- Email: [email protected]
On numerical nonlinear analysis of highly flexible spinning cantilevers
Computers & SrrurrurPs. Printed in Great
Vol
13. pp
357-362.
0045-7949/81/010357~2.~~0 Copynphl (D 1981 Per&mm
1981
PressLtd
Britain AllriphIsreserved
ELASTIC-PLASTIC ANALYSIS WITH NASTRAN USER ELEMENTS L. A. LARD A. 0. Smith Corporation, Engineering Systems, 8797 North Port Washington Road, Milwaukee, WI 53217, U.S.A.
(Received25 April 1980) Abatrad--Described
herein are two user-developed elements that have been implemented in the elastic
plasticcapability of N ASTRAN. The first clement isa thin-walled beam. The second is a flat, triangular shell element. The elements ate shown to give good agreement with plastic limit analysis for five sample test aroblems. ADDlicationis made to static crush of auto frames. Both beam and shell models are investigated ;nd good &&p&son
is obtained with test data.
Finite clement analysis is now widefy accepted by industry in the design phase of product development.
Most often an elastic, static analysis is sufficient. Many times, some dynamic aspect of the problem is also considered. With increasing frquency, even the elasticplastic characteristics must be considered. Because of the various analytical capabilities required, a large, general purpose finite clement program is necessary. There are many candidates and no attempt will be made here to list or compare them. NASTRAN was selected because of its excellent overall capabilities and because of the availability of the source code without lease. licensing or other restrictions. NASTRAN Level 15.5 was the last “free” version of the code. Furthermore, since the elastic-plastic capability has remained virtually unchanged since the release of NASTRAN 15.5, this code may be linked with later versions. The writer has successfully Finked this source code and the numerous changes described here into Version 5OA of the MacNeal-Schwendler proprietary NASTRAN producing a version of MSCjNASTRAN that executes in approximately one-half the time required by NASTRAN 15.5. The principal weakness in the NASTRAN elasticplastic capability-Rigid Format G-is the element library which is Iimited to only membrane behavior in shell elements and rod behavior in beam elements. This means present NASTRAN beam and shell elements will not yield in bending. Consequently, it was decided to develop and add to NASTRAN two new elements which will yield in bending: a beam element and a triangular flat shell element. These elements were added to NASTRAN as user developed dummy elements. Inclusion of these elements incorporate consideration of geometry changes. Details for adding elements to NASTRAN arc documents in the NASTRAh’
Fro~rui?ri~ler.s Manual [I].
DESCRlPTlOh
AND VERIFICATION
elastic-plastic beam element added to NASTRAN is the CDUM 3 clement. It is an element which can be used to model thin-wall beam members. The input data includes the connecting grid points and an o~entation vector. The input section property The
357
data is in the form of X, Y nodal coordinates and thickness data in sequential order around the thin-wall section. To describe abrupt thickness changes the nodal coordinates may be repeated and the new thickness given. Backtracking can be easily e&z&d so that sections such as H or I shapes can be described. The CDUM3 elastic-plastic beam does not use the hinge concept for plastic coflapse. Instead, standard
finite ekment fo~~tio~ were used with numerical integration for calculating the stibss matrix. Hinge formation and plastic collapse occur as a consequence of reaching the yield strain at sufficient integration points. To obtain a beam clement capabk of plastic collapse under these circumstances, the usual beam shape function had to be abandoned and a shape function with internal degrm of freedom and greater flexibility was used. The result is an clement difierent from the usual Euler beam element. Comparisons thus far between CDUM3 models and Euler beam models in static analysis show displacements agree to three significant figures while the maximum stresses agree to within 1.59/,. This is generally adequate for most engineering purposes. Verification of the plastic collapse capability of the CDUM3 beam element will be demonstrated by correlation with limit analysis in two test problems. The first is a cantilever beam