H
I LL IN OI
S
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
PRODUCTION NOTE
University of Illinois at
Urbana-Champaign Library
Large-scale Digitization Project, 2007.
Snap-Through and Post-Buckling
Behavior of Cylindrical Shells Under
the Action of External Pressure
by
Henry L. Langhaar
PROFESSOR OF THEORETICAL AND APPLIED MECHANICS
Arthur P. Boresi
ASSISTANT PROFESSOR OF THEORETICAL AND APPLIED MECHANICS
ENGINEERING EXPERIMENT STATION BULLETIN NO. 443
© 1957 BY THE BOARD OF TRUSTEES OF THE
UNIVERSITY OF ILLINOIS
2050-4-57-62247
UNIVERSITY
SF PRENOIS
ýý pR ESS ý:
ABSTRACT
This report treats the buckling and post-
buckling behavior of a cylindrical shell that is
subjected to uniform external pressure p on its
lateral surface, and an axial compressive force F
(Fig. A). The force F varies with the pressure p
in such a way that F = Xa2p, in which a is the
mean radius of the shell and A is a dimensionless
constant. If the shell is immersed in a fluid at con-
stant pressure p and if the force F results only
from the pressure p on the ends, A = r.
The ends of the shell are assumed to provide
simple support to the cylindrical wall. Accordingly,
the radial and circumferential displacement com-
ponents of the middle surface of the wall vanish
at the ends. If the ends of the shell are free to
warp, no other constraint is imposed on the de-
formation. If the ends of the shell are rigid, the
axial displacement is constant at either end. Both
of these cases were investigated. For generality,
the shell was considered to be reinforced by several
rings or hoops.
Only geometrically perfect shells were studied;
that is, initial dents and out-of-roundness were not
taken into account. Only shells with a linear stress-
strain relation were considered.
If the axial force F is not too great, the shell
assumes a fluted form when it buckles. This form
is illustrated by Fig. B, which is a photograph of
some of Sturmn's test specimens (7)*. The number of
Fig. B. Front Views of Buckled Cylinders
flutes in the buckled form is influenced strongly by
the ratio L/a, in which L is the length of the shell.
Fig. C illustrates several forms of cross sections of
cylindrical shells that have been buckled by ex-
ternal pressure.
When the axial force F predominates, the
buckled shell assumes a form in which diamond-
shaped facets occur (1. Art. 85). This type of
buckling was not considered in the present study;
the axial force F was assumed to be so small that
the fluted pattern occurs. The admissible range of
F was not determined, but the fluted pattern usu-
ally occurs if A does not exceed 7r.
* Numbers in parentheses, unless otherwise identified, refer to the
References at the end of this report.
1. -- L 0
Fig. C. End Views of Buckled Cylinders
Fig. A. Forces Acting on Shell
CONTENTS
I. PRELIMINARY CONSIDERATIONS
1. Introduction
2. Notations
II. POTENTIAL ENERGY OF A SHELL WITH FLEXIBLE ENDS
3. Membrane Strains
4. Fourier Analysis of Displacement Components
5. Membrane Energy
6. Potential Energy of External Forces
7. Elimination of u0 from the Increment of Total Potential Energy
8. Elimination of u, from the Increment of Total Potential Energy
9. Elimination of u2 from the Increment of Total Potential Energy
10. Elimination of ua from the Increment of Total Potential Energy
11. Further Simplification of AV
12. Strain Energy of Bending of an Elastic Cylinder
III. POTENTIAL ENERGY OF A SHELL WITH RIGID ENDS
13. Shell with Rigid Ends
IV. PRESSURE-DEFLECTION RELATIONS
14. Load-Deflection Curves
15. Tsien Critical Pressure
16. Effect of Assumptions on the Tsien Critical Pressure
17. Potential Energy Barriers
18. Numerical Example
V. SUMMARY
VI. REFERENCES
VII. APPENDIX
FIGURES
1. Pressure-Deflection Curve
2. Rectangular Coordinates of Shell
3. Arc of Buckled Cylindrical Shell
4. Increment of Potential Energy versus Deflection Parameter
5. Pressure-Deflection Curve for Ideal Shell
6. Intersecting Pressure-Deflection Curves
7. Pressure-Deflection Curve for Imperfect Shell
8. Statistical Distribution Curve for Imperfect Shells
9. Determination of Tsien Critical Pressure
10. Graphs of f(p) and <^(p)
11. Potential Energy Barrier
1 2. Buckling Coefficient K versus Deflection Parameter W,
1 3. Buckling Coefficients for Cylindrical Shells Subjected to
Hydrostatic Pressure
14. Potential Energy Barriers Separating Buckled and Unbuckled Forms
TABLES
1. Values of K1, K2, . . ., Ks8 for v
0.30
2. Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Values of Coefficients for Computing Buckling
Coefficients K,t, Ki, and K,,»,, (a/h = 1000)
Coefficients Kst, Ki, and K,,,, (a/h = 100)
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
Loads - n =
..... E
I. PRELIMINARY CONSIDERATIONS
1. Introduction
Experimental data on the collapsing pressures
of cylindrical shells have been obtained by Fair-
bairn, Carman, Jasper and Sullivan, Saunders and
Windenburg, Windenburg and Trilling, Sturm, and
numerous other investigators.(12, ,4 5, 6, 7) Theoret-
ical studies of the problem have been performed by
Bryan, Southwell, Cook, von Mises, Donnell,
Sturm, and others.(8' 9, 10, 11, 12, 7) The history of
these theories (to 1948) is contained in the work
of Batdorf.(13'
Von Mises and most of the subsequent investi-
gators implicitly based their analyses on the gen-
eral principle that a motionless conservative
mechanical system becomes unstable when the
value of its total potential energy ceases to be a
relative minimum. The theory of buckling based on
this principle is sometimes called the "infinitesimal
theory" since investigations of relative minima
require only infinitesimal variations. The buckling
load determined by the infinitesimal theory has
been designed by Friedrichs(14) as the "Euler criti-
cal load," since Euler employed the infinitesimal
theory in his study of columns. The Euler critical
loads for elastic cylindrical shells that are subjected
to external hydrostatic pressure are in close corre-
lation with experimental data, provided that the
shells are long in comparison with their diameters.
However, the Euler critical loads are much too
high for short thin shells.
In 1938, von KArmAn and Tsien(15) called at-
tention to the fact that an ideal shell can be in a
state of weak stability, such that a small blow or
other disturbance causes it to snap into a badly
deformed shape. Simple examples of this type of
equilibrium are common. The equilibrium of a coin
that is balanced on its edge is stable, but such
weak stability is usually unsuitable for engineering
design. Similarly, if the center of gravity of a ship
is so high that the slightest push will cause the
ship to capsize, the border line of stability has been
reached. However, this condition has no significance
for the design of hulls. Analogously, the Euler criti-
cal load of a shell loses much of its significance
when snap-through can occur. We merely know
that the Euler critical load is the upper bound of
the load that will actually cause failure.
In this investigation,
the occurrence of weak
stability is manifested by
the conclusion that the
pressure required to
maintain a buckled form
is frequently much less
than the Euler critical
pressure. A pressure-
deflection curve for an
ideal elastic cylindrical
shell that is loaded by
external pressure has the
general form shown in
Fig. 1. Pressure-Deflection
Fig. 1. The falling part Curve
of the curve (dotted in
Fig. 1) represents unstable equilibrium configura-
tions. Also, the continuation of line OE (dotted)
represents unstable unbuckled configurations. Actu-
ally, the shell snaps from some configuration A to
another configuration B, as indicated by the dashed
line in Fig. 1. Theoretically, point A coincides with
the Euler critical pressure E, but initial imperfec-
tions and accidental disturbances prevent the shell
from reaching this point. To some extent, point A
is indeterminate, but it is presumably higher than
the minimum point C, unless the shell has excessive
initial dents or lopsidedness. In this report, a
hypothesis of Tsien(19' is used for locating point A.
The pressure at point C is the minimum pressure
under which a buckled form can persist. Thus,
ILLINOIS ENGINEERING EXPERIMENT STATION
if the shell is in a buckled state, and if the ex-
ternal pressure is gradually relieved, the shell will
snap back to the unbuckled form when the pressure
at point C is reached.
An analysis of the post-buckling behavior of a
structure determines the buckling load automati-
cally. For example, an analysis of the form of a
buckled column reveals that there is no real non-
zero solution unless the load exceeds a certain
value, the Euler critical load. Accordingly, in prin-
ciple, the nonlinear theory of equilibrium obviates
the need for a special theory of buckling. However,
as a practical expedient, it is usually easier to de-
termine the Euler buckling load of a structure by
solving a linear eigenvalue problem than by calcu-
lating the bifurcation point of a curve in configura-
tion space that represents all equilibrium configura-
tions.
Problems of post-buckling behavior of elastic
shells may be approached in two different ways.
On the one hand, we may seek to solve the equi-
librium equations and the compatibility equations,
in consistency with given boundary conditions.
However, in the large-deformation theory of elas-
ticity, the compatibility equations are an extremely
complicated set of differential equations, represented
by the vanishing of a Riemann tensor.(20) As Dr.
C. Lanezos once remarked, "We could not hope to
solve the general compatibility equations, but for-
tunately we already know their general solution.
It is merely an arbitrary displacement vector. We
should be happy that we know this solution, and
we should make every possible use of it."
When the components of the displacement vec-
tor are adopted as the dependent variables in a
shell problem, only the equilibrium equations and
the boundary conditions remain to be considered.
The equilibrium equations may be derived by bal-
ancing forces on a differential element, but, in
large-deformation theories, the rotations of the ele-
ments introduce a complexity into this procedure.
Consequently, the equilibrium equations are ob-
tained most readily in terms of the initial coordi-
nates by applying the Euler equations of the
calculus of variations to the potential energy in-
tegral. Unfortunately, in most shell problems, the
equilibrium equations are too complicated to be
solved rigorously. Instead of tackling the equilib-
rium equations directly, we may revert to the po-
tential energy integral and apply approximation
methods of the calculus of variations. This pro-
cedure was employed in this investigation. The
theory is accordingly founded on the well-known
principle that all states of equilibrium - stable and
unstable - are determined by the stationary values
of the potential energy. The stable states corre-
spond to relative minima of potential energy.
The potential energy
of the shell is the sunm of 1Z
four parts; namely, the
membrane strain energy,
the strain energy of bend-
ing, the strain energy of
reinforcing rings, and the
potential energy of ex-
4-t 1 f A i l
teina oUtrces. Ar c es
to 13 inclusive are de- Fig. 2. Rectangular Coordi-
to 13, inclusive, are de- nates of Shell
voted to the derivation of
the potential energy expression.
In the development of the theory, the axial,
circumferential, and radial components of displace-
ment of the middle surface (u, v, w) (Fig. 2) are
approximated by three terms of Fourier series
(Eq. 11). By using the assumption that the shell
buckles without incremental hoop strain on the
middle surface, the Fourier coefficients vi, v2, vS,
wo, w2, W3 are all expressed as functions of w,. Sub-
sequently, w, is replaced by a more convenient
parameter W, defined by TV = (n - 1/n) wi/a,
where n is the number of waves in the periphery of
the buckled shell. It is assumed that W = Wo cos
rTXiL, where x is an axial coordinate with origin at
the center section of the shell, and Wo is a constant
that must eventually be chosen to minimize the
buckling pressure. The Fourier coefficients Uo, u,,
u2, us are determined by the calculus of variations
to. minimize the buckling pressure. Accordingly,
these are finally expressed as functions of TV.
2. Notations
a = mean radius of the shell
L = length of the shell
h = thickness of the shell
r = a/L
I = moment of inertia of the cross sec-
tion of a reinforcing ring about its
centroidal axis
p = pressure on the lateral surface of the
shell
F = axial force that acts on the shell
(Fig. A)
X = a constant, defined by F =Xa2p
n = number of complete waves in a cross
section of the buckled shell
Bul. 443. BEHAVIOR OF CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE
E = Young's modulus
v = Poisson's ratio
= n 2r 2 = 0.295804nL/a, ifv =
0.30.
x = an axial coordinate with origin at the
center section of the shell
0 = an angular coordinate (Fig. 2)
u,v,w = axial, circumferential and radial dis-
placement components of the middle
surface due to buckling (Fig. 2)
V = total potential energy of the shell
(strain energy plus potential energy
of external forces)
AV = increment of potential energy due to
buckling (Eq. 100)
U, = strain energy of a reinforcing ring
(Eq. 99)
Ub = part of the strain energy of the shell
that results from bending
K = constant in the buckling formula, per
=KEh/a
Ki = value of K determined by the infini-
tesimal theory of buckling
Kst = value of K determined by the snap-
through theory of buckling (Tsien's
theory)
Ki, K2, . . ., K1 = functions of n and v, defined by
Eqs. (39), (47), (58), and (67), and
tabulated in Table 1
a,,a2,b1,b2,b3,c1,c2,C3 = functions of n, v, and r, de-
fined by Eqs. (72) and (98), and tabu-
lated in Tables 2-20
B1, B2, Ba = constants defined by Eq. (101)
Wo = A parameter defined by Eq. (36). Wo
is a measure of the deflection due to
buckling.
7rx
W = Wo cos L
LPrimes denote derivatives with respect to
Primes denote derivatives with respect to x.
II. POTENTIAL ENERGY OF A SHELL WITH FLEXIBLE ENDS
3. Membrane Strains
In this article expressions for the membrane
strains of the shell in terms of the displacement
components of the middle surface of the shell are
derived.
The shell is referred to rectangular coordinates
(x, y, z), such that the x-axis is the geometrical
axis of the cylinder (Fig. 2). The positive x-axis
in Fig. 2 is directed toward the reader. The circle
in Fig. 2 represents a cross section of the middle
surface of the unbuckled shell. The origin of x is
taken to be the middle section of the shell.
When the shell buckles, the particle that lies at
point (x, y, z) on the middle surface is displaced
to the point (x*, y*, z*). In terms of the axial,
circumferential, and radial displacements compo-
nents (u, v, w) and the angular coordinate 0 (Fig.
2), the coordinates (x*, y*, z*) are given by
* = x + U
y* = a sin 0 + v cos 0 + w sin 0
z* = a cos 0 - v sin 0 + w cos 0
The displacement components (u, v, w) are fu
tions of x and 0. In deriving Eq. (1), we have n
lected the fact that the deformation bef
buckling alters the radius slightly.
If x and 0 take infinitesimal increments dx
dO, the coordinates (x*, y*, z*) take increme
(dx*, dy*, dz*). These increments are obtained
differentiation of Eq. (1); hence
dx* =
dy* =
dz* =
(1 + ux) dx + uod6
(vx cos 0 + wx sin 6) dx
+ (a cos 0 + ve cos 0 - v sin 0
+ we sin 0 + w cos 0) dO
(-v sin0 + w cos ) dx
+ (-asin - vesinO - vcosO
+ we cos 6 - w sin 6) do
where subscripts x and 0 denote partial derivates.
Consider two differential vectors (dx*, dy*, dz*)
and (Sx*, 8y*, Sz*), the first being the increments
of (x*, y*, z*) when x alone receives an increment
dx, and the second being the increments of (x*, y*,
z*) when 0 alone receives an increment dO. Setting
dO - 0 in Eq. (2), we obtain
dx* = (1 + ux) dx
dy* = (vx cos 0 + wx sin 0) dx
dz* = (-vx sin 0 + w, cos 0) dx
Setting dx = 0 in Eq. (2), we obtain
x* = uodo
by* = (a cos 0 + ve cos 0 - v sin 6
+ we sin 0 + w cos 0) dO
bz* = (-a sin 0 - ve sin 0 - v cos 0
+ we cos 6 - w sin 0) dO
The squares of the magnitudes of the vectors
(dx*, dy*, dz*) and (bx*, by*, az*) are
(ds*)2 = (dx*)2 + (dy*)2 + (dz*)2
(5s*)2 = 2 (y*)2 + (6z*)2
Accordingly, Eqs. (2') and (2") yield
(1) (ds*)2 = [(1 + uX)2 Vx2 + Wx2] (dx)2
(Bs*)2 = [uo2 + a2 + e + v2 + w02 + w2
+2 ave+ 2aw + 2vew - 2vwe] (dO)2 (4)
The initial magnitudes of the vectors (dx*, dy*, dz*)
and (ax*, by*, bz*) -that is, the magnitudes of the
line elements before buckling-are
ds = dx,
6s = adO
Consequently,
(ds*) = 1 + 2ux + u'2 + vx2 + wx2
S= 1 + 2 aV + w
S(e + w 2 + v - we
Since the material will not admit large strains,
the ratios ds*/ds and Ss*/bs are approximately
equal to unity. Therefore, the additive terms in-
volving, u, v, w on the right sides of Eqs. (5) and
(6) are small compared to unity. Accordingly,
ds*/ds and Ss*/ls are closely approximated by
Bul. 443. BEHAVIOR OF CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE
binomial expansions of the square roots of the
right sides of Eqs. (5) and (6) in which only terms
to the second degree are retained. Thus, we obtain
ds* 1 1
ds = 1 + uý +-2-v' + -wý2
Ss* vo + w 0u- 1 ( v - we 2
as + a + 2a2 2 \ a /
The shell is already strained before it buckles.
