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Application to Cylinders and Spheres 1 Dept, of Tic""' MI8 —1? 1 ME I : 1 L . , 8 . ! | | 5 J .A3 9 NTACT STRESSES FOR CLOSELY 77-48 ll FORMING BODIES-APPLICATION TO CYLINDERS AND SPHERES FINAL REPORT UNDER CONTRACT DOT-OS-40093 DECEMBER 1976 This document is available to the U.S. Public through the National Technical Information Service Springfield, Virginia 22161 Prepared for U.S. DEPARTMENT OF TRANSPORTATION Office of the Secretary Office of University Research Washington D.C. .20590 NOTICE This document is disseminated under the sponsorship of the Department of Trans- portation in the interest of information exchange. The United States Government assumes no liability for its contents or use thereof. m , 0.6 A A ; /] no, Technical Report Documentation Page „ -f 1- 1. Report No. 2. Government Accession No. 3. Recipient s Catalog No. fj)0 !l "TC T DOT-TST-77-48 /3 1 4. Title ond Subtitle 5. Report Date CONTACT STRESSES FOR CLOSELY CONFORMING December 1976 6. Performing Orgonizofion Code BODIES— APPLICATION TO CYLINDERS AND SPHERES 8. Performing Orgonizofion Report No. 7 . AwtKo^*) MEAM Report 76-1 W. Woodward and B. Paul 9. Performing Organixotion Nona and Address 10. Worii Unit No. (TRAIS) Department of Mechanical Engineering and Applied Mechanics— University of Pennsylvania 11. Controct or Gronf No. 111 Towne Building D0T-0S-40093 Philadelphia, Pennsylvania 19174 13. Type of Report ond Period Covered 12. Sponsoring Agency Nome and Add/ess Department of Transportation Final Report Program of University Research Office of the Secretary Id. Sponsoring Agency Code Washington, D. C. 20590 0ST/TST-60 15. Supplementary Note* OST Technical Monitor: Clifford Gannett, FRA, RA-43 V6. Abilroct Since worn wheels and rails contact conformally, the existing contact stress theories for nonconformal contact are not adequate. In this report a general numerical method of solution for three dimensional , frictionless conformal , elastic contact problems is presented for the first time. The method is used to analyze the conformal contact of a sphere indenting a spherical seat and a cylinder indenting a cylindrical seat. The results of the sphere-spherical seat problem compared well with experimental data. Re- sults of the cyl inder-cyl indrical seat problem were in close agreement to a known analytic solution of this problem. For both analyses, results com- pared favorably with Hertzian theory for problems with small contact regions. A method is given for defining the boundaries of the large contact re- gions, and for solving the associated governing singular integral equation of the first kind. A general iterative procedure is developed which con- verges to the true three dimensional contact region. In addition the solution to a non-Hertzian contact problem with a multiply connected contact region is solved; namely, the case of two spheres in contact where one of them has a surface defect or pit. 17. Key Words 18. Distribution SloUmtnt Rail wheel interaction, contact Document is available to the public stress^ conformal contact, elasticit, through the National Technical non-Hertzian contact, pitted sphere Information Service, Springfield, Virginia 22161 19. Security CUttil. (of this report) 2D. Security Clossif. (of tKis poge) 21* No. of Peges 22. Pnc. Unclassified Unclassified 272 i nept e» «~72> » DOT F 1700.7 R^mAkIIw .1 c.nyl.lW peg* L EXECUTIVE SUMMARY 1 . Introduction This is the final report for Phase I (first year) of a two year effort on Contract D0T-0S-40093, "Improved Wheel and Rail Performance via Control of Contact Stress." The general state of the art prior to the beginning of the project has been summarized in the report "A Review of Rail -Wheel Contact Stress Problems," by B. Paul, FRA-OR&D 76-141, PB251 238IAS , April 1975 The present report gives the detailed mathematical theory of a new approach to the higherto unsolved problem of finding the stresses between two closely fitted or "worn-in" metallic surfaces, such as a moderately worn wheel and rail Before applying the general technique to the wheel -rail problem it is essen- tial to check its validity with simpler geometries such as closely fitting cylinders and spheres, where previous experimental and approximate analytical solutions exist. 2. Problem Statement The overall objective of the contact is to generate a method for calculating the contact stresses between arbitrarily profiled wheel and rails. In this report a general approach to the problem is formulated, and applied to two specific geometries: (a) a cylinder pressed against a closely fitted cylindrical seat and (b) a sphere pressed against a closely fitted spherical socket. In addition, the stress concentrations induced by the presence of a small defect such as a corrosion pit are calculated for the case of a sphere. 