with an unequal-leg channel cross-section as shown in Fig. 1. The loading is a combined compression and bending. This simple example arose from an actual application to auto frame crush. This was a rear siderail section and the crush load was an eccentric compression through the rear bumper. The problem was converted here to one with compression and linear bending moment to more fully test the plastic collapse capability of the element. The theoretical limit analysis load was calculated from equilibti~ after the plastic neutral axis was located by trial and error. The load-deflection curve calculated using NASTRAN Rigid Format 6 and the CDUM3 element is shown in Fig. 1. It is seen to give excellent agreement within l”, of the prediction of limit analysis. The second example is shown in Fig. 2. It is a simple planar frame with a channel cross-section. This “building bent” problem is seen in many civil engineering texts on limit analysis. It is something of a classic because this deceivingly simple problem has an unusual
L A. LARKIN
358
Fig. 1, Plastic collapse of thm-walled cantilever beam. failure machanism. There is no hinge at Grid 2 and segment 123 rotates clockwise about Grid 1 in the collapse mechanism. Segment 45 rotates clockwise about Grid 5 and segtmnt 34 rotates counterclockwise. The plastio limit analysis curve was obtained from multipte NASTRAN static analyses using CBAR elements and inserting hinges at the CBAR ends as plastic hinges formed. NASTRAN Rigid Format 6 successfully -solved this problem predicting collapse at a load 8% above the limit analysis as shown m Fig. 2. The triangular, flat shell element added to NASTRAN is the CDUM4 element. This element was formed by combining the membrane behavior of the constant strain triangie with the bending element published by Razeiey, Cheung, Irons and &nkiewicz [4_ Because plasticity is involved, the integration of the stiffness matrix is done numeri~y using a com-
bination of Gauss and Newton-Cotes numerical integration schemes. The PrandtJ-Reuss stress-strain law was implemented. Ideal, perfect plasticity may be used or the stress-strain curve may be table input. Ver&ation of the CDUM4 shell eIement will be shown by correhtion with three sample probiems for which both dastic soiuttons and plastic Iiit analysis solutions are known. The i&t, a rectangular cantilever beam loaded by in-plane forces, will test only the membrane properties of the element. The CD tiM4 is a constant strain triangle in this situation and it behaved as expected, i.e. being on the stiff side of what beam theory, with shear deformations included, would predict. Figure 3 shows the load~e~tion curve from a NASTRAN Rii Format 6 pi&s& coRapse analysis of the model shown. From the elastic analysis. the load to
12000 f
__
-------v.“--f-
XL
Pfastlc
T,
Drsplacement
iit
“;nla
? -
rnchc--
Fig. Z Plastic collapse of planar frame.
Llzllt
linaiys1s
Elastieplastic
analysis with NASTRAN user elements
Plastic
RF6
1.0
with
Limit
Load
CDUY4
2.0 Beam
359
3.0 End Deflectzon
4.0
5.0
6.0
(/'y
Fig. 3. Plastic collapse of cantilever beam of CDUM4 elements initiate yield is determined. Thii is thm applied and incremented to failure. It is seen the NASTRAN collapse load underestimates the limit analysis collapse load by about 7%. The yielded elements formed two symmetric fields at the wall support in a pattern
exactly like that expected to form a plastic hinge. The second test problem is a square, simply-supported, laterally loaded plate. This problem will test only the out-of-plane or bending properties of CDUM4. Studies for both uniform and concentrated loads
1.2r Upper
Sound
Solution
0.8.
W6
a
wlrh
CDUU4
f
I/.
1.0
2.0
3.0 W/W0
Fig. 4. Load-deflection
4.0 Dilacnlion1r.a
5.0 Center
6.0 Displacement
for a square, uniformly loaded. simply supported plate.
L.A.
360
LARKIN
and stresses damp out rapidly. the mtintte shell can be approximated by a long. finite shell. The radial displacements from an elastrc analysis are plotted m Fig. 5 as a function of the distance from the load pomt. Good agreement IS seen with elastic theory from Timoshenko [-I]. The load-detlection curve at the applied load rmg obtamed from a NASTRAN Rrgtd Format 6 anaiqsrs is shown in Fig. 6 .Again. good agreement is seen between the NASTRAN collapse load and the limrt analysts collapse load obtained by Drucker [j].