When buckling occurs, line elements in the x and 0
directions receive incremental strains, Aex and Aoo.
According to the customary definition of strain,
these increments are
ds* - ds
ds
Consequently, by Eq. (7),
A, = ux + 1 12 1 2
2
as* - as
A v + w 002 1 (v- wo 2
a 2 2 + a
a 2a2 2 a
The shearing strain yxo is defined by yxo = cos 0,
where 0 is the angle between the vectors (dx*, dy*,
dz*) and (5x*, 5y*, az*). Therefore,
dx*8x* + dy*Sy* + dz*8z*
TYo = ds*Ss*
Thus, by Eqs. (2') and (2")
o= d** [(1 + ux) u0 + (vx cos 0 + w, sin 0)
• (a cos 0 + vo cos 0 - v sin 0 + wo sin 0
+ w cos 0) + (-v,. sin 0 +w cos 0) (-a sin 0
- ve sin 60 - v cos 0 + wo cos 6 - w sin 0)]
Since ds = dx and as = adO, this equation reduces to
1 ds 8s
o = a ds* as* [(1+u)u + av,
+ v (vo + w) - wx (v - wo)]
Expanding the reciprocals of the right sides of
Eq. (7) by the binomial theorem, we obtain, to
first degree terms,
ds/ds* = 1 - u, s/s* = 1 - vo+
a
Only the first degree terms are needed in these ex-
pansions, since the first degree terms lead to second
degree terms in the preceding formula for 7-0.
Eliminating ds/ds* and 5s/ds*, we obtain, to
second degree terms
Uo " - wo O
'xo = a-+ v - w a
e ve + w (9)
a a
The axial displacement component u is evi-
dently small compared to the radial component w.
Consequently, the term u02/2a2 will be discarded
from Eq. (8). Also, the terms uxv and uo(vo+w)/a2
will be discarded from Eq. (9), since they are
small compared to the respective additive terms
Vx and uo/a. A comparison of the relative magni-
tudes of v and w is difficult. It has been found
that the quadratic terms in v exert a predominant
effect in some problems of buckling of rings. Con-
sequently, all the quadratic terms in v and w will
be retained.
Eqs. (8) and (9) merely give the incremental
strains due to buckling. The strains just before
buckling are denoted by f,(0) and eo(O). The initial
shearing strain is evidently zero. Consequently,
when the quadratic terms containing u are neg-
lected, the complete formulas for the strain com-
ponents are
1 1
E = ,(o) + u + Wa x + w ( w 2
2 2
o = o0o + ) + W ±w 1 v - W (10)
a 2 a
Us - ( v - We W
Yx0 - a + - W - a
4. Fourier Analysis of Displacement Components
Equations (10) express the membrane strains in
terms of the displacement components of the mid-
dle surface of the shell. In this article, the displace-
ment components (u, v, w) of the middle surface are
expressed in the form of Fourier series in 0. Also,
by the assumption that the shell buckles with zero
incremental hoop strain, the coefficients in the se-
ries for the v and w displacement components are
expressed in terms of a single parameter wi.
In view of the fluted pattern that a buckled
cylindrical shell adopts, the functions u, v, w may
be represented by Fourier series, as follows:
u = U0 + ul cos nO + U2 cos 2nO
+ us cos 3n6 + . . .
v = v, sin nO + v2 sin 2n6 + va sin 3nO + . .. 11)
w = wo + w1 cos nO + w2 cos 2nO
+ w3 cos 3n6 + .
Here, n denotes the number of complete waves
in the periphery. The coefficients ui, vi, wi are func-
tions of x alone. Only the terms to 3n6 will be
retained in Eq. (11).
The membrane strains that accompany buck-
ling are small, since large membrane strains cause
ILLINOIS ENGINEERING EXPERIMENT STATION
excessive strain energy. This fact is exemplified
if we deform a piece of sheet metal in our hands.
Although we can bend it easily, we cannot stretch
it noticeably. This circumstance implies that the
middle surface of a buckled cylindrical shell re-
mains approximately developable, since a wide
departure from a developable form would require
large membrane strains. Loosely speaking, the
"easiest" way for a shell to buckle is that which
entails the smallest membrane strains. Conse-
quently, we introduce the assumption that the
incremental hoop strain Aco that accompanies buck-
ling is zero. This assumption does not exactly yield
minimum strain energy, since the axial strain ex
and the shearing strain 7y0 are then too large in
some regions-particularly the end regions of the
shell. Consequently, the buckling pressure that is
obtained with the assumption Ae6=0 is slightly
too large, both for the infinitesimal theory and
the snap-through theory. The termination of the
series in Eq. (11) after the third terms also raises
the computed buckling pressures, since this ap-
proximation, like the assumption A0o=0, implies
artificial constraints on the buckling pattern.
Eq. (11) yields
ve + w Wo
S"- W= -- W + Oai Cos nO a2 COs 2nO
a a
+ as cos 3nO6 (12)
V --We- = 1 sin nO + 012 sin 2nO + 03 sin 3n0
a
where
invi + wi
a
vi + inwi
(13)
As was remarked previously, the term uo2 will
be dropped from Eq. (8). Then Eqs. (8), (12), and
(13) yield
A = w--° + ai cos nO + c2 cos 2nO + as cos 3n6
a
+ ± (fi sin nO + f sin 2nO + 03 sin 3n0)2
With the trigonometric identity,
sin ino sin jn = 2- [cos (i - j) nO - cos (i + j) nO]
we obtain, after regrouping terms,
-O = + (01 + f2 + + 3)
1
+ - (2ai + 0f02 + 203) cos nO
+ 2 - 3 + 13) cos 2nO
+ (2a3 - 1if2) cos 3nO
- (1 ft2 + 0103) cos 4nO
12 32(33 cos 5nO - 032 cos 6nO
1 (14)
Necessary and sufficient conditions for AEo to
vanish are that each coefficient in Eq. (14) vanish.
Hence,
033 = 0, 2 = 0, a3 = 0, a, = 0
a2 - 2 = 0,
These conditions yield
Wo + 1 a312 = 0
Wl
v1 - , v2 = -2-nw2,
n
Wo =
V3 = 0
(15)
(n - l/n)2 w12
W°
W2 = 4n-1' w3 = 0
W -2 = n (n - 1/n)2 w,2
1 n ' - 2a (4n2 - 1) '
V3 = 0
S= - (n - 1/n)2 ,2 (16)
4a
-(n - 1/n)2 w2
w2 = -4a(4n2- 1) ' w3 0
Eq. (16) expresses the coefficients in the v and
w equations (Eq. 11) in terms of wi. Since the
curve of a buckled cross section cannot intersect
itself, the admissible values of wi are restricted to
a finite range. If wi lies outside of this range, 6
ceases to be a regular parameter for the buckled
cross section.
Eqs. (10), (11), and (16) yield the following
expressions for the strains (where primes denote
derivatives with respect to x):
ex = ex(0) + u'0 + u't cos nO + u'2 cos 2n0
1
+ u'3 cos 3n0 + - (v' sin nO
+ v'2 sin 2nO)2 + (w'o
+ w', cos nO + w'2 cos 2nO)2
cc = e/»)
79 = -- ul sin nO - -- u2 sin 2nO
a a
(17)
Ot =i
Bul.443. BEHAVIOR OF CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE
3n
--- u sin 3n0 + v' sin nO
a
+ v'2 sin 2nO - (w'o + w'l cos nO
+ w'2 cos 2no) ( ---nw-) sin nO
With the trigonometric identities,
cos nO cos 2nO = - (cos nO + cos 3nO)
1
sin nO sin 2n0 = I (cos nO - cos 3n0)
sin nO cos 2n0 = (sin 3n0 - sin nO)
2
these equations yield
e. = [(o) + u'o + ± v'2 + '2
+ - w'o2 + 4 w'12 + - w'22
+ U'l + V'lV2 + W'oW'I
+ w'w'2] cos no
+ [u'2 - '2 4+ 12
+ w'oW'2] cos 2n0
+ ['3 - V'i'2 + w'1w'2 cos 3nO
+ - V4- v'22 + - w'22 cos 4nO
7 = - Ul + V' - W'o01
+ - w'201i sin nO
+ [ 2n 1 , 1
a 2
w'1i13 sin 2nO
1 3n
w'2/31 sin 3n0 - -- us sin 3n0
s2 a
(17)
J
Eqs. (18) are of the form,
Ex = ex() + Co + C, cos nO + C2 cos 2nO
+ Ca cos 3nO + C4 cos 4nO (19)
0xo = Si sin nO + 82 sin 2nO + S3 sin 3nO
where C and S indicate coefficients of cosine and
sine terms respectively.
Eqs. (16) and (18) yield
1
Co = u'o + (1 + 1/n2) w'12
(n - 1/n)4 w1w'12 2+ 4n' 1
16a2 2+ (4n2 - 1)2
C, = u' (n - 1/n)2 [2 + 3 ,
C1 = 4a 4n2-- w- w
C2 = u'2 + (1 - 1I/n2) w'12
(n - 1/n)4 w12w'l2
4a2 (4n2 - 1)
C = ui (n - 1/n)2 wiw'i2
4a (4n2 - 1)
- = (n - 1/n)4 W12w'2
16a2 (4n2 - 1)
n Wi1
S= - -n
a n
+ (n - /n)2 -
4a L
4n2 1 wiw'
2n (n - 1/n) (2n2 + 1) wIw'I
a 2 (4n2 - 1) a
1 3n (n - 1/n)a w12w'1
a - 3 + ( 1 a2
a 4 (4n' - 1) a
(18)
5. Membrane Energy
In Sections 3 and 4, expressions for the mem-
brane strains were derived in the form of Fourier
series; and the coefficients of the series for the
displacement components v and w were expressed
in terms of the single parameter wi. We now
proceed to develop an expression for the increment
of membrane energy due to buckling in terms of
the parameter wi and the coefficients uo, u1, Us, us
of the Fourier series for the displacement com-
ponent u.
The membrane energy is (16)
U-, Eha f L12 +
1- v2 Jo
+ 2veeo + - (1 - v) 7x02 do
(21)
in which E is Young's modulus, v is Poisson's
ratio, and h is the thickness of the shell. Eqs. (19)
and (21) yield
U = EhaL22 (e + C)2 + CL2 + 2
U 1 - 2 (eI(0) + C)2 22 C32
+ C 2 + 2 (fo(o))2 + 4v (eý(0) + Co) oe(O)
ILLINOIS ENGINEERING EXPERIMENT STATION
12
+ -I(1 - V) (81' + 82' + 832) dx
in which L is the length of the shell.
The membrane energy just before buckling is
U.() = Eha fL 2[2(x(o))2+2(e(°))2+4VEx(0)z
where ki, k2, . . . are functions of n only, and T,
X, T respectively represent the integrals that con-
tain ui, u2, and us. These integrals are
+= 2u'I - u w'l w'
2- a- a
2 (1 - v) k1]o -- (I -- 1 dx
L12 2
X = u'22 + 2 (1 - ) u22
+ kilu'2W12 + kl2U'2 W1 2 W12
U2 W1
+ (1 - v) kis 2 W - w'i/ dx
T= u'+2 U32
a a J
fL~ [ Q/ \
+ k14u'3 s-- w' 2
a
(26)
- (1 - v) k 1 2 w',u dx
a \ a J
The constants ki are defined by
S=3n4 + 2n2 + 3
16n4
(n - 1/n)
k2 = 25 n ) [2 (32n4 - 12n2 + 3)
+ 17 (4n2 - 1)2]
k= (n [96 + 44
16n2 (4n2 - 1)2 [96n6 44n4
- 17n2 + 5]
k4 =1+ 1
n2
k, = (n - I/n)4 2+ (4 2 1)2
k6 = 1/n2
S (n- i/n)2
7k = 4n (n2 -)2 (60n6 - 108n4
+ 45n2 - 6)
ks = ( 1)2 (32n4 - 24n2 + 5)
j (23)
Bul. 443. BEHAVIOR OF CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE
(n - 1/n)2 (8n12 + 1)
2 (4n2 - 1)
ko n (n - 1/n) (8n2 - 3)
2 (4n2 - 1)
S= 1 (1 - 1/n2)
k - (n- I/n)4
2 (4n2 - 1)
km_ (n2 - 1) (2n2 + 1)
n4n2 -1
k_ - (n - I/n)2
2 (4n2 - 1)
3n (n - 1/n)3
4 (4n2 - 1)
(27)
The notations Ki, K2, . . . , are reserved for cer-
tain combinations of the quantities ki, k2, etc.
6. Potential Energy of External Forces
The potential energy of the external forces
consists of two parts, the potential energy of the
axial force F and the potential energy of the lateral
pressure p. If the ends of the shell are rigid, the
Fourier coefficients ui, u2, us vanish at the ends.
Then the increment of potential energy of the
axial force is simply 2Fuo(L/2).
If the ends of the shell are not rigid, the poten-
tial energy of the force F depends on the way in
which the axial load is distributed. We assume
that it is distributed so that the axial stress o-x in
the cylindrical wall is constant at the ends, x=
+L/2. Then the increment of potential energy of
the force F is
ADQF = -2 fahhx(L/2)u(L/2)dO
Since ax(L/2) is constant, F = -27ahax(L/2). Con-
sequently
AQF = F f u(L/2)dO
7r Jo
With Eq. (11), this yields AQF=2Fuo(L/2). This
is the same result that was obtained for a shell
with rigid ends.
Since F = Xa2p, the preceding formula yields
ADF = 2Xa2puo(L/2)
Since, by symmetry, u vanishes at the center sec-
tion, x = 0, this equation may be written in an in-
tegral form, as follows:
L/2
AUF = 2Xa2p u'odx
(28)
To calculate the potential energy of the lateral
pressure p, we must determine the area A* of a
cross section of the buckled shell. The intersection
of the plane x = constant with the middle surface
of the buckled cylindrical wall is represented para-
metrically by y* = y*(O), z* = z*(O). These functions
are given explicitly by Eq. (1).
The area enclosed by the curve y*=y*(O),
z*=z*(0) is
A* = - y* do
0- a8 o
(29)
The sign on the right side of this equation is nega-
tive, since the positive sense of 0 runs clockwise.
Eq. (29) is a special consequence of Green's the-
orem.
By Eq. (1),
y* = a sin 0 + v cos 0 + w sin 0
z* = -a sin 0 - v0sin 0 - v cos 0
80
+ we cos 0 - w sin 0
With Eqs. (11) and(15), these equations yield
y* = (a + w0) sin0 - w1 cos 0 sin nO
n
+ wi sin 0 cos nO - 4n2 -nwo 1 cos 0 sin 2n0
+ 4n2 - sin 0 cos 2nO
= (a + w0) sin 0 - wo sin 0 cos 2n0
+ (n - 1/n) wi cos 0 sin nO
Consequently, if n>2, Eq. (29) yields
1 2TW3
A* = 7r (a + wo)2 - ir(1 -1/n2)w2 - 2(4n2-1)
By means of Eq. (16), w0 may be eliminated from
this equation.
The effect of the deformation before buckling
on the incremental cross-sectional area will be neg-
lected. Then the increment of cross-sectional area
due to buckling is AA =A*-ira2. Consequently,
AA = [ - k6wi2 + ke
where
1
k16 = (n2 - 1)
k17 = (n - 1/n)4 - 1 1 (31)
Although Eq. (30) has bee 2(4n derived - 1
Although Eq. (30) has been derived for n>2, it
ILLINOIS ENGINEERING EXPERIMENT STATION
remains valid for n = 2, as we see by carrying out
the integration specifically for n = 2.
The increment of potential energy of the lateral
pressure due to buckling is approximately
A, = fAAdL/2
Ja
Eq. (32) implies the approximation that the axial
displacement u does not influence the work of the
lateral pressure p when the shell buckles.
