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.
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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 = T 1
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.
T
CT (ds cos0 t)
xy
Figure 1 Gradient on Boundary 7
From a free body diagram of the infinitesimal
element in Figure 1, the following equations are devel oped.
T ds t = i ( (J cosO + T sinA) ds t x xy
+ j ( T xy> cosQ 4“ CTy sin0) ds t
T = ( Stresses are related to the Airy's function according to the equations CT^c d . 0"^= and dy' T e l dy 4- (-3^M )(-dx) ds ^x^y ds 4- j (-^ M ) d y 4- (-dx)j Sx^y ds Sx7 ds J T = ( i d - j d ) dM dy dx ds k x T = ( i d 4- i d ) dM dx dy ds = V ^ l = d ( V m ) ds ds Therefore P2 _ _ k x T d s . (2.4) V M b/p2 = V M b/pl * £ The value of V My can be set equal to any constant at point 1 without affecting the value of the stresses. (Stresses are calculated from second deriva tives of the stress function, or first derivatives of the gradient.) When different values are assigned to the gradient at point 1, the difference is absorbed in the second and fourth functions of Michel1's set. In car tesian coordinates these functions are Mg = C2 x and m 4 = c4 y. Positive Definiteness of Matrix The matrix involved in the least squares anoroach is n | ( V ft • V fj) ds j = 1,2,....n The integrand of the above expression forms a matrix which is denoted by F. The vector of constants c^ are assumed to be linearly independent which implies F must be nonsingular. Matrix F can be written as (i • ^7 fu)(t • V fvH(j • V fu)(j • V fv) ds v = 1,2,3,...... n . Thus the matrix F is the sum of two matrices, say Gt G and H where G represents the i components of the gradients and H the j components of the gradients. i.e., F = G*- G + Any matrix R is at least positive semi-definite (Hohn, 1966, p.344). Matrix F then must be at least positive semi-definite since Xt F X = Xt(Gt G)X * Xt(H^ H)X _ 0. But a symmetric nonsingular matrix cannot be positive semidefinite, and by assumption F is nonsingular. Therefore F must be positive definite. The proof outlined above is based on the premise that exact integration takes place. This, of course, is not the ease. Some difficulties can be encountered when numerical integration is used. As an example the integral t o of G G is written out for four functions and three boundary points. Typically the gradient of function r evaluated at point s is denoted by grys. *181/1 *281/2 *381/3 *181/1 *182/1 *183/1 *184/1. *182/1 *282/2 *382/3 *281/2 *282/2 *283/2 *284/2 UI83/1 *283/2 *383/3 *381/3 *382/3 *383/3 *384/3 *184/1 *284/2 *384/3 The u 1s are weight factors dependent upon the integration scheme employed. For example, if Simpson1s one-third rule were to be used, would equal h/3, ug would equal 4h/3 ; 10 and ty would equal , where h is the arc length interval. Now F =s Gt G 4- Ht H t G Ht G 4 x 6 H 6 x 4 (2.5) The j components of the gradients can be shown to be linear combinations of the i components of the gradients. The 4 x 6 matrix of the equation (2.5) has a rank of only three. A violation has thus been created since " The rank of the product of two matrices cannot exceed the rank of either factor." (Hohn, 1966, p.127) For the example, F would have a rank of three which would make it singular, contrary to the conditions set down. To avoid this problem, at least as many boundary points as functions must be used. Comparison of Galerkin and Least Squares In the least squares procedure described, a complete set of functions which satisfy the biharmonic equation is used, and boundary conditions are enforced through determining values of the arbitrary constants. The Galerkin procedure, on the other hand, uses only functions which automatically satisfy the boundary conditions, i.e., yn = V a^ f^ where each f^ satisfies boundary conditions„ The constants are solved for by integrating the product of the Euler