The use of the NASTRAN conical shell element in static, eigenvalue, and direct transient analyses is demonstrated. The results of a NASTRAN static solution of an externally pressurized ring-stiffened cylinder agree well with a theoretical discontinuity analysis. Good agreement is also obtained between the NASTRAN direct transient response of a uniform cylinder to a dynamic end load and one-dimensional solutions obtained using a method of characteristics stress wave code and a standing wave solution. Finally, a NASTRAN eigenvalue analysis is performed on a hydroballistic model idealized with conical shell elements

Goeller, J. E.

Messick, W. T.

Yang, J. C. S.

English

NASA Technical Reports Server

TRANSIENT ANALYSIS USING CONICAL SHELL ELEMENTS
By Jackson C. S. Yang, Jack E. Goeller,
' and William T. Messick
Naval Ordnance Laboratory
SUMMARY
The use of the NASTRAN conical shell element in static,
eigenvalue, and direct transient analyses is demonstrated. The
results of a NASTRAN static solution of an externally pressurized
ring-stiffened cylinder agree well with a theoretical discontinuity
analysis. Good agreement is also obtained between the NASTRAN
direct transient response of a uniform cylinder to a dynamic end
load and one-dimensional solutions obtained using a method of
characteristics stress wave code and a standing wave solution.
Finally, a NASTRAN eigenvalue analysis is performed on a hydro-
ballistic model idealized with conical shell elements.
INTRODUCTION
/
One of the principal areas of interest at the Naval Ordnance
Laboratory is high-speed water entry of naval ordnance. In order
to achieve stable water entry (no broaching) at low entry angles
off the horizontal, the nose is frequently made blunt so that _
the impact force is nearly axial. The rise time of the impact
force can be quite small, depending on the entry angle. Hence, :. ,.
a transient analysis of structural response is required. This
paper deals with an analysis of a ring-stiffened hydroballistic _ ..:
model which is designed to impact the water at very high speeds. :_'
The NASTRAN conical shell element appeared to be useful since i :_.
many of the models tested at NOL are axisymmetrio, monocoque i ,_._._J
structures of contour shape which are exposed to external _'_._?
pressure and axial and transverse loads However, there has :"-_:_:,
been little reported use of this element. Reference 1 illustrates ,_ ._-
its use in a modal analysis of a rlng-stiffened shell and
demonstration problem 1.5 {reference 2) is a static loading of
a uniform cylindrical shell. In order to gain confidence in the
use of the conical shell element before modeling the hydroballistic
model, simple structures were analysed and the results uompared
to theoretical values. NASTRAN runs were made on the CDC 6400
computer at NOL using Level 15. i. i.
1974006473-130
https://ntrs.nasa.gov/search.jsp?R=19740006482 2020-03-17T14:15:35+00:00Z
SYMBOLS
A cross-sectional area of shell
E modulus of elasticity
Fo constant end load on cylindrical shell
!
h thickness of cylindrical shell
L length of cylindrical shell
i I m mass per unit length of cylindrical shell
."."1 Mx axial moment per unit length
R radius of neutral axis of circular cylinder
t time
• rise time of force pulse
u axial displacement
x axial distance /
-/
6 displacement of circular shell at junction with ring ,
stiffener
u poisson'8 ratio
o% hoop stress ._;:_.
°x axial stress ,,,:,-
_r rth eigenvalue in circular frequent.2 ;,_
EXTERNALLY PRESSURISBD RING-PTIFFENED CYLINDER >
The ring-stiffened 8hell section of a hydroballiatic model
was analysed to determine the stresses when it i8 exposed to ,.. .;, -
external pressure during launch in the gas gun. A midsection '}" .....consis_ing of three typical bay8 was analyzed. The shell was
assumed to be olsmped at ea_h end (no pressure applied at ends), t