The total increment of potential energy of the
external forces due to buckling is A2 = A2F+A±,..
Consequently, Eqs. (28), (30), and (32) yield
,L/2 2 r fLI2 [ W1 2
ASI = 2Xa'pf u'odx + 2ra'p 2 -k16 --
+ k17 W -)4] dx (33)
7. Elimination of u0o from the Increment of Total
Potential Energy
The increment of total potential energy due to
buckling is AV= AU7+A+A7Ub, in which AUm is
the increment of membrane energy, A2 is the in-
cremental potential energy of the external forces,
and AUb is the incremental strain energy of bend-
ing. The slight axial bending that exists before
buckling will be neglected. Accordingly, AUb is
approximated by the total strain energy of bend-
ing Ub. A detailed analysis of the term Ub is devel-
oped in Section 12.
Since the term Ub does not depend on the axial
displacement u, Eqs. (23) and (33) yield
rEha rL/2 2
AV =-- 2u'o2 + k4U'ow 12
+ ksu'o (wi/a)2 w'12 - E2XaP --I u']o dx
+ 2Xa2p 0 u'odx
+ terms that do not contain u'o.
By the principle of minimum potential energy,
the axial displacement u provides a minimum to
AV. Consequently, u'0 minimizes the integrand in
the preceding equation. A necessary condition for
the value of the integrand 0 to be a minimum is
ao/au'o= 0. Furthermore, this condition is suffi-
cient to insure a relative minimum, since
2 _ 4irEha
au'o2 1 - V2
Consequently,
u' - k4w'12 - f k5 (wi/a)2 w'i2 (34)
Eliminating u'o from AV by means of Eq. (34),
we obtain
AV Ub + rEha fL 12 1 4
1- V2 0 k8
_4 a
1 r, 1± i-, Xap "11
+ (1 - V) k 1+V6 k4 X w'l
+2(1 - 7r Eh I
-1 (i--)k7T- + k5aP(W21w/12
2 1 L7± Eh I (-a )
I (/1 I w 2
-2 (1- k16 ap w 2
+2(1 () -
+ 2 (1 -) k17 ap wi' dx
+ rEha ( X + T)
I -wl V,
It will be assumed that
Wo cos --
W _ L
a n - 1/n
>(35)
7rX
Lr
- w cos Lx
L
in which Wo is a constant. Observations of buckled
cylindrical shells suggest that this is a reasonable
assumption. Since this assumption implies an ar-
tificial constraint, it possibly raises the computed
buckling pressure, but it cannot lower it. Eq. (36)
yields
2L 2( 34a 8W o
Sw14w'14dx = 256 (n - 1/n)8 L3
L w1w'4 dx =
7ri4a6W6 0
32 (n - 1/n)6 L'
fL/2 7r2a2Wo2
w'12dx =
fo 4 (n - 1/n)2 L
foL/2 2= 44 W40
do 16 (n - l/n)4 L
Bul.443. BEHAVIOR OF CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE
IL/2 22 a6 W6
wdx 32 (n - 1/n)6 L
fL/2 a2L Wo2
wxo 4 (n - 1/n)2
fL/2 3 a4Wo4 L
4do 16 (n - 1/n)4
Consequently, by Eq. (35),
AV = Ub + K6
I L
Sa4Xp
- K,4
L
- Ka2pL] Wo2 + [K1Eah
-K7 Eah a4Xp 2]
-K L K K--- - KnapL Wo4
I EaCh +K8 Ealh WL
KEah 8+rEha (q,±X-T)
+ 223- w + (+X+T)
in which
- 1 k
K 16 (1 - 2) (n - /n)4
37r5
256 (1 - ,2) (n2 1)2
3r (k2 - k2)
2 256 (1 - V2) (n - 1/n)8
517~
65536 (1 -2) (4n2 - 1)2
7r (ks - - k4k)
K6 --
32(1 - v2) (n - 1/n)6
n 2n - 1)2 + 4n2 -1 + 2n21
256 (1 - V2) (n2 -)2
K r k 7r2 (n 2 +
8 (n - 1/n)2 8 (n2 - 1)2
32 (n - 1/n)4
r2 [ 4n2+1 1
128 (4n2- 1)2
K6 = -3k
8 (1 + v) (n - 1/n)2
(37)
K7 =
73
8 (1 + P) (n2 - 1)2
73k7
32 (1 + v) (n - 1/n)4
73 [-F - n2
32 (1 + v)L (4n2- 1)2
2n2 - 1 3n2 - 4
+ 2 (n2 - 1) (4n2 - 1) 4 (n2 - 1)2
K8 -
64 (1 + v) (n - 1/n)6 512 (1 + v)
(4n2 1)2 4n2 - + 2]
Ž(38)
S= rki
16 - 2 (n - 1/n)2
Ku = S3rkn
7 8 (n - 1/n)4
128 2 (4n2 - 1)
4 rn2
4 (n2 - )
(39)
These constants have been tabulated (Table 1).
8. Elimination of u, from the Increment of Total
Potential Energy
Since the origin of x is the mid-section of the
shell, the displacement component u is an odd
function of x. Therefore, u\ is an odd function of
x. By the principle of minimum potential energy,
this function provides a minimum to V'. When w,
is eliminated by means of Eq. (36) the equation
for \IV becomes
L/2 7r2 3 . 2 3 Trx
= L i2 k9 2o u'l cos - sin
-- ( -- e . 7) 7 1 7r.
- (1 - v) u sin L- + (1 - v) --kioul
L L~x 1 L L2]
sin - cos -- + (1 ) 2 U dx
LC L 2 a 2
where, for brevity, w - 1/n
n - 1/n
With the
7rx
cos -
L
lrX
sin L-
L
(39)
trigonomet]
sin2 -)
cos 2
ric identities,
1 ( -x
= ( COS
4 L
1 (. rX
cos 3rx
L)
+
this yields
fL/2 7r 2 2aW ( rX
I= [ u'12 -ku' cos4
. 3rx
- sin ,
-cos-)
ILLINOIS ENGINEERING EXPERIMENT STATION
+(1-v)-- u sin + (1-v)-) 0 sll
. 1 n2 ,1|
+ sin 37rX kiou + I (1 - v) 2- u1 ]
Since ux(0)=0, integration by parts now yieh
= /2u k2u12 + 2A inrx
J0 [ U L L
L/2B . 37rx
+ u1 sin I dx
in which
A = (1 -v) k ~o - 3k9
2 (1 - v) 7r
a 2
7 n i1-
k^n_ -T-
For integrals of this particular type, Eu
equation of the calculus of variations is
necessary and sufficient for a relative minimui
Euler's equation for I is
, A . rx B . 3wx
u " - k u2 = - sin L- + B- sin B --
LThe general odd solution L Li
The general odd solution of Eq. (42) is
ui = CL sinh kx -
K10 =r (1 ) k0
16 (n - I/n)a
7r (1 - ) n (8n2 - 3)
32 (4n2 - 1)
KI - (1 - v) rn
2 (n2 - 1)
(47)
r = a/L
(40) Then, by Eq. (41),
A = (K1o - Kor2) Woa - KnWo
B = (Klo + 3Kor2) Wo3
kL = n 1 - V
2-kL =1 =r 2
(48)
Accordingly, Eq. (46) yields
1 _ -_ = WO2f( + [f2) r2f()] W04
+ [f4(ý) - r2f%(ý) + r4f6()] W0' (49)
where
ler's
both
m.(17)
(42)
AL . xrx BL . 3rx
- sin -sin (43)
a L /3 L
in which C is an arbitrary constant of integration,
and
a = 2 + k2L, = 97r2 + k2L (44)
Substituting Eq. (43) into Eq. (40), we obtain
1 A2L B2L (45)
- kC2L' sinh kL -44) 4
If there are no constraints at the ends that
affect the axial displacement u, the constant C
must be chosen to minimize I. Obviously, this
condition implies that C = 0. Then,
AA L B2 L
4a 4/3
- rKll2 -- Klo2
l (1) - 4 V) a 4(1 -p2)
2 rKjoKn K=rKKo
2(1-v')a ) 2(1 -V2)
-- = rKgKn --rK 9
f -2(1- V) fa ( -) 2)
9. Elimination of u2 from the Increment of Total
Potential Energy
Like ul, U2 is an odd function of x. This func-
tion must be chosen to minimize X. When wi is
eliminated by means of Eq. (36), the equation for
X becomes
X = f u's2 + 2(1 - v) n 2(2
Jo I a
+ T6224 (1 - 1/n2) (2 1 - cos 2r
S(n - 1/n)4W4 (1- 47rx] U
4n2 - 1 L ]
(1 - v) (n2 - 1) (2n2 + 1)2
2L (4n2 - 1) WUM2
sin 2L} dx
(51)
Evidently, A is negative and it reduces AV. This
circumstance might have been anticipated, since
the introduction of ul effectively gives the shell
additional degrees of freedom.
Let us set
K, = * k r3n (8n' + 1) (47)
8 (n - 1/n)3 16 (n - 1) (4n2 - 1)
Integration by parts yields
L/2 2A . 2xx
X = u'2 + 4k2u22 - - u2 sin --
[o L L L
2B . 47rx 1 r (1
L u2sin dx + r2(1
- 1/n2) W2us (L/2)
I
Bul. 443. BEHAVIOR OF CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE
where
A [(1 - ') (n2 - 1) (2n' + 1)
4 L 4n2 -
+ r2r2 (1 - 1/n2)] W
(53)
B 3r2 (n - l/n)4 4
8 4n2- 1
Sn 1-v
k = , r = a/L
Euler's equation for the integral X is
f A . 27x B . 4rx
u 2 - 4ku2 = -L sin L L sin -L-
The general odd solution of this equation is
AL . 2rx BL . 4rx
u2 = CL sinh 2kx + AL sin 2r- +-BL sin 7- (54)
4a L 47- L
where
a = 72 + k2L2, 7 = 4r2 + k2L2 (55)
Substituting Eq. (54) into Eq. (52), we obtain
X = kL2C2 sinh 2kL - 2L - 2L
16a 167
irACL 2rBCL
- - sinh kL + sinh kL
a2
+ 2L (1 - 1/n2) c2C sinh kL (56)
If the ends are not constrained against warping,
C must be chosen to minimize X. Then Eq. (56)
yields
A2L B2L 1 ra2 2 n
S= 16a 167 8k 2L2
- rA + 2--- tanh kL (57)
a 7
Set
7r (1 - P) n2 (2n2 + 1)
12 4 (n2 - 1) (4n2 - 1)
73 73
K = 3 K18 = 2-
K3 4 (n2 - 1)' 8 (4n - 1)
Then, by Eq. (53),
A = (K12 + Kiar2) Wo2 B = Ks8r2W4
1 n 1 - (59)
2 2r 2
Consequently, to sixth degree terms in Wo,
r X
- TL - WO'41(i) - Wo02(6 ) (60)
1 - v2 L
where
1() = 7r (K12 + Kiar)2 (61)
16 (1 - v2) a
ra tanh 2 7rTr2
+ 16(1 - 2) 2 (n2- 1)
K12 + K1ar2 2
a I
ra Kisr2 tanh 2 rr2
4 (1 - v) y L 2 (n2 1)
K12 + K3r ]
a J
(61)
Numerical calculations show that the eighth de-
gree terms in Wo are entirely negligible.
10. Elimination of us from the Increment of Total
Potential Energy
The Fourier coefficient u3 is an odd function
of x that must be chosen to minimize T. When w,
is eliminated by means of Eq. (36), the equation
for T becomes
12( I 9 9 1 ao , .1 rX irx
T = u'2+9k232+ 2 U'3 sin2 7 cos TL
+ (1- v) k-- s cosr sin dx
L 3 L Li
With the trigonometric identities,
* (irx x _X 1/ vx 3rx\
sin2 cos - = I cos r- cos L/
iX \. irx 1 ( lr 3X )
cos2 sin 1- sin -- + sin-3rx-
iL in 4 +iL L/
this equation yields
= L/2 - 2kl4a 2C3 / rX
= L [,'3 + 9k2U3 + 4L2 U3 (cos
3rax\ .rkW3 ( . rx
- cos --1 + (1 - V) 4L-- usin L
3rx
+ sin 3- )] dx
Integration by parts now yields
T = fL 'a2 + 9k2u2 + -- U3 sin
J0 ~ ±~ ± L L
2B . 3rix -
- us sinL ] dx (62)
where
A = 3 [r2k14r2 + (1 - P) kis]
8 (63)
B = 78 [3r2kl4r2 - (1 - v) k15], r = a/L
Euler's equation for the integral T is
- 9k A . s x B . 3rx
" - 9 ku = - sin L L sin L
ILLINOIS ENGINEERING EXPERIMENT STATION
The general odd solution of this equation is
AL . xrx
us = CL sinh 3kx - a sin L
+ BL sin 3rx.
9a L
where C is a constant of integration, and
a = 72 + k2L2, 5 = 7r2 + 9k2L2 (
Substituting Eq. (64) into Eq. (62), we obtain
3 A2L
T = 2 C2kL2 sinh 3kL 4- AL
To minimize T, we set C= 0. Then,
A'L
4T-
B2L
36a
B2L
36a
The following notations are introduced:
Krk = k __24 -=7n
8 (n - 1/n) 16 (n2 - 1) (4n2 -
K5 = ir(1 - v) ki _ 3r(1 - v) n
S- 8 (n - 1/n):1 32 (4n2 - 1)
Then, by Eq. (66),
where
T7 T = - Wg(6)
1 - V2 L W
gr [ (K14r2 + KI))2
g 4(1 - v2)
(3Ki4r2 - Ki5)2
9a
kL n 1-v
2 2r \ 2
11. Further Simplification of AV
In Sections 5 to 10, we have developed ex-
pressions for the various components of the in-
crement of the total potential energy due to buck-
ling. In this article, we proceed to further simplify
the expression for AV (Eq. 38).
Set
h 1 1 _
p=KE -, r =a/L, i =-- \
a
Then Eqs. (38), (49), (60), and (68) yield
AV Ub
AhL= - L + (bl - Kai) Wo2
EahL EahL
+ (b2 + Ka2) W04 + b3Wo6
in which
a, = K4Xr2 + Kiu
a2 = - K5Xr2 + K17
b = f + Ker2
b, = f2 - (K, - f,) r2 + Kir4 - 0i
b3 = f4 + (Ks - f5) r2
+ (K3 + f) r4 - 2 - J
An eighth degree term in Wo has not been included
in Eq. (71), since numerical calculations show
that it is very small. The constants (ai, a2) come
from the potential energy of the external forces,
and the constants (bi, b2, bs) come from the mem-
brane energy.
The constants Ki, K2, . . . have been tabulated
for v = 0.30. (Table 1). With these data, the con-
stants a,, a2, bl, b2, b3 have been tabulated for X = 7r.
(Tables 2 to 20). It is to be noted that X affects
only at and a2; it does not enter into bi, b2, bN. In
the construction of the tables, it was convenient
to treat ý as an independent variable. The corre-
sponding values of L/a have been included in the
tables.
12. Strain Energy of Bending of an Elastic Cylinder
In Eq. (71), AV is expressed in terms of the
single parameter Wo and the strain energy Ub due
to bending. By the theory
of this article, Ub is also
expressed in terms of Wo.
Also, the strain energy of
reinforcing rings is ex-
pressed in terms of Wo.
Thus, the increment of the
LoLai potential energy is
reduced to a function of Fig. 3 Arc of Buckled
the single parameter Wo.
