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BUCKLING AND DEFORMATION OF HEATED CONICAL SHELLS By LUKANG CHANG A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA June, 1967 ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to Dr. S. Y. Lu, Chairman of the Supervisory Committee, for suggesting this subject and providing invaluable guidance throughout the entire period of this research. It was by his constant encouragement that this work was made possible. He would also like to thank Dr. W. A. Nash, Chairman of the Department of Engineering Science and Mechanics, for his valuable suggestions and financial support throughout the author's graduate work. Gratitude is also expressed to Dr. I. K. Ebcioglu, Professor of Engineering Science and Mechanics, and Dr. R. G. Blake, Associate Professor of Mathematics, for serving as members of the writer's supervisory committee. Final thanks go to the National Science Foundation for sponsoring this research. TABLE OF CONTENTS ACKNOWLEDGMENTS . . . LIST OF FIGURES . . . NOMENCLATURE . . . ABSTRACT .. . . . CHAPTER I. INTRODUCTION . . II. BASIC RELATIONS AND FUNDAMENTAL EQUATIONS 1. Geometry of Shells 2. Basic Equations . 3. Transformation of Coordinates . III. METHOD OF APPROACH . . 1. Boundary Conditions . 2. Thermal Buckling . 3. Postbuckling Behavior . IV. SOLUTION AND NUMERICAL RESULTS . 1. Critical Temperature . 2. Minimum Temperature . 3. Deformation . . V. CONCLUSIONS . . APPENDIX A. EQUATIONS FOR CASE I . . B. EQUATIONS FOR CASE II . . C. COMPUTER PROGRAM FOR NONLINEAR ANALYSIS Corresponding Symbols Used in Computer Program . REFERENCES . . . BIOGRAPHICAL SKETCH . . . Page ii iv vi viii 1 8 8 14 21 23 24 26 28 31 31 49 57 66 72 102 118 118 148 151 * . . * LIST OF FIGURES Figure Page 1. The cross section of a shell segment . .. 11 2. Geometry of conical shell . .... .25 3. Critical temperature versus radiusthickness ratio at H/R = 2 . . ... ...... 40 4. Critical temperature versus radiusthickness ratio at 0 = 100 . .. .. 41 5. Critical temperature versus semivertex angle at H/R = 2 . . .. 42 6. Variation of critical temperature with meridional temperature index (0 = 100, R/h = 300, H/R = 2). 43 7. Critical temperature due to circumferentially non uniform eating (0 = 100, H/R = 2) . ... 50 8. Temperature variation T1 as a function of deflection coefficient ratio (ai/ao) at 5 = 100, R/h = 450 and H/R = 2 . . . 58 9. Temperature variation T1 as a function of deflection coefficient ratio (a/aI) at B = 100, R/h = 900 and H/R = 2 . . . 59 10. Minimum temperature versus radiusthickness ratio at H/R = 2 . .. 60 11. Variation of minimum temperature with meridional temperature index (0 = 100, R/ = 300, H/R = 2) 61 12. Deflection versus temperature (5 = 100, R/h = 450, H/R = 2) . . . 63 13. Deflection versus temperature for axisymmetric case (6 = 10, R/h = 450, H/R = 2). . 64 14. Deflection versus temperature (U = 10, H/R = 2) . 65 LIST OF FIGURES (Continued) Figure Page 15. Comparison of critical temperature with minimum temperature (0 = 10, R/R = 2) . 68 16. Photograph showing buckling of a heated conical shell 70 NOMENCLATURE E = Young's modulus F = dimensionless strain function H = night of conical sell R = mean radius defined in Fig. 2 T = temperature gradient in the middle surface Tl, T2 = uniform temperature rise a a. = coefficients of tne deflection functions defined in Eqs. (IV3) and (IV15), respectively / I b b. = coefficients of the deflection functions defined i' in Eqs. (IV30) and (IV37), respectively a, a. = coefficients of the deflection functions defined in Eqs. (IV47) and (IV50), respectively e e, = meridional and circumferential strains in the middle surface e r = shear strain in the middle surface g, k = temperature distribution factor defined in Eqs. (IV9) and (IV29), respectively h = walltnickness of conical shell I = length of conical shell mi, mi = numbers defined in Eqs. (IV53) n = number defined in Eq. (IV16) r, p = surface coordinates u, v, w = dimensionless meridional circumferential and inward normal displacements x, r = length index and meridional distance defined in Eq. (1136) and Fig. 2 x = value of x at the large end o y = thermal expansion coefficient B = semivertex angle V1' Y2 = numbers defined in Eqs. (IV26) and (IV41), respectively e e = meridional and circumferential strains erl = shear strain S= number defined in Eq. (IV61) 8 = transformed coordinate defined in Eqs. (1131) Kr, k = meridional and circumferential changes of curvature in the middle surface Kr = twist of the middle surface A = (lv)/2 9, 7 = number of half waves in meridional and circumfer ential direction, respectively v = Poisson's ratio T, T2 = temperature coefficients defined in Eqs. (IV9) and (IV29) V2,V4 = operator defined in Eq. (1138) Superscripts = prebuckling state = additional qualities during buckling Subscripts cr = critical values min = minimum values max = maximum values I, II = condition in Cases I and II = partial differentiation with respect to the variables following the comma vii Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy BUCKLING AND DEFORMATION OF HEATED CONICAL SHELLS By LuKang Chang June, 1967 Chairman: Dr. S. Y. Lu Major Department: Engineering Science and Mechanics The instability of truncated conical shells due to thermal loadings was studied. The basic governing equations were derived by use of the large deflection theory, and Galerkin's method was em ployed to integrate the equilibrium equation. Two cases were considered in this investigation. In one case the temperature varies along the generator and the conical shell is restricted completely at both ends, while in the other case the temperature changes in two principal directions and the cone is con strained along the perimeter but is free of resultant longitudinal force at boundaries. The edges are simply supported. The relation of the critical temperature to the geometric parameters were evaluated. Three parameters were used, namely: the radiusthickness ratio, the radiusheight ratio, and the cone's vertex angle. The radiusthickness ratio has the most significant effect on the value of critical temperature. A higher buckling tem perature was found by nonlinear analysis when compared to that obtained by linear analysis. viii CHAPTER I INTRODUCTION Thinwalled shells have many applications as principal elements of structures. One of the main advantages is weightsaving. Among the various types of shells, the conical and cylindrical shells are most frequently used in structures such as space vehicles and reactors. The main advantages of their applications are to reduce the drag and they are easily manufactured. Similar to a slender bar, a shell will deform when a load is applied. An important phase in the design of thinwalled shells is a study of their instability; that is, to deter mine the limit of the applied loadings, mechanical and thermal, before the shells become unstable. If an axial load is applied to a column and if it is gradually increased, a condition is reached in which the equilibrium state becomes unstable and a small lateral disturbance will produce a deflection which does not disappear when the lateral force is removed. Von Karman and Tsien [1] noted that some structures, espe cially the shelllike structures, may experience a state of weak stabil ity such that small blows or other disturbances cause them to snap into badly deformed shapes. Since the early twentieth century, the buckling problem of cylindrical shells, based on the assumption of infinitesimal deforma tions has been studied by many authors. Among them, Donnell [2] published his wellknown Donnell's eighthorder linear equation of shells. In the derivation of this equation the assumptions are made that the wall thickness is small compared to the radius of the cylin der, that the distortion to transverse shear is neglected, and that the deformation