NAVAL POSTGRADUATE SCHOOL · TECHNICALREI i NAVALPOSTGRADUATESi MONTEREY,CALIFORNIA93940...
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TECHNICAL REI NAVAL POSTGRADUATE Si MONTEREY, CALIFORNIA 93940 NPS-59Zfi24031 r NAVAL POSTGRADUATE SCHOOL Monterey, California RADIAL PRESSURE ON A GUN LAUNCHED MOTOR CASE DUE TO SLUMPING PROPELLANT David Salinas and Robert E. Ball March, 1974 Interim Report for Period July 1973-September 1973 Approved for public release; distribution unlimited Prepared for: Commander, Naval Weapons Center, China Lake, California 93555 FEDDOCS D 208.1 4/2: NPS-59ZC74031
Transcript of NAVAL POSTGRADUATE SCHOOL · TECHNICALREI i NAVALPOSTGRADUATESi MONTEREY,CALIFORNIA93940...
TECHNICAL REI i
NAVAL POSTGRADUATE Si
MONTEREY, CALIFORNIA 93940
NPS-59Zfi24031
rNAVAL POSTGRADUATE SCHOOL
Monterey, California
RADIAL PRESSURE ON A GUN LAUNCHED
MOTOR CASE DUE TO SLUMPING PROPELLANT
David Salinas
and
Robert E. Ball
March, 1974
Interim Report for Period July 1973-September 1973
Approved for public release; distribution unlimited
Prepared for:
Commander, Naval Weapons Center,China Lake, California 93555
FEDDOCSD 208.1 4/2: NPS-59ZC74031
NAVAL POSTGRADUATE SCHOOLMonterey, California
Rear Admiral Mason Freeman Jack R. BorstingSuperintendent Provost
The work reported herein was supported by the Naval Weapons Center,China Lake, California.
Reproduction of all or part of this report is authorized.
This report was prepared by:
WJOASSIELEDSECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)
REPORT DOCUMENTATION PAGE READ INSTRUCTIONSBEFORE COMPLETING FORM
I. REPORT NUMBER
NPS-59Zc74031
2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER
4. TITLE ("an. Subtitle)
RADIAL PRESSURE ON A GUN LAUNCHEDMOTOR CASE DUE TO SLUMPING PROPELLANT
5. TYPE OF REPORT & PERIOD COVERED
Interim Report for PeriodJuly 1973 - September 1973
6. PERFORMING ORG. REPORT NUMBER
7. Al)THOR(a)
David Salinas andRobert E. Ball
8. CONTRACT OR GRANT NUMBERfs.)
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Naval Postgraduate SchoolMonterey, California 93940 Code 59Zc
10. PROGRAM ELEMENT. PROJECT, TASKAREA ft WORK UNIT NUMBERS
Work Request No. 2-3128
II. CONTROLLING OFFICE NAME AND ADDRESS
Naval Weapons Center, China Lake,California 93555
12. REPORT DATE
22 March 197413. NUMBER OF PAGES
33M. MONITORING AGENCY NAME ft ADORESSfi/ different from Controlling Office) 15. SECURITY CLASS, (of thia report)
UNCLASSIFIED'15a. DECLASSIFI CATION/ DOWN GRADING
SCHEDULE
16. DISTRIBUTION STATEMENT (of this Report)
Approved for public release; distribution unlimited.
17. DISTRIBUTION STATEMENT (of the abatract entered In Block 20, If different from Report)
18. SUPPLEMENTARY NOTES
19. KEY WORDS (Continue on reverae aide It necaaaary and Identify by block number)
PropellantGun- launched rocketStress analysis
20. ABSTRACT (Continue on ravaraa aide If necaaaary and Identify by block number)
An investigation was undertaken to determine the radial pressuredue to the slumping propellant in a gun-launched rocket motor casesubject to high axial g loading during launch. Using a Rayleigh-Ritz formulation an approximate solution was obtained.
