THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING VIBRATIONS OF A CIRCULAR PLATE ABOUT A POST-BUCKLED STATIC STATE Bertram Herzog A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Engineering Mechanics M 1961 MayP 1961 IP-515

Doctoral Committee: Professor Ernest F. Masur, Chairman Associate Professor Joe G. Eisley Professor Bruce G. Johnston Doctor Roger D. Low Professor Jesse Ormondroyd

ACKNOWLEDGMENTS An interest in the post buckling behavior of structures was stimulated by Professor E. F. Masur and it is to him that the author wishes to express his gratitude for suggesting the problem studied in this dissertation. For the advice and guidance received throughout the course of this investigation the author thanks Professor Masur and the members of the committee. Financial support was received, at various times during the author's recent graduate work, from the National Science Foundation and the Ford Foundation in the form of fellowships. The calculations performed would not have been possible without the use of the facilities of the University of Michigan Computing Center. The aid received from the Industry Program of the College of Engineering in preparing the manuscript is appreciated, ii

TABLE OF CONTENTS Page ACKNOWTEDGMENTS..................................... ii LIST OF TABLES................. iv LIST OF FIGURES.,................................ CHAPTER I. INTRODUCTION.......................................... 1 II. FORMULATION OF THE PROBLEMT,.*..9..................... 6 III. THE PERTURBATION SOLUTION,................... 16 IV. TE POWER SERIES METHOD........................ 28 V. RESULTS AND DISCUSSION,........................ 35 APPENDICES APPENDIX A. THE DYNAMIC BOUNDARY CONDITIONS FOR THE CASE n = 1.... 43 B. THE VIBRATIONS OF THE SIMPLY SUPPORTED CIRCULAR PLATE WIT EDGE COMPRESSION........................ 49 REFERENCES,.. * * *.. * * *..... * * *.............. *. 56 iii

LIST OF TABLES Table Page I Frequency Parameters p. for n = O0 n = 1 for Various Values of X, XS........................-. 52 II Roots of the Characteristic Equation for Vibration of the Circular Plate with Edge Compression-Linear Problem, n = 0.......5.......................,., 53 III Roots of the Characteristic Equation for Vibration of the Circular Plate with Edge Compression-Linear Problemy n = 1..,*............,...........,. 54 IV Values of Buckling and Frequency Associated Parameters for n = 0,.,.,......,..*............. 55 iv

LIST OF FIGURES Figure Page 1 The Relation Between the Frequency Parameter p. and the Load Parameter X...,.............58...*, 38 2 The Relation Between the Frequency Parameter L. and the Load Parameter X for Small X,........ 39 3 The Modes of Vibration Plotted for Half the.Plate... 40 4 The Relation Between XS and X,..................... 41 5 The Relation Between XS and X for Small lX......... 42 ~

I. INTRODUCTION Modern engineering structures have been experiencing a very rapid decrease in their "thickness" dimension as severe weight (and other) limitations have been imposed during recent years. As a consequence many structures are used in a post buckled state, that is to say the loads sustained are greater than those predicted in the usual "Euler column" sense. The motivation for this state of practice is obvious, however, a safe road to design remains to be paved. In addition, structures loaded in this manner are frequently expected to survive an environment of dynamic forces while subjected to these high static loads. The purpose of the present study is to determine the dynamic characteristics of such a structure. The results of this study are in the form of the natural frequencies and shapes of the modes of vibration of a circular plate as the function of a load parameter. The free vibrations of elastic bodies or structures about an equilibrium configuration have been studied extensively. The natural, frequency of vibration and the shape of the mode of vibration are the most important features which are obtained out of the solution of an eigenvalue problem. If such a body or structure is first preloaded statically, then the resulting frequency and mode of vibration exhibit interesting features. In general, a tensile system of stresses or forces causes an increase in the frequency of vibration, while compressive forces serve to decrease the frequency of vibration. The initial loading affects the effective stiffness of the structure and in the case of a compressive cl1, I

loading the effect is such as to reduce the stiffness to zero, which is indicated by zero frequency of vibration, and thus buckling of the structure in the conventional sense occurs. Indeed, a dynamic approach to the evaluation of buckling loads predicts buckling when the frequency of small vibrations about the static configuration goes to zero. The most straightforward example of such a problem is the lateral vibration of an elastic bar which is axially loaded. () The mode shape is sinusoidal for a simply supported bar, and the square of the frequency of vibration is linearly related to the axial force (or an associated loading parameter). Lurie (2) has discussed several examples related to vibration and structural stability and cites both theoretical and experimental results. He shows that, in general, within the framework of linear theories, whenever the mode shape of buckling and of vibration in the presence of axial loads is the same, then the interaction curve between the square of the frequency and some monotonic increasing load parameter will always be linear. Massonnet(3) discusses this same subject extensively, but frequently has to resort to approximate means, such as the Rayleigh-Ritz method, to solve the problem. These problems are solved within the framework of a linear theory. The buckling problem is phrased as an eigenvalue problem of the linear theory where the eigenvalue is associated with the critical load and a buckling mode of undetermined amplitude is obtained. These results imply no lateral deflection (the trivial solution) or that buckling occurs suddenly and with uncontrolled amplitude. That this paradoxical situation never arises in reality is explained by the presence of some imperfections, either in the structure or in the

loading system, which always insure that the structure deflects laterally as the load reaches the critical value. Even a crude experiment with a simple column shows that the structure does not collapse violently as the critical load is reached; however, the deflections do become large. The effects of these large deflections are not fully included in a linear theory. In order to discuss such phenomena more adequately, an improvement in the theory is made which results in nonlinear differential equations. In the case of a plat such equations were given by von Karman.(4 The solid circular plate is the structure to be investigated here. The linear equations of the classical theory of plates have been solved extensively.(56) The buckling of a circular plate was first studied by Bryan.(7) Federhofer(8) studied the problem of the vibrating clamped edge plate subjected to edge loads and presented extensive results of the interaction between compressive (and tensile) forces and the frequency of lateral vibration of the plate. * There exist relatively few solutions for the nonlinear equations for plates, introduced in 1910 by von Karman. The problem is particularly difficult for rectangular plates where several approximate (9) methods have been introduced, in particular by Marguerre.) Bisplinghoff and Pian(lO) treated the case of vibration of a rectangular plate of infinite length and some cases for plates of finite length. For the circular plate, however, several more solutions are available. Way(ll) solved, by power series methods, the problem of a circular plate subjected to lateral load. Friedrichs and Stoker(12'l3) used perturbation and * During the course of the present investigation it was necessary to solve such equations for a simply supported plate. Results similar to those of Federhofer for the clamped edge plate are presented in the Appendix. B.