Equation of the functional describing the problem with the individual functions. For example, suppose a physical situation is described by the extremal of i (y'2 4- x^y2 - 2xy) dx = v(y). The Euler 2 equation is y" 4- x y - x = 0 = L(y). Galerkin proposed the arbitrary constants be solved for by integrating n f^ dx = 0, i.e., by orthogonalizing the Euler equation with the original functions. Convergence has been proven for Galerkin6s method if the set of functions is complete. The finding of a complete set which satisfies boundary conditions can be somewhat troublesome - especially when more than one dimension is involved. Naviers solution for the simply supported plate is an example of Galerkin’s method. CHAPTER 3 EXAMPLE PROBLEM Statement of Problem The problem worked for this thesis is taken from Frocht (1948). The example was chosen because of the singularities in geometry and loading pattern. The stress field produced by these singularities is not easily represented by a series of functions. 180 lb 1.885 thickness = 0.255" 180 lb Figure 2 Example Problem 12 13 Numerical Integration Schemes Two different integration schemes are employed in the program for the procedure. First, to calculate the gradient on the boundary using Equation (2.4), the trapezoidal rule is used. For linear loading, patterns this method will give exact results within truncation error. For singular loads as in this problem, an approximation to the loading nattern must be made before applying the trapezoidal rule. Figure 3 Equivalent Loading Representation T is the traction per unit of arc length. This implies (h)(t)(ds)=T ( h ) (0.255") (1.414 w") = 180 lb h = 180/ (0.255) (1.414 w ) Ib/sq. in. (3.1) 14 For the integration of Equations (2.3) which leads directly to a system of linear simultaneous equations, a more precise method of integration is employed. The in tegral of the product of the gradients involves higher order polynomials. For example, suppose twenty functions are needed for convergence. Mgg = (20x^y - 20xy^) i 4- (5x^ - 30x^y^ + 5y^) J. An eighth power polynomial would be the integrand. Because higher order polynomials are usually involved in the integration process, Gauss8 quadrature is used as the numerical integration scheme. Gauss attempted successfully to find an uneven spacing pattern which would increase numerical integration accuracy for the same number of points used by an even spacing integration scheme. Very simply, if five equally spaced points are used as the basis of a finite difference integration scheme, fourth order polynomials will be integrated exactly. If five- point Gauss quadrature is used, (2n - 1 ) or ninth order polynomials will be integrated exactly (Scarborough, 1958). Ten-point Gauss quadrature is used in the computer program for this thesis. Sixteen-point quadrature would have given the required accuracy to handle all integrals exactly in the Boussinesque formulation of the example problem. The effect which this more accurate integration scheme would have had was not investigated. Possibly the 15 somewhat fuzzy factor of computer truncation error would obscure this inquiry. Figure 4 show the eight arc lengths which were divided into ten-point intervals. The entire set of the ten points is interior to its interval. .9425) e Figure 4 Boundary Point Distribution The following formula is used to calculate the Gaussian points given the end noints a and b: x = ut (xa - xb ) + (xa + xb)/2 (3.2) where u^ = - u ^q = -.48695326 u2 = - Ug = -.43253168 Ug = - Ug = -,33970478 u4 = - uy = -„21669770 ug = -ug ■= «.07443717 The weight factors R associated with the integration procedure are contained in the computer program which is located in the Appendix. Finally, the Gauss quadrature formula is f(s) ds = (b-a) [ Rjf(xi,yi) 4- R g f ( x g ^ ) 4- . . . The ten point intervals were chosen smaller near the point of application of loading. This enabled the