DD i JAN 73 1473 EDITION OF 1 NOV 65 IS OBSOLETES/N 0102-014-6601
|
UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (Whan Data Entered)
TABLE OF CONTENTS
NOTATION iii
INTRODUCTION 1
ANALYSIS 2
RESULTS AND DISCUSSION 11
CONCLUSIONS 17
REFERENCES 19
COMPUTER PROGRAM LISTING 20
n
NOTATION
a maximum rigid body acceleration of the projectile
A, B material constants, eq. (8)
C. constants due to body forces
e dilatation
e..e..eOQ normal strain componentsrr zz 98
E Young's modulus of elasticity
f body force in the z direction
F.
.
coefficients of eqs. (21) from Rayleigh-Ritz method
G shear modulus
L length of propel 1 ant
r, r. , r radius, inner radius, outer radius
T total potential energy
u displacement along r axis
U elastic strain energy
V potential energy of external forces
w displacement along z axis
a,B,Y><5tn,x,s,c Rayleigh Ritz undetermined coefficients
Y shear strain component
p weight per unit volume
a »o » ,,,normal stress components
zz rr 68
x shear stress component
in
INTRODUCTION
The launch phase of a gun-launched rocket is associated with high
acceleration loading of up to 10,000 g. As a result of the high accel-
eration the motor case portion of the rocket, which contains the
propellant, is subjected to axial and lateral loads. The lateral loads
are associated with the deformation of the contained propellant. This
"slumping propellant" behavior is the concern of the present investi-
gation.
ANALYSIS
The propel 1 ant-motor case system may be idealized as a two-dimensional
system because of axial symmetry. With cylindrical coordinates, r and z,
a typical section is shown below.
ro
fore +-n~^ /7_l propel lant
z, w
aft
ITmotor case
u = displacement in the r direction
w = displacement in the z direction
r.j= inside radius of propel 1 ant
r = outside radius of propel 1 ant
r, u
L = length of propel lant
The particular problem of the unbonded
propel lant is considered here. The
stiff motor case shell is assumed to
completely restrain axial displacement
at the aft end (r,0) as well as radial
displacement along the outer radius
(r ,z) with the resultant boundary conditions,
w(r, 0) = x rz (r,0)= (la)
u(r ,z) = x rz (r ,z) = (lb)
The remaining sides (r, L) and (r. ,z) are stress free, that is
_JiL
oz(r,L) = T
rz(r,L) =
ar(r
i,z) = T
rz(r.j,z) =
(lc)
(Id)
The acceleration of the propel lant subjects every particle to the body
force
fz
= "pam (2)
where p is the weight/unit volume, and a is the maximum acceleration
in g's.
A solution to the problem is obtained by the Rayleigh-Ritz method.
According to this method, an approximate solution satisfying the displace-
ment boundary conditions is formed. The unknown coefficients of the solution
are determined by minimizing the total potential energy of the system.
Here the displacement fields
u(r,z) = (L-z)(o + Br + yr 2 + 6r 3)
= (L-z)f(r) (3)
w(r,z) = z(L - |)(n + xr + sr 2 + cr 3) = z(L - f)g(r)
are assumed. The linear z term in the u displacement and the quadratic z
term in the w displacement were chosen to satisfy the following conditions:
i) Results from the Rohm and Haas finite element solution,
ref. 1, Appendix B, for the accelerated, unbonded propel 1 ant
show o and o to be linear functions of z. Note that Equations (3)
give this condition, i.e.
„ - E |3u . .,/u,
aw\| _ E
°r " MF[J7+"lr
+"iz~j|~ M^
e E [iw fau +u\l E
z l-v^az Ur r 1-v2
L \ U
|H(^+ g (r)}](L-z)
ii) The propel 1 ant is assumed to be nearly incompressible, and thus
linearly varying compression along the z coordinate causes the pro-
pell ant to move towards the centerline since it is restrained by the
case from moving outward, i.e. u ^ for all z,and u = for z = L.
