ON THE GENERAL SELF-EQUILIBRATED EIID LOADING OF A SOLID LINEARLY ELASTIC CYLINDER Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY ‘ JAMES LOUIS KLEMM 1969 IMC- This is to certify that the thesis entitled On the General Self-Equilibrated End Loading of a Solid Linearly Elastic Cylinder presented by James Louis Klemm has been accepted towards fulfillment of the requirements for Ph. D. degree in MGChaniCS @gm Major professor Date lQ/3O /é? / 7 0-169 LIBRARY Michigan State University sarcoma av " "DAB & SUNS' WE’LEIIIQIILJEE- ABSTRACT ON THE GENERAL SELF-EQUILIBRATED END IDADING OF A SOLID LINEARLY ELASTIC CYLINDER By James Louis Klemm The problem studied is that of a semi-infinite cylinder with the long sides free from stress and a self-eauilibrated, but not necessarily axisymmetric, load applied to the finite end. The method of biorthogonal vectorial eigenfunctions is applied to a formulation of the problem, within the classical theory of elasticity, using the stress equations of equilibrium and the Beltrami-Michell equations of compatibility. The axisymmetric case is shown to unify the classical work on axisymmetric self-equilibrated torsion end loading, with the more recent work on torsionless self-equilibrated end loading. The first few eigenvalues are calculated for various indices of theta dependence, and the numerical solution of two illustrative problems is presented. ON THE GENERAL SELF-EQUILIBRATED END IDADING OF A SOLID LINEARLY ELASTIC CYLINDER By James Louis Klemm 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 Material Science 1969 / t“ (7' {FtyzvfltJ /- 7 0 To the memory of my father, Paul O. Klemm ii ACKNOWLEDGEMENTS The author wishes to acknowledge his debt to the National Science Foundation for their sponsorship of the 1967 Summer Institute in Applied Mathematics and Mechanics, which brought him to Michigan State University for this study, and for their partial support of this project through research grant GK 1325. The author wishes to express his thanks to the following individuals. Dr. D.J. Montgomery, Chairman of the Department of Metallurgy, Mechanics and Material Sciences, for his support of these studies, Dr. R.W. Little, for Suggesting the problem and serving as advisor and mentor, and Committee members Dr. J.S. Frame, Dr.D,H,YQYen and Dr. L.E. Malvern. Finally the author would like to express his thanks to Mr. E.Graber for writing some of the programs used and reading some of the more tedious parts of this work. iii CHAPTER I: INTRODUCTION TO THE PROBLEM Section 1.1: Origin and Scope of the Problem The problem studied is that of the deformation of a semi- infinite elastic solid cylinder due to a self-equilibrated end load. The long sides are assumed to be free from stress. All analysis is undertaken within the classical theory of elasticity. The problem arises naturally in either of the following two ways: as a difference problem for the general loading of long cylinders or as an example in the general discussion of St. Venant's principle. Solutions are known which can treat the side loading of a long cylinder (see, for example, Lur'e [24, Chapter 7]). These solutions carry with them, however, Specific end conditions with control only over the resultant force and resultant moment applied to the ends. The difference problem for a semi-infinite cylinder, between general loading problems and the class of problems amenable to treatment using these solutions, is a self- equilibrated end loading problem. In the review of the literature that follows it will be seen that a number of peOple have studied this difference problem for axisymmetric loadings, but that much remains for study in the non-symmetric case. St. Venant encountered this difficulty in his Treatise on Torsion [28], since he could satisfy the end conditions, using his solutions for the torsion of long rods, only up to the pre- scribed resultant force and the prescribed resultant moment. He asserted, therefore, that this failure to satisfy the end conditions exactly had little effect apart from the ends of the rods and this assertion, along with several variations of it on the part of others, became known as St. Venant's principle. As contributions to the study of the principle made by Thomson and Tait [34], Bousinesque [5], and others seemed to give the principle some validity, it became widely accepted, if only out of necessity. In 1945, however, von Mises [38], raised the question as to whether these more general assertions being justified by appealing to "St. Venant's principle" were indeed true, and pro- duced counterexamples to prove that for certain kinds of loadings, the effect away from the area of loading was much greater than was widely believed. Furthermore, he proposed, as sufficient, a criterion for assuring the rapid decay of stress away from the region of loading. Since the publication of his paper the study of problems connected with the statement of such a principle, the range of validity of a given principle, and the study of the St. Venant boundary region (the region of the solid where the distribution of the loading is not negligible) have been among the major areas of interest in the theory of elasticity. Of the papers that followed that of von Mises, Sternberg [31], who first proved the correctness of von Mises' formulation of the principle, Erin [ll], Boley [4], Dzhanelidze [10], Donnel [9], Toupin [35], Keller [19], Knowles [20], and Roseman [27] have presented papers which attempt to give a more precise statement of the conditions governing the rate of decay of stress and estimates of the orders of magnitude for the re- sulting stress within the body. The difference problem, for the case of self-equilibrated stresses, is related to these studies in the following way: The stresses are known to decay as one moves away from the vicinity of a self-equilibrated loading, and their entire effect is confined to the St. Venant boundary region. Thus the difference problem is a St. Venant boundary region prob- lem. The investigations mentioned above provide estimates for the effective width of this region and give bounds on the stresses that such a loading provides in the interior of the body, often for quite general geometries, but within restricted geometries it is sometimes possible to solve the St. Venant's problem, itself, and obtain the exact stresses that a given self- equilibrated loading provides at each point of the body. The formulation and solution of the St. Venant boundary region problem for the end loading of a semi-infinite cylinder is presented in this thesis and numerical results are reported for two particular problems as an indication of the rate of convergence of the series of eigenfunctions which constitute the solutions. When diSplacement end conditions (or diSplacement conditions in conjunction with stress end conditions) are prescribed, one faces difficulties of two sorts: First, the conditions for the existence of decaying solutions for these kinds of boundary conditions are known only for the end loading of a semi-infinite two-dimensional strip (see Gusein-Zade [15, 16]) and these con— ditions are quite different from the corresponding conditions for stresses. Second, when diSplacement end conditions are pre- scribed in conjunction with stress conditions, one is known to be liable to the generation of stress singularities (see Williams [42]). Although these difficulties have not yet been resolved, this thesis does present a way of formulating two of the fundamental mixed problems in terms of stress-type variables and provides a solution in terms of a series of eigenfunctions, whose coefficients may be chosen directly by a biorthogonality relation derived in this study. Section 1.2: Progress on the Class of End Loading Problems for a Cylinder Problems of axisymmetric end loadings of cylinders have been studied by a variety of methods: Purser (see Love [23, Section 226B]) obtained the eigenfunction expansion governing the axisymmetric torsion end loading problem. Prokopov (see Lur'e [24, Chapter 7]), studied the solution of axisymmetric end loading problem in a diSplacement formulation using the Papkovich-Neuber formulation and calculating