consists of a large number of waves in the circumfer ential direction. Three equations of equilibrium in the normal, merid ional and circumferential directions are combined, and are represented by one equation through the mathematical technique. Unfortunately, the results predicted by the linear classical theory do not agree with the experimental evidence. In order to explain these discrepancies, von Karman and Tsien [3] made a very important advance in the understanding of the problem of buckling of cylinders subjected to axial compression by assuming a diamondshaped deflection pattern. They stated that the buckling phenomenon of curved shells could be in general only by means of tne large deflection theory. They applied this nonlinear theory and tne concept of "snapthrough" phenomenon to study the postbuckling behavior of the cylindrical shell under axial compression. The work of von Karman and Tsien has been extended and refined by several authors [4,5,6]. In 1950, Donnell and Wan [7] introduced the concept of imper fection in the analysis of von Karman and Tsien. Their work was ex tended by Loo [8], who, based on the finitedeformation theory together with a consideration of initial imperfection, studied the buckling of cylindrical shells due to torsion. A continuation of work done by Loo was later investigated by Nash [9]. Cylindrical shells subjected to hydrostatic loads have been studied by Kempner and CrouzetPascal [10] and Donnell [ll]. The problem of pressurized cylindrical shells under axial com pression or bending has been studied by Lu and Nash [12] on the basis of nonlinear finite deflection theory. The results of these finite deflection studies are in good agreement with the experimental evidence. However, recent papers [13,14] report that when more terms are used in the deflection function the minimum buckling load becomes smaller, and there is no way to predict how many terms should be taken for the approximation. The problem of stability of conical shells subjected to mechan ical loadings has been studied by several authors in recent years. The axisymmetrical buckling of circular cones under axial compression was examined by Seide [15]. The buckling loads corresponding to hydro static pressure were investigated by Hoff and Singer [16] and Seide [17]. Mushitari and Sachenkov [18] suggested a transformation of coor dinates to study the buckling of simply supported cones subjected to axial compression and normal pressure, and the buckling of internally pressurized cones under axial compression was later investigated by Seide [19]. In all of the abovementioned references, the prebuckling stresses are assumed unchanged during buckling, and the additional middle surface forces depend on normal deflection only. The large deflection theory is likely to reveal a "snapthrough" type of behavior for conical shells. Schnell [20] used the energy method to solve the nonlinear buckling problem under axial compression and internal pressure. The results gave a better agreement with the experiment than those obtained from the linear theory. The study of nonlinear axisymmetric buckling of the conical shell was also examined by Newman and Reiss [21] and Famili [22] by the use of finitedifference approximation. The upper and lower buckling loads have been precisely discussed in [22], and Famili's results are seen to be between the results of Niordson [23] and Bijlaard [24], in which Niordson's work is considered as upper bound and Bijlaard's work is considered as lower bound. The buckling problem of cylindrical shells subject to a uniform temperature gradient has been studied by Hoff [25] for simply supported ends and Zuk [26] for clamped boundary conditions. In these works, the cylinder is restrained circumferentially at both ends but is free from constraint in the axial direction. The longitudinal stress is assumed to be zero, and the hoop stress varies in the axial direction and rapidly decreases as the distance from the edge increases. Hoff approx imated the hoop stress by an infinite cosine series and solved the problem by the use of Donnell's equation of thin shell, while Zuk pre sented the hoop stress by a cosine function and solved the governing shell equation by Galerkin's method. This problem was later analyzed by Sunakawa [27] for various conditions of constraint. He used the axisymmetrical condition in the state prior to buckling, and the prob lem of finding the hoop stress is therefore much simplified. Similar to [25], the buckling of thinwalled, cylindrical shells under circumferential temperature gradients was examined by Abir and Nardo [283. The above analysis is performed with the aid of Donnell's equations. The variation of the thermal stress with the circumferen tial coordinate is represented by a Fourier series and the stresses are assumed constant through the thickness and in the axial direction. Hoff, Chao and Madsen [29] have also investigated the buckling of a cylindrical shell heated along an axial strip by means of Donnell's linear equation of cylindrical shells. It is assumed in [29] that the cylindrical shell is very long in the axial direction and thus the axial stress caused by heating has constant values over a substantial length of the heated strip as well as across its entire width; and out side the heated strip the thermal stress is assumed to be zero. The problem was solved for the heated as well as the unheated regions. The radial displacements, the slopes, the bending moments, and the transverse shears are taken to be the same at the boundary for two different regions. In a later report [30], the same thermal buckling problem was solved by using the actual temperature data obtained from experiments. This is in contrast with the exact solution of [29] in which use was made of an idealized representation for the temperature variation in the circumferential direction. The results were found to be in close agreement with those of the experiments,provided the heat strip was not very narrow. The problem of thermal buckling of conical shelil heated along a generator was recently studied by Bendavid and Singer [31]. The solution of this problem is obtained by a Rayleightitz approach in conjunction with displacement functions modified by a shape factor. In this analysis, a hyperbolic axial thermal stress distribution is assumed, the shell is simply supported and is taken long enough so that the edge effect of the thermal stress may be neglected. The radius of the shell is also assumed large compared to the thickness. The hoop stress induced by the thermal load may also be disregarded. The axial stress is taken in correspondence with the assumed temperature distribution, and the circumferential displacement is chosen to be zero at the edges during heating. The purpose of the present investigation is to study the buck ling and postbuckling behavior of conical shells under presumed temper ature distributions. It is well known that finitedeflection analysis on shells of other geometries has been closer to experimental results than classical smalldeflection analysis, and it is also noted that during buckling the normal deflection of the shell is no longer infinites imal. In the present study, therefore, the terms due to large deforma tion are included in the basic equations. Two cases are considered in this study. In one case the tem perature varies along the generator and the conical shell is restricted completely at both ends, while in the other case temperature changes in two principal directions and the cone is constrained along the circum ference but is free of resultant longitudinal force at boundaries. The conical shell to be treated is assumed simply supported and free from other loadings. The shell is considered to be elastic and isotropic. Its material properties are taken to be independent of temperature, and no dynamic