A square plate with clamped edges under a concentrated load was modeled using NASTRAN and ASKA finite element computer programs. Deflections were computed for various width-to-thickness ratios (b/t) of the plate element, and were compared against the classical theory to determine the b/t limitations. A cylinder with simply supported ends was modeled using NASTRAN and STAGS computer programs for buckling analysis. The models were subjected to a uniform radial pressure loading. Several parameters were changed, and the effects of those variations are presented. Utilizing these data, a model which will produce results comparable to published empirical data can be constructed and processed for a minimized cost

Shah, L. C.

English

NASA Technical Reports Server

NASTRAN STATIC AND BUCKLING ANALYSIS -
COMPARISON WITH OTHER LARGE-CAPACITY PROGRAMS
by Lalit C. Shah
Rock-_'cllInternatJonal
B-I Division
Los Angeles,Califol_ia
!
i
SUMqARY
i
a
A square plate with clamped edges under a concentrated load was modeled
[ using NASTRAN(referencei) and ASKA (reference2) finiteelementconjurer
i ' prograns. Deflectionswere computedfor variouswidth-to-thicknessrati,Js; ' (b/t)of the plate element,and were comparedagainstthe classicaltheoryto
! determinethe b/t limitations.
I A cylinderwith simplysupportedendswas modeledusingNASTRANand
STAGS (reference3) computerprogramsfor bucklinganalysis. The models'.¢ere
I subjectedto a uniformradialpressureloading. Severalparameterswere
changed,and the effectsof thosevariationsare presented. Utilizingthese
} data, a modelwhich will produceresultscomparableto publishedempirical
i data can be constructedand processedfor a minimizedcost.l
f ,| m'KfIC ANALYSIS
'/
:2
The user c_f finite element computer programs has numerous limitations to ,_
"_",. be considered when constructing a mathematical model of the structure to be _."
, analyzed. Althoughconsiderableinformationis availableconcerningthe plate
,_,_ elementaspectratio (a/b) (i.e,,length-to-widthratio),the effectof vary-
,.: ing the width-to-thicknessratio (b/t)has not previouslybeen presented, ii
[_, _ The effectof varyingthe plate elementb/t ratiowas investigatedfor the
_ " ! NASTRANand AS}_ finiteelementcomputerprograns.
_i_ This investigationutilizeda squareplate with clampededges. Two ele-
ments,CTRIAZand CQUAD2,availablein NASTRAN,were used in two separate -_
!_'. models. One tri:_le-plateelement,TRIB3, availablein ASKA, was used in
_,,_., the thirdmodel. These models,shown in FigureI, were IS2.4cm (60 inches)
_c,_,',- squareplateswith variedthicknessto achievethe b/t ratio desired• The
69 ""t
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.. . , -
]974006473-075
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ibasic model mesh size was selected based upon previous experience. One model "
with mesh size reduced by a factor of 2 was processed, and the results were
compared to verify that the basic model mesh size was valid.
• One loading, which consisted of a concentrated load applied in the _eo-
metric center normal to the plate, was selected due to its ideal checks for
the bending characteristics of any plate element. This loading was applied to
each model processed.
, The results of the two NASTRAN models and the ASKA model are summarized
in table I. The resulting computed deflections for the three models are tabu-
_ fated for the various b/t ratios investigated. Included in this table are the
theoretical deflections based upon classical equations (reference 4). These
[ deflection data are presented graphically in figures 2 through 4 The plot of
the percentage di fference between computed deflection and theoretical deflection
._ _ is shown in figure 5 for the three models investigated. The t_,'o NASTK_ plate
t_T J,'%
, elements, CTa,_, rand _,_c_"_n°-,break deem. in regions of b/t less than five. The
i ASKA element, TRIB3, is quite consistent, even for extremely low values of b/t.
i It is apparent that a limitation on the value of b/t exists for the NASTR/hN
plate elements. This limitation should be considered along with the aspect
_. ratio (a/b) limitations _hen constructing a model for the ;_ASI'RXq computer
program.