the first few eigenvalues, along with the values of some useful parameters. Valov [37] used a Fourier-Bessel expansion with the Papkovich- Neuber potentials to study two mixed problems, obtaining an infinite system of linear equations, whose regularity he also investigated. Horvay and Mirabal [17] applied the calculus of variations to the study of axisymmetric normal and shear end loading. Alekandrov and Solov'ev [2] reduced the Papkovich-Neuber formulation of three-dimensional axisymmetric problems into problems of a complex variable, and Chemeris [8] to integral equations. Mitre [25] studied the transversely isotropic cylinder of finite length using a single second order Love-type strain function. Little and Childs [22] obtained a vector biorthogonality from Love's strain function by use of the calculus of residues. Swan [32] integrated the displacement equations to obtain the governing eigenfunctions for the case when the sides of the cylinder were free from stress, but expanded the solution obtained in terms of a complete set of orthonormal functions and lost computational efficiency due to the slow convergence of series of Bessel functions. Approximate methods include work by Warren and Roark [39], who employed an aggregate of exact solutions obtained from a displacement formulation and chose the coefficients for two torsionless stress - stress end loadings by a least squares technique. The list of papers presented above is not complete, but does indicate the scope of interest and the variety of methods which have been applied to this class of problems when restricted to being axisymmetric . Relatively few papers, however, report on methods of attacking the solution of non-axisymmetric three dimensional problems. Muki [26] (in Japanese) extended Sneddon's axisymmetric solution for a half-plane to non-symmetric loadings, and Alexandrov and Solov'ev [3] have extended their formulation of axisymmetric problems in terms of complex variables to that of non-axisymmetric ones. Flfigge and Kelkar [12] have proposed a way of solving the general displacement loading of a semi-infinite cylinder, but the method is presented in detail only for the axisymmetric case. The paper derives the following generalized vector orthogonality relation for the axisymmetrically loaded solid cylinder: 1 ‘1 a T— a _ _ m n (km-18%) yn S ymdr " men u1h2n(0)h2m(0) where km and kn are, respectively, the mEll and nEh eigen- values with corresponding eigenvectors ym and y“, S is a weighting matrix, ”I a constant and hm, hn are functions whose form is known but which depend on the eigenfunction chosen. The coefficients desired are the constants d occurring in an eXpan- n sion of the form but the orthogonality relation cannot be used directly to isolate Ch . . . . the m—_ coeffic1ent Since the boundary data involves all of the eigenfunctions,simultaneously, and the right hand side remains valid only when the functions, h, have a prescribed relationship to which eigenfunction is used. No numerical work was included, (not even to the extent of tabulating eigenvalues) and there is no indication that the authors were aware of the computational difficulties, including the possible necessity of solving a truncation of an infinite system of equations, frequently associated with this class<1f problems. Finally, D.R. Childs [7] claims to have obtained a general solution for three dimensional problems of elasticity in terms of a Single biharmonic function. The reviewer, B.E. Gatewood [13], suggests that the method, at best, could possibly apply to some axisymmetric problems. The author presents no example or argument to indicate that his solution is sufficiently general to be of use in the Study of boundary value problems. Section 1.3: The Equations of Compatibility The three equations of equilibrium for an elastic body may be written as three equations involving the three unknown displacements in the coordinate directions or as three equations in the six independent stresses. It is therefore necessary that the six small strains or the six independent stresses satisfy three equations,in addition to the equations of equilibrium, to assure that they give riseto a single-valued displacement field for a simply-connected body. This information is, in each case, carried in a set of six partial differential equations, known as the St. Venant equations of compatibility for the small strains case, and the Beltrami-Michell equations of compatibility for the stress case. An outline of a way of deriving the latter set of equations from the former may be found in any text on elasticity. The fact that there are obtained six equations of compatibility in the general case, when it is known that the system carries the information of three independent equations has apparently been reSponsible for delaying the study of the stress formulation of three dimensional problems of elasticity. The dependence of the system has been studied by Southwell [30], Washizu [40] and Grycz [l4], obtaining, at best, a system of three conditions which must be satisfied on the boundary of the region and three different conditions which must be satisfied in the interior. It is only in the case of ”plane" problems, when the number of independent stresses reduces to three, the number of equations of equilibrium to two and the number of compatibility equations to one, that the stress formulation has widely been used. Section 1.4: Method of Analysis The present work follows that of Smith [29], Johnson and Little [18], and Little and Childs [22] in its use of biorthogonal vectorial eigenfunction expansions. (The relationship between the eigenfunction solution and that obtained by a Fourier-Laplace transform is discussed, for the case of a semi-infinite strip, by Buckwald [6].) In particular, it follows the approach of Johnson and Little in its formulation of the problem, but is somewhat more complex in its details due to the increase in the number of variables. Two fundamental mixed boundary value problems are studied, the first with prescribed normal stress and prescribed tangential displacements applied to the end, the second with prescribed nor- mal displacements and shear stresses. Both have the long sides free from stress. Variables are introduced, for each of these two cases, which are of the order of stresses with regard to differentiation; carry the information of the displacement boundary conditions; allow each problem to be stated, independently, in terms of a three-vector second order partial differential equation; and integrate certain of the Beltrami-Michell equations. As might be expected by the number of conditions that these variables were required to satisfy, to find them was a major step in the solution of this problem. The vector partial differential equations are reduced to ordinary differential equations by a separation of variables. (The validity of separation of variables for the equations of elasticity in this geometry was discussed by Tsay [36].) It is found that a generalized biorthogonality, of the kind of Langer [21], can be found which would permit direct calculation of the generalized Fourier coefficients of the eigenfunction expansion, if it could be assured that the problem under consideration was both nonsingular and involved a decaying solution. A six component vector is formed consisting of the variables of the two mixed problems described above, and it is shown that this vector satisfies a first order partial differential equation, whose constituent equations come from the equations of equi- librium, the equations defining the new variables introduced, and equations obtained by integrating certain of the Beltrami-Michell 10 equations. A generalized biorthogonality relation is obtained for this six-vector and through use of the first order vector partial differential equation, it is rendered useful even in higher orders of theta dependence. Finally, numerical work for certain trial total stress problems is performed by substituting the boundary data into the six-vector, using the generalized biorthogonality, and solving a finite system of linear equations, obtained from an infinite system of linear equations by