ICASPAPERNO.„10

InfluenceofDiscontinuityStressonMain

PropellantTankageofa SpaceShuttleOrbiter

byHansR. Meyer—Piening,Head,Section"DynamicStability"

ERNO,RaumfahrttechnikGmbH,Bremen,WestGermany

and

JohnH. Dutton,ProjectStrengthEngineer McDonnellDouglasAstronauticsCompany—East

St. Louis,Missouri,USA

TheEighthCongress01the

InternationalcounciloftheAeronauticalsciences

INTERNATIONAALCONGRESCENTRUMRAI—AMSTERDAM,THE NETHERLANDS AUGUST28 TO SEPTEMBER2, 1972

Price: 3. Dfl.

á

INFLUENCEOFDISCONTINUITYSTRESSESONMAINPROPELLANT TANKAGEOFASPACESHUTTLEORBITER

HANS-R.MEYER-PIENINGErnoRaumfahrttechnikGmbh,Bremen,WestGermany

andJOHNH. DUTTON

McDonnellDouglasCorporation,St. Louis,Missouri,U.S.A.

Abstract AT1

A general closed form solution is given for the calculation of local

stresses in cylindrical axial load carrying pressure vessels which

are heavily stiffened in the axial direction and circumferentiallyrestrained by closely spaced ring stiffeners. The ring stiffeners

are treated as discrete stiffeners and are allowed to have a temper-

ature variation throughout the radial extension of the stiffener

and different material properties compared with those of the

vessel.

Numerical results are presented for representative L02 andLH2 main propellant tank geometries and loads for a fully reusa-ble Space Shuttle Orbiter configuration studied by the McDonnellDouglas Corporation (MDC) under a NASA-funded Phase BContract. A parametric study is included in which stiffener spac-ings and cross-sections are varied.

List of Symbols

A Ts

wp

a

F( 4)

ex. aci,

ax(s)

F, =

Y

temperature difference of area Ai of main ring

stiffener

temperature difference of tank wall

axial displacement

radial displacement

coefficient of displacement function. equation (30)

axial coordinate

radial coordinate

thermal expansion coefficient

parameter. equation (31)

parameters, equations (32), (33)

axial and circumferential strains

Poisson's ratio

axial and circumferential stresses in shells

axial stress in axial stiffener

coefficients, equations (34), (42)

differentiation with respect to (x/a)

a

A, As

c, ch

C1- C4

Dx

Dm

e, e'

EX, EMX

Kx

95

Mx

xNyb

Qx

7, sh

tit

radius of neutral surface, (figure 2)

cross section area of ringstiffeners, (figure 3)

lateral thickness of axial stiffener, (figure 3)

abbreviations, equation (35)

coefficients of displacement function, equations

(28), (29)

extensional stiffness, equation (9)

extensional stiffness, equation (8)

extensional stiffness, equation (13)

coefficient, equation (17)

resulting extensional stiffness, equations (26), (27)

radial distance from neutral surface, (figure 2)

Young's modulus

abbreviations, equation (35)

radial extension of axial stiffener from

middle surface of tank wall

resulting moment of inertiabending stiffness, equation (18)

bending stiffness, equation (10)

bending stiffness, equation (14)

spacing of main ring stiffener (figure 3)

distributed bending moment

prescribed axial line load

resulting axial line load, equations (26), (27)

circumferential line load

internal pressure

shear force of shell

inner radius of ring stiffener (R= a -e+ t/2)

spacing of axial stiffeners

abbreviations, equation (35)

wall thickness of tank, figure (4)

web thickness of main ring stiffener

I. Introduction

The main liquid oxygen and liquid hydrogen tanks of a fullyreusable Space Shuttle Orbiter configuration (figure I a) studiedby MDC, a NASA-funded Phase B study contract, were designedintegral with the primary body structure. An orthogonally stiff-ened pressure vessel design for these tanks (figure 1b) providedhigh axial buckling efficiency, discrete support for the thermalprotection system, and frame continuity with upper frames sup-porting longerons resulting an overall highly efficient structuralarrangement.

