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RESEARCH INVESTIGATIONS OF BULKHEAD CYLINDRICAL
JUNCTIONS EXPOSED TO COMBINED LOAD, CRYOGENIC I ~ TEMPERATURE AND PRESSURE
PART IV THEORETICAL ANALYSIS OF THE BUCKLING PROBLEM
Tsu-Te Wu H. Wagner
GPO PRICE $
OTS PRICE(S1 S
Hard copy (HC) 2- D Microfiche (MF) 75
Techn ica l Report Con t rac t No. NAS8-5199
P r e p a r e d for
NASA Huntsv i l le , Alabama
Nat iona l Aeronaut ics and S p a c e Administration Washington 25, D. C.
G g e C. Marshall Space F l i g h t Cen te r
i
.
Department of Engineer ing Sc ience a n d Mechanics Engineer ing and Industrial Exper iment Station
University of F lo r ida Gain e s v i 1 1 e, Flo rid a
Augus t , 1964
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E X A R C H INVESTIGATIONS OF BULKHEAD CYLINDRICAL JUNCTIONS
PART IV THEORETICAL ANALYSIS OF THE BUCKLING PROBLEM EXPOSED TO COMBINED bUD, CRYOGENIC TEMPERATURe
Tau-% WU H. Wagner
Tbchnlcal Report Contract No. NAS8-5199
Prepand f o r George C. Marshall Space Fl ight Centar
NASA Huntsvine, Alabama
National Aeronautic8 and space Adndnistra t ion Washington 25, D. C.
Department of Engineering 5olenao and Mechanics Engineering and Xndustrlal Experbent Station
Univereity of Florida Oeinesvl l le , Florida
Reproduction i n whole or In part I s permitted for any purpose of the United States Ooverment. A l l other pubushing right8 are reserved.
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TABLE OF CONTENTS
A C K N C W L E D G M E N T S . . . . . . . . . . . . . . . . L I S T O F T A B L E S . . . . . . . . . . . . . . . . L I S T OF FIGURES 0 . 0 0
ABSTRACT . . . . . . . . . . . . . . . . . . .
CHAPTER
I. INTRODWC TION . . . . . 11. ANALYSIS . . . . . . . . . . . . . . .
1. Basic Equations . . . . . . . . . Existing Prior to B u c k l i n g . . .
3. Boundary Conditions . . . . . . . 2. N o r m a l Component of Stress
4. The Method of Approximate Solutions
111. APPLICATION OF A SPECIAL PROBLEM. . I V . CONCLUSION A N D DISCUSSION . . . . . .
LIST OF REFERENCES . . . . . . . . . . . . . . APPENDICES
A. DERIVATION OF' DIFFERENTIAL OPERATORS A AND D . . . . . . . . . . . . . . .
B. DERIVATION OF ADDI'JXONAL F O X E COMPONENTS I N THE BUCKLED SHELL (Tq T e ANDTqe 1 . . . . . . . . . . . . .
P a g e
ii
V
vi
v i i
X
1
4
4
7 9
10
24
26
39
41
43
iii
Page 4
TABLE OF CONTENTS (continued)
Page
C . DERIVATION OF THE PARAMeTERS OF THE CHANGES OF CURVATURE (K1 AND K2) . . 45
Do COEFFICIENTS fn j , hnj, qnj, gnj, . . . . . . . . . . . . %j a n d e n j 46
BIOGRAPHY..... . . . . . . . . . . . . . 61
iv
Page 5
LIST OF TABLES
T a b l e
1. B u c k l i n g Fogces for X n t e r n a l P r e s s u r e p = Psi. . . . . . . . . . . .
2. B u c k l i n g F o r c e s f o r I n t e r n a l P r e s s u r e p = PA. . . . . . . . . . . .
3. B u c k l i n g F o r c e s for I n t e r n a l P r e s s u r e p = 1Psi . . . . . . . . . . .
4. B u c k l i n g F o r c e s for I n t e r n a l P r e s s u r e p = 2 5 ~ S i . . . . . . . . . . .
5. B u c k l i n g Forces fo r I n t e r n a l P r e s s u r e p = 35PSi . . . . . . . . . . .
6. B u c k l i n g F o r c e s for I n t e r n a l P r e s s u r e p = 4 5 P S i . . . . . . . . . . .
7 . Buckling F o r c e s f o r I n t e r n a l P r e s s u r e p = 55psi . . . . . . . . . . .
