ABSTRACT A COMPARISON OF FINITE ELEMENT AND FINITE DIFFERENCE METHODS IN ELASTOSTATIC PROBLEMS by Nicholas P. Dario Four objectives of this thesis are: to compare finite element and finite difference solutions to elastostatic problems, to present an apparently different formulation of the Navier equations in finite difference form and to demonstrate their applicability, to formulate and apply the axially symmetric linear strain triangular ring stiffness. matrix, and to present solutions for simple composite bodies. For the sake of completeness, finite element stiffness matrices are derived for plane and axially symmetric problems. Both constant and linearly varying strain triangles are considered. Nodal point forces associated with boundary tractions are treated in detail. The constant and linear strain triangles as well as the constant strain triangular ring have been presented by other authors. The linear strain triangular ring has been mentioned by other writers but has apparently not been specifically presented prior to this. Furthermore, the present author is unaware of earlier published applications of this stiffness matrix. Finite difference expressions associated with the Navier elas- ticity equations are derived in a more general form which allows con- sideration of anisotropic materials. This is done by simply replacing \Ill|| v . . . w .. . . . . so p.. . ...I v. . pm 0‘ o. ‘. . I “4.. L O~ r\~ 9.x hr” . . .\_ v o . . . .4. — .. A t w . o s E s . ”I .HH .3 .t (C .u s. q . .t :3 .5 ‘3 wk. .. an .. ... I It” it“ .3 .m r. .t ... .. .a ,... . . .u x: "n. ... Gs .. J. :3 t Lu .m. C. at .u. . . .‘. .. .3 ¢ . .. . C .C . ~ 9 t . . ... 7» rt v; 5% u... . t .. I u. .v. n. .u of .n a“ n C. .3 vs nu ok 9- t L1 P. .r‘ mm a. .. . . . 5» Z. .3 .t _ v .3 . u - . .h‘ .b. t (a .. . rs «I. L. , . .2 u . _. .. . o .. .w. ,. . 2 ~ Vt n a . a 5n .3 CL .. s e ‘ a ”1 . . .3 .r» . . . . . 4 . .. t .r» « ~ . a \ a . a on; L a .. . . .c« .r. i . . . . . ‘ a . ”u. o s . u .. . .1 3‘ .4. . p. LL 2. 1‘ v. a. .. .o. .. .. .t .. .. .u .. ... .... .n .. b. .1 .‘ o. u .3 Ly» .. . h h .. n “ ..... .. .. ..:. .. .. .3. ~k ..... ... . .. .. u}. . . .. is bx .... .. .uu NICHOLAS P. DARIO derivatives by appropriate difference expressions. These are also derived by what is believed to be a different method involving the equilibrium of a material element. Inherent in the procedure is the necessity of making assumptions of the strains in terms of displacement differences. The method has the advantage that static boundary con— ditions are readily derived as well. The comparison of the methods is given in terms of Specific applications. Both plane stress and axially symmetric examples are included. In each category, a problem with a well—known elasticity solution is treated so that comparisons can also be made with the so- called "exact" solution. An application involving a simple composite body is also presented. The investigation demonstrates the ability of the finite element and finite difference methods to give equally good results in displace- ment analysis. Agreement with elasticity solutions is excellent for each method. However, the stresses which result from the finite difference analysis and the finite element analysis using constant strain triangular elements are generally less satisfactory than those obtained in the finite element analysis which employs linearly varying strain triangular elements. This is especially true at boundary points. Displacements in simple composite bodies treated are also very compar- able for the various methods. Interfacial stresses, however, were more erratic for the finite element solutions than for the corresponding firuxe difference solutions. The smoother variation of the difference sollndons is believed to be more realistic. PLEASE NOTE: Some pages have small and indistinct type. Filmed as received. University Microfilms A COMPARISON OF FINITE ELEMENT AND FINITE DIFFERENCE METHODS IN ELASTOSTATIC PROBLEMS BY \ Nicholas PfUDario A THESIS 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 1969 a“ ACKNOWLEDGEMENT The numerical work in this thesis was carried out in part at the Michigan State University Computer Center. The balance of the work was performed at General Motors Institute in its Computer Laboratory. The author takes this opportunity to thank these groups for their cooperation in this effort. I am especially indebted to Walter M. Arnston of the GMI Computer Services Department for many hours of assistance in debugging programs written for this research. Special thanks go to Mrs. Janice Walton for her patience in the preparation of numerous sets of tedious data for the computer programs. Sincere appreciation is extended to Professor William A. Bradley for suggesting this tOpic and for his guidance and assistance in all phases of this research. Finally, I am extremely grateful to Mrs. Frances Abel for many hours of typing in the preparation of this final thesis. u \ .- -‘ \ ~¢ '. v. ..' ,.‘ .52 F) r J ' ( n . Lad o R J L” w L-- r~ s.-- . . t O '5'“ . U' 0 Y“ LA“ "-J 'V TABLE OF CONTENTS ACKNOWLEDGEMENT . . . LIST OF TABLES. . . LIST OF FIGURES . . . . . . . . . LIST OF SYMBOLS . I. INTRODUCTION. 1.1. Remarks . . . . . . 1.2. Previous Developments 1.3. Present Investigation . II. FINITE NNNN J-‘UJNH NNNN mNO\U1 2.10. III. FINITE uwwwwu O‘U'IJ-‘UJNH DIFFERENCE METHOD. General Remarks . . . . . . . . Differential Equations for Plane Stress . . Differential Equations for Plane Strain . Differential Equations for Axially Symmetric Problems. . . . . . . . . . . . . . . . . . . Finite Difference Equations for Plane Stress. . Finite Difference Equations for Plane Strain. Axially Symmetric Finite Difference Equations Alternate Derivation of Plane Stress Difference Equations . . . . . . . Alternate Derivation of Axially Symmetric Difference Equations . . . Finite Difference Stresses. ELEMENT METHOD . General Remarks . . . . . Direct Stiffness Method . Constant Strain Triangle. . . . Linearly Varying Strain Triangle. Constant Strain Triangular Ring . Linear Strain Triangular Ring . iii Page ii vii xii NH 12 15 l8 19 23 35 48 50 SO 52 64 7O 79 88 ‘ v \ i1 --'¢| ... ‘.- . - _ _‘__ . -‘QO . \ . -.M I . _\ ‘~ ‘ . ._. , p -,~ ‘.. -.- l I ‘x . ‘ I ~ IV. PLANE STRESS APPLICATIONS 4.1. Cantilever Beam . . . Elasticity Solution . Finite Difference Solution. CST Solution. LST Solution. Further Comparisons Concluding Remarks. 4.2. Composite Plate . Finite Difference Solution. CST Solution. . . . . LST Solution. Comparison of Solutions Concluding Remarks. V. AXIALLY SYMMETRIC APPLICATIONS. 5.1. Thick Cylinder. . . . Elasticity Solution . Numerical Solutions . Concluding Remarks. . 5.2. Composite Solid Cylinder. Finite Difference Solution. CSTR Solution . . . . LSTR Solution . Comparison of Solutions Concluding Remarks. VI. CONCLUSIONS AND RECOMMENDATIONS . 56.1. Conclusions . . 6 2 Recommendations BIBLIOGRAPHY. . . . . . . . . . . . . . APPENDIX A. ADDITIONAL RESULTS . APPENDIX B. COMPUTER PROGRAMS. . . . . iv Page 95 95 97 98 101 106 109 121 124 126 133 133 143 172 177 178 178 180 187 187 191 200 207 215 232 236 236 237 241 246 259 1"" ._ . .. . ... .. I. .. ,.. ». any so» .I- .3. Q - a 0“. .v. w ‘1‘ 4.10 4.11 4.12 4.15 4.16 4.17 4.18 LIST OF TABLES Beam Deflections, 65 Point Configurations. Beam Deflections, 225 Point Configurations . . . . . Beam Flexural Stress, 225 Point Configurations Boundary Flexural Stress, 225 Point Configurations . Beam Shear Stress, 225 Point Configurations. Beam Longitudinal Displacements, 225 Point Configurations . . . . . . . . . . . . . . . . . . Composite Plate Top Edge v—Displacements Composite Plate Top Edge u—Displacements Composite Plate Horizontal Interface v-DiSplacements Composite Plate Horizontal Interface u-Displacements Composite Plate Vertical Symmetry Axis v-Displacements Composite Plate Horizontal Interface Stress, oX (Matrix). Composite Plate Horizontal Interface Stress, ox (Stiffener) . . . . . . . . Composite Plate Horizontal Interface Stress, 0y (Matrix). . . . . . . . . . . Composite Plate Horizontal Interface Stress, 0y (Stiffener) . . . . . . . . . . . Composite Plate Vertical Interface Stress, 0y (MatriX) O O O C O O O O C I O O 0 Composite Plate Vertical Interface Stress, 0y (Stiffener) . . . . . . . . . . Composite Plate Vertical Interface Stress, Txy (MatriX) o o o o I o o o o o o o o o o o o o o Page 110 111 113 115 119 122 152 153 154 155 156 163 164 166 167 169 170 173 l' 4. 01' l.’ . v u. . o — o-.. . — . t n .. -.-. - -. . h 0 .. ‘ V s - v ' u .. ' ~ u I"' \ Table 5.12 5.13 Composite Plate Vertical Interface Stress, T (Stiffener). . . . . . . xy Thick Cylinder Radial Displacement . . . . . . . Thick Cylinder Radial Stress . Thick Cylinder Circumferential Stress. Composite Cylinder End w-Displacement. . Composite Cylinder Horizontal Interface w-Displacement . Horizontal Interface Stress,or (Matrix). Horizontal Interface Stress, or (Stiffener). Horizontal Interface Stress, oz (Matrix) Horizontal Interface Stress, oz (Stiffener). Vertical Interface Stress, oz (Matrix) Vertical Interface Stress, oz (Stiffener). Horizontal Interface Stress, 06 (Matrix) Horizontal Interface Stress, o (Stiffener). 0 vi Page 174 183 185 188 216 218 223 224 226 227 229 230 233 234 Lu, .‘1 LIST OF FIGURES Figure Page 2.1 Rectangular Mesh . . . . . . . . . . . . . . . . . . . . 15 2.2 Mesh Point and Neighboring Points. . . . . . . . . . . . 15 2.3 Rectangular Mesh . . . . . . . . . . . . . . . . . . . . 19 2.4 Mesh Point and Neighboring Points. . . . . . . . . . . . 20 2.4 Material Region Around a Mesh Point and Associated Cartesian Stresses and Body Forces . . . . . . . . . . 24 2.6 Outside Corner Boundary Point. . . . . . . . . . . . . . 29 2.7 Vertical Boundary Point. . . . . . . . . . . . . . . . . 31 2.8 Horizontal Boundary Point. . . . . . . . . . . . . . . . 32 2.9 Inside Corner Boundary Point . . . . . . . . . . . . . . 33 2.10 Cylindrical Volume Element . . . . . . . . . . . . . . . 35 2.11 Circumferential Stresses . . . . . . . . . . . . . . . . 36 2.12 Circumferential Stresses . . . . . . . . . . . . . . . . 36 2.13 Material Region Around a Mesh Point and Associated Cylindrical Stresses and Body Forces . . . . . . . . . 37 2.14 Composite Material Region. . . . . . . . . . . . . . . . 48 2.15 Material Region. . . . . . . . . . . . . . . . . . . . . 49 3.1 Constant Strain Triangle . . . . . . . . . . . . . . . . 64 3.2 Normal Boundary Traction-CST . . . . . . . . . . . . . . 68 3.3 Linear Strain Triangle . . . . . . . . . . . . . . . . . 70 3.4 Normal Boundary Traction-LST . . . . . . . . . . . . . . 76 vii II|.| . . .. .3 .. .5 .2 .2 ,. .. a . . . p . . a a t a r. us - s. .‘w a. q“ u. .u an an a. .. u. ~¢ .1 2. .f s. .1 .n .n. .3 .4 . a . . .4. . x x . . ~ .\ ~ .r. . . .1 W2 . . . s .. . . . no. .Di . . . . 0A.. ' ~ .-u .u) y . r11 .- ¢ ~ .16 A- .21 . A an... .14 « V :J (h... .‘c ‘ . v o o o .14 1 .51. F. \1 AI.- p‘a « a «A V A . d o c .. a . t I o ~ 0. c. Q. 9. Q. s. a. h V o 1 I a. . u. ‘. Figure Page 3.5 Constant Strain Triangular Ring Segment. . . . . . . . . 79 3.6 Cross Section of Ring Element. . . . . . . . . . . . . . 79 3.7 Axial Boundary Traction-CSTR . . . . . . . . . . . . . . 84 3.8 Linear Strain Triangular Ring Segment. . . . . . . . . . 88 3.9 Axial Boundary Traction—LSTR . . . . . . . . . . . . . . 94 4.1a Cantilever Beam. . . . . . . . . . . . . . . . . . . . . 96 4.1b Cantilever Beam-LST Finite Element Configuration . . . . 96 4.2 Cantilever Beam - Finite Difference Configurations . . . 99 4.3 Finite Difference Beam Deflections . . . . . . . . . . . 100 4.4 Finite Difference Flexural Stress. . . . . . . . . . . . 102 4.5 Cantilever Beam-CST Finite Element Configurations. . . . 103 4.6 CST Beam Deflections . . . . . . . . . . . . . . . . . . 104 4.7 CST Flexural Stress. . . . . . . . . . . . . . . . . . . 105 4.8 Alternate CST Beam Configuration . . . . . . . . . . . . 107 4.9 LST Beam Deflections . . . . . . . . . . . . . . . . . . 108 4.10 Beam Deflections . . . . . . . . . . . . . . . . . . . . 112 4.11 Beam Flexural Stress . . . . . . . . . . . . . . . . . . 114 4.12 Top Boundary Flexural Stress . . . . . . . . . . . . . . 116 4.13 CST Boundary Flexural Stress . . . . . . . . . . . . . . 118 4.14 Beam Shear Stress. . . . . . . . . . . . . . . . . . . . 120 4.15 Longitudinal Free End Displacements. . . . . . . . . . . 123 4.16 Composite Plate. . . . . . . . . . . . . . . . . . . . . 125 4.17 Composite Plate - Finite Difference Configurations . . . 127 4.18 FD TOp Edge v-Displacements. . . . . . . . . . . . . . . 128 4.19 FD Top Edge u-Displacements. . . . . . . . . . . . . . . 129 viii o . ...'n :ALDA: 'h. . D '5. .‘ c ‘5: I. n I ..v ~. §.~? .‘n ~ , '4. .I ‘ . ui‘ ‘1‘. n .'4. \- 51. ’1 . 1 .‘.‘ ‘- . .\ .- ‘. !.~ J‘, ‘\ s . ' l u ‘- Q “F ‘1 l '1 ‘1‘. s. -. Figure 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 4.35 4.36 4.37 4.38 4.39 4.40 4.41 4.42 4.43 4.44 FD Horizontal Interface v—Displacements. FD Horizontal Interface u-Displacements. FD Vertical Symmetry Axis v—Displacements. Composite Plate - CST Configurations CST Top Edge v-Displacements CST Top Edge u—Displacements . . . . . . . CST Horizontal Interface v-Displacements CST Horizontal Interface u—Displacements CST Vertical Symmetry Axis v-Displacements Composite Plate - LST Configurations . LST Top Edge v-Displacements . LST Top Edge u-Displacements . . . . . . LST Horizontal Interface v-Displacements LST Horizontal Interface u-Displacements LST Vertical Symmetry Axis v-Displacements FD, CST, LST Top Edge v. . . . FD, CST, LST Top Edge u. . . . . . FD, CST, LST Horizontal Interface v. FD, CST, LST Horizontal Interface u. . . FD, CST, LST Vertical Symmetry Axis v. . . Composite Plate - FD Stress Distribution . Composite Plate - CST Stress Distribution. Composite Plate—LST Stress Distribution. Composite Plate Horizontal Interface ox. . Composite Plate Horizontal Interface Cy ix Page 130 131 132 134 135 136 137 138 139 140 141 142 144 145 146 147 148 149 150 151 158 159 160 165 168 Figure 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.22 Composite Plate Vertical Interface 0 Composite Plate Vertical Interface TX Pressurized Thick Cylinder . . . . Thick Cylinder, Finite Element and Finite Difference Configurations . . . . . . . . Thick Cylinder Radial Displacements. Thick Cylinder Radial Stress . Thick Cylinder Circumferential Stress. Composite Cylinder . Composite Cylinder — Finite Difference Configurations. FD Cylinder End Axial Displacements. FD Horizontal Interface Axial Displacements. Extrapolated FD Displacements. . . . . . Composite Cylinder - FD Stress Distribution. FD Corner Stresses . Composite Cylinder - CSTR Configurations . CSTR Cylinder End Axial Displacements. CSTR Horizontal Interface Axial Displacements. CSTR Extrapolated Displacements. Composite Cylinder — CSTR Stress Distribution. CSTR Corner Stresses Composite Cylinder - LSTR Configurations LSTR Cylinder End Axial Displacements. LSTR Horizontal Interface Axial Displacements. LSTR Extrapolated Displacements. Page 171 175 179 181 184 186 189 190 192 194 195 197 198 199 201 202 203 205 206 207 208 210 211 212 \ '4 . ‘ J . 1‘ . I . rv .. . n . c J . o .n .¢ ”I. um... I!» us. .r.. .I. o... u'u nun '1 Ix . V. v. 7‘ v. o I.... & . ~ s t . . \ Q y n \c “H \H . . v. . .. .V. U s .u .g _.s .\C _. _.~ .. .u »\.~ \ . u an .- .I n. ... ..~ t 5" rd Fl: ... A. !.v A. 'L !& v M on 50 .Id .c, . . . o . _ .6 ~ . a u u u u . ~.. . . ..4 . g n c | v I n o c I O o S c a t n.\ o . . n u . . . Q . m u(- V ..V .91. ~ »I- PG. . s ‘ t . \ ..c ... . . .1. L. b. x... in. -3. us... -. I. 1.. .3. A... f. |.. in. o‘d Figure 5.23 Composite Cylinder - LSTR Stress Distribution. . . . 5.24 LSTR Corner Stresses . 5.25 Comparison of End Axial Displacements. 5.26 Comparison of Horizontal Interface Axial Displacements 5.27 Comparison of End Radial Displacements 5.28 Comparison of Horizontal Interface Radial Displacements. 5.29 Horizontal Interface or. . . . . . . . . . . . 5.30 Horizontal Interface 02. 5.31 Vertical Interface oz 5.32 Horizontal Interface 06' 6.1a Square Plate with a Circular Hole - Finite Difference Approximation for Present Analysis 6.1b Square Plate with a Circular Hole - Possible Finite Difference Approximation . . . . . . . . . A1. FD Cylinder End u-Displacements. A2. FD Horizontal Interface u-Displacements. . A3. FD Axial Displacements . . . . . . . . . . . . . . A4. FD Radial Displacements. A5. CSTR Cylinder End u-Displacements. A6. CSTR Horizontal Interface u-Displacements. A7. CSTR Axial Displacements . A8. CSTR Radial Displacements. A9. LSTR Cylinder End u-Displacements. A10. LSTR Horizontal Interface u—Displacements. All. LSTR Axial Displacements . A12. LSTR Radial Displacements. xi Page 213 214 217 219 220 221 225 228 231 235 239 239 247 248 249 250 251 252 253 254 255 256 257 258 5U NI 13 h, k, r N, S,E,W LIST OF SYMBOLS Rectangular Cartesian coordinates Cartesian normal stress components Cartesian shear stress components Cartesian normal strains Cartesian shear strains Cartesian displacements Cartesian body force components Cartesian boundary traction resultants Polar cylindrical coordinates Cylindrical normal stress components Cylindrical shear stress components Cylindrical normal strains Cylindrical shear strains Cylindrical displacements Cylindrical body force components Cylindrical boundary traction resultants Modulus of elasticity (Young's modulus) Poisson's ratio Shear modulus General elastic constants Mesh dimensions for difference approximations North, south, east, and west designation xii NE SW SE W [k] [ka] [K]. [111 p. q [f] dV da ds m. [17*] Northeast Northwest Southwest Southeast Displacement functions Generalized displacements Generalized forces Work Element stiffness matrix Generalized element stiffness matrix Overall structural stiffness matrix Boundary traction Matrix of element nodal point forces, or force intensities Element of volume Element of surface area Element of arc length Matrix of overall structural nodal point force or force intensities Weighting functions xiii . . II .c a . . x . v. .. .- . ¢ .6 a . .Q 5 Q Q . . .7 . «.5. .._ . . .. .__ .f f i I. 4. ~.. . _ . . . t . a Q Q s u u v. .u 2. . s .t s... s a.» mu 6 \u .r‘ ‘s . a . . . 9.; FA . . 3 .. . : a .. a .2 .. n. .. .: é. .. .. .. ~ _- .t .. . . . . a _. v. . . .fi .- x a . . .. .. .1 a. . . .3 Ex . . .t. at. I. INTRODUCTION 1.1 Remarks A fundamental problem of mechanics of deformable bodies is the determination of the state of stress and deformation in arbitrary three dimensional solids. Of particular importance are two dimensional situations involving plane stress or plane strain. The literature of the classical theory of elasticity contains exact solutions to many of these problems. These are restricted for the most part to two dimen- sional problems involving simple geometry and boundary conditions. In more complicated problems, it is necessary to resort to approximate methods of solution. Two approximate procedures which have found widespread appli— cation in recent years are finite difference and finite element methods. Finite difference methods involve mathematical approximations. The governing differential equations and related boundary conditions are replaced by difference expressions. These relate discrete values of approximating functions at a finite number of points. The result is a system of linear algebraic equations which is solved by standard numer— ical procedures. Finite element methods refer to a class of approx— imate procedures in which the actual body or structure is replaced by an assemblage of carefully chosen elements connected at a finite number of points called nodal points. In the stiffness method for example, an assumption of the strain distribution in the element is . . . 4.6 821.". S . - ...- . ,..- Q." - 5“.06-ID~ .' ' b D. """ I'ID' ““‘5 '6... I-M. . .. .~ . ' U I" In '1. K (I C 0 O u.- ..‘ 1-5... ~'-- o|.‘:'-.tb. . I " ‘ r . ‘ ... .‘e..: - s c .e ' ; a. u.“-- .- . ."~: §. Y .Io ".e "r- “ a Q. ‘4 “1. .. ‘ ‘~us [. . -'~':.-. “. '1' . ‘5.£- .- . "yes ‘ ,_‘ § 11 a; ' _ ' 1 u -a s - ‘ K H L ,‘r (1" '9. . \‘. ‘ . -.‘\:...‘ ‘ “Gl F 1'- bl . u. s, ”p 2 made with strains related to the displacements of element nodal points. Equilibrium conditions are then satisfied at the nodal points. There is no need for approximating the governing equations as is the case with finite difference methods. The approximation,on the contrarg is of a physical nature. The procedure results in systems of equations which relate nodal point displacements to nodal point forces through stiffness or flexibility influence coefficient matrices. These are linear alge— braic equations which are likewise solved by standard numerical procedures. 1.2 Previous Developments The first application1 of finite difference methods is apparently due to C. Runge [1]2. He used the method in the analysis of torsion problems. L. F. Richardson [2] made further progress by applying an iterative procedure to obtain the stress distribution in dams. H. Marcus [3] and later H. Hencky [4] were successful in applying finite differences in the analysis of plate bending problems. R. V. Southwell [5] and his students are responsible for many applications in recent times. The finite element methods are a generalization of well known structural procedures which were originally developed in conjunction with aircraft structural problems. They are related to the so called "matrix methods of structural analysis" advanced by Langefors [6] and Argyris [7]. 1Timoshenko and Goodier, "Theory of Elasticity," Page 461. 2The numbers in square brackets refer to references listed in the Bibliography. . -0 O or ‘— p ‘. ..,.... o . 'U " —' h ,. -. ..--~ . . u - . Iz' ‘ -5 .- an. u r I .- anwnvo. .1 ~‘ .‘_ u . \ 41“ ‘.. I t;- L.’ .- -y- ! “'; - . . g r . .s . 0. av . .. us ...-....- _-; ‘n .. o~o-‘ A“ . ‘1 3 In recent times, application of these methods to continuum problems and other structures has been extensive. Their increasing use and develop— ment is closely related to progress made in digital computation and generally greater availability of digital computers themselves. The first achievement in the area of finite element methods is due to Hrennikoff [8]. He developed a framework analogy for plane stress problems in which the actual body is replaced by a lattice of beam elements. The procedure was subsequently improved by McHenry [9] after which Parikh and Norris [10] generalized the method by including bending. A most sifnigicant achievement is due to Turner, Clough, Martin, and Topp [11]. They presented a triangular plate element stiffness matrix which could be used in the analysis of plane stress problems. This element is assumed to be in a homogeneous state of strain and the displacement field is a linear one. This matrix has been used extensively and is directly responsible for many advances which have occurred during the past ten years. Argyris [12] has given this matrix a different form, one which he calls the natural or invariant stiffness. The original work of Argyris and Kelsey [7,13] demonstrates the capability of the methods to account for initial strains of a thermal or misalignment nature. Turner, Dill, Martin, and Melosh [14] consider the large deformation of heated structures. Argyris [15] discusses initial strains due to plasticity and thermal effects. DeVeubeke [16] introduced a plane stress triangle plate element for which the strain variation is linear and thus the displacement field is quadratic. This element has been used by some writers including Argyris [17, 15] and Felippa [18]. It has proven to be very useful for problems involving stress concentration. Felippa [18] has discussed . . v . p I: ..| IIJ1 . . .d L. L. I . v. n v. n I“ O ._ 5. F.- ~— I: . . u. 1 | a o \. .3 . PA u .. A v C S. t .N. .1 1 u. .f . n .3 .. v” . v. . T .. . . .7 .. s. t . . ‘1 .o no .. .v . .1 .0. u . 2. —. p. . I : .Aq ~. out ,-4 ..J . v . n0. . u n . l .r. . . ‘0. m. i L I .N. . . nu. . . .\ .3 o . v . .. .- . » . v . .3 .3 . A v. .\. . .§ .: u. . . u A .y. x. u D. u . r. O s v . v .u. . . ... on. . . a a .. . u .9... f. s. s . . v .1 “\N . In - _ . . . . . o ‘% . . . . .u . . n . . .. . r . .1 .~. a” .v. . . . v . .. v . L. .1- ..un. . . a .4 t . \K Mo“. «K 2.... T. .. . u . vi. ~ . . .: . a . . .v.. . . . . . .u.~ \. . ~ . .I. h . J .2 .. .. ;. .... u. ... . .. . . 2 . . . ! 4 other refinements to these stiffness matrices involving quadratic and higher order strain variation. Finite elements have also been used in the analysis of plate and shell problems. Among the many contributors in this regard are Melosh [l9], Argyris [20], Schmit [21], Clough [22] and Zienkiewicz [23]. Argyris [24] has demonstrated the applicability to large displacement problems as well. Wilson [25] and Rashid [26] have worked out stiffness matrices for axially symmetric ring elements. Argyris [24, 27] has used a tetrahedron element in the analysis of three dimensional problems. A number of writers have discussed the dynamic problems involving the determination of natural frequencies and natural modes of oscillation for various systems [28, 29, 30]. Felippa [18] gives a detailed account of nonlinear analysis including the formulation and solution of elasto- plastic problems. Chang and Taylor [31,.32] demonstrate the usefulness of the method in linear viscoelastic problems which arise in nuclear reactor work. 1.3 Present Investigation The objective of this present work is to compare solutions of elastostatic problems obtained by finite element and finite difference methods. Included are some examples which have known solutions. Thus, comparisons are also made with the exact elasticity solution in these cases. It was of primary interest in this dissertation to obtain solutions to problems involving composite materials. Exact solutions to such problems are not generally available. In these situations, the results of several approximate solutions are compared with one another. . a. nu- . v ..........C '.' "' 0... ...c ,3 .. -. A O ‘ .. . o... . ' y‘ . "‘ 5 Included in this investigation are formulations of the finite difference and finite element methods for plane stress or plane strain and axially symmetric elastostatic problems. Particular applications pertain to plane stress and axially symmetric problems only. Finite element formulations are given for both constant and linearly varying strain elements. Finite difference problems are formulated in terms of displacement (Navier) equations of equilibrium. This is in contrast with the usual stress function approach which has been used so often in the analysis of plane problems. ... . A “C ........ . -n "I.“ .i.. ..-..t .\ Q s... 9. p. ; 'IIL - n e a: .‘v . t :- . II. FINITE DIFFERENCE METHOD 2.1 General Remarks The analysis of elastostatic problems by finite difference methods is a two step procedure. The first step involves obtaining finite difference expressions for the governing partial differential equations and associated boundary conditions. These difference equations relate discrete values of an approximating function at a finite number of points. A mesh of lines is then superimposed over the domain of the boundary value problem forming a set of nodal points. A finite system of linear algebraic equations is obtained by writing difference equations for each nodal point of the system. The solution of these equations comprises the second step in the finite difference procedure. The equations are characterized by the existence of a relatively small number of non-zero coefficients. The coefficient matrix is said to be sparsely populated. It is therefore possible to deal with truly large systems involving as many as 1000 equations. The solution can be obtained by iterative procedures or by a modified Gauss elimination technique. It is possible to achieve the first step in a number of ways. One of these is to simply replace the governing differential equations and related boundary conditions by appropriate finite difference expressions. This would be the most direct approach if boundary n .' ,-..p.¢¢. 3. -u- ...aoa - A v .n... a'to . ' '...._ Dav--0 ' --.£.o£ .-- . ........._b . ‘~-- |.. a :"‘--.1. .MV v--..-.. _ '\ ""*~'»..-.. 0 " -0 ...~‘ .‘ "\\v\ "‘ "hbu_‘. ’ D W 0‘-..‘ ‘ . b. ' f .... u.._' “" -~..:. ‘- -...‘, ..'. ; tv~fi:‘ -.: s.- ‘v ““0. _ .-. . 't— . n . ..J . a .. . ' “ ‘. “ c ' NII' » F... a!“ .- :";. . s.‘ 3‘ ‘ -, § - ‘S \c u' i: '5‘ ' ‘VI I - 7 conditions are known in advance. A second approach involves a vari- ational principle whereby potential energy is difference form. Letting the total potential ary value results in both governing equations expressed in finite energy take on a station- and associated boundary conditions in the form of finite difference eXpressions. Still another approach involves writing equilibrium equations for material regions corresponding to interior and boundary points. Approximate expressions for stresses are used along with any externally applied loads resulting in finite difference expressions for both interior and boundary points. 2.2. Differential Equations for Plane Stress Consider first the state of stress in boundary forces which are applied parallel to are uniformly distributed over the thickness. a thin plate loaded by the plane of the plate and For convenience, the mid- plane of the plate is taken to be the x-y plane. If the stress compo— nents oz, Tyz, and sz are zero at every point in the body, the state of stress is called plane stress. Thus, the state of stress in such a body is completely specified by the stress components ox, oy, and T . xy The equilibrium of the force system is expressed by the equations 30 at —+———’EZ+X-0 8x By 30 31 +——"1+Y=0 3y 3x (2.1) where X and Y are body force components reckoned per unit of volume. For most applications, the orthotrOpic constitutive relation- ships are sufficiently general. For the case of plane stress these become (2.2) Txy 8 C33ny In the case of isotropic behavior, the elastic constants are (2.3) C33 ' 2(1 + v) with E the modulus of elasticity and v Poisson's ratio. The strain-displacement relationships are Bu E '— x 3x (2.4) 3333 t Y 8y .2231 ny 3y 3x where u and v are continuous displacement functions in the x and y directions respectively. The three strain components ex, Ey’ ny cannot be specified independently since they depend on two functions u and v. By differentiating the equations (2.4) it is possible to show that the strain components must satisfy the equation 323x 325 BZYX 2 3x2 x y 3y which is called the compatibility equation. Thus if a stress or dis- placement field is assumed, it is necessary that equation (2.5) be satisfied in order to assure continuity of deformation. Then at each point in the body,the equilibrium of the stress field is expressed by equations (2.1). These can be expressed in terms of strains by introducing equations (2.2). Thus 36X 8c ayx C115?