This assumed displacement field is not valid for the bonded propellant.
The equation for w satisfies the displacement boundary condition w(r,0) = 0.
The displacement boundary condition, u(rQ,z) = is satisfied by setting
2 3
a = -erQ
- YrQ- 6r
Q(4)
The total potential energy of the system, T, resulting from the displace-
ment fields given by equations (3) must be formed. Denoting the elastic strain
energy as U and the potential energy of external forces as V, we have
T = U + V (5)
Determination of the elastic strain energy U proceeds as follows.
According to the strain-displacement relations
ef
- f£- (L - z)(B + 2Y r + 35r2
)
ee=£= (L - z)(* + b + yr + 6r')
e , " It- = ( L " zMn + xr + er2
+ ?r3
) (6)Z o z
_ 8U 3W _ / . a _ . „ 2. x _
3\
y = «— + r— = -la + 3r + yr + or ;T rz iz 3r ''
2
+z(| - z)(x + 2cr + 3cr )
and the stress-strain relations,
or
= Aer
+ B(ez
+ e )
ar
= Aee
+ B(er
+ ez
) (7)
az
= Ae2
+ B(er
+ eQ
)
x = Gyrz Wz
where
!l + v|(1 - 2v)
R - vE..B "
(1 ? v)(1 - 2v)
and
(8)
G ° 2(Wv) (9)
Substituting equations (6) into (7) yields stress-displacement
equations. The elastic strain energy of the system is given by
u.I•ij
eij
dV =i (°r
er
+ Ve +°z
ez
+ Vrz* dV (,0)
where the integration is over the volume of the propel 1 ant.
independence with the e coordinate gives
The
ri
L
U = 2tt '
\ (arer
+°e
ee
+°z
ez
+ Wrz* r dr dz
ro°
For convenience, let
(11)
ua
- , °je„ r dr dzr r
ub
= i- o Qe_ r dr dz (12)
uc
, o e r dr dz
ud
= , ir2Yr2
r dr dz
then
u = u + u. + u„ + u
.
abed (13)
Substituting the stress-displacement, and strain displacement relations
into equation (12), and integrating, we obtain
u =_*L~T
-j-if^+^ioC +y0\)-»-
A
4(4y8«S H-i?f
X+3<S^
+ A*X + /aO +-4! (5^ + 35X^/9^rt.
/l
(14)
UL = *L
3v 4
+ |_-y£/i5
-*-i<5
xA6l
u. =
3
3
Jl,
A^^+A^X^+^X^+^O+^C^S+mO
«.7 a j
+^(45n + 37TX-»-A/9^^o< C )^ A S (4JX + 3tff4 v l 5 -n
J a;
(15)
(16)
U J =
+yx) ^--0!(3^c+aT5 +iA) + 2.'f3Tf<7
*. (17)
-»il
The potential energy of external forces V, associated with the inertia
body force f given by equation (2) is,
V = - f * w dV = - 2tt
ro
L
• •
p *m' w ' r dr dz
• i
rl
(18)
Substituting the expression for w from equation (3) into equation
(18) yields
irL n 2 2 3l
lt2 5
-,
V = - ^ p am * [nr + f xr + \ sr + f tr ]m
l
The total potential energy of the system is obtained as
(19)
T = U + UK + U + U . + Vabed (20)
The theorem of minimum potential energy states that equilibrium is
associated with a minimum of T. Here T is a function of the
independent parameters B,Y»<$»n,S and c,. The minimum of T is obtained by
taking the partial derivative of T with respect to each of these seven
coefficients. This yields the system of 7 linear algebraic equations
in 7 unknowns,
Fn ^12
F2 iF2 2
17
27
^ <c?