effects are considered. The principle of stationary energy is used to find the governing equations of equilibrium subject to thermal loading. Hence, the buckling and minimum temperatures are found from these equations. I The deflection function is first assumed to satisfy the simply supported boundary condition, the stress function is then found from the compatibility equation, which is carried out by the use of the nonlinear deflection theory. The solution of this problem is obtained by applying the Galerkin method to the equilibrium equation. Numerical examples of finding the critical temperatures for different geometrical parameters of the conical shells are given in Chapter IV. The deflectiontemperature relations after buckling are plotted in Figs. 1214. The computation was carried out on the IBM 709 Computer and tne general programs were written in Fortran IV language. The detailed expressions of the symbols used in Chapter IV are explicitly given in Appendices A, B and C. In Appendix C, the expres sions are presented in Fortran language. Since the capacity of the IBM 709 Computer was too small for the program shown in Appendix C, the program was therefore divided into three parts during the numerical calculations. In order to check the magnitude of the buckling temperature, a test was performed by applying heat uniformly over the surface of the conical shell. CHAPTER II BASIC RELATIONS AND FUNDAMENTAL EQUATIONS 1. Geometry of Shells A surface is a locus of points, whose coordinates can be described as a function of two independent parameters xl and x2. In shell theory a special type of curvilinear coordinate system is usually employed. For any point on the surface, the coordinates are of the type X1 = X1(x 1x2), X2 = X2(xl,x2) and X3 = X3(x1,x2), in which (Xl X2, X3) are rectangular coordinates and the parameters xl and x2 are surface coordinates. The position vector of any point on the surface can be ex pressed as: = X, X.e X, (111) The distance ds between points with the surface coordinates (xl, x2) and (l1+dx1, x2 + dx2) is determined by ds doJ = W,) Z9d ,(XXz) C, (WXe) (112) where A, B and C2 are the fundamental magnitudes of the first o o o order. They are zal 0 1o,= 2aX al dXLI acP d3t d X For orthogonal surface coordinates of and a  1 2 vanishes, and the magnitudes are A and C respectively. 0 The unit normal vector of the surface is / C The fundamental magnitudes of second order are defined as: 1 t= :;' M/= ax,.Xz" /PC p / Ae C0 aX. aX, z d X, dZXL a . aXt dXz 91dti d36 9X3 Ox, d AI (113) (114) d j   A.c, N O~x fC 2 a'x. axf SX, aX1 aX, 9 7Z, dj~z aX1 .)X2 .1 Xt aaiz aiit a~cJ a i aX, (115) It is shown in differential geometry that the lines of prin cipal radii of curvature coincide with the coordinate lines if, and only if the coordinates are orthogonal and thus Mo = 0. In this case, the principal curvatures are / These equations affix signs R1 and 2. If the lines of the theorem of Rodrigues is  ^ an / o Let z denote the Let +z denote the Positive z is measured in /Va SCc (116) to the principal radii of curvature principal curvature are coordinate lines, expressed as follows: an9 d 9><2~ c;R 0'^ (117) normal distance from the middle surface. the sense of positive normal n of the middle surface as shown in Fig. 1. To any set of values of (xl,x2,z) there corresponds a point in the shell. The position vector of a point on the middle surface is LO and the position vector of the corresponding z from the middle surface is ?  z from the middle surface is R . we have (118) v Fig. 1. The cross section of a shell segment. Hence by differentiation, .0 ;CII) (119) Substituting Eq. (117) into Eq. (119), the following relations are obtained, di PX1I tl4 .4~c From Fig. 1 point at distance z O)6 dl 3Ix, (1110) z a+i 4 By referring to a surface parallel to the middle surface at a distance z, we have .~  CRXz + + },^ * xe as~ (II11) 2 1c = ,4/ Z) */ r where A=,4 ( .l, ), d= Co'l  The straindisplacement relations are approximated as follows [32]: // / dA 611= AjCi a : / )2 Z 2 ? z ) d ^Xz ar I (acz ,) /CA (IT12) *J/dw\ d~kr )_ L ,,I C C at / C cC? ~ d~ d44 r~ A ax C Co 91 \. xdxl) (1113) where u, v and w are the displacements of a point in the middle surface in xl, x2 and z directions, respectively. W is taken inward normal to the middle surface as positive. Eqs. (1113) are obtained under the assumption that the t verse shears vanish. In small deflection theories all quadratic in the straindisplacement relations are neglected. Letting el, e2 and el2 denote the values of e1, e2 and on the middle surface, respectively, we find [32]: S~o .Co j( ) (i x/ e~ 1. ( ; , C I / C >/ / ,o ( 0,4o _x// Z rans terms e12 (1114) The changes of curvatures H' K2 and K12 on the middle sur face have the following forms, ? W f /__ w 2' ( /v^ ) d (1115) 2. Basic Equations A truncated conical shell as shown in Fig. 2, on page 25 is considered. The distance r is measured along the generator from the vertex to a point on the middle surface, and B is the semi vertex angle. Thus, the surface coordinates are taken as: xI = r, x2 = c. A point on the surface can be represented parametrically as: X, = r.g/c t.oscO X2 = ~~= (1116) S s Substituting Eqs. (1116) into Eqs. (113), (115) and (116), we have ,to = i, c,,= r.,, ,= o IZ= r ,I (II17) Referring to the above equations, the straindisplacement relations for a conical shell on the middle surface are obtained from Eqs. (1114), er= er r) r9 eo 7. " and a comma, followed by subscripts indicates differentiation with respect to subscripted variables. From Eqs. (1115), the changes of curvature are found ^r= T rr r1 "v =f~ W7 (1119) M2 = 9 ) 0,.^^ ^Y The conical shell is assumed to be perfectly elastic and the material properties are independent of the change of temperature; the stressstrain relations are given by the following: =  / / 7 T = l I (er) / T (II20) E where E, 0, v, and T represent Young's modulus, thermal expansion coefficient, Poisson's ratio and the temperature difference, respec tively. The equation of equilibrium can be obtained by applying the principle of stationary energy. The strain energy of the shell is written as: L/= Um + rZ + 7r (1121) where U and Ub are the membrane and bending strain energy, respec tively, while UT represents the part of the strain energy that results from heating. The expressions of the strain energy are given as S= F ff(ze e, *,2A)e eg o r = z ffx ,,Z# X (,vi e,. O..^L^,r, Ulm V09.) D z a 19, Ur, _1/fe, Te,, 0) hT6) /7 c XJrA9I (1122) where h is wall thickness and 6h/z 7=L or~ h/i 41zd~  /3 /Z7 (/Jz) (1124) If there is no external force, and equilibrium exists, the principle of stationary energy requires that the strain energy U satisfies the Euler's equations of the calculus variation. Symbol ically, this can be expressed as tr = o The functional U has the following form U = f/(r, U r U. .7 9Pr. '< ^ he ur er '' rr M ru artio, orr )(1 ))II25) The Euler equation for U1 is Sdr ( ( )A O Srr + )4^ (_ _____ SY6( ( oU'79 Cy V'z^^ (1126) (1123) 18 the Euler equation for v is 4 2^ ^ / )_ ___ ___ 9rr a go drr (1127) rc 9r a ) )=o and the Euler equation for W is or ) d 4 4 t 4 ) = o (1128) od4rOP l wrjf The Euler equations for u and v find the two equilibrium equations in the middle surface, which correspond to the equilibrium state in meridional and circumferential directions, respectively. In order to reduce the number of unknowns, the Airy stress function 7 is defined to satisfy the equations of equilibrium in the u and v directions O\ r F, r r+ (rAid F) FqPco (1129) y = ~rr ^r^t^M~fi ), C^J r 19 By substituting Eqs. (1118) into Eqs. (1120), the stress displacement relations are obtained, 2 /V " (1130) We introduce the following dimensionless notations, After substituting Eqs. (1129) into Eqs. (II30) and eliminating the displacement variables U and 7, the compatibility equation is found as the following: V*4' = (dc j)lrrJ r )re> 'A^ S(Cw;e) ; e) r ~ 7jr [L (Tr + ~ ,r) + J (1132) where Sz 2i v( = ) + L f 1 7 L7 +) aT O " By substituting Eqs. (1118), (1119), (1120), (1129), (II30) and (1131) into Eq. (1122) and using relation (1128), the equation of equilibrium in normal direction is found to be of the following form: 3 i' " y = A A"')Frr ,_,., + ( Q r (~ _2 F I" _/ Q), r x /J'789 (1134) where Mr M, = ,_/,~B D/ =ZLL (1135) (IT33) 3. Transformation of Coordinates For convenience, the following transformation is introduced x =i 7 (1136) By substituting Eqs. (1136) into Eq. (1132), the dimension less compatibility in the newly defined coordinates is C Z = e& C^,)e(u xW.