For an elastic cylindrical shell, the strain en-
ergy of bending, per unit area of the middle sur-
face is (18)
D [K,2 + K02 + 2VKKo + 2 (1 - v) T2] (73)
where K. is the change of curvature in the lon-
gitudinal direction, K0 is the change of curvature in
the circumferential direction, and r is the local
twist. The flexural rigidity D is defined by
Eha
D =
12 (1 - v')
To express Ko in terms of the displacement
components (v,w) of the middle surface, we refer
(71) to Fig. 3. The arc in the figure represents a part
of the cross section of the middle surface of a buck-
led cylindrical shell. Since, by assumption, there
(72) is no incremental hoop strain due to buckling,
ds* = ds = adO. Also, by Fig. 3, Rdo = ds, where
*
(74)
Bul. 443. BEHAVIOR OF CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE
R is the radius of curvature of the arc. Further-
more, we see by Fig. 3,
sin 0 = - dz*/ds, cos 4 = dy*/ds (75)
Differentiating these equations, we obtain
d4 dez* 4de sin - (76)
ds cos 4 - ds2 ' ds d2 =
By geometry, 1/1 = d4/ds. Consequently,
1 - d2z*/ds2
R cos 4)
_ -d2y*/ds2
sin 4
Introducing the parameter 0 we obtain by Eq. (77)
1 Zoo* 1 Yoe*
a ye* a zo*
Since (yo*)2 + (zo*)2 = a2, this equation yields
Ze*yoe* - yo*zoo*
By Eq. (1),
yo* = a cos 0 + vo cos 0 - v sin 0
+ We sin 0 + w cos 0
zo* = - a sin 6 - vo sin 0 - v cos 0
+ we cos 6 - w sin 0 .(79)
yoo* = - a sin 0 - 2ve sin 0 + oo cos 6 - v cos 0
+ 2 we cos 0 + Wee sin 0 - w sin 0
zoo* = -a cos - 2ve cos 0 - vooe sin 0+v sin 0
- 2we sin 0 + Wee cos 6 - w cos 0
Eqs. (78) and (79) yield
1 1 1
R -3 f[a + (vf + w)] [a + 2 (v + w)
- (woo + w)] - (v - we) [(DVe + we)
- (v - we)]}
By Eqs. (12) and (16),
(80)
vo + w = wo - Wo cos 2n0
v - we = (n - 1/n) wi sin nO
Wee + w = Wo - (n2 - 1) w cos nO
- Wo cos 2n0
voo + we = 2nwo sin 2nO J
Since Ke0= 1 - Eqs. (80) and (81) yield
R aa
1 3 1 W2
cos 2n0 + ± n (n - 1/n) WoWi cos 3n6
2
+ - Wo2 cos 4n}
2- 1
With Eq. (16), this yields
1
a Ko = nW cos nO + - nW3 sin2no cos nO
2
+ 1 W4 sin4 nO
4
where W = (n - 1/n) wi/a
Eq. (82) yields
Da K2dO = -- Eh n2W2 - 12 W4
2 j e 24 (1 - V2) a 4
1 1.5 7
+ - n2 W6 + T__ WI
Subsequent calculations show that rarely, if ever,
1
does W exceed the value I. Consequently, the
eighth degree term in W is quite negligible, and it
will not be included in the subsequent analysis.
Eq. (36) may be written
rx(
W = W cos (84)
Eqs. (83) and (84) yield
1 D L/2 2 2f d rEn2hL [W2
2 D L/2d o 48a (1 - v2) 0
+ -W4+ 256 W06 (85)
Eq. (85) represents the principal part of the
strain energy of bending. Since the longitudinal
curvature is a minor effect, we shall use the ap-
proximation,
(86)
Kx - XX
Then, by Eq. (11),
Kx = - (W" + wl" cos nO + w2" cos 2n0) (87)
Consequently,
1 fo2T r-Eh a w
I Da f2dO = 24 (1 -iE 2) (2w"o2 + w"& + w"22)
With Eq. (16), this yields
1 Daf 2Kd 7rEhSa \ W"2
2 7o 24 (1 - v2) (n -/n)2
+ 2 +
+[ 2 (4n2 1)2
[W'14+2WW'2W"+±W2W"2]} (88)
Eqs. (84) and (88) yield
1 DafL/2 dx K2dO = d 7r-Eh3a3 W°2
2 Da -/2 48(1- 2)L3 (n-l/n)2
1+ 2
+ 8 [2
ILLINOIS ENGINEERING EXPERIMENT STATION
+ (4n - 1)2 W0 (89)
Eqs. (84) and (87) yield
2vKxKodO = - [ w"oW4 + w"1 (nW
+I nW" - w"2W4]
With Eq. (16), this yields
2VKxKodO = -2rv' (W'2 + WW") W4
+ -1 (nW + nW)
W4 (W2 + WW")
+ 16(4n2-1) (W' + WW")
Eqs. (84) and (90) yield
1 rL/2 2r 73rvEha n2
-Daj dxo 2vKxKodO-- 24(1_-2)L 2-
2 J-L/2 Jo 24(1-v)L n I'-1
(90)
1
By Eq. (16), wo = - 4 aW2. Consequently,
2a (1 - v) 2 dO = 27r (1 - ) a W'2
+ 3 '2 + - W4W'2)
4 32
Hence, by Eq. (84),
L/2 2r a 2W=2
2a (1 - v) E/dxf T2d0 = r(1 - ) L- W
+± W04± W'6 (96)
By Eqs. (73), (85), (89), (91), and (96), the
total strain energy of bending of an elastic cylin-
drical shell, excluding the strain energy of rein-
forcing rings, is determined by
Ub h2
EaL = a 2 (1 c2Wo2 + c3Wo4)
where
(97)
r = a/L
S 32Wn2 W 1 [3
32(n2-1) 64 L
2 1 w
4n2-l1 j o
(91)
The twist r is defined by T = a4/ax (Fig. 3).
Since sin 4 = - az*/as and cos ( = ay*/as, (see
Eq. 75),
SCOS 4 = -
T sin 4r = --
Hence,
02z* 1
9 - zxo*,
8xas a
a2y* 1
8xas a e
T cos2 b = - yo*zo*, 7 sin2 4 = -a- zo*y~o*
Adding these equations, we get
a2r = zo*yxo* - yo*zxo* (92)
Eqs. (1) and (92) yield
a2r = a(v' - wo') + (vo + w) (v' - we')
- (ye' + w') (v - wo) (93)
Consequently, by Eqs. (16) and (82),
aT = aW' - - Ww'o + 3 W'wo sin nO
+ 1- (Ww'o - woW') sin 3n0 (94)
Hence,
2a (1 -) 2(1) = [(aW'
3 3 2
32 w + )
+ (Ww'1 - woW')2] (95)
cl- - n + + 22 r 1
+ ]
S(n - )n2 -
c -256 (1- 2) 2
3 (4n2- 1)2
c3 = 12288(1 - ) 5 n+2 5
+ -8 1
'4n 2 - 1I/
'(98)
The constants cl, c2, c3 have been tabulated (Ta-
bles 2 to 20).
The strain energy of a reinforcing ring is
U, = 1 a "EIKo2dO
in which I is the moment of inertia of the cross
section of the ring. Consequently, if I is constant
for a ring, Eq. (83) yields
Ur = El [n2W2 + 1 n2W4 + n2W6]
2a 14 32
(99)
The eighth degree term in W has been discarded
from Eq. (83).
Eq. (84) determines the value of W for any
ring. The strain energy of all reinforcing rings is
represented by E Ur, where the sum extends over
all rings.
Bul. 443. BEHAVIOR OF CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE
It is assumed in the derivation of Eq. (99) that
the centroidal axis of the ring coincides with the
middle surface of the shell. Actually, this condition
rarely occurs in practice. Possibly Eq. (99) can be
retained for off-center rings if El is modified to
take account of an effective width of shell that
acts with the ring. However, this problem is not
examined in the present investigation.
Eqs. (71) and (97) yield
AV
Eahl = (B1 - Kai) W02 + (B2 + Ka2) W04
Ea hL
IUr
+ B3Wo6 + - -
EahL
in which
h2 h2
B1 = b1 + cl -- B2 = b2 + c2 -a2
P- 1 h2
3: - 3 3 - t'3
(100)
(101)
The constants a,, a2, bi, b2, b3, C1, C2, C3 have been
tabulated (Tables 2 to 20). The last term in Eq.
(100) representing the effect of reinforcing rings,
is a sixth degree polynomial in Wo, in which only
even powers occur (see Eq. 99).
III. POTENTIAL ENERGY OF A SHELL WITH RIGID ENDS
13. Shell with Rigid Ends
In the preceding analysis, it has been assumed
that the ends of the shell are free to warp out of
their planes; that is, that the ends of the shell im-
pose no restrictions on the axial displacement u.
If the ends of the shell are rigid plates, the Four-
ier coefficients ul, u2, u3 (See Eq. 11) vanish at
the ends. Accordingly, the functions T', X, and T
must be modified. Numerical computations show
that the effect of u3 is quite small, and conse-
quently the function T will be discarded.
Eqs. (40), (41), (42), (43), (44), and (45) re-
main valid. However, the constant C must be
chosen so that u, vanishes for x = L/2. Conse-
quently,
where
(() = 1K2i(ýl
1 - v2
27rK0oK,1 (F2 - F1)
1 2
2, KK2r 11 (F, + 3F2)
4(ý) =
rK102 (F1 - 2F2 + F3)
1 - v2
2rK9Kio (F1 + 2F2 - 3F3)
(-) = -- --- 2
rK92 (F1 + 6F2 + 9F,)
1 1 2
A B kL
C = - csch -
a 2
Eqs. (45) and (102) yield
LA2
4a
+2(A - B 2L coth
in which S is defined by Eq. (48). Consequ
T = A2Fi (F ) - 2ABF2(ý) + B2F3(2 )
Lwhere
where
2 coth - (2 + 2)
(72 + 4ý2)2
F,( - 2 coth ý
(7r2 + 4ý2) (97r2 + 4(2)
F 2g coth ý - (2 + 9 7rV/4)
F3() (9r + 4ý2)2
Hence,
- L = W1(Q) + [2()
+ 2 34(ý)] WT' + [14(f)
- r2(() + r^46(ý)] W06
It may be shown that the expression on the
right side of Eq. (104) is a negative definite quad-
ratic form in A and B. Consequently, I always
reduces the membrane energy. For a shell with
rigid ends, the functions #ik, . . . e replace the
functions fi, . . . f6 of Eqs. (50) and (72).
(103) Turning attention to the function X, we ob-
serve that Eqs. (54) and (56) remain valid. The
ently, end condition, u2 = 0, obviously requires that
C = 0. Consequently, Eq. (56) yields
(104)
Hence,
A2L
X- 16
16a
B2L
167'
(108)
(109)
1 - V W
where
S(105) X ) = r (K12 + Ki3r2)2 (110)
(105) X() 16 (1 - 2) (2 + 4ý2)2
Accordingly, the function X, replaces the function
1, of Eqs. (60) and (72). The functions 02 and g
are discarded from Eq. (72). The second term in
Eq. (108) has been neglected, since B3 is of eighth
degree Wo.
With the above modifications, the preceding
theory applies for a shell whose ends are rigid
(106) plates.
(107)
Fi(ý) =
IV. PRESSURE-DEFLECTION RELATIONS
14. Load-Deflection Curves
For a given shell and a given value of the pres-
sure p, we may plot a graph of AV/(EahL) versus
WTV by means of Eq. (100). The forms of the
graphs, corresponding to several values of p, are
illustrated by Fig. 4. The pressures indicated on
the curves are such that pi < p2 p', the shell will maintain a buckled
form if it is initially forced into that condition by
external disturbances. Since initial disturbances
and imperfections always exist, von KArman and
Tsien(15) originally conjectured that pi is the maxi-
mum safe pressure.
Tsien(10) later concluded that, although pi is the
greatest lower bound for the pressures at which
buckled configurations can persist, there is little
danger of a shell passing into a buckled configura-
tion unless, in doing so, it loses potential energy.
In Tsien's words, "The most probable equilibrium
state is the state with the lowest potential energy.
- This principle of lowest energy level is verified
by comparing experimental data with theoretical
predictions. However, in view of the prerequisite
that arbitrary disturbances of finite magnitude
have to exist, the buck-
ling load determined by
this principle may be
called the 'lower buckling
load.' The classic buck-
ling load that assumes
only the existence of dis-
turbances of infinitesimal
magnitude may be called
the 'upper buckling load.'
Of course, by extreme
care in avoiding all dis-
Deflection turbances during a test,
Fig. 7. Pressure.Defletcion the upper buckling load
Curve for Imperfect Shell
can be approached. The
lower buckling load, however, has to be used as a
correct basis for design."
According to Tsien's reasoning, the pressure P2
(Fig. 4) is the maximum safe pressure. This is the
pressure at which the potential energy of the un-
buckled form ceases to be an absolute minimum.
Fig. 8. Statistical Distribution Fig. 9. Determination of Tsien
Curve for Imperfect Shell Critical Pressure
It will be designated as the "Tsien critical pres-
sure." The curve in Fig. 4 that corresponds to p2
is tangent to the axis of Wo at a point to the right
of the origin.
The buckling pressure of an imperfect shell
poses a statistical problem. Load-deflection curves
for imperfect shells have the general form shown
by Fig. 7. This figure is to be contrasted with the
load-deflection curve for an ideal shell (Fig. 5).
Donnell has emphasized that the designer is con-
cerned principally with the maximum pressure on
the load-deflection curve (denoted by p, on Fig.
7). Since the falling part of the curve (dotted in
Fig. 7) represents unstable equilibrium configura-
tions, the maximum point lies at the boundary of
the stable range. Therefore, ps is the Euler critical
pressure for the imperfect shell. This pressure may
be expressed conveniently as a fraction f of the
Euler critical pressure p4 for a perfect shell; that
is, f = p5/p4. Since p5 depends on initial imperfec-
tions in the shell, tests of a large number of shells
with the same dimensions would lead to a statistical
distribution curve of the general form shown in
Fig. 8. The ordinate 0 of this curve is defined by
the condition that (df is the probability that a
random shell will fall in the interval (f, f + df).
The specification of a safe pressure is somewhat
arbitrary. Under some circumstances, an operating
pressure would be considered safe if 95% of all
specimens would fail above that pressure. In other
cases, the safety limit might be raised to 99%, or
some other value. Tsien implied that his definition
of the lower critical pressure provides a value that
lies near the maximum safe pressure (Fig. 8). At
present, this conclusion is largely conjectural, but
since Tsien's critical pressure affords a ready em-
pirical criterion for safe design of shells, it has
been charted by Euler critical pressure (Fig. 13).
The Tsien critical pressure is determined by the
equation AV = 0. If we select the intercepts of the
curves of Fig. 4 with the wo-axis, we can plot
the resulting relation between p and Wo. The graph
Bul. 443. BEHAVIOR OF CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE
has the general form shown in Fig. 9. The mini-
mum ordinate of the curve is the Tsien critical pres-
sure, and the corresponding value of Wo determines
the deformation of the buckled shell, if the applied
pressure equals the Tsien critical pressure. The
intercept of the curve with the p-axis is the Euler
critical pressure. Although Fig. 9 looks like Fig. 5,
the two curves are distinct, since they are derived
by different formulas. Fig. 9 is not a graph of
equilibrium configurations; it merely serves to show
how the Tsien critical pressure may be computed.
If there are no reinforcing rings, and if the shell
is elastic, the equation AV = 0 yields
B, + B±2WO2 + B3W04
a, - a2Wo2
(112)
The form of Eq. (112) remains valid for a cylinder
that is reinforced by elastic rings, if the coefficients
bi, b2, b3 are modified suitably (See Eq. 99).
Plotting K versus Wo by means of Eq. (112),
we obtain a curve that is essentially equivalent to
Fig. 9, although the ordinate is K rather than p.
The intercept of the curve with the K-axis is de-
termined by setting Wo = 0 in Eq. (112). Conse-
quently, the value of K corresponding to the Euler
critical pressure is
Ki = Bi/ai (113)
The notation K, denotes the value of K that is
obtained by the infinitesimal theory of buckling.
The value of K that corresponds to the Tsien
critical pressure, denoted by K,t (the subscripts
"st" denote snap-through) is the minimum value
of the function defined by Eq. (112). To minimize
K, Wo2 must be a root of the equation,
W° = -- [1 + a +2B - 2aio (114)
Although Eq. (114) may be solved by the quad-
ratic formula, it is usually poorly conditioned for
this type of solution, and it is most easily solved
by iteration. The procedure is to obtain an ap-
proximation of Wo2 by neglecting the fourth degree
term on the right side of Eq. (112), and to use
this approximation to refine the first approxima-
tion. The process of refinement may be iterated,
and it converges quite rapidly. The value of Wo2,
determined by Eq. (114), must be substituted into
Eq. (112). Thus, the Tsien critical pressure is de-
termined. It is represented by p - KctEh/a, where
the subscripts "st" denote "snap-through." The
Fig. 10. Graphs of f(p) and 0(p)
buckling coefficient Kst is plotted versus WTV in Fig.
13 for values of a/h of 100 and 1000 and X = 7r.
16. Effect of Assumptions on the Tsien Critical
Pressure
It is well known that arbitrary assumptions
about the deformation of a structure cause the
computed value of the Euler critical load to be
too high. The same conclusion applies for the Tsien
critical pressure. To verify this assertion, we ob-
serve that the potential energy V is a functional
of the displacement components (u, v, w) and the
pressure p. Let Class I be the set of all continuous
differentiable functions (u, v, w) that satisfy the
forced boundary conditions.