+C12—13X +C33—X-ay +X=0 36X 36 Byx C21 5;—-+ C22 S§Z-+ C33 —3;X-+ Y = 0 It is now possible to eliminate strains through the strain—displacement relationships (2.4). The result is 32u azu 32v C -——— + C ——— + (C + C ) + X = 0 11 33 12 33 3X2 ayz axay (2.6) 82v 82v 32u C ———-+ C ———-+ (C + C ) ————-+ Y = 0 22 33 21 33 ayz 3X2 Bxay Equations (2.6) are a generalization of the Navier plane stress equa- tions. They reduce to the Navier equations for isotropic materials. It is only necessary to replace the constants C11, C12, C21, C33 by expressions (2.3). The result is o - ._. o r C a .. O .. a - o - O - "' A~’ ""‘ 4‘ . ' “v on... ...,. - . ‘~-~~. a- - - .4.— ‘ O h.“ _. . ‘ '- ...n.-.‘.~‘ . 1‘ w-..‘0 . ‘-v..“‘. :A-p h. ‘..‘ [In 7 C 4 O , u I!- (In (11 (Yr .4 I — —' “i". A O l.. l1 ..‘ ‘ ..‘ a: 7' -‘ ““4 . .‘ _ “kt . “ .“'.‘;-“. I ““‘5 '. > a : “F ‘n M316 .. ‘. [1‘ 10 2 2 2 E 3 u + 2(1E+ ) 8 u + 2(1E- ) g g + X = 0 1 - v2 8x2 v Byz V x y (2.7) 2 2 2 E 8 v E 3 v + E 3 u + Y = O + V2 3y2 2(1 + V) 3X2 2(1 " V) Bxay These last equations are also presented by Sokolnikoff [33] in the indicial notation and in terms of the Lamé coefficients. Thus it is seen that the plane stress elasticity problem can be formulated in terms of two second order partial differential equations in displacements. In principle, one would hOpe to be able to find dis- placement functions u and v which satisfy equations (2.6) or (2.7). Generally, this is a formidable problem so that one is forced to resort to approximate methods of solution. Finite difference expressions for these equations will be presented in a subsequent section of this chapter. 2.3 Differential Equations for Plane Strain Although very different in principle, the plane strain formu- lation closely resembles that for plane stress. A body is said to be in a state of plane strain parallel to the x—y plane if the displacement component perpendicular to this plane is zero for all points in the body and if the remaining displacements are independent of the z coordinate. Thus, the relationships u = u(x,y) v(x.y) (2.8) < I w E 0 .;-o‘ "‘ .' ‘t 5”: . u... 1: re; «- us Pvt 5. .LE 5,: r' I.) ‘~ 11 define the state of plane strain. It follows from the strain-displace- ment relationships that _ fifl.= ez _ 32 O y = a” + §3-= 0 (2.9) yz 3; 32 = 22. 2!. sz 82 8x ll 0 The non—zero strain components are ex, Ey’ and ny° The orthotropic constitutive relationships for plane strain are Ox = C11E): + C12€y 0y = C21€x + C22€y (2.10) oz = C31€x + C32€y Txy = uquy The two remaining shear stresses vanish throughout the body in view of equations (2.9). For isotropic materials, the elastic constants are _ (1 - v)E 22 ' (1 + v)(1 - 2v) ll c = c = c = c ”E 12 21 13 31 = (1 + V)(1 - 2v) (2.11) E can = 2(1 + v) The equilibrium of the force system for the case of plane strain is likewise expressed by relationships (2.1). Using the strain- ‘.‘l .0'-'-' ;;,.3-K-C... - d \ I .uuonvu. . n u. t. I‘I . ‘ ‘v . '° .- -~n.., , l I. - . . '-‘. . . ‘ Q . ‘A U ‘4‘ . .4 .. L". p. x, 9.. ‘ . "..,_p-.‘ s ‘ V ‘1‘ ‘- . cul- ‘1 A . ‘-. a , ‘- ‘tl \ . . -I . v n“ I ‘L .6“ ”5“ v... in ’ l.‘~' .'- 5. .‘I ‘v ‘v’u . 'ut 9: .5‘ . LE . . Ez'ri‘s: U = I. i a 12 displacement relationships and the constitutive relationships, (2.10) above, equilibrium equations in terms of displacements similar to equations (2.6) can be written for the case of plane strain: 32u 32u 32v C ———-+ C ———-+ (C + C ) + X = 0 1 an 12 an 1 3x2 3y2 axay (2.12) 82V 82v 82u C ———-+ C ——— + (C + C ) ——7—-+ Y = O 22 MM 21 RH , 3y2 3X2 axdy For isotropic materials these become (1 — v)E 82u + E 82u + E 82v + X = 0 (1 + v)(1 - 2v) 8x2 2(1 + v) Oyz 2(1 + v)(1 - 20) Dxny (2.13) _ 2 2 2 (1 v)E 3 v + E 8 v + E a u + Y = 0 (1 + v)(1 - 2v) Byz 2(1 + v) 3x2 2(1 + v)(1 - 2v) Bxay The above are the Navier equations for plane strain. They are likewise presented by Sokolnikoff [33] in indicial notation. 2.4 Differential Equations for Axially Symmetric Problems The state of deformation in a solid of revolution is called axially symmetric if the displacements are the same in all planes which pass through the axis of revolution. Thus, the circumferential displace— ment vanishes at each point in the body and the remaining displacement components depend only on the radial and axial coordinates. These ideas are expressed by the relationships u = u(r, z) w = w(r, z) (2.14) v E 0 I II .C v. v. . u. . . .. .t .. .h .3 r. .. u“ m «L. 9. u.‘ D; .5; .Q o . .-u | D s t a r u rd 4 7 -~ .t tv I 7. r %. a film . s .4. .vs ,k ~ ‘ 4 .r» a .t «4‘ ‘s A... . u o . o . s u . u .-s . a at v . u . b .,. I‘M . . .I .3 u . . 13 Here u and w are the radial and axial displacements respectively. The stress-equations of equilibrium for axially symmetric problems are 30 31 o - o r rz r 6 at + 32 + r + R - 0 (2.15) ST 80 T r2 +.__E.+ r2 + z = 0 3t 32 where R and Z are the radial and axial components of body forces reckoned per unit volume. Once again, the orthotrOpic constitutive relationships are of suitable generality for most problems. 0 = C c + C1262 + C135 r 11 r 8 Q [I C e + C2262 + C236 (2.16) z 12 r 6 0 II C E + C23€z + C336 0 13 r 8 T = C e rz Mu rz The elastic constants for isotropic materials are much the same as those in the plane strain formulation. _ (1 — v)E 11 22 33 ' (1 + v)(1 - 2v) vE C12 ‘ C21 ’ C23 ’ C32 ’ C13 ‘ C31 ' (1 + v)(l - 2v) (2.17) = ___ll____ an 2(1 + v) The strain-displacement relations in cylindrical components -.. ~ “caovi .9 ,. -s 7" st .;- . o. “" t-ozt. .__ :s.‘ ~o "5 P . » 14 applied to the axially symmetric case become 8 = 22. r 8r 8w - Ez ‘ 32 (2.18) c =-3 6 r -.§2 22. er _ 82 + r When equations (2.18) and (2.16) are introduced into equations (2.15), a set of equilibrium equations is obtained in displacement components. 2 2 €116:—ll +%§}ri)+cwa_—E'C33u— 3:2 322 r2 32w 1 3w + C +C + C -C ———+R=0 ( 12 an) Braz ( 12 23 ) r 32 (2.19) 2 2 2 1+1; (3w {5%) +C228w+(C12+CLm) 213812 3r2 822 1 Eu _ For materials which display isotrOpic behavior, these become the axially symmetric Navier equations. __(1 - v)E (aZu + l_§2) + E 32u _ (1 — v)E 1i_ (1. + V) (1. " 2V) 31:2 1‘ 81‘ 2(1 + V) 322 (l + V) (l — 2V) r2 + E 32“ + R = o (2 20) 2(1 + v)(1 - 2v) Braz ° _j;__g 32w +’l.§wj (1 - v)E 32w + E a2u 2(1 + v) [BIZ r 3r (1 + v)(l - 2v) 822 2(1 + v)(1 - 2v) Braz E l Bu _ + 2(1 + v)(l - 2v) r z + Z - O a ‘ '.-n" .,‘ ,.....s -- I y .- . .' -...:'.-.-. . “--‘ I ‘ ‘A. 1 1 n- H. 15 2.5 Finite Difference Equations for Plane Stress Consider the arbitrary domain of Figure 2.1 corresponding to the plane of plane stress or plane strain. A mesh (rectangular in this case)3 is superimposed over the actual domain. The points of intersection of these lines within the domain are called mesh (nodal) points. The points of intersection with the boundary Figure 2.1 are called boundary points. It is Rectangular Mesh usually convenient to use a uniform spacing with equal magnitudes in both directions. However,in certain cases it is desirable to select a different x and y spacing, while in still other instances, a nonuniform spacing is useful. The latter is particularly ,_“__ NW " _ true where stress concentration is f NE involved or in the neighborhood of irregular boundaries. Only uniform spacing is treated in this work. "“‘fi‘ —""".""7" £1 to (1 # h”) 04 In Figure 2.2, let the x and y spacing be h and k respectively. SW 8 SF r._-..__ h +q .- h .— -" The mesh or boundary points in the Figure 2.2 Jimmediate vicinity of an arbitrary Mesh Point and Neighboring “Bah point 0 are shown. Points \ Some formulations may dictate the use of other mesh configurations (e.g. oblique, polar, or arbitrary curvilinear meshes). 16 The partial derivative with respect to x of a function u(x, y) can be approximated at the point 0 by the first divided difference it: uE—uw = -—-————- (2.21) 3x 0 2h In a like manner u - u Bu N S 3y 0 2k . Approximations to second derivatives can be established in much_the same u - u way. At the point 1, midway between 0 and E, gfi-z -§—E——Q’ and at the au u0"“w point 3, midway between 0 and W, SE-z __—h———' Thus the second partial derivative of u with respect to x can be approximated by differences in the approximate first partial derivatives. Bu _ Bu 2 3x1 3x3 uE — 2u + .211. ._.___h____= 20 u“ (2.23) 2 3x 0 h 2 The finite difference approximation to g—E-obtained in much the same 3y2 way is 2 u - 2u + u a u z N 0 S (2.24) 2 2 3y 0 k 2 To obtain the second mixed partial derivative gxgy , the first Partial derivative of u with respect to y is approximated at points E and W. That is pt A (a V.‘ Then 82u = 3x3y0 When equations 17 “NE ' “SE 2k “NW _ "8w 2k 22. _ 32 §__(§2) 2 By E By W = ”NE ’ “SE ' ”Nu + usw (2 25) 3x 8y 0 2h Ahk ' (2.23), (2.24), and (2.25) are introduced into the govern- ing second order partial differential equations, the finite difference expressions are obtained. Corresponding to (2.6) for plane stress are 2C11 C33 C11 C3 [ 2 -——J u - -——-(u + uw) — ——- (u + u ) h2 k2 o h2 E k2 N s (C12 + C33) ( + ) - x (2 26) 4hk VNE vNw vSE sz “ o ' 2C33 2C22 C33 C22 0 - -——-(vE + vw) — ——— (vN + vs) h2 k2 h2 k _ C21 + C33 ( _ _ + ) _ Y 4hk uNE uNw USE ”3w 0 Here X0 and Y0 are body force components applied at the mesh point 0. If the become same spacing is taken in the x and y directions (2.26) 8 - _ (Cll + C33) uO 4C11(uE + uw) 4C33(uN + us) .. _ ._ 3 2 (C + C33)(v v v + vsw) h XO (2.27) 12 NE NW SE l I. .l _l u z - 18 8(C22 + C33) vO - 4C33(vE + vw) - 4C22(vN + vs) — — — = 2 (C21 + C33)(“NE uNw USE + “sw) h Y0 These same equations for isotropic materials become 8(3 - v)uO - 8(uE + uw) - 4(1 - v)(uN + us) - (l + v)(v - v - v + v ) = 8 l—:—BE- h2X (2 28) NE NW SE SW E O ' 8(3 - v)vO - 4(1 - v)(vE + vw) - 8(vN + vs) - (1 + v)(u - u - u + u ) = 8 l—:—33 h2Y NE NW SE SW E O The relative magnitudes of the coefficients in the above equations becomes more apparent when a particular value of Poisson's ratio is assigned. Taking v = 1-in equations (2.28) one obtains 4 30h2XO 88u0 - 32(uE + uw) — 12(uN + us) - 5(vNE — va — VSE + VSW) = _-ET_—— (2.29) 30h2YO 88vO - 12(vE + vw) - 32(vN + vs) - 5(uNE - uNw — uSE + USW) = ———ET~— 2.6 Finite Difference Equations for Plane Strain The development of finite difference equations for the case of Plane strain involves the same concepts presented in the previous section. One simply introduces the partial derivative approximations (2.23), (2.24), and (2.25) into the appropriate differential equations. When this is done for the isotropic relations (2.13) with equal spacing, h, in the x and y directions,the result is 19 8(3 - 4v) uO - 8(1 - v)(uE + uw) - 4(1 2v)(uN + us) 8(1 + V)(l - 2V) hpx - (v - v - v + v ) = E 0 NE NW SE sw (2.30) 8(3 - 4v)v0 - 4(1 — 2v)(vE + vw) - 8(1 v)(vN + VS) = 8(1 + v)(l - 2v) hZY - (u - u - u + u ) E 0 NE NW SE SW These results are more readily compared with corresponding plane stress equations for a particular choice of Poisson's ratio. Again using 1 v = z-one obtains 2 30h X0 96u - 36(uE + uw) — 12(uN + uS) - 6(vNE - v — v + v ) = E O (2.31) 2 30h YO 96vO - 12(vE + Vw) - 36(vN + vs) - 6(uNE - u - u + u ) = E NW SE SW 2.7 Axially Symmetric Finite Difference Equations The development of Z finite difference equations for axially symmetric problems is quite similar to that for plane - >_‘._--.-. 1 stress or plane strain. Figure 2.3 - ———-——'—'4 1 . _ 1 .---,__.___._._*—_q - _ .... _ .41] O O - .' A '- — --. represents one quarter of the cross ] section of a solid of revolution. The o ~»—o——- —~.«--—( - +..~———.—.—..4 z axis is the axis of symmetry. A ' ' ' f ' ? rectangular mesh has been superimposed over the region. Figure 2-3 Rectangular Mesh at. o"' 5.... 13.. ;_...... w’..$.t. --~ ‘3 . —. .u..¢— "’9‘ -~- 0 a ) ‘ ”0~~:.—~-. .. "OI. v u .. ' “.-~....,. _ ‘v -. .-_ .v . L A typical mesh point 0 with mesh or boundary points immediately around it is shown in Figure 2.4. of a function u(r,z) up to and including the second order are approximated by the following difference expressions: Partial derivatives 20 ,.NW N; r ] NE k 0 E W k V... sw g s SE 1. h iJ—t‘-[1 .— Figure 2.4 Mesh Point and Neighboring Points 22. 2 ”E - “W 3r 0 2h .22 1 UN - “S 82 0 2k 82u 2 Us ’ 2Uo + uw 8r2 0 h2 2 u u - u + u 8 u 2 NE NW SE SW Braz 0 4hk - + azu 2 ”N 2U0 US 822 O k2 (2.32) (2.33) (2.34) (2.35) (2.36) These finite difference approximations are introduced into the differ— ential equations (2.18). (uE - 2uO + uw) Cll (uE - uw) (uN - 2uO + us) ‘ C11 + r 2h + cu“ - C33 h2 o k2 + (c + c ) ("NE wNw wSE + wsw) + C12 ' C23(WN ' ws)+ R 11 an 4hk r 2k ..n '.-'oa . a (wE - 2w0 + ww) ClW (wE — ww) (wN - 2wO + wS) C44 + r 2h + C22 h2 0 k2 + (C + C ) (uNE “NW USE + usw) + C23 + C44 (uN - us) + Z = 0 12 1+4 4hk to 2k 0 Upon simplification, the above reduce to h2 h2 h h (2C11+ 201414 2 + C33 2 ) L10 - C11 (1 + 317—) LIE - C11 (.1 - 21' ) Uw k r0 0 0 + - C h----(u + u ) - C12 Cu“ h-(w - w - w + w ) 1.14 kg N s 4 k NE NW SE sw C - C 12 23 D. L _ = 2 2 k to (wN wS) h R0 (2.37) 2(C +C 12-) -C (l+—h—) C (l h)w 22 1+4 2 w0 1+4 2r wE 1+1. 21- w k 0 O C + C h 12 an h ' C22 k2 (“N + ”3) ' 4 k (“NE ’ “Nw ' SE + ”sw) C23 4‘ Cm. h h 2 - -—-- (u - u ) = h Z 2 k rO N S 0 When equal spacing is taken for the r and 2 directions, k = h, one obtains (2C11+2Cw+c33—) u0"C11 (1+ '27) uw — w ) = 1.211 (2.38) :3 .0 . . .n .5 ._ . 9.‘ ‘ rulv . a . ’1 , . A .~ H i . 1... Q. r. .3 2i ., c... . w. . . .d» nu. .uJ . . .u. .. 4. s a \ it . ..s .1 .3 .l- n‘ .F» .Iv (I \ .51. cl. 6 L h h 2(022 + Cut.) wo _ can” + ZrO) wE ' can” ' 7;) ww C + ) C12 + Cut. ( 22(WN ws. ' 4 “NE ‘ “Nw ' u3E + “514) C23 + Cut. h_( ) _ 1122 2 r “N “Is ‘ o For the case of isotropic elasticity, the coefficients Ci are given 3 by equations (2.17). The resulting finite difference relationships are 112 h h 8[(3 — 4v) + (1 - v) ——-] uO - 4(1 - v)(2 + ;—) uE - 4(1 - v)(2 - ;—) u 2 r0 0 0 W — 4(1 - 2v)(uN + us) - (wNE — wNw - wSE + WSW) h2 = 8(1 + v)(l — 2v) E—-R (2.39) 0 8(3 - 4v) wO - 2(1 - 2v)(2 + $—) WE - 2(1 - 2v)(2 — EL) w 0 O W -8(1 - v)(wN + wS) - 2 gg-(uN — us) - (uNE - uNW - uSE + USW) 1'12 = 8(1 + V)(l - 2V) -E—- 20 1 Finally for the specific case in which Poisson's ratio is taken as Z. these become .. ._ V o c. o. u U n I. .o . q ‘ Q Me . I; O .1. .n - . :1» r: u . ._...c. f .. .t .t t .—4. 7. av . t .u. .4: o. s H. . . . .t . . . c at Y . . ... .n. . «a. . a .3 o o o.“ .. J . ~ A L - 3‘ (I . .1. .¢ A VA 95 .a& .< ‘ . v w .—~ -\~ g » n .I. .~ . . . . 3. .a \LN . . . S. l . 7.. .. . . § QR .o W . ..§ 55 .~ Q -‘ Vs \v‘» ‘\ 23 112 h h (16 + 6 2) uO - 3(2 + r ) uE - 3(2 - ;—) uw - 2(uN + us) I 0 O 0 1'12 - (wNE - WNW - WSE + WSW) - S'E- RO (2.40) h h h 16wO - (2 + r ) wE - (2 - ;—) ww - 6(wN + wS) - 2 ¥—-(uN - us) 0 O 0 1'12 ' (“NE'uNw'USE+“sw) =5E'Zo 2.8 Alternate Derivation of Plane Stress Difference Equations Finite difference equations corresponding to the previously mentioned situations can also be derived from the equilibrium of a material element in the neighborhood of an arbitrary mesh point 0. This method has the advantage that boundary conditions can be derived in exactly the same way. This is important in the case of certain ques- tionable situations such as corners where boundary conditions are not immediately apparent. Inherent in this procedure is the need to make assumptions regarding the strain approximations to be used. Corresponding to dif- ferent choices for these strain approximations are somewhat different finite difference equations. Figure 2.5, on Page 24 illustrates the rectangular region around the mesh point 0. The x and y dimensions are h and k respectively. X 0 and Y0 are body force components per unit volume assumed to act at the point 0. In the case of boundary points, to be discussed later, X0 and Y0 may be components of the static resultant of boundary tractions. The normal and shear stresses are designated in the usual way and sign 24 SW 1 i "y y SE I y I F1F---- h ---iF+ I .__... .._.... _ _ ...._. __ SW S SE Figure 2.5 Material Region Around a Mesh Point and Associated Cartesian Stresses and Body Forces .npn‘n. I -as ,.u‘vn- n-fl ‘ u "'|O-.|-.I : .. 4. in, ‘ . "f‘D-D. ........,‘;_t . "«‘-D.. . _ u --OQ‘. . w...bg‘.e a - . u. u I o c "1‘ . o 2. 25 conventions of classical elasticity are employed. The four quadrants around the mesh point 0 are designated as northeast, northwest, south— west, and southeast beginning in the upper right hand corner and pro— ceeding counterclockwise. Normal and shear stresses corresponding to these regions are given superscripts accordingly. In deriving finite difference expressions corresponding to previously presented results, it is necessary to express stresses in terms of displacements through the constitutive relationships and approximate expressions for strains. For the isotropic materials, the constitutive relationships (2.2), with isotropic elastic coefficients (2.3) and strain-displacement relationships (2.4) become P Du 8v Ox — 2 (3X + V 8y) 1 _ y 1 _ V2 3y 3X E 3v Bu Txy - 2(1 + v) (3x + By (2°41) E Bu 8v T (—— yx = 2(1 + v) 8y 3x The displacement gradients in equations (2.41) are approximated in Various ways to give the following stress expressions: NE yx SW T 9.. XY SW E E vNE " vE + vN ' vo [ + v 2k 1 - v2 E ( N + uNE ' uN + uE " U0 1 _ V2 k 2h (2.42) E [ E v0 + uNE ' uE + uN ' “0) 2(1 + v) 2k E (”N u0 + vNE ‘ vN + vE ' V0] 2(1 + v) 2h E u0 — “w vN ' v0 + vNw ’ Vw ( + v ) 2 h 2k 1 — v v - u - u + u - u E ( N + v N NW 0 w) 1 _ v2 k 2h (2.43) E (V0 w + uN ' uo + uNw ' uw) 2(1 + v) ) 2k 11 u v - v + v - v _E, { 0 + N NW 0 w) 2(1 + v) \ 2h 11 - v - v + v - v E 0 uw o s w sw { h + V 2k ) l — v2 E (,0 u0 ’ uw + u3 ‘ ”5w 1 +V 21] J l - v2 _ (2.44) E {Y w + ‘10 us + “w usw) 2(1 + v) [ 2k u u v - v + v - v E { s + 0 w 3 SW) 2(1 + v) ( 2h v.” .x. 2.. .x. x .‘. vk" ‘~.; In A -x"-.‘ WE I . \ = I . n. . a , 1 at I n). n.. 1‘: we. .5. :C n? » c . t \ a . Tr.— «u. h\1. n]. Y c . L. .\\. x ml“ . . .le \Ar/S P.» 1" ~ :1 a u \ .... . . . v4". .‘ OSE = E [ E O + v E SE 0 S) x 1 _ v2 h 2k SE E vo ' vs uE ' uo + “SE " us o=———(——— 2. J y 1-V2 (2.45) TSE = E (VE ' vo + uE ' "SE + u0 ' “5) xy 2(1 + v) h 2k TSE g E (“o ‘ us + vE ‘ vo + vSE ' vs) yx 2(1 + v) k 2h The equilibrium of an arbitrary element requires that XF = O x 2F = 0 Y If the element is of thickness t, these equations become k NE k NW k SW k SE t 2 x - t 2 Ox - t 2 0x + t 2 x h NE h NW h SW h SE + t 2 Tyx + t 2 Tyx - t 2 Tyx - t 2 Tyx + thk X0 - O (2.46) h NE h NW h SW h SE t20y+t20y-t20y-t20y k NE k NW k SW k SE _ _ _ _ _ _. + _—-_ + t 2 Txy t 2 Txy t 2 Txy + t 2 Ixy thk Y0 O The stresses (2.41) through (2.45) are next substituted into equations (2.46) and after some simplification with k = h one obtains 1 - v2 + + + 1 - v2 + + + E E E l - v2 1 - v2 1 - v2 E 1 - v2 E 1 - v2 E 1 - v2 28 [4(u0 ‘ UE) + 2v(v0 + vE - V — + (l - v)(vO + v - [4(u0 - uE) + 2v(vS + v - SE + (l - v)(vE + v [4(u0 - uw) + 2v(vw + v - v [4(uO - [4(V0 - VN) + 2v(uN + u - u [4(VO - VS) + 2v(uE + u [4(v0 - VS) + 2v(u0 + uS - uw - USW) + 2(1 [4(v0 - O + (l - v)(vO + v uw) + 2v(vN + va - + (1 - v)(va + O E + (l - v)(uO + uE SE + (l - v)(uS + u + (1 - v)(uw + u vN) + 2v(uw + uNW - + (l - v)(uN + u 0 SE S S vo vw uNE) + 2(1 - v)(vO - v ) UN “'11 O + 2(1 - v)(uO - uN) VNE)] vE) + 2(1 - v)(uO - us) vO - vs)] - VSW) + 2(1 - v)(uO - us) vw ' sz)] - vw) + 2(1 - v)(uO - uN) vN - v0)] = 8h2xO (2.47) ‘U - us) + 2(1 NE )1 SE ' uo ' “E)] 0 U0 NW sw " ”s)] - uN) + 2(1 u0 - UN)] U E v)(vO - VE) v)(vO - vw) v)(vO - vw) 2 8h YO . u... Ila Q g "I aN- .A. c. .3 ‘1. .3 7: .r a“ a. r. a“ a. ..A .V» .‘. .‘a o. 4. . w. J. .. .. . .4 Q- .~. . . r. 2 . . ... on. .. .1 .,. at, s. .. vb. nlw ”a Q ““cr.v~ 29 Equations (2.47) are expressed in a form which allows certain boundary conditions to be derived. These will be discussed subsequently. After considerable simplification, equations (2.47) can be reduced to the following: 8(3 - v) uO - 8(uE + uw) — 4(1 - v)(uN + us) - (l + v)(v - v — v + v ) = 8 l - V2 h2X NE NW SE SW E 0 8(3 - v) vO - 4(1 - v)(vE + vw) - 8(vN + vs) - (l + v)(u - u - u + u ) = 8 l—:—23 h2Y NE NW SE NW E 0 These results are identical to those which were obtained in section 2.6 and labeled equations (2.28). Next, consideration is given to the development of static boundary conditions. Boundary expressions treated here are restricted to rectangular boundaries which are parallel to the coordinate axes. Thus, one is able to deal with points on vertical or horizontal boundary surfaces as well as 90° corners. y I 3? Other boundary conditions can be approxi- 0 x mated using sufficiently small spacing. 1 :w E ' O The first situation treated h here is the 90° outside corner. In . I Figure 2.6, point 0 is such a corner SW" - m 3 rd in hJ point formed by the intersection of ' Figure 2.6 vertical and horizontal boundary ' P ' t surfaces. The x and y spacings are Out31de Corner Boundary 01n V... 5 LP} I (3.-) ..,.. ul O'n-OQ ‘H: D. ...~HUI :r'!::‘ s. < ’--v-.« ¢ .1 c. ' ,‘ '“-- Q , .ca.“ .. O - l x ‘I k '4 ‘ . . a . I ‘- D - \ r. '0'“ L n . . I s ' ‘ I ‘n m. - “s' 9 .0 . I {Le '3'. "u, . . 30 both h. i0 and E0 are the components of the resultant of boundary tractions. The finite differences expressions for such a point follow immediately from equations (2.47). One need only use the left hand side terms of these equations which pertain to the southwest corner. The expression in the third bracket in each of equations (2.47) is pertinent. The result is E I—:_:; [4(uO - uw) + 2v(vw + v0 - vS - VSW) + 2(1 I C v A C O l C U) v I < < V I a) X + (1 — v)(vO + VS W - SW __E__2_ [4(v0 .. VS) + 2v(u0 + us - uw - USW) + 2(1 - \2)(vO - vw) 1 - v I C I (I) v-< + (l - v)(uw + u0 SW - uS)] - 0 Upon simplification, the above become 2(3 - v)uO - 4uw - 2(1 - v)uS + (l + v)vO - (l - 3 )vw _ 2 _ 8 3>——J1— x (2.48) + (l - 3v)vS - (l + v)vSW = E 0 2(3 - v)v0 - 2(1 - v)vw - 4vS + (1 + v)uO + (1 - 3v)uw 1 - 3 ) — (1 + ) = 8 lei—XE-f ' ( V us V “sw E 0 Then for the Special case with v = %-one obtains 30)?O ZZUO - 1611“ -’ 6118 "I’ 5V0 - Vw + VS -- Svsw a __...E_.._ ZZVO - 6vw — 16vS + Suo + uw - US — SUSW = ——ET- \ .5. "r-; .- -‘ L... -n- DA-.§.~M:I . q.‘..ho.- ‘85 ‘ s . . .0“ a..- o'.-o.' ~ .) . . us .. w s p. .. I D. . x t h... . V «Q .P. N» K I C V. “a M t 0| .5 a . 1A . . E 0%.. :— nu ~ L o S 0 ~ .. u E . EH.“ HIV a: E ‘ H”... 5 .\~ 9|» 5 N u .Pu . 9. 1 a .2. a; rt o.c ”a“ P. Q . § ‘U- 31 Consider next a point 0 on .--*W y a vertical boundary surface. Such a I point is illustrated in Figure 2.7. Once again 20 and E0 denote components of the resultant of boundary tractions. The spacing of mesh points is h for both directions. 1 E¥i S I" .m . I" . __. f I The material around the mesh point 0 includes the northwest Figure 2.7 and southwest regions. The Vertical Boundary Point expressions which correspond to these regions are obtained from the third and fourth bracketed terms in equations (2.47). These terms must be added to obtain the left hand sides of finite difference equations which pertain to the boundary point 0. When this is done the result is E 1 - v2 [8(uo - uW) + 2v(vN - VS + va - VSW) + 2(1 - v)(2u0 - UN - Us) + (l - v)(vS - vN + va - vsw)] = 8XO (2.50) E 1——-—\-)-2— [4(2Vo - VN -' VS) ‘I" 2V(US " UN ‘I' UNw - US”) + 4(1 " V)(V0 — vw) + (l - v)(uN - u + u + u )] = 8Y S NW SW 0 These can be simplified to some extent and upon doing so one obtains C55 . .5. .-' ......3 c“ ...t K—“‘ 2-’ c- uaos >4 -~‘ .3 a find n“ 2“ [3r- 32 4(3 - v)uO - 8uw - 2(1 - v)(uN + us) - (1 - 3v)vN + (l - 3v) vS 1 - v2 - 8 E X0 + (1 + v) v — (1 + v)vSW NW (2.51) 4(3 - v) v0 - 4(1 - v)vw - 4(vN + VS) + (l - 3v)uN - (l - 3v)uS I (D t-< + (l + v)uNW - (l + v)uSW — -—Er——- 0 . 1 Then for cases where Poisson's ratio is taken to be 23these become 30)“:O 44u0 - 32uw - 6(uN + us) - vN + VS + SVNw - SVSW = -E——- _ (2.52) BOYO 44v0 — 12vw - 16(vN + vs) + uN - uS + SuNw — SUSW = -iT—- The case of a point 0 on a y § 0 horizontal boundary surface is treated I 4} X0 in almost exactly the same manner as + the preceding. This situation is h illustrated in Figure 2.8. The i 4 SE P .. notation is identical to that used SW a h '1 II N previously with regard to other "‘ I‘ boundary points. Figure 2.8 In this case, the material Horizontal Boundary around 0 is in the southwest and Point southeast regions. Corresponding to these are the second and third bracketed expressions of equations (2.47). The I , . .. . . a I. I .. .\4 /I L. v.. .Os CI I d P‘ L3 F. u it . I . J r. . .9. a. .y. .. .Ly .L r. 0.. b.» .n P. . .F‘ .. o —. . c . sm .1. I r .. . I “IV ’I‘ E . a .. Inn 95 NIH c _ . I .0 I n! ‘15 I 7‘ .\ Q. . v u w H d v A .\.A p». r o . a V I. ’14 {t P V. 33 resulting finite difference equations for such boundary points are 4(3 - v)u0 - 4(uE + uw) - 4(1 - v)uS + (l - 3v)vE - (1 - 3v) vw - (l + v)v + (1 + v)v = 8 l—:—23- E SW SE E 0 (2.53) 4(3 - v)vO - 2(1 - v)(vE + vw) - 8vS — (1 — 3v)uE + (l - 3v)uw - (l + v)u + (l + v)u = 8 l—:-23 E SW SE E O l . Then for v ='Z equatlons (2.53) become 303EO 44uO - 16(uE + uw) - 12uS + vE - vw - SVSw + SVSE = _ET—- _ (2.54) BOY0 _ _ .. + _ = 44vO 6(vE + vw) 32VS uE uw 5uSW + SuSE E Finally, the finite difference . NW equations for a typical 90° inside ! h corner are discussed below. An inside corner with material in the I W northwest, southwest, and southeast h regions is illustrated in Figure 2.9. With material in these three regions, ' &_. . " I 3W ‘ all of the bracketed expressions in I t I'- h w ' .- equations (2.47) are used except the Figure 2.9 first. When these expressions are Inside Corner Boundary summed and simplified, finite Point difference equations for a typical corner point are obtained. +~~~ Ls 2.. x 34 6(3 - v)uO - 4u - 8uw - 2(1 - v)(uN + 2uS) (l + v)v0 + (l - 3v)vE E - (1 - 3v)v + (l + v)(v - v + v ) = 8-l—:—Xi E N NW SW SE E O (2.55) 6(3 - v)vO - 2(1 — v)(vE + va) - 4vN - 8vS - (l + v)uO - (l - 3v)uE +(l-3v)u +(l+\))(u -u +u )=81———32—§ N NW SW SE E O 1 When v = z-these become 66u - 16(uE + 2uw) - 6(uN + 2us) - 5v0 + v - v 0 E N 30XO + 5(vNw — vSW + VSE) = “ETE— (2.56) 66vO - 6(vE + 2vw) - l6(vN + 2VS) - 5uO - uE + uN 30?0 + 5(uNw - uSW + USE) =-—E—— Equations (2.29), (2.48), (2.51), (2.53), and (2.55) form a set of finite difference expressions which can be used to treat a wide variety of plane stress problems. Equations of this type are written for each point in the domain of the physical problem. Equations (2.29) are for interior points whereas the others pertain to boundary points. The resulting system of linear algebraic equations is then solved for the unknown displacements. These equations are not without restrictions. As formulated, they assume the material to be homogeneous and isotropic. Anisotropy can be considered by reformulating the various equations in terms of ..; -V"€' ., '.-5 DD-.. . . .u.».... f ..- - .. . ., \ ...-~_L'....'.‘. -. -~-o--no _ v‘\ .-—..3-s‘.v | . ;"‘9— .;’r- -t-.-.-,_. .. V'“. ' .CL. ; 1..., _ v 3 . en, \_ “K .._ ._ ‘ \ v . ' V_.~. ‘ y-.,‘b-:- ' ‘ ~v \“ g . “\‘_"" Vs- . -_‘A - v t."‘...-\- “‘..~' .‘.‘ -- V .. h“? on, ; V~_ - “'«t‘ P: s ._ ‘- -. « ‘s ‘. »-Lt ‘.‘ L\ - . A- o . _' . '5. V . k ‘— “‘ ‘ C - ‘k V' 31 I w '1 ('7' *1 35 the proper elastic coefficients, as done for example with regard to interior point equations in section 2.2. The ability to handle non- homogeneous materials in general and specifically the case of material composites was of main interest in this investigation. This can be accomplished by simply allowing for different elastic properties in the 4 regions around an arbitrary point. This idea is discussed further with regard to axially symmetric problems (page 48 ). 2.9 Alternate Derivation of Difference Equations for Axially Symmetric Problems As was true for the plane stress (or plane strain) problem, it is convenient to derive finite difference equations for the axially symmetric problem by applying equilibrium considerations to a material element. Boundary conditions can be worked out at the same time with little extra effort. 2 The axially symmetric problem is generally a three dimensional AO FEET 112-.51 “ I ———.-— situation insofar as stress and strain ‘ ' l. 0 L r-- are concerned. Figure 2.10 displays ~ g I the typical element of volume in cylindrical coordinates. The increments in the coordinates r, 9, and z are taken as h, A8, and k respectively. The cylindrical stress components Figure 2_10 are Or’ 0 oz and Trz' The shear stresses 0’ Cylindrical Volume Ire andTez are identically zero in view of Element a... T. . . s ‘ “x ca '- L. s . “u .\ L. .1 s. . . . . ..— an w . 36 the axial symmetry. The development of finite difference equations is quite comparable to the work involved in the plane stress problem. The existence of a third normal stress, namely 0 is a major difference. 6’ The stresses are more readily displayed in several ~\_ illustrations for this case. For example, Figure 2.13, page on 35, is a view of the element of Figure 2.10 corresponding to the radial-axial plane. The regions 0 around an arbitrary mesh point 0 0 are again denoted by northeast, northwest, southwest and southeast. The stress components acting in , , Figure 2.11 these regions are superscripted accordingly. The circumferential Circumferential Stresses stresses are displayed in Figure 2.11 These are generally different in NW N1 the four regions around 0. However, 06 \ \ 00 as shown in Figure 2.12 these do not vary with 0. A0 The stress-strain law for axially symmetric (9%) I! NE problems was presented in ’l () section 2.4. It is repeated Figure 2.12 here for the isotropic case. Circumferential Stresses Z 37 I 1 N” _ __ __ __ #— __ _ _ _ .. f— ‘1 | I NW . ONE l 0. z I I TNW TNE I l zr' zr I k | .NW NE I r ‘ u ‘ NE I | ZO r I NW L k Trz RO _ -._r___- - - .._+E——Ji I SW TSE I 0r rz SE I I r I sw Y Trz l ISW I TSE l I zr OSW zr SE 2 0z I L. I‘— “ “* .1 Figure 2.13 Material Region Around a Mesh Point and Associated Cylindrical Stresses and Body Forces " re nt,., " "‘“"~4...., '1 'LI 1 38 E 0r = (l + v)(1 — 2v) [(1 - v)€r + V(CO + E:2” _ E 6 - (l + v)(l - 2v) [(1 — v>€6 + v(ez + er)] (2.57) - E [( z _ (1 + v)(l - 2v) 1 — v)ez + u(er + 66)] E Trz — 2(1 + v) er Introducing the strain-displacement relationships, (2.5), these become _ E _ Lu 2 E 0r - (1 + v)(1 — 2v) [(1 V) 8r + v r + V 82 E u Bu 8w 7 GO _ (l + v)(l - 2v) [(1 - V) r + V 8r + V EEJ (2'58) _ E £933. is 2 Oz - (l + v)(l — 2v) [(1 v) 82 + v 8r + V r. rz 2(1 + v) Br 82) zr 2(1 + v) 82 Dr Proceeding in much the same way as was done for the plane stress problem, the following approximations for stresses are used: «a. p. .I' ZI sk fiK. .‘v .5. I4 "S . 