* XWs (.ex  (W< ) f (i (U, a?6tr (1137) where 71= V7 77 2 vz=e (a The dimensionless equation of equilibrium after the transforma tion becomes + (~eQo x6 )(x,x ;Cx x)2 e Fe)(1o(6)] (1139) (1138) *o= ~^o/a, ) +96z 22 It is noted in Eq. (II34) that the thermal moment due to the temperature gradient across the wallthickness is disregarded because the temperature is assumed uniform throughout the thickness. Eqs. (1137) and (II39) are the basic equations for the buckling of conical shells due to the temperature changes. CHAPTER III METHOD OF APPROACH Since the governing equations of the buckling problem have been formulated, our next task is to seek the solutions of the established differential equations together with the boundary conditions. To deter mine the critical temperature at which the conical shell becomes un stable, the deflection is infinitesimal and the linearized relations in Eqs. (1137) and (1139) are used. For the buckling of shells, the two coupled nonlinear equations (1137) and (II39) will be solved simultaneously. It is virtually impossible to obtain an exact solu tion, and only approximate numerical methods will be sought. One of the powerful methods is the method of Galerkin [33,34], which is not only closely related to the variational method but also parallels Ritz's method. The Galerkin method is briefly explained below. A differen tial equation can be expressed in the form L (P) = 0 (1111) if L(P) exists in a twodimensional domain, an approximate solution of P is assumed in a series form, i.e., /0 (X, ) = ." a4' (,y) (1112) '= where P is the approximate n functions which satisfy the are constant coefficients. (IIIl), we find that L(P ) imize M, Galerkin applied a solution of P, the f.'s are appropriate given boundary conditions, and the a 's By substituting Eq. (1112) into Eq. = M, where, in general, M / 0. To min set of orthogonal conditions, II (4w) A x4,)dl~o1XK (1113) i = 1, 2, 3, ... n where 0 represents the twodimensional domain and the coefficients ai can be determined by solving the n algebraic equations in Eq. (1113) simultaneously. I. Boundary Conditions The cone, as shown in Pig. 2, page 25, is considered simply supported and has zero circumferential strain at the edges. The con ditions for normal deflection at the two ends are expressed as: W = 0 S= O and X = X. w;x (i/a)7A. =i O at X = 0 and X = X. (1 114) (111I5) When the edges of the shell are completely restrained in circum ferential direction, the following two conditions are obtained: At x = 0 F.x ('>' ,e r = 0 (116) X3 0 k Q) * R I Fig. 2. Geometry of conical shell. and at x = x o ExX (>& bjf e exo r = o (III7) The end conditions along the direction of the generator will be discussed in the individual cases to be considered. 2. Thermal Buckling When Eqs. (1137) and (1139) are solved simultaneously, the coupled relations are nonlinear. Now we consider only the linear terms in Eqs. (1137) and (1139) as well as in the conditions of constraint. This problem will be approached by two steps: the deflection and the stress function in the prebuckling state are found first; thereafter, the deflection functions and the stress function during buckling are considered. The compatibility equation and equation of equilibrium in the prebuckling state are written, respectively, in the following forms: 2 *= ( tcoft) e 3 5,x kx) o VTf (Ir a) and 2pv)'7 (A^ )e3 ^xx ^ ) L.x The operators V2 and 4 are defined in Eq. (1138), and the super script (') is used for functions in the prebuckling state. The deflec tion function w',which is assumed in a series form, satisfies the simply supported boundary conditions given in Eqs. (III4) and (III5). The solution of the stress function F', which is the sum mation of the general solution of the homogeneous equation, and a particular integral, satisfy Eq. (III8). The arbitrary constants of the general solution of F' are determined in accordance with the con ditions of constraint. By applying the Galerkin method to Eq. (III9), the coefficients of the assumed deflection function w are then found. When Eq. (III8) is substituted into Eq. (II37), and after eliminating the nonlinear terms, the compatibility equation becomes v "= co ) e (t x) (InI10) where Zt" = T = additional displacement during buckling and F = F /r = additional stress function during buckling. When relation (III9) is used in Eq. (1139), and the terms contain ing the products of the additional stresses during buckling found from F" with the derivative of w, are neglected, the equilibrium equation is S4X x ?v c 2x ^ )(A (e f6s ) W x a 6,' V (IIIll) In a similar manner to the determination of prebuckling stress function F', F" is then found corresponding to the assumed deflection function w" and the conditions of constraint. After applying the Galerkin method to Eq. (IIIll), a set of linear algebraic equations is obtained. The critical temperature can be determined by minimizing the results obtained from these algebraic equations with respect to the wave numbers along generator and circumference. 3. Postbuckling Behavior The basic equations used for postbuckling problems are Eqs. (1137) and (1139). After subtracting Eqs. (1118) and (III9) from Eqs. (II37) and (1139), respectively, it is found that e "= e )(,;; ^. ) 0~, ( Ole3F X(. x (zt; 6 ) ( (11112) r y,[ ade , jJ ,,^ D 17 kr = e L x X e X 49 A' s (III13) where (III14) so" = v " ' The above equations have the same expression as defined in the previous section; however, the relation between F" and w' is different from the linear case. The deflection function w", which is assumed in a series form, satisfies the boundary conditions. In a similar manner to the deter mination of the stress function in the linear case, the stress function during buckling F" is found to satisfy the compatibility equation and the conditions of constraint. Since Eq. (III12) is nonlinear, F' should be a nonlinear function of the coefficients of the deflection function w". By substituting the deflection function w and its correspond ing stress function F into Eq. (III13), and integrating it by the Galerkin method, a set of nonlinear algebraic equations is obtained. 30 The temperature can be determined by solving the algebraic equations, and the minimum temperature is determined by minimizing the temper ature T with respect to the number of waves in axial and circumfer ential directions. CHAPTER IV SOLUTION AND NUMERICAL RESULTS 1. Critical Temperature Two different temperature gradients and their corresponding conditions of constraint are considered in the solution of critical temperature. Case I In this case, the conical shell is subjected to a meridional temperature gradient and restricted completely at both ends. Thus, in addition to the boundary conditions expressed in Eqs. (1114) (1117), another boundary condition in meridional direction is iXo Lxo f erc = u.