equivalent load representation of Figure 3 to be spread over arc lengths be and cd of Figure 4. In addition, more points are located near the region were a large gradient in the stress field would be expected. The effect of the choice of boundary points is investigated in the next chapter. Equation Solver A descent-to-the-center equation solver is used to solve the simultaneous equations (2.3). The method is especially well suited for ill-conditioned matrices and is limited to positive definite, symmetric matrices. (Booth 1957 ) 17 The basis of the method is an error function S where S = (1/2) Xt A X - X, X is the vector of unknowns, A is the symmetric matrix, and B is the constant vector. This error is minimized in a definite number of steps, i.e., not by an iterative procedure. The value of X corresponding to a minimum error is the solution. The descent-to-the-center method is analogous to the Gram-Schmidt process of matrix algebra. The error describes a set of concentric hyperellipsoids. The common center of these ellipsoids is the solution. By mathematically pro ceeding to the center along vectors, each of which is orthogonal to all previous descent vectors. The solution is obtained in a number of steps equal to the number of equa tions being solved. Since the matrix is symmetric, only slightly more than half of it need be stored. When solving the equations by descent to center, the matrix is not destroyed. If m equat ions with m unknowns are stored, a number n (less than m) of equations can be solved if desired. An escalation scheme is employed in the solver which avoids the necessity of all computations being made each time a larger number of equations is soIved. Each solution produces an error S which is calculated. A convergence analysis is made in the next chapter based on the value of (S„' ui - Sram-iz i)/Sm * m . 18 Calculation of Stresses Once the arbitrary constants have been calculated the stresses follow by definition of Airy's stress function from the equations A stress solver is included in the Fortran program, It first calculates stresses in polar coordinates, and then converts these to cartesian via the stress transformation law. Principal stresses are calculated from the cartesian stresses by using Mohr's Circle relationships. Symmetry of Problem The boundary and loading patterns are symmetric with respect to the x and y axes. This symmetry implies that only even angled cosine functions are needed to rep resent stresses, i.e., r^, r^cos 26, r^cos20, r^cos48, r^cos 40, etc. In addition, the functions rcos0 and rsin0 are included in the computer program to allow for an arbitrary selection of the gradient at one point on the boundary. 19 Y Figure 5 Symmetrical Stresses For example, in Figure 5 the stresses at point a should equal the stresses at point b . The following relationships are obviously true. cos A 4 cos(1RO-Q) cos 2A = cos 2(180-A) sin ft 4 sin(180-ft) sin 20 4 sin 2(180-ft) When calculating stresses by equations (3.3), second derivatives with respect to ft are involved for (7r andC7q . Thus if cosine functions are used to compose Airy1s stress function, symmetric stresses will result. 20 Comments on Programming The boundary data input data consist of end points for Gauss quadrature intervals. The program computes the ten interior points for each interval for straight line boundaries. A subroutine for curvilinear boundaries can be developed through use of a Newton-Raphson type search. Tractions are read in in terms of cartesian coor dinates. The gradient of the stress function is set to zero at x = 0.9425 and y = 0.0. The Boussinesque tractions (explained in the next chapter) are calculated by a sub routine. The program transforms all tractions to polar coordinates6 The gradients of Michell8s functions are generated from generic form in polar coordinates. Stresses are calculated in terms of polar coor dinates and then transformed to be