The resulting structure is relatively flexible with regard to cir-

cumferential stretching in between the ring stiffeners and rela-

tively stiff with regard to axial bending of the wall.

PHASEB SPACESHUTTLEORBITERCONFIGURATION

LONGERON

UPPER FUSELAGEFRAMES

FUSELAGE SHEAR PANEL

INTEGRAL MAIN BOOSTPROPELLANT TANKS

Figur. la

u' w"=—a z

a-

a + z

(compare figures 2 and 3).

(10)

3Eli ah (e + e"

s 2 3

ORBITERMAINBOOSTPROPELLANTTANKCONFIGURATION

LONGITUDINALSTIFFENERS

LONGITUDINALBUTT WELD

B-B (BETWEEN RINGS)

RING- /LONGITUDINAL ----1-

CENTERLINE STIFFENER _ 6 IN.

_I.WEB

\- \CENTERLINE WEB

\COMMON BULKHEAD

LH2

20A-A

Figure lb Figure 3

DIMENSIONSANDCOORDINATES

This results in high bending stresses in the outer fibers of the

axial stiffeners at each ring stiffener. Careful attention to detaildesign to minimize structural weight impact was required in thoseareas where high compressive stress could cause local crippling

of the stiffeners.

The derivation of the equations is based on Fliigge's derivation,

given in reference I for cylindrical pressure vessels with closelyspaced ring stiffeners. The equations are modified to account for

different bending and stretching properties in the axial and cir-cumferential direction.

For the evaluation of bending stresses in the axial stiffener,

the stiffeners are treated as being in a discrete arrangement.

II. Basic Equations

The basic equations are derived in accordance with the deriva-tion given by Frtigge in reference I. lt is assumed that the bend-ing stiffness does not vary along the axial and circumferentialdirection of the vessel and that the applied load is axisymmetric.

For the region of maximum vehicle axial stresses, the variations ofstiffenesses can be assumed to be negligible, as can be the varia-tions of stresses in the circumferential direction.

The axisymmetric strains are given by

The stresses are given by

•

(Ex + v(0)

1 -

a d - 9

•

((uS "x)1 -

In the axial stiffeners 0,5 is equal to zero, and therefore,

ax(s)= E E - z Z‘)a a2

For the net running load in axial direction one gets

-e +112 (a, z) b= Rx= f dz + — f ox(s)- dz

-e - a s -c+ t/2 a

—C w ti

z) (a + z ) dz , a+z

a(1 - v2)-e - t/2 a a-

Eb u' w"+ — f (- - z ) (a + z) dz

a 's -0+ C2 a a2

ax

(1) or finally,Dx D Kx

Nx = —a + w+ "— w(2) a a3

where

t(1 -e/a) bh e'-e t e tDx= E

_ s 2a 2h a 4a

CROSSSECTIONOF AXIALLYSTIFFENEDSHELL

••111.

Flgurs.2

2

ay

E tD =

1- v2

E t3Kx. (— - aet)[1- —(1-v21-171 - (a-2e)

12 2s t5s

For the circumferential direction the corresponding expressionbecomes

2)-e + t/2

-(co + v (x)(a + z)dz f

-e-t/2

where

1A = h + t [

h(1 _,2)2

-e + t/2 a f z2 I--= f (w + 11'— v z(a+ z)—)dz ,.. 1

a a2 B = an + n ——+ tat --e-t/2 4 b(1-v- ) 2

(20)

orKcs

= v—U‘ +—NV -NV"aaa3

an2 n3 at2 t3s1 i

(12)- 2 + 3812 b(1_02) 2

where

Et= (1 e/a)

1-v•

(13)

The numerical evaluation related to Space Shuttle Orbiter tank

geometry reveals that the term 4A/B2 is very small compared

with unity (2:0.2 percent) and, hence, e may be obtained from

E t3 2—(-12 + C t - eat)1_02

(14)(21)

For the bending moment the following expression is

derivede' = h - (22)

-e+ t/2 e'