showed good convergence to exact elastic values and agreed with the results obtained by the original investigators [2]. A NASTRAN Rigtd Format 6 plastic collapse analysrs was made for the case of a simply-supported. square plate under uniform load. Figure 4 shows the load-deflectton curve obtained. Although an exact hmtt analysis solution is not available, close upper and lower bounds have been obtained by Hodge and Belytschko [3]. Figure 4 shows close agreement between the NASTRAN collapse load and the bounds from limit analysis. The third test problem IS an mtimte. ctrcular. cylindrical shell with an axially symmetrtc rmg load. There are both large hoop stresses and localized bending around the applied ring load. so thts problem WIII test the combined bending and membrane capabilitres of the CDCJM4 element. Because displacements
RF24
4PPLICATION
TO .AtiTO FRAiiE
with
CD""4
a * Cylxnder
Radius
t = Cylrnder
Th1chness
= Poisson's x = distance
-0.1
L
Bx - Distance
Fig. 5, Displacements
CRUSH
Since the mass of the auto frame ISa small II acuun tit the total vehicle mass. the crashworthiness of an auto can be studied from data extrapolated from quasrstatic crush of the frame. Lsmg this same argument.
from
Load
rut10 from
load
Point
from static analysis of ring loaded cylindrical shell.
2.0-
1.0
2.0
3.0 D~menslonless
4.0 Radial
5.0 D~aplacement
tJ.0 - N/WY
FIN. 6. Plastic collapse of ring loaded cylindrical shell
7,
5.0
Elastic-plasticanalysiswithNASTRAN crush tests can be made on isolated frame parts which is a convenient mode of testing in the laboratory. The first example of static frame crush is on the front-most part of the frame known as the “front horn structure”. A NASTRAN Rigid Format 6 plastic collapse analysis using CDUM3 beam elements and a one-half model of the structure is shown in Fig. 7. NASTRAN deformed plots have been superimposed to illustrate the progressive collapse under a longitudinal crush load. It is interesting to note from the
user elements
361
deformed plots the obvious formation of plastic hinges. Plastic hinges are not used here as analysis tools but arise simply as a natural consequence of highly locaked plastic flow. The second example exammes that part of an auto frame called the Worquebox~.It is just behind the front horn structure. Figure 8 shows a detailed shall model of one-half the strufzture. That part assumed plastic is indkated in Fig. 8 and was modeled with 402 CDUM4 shell elements. The elastic part was modeled with 140 CQUADZ elements and 48 CIRIA2 elements.
those static
Fig. 7. Plastic collapse of auto frame front horn structure
Fig. 8. Shell element model of auto frame torquebox NASTRAN
CDUM3
Bar Node1
AXMI
hrtusl
Crush
Load
0.8
l.0
1.2
1.4
1.6
Test
1.8
Data
2.0
Fig. 9. Torquebox plastic collapse.
2.2
2.4
2.6
2.8
3.0
361
L. A.
hRKlN
Figure 9 shovvs the load-deflection curve from a N QSTRAN plastic collapse analysis of this shell model and compares it with test data. Also plotted in Fig. 9 is the NASTRAN collapse analysis of a CDUM3 beam model of this same structure. It is seen the test data Indicates the actual structure is very flexible. This is due to the added deformation of the support and loading tixtures. Correcting this by shifting the test data to the left so the linear part matches the linear part of the calculated curve gives the “modified data” curve. Good agreement is seen between the shell model and the modified test data. The load-deflection curve m Fig. 9 for the CDUM3 beam model shows a definite collapse load only 5”” above test results. Houever. the overall structural behavior predicted by the CDLM3 model is much stiffer than test data. Lnderstandably, the beam model seems less able to predict the details of the local deformation and yielding than the shell model. In conclusion, thin-ualled beam and triangular shell elements have been added to the elastic-plastic capability of NASTRAN. These have been verified by correlation Rith known solutions. Both new elements are seen to have useful roles in practical, industrial applications such as auto frame crush calculations.
This work is viewed as a first major step in extending NASTRAN
capability.
Areas of interest
for future
uork include the metal forming problem, development of solid elements and alternative forms for the stressstrain relationship.
REFERENCES 1. The A.4STRRdA Programmer‘s
Manual, NASA W-223 Scientific and Technical Information Office NASA. Washington. D C (1973 ). 2. G. P Bazeley. Y K. Cheung. B X4 Irons and 0 C Zienkiewcz. Triangular elements in plate bendingconformmg and non-conforming soltmons. Proc .Air Force
Firsr Conj
\larrl.u .Clerhod> Srructural
Meclr
.
4FFDL-TR-6640. pp 547-576 Wright-Patterson 4 1. B., Ohro (1966). 3. P. G. Hodge. Jr. and T. Belytschko, Numertcal methods for the limit analysis of plates. J Appl. Me&. 35.796802 (1968). 4. S. Ttnwshenko and S. Womousky-Krteger, 2%wr o/ Plares and Shells. 2nd Edn. McGraw-Hill. New York (19591. 5. D. C. Drucker, Limit analysis of cylindrrcal shells under axially-symmetric loading Proc First Midwesrern Conf’ So/id Mech. (1953).
