Buckling and deformation of heated conical shells.


Material Information

Buckling and deformation of heated conical shells.
Physical Description:
viii, 151 leaves ill. : ; 28 cm.
Chang, Lu-Kang, 1938-
Publication Date:


Subjects / Keywords:
Elastic plates and shells   ( lcsh )
Buckling (Mechanics)   ( lcsh )
Deformations (Mechanics)   ( lcsh )
Thermal stresses   ( lcsh )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )


Thesis--University of Florida.
Bibliography: leaves 148-150.
Statement of Responsibility:
Lu-Kang Chang.
General Note:
Manuscript copy.
General Note:

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
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oclc - 13645121
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Full Text






June, 1967


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.





ABSTRACT .. . . .



1. Geometry of Shells

2. Basic Equations .
3. Transformation of Coordinates .


1. Boundary Conditions .
2. Thermal Buckling .
3. Postbuckling Behavior .


1. Critical Temperature .
2. Minimum Temperature .
3. Deformation . .






Corresponding Symbols Used in Computer Program .





















* .
. *


Figure Page
1. The cross section of a shell segment . .. 11

2. Geometry of conical shell . .... .25

3. Critical temperature versus radius-thickness ratio
at H/R = 2 . . ... ...... 40

4. Critical temperature versus radius-thickness 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 radius-thickness 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

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


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. (IV-3) and (IV-15), respectively
/ I
b b. = coefficients of the deflection functions defined
i' in Eqs. (IV-30) and (IV-37), respectively

a, a. = coefficients of the deflection functions defined
in Eqs. (IV-47) and (IV-50), 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. (IV-9) and (IV-29), respectively

h = wall-tnickness of conical shell

I = length of conical shell

mi, mi = numbers defined in Eqs. (IV-53)

n = number defined in Eq. (IV-16)

r, p = surface coordinates

u, v, w = dimensionless meridional circumferential and
inward normal displacements

x, r = length index and meridional distance defined
in Eq. (11-36) and Fig. 2

x = value of x at the large end

y = thermal expansion coefficient

B = semivertex angle

V1' Y2 = numbers defined in Eqs. (IV-26) and (IV-41),

e e = meridional and circumferential strains

erl = shear strain

S= number defined in Eq. (IV-61)

8 = transformed coordinate defined in Eqs. (11-31)

Kr, k = meridional and circumferential changes of curvature
in the middle surface

Kr = twist of the middle surface

A = (l-v)/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. (IV-9)
and (IV-29)

V2,V4 = operator defined in Eq. (11-38)


= prebuckling state

= additional qualities during buckling


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

Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy



Lu-Kang 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 radius-thickness ratio, the radius-height ratio, and the cone's

vertex angle. The radius-thickness 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.




Thin-walled shells have many applications as principal elements

of structures. One of the main advantages is weight-saving. 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

thin-walled 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 shell-like 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 well-known Donnell's eighth-order 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 diamond-shaped 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 "snap-through" 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 finite-deformation 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 Crouzet-Pascal [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


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 above-mentioned 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 "snap-through"

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 finite-difference

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


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 thin-walled, 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 Rayleigh-titz 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 finite-deflection analysis

on shells of other geometries has been closer to experimental results

than classical small-deflection 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 deflection-temperature relations after buckling are

plotted in Figs. 12-14. The computation was carried out on the IBM

709 Computer and tne general programs were written in Fortran IV


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


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.



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, (11-1)

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) (11-2)

where A, B and C2 are the fundamental magnitudes of the first
o o o
order. They are


1o,= 2aX



d X

For orthogonal surface coordinates

of and a -
1 2

vanishes, and the magnitudes

are A and C respectively.

The unit normal vector of the surface is

/ C

The fundamental magnitudes of second order are defined as:

1 t= :;'

M/= ax,.Xz"


-p /
Ae C0



d X,

a .




d36 9X3
Ox, d AI



d j
- --

N -O~x fC


9 7Z,


.1 Xt

a i



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



to the principal radii of curvature

principal curvature are coordinate lines,

expressed as follows:

an9 d
9><2~ c;R 0'^


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



Fig. 1. The cross section of a shell segment.