Figure 1 depict8 the finite elements used in synthesizing the _
NASTRANmodel of the ring-stiffened cylindrical shell The _ ' _
overall model had _6 rin9_ (or grid circles) and 95 elements, i "yielding a total of 471 degrees of freedom for each har onic. _" r _ ' '. _"
.1.2-6 ) "_"°'_
Sl nnmumlmml
1974006473-131
-, 4%,
The zero th harmonic was used in the problem since the ]oading :
_' was axisymmetric. The material for the model is aluminum.
I' Element forces, bending moments, stresses and ring point
deflections were obtained for an externally applied pressure _
: load of 6.894 (I06) N/M 2. To gain further insight in using _
conical shell elements, multipoint constraints were applied to
; the ring at the junction between the ring stiffener and the •
; cylindrical shell. This eliminates the overlapping of material
from the stiffener and the cylindrical shell, see Figure i. The
number of elements was also varied to observe the effect of :!_
element size.
A discontinuity analysis was done on the ring-stiffened
r cylinder in order to assess the accuracy of the NASTRAN results. _
[ The rings were considered to have a thick-walled Lam_ stress _distribution caused by the discontinuity shear force at the
ring-shell Junction. The short shell between rings was considered _
a beam on an elastic foundation so that displacement and slope _ "
relations from reference 3 were used. By using compatibility of _
slopes and displacements at the Junction, the discontinuity shear _
and bending moment were obtained. Axial and hoop stresses were _
calculated from _
The significant results fz_ the N_TRAN static analysis
(rigid format one) and the discontinuity analysis are presented
in Figures 2 through 5. Figure 2 shows a comparison of the radial
displacement along the center bay. Figure 3 shows the hoop stress .....
at the inside and outside wall. The agreement is quite good. :. .,.
Figures 4 end 5 show the co_parlson of the axial bending moment
and axial stress. The agreement is quite good in close proximity
to the stiffener, but _adually deviates near the midspan of the
bay. This might possibly be Impx_vod by using smaller elements.
The ring size was very .tll:_rtent. around the stiffeners because
of the rapid attenuation of the bending moment. Very few
differences were observed when HPC was used at the Junctions
of the shell and the stiffene:s. This indicates that the
overlapping does not have too much effect on the results for the /
shell-to :in9 thickness :at/o used in this problem.
1_7
i
1974006473-132
", _,?
4r
RESPONSE OF UNIFORM CYLINDER TO DYNAMIC AXIAL LOAD
In a typical water-entry body, the structure is exposed
. to transient loading and the body must be considered as free-free.
Before proceeding to actual modeling of the complex structure
" involving stiffening rings, etc., several simplified structural
models were investigated and comparisons made with known classical
solutions. Rigid format nine, "direct transient analysis," was
used in the NASTRAN program. Figure 6 depicts the finite elements
used in synthesizing the NASTRAN model of the cylindrical shell.
The overall model had 21 rings (or grid circles) and 20 elements,
yielding a _ct:l of 42 degrees of freedom.for each harmonic. The
.'. zero th har_i_ was used in the problem slnce the loading was
' axisymmetric.
b
_ The transient dynamic stresses, element forces, and
_ deflections were obtained for selected eZements and rings. Three "
_ _ cases of dynamic loads were applied to one end of the cylindrical
different rise times and a trapezoidal pulse. These dynamic
loads are specified on TABLED1, TLOADI, and DAREA cards. The
structure was considered as having free-free boundary conditions.
Comparisons of the NASTRAN results were made with one-dimensional
stress wave code which uses the method of characteristics and