Vol
13. pp
357-362.
0045-7949/81/010357~2.~~0 Copynphl (D 1981 Per&mm
1981
PressLtd
Britain AllriphIsreserved
ELASTIC-PLASTIC ANALYSIS WITH NASTRAN USER ELEMENTS L. A. LARD A. 0. Smith Corporation, Engineering Systems, 8797 North Port Washington Road, Milwaukee, WI 53217, U.S.A.
(Received25 April 1980) Abatrad--Described
herein are two user-developed elements that have been implemented in the elastic
plasticcapability of N ASTRAN. The first clement isa thin-walled beam. The second is a flat, triangular shell element. The elements ate shown to give good agreement with plastic limit analysis for five sample test aroblems. ADDlicationis made to static crush of auto frames. Both beam and shell models are investigated ;nd good &&p&son
is obtained with test data.
Finite clement analysis is now widefy accepted by industry in the design phase of product development.
Most often an elastic, static analysis is sufficient. Many times, some dynamic aspect of the problem is also considered. With increasing frquency, even the elasticplastic characteristics must be considered. Because of the various analytical capabilities required, a large, general purpose finite clement program is necessary. There are many candidates and no attempt will be made here to list or compare them. NASTRAN was selected because of its excellent overall capabilities and because of the availability of the source code without lease. licensing or other restrictions. NASTRAN Level 15.5 was the last “free” version of the code. Furthermore, since the elastic-plastic capability has remained virtually unchanged since the release of NASTRAN 15.5, this code may be linked with later versions. The writer has successfully Finked this source code and the numerous changes described here into Version 5OA of the MacNeal-Schwendler proprietary NASTRAN producing a version of MSCjNASTRAN that executes in approximately one-half the time required by NASTRAN 15.5. The principal weakness in the NASTRAN elasticplastic capability-Rigid Format G-is the element library which is Iimited to only membrane behavior in shell elements and rod behavior in beam elements. This means present NASTRAN beam and shell elements will not yield in bending. Consequently, it was decided to develop and add to NASTRAN two new elements which will yield in bending: a beam element and a triangular flat shell element. These elements were added to NASTRAN as user developed dummy elements. Inclusion of these elements incorporate consideration of geometry changes. Details for adding elements to NASTRAN arc documents in the NASTRAh’
Fro~rui?ri~ler.s Manual [I].
DESCRlPTlOh
AND VERIFICATION
elastic-plastic beam element added to NASTRAN is the CDUM 3 clement. It is an element which can be used to model thin-wall beam members. The input data includes the connecting grid points and an o~entation vector. The input section property The
357
data is in the form of X, Y nodal coordinates and thickness data in sequential order around the thin-wall section. To describe abrupt thickness changes the nodal coordinates may be repeated and the new thickness given. Backtracking can be easily e&z&d so that sections such as H or I shapes can be described. The CDUM3 elastic-plastic beam does not use the hinge concept for plastic coflapse. Instead, standard
finite ekment fo~~tio~ were used with numerical integration for calculating the stibss matrix. Hinge formation and plastic collapse occur as a consequence of reaching the yield strain at sufficient integration points. To obtain a beam clement capabk of plastic collapse under these circumstances, the usual beam shape function had to be abandoned and a shape function with internal degrm of freedom and greater flexibility was used. The result is an clement difierent from the usual Euler beam element. Comparisons thus far between CDUM3 models and Euler beam models in static analysis show displacements agree to three significant figures while the maximum stresses agree to within 1.59/,. This is generally adequate for most engineering purposes. Verification of the plastic collapse capability of the CDUM3 beam element will be demonstrated by correlation with limit analysis in two test problems. The first is a cantilever beam with an unequal-leg channel cross-section as shown in Fig. 1. The loading is a combined compression and bending. This simple example arose from an actual application to auto frame crush. This was a rear siderail section and the crush load was an eccentric compression through the rear bumper. The problem was converted here to one