-4power series methods to solve the problem of the simply supported circular plate subjected to compressive radial loading (in the plane of the plate). They treated only the axially symmetric case in a very exhaustive manner. A nonsymmetric version is beset by considerably more mathematical difficulties. The methods of these writers (14) were applied by Bodner(1) to a clamped edge plate for the same type of loading. Bromberg(5) used the methods utilized by Friedrichs and Stoker to study the effect of very large lateral loads which give rise to certain instabilities. Keller and Reiss( applied numerical methods to the problem discussed by Friedrichs and Stoker. Similar problems are studied by Alexeev(l7) and as a special case in a paper by Panov and Feodossiev. (18) Masur9), in a paper published in 1958, utilized a stress function space together with a minimum energy principle to obtain a sequence of solutions with error estimates for the post-buckling behavior of a plate using the von Karmarn equations. In a recent paper, Massonnet(20) considered the effects of initial curvature on the natural frequencies of vibration of an edgecompressed, clamped edge, circular plate. He solves the static problem by the. method of Friedrichs anid Stoker and then assumes that the mode shape of vibration is the same as that of the static problem, and utilizing the Rayleigh-Ritz method obtains the approximate frequency of vibration. The present study is concerned with the linearized vibrations of a circular plate rela aative to a static buckled configuration which is governed by the von Ka/rma/n equations. The plate is subjected to radial displacements which are the cause of the buckling and post buckling

equilibrium behavior. Although these boundary conditions are different from those of Friedrichs and Stoker, they are nevertheless mathematically equivalent for the static problem. It is here possible, however, to treat the problem of nonsymmetric vibrations relative to a symmetric buckling or static configuration.

II. FORMULATION OF THE PROBLEM For a preliminary consideration of the differential equations governing the pre-sent problem consider the xy plane of a cartesian coordinate system to be the middle plane of the plate. The z direction is the direction of the lateral deflection. Such a plate may be subjected to membrane forces in the plane of the plate and lateral loads in the z direction. The thickness of the plate is h. In the absence of body forces in the x and y directions there are two relevant differential equations due to von Karman. (4) The Equation (2.1) represents the equation of lateral equilibrium while compatibility is expressed by the Equation (2.2). DAw - ciacjpFpij = P (2.1) Eh.F -= Fiijj = - 2 cicj aww ij - Eh(7w,,y - W xywy) (2.2) where A represents the Laplacian operator, F is the Airy stress function, W is the lateral deflection of the plate and p is the load per unit area applied to the lateral surface of the plate. Further, the flexural rigidity is D = Eh3 12(1-v2) where E is the Young's Modulus of Elasticity and v is Poisson's Ratio, -6

-7and 0 1 cij = 1 Inherent in the utilization of these equations is the inclusion in the strain displacement equations of nonlinear terms involving the derivatives of w with respect to x and y. For a moving plate the inertia terms which are due to the motion of an element of the plate in the plane of the plate are neglected in comparison to those due to the lateral motion.* These inertia terms, which in actuality represent body forces, may, in this case then, be treated as the lateral load 2 p = - ph - (2.3) where p is the mass per unit volume. Only small amplitude harmonic vibrations with respect to the static configuration of larger amplitude are considered. Consistent with this assumption the following partitioning of the" stress function F, the displacements, strains and other quantities is proposed F = FS + E*FDeji t -S w = w + ci~De i t ~~~~S D ico~t ~(2.4) eij = eij + *eije N.. = NS. + E*Nj..eOt ij 13 1i where eij and Nij are the cartesian components of the membrane strains and stresses, while cu is the circular frequency of vibration and. e* is an arbitrary, small parameter. * For a discussion to this point see references (21) and (22).

The membrane stresses N.i are derivable from the stress function F by ij = jC:aCjF lap (2.5) The membrane strains eij are related to the stresses by eij = 1 [(l+v)Nij - vNkkij]] (2.6) Ehij where 5ij, the Kronecker delta, has values r1 o~ ij = o 1 Substitution of these quantities in Equations (2.1) and (2.2) and retaining only those terms which contain e* to the power of one or less yields two sets of differential equations, one governing the static problem and the other governing the dynamic problem. These are -S S _S Eh _c -W a' J(2.7) D6w - Sa.p( pijD D+ F.opi) = DH~jD - cicjg(FcBWtij + F,cW,ij) -D -hc -8 -D (2.8) AFD = -, Eh ciacj W, c, ij Since all detailed discussions of this plate are for a solid circular one, of outside radius R, the problem is rephrased in terms of the polar coordinates. The static configuration is assumed to be axially symmetric. However, permit a nonsymmetric dynamic configuration. In particular, all quantities are chosen in the following form* without any *Henceforth, unless otherwise noted, a summation symbol not having the summation limits specified is intended to be summuned over n from O to oo.

significant loss of generality: -S -S -D -D u = u (r) u = n(r) cos nQ s = S(r) = v = vD(r) sin nQ + (R)2 B*r n R = wS(r) 7 = wD(r) cos nQ (2.9) FS = FS(r) FD = FD (r) cos n where the u and v variables represent the displacement components in the radial and tangential directions respectively. Simultaneously it is advantageous to render all pertinent quantities in these equations dimensionless, and for this purpose let r x - S R -S R -D Y Y R -S R -D V = -2v vn = vn y n W =.wD (2.10) W Y Y FS FD,p FS on n 2 2 2 where y = h2/12(-v2 By the use of the above expressions the differential equations for the static case become, V4W 1 (Otw ) = 0 (2. 11a) and V47 =-1 (WW1)t (2.llb) 2x where V2() = [x( )t]' and primes designate differentiation with x respect to x.

-10The dynamic case for the nth mode* is governed by x x x 1 (ntW1)' + n2 nW = nwn x x,2 and (V2 - n2) [ (W'wn') - 2 1w] (2.12b) 22 (n) phR2 Wn where,(n) = D In order to be able to state the boundary conditions clearly, we rewrite all of the quantities involved, including moments, stresses and strains, in terms of the non-dimensional quantities. This is a matter of formal substitution and the results are listed below. As an example, consider in detail the radial strain and the radial stress. The radial strain for the dynamic configuration is defined as eD _ D. rr Zr br br where as before the superscript D refers to the dynamic configuration and the superscript S refers to the static one. In terms of the dimensionless quantities define a strain as du -n dW dwn n ex = X If + d d) cos nQ = e cos n xx sx dx xx and hence eD = (-)e rr xx Consider now the definition for the radial stress component Nrr. The static membrane stress (as obtained from the stress function FS), is 1 dFS rr r dr * Separation into modes is possible because of the assumed axial symmetry of the static solution.