truncation. CHAPTER II: THE ANALYTICAL FORMULATION Section 2.1: The First Fundamental Mixed Boundary Value Problem The problem studied here, and throughout this study, is the equilibrium of a semi-infinite, linearly elastic, solid cylinder. The radius is taken to be one and the cylinder is to be described with a polar cylindrical coordinate system, with the positive z-axis along the axis of symmetry of the cylinder and the finite face along the plane 2 = O. The dis- placements in the r, e, and 2 directions will be denoted by u, v, and w, reapectively. The first fundamental mixed problem denotes the follow- ing mixed boundary value problem: The long side, r = l is free from stress and on the end 2 = O, the normal stress 022’ and the two tangential displacements u and v are prescribed. Furthermore, it is assumed that these end conditions have been prescribed in such a manner that the resulting end loading is self-equilibrating. The equations governing the equilibrium of an elastic body may be written in polar cylindrical coordinates as follows: 11 12 Equations gf Equilibrium (3 equations, 6 unknowns) 1.1) aczz/az + L Trz + 1/r a"'ez/ae = 1.2) L Orr - Cee/r + l/r 8Tre/ae + 8Trz/az = 0 1.3) l/r BOee/ae + aIrrz/az + L TrB + TrB/r = 0 where 1.4) L = a/ar + l/r The Beltrami-Michell Equation gf Compatibility (6 equations, 6 unknowns) 2 1 2 2 _ 1.5) v 022 + 1w 5 K/az - 0 2 2 2 air, 1 135 . + —— - - ——-— ——— = 1 6) V Orr 2 (gee Orr) 4/r 89 + 1+v 2 O r ar 2 2 2 a 1.7) ‘v 069 - 2/r (gee - arr) + 4/r Tre/ae 1 as 2 it ___ 1 = ‘ + l+v ( /r ar + 1/r 2) 9 89 1.8) V2 'r - 2/r2 else (0 - o ) - 4/r2 1' r9 ‘ 96 rr r9 1 a_ AK +-—-— 1 A_ = o l+v 5r ( /r as) l 9) ( 2 - 1/ 2)T - 2/r2 i1fl§ +'-l- i:&_ = O ' V r rz 59 l+v araz 13 am 2 2 2 2 rz 1 §_K . - 1 + 2 -———- + —-— 1 ——- = o l 10) (v /r )Tez /r 89 1 /r aeaz where v is Poisson's ratio, . = + = + 1 11) K Orr + 089 022 R 022 represents the first invariant of the stress tensor and 2 1.12) 9'2 = 5 /ar2 + l/r a/ar + l/r2 232/592 + 52/522 represents the Laplacian operator in this coordinate system. The definition of small strains is as follows 1.13) err Br , 599 r + l/r 59 , 822 OZ = -1- a'li + LN = l 3.. w a! 1.14) 61:2 2 (62 at) 3 e92 2 (l/r 89 + 52) -1 fix V Au __ - _+1 .1... 6r9 2(ar r /r 59) with u, v, and w, being the displacements in the r, e, and 2 directions, reSpectively. Hooke's law may be written 1 1 . = -- .. + , = —- - l 15) err E [Orr v(oee 022)] €99 E L099 \J(Orr + 022)] e = l L0 ' V(o + o )1 22 E rr 69 =.lt! = ii! = Iii 1°16) erz E Trz ’ 692 E Tez ’ ere E Tre 14 where E is Young's modulus. Two more useful equations are obtained as follows: Adding equations 1.5), 1.6), and 1.7) yields: f) 1.17) v‘K = 0 and using this equation with the sum of 1.6) and 1.7) gives 2 __1__a_ =. 1.18) v R 1w 2(11+om) 0 From equations 1.13) and 1.15) one obtains l .1 ,+ bl=_r]_- _2 1 9) LJ 1/r 89 E L( v)R v 022] Define a new variable, T, by 51' 1.20) L 'r - l/r ”liz— + 511: - 92 89 62 I 0 Subject to the boundary condition lim T = 0. Now z—m - 5T \ E 62 56 52 E 59 az gu‘ SO 2 at - - E 82 aear oz E 56 oZ 59 2 + 1/r DAR- 598? 15 Subtracting gives 51' 29.2). -32. - = c. at; _ E (l/r 59 L T ) az (1/r 59 LV) ez or 1.21) 3%1’1 T = l/r 3% - Lv Equations 1.19) and 1.21) show that the displacements u and v are specifuxiup to rigid displacements by the variables R, 022 and T. There- fore, a formulation is sought of the fundamental mixed boundary prob- lem, with normal stress, 022 and tangential displacements u and v prescribed in terms of the three variables R, 022, and T. Differentiating the first equation of equilibrium (1.1), and substituting the result in the Beltrami-Michell equation, equation 1.9), gives 2 2 2 a. i? a T92 1 3:3; 1_ B Trz a Irrz ’r(~_z_)‘1/r§5§§"_259+2 2"+ 2 o o r r 59 52 2 15.-.. = +1+w aroz (R + Ozz) O or ‘1‘ 6‘1“ 1 Q_ i 5 rz rz __ _ -_ —-— —-———=O 1+v 5% (R vozz) + l/r e9 ( L fez + l/r 89 ) + 522 Applying equation (1.20), the definition of T, integrating with respect to z, and using as the boundary condition that the solution is zero at infinity, yields 16 122) LL(R-vo)+1/rh‘i+&’==o ' l+v ar 22 59 OZ or 1 22') 5022 = 1+V hI.+.li2.E:££.+ l/v i3 ° ar vr 59 v 52 Ar ° Now Operating on the first equation of equilibrium, equation 1.7), with Operator l/r ED , and Substituting the result in the Beltrami-Michell equation, equation 1.10) gives _ 27; _ __r_7; ._9_z_ l/r a/ae (AZ ) a/ar (l/r ae ) + a/ar L Tez + 2 52 1 £12— +T+j /raeaz(R+° )-0 Again using the definition of T, equation 1.20), and integrat- ing as above gives 5T 52._ _l_ d. _ _ 95 = 1.23) ar liv l/r 59 (R vozz) 52 O or 50 at 1.23') 1/r—Z—§=_1+_yal+l+_2__ea+lla3, 39 v ar v 52 vrae Operating on equation 1.23) with L and on equation 1.22) with l/r'ga and adding the result yields 5T 2 T Lag-L—Q-z-+1/rza-l+1/r rz=0 ar 52 17 Applying the definition of T, equation 1.20), one finds that T is harmonic, i.e., 1.24) V T II 0 At this Stage a system of three second order differential equations, equations 1.18), 1.5), and 1.24), have been obtained describing this fundamental mixed problem. Only three boundary conditions (finiteness at r = 0 will have the effect of an additional three boundary conditions) are needed for the solution of this problem. Having four first order equations in the five fundamental variables, R, 022, T, Trz’ and T92, if the procedure of Johnson and Little [18] is to be followed, one must fill out this set of first order equations to a set of six. What would first suggest itself would be to attempt to apply the program established above to the other Beltrami-Michell equations. The usefulness of the second and third equations of equilibrium and the Beltrami-Michell equations 1.6), 1.7) and 1.8) (along with the limitations to their use) will be seen in the consideration of boundary conditions. This leaves the first equation of equilibrium, equation 1.1), the Beltrami-Michell equation 1.5), and the derived first order equations 1.20), 1.22) and 1.23). These first order equations do not extract further information from equation 1.5) indicating, perhaps, that the information is independent from that carried by equations 1.1), 1.9) and 1.10) and their derived equations. 18 The missing information is obtained directly from con- sideration of the fundamental mixed problem dual to the one now begin studied. Writing equations 1.18), 1.5) and 1.24) as a single vector partial differential equation gives 2 1.25) L a/gr {1’} + l/r2 52/292 {I} + [u] 5—2-{1'} = 0 52 where R D v -1 O 1.25a) {E} =(Kozz , 1.25b) [u] = TIC 1 2+v 0 LT 0 0 l+v From the separation of variables 1.26) {?1 = $©<9>2 one obtains —-s m —-o lme -a Z 1.263) [f] = Z mm(r)e e m=O where m is an integer, a is an undetermined parameter, and {mm} satisfies 2 -—s -9 2 -—s 1.26b) L d/dr {Rm} - 95 {Rm} + am [U]{mm} = o r Substituting 1.263) into 1.25) one obtains the form of the solution for the variables R, 022’ and T. 19 Thus 00 ime -Q/ 2 1.27) R = 2: R (r)e e m m=0 m m , - z o = E o (r)elme e m 22 zzm m=0 a: , - Z T = 2 Tm(r)elme e m m=0 where R , 0 , and T , are components of the vector {$ ], and m zzm m m 1.28a) Rm = -[ (Am + 2(1+v)Bm)Jm(o/mr) + Bm ° amr ° Jm+1(amr)] 1°28b) Ozzm = Am Jm(amr) + Bm . O’mr . Jm+l(amr) 1.28c) Tm = -1 Cnu Jm(amr) Boundary Conditions The vanishing of stress along the sides of the cylinder gives rise to three boundary equations for the determination of the unknown constants in the equation 1.28). All three of these equations may be obtained by use of the equations of equilibrium in conjunction with certain of the Beltrami-Michell equations. 