also a one-dimensional closed form standing wave solution. These
latter solutions do not include the effect of hoop stress as the i
NASTRAN element does.
The standing wave solution was obtained for a fr_.e-free bar
loaded at the end x - 0,by a force which is a ramp to time to
and a constant FO after 4_ The displacement of the bar isgiven by (see referenc
O< t< t o
i
2_'0 r,m sin _rt r_x -__ _
" EEE %3 col; (-£-')
1974006473-133
........ -............. _, ........_& ._ ..... ................. --,,.
" RE_ODIIc-IRI+_! ¢ 0_: ltll_C)PIC_,ItlAL PA(,[ IS PC)(SR
i
2Pc = sin _rt- sin '_r,_+-to)2: cos (_1
_-,:. _to r= 1 _r 3
where _r = r_ _/-_-_
+_ mLZ
! the first term is the rigid body motion. The second term may be
I Looked upon as the static deformation; the series represents the
: _armonlc oscillation terms. The stress can be computed from
ax - E ___u
_x
, Figure 7 shows a comparison of displacement at the end
_++++' (x - 0) where the force i8 applied and at the midspan (x - 152.4
+ m). The NASTRAN solution follows the simplified theoretical
_++ ++:_olution reasonably well. Figure 8 shows a comparison of the
_+++. _xlal stress at the neutral axis of the first element and the
_i _: _dspa_ element. Thl8 axial stress was used since the theoretical
:_i++_'_*_,_olutlon ignores bending stress. The comparisons are, in general,
<.._; _ot bad. There appears to be some long-time effect, but this
_j_:o.,_ight be due to the rather large element size used in the NASTRAN
i:_olutlon._-- Figures 9 and I0 show similar comparisons, except a /
risetimewasusedontheloadingf.ctlons.Piguren
lhows a comparison of axial stress at the first element and
element for a trapezoidal loading pulse. Again, the
with the one-dimenslonal stress wave theory is
reasonably good. _,
EXGENVALUE ANALYBZ8 OF HYDROBALLISTIC MODEL I /
I++++
The hydroballistim model and the finite element discretization _+,*
shown in Figure 12. The length of the model is 345.12 ca and _++
body diameter is 34.29 _n. Xt ¢onslsts of a thick-walled
nose 8eotlon and rlng-stlffened aluminum mid and tail --
mctions. The aluminum skln is .794 om thick. Four equally ++
fln8 a_e atta_he_ to the midsection and four to the tall
lectlon. +
The blunt nose of the model causes the load at water impact
be nearly axial. _e_efore, the axial mode of vibration o_
model was examined. Figure 12 shows the 74 rings and 74
shell elements used to represent the model. The neutral *
of some of the elements has been moved insteaa of retaining _
original positions and using llll_'sto connect the rings. , +_+_
129 "_
1974008473-134
t.f
4
The reason is that in a static analysis of a clamp band which
- had MPC'd rings at discontinuities in section, an ill-conditioned
stiffness matrix was obtained. The "epsilon sub E" check yielded
values on the order of one. Since the changes in stiffness of
the hydroballistic model sections were even more severe, it was
, thought that for a first solution, a slight loss in accuracy from
modifying the model would be acceptable.
The eigenvalue analysis was performed for axial vibration
by allowing radial and axial displacement and axial rotations at
each ring. Thus, a 222-degree-of-freedom model was analyzed.
The first ten modes of vibration are given in Table I. The
' fundamental mode of 282 cycles per second seems reasonable.
Shifting the neutral axes of some shell sections yielded a
i well-conditioned matrix with the "epsilon sub E" check having
a value of 3 x 10-13.
The transient response to an axial impulse will be obtained
and the results compared to the data obtained from the instru-
mented hydroballistic model with a water-entry velocity of 305
meters per second.
CONCLUSIONS
' The results obtained from using the NASTRAN conical shell
element agree well with theory. Using a 90-degree orientation
for the conical shell element and an MPC for representing a ring
