research investigations of bulkhead cylindrical …

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

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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

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.

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

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

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

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

!-

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

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

Principal curvature of middle surface, see equations ( A 1 0 )

Resulting axial force (see Figure 4)

ix

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

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 .

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.

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

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.

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

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

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

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

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 *

' . 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

I 12

, Substituting the relations (22) into equation (31, I

w e have

I I Theref ore,

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

14

i where

I Therefore, applying the operator A n [equation (2611 x

~ on the function [equation (2811, w e obtain I

15

*L Applying the operator D ( - - - - - - 1 on 9 ( - - - - - - ) w e

have

* Applying the operator A on $ [equation PO)]

w e have

16

.*- Performing D on , one finds

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

19

Substituting equations (33) and (34), and relations

(1) and (221, into equation (21), we obtain

20

and (37) become

21

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

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).

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

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.

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

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.

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

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

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

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

32

f Liquid

Figure 1. General sketch

I

3 3

I /i

/ I

Figure 2. Truncated shell geometry and load

\

Figure 3 . E l e m e n t of the middle surface

34

I . I

E l

4 I I

Figure 4 . Sketch of equilibrium conditions

36

Bu Id head

/1 / '

/ ! ' ,

/ I

'\ I /' ' $7

Cylinder

,p

c 7 Liquid

Figure 5. Sketch of the upper-end support of the bulkhead

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

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

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

I

I I

I

APPENDICES

I

42

We arrive at

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

Thus w e arrive a t ,

I

I I I I I I I I I I

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

APPENDIX D

hnjl qnjl COEFFICIENTS fnj ,

nl d and e gnj' nj

1. Part I

46

48

49

2. Part I1

50

51

52

I

I -

53

54

4. Part IV

55

56

5 . Part V

57

6. Part VI

5 8

59

60