10) First boundary condition: Trz = 0 along r = 1. Because of the requirement that the solution should de- cay to zero as 2 tends to infinity, it suffices to require that 20 BT 1‘2 52 II H r which, by equation 1.22), is assured by do _.1_ £111- __Z.a bl = 1.29) 1+v (5r v *r ) + l/r 59 r=l O or, for the eigenfunction associated with the index m, of theta dependence, dR dozzm m __g__ . = ( dr - v dr ) + 1m Tm 0 l 1.29a) I1; Using equations 1.27) gives 1.2%) [m Jm(ozm) - 07m Jm+1(qm)]Am 2 _ + [(am+2m)Jm(am) - (m+2)am Jm+l(0m)]Bm - m Jm =0 68 5r r 21 Adding this to the Beltrami-Michell equation, equation 1.6), yields 2 5T 5r T ar 592 522 562 ar r + Ils‘a—i (R + o zz) = 0 5r Using equation 1.17) gives 2 a 0 rr _ 1 2 2 2 —— _ FC [l/r 5/5r + l/r 5 /ae ](vR - 0 > 2 22 52 BT92 _ EZEE +.:£E) 1 Q35 EL. + 52 (1/r 59 5r r 1+v 2 rz 5r , and the first boundary condition, one Using equation 1.1) to eliminate , equation 1.23) BT92 89 obtains the second boundary condition to eliminate l/r 1.30) ‘—- W[1/r 5/5r + 1/r2 32/892](g?z - vR) 1 a: +1--+-v_ 2(R-Vozz) 5z 21L 2 253 - 2/r Bear I:; 1/ 595 (R - vozz) _ 1 = 0 or 1 2 2 02 1.30a) Ii; [l/r d/dr - m /r ](qzzm - vR m) +£ l+v (Rm - voyzm) 22 This condition may be written as follows: 2 1.30b) [(m(m+l) - am)Jm(am) - am Jm+1(ozm)]Am + [(-92-2vm(m—1)+4m2)J (a ) + (m(m-1)-2v-az)a J (a)]B m m m m m m+1 m 2 + [-2m Jm(am) + ZQOJm+1(am)]Cm — 0 30) Third boundary condition: Tre = 0 on r = 1- Operating on the third equation of equilibrium, equation 1.3) by the operator - g; + l/r and the second equation of equilibrium, equation 1.2) by 1/r 5/59 and adding, one obtains 2 2 , -LEEQ-LB_LLQ_11_&_+::9+L(_LT +2., -lllm) 5r 2 862 r 5r59 r2 52 92 r 92 r 59 50 50 -1/2 ”+L—9—o 59 r2 )9 Adding this to equation 1.8), gives 52¢ 2 2T arr .___%§._ l.a_3_ +1/r2 a_R +.§_ (_ -L T +.__Q§ _ l/r .____) az r5r59 as 92 r 59 D_. = +1Tv5 (1 /rL 89 K) 5T so using equation 1.20) to eliminate 1/r -S%E , gives 2 5 T 2 5T rQ_ -1 5_ 5_ _ 5 T 5_ 82 322 l+v 5r [1” as (022 mm + 622 + 2 r I a I 23 and by use of equation 1.23), this may be written as follows: 0’ .1 2 r9 = -1 x _ a T 82 -—-1,N La/ora/r a/ae(oZZ vR))] + 522 + 2 :1; [5; - TIC l/r a/ae (R - vo ZZ>1 Using equation 1.24) the boundary condition reduces to 2 2 _l-a_ _ 1 2311 2.311 LT 1°31) l+v 5r [l/r 8/69(OZZ VR)J + r 5r + r2 592 + 822 2 i. __- 1 - = O + 1+Va [ /r 5/59 (R vo 22)] r = 1 or 2 d 2m2 2 1.3la) -r-a—;(Tm) -—2Tm +a/mmT +JL [1/1’ -—((2- ”Rm + (1- -2v>o Zm) im 1 . - _— —2—((2-v)Rm + (l-Zv)ozzm)] r The final form becomes, 1.31b) [—m(m-1)Jm(am) + m am Jm+1(am)]Am + [(-a:m + 2vm(m-l) - 4m(m-l))Jm(am) 2 + (m + 5m - 2vm)o/m Jm+1(am)]Bm + [(2m(m-1> - gimmep + Zam J “lemme," = 0 24 Alternatively, the last two boundary conditions can be obtained from the definition of small strains in polar co- ordinates and Hooke's law as follows: 0' 2 ) Consider HI: + "ill-4 0/ ml? 1 =— - + E [099 V - 4-1— (11— (o - MD) 522 dz 5r 1' 09 1+\) 51' r 59 22 which reduces to the condition 1.31) by the same argument as in 30)- All three equations of equilibrium have now been used, and the Beltrami-Michell, equations 1.9) and l.10),have been integrated. It may be seen that the equations of equilibrium reduce 1.6) and 1.8) to a simpler form, but leave the second z—derivatives of Orr and Tre’ respectively, so they become appropriate for specifying boundary conditions, but will not be used further. This leaves only equations 1.5) and 1.7) to consider. Equation 1.7) will not be used directly, but only in its sum with equation 1.6). This focuses attention on the use of equations 1.5)_ 1.7) and (or) 1.18) for the further informa- tion needed to complete the study of the end loading. Con- sideration of this case, however, will be completed first. 26 The Transcendental Equation Equations 1.29b), 1.30b), and 1.31b) give three homogeneous equations for the coefficients Am, Em, and Cm. The necessary and sufficient condition that these three equations should have a non-trivial solution is that their determinant should vanish, which gives rise, after appropriate simplification, to the transcendental equation for a m 6 a 22 222 3 1.32) (qm - Zamm(m-1) + 4(1-v)ymm (m-l) + amm (m -1))Jm(am) + (-2a:(m+1) - 4(1-v)q:m(m-l) + Aqim2(m-l) 2 2 3 2 2 - 1- _ _ y _ 8< oamm (m 1) 29mm (m 1))JmJm+1<1-m2> . 2 2 2 8) aozzm' = - imv - 1m amj T - amj ij o - o 2 . 5r 1 V m] (l-m )(l-v) m1 (1-v2)(1-m2) \J 01 . C' . + m] zzm] 2 2 (l-v )(l-m ) 2 T g - a T a U 2.9) B m] = am} mj +,lWSZ'V) R +.£m11'2V2 o at 2 2(1+v) mj 2(1+v) zzmj 2(1-m ) . 2 . 1m 0 . R 1m v a . + m3 mi - m1 2 2 szmj 2(1+v)(1-m ) 2(1+v)(1—m ) Substituting 2.7), 2.8), and 2.9) into 2.6) one finds that 31 2 2 -1 (m) = U" “mi —<1) _ O’mi -.—<2> 3k (1_V2)(1_m2) Ymk (1_v2)(1_m2) mk 2 *(1) "15.2.21-(3) m CYmi —(3) —(1) Link 2 (1+v) Ym + mk +'Ymk 5r mj 2(1+v) (1-m2 ) v2 02 a2 im v a +( mi -<1> O’mi 37(2) _ mi -<3> (1-v2)(1-m2) mk +(1-v2 )(1 m 2) mk 2(1+v)(1-m2) mk g1 2 2 (3) < > “(123 im - v — — 2 + 2— (I-W) Ymk + Ymk - 5r 4.) szmj + (gm go) _ i‘“ " C"mi -<1> _ _i___mv .—<2> l-v mk (1-m2)(1-v) Ymk 1- -v Ymk im 2 2 _ O’mi —(2> 0’m m: —<3> 2 Ymk 2Ymk (l-m )(l-v) 2(1-m2 ) —(3) r=1 _. .___.) T . ] Ymk 5r m] r=O The adjoint boundary conditions are taken as follows: “(1) 2 2.10) $11)- aYm - 1m i_;_2‘—(3) _ v amk 2(1) mk 5r 2 (1+v) (1_v2)(1_m2) mk _ 2 2 O’mk —-(2)+ ‘m 0’m —(3) _ 2 2 mk 2 Ymk ’ 0 (l-m )(l-V ) +2(1+\))(1-m ) r=1 32 51(2) 1 V2 0(2 2.11) W2)- __EL +__ it" $_-2y_1 1(3) + mk WI) mk at 2 (IN) mk (1_v2) (1_m2) Ink 2 im + V “mk -(2) _ " “am 2‘3) 2 2 Ymk 2 mk = O (l-v )(l-m ) 2(l+v)(l-m ) r=1 -(3) . BY __ Fl) _ Eli-(3+ W3) _ mk 2.12) 1-: Ymk _V Ymk+ Ymk ___M . 2 . 2 2 1m V dmk “(1.) 1m O’mk _(2) ka _(3) - 2 Ymk - 2 Ymk - ——-—-2 wmk (l-m ><1-v) (l-m )(1-v> 2(1.m ) r=1 As a result of this choice of boundary conditions, equation 2.6) takes the form 2 _ 2 (m) _ (“mi dam) 1 ( -v 1“”- _g(2) -(3) R (1 _ m2) 1+v l mk l-u mk 2m Ymk mj 2 1—(1).L_(2) _ im v “(3) + l+v (l-v Ymk + 1- -v Ymk 2 Ymk szmj -1mv-(1)_iri_-(2) <3) +(1-v‘y mk l-v 2 ink) ij r=l A biorthogonality relation of the following form is therefore obtained 2 2 1 a + a 2.13) (am - amk){f{wmk} [U]{mmj}dr o 1 l v -(1) -(2) in1-(3) 2 [IN (l-v Ymk +—1-v Ymk 2 mk ij ' (ha) ._1_. -<1)_ -(2) 1m __\3. (3) + l-l-v (1-:Y Yank l-v Ymk 2 Ymk) szmj im v§(1)+_i__m—(2) _ -(3) = + (l'v .nk+1-v Ymk + 2 Y mk )ij]r=l} O 33 It remains to show that the boundary conditions 2.10), 2.11), and 2.12) determine the same transcendental equation as 1.32). Substituting the form of the adjoint functions into the boundary conditions one obtains three simultaneous linear —4~k _., __ equations in the unknown coefficients A , B2 , and Ck : mk mk mk 02 q k 2 -* 2.14) [(4n - In ) a J (a )+ a J (a ) A 2 (l-v)(1-m ) mk m mk mk m+l mk J mk 2 F a 2 2 mk 2 —':': + L(-Q/ + ) 0/ J (CY ) + (m ' )0 J (CY )] B 2 2 mk (1_v) (l-m ) mk m mk (1-v2) (l-m ) mk m+l mk mk 2 f' a m 0, “‘1’ ___k__m :2- _ + L(2 v +1 m2) 2(l_-—-i—v—) Jr11(Olmk)] Lmk — O V dmk 2 -‘>'r 2.15) (-m + 2 ) O{ml-(\Imwzmk) + C1[kam+l (amk)] Amk (I-V)(1-m ) U 2 2" dmk ‘ ( ) + -a +~20Avhn- )0 ~1 0 mk (l-v)(1-m2) mk m mk 2 V 0’ ‘ . + (m - 2(l+\)) + mk 2 )aikJm+1(Q/mk)] 32k (1-V)(1-m ) l 2 . V {I m C! k mk -* + - - m = [(1 2" 1_mz 22) 2(1+\1) J m(qm 12] ka 0 02k 2.16) [1’92 (1 + k2)m 0/ J (am 131A dk -v 1"“12 mmk mk 02 2 1+») k +[ f-v (V - 1Tm2)m ykam (CY mk) 1+\) mk 2 21' —‘k 3‘ l-v (1 + 1 2) m O’kam+1(‘” mk)i Bmk -m J 2 4_r_ _ o’mk . 2 -* _ L m m akamb’mk) + O’kam+1(O’mk) ka ’ O 34 Proceeding as before, to obtain the transcendental equation determining the 's by setting the determinant of the O’mk coefficients equal to zero, one obtains again the equation 1.32). For m = l, as observed above, the derivation given does not hold beyond equation 2.6). This presents no difficulty, but the subsequent equations must be modified, as follows: de. 602211 2.71) .__lar — .51“ + R11. - v Ozzlj _ _i__ - _2 2'81) le ' 1+v R13 1+5 Gzz1j . 2 . 2 2 3T1. 1(2-011.) 1(1-Zv-v +211.) 2.9) -+—l=————-J—R,+ Jo , 1 ar 2(1+v) lJ 2(1+w) 221] _ 1:111- 5:2 Substituting 2.71), 2.81), and 2.91) into 2.6), yields .0). —<1> —(2) igl-l-(3) O___z__1_zl Ijk — [Kv Ylk +~Y1k Ylk )L Br —<1> 1 31(3) 2 .4_. + gnaw—(3)31... _ R 1k 2(l+w) 1k 5r 1+v or lj i(1— —4v- v2+v a2 1) "(1) (2) -(3) +(“"’ 1k +Y1k + 2(1+v) 1k “(22 (3) 2’“’1k +3» aYlk ) O ] -«————— 1 ' 5r +v 5r 221]. r=l Adjoint boundary conditions for the case m = 1 are as follows: 35 -m + w) _ 11221 -<3) = 2.101) v‘i’lk 1k 2 Ylk _ O r—l . 2 —(1) —(3) 2 11 ) 2211) + iffilflbg.qf3> _ 3312. _ .1. 