This dissertation has been 63—6186 microfilmed exactly as received H U L B E R T , Lewis Eugene, 1924- THE NUMERICAL SOLUTION OF TWO- DIMENSIONAL PROBLEMS OF THE THEORY O F ELASTICITY. The Ohio State University, Ph.D., 1963 Engineering Mechanics University Microfilms, Inc., Ann Arbor, Michigan THE NUMERICAL SOLUTION OF TWO-DIMENSIONAL PROBLEMS OF THE THEORY OF ELASTICITY DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Lewis Eugene Hulbert, B. Sc., M.Sc. >/ u V U u The Ohio State University 1963 Approved by Advislr partment of engineering Mechanics ACKNOWLEDGMENTS I would like to express my appreciation to my major adviser, Francis W. Niedenfuhr for his guidance and encouragement during the course of research on this dissertation. I would like to express my gratitude to the Numerical Computation Laboratory of The Ohio State University for providing the computing services necessary for performing the research. I would also like \,o express my gratitude to Battelle Memorial Institute for the financial assistance that they provided me under their education program. Finally, I would like to thank my wife, Beatrice, for her understanding, patience, and encouragement, without which this dissertation could not have been written. il CONTENTS Page ACKNOWLEDGMENTS............................................. II LIST OF TABLES . ......................................... v LIST OF ILLUSTRATIONS........................................ vltl INTRODUCTION ............................................... 1 Chapter 1. DEVELOPMENT OF THE MATHEMATICAL EQUATIONS OF THE FIRST FUNDAMENTAL PROBLEMS OF THE PLANE THEORY OF ELASTICITY .............................. 6 The equations of the three-dlmensional theory ol‘ elasticity.................................... 6 Plane s t r a i n ....................................... 9 The generalized plane stress state ................ 13 Equations of the plane problems in terms of stresses....................................... 19 Displacements in multiply connected regions. Michell's equations.............. 2h REFERENCES ........................................... 55 2. THE STRESS FUNCTION FOR PLANE P R O B L E M S .............. 57 Reduction of the plane problem to a problem without body forces.............................. 57 The series solution of the biharmonic equation in polar coordinates ............................ 68 The stress function in multiply connected regions . 85 The symmetric stress functions for multiply connected regions ................................ 91 REFERENCES ............................................. 115 iii CONTENTS (Continued) Chapter Page 3. SATISFACTION OF THE BOUNDARY CONDITIONS. THE POINT MATCHING APPROACH............................... 116 The point matching approach ......................... 119 The classical approach for satisfying boundary c o n d i t i o n s ......................................... 129 REFERENCES ............................................. 132 h. NUMERICAL RESULTS....................................... 13^ Circular holes in an Infinite plate arranged with cylindrical symmetry ......................... 137 Circular holes arranged with cylindrical symmetry In a finite circular plate .............. 172 Problems involving noncircular holes ............... 202 Problems involving infinite rows of holes ........... 211 Torsion problems ................................... 219 REFERENCES ............................................. 228 5. THE COMPUTING PROGRAM................................... 230 Program I ............................................. 231 Program I I ........................................... 2bh Program I I I ........................................... 2^6 BIBLIOGRAPHY ......................................... 276 AUTOBIOGRAPHY ......................................... 280 iv LIST OF TABLES Table Page 1. Boundary values of ^-/p on the outside hole of three holes In a line in an infinite plate loaded in uniaxial tension normal to the line of centers of the h o l e s ................................. lLl 2. Boundary values of ®£>/P on center hole of three holes In a line in the infinite plate .loaded In uni­ axial tension normal to the line of centers of the h o l e s ............................................. l!+2 3. Values of ^x/p and Oz/p on the x-axis between the center and outside holes of three holes In a line in the infinite plate loaded in uniaxial tension normal to the line of centers of the h o l e s ...........lUU U. Boundary values of ^J^/p on the semicircle of the basic symmetry element for two, four, and six holes symmetrically arranged along the arc of a circle In the infinite p l a t e ......................... 