^ dy = 0 (IVl) o o By substituting Eqs. (1120) and (1129) into Eq. (IV1), the dimen sionless condition of constraint expressed in terms of stress function F is obtained as: +Y.A x 4 ie 771Ir] = (IV2) Since the temperature field considered in this case varies only in the meridional direction, the prebuckling deflection and stress function are therefore taken to be axisymmetric, i.e., they are independent of e The functions with subscript I are correspond ent to Case I considered here. The deflection function in the prebuckling state is assumed to satisfy the simply supported boundary condition, and in accordance with the assumption of Mushitari (181, the prebuckling deflection is chosen in the form: X TX (IV3) in which A  (IV4) In this study, i is taken as an odd integer, M = 5 and v = 0.3. The coefficients a' (i = 1, 3, 5) are in terms of the temperature gradient and will be determined later. According to Eq. (III8), the prebuckling stress function can be written in the following form: j; = l e j j A (IV5) In the above equation, FI and F are the particular solutions of 1 2 F They are due to deflection wl and the temperature gradient T, I respectively, i.e., Ve ,=(,. 0 ., J )4 (IV6) and V = 17 7 (IV7) Substituting Eq. (IV3) into Eq. (IV6) and integrating it, we find e,,',l) 2 (H7X 7. \'rX a i 7 E cos  ) (IV8) where c. and d. (i = 1, 3, 5) are found as functions of x and are given explicitly in Appendix A. The temperature gradient is chosen as an exponential function of x, 7 = T; + e (IV9) where T1, 1 and g are constants. If g vanishes, Eq. (IV9) implies that the distribution of temperature is uniform over the surface of the conical shell. Integrating Eq. (IV7), the function F is obtained as: F, =LC [ t,) (Z^ ]^3e (dvo>) Substituting Eqs. (IV8), (IV10) and (IV5) into Eqs. (III6), (III7) and (IV2), we obtain three algebraic equations. By solving these three equations, and the arbitrary constant A' 0 in Eq. (IV5) is considered zero, the constants A' (i = 1, 2, 3) are found as: = a. ( c, .dT;, a5 / a u i = 1, 2, 3 (IV11) By substituting Eqs. (IV3) and (IV5) into Eq. (III9), it is found that D V 4;e a. ^( ^xx4 ) = 4 (IV12) Applying Galerkin's integration to Eq. (IV12), and if only three terms of the prebuckling deflection are taken in Eq. (IV3), the following three algebraic equations are obtained: tnr^."t Xo f" r)^ An ]d7A(  0 o 0 i = 1, 3, 5 (IV13) Solving the above three equations simultaneously, the coefficients a. (i = 1, 3, 5) are obtained, which vary linearly with OT1 and S1. They can be expressed as: a. ai (xo. o (f ,,7 ,4,, ,2 As, 7" / /? i = 1, 3, 5 (IV14) The six unknowns Ai (i = 1, 2, 3) and a. (i = 1, 3, 5) can be deter mined by solving Eqs. (IVll) and (IV14). They are found to be r functions of xo, TI, r, , and 0. Therefore, F is determined in accordance with the temperature distribution T. The expression A' and i a' are given in Appendix A. An additional deflection function during buckling which satis fies the simply supported boundary condition is assumed as: ,I" A 3X #,JF Y (IV15) where p and 1 are the numbers of half waves in meridional and cir cumferential directions, respectively. By substituting Eq. (IV15) into Eq. (III10) and letting 7 (IV16) S2_4('3 we obtain lP = ( .1A R )[e A X ( hk (IV17) The additional stress function during buckling is the summation of two solutions: one is the homogeneous portion of Eq. (III10), and the other is a particular solution which satisfies Eq. (IV17). This stress function can be expressed as: iX ( x . 1=(A'.e ,, / ?, e (IV18) where ci and d" (i = 1, 3) are found as functions of 71, p and x The constants A, A and A" are determined in terms of ai o 12 3 (i = 1, 3) from the following conditions of constraint:: =,X, )C/% ) ,,X A/Se = 0 at x = 0 and x = x (IV19) o and o I) x E / (i.) 1 Since it has been noted that the homogeneous solution of the stress function F has been already considered in the prebuckling state, we can say that the homogeneous solution of function F" can 2 be neglected without significant error [18]. When substituting Eqs. (IV3), (IV5), (IV15) and (IV8) into Eq. (IIIll), it is found that S" "J0 * e C r ), ' +e =L; . (IV21) By applying the Galerkin method to the above equation, a set of alge braic equations is then established: e c . ) Q csnZdXe 6 = o o o 0 0 i = 1, 3 (IV22) After integration, Eq. (IV22) can be briefly expressed in the following forms: C i' (Ca3 7'ajcO77 Se 0 (IV23) (Cj1+oi jjl ~'7; f,/2)a(C$(Cs3 C d37 WJ De T1 q) aoel (IV24) Eqs. (IV23) and (IV24) are two linear homogeneous, algebraic equations and have a nontrivial solution only if the deter minant of the coefficients of a' and a# vanisnes. This requirement can be expressed as EC,](3 V,Idlg 4 oeT,7[e o (W31 ofjI ro/7&r =0 where '7 Z (IV25) (IV26) Eq. (IV25) is a secondorder algebraic equation of T1 and yl. For given values of yl and g, two solutions of TI are determined from Eq. (IV25), and T1 is found as a function of the number of half waves p and 1. After minimizing the solution of T, we find two values of the critical temperature. One of them is dis regarded because it is physically impossible. It has been observed that the buckling hoop stress is local ized near the fixed edges; in other words, the hoop stress is high near the supports and it is low in the middle of the shell unless the shell is very short and thick [25]. This means that high average thermal hoop stress exists only when the shell is very stable, and the shell that buckles easily does not develop the hoop stress. For the buckling problem, it is found that the thermal hoop stress con verges rapidly when a series in Eqs. (IV3) and (IV15) are used. It is also noted that the meridional compression has a much higher effect on the buckling temperature than the hoop stress does. There fore, the value of the buckling temperature will not change signif icantly if more terms in Eqs. (IV3) and (IV15) are taken. Because the buckling is mainly caused by axial thermal compression, the con ical shell is expected to buckle in multiple wave patterns with nearly the same wave length in both principal directions. The numerical cal culations are thus made by taking W/n = (H tan 0)/nR in Eq. (IV25), where H and R represent the height and mean radius of the conical shell as shown in Fig. 2, page 25. The results of the problem were obtained from the IBM 709 Com puter, and are illustrated in Figs. 3 6. The relations presented in Figs. 3 5 show the effects of a conical shell's geometrical variables 4 3 2 *oP=10 = 300 0 1 11 0 200 400 600 800 1000 R/h Fig. 3. Critical temperature versus radiusthickness ratio at H/R = 2. H/R = I I I 600 Fig. 4. Critical temperature versus ratio at = 10. radiusthickness V 2 0O 0 I I 200 I I 400 I I , I 800 R/h 1000 "I I I 4 p I I I o ICf 200 300 Fig. 5. Critical temperature versus semivertex angle at H/R = 2. at H/R = 2. 1.6 1.2 0 0.8 0.4 0 ( T = O.1 T, T. = # , T =T, + e9' 0.2 0.4 0.6 Fig. 6. Variation of critical temperature temperature index (B = 100, R/h = with meridional 300, H/R = 2). 0.8 1.0 D on the critical value of uniform temperature rise (i.e., T1 = 0). In Fig. 6, the variation of the critical temperature with the tem perature index g of Eq. (IV9) is depicted at a different 71 /T ratio. Case II In this case the conical shell is subjected to meridional and circumferential gradients and restricted circumferentially at both ends. Thus, in addition to the boundary condition expressed from Eqs. (III4) to (III 7 ), other boundary conditions are f Oac/ 9 =o at X =0 and X = o (IV27) 0 or in another expression, j ( x F, )W0O = 0 at X=0 and X=X (IV28) o The above conditions are applied to a shell which is unstrained in compression but is restrained in bending. If the temperature gradient has the form (j = positive integer, 0 < 9 < 2n) the conical shell is hotter at one side than the other. As the index j increases, the heated portion becomes narrower and the buckling behavior is closer to the case under compression [35]. In the present study the temperature distribution is taken as: ;I Tn T tz e= T (IV29) with k = 1/2 sin B. The subscript II is used to indicate the functions associated with Case II. Both T2 and 72 are taken as constants. The normal deflection in the prebuckling state satisfies the simply supported boundary condition and is assumed in the form: = .o (IV30) i'/ In this analysis, it is assumed that the prebuckling deflection w' is axisymmetric and with i = 1, 3, 5, takes the form, W3 ., ; ) (IV31) In a similar manner to the determination of stress function F in Case I, the stress function FI prior to buckling can be found in the following form: *== 3 p < +S (IV32) where F is the homogeneous solution of F and is found as: 5 () A 2 49 2) V 6 j x ? 