Mx = f ax(1 + z/a)zdz - f ax(s)(1+z/a)zdz

e-t/2 -e+ t/2

E -e+ t/2 za, z2 z2 woo

= _ „ ff

-e-t/2 a a2

e' u I ul 2 w o w

E f [-z+ —zz3 a3 1dz-e+ t/2 a a2 a2

wh ch results in

Mx = Dm u' + E7.w" + w

where

(15) Now the following equations have to be solved. For equilibrium

of radial forces, see figure 4a and 4b,

SIGNCONVENTIONFORSHELLELEMENT .771

clx

+ — idz \/'z3w" vzw ,

.•°a3 a./.••

a oe

+ PrR

//

II Qx+dQx

(16)/

E et e2t t3

Dm =_1(__ _ _ _ _ )1.1 112a2 2s

1-v2 a a2

b(1-v2) h(e' -e)(e')3+ e3 et2 t2 I + (17)

s 2a 3a2 4a2 8a

E 9 t3 e3t et3K = 1(e-t+ - ) [1

b (1-v2)11

1-v2 12 a 4a 2s

Eb et2 3 e2t2 t4 (e,)3 + 03 (e,)4 _ e4

++ 1- 1 (18)s 4 8 a 64a

3 4a

The coefficient Dm, equation (17), has to vanish since for

w" = w = 0, Mx is required to be zero for all values of u'.

From equation (17), i.e., from the condition Dm = 0, the value

of e can be calculated. After replacing e' by li - e the evaluation

leads to a quadratic equation:

e = — (1 - / -

-B2

(19)

F igure 4a

EQUILIBRIUMOF RADIALFORCES

dQx(Mx+dMx) (11'+4x)

PIG 0:015713/251:tzt) NoQx Mx

Pr t t tFixItw Pr I w + dw

d ._ x

Figure4b

3

(34)

x = 0: w' = 0

w" = w(x = 0)

x = L/2 w' = 0

= r . w(x= L/2)

where w(L12) is identical to the radial deflection of the ring

stiffener and r and -= are factors to be discussed later which

relate the deflection of the tank wall to the resulting radial lineload Ox.

Mx" + allo - 171xw" pra2

where

D DKx= — u' + v-w w"a

a a3

Do KasIfN = v u + w - w

ck a a a3

For reasons of simplicity the following abbreviations will be

introducedM= -w"+ v-D wx a2 a

or, using Equation (7),

Mr .. K m,w +yeDw2

Elimination of u' from the equation for i ix and No yields

Do _ KA Dd, Kx= v —a N + wD (1-v2 -t)-vw,,(." )aN

w Dx x Dx a2 Dx a2

and after substitution in Equation (23) one finds

Ko DoKx e ,- w"" - [Rx+ v (- - ) - v - D1w" + D (1-v-- ) w (26)a2 Dxa2 Dx a2 a

EX = eaL/2a, Emx = e-aL/2a

= cos OL/2a, = sin pL/2a

ch = cosh pL/2a, sh = sinh AL/2a

dMx _ dw— - - - Q, = 0dx A dr A

or

(35) 11

Then the four constants Ci - C4 can be found by solving the set

of equations represented in table I.

From figure 3 the following relation, already contained in

equation (23), can be found

pr a2 _ v Dx aNx Mx' - Nxw' - a Qx = 0

Introduction of equation (16) into equation (37) yieldsEquation (26) is of the form

EI w"-N w" =1.:• (27) - - v - D w' - Fix w' = aQxa 2 a

At x = L/2 and x = 0 the slope w° is required to vanish, see equa-

tion (34), and, hence, equation (38) becomes

(28)a 2

aQx (39)

for the case:1Z1x < 2 V V.EI

andax/a- -a x/a_

w = wp + Clecosh x/a + C2ecosh flx/a

Owing to the assumed linear elasticity of the ring stiffener the radial line load Qx is directly related to the radial displacement w.