Hence by differentiation,




Substituting Eq. (11-7) into Eq. (11-9), the following relations

are obtained,



From Fig. 1

point at distance





z a+i


By referring to a surface parallel to the middle surface at a distance

z, we have

-.~ -

+ + },^
* xe as~


2 1c
= ,4/ Z) */ r


A=,4 ( .l, ),

d= Co'l -

The strain-displacement relations are approximated as

follows [32]:

// / dA
611= AjCi a

: / )2
Z 2 ?

z )
d ^Xz

ar I (acz ,)



d~kr )_
L ,,I C C at /

C cC? ~ d~ d44 r~
A ax C Co 91 \.



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. (11-13) are obtained under the assumption that the t

verse shears vanish. In small deflection theories all quadratic

in the strain-displacement 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





The changes of curvatures H' K2 and K12 on the middle sur-

face have the following forms,

? W-- -f /__
w 2--' ( /v^ ) d-


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 = ~~= (11-16)

S s

Substituting Eqs. (11-16) into Eqs. (11-3), (11-5) and (11-6),

we have

,to = i, c,,= r.,,

,= o IZ= r ,I (II-17)

Referring to the above equations, the strain-displacement relations
for a conical shell on the middle surface are obtained from Eqs. (11-14),

er= -er r)

eo-- 7. "-

and a comma, followed by subscripts indicates differentiation with
respect to subscripted variables.
From Eqs. (11-15), the changes of curvature are found

^r= T rr

"-v =f~ W7 (11-19)

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
stress-strain relations are given by the following:

= -- / -/ 7-

T = l I (er) / T (II-20)


where E, 0, v, and T represent Young's modulus, thermal expansion
coefficient, Poisson's ratio and the temperature difference, respec-
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 (11-21)

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 V-09.)
D z a 19,

Ur,- _1/fe, Te,, 0-) hT6) /7 c XJrA9I


where h is wall thickness and

7=L or~



- /3
/Z7 (/-Jz)


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. -'< ^

h-e ur er '' rr M ru artio, orr )(1 ))II-25)

The Euler equation for U1 is

Sdr ( ( )A O Srr

+ )4^ (_ _____
SY6( ( o-U'79 Cy V'z^^




the Euler equation for v is

4 2^ ^ / )_ ___ ___
9rr a go drr

rc 9r a ) )=o

and the Euler equation for W is

or ) d 4 4 t

4 )- = o (11-28)
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

O\ r F, r r+ (rAid F) FqPco

y = ~rr

^r^-t^M~fi ), C^J- r


By substituting Eqs. (11-18) into Eqs. (11-20), the stress-

displacement relations are obtained,

2 /-V "


We introduce the following dimensionless notations,

After substituting Eqs. (11-29) into Eqs. (II-30) and eliminating
the displacement variables U and 7, the compatibility equation is found

as the following:

V*4' = (dc j)lrr-J r )re> 'A^

S(Cw;e) ; e) -r

~ 7jr [L (Tr + ~ ,r) + J



S-z 2-i

v( = ) + L f 1
7 L7 +) a-T O "

By substituting Eqs. (11-18), (11-19), (11-20), (11-29), (II-30)
and (11-31) into Eq. (11-22) and using relation (11-28), the equation of
equilibrium in normal direction is found to be of the following form:

i' "- y = A A"-')Frr ,_,.,

+ ( Q r (~ _2 F I" _/ Q), r
x /J'789



M, = ,_/,~B




3. Transformation of Coordinates

For convenience, the following transformation is introduced

x =i- 7


By substituting Eqs. (11-36) into Eq. (11-32), the dimension-

less compatibility in the newly defined coordinates is

C Z = e& -C^,)e-(u x-W.* X-Ws (-.ex

- (W< ) -f (i (U,- a?6tr



71= V7 77

vz=e (a

The dimensionless equation of equilibrium after the transforma-

tion becomes

+ (~eQo x6 )(x-,x -;Cx x)-2 e -Fe)(1o-(6)]


*o= ~^o/a,- )



It is noted in Eq. (II-34) that the thermal moment due to the

temperature gradient across the wall-thickness is disregarded because

the temperature is assumed uniform throughout the thickness. Eqs. (11-37)

and (II-39) are the basic equations for the buckling of conical shells

due to the temperature changes.



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. (11-37) and (11-39) are used. For the buckling of shells, the

two coupled nonlinear equations (11-37) and (II-39) 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 (111-1)

if L(P) exists in a two-dimensional domain, an approximate solution

of P is assumed in a series form, i.e.,

/0 (X, ) = ."- a4' -(,y) (111-2)

where P is the approximate
functions which satisfy the

are constant coefficients.