2:12. _ o ' 1 1k 2(1+v) 1k 5r 1+v 5r r=1 2 2 ' 1-4v-v +v a ) _ -(1) -(2) 1‘ 1k +(3) 2'121) v Y1k +Y1k + 2(l+v) Y1k *(2) ‘(3) _ BY1k + iv aYlk = 0 3r 1+v 5r Equation 2.6) for m = 1 takes the form 2 2 ' _ f" _I(1) = 1("’11 0’1k) 470) . R + _, w) . O jk 2(l+v) L 1k lj 1k zzlj =1 and the biorthogonality relation for m = 1 takes the form 2 2 ”'1 + - 2.131) (alj - alk)i\£{11k} [U]{mlj}dr - __i__ _-<3) , —<3> . = 2(l+v) [Ylk R1j+ VYlk Ozzlj]r=1} 0 Substituting the form of the adjoint functions from equation 2.4), with m = 1, into the boundary expressions 2.101), 2.111) and 2.121), one obtains three linear equations —* in th k A 3* d —* ' e un nowns lk’ lk’ an C1k . —'k 2 -'2‘: 2. + 1 -2 + 141) (1+V)a1kJ1(a1k)A1k ( +”)[ O‘11 - 0/2.] (01 m: m m m m m m+l m m 2 + Hot: - 3111(m-1) )ame(cvm) + (2m-1)ame+l(am)] Fm = O 41 3.16) [(1-2v)amm(m-1)Jm(am) - (1-2v)aim Jm+1(am)]Dm 2 2 m 2 2 + [(3/3] - 3amm(m-1) + Marni) Jm(am) m 2 2 2 + (-202m + amm - m(m -1))Jm+1 m m (W )] F = 0 As before, thevanishing of the determinant of this system gives tfluzvalues of a for the system to have a non-trivial solution for m Dm’ Em, and Fm, and again, the resulting equation for am is that of equation 1.32). The representation of the displacement From equations 1.14) and 1.15), one may note 2 3T - Lfl+LM=M LT +1/I‘ .___.QE. .2... (Idu+li‘!_ ar r2 R62 E rz AB 52 r 59 and by 1.19), one has 2 5T A0 L §!_+ 1/r2 §_E.=.§Ll__l (1 T + 1/r -fi§) - l €1-v)ag - 2v —f§£> Br 2 E rz 89 E 52 OZ 86 so ELafl+1/zaz—w)=1+v L +1/i:Q-Z-+(1 ( -i/Ez9. Br r 592 ( ) Trz r 59 -v) [Q r 59 Section 2.4: The Adjoint Problem 11 The adjoint equation of equation 3.9) for a given index, m, of 9 dependence is the following: 4.1) 42 Again, the computation is facilitated if one first solves for ~ _ <1) .,(2> gm—gm +151“ (1) <2) (3) then for gm , gm , and, finally g . 4.2a) 4.2b) 4.2c) 4.3) m Solving the system 4.1), one finds En?) = [2(1w)gmr Jm+1(0’mr)16:: + [-4(1+v)m Jm(omr) + 2(1+w)qmr Jm+1(amr)]E: Er?) = iii-Ulwmmr Jm+1(amr)]5:: .4- + {-4 (1+v)m Jm(cymr) + 2(1+u)o[mr Jm+1(amr)]Em} 35,3) = [(ozmr)23m(amr)]6: + [((omr)2 + mummmrnfi: 7': + [(er)Jm+1(amr)]Pm Letting (m) _ 2 2 1 s a 131. (am,- - or...) gigmkitvzunmjidr 43 and using the differential equations 4.1) and 3.9) and integrating by parts, one finds -¢ + —o 4.4) IJ‘E’= [13,, ,1 mm,“ 41‘1th [vamp -[§;{Emk1]+ - m1] :2 where the boundary conditions on the fi's are given by equa- tions 3.11), 3.12), and 3.13). For m # O, 1, one has from the boundary conditions of equation 3.9) 4.5) T = O rzmj 4 2 2 5’ - 4.6) [am]. + m (m -1)]—;—f—m—1 = T331:(aijfin2(m2-1))-(1-v)m(m2-l) x x)]—--—1 +.I;;{(m+1)(m- v> - g2,(1-v)m(m2-1) (2m+1)]Q m] In] + 1mm: 1.24m (m2 -1)) - 2012 m(m 2-1)]—-§3‘31 = o Qm a 4.7) [am jz-hn (m2-1)]T =-—[(1-v)m(m- 1)(ozmj—a-l+m(m+1))] ezmj +-1:31-<1-v)m(a:j-m2 + (1-v)m2(m2-1)]QmJ 4 m2( + [a _ _ m2_1)]_:§§"_li 44 Because of the form of the boundary conditions, the treatment of the biorthogonality is not as straightforward as it has been found to be in the previous cases and the algebra associated with these results becomes quite formidable. It is convenient, however, to consider the following quantity: 4.8) [0; + m2 (m 2-1)]1J("‘) =[21-i-((a:JJ+m.22(m -1)) - am J.(1- -2v)m(m -1)+(1'V)m2 0“ 1)(m1))EJka) —(2 ) , a; +‘I$;(az.(1-v)m(m- 1) + (1- v)m2 (m2 '1)> (€22) ' g: —) + (a:lj + m 2(m 2.1))523)]£‘El at +[_1L ((m+1)(m- -v(m-1))(am j+m2 (m2 -1))-Q: j(1-v)m(m 2-1)(2m+1))g(1) ‘(2) 6 +211; (-(1-v)m(a:j-m2(m2-1)) + (I'V)m2(m2'1)) (Edi) - k _) 5r —(3) _ 1 agmk + (c12j4m2 (m2 -1)) (11?; ngJk) - {1, E‘2>+ Em- -ar —-)]QmJ + i [fm(a:j + m2(m2—1)) - Zaijm(m-1))E2l) - 1(Q:J + m 2(m 2—1))E(2) 55(2) 8* 2:1 with m“ )1 92m] Taking as boundary conditions 45 1 4 2 2 2 2 2 2 — 4.9) {#‘O’mkm (m -1))-amk<1-v>m+<1-v>m (tn-Wm ”Mug? “(2) . 2 2 2 _ _ acuk + fiamka-wmm-DHl-vm (m -1>]) 2 _ - amk(1-v)m(m2-l)(2m’1)]gé:) . 4 2 2 2 2 2 agfii + 1_-_:\)_[_(1_v)m(amk+m (m -1))+(1-v)m (2W1) (m ‘1)] (Erik) - _ar: ) 7(3) 4 2 2 - —1 ' - —3 55k +tamk+m 1[:£c;k>-fién€i’ mi; -~—-—] 4.11) [m(a:k + m2(m2-1)) - Zaik m(m2-1)]Eé:) . 2 2 — _ - 1[a:k + m (m -1)]g£:) —(2) _ BE - i[(q:k + m2(m2-1)) - 2m2(m2-1)][:géi) - —g§E—:] = 0 These boundary conditions give 46 4 2 2 (m) _ 4.12) [amj +'m (m -l)]Ijk _ _ 2 2 1 2 _ _ 2_ -(1) ‘ (amj ‘ amk){[:1-w(am j + “r1119 (1 ”m(m 1))ng 2) BC<2> 2 3 ___ - _ ( _ mk -( ) + ma v)m(m1 1)(Em k at ——)+ (amfl 2km.” JB’Q‘EI a.» +[——((oz:J .lim k->m(m2 -1><2m+1))E(1) 5E9) +'—-‘( (l-v)m(oz2 +~a2 ) (Q(2)- A)) 1+v mj mk mk 5r 2 2 53(3 ) ‘(3) _ (1) _ _1__m (2) _ ___— (amj + O’mk) (ka l-h) Qm ng ar )] Qmj - _ 1) -(2) i(m(a: j +-a mk) 2m(m21))gmk + (a: j + amk)gmk 2 2 (2) 53:11:) ‘ 62 + (amj + ozmk) (ka " 5r -——k'afl 5r F1} and one obtains a biorthogonality relation of the form 2 1 4 2 2 n 4.13) (amj - dip {Law + m (m -1>1$ka}+[v21{nmj)dr - [01:1]- + m 2(m 2-1)]I§m) = o It may appear that equation 4.13) presents a biorthog- onality relation that is completely unusable. A similar prob- lem, however, will be seen to arise in the treatment in the next section,and there a means of rendering such an expression useful for the solution of specific boundary value problems will be 47 deve10ped. As the numerical work of this present study will be confined to that development, these methods will not be applied here to equation 4.13). Substituting the form of the adjoint functions from equations 4.2) into equations 4.9), 4.10) and 4.11) one obtains O C 0 fl 1 three linear equations in the unknowns D , E , and F mk mk m k: 4.14) [(ozgk + 2(1-v)a:km(m-1) + aikm2(m2-1) (3-2v>)Jm>+2<1-v>aikm2(m+1)(m2'1>)Jm(qu) 3 4 2 2 4 2 2 +->+4(1-v)amkm(m+1)(amk+m2)+2amk)(amk+m (m -1)> +‘(a:k(a:kfm2(m2'1))+20mk(m+1)(m-V(m‘1))(a:k+m2(m2-1)) + 2(1-\,)Q/:]km(m-1)-2(l-V)Q/:1km(m2_1) (2m+1)-2(1-\))Q’mkm2 (m2_1)2)Jm+1(O’mk)-Jfi:k 2 4 2 2 4 —:< _ + [’ka('°’mk+m (m '1))Jm(amk)“mk(m+1)(“mkmm’n)Jm+1(0'mk)]ka ’ O and 2 4 2 2 fi'r 5 3 2 + (Zamk(m+1) ~2m1mk(m -1))Jm (amk)]Dmk +1 6 4 2 2 2 + [(ormk - éamkmon-l) + Bamkm (m -l))Jm(0/mk) = O 5 3 2 -* + (2C)’trlk(m 1) - 20{mkm(m -1))Jm+1(amk)]Emk Taking the determinant of these equations gives an expression which is equivalent, again, to equation 1.32). 49 For m = 0, one has that the boundary conditions 3.11), 3.12), and 3.13) reduce to 4.5) “r ,= 0 o r203 8T . 6Q 4.6 ) rzo] = _l_ oj.+_ Q 0 5r l+v 5r l+v oj 5T . 4.7) __e_zgl= 0 ar ezoj and equation 4.4) becomes (2) 4 8) 1(0) = C(1)+ g(3) 3391+? Ea) _ BCOk '1' ° D jk IT» ok ok at ok 5r ezoj —<1)-(3>_ A3k) _O__ + (1W Cok+ Q0 ar —)on]r=l Thus the adjoint boundary conditions become 4°90) 1+v 3(1)+ Qéi) - 0 r=1 (2) BE(2) 4.10 ) 23 - o o ok ar r=1 “(3) -(1) —(3) agok 4'110) l+v on+ gok ar which gives 1(0) 4.120) Ijk =0 hence one has the biorthogonality relation 50 2 2 1-0 + "‘ _ 4.130) (aoj - aosz{gok} [v2]{n0k}dr — 0 O . . - 1 —’2 — 3 Substituting the forms of gék)’ Qék)’ and gék) from equation 4.2) into the boundary conditions 4.90), 4.100) and 4.11 ) one obtains three linear equations in the unknowns o * 2* d F* Dok’ ok’ an ok' 2 -* 2 —* + 4'140) [aokqo(aok) + 20[oleWoknDok + [aokJo(aok) zaojl(aok)]Eok + _* - aokJ1(aok)Fok ‘ O 4 15 [ 2 2 6* ° 0) aokJo(aok) - O[oJlm’okH ok + 2 + 2 J ( ) E* = o {-aokJo(aok) aok 1 aok ] ok 4.16 ) [-a2 J (a ) +(c1r3 + 2v 0 )J (a )]B* o ok 0 ok ok ok 1 ok ok 2 3 -* + {-aokJomok) + (aok + 2V o[ok)Jl(aok)]Eok + E 2 J ( + J ) 5* - o <1’ok o CYok) do 1(aok 1 ok — and the determinant of these three equations gives the same equation as 1.32), when we substitute in that m = 0. Setting m = 1, one has 4.51) Trzlj 8T . . . rzl] =._l_ 1] ._g_ . 921] 4'61) 5:- 1+9 5r + 1+v Q1j+ 1 ar 51 = _. l-v ezlj 4'71) Tez1j 11+» 1j'+ 5: Then 5Q . 1(1) ‘(1)+ (3) 13 4'81) Ijk =[(1+\1C1k gm) 5r aE0) (1)+ (2) +1( g1k +g1k - 5r .___) Tezlj at“) if” =1 -(1) _. 