BUEKI,INGANALYSIS
Buckling analysis is an eigenvalue problem which may result in very high
computer processing costs to achieve a valid solution. This report presents
an investigation into the various modeling parameters that affect the solution
, and the computer cost. The result_ of this study reveal an approach to achier-
, ing a valid solution for minimized computer cost.
:_ .... This investigation consadered a cylinder under uniform radial pressure
,, loading. According to Do_nell's equation, under uniform radial pressure, the
. ,s
:,:_. buckling stress of the cylinder is:
K 2
•,,i! = y t :
' ' °cr 12( )
i For moderately long cylinders, this equation gives quite good correlation with
test data (reference5). For this investigation, a data point was selected
, where the test result and the preceding equation value practically coincide.
This cy]inder model is shown in figure 6. The cylinder was modeled for
70
i
1974006473-076
ii';. NASTRP_N,a finite element computer program, and for STAGS, a finite di florence .
_" computer program. Essentially, the same parameters were varied for both models
_}i _, in determining effects the solution validity and the computer costs.upon
_:l-__ The results of the SASTK_Nand STAGSmodels are presented in tables 11
_21t5.'"
_ and Ill, respectively. These data are presented in figures 7 through 9.
_.: Appendix A contains the mode shapes for all the m.,de!s st,,,liod in this
_ investigation. [
ii A significant parameter in modeling for either NAS'rR_Nor STAGSis the
circumferential spacing of grid points which determine the number of elements
per half wave-length. As indicated in figure 7, an extremely narrow range of
j circumferential spacing may be considered in modeling in order for NASTP&_/
buckling analysis to achieve valid results. 17ae NASTRANmodel that is very
fine is equally as erroneous as a model that is very coarse. These models
that are outside this narrow band of acceptable circumferential spacing pro-
duccd results that deviated from the theoretical value by up to 70 percent.
, The inproper selection of the circumferential spacing for the STAGSprogram
' can result in extremely high errors, over 3,000 percent, as sho_a_ in figure 7.
The results from the STAGSmodels indicate that the error percentage is directly
' related to the coarseness of the model, and as the circumferential spacing is
, reduced, the computed value approaches the theoretical solution. For this
, particular cylinder model to achieve a valid solution, the STAGSmodel, required
i a 3-degree circumferential spacing, whereas the NASTRANmodel required a
10-degree spacing.
The aspect ratio of the plate element.° was considered as an important
parameter in this ix,vestigation. Although most of the models utiiized a
constant number of uniformI/ spaced longitudinal cuts, a few were processed
using nonunifo_n_Ty spaced longitudinal cuts to determine the effects of vary- _;_.
ing the aspect ratio. It was a surprise to learn that the results did not
change appreciably. Apparently the aspect ratio of the plate elements is not a "_"
critical parameter for NASTR/_ buckling analysis. The data presented in fig- "
ure 8 for extremely low and extremely high aspect ratios are related to the very
7 '. coarse and the very fine circumferential spacing models, respectively. There- -'_
fore, the most probable reason for the results is due to the circumferential _;':
spacing. ,"
• The NASTRANmodels were processed on IBM370/165, and the STAGSmodels ,:
were processed on CDC 6600 computer system. The resulting machine time data '
[_ . are presented in tables iV' and V. This information is converted to machine ;.
| I cost in dollhrs and presented in figure 9, Even the very fine model used in
_. STAGSto achieve a valid solution resulted in less computer cost than any of• m_ the NASTP,u_models processed. This may be partly attributed to the two corn- -
if. I purer systemsused in the investigation.Althoughfigure9 presentsthe com-
• puter cost, a significant part of the total cost for buckling analysis is the
man-hours required to construct the model and prepare the data. Also, the _;",
,_-. ..
1974006473-077
INASTR_N program provides a lot more flexibility in modeling as compared to the
" STAGS program. The total cost data for this investigation are not available,
but it is estimated that for a typical problem, the total cost would be nearly
equal for these two progrmxm.
CO_I(,.USIONS t_
i
]
In the statZc analysis investigation, it was determined that the NASIF_C_ $'
plate element has a width-to-,.hlcknessratio (b/t) limitation, as well as an
aspect ratio limita_Jon. These are both important parmneters to be considered
in modeling thick plate st_xctures. Extra care should be e,,e._1_edto avoid
large aspect ratios and/or small (less than five) width-to-thich_ess ratios.(
The investigationdid indicate that ASKA element TRIB3 is consistently valid
for extremely low _alue_ of b/t. For those structares whose configuration
requires modeling to b/t values less than five, it is recommended that they
get processed using the ASKA program or use solid elements available in
NASTRAN.