39 ONE = E (1 _ v) uE ' u0 + uE + u0 r (1 + v)(1 - 2v) h V h 2(r + —) 0 2 + v wN - wO + wNE - wE 2k U U 'U W ‘W NE _ E _ _9_ E o N o 06 - (1 + v)” _ 2v)[(l v) to + O ———————h + v—————-—k ] (2.59) NE = E (1 _ ) wN ' wo + uNE ’ uN + “E ' U0 02 (1 + v)(1 - 2v) V k V 2h uNE + uN + E + u0 + v h 4(r0 + 29 NE = E wE ' w0 + uNE ' uE + uN - uo Trz 2(1 + O) h 2k NE = E uN ' uo + wNE ’ wN + wE ’ wo Tzr 2(1 + v) k 2h Similar sets of stresses exist for the northwest, southwest, and south- east regions. The pertinent equilibrium equations for the axially symmetric C388 are It is significant that the o stresses have a radial component and hence 8 must be accounted for in the radial force equation. In this connection, \ .,.,.'-.. up: ‘ I _.. Juro~nn TE :1 ' r t h I" ,‘ \l \ . h... ‘ ‘- ‘nsq ‘ . “s, h“ _., E'd:~ ,, u L, o . ‘. ’. s ‘ 1 . a 40 the approximation sin %§-= %2_ is used. (OISE + 03E) 1;- (r0 +391“) — (a:Iw + i”) 12‘- (r0 - 125m - 2(03E + ogw + 08w + 03E) g-%-%§-+ ROrOAth = 0 (ONE - 02E) % (to + "him + (012” — 0:”) % (r0 - $239 (2.61) + (12: + 1:5) % (to + gme - (9:: + rig) g (r0 - gm) + ZOrOAth = o The quantities R0 and Z0 are body force components assumed to act at the mesh point 0. Introducing the stress approximations of the form (2.59) into equations (2.61) yields E (1 _ V) uE - uo + v uE + u0 (l+v)(l—2v) h 2( +g) r0 2 W ‘W +W "W N NF E k h +" 2k ]§(r0+§ u - u w - w + w - w E N 0 NE N E o h h +2(l+v)[ k + 2h ]§(ro+4) U U “U W ‘W _ E _ 0 E o N 0 pg (1+v)(1-2v)[(1 V>¥3+V h 4'“ k J4 u - uw u + uw _ E (l-v)—O———+\)—2—-———— (1+v)(l-2v) h - 2(rO-f) .0— 41 + E uN ' “0 + w0 ’ ww + wN ’ wNw g.(r ‘g) 2(1 + V) k 2h 2 O 4 _ E (1 _ v) EQ_+ u0 ’ uw + wN ' w0 EE_ (1 + v)(1 - 2v) r0 h *V k 4 u - u + _ E (1 _ V) o “w + o “w (1 + v)(1 - 2v) h V 2( _ 3) r0 2 w - w + w - w o s w sw k h + V 2k ]'2 (r0 ' 2) _ E uo ’ us + wo - ww + ws " wsw h_( h) 2(1 + v) k 2h 2 r0 4 _ E (1 _ v) :g_+ u0 ' ”w + w0 w3 g5_ (1 + v)(1 — 2v) to h V k 4 + E (1 _ v) uE - o + uE + u0 (1 + v)(1 - 2v) h V 2(r + g) 0 2 w - w + w - w 1 0 s E SE k h _ E u0 ' “s + wE ' wo + wSE ’ ws h.( 3) 2(1 + v) k 2h 2 r0 4 U U - U W W E _g E 0 o s 1 33_ = (l + v)(l _ 2v) [(1 - v) to + h + v R J 4 + R rohk 0 (2.62) E (1 - ) wN - w + uNE - UN + uE - 110 (1 + v)(l - 2v) V k V 2h uNE + uN + uE + u0 h h + v -‘(r + z) 4(r + h) 2 O o 4 + E wE ' wo + uNE " uE + uN ' u0 §_( +ug) 2(1 + V) b 2k 2 r0 2 + E (1 _ v) wN " w0 + uN ' uNw + u0 ' uw (l + v)(1 - 2v) k V 2h uNw+uN+uw+uOh h + v -— (r - —0 Mr _R) 2 o 4 0 4 E wo - ww + uN - u0 + uNw " uw 5_( _ g) 2(l+v) h 2k 2 1”o 2 E (1 v) w0 " wS + u0 ' “w + uS ‘ usw (1 + v)(l - 2v) k V 2h uSW + uS + uw + 1.10 h h +v h -(r --) 4(r _._) 2 O 4 O 4 _ E wo ' ww + u0 ' uS + uw ‘ usw 5_( _ g) 2(l+\)) h 2k 2 r0 2 w - w u - u + u - u _ E (1 _ v) o S + v E 0 SE S (1 + V)(1 - 2v) k 2h USE + uS + uE + 110 h h + v —-(r + —) 4(r + E) 2 O A O 4 w — w u - u + u - u E E 0 E SE 0 S k h + 2(1 +'v) [ h + 2k 1'2 (‘0 + 29 + Z0‘0hk ‘ 0 Once again the, terms corresponding to the 4 regions around 0 have been kept together to facilitate the consideration of boundary conditions and the governing equations for fiber reinforced composites. With k = h and a considerable amount of rearranging, equations (2.62) can be expressed in the following way: 43 0 r0 E [[80 - 4v) + 2(5 - 14v) 11--+ 8(1 - v) hi] (1 + v)(l - 2v) r 2 U0 - 8(1 - v)(2 +:—) uE - 2(1 - 2v)(4 +-1rl—) u 0 O N h h +4+(l-6v)—-w—4+(l+2V)—]W~ [ r0]0[ rO Nb - [4(1 - 4v) + (1 - 6v) 1:3wa + [4(1 — 4v) + (1 + 2v) 26]wa O r2 E h h2 + (l+v)(1 _ 2v) §[8(3 - 4v) - 2(5 - 14v) r—+ 8(1 - v) —-]u0 O - 8(1—v)(2-:—) uw-2(l-2\))(4-:;—) u 0 O N h h _[4_ (1—6v)?;]w0+[4- (1+2V) F6]WNW +[4(l - 4v) - (l - 6v) $3] ww - [4(1 - 4v) - (l + 2v) 2;]wa 2 RE h h + (1 + V)(l- 2V) {[8(3 " 4V) - 2(5 - 14V) ;(—)'+8(l - V) :]u0 -8(1-v)(2-$—) uw- 2(1-2v)(4—-11-1_——~)u O O S h h + [4 - (l - 6v) ?(-)-JWO -[4 — (l + 2v) :8]WSW - [4(1 - 4v) - (l - 6v) %(—)-:lww + [4(1 - 4v) - (l + 2v) $3]wa . .__.__— o . . - V I i r Y 44 O r E h 1.12 + (1 + v)(l _ 2v) [[80 - 4v) + 2(5 - 14v) I.— + 8(1 - v) 7] 110 O h h - 8(1 - v)(2 +?6) uE — 2(1 - 2v)(4 +?(;) uS -[(4+ (1 - 6v) £—]w0+[4+ (1+2v) E—JWSE O O +[4(1-4v)+(1-6\)) h—Jw -[4(1-4v)+(1+2v)D—wa rO E rO S _. 2 - 32h RO (2.63) E 8(3-4v)+4(2-3v)9— w -4(1-2v)(2+9—)w (l + v)(1 - 2v) r0 0 r0 E h h- - 4(1 - v)(4 +—r—(;) wN+[4 + 2(1 - 3v) EJUO - [4 + 2(1 + v) g—J uNE + [4(1 - 4v) + 2(1 - 5v) %-] uE O O - [4(1 - 4v) + 2(1 - v) %-]uNf 0 E h h + (1 + V)(]. _ 2V) g[8(3 - 4V) - 4(2 - 3V) r—OJWO - 4(1 - 2v)(2 - 11;) ww h h - 4(1 - v)(4 - QWN - [4 - 2(1 - 3v) r—O]u0 F b ' h + 4 - 2(1 + v) gluw- [Ml - 4V) - 2(1 - 5v) 'r—O’Juw r + L4(1 - 4v) - 2(1 - v) 327—]11Nf O rllll‘h L. . 7‘ .4 a A.» 5 a .43 a ,. - P~1 fa» " A y .0 . .0. a . 45 E + (1 + v)(l - 2v) W h h [8(3 - 4v) - 4(2 - 3v)-;;J‘w0 - 4(1 - 2v)(2 — ;S) w h h -4(1-V)(4-¥) WS+[4-2(l-\)) §]u0 h h - [4 - 2(1 + v) $6.]USW + [40. - 4v) - 2(1 - 5V) EJLLW h - [4(1 - v) - 2(1 - v)-¥S] uS E [8(3 - 4v) + 4(2 - 3v) g—{lwo - 4(1 - 2v)(2 + 2—) WE + (1 + v)(l - 2v) 0 O h h -' 4(1-V)(4 +E‘S) WS - [4 + 2(1 - 3V) a] [10 h h ' + [4 + 2(1 + v) 1:6]uSE - [4(1 - 4V) + 2(1 - 5V) r—O'JUE + [4(1 - 4v) + 2(1 - v)-%—] uS = 32h2ZO 0 For an ordinary mesh point the bracketed terms on the right hand sides of equations (2.63) can be added to obtain a more simplified form. h2 h h 32[(3 - 4v) + (l - v) -—{]u0 - 16(1 - v)(2 + :fDUE - 16(1 - v)(2 - ;f) uw 2 r0 0 O - 16(1 - 2v)(uN + us) + 2(1 + 2v)-%— (wN - wS) - [4 + (l + 2V)%—]WNE 0 O +-[4 - (1 + 2v) g—' h 0 WNW—[4 - (1+ 2v) TJWSW O + [4 + (1 + 2v) %—]WSE = 32 (1 + “I? ‘ 2V) h2R0 (2.65) 0 -- 1 4 “- ‘ .-4 4. -“ ‘- . ~ .' v. tv . Aa.fl:' , 4.. . .. ‘u" — y 4 fan 4.. - (. A - ‘ l .1‘ “I’ it -. ‘ 39‘ .. 46 32(3 - 4v) wo - 8(1 - 26)(2 + E—) wE - 8(1 — 26)(2 — 4}) w 0 O W -32(l - v)(wN + wS) - 4(1 - v) {-E—(uN - us) - [4 + 2(1 + v)E—]UNE O O + [4 - 2(1 + v) 2—- NW'— [4 - 2(1 + v) %-] usw O O |.____J C +-[4 + 2(1 + v) %—-]uSE = 32 (l + V>él - 2v) h2ZO . . l . For a particular ch01ce of P01sson's ratio, say v = Z3one obtains (16 + 6-23) u - 3(2 + h—-) u - 3(2 - h—) u - 2(u + u ) 2 o r E r N N S r O O 0 3h 3h 3h + 4 r (wN _ WS) - (1 + 8 r ) wNE + (1 -.§ r ) wNW 0 0 0 3h 3h __ h2 -(1-—§——r) SW+(1+§-——r)wSE-5—E R0 (2.66) 0 0 16 -(2+-h—)w —(2-1—) -6( + ) wo r E r ww wN wS 0 0 3h 5h 5h ’Zr (UN—US)-(1+gr)UNE+(1-§r)uNw o o o - (1 - é-h- u + (1 + 2-E—) u = 5 DE-Z 8 r0 sw 8 r0 SE E 0 Now comparing these equations with equations (2.40) in section 2.7 it becomes apparent that the two sets of equations are not identical. However, if one makes the additional assumptions that \. u. . .4 .. .. I. . .3 x . . . L. L . . . "6. .mL. 4 n a .1 .4 g V ."< s. .. .u . . . . c L. 6,. 4 4 \ . . . . v . 4 . . . vi Cm Z. «t .14. ‘ 3‘ . .1 .a .t I? .4 . . .r. g. .‘x a 6 . .. 2‘ >4 .4 ~ .: . .. ... ... . v. .~ a“ m. .1 . ... >4 .. FL .1 .~ .. .. 4 . .1 ... .. . . . mu 6.. .3 .1 0L .C 6 5 LL :C Wu. K . 2. .. a u 7 PL .6 4 .9» 47 NE + NW SE + SW NE + NW SE + SW equations (2.66) can be shown to be identical to the previous results obtained from the equations of elasticity. It is unnecessary to do this, however, since equations (2.65) or (2.66) simply represent a slightly different set of finite difference equations which should give equally good results. These equations have been used exclusively throughout this investigation. In the previous section which dealt with plane stress problems, a certain amount of detail was included with regard to boundary condi— tions. It is possible to do much the same thing here for points on boundary surfaces of the solid under consideration. However, as these equations have become more involved, it is advantageous to let the com- puter do the calculations rather than derive explicit expressions for boundary points. Thus equations (2.63) or their equivalent are pro- grammed with each bracketed term a subroutine. Equations for the inter— ior and boundary points of the problem are generated by calling appro— priate subroutines. For example, for a point on an outside corner, only the first bracketed terms are required in the two equations (2.63) to form the left hand side of such relationships. Thus only the subroutine which computes the coefficients corresponding to the first bracketed F.‘ -v'- .A: 0.... a l \- «nr- O~P. \ 1. _...-4». .-. - . O~ o. A. r- b- iOv-o. ‘7 I- u A.) .1». _'_‘ ' ’ b.- .0 s... ‘ f~au5 ; Oh. ' V - s “‘5 . \ 4.. 1._ ‘,, . ~t:..‘--. :-.. - .‘- ‘A ‘\u . ? “'C - , c ., t I”; yL1‘t 1 ‘M‘S A. '4 - . ;w. . . . t53~ ‘u r p. 48 terms would be called. In the case of a typical mesh point, each of 4 subroutines must be called to generate the coefficients since all bracketed terms of equations (2.63) are involved. In a like manner, one is able to treat composite materials. As shown in Figure 2,14, the region E=5X105psi N around a particular mesh point in such a composite may consist of two different materials. Finite difference equations for such a point are obtained by calling the 4 subroutines discussed above but in this case using the appropriate elastic constants for Figure 2.14 the 4 different regions around the Composite Material R9810“ mesh point 0. 2.10 Finite Difference Stresses As discussed earlier, the finite difference equations derived by the alternate method were used exclusively in this work. The interior and boundary point equations form a system of equations which is solved by standard numericalprocedurestx>yield the displacements for the mesh points of the problem. The stresses corresponding to this alternate procedure are expressed by equations (2.42) through (2.45) for plane stress problems and equations (2.59) for axially symmetric problems. It is necessary to exercise care in the interpretation of these stresses. The reason for .. ., ..""- A4. bbtnfi—rv . "“o\ D In H» . “v g.__' .“ '~cu. . - ‘. "‘v-§ ' -.‘_‘ .‘ :I. ..‘ \ " MIC _ E “9." - ‘ .2 ,. ‘41.... . ‘h‘ ._>- -..: _ : . 49 this rests with the fact that a given region is generally associated rSE SW with 4 different sets of stresses . Typical corresponding to the fact that it Region pertains to as many as 4 different mesh points. Thus for example, bNE NWJ the material region in Figure 2.15 1 2 is simultaneously the NE region Figure 2.15 for point 1, the NW region for Material Region point 2, the SW corner for point 3, and the SE corner for point 4. It is, therefore, suggested that a set of mesh point stresses be defined in terms of these material element stresses. For this purpose, the mesh point stresses are assumed to be the average of the stresses in the material regions around the point. Previous experience suggests that this averaging technique would give good results at interior points but perhaps less satisfactory results for boundary points. -....; P. ,- 7.....‘-‘ . ' . "~n \ ':":~‘..L L... “-A u: IaJi ..-C._ '~ ~ -. >- “‘.‘ "- .>-"‘ Jts._.\. .- D 4 ‘Q l . ,v ‘ ~ ." L 9- s ‘v i I u. ,- c,‘ h - 5‘: ‘4‘ h". \ ‘. u 7 ~\ ‘ :-, a“, p-. 5 c. F;' ‘s . ‘\ III. FINITE ELEMENT METHOD 3.1 General Remarks Finite element methods represent a large class of approximate procedures in structural mechanics. Basically, these methods involve replacing the actual structure or continua by a model consisting of a finite number of carefully selected elements which are connected at a finite number of points. The approximation is thus a physical one in constrast to the mathematical approximations of finite difference methods. Finite element methods can be classified according to the behav- ior of elements in the model. A compatible element or a displacement model is one satisfying compatibility but not equlibrium. An equilib- rium element or equilibrium model is one which satisfies equilibrium but not compatibility. A mixed model is one satisfying neither equilibrium nor compatibility. These classifications are discussed in detail by de Veubeke [34]. One of the chief purposes of finite element methods in structural analysis is to develop relationships between generalized loads and generalized displacements through the elastic and geometric properties of the element. A second classification of finite element methods is based on these relationships. In one approach, this is done through the so called flexibility matrix by which generalized displacements are derived from generalized forces. The other approach derives the 50 v‘ I 11‘ Fn .2 .. a .. a . .t >\ , . 1... 6 . . .. . . ... t .t C. .1 _. 4: s 5 ax .\ re .3 s”. I . .‘.. J.‘ O \ s~ fins. 4 .§ be “P“ .P‘ Lu .4“ L; .\u n\. Ht . 1p. 6 . . : «N. . 4 a _ 4v. 5 u. , u . n . . . H. .. . ml .T .. p“ “I u a 4 .C o ., . . . .va .4 M G. 42% 2541“ .r\ Hi > 1 u 4 I .4 . .im .W. AN" 3.. ._ .. 1.. .v‘ .9. In a: .v x .5 s v 4 .qn —. n. .6 s vs u .u .fi .h. n. o. .v: .H v. .u. He. 4. .. ~p. nu . n. in .f r‘ on v. .91. q .. I .. . 51 generalized forces from the generalized displacements through the appro— priately named stiffness matrix. A third classification of finite element methods is based on the method of solution. The matrix displacement method treats displacements as unknowns whereas the matrix force method treats forces as unknowns. A mixed method of.solution is also possible with some unknown forces and some unknown displacements. As pointed out by Felippa [18], the direct stiffness method is perhaps the most powerful and fully developed of the finite element methods. The direct stiffness method employs a displacement model and treats displacements as unknowns. The word direct is used to indicate the way in which the overall structural stiffness matrix is assembled from the individual element stiffness matrices prior to imposing dis- placement boundary conditions. Thus in the direct stiffness method, the overall stiffness matrix is obtained by a simple systematic addition of element stiffnesses. Argyris [27] and others accomplish this by means of transformations involving location or "Boolean" matrices. The latter procedure seems to be less efficient and thus has been avoided by many investigators. In the present investigation, the direct stiffness method is used exclusively. Thus no further reference will be made to other finite element methods. Furthermore,the philosophy of the method has been thoroughly treated by a number of writers [35] so that only a brief description of the basic steps will be presented here. The specific discussion and examples will relate to two dimensional problems and triangular elements. It is a relatively simple matter to extend the work to three dimensional situations. _ _ . . .6 A .n q_ .\_ . ‘ Op” - o u u a . .4. .r. . .t .1. . s .h. Qx . — n u .C .C L. . . .0 o p , . c 1. . . . u . .p. .ps :- —u *5 IE .. .3. .t .u ‘1 ; . .t I Q u. H. :— u.‘ Q Q .4 . .p. 52 3.2 Direct Stiffness Method The behavior of the actual structure or continua is assumed to be approximated by a discretized structure consisting of an assemblage of carefully chosen elements connected at a finite number of points. These may be beam elements in the case of frames, triangular plane stress elements for certain two dimensional continua, quadrilateral plate elements for plate flexure, or tetrahedra in the case of three dimen- sional continua. Other physical problems may dictate use of still different elements or possibly combinations of these elements for truly complex structures. Inherent in the procedure is the assumption of element displace- ment modes. These displacement modes must satisfy internal compat- ibility and should insofar as possible maintain compatibility of dis- placements across element boundaries. The number of displacement modes used must agree with the number of degrees of freedom of the element nodal point system. Thus for a plane problem in rectangular Cartesian coordinates, with n nodal points, the equations 2n 110930 = X Ui(X.y) ai i=1 (3.1) 2? V(X.y) = V (X.y) a 1:1 1 1 define the assumed displacement field in terms of independent displace- ment functions U1 and V1 and generalized displacements ai. These can be expressed in matrix notation as well. 53 mm = tu1t [a] (3.2) mm = [V(X.y)]t [a] As stated above, the dimension of [a] is 2n and it agrees with the number of degrees of freedom for the element nodal point system. The next step is to express the nodal point displacements in terms of the generalized displacements. This is done by evaluating (3.1) or (3.2) at the nodal point coordinates. 2n “1 = 1:1 Ui(xjo yj) 0-1 j = l, 2, . . . n (3.3) Zn v3 = 121 V1 (xj, yj) Q1 The matrix notation is more compact and allows the Zn equations of (3.3) to be written as [U] = [A110] (3.4) where [u] and [a] are column matrices defined by = < 111 u2 . . . . u v v . . . . v > (3.5) (3.6) The matrix [A] is a square matrix whose rows are formed by evaluating tflle assumed displacement functions at the nodal point coordinates. Thus .: .. . .Fe L. «I. .5 an chi .- Q 5... A... n a... 54 010:1. y1)U2(x1. yl) . . - . . . U2n(x1. yl) Ul(x2, y2) U2(X29 Y2) 0 ° 0 0 0 0 U2n(x2’ y2) [A] = U1(xn) yn) U2(xn’ yn) ' U2n(xn’ yn) (3.7) Vl(x1, yl) V2(x1, yl) . . . . . . V2n(x1, yl) 3 Vl(xn. yn) V2(xn. yn) . . . . . . V2n(xn. yn) ’ The generalized displacements are then expressed in terms of the nodal point displacements. This is simply an inversion of equation (3.4). [a1 = [A‘lnu] (3.8) In the next phase, the strains and stresses are evaluated. The strains are given by the matrix relationship [6(X.y)] = [D(X.y)][a] (3.9) Where [C(x,y)] is a column matrix defined by H9 1. u a .. can..- . . V Ur a. ‘Y; 'C ‘ n o w ‘w. 5.)! . s 55 [e(x.y)]t= (ex ey yxy> (3.10) and the matrix [D] is formed by appropriate differentiation of the dis- placement functions U and Vi' For example, in the case of plane stress 1 or plane strain 3U1 8U2 auzn 3x 8x 8x 3V 3V 3V 1 2 2n GUI 8V 3U2 8V2 8U2n + 3V2n L-By 8x 3y 8x 8y 3x The stresses arise from the constitutive relationships and can be written in matrix notation as [0(X.y)] = [C][e(X.y)] (3.12) or in view of equation (3.9) [0(X.y)] = [C][D(X.y)][a] (3.13) In the above, the matrix [C] is the matrix of material properties and [0(x,y)] is the matrix of stress components given by [0(X.y)]t = ( ox(X.y) oy(X.y) Ixy(X.y) > (3.14) - I ~ I A F . , . . . r . 4 . n». .: . a .u- 0 5 c a I. . a o .n. O s Tlu .: “a .fi. h L. up; .I .. . a. u A... .c .3 .. . .u» a .. .4. v n r . .. . n u I u 6 a .6 u t .41: n it o s 4 a n u. 56 It is quite possible to treat very general material characteristics including orthotropic elasticity and elasto-plasticity. For isotropic plane stress the matrix [C] becomes _ l v 0 T E [c] = —————-—— v 1 0 (3.15) l - V2 1 — v L_0 0 2 . where E is the modulus of elasticity and v is Poisson's ratio. A generalized coordinate stiffness matrix, [kc], is derived from the principle of virtual displacements. It is necessary to equate the virtual external work to the virtual internal work. Associated with a set of virtual displacements 6u(x,y) and 6v(x,y) are virtual strains [5E(X,Y)]. The virtual internal work for a differential volume, dV, in the element is d<6wi> = [68(x,y)]t [0(X,Y)] dv (3.16) Using relationships (3.13) and (3.9) this last result becomes d(6wi) = [ant [D]t [C][D][a] dV where [do] is the column matrix of virtual generalized displacements associated with 6u(x,y) and 6v(x,y). The total internal work is the volume integral of the above expression awi = J [éalt [01‘ [C][D][a] dv vol . N ..u~: lp“ - .‘. . .v v. hubt‘. . ~ s..- .-‘ In» L‘” 11" '5‘ (IV v Hg. .‘n , 'v-E .-. .. I-An. 'l H ‘WI‘; ' IT' ' 1 ‘Su V"; 5'. . .» 57 Since [a] is independent of position éwi = [6a]t I [D]t [c][n] dV [a] (3.17) vol Now associated with the generalized displacements [a] are generalized forces [8]; the product of the generalized displacements and the generalized forces yields external work. Thus the virtual external work is awe = [6a]t [s] (3.18) Since the virtual internal work equals the virtual external work during any virtual displacement, it follows that [éalt [a] = [dalt I [01‘ [c1191 dv [a] vol However, virtual displacements are arbitrary displacements consistent with the kinematic constraints. Thus the above equation implies t [B] =[ [D] [C][D] W [a] (3.19) vol From the definition of the stiffness matrix [ka], that is [8] = [kalta] (3.20) one may conclude that [k ] = I [D1t [c1101 dv (3.21) a vol . O. AD‘ I my area. '33:; "..' “"' —~‘. ~..;,: “--. . y“ ‘H..-._ ~ - . ,, 4.. ‘ . ‘ “'fna'JC QA, 58 The nodal point stiffness matrix [k] relates nodal point forces to nodal point displacements. In view of equation (3.8), the matrix [k] is derived from [km] by a standard coordinate transformation. [k1 = [A‘llt [ka][A-1] (3.22) Having given a general procedure for working out element stiff— ness matrices, attention is next given to the problem of assembling the overall stiffness matrix of the discretized structure. In the direct stiffness method this is a fairly routine procedure. Involved is the concept that the overall stiffness matrix relates applied loads at the nodal point of the assembled structure to the resulting nodal point displacements. A particular coefficient of this matrix associated with a specific nodal point of the complete structure is the algebraic sum of corresponding stiffness coefficients of elements which have this same point a nodal point of the element. In other words, stiffness coeffi- cients of the complete structure are obtained by summing the stiffness coefficients of elements surrounding a particular point of the complete structure. This is an essential feature of the direct stiffness method. It allows the structural analyst to identify and store only the non-zero stiffness coefficients of the complete structure. This is a signifi- cant achievement since stiffness matrices are generally very Sparsely populated. By storing only the non-zero coefficients, one is able to consider much finer discretizations of the actual system. Not to be overlooked, however, is that prior to adding these element stiffness coefficients they must all have been referred to the same global coordinate system. ..1 n nebso -- a \ gaadnnn H I 'l. to .,: yo n-‘. " :05 4.A5auL .. 4...: 0-. u... ‘ my. U) n- ,. ~- u,‘“. .1. (II 59 The next part of the analysis is the determination of element nodal point forces. These are generally the result of internal stresses arising from temperature changes or perhaps imperfections, body forces, and surface tractions. Although it is possible to account for each of these, for example as in reference [25], only the nodal point loads arising from surface tractions are considered here. As pointed out by Archer [36], these forces must not only be statically equivalent to the distributed boundary forces but they must furthermore be kinematically consistent with the assumed diSplacement field corresponding to elements on the loaded portion of the boundary. Thus it is required that the virtual work done by the actual loads, be the same as that done by the nodal point forces. For convenience, the boundary tractions are considered to have components in the coordinate directions. These are designated by p(s) and q(s) with p(S) the x component and q(S) the y component. It is assumed that these have been integrated over the thickness. During a virtual displacement, the work done by the actual force system is 6W = I [p(S) 6n + q(S) 6v ] ds (3.23) 1 B B B where uB and VB are boundary displacement components of the loaded portion of the element. In view of equations (3.2) these boundary displacements are C. II [uBnal (3.24) 4 l - [VBlla] .bs I I .r. w. h . . , . . V. .3 Vt Le .- . .. . .uc 4A :- 0 I n‘. n Q Q o .. : -..... .. a ,. o u an .‘n .n i . .uu . 05 n . A» II: o 9 a .\v a _ 60 where [UB] and [VB] are row matrices of displacement functions which have been evaluated on the loaded boundary. Thus equation (3.23) becomes 6w1 - [B < p(s) [U31 + q(s) [v3]) ds [6a] But the generalized displacements [a] are related to nodal point displace- ments by equation (3.8). Writing the above virtual work accordingly -1 5W1=J (p(s) [UB] + q(s) [VBD ds [A ][6u] (3.25) B It is required that equation (3.25) be equal to the virtual work of the statically equivalent nodal point forces acting through the same virtual nodal point displacements. The work of the external nodal point forces is éwz = [th [(Su] (3.26) where [f]--- as (3.27) B A more direct approach to obtaining these loads is to express the boundary displacements in terms of nodal point displacements. “H- V . ~cl‘v‘ ’ ._ lt..~.: - . I- "a v-A ON- 61 In general u(x,y) = 1M (3.30) [u]t= The matrices [M] and [N] consist of weighting functions which relate element displacements to nodal point displacements. As used in (3.29), these weighting functions have been evaluated on the loaded boundary. The virtual work of the boundary traction is then 6W1 . [B p(s) [MB]t [dux] ds + I q(s)[NB1t [Guy] ds (3.30) B Furthermore, the virtual work of the nodal point forces is 5W2 - [ix]t [Gux] + [fy]t [any] where Z . i . . u. . .1. I 1 4 .1 1L Ill. v ¢ . _ a r r; A\ 9L 1 .t S .3 . . t .3 « s? 51w “w .m. ..... 1‘ E ah. I- I L I .K a .n: . a» o\ «MW HIM it q .t :1.- \ In tam V .11» 1'1 2 u ‘ .. C . .o s .. r 1. 3.: an ~\.. ). . an t e .. A . .6 .1 .3 .u. a b . sin y a u s .3. R1. .9 s A: .F» t 5 ul- \ t 111- .IIl-l— ‘- “ w: 62 f [f1-[ x] f Y Equating owl and 6W2 and noting that [éux] and [Guy] are arbitrary, one obtains [fx] [B p(S)[MB] ds (3.31) [fy] [B q(s)[NB] d8 The nodal point forces for the complete structure are clearly the algebraic summations of element nodal point forces. Thus if several elements join at a boundary point j, the nodal point forces of these elements which correspond to j are added to obtain the nodal force of the complete structure. Having assembled the complete structural stiffness matrix [k] and the nodal force vector [F] one is led to the matrix relationship [F] = [Klls] (3.32) In equation (3.32) [s] is a column matrix of nodal point displacements of the complete structure. In two dimensional problems the dimension of 1%] is twice the number of nodal points of the assembled structure. The object now is to determine these displacements. This requires the inversion of equation (3.32). Since [R] pertains to the unrestrained structure, it is a singular matrix and thus cannot be inverted until kinematic constraints have been imposed. This renders the structure externally stable and has the effect of reducing the size of the --ov‘ ~...- 5- V- 63 matrices in (3.32). One may then write [F] = [Klls] (3.33) where [K] is non-singular. The solution is 1s] = no"1 [F1 (3.34) and it is obtainable by various inversion techniques. The use of the Gauss Seidel successive over-relaxation technique is discussed by Clough [35]. A modification of this procedure has been used throughout this investigation. A simplification of Gauss elimination applicable to band matrices has been discussed by Tocher [37]. The final step in the analysis involves the calculation of stresses. As indicated by de Veubeke [34], the best stresses in a dis- placement model analysis are those derived from the displacement field. In the case of linear displacement fields, however, care must be exercised in interpreting element stresses. Wilson [38] has proposed an averaging technique which appears to give fairly good results. Accordingly, the stresses follow from equation (3.13). That is [0(x,y)] = [C] [D(x,y) ] [a] The generalized displacements [a] are eliminated using equation (3.8). Thus element stresses are expressed by the relationship [o(x.y11 = 1c110fi - ) \(16 l (3.37) (011) 2 v(x,y) a [v(x,y)]t [a] = < O 0 0 l x y > < . g 105/ Evaluating these at the nodal point coordinates gives the specific form of equation (3.4). u, '1 o o o o o7 011 / \ u2 1 x2 y2 0 0 0 a2 u3 = 1 x3 y3 0 0 O < a3 g a v1 0 0 0 l 0 0 a“ (3.38) K v2 0 O 0 1 x2 y2 \ as / v3 L0 0 O 1 x3 y3 as or In] - [Alla] The inverse of [A] is easily obtained by partitioning the matrix. Thus 7:6 29': . '3.). 4.. 13A,: 1‘, I ‘3: 111:5 x2y3'x3y2 0 yz-y3 Y3 -l [A ] = -—-————- x -x —x ny3 x3y2 3 2 3 0 O O 0 L O O The matrix of strains (3.9) is an ‘ 3x ['0 1 3v 1e1= 3; )= o o Bu 3v 3§-+ 5;") L_o o and thus ' 0 l 0 [D(X.y)] = 0 0 0 L 0 0 1 66 o o o 0 o o x2y3-x3y2 O yz-ys y3 X3-X "X3 0 o o l 0 o 1 o 1 o . W o o” o 1 1 o] 0 0 0 0 (3.39) _y2 x2 “1 92 1% (3.40) The matrix [Q] of elastic properties is (3.15) for plane stress problems. For plane strain problems [C] ' (1 + v)(1 - 2v) E (3.41) at" 51:: I n . In“ p H' 'J‘h‘a, I! \ 67 The generalized coordinate stiffness matrix becomes 111-] 1111‘ 11211111 1v 0 vol and since [D] and [C] are independent of position Ix y - x y Ih 110,1 - 1111‘ 1c11111 [ 1 av = 2 3 2 3 2 1111‘ mm V0 In the above, -% |X2Y3 - x3y2| is the area of the triangle and h its average thickness. The nodal point stiffness matrix is lx2y3 ' x3Y2] 1 ‘ 1111‘ 1c111111A’ 1 (3.421 1111 - 11 11511 with the required matrices given by equations (3.39) and (3.40). The elements of [k] can be written out explicitly from (3.42) without too much effort. The final result involving 36 coefficients is presented in reference [14]. In other situations involving more complicated displacement fields this becomes impractical. In some cases it is pre— ferable to work out the matrices of (3.42) or its equivalent and to the perform the matrix multiplication on the computer. As an example of nodal force calculations determined from boundary tractions consider the situation illustrated in Figure 3.2. Shown in the figure is a portion of the curved boundary with a normal traction p(y). The triangular element 1-2-3 can only approximate the boundary along 1-2. Clearly, only x-components of nodal point forces are involved in this example. The first equation of (3.31) is v. | ° I .' a I . ' F l ..‘.., 1.5.6 ..- . u ‘W '-e ..A in D" U»- 1......1"c ‘b‘fibqv u- 31011 D:.‘-£ u~ 5.d“~ "1 . u. ""1-.. tn: n‘» 7:115 68 11x1 - I p(y) 1MB] ds B where [MB] is the column matrix of weighting functions evaluated on the boundary. To obtain these 3 functions consider the first of equations (3.36). u(x,y) = al + azx + 03y Figure 3.2 Normal Boundary Traction-CST The boundary here is identified by the equation x = 0. Thus U = U(O,Y) a (I B + 03y (3.43) 1 Evaluating the latter at points 1 and 2 on the boundary gives -h 111 =01 +a3( ‘2‘) u - a + a (—) These last equations can be solved for al and dB to give 4. 11 U2 'n..;O‘ av- HVOGQQUIC . : 251:1 ”:2 I4. ‘ a")- :36 r 69 Equation (3.43) can then be written as -_l.__x l x u13'(2 h)“1+(2+h)“2 The matrix of weighting functions is 1111t ll ./\ NIH _.X h o > r~:1|1—-1 + =1'1~< B Then a 2 NIH + =~1~< 2 [fxlt [1 M10 [MB]t ds = p(y) (%_% L1 2 and the result is E 1 f1 = [2 (§-%)P(Y)dy x 1 "2 h 12 = F (%+{’;)p(y)dy x -2 2 f = 0 For the special case p(y) = p, a constant =93 f1 2 x gm!— f2 2 X Also, if p(y) is a linear variation expressed by (3.44) (3.45) 0) dy (3.46) (3.47) 3‘ is {he 70 P + P1 P2 - P1 p(y) ' —3—§———' + -—-——- y one obtains h fl 6(92 + 2P1) x (3.48) h f2 . 3(21’2 + pl) x y, if p(y) is quadratic with the form 9 - p p + p - 2p P(Y) = P + ‘3————l' + l 2 0 y2 where 0 h 2 h 2 the load intensity at the origin h fl =-g (29, + 9,) x (3.49) h fzx - ‘6' (Zpu‘i' P2) inearly Varying Strain Triangle In order to maintain compatibility between triangular plate ts and beam segments, aeke [34] introduced a lized plane stress element involves a quadratic :ement field. Such nent is displayed in 3.3. The origin of Figure 3.3 1ates is placed at point 1 Linear Strain Triangle 1 loss of generality. .