8';ee 2 x (IV33) /~~.'=8,(xO4 )i8X40 8 e B (i = 1, 2, 3, 4) can be determined by the conditions of constraints given in Eqs. (116), (III7) and (IV28). They are found to be func r tions of xo, T2, 72 T and b' with i = 1, 3, 5. The functions F3 and F1 in Eq. (IV32) are the particular solutions of the stress func 3 4 tion FI which correspond to the deflection function w' and the tem perature gradient T, respectively. They can be expressed in the follow ing forms: + A cos ) (IV34) and '= K/' rrz) (IV35) The b 's (i = 1, 3, 5) are found by substituting Eqs. (IV32), (IV33), (IV34) and (IV35) into Eq. (III9) and integrating it by the Galerkin method. The coefficients B' (j = 1, 2, 3, 4), gi, hi r and b' (i = 1, 3, 5) are functions of x T2 and which and b and i, which are expressed explicitly in Appendix B. It is noted in the above analysis, that the prebuckling hoop stress is independent of the temperature gradient provided the temper ature function is chosen as in Eq. (IV29), since (IV36) /F<, X 4 "X = 0 During buckling, the additional deflection is assumed in the form: AP JA/Trx (IV37) It has been mentioned in the previous case and reference [18] that the homogeneous solution of the stress function during buckling can be neglected without significant error. Therefore, only the par ticular solution of Eq. (III10) is considered as the additional stress function during buckling. By substituting Eq. (IV37) intb Eq. (III10) and assuming that the stress function has the form , (A.) X AP ,, r k:'^s ^ ) cOn' s e (IV38) and g" and h" are found by tne use of the Galerkin method. 1 1 Substituting Eq. (IV38) into Eq. (III10), we have Sr(/.A)X > r'0f xr C05 JP J^ S// 'k X'< A IX x rrX 7 0 x M cos$Bes ikJ= 0 (IV39) In order to determine the coefficients g' and h (i = 1, 3), the following approach is employed; Eq. (IV39) is integrated by the. use of the Galerkin method in the circumferential direction, which provides the following relations: 4 $Q e Cos nwo( = O (IV40) o After integration, the coordinate parameter 8 vanishes in the above equation, and it becomes a homogeneous equation in terms of x only. The coefficients g" and h' (i = 1, 3) are obtained from Eq. (IV40) by comparing the corresponding terms of x. They are found to r1 be functions of L, 1, xo, k, cot 0, and are given in Appendix B. Substituting Eqs. (IV37) and (IV38) into Eq. (III11), two algebraic equations are obtained by applying the Galerkin method to Eq. (III11). For this case, w = wi, F = F F = F and I II w = w + wi in Eq. (III11). If the following notation is intro duced 2 = Trz/T (IV41) the two algebraic equations can be briefly expressed as: (,, c ~Zo / )0 /, (Z3 z )e ,,3 = 0 (IV42) SA 0/ Z', )461 A ('33 433 T'^. ? 0 (IV43) For a nontrivial solution, the determinant of Eqs. (IV42) and (IV43) is zero. After solving the determinant, the term 72 of the temperature gradient can be written as: O(TZL = O t i(l. Za ^, (IV44) In a similar manner to the determination of critical temperature in Case I, the critical temperature in this case is found by minimizing the wave numbers i and 1 of Eq. (IV44). The crit ical temperature at different magnitudes of y2 is shown in Fig. 7. The details of expression used in this case are given in Appendix B. 2. Minimum Temperature After buckling the normal deflection becomes finite; thus the seconddegree terms of the derivatives of normal deflection should be included in the geometric (straindisplacement) relations. These rela tionshhips are expressed in Eqs. (1118), and the nonlinear equations (1137) and (1139) will be used for the solution. Since the thermal stresses in the shell depend on the boundary restraint, unlike the 12 s^R / h =150 O x 6 8 3R/h=300 R/ h =450 4 R /h = 900 I I" I !lI 0 0.2 0.4 0.6 0.8 1.0 T, =;T2/2 Fig. 7. Critical temperature due to circumferentially non uniform heating (B = 100, H/ = 2). case under external loading, the average membrane stresses will be dif ferent before and after deformation. The nonlinear effect on the value of the temperature gradient to maintain equilibrium after buckling, is here investigated. In the present nonlinear analysis, the temperature change and boundary conditions are the same as considered in Case I in the linear analysis of the buckling problem, in which the conical shell is subjected to meridional temperature gradient and restricted completely at both ends. In order to compare the value of buckling tem perature with the minimum temperature in equilibrium state after buck ling, the deflection functions used in the nonlinear analysis are basically the same as those assumed in the linear case, but only two terms are used in the prebuckling state as an approximation. Another boundary condition in addition to Eqs. (III6) and (III7) is, J e/X = o (IV45) 0 0 The above condition implies that the average length of the cone is unchanged during heating. Eq. (IV45) can be expressed in terms of the stress function F and the normal deflection w, /ffSlft if fe //16,^ vS x 7 J^.^ree o o X Y C/( =e (IV46) It can be observed from the above condition that for the same temper ature rise, the average stress is less than that considered in the linear case, because the additional nonlinear term in Eq. (IV47) is always negative. The normal deformation prior to buckling is chosen as: AP a, 4, at 37r iv^'1 In a similar manner to the determination of the prebuckling stress function in the linear analysis, the stress function prior to buckling is j, A A, zx Aft 2eX yC x^/^ At If 4^^< ^ ^ (IV48) The boundary condition and the temperature distribution in this non linear analysis are the same as in Case I; therefore, by setting the terms to correspond to the third term of the deflection function, w equals zero, the following relations are obtained: A, A A, r = A g /6 = l ,A = (IV49) AZ = 3t Z ,, =A The additional deformation during buckling is assumed the same as in Eq. (15), i.e., r = ae 4x / ,'_ r X ) co. 34 (IV50) The additional stress function during buckling has the form: A= = A x ; A ,, F p q, 4?/ x 4 et e e F (IV51) The particular solution of F" has been found as Xa5 / ,,, + K d IA r,,/ )x X ^'i ^<6'sAx) XK A1()X 2 AX) ) (K 2 Co (~r.Z ct'I )K, (Az 6r O tX#Xsl S 4gr sixV, x zs Ar e t3 r ;I cKs 6 r A >X ^ 7 fts MX) X #Kr ,eJz zm3X 9 .4 .4 Ii2f;x] f aI6e2AXJk'o x ,, a, a., xK/ coS a.t,x /.izj. ) Kao Kh a e4 v r.s, /r4 4 ,,, C,, e"K , i 'u48 CoS 2M9fJ CoSo C1 '.KM9 MC2/9 Y f3o5/)o ') (IV52) in which ' M, = i r ;C a~ i is an odd integer By substituting Eqs. (IV51) and (IV52) into Eq. (III12) and com paring the corresponding coefficients, Kj (j = 1, 2, ... 49) are A? 7r =' 7C< (IV53) obtained as functions of x , Fortran language in Appendix C. By applying the relation (IV46), the boundary conditions U, . 8 and 1 They are given in h 1* (IV14) to Eqs. (III6), (III7) and during buckling are CF.xx ( a ",, ) = 0 at x = 0 and x = x By substituting Eqs. (IV50), (IV51) and (IV52) into Eqs. (IV54) and (IV55), we obtained three algebraic equations: AI, 'A and A3 which are then obtained by solving these three equations in the following forms: A A" A, 2 A, = A + A. Q a 2 ) ,4 = a, / %,o a, a, 1 ,, ,,' (IV56) Substituting the expressions in Eq. (IV47) (IV52) into the equation of equilibrium (III13) and assuming the error is 9q, and * I 2 fln V,9,F~ ~FJ~yijd (IV54) = 0 (Iv55) two equations are obtained by applying the Galerkin method: f4f cos _4:ot O 0 The above two equations can be expressed in the following forms after integration: tyh As 4 a, a3 4 Cas R/=,, O A "2 A . i a, + a,", (IV57) (IV58) (IV59) T,)S.z ( rS6 A",3s7 r/, ( S6 0,dL W1 At "2, A A 3. 