P a g e
28
28
29
29
30
30
31
V
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LIST OF FIGURES
Figure Page
. . . . . . . . . . . . . 1. General Sketch 32
2. Truncated Shell Geometry and Load . . . 33
. . . . . 3. Element of the Middle Surface 34
4. Sketch of Equilibrium Conditions . . . . 35
5 . Sketch of the Upper-End Support of the Bulkhead. . . . . . . . . . . . . . . . 36
6. Buckling Axial Force vs Number of Circumferential Buckling Waves for Various Internal Pressures . . . . . . . 37
7. Critical Axial Force vs Internal Pressure 38
vi
Page 7
!-
NOTATIONS
a
b
t
An, Bn
k
9, * V
Major axis of meridian e l l i p s e or r ad ius of c i r c u l a r c y l i n d e r
Minor axis of meridian e l l i p s e o r he igh t of dome
Constant she l l th ickness
C o e f f i c i e n t s of series expansion for radial deflec- t i o n and stress funct ion , r e spec t ive ly
Lam6 parameters def ined by equat ion (A71
*
P r i n c i p a l r a d i i of curva ture of the middle surface of a s h e l l equal t o kav3 and kav, r e s p e c t i v e l y
Radius of t n e p a r a l l e l circle of a she l l
Middle su r f ace coord ina tes 1
(1 + (k2 - 1) sin2c) ])i
* The le t ter "A" r e f e r s t o a n equat ion of t h e
Appendices.
v i i
Page 8
E
/u
1
m
P
pe
W
V 2 = A
D
Young I s modulus
Poisson's ratio
number of buckle waves in the circumferential direction
Total axial tensile load per unit circumferential length at the shell equator
Axial tensile load per unit circumferential length at the shell equator due to liquid load and its inertial force
Internal pressure
Hormal load per unit middle surface area
Additional force components per unit length in the buckled shell
Membrane force components per unit length prior to buckling
Normal buckling displacement of a meridian point
Altitude angle of the truncated edge (see Figure 2)
Stress function defined by equation (2)
Laplacian operator defined by equation (A41
Differential operator defined by equation (A51
viii
Page 9
Principal curvature of middle surface, see equations ( A 1 0 )
Resulting axial force (see Figure 4)
ix
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CHAPTER I
INTRODUCTION
The desire to reduce structural weight in missiles
and space vehicles in order to increase payload hae
resulted in thin-walled shell structures subject to
elastic and inelastic buckling as one primary mode of
failure. In many instances an adequate buckling analysis
or experimental information is difficult to obtain. Con-
sequently, very crude but hopefully conservative approxi-
mations or idealizations are employed for analysis of the
design. Often designers even avoid entirely particularly
lightweight configurations because of the complete lack
of experimental or theoretical information on potential
instability problems. The results of the above situation8
may be either the choice of a design which may not be
near optimum or, perhaps even worse, a very expensive
static test or flight test failure resulting in costly
delays and vehicle modifications.
1
Page 11
2
2,,"This paper discusses a s t a b i l i t y problem of a
m i s s i l e l i q u i d p r o p e l l a n t tank.1 Pr imar i ly , t h e buckling
of missile s h e l l s c o n s i s t s of two factors:
,U" I
cl-'
(a) I n t e r n a l pressure. I n shallow s h e l l s of
r evo lu t ion ( i .e. where t h e r a t i o r ad ius of c y l i n d e r "a"
over height of dome "b" i s l a r g e r t h a n f i 1 , t h e i n t e r n a l
p re s su re can produce c i rcumferent ia l compressive stresses
of s u f f i c i e n t magnitude t o cause e las t ic buckling of t h e
knuckle p a r t of t h e shell . I n 1964, Adachi and Benicek
[ l J made an experimental i n v e s t i g a t i o n on buckl ing of *
bulkheads w i t h toroidal t r a n s i t i o n s between s p h e r i c a l
caps and c y l i n d r i c a l walls under i n t e r n a l pressure .
