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AIRY's FUNCTION by a MODIFIED TREFFTZ's PROCEDURE By Airy's function by a modified Trefftz's procedure Item Type text; Thesis-Reproduction (electronic) Authors Huss, Conrad Eugene, 1941- Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 27/09/2021 16:34:14 Link to Item http://hdl.handle.net/10150/318170 AIRY'S FUNCTION BY A MODIFIED TREFFTZ'S PROCEDURE by Conrad E„ Huss A Thesis Submitted to the Faculty of the DEPARTMENT OF CIVIL ENGINEERING In Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE In the Graduate College THE UNIVERSITY OF ARIZONA 19 6 8 STATEMENT BY AUTHOR This thesis has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED APPROVAL BY THESIS ADVISOR This thesis has been approved on the date shown below: DR. „ _ _ Date Professor of Civil Engineering ACKNOWLEDGMENT I wish to express appreciation to my thesis advisor, Dr. Richmond C . Neff, for his guidance and essential sugges­ tions which made possible the compilation and completion of this thesis. May I also acknowledge the assistance of Mr. Melvin L. Callabresi and Dr. Ralph M. Richard of The University of Arizona in making available their finite element computer program. I furthermore wish to thank my wife, Dixie, for her help in the preparation of this manuscript. TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS oeeoo 00000000400900 0 a««eae 090009 LIST OF TABLES,.,.oo.00000..0.ooo.o.o.o0..0.0.0.o.oo.vii ABSTRACT0000000.00000000.0.000.00.0000..0000000.0O.00VXXX CHAPTER 1 INTRODUCTION ____ 1 Airy's Stress Function.................. 1 Boundary Conditions Through Least Squares 2 2 FORMULATION. ..... 4 Error Function.......................... 4 Calculation of Gradient on Boundary..... 6 Positive Definiteness of Matrix......... 8 Comparison of Galerkin and Least Squares 10 3 EXAMPLE PROBLEM......... 12 Statement of Problem ..... 12 Numerical Integration Schemes........... 13 Equat ion SoIver......................... 16 Calculation of Stresses ................. 18 Symmetry of Problem..................... 18 Comments on Programming. ....... 20 4 BOUSSINESQUE FORMULATION.................... 21 Convergence Rate of Equivalent Problems. 21 The Boussinesque Function...... 22 Superposition Procedure................. 24 Effect on Convergence................... 27 Effect of Boundary Point Selection...... 31 5 COMPARISON OF RESULTS....................... 37. R* in x t e E1 cm on t.......................... 3 7 Photoolastxcxty......................... 37 Comparison of Boussinesque and Singular Lo ad x ng............. 40 Suggested Extensions.................... 40 iv V TABLE OF CONTENTS— Continued Page APPENDIXOOOeQOOOOQOOe 000000000006000000000000000000 A2 REFERENCES....o............ ....... .... ............. 53 LIST OF ILLUSTRATIONS Figure Page 1 Gradient on Boundary....................... 6 2 Example Problem............................ 12 3 Equivalent Loading Representation.......... 13 4 Boundary Point Distribution................ 15 5 Symmetrical Stresses....................... 19 6 Concentrated Load on Straight Boundary..... 23 7 Square Ha 1 f Plas^e.....o................. 24 8 Complementary Half Plane................... 25 9 Boussinesque Boundary Tractions............ 25 10 Superposition of Stress Fields„............ 26 11 Boundary Point Intervals - Boussinesque.... 32 12 Node Point Location........................ 35 vi LIST OF TABLES Table Page I Boussinesque Convergence.......................... 29 II Equivalent-Loading Convergence................... 30 III Comparative Stresses for Three Boundary Point Patterns.................. 3^ IV Node Point Coordinates........................... 36 V Finite Element Compared With Least Squares....... 38 VI Photoelasticity Compared With Least Squares...... 39 VII Demonstration of St. Venant's Principle.......... 4l vii ABSTRACT Two-dimensional linear elasticity problems are math­ ematically described by Airy8s equation, V = 0. A least squares procedure using Airy’s formulation is presented herein. By using a complete ( or closed ) set of functions, each of which satisfies the biharmonic equation, and by en­ forcing approximate compliance at the boundary through a least squares approach, convergent solutions are found. An example is analyzed to demonstrate the practicality of the method. The Boussinesque function is used to represent sin­ gular loads. Use of the Boussinesque function accelerates convergence of the solution. A convergence criterion is established in the "descent to the center" equation solver. Results are compared with the finite element method and photoelasticity. viii CHAPTER 1 INTRODUCTION Airy*5 Stress