3  O., a,; 58 4 as3 S d3 'o = 0 (IV60) In an attempt to solve Eqs. (IV59) and (IV60) simultan eously, it is necessary to define A P; s a, o= a3 (IV61) 6L, a z w7 )Sj rS aj CI (7r ) 54 By substituting Eq. (IV61) into Eqs. (IV59) and (IV60), and solving for the constant TI, we find S= V/ V, (IV62) r where vl, v2 and v3 are functions of u, 0, 8, v1 and C. If the geometry of the conical shell E, B and the temperature ratio yl is given, the value of OT1 can be plotted versus C for fixed values of p and 1. Examples are given in Figs. 8 and 9. The critical temper atures are then found from the minimum value of these curves. The temperatures should be equivalent to the values found from the relation bT~.1/ = 0. The numerical results for this case are illustrated in Figs. 10 and 11. 3. Deformation The relations between the temperature rise and the deformation after buckling can be obtained by solving Eqs. (IV59) and (IV60) simultaneously, in which al and a3 are expressed in terms of T and y1' Numerical examples are given for symmetrical (n = 0) as well as unsymmetrical cases. If only the first term of the deflection function w" is taken, i.e., a3 is equal to zero in Eq. (IV50), then Eq. (IV59) can be written as: a, W101TIP;( &r, ) RJ 4R9,] =0 (IV63) Fzs ( a,/ ') x 0 Fig. 8. Temperature variation T1 as a function of deflection coefficient ratio (a/a1) at 8 = 100, R/h = 450 and H/R = 2. (/=19, tl=29)  (u= I 7.= 26 ) (1=13, t?=20) 1.5 2 2.5 3 (a;/a;') x10 Fig. 9. Temperature variation T1 as a coefficient ratio (a"/a1) at H/R = 2. function of deflection 8 = 100, R/h = 900 and b 4 1 3 S\20 (I I I 101 1 0 200 400 600 800 1000 R/h Fig. 10. Minimum temperature versus radiusthickness ratio at H/R = 2. 1.6 1.2 X i 0.8 0.4 0.4 ST = O. IT. T =T. +T, e9X I I 0.2 I I 0.4 I I 0.6 I I , I Q8 1.0 Fig. 11. Variation of minimum temperature with meridional temperature index (6 = 100, R/h = 300, H/R = 2). ,,,, lI I II i iII I The above equation can be solved for the magnitude of the deformation a in terms of T1 and yl. From uniform temperature distribution, is equal to zero. Figs. 12 and 13 give the value of Ti versus (w")max for different combinations of wave numbers, while AT = T (T)r versus (w") curves are plotted in Fig. 14. max Since tne numerical calculation is very cumbersome, the high speed electronic computer is therefore employed. The solution pro grams were written in Fortran IV language, and the numerical work was carried out on tne IBM 709 Computer at the University of Florida Computing Center. 63 8 8 N 0 0 2 4 V4 2 0 0 I 2 3 4 5 T. ,x 103 Fig. 12. Deflection versus temperature (B = 100, R/h = 450, H/R = 2). 0.7 75//o C 0.5 0.2 5 A I 1.5 2.0 2.5 3.0 yT., x 103 Fig. 13. Deflection versus temperature for axisymmetric case (0 = 10, R/h = 450, H/R = 2). rj 6 " 0 / 0 L// 2  I I I I I 0 0 I 2 3 4 5 o[T,(T.)cy] x 10 Fig. 14. Deflection versus temperature (8 = 100 HA = 2). CHAPTER V CONCLUSIONS Numerical examples have been given in the previous chapter for different cases. It has been found in Figs. 3, 4, 5 and 10 that the radiusthickness ratio has the most effect on critical temperature, while the change of ratio H/R and the semivertex angle of the cone B vary the temperature only slightly. It can be observed from Figs. 3 and 7 that for the same geometrical parameters, the critical temper ature in Case II is almost five times higher than in Case I. This implies that when the shell is longitudinally restrained, the axial stress plays a more important role than the hoop stress during buck ling. Since the thermal stresses depend on the condition of constraint and temperature distribution, the thermal buckling problems have to be treated individually for each case. The two cases (Cases I and II) con sidered in this study are those under which the shell is the most likely to buckle. Figs. 12 to 14 give the deformationtemperature relations. These figures indicate that the deformation is proportional to the temperature, the ratio R/h, and is inversely proportional to the numbers of waves in two principal directions. It can be seen from the condition of constraint Eq..(IV44), that for the same temperature, the thermal stresses decrease with the increase of normal deformation. However, in the axisymmetric case, the deformation after buckling is very sensitive to the increase of temperature, and it is also noted in Fig. 13 that point A should be the critical temperature for this case. The buckling of shells subjected to external loadings has been studied by many authors in the past. They found that, in general, the buckling loads obtained by the use of the classical theory were larger than those found by means of the finitedeflection theory. It is interesting to note that in the present investigation a higher buck ling temperature was found by nonlinear analysis than that obtained by linear analysis. This phenomenon can be explained by the condition of constraint Eq. (IV1). When the large deflection theory is considered, 2 the nonlinear term (w ) appears in the condition of constraint as expressed in Eq. (IV44), since (w ,)2 is always positive; therefore, it can be seen from Eq. (IV44) that during buckling a certain amount of thermal stress is released by the consideration of this term. Conse quently, in the use of the large deflection theory, a higher critical temperature is obtained in the buckling of shells. Thus, for the post buckling case, the shell will remain stable at critical temperature and becomes unstable when the temperature reaches minimum temperature. A comparison of critical temperature with minimum temperature at 8 = 10 and H/R = 2 is given in Fig. 15. In order to compare the experimental with the theoretical results, a test is performed in this study. The truncated cones under the test were fabricated from a flat brass sheet with a thickness of 0.005 inch, and the coefficient of thermal expansion is 10.4 x 105/F. d(T,)cr o'(T. )mi I I I I 200 I I 400 I I 600 I I 80o 1000 R/h Fig. 15. Comparison of critical temperature with minimum temperature (0 = 100, H/R = 2). 4 3 70 69 R The dimension of the conical shells used in the test are: = 500, = 2 and 8 = 15 as shown in Fig. 2. The brass conical shells .R were mounted so that they were prevented from lengthening by two rigid plates, which were held in place by four 1/2inch screw rods. At the two ends of the conical shell, two rings were made and mounted on the plates. The cone was then fixed on the rings by screws. Heat was provided by infrared lamps, which were placed inside the shell and were designed for uniform temperature distribution over the surface of the shell. Thermal papers and thermal couples were both used to measure the temperature. Three cones were made and tested; the buckling temperatures were found to be 1200F, 130F and 135F, which are higher than the theoretical value (98 F). This is primarily because the buckling was determined visually. It is therefore reasonable to believe that the buckling occurs before it can be observed; in other words, the actual buckling temper ature should be lower than that found experimentally. As temperature increases after buckling, the deformation increases, too. Fig. 16(A) shows the conical shell beginning to buckle, while Fig. 16(B) shows the buckling pattern of the cone as the temperature continues to increase after buckling. (A) (B) Fig. 16. Photograph showing buckling of a heated conical shell APPENDIX A EQUATIONS FOR CASE I / '. 1  i = yt,., . I A , tZ I 2 I j a 7"z).. ly A (m ? A 2) . =  14 2 ( /3~ ) A )  (A2A> )M ](yf 7. A'7z""+ ,Z 2 'A1t /6 4 r A ; ) ,4= i3 ) (2 )Z A , ( z Z Ut O' J f ", .)( 2+,9 ) Ai = ( 2zJ 3y ,(', 2 , A3 = 2 \ /2A / , $5 = A /A' + 2( 4/A Ab = (,+ )A ;L, + 2 (. x)n, c. m.