(21)
F 71 F72 F77 5 C 7
The explicit expressions for the F. . and C. (i.. = 1, 7) coefficients
are
Wl>*) = A^Jl 3-siA? -H/l>; +A9 *Z +*aI JU 2i )
F(a,l)-F(i^)
30
F(i,5j = F(s,0 =
F(lJ7) = F(7,0 =
fm = F fc*) -
FC3.7) = F t7/3) - B {f4 /«/ + i4 Jit- £ *!)
P(4,4)= A{^-^M
F(4,5)= F(5>)= A-'i|(«»-Al)]
F(4.0 = F(vO = A-^U.4 -/l*)}
10
FO,7) = F(7,4)= AUK4 -*?)}
F(s/S)=A{iK4-^)heLl{|«-^)}
F(s>)=F(fe/s) = A{|=(^-/)n}+6L^^(^-^)}
F(s/7) -FUs) = A\i (/iL- J»t)] + GL
l
[f ( .l,*-**))
F<«,*)- Ali(^-A tMi + GLl{±(<-A!)j
F(4l7) = F(^ 6)=A{|(^-lpj + 6Li {||(.a|-/i?)}
F(7,7)= A{i(A«-JlJ)} + 6L*{|(/i;-AS)i
C,-C,-C,-0
C 4 - -/<**. (A.1-*!) Cs=-aj>a^(/i?-vj?)/3 .
C6= -/a^CA^-yi^/a. C7 =-ij> «-w (A
s--f)-)/s.
A computer program was used to obtain the solution of the problem. A
large number of problemswere solved for various values of the physical
parameters, r . , r , L and a , and the propellant properties u and
E. A listing of the program is given in Appendix A.
DISCUSSION OF RESULTS
It has been noted by other investigators that numerical difficulties
arise in the analysis of propellants when Poisson's ratio approaches 0.5
11
(ref. 2). To circumvent this difficulty Hermann (ref. 3) gives a
special variational theorem for nearly incompressible materials. In
the present investigation the numerical problem was overcome by using
a CDC 6600 computer with double precision. This results in approxi-
mately 32 significant digits, sufficient for Poisson's ratio to
0.4999999.
A study was conducted on the effect of Poisson's ratio on the
propellant stress for a propellant with 1.5 inch outer radius, 0.5
inch inner radius, 9.5 inch length, 500,000 psi Young's modulus and am
equal to 8000 g. Poisson's ratio was varied between 0.45 and 0.4999999,
The computer program results for ar at the propel! ant-Case interface at
the base are given in Table 1.
Table 1
V or(psi) Av Ao e
.45 -1589 -2.0 x 10
.48 -1691 .03 -112 -8.9 x 10
.49 -1771 .01 - 80 -2.1 x 10
.499 -2513 .004 -616 -2.8 x 10
.4999 -3744 .0009 -1231 -4.0 x 10
.49999 -4469 .0009 -725 -4.8 x 10
.499999 -4613 .00009 -144 -5.0 x 10
.4999999 -4630 .000009 - 27 -5.0 x 10
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j- fc--:^
6^ :!!! R4*>i ^B^it'0 i'fi in IHl !
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i I ! ' ! : M i:
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2 !_____ ____=___=—___^_—— •
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• ^~--—r-^-=-=
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Figures 1 through 3 are curves showing these results; the first
figure being a direct plot of stress o versus Poisson's ratio v.
Figures 2 and 3 show more clearly that the stress approaches a finite
value of -4650 psi as Poisson's ratio approaches 0.5. In Figure 4 the
dilatation e = e + ew + e . at r = r and z = is plotted against9 r z o
Poisson's ratio. The resulting curve verifies that the incompres-
sibility condition e approaches zero as v approaches 1/2 is satisfied.
In addition to the study on the effect of Poisson's ratio on
propel 1 ant behavior, analyses were made to determine the effect of
other parameters to behavior. The results obtained show that the
radial stress varies linearly with respect to acceleration, radii,
length and Young's modulus of elasticity.
