orbiter - icasweb web bulkhead lh2 20 a-a lb 3 coordinates the detail those crippling stiffeners....
TRANSCRIPT
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