'_" The buckling analysis investigation revealed that modeling requirements
are quite differen_ from static mmlysis. The conventional rules for static !
analysis modeling are neither sufficient nor applicable for buckllng analysis, i
Although the effect of varying aspect ratio is negligible, the effect of vary-
ing the nu_er of elements per Imlf wavelength is very critical to both a valid
solution and the computer cost. The cost of performing a valid buckling analy-
sis is v_ry high measured by static analysis standards. Although STAGS com-
puter cost is quite low, the man-hour cost is quite high, compared to NASTRAN
costs. The evaluation of the buckling analysis performed in this investigation
has revealed that it is very difficult to separate a reasonable solation from
an erroneous solution. The NASTP_N models indicate an extremely narrow band
of circumferential spacing (nu_er of elements per half wavelength) may be
selected for a valid solution, whereas the STAGS 1_dels indicate the finer
, models produced an acceptable solution.
_ Üd)'}_" • ACI_3WLEDGE_.NT
The author expresses his gratitude to Mr. W. D. Mock and blr.Y. N. Tsou
for the assistance in carrying out this work.
iJ,
-: |
72
r,
1974006473-078
[ !
F
!
SYMBOLS
8 Circumferential spacing
Length of _vlinder element
i a Length of plate
!
! b Width of plate
? 1 t Thickness of plate or cylinder
: R Cylinder radius
i Pcr Critical buckling load - program output
i i P Critical buckling load- theoretical value
i o Critical buckling stress -theoretical value
Cr
REFERSNCES
l i i. The NASTRA_ User's Manual (Level 15), NASA SP=222(01), June 1972.
2. ASKA User's Reference Manual, ISD Report No. 75, institute for Statics and .
Dynamics, University of Stuttgart, 1971.
3. Almroth, B. 0., Brogan, F. A., Meller, E., and Zele, F., User's Manual for .;&
STAGS Computer Code, LMSC-D266611L, Lockheed Mis_iles _ Space Co., Inc.,
%pril 1972. !
._ii,, 4. Timoshenko,S.;and Woinowsky-Krleger,S.: Theory of Platesand Shells. Second
ed., McGraw-Hill Book Co., Inc.,1959. .::
5 Gerrad, George, and Becker, Herbert, Handbook of. Structurai Stability,
Part Ill, b:#_.ATN 3783, August 1957. _
73 _:_
°%
1974006473-079
i: i
• 1
}
Table I
STATIC ANALYSIS - DEFLECTION DATA
'_ Plate
' Theore t ica 1
; : Thicknes s NASTRAN N3.S[R/_N ASKA
. t (cm) b/t CQUAD2 CTRIA2 TRIB3 (Ref 4_
i
1,27 20,0 !5.0C x 10 "i 13.64 x 10 "1 13.39 x 10"1 14.91 x 10-1
T
[ 5.08 5.0 24.0 x !0 -3 22.07 x 10 -3 20.9 x 10 -3 23.37 x 10 -3! P
2.92 x 10
t 10.16 2.5 3,2 x 10 -3 2.97 x 10 "3 2.62 x 10 -3 -3
i 15.24 1.67 10.39 x 10 -4 9.73 x 10 -4 7.82 x 10 -4 8.61 x 1{)-4
i 20.32 1.25 4.93 x 10 "4 4.65 x 10 -4 NA 3.63 x 10 -425.4 1.00 2.87 x 10 -4 2.72 x 10-4 16.74 x 10 -5 18.54 x 10 -5
1 "
71+
1974006473-080
• w
f
: .i
Table I I " i
4|i 'BUCKLINGANALYSIS- EIGFANALUEDATA NASTP_%_DDEL
Mesh Size [Wegree
: Shell 0, deg f, em Per/P of Freedom*
L , •
? 90 2 2_ 4 1.6434 T1
90 5 25.4 i.5409 TI
-j
90 9 25.4 i.3614 R.
r
t 90 10 25.4 1.0390 R2i
_, 90 11 25.4 .7844 R2
i 180 20 25.4 .5568 R3
i 180 30 25.4 .3079 R3
t L_ , •
' : *Ei,-envectors are normalizea with respect to this degree
of freedom.