It has been found that there exists a pressure
p' for a given shell, such that a buckled configura-
tion will persist if p >p'. The buckled configura-
tion, being stable, provides a relative minimum
to V among functions of Class I. This minimum
of V depends only on p; hence, it will be denoted
by f(p). (See Fig. 4.)
It has been found that there exists a pressure
p" (the Tsien critical pressure), such that f(p) > 0
in the range p' < p < p", f(p") = 0, and f(p) < 0
in the range p > p".
Let Class II be a given subset of Class I, as
determined, for example, by assumptions about the
nature of the deformation pattern. We have em-
ployed two assumptions of this type: (1) the shell
buckles without incremental hoop strain. (2) The
function w, is represented by a single term of a
Fourier series in x (see Eq. 36). When the func-
tions (u, v, w) are restricted to Class II, the mini-
mum value of V is 0p(p). If our assumptions are
good, 0(p) differs but slightly from f(p).
Since Class II is a subset of Class I, q(p) f(p).
Consequently, the graphs of f(p) and 04(p) have
ILLINOIS ENGINEERING EXPERIMENT STATION
the general features shown by Fig. 10. The essen-
tial characteristics of these functions are that 0
and f are positive for small values of p and nega-
tive for large values of p, and that the curve repre-
senting 01(p) lies above or in contact with the
curve representing f(p).
The Tsien critical pressure is the intercept of
the graph of f(p) with the p-axis (Fig. 10). Evi-
dently if 0(p) is used as an approximation for f(p),
the computed value of the Tsien critical pressure is
too high.
17. Potential Energy Barriers
The maximum on the curve corresponding to p2
(Fig. 4) represents a potential energy barrier that
the shell must cross to arrive at the buckled form,
if the pressure is exactly equal to the Tsien criti-
cal pressure. Therefore, it serves as a rough indica-
tion of the imminence of snap-through. The value
of this maximum may be derived from Eq. (100).
For brevity, Eq. (100) is written as follows:
y = ax - bx2 + cx3
& = EahL' x = Wo (a)
a = B1 - Kai, b = (B2 + Ka2), c = B3
The graph of y is as shown in Fig. 11.
The maximum value of y is determined by dif-
ferentiation with respect to x. Thus,
a - 2bx + 3cx2 = 0 (b)
The roots of Eq. (b) are
b [1 < 3ac
x = 1 - 1 b2
Sc(c)
X2 = I ( 3ac+
=3c 1t 1 b2
Fig. 11. Potential Energy
Barrier
The root x1 provides the
maximum, and the root x,
provides the minimum
(Fig. 11). Since the value
of the minimum is zero,
a - bx2 + cx22 = 0
Eqs. (c) and (d) yield
b = 2 \/ac
Consequently, Eq. (c) yields
(d)
(e)
(f)
x2=
Eqs. (a) and (f) yield
4c
ymax. 27 x X2
In terms of our previous notations, this equation
yields,
(AV 4 hL B-
Eamax3 27 a2 B3W
(115)
where 1Wo is the root of Eq. (114) that corresponds
to the point of tangency with the x-axis (Fig. 11).
Eq. (115) is plotted in Fig. 13 for a/h = 1000,
X = and for n= 2 to 20.
1 8. Numerical Example
Consider a shell with the following proportions:
a/h = 100, L/a = 0.6010. The value L/a = 0.6010
is selected to coincide with a tabulated value. This
condition is unimportant. If a selected value of
L/a does not appear in the tables, interpolation
must be used.
(a). Euler Critical Pressure for Shell with Simply-
Supported Ends and No Axial End Constraint.
The equilibrium pressure corresponding to any
given state of deformation is represented in the fol-
lowing form: p = KEh/a. The constant K is evi-
dently equal to the compression hoop strain that
exists in the unbuckled shell at pressure p. The
value of K corresponding to the Euler critical pres-
sure is denoted by Ki. By Eq. (113), K1 = B,/a,,
where B, = b + ch2 /a2. Accordingly, in this ex-
ample, B, = bi + 0.0001 c,. The constants al, bi,
c, for a shell with simply supported ends have been
tabulated (Tables 2 to 20). The number of lobes
in the buckled form must be determined by trial
to minimize Ki. For very long shells, n = 2. In
general, n increases with decreasing L or h. In the
present example, L/a is small, but h/a is large.
Therefore, a moderate value of n - for example,
n = 10 - might be estimated. It is found by several
trials that the value n = 9 actually provides a
minimum to Kj. For n = 9, Table 9 yields (with
A = r) a, - 0.9327, bi = 0.0006329, c, = 10.45.
Consequently, B, = 0.001678. Accordingly, Eq.
(113) yields Kj = 0.001678/0.9327 = 0.001799.
The condition A = 7r indicates that a uniform
hydrostatic pressure acts on the ends of the shell.
If A = 0, the axial force due to the pressure on the
ends is removed. Then a1 = 0.7952, as noted at the
bottom of Table 9. Since bi and c, are independent
of X, B, has the same value as before. Thus, if
1 a
xx 3 c
Bul. 443. BEHAVIOR OF CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE
A = 0, then K, = B,/a, = 0.001678/0.7952 =
0.002110. Accordingly, in this example, the hydro-
static pressure on the ends reduces the Euler criti-
cal pressure about 15%.
For the case A = 0, von Mises(1' derived a for-
mula that may be put in the following form:
Ki = 1 - + h22
S 1)(1 + n2L 12(1 - P) a
n -2n2-
. n + n L2 (116)
1+ n2 L
In the present numerical example, von Mises' for-
mula yields K, =0.001900. This result is about 10%
lower than that computed by the present theory.
There is seemingly a systematic deviation between
the present infinitesimal theory and von Mises'
theory, for thick shells that are short compared to
their radii. However, in all cases, the Tsien crite-
rion yields lower values than the von Mises' theory.
(b) Pressure-Deflection Curves for Elastic Shell
with Simply-Supported Ends and No Axial End
Constraint.
The pressure-deflection curve is essentially a
graph of K versus WTV,, where K is defined as above.
This curve may be plotted by means of Eq. (111).
Setting n = 9, we obtain from Table 9 (with A =X ),
a, = 0.9327, a2 = -1.270,
bi = 0.0006329, b2 = -0.02334, bN = 0.3077,
c, = 10.45, c2 = 14.89, ca = 0.1124.
The b's and c's are independent of A. The quanti-
ties B1, B2, B3 are determined by B, = bi + ch2/a2,
B, = b, +c, h'/a2, B3 = b + c+. h2 /a2. In the pres-
ent example, h2/a2 = 0.0001. Hence,
B, - 0.001678, B2 = - 0.02185, B3 = 0.3077.
Substituting these values of a, a2, B, B,, B3 into
Eq. (111), we obtain an equation whose graph is
shown in Fig. 12. Since the curves corresponding to
n = 8 and n = 9 intersect each other, the curve for
n = 8 is also plotted.
The intercept of the curve for n = 9 with the
vertical axis is the Euler critical hoop strain,
Ki = 0.001799. The minimum value on the curve
for n = 9 is Kmin = 0.001170. The minimum value
on the curve for n = 8 is Kmin = 0.001080. This
is the lowest minimum that occurs for any value of
n. Therefore, it determines the lowest pressure at
which the shell will maintain a buckled form, if it
is perfectly elastic.
2400
a/h =/00, A7 z-
* denotes rigid ends (Article 13) n=9
* denotes Tslen coefficient
2200 _____
1800
oo-\\ \ L/o 0.6010 I
600oo -
1400 - -- -- - - --- f - ---
1200 ._"
/000 - ----
L/a /59 n-8
600 --._
400-
U .05 JoIU .IS
Wo
Fig. 12. Buckling Coefficient K versus
Deflection Parameter Wo
Hence, in this example, the lowest pressure at
which the elastic shell will maintain a buckled
form is 60% of the Euler critical pressure.
The value of K corresponding to the Tsien criti-
cal pressure has been denoted by K,t. For this case,
Kt = 0.00133. This result may be obtained from
Eq. (112). The Tsien critical pressures have been
marked on the curves of Fig. 15. The Euler critical
pressure for this case (see above) is Ki = 0.001799
which is 35% higher than Kt.
(c) Effect of Rigid Ends.
If the ends of the shell are hinged, but the axial
displacements are constrained by the action of
rigid end plates, the buckling pressure is increased
significantly. Eqs. (111) and (113) remain valid,
but the coefficients b,, b,, b, are changed. The con-
stants a,, a,, c, c2y, C3 are not altered.
The constants b1, b,, b, have not been tabulated
for a shell with rigid ends. Consequently, their
ILLINOIS ENGINEERING EXPERIMENT STATION
values must be computed by means of Eqs. (105),
(107, (110), and (72).
The constraint imposed by rigid end plates
generally increases the number of lobes in the
buckled form. Trying n - 10, we obtain by Eq.
(48), e = 1.778. Hence, by Eq. (105),
Fj(ý) = -0.003675, F2(Q) = 0.001648,
Fa(Q) = -0.002098.
Using the values of the K's from Table 1, we ob-
tain by Eq. (107),
,i = -0.0001565,
3a = 0.000382,
P5 = 0.02203,
IP2 = 0.005605,
T4 = -0.05900,
IP = -0.006755.
Eq. (110) yields X1 = 0.002364.
In Eq. (72), the functions fl, f2, . . . are to be re-
placed by i,, !2, . . ., respectively. Also, X, replaces
0,. The functions ,2 and g are discarded since they
are negligible.
Hence, bi = 0.0006858, b2 = - 0.01210, b3 =
0.2929. Interpolating values from Table 10, we
obtain
al = 0.904, a2 = - 1.27, c, = 11.68, c2 = 15.14,
c3 = 0.139.
Since B, b, + ch2/a2, B 2= b2 + ch2/a2, and
B, = b + ch2/a2, B -= 0.00185, B2 = - 0.0106,
B3 -= 0.293. Eq. (111) now provides a graph of K
versus Wo (Fig. 12).
It is necessary to repeat the calculations for
several other values of n. It is found by trial that
the value n = 11 provides the lowest buckling pres-
sure. The curves corresponding to n = 10 and
n = 11 are plotted in Fig. 12. It is seen that these
curves are significantly higher than the curves
obtained for a shell without axial end constraints.
Similar calculations have been performed for
L/a = 1.159 and the results have been plotted in
Fig. 12. All the curves for L/a = 1.159 are lower
than the corresponding curves for L/a = 0.6010.
V. SUMMARY
A theory, based on an energy analysis, has been
developed for the snap-through and post-buckling
behavior of simply-supported ideal shells under
the action of external pressure. The principal re-
sults of the theory are given: (a) by Eqs. (71),
(72), (97), (100), (101), (111), (112), and (113)
for elastic shells whose ends are free to warp out
of their planes, and (b) by Eqs. (72), (105), (107),
(110), (111), (112), and (113), and the modifica-
tions indicated in Article 13 for elastic shells whose
ends are rigid plates.
IQ
0ý
The main results of the computations are pre-
sented in the form of tables and graphs. Tables
1 to 20 list the parameters needed for calculation
of the buckling coefficient K given by per = KEh/a.
The use of Tables 1 to 20 is illustrated by a nu-
merical example (Article 18). Table 21 gives values
of K, for elastic shells whose ends are free to warp
out of their planes, as determined by the infinites-
imal theory and the Tsien snap-through theory for
various values of L/a and A, and for a/h = 1000.
For no axial pressure (A = 0), some values of K
Log,0IOL/o
Fig. 13. Buckling Coefficients for Cylindrical Shells Subjected to Hydrostatic Pressure
ILLINOIS ENGINEERING EXPERIMENT STATION
0/h /lO00o , =7-
500 -
25C _ - _ _ __ - __
00
-=10 \
Sn=5
50
250 - -
#. - - -- - -- -- -- -- - -
.10 .25 .50
50 /0 25
50 /00
Fig. 14. Potential Energy Barriers Separating Buckled
and Unbuckled Forms
as calculated by von Mises' theory are given for
comparison. Table 22 lists similar values of K for
a/h = 100.
Discrepancies between von Mises' theory and
the present infinitesimal theory are greatest for
short thick shells. Apparently, the trouble lies in
the assumption that the shell buckles without in-
cremental hoop strain. Von Mises did not make
this assumption.
For elastic cylinders whose ends are free to
warp out of their planes, the Euler buckling coeffi-
cient (Infinitesimal Theory) and the Tsien buck-
ling coefficient (Elastic Snap-Through Theory) are
plotted versus L/a in Fig. 13 for a/h = 100 and
a/h = 1000 with X = r. Some of the data of Fig.
13 are reproduced in Tables 21 and 22. For long
slender cylinders (see Tables 21 and 22), the Euler
buckling coefficient is only slightly higher than the
Tsien coefficient. However, for relatively small
values of L/a (say, L/a = 0.6010), the Euler coeffi-
cient may be 30 to 35% higher than the Tsien
coefficient. In the numerical example of Article 18,
the Tsien coefficient is approximately 14% higher
than the minimum pressure under which the elastic
shell will maintain a buckled form. Prevention of
end warping raises the critical pressure (Fig. 12).
For A = 0 (no end pressure), all the critical
pressures as determined by the different theories
are raised. The effect of axial compression is great-
est for small values of L/a. It becomes insignificant
for very large values of L/a (Tables 21 and 22).
The negative slopes of the load-deflection curves
(Fig. 12) denote a condition favorable to snap-
through. The potential energy barrier that the shell
must overcome to snap-through is discussed in
Article 17. Fig. 14 is a chart that shows these bar-
riers for a/h = 1000 and X = 7. The curve is dis-
continuous because of sudden changes in n. The
dashed curves have no significance; they merely
outline the region in which the discontinuous curve
lies. The points of discontinuity correspond to the
cusps on curve 4 of Fig. 13. For example, if L/a =
0.6 and a/h = 1000, Fig. 14 shows that n = 17
and 1012 A V/Ea3 = 11.5. Hence, if a = 20 in. and
E = 30,000,000 psi, AV =- 2.76 in.-lb = 0.23 ft-lb.
This result means that only 0.23 ft-lb of work must
be supplied from the outside to cause snap-through.
Accidental disturbances might easily supply this
much energy. Imperfections are perhaps a more
frequent cause of snap-through than accidental dis-
turbances, although submarine hulls may be sub-
jected to damaging shocks.
VI. REFERENCES
1. "Theory of Elastic Stability," by S. Timoshenko,
McGraw-Hill Book Co., Inc. New York and London,
1936.
2. "The Resistance of Tubes to Collapse," by William
Fairbairn, Philosophical Trans., Vol. 148, pp. 389-
413, 1848.
3. "The Collapse of Short Thin Tubes," by A. P. Carman,
Bulletin No. 99, Eng. Exp. Sta., Univ. of Ill., 1917.
4. "The Collapsing Strength of Steel Tubes," by T. Mc-
Lean Jasper and John W. W. Sullivan, Trans.
A.S.M.E., Vol. 53, APM-53-17b, pp. 219-245, Sept.-
Dec., 1931.
5. "Strength of Thin Cylindrical Shells Under External
Pressure," by H. E. Saunders and D. F. Windenburg,
Trans. A.S.M.E., Vol. 53, APM-53-17a, pp. 207-218,
Sept.-Dec., 1931.
6. "Collapse by Instability of Thin Cylindrical Shells
Under External Pressure," by D. F. Windenburg and
C. Trilling, Trans. A.S.M.E., Vol. 56, APM-56-20,
pp. 819-825, Nov., 1934.
7. "A Study of the Collapsing Pressure of Thin-Walled
Cylinders," by R. G. Sturm, Bulletin No. 329, Eng.
Exp. Sta., Univ. of Ill., 1941.
8. "Application of the Energy Test to the Collapse of
Long Thin Pipe Under External Pressure," by G. H.
Bryan, Proc. Cambridge Phil. Soc., Vol. VI, pp. 287-
292, 1888.
9. "Collapse of Tubes," by R. V. Southwell, Phil. Mag.,
pp. 687-698, May, 1913; pp. 503-511, Sept., 1913; pp.
67-77, Jan., 1915.
10. "The Collapse of Short Thin Tubes by External Pres-
sure," by Gilbert Cook, Phil. Mag., pp. 51-56, July,
1914.
11. "Der kritische Aussendruck zylindrischer Rohre," by
R. von Mises, Vol. 58, V. D. I. Zeitschr., pp. 750-
756, 1914.