2- -v —(2)+ 1 -v1- —13) _ r +lit-h g1 ng 1+v 5r ——+ g1k ar —_)Qlj]r ='O and the boundary conditions become 4.91) 1Tb (1) +E(3)- +Efi) 33:11:) 0 These boundary conditions give I(1) 4.121) 1jk =0 and hence one has the biorthogonality relation 52 1 2 2 + -o 4.131) (alj - 01k) f{31k} [V2]{n1j}dr — o o . . -(1) -(2) -(3) Substituting the forms of glk , glk , and glk from equation 4.2) into the boundary conditions 4.91), 4.101) and 4.111), one obtains three linear equations in the unknowns 5* E* d 5* 1k’ 1k’ a“ 1k° 4 14 ) E 2 J ) + 2 J ( ) 6* + [ 2 J ( + 2 J ( ) E* ' 1 0‘1k 1(alk 0’1 2 0’1k 1 1k 0’1k 1 0’1k) C’1k 2 0’1k 1 1k fl + [alkJ2(alk)]F1k — O 4 15 ) [-Zdz J (a ) + (03 + 4v 0 )J (O )]B* ' 1 1k 1 1k 1k 1k 2 1k 1k + [ 2 2 J ) + ( 3 + 4 J E* ‘ O’1k 1(“1k Oi1k 01k) 2(0’111)1 1k + 2 ) + 2 _* - o E”alkJ1m1k alkJ2(alk)]F1k ' 2 —* “'161) ['aikJ1(a1k) + “’1kJ2(a1k)]D1k 2 -—* + [alkJ1(alk)]E1k ' 0 Section V: The Six-Vector In the previous sections a system of six first order equations has been deve10ped consisting of 1.1), 1.20), 1.21), 1.23), 3.1) and 3.2). This system may be written in vector form as S3 w115r1?1 + I/r 1w211f1 + I/r §-1w311?1 + §-1w411f1= 5.1) where (R 1 zz 5.1a) {T} ={ T T P rz T92 tQ J #0 0 5.1c) [W2] = L 5.1e) [WA] = A form of solution provided by a separation of variables is assumed 5.2) and it is found that equation {1} = meJ 5.1b) [W1] 1 1 , 1 1J 1 0 00 0 -1 10 O l l O F 5.1d) QB OH Z {?m(r)}eimee-q§ (“=0 _1_ l+v 1 1w31 = satisfies the ordinary differential 54 d 1 -+ 5.3) [W1] $112,} + ; [W2]{fm} + ri—nl [w3]{?m} - amwflffm} = 0 As the equations of this system were derived by integrating the second order systems of the two previous fundamental prob- lems, it is not necessary to re-solve them, nor is it necessary to re-derive the transcendental equation, as the boundary con- ditions remain the same as before. The adjoint equation 5.3) is as follows: 5.2.) {141+ L1; 1 + l/r 1w 1+1; 1 1 5r m 2 m -i”-‘1w1+1‘g'1-&1w1’f§1=o r 3 m m 4 ‘ m It is necessary, however, to solve this system of equa- tions, as no assumption is being made as to a relationship be- tween the adjoint functions of this problem and those of the previous ones. Furthermore, the adjoint boundary conditions will be specified for the convenience of the current problem, so it will be necessary to again check that the choice of eigen- values so determined will be compatible with those of the pre- vious problem. Solving the system of adjoint equations, one finds that 5053) Eff,” = {m Jm(amr)]Z3T,, + [2(1W)amer+1(0/mr)]5: 55 5.5b) E<2> * 2 z‘c [-amer+1(amr)]am + E-(amr) Jm(amr)+2m CYmer+1(Q’mr)]c§n 5.5c) Eé3) i{[m Jm(amr)-amer+1(amr)15: _* + [-2(1+V)amer+1(er)]Ch} ll 5.5a) £14) In [2(1+v)amer(amr)]E; 5.5e) gés) i{[amer(amr)]5: + £2(1+v)amer(amr)]E;} '-(6) —* 2 -* 5.5f) gm [erJm(amr)]ah +-[ (amr) Jm+1(amr)]Ch The boundary conditions for the stresses are as follows: The condition Trz = O on r = l is used directly. The con- ditions c = 0 on r = 1 and T = O on r = 1 are taken rr Oz as they are found in equations 1.30) and 1.31) and the r-derivatives are eliminated through the use of the equation 5.1). The following boundary conditions are thereby obtained 5.6) 1' =0 1 2 2 1 2 2 5.7) T:;{(1-v)m + vm + Q’mj3ij +-l+v[(1-v)m - m - amjv]ozzmj l-v Q = O + ' + 2‘ -—— 1m ij. 1m a 1+v amj mj r=1 ijezmj 56 (mv + (1- -v)m2 )Rm mj +'—l— (m ' (I'V)m2)o 5'8) 1+v zzmj l+v , 2 2 l-v _ + 1(01mj - m )ij - 2 i amTezmj +—1+\)yijmj r=1 — 0 Further, these boundary conditions may be expressed in the form 5.9) . = 0 rsz =1 5 10 ( 2 1 “——£—-—[ 2 m + ( 2 l ' ) “m1 m ' ”9sz F1 - 20w) “m “m m ' ”Rm 2 1 2 2 O[mj + 2(1+v) -(’Ym_j\)m - m(m -1)]Gzzmj - 2 mj r=l 2 _ 2 2 5.11) (l-v)amj(m -1)Qmj - {-amj + (1-v)m(m -l)]RmJ r=1 + [0: jv + (1- -\))m(m2 -l)]ozzmj + i(1+\J)[-O’2.m + m(m2-1)]T m] mJ r=1 Letting mJ I_1—~ + 5.12) Ijk =(a . - amk) I {gmk} [W4]{?mj}dr O expanding, using the differential equations 5.3) and 5.4), and integrating by parts, it is found that: 13'0“) 5.13) =[ig mk} [w111fm j}]::: 57 For m # 1, one obtains, using 5.9), 5.10), and 5.11) in equation 5.13), that (m) 2 (1) ‘12) 5.14) mJ(m2 -1)Ij =[Qymj(m -1) (1+v 8 mk +gmk) + 2—"2-(1w) [am .m+\2m(m 2-1)]g(2)+ f1—1-afu+.(1-v>m . 1’" Thus the boundary conditions become: ' (1) (2) —(5 ) 5.15) mk(m 221)(1+v2 mk + gmk)+ +2?1:;){22k m+-vm(m2 -1)]gmk l 2 + 1:32-2mk + (l-v)m(m2 -l)]g(6) = O 5.16) mk(m 2—‘—"—-1)( (1) +—(2)) - I“ g mk k [5:1 kvm + m(m 2-1)]g(5k) 2(1+v) +'I%;{02k v + (1-v)m(m 2-1)]g(2) = 0 2k am 5.17) mk(m 2-1)-(3) - g(2) + lgli—l[-q2 km + m(m2-1)]g(2) = 0 Under these boundary conditions 58 2 (m) _ 2 (1) ‘12) 5_18) amj(m -1)Ijk — (amj - amk) [:Qm -1) (1+v g mk + gmk) 1 +2—L— j Otmk)g(2 ) ' fi(€1.amk)g(2))R 2 (1+v) (O’m mk m1 Rm 2 —(1)+—(2) ivm ‘—(5) v —(6) ._ (21 -13> _ “m1 + Cym —(5> , iilixzm '-<6> T r20 0" ')g 2 gmk l-v mgmk) mn F1 and one has the biorthogonality condition for m # 1. [I .. 2 5.19) (amj(1- amk) . j {gmk} (Ci/m(m -1))[w4J{?mj}dr , E(D+-Q)_1L__ «a 222% "2“ (2m 1)( gmk) 2(1+v) O’mk gmk l-v amk gmk mj -m «w —w> + (2(1+v)gm__ +1+v gmk)22mj ij 2 iL-a>_%m mv%k%m,v% k-wg + ((m 21)(1+\) mk gmk) 2 2(1+\)) gmk l-vm gmk zzmj mk mk gmk 1-\) mi 2 (201121)?” + 1/2 0/ 2(5) + 2.112). m €5,122) T ( l-v gmk 1/2‘(5)+LM1_+V_1‘(6) T ”0 .0 g mk )2 mj mj r=1 This form is analogous to that of equation 4.13) of the previous section, and, for m > 1, it is developed to render it usable for total stress end loadings, as follows: Using the differential equation 5.3) one can write 59 1 —o + 2 5.20) J; {g k} amjon -1)[w4]{fmj}dr 1 -o + 2 r d "' l =£ {gmk} (m -1)3L[W1] a"; {fmj} + ; [W2]{?mj} + :9 [w3]{?mj}} dr and since, by equation 1.1) T =-— (1.0 . . T . m] zzm] rsz r 92m] equation 5.19) for m > 1 takes the form 1 5.21) (amj - 0111190; {gmk}+(m2-1)[[w1] S-flfmj} +11; [w2]{'fmj} fl [w3]{fmj}] dr ‘+(1) ‘(2) -([(m2-1)(1-+-5gmk gmk _ ivm a _, a v _ (m2_1)(_;_-<1> gm) _ __.___Ln_}_<_ g6) +____ mk gun) O . zzm] > + im(o‘rnfimrnk) -(5)_ L325 -<6> 2(1+V) gmk l—v gmk ij + l-T-v—gmk k 2(1+\)) mk l-v gmk + imv ”(5)+ ___ g(6) £9 (2 2(l+v) +1- k)(P Trzmj + r feznj) +( :1] Having specified the boundary conditions of the adjoint problem so as to reduce equation 5.14) to a desired form, it is necessary to see that the series of eigenvalues so determined is the same as before. One therefore Substitutes the form of the 60 adjoint functions, given in equations 5.5) into the boundary conditions of equations 5.9), 5.10), and 5.11) to obtain three —* linear equations in the unknowns (7 -* —* mk’ Bmk’ and cmk _1 3 2 _ _ 2 2 ‘* 5.22) 1:;[(-amk+(1-v)ormkm(m -1))Jm(amk) (1 v)o/mk(m -1)JM+1(oz k)]dmk 2 -—* + WOO/:13 + (2-v)o/mkm(m -l))Jm(crmk)]8mk mk + U-aikonZ-l) - a3 m - amkvmmz-anmomk) + fi—vwgkfiLwaikmmZ-IHZ(l-v)aik(m2-1))Jm+ (a )1? 1 3 5.23) fjsi(amkv+(1-v)amkm(m2-l))Jm(amk)-(l-v)a:k(m2-1)Jm+l(a k)]a? + -—l—~—{(a3 vm + (1-2 ) m(m2 l J )'H* 2(1-w) mk V amk ‘ )) m(o’mk 36mk + [(-a:k(m2-1) + aikvm + amkm(m2-1)Jm(amk) 1 4 2 2 2 2 ‘3" + fi('amkv+(1“’)amkm(m -l)-2v(1-\))ozmk(m '1))Jm+1(°’m >10 5.24) -_—i(-oz:l kmmm k2m(m -1))Jm (am knzlk [ 2 2 2 —* + 2(1+v) ( 0: k+'20mkm(m -1))Jm(amk) - 2amk(m -1)Jm+l(amk>16nk +[ 0,3 13 Mo k) + 1 jeakm-mak m(m2- -_1) 2(1- mm k«112-1)»: m+lgok + 1/2 gok Toj r=l _ 0 Furthermore, the problem and its biorthogonality relation, for m = O, decouples into an axisymmetric torsion problem and an axisymmetric torsionless problem; one part of the biorthogonality determining the single series of constants associated with the axisymmetric torsion problem, the other determining the two series of constants associated with the axisymmetric torsionless problem. The eigenvalues of the former problem are the real eigenvalues; the latter are the complex ones. For m = 1, the boundary conditions on the stress functions can be expressed as 5.9) 'r . l rzlj r=1 5.10 - 10+»le 1) 13' " Ozzlj r=l r=l 62 a ' . l-v 5.11) T. =-‘1‘1T.-1 Q. 1 9213 F1 2 11 MM 13 F1 Setting (1) 1 1 + ~ expanding, using the differential equations 5.3) and 5.4), and integrating by parts, it is found that _(1) + (1w)g(2) (1) = (2) _ 5.131) 1jk [dang-1k o 1(g1k zzlj “(3) __l “(5) fl “(5)... "(6) + 1 glk + 2 81k )le + (2(1w) g1k +81k)Q 13°] r=l One therefore takes as boundary conditions ‘(2) _. 