read out in terms of principal stresses and cartesian stresses. The program is heavily documented with comment statements and is contained in the Appendix. CHAPTER 4 BOUSSINESQUE FORMULATION Convergence Rate of Equivalent Problems The convergence of certain mathematical series can be accelerated by breaking the series into two other series. For example, the series (Kantorovich and Krylov, 1964, P. 78) OQ y l/(n2 + 1) takes 1000 terms to get accuracy n=l of 0.001. This series can be broken into two series. l/(n^ + 1) = y r 1/n^ - l/n^(n^ + 1)"] n=l n=l The summation of the first term on the right hand side of the above equation is known to be 'TT /6. The second term converges to three decimal accuracy in eight terms, a vast acceleration. This process can be continued until convergence can be obtained with just one term. ) l/(n2 + l) = ) £l/n2 - 1/n^ 4- 1/n6 - 1/n® 4- l/n^O . l/n^(n2 4- 1)] The summation of the first five terms is known. The last summation can be computed to three decimal accuracy with just one term. 21 22 A similar idea can be used in the least squares procedure. Very likely the singular loading pattern of Chapter 3 causes stress patterns which are difficult to represent by a small number of functions. In this chapter the problem is broken down into three problems, and the solutions are then superimposed or added. The stress field for a point load on the half plane has been known since the late nineteenth century. Through superposition this function is used to accelerate convergence. The Boussinesque Function The stress distribution due to a load P acting on a semi-infinite plane is given by (Timoshenko and Goodier, 1951). CL = - 2 P cos 7T t r °"@ = o, TrS ^ (4.1) Figure 6 on the next page shows the loading for this function. i 23 P y X Figure 6 Concentrated Load on Straight Boundary 24 Superposition Procedure In the half plane the example problem can be drawn as shown in Figure 7. Since T rq and G q equal zero, there will be no stresses on the planes 1-2 and 1-4. To use the Boussinesque function, the value of P* (not P) must equal 180 lb. The value of P can befound by integrating the vertical component of force alongthe arc ab. P* = 180 lb sr 2 P I (cos 6) cos 0 r dA 77* t J a r P = 360 7T lb ( 77* ♦ 2) This value of P must be used to insure P* = 180 lb. Ill III Free Body Diagrams Figure 7 Square in Half Plane 25 A force P must also be applied at point 3 as shown in Figure 8. When Figures 7 & 8 are superimposed. Figure 9 results. Free Body Diagram P P* Figure 8 Complementary Half Plane Bouss inesque tractions Figure 9 Boussinesque Boundary Tractions 26 The stresses throughout the square of Figure 9 can be calculated by superimposing the Boussinesque stresses of equations (4.1). The Boussinesque boundary tractions can be calculated by first calculating the stresses of equations (4.1) at the boundary - and then using the stress transformation law to find the stresses acting on the boundary. By reversing these Boussinesque tractions, Figure 10 results. Reversed Boussinesque Tractions Figure 10 Superposition of Stress Fields 27 Because the stress fields are linear, they can be superimposed. As previously mentioned, the stresses in Square I of Figure 10 are easily calculated. The stresses in Square II can be obtained by the least squares pro cedure. Since the stress field of Square II is more nearly uniform than that of Square III, fewer functions are needed to represent it. Effect on Convergence Elsgole (1962, p.148) says the following as to the convergence of direct methods in the calculus of variations: Having computed yn (x) and yn4.l (x), we compare them at some points of the interval (x0 , xl). If with the degree of exactness required for a given purpose their values coincide, then we consider the solution of the variational problem equal to . yh('x). Figure 12 shows the locations of the points where stresses were calculated to test for