Table 111 '
BUCKLINGANALYSIS- I!IGENVALUEIIA'I'ASi'AiiS _I)I)I:L ,,
l_gree _
'- Mesh Size Per/p , ,,) ,Shell • O, deg |, cm of Freedom*
90 3.1 4.24 1.08 T1 ,.
90 3.1 9.53 1.09 ,,.T1 .;;'.___(. 1 05 i II "
90 3.1 38.1 •37 T1 ::i,_
90 5.3 4.24 I.27 T1 _:,
90 11.35 13,7 2.78 "t'1
90 32.5 12,7 29.9 T1
' " | i
• Eigenvectors are nomalized with respect to this degree
of freedom. _
75 i:,}
' my ,wlnmIBqm i
] 97400647:3-08 ]
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)
' I
I_
. )
'L'able IV
BUCKLINGANALYSIS - MACItINETIME DATANAS'I_L'_ _DI)tiL It_Xl370/105
Number of CPU "rime ..... Cham_el TiJne ....
; Grid Points (sec) (sec) Billing Units
184 222. 432 114. 732 .8. )015
_, 76 80. 208 93.438 12. 9700
44 57.732 91.212 I0.4553
: 40 63.23 102 72 12.1796
i 40 54.63 93.75 i0.8100
i 36 58.398 104.118 ii.5798
"t " 28 ...... 43.662 95.358 9.6276
I
Table V
BtrKLING ANALYSIS - M_2111NETIME LIATASrAGS MODELCI)C 6600
. i
Number of CPU Time i/O Time
Grid Points (see) (sec) System _c
'300 46.228 105.800 72_.678 '
180 18.469 40.9 28.694 ._
150 19.176 48. 626 31. 332
'_'" 90 11.312 35.816 19. 766
60 7. 858 30. 624 1 _,.514
36 3.227 21.!62 8.517 "
;;_ 20 2.036 23.259 7.85
'" i i
I
1974006473-082
+,"
• k
[ ' _.,L_ +, • .
_r
iI +++++++ + +-_ + + _-_T'.,_.I:+:+/_,+/ + " "+'"' i., u I ; !- , ++_.-.+.._+/_+ _ , + .++I + _ #, .'_,," +_ +'_+ + '+ ,_- , , ,, , , , _ \ , I _ I ," /
, : tsz.4 __+,._+.+_+_. t.4 _. > ,+, ,+52.4
_+ /,,- +, ++,-.
._ _, I'i +' ,
_-------152.4cm_ b---152.4cm------_ N---152.4_+m,----.---_
NASTRAN-CqUAD2 NASTRAN-CTRiA2 A._KA-TRIB3
ALL EDGESCLAMPED ALL EDGESCLAMPED ALLEDGESCLAMPED
Figure 1. Static analysis - basic model geometry.
m'°l 1
I.O-
DEFLECTION,
CENTIMETERS
Ixl0"I
+
IxlO2
• t
+_+++
+
"+:+" lxlO3
' +"_ :+ +I
,:+.,++++
..¢++.
+ IxlO-4+. + I I I I I I I I I0 2 4 6 B l0 12 14 16 18 20
btl
k Figure 2. Deflection vs thickness ratio, _TRAN - CQgAD2.
77
" I I -'_
1974006473-083
= =.
m
i,,4'
I0.0 i
,'
l.o , ii
: DEFLECTION,
CENTIMETERS _i
ixlO-1
lx10-2
lx10"3
lxlO-4 ----' I I I i ,, I I I
O 2 4 6 8 I0 12 14 16 18 20
bit
"" Figure 3. Deflection vs thickness ratio, NASTRAN - CTRIA2.
lO.O
-_ 1.0
DEFLECTION,
• ' CENTIMETERS
, lxlO"1
;,; L_
;_=:_ /_ lxlO"2
_"/":L ixlO"3
l/k,.t.
lxlO-4 I I I I I I_ _ , .