Function Airy in 1862 showed that the biharmonic equation, ^7 = 0, mathematically describes two-dimensional problems in infinitesimal, isotropic, linear elasticity. If a func­ tion can be found which satisfies the biharmonic equation and the boundary conditions for a specific problem, the solution is in hand. The determination of such functions for boundaries which are not geometrically nice is difficult. Closed form solutions are elusive if not non-existent. Michell set down for elasticians an infinite number of functions prefixed by arbitrary constants which individ­ ual ly satisfy the biharmonic equation. Because this set is a relatively complete set, it describes any plane elasto- static problem. For simply connected regions the set reduces to (Timoshenko and Goodier, 1951, p. 116) 0 0 0 0 M = Cj[r2 + ^ c (4n-2)rncos ne + ^ c(4n-l )rn*2cos n6 n=l n=l 0 0 0 0 « + YU C(4n) rn sln n6 H I c (4n+l)r sln n0 n=l n=l 1 The arbitrary constants$ of course, must be solved for from boundary conditions. The subscripts of the constants indicate the order in which the functions are computed. By using MichelI's functions, the solution of the biharmonic differential equation reduces to the determi­ nation of arbitrary constants. Although the set contains an infinite number of elements, as increasingly more func­ tions are made to satisfy approximately the boundary con­ ditions in a meaningful way, the set converges toward solution. By taking a large enough finite number of these functions in the proper order, convergence to any number of significant figures is theoretically possible. The purpose of this thesis is to set down a practical procedure for solving plane elastostatic problems using Michell’s set of functions. The solutions will be restricted to simply connected regions, and stress tractions only will be allowed on the boundary. Boundary Conditions Through Least Squares As already mentioned, the determination of the constants must be done in a manner which insures convergence, and more importantly, convergence to the proper solution. For a complete set Mn (x), convergence in the mean to the desired function, say F(x), is guaranteed by definition of a complete set (Churchill, 1963, p.61): To use convergence in the mean to satisfy boundary condi­ tions, the value of the actual function on the boundary must be calculated and a line integral employed. The value of the function on the boundary is denoted by M y . P U m 1 ( Mb - Mn(s) )2 ds = 0 (1.2) n go o The next step is to define an error function in terms of the above integral, and then this error function must be minimized. As will be shown, when integrating numerically, the number of points on the boundary used must be at least as large as the number of functions used. CHAPTER 2 FORMULATION Error Function An error function is defined as E = T1 <f> ( V Mn - V Mb) . ( V M n - V Mb) ds (2.1) where V My is the gradient of Airy1s function on the bound­ ary and V M n is the gradient of the first n functions of Michel11s set. The error E depends on how well the arbitrary constants are determined and must approach zero uniformly as n approaches infinity by definition of convergence in the mean. To insure a minimum error for a finite number of func­ tions, E is differentiated with resnect to the arbitrary constants cj. r _ d E = 0 o ( V Mn - V My) • ( V f j) ds j = 1,2,3 n (2 .2 ) it As the- number of functions approaches infinity, the vector V fj becomes arbitrary. By the fundamental lemma of the calculus of variations, since V fj is arbitrary, the quantity ( V Mn - V 1%) = 0. This lemma as stated by Elsgolc (1962) is as follows: "If G(x) is continuous in xQ £ x £ , and G(x) N(x) dx = 0 where N(x) is an almost arbitrary function, then G(x) = 0 in xQ £ x £ x^." Thus in the limit V = V My, and the boundary conditions are satisfied. Since Airy1s function satisfies equilibrium and compatibility, the procedure will produce the unique and, hence, correct solution. Equations (2.2) can be rewritten as ^ ^ CJ ^ ^ f V t1s = ^ ^ Mb • V f.) )ds j = 1,2,3 n (2.3) For each value of j there is an equation. The integral on the left side of the equation represents a matrix which is symmetric. The unknowns are the constants c ^. If n functions are used, n simultaneous equations must be solved. Calculation of Gradient on Boundary The actual gradient on the boundary ( V 1%) can be calculated to within an arbitrary constant (Allen, 1954). The following notation is used: T is the stress traction per unit of arc length, n is the vector normal to the surface S, M is Airy's stress function, and t is the thickness of the shape.
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