The corresponding non-dimensional stress can be defined as 1 d(D x dx and as a consequence the relationship among these two stresses is N = T rr R2 xx Similarly the various dimensional stresses and strains are as listed below with their relationship to the dimensionless quantities. NS =D i D T Nrr -R- X R- xx NS D off D NSr = Tr (2.13) ND X (1 nt n) cos n D i R2 Lj r - I ^n" c08 ne - 0 R2 ND D02 nitn, n n rDe = n( - Tsi n = t sin nQ Consequently the stress strain relationships among the dimensionless quantities are: e - o-SG =2 t1 V n E = T - vT _ = -S Egg Tgg - vT1 Ott ( -, tn -S ex E - O (2.14) ex - vt - 1 n Zn _ n" e = tnx - vtnX = -n ( n- v n = E =n 1 T'n T x en = Et 2( v)n(0 re G x8 x~~~~~~~~~~~(.-4

-12since G = E/2(1+v). Further the relationships among the bending moments are: MS = -D()2(W" + + wI) = D(R)2 M 4rr R x R XX 8 = )-D )2C W' + VW") D() M MS = 0 = M rQ xO MD = D X [wn" + V( wn - w )] cos n (2.15) - (7)21 ~ n - D(5) E mn cos nQ = D( [x wn wn + vwn] cos ne = D(R) Em9 cos ne D(l R Z2 ~n wn MDr = D(1-v)()2 Xn(- + — ) sin nQ J-'G' *R ^-i x x The appropriate boundary conditions for a circular plate must now be considered. It is convenient to consider the relevant conditions for the static equations apart from those associated with the differential equations governing the vibration motion. Let the plate be simply supported at its circumference. The usual interpretation of "simply supported" is to consider that both the deflection at the support and the radial resisting moment offered by the support are zero. However, a theory that includes membrane effects must in addition specify a restriction, at the boundaries, upon the membrane displacements or

-13stresses. Thus for the static problem a radial displacement at the support is specified. Alternately, the radial membrane stress can be specified, which is indeed the manner in which Friedrichs and Stoker(13) chose to state their problem. However, since this investigation is primarily concerned with the dynamic problem a displacement condition appears more appropriate. The effects upon the static problem depending upon the nature of the membrane boundary condition specification are equivalent for either of the above cases. Physically the most realizable- situation is to specify a zero radial displacement of the plate at the edge and then consider the effect of uniform heating of the plate (the supporting structure being assumed rigid in the usual sense as well as with respect to temperature changes). Thermal effects would require.a additional term in Equation (2.2) and make the boundary condition homogeneous, which again is mathematically equivalent to the situation chosen here. Thus the boundary conditions governing the solution of the static problem are B1(W) - W(1) = O (2.16) B2(W) = (w" - W,) )x (2.17) and u (1) - te (2.18) Here uE is the magnitude of the radial displacement which is required to cause the plate to buckle in the usual or linear, Euler, sense. The term X is a parameter determining the extent to which the post buckling domain is penetrated. This third condition is conveniently rephrased in terms of the stress function ~. Utilizing the relationship between

~14u and e for a symmetric configuration, one finds eQ -S S u = x eQ Consequently, by Equation (2.14) B3(0) - x[x( t )' + (l-v)(')] = - UE (2 19) where UE = - E The boundary conditions for the dynamic equations require more careful consideration as a consequence of the permitted occurrence of nonsymmetric modes. Again the notion of simple support implies that (wn) = 0 (2.20) and B2(w) = 0 (2.21) Consistent with Equation (2.9) the prescribed membrane displacements at the boundary take the form u(l,Q) = An cos nQ (2.22) v(l,Q) = B sin n + B*x where the A's and B's are specified. It appears then that a specification of An and Bn provides the necessary number of conditions required of the problem. For the axially symmetric case, n = 0, v is identically zero (plus the possibility of rigid body motion) and just one condition needs to be imposed to specify the problem completely. However, a notable situation exists when n = 1. It can be shown** that in this case, also, just one additional condition is sufficient to specify the problem completely. This and other details for the case, n = 1, are carried out in detail in Appendix A.

-15The situation for n = 0 permits choosing the boundary condition which is similar with that for the static case, i.e. zero radial displacement, which in turn is synonymous with e (1) = 0 (2.23) Furthermore, this same condition is admissible and sufficient for n = 1. These are the only cases for which computations are carried out. Consequently the third boundary condition used for the dynamic case is B4(0n) e e(l) = [ -v1 x -n n)] = ~ (2.24) with n = 0, 1.

III. THE PERTURBATION SOLUTION In recalling the differential equations and boundary conditions governing the problem, we consider first the static case. It is required to solve the following differential equations v4W - (*'W1)' -= 0 (2.11a) x v4* = - (W'W )t (2. 11b) 2x and the associated boundary conditions** B1(W) = 0 (2.16) B2(W) 0 (2.17) B3(* ) -XUE (2.18) It is convenient here to partition the stress function such that " = Xe + 0 (3.1) where the function 0o satisfies the differential equation V4,o = (3.2) and the boundary condition B3(o) = UE (3.3) Consequently, the function t satisfies the differential equation =4 = - 1 (W'wi)' (3.4) 2x and the boundary condition B3(,) = 0 (3.5) It should be understood that the regularity requirements at the origin are being considered implicitly. -1L6n

-17The first of these equations will be recognized as the usual problem of plane elasticity whose well-known solution for the solid disk is = Tx2 (3.6) ~ 2 UE where T = UE l-v and where UE is found in Equation (3.17). In order to proceed to the topic of this thesis, it is first necessary to reproduce the solution previously obtained by Friedrichs and Stoker. There are two reasons that necessitate the repetition of this work. First, although these writers have completely solved this problem, their results are presented in such a form as not to permit direct application to the present work. Secondly, they were able to simplify the problem by a substitution of variables for W and for the stress function thereby reducing the order of the differential equations and making them directly integrable, at least in part. This is not possible here, since in the dynamic equations, due to the inertia term, w will appear explicitly and hence neither the integration nor the substitution of variables appears to be possible. Consequently, it is necessary to proceed with a solution in terms of W and 0 following, however, the example set forth by Friedrichs and Stoker. Assume the functions W, O, and X to be expandable in perturbation series:* * It can be shown that the other terms vanish.