convergence. As mentioned before, the equation solver in the computer program operates on a matrix without altering it. To make use of this option, a matrix is generated using more func tions than is anticipated to be needed for convergence for a specific problem (based on experience). As a first trial, a sub-matrix and sub-vector are operated on. 28 For example, in the Boussinesque formulation, a matrix based on 31 functions was generated. First, the 1 x 1 matrix was solved, then the 2 x 2, etc. When the equation solver is operating on n equations, it uses the \ last vector calculated during the solution process of (n - 1) equations as a first descent vector. Very likely then, the process will take fewer than n vectors to get within a certain tolerance of the solution. A termina tion test is included in the subroutine based on the value of VAV „ 00000001 (Sn=,^) where V is the last calculated descent vector, A is the matrix, and S is the descent error calculated from the previous solution. At times, several vectors must be calculated, but usually a number less than the number of equations.. The entire example problem including the 31 sets of simultan eous equations took less than four minutes on the IBM 7072 digital computer. To avoid calculating stresses at a number of ran dom points each time an increased number of functions is used, a printout command is based on the error S associated with the descent-to-the-center process. The value Q = (Sm i - Sn)/ Sn is the actual controlling quantity. For example, printouts were made for Q = 10“^, 10"^, etc. 29 TABLE I BOUSSINESQUE CONVERGENCE Node 11 Functions 18 Functions 31 Functions S1sx Sigy sigx Slgy Si§x Sigy psi psi psi psi psi psi 73 228 -783 228 — 782 229 -782 75 160 -617 159 -616 160 -615 77 81 -398 80 -396 79 -394 79 12 -120 6 -122 3 -119 8 fr-8 00 61 219 -881 220 00 221 -882 63 175 -778 175 -777 176 -778 65 92 -559 92 -558 94 -557 67 19 -303 20 -301 21 -300 49 159 -110 160 -110 160 -110 51 107 -981 108 -982 108 -983 53 13 -746 14 -745 14 -747 55 ” 68 -481 -67 -479 -67 -479 37 95 -1404 96 -1406 95 -1407 39 17 -1236 18 -1238 17 -1239 43 -198 -653 -196 -654 -198 -655 31 -310 -815 -307 -818 -310 — 820 23 -469 -1119 -468 -1122 -470 -1126 9 -874 -3523 -881 -3520 -881 -3518 30 TABLE II EQUIVALENT-LOADTNG CONVERGENCE Node 22 Functions 59 Functions 66 Functions sisx Sigy Sigx Sigy Sigx Sigy psi psi psi psi psi psi 73 231 -784 231 -784 230 -783 75 162 -616 160 -614 160 -615 77 80 - -391 79 -392 79 -393 79 -1 -113 5 -120 5 -119 61 222 —884 222 -885 222 -884 63 177 -779 178 -779 178 -779 65 95 -559 95 -555 94 -555 67 26 -299 20 -295 20 -295 49 160 -1103 158 -1102 158 -1102 51 106 -982 106 -986 106 -985 53 8 -742 15 -750 15 -750 55 - 74 -481 -61 -480 -62 -480 37 95 -1420 88 -1411 88 -1411 39 19 -1245 9 -1239 9 -1238 43 -200 -638 -209 -658 -209 -658 31 -297 -791 -322 -808 -321 -808 23 —462 -1145 -448 -1144 -447 - m i 9 -895 -3619 -942 -3745 -946 -3746 31 In the Boussinesque formulation, at the twenty- ( two points checked for stresses, all stresses calculated with 18 functions agreed with those calculated with 31 functions within 3 psi. Table I contains these results. With just eleven functions, very satisfactory results were also obtained, and these are also contained in Table I, In the equivalent loading representation, all stresses calculated with 59 functions agreed with those calculated with 66 functions within 2 psi. These results are contained in Table II, With just twenty-two functions, very satisfactory results were obtained, and these are also contained in Table II. Effect of Boundary Point Selection If a high-enough Order integration scheme is used, the choice of boundary points should have no effect on the determination of the arbitrary constants. If at least as many points as functions are used, the matrix generated seems to be assured of being nonsingular. Figure 11 shows the three different point patterns