(b) Axial tens ion . In deeper bulkheads of revo-
l u t i o n (i.e. a/b S C ) , t h e i n t e r n a l p re s su re a c t i n g on
t h e concave su r face produces predominantly t e n s i l e
stresses without introducing t h e p o s s i b i l i t y of dome
c i r cumfe ren t i a l buckling. Then, the main cause of
buckl ing of these k ind of s h e l l s i s a x i a l t e n s i l e forces.
This paper, however, d i scusses buckling of t h e
knuckle of t h e j o i n t between a c y l i n d r i c a l s h e l l and a
* N u m b e r s i n bracke t r e f e r t o t h e l i s t of re ferences .
Page 12
3
semielliptical bulkhead under axial tension. Since the
ratio of the length of major to the length of minor axis -\
of the shell considered is f i , internal pressure cannot
produce circumferential compressive stresses to cause
elastic buckling. The buckling of this type of shell is
mainly due to axially tensile load (including liquid
load and its inertial force). Viewed from testing patterns,
the buckling of a shell of revolution under axial tension occurs primarily at the region where its curvature is
largest. Thus, for simplicity, this type of shell may
be replaced by a truncated semielliptical shell under
axial tension to examine its buckling behavior by intro-
ducing adequate boundary conditions.
In 1963, Yao made the investigation on buckling of
a truncated hemisphere under axial tension [21 .
similar basic idea is employed in this paper, but the
problem discussed in this paper is quite different from
that of Yao's paper.
A
In the analytic work, Vlasov's small deflection
theory and Galerkin's method are used. The numerical
execution was performed by means of an IBM 709 computer.
Page 13
CHAPTER I1
ANALYSIS
Examing buckl ing t e s t i n g p a t t e r n s , w e know t h a t a
shell of revolu t ion under axial t ens ion , as shown i n
F igure 1, w i l l buckle primarily a t i t s knuckle part.
Thus it may be possible t o examine the buckl ing behavior
of a t runca ted s e m i e l l i p t i c a l she l l of revolu t ion (shown
i n F igure 2) i n s t e a d of that of a complete semielliptical
one by in t roducing adequate boundary condi t ions .
1. Basic Equations
"he fol lowing geometric r e l a t i o n s are introduced:
1 V "
[ l + (kz - 1) s i n Z
3 R,, = kav
R2 = kav
Ro 2= kav s i n $
4
Page 14
5
(a) Equilibrium equation. Using the middle surface
coordinates 9 and 8 , as shown in Figure 3, we know that
the equilibrium condition in the direction normal to the
surface for an elliptical shell element can be expressed
by the following equation ( 3 1
where
9 = stress function
For derivation of these equations see Appendix A.
Page 15
The additional force components in the buckled
shell wall are given by
For derivation of these equations see the Appendix B,
Equations (5), (6) and (7) approximately satisfy
the equations of equilibrium for a shell element in both
the meridional and the circumferential directions.
(b) Compatibility equation. The stress function
$
compatibility equation [3]
and the normal deflection hr are then related by the
Page 16
7
2. Normal Component of Stress Existinq Prior to Buckling
In dealing with the problem of stability of a
semielliptical s h e l l of revolution, one must take into
account the normal component of stress existing prior to
buckling.