W = eW1 + 3W3 + ce55 + **** 0 = e2k2 + e404 + ~606 +... (3.7) X = X+ + EXX + 4 +.... Here e is the perturbation parameter which will be chosen as a monotone increasing function whose direct significance will be fixed at a later point in the development. It will eventually be related directly to the amount of deflection in this problem. On the other hand, it could represent the amount of radial displacement or, indeed, it could be any other monotone increasing function. Substitution of these perturbation expansions in the differential equations and boundary conditions yields a sequence of differential equations, with associated boundary conditions, when coefficients of like powers of e are equated. Consider now the various differential equations. It will be recognized that associated with cE we obtain the equation whose solution is the function $o. For e1 the differential equation is Vl1 Xo(&WX ) = o (3.8) 1 x or alternately by Equation (3.6) L1(W^) ^w + XT j (x)' = 0 (.39) and the boundary conditions are B1(W1) = 0 (3.10) B2(W1) 0 (3.11) This is seen to be the linear eigenvalue problem for the buckling of the

-19plate subjected to compressive edge traction or displacement (X T represents the eigenvalue of the problem). There exists an infinity of eigenvalues and the solution to the problem may be represented as a development in the form of the associated modes or eigenfunctions* as 00oo oo00 W = A(m = A( l[Jo(mX) - Jo ()] (3.12) m=l m=l This solution automatically satisfies the boundary condition B1. The eigenfunctions must satisfy the associated characteristic equation which is obtainable from the boundary condition B2, The characteristic equation for the mth mode is (1 + v) J1 (am) + amJ2(am) 0 (3.13) where Jp(x) is the Bessel function** of the first kind and of order p and am is a root of the characteristic equation. There is an infinite number of such roots. However, am is related to X(m) and T through the differential 0 equation by X(m)T a2 (3.14) o m Consequently the characteristic equation determines X)T. Since the interest here centers around the first buckling mode, i.e. the symmetric one, only the lowest eigenvalue and its associated eigenfunction is * Superscripts in parentheses are intended to identify the variable and not to act as an exponent. Wherever possible, however, parentheses will be omitted whenever there is no possible confusion and will be included only to avoid confusion in isolated cases. n iew of the varying defintons for the various Bessel unctions. the ref. (23) is used as a standard throughout this thesis to avoid possible confusion,

-20considered. Consequently W1 = A [Jolx) - Jo(l)] (3.15) and (1)T = (3.16) It is convenient to choose X = (1) = 1 0 0 whereby T = a 1 Furthermore, let A(1) A =1 This choice governs the selection of e. Having obtained T, the value of UE is now determined, UE = (l-v)T = (l-v)al (5.17) The above-mentioned eigenfunctions form a complete set of functions satisfying the boundary conditions at x = 1 and the regularity conditions at the center of the plate and may be utilized in developing expressions for other functions by the familiar expansion property of eigenfunctions. This property will be utilized extensively in the subsequent paragraphs. For e2 the differential equation for <2 is obtained, V4 =- L (wfwt), (5.18) 2 2x 1l) (3.18) in which W1 represents the function just obtained. For a detailed discussion of systems of eigenfunctions and their properties see ref. (24) or (25).

-21The associated boundary condition is B3(2) = 0 (3.19) The equation is partially integrable and upon using the regularity conditions at the origin the following expression in closed form is obtained through the use of familiar recursion relations, reference (23), 2 (1! ) = - 1 [I(ax) - Jo(alx) J2 (alx)] (3.20) x 2 4x Further integration of this expression seems impossible except by substitution of an infinite series. When this is done and another integration performed, the result is 1 2' 1 X (-1)(l + C (3.21) x 2 4r= r' ((2+r).) (lr)' Further integration is not necessary since the function 02 will not be needed explicitly. The constant C can be determined from B3. For c3 the differential equation governing W3 is obtained L1(W3) = F3(x) (3.22) where F (x) = - ('Wt)I + F(x) 3 x o 1(.25) (3.23) 2 T(xW)' + F(x) x 3 and where F3(x) = - ( W) (3.24) 3 x 21

-22The associated boundary conditions are B1(W3) = 0 (3.25) B2(W3) = o (3.26) The differential equation here is nonhomogeneous, but the associated homogeneous equation is identical with Equation (3.9). This homogeneous system has the nontrivial solution W1. Hence the nonhomogeneous differential equation may have no solution or, if it has a solution, then this solution is not unique inasmuch as there can be added to it any arbitrary* multiple of W1. In order that the above nonhomogeneous equation possess a solution, the right hand side must satisfy an orthogonality condition which is, in this case, 1 F3(x) Hlx dx = 0 (3.27) o This orthogonality condition serves to determine the coefficient X2, 1 1 f F*H x dx fI'iH'H dx - = 31 - 3 ~ o 1 1 (3.28) T f x(xH1)'Hldx TI x.Hidx o o1 The particular solution can now be constructed and the procedure continued to determine further perturbation coefficients and functions. However, as will become apparent in the sequel, there is no need to pursue the solution of the equilibrium problem beyond this point. Turning now to the subject of this thesis, which is the vibration of the plate in the presence of the static or initial configurations and the associated equilibrium system of stresses discussed in the previous paragraphs, it will be noted that the method of solution in the * While the choice is theoretically arbitrary, the specific value is selected on the basis of convenience of computation.

253dynamic case has much in common with that used above and hence only the essential points are presented. The equations governing the motion of the plate are (2.12a), (2.12b) and, as a matter of convenience, they are presented again. By substitution of (3.6), Equation (2.12a) is further simplified to (V2 n2 2 2(V2 )wn + XoT(V2 _ )n ) ( wn) + Ot x7x x As pointed out in Chapter II, the boundary conditions are B1(wn) = 0 (2.21) B2(wn) = 0 (2.22) and B (0n) = (2.24) if attention is restricted to the cases n = and n = 1. In general, the functions w, 0 and pt are expanded in perturbation series utilizing the same parameter c as in the static case. n = n + C2w + n = 1 + e + *. (3.30) ~n n In = k + E2 +.. Upon substitution of these perturbation expansions* in the differential * The fact that wn and pn are even expansions in C and that on is an odd expansion may be easily verified upon substitution in the relevant equations. For the sake of brevity these steps are omitted here.

-24. equations and associated boundary conditions, a new sequence of differential equations is obtained whose solution will follow very similar methods to those presented for the static case. It will also become apparent that the partitioning of the stress function &* makes this future work similar to the previous paragraphs. For c~ the differential equation is W) = (V wn + XoT(V n2 n nn (31) L2(wx) 2( - n22 oWo and the associated boundary conditions are Bi(wn) = 0 (5.32) B2(wn) = 0 (3.33) The solution of this equation is 00 n= a (m) nmw (3-34) o 0 0 m=0 where nm nm - nm \ Jn2 rmr O = Jn( x) - n x2 In(l x) (3.35) 2 I(nm) which immediately satisfies B1 and where In is the modified Bessel function. This eigenvalue problem is governed by the characteristic equations obtained in the usual fashion from the boundary conditions, B2. In(nm)[(2n + 1 + V)1 v Jn+l( ) - (n1)2 Jn+2 )],nm. n m2 rim (3 36) + n(2)[(2n + 1 + v)P Inl( ) + ( )2 I+2( ) = 0 ( 6 re an ae late to (i.e. ) n by\ where P, and pp are related to X T (i.e., a,) and pinm by