The results of this analysis were compared with two other sources;
a Rohm and Haas finite element analysis (ref. 3), and an exact
solution for an incompressible material with zero interior radius.
The present formulation resulted in a 20 percent higher radial stress
than the finite element program, and 2 percent higher than the exact
solution, at the propel 1 ant-case interface at the base.
CONCLUSIONS
A computer program was developed for the determination of the
stress distribution due to a slumping propel 1 ant contained within a
launched rigid motor case. The program was initially incorporated
into the SATANS finite difference program for the calculation of the
elastic static buckling load of the motor case subjected to high
acceleration loading. The stiffening of the motor case, which is
predicted by theory, was obtained in the SATANS analysis; a favorable
structural response.
17
An unfavorable effect on structural integrity is associated with
the additional contribution to yield from the propel lant radial stress.
For a motor case of 2.5 inch outer radius, zero inner radius, 1/8 inch
thickness, 20 inch length, and a maximum acceleration of 8,000 g,
the radial stress of approximately 10,000 psi gives a hoop stress of
100,000 psi. The contribution to yield of this stress component is
of significant magnitude. That is, the propel lant has a beneficial
buckling stiffening effect but an adverse affect on yield.
The present analysis is based on the linear elastic theory of
solids and, therefore, the results are not valid beyond the initial
yield point of the propel lant. Because it is difficult to obtain the
material properties of propellants (Young's modulus, Poisson's ratio,
and initial yield), definite assertions regarding the structural
integrity of the propel lant are not given here; however, the stress
results obtained in some of the analyses show that additional study
regarding this concern may be warranted.
18
REFERENCES
1. Ball, Robert E., and Salinas, David, "Analysis of a ThreeInch Gun Launched Finned Motor Case," Naval PostgraduateSchool, NPS-57Bp-72011A, January 1972.
2. Thacker, J. H., "Deformation of Case-Bonded Propel lants UnderAxial Acceleration," 20th Meeting Bulletin, JANAF-ARPA-NASAPanel on Physical Properties of Solid Propel lants, I,
Nov. 1961, 81-90.
3. Hermann, L. R., "Elasticity Equations for Incompressible and
Nearly Incompressible Materials by a Variational Theorem,"AIAA Journal, Vol. 3, No. 11, Oct. 1965, 1896-1900.
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26
DISTRIBUTION LIST
No. Copies
1. Defense Documentation Center J2Cameron StationAlexandria, Va 22314
2. Library 2
Code 0212Naval Postgraduate SchoolMonterey, Ca 93940
3. Naval Weapons CenterChina Lake, Ca 93555
Attn: John O'Malley 1
Ray Miller, Code 457 1
Howard Gerrish, Code 4545 1
Bob Dillinger/Dwight Weathersbee, Code 4573 1
4. Naval Weapons LaboratoryDahlgren, Va 22448
Attn: Ken Thorsted, Code GL 1
5. Naval Ordnance Systems CommandNavy DepartmentWashington, D. C. 20360
Attn: J. Murrin, Code ORD 331 1
Z. Levensteins, Code ORD 032 1
Code 035 1
6. Naval Air Systems CommandNavy DepartmentWashington, D. C. 20360
Attn: Code 320 1
7. Asst. Prof. David Salinas, Code 59Zc 2
Mechanical Engineering DepartmentNaval Postgraduate SchoolMonterey, Ca 93940
8. Assoc. Prof. R. E. Ball, Code 57Bp 2
Aeronautics DepartmentNaval Postgraduate School
Monterey, Ca 93940
27
9. Chairman, Code 57BeDepartment of AeronauticsNaval Postgraduate SchoolMonterey, Ca 93940
10. Chairman, Code 59NnDepartment of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 93940
11. Dean of ResearchCode 023Naval Postgraduate SchoolMonterey, California 93940
28
J J.44
5 6853 01068088 7