In the following discussion it will be assumed that an addi-ax/a - -a x/a -+ C3e sinh ox/a + C4e sinh p x/a (29) tional ring stiffener will be attached to the tank wall at x = 0,

0 in equation (34). The cross section of the ring stiffener

for the case: 315 > 2 V D•EI at x = L/2 is A = Ai and that of the ring stiffener at x = 0 isAs, whereby As < A.

In the above expression the terms wp, a, t3 and jEt-are given as

Pr a2 v(Dcb/Dx)a

D(1 - v2 Do/Dx)

The solution of Equation (27) is known to be

a x/a -ax/aw = wp + Cle cos /3 x/a + C2e cos ifix/a

+ C3e sin flx/a + C4e-ax/aax/a sin ax/a

wp -

The radial line load acting at the ring stiffener is twice the load

given by equation (36), since Qx is transferred to the ring stiffener

(30) from both sides of the shell

a 2 =3Tx4E14E1

1EA(31)

Qx x = L/2 = w(x = L/2) —R2 (40a)

31-x -2and = -/324E1 4E1 1

EAs

Qx 1x 0 = W(x = 0) R2

(32), (33)

(40b)

The four boundary conditions are Finally, Equations (39) and (40) yield the condition

4

TABLEI

SETOF EQUATIONSFORDETERMINATIONOF C1—C4

CASE I, Nx< 207

-a 13 fi

(—ag — flii-)EMx (a.; + AT); (—ar + /3F)EM x C2 0

–AB ----- –BA –BA c3 p

[(–AB–f)T + BAFIEM, [(AB–f)ii – BAFIE, [(–AB–r)i- – BAIEM, C4 r • iv-p

1 a

aF — /3W)Ex

. AB – L-7,

- 1(AB—1-).,_ BA.S-1Ex

where AB = a3 —342 BA = p3 — 3.2p

. CASE II , Nx > 2 rfr .71-

a — a

(ach —fish)Ex (—ach + ifsh) EMx

AB –Z.,= –AB – ---

RAB–f)ch + BAsh)Ex [(–AB–Dch+BAshlEMx

/3 0 Cl

0

(ash + ITch)Ex (–ash + Fch)EMx cn

4 =

BA BA C3

wP

[(AB–f)sh + BAchlEx [(–AB–f)sh + BAchlEMx—

C4

Pv7ip

where AB = a3 + 3432and BA = 3.2fi

w for zero temperature difference between ring stiffeners and tank wall.

EAaa2 (x = L/2)

2 R2 (x = L/2)

K —EAsa,w " (x = 0)

2 R2w(x = 0)

from which the values of r and 2-7.,in equation (34) and in table 1are directly obtained to be

E'Aa3 —E As a3r = _

2KR2 2KR2

where, for Case I, (Rx < 2 1 D EI )

cos /3x/a + C2 e—axia cos f3x/awp + C1 ex/aw = (28)

+ C3 eax/a sin fix/a + C4 e—ax/a sin fix/a

w" = C1 [(a2-132) cos fix/a — 2 cz0 sin /3x/ale ax/a (44)

C2 [(a2—/32) cos fix/a + 2a/3 sin Px/al e —axia

C3 [(a2-02) sin /3x/a + 24 cos fix/ale ax/a

C4[(a2—/32) sin /3x/a +- 2a/3 cos /3x/al e —ax/a

(42a, b)

u' 51x Kx

a Dx a a

and for Case II (Nx > 21 13El )

w = wp + C1 eaxja cosh #x/a + C2 eax/a cosh #x/a

(29)

+ C3 eaila sinh #x/a + C4 e—axia sinh #x/a

w" = C1 [(a2 F32) cosh # x/a + 2 a# sinh ax/al eax/a (45)

C2 [(a2+ #2) cosh #x/a — 2aF sinh Fx/ale --ax/a

C3 ((a2 + #2) sinh Fix/a + 2a# cosh -fix/ale axla

C4 [(a2 + #2) sinh #x/a — 2a#cosh r3x/al e —ax/a

From Equations (1) through (5) the following formulas are de-

(43) rived

In equation (41) the function w, according to equation (28) or(29), and its third derivative with respect to (x/a), wm, has to beintroduced in order to find the fourth equation in table I. Fromthe set of equations given in table 1 the four constants C1 - C4can be obtained. For the case As = 0 the calculations can besimplified by setting C1 = C2 and C3 = -C4, which for this casesatisfies the equations contained in row 1 and row 3 in table 1.This special solution corresponds to hyperbolic functions insteadof to the assumed exponential functions.