151 5. Boundary values of ^ / p on f*0 for seven and nineteen pressurized holes in the infinite p l a t e ........ , . 160 6. Boundary values of ^o/p on 0 f°r seven and nineteen pressurized holes in the infinite plate . l6l 7* Boundary values of /^/p on ^2,0 an(^ °^ nine~ teen pressurized holes in the infinite plate.......... 163 8. Values of the stresses calculated at selected interior points of the seven-hole problem .......... 170 9. Values of the stresses calculated at selected interior points of the nineteen-hole problem .... 171 10. Boundary values of ^ c !P on ^he intersection of the x-axis with the inner and outer boundaries of the eccentric hole problems ............................ 176 v LIST OF TABLES (Continued) Table Page 11. Boundary values of ^q /v on the hole boundaries for three and four holes in a circular plate with ratio of hole radius to exterior radius of 0 . 3 0 ........................ .................. 180 12. Boundary values of <^£)/p for four holes in a circular plate with ratio of hole radius to exterior boundary radius of 0 . 2 5 ................ 181 13* Boundary values of the ^“o/v on the exterior boundary of problems involving three and four holes In a circular p l a t e .......................... 182 lU. Stresses on the central hole of problems Involving seven circular holes in a circular plate .......... 185 15. Stresses on the exterior boundary of problems involving seven circular holes in a circular p l a t e .............................................. I85 16. Stresses on the eccentric hole of problems involving seven circular holes in a circular plate ........ 186 17. Boundary values of Rv^-./p for problems Involving six circular holes in a circular plate loaded by six concentrated tensile forces .............. 197 18. Boundary values of C^/p at points of the boundary of a pressurized star-shaped hole in the infinite plate as a function of x ........................ 207 19. Boundary values of C^/p for an infinite row of pressurized holes as calculated by the computing program and given by H o w l a n d .................... 2lk 20. Boundary values of <Jg /p for two rows of pressurized holes in the infinite p l a t e ...................... 217 21. Boundary shear stresses on the eccentric holes of circular tubes with longitudinal circular holes as a function of 0 ~ ~ @ ...................... ^ 5 vi LIST OF TABLES (Continued) Table Page 22. Boundary shear stresses on the center holes of circular tubes with longitudinal circular holes as a function of OL ~ 7? - O .................. 226 23. Boundary shear stresses on the exterior boundary of circular tubes with longitudinal circular holes as a function of Oi - ft - O .................. 227 2 b . Card input to Program I ................... 232 25. Card input to Program I I I ........................... 2 U7 26. Function table for subroutine DPHI0 ................. 253 27. Function table for subroutine DP H I D X .............. 25*+ 28. Function table for subroutine DPHIDY ............... 255 2 9. Function table for subroutine DPIEDX2............... 256 30. Function table for subroutine DPHDY2 ............... 257 31. Function table for subroutine DPHDXY ............... 258 vii LIST OF ILLUSTRATIONS Figure Bage 1. Illustration of plane strain ........................ 9 2. Illustration of generalized plane stress ............ lL 3. Illustration of the method of introducing a cut i n a r i n g ......................................... 25 4. The physical interpretation of dislocations ....... 29 5. The boundary coordinate system ...................... 37 6. A line integral circuit around a simply connected r e g i o n ............................................. 7- Line integral circuits enclosing a h o l e ............ 18 8. Calculation of dislocations in the general multiply connected region ........................ 51 9. The circuit for evaluating dQ ...................... 73 10. Two cases for which the origin is external to the re g i o n ......................................... 73 11. The circular path for evaluating the dislocations . 79 12. A region containing an arbitrary number of arbitrary circular holes ..................................... 86 13. A group of holes with cylindrical symmetry ........ 92 ll. A row of holes with translational symmetry ........ 92 1 5* A point set with cylindrical periodicity............ 9^ 16. A set of points with cylindrical s y m m e t r y .......... 98 17. The angular functions , for cylindrical symmetry . 101 18. The functions for cylindrical symmetry.............105 viii LIST OF ILLUSTRATIONS (Continued) Figure Page 19• f°r translational s y m m e t r y .........................110 20. The doubly periodic array of cooling holes ......... 121 21. Three holes in uniaxial tension