(III-l), 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. (111-2) into Eq.

= M, where, in general, M / 0. To min-

set of orthogonal conditions,

II (4w) A -x4,)dl~o1XK


i = 1, 2, 3, ... n

where 0 represents the two-dimensional domain and the coefficients

ai can be determined by solving the n algebraic equations in Eq.

(111-3) 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 1-14)


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 (11-6)



*--- R


Fig. 2. Geometry of conical shell.

and at x = x

ExX -(>& bjf e exo r = o (III-7)

The end conditions along the direction of the generator will

be discussed in the individual cases to be considered.

2. Thermal Buckling

When Eqs. (11-37) and (11-39) are solved simultaneously, the

coupled relations are nonlinear. Now we consider only the linear terms

in Eqs. (11-37) and (11-39) 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:

*=- ( tcoft) e 3 5,x kx) o VTf (Ir -a)


2pv)'7 (A-^ )e3 ^xx ^ ) L.x

The operators V2 and 4 are defined in Eq. (11-38), 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. (III-4) and

(III-5). 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. (III-8). 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. (III-9),

the coefficients of the assumed deflection function w are then found.

When Eq. (III-8) is substituted into Eq. (II-37), and after

eliminating the nonlinear terms, the compatibility equation becomes

v "= -co ) e (t x) (InI-10)


Zt" = -T = additional displacement during buckling


F = F- /r = additional stress function during buckling.

When relation (III-9) is used in Eq. (11-39), 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

x ?v c 2x -^ )(-A (e- -f6s ) W x a- 6,' V


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. (III-ll), 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. (11-37) and (11-39). After subtracting Eqs. (111-8) and (III-9)

from Eqs. (II-37) and (11-39), respectively, it is found that

e "= e )(,;; ^. ) 0~, (-
Ole3F X(-.

x (zt; 6 ) (-


r y,[ ade -, jJ ,,-^
D 17 kr = e L x

X e X 49 A' s-




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. (III-12) 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. (III-13), and integrating it by the

Galerkin method, a set of nonlinear algebraic equations is obtained.


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.



1. Critical Temperature

Two different temperature gradients and their corresponding

conditions of constraint are considered in the solution of critical


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. (111-4) (111-7),

another boundary condition in meridional direction is

iXo Lxo
f erc = -u.^ dy = 0 (IV-l)
o o

By substituting Eqs. (11-20) and (11-29) into Eq. (IV-1), the dimen-

sionless condition of constraint expressed in terms of stress function

F is obtained as:

+-Y.A- x 4- ie 771Ir] = (IV-2)

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 (IV-3)

in which A -- (IV-4)

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. (III-8), the prebuckling stress function can be written in the

following form:

j; = l e j j A


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,
respectively, i.e.,

Ve ,=(,. 0 ., J )4 (IV-6)


V = 17 7 (IV-7)

Substituting Eq. (IV-3) into Eq. (IV-6) and integrating it, we find

e,,',l) 2 (H7X
7. \'rX
a i 7 E

cos -- ) (IV-8)

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 (IV-9)

where T1, 1 and g are constants. If g vanishes, Eq. (IV-9) implies

that the distribution of temperature is uniform over the surface of

the conical shell. Integrating Eq. (IV-7), the function F is

obtained as:

F, =-LC [ t,) (Z-^ ]^3e (dv-o>)

Substituting Eqs. (IV-8), (IV-10) and (IV-5) into Eqs.

(III-6), (III-7) and (IV-2), we obtain three algebraic equations.

By solving these three equations, and the arbitrary constant A'

in Eq. (IV-5) is considered zero, the constants A' (i = 1, 2, 3)

are found as:

= a. ( c, .dT;, a5 / a- u

i = 1, 2, 3 (IV-11)

By substituting Eqs. (IV-3) and (IV-5) into Eq. (III-9),

it is found that

-D V 4;-e a.