- . . If “a“: in: 51.5». 05.1.; o.« A btlb ~- Av.“_"‘. v-.d.utu W» ‘ .. :‘?5. (E ‘5” t 71 lal point system includes the triangle vertices and the midpoints triangle sides. The quadratic displacement field is expressed by the relationships 2 2 = + + + u(x,y) a1 ozx 03y aux + osxy + a6y (3.50) v(x,y) = a + a x + agy + a x2 + allxy + any2 7 8 10 dal point displacements follow from the matrix equation T... A- .— .01.. [U] = [A][a] ed by evaluating equations (3.50) at the nodal point coordinates. tter can be written as ux A11:O 1 ox = ’ (3.51) ' uy 0 'A22.J oy [u ]t . < u u u u u u > x 1231156 ]t [u = < vlv2v3vuv5v6 ) L's: [A ‘ne 13.1": ' Ale utr V ,-, A ”a "at-"339m h ‘Rn. I‘cd‘t: : to write 72 Also [A22] = [All] with ’1 o o o o o = 2 2 [A11] 1 x2 y2 x2 x2Y2 y2 2 2 1 x3 y3 x3 x3Y3 y3 x2 y2 x2 x2Y2 y; 1 ‘2" “2" z— 4 .— 1 xs'xz y3-y2 (”‘3""2)2 (X3'X2)(y3'y2) (3'3'3'2)2 2 2 4 4 4 x3 Y3 (5'3’3’2)2 x3Y3 Y3 1 T 2— 4 4 T J . The inverse relationship of (3.51) is (3.53) It is not suggested that an explicit expression for [A_1] be worked out. The matrix [A—I] can be determined numerically within the computer Program by simply calling a matrix inversion subroutine. Another alternative is suggested by de Veubeke [34] whereby displacement components u(x,y) and v(x,y) are expressed in terms of nodal point dis- Placements through a set of 6 weighting functions. Thus it is possible to write u(x,y) = ulwl + uzw2 + u3w3 + 111qu, + usw5 + uew6 v(x,y) =1 vlwl + vzw2 + v3W3 + vkw1+ + vsw5 + v6W6 The “1. 1 = l, 6, are functions of x and y. For further details, the reader is referred to reference [34]- 1111.11. 1.15 the [3111,11] Y ‘ [he h._a: 53635 ‘1le The matrix of strain components follows from the partial derivatives of equations (3.50). [8(X.y)] = Lu 3y a]. 8x 73 Thus r0 1 0 2x y 0 0 O O 0 L0 0 l O A x 1 0 2x y 0] and the matrix [D(x,y)] corresponding to (3.11) is [D(x,y)] - 0 0 0 0 0 x 2y 2x y 0 (3.54) The matrices of elastic properties are (3.15) and (3.41) for plane stress and plane strain respectively. The generalized coordinate stiffness is 11,1 = [vol 1111T mm «W (3.55) However, in this case, the integrand is a function of position, as in equation (3.54) . The calculations thus become rather involved. At this Pointz1a more general matrix of linearly elastic properties is intro— duced . Thus [C] - allows arbitrary anisotropy for the two dimensional case. — C11 C21 C L 31 12 C13 22 C23 C 32 33 With (3.56) this , . v: . A' a. u '74 value of [C], the integrand in equation (3.55) can be written as where 111,11 - 1Q“! - 111,21 - Q11 Q 1111T 1c11111 - 21 [o o o 0 C11 C13 0 C31 C33 0 2xC11 ZxCl3 0 xc31+7C11 yC13+XC33 [o 2yc31 2yc33 [o o o 0 C13 12 0 C33 C32 0 2xCl3 2xCl2 0 yc13+XC33 yC12+XC32 [ O 2yC33 2yC32 F 0 u 0 u as] L1, 0 21 ‘21 0 ZXCJl 2XC33 0 xc21+7C31 xc23+YC33 L O 2yC21 2yC23 0 C33 C32 0 c23 C22 0 231C33 211C,2 O xC23+7C33 xC22+7C32 _0 2yC23 ZyC22 Q12 Q22 0 2xCll 2xC3] 2 4x C11 2 2xyCll+2x C3, 4xyC31 0 2xCl3 2xC33 2 4x C13 2 2xyC13+2x C33 4xyCn 2 4x CH 2 2x C21+2xyC31 4xyC21 2 4x C33 2. 2x £23+2xyC3, 4xyC7, (3.57) 0 O yCll+XCH ZyCH yC31+xC33 2yCH 2 2xyC11+2x C13 4xyCH 2 2 2 y C11+xy(Cl3+C31) + x C53 2y C13+2xyC33 2 2 2y C31+2xyC33 by C33 J 0 0 xC12+yC13 2yC12 xC32+yC33 Zyc32 2 2x C12+2xyCl3 4xyC12 2 1 2 2 x C32+XY(L12+C33) + y c13 2y C12+2xyc32 2 2 2xyC32+2y C,3 4y C3? (3.58) 0 I) yC,l4xC.‘ Zyugg y(la l+x(:', ‘ 2y(17 1 . . 2 1 2xyL§l+2x C3, 4xy(.H 2 -- 1 2 x2C23+xy(C?)+C33)+y C31 2xyC21+2y C33 2 2 2y C21+2xyC2, 4y C7, 0 0 xC32+yC33 2yC32 xC22+yC23 2yC22 2 2x C32+2xyC33 4xyC3? 2 1 2 2 X C22+xy(L23+C5?)+y C31 ZXYC23+2Y C32 2 2‘ 2xyC22+2y 023 4y £92 I r " X‘f :. ‘ . I I :. I 13 Ea. -- trialg; Eatflx 75 Each of these coefficients must be integrated over the element volume. What appears to be a very formidable problem is not as bad as it would seem. Actually there are only 6 different integrals involved. These are the following: X ‘23 II 3‘ (x2 + x3)h UH>1 (yz + y3)h ofl> (3.59) 17 I J 1 I I 1.3.1.1.}..2M1W1H f I f In each of these expressions, h represents the average thickness of the triangular element and A is its area. Thus the generalized stiffness matrix is of the form ' dV 2 dV [k] g I [311+312 _J dv =[IQl_1_d__:_[Q12d__J vol 1 Q21 V 1 Q22 V The nodal point stiffness matrix is obtained by the coordinate transformation (3.22). Thus 1 . ’P"!Ii~ '\ uteeib- ~.'~74. .- ‘I-‘v an... 1.2% 3;; : I‘d -.,. V‘- L».- . 1 a-.. .4. iv“ ..‘: 1v. '1 “:9. .‘.."L. - .4 , ‘ .v a “=“ ,4. u: r-v :' IT‘ (I 7-“ ,-‘ LA) - LII [he 501 This same matrix is presented by Fellippa [18]. He derives the matrix in a very elegant way using the novel idea of area coordinates. It can be expressed in an integrated form as a result of the approach taken. Its form is readily adapted to computer programming and consequently was used by the present author in this investigation. A typical example of the determination of nodal point forces is considered next. The situation is illustrated in Figure 3.4 where ‘1' p(y),( a distributed load or traction ,’ acts normal to the boundary. The triangle side 1-4-2 of length h only approximates the curved boundary. The origin of coordinates is the mid point of this side. From the first equation Figure 3-4 of (33.50) Normal Boundary Traction-LST u(x,y) a1 + 02x 03y aux asxy 06y The boundary in question is characterized by x = 0. Thus 3 2 uB a1 + a3y + 06y (3.61) ‘ I - ...,.1 ”Us“; ' 1 .." “‘l“.‘ (A! 77 The boundary nodal point displacements are -h -h 2 u1 - a1 + a3 ( 2) + 06 ( 2) u“ = o1 h h u2 - 01 + 0‘3 (2) + a6 (2) Solving these equations for al, a3, and as one obtains 1 4 _ u2 ’ u1 a3 - h 1.11 + u2 - 2uu a6 = 13 2 Intrrnducing these results into equation (3.61) yields + (.gXE.+ Z. 2 2 4 2 “B = (.1. _ z_)u1 1 (1 _ _z_) u h) “2 h2 h hZ “ h2 Thus the matrix of weighting functions is 2 2 2 m§‘=(bL-x 9L+% 0 1_fl_o o) h2 h h2 h2 Then from the first of equations (3.31) [31‘ = [B p(y) [1131‘ ds (3.62) (3.63) 78 one obtains h 2 2L 1 f1 3 I h P(Y) ( 2 - h) dy x -—- h 2 h 2 2 f, = [h p(y) (1—+1>dy x "'2' .11 2 4 f = [ pst situations. r 11 12 Oz _ C12 C22 06 C13 C23 T 0 0 rz 0r [0(r.2)] = [C][e(r.2)] F01? Zisotropic materials the constants are C11 = C22 = C (l - v) E 13 23 33 v E 33 ’ (1 + v)(l - 2v) C12 = C13 = E can 2 2(1 + v) (1 + v)(l - 2v) 'T he generalized stiffness matrix is [kc]: 1 vol [D]t [CHD] dv an In the axially symmetric case,it is expressed by (3.72) (3.73) Can 83 The integrand is independent of 8 so that [kc] = 2n J [D]t [C][D] r dr dz (3.74) area It is possible to ignore the numerical factor 211 in (3.74). In the equation [F] = [K][s] for the assembled structure, the factor 211 appears on both sides and can than be cancelled. The matrix multiplication under the integral results in 1111t 1c1w1r = (3.75) 1_C33/r (C13+C33) C332” 0 0 23 ' (C13+C33) ((311"':2(313"'(333)r ((313%33)z 0 0 (Clz‘l‘cza)r (3332/r (€13+(333)z (3111+r+ca3""2/r 0 Cuur C23Z 0 O 0 0 0 0 O 0 er O er 0 LC2 3 (C12+Cz3)r C232 0 O 0221* J Thus the expressions in (3.75) must be integrated over the cross sectional area Of the ring. This can be done quite easily by means of numerical integration formulas. After doing this, the nodal point stiffness matrix can be evaluated from the formula [k] = [A‘l]t 1 [D]t [c110] r dr dz [A_1] (3.76) area N Ote’ the constant 211 as mentioned earlier is not included here. 84 Since the elements employed here are circular rings, the concept nodal point force must be generalized. It is convenient in this 'd to speak of nodal circle forces. These are then forces on circles 1dius equal to the nodal point radius. Thus the problem of replacing .ary tractions by nodal forces involves determining nodal circle loads. of the general aspects of the problem are considered here. For convenience, consider 'e 3.7 where a traction q(r) is assumed :t normal to the area shown. 21 .s l and 2 are on one side of 1ndary element. Again for 1 1 1 bq(r) :nience, the line 1-2 is taken ‘ rpAAG ,//P\‘&;7“n , .5: ' 1 horizontal. O 2 h : /%\\\- — / da 42 h The work of the applied 5- r .Symmetry Axis ng during a virtual displacement Figure 3.7 is Axial Boundary Traction-CSTR 6W1 = J 6w q(r) da (3.77) B iSplacement w(r,z) can be related to nodal point displacements through rix of weighting functions as was done in equations (3.28). Thus mm = 1N (3 79) z 1 2 . . . . n . 16 last expression n represents the number of nodal points for the 311C . , .s .-: 0 ‘7 g .: § , L - . it» I I n. ’b «__ A...:“ 1 » - “3'. n St; the CO 53‘ (‘0 .~‘ 1 85 Evaluation of (3.78) on the boundary leads to w =1N1t1u1 (380) B B 2 ° In view of (3.80), equation (3.77) can be written as t . 5W1= 13 q(r) [NB] da [éuz] (3.81) The work done by a set of nodal circle forces with intensities The components of equation (3.84) are then h 1 r0 r 211(r0+§) fl =J (1(I') (E-h—+E) da 2 B (3.88) h 1 r0 r 211(r0-E) f2 =1 q(r) (2_h_+h) da 2 B f3 = O z I f q(r) is a constant of magnitude q, mere [f 1 '9'» 87 r+E f =fl1°61 1z 2 +g r0 2 (3.89) h h ro'E f =L1 1 2 2 h z r + —- 0 2 . q1 ' qz h a linearly varying load, q(r) = q1 + ——Hh__— (r - r0 -'§) qlh h qzroh T‘ro+z>+—d—— f = l h 2 r0 +‘§ (3.90) qlroh + qzh (r _ h) f _ 6 3 0 4 2‘ _1 2 r0 2 In assembling the overall structural load vector, it is emssary to perform the division indicated in equations (3.89) and ) . The form of (3.88) can be adhered to by defining a structural (zircle load as the total load on a circle of the prescribed radius. 2<>rresponds to the left hand sides of equations (3.88). However, tine numerical factor 2n also appears in the generalized stiffness i, it can be cancelled as indicated previously. Thus, the behavior éassembled structure is characterized by the equation [IF] = [K118] (3-91) [1?] is the column matrix of assembled structure nodal circle force E“tunes. The elements of [rF] are obtained by multiplying the elements 1 by the corresponding nodal circle radii. “'uG; H 1 . "P‘fin - kknz‘b‘ t: “"v, - VC‘.‘ P I ’ 1.. CI “5 '14“ .r J31]: ~ 51mm -1 Ekaiu 3.6 Linear Strain Triangular Ring The procedure for developi 88 mg the stiffness matrix and associated nodzil force vector for such an element has been essentially covered in the previous articles . The notion of a quadratic strain variation was treerted in article 3.4. The important aspects of an axially symmetric ring element were discussed in article 3.5. Consequently only a few of ‘tlie pertinent results are included here. A portion of the finite element is show in Figure 3.8. The nodal point system c30!‘£S:I.sts of the triangle vertices and the midpoints 015 its sides. The nodal POint coordinates are 1 = 1,2,...6 z j - 1,2,...6 The assumed displacement f “n C t ions are u(r,z) = a1 + azr + o3z w(r,z) = a7 + oer + ogz Figure 3.8 Linear Strain Triangular Ring Segment 2 2 + our + asrz + 062 (3.92) 2 2 + alor + allrz + alzz The nodal point displacements follow when equations (3.92) are E: valuated at the coordinates (r I: adial displacements is j’ 21). For example,the column matrix of I.) 111 1'1 r1 21 112 1 r2 22 U3 = 1 r3 23 1.11+ 5 1 r1+ 2“ us 1 r5 z5 or in short [ur] = [A11] [at] Similarly [uz] = [AZZJIaz] and [A22] = [A11] The column matrices in the 89 2 2 1 r1 r121 z1 / 0‘1 2 2 r2 r2z2 22 02 2 2 r3 r3z3 Z3 fi 013 (3.92) 2 2 ru r424 24 an 2 2 r t Z Z 5 5 5 5 \ Cl5 2 2 r6 r626 zeJ “6 above relationships are u“ U5 U6 > (3.94) wl+ wa6 > a“ 015 016 > (3.95) “10 “11 “12 > Hizh .1e 511d 1:16 '1 «1: derived 90 Then r1 ' ' A ‘ O a 11 1 r [u] = mm = —————— O : A22 oz (3.96) I The inverse transformation is -1 ' All 10 Ur [a] = _—T:l_ (3.97) 0 :A22 uz with -1 -1 [A22] 7 [A11] It is again desirable to compute the inverse matrix numerically. The Strains are given by 1 e(r,z) = 4 5" = [D(r,z)][a] P. r Bu 3v —_ \3—z_ 3r The matrix [D(r,z)] consists of displacement function gradients and is derived from equations (3.92)- 1.21 = 11'- 1.1! .31.: 91 (3.98) 0 1 0 2r 2 O 0 0 0 0 0 O D(r,z) = 0 O 0 O O 0 0 O 1 0 r 22 2 ~1- 1 5 r z -z—- 0 o 0 o o 0 r r r 0 O 1 0 0 22 0 1 0 2r 2 O J L. The stresses are related to strains through the constitutive relationships (3.72). The generalized stiffness matrix is 12 [Rd] = 1 1111 [c1101 dv vol The integrand is of the form [D] The matrices [Qij] i, j = 1, 2 are presented on the following page. ’— 92 C C +C C 2 11 11 12 11 z 22 r 2C12+c11 (CHER) F c117 r r2 r 2 z z 2(C11+C12) (Chi-C12); 3((3111'912)r 2((311"'(312)2 (C11"‘c12)r—' 2 3 z z 2 (Q1 1] . Cll;; +Cuu (2C12+Cll)z (C11+C12) ; +Cuur Cll :; +2Chuz 2 2 (5C11+4C12)r 3(C11+C12)rz (2C12+C11)z 2 2 1 23 2(Cl)+Cx2)z +C~hr ( C11+L)2) ;—+2C““rz :2 (Symmetric) C —~ +4C :2 llr2 '0“ [012] -10211 (3.99) ' fi 12 z 0 0 —;—- 0 C12 2C12 r 0 0 2C12 0 2C12r 4Clzz [Q .. 2 i 21] 0 C1... c12 r 2c“: mum“): 2c12 r 2 0 O 3Clzr 0 3C12r 6C12rz 2 2 O Cunt 2Cx22 2Cuhr (2C12+Chh)rz 4C122 22 2 2J 0 zcuur CIZF— “Cunt: (C12+2C““)z 2C12 f_ J r 4 0 0 0 0 0 0 1 ck“ o 2C“hr ch“: 0 : [(2 l _ 22 CH 0 Cur ZCHr ’ 2 1 “Cunt ZCuurz 0 2 2 Cllr +Cuuz 2Cllrz 1 (Symmetric) “C1122 1 TheSe expressions assume orthotropic behavior which is identified by the matrix of elastic properties in equation (3.72). WC ~- 1‘. t ...' q" 1...“-- ...., .r‘ 1.1!. 5.. ,.”;a .“_su "n 9 Nth. L 5‘"; ‘0- ‘ t' . 1 It- As 5... k. “V .. 5N. ‘. in te 93 To completely formulate the generalized stiffness matrix, [kal’ the coefficients of equations (3.99) must be integrated over the element volinne. The axial symmetry quickly reduces this to an area integration ave]: the triangular cross section. The latter is most efficiently per- for1nefl by numerical integration formulas. The nodal point stiffness matrix is 1 [k] = 1A'11T 1ka11A‘ 1 1.. When: ‘these element matrices are referred to the same global coordinate ' 3X88, the overall structural stiffness matrix is easily assembled by the direct stiffness approach. Nodal circle force intensities are determined as in article 3.5 for? tihe constant strain ring. From the second of equations (3.92) w(r,o) = a7 + oar + alorz (3.100) In terms of nodal point displacements wl, w2 and w this becomes 1, 2 I‘ r r _ 0 0 _1_ _2_ __1__ 2 w(r,0) — L( 2h + 2h?) + ( 2h hZ] + 2h2 r ] wl (3.101) rro r3 1 r0 1 + (2h-+ ———J + ( 2h.- -—3 r + ----—-r2 w2 1. 2h2 112 2h2 2 2 r 2r + (1 -—3)+——° ——l—r2 w 112 112 112 “ —_- ‘ '0‘! ‘- ...:b ‘-I . . ..'- - ‘ ..s-.C 1 V ‘- £.. do v v--— .. .t:. . . . . 0-1. . f .1....: ..‘..'.I J... .H‘sto..‘ The notation here is of Figure 3.9. The nodal Le radii are r +2 0 2 ’ h and r0 - 2 for r1, r“, and aspectively. Using the procedure of :le 3.5, one is led to the )wing nodal circle force isities for the case = q, a constant: 1 f1 ‘3“ Z 2 £11 =§qh Z 1 f2 "6“ 94 Figure 3.9 Axial Boundary Traction-LSTR (3.102) IV. PLANE STRESS APPLICATIONS The various finite element methods have been widely applied to plane problems in continuum mechanics. Finite difference methods have likewise proved to be extremely useful in plane stress analysis; however, most of the applications have involved the stress function approach. Treated first are several plane stress applications. The emphasis is placed on the comparison of the methods involved. Two fi— nite element solutions are presented in each case. These involve the direct stiffness method using linear strain triangles (LST) in the one solution and constant strain triangles (CST) in the other. One finite difference solution is given as obtained by the displacement formulation (FD). Hereafter, the abbreviations LST, CST, and FD will be used to indicate such solutions. In some examples, the elasticity solution is also available. The designation "exact" when used will refer to the elasticity solution. 4.1 Cantilever Beam As a first example, consider the cantilever beam of Figure 4.1-a, Page 96. The beam carries a parabolic load on its end whose resultant is 1000 pounds. Its dimensions in inches include a length of 6, a depth (If 2, and a thickness of 1/2. The assumed material properties are: Y(Jung's modulue E = 107 psi and Poisson's ratio v = %u 95 96 191:!“ ll H u NIS‘ ll 1... P (resultant) = 1000 lb I § c = .. .. .. {—x c V P‘s 2 = 6 V Figure 4.1a Cantilever Beam Free End Loads Fixed End Loads 12.5 1 - ‘ 1 1 .5 350. 135 . 1500' 0 a o o (1 3000. 275. ' 4' ‘ 5 5 5 275. 9 o o a 1» 3000. 350. c c a : 1350' 1500. 12.5 112.5 65 Nodal Points Figure 4.1-b Cantilever Beam - LST Finite Element Configuration 97 This same problem, with somewhat different properties, was dealt with in a paper by Argyris [15]. The author compares the end deflection obtained from LST and CST solutions with the exact solution. He con— cludes that the CST solution is unsuitable for this situation. Elasticity Solution. The cantilever beam is a classical problem of the theory of elasticity. Its solution may be found in any of the popular texts. For example, Timoshenko and Goodier present the solution for two different sets of displacement boundary conditions. The boundary conditions of the present problem have been chosen so as to allow free warping of the supported end. Thus in Figure 4.1-a,it can be observed that u(fi, 0) = v(R, 0) = 0 and u(f, c) = u(l, -c) = 0. The latter prevent rigid body rotation. Following the approach of Timoshenko [39], the stresses ny o = - x I o = 0 Y =_£_ 2-2 Txy 21 (C y ) are valid at points sufficiently distant from the supported end. The displacements are 2 3 3 2 2 2 = _ Px z _ Ez_. Ez_._ Es_._ £1... £2_ u(x,?) 2E1 6EI + 616 [610 231 v 6131) 2 3 2 2 2 = ny Px _ Pc Pc P2 v(x.y) 2E1 + 6E1 1316 + V 6E1 + 251) x P22c P£2c P23 + v ) 3IG 6E1 3E1 + 1 98 where I represents the area moment of inertia of the beam cross section and G is the shear modulus. It is clear that the above stresses are strictly correct only when the load is distributed parabolically over the free end. However, any statically equivalent end load will produce these stresses at points sufficiently distant from the free end. In the finite element and finite difference methods, distributed loads are replaced by suitable concentrated forces. Thus in Figure 4.1-b, a set of end loads corre- sponding to parabolic loading on LST elements is shown. Similarly, fixed end loads are imposed corresponding to parabolic transverse loading and linear longitudinal loading. The presence of these loads insures the validity of the above stresses throughout the beam. Finite Difference Solution. The finite difference solution was worked out for 3 mesh point systems. This was done to give some indica- tion of the convergence to the exact solution as the number of mesh points is increased. The three configurations chosen for this purpose are shown in Figure 4.2. The mesh point spacing has been successively reduced by a factor of 2. Accordingly, the mesh point systems consist of 21, 65, and 225 mesh points respectively. The end loads are also shown. Figure 4.3, diSplays the beam deflections obtained using these three approximations. As would be expected, the results are not Particularly good until a large number of mesh points is used. Thus the 225 mesh point approximation gives a maximum deflection of 21.237 X 10..2 in. The error here is 3.47%. It is apparent that conver- gences to the exact solution occurs from below. 99 Free End Loads Fixed End Loads 3000 . --* J [187.5 187.5 625. I [625. J 3000. 187.5 ‘ 187.5 21 Mesh Points Free End Loads Fixed End Loads I 54.6875 54.6875‘fT 51875. 265.625 1 265°625 2250. 359.375 359-375 J 265.625 2250. 265.625] 41 1875. 54.6875 65 Mesh Points 54'6875 Free End Loads Fixed End Loads 14'6“ 14.648 .5. 750 80.078 ] 80'078 .4. 2250. 138.672 750. u 1 122-2:: 185.547] ° 173.828 750 173'823 1 138.672 ‘ 756 138-672 80.078 2550 80‘078 4 64 . 14.648 ° -¢—-750. 225 Mesh Points Figure 4.2 Cantilever Beam Finite Difference Configurations 100 mdoauooamon soon monouomwwn ouaaam .m.a muswam ”muzquVUszHomoau-x comm no.m no.3 on.m no.m no." ao.o 11!”. 1| d # 1-1 1 ,, . u 77/4 m lun.m .77, a on 328.. :3: m. D /. u 22:: :8: mm 0 / a mazmad Imus mmm 1. 111 zamansam >~Huuamcsm . III a u . mimmé v momem.m (SBHONI) 14011331330 101 The flexural stress variation on the particular cross section x = %-£ is shown in Figure 4.4, page 102. The pattern here is quite com- parable to that which occurred for deflections. Very good stresses are obtained using the 225 mesh point approximation. The variation is essentially linear with a maximum value of 14,620 psi. This last result is 2.54% lower than the exact stress of 15,000 psi. CST Solution. The finite element solution using constant strain triangles (CST) was carried through using the 3 configurations of Figure 4.5. These include 21 nodal points with 24 elements, 65 nodal points with 96 elements, and 225 nodal points with 384 elements. The locations of nodal points corresponds identically with the locations of mesh points for the previous finite difference solutions. The nodal point loads are the same as those for the finite difference solutions. The deflection curves for these three approximations are plotted in Figure 4.6, page 104. The results compare quite closely with those obtained using the finite difference method. The lower order approxi- mations are somewhat of an improvement over the corresponding FD solu— tions. The 225 point solution, however, is slightly worse. Thus for example, the end deflection is found to be 2.219 X 10_2 in. for the 225 point configuration. This is less than the exact value 2.3175 x 10’2 in. by 4.26% The flexural stresses, obtained by averaging element stresses, are diSplayed in Figure 4.7 on page 105. They are generally comparable to FD Stresses at interior points but are poorer indications of the true State of stress along boundaries, particularly at the lower order apprOX1-Illéi‘tions. Even the 225 nodal point approximation gives rather 102 mmmuum Housxmam moamuowmao muacam .q.q madman anIUZkuHmZHamooun> DD.H mu. om. mm. no.0 4 u a a HwH.HuD.Q mHZHDL 1mm: am mu meHDL Imut mm no mHZHDm Imux mmm .+ mO§UO.m zouhnqom >~HDHHmcqu + mu 0 taauo.a + nu +. rmAwm.H rm mo.m (I8d3853815 WUHDXBWJ 9/1'S=X lU 103 Free End Loads Fixed End Loads I .I__3000. 187.5 ‘ T 187.5 625. 1 625. ‘ 3000. ‘- 187.5 1 '187.5 21 Nodal Points Free End Loads Fixed End Loads 54 . 6875 54 .6875 *1875. .625 265. 625 ‘ 2250. 359. 3751 359.375 2 - 65 625] 2250. .625 4—1875. 54.6875 54.6875 65 Nodal Points Free End Loads Fixed End Loads ““648 + 750. 80.078] 80.078 138.672 1 750. 173-828] 73. 750. 185.5474 173-828 138.672 . 750. 80.078 750. 1 2250. (“648 750. 225 Nodal Points Figure 4.5 Cantilever Beam — CST Finite Element Configurations 104 683338 58 98 .96 9:53 AmuxqusmamzHamuao-x comm oo.m oo.r an.m oo.m on." no.a 4 4 11 fl magma. m / " mthad Jena: am emu nu I]: 2236. 3:8: mm Bu 0 o mazmum scan: mmm emu 1. n amuaaamm >~Hmmamm3u . a u m.-. m. a u tr m mama... (831-13111) N01103‘UBD 105 mmouum Hausxmam emu .~.q muswfim mmUIQZkuHmZHDmoDUI> no; mm. mm. mm. 85 + u q unuanD.D mu mn+uo.m O O .548; + + rn+um.fi $.28“. .EBz 3.59 D 328“. .anz 3.98 w BEE E8: BEE . onkzdom >wHuH~mm4m , Ex. .64 (IBdISGBHLS WHBHXBWJ 9/1~Q=X lH 106 poor boundary stresses. For example, on the cross section x = g 9. the maximum stress is 12,625 psi, some 15.87. less than the exact value. It should be mentioned, however, that the element layout affects the results to some extent. Thus for example, the layout of Figure 4.8 on page 107 gives somewhat better boundary stresses but less satisfactory interior stresses and deflections. The maximum stress on the section x = g- 2. is 13,912 psi compared to 12,625 for the former layout. The end deflections are 2.2004 X 10”2 in. and 2.219 X 10-2 in. respectively. It should also be emphasized that in some cases it may be prefer- able to derive boundary stresses by extrapolation rather than by averaging element stresses. This last point is a subject in itself. It will not be pursued further in this work. LST Solution. Only a single solution involving linear strain triangles (LST) is considered in detail. The element layout and cor- responding nodal point loads are seen in Figure 4.1-b on page 96. Very excellent flexural stresses and deflections are obtained for this configuration. In Figure 4.9, page 108, deflections are seen to be nearly identical to the exact deflections. In fact, the end deflection is 2.319 X 10-2 in. or just .0652 higher than the elasticity SOIUtion. The higher value results from the fact that it is not possible to exactly represent the prescribed distributed forces. The theoretical stress variation is a linear one with respect to bOth X and y coordinates. Thus in using the linear strain triangles, one WC>uld expect to obtain stresses which are nearly exact. As seen in Table 4,3, page 113, this is precisely the case. There is essentially no difference between the LST and the exact stresses on the 107 coaumuswwucou Boom Hmu mumauoud< .m.q muswam mucwom Hmvoz mNN cam vmxwh m.¢ madman aw mm made man wawvooq can «mum 108 no.m msoauumawwa scum 9mg .m.a muswam mmuIQZkupmzHomomuux oo.m om.r mo.m on.m an." om.m moemo.m mezHom amomz mm-~ma .T ZDHhaaow semuHHmmam . moeum.fi m mm.m (83H3N13NOI1331330 109 cross section x = %-£. The only noticeable deviations from the exact theory occur near the supported and free ends. Further Comparisons. In order to facilitate comparison of the three approximate solutions discussed above, the previous results are tabulated and plotted in a different form. Other results including shear stresses and longitudinal displacements are presented as well. Starting with deflections, Tables 4.1 and 4.2 on pages 110 and 111 list deflections corresponding to the 65 and 225 point configura— tions respectively. In Table 4.1,it may be observed that CST deflec- tions are somewhat better than FD deflections at the lower order approx- imations. In Table 4.2, the reverse is seen to be the case at the Inigher order approximation. The results in Table 4.2 are graphically illustrated in Figure 4.10. Clearly, the LST solution using only 65 Inodal points is superior to CST and FD solutions involving almost 4 times as many points. The flexural stresses on several cross sections near the fixed Enid are presented in Table 4.3, page 113. It is evident that the LST snolution is best in each case. The FD solution is more satisfactory tfllen the CST solution for this configuration. The stresses for the (tress section x = %-£ are also presented in graphical form on page 114. Itis instructive to study the stress distribution on a free boundary. FVDr example, the theoretical tensile stress on the top of the beam ‘Laries linearly with the horizontal coordinate x. The corresponding approximate results are listed on page 115 and plotted on the page which follows. It is clear that none of the solutions is exceptionally good e"erywhere on the boundary surface. The LST solution is excellent for heEgions somewhat removed from the free end. The FD solution is 110 Table 4.1 Beam Deflections 65 Point Configurations X-Coord. Finite Diff. C.S.T. Elasticity L.S.T. 0. .01905 .02001 .023175 .02319 .5 .01673 .01758 .020350 .02035 1.0 .01443 .01517 .017562 .01757 1.5 .01220 .01285 .014850 .01486 2.0 .01006 .01058 .012250 .01226 2.5 .00805 .00850 .009806 .00980 3.0 .00619 .00651 .007537 .00757 3.5 .00452 .00481 .005500 .00551 4.0 .00306 .00322 .003725 .00373 4.5 .00185 .00202 .002250 .00225 5.0 .00091 .00097 .001112 .00112 5.5 .00028 .00039 .000350 .00036 6.0 .00000 .00000 .00000 .00000 111 Table 4.2 Beam Deflections 225 Point Configurations *65 Point Configuration X-Coord. Finite Diff. C.S.T. Elasticity L.S.T.* 0. .02237 .02219 .023175 .02319 0.25 .02101 .02083 .021760 0.50 .01965 .01948 .020350 .02035 0.75 .01830 .01814 .018949 1.00 .01696 .01680 .017562 .01757 1.25 .01564 .01550 .016194 1.50 .01434 .01420 .014850 .01486 1.75 .01307 .01295 .013533 2.00 .01183 .01171 .012250 .01226 2.25 .01063 .01052 .011003 2.50 .00947 .00936 .009800 .00980 2.75 .00835 .00826 .008642 3.00 .00728 .00719 .007537 .00757 3.25 .00629 .00620 .006488 3.50 .00531 .00524 .005500 .00551 3.75 .00443 .00437 .004577 14.00 .00360 .00354 .003725 .00373 4 .25 .00285 .00282 .002947 4 . 50 .00217 .00213 .002250 .00225 4 . 75 .00158 .00157 .001636 5 . 00 .00108 .00105 .001112 .00112 5 . 25 .00066 .00066 .000682 5 . 50 .00034 .00032 .000350 .00036 S . 75 .00012 .00013 .000121 6 - 00 . 00000 .00000 .000000 . 00000 112 maoauooawon swam .oa.q ouamwm ”WMIDZHuuhmZHDmDounx oowm - oo.m ou.¢ am.m mo.m mu.a mm.m .,/u masuo.m /a 9260 1mm: mmméu D /.. 323.. 58a: mmm¢mu o /u 32:: 4:82 3-5.. + /... .3538... 58536 . /... /...: l... .. mgumé m. mmm (S3HONI)N011331J3U y-coord. - .50 -1.00 y-Coord. - 1.00 y-Coord. - .25 - 1.00 113 Table 4.3 Beam Flexural Stress 225 Point Configurations Finite Diff. - 6.4 3650. 7293. 10906. 14620. Finite Diff. - 10. 4062. 8111. 12057. 15926. Finite Diff. - 19. 4386. 8773. 13039. 16684. 3643. 7251. 10840. 12625. 4082. 8063. 11987. 13803. 4395. 8765. 12981. 14595. Elasticity 3750. 7500. 11250. 15000. Elasticity 4125. 8250. 12375. 16500. Elasticity 0. 4500. 9000. 13500. 18000. L.S.T.* - .3 3748. ** 7495. 11250. ** 15004. L.S.T.* - .6 3962. ** 7926. 12240. ** 16554. L.S.T.* - .3 4604. ** 9207. 13487. ** 17767. * 65 Point Configuration ** Interpolated Result 114 mmouum amusxoam Boom .HH.¢ unawam DD.“ mu. thHDm Imwz mmmuuuzumummHn uHHZHL mHZHDm Ammo: mmM1ZDHHDJQm Rmm mHZHDm ammo: mwnonwsqmm “m4 zDHHDJDm >MHQH~mm4u AmMIQZwaHmZHDmDDUI> x+om am. 1 mi 1.9 $01 .61 .mo.a mm. " ,mo.m .wo.m (1881583818 1030x313 8/1*Q=X lH 115 Table 4.4 Boundary Flexural Stress 225 Point Configurations x-Coord. Finite Diff. C.S.T. Elasticity L.S.T.* 0. 365. 317. 0. 246. .25 722. 711. 750. 852. ** .50 1441. 1260. 1500. 1458. .75 2166. 2147. 2250. 2232. ** 1.00 2893. 2527. 3000. 3005. 1.25 3621. 3586. 3750. 3765. 1.50 4351. 3792. 4500. 4524. 1.75 5081. 5022. 5250. 5263. ** 2.00 5813. 5056. 6000. 6002. 2.25 6546. 6457. 6750. 6739. ** 2.50 7280. 6322. 7500. 7475. 2.75 8015. 7893. 8250. 8240. ** 3.00 8751. 7587. 9000. 9004. 3.25 9488. 9329. 9750. 9768. ** 3.50 10225. 8852. 10500. 10532. 3.75 10964. 10766. 11250. 11269. ** 4.00 11702. 10117. 12000. 12005. 4.25 12440. 12201. 12750. 12745. ** 4.50 13175. 11379. 13500. 13484. 4.75 13905. 13621. 14250. 14244. ** 5.00 14620. 12625. 15000. 15004. 5.25 15304. 14952. 15750. 15779. ** 5.50 15926. 13803. 16500. 16554. 5.75 16450. 15957. 17250. 17161. ** 6.00 16684. 14595. 18000. 17767. * 65 Point Configuration ** Interpolated Result mmouum Hmuaxoam mumvabom doa .NH.¢ anamflm mmuIDZHuwPCZHanQutx on.m oa.m no.5 ou.m on.m so. me. ‘w. .- 3 SEE 1mm: www-muzmmuts EH22 D O 328.. .682 mmmquSém ems O 1 .3281 émoz 8:28:33 5.. + O .8238 53221.3 . 1 m9 0 6 3 n O .0. 0 Al. N. O a O ‘ u a a: O t q- o m 0 .. emu at ..nu.m Um. H mum.