-25(nm)2 nm2 2 = (1) p ) nm (prm)2(pm)2 (33 ) The functions wnm obey the usual orthogonality conditions which, in this case, are nrm rs (s8 w w x dx = 0 (5.38) and 2 n22 rim 2 2 [( _ n ) w + XoT(V - A) wm]wo xdx = 0 (3.39) O 0 x x 0 It should be noted that, for a specific value of n, representing the number of nodal diameters, there exists an infinity of roots of the above system which are designated by the index m. The index m represents the number of nodal circles appearing in the vibration pattern of the plate. A similar system for the case of a clamped edge plate was (8) solved by Federhofer in 1935. (8) For c a differential equation governing the function 01 is obtained which is (V 2 -n2)2 = - g(x) (3.4o) x where gl(x) = i (Wwn') _ n2 Wlwn (3.41) x 10 x2 o and the associated boundary conditions are B,(n) = 0 (3.42)

-26This is a problem of integration; however, the form of the function gl(x) is such that a numerical integration is evident except for n = 0. For e2 a differential equation governing the deflection function w2 is obtained which is L2(W2) = f2(x) (3.43) where rm run f2(x) = 2 Wo + f2(x) (3.44) and where 2 2T(V X 2) wnm + 1 (Ofwnml)I f*(x) -X T 9w)( l) ~~~2 0x 2x (3.45) i2- " w lm +1 m-' x x and where associated boundary conditions are B1(w nM) = 0 (3.46) 2- <2 = 0 (53.47) By the now familiar process, f2(x) must satisfy an orthogonality condition in order to obtain a solution for w2 and out of this relation nm emerges 2 J f f2(X)wo xdx 2 - (3.48) 2 1 f wo xdx For the special case, n = 0, numerical results are readily obtained. In particular the value of the rate of change of frequency with respect to the load parameter is desired in the neighborhood of the condition of linear buckling which is lim d. = lim (d )/(d) = (/49) c-where p defined by Equation ( 0). where ~2 and ko are defined by Equation (3.30).

-27In pursuing the analytical steps outlined previously for the symmetric mode of vibration it becomes apparent that 0 becomes a multiple of P2 and hence from Equation (3.48) and by the use of Equation (3.21) f xH'H'dx k = 2X2T 0 1 1 (3.50) 2o 1 flxHIH1dx which is readily evaluated. Thus finally >2/X2 = 49.29 (3.51) In closing, then, it is evident that when the critical or linear buckling condition is reached the frequency of vibration is zero and as the plate proceeds into the post buckled condition the square of the frequency increases initially in a linear fashion with the buckling parameter. Further calculations with the perturbation method become exceedingly cumbersome and are abandoned in favor of the power series method of the next chapter. However, the present result is exact as e approaches zero since higher order expansions vanish for this condition.

IV. THE POWER SERIES METHOD Another possible means of solving the system of differential equations presented here is to develop the solution in terms of a power series. Again we borrow the results of Friedrichs and Stoker for the solution of the static problem. The phrasing is slightly different and some of the numerical computations used are only minor variations of theirs. The system of differential equations for the static case is VW7 - W ('W')' = 0 (2.11a) v4 1= - (w'w)' (2. llb) 2x and the boundary conditions associated with them are B1(W1) = 0 (2.16) B2(Wl) = O (2.17) B3(0) = -XU (2.19) Let the functions W and < be expressed in power series of the coordinate x. These series are even functions in x. 00 W = X amx2m (4.1) m=O 00 = bx 2m (4.2) m m=O It is easy to show by substitution in Equation (2.11a) and (2. llb) that the coefficients ao, al, bo, and b1 are, at this point, arbitrary and -28

-29that all of the other coefficients of the two series will be given by the following recursion relationships. am - i E i(m-i) a.bi (4.3) =2m(ml) im-1 i=l m> 2 m-1 bm = i i(m-,) aiami (4.4) 4m2 (m-1l) i=L In terms of the coefficients in the power series, the previously stated boundary conditions become 00 a 0 o (4.5) It is evident that Equation (4.5) serves only to determine 0 / m(2m - 1 + v) a = 0 (4.6) ) m(2m - 1 - v) bm= -2E (4.7) It is evident that Equation (4.5) serves only to determine the coefficient a0 after the others have been computed. The two equations (4.6) and (4.7) must be solved simultaneously for the values of a1 and b1. The value of the coefficient bo remains arbitrary and this is as might be expected since the stress function 3 can in general be varied with respect to as much as an arbitrary linear function of the cartesian coordinates without affecting the stresses. Hence, in principle, once the value of X is specified, it is possible to obtain all of the necessary coefficients in order to be able to describe the complete solution for the static case.

50 — For the dynamic case the differential equations to be solved are x2 X X2 (V 2 )2 n 1,2 n n (2.12a) -1 (n'W')' + n2 - nwn 0 x x2 (V2 n22 0(W'wn)' n2 W"wn] (2.12b) (V2 - 22 =. [_ (Wwn:) - W ] (2.12b) X x x2 The associated boundary conditions are Bl(wn) = 0 (2.20) B2(wn) = 0 (2.21) B4(0 ) = 0 (2.24) The solution of the above differential equations may be expressed as power series in x. Upon calculating with the power series by the usual methods, it becomes clear that the solutions which are not singular at the origin are 00 w(n) = xn cn)x2 (4.8) m=0 and c0o (n) = n n)2m (49) m=0 The recursion relationships for all values of n that evolve from this system are, after dropping the superscripts for c and d, * For n > 2 the number of boundary conditions increases to four.

-531m 16m(m-1)(m+n)(m+n1) m-2 m + (2m+n-2) 2i(n+2m-2i)(aidm-_i +biCm-i) i=l m - n 2i(2i-l)(aidmi bicmi)} Li~ i=l (4.10) m>2 m d =- 1 {(2m+n-2) I 2i(n+2m-2i) aimi 16m(m-1) (m+n)(m+n-l) i=l 2.(4. *11) - n 2i(2i-l) aicmi} while the coefficients co, cl, do, and dL remain, at this point, undetermined. In terms of these power series, the boundary conditions for the dynamic problem assume the following form: 00? Cm =0 (4.12) m=O 00 [(2m+n)(2m+n-l+v) - vn2]cm = 0 (4.13) m=O 00 [(2m+n)(2m+n-l-v) + vn2]dm = 0 (4.14) m=l It is readily verified by examination of the recursion relation in the cases when n 0 and n 1 that d will not appear in the problem and this is again plausible in view of the remark made in connection with b0.