3. Results Achieved Since worn wheels and rails contact conformally, the existing contact stress theories for nonconformal contact are not adequate. In this report a general numerical method of solution for three dimensional, frictionless, conformal , elastic contact problems is presented for the first time. The method is used to analyze the conformal contact of a sphere indenting a spherical seat and a cylinder indenting a spherical seat. The results of the sphere-spherical seat problem compared well with experimental data and are significantly more accurate than those of a previously published attempt to solve the problem. Results of the cylinder-cylindrical seat problem were in close agreement to a known approximate solution of this problem and agree well EXEC. -1 with an existing photoelastic experiment. For both analyses, results compared favorably with Hertzian theory for the limiting case of small contact regions. A method is given for defining the boundaries of the large contact regions, and for solving the associated governing singular integral equation of the first kind. A general iterative procedure is developed which converges to the true three-dimensional contact region. In addition the solution to a non-Hertzian contact problem with a multiply connected contact region is solved; namely, the case of two spheres in con- tact where one of them has a surface defect or pit. The method developed was capable of detecting extremely steep gradients in stress at the defect. 4. Utilization of Results Better understanding of the contact stress distribution at the interface of wheel and rail could lead to substantial advances in the solution of several key problems in railroad technology. Examples include wheel screech, flange impact noise, wheel and track wear and fatigue failures, deterioration of ride quality and possible derailment due to lateral and longitudinal slippage, increase of headway (and loss of economic capacity) due to adhesion limits in braking and acceleration. The results of the research has potential for wheel and rail designers, and those doing research and development work in the areas of wheel and rail failures, rail-car dynamics, ride-comfort, and safety. 5. Concl usions The objectives set for Phase I of the research have been achieved. In addition to the generation of a comprehensive survey report on wheel -rail contact stress phenomena, the work reported on herein successfully tested the validity of a new method for finding contact stresses between closely fitted curved surfaces such as cylinders and spheres. This work is a necessary precursor to the solution of the more complicated geometrical configuration of wheel-rail interfaces, which is the subject of Phase I of this research project. EXEC. -2 CONTACT STRESSES FOR CLOSELY CONFORMING BODIES—APPLICATION TO CYLINDERS AND SPHERES William Woodward and Burton Paul Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania Philadelphia, Pennsylvania TABLE OF CONTENTS Chapter 1. INTRODUCTION 1 2. FORMULATION AND SOLUTIONS FOR NON-CONFORMAL CONTACT PROBLEMS 9 2.1 Introduction 9 2.2 Governing Equation for Non-Conformal Contact Theory 10 2.3 The Simply-Discretized Method of Singh and Paul 17 2.4 Functional Regularization 22 3. FORMULATION OF CONFORMAL CONTACT PROBLEMS 25 3.1 Introduction 25 3.2 Assumptions in Conformal Contact Theory ... 26 3.3 Formulation of Conformal Contact Criterion . 28 4. GENERATION OF INFLUENCE FUNCTIONS 42 4.1 Introduction 42 4.2 Influence Function for a Point Load on a Half Space 43 4.3 Influence Function for a Line Load on a Plane 46 4.4 Numerical Generation of Influence Functions . 50 4.5 Influence Function for a Point Load on a Sphere 53 Chapter 4.6 Influence Function for a Point Load on a Spherical Cavity 65 4.7 Influence Function for a Cylinder under Concentrated Line Loads 71 4.8 Influence Function for a Line Load on a Cylindrical Cavity 82 5. CONFORMAL ELASTIC CONTACT OF A SPHERE INDENTING A SPHERICAL CAVITY 92 5.1 Introduction 92 5.2 Formulation 93 5.3 Numerical Procedures 107 5.4 Numerical Results 113 5.5 Conclusions 128 6. CONFORMAL ELASTIC CONTACT OF A CYLINDER INDENTING A CYLINDRICAL CAVITY 132 6.1 Introduction 132 6.2 Formulation 134 6.3 Numerical Prodecures 142 6.4 Numerical Results 145 6.5 Conclusions 153 7. CONTACT STRESSES FOR MULTIPLY-CONNECTED CONTACT REGIONS 156 7.1 Introduction 156 7.2 Formulation 158 7.3 Pitted Sphere Geometry 163 7.4 Numerical Solution Procedure 168 7.5 A Numerical Example 173 7.6 Conclusions 183 8. CONCLUSIONS 185 Appendix A. DOMINANT SINGULARITIES IN THE STERNBERG INFLUENCE FUNCTION FOR A POINT LOAD ON A SPHERE 138 B. DISPLACEMENTS ON A CYLINDER UNDER TWO DIAMETRICALLY OPPOSED LINE LOADS 199 C. SINGULARITIES IN THE INFLUENCE FUNCTION FOR A CYLINDER UNDER TWO DIAMETRICALLY OPPOSED LINE LOADS 204 Appendix D. DERIVATION OF SURFACE DISPLACEMENTS FOR A CYLINDRICAL CAVITY UNDER TWO DIAMETRICALLY OPPOSED LINE LOADS 210 E. SINGULARITIES IN THE INFLUENCE FUNCTIONS FOR A CYLINDRICAL CAVITY UNDER TWO DIAMETRICALLY OPPOSED LINE LOADS 217 F. DERIVATION OF THE PROFILE FUNCTION FOR CONFORMAL CONTACT OF A SPHERE AND SPHERICAL SEAT 222 G.
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