“ (ISdlSSBHlS 1930x313 AUHUNHUB 117 consistently lower than the exact solution (except at the free end). Its variation is essentially linear. The CST solution is very erratic, at times decreasing with increasing x. This is essentially the result of the choice for an element layout. As mentioned earlier, the con- figuration of Figure 4.8 gives somewhat better boundary stresses. In Figure 4.13 it is apparent that the stress variation using this alter— nate arrangement of triangles is less erratic. The former arrangement is clearly scattered on either side of the latter. Apparently a best fit curve is a better indication of the actual stress variation for CST configurations. Again it should be kept in mind that these are not the best boundary stresses. For truly accurate results, one would resort to extrapolation. Further insight into the relative merits of the methods dis- cussed is gained by examining the shear stress distribution. The exact shear stress varies parabolically with y and is independent of x. Shear stresses corresponding to the various approximations are tabulated on page 119 and graphically presented in Figure 4.14. None of the shear stresses conform very closely with the exact theory. Particularly poor results occur on the free top and bottom boundaries. Certainly some of this difficulty arises from the use of constant and linearly varying strain elements in a parabolic stress field. More satisfactory results would certainly be obtained if the element size was further reduced. The FD solution is consistently better than either the LST or CST solutions. The LST solution is poor because only one element is placed above and below the neutral surface in this configuration. In doing this,one is trying to approximate a parabola by a single straight line. The use of more points in the 225 point CST and FD solutions allows for mmouum Hmuaxoam humvaaom emu .mH.¢ ounwum mmUIDZHHUHmZHQmUQUnx Do.m Do.m no.9 ou.m no.m no." no-0 W i 1 d J 118 G 8... ”$53.: emu SHE mmm 0 0 8188 am... £58.: :3 ES... mmm + m K 23:35 .5355“ . m o rogum.a (138)883813 1UHHX31J AHUUNHUB ¢Q+um.u 119 Table 4.5 Beam Shear Stress 225 Point Configurations Pg. y-Coord. Finite Diff. C.S.T. Elasticity L.S.T.* 0. 1508. 1487. 1500. 1788. - .25 1409. 1378. 1406.25 - .50 1115. 1088. 1125. 1055. - .75 641. 611. 656.75 - 1.00 344. 345. 0. 240. _a x 12” y-Coord. Finite Diff. C.S.T. Elasticity L.S.T.* 0. 1559. 1560. 1500. 1715. - .25 1453. 1432. 1406.25 - .50 1128. 1099. 1125. 993. - .75 599. 560. 656.25 - 1.00 275. 249. 0. 207. y-Coord. Finite Diff. C.S.T. Elasticity L.S.T.* 0. 2053. 1570. 1500. 1747. ~ .25 1932. 1952. 1406.25 - .50 1599. 1172. 1125. 1048. - .75 962. 946. 656.25 1.00 531. 93. 0. 247. 120 mmouum ummnw Boom .Qa.q unawam mmMIDZwahmzwamoout> no.“ mm. mm. J“? on. ‘1 1d 0+ mezHom 1mm: www-mozummccHu mszHc mu mHzHom ammo: mmm-on»:3om emu Au mthom ammo: mmnonbaaom ems +. onesuom >HHqumm3m . l o ODD" fig; D m '4 9/1-5' Doom X 18 (1883583318 HUBHS 121 a better approximation to such a function. Additional analysis using more nodal points in an LST solution has shown this to be the case. Further examination of the results in Table 4.5 indicate that the end stresses are also poor. This is additional evidence of the fact that care must be exercised in interpreting such results. Extrapolation would certainly be advisable in view of the relatively good interior stresses for the CST and FD solutions. The last results presented here involve the longitudinal displacements (u) for points on the free end of the beam. These are [>resented on pages 122 and 123. They differ little in pattern from zilready presented transverse displacements. The FD displacements are silightly better than the CST displacements, but both fall short of the eo on.“ am. am. as. am. mo.o W a - J a 0.0 \a\ 328.1 :8: mmmuts BEE D 223.1 ammo: mmmleSSm emu O mimomn 828.. ammo: 3-23538 5.. + m 28538 1123558 . m. a 1 “U D 3 n" 3 \ N o \\ u \\\ mEuo...m .\-\ H \\ m \\ \\ B \\ +\ mined 124 4.2 Composite Plate The second numerical example considered here is that of a composite plate. One quarter of the symmetrical arrangement is shown in Figure 4.16 on the page which follows. The plate has a length of 4 in., a width of 2 in. and is of unit thickness. A uniformly distri- buted load of 1200 psi acts on the top and bottom ends. As shown in the figure, the plate consists primarily of a rather flexible material (perhaps plastic) molded around a rather large rectangular stiffener (perhaps aluminum). The ratio of elastic moduli is taken to be 20. Throughout the analysis, the continuity of displace- ments across the material interfaces is assumed. A more ambitious and realistic example allows the stiffener (inclusion) to be more fiber like and thus completely surrounded by the matrix. Many such fibers could conceivably be present. This would of course involve a three dimen- sional or axially symmetric analysis. A more practical example is treated in Chapter V of this thesis. Involved is a reinforced cylinder Which is analyzed using axially symmetric programs. In the present example, the solution is carried out using two finite element models and the finite difference method. Several dis— Placement curves are plotted correSponding to each method to demonstrate convergence of the approximate solutions to the true solution. Of Primary interest is the stress distribution along the interface of the tW0 materials, as it occurs in both the matrix and the inclusion. 125 1200 psi 5 :11 ll \ x 105 psi Horizontal Interface \\ Q. “4 tu\\ u u 85;“ .\;:\ \W ‘0» ‘ 1‘; \ / 1' .\\ \ \\ .;\;:§§}, 515.1 f . t/Vertical Interface __‘______—_dvur Lar— Figure 4.16 Composite Plate 126 Finite Difference Solution. The FD analysis employed here consisted of 4 different mesh arrangements ranging from a 45 mesh point configuration to a 231 mesh point configuration. These 4 mesh arrange- ments are identified in Figure 4.17, page 127 . It is of particular importance in situations of this kind to know what deformations occur. One might ask for example how the total maximum extension compares with that which would occur if no stiffener were present at all. In Figure 4.18, page 128,the vertical displacement of the top edge is plotted for the 4 configurations mentioned above. Clearly, displacement convergence is very rapid. Since no exact solu- tion is available, one can only make comparisons with problems for which solutions are available. Thus the maximum displacement of 3.26 X 10-3 in. derived from the 231 point configuration seems very reasonable when compared with the value 4.8 X 10-3 in. which occurs in a homogeneous solid plate having E equal to 5 X 105 psi. The horizontal displacement of this same edge is shown in Figure 4.19. Again, very rapid convergence is displayed. It is evident from both Figures that relatively good displacements are obtained with even a very course mesh spacing. Additional evidence of the nature of the displacement solution is given by Figures 4.20 and 4.21 on pages 130 and 131 respectively. These are plots of the displacements of points on the horizontal interface. It is interesting to note that displace— ments are generally overestimated in the stiffener and underestimated in the matrix. A final set of curves is plotted in Figure 4.22. These involve vertical displacements of points on the vertical symmetry axis. Once again the excellence of the displacement solution is displayed, even at the crudest approximation. 127 45 Mesh Points 91 Mesh Points '~< —’- 120.— -—- 120. -——- 120. -—-— 120. -¢-120. -4-'120. —-,- 120. -—-—-120. "F- 120 153 Mesh Points 231 Mesh Points Figure 4.17 Composite Plate - Finite Difference Configurations -"'120. -" 60. 128 Do. muaoaoomammwa|> omen now am .ma.c ouawfim AmquzHHMHCZHomoomix ma. om. mm. no.5 . « DD.N SB... :8: m: E SE... :3... 8 0 law...“ .58.. :3: mm“ + :5: 1mm: “mm . & and .om.m 80+3“(S3HONI)1N3H3081JSIU-A 3003 AOL muaoamomammanna mwum nos am .mH.q munmfim mmutuzmwuHCZHDmnomux .H ma. mm. mm. no.0 111 4 a J “30.0 129 0')! HZHDm rmuz mt HZHGm rmuz Hm HZHDm 1mm: mm” HZHDm Imwz Hmm 3 +013 1 1 mm. mm. on. « 80+3'(S3HONIJ1N3H3DHWdSIU-fl 3003 dfll 130 mucoaoumaamfinn> mommuouaH aqueouauom am .om.q muamam AmeDZkuHmzHDmomuux on." ma. mm. mm. no.o 1 J a 4 3.9.0 u..\\\ -\\- u.\ JD". 58.. :3: m... D ES... :3: 5 0 53.. 2mm: mm; + .281 zmuz "mm . low. (T 8\ 5m. + car 0+3“(S3HDNIJLN3H3381dSIflrfl 8 131 mucoamomaamfinls oomwumucH Hmuoouwuo: am .HN.¢ ounwfim AmuIDZHHuHmZHDmooaix no.3 ma. om. mm. oo.o d 44 1 d1 -DD.D . n“ - MM VIAW.1+ 1 uu|l\\\ SHE :3: 2. D \ ES... :3: S D Em. $3.. :8: mma + ezHu: :mu: 3mm . .G as 1 .\ .84 \-.\. \\ 3. as... EU+3N(S3H0NI]1N3H3081dSIU-A 132 mucoaoouaamaol> mfix< huuoafihm Hmofiuuo> am .N~.c ounwam DOA... mm.“ nmuruzd uHmZHDmDDDt> 00.“ um. hZHDL 1mm: mt HZHDL rmu: Hm HZHDm 1mm: mm“ HZHDL 1mm: “mm 0» won 3 3 1 1¢ 85 “.56 i D D ,4 3 9’ cu 80+3' (S3HONI) 1N3H3OH'HSIU-A 109m 133 CST Solution. The analysis with CST elements was carried out using the three layouts of Figure 4.23 on the following page. The loca— tion of nodal points is not generally the same as that used in the FD solution. However, the nodal point system is identical to that used in the LST analysis. The convergence of displacements for the CST solutions to an apparent true solution is illustrated in Figures 4.24 through 4.28 which follow. The 25 point nodal point solution is a rather low order approximation and evidently gives very poor results. There is not an appreciable difference between the 81 and 169 point solutions indicating fairly rapid convergence beyond this point. The maximum extension derived from the 169 point solution is 3.2551 x 10'3 in. The best FD solution gave 3.260 X 10-3 in. LST Solution. The three LST arrangements are illustrated in Figure 4.29, page 140. As mentioned earlier, the corresponding nodal point systems are identical with those used in the CST analysis. The top edge displacements are given in Figures 4.30 and 4.31. These results demonstrate that convergence to the true solution occurs from below. It is interesting to note that at certain points, the 25 nodal point CST solution is better than the 25 point LST solution insofar as horizontal displacements are concerned. However, in general, these LST displacements are higher than both FD and CST displacements and thus apparently more satisfactory. The maximum extension for the 169 point LST solution is 3.2745 X 10-3 in. The comparable diSplacements for the best CST and FD solutions are 3.2551 X 10.3 in. and 3.260 X 10..3 in. respectively. ¢ 75. .__.150. --’-150. 150. -'— 4,.1 25 Nodal Points 0 Y -1I-150. --’-150. -‘-'150 .-—c-150. —a-—150. -o— 75. |_ "i -—’-100. -'50. [‘2 4 1 1" 81 Nodal Points 169 Nodal Points Figure 4.23 Composite Plate - CST Configurations 135 on. muaoaouoaamun-> mwvm now Hmo .¢~.c mesmem MmMIDZHuuhmZHDmDDUIx e me. um. mm. uu.u emu ezeut mm 0 .um.m emu ezeut S + emu ezeut mm: . G Jn=H.m .um.m 80+3*(S3H3NI)1N3H3081d810~h 3003 d01 136 an. H mm. mm. mm. om.m .1 1 J- 4 - u.-D.D emu ezeud mm o emu ES... 8 + emu HzHom mma . 0 Jam. ‘\\ \\ 5W5. \\ a. .\\x\\\. muamaoomaamwals owvm Q09 Hmo .mm.q ousmfim mmuIDZHVMMCZHumUUQIx .a 80+3-(S3H3NI)1N3N3081d810-0 3003 d01 137 on. mucoamomadmeol> moweeouaH Hmuaoufiuom Hmo .oN.q ouowfim emuIUZHWMHCZHDmmmDI> me. um. mm- uu.u 4 e u 00.0 hikiIIIICIllllllll III 1 so.\ emu ezeut mm no .um. emu ezeut em +. emu ezeui mue . . 400.“ .um.e 80+3'(S3HONI)1N3H3081ASIU~A 138 DD.« muauaoooanmwnlo oomwuoucH Hmuaonfiuom Hmo .NN.¢ ouaweh mmuerwaPIZHDmDDusx me. um. um. uu.u q . . 4 .ahu.o on ma. emu ezeut mm 0 \ emu $3.. 8 + Hmu HZHDm mmw ..fi \ \\ em. 80+3“(83HON111N3H3081d810-0 139 Do.m mucoaoomammeol> mfix< %euoaazm Hmoeueo> Emu .mm.¢ oeswem mmuIQZwaHGZHDmDDDIx om.” DD.H om. DO.D \ emu ezeud mm 0 .uu.e emu ezeut m + emu ezeut mme . x \\Q \\ \ \\i\. \\ JDD.W ioo.m 80+3~(S3HONI)1N3H3081d810 A mm.mmll.l 00.00411 mm.mm.fi .I 0000* mm.mmH llll 00 . 00" mm.mmal|.ll 00.00 Ill mm.mma all 00.00cll mm.mm7lll 25 Nodal Points mm.mm .Al Y .Omu‘l .oom III-l. ‘ L .uueLI .uumllu .ooa ill 69.... IIIII. . 03 AH . oomnkll. .omnll ‘ I}, .P\/ ‘ ‘ WW4? ‘ 169 Nodal Points 81 Nodal Points Figure 4.29 Composite Plate — LST Configurations 141 muaoaoomammwol> owpm ace emu .om.c meDmHm emuzquuuPCZHnmoauix uu.e me. um. mm. uu.u emu ezeut mm 0 summ em... SE: 8 + emu ezeud mm: . €0+3fl(S3HON111N3H30016810-A 3003 d01 10m.m 142 meooaoomaamenls owvm ace 9mg .Hm.¢ oeswfim emu:QZHuupm2HDmoounx uu.e me. um. mm. uu.u 1 . i i .._..u.u imam. 9 em... ezeue mu 0 em.. .53.. mme + emu ezeut mm: . Jam. t .2. n .8. e 80+3'(S3HONI)1N3H3081d810-0 3003 d01 143 The horizontal interface displacements are displayed in Figures 4.32 and 4.33. The lowest order solution involving 25 nodal points is quite poor. The 81 and 169 point solutions are much more comparable. The vertical symmetry axis displacements are seen on page 146, Figure 4.34. These do not vary appreciably from the lower to the higher order approximations. Comparison of the Solutions. For further comparison of the solutions already discussed, the displacements and stresses correspond- ing to the highest order approximations are considered next. Stresses at the lower order approximations are not likely to be as meaningful and are consequently not examined here. The stresses which are used for this purpose are nodal point stresses obtained by averaging appropriate element stresses. It was pointed out earlier that these may not always be accurate on boundary surfaces. The comparisons, however, are never— theless felt to be largely meaningful. Beginning with displacements, the plotted information of pages 147 through 151 pertain to displacements of points on the top edge, the horizontal interface and the vertical symmetry axis. These results are also listed in Tables 4.7 to 4.11 beginning on page 152. The top edge displacements, Figures 4.35 and 4.36, are essentially the same for the three solutions with the LST results slightly higher in each case. This is also the case insofar as the vertical symmetry axis displacements are concerned (Figure 4.39). Noticeable variations be- tween the three solutions are apparent with regard to the horizontal interface displacements. The largest variations are seen to occur for the vertical displacement component (Figure 4.37). The LST 144 mucoaoomaameal> mommeouaH awesomeeom 9mg .Nm.¢ oeomwm FmUIDZHUMHCZHDmDDusx uu.u me. um. mm. uu.u « 1 4 1|: 00.0 3\ emu ezeui mm 0 ium. em._ ezeut S + emu ezeui mme . G \ uu.u .\ Q .83 €0+3“(S3HONI)1N3N3003dSIU-A 145 on.“ madmamumaamanls mommumucH Hauaouwuoz Ema .mm.q muawfim nmurQZHuuHmZHamooDcx mu. om. mm. on. u\ 5.. :5: mm 0 5.. E8 3 + qu HZHDm mww v \x: 1 D .n-D.D cw. am. am. 1 Dr. 8043'[SBHONIllNBHBOUWdSIU-fl 146 muamamomanmanl> mwx< muuwaahm Hmuwuum> qu .Qm.o mpswfim flmerZHUUHCZHDmDDsz Do.m om.a oo.a ‘1 4 Id 5.. 55.. mm 0 5.. 55.. 5 + 5.. SHE mm: . 5cm." goo.m noo.m a3H3N11lN3HBOUWdSIU A a. €D+3fil 147 an? H mu. > mwum ace am. .950 .am .mm.q musmfim AmNIDZHuuhczHDUDDUIx om. mm. Do.n. on. FEE 5m 0 5a 55.. mm: + 5.. 55.. mm: . 'd +nmm.m n=u.m .nxm.m 80‘3“(53H3N1)1N3H30UWdSIU-A 3303 dUl 148 mu. 2 mwum mos emu .amo .om .om.¢ ousmfim nmu102kuwc2HDmnouux um: Q... :35. :5 0 5.. .55.. mm. + 5.. SHE mm: . d \ 4| \\ \\‘ d \\ 9n. u as mm: mm. ms. hzu.u €U+3fi(S3HONIILN3H3OUWdSIU-fl 3903 dUl 149 > mummumuaH HmuaoNHuom .954 .950 .nm .hm.q muswfim AmquZwaHmZHamoaun> no.“ mu. mm. mm. no.0 J 4 q 4 .30 an. ESQ 5m 0 . om. 5a 5.5“. mm: + 5.. SHE mm: . Anzu.fi . um... 80+3'(S3HDNI)1N3H3UHWdSIU*h 150 on.“ a mumwumucH Hmuconwuom qu .Hmo .am .wm.c madman mmurozmwuHCZHDmDDonx mm. mm. DO.D a... 5.8.. .mm 0 5o 55.. mm: + HmJ HZHDm mme . 4 Id 3.0.0 H 40m. 1 Dr. 80+3'(S3HONIJLN3H3OH1JSIU-fl 151 > mfix< shamaamm Hmofiuum> Ema .950 .am .mm.¢ muswfim ”murozc uHIZHDmDourx ao.m om.. no.. om. ‘ oa.mm a + . . . > .5 1i. > 5 h. O+< .c C c dlc . A _ o“. anam 0 .8..m 5.... 55.. mm: + m... 5.. 55.. mm: . m m H 1... N 1. nu N m .oo.m3 mm u 3 + U B -85 152 Table 4.7 Composite Plate Top Edge 3 v-Displacements X 10+ 231 Point 169 Point 169 Point x-Coord. Finite Diff. x-Coord. C.S.T. L.S.T. 0. 2.7391 .7342 .7469 .1 2.7477 .08333 .7446 .7546 .2 2.7730 .16666 .7581 .7736 .3 2.8139 .25 .7909 .8024 .4 2.8681 .33333 .8264 .8414 .5 2.9324 .41666 .8767 .8906 .6 3.0027 .5 .9284 .9454 7 3.0741 .58333 .9876 .0035 .8 3.1424 .66666 .0462 .0633 .9 3.2043 .75 .1045 .1208 1. 3.2600 .83333 .1594 .1747 .91666 .2093 .2264 .2551 .2745 153 Table 4.8 Composite Plate Top Edge u—Displacements X 10+ 231 Point 169 Point 169 Point x-Coord. Finite Diff. x-Coord. C.S.T. L.S.T. 0. .1 .1214 .08333 .1004 .1030 .2 .2402 .16666 .1996 .2046 .3 .3541 .25 .2954 .3024 .4 .4605 .33333 .3876 .3961 .5 .5576 .41666 .4736 .4831 .6 .6443 .5 .5540 .5639 .7 .7206 .58333 .6264 .6360 .8 .7880 .66666 .6928 .7014 .9 .8495 .75 .7517 .7601 1. .9093 .83333 .8063 .8141 .91666 .8570 .8649 .9065 .9143 154 Table 4.9 Composite Plate Horizontal Interface 231 Point x—Coord. Finite Diff. O. .1604 .1 .1634 .2 .1729 .3 .1909 .4 .2222 .5 .2818 .6 .6577 .7 .8720 .8 1.0194 .9 1.1294 1. 1.2186 v-Displacement X 10+ x-coord. .08333 .16666 .25 .33333 .41666 .58333 .66666 .75 .83333 .91666 169 Point C.S.T. .1603 .1648 .1689 .1830 .1992 .2318 .2776 .5994 .8018 .9442 1.0537 1.1399 1.2125 169 Point L.S.T. .1579 .1596 .1649 .1782 .1979 .2318 .2870 .6168 .8316 .9679 1.0769 1.1630 1.2403 155 Table 4.10 Composite Plate Horizontal Interface u-Displacements X 10+3 231 Point 169 Point 169 Point x-Coord. Finite Diff. x-Coord. C.S.T. L.S.T. 0. .0180 .08333 .0149 -.0158 .0365 .16666 .0296 -.0317 .0556 .25 .0449 —.0484 .0751 .33333 .0595 -.0659 .0933 .41666 .0756 -.0836 .1654 .5 .0861 -.O985 .2189 .58333 .1418 —.1580 .2646 .66666 .1905 —.2089 .3071 .75 .2310 -.2504 .3487 .83333 .2687 -.2887 .91666 .3045 —.3232 1.0 .3389 -.3577 156 Table 4.11 Composite Plate Vertical Symmetry Axis v-Displacements X 10+3 231 Point 169 Point 169 Point y-Coord. Finite Diff. y—Coord. C.S.T. L.S.T. 0. 0. 0. 0. 0. .1 .0186 .16666 .0309 .0305 .2 .0370 .33333 .0607 .0596 .3 .0548 .5 .0880 .0865 .4 .0717 .66666 .1123 .1102 .5 .0877 .83333 .1367 .1318 .6 .1027 1. .1603 .1579 .7 .1167 1.16666 .5884 .5860 .8 .1304 1.33333 1.0189 1.0248 .9 .1446 1.5 1.4585 1.4659 1. .1604 1.66666 1.8871 1.8988 1.1 .4140 1.83333 2.3148 2.3250 1.2 .6727 2. 2.7342 2.7469 1.3 .9349 1.4 1.1981 1.5 1.4603 1.6 1.7202 1.8 2.2316 1.9 2.4849 2. 2.7391 157 displacements are generally the largest for the 3 sets of results. The exception to this observation occurs in the stiffener where displacements are apparently overestimated in general. Regarding the horizontal interface y =-% gives a maximum vertical displacement of 1.2404 x 10-3 in. The CST and , the LST solution FD difference values for this same displacement component are 1.2125 X 10—3 in. and 1.2186 X 10—3 in. respectively. In this connec- tion, the LST solution is apparently best. As a final observation, it should be noticed that FD displace- ments generally represent a slight improvement over the CST displace- ments. However, not to be overlooked is the fact that considerably more points (231) have been employed as opposed to 169 for the CST solution. In any stress analysis, the location of regions of maximum stress and the determination of these stresses is of primary concern. For the present problem, these regions correspond to the material inter— face. The overall stress distribution obtained from the best solution for each of the methods is presented in Figures 4.40 through 4.42 on page 158 through 160. Two sets of stresses are shown for interface points because some stress components are not continuous across the interface. It is evident from these results that maximum stresses indeed occur in the interface regions. In fact, compared to the 1200 psi applied stress very large values of 0y occur in the stiffener. Along the vertical interface, the methods indicate 0y stresses in excess of 3600 psi, or a stress concentration factor greater than 3. The maximum 0y in the matrix is nearly 2300 along the horizontal interface near the corner. The stress concentration factor is approximately 1.9. Composite Plate - FD Stress Distribution -309 -302 -283 .-252 -210 45 408 - 60 T - 23 4 1 1203 1203 1203 1203 1202 1200 1199 1197 1195 1195 1201 10 6 13 13 23 26 26 22 14 4 -1 -234 :239 ..--2'7 5:196, 1-167 -131___Uq :27 _ 5. -62 _ -33_” L_-13 -5 ”1269 1209 1209 ’1207 1205 1202 P~1198 . 1193 1188 1187 " 1196 12 11 22 32 4o 45 45 39 27 13 4 ~180. -177 51:19:... 1 25 -135 ,,._JJJL.- _. _ 291.. :23 _--31 . 2 -5 1223 1224 1223 1220 1215 1206 1194 1162 1171' "1166 “ 1177 (51‘ 14 18 36 53 67 75 1 76 68 51 28 14 137 135 13 122 1129. 40 £3 4 O ' 1 ' ‘. - 1'. . ‘ - _-_-- ¢ _. .15 :25 40 4 1214 1244 1243 1238 1229 1213 5‘1191 ‘L1167 1148 1136 ‘ 140 37 15 22 44 67 86 99 102 93 71 41 23 1 1' KY -93 _-93 -97 .94 :36 __1_ -73, -54 -34 -18_ -6 -2 15 24 49 75 99 118 ; 123 112 87 51 29 .58 -60 -64 -67 _..:96 - _ J, ___1,_, -39 -20 - .L -1 T291 1292 1293 1290 1275 1241 I 1184""‘L1130 r"1086 '1052 7 026 15 23 48 76 107 134 1 143 129 99 57 32 -14 , -18 -26 -38 -46 .t .31 . -1..-.22 :1..- 3. 2 1311 1313 1319 1322 1311 1269 "1180 1102 111045 “ “1003 "J 962 13 18 40 69 108 148 163 143 105 58 31 39 35 ._ -21 _. -1 142 12.424.. .3 _ . 16 ._ 1-11.. . _ 9.- 4 324 1329 1341 1357 1360 1319 1169 1064 1 998 952 903 11 1o 25 52 98 162 188 154 104 52 25 19!- »_4_ 99, . __”83____ fi 53 14 fi_ -4 ' 20 38_ - _,__ 30 . _14 6 1328 1335 1356 1389 1423 15|3 I T140 1007' 941‘ ’ 963 859 6 o 5 22 70 173 ; 219 158 93 39 IS 175 172 162 139 88 34 ' 35 45 29 12 5 1324 133g 135_3__ 3 77 .1506. .1486 fl} _1602 1068 929 881 859 837 ”‘5 -3 -14 -14 12 167 254 150 76 25 4 1 243 245 250 264 304 1? ‘ 319 1328 1356 1410 1514 1198 o -10 -25 -41 .49 318 -102 «12 -16 -5 -2 - - - __ ____+ .-_ ~1500 -1519 .1564 1583 -1361 -1147 1 849 821 816 ‘818 830 1295 1306 1346 1464 1884 374 362 166 75 21 0 50 -30 -c6 2 375 532 1 -793 -766 -674 -499 -770 ~154 -70 .79 - 2 -41 1-'9., __N -7 1291 1306 1367 1534 2019 3691 148 - 589 .698 741 773 817 .7 -1 19 105 311 44 476 358 [199 99 35 9 1 1 ' 1 -268 -238 -153 -33 ‘ 51 71 ‘ _ ?_ , - __J - 1314 1342 1443 1689 1 2249 3483 ‘u1--' ~1" 1“ 1-37 ' 39 - '4 -- 6 “5 283 p ‘57 532 163 , 445 ,584 662 718 782 58 59 383 320 1203 121 52 21 | 1 71 87 128 166 157 135 56 . 32 .1 -10 - -3 1393 1431 1563 1839 2342 3171 ,5, 1 362' 495 *586 "656” 727 257 260 . 261 244 192 156 53 60 I 25 4 7 1 o 1506 1549 1688 1947 . 2349 2893 157 312 431 3;} 1595 661 72 123 344 351 39' 37 251 228 180 123 1 64 33 - 1 3 I 341 335 313 268 200 162 - 1__17 _.'-§J__. ’.18 . 6 2 1627 1669 1798 . 2018 2321 2677 151 ‘1 279 1384 470 ‘538 '1 594 63 118 226 I 305 318 297 200 1 185 '152 109 59 31 1 1 1 l _}66 355 322 .297 1, 197 q 16 10 85 5o 24 8 3 1738 1775 1885 2062 4 2285 2540 14% ' 257 -350 4428 1490 ' 535 50 100 186 , 243 249 224 155 ‘ 145 132 39 50 2] 1 I 362 350 313' _ 256 190 157 J06 1’ g) I 53 27 3 '1328 1859 1950 2089 2253 2411 141 242 I326 [397 451 486 37 76 140 178 176 16 111 1117 1 91 1.8 18 711 1 1 , 349 336 299 244 183 153 6 ”a 1 5, -B 3 1893 1920 1994 2105 2229 2341 1%” 23, 13.0 35“ ‘42 ,5, 24 50 ‘ 93 “5 “‘ ‘03 75 70 l 60 45 25 14 337 324 ; 288 . 236 178 14 105 . 87 1 54 28 10 3 1932 1955 i 2020 1 2114 2215 2302 137 1 226 300 162 407 430 12 25 i 46 I 57 55 5 37 g 35 30 23 13 7 1 332 320 I 284 232 176 148 104 ’ 87 54 28 1o ’ 3 1945 1967 2029 2117 1 2211 229 136 1 224 297 358 01 423 3 6 I 14 1, 18 18 17 9 L 8 8 6 3 1 -———x Figure 4.40 . 0 1119 71 Composite Plate - CST Stress Distribution — . -249 -297 -236 —263 .197 [2198' 1 438 1 .117 1 -71 —42 -19 .3 1208 1207 1207 1206 1 1206 1 1204 1201 ; 1198 1195 1192 1191 1190 6 8 17 24 ; ' 42 1‘ 38 ’ 38 28 20 7 1 1 i r ' -194 -2o1 -185 -182 ' -159 .144 l ‘ 1222 1221 F222 ,- 1219 1 1217 1211 ' 1 ’ ~ _' I -116 1 -92 -67 1 -40 -25 .8 7 ‘3 25 39 1 5° 1 5° 1 1205 ‘ 1197 1 1188 ‘1180 1172 1173 1 1 1 6 6 61 , 52 37 19 1 ' j 1 1 ' 1 1 1 ‘ 1 -122 1:120,5,. :129 415 1-111 ’ ~94 1_:4 ...-65 1 : . .1.;31 . -15 :6 25 1253 1252 1251 1 1243 1713 1715 1194 1171 1149 1113 1119 8 19 17 57 i 7 9? 10 106 1 104 1 87 68 37 , 1 1 1 -54 -51 ~59 420 ":68. ~94 .42 .99 -24__ 1‘ :9. 5m —4 11 1290 1292 1292 1290 1269 1240 1194 1147 11108 1072 1047 9 17 37 57 83 1 111 131 144 132 116 83 1 49 . A 1 1 1 ‘ 1 1 1 34 24 24 ,_1_ 2. -_5 >44 e-1o_. 32 2, 14.5 ‘ .16, 13 6 1317 1318 1327 1338 _1 13 6 1349 1285 1190 1111 11039 1000 963 4 1o 20 4o ‘ 106 162 17 165 125 , 715 44 . 1 1 I l 132 131 122 __118 82 1 90 ~16 1 -1 _ 62 5 +7 20 30 6 1321 1330 1390 1370 1398 1 1‘5] 148‘ 11204 102‘ _ 957 915 887 .2 -2 -2 0 19 40 162 : 252 170 1 113 66 23 187 244 185 256 1 1 302 1323 1329 1347 1381 ‘ 1435 l 1525 24 - -12 .14 .35 1103 . __r 295, _ £54 -28 1 .32 “1 -z_ . -930 11452 -896 -1432 .751 1.1049 .411 834 1126 812 816 820 1308 13 1367 1438 1726 2188 2889 457 206 ‘ 133 59 22 11 - -8 63 125 694 551 1 1 1 -398 -521 .314 -4 7 -73 -238 .15 39 .38 ~61 -30 fl .7 I!" 6 13V 33 180 416 586 650 701 743 15 30 82 139 271 442 468 m m, 25‘ I 155 94 3, 1 1 1 1 53 I 2_5 L11 . , ,151, 45; 50 ,5, ,‘,-_12 ~7 -:1 1464 1413's ““ 1584 *1728 2049 244 117 425 1 523 585 641 37 90 179 711? 3414 293 263 271 162 101, 51 | 344 297 327 276 254 190 100 114 1 77 48 21 10 2 1635 6 8 1748 1914 2109 2382 2657 160 1 753 If? ‘25 493 544 49 95 184 270 305 325 273 197 191 162 132 91 51 1 1 1 258 363 329 297 240 185 142 107 97 59 _38__ ,._.‘_§ 1 1 L, 1813 1824 1903 2001 2146 229a 2365 190 231 302 371 421 1‘470 34 69 136 178 212 185 183 128 120 109 88 66 35 3 344 31 7 117 93 62 38 18 5 7939' 1934 9 o ”2118 ' 2 144 213 280 336 386 418 18 36 85 92 96 94 79 61 60 53 45 32 19 3‘5 329 309 263 217 69 139 103 6 64 u 19 5 952 1959 2010 2072 2150 2228 2255 177 2?? 271 329 379 ‘06 12 6 33 16 47 17 39 J 25 1 17 1 Figure 4.41 0 1193 1 -5 1177 9 1026 26 l1 l0 0 827 -6 26 1' XY y I -301 .315 V302 -276 .232 1‘5 '4‘3 ' ‘ 4'7 ‘9 1 l 1191 ”9. 1190 ”95 ”9., "9., 1700 1199 1203 1207 1203 1700 1219 -5 1 6 11 13 13 '5 § 1‘ 12 fl 1 -1 .17 l '2°‘_.0.,,7391__;L'_‘L7, , :‘355 _ 3ij __7_ ~”9 1 -09 -67 . .19 .9 1 17221 218 ‘ 1221 1217 1215 1207 war $1201 1155’ 1176 '1173 1176 1109 -3 I8 27 35 52 61 64 63 63 96 35 25 -0 ‘1 1 -131 -129 -121 -121 -114 .100 -86 ~69 .49 -30 -I8 -6 ‘5 1253 1252 1250 1292 1236 1224 ‘21! L“'7'2' "TB—o ‘ 1136 1136 ' 1130 0 22 47 7o 91 105 115 119 117 7a 43 7 i -65 -63 . 66 ~80 .74 -63 -70 ~30 -26 ~12 -5 -3 2 1295 1295 1300 1299 12913 1270 ‘fgg 137 1:3; ‘33 "’68: "’2 ‘0" -1 22 38 69 94 100 6 1; 14 4 49 ~62 an '50.. __, l3 f“__, " ° ° 3 1333 1339 1347 1357 1353 Tm 1230 1125 1006 1039 999 963 910 3 11 23 49 ea ”6 193 1 3 166 m 99 5: 11 ‘ \ 1 125 “N124 116 92 61 34 __4__ «la ‘ 21 52 «a 213 12 2 ”3? LI“ [155 1314 1418 1579 I551 25?? 999 94‘ 906 866 MO 5 -7 J I‘: Na ‘7'» IR? 789 176 ”9 I? 74 2311 239 734 .1 757, .7 [3?] L355 1156 I“! ”M 15m. H9] 0 -I5 -77 9 -6(r -‘;u 511 __ Hint 3.3. _ 3711 .11. [:7 . n 4559 4535 -1275 21734 -IH60 442/ .5671 m $2: “71] ".1: ”’,’ ".’“ 1316 1266 1164 1259 1473 21111 3377 ‘ ’* ’ ‘1" -5 -5e .147 >290 .249 59 721 ‘fiq— 4 9 —357 13 -225 122 -5 2‘ -66 .44 . 6 .31 2 .2 2 2 0 1359 1497 423 629 656 716 760 902 29 -u 57 166 73 577 420 925 m 194 155 83 30 5 399 17 101 250 5115 m 1313 ml 53 2 »1 As 11 .7 139:1 1:13 1516 10 0 I953 N95 1712 105 336 M9 533 59'. Ma 709 32 172 275 361. 470 w 500 121 :277 726 170 11'. 62 11. 1 | J: .117 14.3 359 ._ .?D.,3,_.293 . 9! J17. ,__4 92 63 u 15 u -2 1615 1624 1710 1931 2150 2 32 7628 167 W2 363 m 519 569 5'»; 12 93 106 312 329 75 291 13 176 163 132 as u a 9 410 3'0 265 199 M7 I” '28 93 0-, M '7 7 1011 1020 1075 ”796—“9 Wis—WWI '57 315T 1175 457 m 901 1 75 us 197 234 135 113 “3 ”3 10' 70 no 10 -1 399 3136 359 324 267 133 91 .5 .0 PHI—‘9 9 2027 294 302 no " n 7. -19 54 65 76 112 59 15 30 T3 1 388 3" 349 306 267 223 la? I78 ”7 140 “I. l. , ' 1990 1966 2012 2079 ' 2132 2193 2245 154 219 2719 797 791 11:: 31:. '23 -3 5 I 6 ” o 2 19 7-7 27: 1 11. 1 Composite Plate - LST Stress Distribution Figure 4.42 161 Further examination of these stress results indicates that UK is quite small (less than 400 psi) throughout the matrix. In the stiffener, however, large compressive values as great as 1860 psi arise along the horizontal interface. This is clearly the result of the large ratio of elastic moduli for the two materials. Maximum shear stresses also occur along the material interface. The indicated values are approximately 700 psi for the stiffener and 500 psi for the matrix. It is well to recall that due to the sharp corner corresponding to the intersection of the vertical and horizontal material interfaces, the theoretical stress components at the corner are undefined. Since the nodal point stresses represent the average stresses in the neighborhood of any point, they clearly would not reflect this situation. A second point of clarification is in order. Regarding inter- facial stresses, certain components should be continuous across the interface in view of equilibrium considerations. For instance, 0y and Tyx should be continuous across the horizontal interface whereas 0x and Txy should be continuous across the vertical interface. The results in Figures 4.40 through 4.42, however, do not reflect this. The difference is apparently due to the fact that nodal point stresses are indicative of the average stress around a given point. These discrepancies would diminish if smaller spacing and smaller triangles were used in the vicinity of the interface. Along with this, one could extrapolate from the interior points and this should further improve the results. These ideas, however, probably would not greatly improve the situation at the corner. In order to examine and compare these interfacial stresses in greater detail, certain of these are next tabulated and plotted together; 162 Beginning with Ox’ this component should be continuous across the vertical interface but discontinuous across the horizontal interface. The former stresses are relatively small and therefore not considered further. The latter are listed in Tables 4.12 and 4.13. The matrix stresses are shown in the first of these tables and the stresses for the stiffener in the second. These stresses are the average of element stresses around a given point. The averaging is done only for elements in regions of the same material. In Figure 4.44, the interfacial stress OK for the stiffener is rather erratic for the LST and CST solutions. The smooth variation of the finite difference solution seems much more realistic. The erratic tendencies of the finite element solutions are no doubt partly due to the choice of an element layout. It was observed with regard to the cantilever beam of section 4.1 that other configur- ations give rise to improved boundary stresses. It is felt, however, that to some extent, this behavior is a characteristic associated with the use of triangular elements in finite element methods. In view of this, it seems likely that some sort of a best fit curve is more repre- sentative of the true stress state. The normal stress component 0y is listed for the three solutions in Tables 4.15 through 4.18. The first two tables pertain to the horizontal interface whereas the others pertain to the vertical inter- face. These same results are graphically presented in Figures 4.44 and 4.45 on pages 168 and 171. It is apparent from these figures that the three solutions are more comparable than they were for the 0X stress. The finite element 0y stresses display a much smoother variation than 0X. In Figure 4.44, the largest deviations in fly across the horizontal interface obviously take place at the corner. In fact these deviations 163 Table 4.12 Composite Plate Horizontal Interface Stress oX (Matrix) x-Coord. Finite Diff. x-Coord. C.S.T. L.S. O. 243 O. 187 238 .1 245 .08333 244 239 .2 250 .16666 185 234 .3 264 .25 256 251 .4 304 .33333 179 256 .S 12 .41666 302 294 .6 -102 .5 24 -2 .7 - 42 .58333 -105 -114 .8 -16 .66666 - 28 - 55 9 - 5 7S - 32 - 28 l. - 2 .83333 — 7 - 6 .91666 - 6 — 2 164 Table 4.13 Composite Plate Horizontal Interface Stress ox (Stiffener) x-Coord. Finite Diff. x-Coord. C.S.T. L.S.T. —1500 -930 -1554 ~1519 .08333 -l452 ~1585 -1564 .16666 -896 -1275 -1583 .25 -l432 -1734 -l36l .33333 '751 ~186O -ll47 .41666 -1049 -1427 .5 - 411 - 586 165 x6 oomwuounH Hmuconwuom woman wuamomaoo .ma.