From equilibrium equations [4] (see Figure 4)
and relations (l), we obtain the following expressions
for prebuckling forces
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8
During buckling, because of curvature changes,
these finite membrane forces contribute a normal
component.
soidal revolutionary shell can be evaluated to be
Principal curvature changes of an ellip-
For derivation of K1 and K2, see Appendix C. The normal
component is
Therefore, by substituting equations (12) , ( 1 3 , (14)
and (15) into equation (161, and using relations (1)
w e obtain the following expression for the normal
component t
Page 18
3 . Boundary Conditions
We can assume that the upper edge of the trun-
cated semielliptical shell of revolution which dis-
places the complete semielliptical one is clamped and
that its lower edge is simply supported, depending
upon the following factsr
(a) The upper edge of the shell is actually
stiffened by the presence of a skirt which supports
the tank. ($ee Figure 5).
(b) The shell will buckle primarily at the
region where its curvature is largest (see indicated
buckling pattern in Figure 2 1 , and buckling does not
arrive at both edges of the truncated semielliptical
shell of revolution . These conditions may be expressed as
at $=f- at +=$
= 0 leading to
$0
7r - T j = 3 e = o at $= and
or, for equations ( 5 ) , (6) and (71 , to
Page 19
4. The Method of Approximate So lu t ions
A p o s s i b l e method of solving the problem would be
f i r s t t o assume an a r b i t r a r y normal d e f l e c t i o n func t ion
t h a t sat isf ies the normal d e f l e c t i o n boundary condi t ions
(18); then equat ion (8) could be solved fo r the stress
funct ion $ i n terms of a p a r t i c u l a r s o l u t i o n involving
U a n d t h e genera l s o l u t i o n of t h e homogeneous equat ion,
the cons t an t s of i n t e g r a t i o n being determined by the
stress func t ion boundary condi t ions. F i n a l l y , the
normal d e f l e c t i o n func t ion and the der ived stress
func t ion can be s u b s t i t u t e d I n t o equat ion ( 2 ) , which
then would be solved by means of the Galerkin method [sJ . Because it is d i f f i c u l t t o so lve fo r the stress funct ion
i n terms of t h e normal d e f l e c t i o n func t ion , however,
t h i s method of s o l u t i o n can be rep laced by the equi-
v a l e n t process of choosing a r b i t r a r y func t ions t h a t
s a t i s f y t h e appropr ia te boundary condi t ions for both
t h e stress func t ion $ and t h e normal d e f l e c t i o n func-
t i o n and so lv ing both equations ( 2 ) and (8) by the
Galerkin method.
Thus *
Page 20
' . 11
* where &I* and @ are proper functions subjected
to the boundary conditions given by equations (18) and
(19) and s expresses the first variation.
For convenience, the following substitutions
are introduced:
The boundary conditions (18) and (19) then become
at x=O 1 and
Page 21
I 12
, Substituting the relations (22) into equation (31, I
w e have
I I Theref ore,
Page 22
13
S ibstituting the relat ions (22 ) in to equation ( 4 ) , c
have
Substituting the relat ions (22) into equation ( 1 7 1 , w e
have
i? * Suitable expressions for d and 2 which s a t i s f y
the boundary conditions (23) and (24) may be taken as
Page 23
14
i where
I Therefore, applying the operator A n [equation (2611 x
~ on the function [equation (2811, w e obtain I
Page 24
15
*L Applying the operator D ( - - - - - - 1 on 9 ( - - - - - - ) w e
have
* Applying the operator A on $ [equation PO)]
w e have
Page 25
16
.*- Performing D on , one finds
Page 26
17
Substituting equation (29) into equation ( 2 8 ) , we obtain
!
Substituting equations (311, (32) and (351, and
relations (1) and (22) Into equation (201, we obtain
Page 28
19
Substituting equations (33) and (34), and relations
(1) and (221, into equation (21), we obtain
Page 29
20
and (37) become
Page 31
22
t
t
t
. _
.
( 3 9 )
By i n t e g r a t i n g equat ions ( 3 8 ) and ( 3 9 ) , w e g e t t h e
fo l lowing equations:
where j =. 2,3,4 - - - - - - m In t roducing t h e r e l a t i o n P = P, + (where P =
e
external axial load) i n t o equat ions (38), w e o b t a i n the
f o 1 lowing equat ions
Page 32
where J = 2 , 3 , 4 - - - - - ~
For fnj, hnj, Qnj, gnj, dnj and enjt see
Appendix D.