investigated with use of the computer program. Each pattern has eight intervals which are divided into ten point Gauss ian spacings. Pattern A; 1 - 2a, 2a - 3, 3 - 4a, 4a - 5, 5 - 6a, 6a — 7, 7 - 8a, 8a — 1 32 4c 2c (.2, .7425) 4a 6a Figure 11 Boundary Point Intervals - Bdussinesque Pattern B: 1 - 2b, 2b - 3, 3 4b, 4b *» 5, 5 - 6b, 6b " 7, 7 - 8b, 8b — 1 Pattern C: 1 - 2c, 2c - 3, 3 - 4c, 4c - 5, 5 - 6c, 6c » 7, 7 - 8c, 8c - 1 The results of Table III appear to be very good„ But notice should be made of the fact that a large portion of each stress was calculated by the Boussinesque function. In addition, only twenty-one functions were used to deter mine Table III. The boundary may be very substantially determined by all three of the patterns. In other words, 33 this investigation does not demonstrate that the computer program will always or even usually overcome the problem of boundary point selection. However, the use of a very large number of boundary points should make the numerical integration accurate enough so that truncation errors within the computer will dominate the resultant errors. Table III contains the results of the inquiry. Table IV gives the coordinates of the node points listed in Table III. Figure 12 shows the location of the node points. 34 TABLE III COMPARATIVE STRESSES FOR THREE BOUNDARY POINT PATTERNS NODE PATTERN A PATTERN B PATTERN C Sigx Sigy Sigx Sigy Sigx Sigy psi psi psi psi psi psi 73 229 -783 229 -782 228 -782 75 160 -616 160 -615 159 -615 77 80 -395 79 -394 79 -395 79 6 — 120 4 -120 4 -121 61 221 -882 221 -882 220 -881 63 177 -778 176 -777 176 -777 65 94 -558 93 -557 93 -557 67 21 -301 21 -299 20 -301 49 161 -1099 160 -1099 161 -1098 51 109 -983 109 -983 109 -982 53 15 -747 15 -746 15 -746 55 —67 —480 -67 -479 —67 —479 37 96 -1407 95 -1407 96 -1406 39 18 -1239 17 -1239 18 -1238 43 -196 -656 -197 -655 -196 -655 31 -308 -820 -309 -820 -307 -819 23 -469 -1124 -470 -1125 -469 -1123 9 -882 -3520 -880 -3519 -883 -3519 35 23 43 45 55 67 63 65 Figure 12 Node Point Location TABLE IV NODE POINT COORDINATES Node Point X - Coord„ Y - Coord. 73 .14 in. .0 in. 75 .29 .0 77 .44 .0 79 .64 .0 61 .09 .23 63 .19 .23 65 .33 .23 67 .48 .23 49 .075 .43 51 .142 .43 53 .24 .43 55 .345 .43 37 .06 .56 39 .115 .56 41 .18 .56 43 .26 .56 29 .1 .635 31 .21 .635 CM .081 .7125 23 .155 .7125 9 .05 .8425 CHAPTER 5 COMPARISON OF RESULTS Finite Element Figure 12 shows the constant stress triangle mesh used in the computer program of Mr. Melvin L. Callabresi and Dr. Ralph M. Richard of The University of Arizona. The stresses are calculated by taking a simple average of the corresponding stresses of all triangles surrounding a particular node point. Stresses are also calculated at these same node points by the Boussinesque formulation of the least squares procedure. The results are contained in Table V. In general, the least squares stresses are of slightly higher magnitude than those of finite element. Photoelasticity The results are taken from Frocht (1948). Stresses are compared with least squares along the lines y = 0.0 and y = 0.377. The results are contained in Table VI. Values for other locations are not tabulated by Frocht. In general, the least squares stresses are of lower magnitude than those of photoelasticity. 37 TABLE V FINITE ELEMENT COMPARED WITH LEAST SQUARES J Node Finite Element Least Squares SiSx Sigy Sigx Sigy psi psi psi psi 73 203 — 761 229 -782 75 119 — 604 160 -615 77 49 — 3 69 79 -394 79 7 -261 4 -120 61 212 -879 221 -882 63 153 -736 176 -777 65 72 -518 93 -557 67 5 -277 21 -299 49 145 -1095 161 -1099 51 111 -951 109 -983 53 23 -720 15 -746 37 92 -1395 95 -1407 41 -91 -986 -90 -977 29 -156 -1622 -64 -1477 21 -274 -1786 -182 -1898 15 -467 -2420 -477 -2662 9 -637 -3358 -882 -3520 39 TABLE VI PHOTOELASTICITY COMPARED WITH LEAST SQUARES Coordinates Photoelasticity Least Squares SiSx Sigy Sigx Sigy X Y psi psi psi psi • 8 00 CO '.09425". .0 " 260 =876 