-532The boundary conditions thus present a system of homogeneous linear algebraic equations in the three unknown coefficients co, cl, and dl as well as of the eigenvalue p. The explicit form of the characteristic determinental equation associated with this system is, by virtue of the form of the boundary conditions and the recursion relations, probably unattainable. In any case, no particular benefit would be derived from it since a numerical solution will have to be attempted and the appropriate algorithm can be stated in terms of these equations. The solution of this system of equations is to be performed on a digital computer in the following manner. An arbitrary value, such as unity for example, is assigned to the coefficients co, cl and dl. The remaining coefficients in the power series expansion are determined by the recursion relationship. A generic term in the series depends upon co, c1 and di and polynomials in terms of l. Briefly then, each coefficient in the power series is the sum of three polynomials in kyLt the coefficients of these polynomials being the constants co, cl and dl respectively. The various coefficients can be generated and their polynomial representation is substituted into the boundary equations (4.12), (4.13) and (4.14). Theoretically at least, each equation is composed of infinite series aij in Cp having as their coefficients c0, c1 and d1. In matrix notation AB: (4.15) where A = {aij} ij = 1,2,) and B c=

p33This system of equations will admit a nontrivial solution when the determinant of the coefficient matrix A is zero. This determinant is equivalent to a power series in t whose roots represent the required eigenvalues. Clearly the procedure here must be modified to truncate this process. The coefficients of the power series of w and 0 are computed until a particular term becomes less than a designated value. However, convergence at this point alone is not necessarily the final test. Satisfying the boundary conditions yields an approximation to the a.. in that they are now polynomials. It should be noted that the first neglected terxm for instance, provides, in general, a contribution to all the previously computed coefficients in aij. However, the system was sufficiently convergent to insure convergence in this sense. The aij are polynomials in p of order m/2 to the nearest integer when m terms are taken in the power series. It is quite clear then that the determinant when expanded is capable of yielding a polynomial whose order is three times that of the order of the a... This determinant may be expanded algebraically, by the computer, producing a polynomial characteristic equation which must be solved for the roots, by standard methods. Generally, about sixteen terms in the power series produced sufficiently convergent terms in the aij and it sufficed to take five or six terms in the characteristic equation to obtain the roots. For very large penetration of the post buckling domain the size of the numbers involved and the number of terms required for the necessary

accuracy required a modification of this technique due to the limitations imposed by the available computer. The modified procedure consisted of an iteration procedure one step earlier in the above process. The aij were evaluated using trial values for \i and then checking to see if the determinant is zero. Computing time of this iteration procedure was approximately eight times longer than by the first method. Weinitschke(26) who faced a similar computation in connection with a problem in shells chose to use several power series expansion each being valid in part of the region and then matching them together. By this means fewer terms in each series were required for convergence.

V. RESULTS AND DISCUSSION The penetration of the post buckling domain is measured by X, a ratio of the edge displacement to that required for the initial instability. Friedrichs and Stoker(12,3) use a parameter XS based upon a stress ratio. For convenience the results here are expressed in terms of X; Figures 4 and 5 show the relation between the two parameters. The main result offered here is the: relation between. (the squared frequency parameter) and X. Figure 1 shows the relation for a symmetric mode (n = O) and the first mode having a nodal diameter (n = 1). The details of this relation in the vicinity of X = 1, the early stages of penetration of the post buckling domain, are shown in Figure 2. In Figure 3 the shapes of the modes of vibration are depicted. The data used in plotting Figures 1, 2 and 3 and further information are given in Table I. The fact that i increases linearly with X in the vicinity of initial instability for the symmetric mode as shown by the power series analysis is borne out by the perturbation analysis. From Figure 2 the numeric value of the slope is fifty which compares rather well with the results of Equation (3.49), i.e. 49.29. It is interesting to observe that after some increase in X the frequency of the nonsymmetric mode is lower than that of the symmetric mode. Examination of the modes for the symmetric case shows that at a value of X between 13 and 19 a nodal circle appears for the lowest frequency. Near this value of X * All calculations are based upon the value of Poisson's ratio v =.318. -55

.36the frequency of the axially symmetric mode increases less rapidly and eventually becomes again less than that for n = 1. This behavior of the frequencies is reasonable inasmuch as the nonsymmetric mode is essentially inextensional while the symmetric mode is essentially extensional and consequently the frequencies of the symmetric mode can become greater than those for n = 1. Upon the appearance of the nodal circle in the n 0 -O mode, this mode also becomes essentially inextensional and the frequency increases at a lesser rate while X is increased. Under these circumstances the frequency can, and does, become less than that for n = 1. Apparently the frequency reaches an asymptotic value as X is increased.. It should be noted, however, that as X becomes large the accuracy* of the results becomes less certain but in any case they also become less meaningful because by the time X reaches values approaching o the equilibrium configuration is in, what Friedrichs and Stoker call, the asymptotic range. In this range the plate has been stretched as a membrane except for a narrow boundary layer at the edge where large bending stresses occur. That a plate could reach such a state is subject to question on practical grounds, in particular, if one considers the effect of imperfections upon the behavior of the ideal plate considered here. Also the onset of plastic yielding or secondary buckling is likely to invalidate these somewhat academic results. A very significant result is that, in the range of computations, the frequency of vibration does not return to zero for X > 1. This implies " Accuracy was determined by the amount the deflection at the edge differed from zero with respect to the magnitude of the largest deflection. For good results this amounted to one part in 108 while for poor results one part in 104.

-537 that, at least within the limitations imposed here on the nature of the assumed vibration modes (i.e. small amplitude vibration), the positive definiteness of the potential energy and hence the stability of the buckled configuration is unchanged if only expansions up to the second power in the terms representing the additional neighboring deflections are included. It appears likely that this is true also in relation to higher modes. Consequently the experimentally observed phenomenon of secondary buckling(12) cannot be explained in terms of a simple branch point. Research delving into this matter is continuing and attention is directed at the possibility of a discontinuous snapthrough to a position of lower potential energy. This becomes possible when the quadratic form introduced in reference (19) loses its positive definite character. This is indeed the case as pointed out in the paper, reference (19). The problem of secondary buckling therefore shows great similarity with that of buckling of certain types of shells.

-381200 1000 - 800 --- 0 I i! ii i —--------- I 0 20 40 60 80 100 X Figure 1. The Relation Between the Frequency Parameter i and the Load Parameter X.

500 400 /I 300 n~~= 200 100 0 0 2 4 6 8 10 12 X Figure 2. The Relation Between the Frequency Parameter t and the Load Parameter X for Small X.

."4oTHE MODES FOR n =O X THE MODES FOR n = I 2P —----,00 X 7 ~1.004 --— _ 7.11 13.86 I \ / WMAX x=O x=l xO x=s Figure 53 The MoesB of Vibration Plotted for Half the Plate.

ii 9 --- xs l:5 - 3 I I I I________________________________________ I I, 0 20 40 60 80 100 120 140 Figure 4. The Relation Between XS and X.