12. "A New Theory for the Buckling of Thin Cylinders
Under Axial Compression and Bending," by L. H.
Donnell, Trans. A.S.M.E., Vol. 56, AER-56-12, pp.
795-806, Nov. 1934.
13. "A Simplified Method of Elastic-Stability Analysis for
Thin Cylindrical Shells." I-Donnell's Equation, by
S. B. Batdorf, NACA TN No. 1341, June, 1947.
14. "On the Minimum Buckling Load for Spherical Shells,"
by K. 0. Friedrichs, Theodore von Karmtin Anniver-
sary Volume, California Institute of Technology,
Pasadena, California, pp. 258-272, May 11, 1941.
15. "The Buckling of Spherical Shells by External Pres-
sure," by Th. von Karman, and H. Tsien, Jour.
Aero. Sci., p. 43, 1939.
16. "Theory of Elasticity," by S. Timoshenko and J. N.
Goodier, McGraw-Hill Book Company, Inc., 2nd
Ed. p. 157, Eq. r. 1951.
17. "Approximation Methods in Higher Analysis" (In
Russian), Kantorovich and Krylov, Moscow, 1950,
Chapter IV, Article 1.
18. "A Treatise on the Mathematical Theory of Elasticity,"
by A. E. H. Love, Dover Publications, New York,
4th Edition, p. 503, 1944.
19. "A Theory for the Buckling of Thin Shells," by H. S.
Tsien, Jour. Aero. Sci., Vol. 9, No. 10, Aug. 1942.
20. "Finite Deformations of an Elastic Solid," by F. D.
Murnaghan, Art. 4, Chap. 2, John Wiley and Sons,
New York, 1951.
VII. APPENDIX
Table 1
Values of K's for v = 0.30
Subscripts denote number of zeros preceding first significant figure.
n Ki K2 K3 K4 K5 K6 K7 K, K,
2 0.43787 0.0211631 1.44334 0,68539 0.16004 0.33126 0.21035 0.087164 2.8422
3 0.061575 0.0321363 0.40657 0.19277 0.15654 0.046584 0.084115 0.090544 1 5157
4 0.017515 0,0465935 0.19723 0.093213 0.15548 0.013250 0.045660 0.091700 1.0582
5 0.0268417 0.0426701 0.11807 0.055688 0.15501 0.0251760 0.028753 0.092232 0.81969
6 0.0232170 0.0412798 0.079100 0.037263 0,15476 0.0224338 0.019794 0.092518 0.67139
7 0.0217104 0.0568822 0.056878 0.026773 0.15461 0.0212940 0.014467 0.092691 0.56956
8 0.0399290 0.0540245 0.042946 0.020204 0.15452 0.0375116 0.011039 0.092804 0.49506
9 0.0361575 0.0525084 0.033611 0.015807 0.15445 0.0346584 0.0287018 0.092880 0.43805
10 0.0340208 0.0516438 0.027041 0.012713 0.15441 0.0430419 0.0270368 0.092935 0.39296
11 0.0327367 0.0511218 0.022236 0.010452 0.15437 0.0320704 0.0258084 0.092975 0.35638
12 0.0a19271 0.0,79152 0.018613 0.0287479 0.15435 0.0314580 0.0248762 0.093006 0.32609
13 0.0313963 0.0657437 0.015813 0.0274309 0.15433 0.0a10563 0.0241519 0.093030 0.30058
14 0.0310364 0.0642685 0.013603 0.0263916 0,15431 0.0478405 0.0235779 0.093049 0.27879
15 0.0478540 0.0632380 0,011827 0.0255568 0.15430 0.0459418 0.0231153 0.093064 0.25997
16 0.0460605 0.0625006 0.010379 0.0248760 0.15429 0.0445849 0.0227366 0.093076 0 24354
17 0.0447512 0.0619617 0.0,91817 0.0243134 0.15428 0.0435944 0.0224237 0.093087 0.22908
18 0.0437773 0.0615605 0.0281810 0.0238432 0.15427 0.0428576 0.0221613 0.093096 0.21624
19 0.0430408 0.0612568 0.0273359 0.0,34460 0.15427 0.0423004 0.0219394 0.093103 0.20477
20 0.0424754 0.0610235 0.0266155 0.0231075 0.15426 0.0418727 0.0217500 0.093110 0.19446
Table 1 (Concluded)
n Ko K K, K Ku K14 K15 Ku6 K17 Kis
2 0.26572 0.73304 0.43982 2.5839 0.086128 0.027489 1.04720 0.071177 0.25839
3 0.40644 0.41234 0.33576 0.09895 0.020763 0.017671 0.88357 0.072579 0.11074
4 0.54541 0.29322 0.30718 0.51677 0.0282027 0.013090 0 .83776 0.073047 0.061520
5 0.68375 0.22907 0.29502 0.32298 0.0240780 0.010412 0.81812 0.073259 0.039149
6 0.82178 0.18850 0.28868 0.22147 0.0223231 0.0286504 0.80784 0.073374 0.027103
7 0.95964 0.16035 0.28493 0.16149 0.0214493 0.0,74009 0.80176 0.073442 0.019876
8 1.09740 0.13963 0.28254 0.12304 0.0,96502 0.0,64680 0.79786 0.073487 0.015199
9 1.2351 0.12370 0.28091 0.096894 0.0467496 0.0257446 0.79522 0.073517 0.011999
10 1.3727 0.11107 0.27975 0.078299 0.0349059 0.0251671 0.79333 0.073539 0.0297137
11 1.5103 0.10079 0.27890 0.064596 0.0,36778 0.0246953 0.79194 0.073555 0.0280244
12 1.6479 0.092271 0.27826 0.054207 0.0328282 0.0243026 0.79089 0,073567 0.0267405
13 1.7854 0.085085 0.27776 0.046140 0.0,22216 0.0239706 0.79007 0.073576 0.0257419
14 1.9230 0.078943 0.27736 0.039752 0.0317769 0.0236862 0.78943 0.073584 0.0249499
15 2.0605 0.073631 0.27704 0.034605 0.0314435 0.0234399 0.78890 0.073590 0.0243112
16 2.1980 0.068992 0.27678 0.030398 0.0311886 0.0232245 0.78848 0.073595 0.0237886
17 2.3355 0.064905 0.27656 0.026915 0.0499038 0.0230345 0.78812 0.073599 0,0233556
18 2.4730 0.061276 0.27638 0.023999 0.0483393 0.0228656 0.78783 0.073603 0.0229929
19 2.6105 0.058032 0.27623 0.021532 0.0470878 0.0227146 0.78758 0.073606 0.0226859
20 2.7480 0.055116 0.27610 0.019427 0.0,60749 0.0225787 0.78736 0.073608 0.0224239
Table 2
Values of Coefficients for Computing Buckling Loads-n = 2
n = 2 X r v = 0.30
L/a a, az bi b2 b6 ci c2 c3
4 6.7612 1.0943 0.06018 0.039680 0.039558 0.3233 0.05798 0.025626
5 8,4516 1.0773 0.06414 0.034165 -0.0a7060 0.0a4230 0.3102 0.05634 0.025624
6 10.142 1.0681 0.06629 0.032065 -0.033504 0.0,2144 0.3032 0.05553 0.025622
7 11.832 1.0625 0.06759 0.031134 -0.031926 0.031197 0.2990 0.05508 0.025621
8 13.522 1.0589 0.06843 0.046720 -0.031146 0.047193 0.2963 0.05480 0.025621
10 16.903 1.0547 0.06942 0.042788 -0.044744 0.0,3041 0.2932 0.05448 0.025620
12 20.284 1.0524 0.06996 0.041354 -0.042327 0.041494 0.2915 0.05431 0.025620
15 25.355 1.0506 0.07040 0.055574 -0.059597 0.056222 0.2901 0.05417 0.025620
20 33.806 1.0491 0.07074 0.041769 -0.0s3054 0.051996 0.2890 0.05407 0.0%5619
25 42.258 1.0484 0.07090 0.0,7238 -0.061254 0.068235 0.2886 0.05402 0.025619
30 50.709 1.0480 0.07098 0.0,3483 -0.066066 0.063989 0.2883 0.05400 0.025619
40 67.612 1.0477 0.07107 0.061094 -0.061921 0.0,1270 0.2880 0.05397 0.025619
50 84.516 1.0475 0.07111 0.074439 -0.077986 0.075228 0.2879 0.05396 0.0,5619
60 101.42 1.0474 0.07113 0.072111 -0.073838 0.072535 0.2878 0.05396 0.025619
75 126.77 1.0473 0.07115 0.0s8454 -0.071617 0.071046 0.2878 0.05395 0.025619
100 169.03 1.0473 0.07116 0.0s2501 -0.0s5054 0.0s3360 0.2877 0.05395 0.025619
125 211.29 1.0472 0.07117 0.0,9686 -0.0s2122 0.0s1408 0.2877 0.05394 0.025619
150 253.55 1.0472 0.07117 0.0,4119 -0.0,1011 0.096960 0.2877 0.05394 0.025619
200 338.06 1.0472 0.07117 0.0,o9321 -0.0,3034 0.01ol98 0.2877 0.05394 0.025619
If X=0, a = 1.0472 and as= 0.07118. The b's and c's are independent of X.
Bul. 443. BEHAVIOR OF CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE
Table 3
Values of Coefficients for Computing Buckling Loads - n = 3
n = 3 =
L/a a, a2 bi
2.704 0.9664 0.025343 0.021910
3.155 0.9444 0.02318 0.021120
3.606 0.9301 0.03476 0.036956
4.508 0.9134 0.04837 0.033063
5.634 0.9026 0.05709 0.031318
6.761 0.8968 0,06182 0.046536
7.888 0.8933 0.06468 0.043589
9.015 0.8910 0.06653 0,042128
11.269 0.8883 0.06871 0.0s8834
13.523 0.8869 0.06989 0.054292
16.904 0.8857 0.07086 0.051768
22.538 0.8848 0.07161 0.065622
28.172 0.8843 0.07196 0.062307
33.81 0,8841 0.07215 0.0N1113
45.08 0.8839 0,07234 0.073527
56.34 0.8838 0.07242 0.071454
67.61 0.8837 0.07247 0.0s6938
84.52 0.8836 0.07251 0.0s2879
112.69 0.8836 0.07254 0.099986
If X=0, a1=0.8836 and a2=0.07258. The b's and c's are independent of X.
r ' = 0.30
b2 b
-0.027619 0.01077
-0.024485 0.026490
-0.022793 0. 04137
-0.021234 0.021911
-0.035394 0.038630
-0.032639 0. 034443
-0.031450 0.032511
-0.048596 0.031522
-0.043569 0.046506
-0.041754 0.043220
-0.057145 0.041350
-0.052271 0.054352
-0.009335 0.0s1799
-0.0,4494 0.068725
-0. 01421 0.062780
-0.07583 0.061145
-0.0:279 0.075550
-0.07117 0.072294
-0.0s34 0.0s729
Table 4
Values of Coefficients for Computing Buckling Loads - n = 4
n = 4
L/a al a2 bi
1.690 0.9402 -0.09791 0.021769
2.028 0.9089 -0.04568 0.039656
2.366 0.8900 -0.01418 0.035662
2.704 0.8778 0.026266 0.033516
3.381 0.8634 0.03031 0.031548
4.226 0.8542 0.04569 0.046660
5.071 0.8492 0.05405 0.043302
5.916 0.8461 0.05909 0.041812
6.761 0.8442 0.06236 0.041074
8.452 0.8419 0.06621 0.0.4454
10.14 0.8406 0.06830 0.0,2161
12.68 0.8396 0.07001 0.0M8881
16.90 0.8388 0.07134 0.062809
21.13 0.8384 0.07195 0.021145
25.35 0.8382 0.07229 0.075478
33.81 0.8380 0.07262 0.071698
42.26 0.8379 0.07277 0.0N6805
If X= 0, ai=0.8378 and a0 =0.07305. The b's and c's are independent of X.
v = 0.30
b2 b
-0.01273 0.03261
-0.026993 0.01837
-0.024116 0.01112
-0.022562 0.027116
-0.021132 0.023311
- 0.04877 0.021507
-0.032421 0.037797
-0.0A1330 0.034422
-0.047886 0.032687
-0.043275 0.031152
-0.041592 0.045715
-0.056562 0.042399
-0.052088 0.027749
- 0.08585 0.053205
-0.024152 0.001555
-0.0.1322 0. 06495
-0.07545 0.06205
Table 5
Values of Coefficients for Computing Buckling Loads-n 5
=5 X = -
L/a a0 as b1
1.082 0.9676 -0.3429 0.022170
1.352 0.9138 -0.1931 0.021080
1.623 0.8846 -0.1117 0.035895
1.893 0.8669 -0.06262 0.033457
2.164 0.8555 -0.03077 0.032147
2.704 0.8420 0.026678 0.049454
3.381 0.8334 0.03065 0,044068
4.057 0.8288 0.04367 0.042017
4.733 0.8259 0.05151 0.041108
5.409 0.8241 0.05661 0.0t6567
6.761 0.8219 0.06261 0.0M2726
8.113 0.8208 0.06686 0.051324
10.142 0.8198 0.06852 0.065456
13.52 0.8191 0.07060 0.021734
16.90 0.8187 0.07155 0.077115
20.28 0.8185 0.07208 0.073432
27.04 0.8184 0.07259 0.071086
33.81 0.8183 0.07283 0.084452
If X=0, ai= 0.8181 and a2= 0.07326. The b's and c's are independent of X.
b2
-0.02441
-0.01227
-0.026733
-0.023963
-0.022467
-0.021089
-0.034693
-0.032329
-0.0,l280
-0.047586
-0.043140
-0.041530
-0.0s6305
-0.0.2005
- 0.08229
-0.02,4012
-0.061257
-0.075173
v = 0.30
b1
0.09734
0.04999
0.02822
0.01712
0.01096
0.025124
0.022338
0.021213
0.0.6890
0.034191
0.031802
0.0M8942
0.043757
0.041215
0.005027
0.052439
0.067783
0.0N3205
Table 6
Values of Coefficients for Computing Buckling Loads - n = 6
n =6 A = r v =0.30
t L/a a a0 b6i
1.2 0.6761 1.0639 -0.9902 0.023362
1.6 0.9015 0.9519 -0.5249 0.021470
2.0 1.127 0.9000 -0.3095 0.037312
2.4 1.352 0.8719 -0,1925 0.0,3991
2.8 1.578 0.8549 -0.1220 0.032341
3.2 1.803 0.8438 -0.07619 0.031454
4 2.254 0.8309 -0.02234 0.046401
5 2.817 0.8226 0.01111 0.042754
6 3.381 0.8181 0.03083 0.041365
7 3.944 0.8154 0.04212 0.057497
8 4.507 0.8136 0.04944 0.0s4444
10 5.634 0.8115 0.05806 0.051844
12 6.761 0.8104 0.06274 0.028956
15 8.452 0.8095 0.06657 0.063687
20 11.27 0.8088 0.06954 0.061170
25 14.08 0.8084 0.07092 0.074790
If = 0, a,= 0.8078 and a2= 0.07337. The b's and c's are independent of X.