243 .28275 ,0 173 = 679 164 = 625 .5655 .0 36 -214 25 -212 .84825 .0 1 -13 10 -10 .09425 .377 196 -1041 173 = 1004 .28275 .377 51 -609 30 -642 .47125 .377 51 -257 63 = 240 40 Comparison of Boussinesque and Singular Loading Because of the difference in load representation, a difference in stresses is to be expected near the load application point. At points removed from the loading area stresses should be much alike. The comparison serves as an excellent demonstration of St. Venant1s Principle. At the line y = 0, the stresses calculated are all within 5 psi. Near the load point, a stress difference of 12% in the x-direction and 3% in the direct ion is typical. The results of the comparison are contained in Table VII. Suggested Extensions The most serious drawback ofthe procedure as set forth is that it handles only stress tractions. The com puter program does not handle the displacement boundary value problem, or the mixed boundary value problem. If a complete set of functions could be found for Navier1s equations, perhaps a least squares procedure could be formulated for the displacement boundary value problem. Another shortcoming of the Fortran program is its inability to generate Gauss quadrature points for curvi linear boundaries. A NeWton-Raphson type search sub rout ine could be incorporated into the existing program to determine Gaussian spacing for more general boundaries, 41 TABLE VII DEMONSTRATION OF ST. VENANT'S PRINCIPLE Node Singular Loading Bouss inesque Sigx Sigy Sigx Sigy psi psi psi psi 73 230 -782 229 -782 75 160 -615 160 -615 77 81 -396 79 -394 61 224 -884 221 -881 79 8 -118 '3 -119 63 177 -776 176 -778 65 91 -553 94 -557 67 18 -301 21 -300 49 166 -1110 160 -1100 51 114 -989 108 -983 53 17 -739 14 -747 55 -66 -464 -67 -479 37 88 -1426 95 -1407 39 18 -1258 17 -1239 #-4 00 00 43 8 -635 -198 -655 31 -302 -822 -310 -820 8 00 9 -802 -3588 00 -3518 APPENDIX Contained in the appendix is the corrmuter nrogram used to obtain results for this thesis. The main program follows the listing of the subroutines„ 42 TE LO DE INTERMEDIATE THE 6 LOOP CALCULATIONS TOWARD DOES C no n n n no n n no no onnon COMPILE EXECUTE FORTRAN# FORTRAN * P *P +8(IB)*X(J) 4 R(K)= P - C(K) 5 102 KT»K 101 KT«J 0 IB*LN~KT103 C IS TOLERANCEACC VALUE AS FOR ESCALATINGTESTUSED DEFINTE MATRICES D O 11 I=11D O * l *XINE) 0.0 # N E =VINE) 0.0 OF TWICE =ERROR = DESCENT BX**22C SUBPOUT INE 3EUSS (P * T * XX * YY * A , TRACK * TRACY) C THIS SUBROUTINE CALCULATES 30USSINESCUF TRACTIONS C FOR SQUARE IN HALF PLANE DIMENSION XX(3 9) ,YY(89)*TRACX(89)♦TRACY(39) AFC=.5*SORTF(2.) 01=3.141593 DO 1 1=45*67 RB=SORTF(XX(I)**2+(A-YY(I))**2) A1=AFC*( (4-YY(I ) )-XX( I ) )/RB r\ n n n m non 1 TRACY(n*AFC*I12-AFC»52 1 TRACY(n*AFC*I12-AFC»52 2 TRACY(ML)»-TRACY(NJ L- N=45+M A2=-AFC*((A-YY(!))+XX(lJ)/Rfi NFD4a(NFU-1)/4 NFD2= (NFU-3 ML=M+1 M 2 00 = 1 ♦?! .R2«A1*A1*TRACR TRACX(MM)a-TRACy(N) TRACR=(2.*P*(A-YY(I)))/(RB*PI*T*RB) COSN=X5(N)/P TRACX(ML)«-TRACX(N) TRACY(L)*-TRACYfN) T R A C Y ( M M ) » T F ? A C Y ( N ) T12=A1*A2*TRACR A=.?425 RETURN -QT(SN)**2+YS(NP-SQRTF(XS(N )**2) DO N*ItNPSA00 NODES. THE 400 LOOP CALCULATESATRESSES FOR PARTICULAR TRACX(89)*TRACX(1 ) NFU=31 DIMENSION XS{29)*YS(29) STSOl. (XS,YS,C,GDL)SUBROUTINE SINMaYS(N)/P TRACX(1)*-TRAC (45)X TRACY(S9)»-TRACY(1) TRACY(1)=-TRACYf45) TRACX(L)*TRACX(N) TRACX(I)=-AFC*Tl2-4FC*R2 IESO C(41DIMENSION C ) END TRACY (45 ATO - TN MS B MDFE IFCAUTION BE ATANF MUST FUNCTIONS MODIFIED - OTHER STHETaSIGR SIGR=2t*C(l) TRACX TRACX(23)=0* NPS«29 = POINTSNPS NODE NUMBEROF CALCULATED LEAST PROCEDURE. THE SQUARES BY than TIONS AND DIRECTIONS PRINCIPAL THEUSING CONSTANTS THIS IN SUBROUTINE CALCULATES STRESSESX-Y DIREC TAUzC. TRACY(23)=-TRACY(67) 45—M even ( 67) *0 * angled )/?. functions are to be used 45 46 THEDA = 2•#ATANF(Y5(N)/XS(N) ) RBIG=1 • PP a p-R- p RGIAMsPP C THE 47 LOOP CALCULATES least squares stresses from C GENERIC rORM* DO 4 7 L=1*NFD2 YX = L