^( ^xx-4 ) = 4

Applying Galerkin's integration to Eq. (IV-12), and if only three

terms of the prebuckling deflection are taken in Eq. (IV-3), the

following three algebraic equations are obtained:

tnr^."t Xo
f" r)^ An ]d7A( -- 0

o 0

i = 1, 3, 5 (IV-13)

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 (IV-14)

The six unknowns Ai (i = 1, 2, 3) and a. (i = 1, 3, 5) can be deter-

mined by solving Eqs. (IV-ll) and (IV-14). They are found to be
functions of xo, TI, r, --, and 0. Therefore, F is determined in

accordance with the temperature distribution T. The expression A' and

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 (IV-15)

where p and 1 are the numbers of half waves in meridional and cir-

cumferential directions, respectively. By substituting Eq. (IV-15)

into Eq. (III-10) and letting

7 (IV-16)

we obtain

lP = ( .1A R )[e A
X ( hk


The additional stress function during buckling is the summation of two

solutions: one is the homogeneous portion of Eq. (III-10), and the

other is a particular solution which satisfies Eq. (IV-17). This

stress function can be expressed as:

iX ( x .

1=(A'.e -,, -/ ?, e


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 (IV-19)

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
be neglected without significant error [18].

When substituting Eqs. (IV-3), (IV-5), (IV-15) and (IV-8) into

Eq. (III-ll), it is found that

S" "J0

-* e C r ), '

+e =-L; -.


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 (IV-22)

After integration, Eq. (IV-22) can be briefly expressed in the

following forms:

C i' (Ca3 7'ajcO77

Se 0


(Cj1+oi jjl ~'7; f,/2)a(C$(Cs3 C d3-7

WJ De T1 q) aoe-l


Eqs. (IV-23) and (IV-24) 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



'7 -Z



Eq. (IV-25) is a second-order algebraic equation of T1

and yl. For given values of yl and g, two solutions of TI are

determined from Eq. (IV-25), 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. (IV-3) and (IV-15) 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. (IV-3) and (IV-15) 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. (IV-25),

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





= 300

0 1 11
0 200 400 600 800 1000


Fig. 3. Critical temperature versus radius-thickness
ratio at H/R = 2.

H/R = I



Fig. 4. Critical temperature versus
ratio at = 10.










, I




"I I



p I I I
o ICf 200 300

Fig. 5. Critical temperature versus semivertex angle
at H/R = 2.
at H/R = 2.







T = O.1 T,

-T. = # ,

T =T, + e9'




Fig. 6. Variation of critical temperature
temperature index (B = 100, R/h =

with meridional
300, H/R = 2).




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. (IV-9) 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. (III-4)

to (III- 7 ), other boundary conditions are

f Oac/ 9 =o at X =0 and X = o (IV-27)

or in another expression,

j ( x F, )W0O = 0 at X=0 and X=X (IV-28)

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:

Tn T -tz e= T (IV-29)

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 (IV-30)

In this analysis, it is assumed that the prebuckling deflection w' is

axisymmetric and with i = 1, 3, 5, takes the form,

W----3 ., ; ) (IV-31)

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- (IV-32)

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 (IV-33)
/~~.'=8,(x-O4 )i8X40 8 e

B (i = 1, 2, 3, 4) can be determined by the conditions of constraints

given in Eqs. (11-6), (III-7) and (IV-28). They are found to be func-
tions of xo, T2, 72 T and b' with i = 1, 3, 5. The functions

F3 and F1 in Eq. (IV-32) 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 ) (IV-34)


'= K-/' rrz) (IV-35)

The b 's (i = 1, 3, 5) are found by substituting Eqs. (IV-32),

(IV-33), (IV-34) and (IV-35) into Eq. (III-9) and integrating it by

the Galerkin method. The coefficients B' (j = 1, 2, 3, 4), gi, hi
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. (IV-29), since


/F<, X 4 "X = 0

During buckling, the additional deflection is assumed in the



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. (III-10) is considered as the additional stress

function during buckling.

By substituting Eq. (IV-37) intb Eq. (III-10) and assuming that

the stress function has the form

,- (A.) X AP ,,

r k:'^s -^- ) cOn' s- e

and g" and h" are found by tne use of the Galerkin method.
1 1
Substituting Eq. (IV-38) into Eq. (III-10), we have

Sr(/.A)X >- r'0f

xr C-05 JP J-^ S// 'k- X'< A IX

x rrX 7 0
x M cos$Bes ikJ= 0


In order to determine the coefficients g' and h (i = 1, 3),

the following approach is employed; Eq. (IV-39) is integrated by the.

use of the Galerkin method in the circumferential direction, which

provides the following relations:

4 $Q e Cos nwo( = O (IV-40)

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.