2 I —-_____\ e tn B n X ad X r S l X. Figure 5. The Relpation Between I and X for Small X.

APPENDIX A THE DYNAMIC BOUNDARY CONDITIONS FOR THE CASE n = 1 The boundary conditions for the case n = 1 require some special consideration. The condition of "simple" support requires that the lateral deflection as well as the radial bending moment at the outside edge be zero. To complete the boundary conditions, some requirements must be imposed on the membrane displacements or the membrane stresses (indirectly, through the stress function) or a combination of both. In general, two more boundary conditions are required, but due to the obvious symmetry in the case n = 0, only one is required beyond the implications of symmetry. Consider now the details of the case n = 1. For simplicity, the superscripts are omitted inasmuch as this discussion is for a specific case. The stress function for n = 1 is of the form = ir(x) cos 0 while the lateral displacement is w = w(x) cos Q From the strain displacement equation it follows that av - = xe -u and au de W a xx dx x av dx ax -43

.44, The stress strain equations, utilizing the expression of the stresses in terms of the stress function imply that 1 1 e x = [r' - x-r, - V "] cos o and e = [T' V(j] Cs [, (' - Co ] cosQ x [" - v=()'] cos @ where primes mean differentiation with respect to x. Hence au = [(l4r)1 v)" - w w] cos o (A.1) ax x and upon integration u = [1 - v' S W'w'tdx] cos @ + F1() (A.2) Further V = x[I" - v(')t] COS - [^t- VT x Xx (A. 3) - f W'w'dx] cos 0 - F1(0) and thus v = x[It" -v )t1] sin - [1 t- Vf (A.4) - S Wtw'dx] sin - F2(O) + G(x) where dF = F1 (A. 5) dQ

However, - ) +_ _ = - -(l + v) 2e - x _ + x_ ) +ldW 3w _ 1+v) ) exQ x 6Q ax x x dx aQ ax x ax Upon substitution of the appropriate quantities in the above expression and simplifying the following is obtained: (i x + x" - + W W) sin x. x x = d (A.6) + - ( + F2) + X(G) = 0 x d~ x Equation (A.6) must hold for all x and Q and it can be shown that the required necessary and sufficient conditions are x(3 + xr"-3L+ Wtwt -1 Ww) = -2C (A. 7) x x x dFl d- +F2 = 3D1+ 2C sin o (A.8) dQ x )1 = -3D (A.9) Equation (A.7) is satisfied if 1 satisfies the compatibility equation associated with the von Karman system of equations. Equation (A.8) will be satisfied if F2 satisfies dO2 The solution of Equation (A.10) is F2 = - C cos + + C2 sin 9 + 3D1 - CO cos 0 (A.11)

-w46from which it followsr by Equation (A. 5), that F1 = C1 sin @ + C2 cos 8 + C(Q sin @ - cos G) (A.12) The solution of Equation (A.9) is G a= + Dx (A.13) x2 Consequently the displacement components become u(l) ( (f - / W'w'dx + C2) cos 8 - C1 sin @ + C(8 sin - cos Q) (A. 14) v(l) = (g + f W'wdx - C2) sin 8 - C1 cos 8 - 53D + CO cos 8 + -l + Dx x2 where f =- x") (z-v1) 1 ] x (l ) g - [Xr - (1-tV) X --, "(1) - (l-v),(.l) and f + g =i [,, - vt)'] x x x l Consistent with Equation (2.19) (and hence without any significant loss of generality) let C - O C1 = 0 Single valued displacements and regularity conditions at the origin, require that C = 0 and D1 = 0 For n = 1, from Equation (2.22) u(l) = Al cos Q ~v(l) _ ~i ~@+ B*~(A. 5) v(l) = B, sin 9 + B*

..7Consequently D = B* and f - f W'wdx + C2 = Al (A.16) g + f Wtw'dx - C = B1 (A.17) In addition to the conditions imposed by the simple support at the edge it is sufficient to require that f + g 0 (A.18) which by the use of (A.16) and (A.17) implies that Al = B2 (A. 19) and it must also be noted that egg(l) = f + g and hence Equation (A.18) implies e (l) = 0 (A.20) which is the boundary condition used in the main text. For the power series approach 00 ~ 0(1) = alx + ca2 dmx2m53 (A.21) m=l where, for convenience of this discussion, a1 and a2 are the arbitrary constants and do is chosen as unity. It is easily shown that (f + g) does not depend upon a. Further, none of the stresses depend upon al. Indeed, none of the stresses depend upon al, and further it can be shown that al and C2 can be combined as a single constant C To demonstrate this, substitute the expression for r into f and g and the associated equations. Thus from Equation (A.16) C2 + (l-v)a2 + a2 [1 - v(2m+3)5] d - / W'w'dx = Al (A.22)

-w48is obtained and by the use of Equation (A. 17) and 2 - (1-v)c1 +'a2 [(2nr+3)(2m+2) - (l-v)] d + f Ww'dx = B1 (A.23) Let 2 = C2 + (l-v)al In terms of the power series solution f + g =2 X (2i+5-v)(2m+2)dm Consequently (f + g) is independent of al. Further, it can be seen that Equation (A.23) then serves to determine Cp.

APPENDIX B THE VIBRATIONS OF THE SIMPLY SUPPORTED CIRCULAR PLATE WITH EDGE COMPRESSION (Linear Theory) During the course of the present investigation, the linear problem of plate vibration in the presence of radial edge compression was solved. The same problem for a clamped edge plate was treated by Federhofer. (8) The boundary condition here is that of simple support. The relevant equations arose in connection with the perturbation solution and are repeated here for convenience. The differential equation, after separation of variables, is similar to Equation (3.31), (.. _ )2 o + XoT(V- ) o - o6wo o (B.1) and the associated boundary conditions are B1(wn) = 0 (5 32) B3(w) = 0 (3.33) The solution to this equation is 00 n (m) nm 0 a0 wo (3534) m'6 where nm O = Jn( x) - Jn ) In(1 x) (55) O sas s(in (o) which immediately satisfies B1. This eigenvalue problem is governed by the characteristic equations obtained in the usual fashjion from the ^4P

-50boundary conditions, In(m) [ (2n+l+v)Pm Jn+1(m) - (nm)2J ( nm) r,rrmd + (Prm)2I (,nm)] (3.36) + Jn( m)[(2n+l+v) 1 n+l( ) + (1)I+2 = where 1 and nm are related to X T(i.e., a) and ltnm by (nnm )2 ( - m)2 2 2 1 (B.2) n (Pm )2 (gnm)2 Since only the linear problem is considered here subscripts on X and a are omitted (which in Chapter III had meaning in connection with the perturbation solution). For the linear buckling problem Inm t = 0 (B. 3) and it may be shown this is equivalent to nrn K1 = 0 1 (B. 4) P = Oa 2 and Equation (3.36) becomes (2n+l+v)Jn+1(Bn) Jn+2(Pm) = 0 (B.5) where n designates the number of nodal diameters and m the number of nodal circles. The roots of the characteristic equations were obtained for several values of n and m by a high speed digital computer using a step-by-step searching method followed ll by an interval halving method until the roots were reproduced within specified limits In Table 1i

-51 and III roots of Equation (3.356) are tabulated for several values of n, m and a. For the axially symmetric mode the case of free vibration and of buckling was solved for a number of roots. These results are presented in Table IV.