TEMP=YX*THEDA COSA=COSF(TEMP) SIMA=SINF(TEMP) LL=2*L B=LL-1 7 -LL D=LL+1 E*LL+2 NA=2*L+2 NB*2*L+3 SIG R = SIGR « C(N A)* RB!G > C0 S A * Z * B > C(N B)* C0 S A * R GI A N *(E-Z 1**2) STHET=STNET+C(NA)*Z*S*R8IG*COSA+C C CALCULATION OF PRINCIPAL STRESSES COMP 1 *(SIGX+SI GY)/2 * COMP2=SORTF(( SUBROUTINE CNST (X♦Y.TRACX,TRACYtSUMBfU#R♦AL eXX.XS♦ 1GDL) DIMENSION XS(22),YS(22) DIMENSION B(IOOO)♦SUMB(69) DIMENSION XX(41) COMMON KN,NE,LN,ACC,1B DIMENSION U(10).R(10) DIMENSION DELX(89)»DELY(80)»GRADR(69)*GTHET(69) DIMENSION SIN(89)•COS(89) DIMENSION X (89) »Y(89)tTRACXf 89),TR1CY(39) DIMENSION DX(89) •DX.y(fl9) DIMENSION AL(C ) C THIS SUBROUTINE CALCULATES ARBITRARY CONSTANTS FOR C LEAST SQUARESe ACC* *000001 NFU=31 KN = 31 LN = 62 TEST*:. GOOD =#000001 NP =89 NFUPcNFU+1 ZAXD = fUC ZAYD=0#0 C INITIALIZATION OF GRADIENT ON POUNDA9Y DELX< 1 )=0. • DELY(15=0.0 N P1 = N P -1 C LOOP 1 CALCULATES GRADIENT ON BOUNDARY. DO 1 I=1 ♦ NP1 J = I + 1 DX( I ) *X{J ) - X(I ) DY=Y(J)-Y(I) IF(DX(I))99,99,99 96 DXM(I>=SQRTF(l.+(DY/DX(I))**Z.) GO TO 97 99 DXM(I)=1. 48 DX(I)=DY C TRAPEZOIDAL RULE FOR INTEGRATION 97 TRAPY»DXM(I)*(TRACX(I)+TRACX(J))*ABSF(DX GRADRO )«SIN(M) GTHETO J«COS(M) RBIG»P R2*P**2. C LOOP 965 CALCULATES GRADIENTS FROM GENERIC FORM* DO 965 N*1*NFD2 YX*.N TEMP«YX*THCDA SXNA-SINF(TEMP) COSA-COSF(TEMP) NA»2*N+2 NB=2*N+3 A=2*N-1 0B=A+l* C=A+2* D=A+3* GRADR ( NA ) aBB*P.B!G*C05A 5THCT(NA)=-BS*Rn!G*SINA GRADR 300 FORMAT ( 13H STEP .NUMBER , 12* 10(2H, )> 500 FORMAT (13H1VAV/QF L.E. »1PE14.7) 700 FORMAT (1H ) 800 FORMAT(1H /•(1P6E15•7 ) > RETURN END C MAIN PROGRAM C AIRYS FUNCTION BY MODIFIED TREFFTZ METHOD DIMENSION X (?)♦Y(?) ,TRACX(S9)* TRACY(39)*XX(39),YY{8 19»AL(8) DIMENSION U(10),2(10) DIMENSION XS(29),YS(29) DIMENSION CONST(69) DIMENSION ZZ(41) NPS»9 ND»2 9 C READ IN INTERVAL END POINTS READ 11,(X(N),N=1,NPS) DEAD 11,(Y (N),N»1,NP5) C READ IN NODES WHERE STRESSES TO BE CALCULATED READ 11, (XS(N) ,N = 1*ND) READ 11, ( Y S ( N ) ,\’=1,ND) 11 FORMAT (SF10.5) PRINT 710 710 FORMAT (1HC,26H X COORDINATES OF BOUNDARY > PRINT n,(X(N) ,N=1,NPS) PRINT 711 711 FORMAT(1 HO »26H Y COORDINATES OF BOUNDARY ) PRINT 11 , U(6)*-U(5 > Uf7)=-U(4) U(8)»-U(3) U(9)=-U(2) U(lO)r-U(l) C GAUSS QUADRATURE WEIGHT FACTORS R(l)«#03333567 R (2)* e07472567 R(3)=.10954318 R(4)*•13463336 R (5)3•14776711 R(6)*R(5 > R < 7)=R(4 ) R (8)*R(3) R ( 9 ) = R ( 2 ) R (1 0 ) s R (1 ) XX(1 )=X(1 ) XX(12)=X(2 ) XX(23)xX(3 ) X X ( 3 4 ) = X ( 4 ) X X ( 4 5 ) 3 X ( 5 ) XX(56)=X(A) XX(67)eX (7) XX < 7 9)eX(8) X X ( 3 9 > = X I 9 ) VY(1)=Y()) YY(12)=Y(2) YY(23)*Y(3 ) YY(3 4)=Y(4) Y Y ( 4 5 ) = Y ( 5 ) YY(S6>=Y(6) YY(67)=Y<7) YY(78)=Y (3 ) YY(89)3Y (9) M0«0 LEG= 8 C LOOP 313 CALCULATES INTERMEDIATE POINTS* DO 313 L = 1«L £G 1 LL=L+1 YD=Y(LL)-v(L) XD=X{LL)-X(L) A»(X(LL)+XfL) )/2 • P=(Y(LL)+Y(L))/2. DO 313 Melt 10 MO*MO*1 XX(MO)*U(M)*XO+A 313 YY(MO)=U(M)*YD+e A»YY(22 ) T=•255 P=180.*1.5708/1.2854 52 GDL»P CALL 9EUSS CP♦T•XX *vy*A *TRAC XtTRACY) CALL CN5T (XX♦YY♦TRACX•TRACYiCONSTtU♦R#AL♦ZZ tXS»YS• 1GOL) CALL STSCL (XS,YS,Z?,P) STOP END REFERENCES Allen, D. N, de G„, Relaxation Methods in Engineering and Science. McGraw-Hill Book Company, Inc., New York, 1954. ; Booth, A. Do, Numerical Methods. Academic Press Inc., New York, 1957. Churchill, R. V., Fourier Series and Boundary Value Problems, McGraw-Hill Book Company, Inc., New York, 1963, Elsgolc, L. Eo, Calculus Of Variations. Add ison-Wes ley Publishing Comnany Inc., Reading, Massachusetts, 1962. Frocht, M. M., Photoelasticity. Vol. 1. John Wiley & Sons, Inc., New York, 1948. Hohn, F. E ., Elementary Matrix Algebra, The MacMillan Company, New York, 1966. Kantorovich, L. V., and V. I. Krylov, Approximate Methods of Higher Analysis. Interscience Publishers, Inc., New York. n S S T ” Scarborough, J. B., Numerical Mathematical Analysis. The John Hopkins Press, Baltimore, 1958. Timoshenko, S. and J. N. Goodier, Theory of Elasticity. McGraw-Hill Book Company, Inc., New York, 19517 53