b2
-0.05376
-0.02393
-0.01202
-0.026542
-0.023888
-0.022418
-0.021067
-0. 04600
-0.032282
-0.031254
-0.047385
-0.043087
-0.041500
-0.026181
-0.051966
-0.008035
b3
0.3080
0.1386
0.07128
0.04028
0.02445
0.01570
0.027341
0.023356
0.021743
0.039908
0. 036030
0.032594
0.031288
0.045412
0.041749
0.0J7237
Cl C2 C3
6.707 9.160 0.04956
4.663 3.303 0.05000
3.844 1.677 0.05021
3.434 1.083 0.05032
3.198 0.8222 0.05038
3.049 0.6925 0.05043
2.879 0.5798 0.05048
2.773 0.5302 0.05051
2.716 0.5106 0.05053
2.682 0.5014 0.05054
2.660 0.4963 0.05055
2.634 0.4914 0.05056
2.621 0.4892 0.05056
2.609 0.4877 0.05056
2.600 0.4866 0.05057
2.596 0.4862 0.05057
Cl
0.8671
0.8052
0.7664
0.7222
0.6947
0.6800
0.6712
0.6656
0.6590
0.6554
0.6525
0.6502
0.6492
0.6486
0.6480
0.6478
0.6476
0.6475
0.6474
C2
0.1730
0.1519
0.1424
0.1324
0.1275
0.1252
0.1240
0.1233
0.1226
0.1222
0.1219
0.1216
0.1216
0.1215
0.1214
0.1214
0.1214
0.1214
0.1214
C3
0.01262
0.01262
0.01263
0.01263
0.01264
0.01264
0.01264
0.01264
0.01264
0.01264
0.01264
0.01264
0.01264
0.01264
0.01264
0.01264
0.01264
0.01264
0.01264
Cl
1.719
1.532
1.425
1.358
1.281
1.233
1.208
1.192
1.183
1.171
1.165
1.160
1.156
1.154
1.153
1.152
1.152
C2
0.4779
0.3522
0.2959
0.2671
0.2410
0.2289
0.2237
0.2211
0.2196
0.2180
0.2172
0.2167
0.2163
0.2161
0.2160
0.2159
0.2158
C3
0.02260
0.02256
0.02254
0.02252
0.02251
0.02250
0.02249
0.02248
0.02248
0.02248
0.02248
0.02248
0.02248
0.02248
0.02248
0.02248
0.02248
3.248
2.675
2.388
2.223
2.119
2.000
1.926
1.886
1.863
1.848
1.830
1.820
1.812
1.806
1.803
1.802
1.800
1.799
C2e
1.731
0.9347
0.6409
0.5111
0.4459
0.3883
0.3623
0.3518
0.3466
0.3438
0.3409
0.3396
0.3386
0.3379
0.3376
0.3375
0.3373
0.3372
0c
0.03476
0.03489
0.03496
0.03500
0.03503
0.03506
0.03508
0.03509
0.03510
0.03510
0.03511
0.03511
0.03511
0.03511
0.03512
0.03512
0.03512
0.03512
ILLINOIS ENGINEERING EXPERIMENT STATION
Table 7
Values of Coefficients for Computing Buckling Loads - n = 7
n=7 X=
( L/a ai a2 bi
1.0 0.4829 1.1624 -2.009 0.0,3948
1.2 0.5795 1.0522 -1.373 0.022433
1.6 0.7727 0.9426 -0.7400 0.021064
2.0 0.9659 0.8919 -0.4472 0.035292
2.4 1.159 0.8644 -0.2881 0ý.02889
2.8 1.352 0.8478 -0.1922 0.021694
3.2 1.545 0.8370 -0.1299 0.0A1052
4 1.932 0.8243 -0.05671 0.044634
5 2.415 0,8162 -0.029858 0.0,1994
6 2.898 0.8118 0.01559 0.0s9889
7 3.381 0,8091 0,03094 0.0,5431
8 3.864 0.8074 0.04090 0.0,3220
10 4.829 0.8054 0.05262 0.051330
If X =0, ai=0.8018 and a2=0.07344. The b's and c's are independent of X.
b2
-0.08504
-0.05314
-0,02364
-0,01188
-0.026580
-0.023838
-0.022388
-0.021054
-0.0A4544
-0.032255
-0.031239
-0.047345
-0.043050
=0.30
Nb ct c(2
0.6668 12.388 33.43
0.4162 9.098 16.59
0.1875 6.335 5.798
0.09646 5.227 2.817
0.05455 4.670 1.731
0.03314 4.350 1.258
0.02129 4.148 1.024
0.029965 3.918 0.8224
0.024560 3.774 0.7353
0.022370 3.696 0.7016
0.021348 3.650 0.6859
0.038206 3.621 0.6776
0.0.3532 3.586 0.6698
Table 8
Values of Coefficients for Computing Buckling Loads - n 8
n = 8 X = 6 v = 0.30
t L/a al a2 bi
1.0 0.4226 1.153 -2.645 0.022993
1.2 0.5071 1.045 -1.814 0.0?1844
1.6 0.6761 0.9367 -0.9884 0.0,8064
2.0 0.8452 0.8867 -0,6061 0.034012
2.4 1.014 0.8596 -0.3985 0.0a2190
2.8 1.183 0.8432 -0.2732 0.0,1284
3.2 1.352 0.8326 -0.1920 0.047975
4 1.690 0.8201 -0.09641 0.043512
5 2.113 0.8121 -0.03525 0.0,1511
6 2.535 0.8077 -0.0-2024 0.037492
7 2.958 0.8051 0.01801 0.054113
8 3.381 0.8034 0,03101 0.0,2438
10 4.226 0.8014 0.04630 0.0,1012
If \=0, ai=0.7979 and a= 0.07349. The b's and c's are independent of X.
b2
-0.08441
-0.05273
-0.02346
-0.01178
-0.026470
-0.023808
-0.022370
-0.021046
-0. 0,4509
-0.0,2237
-0.031213
-0.047285
-0.043024
Na
0. 8667
0.5411
0.2438
0.1255
0.07103
0.04316
0.02774
0.01299
0.025950
0.023093
0.021760
0.021072
0.034614
Cl
16.13
11.86
8.264
6.822
6.097
5.680
5.417
5.116
4.928
4.828
4.768
4.729
4.683
c2
56.54
27.87
9.537
4.483
2.649
1.852
1.460
1.124
0.9809
0.9263
0.9012
0.8882
0.8761
Table 9
Values of Coefficients for Computing Buckling Loads - n = 9
n = 9 \ = , v = 0.30
L/a a, a2 bi b2
1.0 0.3756 1.147 -3.366 0.022349 -0.08398
1.2 0.4507 1.040 -2.315 0.021448 - 0. 05292
1.6 0.6010 0,9327 -1.270 0.036329 -0.02334
2.0 0.7512 0.8832 -0.7862 0.033149 -0,01172
2.4 0.9015 0.8563 -0.5235 0.031719 -0.026437
2.8 1.052 0.8401 -0.3651 0.031008 -0.023788
3.2 1.202 0.8296 -0.2623 0.046260 -0.022357
4 1.502 0.8172 -0.1414 0.042756 -0.021041
5 1.878 0.8093 -0.06404 0.041186 -0.0,4484
6 2.254 0.8050 -0.02201 0.055880 -0.0ý2225
7 2.629 0.8024 0.0.3333 0.053228 -0.031222
8 3.005 0.8007 0.01978 0.0,1914 -0.047247
10 3.756 0.7987 0.03913 0.07941 -0.043008
12 4.507 0.7977 0.04964 0.0,3856 -0.041462
15 5.634 0.7968 0.05823 0.01587 -0.056022
20 7.512 0.7961 0.06492 0 075037 -0.051914
If X=0, ai=0.7952 and a2=0.07352. The b's and c's are independent of X.
b,
1.0934
0.6827
0. 3077
0.1585
0.08970
0.05453
0.03505
0.01642
0.0%7525
0.023913
0.022223
0.021356
0.0,5839
0.032901
0.031068
0.043941
Cl
20.38
14.99
10.45
8.629
7.714
7.186
6.855
6.474
6.237
6.110
6.034
5.985
5.927
5.896
5.871
5.851
C2
90.02
44.18
14.89
6.828
3.908
2.644
2.024
1.495
1.271
1.186
1.148
1.129
1.111
1.103
1. 098
1.095
Table 10
Values of Coefficients for Computing Buckling Loads - n - 10
n = 10 X =
L/a al a2 bi
0.8 0.2704 1.339 -6.559 0.023302
1.0 0.3381 1.143 -4.171 0.021894
1.2 0.4057 1.036 -2.874 0.021167
1.6 0.5409 0.9298 -1.584 0.035103
2.0 0.6761 0.8807 -0.9876 0.0,2539
2.4 0.8113 0.8540 -0.6634 0.031386
2.8 0.9466 0.8379 -0.4677 0.048127
3.2 1.082 0.8275 -0.3410 0.045047
4 1.352 0.8152 -0.1918 0.042222
5 1.690 0.8073 -0.09624 0.0,9563
6 2.028 0.8030 -0,04437 0.054742
7 2.366 0.8005 -0.01309 0.0s2604
8 2.704 0.7988 0.027217 0.0,1543
10 3.381 0.7968 0.03109 0.0,6407
12 4.057 0.7958 0.04406 0.0,3112
15 5.071 0.7949 0.05467 0.061282
20 6.761 0.7942 0.06293 0.074074
If X=0, ai=0.7933 and a = 0.07354. The b's and c's are independent of X.
bz
-0.1431
-0.08367
-0.05227
-0.02326
-0.01168
-0.026412
-0.023794
-0.022348
-0,021037
-0.034468
-0.0a2217
-0.0al218
-0.047221
-0.043000
-0.041454
-0.0s6006
-0.0A1911
= 0.30
b,
2.3217
1.3467
0.8410
0.3792
0.1953
0.1106
0.06724
0.04323
0.02026
0.029287
0.0A4831
0.022745
0.021675
0.0,7215
0.033586
0.031508
0.044882
Cl
40.02
25.12
18.48
12.89
10.65
9.520
8.871
8.461
7.992
7.699
7.543
7.449
7.388
7.317
7.279
7.248
7.223
C2
330.6
136.6
66.83
22.27
10.02
5.593
3.679
2.742
1.945
1.610
1.484
1.428
1.400
1.373
1.363
1.356
1.352
C3
0.06675
0.06738
0.06802
0.06831
0.06847
0. 06857
0.06863
0.06870
0.06875
0.06877
0.06879
0.06880
0.06881
C3
0.08709
0.08795
0.08880
0.08920
0.08942
0.08954
0.08963
0.08973
0.08979
0.08982
0.08985
0.08986
0.08988
C3
0.1101
0.1112
0.1124
0.1129
0.1131
0.1133
0.1134
0.1136
0.1136
0.1137
0.1137
0.1137
0.1137
0.1137
0.1137
0.1137
e3
0.1334
0.1359
0.1373
0.1387
0.1393
0.1397
0.1399
0.1400
0.1402
0.1403
0.1404
0.1404
0.1404
0.1404
0.1404
0.1404
0.1404
Bul.443. BEHAVIOR OF CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE
Table 11
Values of Coefficients for Computing Buckling Loads - n = 11
n = 11 X = - v = 0.30
t L/a at a2 bi
0.7 0.2151 1.501 -10.40 0.023732
0.8 0.2459 1.335 -7.949 0,022720
1.0 0.3073 1.140 -5.061 0.021560
1.2 0,3688 1.033 -3.492 0.039612
1.6 0.4917 0.9277 -1.932 0.034202
2.0 0.6147 0.8788 -1.210 0.022091
2.4 0.7376 0.8523 -0.8179 0.031141
2.8 0.8605 0.8363 -0.5814 0.046929
3.2 0.9834 0.8259 -0.4279 0.044156
4 1.229 0.8137 -0.2474 0.041830
5 1.537 0.8058 -0.1318 0.057876
6 1,844 0.8016 -0.06907 0.053906
7 2.151 0.7990 -0.03123 0.0s2145
8 2.459 0.7974 -0.01667 0.051271
10 3.073 0.7954 0.02221 0.0N5279
12 3.688 0.7944 0.03790 0.0«2565
15 4.610 0.7935 0.05074 0.0W1057
20 6.147 0.7928 0.06072 0.073361
If X =0, ai=0.7919 and a2=0.07356. The b's and c's are independent of X.
b2
-0.1937
-0.1428
-0.08234
-0.05213
-0.02319
-0.01165
-0.026394
-0.023763
-0.022342
-0. 021034
-0.034456
-0.032211
-0.031215
-0.047204
-0.042993
-0.041454
-0.0.5994
-0.051907
ba
3.830
2.805
1.627
1.016
0.4581
0.2360
0.1336
0.08127
0.05226
0.02450
0.01123
0.025844
0.023327
0.022027
0.038731
0.0a4340
0.021825
0.045906
Cl C2 C3
66.94 822.2 0.1585
48.36 483.2 0.1612
30.37 199.4 0.1643
22.35 97.30 0.1661
15.60 32.14 0.1678
12.88 14.26 0.1686
11.52 7.795 0.1690
10.73 5.006 0.1693
10.24 3.642 0.1694
9.670 2.487 0.1696
9.316 2.002 0.1698
9.126 1.822 0.1698
9.013 1.742 0.1699
8.940 1.702 0.1699
8.854 1.666 0.1699
8.808 1.651 0.1699
8.770 1.642 0.1700
8.740 1.636 0.1700
Table 12
Values of Coefficients for Computing Buckling Loads-n= 12
n = 12 X = xr
, = 0.30
L/a at a2 bi
0.1972 1.4976 -12.395 0.023128
0.2254 1.3320 -9.4730 0.022279
0.2817 1.1372 -6.0362 0.0z1307
0.3381 1.0314 -4.1693 0.038056
0.4508 0.9262 -2.3131 0,033522
0.5634 0.8775 -1 4539 0.031752
0.6761 0.8510 -0.9872 0.049566
0.7888 0.8350 -0.7057 0.045610
0.9015 0.8247 -0.5231 0.043484
1.1269 0.8125 -0.3083 0.041534
1.4086 0.8047 -0.1708 0.056603
1.690 0.8005 -0,09615 0.053275
1.972 0.7980 -0.05112 0.051799
2.254 0.7963 -0.02190 0.051066
If X=0, a1=0.79089 and a02=0.073567. The b's and c's are independent of X.
1,2
-0.1933
-0.1425
-0.08328
-0.05204
-0.02314
-0.01162
-0.026381
-0,023756
-0.022336
-0.021032
-0.034446
-0.022207
-0.0A1212
-0.047187
b,
4.552
3.333
1.933
1.208
0.5446
0.2806
0.1589
0.09665
0.06216
0.02915
0.01336
0.026953
0.023959
0.022412
Cl c2 C3
79.56 1163 0.1886
57.49 683.4 0.1918
36.12 281.7 0.1956
26.58 137.2 0.1976
18.55 45.03 0.1997
15.33 19.74 0.2006
13.70 10.61 0.2011
12.77 6.680 0.2014
12.18 4.758 0.2016
11.51 3.133 0.2019
11.09 2.454 0.2020
10.86 2.203 0.2021
10.73 2.092 0.2021
10.64 2.036 0.2022
Table 13
Values of Coefficients for Computing Buckling Loads - n = 13
n = 13 N = r
v = 0.30
L/a at a2 b, b2
0.7 0.1820 1.4946 -14.558 0.022660 -0.1934
0.8 0.2080 1.3295 -11.1289 0.021938 -0.1423
1.0 0.2600 1.1353 -7.0960 0.0,1112 -0.08443
1.2 0,3121 1.0298 -4.9053 0.036850 -0. 05195
1 .6 0.4161 0.9249 -2.7270 0.032994 -0.02310
2.0 0.5201 0.8764 -1.7188 0.031490 -0.01160
2.4 0.6241 0.8500 -1.1711 0.048132 -0.026381
2.8 0.7281 0.8341 -0.8409 0.044769 -0.023749
3.2 0.8322 0.8238 -0.6266 0.042962 -0.022333
4.0 1.040 0.8116 -0.3745 0.041304 -0.021030
5.0 1.300 0.8039 -0.2132 0.055611 -0.034439
6.0 1.560 0.7996 -0.1256 0.0,2782 -0.032203
7.0 1.820 0.7971 -0.07274 0.051528 -0.0al210
8.0 2.080 0.7955 -0.03845 0.069055 -0,047177
If \=0, ai=0.79007 and a2=0.073576. The b's and c's are independent of X.
b8
5.336
3.907
2.267
1.416
0.6386
0.3291
0.1864
0.1134
0.07292
0.03420
0.01568
0.028162
0.0A4648
0.022832
Cl C2 C3
93.28 1601 0.2212
67.41 940.4 0.2250
42.36 387.2 0.2295
31.18 188.4 0.2319
21.77 61.49 0.2343
17.99 26.71 0.2354
16.08 14.16 0.2360
14.99 8.762 0.2364
14.30 6.124 0.2366
13.50 3.898 0.2369
13.01 2.971 0.2371
12.75 2.629 0.2372
12.59 2.479 0.2372
12.48 2.403 0.2373
L/a a0
0.1690 1.4922
0.1932 1.3275
0.2415 1.1338
0.2898 1.0286
0.3864 0.9239
0.4829 0.8755
0.5795 0.8492
0.6761 0.8334
0.7727 0.8230
0.9659 0.8110
1.207 0.8032
1.449 0.7990
1.690 0.7965
1.932 0.7948
If = 0, a- =0.78943 and a2= 0.073584
Table 14
Values of Coefficients for Computing Buckling Loads - n = 14
n = 14 X = v = 0.30
a02 b b2 b3
-16.8938 0.022289 -0.19283 6.184
-12.9170 0.021668 -0.14213 4.528
-8.2404 0.039568 -0.08305 2.627
-5.7000 0.025896 -0.05188 1.641
-3.1741 0.032578 -0.02307 0,7402
-2.0049 0.031282 -0.01159 0.3815
-1.3698 0.047000 -0.026363 0.2160
-0.9869 0.044105 -0.023744 0.1314
-0.7383 0.042549 -0.022330 0.08453
-0 4460 0.041122 -0.021029 0.03965
-0.2590 0.054829 -0.034432 0.01818
-0 1574 0.052394 -0.032218 0.029462
-0.09609 0.0t1314 -0.0l1208 0.025378
-0.05632 0.0N7788 -0.047162 0.023283
. The b's and c's are independent of X.