The case Nx = 2. b\P—T-Iwill not be discussed, since it maybe regarded as an exceptional case and may be avoided by aslight modification of the value of Nx.

In order to evaluate the resulting stresses, it is suitable to cal-culate u', wand w" for each desired station along the axial coordi-nate x.

From equation (7) we have

5

6

(58)x-ri(x = L/2)

(51)

I Ai A Ti

I AiT(R) = (a + t) aT

= L/2)a–e+t

L/2) = w (x = L/2) – RIQT

and, hence,

(R) -

L = 20 in.

= 83 in.

E = 1.07 107 lb/in.2

v = 0.3

(48) h = 0.6 in.

1) = 0.04 in.

t = 0.055 in.

s = 3.3 in.

A = lAi = 0.26 in.2

Max bending stress at outer fiber of axial stiffener

u' w"(s) = E f—)

x a a2

– Max hending stress at inner shell radius

u' w"ax = (— (e + t/2) a

a –e – t/2

Hoop stress in the shell

IV. Numerical Results

Numerical results are obtained for geometric properties whichare closely related to a preliminary Phase B tank design for a MDC Space Shuttle Orbiter.

The design values at a representative station are:

For the compression side of the tank (due to overall vehicle loads)the axial line load was approximately Nx = -1000 lb/in and theinternal pressure was 30 lb/in2. The temperature difference acrossthe ring stiffener at x = + L/2 was assumed to be T = I 0°F.

Figure 5 shows the obtained variation of axial and hoop

stresses between two ring stiffeners. It can be seen that bending

of the axial stiffener due to the existence of the ring stiffener at

x = + L/2 causes high compression stresses in the outer fiber of

the axial stiffener. This could result in local crippling of the

stiffener.

VARIATIONOF STRESSESVSAXIALCOORDINATE

80C-2-}HOOP STRESS

® HOOP STRESS

40

c,AXIAL STRESS0 „EN. miND

-80

120

1 160;0.6 0.5 0.4

AXIAL STRESS

A = 0.26 IN.2s = 3.3 IN.h = 0.6 IN.h = LO IN.

0.3 0.2 0.1 0

X

Figure 5

In addition to the results for h = 0.6 in. the case of a higher

axial stiffener, h = 1.0 in., has been investigated. This modifica-

tion results in a considerable reduction of the maximum stress

level in the stiffener.

2(a – e + t)2 tR

III. Consideration of Temperature Gradient

In the following discussion it will be assumed that the ring

stiffener, owing to heat conduction, has a certain temperature

gradient. The thermal deflection of the ring stiffener may be cal-

culated from

Radial stress in the ring stiffener

E A w(t. L/2),R(R) =

w-v(eT- )-1

a22

w u' ta

a – e –2

Hoop stress in the ring sitffener

(R) E "(x = L/2)aos –

a – e + t

where Ai designates a certain portion of the ring stiffener cross

section at temperature A Ti, whereas the thermal deflection of the

remaining tank wall is given by

w 4s) = a aT ATs (52)

Since w-r(R) may be different from wT(s), equation (41) has to

be modified in order to match the new compatibility condition

E A Ai ATI

Q, = — (a–e+–)aT ( AT5)1 (53)

2R2 2 Ai

The only difference in table I will be that wp has to be replacedby Tv where

Ai ATi

w

-

= w –(a–e + –t.)aT ( Ai

ATs) (54)P P 2

For the determination of the radial deflection w has to be re-

placed by 7, where

w

-

= w + aT a ATs (55)

and w(x = L/2) in equations (46) and (47) has to be replaced by(x 1/2) where

/ Ai ATi

/ Ai+ aT a ATs (56)

(57)

......