(IV-40) by comparing the corresponding terms of x. They are found to
be functions of L, 1, xo, k, cot 0, and are given in Appendix B.

Substituting Eqs. (IV-37) and (IV-38) into Eq. (III-11), two

algebraic equations are obtained by applying the Galerkin method to

Eq. (III-11). For this case, w = wi, F = F F = F and
w = w + wi in Eq. (III-11). If the following notation is intro-


2 = Trz/T (IV-41)

the two algebraic equations can be briefly expressed as:

(,, c ~Zo / )0 /, (Z3 z )e ,,3 = 0 (IV-42)

S-A 0/ Z', )461 -A ('33 433 T'^. ? 0 (IV-43)

For a nontrivial solution, the determinant of Eqs. (IV-42) and

(IV-43) is zero. After solving the determinant, the term 72 of

the temperature gradient can be written as:

O(TZL = O t i(l. Za ^, (IV-44)

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. (IV-44). 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

second-degree terms of the derivatives of normal deflection should be

included in the geometric (strain-displacement) relations. These rela-

tionshhips are expressed in Eqs. (11-18), and the nonlinear equations

(11-37) and (11-39) 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

6 8

R/ h =450

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. (III-6) and

(III-7) is,

J e/X = o (IV-45)
0 0

The above condition implies that the average length of the cone is

unchanged during heating. Eq. (IV-45) can be expressed in terms of

the stress function F and the normal deflection w,

/ffSlft if

fe- //1-6,^ vS x -7 J-^.^ree
o o

X Y C/( =e

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. (IV-47) 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^^< ^ ^ (IV-48)

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 =

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 (IV-50)

The additional stress function during buckling has the form:

A= = A x ; A ,, -F
p q, 4?/ x 4 et e- e F


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 ')


in which

' M, -= i
r ;C a~

i is an odd integer

By substituting Eqs. (IV-51) and (IV-52) into Eq. (III-12) and com-

paring the corresponding coefficients, Kj (j = 1, 2, ... 49) are

A? 7r
='- 7C<


obtained as functions of x ,

Fortran language in Appendix C.

By applying the relation

(IV-46), the boundary conditions

U, -. 8 and 1 They are given in
h 1*

(IV-14) to Eqs. (III-6), (III-7) and

during buckling are

CF.xx ( a ",, ) = 0

at x = 0 and x = x

By substituting Eqs. (IV-50), (IV-51) and (IV-52) into Eqs. (IV-54)

and (IV-55), we obtained three algebraic equations: AI, 'A and A3 which

are then obtained by solving these three equations in the following


A A" A, 2

A, = A + A. Q a 2 )

,4 = a, / %,o a, a, 1 ,, -,,'


Substituting the expressions in Eq. (IV-47) (IV-52) into

the equation of equilibrium (III-13) and assuming the error is 9q,


* I
2 fln V,9-,F~ ~FJ-~yijd


= 0


two equations are obtained by applying the Galerkin method:

f4f cos _4:ot O

The above two equations can be expressed in the following forms

after integration:


As 4 a, a3 4 Cas R/=,, O

A "2 A .
i a,

+ a,-",




T,)S.z- -( rS6 A",3s7
r/, ( S6 0,dL

At "2, A A 3. 3
- O., a,; 58 4 as3 S d3 'o

= 0


In an attempt to solve Eqs. (IV-59) and (IV-60) simultan-

eously, it is necessary to define

A P;
s a,


6L, a z w7 )Sj rS aj CI (7r ) 54

By substituting Eq. (IV-61) into Eqs. (IV-59) and (IV-60),

and solving for the constant TI, we find

S= V/ V, (IV-62)


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. (IV-59) and (IV-60)

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. (IV-50), then

Eq. (IV-59) can be written as:

a, W101TIP;( &r, ) RJ 4R9,] =0 (IV-63)


( 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



1 3



I I 101 1

0 200 400 600 800 1000

Fig. 10. Minimum temperature versus radius-thickness
ratio at H/R = 2.




i 0.8



ST = O. IT.

T =T. +T, e9X








, I



Fig. 11. Variation of minimum temperature with meridional
temperature index (6 = 100, R/h = 300, H/R = 2).

,,,, lI I II i iII


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.
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.