^^/ ,1 = 4A3 / 2A, 8A ,=Z ( 2 g = (//A ) C, 2 (, *c A) 7,o, m, c, 419 = (, + ) C, m,/ o , Cl 1,4y ('1, ,9) , m IL 47 = (/ 4 A) / + C,/" to = (/+.A) Z / A,, = (I ,/) '3 + m, Cs Az = ( e)Z ; 4,3 = (/, ) C3 AM = (' A) od1  m' o, ) / '  2 (/ A ) M ', C5 Ms d, SCI mj C4 4,, 2A o ^4/9 = e  /I 0q2o = e (3A / )X. ,z4 = (m'f4  $j ^ .s4 ) C A > q9, {(ry r / rt 2 f i ,2 41 y/  M ,) / (m;7 (P W^M.3 / (n3 Ar )[ (f ) l ) S2 A)3A C3 1/d 2CA) X /B = e / / / 6 = C(/,A )C ,,/ > M oI/ f "C 4/r = ( /, A ) e; "cP Sz ,23 = D~' 49 (C, ,3 )L / 2, Ch,' /, ns)J ' ['/A C m, M 3 ) tf 0 = o.5 D ~',9 (4,o A )/fm,' 'm ) VA'; f<,' / mf  (m. '',)[t'/ (s' m )Jj *4zs = 4,i ( m ,s4 3. ( J ( )[ ) ,4 2z = 4 / (A A) ((AJ ,4s = A/.s 's ( ,o ,4 ,, (' )f t ' (1/ 411) (AZ [ r4 az)f t ( )  A (o f /7 w' f ')m =(< 9' 45)(" "4 74 I A3, = A 'A,4/ (46 4,7){L?4(Az 4&;A.W.) ] [ V  743Z = 0.5 1~/ A9s (f,4 /9s ) (n,4 ws )jv)/'\ rtw's i )J f r~/ m)[4 (/ ,/ }, I  "o ,5 15" (,IA /,1 7 gl33 = 8 /; (,4, y ,, ) ,?," (\ / ) )J 435 = 0 /5, 2' 5 s) ul .43. = o.s/.'A,g (4A, 4,) (., ,s) (P,,CX . ) (,,/ ) w = a4 ( / /; )  z  i / ; ra 7 ~? ,s M= A A's (RA. ~,3)/. f,, I . 4o =" o.s ;'A,j (,,,,,,)J  / A; +[rc,' nj); I de/ = 4,A ("i A ~ 2 A5 ) ',J i ~)C " /' 1 i 5  /^2 = ;Af 8 (/ A )mS f AFnrWs / L 7 ) ; /* f Jm A43 = D/i4, (4,7 A/q,&)) fl i ) (YA) W MeI) Av4e = .D'D/s (Cs q7') (/AZ ,; /) S; = 9fD[T//D ^A /w, J2 //,< ,'= mr'/D/' [D 2 i cIre A16 = APC s[( //} //J}., ] e 7 (/o'A "t. X ,49g= (~ ( p 12 ~ ) 2 le7 = /M D[(// 4, /)J .. < / = ( ) (A 4 m ) (7 4; 9<( )A48] Ag,.4 C2 = C,4(' )(/,, e ) I X C5 = C, ,' eA ) C = e C (/ , /o = C= e / Xe . = [(/J3 ) /(e *O ' (e x )], / I ) 74 0 e 151;Z $~ =[b '6 /C) I ds 2C6( ) , C,, = 6C e C,, = (c, aCx.) e C,3 = C 6 e A C, = [,C (, , /e ,C,3 [(,),/T,3 /AJZ] A L[,Ao C, = [ Ar, 4, >] C ,1 r = ('e 3' A/ 0) ( A' A ~(f^7y C, C,e = (C6 Eo (C/,) 4 5] . Cg = C6 (3') '/, A )7o CL, I r [Xo r x= f, 2. d,,= {, + rE^ < /> .l[ ,e y J( )' j ^ C/s = ,r //r,,/ )l/l r (/ /,/ ]7 6 c1 ( /, .7)^ ]j 5 1, I D,= (Cd, C,s)CzCr), 3 = (d CXCz  hV= ($,,4d9)(d2C7) D5 = (d3 dC/,)(CCo) , Ds = Dz D, 0r ., , D = (C,C)(d2 C,) , Dy = (dC, C,6 ) (dz Co) D4 = (d, C:,2)( CO ) , = ( 0o D C4) .ZDo = Dz , .D= Ds 0Do2D, D,6 (C,;7 /)Cz  ,g= ),gV = (D,7 D,6)(Vz l's ) Dg/ = D ,3,8 /9/6 Do= S ,(C7, D,9 03/8 ), / = D.8 D Dzo BZf D .e Dzo D. = (Ceo e9X C'zr T)22Zj'/RZ>,fDazO .V23 = (dC ~ /920' Dt~ =(C~p C/9 ) CZ ~d/I S(D)(D /D .V)a' = (Z2243?23)(Vz l'5) Ds= D8V 1), 2,7 = (C/7 C,)C A, D8 = (D6D3)(D Ds) Dr:= (0,D0 )(DOZ Ds) DZ = D, ( Ds + D Dr C, ) 1,3 = v.,o D,/ , A?6 = D Z D DR3 , Vae= Z) D,.0P2 7 D?9 = V2 VS ?g7, Ape, 74 +A ,g~/r~~prILrl _1Z iE3 = R/9 v'K4 9As 98)f(4'A~)'A (~A~'t:C /rj ,~= VA/ 6A17)('i 2,/A) zJ / 2 '2 7/ El= 34n;/e(A#/)e ~ /JCt/ Im,] A7 7 Z5 ,ey /(, f v ( 3 3 3/ el,o.'A.)x 14l. ,A) ,'X, 7 ,F8 E 011' 1 Z D9 7 3 4F Z12 4 F7 12 , .7? 0= V7 V DO ,2 DZ 7 = DI / ( C'O '/ z 1 AC( D15 r ~2 ~~ I2 SnLI'L]f m"Aj Y,4/ 1sEfE =, 0= '67 P// / F3  h:;z hD 13 iL ?e6 D/s 4 3h76 D11 , = 4 A) J / 3)j f/m 4 /VS /7) I ,w "m ) )J K e;/ 7/ ) / ( / ) /) (rn, t t3) j ; tj /7 Z /A /) ( E,3 = AD 7 /D Z r/ Ee/ = . 5//S I'D /(! R 7)t J t), l) /A ) 27 i D r 224 27 ,","\ /)"Wj y ,/ ,9: r 7 , = 'a D fe/ A )X S7 = i2 /C ) / ') fe Ax 7. 7D,, ,,l ::A. ) (/A',.' A)  J7 c , 3 .s  A) i4. eW3 \)A (/%.AIxj{,,'1) 257J 8E = / ',9J2 F,9 = A23 9f' 2'/ E 2 F ?s #Z2 s L /2' Pe) F,0 = F,,/ rF~z, LEzz = 423 /4Zl, fz/= Ez F4 ,q2 +s D2/4 fg6 Dsz2 fP/2 'E,'3 E7) fZ3 = 14 7 A//28 +yD2/ f, zD /o ^ ^d y VT Ft6 = F2s3 (/8g $9: col g) EZ s = Ezo F, .? F2;; ei,6 (f, , ,,, _ioow,40 7i' h'i 1d 2, E30 = f28 ( ?A7weZ) j ca 3/= (,2 7 F3o) C2:9 F ) ~t~j A = ,5Fj3/$ FLz 7 A/8 [9,)ra Ai n d 42 (,, ,f )J = rS ,, rA ,2< /j' = 0. 5 '(44 R) i/) ) ,4t s ', ) j s = A I r,4 A/s ) ,4z9 f/XA 5 ,^,M'.j [,i { ^,, J/ E, 7= D'C "' // e ,$ ' ..+8 = .t .v < / / < + , / , E'3 = 7 ~L C /W D '> [. :, ') " x ( ,1 '7, rxJ , ,, 1, +z ^ c ) r s J Evo = 39 CAjo E, = Ps,/At /^ +.. .Z D L9 +Da7 /'6 +7) v e= Aj33 ~ AJ', 43= Ajs 4 / E8 "2,4 , fD27 34 7) 4E o= ,y AE .J, E, '3 E, 7~3 #c 8g,/.s / g) 4 " = 3A 7 3 AJ r97 = 15 P C 7 C/ '2 Ddo S '2?d) 4;8 = d'/ f4' A, E49 = A7# jyW ^J7 8 4fisr >yZ7f /E 9) 0.= k=37./ 8 9' "A Z06 7s9? A fSJ) gD/Z Ay'95,7 'A 3 j, = 4" :;j Ec 449 s ) 9= c4^ f ,I I t r/= EL'y .3 = (C,/ E; ) (M=9 Erg) Esj= ( 47 J o ) (fe 9 Ej) ,~ SEg6= E, ( / E,9 cA ) Es8 = (EaE4 )(Eg s ) S .0 65 S 6Ed = E6 4 S/ f 6,/ 7'o z7 ,= , c= ( 77T) RJo a3 A= sFr,( ), , 7,= 6o ( aS =Ego 0("ZF A c (* 2 i4 2 1/f P, 2 4697J 1A = ( T 9 2 / j/CZ (=d6/ ,?o p 0(gD2 r 4o 2 7I ./r= ,,. // 2 2 ' 2 't, ' Ci = < 4 P A V A ) (;f A A yfJ E f n oA ,J ) W5,/ = V," y,,f a4A /7 ^ f' y/ // ^ y1^ 4S2 =A m, v A r,4,A 412t* A I i ' Z) S= A\,,3 Wj A /,r 3j ,'3 gmj *m4Jr,' 4',v/ A 4=f (A> CC,'~, ,', As6 = (^c ,'e d/> 2t < A ,)/ C, d ,4 , Ajr = (/r A c,' w,' d/ As8 = (, a1 ) c,/,' )M'j/3'r 6o =, (i A '(? (/'2A ),/CJ 3', /= / 7' 'e ) jC; a = > (A^)C2 (,/'2 )4'4 c; C = (,A)Cj /m; dA5 A6 = A(< \ )ad 4i y (/,2A ,s)ayz5,, /s = on )<^' ,^c, AT 8= (a t A) C, 0 2(rA)MA C,o Mt t, C ,4, /a (/ W),;A, 7,a = #A) eC z 2( A),3 y orC, 493 = (0'A) Cs MjW 3 / JA / J X o Sr = / & ,.s e S/ 2 2 l Q,= o.25A?4 {7;t/1t)Ai) (2m,/,','t, )(A/) (2mrJ 2 = a. 254 t ) f Z1 A "] ~J A ] 7' ( / 2 ,) "" "31+ (2rt, (', ) J J Q4= c.25R7f(2'Y t)A ,)J +m/ [) ,(d,7)/ J QS = O.Z5/t (A,) i/, (/ f lA O~~~~~~~ f c r '~yA2 /l, Q6 = M~ ,'/8 ,A / 9 (,/ , 39 = o. s 'm, /,9 ('A 2+ /) 2AX. 2 2AXo 2 Q9= 0./25(X ,A e o.,/z5A A mX (/A 'mt, ) 0. 2q Q 0.2= .Z [ orne ,"7x 2e ,a) ~ m wAg t = o.25/,8 A) (A , ')b /)' m," J Q,2 = o.z5 <,"'c , + m,  Qf = /r/ s),d)f((' mJ )1 1/ ) ( mi) / Q/; = o.5, ,)[ ) '(/ a ,,) , ,,,3)[va() 4 1(nm m z,)  SA / "" r "" > 2j7 Mjn 3) (ZM/ nz/ )L (3/) + (Zm, ,n ) P, ,/ a A Q,(, = . ,,/' ;: ; .5,,8> { *, ,', 5/: S 2= ,, ,'?, L _*"  Q2/ = A4s9 I[A 4 e", ] ~ (' s 7JtA a '/7 Qz3 = o.zS7 3 /) A/// (24 '&~// /) /(,2 ) 0>S= '.9ss f 'M W nj,":7 /s r '/f Qz6 = o. 5 g )70 (q )r QZ7= 0.25 ~r, 2 A,3,IA / 2 3 /)2,4 '+ /// f t ," {[ ^j3A/ 2 ;, ,Z^t ,_z} . ,4 Z (3,AI Ops = c, <,_/ ,xoe , ,, 1',, T+*, ( 2 f y, 'i 2 [^ 3\ o ( 2,M, ); JX/ y htt 030= c.54 7 m, mI m,)[(3A/, /,'7t r ; 0 /' (nt,'" c .r /)[(3A/ (,,..,t',,.zr ttj (,,, ( .y>, i ) f 3A Aj ,,j 2 I ,, / 2 I ,l (t w3 +ti /, / ,n, ) ) /, /7t3 ,)J j Qq2 oA t 17 z '/ ,V J , ,t) ,/7/j ,,), J 3\ / ,,,Cm ,Jj S , i L (,51,,,3" /)t,, ( 4 At' ) /r, 9,) J ( s /3 S/2 z / .. 1 , 2 I .. r 27 = ?j 7 3 (/ J_ <,. ,3 ,, )i 3,0 (,s , *, ) j J a.33 = o/w i + , ,. .. M, '/ 2 7 ' / 2 /~r / J / i9 53 A /",.[ ), ,',.J , , Q36 = 5 ( r,.'m")[,A#(,,, ", 7 ^;i J /f 'i ,< ,, o_.2 A 037 = "7. (JA/)/ f/f A / rc',/ _f) ) [a f ,f ,) 2 ," / 2 / I 2 , ? [,(3 ,i)  C ,,, .)J 3A ,)3, ,,, ./3/./) J J( 25) Qje = 0. 5 f [j/ C nj ) 3 ) //C, ) J 9,,  (.3) a39 = aSA7Y (JA\l)/j3/J/ ( j9 .I j#, f;" ^ *I r 2. v 2 21~ ;,,) 1JA S ) 7 3 P, P ,)I LfiAt)/ (>r,!, *'M,) ]JJ (^ = o.25s,4 9(';m, ft' ) [( )() f )f f 7Mr, tS r'//) (mi 3? r,)X / 2 I t / ," ,", ., / L /\A ) "z r,;C .,/ +. C .. /, .t^r L X') r e 7 ~I / rl ,~ ~ ,I ncj )fJ c 05 /r~/ mJ ) (M', (w. m, mt )C3\) 2 J 3Qy = oa/r, (2 zr , IJ%A J 3 ZLh JI 22 4 6 2 )  c., ( A ,s)L3' /f, A rz, {,,,,' (7A J Gs = o.5,74/ 3zQtA&) ///I tr ^ // M f14 t.() J 3 / 4 f i <, e 4/' /xA /JJ ) 1 ) ,7' .. ~' 'j,'M ) (;2 ,. ,. t,') f ,, ,, 21 I / ,> 2 * ." "' 7 ' j, ,,) J m,,, ) 7 y (NJar, ) /42 ( j M. ) ) ))j ] 4 " /< ) 2 Q l // C A ) .( , ) ()( #J 3 j5A44 bWJ _^ (2"i $ 7 053 = 0.25$74 / Z [3 ) ' t J ,' 2/, 'fi).J. + Z' ', ) S= 0 .25;/ / Qj= O.5M /'8(i A1419 J4 o,,=0~A/~;(~(\ 'c;~1/ , = A5 r f m 2 ,z X7= 0.25i47'(3A/) 9 I) (2, ,.,) J t.'J O.s = o.Z5/yS {(2, +rj'M,f3.4'),; <.,. ,r/ ,,)J i). ?N '< z ,) Z / Go= 2A,,' 7L s +, J,7 t' /"J~ s^= Ao)j lrsf{c2A)7'(^, #e)( 0s/ 5 t.Z ,4,,s ( 11 ) [ ;t ) , (4), o t 'A/s, fz;c; j)2 ,"  { ( /.43"t9) J ( Q ,3 = o.z5.",s/l+^ { / ) J,)[/3" //,,<,) yf C^j+f f +<, ,,< /"].. .. / n s ) J Qg42= 2S47s[ (2y. #,') {A /) (,'tr*'j J) 4,e ) V 7/ j LfA + f ) (,,3 ,<,. " qdS = 2/I7 Mfc f Af ) A t) # ,, I Qa4d = 'A rd _;( 2A +/) A c.; J 