-52TABLE I FREQUENCY PARAMETERS pW FOR n 0, n - 1 FOR VARIOUS VALUES OF X, IS X X, ------------- xsn=O n=l 1.00001 100000U 0.000 131.410 1.00009 1.0UU01 0.004 131.412 1.00017 1.uO303 0.009 131.414 1.00034 1.U0005 0.017 131.419 1.00067 1.00010 0.033 131.4Z7 1.00101 1.00015 0.050 131.435 1.00402 1.00059 0.198 131.511 1.00887 1.00129 0.437 131.632 1.01574 1.00229 0.775 131.804 1.02462 1.00359 1.213 i32.02 1.03552 1.00517 1.750 132.299 1.04846 1.00705 2.367 132.623 1.06343 1.0923 3.124 132.998 1.08029 1.01168 3.954 133.420 1.09886 1.01437 4.869 133.886 1.41138 1.05927 20.258 141.735 1,98215 1.13931 48.371 156.161 2.91311 1.26498 94.298 179.920 4.15896 1.42499 155.952 212.109 4.42706 1.45835 169.243 219.088 6.31125 1.68353 262.640 268.598 7.11236 1.77494 302.177 289.862 9.57456 2.04258 421.501 355.833 11.25514 2*21547 499.317 401.278 13*86542 2.47114 610.406 472.235 19.60922 2.99070 794.910 629.174 29.17450 3.76396 9+8.097 886.156 33.88582 4.11459 988.571 1008.043 42.29802 4.70397 1035.067 1213.347 50.15607 5.22066 1061.999 1388.049 61.01744 5.89279 1091.762 1603.025 70.96683 6.47418 1122.798 1774.656

TABLE II ROOTS OF THE CHARACTERISTIC EQUATION FOR VIBRATION OF THE CIRCULAR PLATE WITH EDGE COMPRESSION-LnIEAR PROBLEM, n = 0. a I, "~"~r 2 - 1/2 2 m=O 0.0 2.2274580 2.2274580 4.961569 0.5 2.1656121 2.222583 4.813253 1.0 1.9663810 2.2060495 4.337934 1.3 1.7594769 2.1876378 3.849098 1.5 1.5675703 2.1696259 3.401041 1.6 1.4482550 2.1581109 3.125495 1.8 1.1342579 2.1275669 2.413210 2.0 0.5716016 2.0800789 1.188976 2.0600017 0.0 2.0600017 0.0 m =I 0.0 5.4535042 5.4535042 29.740708 0.5 5.4302508 5.4532215 29.612360 1.0 5.3598667 5.4523547 29.223895 1.3 5.2942696 5.4515402 28.861924 1.5 5.2404103 5.4508624 28.564755 1.6 5.2103511 5.4504824 28.398927 1.8 5.1437857 5.4496359 28.031759 2.0 5.0683233 5.4486605 27.615572 3.0 4.5398584 5.4415361 24.703803 4,0 3.6705019 5.4288658 19.926662 5.0 2.0548660 5.4057816 11.108157 5.3928115 0.0 5.3928115 0.0 m=2 00 8.6125308 8.6125308 74.175687 0,5 8.5979338 8.6124599 74.049359 1.0 8.5539953 8.6122493 73.669139 1.3 8.5133691 8.6120528 73.317583 1.5 8.4802543 8.6118938 73.031049 1.6 8.4618651 8.6118037 72.871920 1.8 8.4213943 8.6116133 72.521791 2.0 8.3759155 8.6113856 72.128238 3.0 8*0702978 8.6098611 69.484143 4.0 7.6216586 8*6075363 65.603703 5.0 7.0022717 8v604'1740 60.248764 6.0 6.1602463 8.5993392 52.974048 7.0 4.9827378 8.5923034 42.813195 8.0 3.1059983 8.5817962 26.655044 8.5739849 0.0 8.5739849 0.0

-54TABLE III ROOTS OF TBE CHARACTERISTIC EQUATION FOR VIBRATION OF THE CIRCUIAR PLATE WITH EDGE CCMPRESSION-LINEAR PROBLEM, n = 1 Ca __ ___ J___ 4/2 RI m O 0.0 3.7310427 3.7310427 13.9206793 0,5 3.6964372 3.7301003 13.788082 1.O 3.5904497 3.7271555 13.382351 1.3 3,4900374 3.7242934 12.997923 1.5 3.4061925 3.7218474 12.677329 1.6 3.3588249 3.7204441 12.496320 1.8 3.2523577 3.7172343 12.089775 2.0 3.1288098 3.7134150 11.618569 3.0 2.1309236 3.6797875 7.841346 3.6306551 0.0 3.6306551 0.0 m = 0.0 6.9641241 6.9641241 48.499024 0.5 6.9460158 6.9639885 48.371974 1.0 6.8914081 6.9635843 47.988901 1.3 6.8407772 6.9632056 47.6~3738 1.5 6.7994036 6.9628938 47.343525 1.6 6*7763914 6*9627209 47.182122 1.8 6.7256276 6.9623320 46.826052 2.0 6.6684310 6.9618944 46.424912 3.0 6.2789499 6.9588226 43.694098 4.0 5.6882904 6.9538944 39.555770 5.0 4.8217636 6.9461792 33.492834 6.0 3.4752368 6.9337776 24.096519 6.9151434 0.0 3.6306551 0.0 m 2 0,0 10 138746 10.138746 102.794170 0,5 10.1262940 10 1386306 102.666754 1.0 10*0890692 10.1385068 102.288096 1.3 10*0546935 10.1383855 101.938358 1.5 10*0267117 10.1382912 101.653723 1*6 10.0111877 10 1382384 101.495807 1.8 9 9770459 10.1381185 101.148473 2.0 9,9387485 10 1379842 100.758874 3.0 9.6829959 10.1370810 98.157313 4.0 9.3130618 10.1357349 94.394726 5.0 8.8144723 10.1338503 89.324523 6.0 8.1635122 10.1312850 82.706868 7.0 7 3193063 10 1277957 74.128439 80 6.2028258 10 1229961 67.791182 10,1054066 0.0 10.1054066 0.0

-55 TABEL IV VALUES OF BUCKLING AND FREQUENCY ASSOCIATED PARAMETERS FOR n = 0 m ~ a I,p | 1/2 0 2.0600017 2.2274580 4.9615690 1 5.3928115 5.4535041 29.740707 2 8.5739849 8.6125307 74.175685 3 11.733336 11.761687 138.33727 4 14.885062 14.9u7513 222*23394 5 18.033223 18.051816 325.86805 6 21.179422 21.195296 449.24056 7 24.324426 24.338273 592.35151 8 27.468644 27.480928 755.20139 9 30.612324 30.623355 937.78984 10 33.755613 33.765622 1140.1172 11 36.898613 36.907774 136.1837 12 40.041390 4U.u49857 1603.9910 13 43.183997 43.191848 1865.5357 14 46.326466 46.'a33817 2146.8225

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