_tiffener yields excellent results. However, using MPC's to
connect discontinuitles in neutral axis radii can lead to ill-
conditioned matrices.
_ Using the conical shell element in static analyses is
_- routine. However, in the process of applying rigid format nine
_irect transient analysis) to conical elements, a number of
._ _ non-standard procedures must be practiced in order to obtain _
......._,- _ the results. The P field in the DAR_hK bulk data card must be
_ ,,. determined by the following formula:
_:_ P - ring ID + 106 • (llarmonic + i)
_'_',': In the case control deck, reference to "grid points" is by the
'_-__V' same formula. In the executive control deck, an alter must be
_i used to switch to Sort 1 output. This enables the output to be
printed in an orderly fashion.
_._. It is hoped that some of the bugs encountered in using
k_" the conical shell element will be eliminated so that more use
_.,_, can be made of it. For axisymmetric structures subjected to
_ loads which may be accurately expressed with a small number of
_"", 150
1974006473-135
% •
j.
harmonics, it is less costly to use this element than to model
with a large number of plates.
ACKNOWLEDGEMENTS
The authors would like to thank Dr. G. Everstine of the
Naval Ship Research and Development Center and Mr. R. Edwards
of the Naval Ordnance Laboratory for their helpful assistance
and suggestions.
REFERENCES
i. Everstine, G., Ring-Stiffened Cylinder, Proceedings of the
NSRDC-NASTRAN Colloqium, held at NSRDC, Washington, D. C.,
12-13 Jan 1970, Paper 3.3
2. NASTRAN Demonstration Problem Manual (Level 15), NASA
SP-224(01), Jun 1972
3. Het_nyi, M., Beams on Elastic Foundation, The University of
Michigan Press, Ann Arb_, Mic_igan_ _967
4. Meirovitch, Leonard, Analytical Methods in Vibrations,
Macmillan Company, New York, New York, 1967
L_
131
1974006473-136
iTable 1 - Eigenvalues for the Hydroballistic Model
, Mode Number Eigenvalue (cycles/second)
' 1 0.0
2 282
•_ 3 644
4 843
t
5 1365
I 6 1692
7 2300
i 8 2582
9 2975
i0 3194
i
g
_, 132
1974006473-137
|
---- - -- " i i mnnn
0.794C/_,_
U --U
_r)
FIGURE1 NASTRANMODELOF EXTERNALLYPRESSURIZEDRING-STIFFENEDSHELL
133 •
f
1974006473-138
"Jr
Ib
• 10 "
(3 ®
_ 5 (3
®
@ <3
0 I I I
2.5 5.0 7.5 10.0
:E
I
Z -5
THEORETICAL
I.--
-!0 e NASTRAN
O -15
Z
10.2CM WZ
i --g
-25 / W
| -30
j DISTANCE,×, CM
FIGURE4 AXIAL BENDING MOMENT
!
-200
-150 _ THEORETICAL
e_ O NASTRAN •!
_o_ -,SO OUTSIDEWALL_ i':_¢i
0
 ,ooIJLI 50
"_ x_ f _ _' INSIDEWALL
r,:__ _ , _,
:, _ 100 :4,
_'"" 150 "'"'
200 , .:
DISTANCE,X, CM
i
FIGURE ,5 AXIAL STRESS ,,_'
'7
]3_ .ii_"
1974006473-140
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1974006473-141
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F THEORETICAL
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c_ x = 144.8 CM o
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• £x 100i
r
_. /i o| 0
i o_ I "t ° t |0 - 0.5 1.0 1.5 -2.0" 2.5 3.0 3.5 4.0 4.5 5.0
150 x = 7.62 CM
O,m, _ .___m ...... _.,-=.4__--_-_.==. AG_O®q_'_
. -_ ,,,- ..,,1,7_o _ -v_,_ooooo_
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t- 50
X w
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TIME, t, (IO)'3SEC
FIGURE 8 AXIAL STRESS
4
1.38
lr
m rJU...... J_..JL_ - J _._.
I
] g74006473-] 43

<F
THLEORETICAL
978,609 N STRESSWAVE
t ® NASTRAN
00.15(10)-3SEC
150 x
= 144.8 CM P@
i _ 50
, 5
' 0
, I
,I
I
J x = 7.62 CM
I150
, ._ , , bx 100 -.
1,"7 "
0 i i
1.0 2.0 3.0 4.0 5.0
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1974006473-145
x = 7.62 CM
150-
z_ 1_
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X
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I I I
0 1 2 3 4 .
TIME, t, (IO)'3SEC
FIGURE 11 AXIAL STRESS
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1974006473-146



Transient analysis using conical shell elements

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