X,'XF',7X,'YF',13X,'XX',13X,'XY',13X, 1 'YY',13X,'UX',14X,'UY') . 300 FORMAT(I4,4X,I4,4X,F8.3,E15.7,3X,I4,4X,F8.3,ElS.7,3X, 1 14,4X1F8.3.E15.7) 350 FORMAT(' NODE MAP LOCAL NUMBERS IN GLOBAL ORDER') 360 FORMAT('0 ',2014) 400 FORMAT(I4,7F15.8) 500 FORMAT(' MATRIX IS SINGULAR EXECUTION ABORTED*******') 600 FORMAT(I4,2F10.5,5F15.7) 700 FORMAT('0 SECONDARY BOUNDARY AT 'pF5.2, 1 ' MESH LENGTHS'IISX,'MESH SUBDIVISIONS'/,55X, 2 'INNER BOUNDARYB',I4,5X,'OUTER BOUNDARY=',I4) 850 FORMAT(10X,8F10.4) 800 FORMAT('0 CORE DATA',//,5X,'POISSONS RATIO =',F5.3, 1 SX,’ MODULUS 3'15‘20010) 900 FORMAT(//,'LOCATION',2X,'IT',6X,'BCT',10X,'FX*',10X, 1 'IN',SX,'BCN',12X,'FY*',10X,'IM',5X,'BCM',12X,'M*',/) 950 FORMAT('0 FIELD POINTS REGION'pI4) 1000 FORMAT('BEAM',I4,/,5X,'POSITION',9X,'UX',14X,'UY',12X, 1 'ROTATION',10X,'FT/FX',12X,'FN/FY',10X,'MOMENT') 1001 FORMAT(' SANDWICH BEAM ANALYSIS') S 4 152 READ(5,DATA1) READ(5,DATA2) READ(5,DATA3) READ(5,DATA4) READ(5,DATA5) WRITE(6,100) WRITE(6,1001) WRITE(6,210) WRITE(6,220) AL1,BT1,GMA1 WRITE(6:220) AL2,BT2,GMA2 WRITE(6,230) WRITE(6,240) EI1,AE1 WRITE(6,240) EIZ,AE2 WRITE(6,101) WRITE(6,245) WRITE(6,350)(XB(J),J=1:NBD) WRITE(5:100) WRITE(6,110) WRITE(6,700) H,MP,KT WRITE(6,800) PR,E WRITE(6,150) WRITE‘61850)(X(J) IJ=11N ) WRITE(6:151) WRITE(6,850)(Y(J),J=1,N) WRITE(6,110) WRITE(6,110) WRITE(6,350) N3=N+2*NBD DO 6 I=1,3 WRITE(6,360)(KJ(J:I),J=1,N3) X(N+1)=X(1) Y(N+1)=Y(1) NT=3*N+6*NBD PI=4.*DATAN(1.D0) CALL SRGEOM(I,X,Y,N,NX,NY,DS,E,PR,A1,A2,A3,A4,AS) D0 5 I‘erT P(I)=BC(I) CALL ASBLE(3,X,Y,KI,KJ,N3,MP,KT,SGN,IT,IN,IM,GM,H,RM, 1 DS,NX,NY,A1,A2,A3,A4,A5,PI) 1 1 CALL CBASB(NBD,N3,AL1,BT1HGMA1,XB EIl, AEl, GM KI, KJ, SGN: IT IN, IM, 1.,1) CALL CBASB(NBD, N3 AL2, BT2 HGMAZ XB, EIZ, AE2, GM KI, KJ, SGNr IT IN IM,-1.,2) CALL DARRAY(2, NT, NT,200,200,GM,GM) CALL DSIMQ00000 3602 0000.0 0 0000.0 0000.0 0000.0! 0000.0 0000.00 0000.0 '6 ‘OGDC3 GBCWO \OCMO GMOGB Fifi”: !°¢3CD (V009 CMOG: HIGH: ‘913CD CMOGD 'OCVOO "¢=¢3 ¢3flVO 0‘30! 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