8- N

0 0 2 4



0 0 I 2 3 4 5

T. ,x 103
Fig. 12. Deflection versus temperature (B = 100, R/h = 450,
H/R = 2).


C 0.5

0.2 5

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


2 -

0 I 2 3 4 5
o[T,-(T.)cy] x 10

Fig. 14. Deflection versus temperature (8 = 100
HA = 2).



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

radius-thickness 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 deformation-temperature 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..(IV-44), 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


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 finite-deflection 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. (IV-1). When the large deflection theory is considered,
the nonlinear term (w ) appears in the condition of constraint as

expressed in Eq. (IV-44), since (w ,)2 is always positive; therefore,

it can be seen from Eq. (IV-44) 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 10-5/F.


o'(T. )mi-











Fig. 15. Comparison of critical temperature with minimum
temperature (0 = 100, H/R = 2).





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
were mounted so that they were prevented from lengthening by two rigid

plates, which were held in place by four 1/2-inch 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 infra-red 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.



Fig. 16. Photograph showing buckling of a heated conical shell



/ '. 1
- -i = yt,., .- I -A ,

tZ I 2 -I
j a 7"z)..

ly A (m ? A- 2)

. = --- 14 2 ( /-3~ ) A )

- (A-2A> )M ](yf 7.


,Z 2 'A1t
/6 4 r A ; )

,4= -i3 ) (2 )-Z
A ,- ( z Z U-t O' J f ", .)( 2+,9 )

Ai = ( 2-zJ -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 ,


('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,

mj C4-

4,, 2A o
^4/9 = e

- /I

0q2o = e

(3A -/ )X.

,z4 = (m'f4

- $j ^ .s4 ) C- -A > q9, {(r-y 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; "c-P 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

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

= a-4 ( -/- /; ) -

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 ~ )


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,

D,= (Cd, C,s)Cz-Cr),

3 = (d CXCz -
hV= ($,,-4d9)(d2-C7)

D5 = (d3 dC/,)(C-Co) ,

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 -

),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

BZ-f D .e Dzo D. = (Ceo e9X C'-z-r
T)22Zj'/RZ>,fDazO .V23 = (dC ~ /920-'

Dt~ =(C~p- C/9 ) CZ ~d/-I

S(D-)(D- -/D
.V)a' =- (Z224-3?23)(Vz- l'5)

Ds= D8V 1),

2,7 = (C/7 -C,)C

A, D8 = (D6-D3)(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 ?g-7,

Ape, 74 +A ,g~/r~~prILrl

iE3 = R/9 v'K4 9As- 98)f(4'A~)'-A (~A~'t:C /rj

,~= V-A/ 6A-17)('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 ~~ I-2
SnLI'L]f m"Aj Y,4/


-=, 0= '67 P// -/ F-3 -- 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 +-y-D2/ f, zD /o ^ ^d y


Ft6 = -F2s3 (/8g $9: col g)

EZ s = Ezo

F, .? -F2;; e-i,6 (f, -, ,,,

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,)r-a -Ai n d
42 (,, ,f )J

= rS -,, rA ,2< /j-'
= 0. 5 '(44 -R) i-/) ) ,4-t 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 / E-8 "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

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 = (Ea-E4 )(Eg -s )-
S .0 65

S 6Ed = E6 4 S/ f 6,/ 7'o -z7

,= ,

c= ( 77-T) 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(g-D2 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 -o-rC, 493 = (0'A) Cs -MjW 3

/ JA / J X o

Sr -= / &

,.s e


2 2 l
Q,= o.25A?4 {7;t/1t)A-i) (2m,/,','t, )(A-/) (2mrJ

2 = a. 254 t -) f- Z-1 A "] ~J A ]

7' ( / 2 -,)
"" "3-1+ (2rt, -(', ) J J

Q4= c.25R7f(2'Y -t)A- -,)J +m/ [-) ,(d-,7)/ J

QS = O.Z5/t (A-,) i/, (/ f l-A

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 / r-c',/ _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 -LfiA-t)/ (>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 0-5 /r~/ mJ )

(M', (w. m,- -mt )C3\-) 2- J

3Qy = oa/r, (2 zr -, IJ-%A J 3 ZLh JI


4 6- 2 )- -
c., ( A ,s)L3' /f-, A rz, -{,,,,' (7A- J

Gs = -o.5,74/- 3zQ-tA&) ///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