N 0 2 4 6 8 IO 12 14 1'_ 18 20
__ bit
_,,,',, Figurc 4 Deflection v._ thickness ratio, ASKA - TRIB3.
] 97400647:3-084
i
a
_r
¢"." i
i
" _ THEORETICAL_/° _,
_. 60- DIFF. PROGRAM VALUE THEORETICAL
% DIFFERENCE (
_,,
_, 50 ,I_ o NASTRAN - CQUAD 2
l_ _ NASTRAN - CINIA 2
• 20
t
---- 10
-2o-I°_ _ --_ . 4-, -_
' ' ' ' $ , I0 2 4 6 8 20
bit
FigureS. % differencevs thicknessratio.
3
20 .i02cm WALl.THICKNESS
0° _0_ .,c_ ;_
/6.20 76.20 --_-
cm @-.8-I .... - cm HIGH ";'_
, _
OF CYLINDER _ S'-_I ..,7" -5
_ 21 _..-_
Figure6. Bucklingnnaly_is- basicmodel geometry.
79 _
_'
m
197400647:3-085
I _"
.#.
s"4 i
,#
J
f STA_C_
/
/ NASTRANl
/
2 /t
ff
Pcr //
/
/P /
! 0 ' ' Idt n
{ IO 0 20 30
_, Figure 7. Ef{ect of varying circumferential spacing.
30' (_ NASTRANI
I STAGS o-- -
Pcr "dF'
P
3-
i_ 0 t ___J t t "
, "-i ......._ 3 - 4 lo
F£gure8. Effecto£ varyingaspectrat£u.
L....... 80 '
1974006473-086
II
!
ate.,.".
NASTRAN.-_I B_LLINOUNIT; _4
_: i SYSTEMSECOND= $ .25
Inn - i:
t _ MACHINE
' _ COST$
Q 50
: |! "
, I
' !
_ [oo 2O0 30O
i NUMBEROF GRID POINTS
Figure 9. Cost comparison between NAST_N and STAGS.
81 _,
1974006473-087
i
\
!
, |
i . Figure A-1. Mode shape - NASTR/_N model. [• i
i i Pcr/P = I. 5409;,- 25. 4 crn; O= 5° .*_'
I
1974006473-088
I
- Figure A-4. Mode shape - NASTRANmdel. :_
I
197400647:3-089
I •
Pcr/P = 0.7844; £ = 25.4 cm; 0 = 11°
M=M
F_ v "5
t
r
t
'2" _.
f
i "
(
I'
i
, Figure A-5. Mode shape - NASTRANmodel.
I :
, Pcr/P = 0.5568; _ = 25.4 cm; 0 = 20 °
; i
2_
Figure A-6. Mode shape - NASTRANmodel.
1974006473-090
J_ Pcr/P -- 0.3079; (_= 25.4 cm; _ = 30°
; -
\
, i.gure A-7. Mode shape - NASTRANraodel.
Per/P = 1.08; _ = 4.24 era; 0 = 3.10
Figure A-8. I¢od¢ shape - STAGSmodel. t
1974006473-091
FigureA-9. _bde shape - STAGSmodel.
Pcr/P = I.II; t = 19.05 cm; 8 = 3.1 °
i!
FigureA-10, _kxleshape - S'rAGSmodel.
t
86
P
1974006473-092
° •
]
• _";_ Pcr/P = 0.37; _ = 38.1 cm; 0 = 3.1 °
)
i. /! Figure A-II. _bde shape - STAGS,,nodel.i
l
Pcr/P = 1.27; _ = 4.24 era; O= 5.3 ° /</
i,
Figure A-12. Mode shape - STACk5model. ._[_
I
1974006473-093
Pcr/P = 2.78; _ = 12.7 crn; 8 = 11.25 °
Figure A-13. Mode shape - STAGSmodel.
Pcr/P = 29.9; ( = 12.7 cm; 8 = 22.5 °
" " FigureA-i4. Y_de shape - STACSmodel,
I
68
i
1974006473-094


NASTRAN static and buckling analysis - comparison with other large-capacity programs

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