5.141) glk - 0 =1 “(1)-(2) (3) mmk ’(5)- 5.151) g1k + (l+v)g1+ i g1k +—2g1k—O LEI—(5)., ‘(6)= 5'161) 2(1-w) g1k +g1k 0 And the biorthogonality relation for m = 1 becomes f 1 lj ' 01k)1~{ {Elk}+tw1]{?1j}dr \ J” r=1 5.171) G: "(5) . ' g1k le 63 Substituting in the form of the adjoint functions, as given in equations 5.5) with m taken to be equal to 1, one obtains three linear equations in the unknowns a71k "81k’ and ES" lk' . + - 2 "* = 0 5 181) [01kJ 2(a1k)]a:k [alkq1(a1k ) O’11y71k + [ alkd1(alk)+2a1kJ2(alk)]31k I O a? + [ -24(1-h))cy lk — ) + 12(1+v)a 1k 1(“1k 1k J2("’1k)1 5.201) [alkJ1(alk)}71k+[2(1+v)a1kJ1(alk)]61k 2 -* = - a J (alk)]clk 0 + [(l-v)a 1k 2 (a lkJ 1 1k) Upon taking the determinant of the coefficients of these equations and setting it equal to zero, one obtains the same result as that obtained when m is set equal to 1 in equation 1.32). CHAPTER 3 The Formulation of Specific Boundary Value Problems With displacement boundary conditions on the end of the cylinder, or displacement conditions in conjunction with stress conditions, one is liable to the difficulties mentioned in Chapter 1, namely, the possibility of stress singularities and ignorance of the physical conditions necessary for a decaying solution. Furthermore, the type of problem whose solution is most desired for engineering applications is one with boundary conditions stated entirely in terms of stresses. For these reasons numerical work was not attempted with either of the biorthogonality conditions developed in considering the fundamental mixed problems of Sections 2.1-2.2 or 2.3-2.4. Such a development was thought necessary, however, because the ability of the auxiliary variables to complete the Specification of a fundamental mixed problem was felt to be a necessary requirement to assure its usefulness in the con- struction of the desired 6-vector. The following problem was considered appropriate for the formulation of numerical methods for the solution of Specific boundary value problems. The long side, r = l, is to be free from stress. At the finite end, 2 = 0, the normal stress, 022’ and two tangential Stresses Trz and T92 may be prescribed subject only to the requirements that they correspond to a 64 65 self-equilibrated loading and be free from stress singularities. A solution is sought which renders the prescribed boundary stresses 0 T , Tezb in terms of a single series of con- zzb’ rzb stants. That is, one seeks . -a .z 6.13) c = (Z 2 a , o , elme e ”U ) zzb . m3 zzm] _ m J z—O . -a ,z 6.1b) 1'er = (2: 2: a , «- ,elme e m3) m J m] rsz 2:0 . -a ,2 6-1C) T b = (X Z 8 . T . elme e mJ ) L 92 m j m] 92m] =0 The numerical solution of a boundary value problem, is accomplished as follows: One first performs a Fourier analysis to divide the prescribed function into functions of r each associated with a different index of theta dependence. Then one selects the appropriate biorthogonality relation for the 6-vector, equation 5.190) for m = 0, equation 5.171) for m = l, or equation 5.21) for m > 1. Then following the method of Johnson and Little [18], the variables 0 are replaced . T . T . zsz’ rsz’ esz b h ' ' d d h y t eir prescribed boun ary ata Ozzmb’ T T e rzmb’ Tezmb' remaining variables are replaced by formal series 6.43) R = Z . m m m j J J 6.4b) T = z a o 66 thus constructing the vector {fmb}, a particular vector of the form of equation 5.1a). One therefore obtains, for each k, a single equation in infinitely many unknowns Lmk{fmb} = amkNmk . th , , , where Lmk IS the k—— biorthogonality relation for the index m of theta dependence,corresponding to using the adjoint . . . th functions of the eigenvalue 0 , and N is the k—— mk mk normalization constant. The set of all these equations for k = 1,2,... constitutes a infinite set of linear equations in the infinite number of unknowns a and may be solved m1,am2,... to any desired accuracy by truncation. The functions of r corresponding to theta dependence with m1< O in the Fourier analysis correspond to the parts of the boundary conditions 900 out of phase with respect to the original axis and may be treated by rotating the axis 900, solving the new problem for m > O, and superposing the Solution. Two trial problems were solved numerically to indicate the rate of convergence of the series expansions. The axisymmetric torsionless problem considered in Little and Childs [22] was programmed using the biorthogonality relations developed in this Study and the results have the same accuracy as given there. The results of the present method are found in Tables 6 and 7. A non-axisymmetric problem with index of theta dependence, m = 2 was programmed. An appropriate Specification was that and Tr Should be entered as real functions and that 0zsz z2b 67 T922b Should be entered as a purely imaginary function, because the behavior of T92 is 900 out of phase, with respect to theta, from 022 and Trz’ as seen in equations 1.1), 1.9), 1.10) and elsewhere. The convergence appears to be somewhat slower than in the axisymmetric case, but of the same order. The results are found in Tables 8, 9, and 10. Roots of the Transcendental Equation 121121 m = 0, n 1 2.6976518 2 6.0512222 3 9.2612734 4 12.438444 5 15.602204 6 18.759055 7 21.911845 8 25.062031 9 28.210443 10 31.357587 11 34.503796 12 37.649288 13 40.794222 14 43.938715 15 47.082846 16 50.226683 17 53.370274 18 56.513658 19 59.656867 20 62.799924 Complex Root +11. .6381471 .8285342 .9674283 .0764211 .1660392 .2421081 .3081733 .3665585 .4188579 .4662104 .5094822 .5493159 .5862151 .6205834 .6527455 .6829654 .7114654 .7384311 .7640179 NNNNNNNNNNNNNNNNHHH 68 = 0.25 3673570 * From Little and Childs [22] + From Watson [41] + Real Root 5 8 11 21 27 3O 62 .1356223 .4172441 .619841 14. 17. 795952 959820 .116997 24. 270112 .420574 .569204 33. 36. 40. 43. 46. 49. 52. 55. 58. .016222 65. 716520 862856 008447 153454 297997 442164 586024 729627 873016 159273 69 \OGDVCJ‘U‘I-l-‘LANp—I H C v = 0.25 Complex Root Real Root 4.2852175 +il.49819612 2.7931415 7.5952965 1.7398515 6.6948248 10.805373 1.9023387 9.9662643 13.985446 2.0249445 13.169015 17.151977 2.1234422 16.346821 20.311147 2.2057668 19.512503 23.465843 2.2764824 22.671322 26.617603 2.3384588 25.825862 29.767330 2.3936165 28.977549 32.915586 2.4433066 32.127236 v = 0.25 Complex Root Real Root 2.1043544 +i0.95922055 4.0882060 5.6799579 1.6129679 8.1137406 9.0352822 1.8244578 11.425432 12.273656 1.9671243 14.650449 15.472033 2.0771376 17.842808 18.651317 2.1670412 21.018853 21.819876 2.2431549 24.185462 24.981772 2.3091822 27.346087 28.139231 2.3674993 30.502666 31.293581 2.4197245 33.656374 \omuoxknwar-i H O \OmVONUTDUONp—a H O 10 11 15 18 21 31 34 Complex Root .2858613 .9912960 .411580 13. 16. 20. 23. 26. 29. 32. 689043 913273 110957 293299 465914 631954 793330 Complex Root .4047845 .2535980 .743891 .064898 .318901 .538323 24. 27. .099690 .269869 737274 923012 +il. .7106445 .8983456 .0255558 .1252664 .2078764 .2785740 .3404296 .3954371 .4449759 NNNNNNNP—‘H +11 NNNNNNNHH 70 v = 0.25 1576914 v = 0.25 .3040193 .7961669 .9646273 .0791555 .1701374 .2464063 .3122987 .3703921 .4223776 .4694370 Real Root 5. .4615179 .826721 16. 19. .488138 25. .836344 .000421 35. 9 12 22 28 32 2957151 083178 296718 666468 160336 Real Root 6. 10. 14. 17 33 4629421 764020 186596 .478662 20. 23. 27. 30. 717023 926895 119542 300877 .474347 36. 642109 71 Table 6 AxiSymmetric trial problem: 042 = 1-2r2 number of pairs of eigenvalues used r prescribed 1 2 5 10 20 0.0 1.000 1.154 1:048 1.000 .988 .995 .1 .980 1.279 .997 .960 .961 .975 .2 .920 1.616 .866 .884 .905 .915 .3 .820 2.058 .705 .800 .803 .816 .4 .680 2.440 .558 .664 .666 .676 .5 .500 2.565 .429 .472 .486 .496 .6 .280 2.229 .275 .263 .289 .277 .7 .020 1.263 .038 .016 .012 .018 .8 -.280 -.436 -.289 -.287 -.281 -.280 .9 -.620 -2.868 -.605 -.605 -.611 -.616 1.0 -1.000 -5.896 -.684 -.885 -.923 -.978 72 Table 1 Axisymmetric trial problem: Trz' = 2.4r - 2.6r2 + 0.2r5 z=o number of pairs of eigenvalues used r prescribed 1 2 5 10 20 0.0 0.000 0.000 0.000 0.000 0.000 0.000 .1 .237 -.840 .347 .271 .242 .237 .2 .459 -1.541 .622 .471 .461 .461 .3 .650 -1.983 .787 .641 .656 .651 .4 .796 -2.095 .856 .806 .802 .798 .5 .881 -1.863 .873 .905 .891 .883 .6 .894 -1.347 .876 .910 .904 .898 .7 .821 -.680 .858 .837 .837 .825 .8 .654 -.063 .754 .675 .667 .659 .9 .383 .255 .479 .417 .406 .387 1.0 .000 .000 .000 .000 .000 .000 Tablg 8 m 0. 1. 0 trial problem: prescribed 0.000 .260 .440 .540 .560 .500 .360 .140 -.160 -.540 -1.000 O' 22 .000 .076 .280 .550 .792 .891 .734 .224 .708 .088 .898 73 = 3 - 4r e=o number of triads of eigenvalues used 0.000 .078 .268 .456 .529 .385 .090 -.316 -.297 .093 1.119 2 0.000 .103 .360 .565 .522 .383 .270 .012 -.315 -.615 -1.420 10 0.000 .270 .417 .540 .522 .470 .306 .074 -.226 -.612 -1.178 74 Table 2 . _ 3 5 m = 2 trial problem: T - 2.4r - 2.6r + 0.2r rz z=o 8:0 number of triads of eigenvalues used r prescribed 1 2 5 10 0.0 0.000 0.000 0.000 0.000 0.000 .1 .237 .809 .032 .357 .278 .2 .459 1.512 .182 .530 .458 .3 .650 2.020 .504 .649 .652 .4 .796 2.273 .955 .788 .798 .5 .881 2.250 1.405 .860 .887 .6 .894 1.972 1.682 .912 .906 .7 .822 3 1.502 1.650 .892 .826 .8 .654 .936 1.269 .618 .667 .9 .383 .392 .640 .313 .366 1.0 .000 .000 .000 .000 .000 Table 19 m r 0. 