¢ wuowwh rMMIQZHWMPIZ 1. 3.171.) urrLPIrcfxf -x ma.s m um. uu.m)- mmmzmctHcmcmc em wmm x immzmcchmlcmu an mms Av flmmzmnanmlcmn em man in HXHmamzlmn Ha smm flu HXHmcmlemu um mma no mmepmchma an _mfl + m“. MMU mm mm .10 his a 13 . E 1 E6 H11 LC U (15318s3318—x 166 Table 4.14 Composite Plate Horizontal Interface Stress 0 (Matrix y ) x-Coord. Finite Diff. x-Coord. C.S.T. L.S.T. O. 1319 O. 1323 1327 .l 1328 .08333 1329 1335 .2 1356 .16666 1347 1356 .3 1410 .25 1381 1413 .4 1514 .33333 1435 1468 .5 1198 .41666 1525 1566 .6 849 .5 1103 1197 .7 823 .58333 834 871 .8 816 .66666 826 858 .9 818 .75 812 847 1. 830 .82222 816 825 .91666 820 822 1.0 827 828 167 Table 4.15 Composite Plate Horizontal Interface Stress oy (Stiffener) y-Coord. Finite Diff. y-Coord. C.S.T. L.S. O. 1295 O. 1308 1316 .1 1306 .08333 1304 1266 .2 1346 .16666 1367 1164 .3 1464 .25 1438 1259 .4 1864 .33333 1726 1473 .5 3740 .41666 2188 2118 .5 2889 3377 168 ho momwumucH Amocouwuom woman mufimoaaou .q<.q shaman inmruzwlmcmzwouomm1x no.2 mm mm. uu.u q a a .U a 6m s1 Q s U 6 flWMZNthcmcmc Hm smm x mmmzmnchmccmu um mma Aw ”mmzmcchmlems cm was As fiXHmsmzlmn cm 1mm an ”XHmemzccmu e6 mas no MXHmcmzcnms p6 mws 1. Av . -ouom 1000? (1821113838197 y-Coord. 1.5 1.6 1.7 1.8 1.9 2.0 Composite Plate Vertical Interface Stress Finite Diff. 136 137 138 141 146 151 157 162 163 148 1198 1602 1413 1319 1269 1241 1224 1213 1206 1202 1200 Table 4.16 Cy (Matrix) y-Coord. O. 169 .16666 .33333 .66666 .8333 .16666 .3333 .66666 .8333 C.S. 177 144 190 160 244 180 1103 1484 1285 1240 1215 1205 1201 L.S. 154 159 157 167 185 194 1197 1504 1238 1228 1212 1204 1200 II .vr—n- n-r-w" ' 170 Table 4.17 Composite Plate Vertical Interface Stress 0y (Stiffener) Finite Diff. y—Coord. C.S L.S.T. 2290 O. 2255 2245 2302 .16666 2319 2289 2341 .33333 2365 2397 2411 .5 2657 2628 2540 .66666 2734 3212 2677 .83333 3368 3761 2893 1. 2889 3377 3171 3483 3691 3740 171 ho oommumusH Hmowuuo> mumam mufimoaaou .me.q ouswam mmruzwlmmewnnnmmuy ) i ,3... .7. . ‘ CC C '1‘ (1‘ C3 C3 OJ [-1 L." 1-1 _ 4 ‘ ”mmZMtchmcmn s6 5mm x ”muzuumemwamn H6 mus Av mmwzmnanmcemn 86 man A“ MXHmsmzcm H6 8mm mu AXHmsmzwcmu Hm mms m6 MXHmemzwnms so mma + 113. rLrL 1 3d]833613*A (. .11.. ll 172 become very apparent as the corner is approached. In Figure 4.45, the stress oy is different in the matrix and stiffener due to the difference in elastic properties. The interfacial shear stresses on the vertical interface are shown in Table 4.18 for the matrix and in Table 4.19 for the stiffener. These are also plotted in Figure 4.46. The results are again quite com- parable for the three solutions. The largest deviations occur at the corner where stresses in the stiffener are indicated to 532, 551, and 721 psi respectively for the FD, CST, and LST solutions respectively. Concluding Remarks. The three solutions seem to have compar- able capability for predicting displacements in a simple composite. The maximum overall extension did not vary appreciable for the three solutions. The situation insofar as stress is concerned is somewhat different, however. The finite element solutions in general and the CST solutions more specifically display erratic tendencies. The finite difference stresses vary in a much smoother fashion and consequently seem more realistic. It is well to recall in this connection that the best FD solution employed 231 points whereas the best finite element solutions used only 169 points. An additional remark is in order at this point. For truly accurate analysis, one would use an arrangement of elements whereby many more smaller elements would be placed along the interface. In this way a better indication of the complicated state of stress would be achieved. This is a very easy thing to do within the framework of the Finite Element Theory. A similar concept in the finite difference y-Coord. 1.2 1.3 1.4 1.5 1.6 1.7 1.8 2.0 Composite Plate Vertical Interface Stress Finite Diff. 37 75 113 155 200 251 310 383 476 318 167 173 162 148 134 118 99 75 45 26 173 Table 4.18 Txy (Matrix) y—Coord. 0. .16666 .33333 .66666 .83333 1. 1.16666 1.33333 1.5 1.66666 1.83333 2. C.S.T. 34 61 128 197 293 398 295 162 162 131 105 64 42 58 118 213 321 425 371 371 198 150 115 64 15 .3- ‘1 174 Table 4.19 Composite Plate Vertical Interface Stress T (Stiffener) XY y-Coord. Finite Diff. y-Coord. C.S.T. L.S.T. 0. 17 O. 39 8 .1 50 .16666 79 70 .2 103 .33333 183 113 .3 160 .5 273 241 .4 224 .66666 468 500 .5 297 .83333 468 420 .6 379 l. 551 721 .7 667 .8 532 .9 448 l- 532 5x 9 oommuoucH Hmowuuo> oumam wufimomaoo .o¢.¢ ouawam Hmmrmzw w n71tum071, 175 oa‘m am.H om.H um. mu. m u w H" I I I .7. 1. n , x(\ I) .113 + A + + HmmzmniHHmlmi Hm Hmm x HmmzmnnHHmchu a6 mmH Aw HmmzmncHHmlen Hm mmH AL HXHmHmzcmn Hm Hmm Hm HmeHmzccmu H6 mmH m6 HXHmHmzccms H6 mmH +. ,4 NA Z (3 C7 C11 ‘3 C7 U1 (IESJ1F;Q3518~AX 176 analysis would involve a transition region whereby the mesh spacing would accordingly be diminished. This effect is not as readily achieved as the comparable concept in the finite element method. No attempt has been made to develop the idea in this research. V. AXIALLY SYMMETRIC APPLICATIONS The finite element methods are not limited to two dimensional problems, and for that matter neither are the finite difference methods. Three dimensional analysis, however, is considerably more involved and generally requires significantly more computer memory. Conse— quently, most of the applications have involved plane stress or plane strain. The axially symmetric applications generally involve a triaxial state of stress. The special character of such situations results in a two dimensional displacement field. As a result, the axially symmetric elasticity problem is almost as readily formulated as the true two dimensional problem. TwO axially symmetric applications are treated in this section. Included are a thick hollow cylinder subjected to internal and external pressure and a composite solid cylinder uniformly stressed at its ends. Both finite element and finite difference solutions are included for each example. The finite element solutions employ the constant strain triangular ring (CSTR) and the linearly varying strain triangular ring (LSTR) discussed in Chapter III. The finite difference solution involves the axially symmetric Navier equations; however, the particular equations used correspond to the alternate derivation of section 2.9. The primary emphasis here as in Chapter IV is on the comparison of the methods involved. In the case of the thick cylinder, comparison is also made with the known elasticity solution. 177 178 5.1 Thick Cylinder The thick cylinder under internal and external pressure is a fundamental problem of the Theory of Elasticity. A portion of such a cylinder is shown in Figure 5.1, page 179. The numerical properties pertinent to this application are also shown. These include outside and inside radii which are 10 in. and 5 in. respectively. The material is assumed to have a Young's modulus E = 107 psi and a Poisson's ratio ”-1- 4. internal pressure p1 is 9000 psi. The finite element solution for a The external pressure designated by po is 15,000 psi. The similar thick cylinder has been considered by a number of other writers. For example, Wilson [25] reports very excellent results using the constant strain triangular ring element. Elasticity Solution. The general elasticity solution for the thick cylinder is presented in the text by Timoshenko and Goodier [39]. Using the notation of Figure 5.1, the stresses are 2 2 _ 2 _ 2 a b (p0 pi) 1 pie pob o = - + b2 - a2 r2 b2 - a2 2 2 _ 2 _ 2 a b (p0 pi) 1 pi8 pob 0 = - ' + b2 _ a2 r2 b2 _ 82 The radial strain is 179 Homcwamo onLH moufluswmoum H.m ouswfim I ‘ Ill! ,Illlllli14 ;AIIIIII 5|. Ea 88H “Ta oooa u on a J IIIIY II] II/ All me< muumaemm . 180 Eliminating the stresses, one obtains 2 2 _ 2 _ 2 l a b (p0 pi) (l + v) pia pob 5r”? °——‘—+ 41...) b2 _ a2 r2 b2 _ a2 Similarly 2_ 2 E =_X (0 +0 ) =___2_2 p13 Pb Z E r 6 E b2_az The displacements are obtained by integrating the strain displacement equations. Thus from the relationships E = 32 r 8r 6 _ a: z 82 one can derive 2 2 _ _ 2 [_ a b (p0 p1) . (1 + v) Pia p b The approximate solutions of the finite element and finite difference methods follow. The results are exceedingly good in each case and thus only one solution for each method is presented. Numerical Solutions. In analyzing the thick cylinder, the configurations of Figure 5.2 on the following page were used. As can be seen, a small axial segment is involved using a total of 105 points in each case. Half of the segment is above the r - 8 plane and half of it 181 =5 r=10 750 _ 5 _ - - 1250 3000 7' '* 5000 1 I t 1500 r + 2500 3000 J ' £ 5000 b i 1 750 i 1250 LST 105 Nodal Points r=5 r=10 1125 1875 2250 3750 2250 3750 -—o- 2250 3750 “4...... 1125 1875 CST 105 Nodal Points 1125 1875 FINITE DIFFERENCE 105 Mesh Points Figure 5.2 Thick Cylinder Finite Element and Finite Difference Configurations t. 182 is below the r - 0 plane. The nodal circle force intensities associated with the external and internal pressures are shown. In the finite element method, these must be multiplied by corresponding nodal circle radii before they can be used in the equilibrium equations. The LSTR solution uses 40 triangular ring elements as shown in the figure. The CSTR solution employs 160 triangular ring elements. As typical indications of the accuracy of the approximate solutions, the radial displacement, radial stress and circumferential stress are summarized and compared with exact results. The specific values used in this comparison correspond to the cylinder mid—plane and are specified as u(r,0), 0r(r,0) and 06(r,0). Beginning with the radial displacements, these are tabulated on page 183. The same information is plotted on the following page, Figure 5.3. The solid curve represents the exact solution. The approximate solutions are shown with appropriate characters as noted in the figure. Clearly, there are no significant deviations from the exact solution for any of the approximate solutions. The radial stress is plotted in Figure 5.4 and tabulated on page 185. The radial stress is compressive throughout the body and ranges from 9000 psi on the inside surface to 15,000 psi on the outside surface. It is apparent that each approximate solution is excellent at the interior points. Noticeable deviations from the exact stresses occur at the inner and outer surfaces for the FD and CSTR solutions. Thus on the inside surface, the error is 4.37% for the FD solution and 6.43% for the CSTR solution. The error in the LSTR solution is only 0.46%. All of these are higher than the exact stress. On the outside surface, the errors are less than 0.5% for all solutions. 183 TABLE 5.1 Thick Cylinder Radial Displacement u(r,0) x 10.-3 r—Coord. Finite Diff. C.S.T.R. Elasticity L.S.T.R. 5.00 -l.1364 -1.l365 -1.1375 —l.1372 5.25 -1.1445 -l.1448 -1.1455 -1.1452 5.50 -l.1548 -1.1548 -1.1557 —l.1555 5.75 -1.1669 —1.1672 —l.l679 -l.1675 6.00 -1.1807 —1.1808 -1.1816 —1.1814 6.25 -l.1959 ~1.1962 -l.1968 -l.l965 6.50 —1.2124 —l.2125 -1.2133 —l.2131 6.75 -l.2301 -1.2303 -1.2309 -l.2306 7.00 —l.2488 -l.2489 -1.2496 -1.2494 7.25 —1.2683 -l.2685 -l.2692 -1.2689 7.50 -1.2887 -1.2888 -1.2895 —1.2893 7.75 -l.3099 -l.3101 -l.3lO7 -1.3104 8.00 -l.3317 —l.3318 -1.3325 -1.3323 8.25 -l.3541 -l.3543 -1.3549 -l.3546 8.50 -1.3771 -1.3772 -l.3778 -l.3776 8.75 -1.4005 -1.4007 -l.4013 -l.4010 9.00 —1.4245 —l.4246 -1.4252 -l.4251 9.25 —1.4489 -l.4491 —1.4496 ~1.4493 9.50 -1.4737 —l.4738 —1.4744 -1.4742 9.75 -l.4988 -1.4990 -1.4995 -1.4992 10.00 -l.5243 -l.5244 . -l.5250 -l.5248 184 muauauuuHamHa HMHnmm novaHHso snags .m.m unawsm nmeuzC umeHDQDOUIm S m m x m m q Jq 11 q * ow. H1 .1 cm. H... a H U I ”U HI 0 J D: . am :6 HI H 3 3 n" 3 N .1 mm. Hm.) I m Hzmzud ES: :3 D m quzmd uHHzHu Eu 0 m musztHa 3.22 + .4. onHzaom >HHuHHmcsu . dom.Hmm fin. 185 Table.5.2 Thick Cylinder Radial Stress or(r,0) r-Coord. Finite Diff. C.S.T.R. Elasticity L.S.T.R. 5.00 — 9392. - 9578. - 9000.0 - 9040. 5.25 - 9735. - 9738. - 9743.7 - 9725. 5.50 -10379. —10375. -10388.4 ~10410. 5.75 -10944. -10948. ~10950.8 —10937. 6.00 -11441. -11434. -11444.4 —11477. 6.25 —11878. 11877. -11880.0 -11867. 6.50 -12265. —12259. -12266.0 —12275. 6.75 -12610. -12608. ~12610.4 -12604. 7.00 —12918. -12913. -12918.3 -12934. 7.25 -l3195. -13194. -l3l95.0 -l3187. 7.50 —l3444. -l3440. -l3444.4 -13448. 7.75 —l3670. -l3669. -l3670.1 -l3667. 8.00 —l3875. -l3872. ~13875.0 —13883. 8.25 —14062. —1406l. -1406l.5 -14056. 8.50 -14232. —14230. —14231.8 —14234. 8.75 -14389. -14388. -l4387.7 —14387. 9.00 -14532. -l4530. -l4530.8 -14534. 9.25 -14665. -14663. -l4662.5 ~14658. 9.50 -14787. -14783. —14783.9 -l4784. 9.75 -14899. —14896. —14896.1 -14897. 10.00 —14952. —14925. ~15000.0 -1507l. 186 ammuum Haacmm “mucfiHso sowna .e.m «uanm AmmxquoHHcZHomoou-m DH m m x m a 4 1 1 «E ww. «I i o a :n+u:.H- no . nu .. .u I U «I rniium. Huh Nu 3 S .5 MW Hzmxud ”1:th 5.. B m quzu3u mquHu Hmu Au uuzumuttHa HquHt +_ yoeua.H- zaHstom >HHUHHmc3u . mw a mega- 187 The circumferential stresses are also compressive throughout the body. They are considerably higher than the radial stresses and range from 19,000 psi on the outside surface to 25,000 psi on the inside surface. The numerical and exact results are listed in Table 5.3 and plotted in Figure 5.5. Once again, the agreement of the numerical re- sults with the exact theory is excellent. The largest deviation occurs with the FD and CSTR solutions on the inside surface. The errors here are 0.43% and 0.7% for the two solutions respectively. The largest error for the LSTR solution is a mere 0.1% on the outside surface. Concluding Remarks. In the preceding discussion, axially symmetric finite difference and finite element solutions for a pressurized thick cylinder were compared with the exact solution from the theory of elasticity. Remarkably good results were obtained by each method for both displacements and stresses. The finite element solution employing linearly varying strain triangles gave slightly better results than the other solutions. The differences in this example, however, were generally very insignificant. 5.2 Composite Solid Cylinder The second axially symmetric problem treated in this work in— volves a solid cylinder which is subjected to a uniform axial end load. Such a cylinder is displayed in Figure 5.6. In this application, the outer cylinder (matrix) is reinforced with a relatively large concentric cylinder(stiffener)of a much stiffer material. The particular case considered here assumes a 1 inch radius for the outer cylinder and a length of 4 inches. The stiffener has a radius of %-inch and its length 188 Table 5.3 Thick Cylinder Circumferential Stress 06(r,0) r—Coord. Finite Diff. C.S.T.R. Elasticity L.S.T.R. 5.00 -25149.7 —25173.9 -25000.0 -25007.2 5.25 -24225.3 -24241.3 —24256.2 —24245.1 5.50 -23586.6 —23589.1 -236ll.5 -23614.5 5.75 -23027.5 -23035.1 —23049.1 —23037.0 6.00 -22536.1 -22536.7 -22555.5 -2256l.6 6.25 -22102.3 -22107.7 -22120.0 —22112.6 6.50 -21717.4 -21718.1 -21733.7 —21733.4 6.75 —21374.4 -21378.8 ~21389.5 —21380.4 7.00 -21067.5 -21068.4 —21081.6 ~21083.4 7.25 ~2079l.7 -20795.4 —20804.9 -20800.3 7.50 —20543.1 ~20544.2 -20555.5 -20554.0 7.75 -20318.1 -20321.3 —20329.8 —20322.2 8.00 -20113.9 -20115.0 -20125.0 -20125.0 8.25 -l9928.0 -l9930.7 -19938.4 -19935.5 8.50 -l9758.2 -l9759.4 ~19768.l —l9766.3 8.75 —19602.8 -l9605.2 —19612.2 —l9605.3 9.00 -19460.3 -l946l.3 -19469.1 -l9468.4 9.25 -l9329.3 -19331.l —l9337.4 -l9335.8 9.50 -19208.9 —l9208.9 -l9216.0 —19213.9 9.75 -l9098.4 -l9097.8 -19lO3.8 -19096.5 10.00 -18975.2 -18968.2 -19000.0 -l9019.3 189 mmouum Hmwusouomasouwo Hovafiamu Joana .m.m ouswum "murDZwawmzmnmoousw DH m n m m 4 CD 14 J 4 j .roi ququu wHHzHu Hm; ququw UHHZHm hmu uuzumemHu MHHZHu zathdum >HHDHthJu won #01 rui fol hPum.m' fur.ml lum.m .ma.m« (ISdloS3818 1UIlN333JHflUHIO 1mm.Hp 190 A 1 1200 psi -5X105psi I [1'1 l Horizontal Interface Vertical Interface +——-—N|o‘ L4— '1 Figure 5.6 Composite Cylinder 191 is 2 inches. The material properties are E = 5 x 106 psi and v = % for the matrix and E = 107 psi with v = %-for the stiffener. The mag- nitude of the applied stress is 1200 psi. The materials are assumed to be perfectly bonded throughout the analysis so that complete displace- ment continuity is maintained across the material interface. Referring once again to Figure 5.6, it can be seen that the stiffener does not extend throughout the length of the matrix. For this situation, there is no exact solution available. Accordingly, only approximate solutions are considered here. As was true for earlier applications, both finite difference and finite element solutions are included in the analysis. Convergence of the various solutions is demonstrated using several approximations for each method. Finite Difference Solution. Four difference solutions were employed in the analysis of the above mentioned problem. The mesh point arrangements used are displayed in Figure 5.7 where %-0f the cylinder cross section is shown. The 4 approximations involve 45, 91, 153, and 231 mesh points. The geometry for each case is identical to that used in the similar plane problem treated in Chapter IV. The difference equations of this analysis utilize nodal circle force intensities (lb/in). The force intensities which correspond to the 1200 psi applied stress are also shown in the figure. The intensity at the symmetry axis is undefined since at this location r = 0. For the purpose of numerical calculations, it is reasonable to use a small finite value of r, say r = .001. The particular equations programmed in this work involve a numerical factor of 5 on the right hand sides. Therefore in preparing data, the intensities of Figure 5.7 must be multiplied by 5. 192 12.500. -" 300. -—’- 137.5 .a— 5556 —* 300. —p— 300. 45 Mesh Points 91 Mesh Points N 2000. ' sh 153 Mesh Points 231 Mesh Points Figure 5.7 Composite Cylinder — Finite Difference Configurations 193 The character of the FD solution is demonstrated in Figures 5.8 and 5.9. These illustrations pertain to axial displacements which occur at the cylinder end and along the horizontal material interface. Addi— tional displacement results are presented in Appendix A, beginning on page 247. In Figures A1 and A2 radial displacements for these regions are plotted. In Figures A3 and A4 , axial and radial displacements are presented for selected points. Referring to Figure A3 , page 249 it should be observed that the axial displacements at r = 0 on the cylinder end seem unrealistic. These are greater than displacements as "E for the point immediately to the right. This discrepancy is believed to be the result of the sigularity at the symmetry axis. Extrapolation from the interior results in a more realistic symmetry axis displacement. The extrapolated results are shown in Figure A3 as well. These are used in further displacement discussions. 0n the end of the cylinder, Figure 5.8, for example, axial dis- placements range from approximately 3.0 x 10-3 in. to somewhat more than 3.4 x 10”3 in. The average deviation between the lowest order solution and the highest order solution is approximately 6%. The variations between the 231 point solution and the 153 point solution are 1% and less. Convergence to an exact solution evidently occurs from below. In Figure 5.9, axial displacements for points in the stiffener are shown as well. These results are comparable to Figure 5.8 in the sense that relatively little improvement is realized in going from the 153 point solution to the 231 point solution. Noticeable deviations occur, however, with regard to the lower order solutions, especially for the 45 point solution. The tendency for displacements to be under estimated in the matrix and over estimated in the stiffener is 194 on. munoaoomaamwn Hafix< mam novawflho am . mx. HmUIUZHHMMmzmemmDum om. m.m ouswflm mm. DD. 0 FZHom rmut mt HzHDm rmw: Hm HzHDm rmut mm" HZHDm 1mm: amm ++oa J j o DD.m D m (U 'TVF 0 CU 80+?“(S3HONI)1N3H3091&SIU~H l haw.m 195 Killlrlllllfik musoaoumHamfin Hmfix< oomwuoucH Hmuaoufiuom am .m.m ouswfim DD.“ mm. AmUIUZHuuHCZHDmQQDIQ om. mm. DD- PZHom 1mm: my RZHom Imuz Hm HZHDm rmut mmu HZHDm Imu: Hmm «+013 a 4 L o L 1 C) C) C) C) am. am.“ nmw.H BU+3*(SBHDNlllNBHBUHWdSIU-H 196 apparent from Figure 5.9. It is of interest to consider the effect of extrapolation on certain of these results. To this end, the Richardson Formula [40] is applied. In the above formula, II and 12 are approximations to some function corresponding to nodal point systems whose numbers are ml and m2. I is the extrapolated value of the function. The result of applying this formula to selected FD displacements is shown in Figure 5.10, page 197 . Both axial and radial displacements are shown. It is evident that the extrapolated results generally repre- sent some improvement. In fact extrapolating between the 91 and 153 point solutions gives displacements which are a slight improvement over results derived from the 231 point solution. A peculiar situation occurs at the upper right hand corner point of Figure 5.10. The radial displacement apparently increases when the number of mesh points is increased. The extrapolations, however, appear to decrease. The stress distribution for the highest order FD solution (231 mesh points) is presented in Figure 5.11. The stresses shown ‘were obtained by averaging the stresses for the material regions around the mesh point. At each point, the stresses are listed vertically in the order or, 02, 0 , and Trz' Two sets of stresses are given for the 0 mesh points on the material interface. Those to the right and above the interface were computed from the matrix stresses. Those to the 1eftzand below the interface were obtained from stresses in the 197 v v v .9778(45) ‘Ti .1360(45)_' .3412(45)FT 3.0751(E) 3.2137(E) .0513(91) .1947(91) .3983(91) 3.0974(E) 3.2349(E) .08ll(153) .2207(153) .4237(153) 3.1071(E) 3.2453(E) .0957(231) .2345(231) .4374(23l) u u .4372(45) .7451(45) -.4511(E) - — .4477(91) .7503(9l) -.4537(E) - .4516(153) .7511(153) -.4548(E) — .4534(231) .7511(23l) v v iv ——o .2654(45) T .3834(45) .259l(45) .2570(E) .3886(E) l .2573(9l) .3873(91) .3162(91) .2525(E) .3929(E) 1 .2530(153) .3909(153) .3410(153) .2493(E) .3977(E) l .2510(23l) .3937(231) .3540(231) u u .0874(45) .3649(45) -.1006(E) - .0974(9l) .3884(9l) -.1075(E) — .1039(153) .4002(153) -.ll43(E) - .1082(231) .4068(231) (45) - 45 Points (91) - 91 Points (153) - 153 Points (231) - 231 Points (E) - Extrapolation ' I Figure 5.10 Extrapolated FD Displacements 3. 3. 3. 4166(E) 4376(E) 4481(E) .7520(E) .7515(E) .751l(E) .3347(E) .3546(E) .3723(E) .3960(E) .4067(E) .4161(E) "f § I: <1 -02I \l in“? f 2. “f“ “F 2 Q ff: sf 0 6 MN 8‘3 E E E jgfi F N N A if “f 196 870 w 15 ‘65 65 409 -57 122 W9 D II 204 90 40 -52 a; 73 62 64 222 In -42 -3‘ 13 670 75 7! 2” I23 -I3 -I7 321 83 6” 7T II US Pl 191 I I9 39 7 -5 365 ll “I 75 I! I7 22! I62 ‘06 52 ID 2 347 452 $32 73 l2 l6 I73 I80 87 59 25 7 m 4:! so. 72 W I! m 97 0 6| 2! 9 m 4|! ‘7 7B 77 r I4 0: 4s 62 29 lo 410 is 69 76 I 7' a 22 63 29 IO :22 m a GI 76 H II I 5 Figure 5.11, Finite Difference Stress Distribution 5 "94 2&8. 2kg. 853“ as?“ atfi- _éEV O H o N O 199 stiffener. In analyzing the data of Figure 5.11, the rather large axial stress in the stiffener is a significant feature. The axial stress is seen to exceed 4800 psi and thus indicates a stress concentration factor of more than 4. Axial stress in the matrix is as high as 1871 psi or more than 50% higher than the applied stress. The presence of the rather rigid inclusion gives rise to large radial and circumferential stresses in the stiffener. These are compressive stresses near the interface which exceed 2400 psi. The corresponding stresses in the matrix are 500 psi tension and less. The shear stresses are greatest in the stiffener and do not appear to exceed 800 psi. As was true for the similar plane problem of Chapter IV, the stresses are undefined at the corner of the stiffener and the matrix. Obviously, neither the difference method nor the finite element method is able to predict such stresses. In view of this, it is instructive to examine in greater detail the corner stresses which were used to obtain the nodal point stresses of Figure 5.11. These stresses are shown below in Figure 5.12 in the order 0 , 0 , o , and . r z 0 I 525 l —19 Trz. It is significant that the 2081 ‘2002 577 . 433 low axial stress in the lower right -108 498 region results in the smaller ? ~ - \A ‘ £71903. -403 é average for the three Tidax/Whjbi 114 41401 ‘- -214 unshaded regions. The axial [>706 .y\ 692 *i/f/s .45”, strsesses in particular are large 1°“ f((" ' ‘ m—w~ - coIllpiared to the applied stress Figure 5,12 (12(30 psi) but are certainly finite. FD Corner Stresses 200 Referring once again to Figure 5.11, two other points are significant. First of all, the stress components for points on the symmetry axis are generally quite unrealistic. The lack of agreement between radial and circumferential stresses is an apparent discrepancy. Much better results are obtained using a quadratic extrapolation from the interior. The second point relates to interfacial stresses. From equilibrium considerations certain of these, as indicated in Chapter IV, should be continuous across the interface. Thus in this case, oz and Trz should be continuous across the horizontal interface whereas Or and Trz should be continuous across the vertical interface. This is apparently not the case. The discrepancies aregreatest near the corner. CSTR Solution. Three CSTR finite element solutions used in this analysis are shown in Figure 5.13. These include a 32 element layout with 25 nodal points, a 128 element layout with 81 nodal points, and a 288 element layout with 169 nodal points. The nodal circle force intensities associated with a uniform axial stress of 1200 psi are shown for each case. These have been multiplied by appropriate radii. Typical displacement curves for the CSTR solutions are plotted in Figures 5.14 and 5.15. The end displacements are shown in Figure 5.14. These displacements range from 3.06 X 10_3 in. to 3.43 X 10—3 in. for the highest order solution. The lower order displacements are consistently less than these. There is not an appreciable difference Iaetween the 81 and 169 point solutions indicating reasonably good con— ‘vergence of displacements. The axial displacement variation along the horizontal inter- I féice is shown in Figure 5.15. The trend in these curves is much the 201 m 0" fig; In N m H N H 3M 1».er ooérl mmémlll 0040.79! ooéolll mmdmlnl oo.om|ll 323+ mm.mml|l o.mmlll 0934' mmdlll z mmélel wmnwém Ill mmémfllll m.NHH IIT- mm.mm.l1l .mm Ilrl mN.om ulll m.nm mm.wa llru mNH.m IITI Figure 5.13 Composite Cylinder — CSTR Configurations 202 Dnr.a muamamouHamaa Hme< sum uoeaaHso memo .eH.m magmas meruzmuuwmzmomomutm me. mm. mm. 85 m 58 ES... mm 0 Jamem Emu SHE 8 + w Emu 58.. mm: . m nu 3 “n 3 N I... e m. m 11,12 mm m. +. U I..-u_.......-i_il..l‘ so \I‘I“ .85 203 no.“ mandamumaaman Hmwx< oommumuaH Haunouauom mamo .mH.m muswfim HmMIUZHuuHCZHanDUIm mm. mm. mm. E8 :5: mm 0 Emu Eu: 3 + mhmu FZHDm mm“ \‘ ‘ [q U ll»!!! 1 0 uu.u am. 00.“ mm.“ 80+3'(SBHONI)1N3H3381&SIU-H 204 same as in Figure 5.14. The tendency for displacements to be over- estimated in the stiffener is apparent. Additional displacement results are presented in Appendix A. Radial displacements are plotted in Figures A5 and A6 for points on the end horizontal interface respectively. In Figures A7 and A8 , displacements for various points are listed on cross sectional diagrams. Of particular interest are certain radial end displacements which are apparently underestimated by the method. Extrapolated displacements corresponding to selected points are listed in Figure 5.16, page 205. Both axial and radial displacements are shown. In the case of the maximum axial displacement for example, the best extrapolated result is a .162 improvement over the 169 point approximation. As was true for the FD results, extrapolation between the two lower order solutions gives results which are comparable to the 169 point solution. The stress distribution for the 169 nodal point CSTR solution is displayed in Figure 5.16. The stresses are listed in the order or, oz, 0 , and Trz for each point. These are nodal point stresses 0 obtained by averaging appropriate element stresses. As was explained in the earlier FD solution, two sets of stresses are shown along the mat— erial interface, one set corresponding to the matrix and the other to the stiffener. The overall situation is quite similar to the FD solution. Axial stresses in the stiffener as large as 4547 psi are observed. Com- pressive radial and circumferential stresses exceed 2600 psi along the horizontal material interface. The maximum observed shear stress is 896 psi. 2.9025(25) 7* 3.0339(81) 3.0670(169) v .2747(25) .2528(81) .2469(169) v v v .0893(25) .3340(25) fl 3.0477(E) 3.2108(E) .1992(8l) .4108(81) 3.0768(E) 3.2313(E) .2239(l69) .4293(l69) u u .4637(25) .7696(25) -.4555(E) - .4563(81) .760l(8l) -.4518(E) - .4529(l69) .7532(l69) v v .3883(25)j .2031(8S) .2505(E) .3891(E) .3890(81) .3028(81) .2451(E) .3934(E) .3924(l69) .3475(l69) u u .0728(25) .3242(25) -.O959(E) - .0937(81) .3785(81) -.1063(E) - .1034(l69) .3979(169) (25) - 25 Points (81) - 81 Points (169) - 169 Points (E) - Extrapolation Figure 5.16 205 CSTR Extrapolated Displacements 3. 3. 1. 1. 4189(E) 4349(E) .7591(E) .7512(E) 3332(E) 3554(E) .3842(E) .4037(E) 206 cg: firm 3%: -2I9 444 ;I52 -I04 -79 -50 '44 -I2 -2 I 97 I I I I I99 II II - . - - - . +%*%% II II 2| 17 I 39 44 3' 3' 12 20 ll -I03 -I43 -I43 -I4I 3a 303 -40 -57 «0 -I4 -I0 2 I2 44 I I II 4% ”IE -I - —l% ' ”3%" 44%;.