Cl C2 C3
108.10 2152.9 0.2565
78.13 1263.9 0.2609
49.10 520.09 0.2661
36.15 252.67 0.2689
25.24 82.14 0.2717
20.86 35.41 0.2730
18.65 18.57 0.2737
17.38 11.32 0.2742
16.58 7.779 0.2744
15.66 4.798 0.2748
15.09 3.559 0.2750
14.78 3.104 0.2751
14.60 2.904 0.2751
14.48 2.805 0.2752
ILLINOIS ENGINEERING EXPERIMENT STATION
Table 15
Values of Coefficients for Computing Buckling Loads - n = 15
n = 15 X=r v = 0.30
t L/a al a2 bi b2 b3 cl c2 ca
0.7 0.1578 1.49031 -19,4030 0.0219918 -0.1926 7.0940 124.016 2835.839 0.294380
0.8 0.1803 1.32592 -14.8381 0.0214514 -0.1420 5.1947 89.6418 1664.536 0.299463
1.0 0.2254 1.13259 -9.4699 0.0383243 -0.08296 3.0135 56.3471 684.529 0.305441
1.2 0.2704 1.02757 -6.5538 0.0351298 -0.05183 1.8823 41.4908 332.244 0.308688
1.6 0.3606 0.92315 -3.6543 0.0322427 -0.02305 0.8492 28.9734 107.647 0.311916
2.0 0.4507 0.87482 -2.3123 0.0311157 -0.01158 0.4376 23.9406 46.1196 0.313411
2.4 0.5409 0.84857 -1.5833 0.0460908 -0.026356 0.2479 21.4086 23.9515 0.314223
2.8 0.6310 0.83274 -1.1437 0.0435719 -0.023740 0.1508 19.9506 14.4137 0.314712
3.2 0.7212 0.82246 -0,85839 0.0422182 -0.022328 0.09700 19.0320 9.76469 0.315030
4.0 0.9015 0.81038 -0.52288 0.059769 -0.021028 0.04551 17.9794 5.85008 0.315403
5.0 1.127 0.80265 -0.30813 0.0t4204 -0.034428 0.02087 17.3216 4.22540 0.315642
6.0 1.352 0.79845 -0.19151 0.0 20846 -0.032197 0.01086 16.9695 3.63096 0 315772
7.0 1.578 0.79591 -0.12118 0.0511447 -0.031207 0.026185 16.7589 3.37054 0.315851
8.0 1,803 0.79427 -0.075527 0.0667869 -0.047156 0.023768 16.6229 3 24099 0.315901
10.0 2.254 0.79234 -0.021845 0.0628178 -0.042971 0.0U1624 16.4637 3 12838 0.315961
12.0 2.704 0.79129 +0.027316 0.0613692 -0.041443 0.038071 16.3776 3.08521 0.315994
If X=0, a =0.7889 and a2 =0.07359. The b's and c's are independent of X.
Table 16
Values of Coefficients for Computing Buckling Loads - n = 16
n = 16 X = = 0.30
L/a al a2 bi b2 b3
0.1479 1.4887 -22.0856 0.021749 -0.1924 8.067
0.1690 1.3246 -16.8920 0.021274 -0.1419 5.907
0.2113 1.1316 -10.7844 0.037308 -0.08288 3.427
0.2535 1.0268 -7.4667 0.0A4503 -0.05178 2.141
0.3381 0.9225 -4.1678 0.031969 -0.02303 0.9658
0.4226 0.8743 -2.6409 0.049794 -0.01156 0.4978
0.5071 0.8480 -1.8115 0.045346 -0.026349 0.2820
0.5916 0.8322 -1.3114 0.043135 -0.023754 0.1715
0.6761 0.8220 -0.9868 0.041947 -0.022340 0.1103
0.8452 0.8099 -0.6050 0.058574 -0.021035 0.05177
1.056 0.8022 -0.3607 0.053689 -0.034466 0.02374
1.268 0.7980 -0.2280 0.0s1829 -0.032194 0.01236
1.479 0.7955 -0.1480 0.051004 -0.0a1205 0.027037
1.690 0.7938 -0.09606 0.065951 -0.047140 0.024288
If X=0, al= 0.7885 and a= 0.07360. The b's and c's are independent of X.
Cl
141.0
101.9
64.09
47.20
32.96
27.24
24.36
22,70
21.65
20.46
19.71
19.31
19.07
18.94
C2 C3
3670 0.3349
2154 0.3407
885.2 0.3475
429.3 0.3512
138.7 0.3549
59.12 0.3566
30.46 0.3575
18.13 0.3581
12.12 0,3584
7.072 0,3589
4.978 0.3591
4.213 0.3593
3.879 0.3594
3.713 0.3594
Table 17
Values of Coefficients for Computing Buckling Loads - n = 17
n = 17 X = . r = 0.30
L/a a01 a2 b
0.1392 1.4874 -24.9387 0.021548
0.1591 1.3235 -19.0764 0.021128
0.1989 1,1308 -12.1824 0.036468
0.2386 1.0261 -8.4375 0.033986
0.3182 0.9220 -4.7139 0,031742
0.3977 0.8738 -2.9904 0.048669
0.4773 0.8476 -2.0542 0.044732
0.5568 0.8318 -1.4897 0.042775
0.6364 0.8216 -1.1233 0.041723
0.7954 0.8095 -0.6924 0.057589
0.9943 0.8018 -0.4166 0.053265
1.193 0.7976 -0.2668 0.051619
1.392 0.7951 -0.1765 0.068890
1.591 0.7935 -0.1179 0.065269
If X=0, a1=0.7881 and a2=0.07360. The b's and c's are independent of X.
b2
-0.1923
-0.1418
-0.08283
-0.05175
-0.02302
-0.01156
-0.026347
-0-023734
-0.022324
-0.021026
-0.034422
-0.032194
-0.031205
-0.017148
bt
9.100
6.664
3.867
2.416
1.090
0.5618
0.3182
0.1936
0.1245
0.05843
0.02680
0.01395
0.027944
0.0 4840
Cl
159.14
115.05
72.34
53.27
37.21
30.75
27.50
25.62
24.44
23.09
22.25
21.80
21.52
21.35
C2 C3
4675.5 0.3779
2743.6 0.3845
1127.3 0.3922
546.36 0,3964
176.11 0.4006
74.74 0.4025
38.24 0.4036
22.55 0.4042
14.91 0.4046
8.483 0.4051
5.824 0.4054
4.855 0.4056
4.432 0.4057
4.223 0.4058
Table 18
Values of Coefficients for Computing Buckling Loads-n = 18
n = 18 X = v = 0.30
Cl C2 C3
178.35 5875.0 0.4236
128.94 3447.0 0.4310
81.08 1415.8 0.4397
59.71 685.86 0.4444
41.71 220.64 0.4491
34.47 93.29 0.4513
30.82 47.45 0.4525
28.73 27.75 0.4532
27.40 18.16 0.4536
25.89 10.10 0.4542
24.94 6.773 0.4545
24.44 5.560 0.4547
24.13 5.033 0.4548
23.94 4.772 0.4549
L/a a0 a2 bi b2 b3
0.1315 1.4864 -27.967 0.021379 -0.1922 10.199
0.1502 1.3227 -21.395 0.021005 -0.1417 7.469
0.1878 1.1301 -13.666 0.035765 -0.08279 4.334
0.2254 1.0255 -9.468 0.033552 -0.05171 2.707
0.3005 0.9215 -5.294 0.031553 -0.02300 1.222
0.3756 0.8734 -3.361 0.047727 -0.01155 0.6296
0.4507 0.8473 -2.312 0.044218 -0.026342 0.3567
0.5259 0.8315 -1.679 0.042473 -0.023732 0.2170
0.6010 0.8212 -1.2682 0.041536 -0.022325 0.1396
0.7512 0.8092 -0.7851 0.056764 -0,021026 0.06551
0.9391 0.8015 -0.4760 0.052910 -0.014417 0.03005
1.127 0.7973 -0.3081 0.0i1443 -0.0s2192 0.01564
1.315 0.7948 -0.2068 0.067923 -0.0a1204 0.028891
1.502 0.7932 -0.1411 0.064089 -0.047137 0.025428
If X = 0, a01 = 0.7878 and a2= 0.07360. The b's and c's are independent of X.
Bul. 443. BEHAVIOR OF CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE
Table 19
Values of Coefficients for Computing Buckling Loads - n = 19
n = 19 X = v = 0.30
L/a ai a2 bi
0.1246 1.4854 -31.1693 0.021237
0.1423 1.3219 -23.8467 0.039015
0.1779 1.1295 -15.235 0.025171
0.2135 1.0250 -10.5576 0.033186
0.2847 0.9212 -5.9065 0.031393
0.3559 0.8731 -4.1669 0.046930
0.4270 0.8470 -3.7536 0.043783
0.4982 0.8312 -2.5842 0.042219
0.5694 0.8210 -1.4214 0.041378
0.7117 0.8087 -0.8832 0.0s6067
0.8896 0.8010 -0.5388 0.052611
1.068 0.7968 -0.3516 0.01l294
1.246 0.7943 -0.2388 0.0N7109
1.423 0.7927 -0.1556 0.044214
If X= 0, at1=0.7876 and a2=0.07361. The b's and c's are independent of X.
b2 b3 cl c2 C3
-0.1921 11.360 198.65 7292.1 0.4719
-0.1416 8.319 143.63 4278.0 0.4802
-0.08275 4.827 90.32 1756.7 0.4899
-0.05169 3.016 66.52 850.6 0.4951
-0.02299 1.361 46.47 273.2 0.5004
-0.01155 0.7014 38.40 115.0 0.5028
-0.0,6339 0.3973 34.34 58.27 0.5041
-0.023731 0.2417 32.00 33.84 0.5049
-0.022321 0.1555 30.53 21.94 0.5054
-0.021025 0.07298 28.84 11.96 0.5060
-0.034417 0.03348 27.79 7.833 0.5064
-0.0a2192 0.01742 27.22 6.334 0.5066
-0.0Al204 0.029905 26.89 5.682 0.5067
-0.047141 0.026047 26.67 5.361 0.5068
Table 20
Values of Coefficients for Computing Buckling Loads -n= 20
n= 20 X = - = 0.30
L/a at a2 bi bba c2 ca
0.7 0.1183 1.4847 -34.5428 0.021116 -0.1920 12.584 220.06 8950.8 0.52288
0.8 0.1352 1.3213 -26,4295 0.038132 -0.1416 9.215 195.11 5250.7 0.53203
1.0 0.1690 1.1291 -16.8884 0.034664 -0.08272 5.347 100.06 2155.6 0.54278
1.2 0.2028 1.02464 -11.7056 0.0,2874 -0.05167 3.340 73.70 1043.3 0.54862
1.6 0.2704 0.9208 -6.5522 0.0s1256 -0.02298 1.507 51.48 334.6 0.55443
2.0 0.3381 0.8728 -4.1669 0.046251 -0.01154 0.7770 42.55 140.6 0.55712
2.4 0.4057 0.8467 -2.8712 0.043412 -0.026336 0.4402 38.05 70.87 0.55858
2.8 0.4733 0.8309 -2.0899 0.042001 -0.023692 0.2678 35.46 40.90 0.55946
3.2 0.5409 0.8207 -1.5828 0.041243 -0.022321 0,1723 33.83 26.31 0.56003
4.0 0.6761 0.8087 -0.9865 0.0s5472 -0.021025 0.08085 31.96 14.07 0.56070
5.0 0.8452 0.8010 -0.6049 0.0s2354 -0.034415 0.03709 30.79 9.014 0.56114
6.0 1.014 0.7969 -0.3976 0.051167 -0.032191 0.01930 30.17 7.179 0.56137
7.0 1.183 0.7943 -0.2726 0.0W6409 -0.0,1203 0.01097 29.79 6.383 0.56151
8.0 1.352 0.7927 -0.1914 0.063798 -0.047135 0.026700 29.55 5.991 0.56160
10.0 1.690 0.7908 -0.09601 0.041576 -0.042963 0.022887 29.27 5.655 0.56171
12.0 2.028 0.7897 -0.04418 0.077653 -0.041439 0.021435 29.12 5.530 0.56177
If 1= 0, at=0.7874 and a2=0.07361. The b's and c's are independent of 1.
Table 21
Coefficients K4t, Kj, and Kvm, (a/h = 1000)
X= 0 X=r
K.t X 10 Ki X 106 K.t X 106 Ki X 106
63.38
50.89
47.83
39.33
36.61
25.71
23.88
22.71
17.77
16.20
15.07
12.34
11.17
9.59
8.576
7.493
6.547
4.466
3.963
3.102
2.611
1.967
1.597
1.094
0.9090
0.8372
0.4479
0.3566
0.3172
91.66
68.80
67.86
52.87
52.15
34.05
33.59
33.95
22.37
21.71
21.76
15.48
15.08
11.65
11.130
8.983
8.532
5.543
5.569
3.843
2.882
2.445
1.715
1.372
0.9958
0.8600
0.6079
0.3795
0.3173
57.50
47.05
44.51
37.21
34.64
24.82
23.06
21.92
17.38
15.85
14.73
12.15
11.00
9.49
8.480
7.427
6.489
4.442
3.941
3.089
2.605
1.962
1.595
1.092
0.9080
0.8366
0.4475
0.3564
0.3172
85.24
65.19
64.29
50.05
50.02
33.14
32.70
33.05
21.98
21,33
21.38
15.29
14.89
11.54
11.03
8.920
8.472
5.517
5.543
3.830
2.876
2.439
1.713
1.370
0.9950
0.8595
0.6074
0.3793
0.3172
-=0
K,. X 106
(Von Mises
Formula)
88.66
65.93
50.70
21,77
11.33
5.356
1.973
0.9728
0.3722
0.4057
0.4982
0.5259
0.6364
0.6761
0.9659
1.04
1.127
1.409
1.537
1.690
2.028
2.254
2.629
2.958
3.381
3.864
5.634
6.761
8.113
10.142
12.68
16.90
22.538
28.172
33.81
50.71
67.61
84.52
ILLINOIS ENGINEERING EXPERIMENT STATION
Table 22
Coefficients Kst, Ki, and Kvm, (a/h = 100)
=0 X 10 X 10
K.1 X 10@ n KiX 106
X =
n K,, X 10«
15 7577.6
14 6707
L/a
0.1690
0.1779
0.1803
0.1878
0.1932
0.1989
0.2113
0.2415
0.2600
0.2817
0.3121
0.3688
0.4917
0.5409
0.6761
0.7512
0.8452
0.9659
1.014
1.159
1.352
1.545
1.578
1.803
1.893
2.164
2.254
2.704
3.381
4.226
5.071
5.634
5.916
6.761
7.888
9.015
11.269
11.832
13.522
16.903
20.284
25.355
33.806
Ki X 106
9273
8454
7684
6969
6310
5175
3693
2134
1935
1498
1334
964.0
712.9
621.3
533.6
423.5
349.8
279.9
222.3
181.1
162.3
148.8
115.3
98.59
84.13
54.23
40.56
32.91
29.22
12120
11020
10000
9055
7432
4813
2500
2268
1663
1481
1039
753.8
648.5
557.4
435.6
360.0
285.2
226.7
183.6
163.9
151.0
116.6
99.41
84.57
54.62
40.76
33.02
29.28
14 12000
6484
5731
3484
1905
1402
1041
882.0
712.0
605.5
506.0
451.7
425.5
361.5
297.0
230.4
186.4
167.1
140.5
113.4
98.93
91.13
83.28
71.08
56.08
42.96
37.22
32.89
4161
3651
2473
1544
1132
904.4
765.1
644.1
547.2
469.5
394.0
340.9
286.1
221.6
181.9
164.4
136.8
111.4
97.62
90.23
69.81
55.32
42.58
37.00
32.79
K,,, X 106
5344
4544
3530
2185
1548
634.7
281.1
180.6
97.46
32.59