1. 0 trial problem: prescribed 0.000 .462 .782 .960 .996 .889 .640 .249 -.284 -.960 -l.778 1 ¢ 92 z=o 9=o number of triads of eigenvalues used .000 .810 .521 .047 .320 .303 .999 .402 .593 .377 .398 75 1 0.000 .035 .137 .332 .584 .772 .773 .509 -.021 -.736 -1.511 ‘°| O\ (r - 4r2) 0.000 .388 .701 .951 1.043 .864 .565 .252 -.324 -1.162 -1.663 10 0.000 .378 .807 .952 .993 .891 .618 .241 4.327 -1.004 -l.720 BIBLIOGRAPHY [1] [21 [3] [4] [5] 161 [71 [81 [9] [10] [111 BIBLIOGRAPHY Abramowitz, M. and Stegun, 1., Handbook of Mathematical Functions, Dover, 1965. Aleksandrov, A.Ya., and Solov'ev, Yu. J., "The solution of the three-dimensional axisymmetric problem in the theory of elasticity by means of line integrals," J. Appl. Math. Mech. 28, 5, 1106-1112, 1964. Alexandrov, A.Ya., and Solovev, Yu.J., "Application of analytic functions of a complex variable to the solution of three dimensional non-axisymmetric problems of the theory of elasticity," Izv. Akad. Nauk SSR, Mekhanika no. 6, 94-99, 1965. Boley, B.A., "Some observations on Saint Venant's principle," Proc. Third. U.S. Congr. Appl. Mech., June 1958; Amer. Soc. Mech. EngrS. 1958, 259-264. Boussinesq, M.J., "Applications of Potentials," Gauthier-Villars, Paris, 1885. Buckwald, V.T., "Eigenfunctions of plane elastostatics,” Proc. Roy. Soc. London (A) 277, 1370, 385-400, February 1964. Childs, D.R., "Solution of the stress-equilibrium and compatibility equations in the presence of body forces and arbitrary temperature fields," Quart. of Appl. Math. 26, 1, 49-56, April 1968. Chemeris, V.S., "Integral equations of the theory of axisymmetric elasticity,” Prikl. Mech. 1, 5, 36-46, 1965. Donnell, L.H., "About Saint Venant's principle," Trans. ASME, 84, E (J. Appl. Mech.) 4, 752-753 (Brief Notes), December 1962. Dzhanelidze, G.Yu., "The Saint Venant Principle," Trudi Leningrad Politekhn In-ta no. 192, 7-20, 1958. Erim, Kerim, "On St. Venant's Principle," Proc. Seventh Int. Congr. Mech I, 28-32, 1948. 76 77 [12] Flfigge, W. and Kelkar, V.S., "The problem of an elastic circular cylinder," Int. J. Solids Strs. 4, 397-420, 1968. [13] Gatewood, B.E., Appl. Mech. Reviews, 22, 6, Rev. 4020, 1969. [14] Grycz, J., "On the compatibility equations in the classical thoery of elasticity," Archiwum Mechaniki Stosowanej 19, 6, 883-896, 1967. [15] Gusein-Zade, M.I., "On the conditions of existence of decaying solutions of the two dimensional problem of the theory of elasticity for a semi-infinite strip,” J. Appl. Math. Mech. 29, 2, 447-454, 1965. [16] Gusein-Zade, M.I., "On necessary and sufficient conditions for the existence of decaying solutions of the plane prob- lem of the theory of elasticity for a semi-strip," P.M.M.: Journal of Appl. Math. and Mech. 29, 4, 892-901, 1965. [17] Horvay, G. and Mirabel, J.A., "The end problem of cylinders," J. Appl. Mech., ASME, 25, 561, 1958. [18] Johnson, M.R., Jr. and Little, R.W., "The semi-infinite elastic Strip," Quart. Appl. Math. 23, 335-344, 1965. [19] Keller, H.B., "Saint Venant's procedure and Saint Venant's principle," Quart. Appl. Math. 11, 4, January 1954. [20] Knowles, J.K., "On Saint Venant's principle in the two- dimensional linear theory of elasticity," Arch. Rat. Mech. Anal., 21, 1, 1-22, 1966. [21] Langer, R.E., "A problem in diffusion in the flow of heat for a solid in contact with a fluid," Toboku Math. Jour., 35, 1932. [22] Little, R.W. and Childs, S.B., "Elastostatic boundary region in solid cylinders," Quart. of Appl. Math. 25, 3, 261-274, October 1967. [23] Love, A.E.H., A_T£eatise on the Mathematical Theory of Elasticity, 4th ed., Dover, 1944. [24] Lur'e, A.I., Three Dimensional Eroblems of the Theory .gf Elasticity, Interscience, 1964. [25] Mitra, D.N., "On axisymmetric deformation of transversely isotropic elastic circular cylinder of finite length," Arch. Mech. Stos. 17, 5, 739-747, 1965. 1261 1271 1281 1291 1301 1311 [321 [331 [341 1351 [361 [37] [38] [391 [40] 78 Muki, R., "Three dimensional problems of elasticity for a semi-infinite solid with tangential load on its surface," (in Japanese), Trans. Japan Soc. Mech. Engrs. (1) 22, 119, 468-474, July 1956. Roseman, J.J., "The principle of Saint Venant in linear and non-linear plane elasticity," Arch. for Rat. Mech. and Anal. 26, 2, 142-162, 1967. St. Venant, A.B., ”Treatise on torsion," Mem. Acad. Sci. Sav. Etrg. 14, 233-500, 1855. Smith, R.C.T., "The bending of a semi-infinite Strip," Austr. J. Sci. Res. 5, 227-37, 1952. Southwell, R.V., "Castigliano's principle of minimum strain- energy," Proc. Roy. Soc., A, 154, 1937. Sternberg, E., "On St. Venant's Principle," Quart. Appl. Math. 11, 4, 393-402, January 1954. Swan , G.W., "The semi-infinite cylinder with prescribed end displacements," SIAM J. Appl. Math., 16, 4, 860-881, July 1968. Thompson, T.R., End Effect 13 a Truncated Semi-infinite Wedge and Cone, Thesis, Michigan State University, 1968. Thomson and Tait, "Treatise on Natural Philosophy," Oxford, 1867. Toupin, R.A., "Saint Venant's Principle," Arch. Rat. Mech. Anal. 18, 2, 83-96, February 1965. Tsay, I.P., "Separation of variables in the equations of the theory of elasticity," Akad. Nauk. Kaz SSR, Vestnik, no. 9 (198) 88-95, 1961. Valov, G.M., "On the axially-symmetric deformations of a solid circular cylinder of finite length," J. Appl. Math. Mech. 26, 4, 975-999, 1962. von Mises, R., "On Saint Venant's Principle," Bull. Amer. Math. Soc., 51, 555, 1945. Warren, W.E. and Roark, A.L., "A note on end effects in isotropic cylinders," AIAA Journal 5, 8, 1484-1486, (Technical Notes), August 1967. Washizu, K., "A note on the condition of compatability," J. Math. Phys. 36, 4, 306-312, January 1958. 79 [41] Watson, G.N., A Treatise on the Theory gf Bessel Functions, Cambridge University Press, 1966. [42] Williams, M.L., "Stress singularities resulting from various boundary conditions in angular corners of plates in extension," J. Appl. Mech. 19, 4, 526-528, December 1952. SUMMARY A semi-infinite elastic cylinder is considered with various types of self-equilibrated, but not necessarily axisymmetric, loadings applied to the finite end and with the long Sides free from stress. The paper offers a formulation in terms of stresses and three auxiliary variables of the order of stresses, and the systematic use of the Beltrami-Michell equations of compatibility. Two of the fundamental mixed problems are each reduced to a second order vector partial differential equation. Separation of variables provides a non-self adjoint ordinary differential equation, so the eigenfunctions are non-orthogonal. A gen- eralized biorthogonality condition is derived for each of these mixed problems, which permits the direct determination of the Fourier coefficients of a vector eigenfunction expansion. Self-equilibrated total-stress end loading problems are here formulated in terms of a six-component vector, involving the three stress variables acting on the finite end of the cylinder and the three auxiliary variables. This vector is Shown to satisfy a first order matrix partial differential equation, whose constituent equations and boundary conditions are obtained from the equations of equilibrium, the equations defining the auxiliary variables, and equations obtained by integration of the compatibility equations. A generalized biorthogonality relation 80 81 is derived for the Six-vector. Total-stress problems are solved using the biorthogonality relation for the six-vector by replacing the stresses defined on the finite end by their prescribed values, replacing the auxiliary variables by formal series expansions, and by solving numerically the truncation of an infinite system of linear equations. Numerical work was performed on two trial problems, one an axisymmetric torsionless problem and the other a truly axisymmetric problem. For the axisymmetric torsionless case it was found that ten pairs of eigenvalues matched the stress within a 10% error and twenty pairs matched within 3%. For the non-axisymmetric, ten triads matched all three boundary con- ditions within a 17% error. The method of analysis is thought to converge rapidly enough to be practicable for the numerical solution of problems of this type.