“- J'Ea—‘+“'u”'- ’ 4“} II I9 34 40 7o 72 72 43 5| :4 2I ‘33: -9I ~9I -45 -4I -44 -54 5:257 -2I -II -I '2: I I I34 I I I I 4 II9 II I I fl. r315 :54 3% %— m - #r—. 49“ 43 I5 24 5I 74 94 no I22 II7 no 49 44 34 I47: I43; I43? I31 43 I.“ '2’ 12.. 3 A; 9 191;. . . 4 I404 344 m II II "l5'-'T - ’3 -4I 2r 35! 45? £ {T 45 -4 I9 27 53 79 "0 I42 I57 I44 I42 I20 42 44 I00 44 54 25 27 -9 -I2 29 44 4o 24 II I5 I513 I520 I53 I52I I509 I403 I244 II44 I074 I 7 94I ' 1;? 19 37 I5 39 3! 2T -51 - {I 4 I4 32 40 47 I42 204 203 I40 I29 43 42 m I94 I79 I04 I37 I45 3; 30: I04 37 43 7 I I I559 I593 I4I4 4 _I472____ I I I092 I004 955 m +‘fi I44 192* r ‘14: “—%I ‘11 ‘7‘" “.25 -24 -37 -9 -I 2 5 30 54 202 344 I74 I09 54 I4 294 374 am 409 1m 4m 73 I57I I504 I404 I449 I709 I020 I29I 337 3;; 347 404 399 444 234 4;! -37 -40 -4 -7 -9 - - - - m 9 I 474 471 44: ~I425 2514 >17 7* 41%? vh- 43 4Q} 22 23 I0 ‘7 I42I I I44 I305 I4I4 I445 MI 352( 5I7 m m 47 I4 I077 -2454 .2352 -2444 ~2074 -200I 442: 24 -I4 -27 73 I57 494 703 432 g" I09 -443 -I79 -340 mi .43 ~95 -II2 37 3:2! 031;: M-L-.JL_- ML _ 243i , “3349 454. I_.__,_ _449___ __47_I 7 .5- __ -244 -543 -504 “-454 ‘24? -25 I7 - 47 T 55 45 on T 44 -I4 50 I37 m 394 404 443 m 427 244 I49 43 7a 73 I79 49 I94 I45 m 9: I4 4 -34 -3I -22 -I0 2044‘ A 2049 ”A 2234, g 2437 ,__02903 H_,3404 -379? 309 #394. 5|: 4"” £2 724 III ”VT WI 75' 135' 53?“ IX 42 7V U 47 44 I54 204 425 554 55I 447 370 an 240 I42 97 47 an :34 309 244 m 4I I0 -4 -4 -5 I I7 I44 253 347 40I 232 I79 I33 44 45 379 W 300 254 no 70 29 I4 0 -2 m 44— 439—40—433— 49% .. 5H 40 92 I49 m 255 I47 I20 90 43 3I 334 244 244 230 :34. 7o 30 I4 3 -2 a 47 73 I03 I04 74 54 44 3I I7 344 254 25I 2Io I40 75 34 I9 4 0 agp_Ja§f__ha+_m .§}_A.g+_J_g}__tg+_a 24 4 35 20 49 I4 4| 4 I4 0 Figure 5.17, CSTR Stress Distribution 5 I I20 -57 1029 , 25 skis I3 -37 -|3 839 M -6 678 23 -5 -5 552 Q Q Q .4 rz 207 The symmetry axis stresses, though more reasonable than the FD stresses, are also apparently incorrect. stresses should be the same here but are not. the FD solution exhibits discrepancies with regard to certain stress components which should be continuous across the material interface. The radial and circumferential The CSTR solution like These discrepancies are more pronounced at the corner. The element stresses which were averaged to obtain the corner nodal point stresses are shown in Figure 5.18. The shaded region is part of the stiffener and the unshaded region is the matrix. The element stresses corresponding to the two materials are averaged to obtain the nodal point stresses of Figure 5.17. These averages are considerably smaller than the maximum element stresses. In view of larger stress predicted to the left and below the corner, the average stresses for the corner in Figure 5.17 seem quite unrealistic. LSTR Solution. employed the three configurations of Figure 5.19. 194 139 1932 I 1913 428 410 . 51 209/ 505 ' -61 1838 \ 1256 482 ‘\ 196 -33 - 648 -.. '\ -~.‘,\\. 7 , --/ i 4:29.49 \ -281 “figgzz530ie. \ 597 7.64; 1x§ ‘5 -25 . ~94 ; -6l I " 7 H ‘I 7' 209 \\ 1‘ 9a ‘67 ‘\ - 466 4 Figure 5.18 CSTR Corner Stresses The finite element analysis with LSTR elements Involved are 25, 81, and 169 nodal point systems comparable to those used in the CSTR solutions. The 25 point layout utilizes 8 elements, the 81 point 50. MIOMOMM (“900300303 0 O O O O MOMOMM O O O O O MOMOMM fl} 1 I 1 I I‘HH ' v v v 7 ¢ v v 4 D p D I I P v.” “WV-‘4' ‘ V ' ‘ ‘ .4“ . , ’ 7‘ '. 4 «I , - - - )‘. Figure 5.19 Composite Cylinder - LSTR Configurations 209 layout 32, and the 169 point layout 72. The axial nodal circle force intensities are shown for each case. They must be multiplied by corre— sponding nodal circle radii for use in the equilibrium equations. The variation of axial displacements for the cylinder end is shown in Figure 5.20. In the illustration, the displacements range from 3.1 X 10.3 in. at the symmetry axis to 3.47 X 10-3 in. at the cylinder periphery. As was true for the previous FD and CSTR solutions, conver— gence to the exact solutions apparently occurs from below. Horizontal interface axial displacements are plotted in Figure 5.21. In this figure, the 81 and 169 point solutions are seen to be very comparable. The 25 point solution is significantly different from' the others. The displacements of selected points, as predicted by the three approximate solutions, are listed in Figures A9 and A10 in Appendix A. It is evident from these figures that all displacements are not consistently underestimated or over estimated by the procedure. In particular, a number of radial displacements display somewhat of an oscillatory convergence. Radial displacements for points on the end and horizontal interface are plotted in Figures All and A12 respectively. Extrapolated displacements obtained from the Richardson Formula are presented in Figure 5.22. Although these are not significantly different from the 169 point solutions, they evidently do provide a further improvement in these results. Extrapolation between the 25 and 81 point solutions gives better results than the 169 point solution. The variation of stresses for the 169 point LSTR solution is Seen in Figure 5.23. The results presented are nodal point stresses ‘Which were obtained by averaging element stresses. The axial stress in the stiffener exceeds 5000 psi. Compressive radial stresses in excess 210 muamamomaauan Hmax< sum nonafiaso mama .o~.m shaman quruzm uhmzHumomDIm. no.“ mu- om. mm. no.9 q 4 + 4! IfiD.m Mn E3 53.. mm 0 +1.in Ema Sn: 8 + m Em... 58.. mm: . w w 3 H 3 N l nu N ml... Macaw mm 0 III u .IIIIIIIIIIIII 1.. 4\.\9II\III\II.I\ m Q1 0\\ \ 4“.“ 211 OD. mucQEmumHmmHa amax< monuwuaH Hmuconwuom mama .HN.m muswwm mmMIDZmewmzwumomuum mm. mm 33 Emma mm 0 Em; SE: 8 + E? SE: mm: . \ U) I‘L‘ .xJJ. CL. .1- C «J J . - . I I (.3 C.) C) CU+3~{QBHQNIllNBHBUHWdSIU—H 212 V v -o: 3.0776(25) .2123(25) .3264(25)fifi 3.0867(E) 3.2467(E) 3.0859(81) .2434(81) .4493(81) 3.1069(E) 3.2649(E) 3.1021(169) .2600(l69) .4669(169) u .4438(25) .6830(25) -.4520(E) .4512(81) ‘ .7540(81) -.4585(E) .4568(169) .7548(169) vi v 4 .2672(25) .3850(25) .3382(25) .2307(E) .3974(E) .2342(81) .3962(81) .37SO(81) .2487(E) .4033(E) .2454(169) .4017(169) .3820(169) u .0856(25) .3609(25) -.1111(E) .1087(81) .4075(81) -.1178(E) .1157(169) .4164(l69) (25) - 25 Points (81) - 81 Points (169) - 169 Points (E) - Extrapolation Figure 5.22 LSTR Extrapolated Displacements 3. 3. 4622(E) 4722(E) .7615(E) .7551(E) .3788(E) .3841(E) .4124(E) .4191(E) Figure 5.23, LSTR Stress Distribution 450 -20I 430 -230 400 -I50 -II7 -74 -50 -I5 4 0 . 1191 1 -I2 2 0 II I5 I7 ll I5 I4 I0 2 .4 I25: 157 -I50 ~I34 42! -I04 -00 -50 -40 -23 -4 -4 I 47 I255 I247 I240 I225 I22I I2I5 II97 IIaI H77 II7e _ - 454 JET 445 -IU“—"*‘132- -II9 407 RIF—I - -77 1547 -I0 22 35 45 43 70 73 7o 47 47 35 25 400 ' "0' '93 '90 I23; .68 -56 225 II? I}? ll: IINg I337 I333 I323 I300 I258 I244 I_ WM» 4 _ ,--° __ _ ._ J m. - -702 .Ivr“—I.Tuv’" 3r -90 ‘ "IF-31 ‘ -W -75 .70 . .41 -2 3o 43 37 II2 I24 I30 I29 I24 I02 72 I 42 I -5I . -44 —43 -55 -45 -34 -39 e 3 II 7 4 42% 1452. 3554-. 2 143.2- -5 IeLL - I399.” I395 ._ I I3 4 . ,st n49 I09. IosI -5 -40 ‘ -45 +42 I"-54 flI -s3 -49 ’ 4% -55 -44 -70 * -71 -3 . 3o 53 I 94 I24 I35 I03 I5I I44 I25 82 42 33 50 35 J '9 -49 -Is I -I4 24 . I07 3I egg | 903 I574 I5aI l582 I5 I544 I453 I I349 mg ____j01|5? _ _ 105;- “4 .' _ __ 7 33 55 “"44”“ “3 “‘ ' "'2’ " 20 ‘ -39' “*157 ’ -34 -55 * -46 I Esra 9 I9 37 74 I29 I95 250 233 I5I I5I 9a 52 I05 I90 l80 I55 II5 94 ; 43 I 54 95 77 4I ' I5 ' ?I__._.__L§B.._._ 1.522-- “J 20 _5._'Q$g_-__,, .119}. L721. 7___ _ ,.!J,&§.___--L|Ol¢ _ '00? __._ _-.?5.3 WC). ".1 I55 I9I I57 I79 I I79 I27 I 38 . -I4 -23 -34 -45 I5 -4 -2 I 24 24 73 22a - 337 I I24 II3 44 I9 I 344 349 340 ' 405 4I4 473 32 ' . I I553 I577 I4I9 ; I5eI I742 I574 I4I5 I I 344 370 375 405 I 422 449 244 -I50 -7I .35 -4 -2 I -20 - - - ”fig:- 442. - .355“- - w _ 9W. 882 949 I 4337w ~2660 ~2T7 - 5 -. -1108 L 38 : 75 2| l4 I0 _ I537 I393 959 II5I -509 2420 4205 . 492 I37 52 I9 -5 .2534 -2444 +2551 -2723 1-2475 I-23I7 -15..) , 35 -I49 -272 -355 I -303 532 93. I -509 -45I -527 -337 I-393 5I -49 .45 -I55 I -07 -49 -45 I ~I8 I me 1459____ J Ia3_4_ ___I2234 33.29 5047 23I 492 I 7I4 73I 704 I333 5 figs I. 4.525 r-5 T -45I ; -334 5I 39I - 22 I 45 52 57 I 54 I55 I5 II3 270 I43 an 509 543 490 20I I55 74 22 447 i 43 II4 I55 557 I54 53 43 -4 I -40 33 275 ;II I747 #795 L19“ 2209 259I __ 32 4'3 22% q_ _ I 5 “.4 5 - 9 _____ I .9, _ 457 - I II4 I72 35V Ita— I % _33 7'7 I—‘Hfi I I34 .1 II4 I 204 372 595 720 755 443 44I 345 I 244 I79 I04 54 377 I 320 303 . 204 I90 23I -H 94 42 I0 .I3 -7 o I 25 243; ,_ 2951____ 3220‘ _ Jgg J}: 7 ”450 7537_ ~ _ 408 454 70 I 35 ‘ 345 T359 ‘4’ 377 4 300 M33? 43 44% - Ta" If ‘ I "99 I ‘92 “ I4 I02 240 m 422 344 305 304 2I9 I54 I5I 55 44 327 m 3I7 257 I95 m 09 m 7I 33 I2 I -I m 3I04 '_3205‘ _,3359____ I 3520“ 2 222 I 3I7 409 492 545 544 132% 525 320 I299 “zirb‘n ""70" 'II 79" I II4 a4 75 ‘I -2 93 I44 220 255 224 I49 I73 I50 I22 99 75 4I 225 255 245 227 I75 I35 II2 II5 75 30 I7 ;I 54: 3240 3275 3335 343I _ _ 3532 ‘577 2I5 _ 305m .5 309 5‘96 _ -_ 5 4__ . ‘ ‘ L - %—-‘ ”233—“'th ““231”- 22r_4r W In * 33— * 30 55 7 -4I 5I 7I 03 I23 04 90 79 so 54 40 22 30 250 259 245 2I2 I73 I47 II5 I05 53 33 24 4 -I5 III 47 4 -I 5 0 0 7 0 2 -3 0 -4 o 3 12!? -73 ~16 2 l189 -57 -7 -I _ ms -56 1032 -69 935 -72 II 854 -52 -2 859 67 587 -I9 rz 214 of 2800 psi and compressive circumferential stresses in excess of 2700 psi exist along the material interface. The maximum shear stress is 934 psi. The results in Figure 5.23 seem very plausible. The only questionable stresses occur at the corner where the two materials join. The decreases in radial and circumferential stresses in the stiffener do not seem realistic. These decreases are the result of averaging carried out over 166 143 elements around the corner. Evidence 2138 2130 460 453 of this point is apparent in 60 311 Figure 5.24 where element S40 —79 2009 1. 1447 stresses are shown. 522 / 226 -52 X 767 Referring once again to Figure 5.23, the agreement of radial and circumferential stresses at the symmetry axis is significant. There is essentially no difference in these stress components for any of the points on the symmetry axis. Certainly, this is further evidence of Figure 5.24 the excellence of the LSTR solution. LSTR Corner Stresses There are, however, discrepancies in certain interfacial stresses. The axial stress should be continuous across the material interface but it apparently is not. 215 Comparison of Solutions. In the preceding discussion of dis- placement solutions, the FD, CSTR, and LSTR results were seen to display similar trends as the number of points was increased. In order to com- pare these results in more detail, a number of displacement and stress curves are included here corresponding to the highest order approxi- mation used for each method. The same displacements discussed earlier are used. The stress comparisons are made in terms of interfacial stresses which are generally the largest stresses for the various regions of the body. Axial displacements for the 3 solutions are listed in Tables 5.4 and 5.5. These same results, presented graphically in Figures 5.25 and 5.26, need little additional explanation. Axial displacements for the 3 solutions are very comparable. The LSTR solution is consistently better than the other solutions. The CSTR solution falls short of both the LSTR and FD solutions. It should again be noted that the axial displacements ar r = 0 for the difference solution are extrapolated results. Radial displacements are plotted in Figures 5.27 and 5.28. The end displacements are practically identical for the 3 solutions. Radial displacements along the horizontal interface differ only slightly. The LSTR solution is again best and the FD solution next best. The stress variation along the material interface is of part— icular interest. This is the region of maximum stress for both the matrix and the stiffener. Consequently, stress comparisons are made in terms of these regions. The interfacial stresses corresponding to the various solutions are tabulated on pages which follow. These same results are presented 216 Table 5.4 Composite Cylinder End + w-Displacement X 10 3 231 Point r-Coord. Finite Diff. r-Coord. 0. 3.06009 0. .1 3.07403 .08333 .2 3.10145 .16666 .3 3.19698 .25 .4 3.18106 .33333 .5 3.23094 .41666 .6 3.28273 .5 .7 3.33222 .58333 .8 3.37573 .66666 .9 3.41084 .75 1.0 3.43700 .83333 .91666 1.0 169 Point CSTR .06694 .07794 .08960 .11678 .12577 .18456 .22391 .26598 .30623 .34236 .37264 .40430 .42934 169 Point LSTR 3.10207 3.11008 3.12732 3.15033 3.18033 3.21873 3.25999 3.30169 3.34268 3.37960 3.41214 3.44154 3.46693 217 muamemuwaamaa Hmax< vam mo comwumuaoo .mN.m muswam ”muruza uwmzHomnom1m no.3 mu. om. mm. no.0 q . 1r 4 q AI DD. m . m E SHE 8m 0 .ommm ES 58... mm: + mm 33 SS... mm: . m U .3 H 3 N l D N nu. Egmw h. .+. U B lama 218 Table 5.5 Composite Cylinder +3 Horizontal Interface w-Displacement x 10 231 Point 169 Point 169 Point r-Coord. Finite Diff. r-Coord. CSTR LSTR 0. .25098 0. .24688 .24535 .1 .25308 .08333 .25439 .24533 .2 .26456 .16666 .25847 .25171 .3 .28617 .25 .27776 .26938 .4 .32339 .33333 .29641 .29399 .5 .39318 .41666 .33674 .33452 .6 .82465 .5 .39235 .40165 .7 1.04466 .58333 .76011 .78139 .8 1.18220 .66666 .97726 1.01132 .9 1.27788 .75 1.11206 1.14399 1.0 1.35295 .83333 1.21204 1.24273 .91666 1.28542 1.31620 1.0 1.34747 1.38196 219 mucmaoumflamwa Hmfix< mommuoucH Hoodonwuom mo comwumaeoo .om.m muswflm mmMIDZHuwHIZHDmmnuim U7 (11 00.0 q a . a E EHE 8m O 58 5.3.. mm: + ES :5: mm: . 00.0 CD+3*t83H3NI)lN3N3UHWdSIU-H 220 madmaooaaaman Hmavmx com «o comfiumaeou .nu.m ouawam MmMIDZkupIZHOm0omcm on.” ma. mm. mm. 85 . 4 4 . use E 23.. Hmm O \n\\ x 38 :EE mm: + v...\\\ ES 58.. mm: . ...\ 1:. 1 10m. 80+3~(83HON1)1N3N3DUWdSIU-fl 221 muooamumaamfin Hmwvmm ovumumucH HMuGONfiuom mo aomfiumaaoo .mm.m muswam HmuIQZHuu~02H0m000'm on.“ ma. om. mm. oo.o fl 1.1 |« 1 uq—D.D E SHE “mm 0 . Emu SHE mm: + ES SHE mm: . . 4mm. \ 40m. a .E. .84 80+3“(83HON111N3H33818810-fl 222 graphically as well. In Tables 5.6 and 5.7 and in Figure 5.29, radial stresses along the horizontal interface are shown. The three methods give very comparable radial stress in the matrix where the stress level is quite low. In the stiffener, the radial stress variation is quite different. The finite element solutions are very erratic and seem less reasonable than the finite difference stresses. Much the same behavior was observed in Chapter 4 with regard to the composite plate example. It was noted there that this behavior depends to some extent on the finite element configurations. Also, one could use smaller triangles around the interface to improve the situation. The axial stress variation along the horizontal interface is given in tables on pages 227 and 228. These results, which are also plotted in Figure 5.30, are more comparable than radial stresses in the same region. The methods predict axial stresses of more than 3500 psi in the stiffener and 1800 psi in the matrix. Axial stress variations along the vertical interface are given in Tables 5.10 and 5.11. In Figure 5.31, this same information is seen to display a similar trend for the three methods. Fairly significant variations in axial stress occur near the corner of the stiffener. Typical of this is the maximum axial stress which is 4831 psi for the FD solution, 4547 psi for the CSTR solution, and 5047 psi for the LSTR solution. The final comparison here pertains to the circumferential stress along the horizontal interface. These stresses are shown in Tables 5.13 and 5.14. The plotted results in Figure 5.32 are more (:omparable in the matrix, where stresses are tensile and relatively ssmall, than in the stiffener where stresses are compressive and rather ZLarge. It has previously been mentioned that FD stresses along the r-Coord. 1.0 223 Table 5.6 Horizontal Interface Stress or (Matrix) 231 Point 388 .0 387 .08333 401 .16666 427 .25 485 .33333 34 .41666 -146 .5 - 55 .58333 - 20 .66666 - 6 .75 - 2 .83333 .91666 1.0 169 Point CSTR 296 378 310 409 319 481 73 -151 - 37 -40 169 Point LSTR 364 369 360 405 414 473 32 -l60 - 71 - 35 224 Table 5.7 Horizontal Interface Stress or (Stiffener) 231 Point 169 Point r-Coord. Finite Diff. r-Coord. CSTR .0 -2319 .0 -l425 .l -2451 .08333 -2616 .2 -2477 .16666 -1752 .3 -2445 .25 —2351 .4 -2103 .33333 -l49l .5 —1803 .41666 -1750 .5 - 950 169 Point LSTR -2537 -2660 -2173 -2735 -2843 -2227 -1108 225 C") no mommuoucH anaconauom .m~.m ouswwm 6 191053 3. (184)39361 NmMIUZHememszwmmm-u m.a ma. um. mm. uu.m fi 11 _. . a_mmm- AmmzmauHememE em 1mm VA HEMZMEEHemWEHmu em mma Av mmwzmutHememems em mud Au HXHmemzwml em amm mw ,moma MXHmemzememo em mma pr < flXHmemzwmema em mma 1. imam a 92.0 mm... .0 .7 x. + Umrv iw r .mmm r-Coord. 1.0 1496 1555 1585 1650 1776 1399 939 896 876 867 868 226 Table 5.8 Horizontal Interface Stress oz (Matrix) 231 Point Finite Diff. r-Coord. .0 .08333 .166666 .25 .33333 .41666 .58333 .66666 .75 .83333 .91666 1.0 169 Point CSTR 1571 1584 1604 1649 1709 1828 1291 921 905 874 871 865 867 169 Point LSTR 1553 1577 1619 1681 1742 1874 1415 968 943 919 882 869 865 227 Table 5.9 Horizontal Interface Stress oz (Stiffener) r-Coord. Finite Diff. 1325 1555 1565 1724 2265 4831 231 Point 169 Point 169 Point r-Coord. CSTR LSTR .0 1621 1637 .08333 1164 1393 .16666 1305 959 .25 1414 1151 .3333 1845 1509 .41666 2481 2420 .5 3520 4206 228 no oomwumuou HmuGONHuom .om.m muawwm ”mmrqugmpmzwmwmmmuw 7......) DD.w Wm. Om. MW LC C 4 . . . mmw I! Z A. o O : A,» XA Ia ‘ DJ 3.1“}! OWN ) 3 mm. 1: II. 1c . x \1 “mmzunueememu ea Mmm u Av Jomum mmuzmunHemwmemu em mpm » mmuzuuaHemcmema em mew A0 axememzwma ea 3mm wt MXHmemZWWRmm em mma m0 MXHmemZWWEmJ an mam + “av luumm A JDDmT (Isa1593a19 WHIXH z-Coord. 0. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 L9 2.0 1203 229 Table 5.10 Vertical Interface Stress oz (Matrix) 231 Point Finite Diff. z-Coord. 199 0.0 199 .16666 198 .33333 198 .5 134 .66666 195 .83333 192 1.0 187 1.16666 175 1.33333 140 1.5 1399 1.66666 1871 1.83333 1609 2.0 1465 1379 1321 1279 1248 1224 1209 169 Point CSTR 259 210 266 314 309 218 1291 1682 1403 1316 1256 1222 1196 169 Point LSTR 213 215 222 215 223 231 1415 1727 1369 1305 1246 1221 1195 z—Coord. 1.0 230 Table 5.11 Vertical Interface Stress oz (Stiffener) 231 Point Finite Diff. 3637 3638 3646 3667 3719 3820 3998 4267 4600 4802 4831 z-Coord. .166666 .33333 .5 .66666 .83333 1.0 169 Point CSTR 3565 3600 3572 3785 3792 4547 3520 169 Point LSTR 3590 3577 3622 3676 4198 5047 4206 o oommuoucH Hmofiuum> .Hm.m ouswwm 231 lumua Juumm ”mmzuuuHemwmu em emm ”mmzuaanmwmemu em mme MmmzmnnHemlmema em mma Juan; MXHmszvmm Hm 0mm mmeHmzwmwmm em 00H mxmwwmxdmpmq ed mmw +C)E]A<>x I. 1.431% A LC U (ISdlaSBdlS WBIXH 232 symmetry axis are unlikely to be valid, and thus extrapolation from the interior is desirable. In Figure 5.32, the FD stress at r = 0 is an extrapolated result. The CSTR stresses in the stiffener exhibit to some extent the erratic behavior of the radial stresses in this same region. Concluding Remarks. The axially symmetric finite difference and finite element formulations are apparently equally acceptable in their ability to predict displacements in simple composite solids. Although no numerical comparisons have been employed, convergence to exact displacements is quite comparable for the three methods. The formulations, however, lead to somewhat different solutions insofar as stress is concerned. Certain stress components are very similar, for example, the axial stress in the axially loaded composite solid. Other stresses, however, tend to be very much different. The radial stresses and circumferential stresses displayed different variations along the material interface. The finite element stresses were more erratic. In this connection, best fit curves seem to be more realistic. r-Coord. 1.0 233 Table 5.12 Horizontal Interface Stress 231 Point Finite Diff. 506 386 399 422 471 265 23 19 l6 14 12 0e (Matrix) r-Coord. .08333 .16666 .25 .33333 .41666 .58333 .66666 .75 .8333 .91666 1.0 169 Point CSTR 337 376 367 404 399 468 238 43 41 22 23 18 18 169 Point LSTR 364 370 378 405 422 469 246 38 28 21 14 10 234 Table 5.13 Horizontal Interface Stress o0 (Stiffener) 231 Point 169 Point 169 Point r-Coord. Finite Diff. r-Coord. CSTR LSTR 0. 161 -1077 -2536 (-2508)* .l -2488 .08333 -2654 —2646 .2 -2467 .16666 ~2352 —2651 .3 -2447 .25 -2444 -2723 .4 —2271 .33333 -2078 —2678 .5 -l401 .41666 —2001 -2317 .5 -1425 -1540 *Extrapolated 235 mu mommuouoH Handoufiuom .Nm.m ouawwm ”mmiuzwemwmszWQUan 00.« mm. mm mm mm 1 — “muzmunHemwmu em emm VA ”mmzmunaemememu Hm mmfi Av MmmzmunHememems ea mas A0 MXHmemzema em Hmm m0 wamemzememw as was no MXHmemzemems ea mma + mww» mMu memu 10 mm mw ”1 B 3 L) C3 L) H C3 U ' (1931883818 Wallwsaslwnaal VI. CONCLUSIONS AND RECOMMENDATIONS Finite element and finite difference methods have been formulated for plane and axially symmetric stress analysis. Stiffness influence coefficient matrices for the direct stiffness method have been discussed. These relate to the constant and linearly varying strain triangle and triangular ring elements. Nodal point forces associated with arbitrary distributed boundary loads were treated for each case. The formulation of the difference method was done in terms of the Navier equations of Classical Elasticity Theory. Difference equations were derived by expressing the equilibrium of a material element. Involved are assumptions of strains in terms of displacement differences. The methods were applied to both plane stress and axially symmetric elastostatic problems. Two examples with well known elasticity solutions provided an excellent basis for comparison for both stress and deformation analysis. 6.1 Conclusions The finite difference and finite element methods have proved to be very capable in the deformation analysis of elastic solids. The finite element solutions employing linear strain triangular elements (LST) were consistently better than either the finite difference 236 237 solutions or the finite element solutions which employed constant strain triangular elements (CST). The improvement was not excessive in every case when a comparable number of points was utilized. In the cantilever beam problem, however, the LST solution for deflections was signifi— cantly better. The finite difference (FD) solutions were comparable to the CST solutions but generally gave somewhat better results. In the analysis of stress, nodal point stresses, obtained by averaging element stresses, were employed. A similar concept was employed with regard to the difference method whereby nodal point stresses were obtained by averaging the stresses for the regions around the mesh point. In this connection, the LST formulations were signifi- cantly better than the CST and FD formulations. This was particularly true along free boundaries. The FD solutions were generally better than the CST solutions and in some cases were comparable to the LST solution. It should be mentioned, however, that in regard to certain interfacial stresses associated with composite materials, both finite element methods exhibited erratic stress variations suggesting the desirability of using best fit curves in interpreting these results. The finite difference method, on the other hand, gave much smoother stress variations and consequently appeared to be more realistic. 6.2 Recommendations The capabilities of the finite element and finite difference methods, in the analysis of elastostatic problems, have been demon- strated for a limited class of applications. The presentation of the difference method in particular was limited to special geometrical situations. In view of this as well as results obtained in this 238 investigation, a number of recommendations are advanced. These follow immediately. Difference Method. The scope of the finite difference method, as presented in this work, included both plane and axially symmetric analysis. The formulation of boundary equations, however, was limited to surfaces parallel to the coordinate surfaces of the problem. It is possible to approximate other situations within the framework of this analysis by treating arbitrary boundaries as a series of broken lines as in Figure 6.1a for example. This technique has not been investigated here. To achieve any degree of accuracy, it would no doubt be necessary to utilize a variable mesh spacing. This last point has not been developed here either. A more suitable approach from a geometric stand point would eliminate the ragged edge as in Figure 6.1b. The development of boundary equations for such situations has not been pursued here and appears at first glance to present difficulties if considered from the equilibrium point of view as in sections 2.8 and 2.9. The method requires further development and application in addition to that discussed above. The possibility of treating 3-dimen— sional problems with simple boundaries presents no difficulties. The procedure of section 2.8 would be applied to the 3—dimensional Navier Equations. Computationally, however, there may arise problems associated with computer memory capability. For example, where each plane stress or axially symmetric equation involves l8 non-zero coefficients, the 3-dimensional equations involve 57 non-zero coefficients. Thus for problems with comparable numbers of mesh points, 239 ‘\ * A -_4...—-.— I i ‘\ Figure 6.13 Finite Difference Approximation for Present Analysis .4- .._ Figure 6.1b Possible Finite Difference Approximation Square Plate with a Circular Hole 240 approximately‘+.22 times as much memory capability is required. Stiffness Method. The plane stress triangle utilizing either constant or linear strain variation has been rather thoroughly investi- gated by various authors. The constant strain triangular ring has been used extensively as well. From the limited scope of this research, the linear strain triangular ring appears to give the same degree of improvement in axially symmetric problems that the linear strain triangle gives in plane stress analysis. It is therefore recommended that additional applications of this element be treated to indicate its full capability. 10. 11. BIBLIOGRAPHY Runge, 0., "fiber eine Methode die partielle Differentialgleichung Au = Constans numerish zu integrieren," Z. Math. Phys., Vol. 56, 1908, p. 225. Richardson, L. F., "The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, with an Application to the Stresses in a Masonry Dam," Trans. Roy. Soc., London, Series A, Vol. 210, 1910, p. 307. Marcus, H., "Die Theorie Elastischer Gewebe Und Ihre Anwendung Auf Die Berechnung Elastischer Platten," Armierter Beton, 1919, p. 107. Hencky, H., "Die Berechnung dunner rechteckiger Platten mit verschwindender Biegungsteifigkeit," Z. Agnew, Math. Mech., Vol. , 1921, p.81, and Vol. 2, 1922, p. 58 Southwell, R. V., Relaxation Methods, Vols. 1, II, III. Langefors, B., "Analysis of Elastic Structures by Matrix Transform— ation with Special Regard to Monocoque Structures," Journal of the Aeronautical Sciences, Vol. 19, No. 7, 1952, pp. 451—458. Argyris, J. H. and Kelsey, 8., Energy Theorems and Structural Analysis, Butterworth, London, 1960. Hrennikoff, A., "Solution of Problems in Elasticity by the Frame— work Method," Journal Appl. Mech., Vol. 8, No. 4., December 1941. McHenry, D., "A Lattice Analogy for the Solution of Plane Stress Problems," Journal Inst. Civil Eng., December 1943. Parikh, K. S., and Norris, C. H., "Analysis of Shells using Frame— work Analogy," World Conference on Shell Structures, October 1962, pp. 213-222. Turner, M. J., Clough, R. H., Martin, H. C. and Topp, L. J., "Stiffness and Deflection Analysis of Complex Structures," J. Aeron, Science, 23, No. 9, September 1956. 241 12. 13. 14. 15. l6. l7. l8. 19. 20. 21. 22. 23. 24. 242 Argyris, J. H., Kelsey, S., and Kamel, H., Matrix Methods of Structural Analysis. A Precis of Recent Developments, AGARDograph 72, Ed. de Veubeke, Permagon Press, 1964. Argyris, J. H., Kelsey, S., "Initial Strains in the Matrix Force Method of Structural Analysis," Journal of the Royal Aero- nautical Society, August 1960, pp. 493-495. Turner, M. J., Dill, E. H., Martin, H. C., and Melosh, R. J., "Large Deflections of Structures Subjected to Heating and External Loads," Journal of Aero/Space Science, Vol. 27, No. 2, February 1960. Argyris, J. H., "Reinforced Fields of Triangular Elements with Linearly Varying Strain; Effect of Initial Strains," Journal of the Royal Aeronautical Society, November 1965, pp. 799-801. Fraeijs de Veubeke, B. M., "Upper and Lower Bounds in Matrix Structural Analysis," A Precis of Recent Developments, AGARDograph 72, Ed. de Veubeke, Permagon Press, 1964. Argyris, J. H., "Triangular Elements with Linearly Varying Strain for the Matrix Displacement Method,” Journal of the Royal Aeronautical Society, October 1965, pp. 711-713. Fellippa, C. A., "Refined Finite Element Analysis of Linear and Non-Linear Two Dimensional Structures," Ph.D., Dissertation, California Univ., Berkeley, 1966. Melosh, R. J., "A Stiffness Matrix for the Analysis of Thin Plates in Bending," Journal of AerOSpace Sciences, January 1961, pp. 44-43. Argyris, J. H., "Matrix Displacement Analysis of Plates and Shells," Ingenieur Archiv., Vol. 35, No. 2, 1965. Bogner, F. K., Fox, R. L., Schmit, L. A., "A Cylindrical Shell Discrete Element," AIAA Journal, Vol. 5, No. 4, April 1967. Clough, R. W., and Tocher, J. L., "Finite Element Stiffness Matrices for Analysis of Plate Bending," Proceedings of Confer— ence on Matrix Methods in Structural Mechanics, Wright Patterson Air Force Base, Ohio, 1965. Zienkiewicz, 0. C., "Finite Element Procedures in the Solution of Plate and Shell Problems," Stress Analysis, edited by 0. C. Zienkiewicz, and G. S. Holister, John Wiley and Sons, 1965. Argyris, J. H., "Three-Dimensional Anistropic and Inhomogeneous Elastic Media Matrix Analysis for Small and Large Displace- ments," Ingenieur Archiv., Vol. 34, No. 1, January 1966, pp. 33-55. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 243 Wilson, E. L., "Structural Analysis of Axisymmetric Solids," AIAA Journal, Vol. 3, No. 12, December 1965. Clough, R. W., and Rashid, Y., "Finite Element Analysis of Axi- Symmetric Solids," Journal Eng. Mech. Div., ASCE, 91, February 1965, pp. 73-85. Argyris, J. H., "Matrix Analysis of Three-Dimensional Elastic Media Small and Large Displacements," AIAA Journal, Vol. 3, No. 1, January 1965, pp. 45-51. Klein, Bertram, "Application of a New Matrix Method to Vibration Analysis of Structures," Journal of Aerospace Sciences, March 1962, pp. 350-351. Flower, M., Severn, R. T., and Taylor, P. R., "Static and Dynamic Analysis of Plates and Shells Using the Finite Element Method," Paper presented at the International Symposium on the use of Digital Computers in Structural Engineering. Dawe, D. J., "A Finite Element Approach to Plate Vibration Problemsf' Journal of Mechanical Engineering Science, Vol. 7, No. 1, 1965. Taylor, R. L. and Chang, T. Y., "An Approximate Method of Thermo- viscoelastic Stress Analysis," Nucl. Eng. and Des. 4, 1966, pp. 21—28. Taylor, R. L., "Methods of Thermoviscoelastic Stress Analysis in Concrete Reactor Vessels," Nucl. Struct. Eng. 1, 1965, Sokolnikoff, I. S., "Mathematical Theory of Elasticity," McGraw- Hill Book Company, Inc., New York, 1956. Fraeijs de Veubeke, B. M., "Displacement and Equilibrium Models in the Finite Element Method," Stress Analysis, edited by 0. C. Zienkiewicz and G. S. Holister, John Wiley and Sons, 1965. Clough, R. W., "The Finite Element in Structural Mechanics," Stress Analysis, edited by 0. C. Zienkiewicz and G. S. Holister, John Wiley and Sons, 1965. Archer, J. S., "Consistent Matrix Formulations for Structural Analysis using Finite Element Techniques," AIAA Journal, Vol. 3, No. 10, October 1965, pp. 1910-1918. Tocher, J. L., "Selective Inversion of Stiffness Matrices, Journal Struct. Div., ASEE, 92, February 1966, pp. 75-87. Wilson, E. L., "Finite Element Analysis of Two Dimensional Structures," Ph.D., Dissertation, California Univ., Berkley, 1963 C 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 244 Timoshenko, S. and Goodier, J. W., Theory of Elasticity, McGraw- Hill Book Company, Inc., New York, 1951. Ralston, A., A First Course in Numerical Analysis, McGraw—Hill Book Company, Inc., New York, 1965. Denke, P. H., "The Matrix Solution of Certain Nonlinear Problems in Structural Analysis," Journal of Aeronautical Sciences, Vol. 23, No. 3, March 1956, pp. 231-236. Melosh, R. J., "Basis for Derivation of Matrices of the Direct Stiffness Method," AIAA Journal, Vol. 1, No. 7, July 1963, Martin, H. C., "Plane Elasticity Problems and the Direct Stiffness Method," Trend in Engineering, Vol. 13, January 1961, pp. 5-8, 19. Allen, D. N. de G., and Wendle, D. W., "The Finite Difference Approach," Stress Analysis, edited by O. C. Zienkiewicz and G. S. Holister, John Wiley and Sons, 1965. Melosh, R. J., "Structural Analysis of Solids," J. Struct. Div., ASEE, 89, August 1963, pp. 205-223. Clough, R. W., "The Finite Element Method in Plane Stress Analysis," Proceedings, ASEE 2nd Conference on Electronic Computation, Pittsburgh, Pa., September 1960. Turner, M. J., Martin, H. C., and Weikel, R. C., "Further Develop— ments and Applications of the Stiffness Method," Matrix Methods of Structural Analysis. A Precise of Recent Developments, AGARDograph 72, Ed. deVeubeke, Pergamon Press, 1964. Davis, C. L., "Iterative Solutions of Plane Elastostatic Problems," Ph.D., Dissertation, Michigan State Univ., 1965. Wilson, E. L., "Matrix Analysis of Non-Linear Structures," Proceeding, ASEE 2nd Conference on Electronic Computation, Pittsburgh, Pa., September 1960. Martin, H. C., "Truss Analysis by Stiffness Considerations," Journal of Engineering, Mech. Div., ASCE, October 1956. Przemieniecki, J. S., "Triangular Plate Elements in the Matrix Force Method of Structural Analysis," (TN), AIAA Journal, Vol. 1, No. 8, August 1963. Irons, B. M., "Engineering Applications of Numerical Integration in Stiffness Methods," AIAA Journal, Vol. 4, No. 11, November, 1966, pp. 2035-2037. 53. 54. 55. 56. 57. 58. 59. 60. 245 Lasker, G. L., "Derivation of an Arbitrary Triangular Plate Bending Stiffness Matrix and its Application to Large Deflec- tion Shell Problems," Ph.D., Dissertation, Michigan State Univ., 1966. Argyris, J. H., "0n the Analysis of Complex Elastic Structures," Applied Mechanics Reviews, Vol. II, No. 7, July, 1958. Gallagher, R. H. Padlog, J., Bijlaard, B. P., "Stress Analysis of Heated Complex Shapes," ARS Journal, 32, May 1962, pp. 700-707. Pian, T. H. H., "Derivation of Element Stiffness Matrices," AIAA Journal, Vol. 2, 1964, pp. 576-577. Przemieniecki, J. S., "Tetrahedron Elements in the Matrix Force Method of Structural Analysis," AIAA Journal, Vol. 2, No. 6, June 1964. Pian, T. H. H., "Derivation of Element Stiffness Matrices by Assumed Stress Distributions," AIAA Journal, (TN), Vol. II, No. 7, July 1964, pp. 1333—1336. Holister, G. S., and Thomas, C., Fibre Reinforced Materials, Elsevier Publishing Co., New York 1966. Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944. APPENDICES Appendix A ADDITIONAL RESULTS Additional results were made reference to in Chapter V with regard to displacements for the composite cylinder problem. These results follow immediately. 246 247 00. a mm. muaoEmomHmmflaln 0cm uwwcwazo 0m ..H< muswfim H00102H00~02H0m000um 0m. 7 J Li (‘1' ‘ I hZHOm :00: my H2H0m 1mm: Hm HZHDm 1mm: mm" szom :00: «mm ‘\ T 4 ~+OCJ l 1 1 mm. 0m. 00. H FOw3“[S3HONI)1N3NBUUWdSIO"O 248 0n0.fi wuaoamomHamHalz oomwuoucH Hmucouwuom am ..~< wuswwm mmmrquuuHmZHum000um me. am. m. mom 53... IE: .9. D SHE IE: 8 O SHE :8: mm“ + .53.. :E: a .m . i J i 0m; 3HJNI)1N3H3381dSIO-fl (‘ 1) ~3~ B 249 3.1833(45) 3.1775(91) 3.1694(153) 3.1622(231) 2.9778(E) 3.0513(E) 3.0811(E) 3.0957(5) 1.6724 1.6965 1.7111 1.7202 3.0569(45) 3.1177(153) 1.7076 1.7675 .2735 .2610 .2555 .2528 .1474 .1465 .1459 .2932 .2772 .1595 .1567 3.1360(45) 3.1947(91) 3.2207(153) 3.2345(231) 1.8414 1.9034 1.9327 1.9486 .3834 .3873 .3909 .3937 .1835 .1804 .1785 .1774 3.2643(45) 3.3432(153) 2.0729 2.1458 1.0338 1.1083 .4212 .4419 (45) - 45 Point FD (91) - 91 Point FD (153) - 153 Point FD (231) - 231 Point FD (E) - Extrapolation «m—n Figure A3. FD Axial Displacements 3.3417(45) 3.3983(91) 3.4237(153) 3.4374(231) 2.2049 2.2507 2.2701 2.2803 1.2591 1.3162 1.3410 1.3540 .5288 .5509 .5610 .5665 250 -.2260(45) -.4372(45) -.6057(45) -.7451(45) -.4477(91) -.7503(91) -.2398(153) -.4516(153) -.6145(153) -.7511(153) -.4534(231) -.7511(231) -.l850 -.3639 -.5130 -.6371 -.3743 -.6459 -.1934 -.3785 -.5261 -.6497 -.3806 -.6516 -.0453 -.0874 -.2465 -.3649 -.0974 -.3884 -.0528 -.1039 -.2832 -.4002 -.1082 -.4068 -.0138 -.0338 -.0706 -.1523 -.0313 -.1578 -.0117 -.0304 -.0714 -.1603 -.0300 -.1617 (45) — 45 Point FD (91) - 91 Point FD (153) - 153 Point FD (231) - 231 Point FD -.0162 -.0361 -.0590 -.1233 -.0358 -.1265 -.0162 -.0357 -.0588 -.1280 -.0357 -.1287 Figure A4. FD Radial Displacements 251 00.H mucosoomaamfialn 0cm umvcwazo 08mm ..m¢ shaman ”muruzmwuwmzwmw000lw 0m. mm. 58 SEE mm 0 SE SHE E + 58 :EE mm: _. :h C: [:1 c-{ FU-}~[SJHONlllNJHJOHWdSIU-n 252 muamEmomHamfiQI: mesmumucH HmDGONHpom «Emu ..o< musmfim ”mmruzwwwwmzwmuummam on; me. E E. m \. Emu SHE mm 0 o \\\\\\ E8 SHE. ..m + memo Hzmuu mme . \ ‘ a. C) Eh: c: _l C) C) '11 DH.” . C) lUWdSlU-fl f \ lNHHl \ | l HHJNI. , .7 if ‘:0~- 3.. f ' . L. 253 2.9025(25) 2.9702(25) 3.0339(81) 3.0848(81) 3.0669(169) 3.1168(169) 1.6038 1.6149 1.6767 1.7359 1.7052 1.7666 .2747 .3114 .2528 .2806 .2469 .2778 .1508 .1609 .1468 .1582 .1446 .1565 3.0893(25) 3.1992(81) 3.2239(169) 1.7894 1.9117 1.9396 .3883 .3890 .3924 .1881 .1798 .1770 3.2400(25) 3.3272(81) 3.3424(169) 2.0349 2.1264 2.1482 .9873 1.0877 1.1121 .4018 .4336 .4418 (25) - 25 Point CSTR (81) - 81 Point CSTR (169) - 169 Point CSTR 3.3340(25) 3.4171(81) 3.4293(169) 2.1929 2.2594 2.2731 1.2031 1.3208 1.3475 .5139 .5505 .5635 CSTR Axial Displacements Figure A7. 254 -.2434(25) -.4637(25) -.6228(25) -.2464(81) -.4563(81) -.6213(81) -.2445(169) -.4529(169) -.6143(169) -.1684 -.3417 -.5230 -.l883 -.3802 -.5253 -.l937 -.3779 -.5272 -.0434 -.0728 -.2102 -.0517 -.0937 -.2556 -.0554 -.1034 -.2796 -.0119 -.0379 -.0369 -.0103 -.0278 -.O666 -.0106 -.0299 -.O679 (25) - 25 Point CSTR (81) - 81 Point CSTR (169) - 169 Point CSTR -.0158 -.0317 -.0752 -.0159 -.0355 -.0565 -.0164 -.0355 -.0601 Figure A8. CSTR Radial Displacements -.7696(25) -.7532(169) - o 6393 -06480 -06484 -.3242 -.3785 -.3979 -.1354 -.1555 -.1606 -.1203 - o 1265 -.1279 255 mucoEmomHamHoI: 0cm Hmvawamo memq ..m< musmwm ”muIDZHLMHmZHOm000vm E. E. .3. 8.4.0 .mm .53 SHE mm 0 Ea... SHE S + mums szou mus - .3. .3 (3 (3 C3 CU-3~[SBHONIllN3HBUHWdSIO-n 256 00. mucoEmomHamHQI: momwumucH Hmucouwuom 09mg ..oa< ouswwm “00:02H00902H0000010 00. ES .53.. mm 0 EB EHE ..m + memJ ezHou mme . a. \ A _L um. mm. um. JHWdSIU-n r. I EO-3N[S3HONI)1N3H3 257 3.0776(25) 3.0990(25) 3.0859(81) 3.1389(81) 3.1024(169) 3.1503(169) 1.6273 1.7008 1.7140 1.7871 1.7358 1.7985 .2672 .2928 .2342 .2733 .2454 .2694 .1397 .1497 .1450 .1608 .1456 .1561 3.2123(25) 3.2434(81) 3.2600(169) 1.8652 1.9631 1.9759 .3850 .3962 .4017 .1836 .1732 .1755 3.3144(25) 3.3615(81) 3.3796(169) 2.1016 2.1611 2.1804 1.0248 1.1290 1.1440 .4192 .4462 .4520 (25) - 25 Point LSTR (81) - 81 Point LSTR (169) - 169 Point LSTR Figure All. LSTR Axial Displacements 3.3264(25) 3.4493(81) 3.4669(169) 2.2481 2.2837 2.3002 1.3382 1.3750 1.3820 .5496 .5676 .5828 258 -.2297(25) -.4438(25) -.5831(25) -.2463(81) -.4512(81) -.6180(81) -.2467(169) -.4568(169) -.6l69(169) -.1857 -.3721 -.5503 -.2007 -.3873 -.5253 -.1991 -.3860 -.5314 -.0476 -.0856 -.2328 -.0588 -.1087 -.2482 -.0582 -.1157 -.3004 -.0108 -.O331 -.0555 -.0115 -.0277 -.0704 -.0105 -.0291 -.0726 (25) - 25 Point LSTR (81) - 81 Point LSTR (169) - 169 Point LSTR -.0145 -.0330 -.0575 -.0164 -.0365 -.0588 -.0165 -.0359 -.0579 Figure A12. LSTR Radial Displacements -.6830(25) -.7540(81) -.7548(169) -.6729 “06497 - 06540 -.3609 -.4075 -.4l64 -.l419 -.1620 -.l648 -.1514 -.1306 -.1340 Appendix B COMPUTER PROGRAMS The computer programs which were used in the analysis of axially symmetric problems by the difference method and by the LSTR finite element method are presented on the following pages. Included is a brief description of each program, the program itself, and sample data. The output of each program tends to be excessive and consequently is not presented. 259 FINITE DIFFERENCE PROGRAM FOR AXIALLY SYMMETRIC ANALYSIS The axially symmetric finite difference analysis involves 3 main steps: (1) generate equilibrium equations, (2) solve the system of algebraic equations for displacements, and (3) calculate stresses. It is possible to obtain mesh point loads directly as part of the computer analysis, but this is not a feature of the present program. Mesh point loads must be computed outside the program and are thus handled as input information. The equations of section 2.9, namely (2.62), are the equilibrium equations related to the first step in the analysis. In the original program (included here), these were coded in a more simplified form . l . . . . . w1th h = k and v =-— It 18 not difficult to remove these restrictions. 4. The coding of the equations is accomplished in terms of 4 subroutines called COFNE, COFNW, COFSW, and COFSE. Then for each mesh point, one or more of these routines is executed depending on whether the point is an interior point or any of a number of types of boundary points. This necessitates the classification of mesh points and the assignment of a coding number for each type. The classification used here is presented in Table B1 on pages 262 and 263, In certain applications involving symmetry with respect to the r - 0 plane, equilibrium equations for points in the r - 0 plane are obtained from two additional subroutines called SYMNSE and SYMNSW. 260 261 The input information to the computer consists of mesh point identification numbers and material properties. Generally speaking, it is also necessary to provide information indicating which points are immediately around a given mesh point. For rectangular applications, this is done automatically in a subroutine called COORD. The coefficient matrix for the system is very sparesely popu- 1ated. In fact, no row of the matrix has more than 18 non-zero entries. The equations are solved by a modified Gauss Seidel Iterative procedure in which only non-zero coefficients are stored in the computer memory. A location array is also required to identify the displacement associated with a given coefficient of the matrix. The final step in the analysis involves the computation of stresses. For each interior point, for example, 4 sets of stresses are computed corresponding to equations (259 ). The stresses for the mesh point are taken to be the average of these sets of stresses. The simplified computer program for this analysis follows beginning on page 264. A list of Fortran Symbols used is given on pages 274 and 275. Sample data for the program is presented on page 276 and 277. 262 Table B-1 Finite Difference Mesh Point Coding Code Description 1 Ordinary interior point 2 Right vertical boundary point 3 Top horizontal boundary point 4 Corner point 5 Point on symmetry axis 6 Horizontal boundary point on symmetry axis 7 Center point with r-0 a plane of symmetry 8 Right vertical boundary point with r-6 a plane of symmetry NW NE SW SE [SW 3 SE NE SE SE 9T7 A 7 SE Code 10 ll 12 13 14 15 263 Table B—1 (Continued) Description NW NE Interior point with r-O a plane of symmetry . r SE ; sw Corner point - [a NE Left vertical boundary point SE Left vertical boundary point with r-6 a plane NE of symmetry r SE Corner point Corner point In Bottom horizontal boundary point [NW 1NE ] O 100 51 47 41 36 74 7O 88 87 39 89 99 31 264 AXIALLY SYMMETRIC FINITE DIFFERENCE PROGRAM PROGRAM AXSFD FINITE DIFFERENCE PROGRAM FOR AXISYMMETRIC ELASTOSTATIC PROBLEMS DIMENSION RRIZ31I9EI23194) COMMON 5(462.18),NC(23199)9NTYP(231)yUI462I9NyN29HyCly lSRRI23l)oSZZIZBIIoSTTIZBI)oSRZ(231),KAD(23l)oSZRt23lI FORMAT STATEMENTS FORMATIBFIOol) FORMATI91594E18ob) FORMATIBEIDobT FORMATI91594F805) FORMATIIbIST READ IN DATA READ 369N9NOUT9KDAT IFIEOF960T71970 READ 479H9C1 PRINT 369N9NOUT9KDAT PRINT 479H9C1 N2=2*N READ 369INTYPIITVI=19NI PRINT 369(NTYPIII9I=19NT GO TOIBTQBSIKDAT GENERATE MESH POINT DATA CALL COORD READ IOOQIIEIIQJIQ J=Io4)g I=IQNT GO TO 89 DO 39 I=19N READ 419INCIIQJ)9J=IQ9TvIEIIQJI1J=194I DO 99 I=19N PRINT SlyINCII9JTyJ=Ig9TQIEIIgJI0J=l’4) READ 479IRRIII9 I=19N) PRINT 479(RRII’9 I=11NT INITIALIZE COEFFICIENT ARRAY DO 31 I=19N2 DO 31 J31918 SII’J)=OQ DEFINE COEFFICIENTS DO 60 J=19N KEY=NTYPIJ7 I=2*J’l II=2*J R=RRIJT GO TOI1929393919495969594919591929I)'KEY IF Xmo~o~c H mm. m« Or m4mor~m0~oomm¢mor~mo~ Hr-ir—T HHr-m NNNN (MIN-o, N P O‘HHNMd'mC)T‘OO‘OO H 0 mm. (C CC CC CC CC CC. 7: «L mN. ms. H00. 0. CC CC CC' r-‘H .c .c .c .c .o .o .o .o .0 .0 who Hoo. m. oH mN. CC (C CO c C C C moH .o .o .o .o .o .o .o .o .o co m. oH mm. or. H00. .0 .o oo oo .0 m. m. m. m. .0 m. m. m. m. CC' CC CC c H c H .c .c .c .c .o .o .o .o .o .o .c .H mm. 2.. ”co. m. .H o 04 on .3. 2. 0 mm 3 mm mm 0 on om mm CC CC CO on wm Fm om #m mm mm Hm mm mm RN ON CC CC m.H H00. m. oH mm. m». OOOOO CC‘ CO CO c H c H or. .c .c .0 .c .o .o .o .o co .o W. cm mm. ms. H00. m. 0: on an hm on mm am mm mm Hm om 0N mm rm r-‘C‘O ¢¢ me N¢ HJ om mm hm on «m mm mm Hm (CC) f‘ r'C C) “x IDOL C'C) o N (2 H0000. mm. who Hon. \0 .H mm. COCO mé .aaw ma Nd Od an an Fm mm an fl. HOG (»CO C) v-‘C’O o o o o o CzC)CCOr-'HH 0000000 CC OOOOO - m.r Hoe. m. .H mm. mu. Hoo. m: a: m: N: Ha ca om mm pm om mm cm mm mm AXIALLY SYMMETRIC FINITE ELEMENT PROGRAM USING LINEAR STRAIN TRIANGULAR RINGS Like the finite difference program, the finite element program has 3 main parts: (I) generate stiffness coefficients, (2) solve the system of equilibrium equations, and (3) compute stresses. Nodal forces are determined outside of the program and thus become part of the input data. The computer program consists of the main program and several subroutines. The main program is essentially involved with reading in the element and nodal point data and initialization. An array (LOC) of location elements is also defined in the main program. The location array indicates the position of non—zero elements in the stiffness matrix. This is necessary in this procedure since only non—zero stiff— ness coefficients are stored. The subroutine STIFl generates the stiffness coefficients. The subroutine makes use of several other routines in the process. The quanties in equation (3.99) are formed in the subroutine Form B. The integration of these expressions over the volume is per- formed numerically in the subroutine NUMINT. A quintic formula discussed by Felippa [18] is used in this connection. The displacement transformation matrix of equation (3.96) is then generated. The inversion of (3.97) is accomplished by calling a 278 279 standard matrix inversion subroutine. Since such routines are available on all modern systems, the program is not listed here. The routine STIF2 is then executed many times to complete the formulation of the overall stiffness. The equations of the system are solved iteratively in the sub- routine GSORP. Involved is a modification of the Gauss-Seidel over relaxation procedure. Relaxation factors of 1.6 to 1.9 seem to give the most rapid convergence for such problems. The final step in the analysis involves the determination of stresses. Element nodal point stresses are calculated in the subroutine STRSA. The strains are obtained first from the product of the matrix [D] in equation (3.98) and the column matrix [a] of (3.97). The matrix [D] is generated in still another program called FORM G. Element nodal point stresses follow from the stress-strain law. Nodal point stresses for the overall structure are obtained by simply averaging the element nodal point stresses. The finite element program including the above mentioned sub- routines is presented next. A listing of Fortran program symbols used begins on page 290. A sample data set is shown on page 292. {'3 301 102 104 151 149 152 600 150 113 201 202 280 AXIALLY SYMMETRIC FINITE ELEMENT PROGRAM LINEAR STRAIN TRIANGULAR RING ELEMENTS PROGRAM LST FINITE ELEMENT PROGRAM-LINEAR STRAIN ELEMENTS AXISYMMETRIC ELASTOSTATIC PROBLEMS COMMON RIIb9IoZIIb9)yEITZTQPRITZI9N1I72I9N2(72)9N3I72I 19N4I72I9N5I72I9N6I7219NMATI72IQFI33BI9LLI33BI9LI(12), 25(338950)oLOCI16995DIOSIGZI16992I,SIGT(16992)9 39LTI338)9KEYI338I9UI338I1KADI16992I9SIGRT16992IQ 4SIGRZI169,219TI293381oCONgKEX9NCY,BETA,M,MP,M29KIT FORMAT STATEMENTS FORMATI55H FORMATIBIIO) FORMATI8F10.6) FORMATIIZI5) FORMATISFloclI FORMATI4OIZI INPUT ELEMENT AND NDDAL POINT DATA READ 301 IEIEOF96OI1009600 READ 1029M9MPcIP PRINT 301 M2=2*MP READ 10418ETA PRINT 104vBETA READ 1029KEX9NCYyKIT PRINT 1029KEX9NCY9KIT READ 1049(RIII9I=19MP1 PRINT 1049IRIIIQI=19MPI READ 1049IZIII9I=19MPI PRINT 1049IZI 91:19MPI READ 1519(N1 N2I119N3IIItNQTIIoN5III9N6II 9 I I ( I=19M) PRINT 1519(Nl 2(119N3(I)9N4TII9N5(I)9N6 I I v )9 I I)! I=19MI READ 1049TFI PRINT 104vIF READ 1529INMA PRINT 1529(NMA 9 I=19MI READ 1499IEIII9P II91=19MI PRINT 1499IEIII9P IIIOI=19MI READ ISZOILLTII9 I=19M2I PRINT 1529(LLIIT! I=19MZI INITIALIZE STIFFNESS AND LOCATION ARRAYS DO 201 IA=19M2 DO 201 JA=1950 SIIA’JA)=000 DO 202 IA=19MP DO 202 JA=1950 LOCIIAvJAI=O INITIALIZE NODAL POINT STRESS ARRAYS DO 107 I=19MP DO 107 J=192 KADII’J’30 19M2I =19M2) I 1 I N ) = I I T 9 I=19MI I ( R 1 I I 9 )9 (I) TII R s") 107 209 210 230 250 270 290 220 224 211 288 278 212 285 311 284 281 SIGRI I,J)=OO SIGTII'J)=OO SIGZII9JI=Oo SIGRZII9JI=0o DEFINE LOCATION ARRAY DO 212 I=19MP DO 209 J=19M2 LTIJI=0 DO 211 N=I9M II=N1INI I2=N2INI I3=N3INI I4=N4INI I5=N5INI I6=N6INI LIIII=2*11‘1 LIIZI=2*I2*1 LII3I=2*I3-1 LII4I=2*I4-1 LII5I=2*I5-1 LII6I=2*16‘1 LII7I=2*II LII8T=2*I2 LII9I=2*I3 LII10)=2*I4 LIIIII=2*I5 LIIIZ)=2*I6 IFIII‘II 21092209210 IF (IZ‘II 23092209230 IF II3'II 25092209250 IF II4-II 27092209270 IF II5'II 29092209290 IF (Ib‘II 21192209211 DO 224 IS=1912 L1=LIIISI LTILII=1 CONTINUE KEYIII=0 DO 278 JA=I9M2 IFILTIJAII27892789288 KEYIIIzKEYIIT+1 K1=KEYIII LOCII9K1I=JA CONTINUE CONTINUE PRINT OUT LOCATION ARRAY IFIIPI28592849285 DO 311 JA=19MP KK=KEYIJAI PRINT 2799 KK FORMAT (6I10I PRINT 2799 ILOCIJA,ITI9 IT=19KKI CONTINUE DEFINE STIFFNESS ARRAY CALL STIFI O 121 111 110 122 100 12 10 15 282 PRINT OUT STIFFNESS ARRAY IFIIPIIZI91229121 DO 111 I=19M2 PRINT 1109ISII9JI9 J=1950I FORMATI8F1505I GAUSS SEIDEL OVER RELAXATION PROCEDURE CALL GSORP CALCULATE STRESSES CALL STRSA GO TO I STOP END SUBROUTINE STIF1 DIMENSION SEEI397I9MTI493I9LI6I9RNI6I9ZNI6I9XI(15I9 1AI696)9DI696I9BI12912)9LIPI6I9NIPI6) COMMON RI169I9Z(169I9EI72I9PRI72)9N1(72)9N2I72)9N3(72) 19N4I72I9N5I7219N6I72I9NMATI72I9FI338I9LLI33819LI(12,9 23(338950)9LOCI1699SOI9SIGZI16992I9SIGTI16992I9 39LTI338I9KEYI33BI9UI33BI9KADI16992)9SIGR(16992)9 QSIGRZI16992I9TI29338)9CON9KEX9NCY98ETA9M9MP,M29KIT READ 129IISEEII9JI9I=193I9J=197I FORMATI6F13o8I READ 109IIMTII9JI9J=193I9I=I94I FORMATIIZIS) DO 400 N=19M I1=N1INI 12=N2INI I3=N3INI I4=N4INI I5=N5INI 16=N6INI C11=EINI*IIo-PR(N)I/Ilo-PRIN)‘2.*PRINI**2I C12=EINI*PRINI/Ilo'PRINI-2o*PRIN1**2) C44=EINIII2.+2.*PRINII C=C11+C12 LIII=NIINI LI2I=N2INI LI3I=N3INI LI4I=N4INI LI5I=N5INI LI6T=N6INI DO 15 I=196 J=LIII ZNIII=ZIJI RNIII=RIJI PERFORM NUMERICAL INTEGRATION CALL NUMINTIRN92N9XI9SEE9MTI CALL FORMBIB9XI9C9C119C129C44) FORM DISPLACEMENT TRANSFORMATION MATRIX DO 72 1:196 AII’lI:lo AII92)=RNIII AII93I=ZNIII AII94I=RNIII*RNII) 72 291 421 833 411 400 50 35 283 AII95I=RNIIT*ZNII) AII96I=ZNIII*ZNIII DO 291 JG=196 DO 291 KG=196 DIJG9KGI=AIJG9KGI CALL MINVID969DET9LIP9NIPI D IS NOW THE INVERSE OF A FORM DT*B*D LII1I=2*11-1 LII2I=2*I2-1 LII3)=2*13-1 LII4I=2*I4-1 LII5I=2*I5-1 LII6I=2*16-I LII7I=2*II LII81=2*IZ LII9I=2*I3 LIIIOI=2*I4 LII1II=2*I5 LIIIZI=2*I6 DO 411 JA=196 JT=LIJAI DO 833 I=19MP IFIJT-1183394219833 CALL STIF2IJA9I9D9B) GO TO 411 CONTINUE CONTINUE CONTINUE RETURN END SUBROUTINE NUMINTIX19Y19XI9SEE9MTT DIMENSION X1I619Y1I6I9XI1519YIISI9XM(15I9XI(15I9XXI15) DIMENSION SEEI397I9MTI493I COEF=XII2I*IY1(31-Y1I1))+XIIII*IYII2)-Y1(3)1+X1(3)* 1(Y1I1)-Y1(2)) COEFBCOEF/Ho XXI1I=0225 XXI21=o13239415 XXI3I=XXI21 XXI4I=XXI21 XXI5I=912593918 XXI61=XXISI XXI7I=XXI5I DO 50 I=1915 XIIII=00 DO 75 K=I94 L=MTIK9II M=MTIK92I N=MTIK93I DO 2 I=197 XIII=X1ILI*SEEI19I1+XIIM)*SEE(29II+X1INI*SEE(3,II YIII=Y1ILI*SEEI19II+Y1IM)*SEE(29II+Y1INI*SEE(39II DO 35 I=197 XMIII=XXIII*XIII 100 75 150 60 284 DO 100 I=197 . XII1I=XIIII+XMIII XIIZ)=XII2I+XMIII/XIII XII3I=XII3I+XMIII/IXIII**2I XII4I=XII4I+XMII)*YIII/XIII XIISI=XII5I*XMIII*YIII/IXIII**2I XII6)=XIIbI+XMIII*YIII**2/IXII)**2I XII7I=XII7I+XMIII*XII) XII8I=XII8I+XMIII*YIII XII9I=XII9I+XMII)*XIII**2 XII10I=XII10I+XMIII*XIII*YIII XII11I=XIIIII+XMIII*YIII**4/XIII**2 XIIIZI=XII12I+XMII)*YIII**2/X(II XII13I=XII13I+XMIII*YIII**3/XIII**2 XII14I=XIIIQI+XMIII*YIII**3/XIII XII15I=XIIISI+XMIII*YIII**2 CONTINUE CONTINUE DO 150 I=1915 XIIII=XIIII*COEF RETURN END SUBROUTINE FORMBIB9XI9C9C119C129C44) DIMENSION BII2912)9XII15) DO 60 1:1912 DO 60 J=1912 BI I,J,=OO BII9II=CII*XII3I BII92I=C*XII21 8(193I=C11*XI(51 BI194I=I20*C12+C11I*XII1) BII9SI=C*XII4I BI196I=C11*XII6I BI199I=CI2*XII2I BI1911I=C12*XII1I BII912I=20*C12*XII4I 8(292I320*C*XII1I BIZ93I=C*XII4I 8(294T=3o*C*XI(7I BIZ95I=2.*C*XII81 8(296I=C*XII12) BIZ,9I=2.*C12*XI(1I BIZ9III=20*C12*XII7I 8(2912I340*C12*XII8I BI393I=C44*XII1I+C11*XI(6) 8(394I=I20*C12+C11I*XII8I BI395I=C44*XII71*C*XII12I 3(396)=2o*C44*XI(8I+CII*XII13I 8(3981=C44*XII1I BI399I=C12*XII4I BI3910I=2o*C44*XII7I 3‘39111=IC12+C44I*XII8) BI3912I=20*C12*XII12I BI4,4)=I5o*C11+40*C12I*XI(9) 66 221 285 BI495I=3.*C'XII10I BI496I=I2oPC12+C11I*XII15) BI499I=30*C12*XII7I BI4911I=3o*C12*XII9I BI4912I=6o*C12*XII10I BIS,5I=2.*C*XII15)+C44‘XII9) 8(596I=C*X1I14)+2.*C44*XII10I BI598I=C44*XI(71 BIS,9I=2.*C12*XII81 BI5910I=20*C¢4*XII91 8(5911I=(2o*C12+C44I*XII10) BIS9IZI=4o*C12*XII15I BI696I=C11*XI(11)+4.*C44*XI(15) 8‘698I320*C44*XII8) 8‘699)=C12*XI(12I BI69IOI=4.*C44*XII10) 8(6911)=IC12+2o*C44I*XII15) 8(6912)=2o*C12*XII14I 8(898I3C44*XI(1I BIB9IOI=20*C44*XII7I 8(8911I=C44*X1I81 BI999I=C11*XI(1I BI9911)8C11*XII7I 8199121=2o*C11*XII8) BIIO910I=40*C44*XII9I BIIO9III=2o*C44*XIIIOI BIII9III=C11*XII9I+C44*XI(15I 8(119IZI=Z.*C11*XIIIOI BIIZ9IZI=4.*C11*XI(15I DO 66 I=I9II J1=I+1 DO 66 J=J1912 BIJ9II33II9JI RETURN END SUBROUTINE STIFZIM99I9D9BI DIMENSION DI696I9BI12912) COMMON RI169I9ZI169)9EI72)9PR(72I9N1(72I9N2(72I9N3(72) 9N4I72)9N5I7219N6I7ZI9NMATI72I9FI338I9LLI338I9LI(12,9 25(338950)9L0CI169950):SIGZ(16992I9$IGT(1699219 9LTI338)9KEYI33819UIB3BI9KADI1699219$IGRI1699219 4SIGRZI1699219TI29338I9CON9KEX9NCY9BETA9M9MP9M29KIT DO 221 J=I9M2 DD 221 KP=192 T!KP,J)=O. DO 222 J=1:6 J6=J+b K=LIIJ6) L=LI(J) DO 222 KI=196 K16=KI+6 DO 222 KJ=196 KJ6=KJ+6 Tt19L1=T119L)+D(KI,M9)*B(KIoKJ)*D(KJ,J) TIl,K)=T(l9K)+D(KI9M9)*B(KI,KJ6)*D(KJ,J) 222 240 317 339 382 386 10 11 12 377 314 376 332 345 347 396 348 346 355 394 329 328 286 TI29LI=TI29LI+DIKI9M9I*BIKI69KJI*DIKJ9JI TI29KI=TI29KI+DIKI9M9I‘BIK169KJ61*DIKJ9JI J2=2*I J1=J2‘1 DO 240 K=1950 KISLOCII9KI IFIK1119291 SIJ19KI=SIJ19KI+TII9KII SIJ29KI=SIJ29KI+TI29K1I RETURN END SUBROUTINE GSORP COMMON RI169I9ZI169I9EI72I9PRI72I9N1I7219N2172I9N3ITZI 19N4I72I9N5I72I9N6I72I9NMATI72I9FI338I9LLI33819L1(1219 2SI338950I9LOCI16995019515211699219SIGTI1699ZI9 39LTI338I9KEYI33819UI338I9KADI1699219SIGRI1699219 4SIGRZI16992I9TI29333I9CON9KEX9NCY9BETA9M9MP,M29KIT CON=1.0*IIO.**I'KEXII PRINT 3399 CON FORMAT (23H CONVERGENCE CRITERION= E20.81 SEIDEL ITERATION PRINT 3869BETA FORMAT I19H RELAXATION FACTOR= E20o8I READ 109INIT FORMATIIOISI IFIINITI 119377911 READ 129IUIII9 1819M2I FORMATI4E20o8) GO TO 376 DO 314 I=19M2 UIII=0o0 KK=0 DIFF=OQO KK=KK+1 IFIKK’NCYI 34693469345 PRINT 347 FORMAT IZIH CYCLE LIMIT EXCEEDEDI PRINT 396 FORMATI30H THE CURRENT DISPLACEMENTS ARE) PRINT 3489KK9 IUIII9 I=I9NMI FORMAT I1H09159I4E15.8II GO TO 1 DO 330 I=19M2 IFILLIIII33093559330 KJ=II+1)/2 UBAR=FIII NUMSKEYIKJI DO 328 J=19NUM N=LOCIKJ9JI IFIN-II32993949329 DIAG’SII9JI GO TO 328 UBAR=UBAR-SII9JI*UINI CONTINUE 364 336 330 306 305 304 303 333 335 334 337 338 44 100 287 UBAR=UBARIDIAG U1=UIII+BETA*(U8AR-U(III DMAX=ABS (UI'UIII) UIII=U1 IFIDMAX-DIFFI 33093369336 DIFF=DMAX CONTINUE KTEN=KKIKIT IFIKK-KIT*KTENI 30393069303 PRINT 3059KK FORMATI17H ITERATION COUNT=I10III PRINT 3O49IU|II9 I=19MZI PUNCH 3049IUIII9 I=19M2I . FORMATI4E2008I - CONTINUE TEST FOR CONVERGENCE IFIDIFF-CDN) 33393329332 PRINT 3359KK ”5 FORMAT I25H THE NUMBER OF ITERATIONS9/IIIIOI) ‘ PRINT 334 FDRMATI42H NODAL POINT R*DISPLACEMENT Z-DISPLACEMENT) DO 337 I=I9MP IO=2*I-1 IE=2*I PRINT 3389I9UIIOI9UIIEI FORMATIIIZ9ZE20.81 PUNCH 3049IUIII9 I=19M2I RETURN END SUBROUTINE STRSA DIMENSION LIP1619NIPI6I9LI6I9AI696)96(4912I9BI12I DIMENSION EPI4I9STRT4I9CEI4941 COMMON RII69)9Z(16919EITZI9PRI7219N1I72)9N2(72I9N3(721 19N4172)9N5I72I9N6I72I9NMATI7219F1338)9LLI33819LI(12,9 25(33895019LOCI169950,9SIGZI16992I9SIGTI1699219 39LTI338I9KEYI338I9UI338I9KADI16992I9SIGRI1699219 4SIGRZI16992I9TI29338I9CON9KEX9NCYgBETApMyMP9M29KIT PRINT 44 FORMATI17H ELEMENT STRESSES/l/I DO 926 I=19M PRINT 10091 FORMATIle ELEMENT NUMBER=I51 LIII=NIIII LIZI=NZIII LI318N3II) LI4I=N4III LISI=NSIII LI6IBN6III DO 10 J=196 LYSLIJI AIJ’II‘FIQ AIJ92I=RILYI AIJ93I=ZILY1 AIJ94I=RILYI*RILYI AIJ95)=R(LY1*Z(LY) 10 20 30 77 51 52 61 62 37 38 39 17 71 926 288 AIJ96I=ZTLY)*ZILYI CALL MINVIA969DET9LIP9NIP) DO 20 N81912 BTNI30. D0 30 N8196 DO 30 J8196 LYBLIJI JE=2*LY JO=2*LY-1 BINIIBINI+AIN9JI*UIJ0I K=N+6 BIKIBBIK1+AIN9JI*U(JEI DO 77 K=194 DO 77 J=194 CEIK9JI=Oo PO=PRIII Y=EIII CEI191I=Y*I10"POI/I10-PO"20*P0**2I CEIl921=Y*PD/(19-PO-2.*PO**ZI CEI29118CEII9Z) CEI494I3Y/(20+20*P01 CEI292)=CE(191) CEI393I=CEII9II CEIZ93I=CETI9ZI CEI39ZI8CEII92) CE(193)=CE(19ZI CEI3911=CEII921 KAY=NMATIII DO 71 N=196 LY=LINI CALL FORMGIG9LYI DO 51 J=194 EPIJI=0o DO 52 J=194 DO 52 K81912 EPIJI=EPTJI+GIJ9KI*B(K) DO 61 J=194 STRIJI‘O. DO 62 J=194 DO 62 K8194 STRIJI=STRIJI+CEIJ9K)*EP(K) GO TO (37938I9KAY JZ=1 GO TO 39 JZ=2 SIGRILY9JZI=SIGRILY9JZ)+STR(11 SIGZILY9JZI=SIGZILY9JZ)+STR(2I SIGTILY9JZ)=SIGT(LY9JZI+STR(3I KADILY9JZI=KADILY9JZI+1 SIGRZILY9JZI=SIGRZILY9JZI+STRI4I PRINT 179LY9ISTRINKI9 NK=194I FORMAT111094F20o2I CONTINUE CONTINUE 461 325 326 327 328 512 511 513 11 12 10 289 PRINT 461 FORMATIZIH NODAL POINT STRESSES/l/I DO 511 I=l9MP IFIKADII911I32593269325 SAD=KADII911 SIGRII91I=SIGRII9IIISAD SIGZII9II=SIGZII9IIISAD SIGTII9lI=SIGTII91IISAD SIGRZII91I=SIGRZII9IIISAD IFIKADII921132793289327 SID=KADII92I SIGRII92I=SIGRII9ZIISID E SIGZII92I=SIGZII92IISID SIGTII9ZI=SIGTII9ZIISID SIGRZII921=SIGRZII9ZIISID PRINT 512919SIGRII9119SIGZII9lI9SIGTII9119SIGRZII91) FORMATI1594F2092I . PRINT 5139SIGRII9219SIGZII9219 SIGTII9219SIGRZII92I r FORMATI11X94FI992I RETURN END SUBROUTINE FORMGIG9LYI DIMENSION GI4912I COMMON R1169I9ZI169I9EITZI9PRI72I9N1172I9NZITZI9N3I72I 19N4I72I9N5(72I9N6I7219NMATI7219FI338I9LLI33819LI(1219 25I338950I9LOCI16995019SIGZI1699219SIGTI1699219 39LTT33819KEYI338I9UI33819KADI16992I9SIGR(16992)9 4SIGRZI1699219TI29338)9CON9KEX9NCY9BETA9M9MP9M29KIT X=RILYI Y=ZILYI IFIX-.001)11912912 X=.001 DO 10 13194 DD 10 J=1912 GIIQJI=00 GTIVZIglo GIIQ4I=ZQ*X GI195I=Y 6‘299)=lo GIZ911I=X GIZ912’320*Y GI3913=lo/X 6(3'2I310 GI393I=YIX GI394I=X C(395I=Y GI396)=Y*Y/X GI4QBI310 6(495)=X GI4pb’=Zo*Y 614981=1o GI4,10I=2.*X GI4911I=Y RETURN END R(I) 2(1) N1(I) N2(I) N3(I) N4(I) N5(I) N6(I) PR(I) E(I) NMAT(I) F(I) U(I) LL(I) S(I,J) LOC(I,J) SIGR(I,J) SIGZ(I,J) SIGT(I,J) SIGRZ(I,J) FORTRAN PROGRAM SYMBOLS AXIALLY SYMMETRIC FINITE ELEMENT PROGRAM Radial Coordinate Axial Coordinate Element Nodal Point Numbers Poisson's Ratio Elastic Modulus Material Indicator for Composites Nodal Point Force Array Nodal Point Displacement Array Displacement Boundary Condition Array Stiffness Coefficient Array Location Array Radial Stress Array Axial Stress Array Circumferential Stress Array Shear Stress Array 290 M MP BETA CON NCY KIT A(I,J) c11,c12,c44 CE(J,J) SEE(I,J) MT(I,J) T(I,J) EP(I) STR(I) G(I,J) IP INIT 291 Number of Finite Elements Number of Nodal Points Relaxation Factor Convergence Criterion Number of Cycles Allowed for Iterative Solution Cycle Print Interval Indicator Transformation Array from Generalized Displacements [a] to Nodal Displacement [u] Elastic Constants Elastic Constants Numerical Integration Coefficients Numerical Integration Nodal Point Numbers Temporary Equilibrium Equation Array Element Strains Element Stresses Displacement Gradient Matrix Indicator to surpress printing stiffness coefficients. If IP is non-zero, coefficients are printed. Initial Displacement Indicator If INIT is zero, initial displacements are set to zero. Otherwise, displacements are read in. 292 m m o m N a o m a 0 ¢ H ooo~¢>o>a HmomNHoHo HmomchH. HmomNHcHo oaowfiboho HmOONHcHo HmooNHcH. HmomNHcH. ooomchowo homHoomOo OONcHONeo 00N¢H0h¢9 OONGHOFQo hmmHoOmOo OONcHoseo 00N3H0h¢o oON¢H0heo hmmHommOo mnnmmmmm. mmmmmmnm. mmmnnmmMo _ H o H o H o H o H H c o o o C o o o o H o o c o o o o o o H o o c o o o o o o H o o o o o o o o o H mN. .oom mN. .OOm mN. .ooocH mNo .ooooH mNo .oom mNo ooom mNo .oom mNo .oom H H N N H H H H .0 .o co co co no co co .0 co 90 .0 .o no .0 .o co .o oo oo co co .0 .0 CO .0 .0 Co Co Co .0 .0 .0 0° 0° 00 CC QC .0 00 0° Co Co Co Co Co .0 .0 oo oo oo oo oo .o oo .o .o .c .0 cc .0 oo .o oo oo oo 9o co oo oo .00H .0 .00m .0 .00H .0 .00H .0 oo .o tH 0N 0H mH mN mH 0H «N mH mN nN MH mH NN NH nN HN mH 5H 0H NH HN HH mH NH 0 5 HH H MH 5 N m H m MH 3 ¢ 0 m m mH o 0H 3H m mH MH 0 o .0 .0 .0 .C m. m. m. m. m. 9H 9H .H oH .H moH moH moH moH moH .N .N .N oN .N .H ms. m. mm. .0 .H ms. m. mN. .o 9H m». m. mN. .o 9H ms. m. mN. .0 .H mp. m. mNo oo o¢ com 5 o.H mN o (*40 mqaz