•

.

.

•

.

..

....

In figure 6 the effect of a reduction of the spacing of the axial

stiffener is shown. The stress reduction is 'midi less pronounced

compared with the effect of increased stillCner height I figure I.

if equal amount of additional structural weight is considered.

VARIATION OF MAXIMUMCOMPRESSIONSTRESSINAXIAL STIFFENER VS STIFFENER SPACING

-80

1) AXIAL STRESS(SEE FIGURE 5)X = L/2

A = 0.26 IN.2

h = 0.6 IN.

120

1400

Figure 6

In figure 7 the influence of a reduction of the cross section area

of the ring stiffener at x = + L/2 is demonstrated. Less circumfer-

ential stiffening yields considerably reduced bending stresses in

the axial stiffeners.

INFLUENCE OF RINGSTIFFENEDCROSSSECTION ANDTHICKNESSOF AXIAL STIFFENER

40

-1600.6 0.5 0.4 0.3 0.2 0.1 0

X

Figure 7

The effect of increased thickness of the axial stiffener is also

shown in figure 7. Fifty percent increase of the thickness reduces

the maximum stress from -103.19 ksi to 78.38 ksi. The first re-

duction of the maximum stress is accompanied by a weight sav-

ing: additional reduction requires a considerable amount of addi-

tional material.

In figure 8, the effect of an additional ring stiffener at station

x = 0 has been investigated. This stiffener has a cross section area

of As and no temperature variation throughout its cross section,

since the radial extension of the stiffener is assumed to be small.

It can be seen that the additional ring stiffener reduces the maxi-

mum stress in the axial stiffener.

-1600.5 0.3 0.2

X00.10.4

INFLUENCE OF AN ADDITIONALRING STIFFENER AT X = 0

40As =

0.0 IN.2

0.04 IN.2

0.08 IN.2

0.13 IN.2

s = 3 IN.h = 0.6 IN.b = 0.04 IN.A = 0.26 IN.2

Figure 8

Finally, the distribution of the radial deflection is presented

in figure 9. For these curves the stiffener spacing is 3.0 in. The

smoothing effect of more rigid axial stiffeners is clearly revealed.

RADIAL DEFLECTION OF TANK WALL

0.20.3 X

0.1 0

Figure 9

0.40.6 0.5

$ = 3 IN.b = 0.04 IN.h = 0.6 IN. A = 0.13 IN.2

h = 0.6 IN. A = 0.26 IN.2

h = 1.0 IN. A = 0.13 IN.2

h = 1.0 IN. A = 0.26 IN.2

(I)

0.50

0.45

0.40

0.25

IU 0.35 u-

uJ 0

.4-6.4 0.30ce

STRESS

-KSI

100

2.0 2.5 3.0 3.5

SPACING OF AXIAL STIFFENER - IN.

4.0

s = 3.3 IN.h = 0.6 IN.b = 0.04 IN., A = 0.26 IN.2

- b = 0.04 IN., A = 0.13 IN.2 b = 0.06 IN., A = 0.13 IN.2

V. Conclusion

It has been demonstrated that heavily stiffened pressurized

tanks require a detailed study of their discontinuity stresses. The

"smearing-out" technique is not applicable for such structures

and may lead to erroneous results. Accounting for axial load

coupling ("beam column effect") was included and found to be

important.

For the tank structure treated in this paper it had been con-cluded that a local increase in thickness (and eventually in height)is required for the axial stiffeners in the vicinity of the ring stif-fener. In addition, the ring stiffener should be designed to re-strain the tank from radial expansion as little as possible. Thisrequirement creates no severe design problems, since the ring

stiffeners are designed to introduce mainly shear loads into thetank structure.

An additional ring stiffener at x = 0 results in a comparativelyhigh weight penalty. The same is true for reduced spacing of the

axial stiffeners. Effect on total structural weight was less than

I 00 pounds, due to proper local design detail resulting from this

study.

References

(1) W. FlUgge, Stresses in Shells

Springer-Verlag, Berlin/

Heidelberg/New York, 1967

8