For a nontrivial solution for An and Bn, the
determinant of their coefficients in equations (42) and
(43 ) must vanish, yielding an algebraic equation in the
following f o m
where fi(m) (i = 1, 2, 3) are rational functions of m.
Equation (44) can be solved for Pe, i.e.
The minima of P, for various p are the critical values of
the axially tensile loads (Pe).
Page 33
CHAPTER I11
APPLICATION OF A SPECIAL PROBLEM
The general characteristics of the special
semielliptical shell of revolution made of aluminum
alloy 2219 are (see
a =
b =
t =
h =
1 =
=
E =
P =
Figure 1):
198 in
140 in
0.163 in
68 in
cos $0
0.647
10.6 x lo6 lb/in2
0.33
B y substituting the above values into equations (42)
and (43) and taking one term in the summation (n = 2) we
find that the determinant of the coefficients of equations
(42) and (43) yields
24
Page 34
25
I
Putting internal pressure p equal to 0 , 5 , 1 5 ,
2 5 , 3 5 , 45 and 55 psi respectively, w e g e t the buckling
loads Pe f o r various number of waves (m) ( see Tables 1
through 5 ) .
The minima of these buckling loads are the
c r i t i c a l t e n s i l e loads for internal pressure 0 , 5 1 1 5 ,
2 5 , 3 5 , 4 5 and 55 p s i respectively.
The pertinent numerical execution was performed
by means of an IBM 709 computer.
Page 35
CHAPTER IV
CONCLUSION AND DISCUSSION
It can be observed from Figure 5 tha t the
c r i t i ca l load of t ens ion increases w i t h the increase
of i n t e r n a l pressure . This fact i n d i c a t e s t h a t i n t e r n a l
p re s su re w i l l decrease the degree of buckl ing of this
type of shel l .
Figure 7 shows tha t there is an approximate l i n e a r
r e l a t i o n between c r i t i c a l t ens ion load and i n t e r n a l
p re s su re i n t h i s p a r t i c u l a r problem. From the equa-
t i o n (44) w e know tha t , for m = mot a l i n e a r r e l a t i o n
between P, and p holds. mo i n d i c a t e s the argument f o r
which P e is minimal for a given p. Actual ly , f o r a l l I
I P the number mo is p r a c t i c a l l y the same, t hus g iv ing
the above mentioned l i n e a r r e l a t i o n shown i n Figure 7.
The subject problem is an i n t e r e s t i n g one, as
it is a p a r t i c u l a r example t h a t an e las t ic system i s
buckled by t ens ion , although the real cause of the
i n s t a b i l i t y i s the compressive hoop stress. Cer t a in ly ,
I 26
Page 36
27
there remain many areas f o r future s tudies , among them
determination of the buckling load by large de f l ec t ion
theory and consideration of i n i t i a l imperfections.
Page 37
28
TABLE 1
BUCKLING F O X E S FOR INTERNAL PRESSURE p = Opsi
m P, lb/in
10 20 30 40 50 60 70 80 90 100 110
11782 4157 2953 2343 2029 1871 1804 1794 1823 1883 1966
TABLE 2
BUCKLING FORCES FOR INTERNAL PRESSURE p = 5psi
m P, lb/in
10 20 30 40 50 60 70 80 90 100 110
13640 5413 4203 3593 3279 3121 3054 3044 3073 3133 3216
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29
TABLE 3
BUCKLING FOXES FOR INTERNAL PRESSURE p = 1SPsi
m P, lb / in
10 17357 20 7 9 2 3 30 6 7 0 5 40 6094 50 5779 60 5 6 2 1 70 5554 80 5 544 90 5573
100 5633 110 5716
TABLE 4
BUCKLING FORCES FOR INTERNAL PRESSURE p = 25psi
m P, lb/ in
10 20 30 40 50 60 7 0 80 90
100 110
21074 10434
9206 8 5 9 4 8279 8121 80 54 8044 8 0 7 3 8 1 3 3 8 2 1 6
Page 39
30
TABLE 5
BUCKLING FORCES FOR INTERNAL PRESSURE p = 35 P s i
m P, lbbin
10 20 30 40 50 60 7 0 80 9 0
100 110
24791 12944 11707 11094 10779 10621 10554 10544 10573 10632 10716
TABLE 6
BUCKLING FORCES FOR INTERNAL PRESSURE p = 45psi
m P, lb/ in
10 20 30 40 50 6 0 70 80 9 0
100 110
28507 15455 14208 13594 13279 13121 13054 13044 13073 13133 13216
Page 40
31
TABLE 7
BUCKLING FORCES FOR INTERNAL PRESSURE p = 55psi
rn P, &/in
10 20 30 40 50 60 70 80 90 100 110
32224 17966 16709 16094 15779 15621 15554 15544 15573 15633 15716
Page 41
32
f Liquid
Figure 1. General sketch
Page 42
I
3 3
I /i
/ I
Figure 2. Truncated shell geometry and load
Page 43
\
Figure 3 . E l e m e n t of the middle surface
34
Page 44
I . I
E l
4 I I
Figure 4 . Sketch of equilibrium conditions
Page 45
36
Bu Id head
/1 / '
/ ! ' ,
/ I
'\ I /' ' $7
Cylinder
,p
c 7 Liquid
Figure 5. Sketch of the upper-end support of the bulkhead
Page 46
I .
(x102)
300
2 5 0
lb 'e (in)
200
150
100
50
~-
37
I
psi p = 55
/ psi p = 45
p = 35
- psi
P = 1 5 ~ ~ 1
/
psi p = 5
Figure 6. Buckling axial force vs number of circumferential buckling waves for various internal pressures
Page 47
I
I I I I I I I I I I
E 0 4 F Q) k
E -4 I I 1
k 0 0 0 0 0 0
0 u
0 n 0 0 0 Q) L n 0 In
pc d 4 N Y
38
cv c
7-1 \ 2
3 0
3 n
3 d
3 ?
3 N
0 rl
0
k u h
a, pc a - Q) u Lc 0 W
rl Id
rl Id U -4 +J
-d k u .
I-
a, & 3 P
Page 48
LIST OF REFERENCES
1. Adachi, G., and Benicek, M., "Buckling of Torispherical Shells Under Internal Pressure," (SESA paper No. 884, May, 1964).
2. Yao, J. C., "Buckling of a Truncated Hemisphere Under Axial Tension, I' (AIAA Journal, October, 1963).
3 . Novozhilov, V. V., The Theory of Thin Shells," trans. from Russian by P. G. Lowe (P. Noordhoff, Netherlands, 1959), pp- 85, 8 7 , 88, 264.
4. Timoshenko, S . , and Gere, J. M., "Theory of Elastic Stability, " (McGraw-Hill Book Co. Inc., New York, 1961).
5. Wang, C . J., "Applied Elasticity, " (McGraw-Hill Book Co. Inc., New York, 19631, p. 162.
39
Page 49
I
I I
I
APPENDICES
I
Page 52
APPENDIX B
DERIVATION OF ADDITIONAL FOFCE COMPONENTS IN THE BUCKLED SHELL (Tq , Te and T p I
From reference [3], p. 8 7 , w e have
The -4 parameters for e l l i p t i c a l revolutionary
s h e l l s are determined by the expressions
and the Codazzi-Gauss conditions [l] reduce to the
re la t ion
43
Page 53
Thus w e arrive a t ,
I
I I I I I I I I I I
Page 54
APPENDIX C
DERIVATION OF THE PARAMETERS OF THE CHANGES OF CURVATURE (K1 AND K2)
Rejecting the displacements U and V, in the
formulae for the parameters of the changes of curva-
ture, one finds [ 31
Substituting the following expressions
into the above equations and using the relation [ 3 )
one has
Page 55
APPENDIX D
hnjl qnjl COEFFICIENTS fnj ,
nl d and e gnj' nj
1. Part I
46