with compression and linear bending moment to more fully test the plastic collapse capability of the element. The theoretical limit analysis load was calculated from equilibti~ after the plastic neutral axis was located by trial and error. The load-deflection curve calculated using NASTRAN Rigid Format 6 and the CDUM3 element is shown in Fig. 1. It is seen to give excellent agreement within l”, of the prediction of limit analysis. The second example is shown in Fig. 2. It is a simple planar frame with a channel cross-section. This “building bent” problem is seen in many civil engineering texts on limit analysis. It is something of a classic because this deceivingly simple problem has an unusual
L A. LARKIN
358
Fig. 1, Plastic collapse of thm-walled cantilever beam. failure machanism. There is no hinge at Grid 2 and segment 123 rotates clockwise about Grid 1 in the collapse mechanism. Segment 45 rotates clockwise about Grid 5 and segtmnt 34 rotates counterclockwise. The plastio limit analysis curve was obtained from multipte NASTRAN static analyses using CBAR elements and inserting hinges at the CBAR ends as plastic hinges formed. NASTRAN Rigid Format 6 successfully -solved this problem predicting collapse at a load 8% above the limit analysis as shown m Fig. 2. The triangular, flat shell element added to NASTRAN is the CDUM4 element. This element was formed by combining the membrane behavior of the constant strain triangie with the bending element published by Razeiey, Cheung, Irons and &nkiewicz [4_ Because plasticity is involved, the integration of the stiffness matrix is done numeri~y using a com-
bination of Gauss and Newton-Cotes numerical integration schemes. The PrandtJ-Reuss stress-strain law was implemented. Ideal, perfect plasticity may be used or the stress-strain curve may be table input. Ver&ation of the CDUM4 shell eIement will be shown by correhtion with three sample probiems for which both dastic soiuttons and plastic Iiit analysis solutions are known. The i&t, a rectangular cantilever beam loaded by in-plane forces, will test only the membrane properties of the element. The CD tiM4 is a constant strain triangle in this situation and it behaved as expected, i.e. being on the stiff side of what beam theory, with shear deformations included, would predict. Figure 3 shows the load~e~tion curve from a NASTRAN Rii Format 6 pi&s& coRapse analysis of the model shown. From the elastic analysis. the load to
12000 f
__
-------v.“--f-
XL
Pfastlc
T,
Drsplacement
iit
“;nla
? -
rnchc--
Fig. Z Plastic collapse of planar frame.
Llzllt
linaiys1s
Elastieplastic
analysis with NASTRAN user elements
Plastic
RF6
1.0
with
Limit
Load
CDUY4
2.0 Beam
359
3.0 End Deflectzon
4.0
5.0
6.0
(/'y
Fig. 3. Plastic collapse of cantilever beam of CDUM4 elements initiate yield is determined. Thii is thm applied and incremented to failure. It is seen the NASTRAN collapse load underestimates the limit analysis collapse load by about 7%. The yielded elements formed two symmetric fields at the wall support in a pattern
exactly like that expected to form a plastic hinge. The second test problem is a square, simply-supported, laterally loaded plate. This problem will test only the out-of-plane or bending properties of CDUM4. Studies for both uniform and concentrated loads
1.2r Upper
Sound
Solution
0.8.
W6
a
wlrh
CDUU4
f
I/.
1.0
2.0
3.0 W/W0
Fig. 4. Load-deflection
4.0 Dilacnlion1r.a
5.0 Center
6.0 Displacement
for a square, uniformly loaded. simply supported plate.
L.A.
360
LARKIN
and stresses damp out rapidly. the mtintte shell can be approximated by a long. finite shell. The radial displacements from an elastrc analysis are plotted m Fig. 5 as a function of the distance from the load pomt. Good agreement IS seen with elastic theory from Timoshenko [-I]. The load-detlection curve at the applied load rmg obtamed from a NASTRAN Rrgtd Format 6 anaiqsrs is shown in Fig. 6 .Again. good agreement is seen between the NASTRAN collapse load and the limrt analysts collapse load obtained by Drucker [j].
showed good convergence to exact elastic values and agreed with the results obtained by the original investigators [2]. A NASTRAN Rigtd Format 6 plastic collapse analysrs was made for the case of a simply-supported. square plate under uniform load. Figure 4 shows the load-deflectton curve obtained. Although an exact hmtt analysis solution is not available, close upper and lower bounds have been obtained by Hodge and Belytschko [3]. Figure 4 shows close agreement between the NASTRAN collapse load and the bounds from limit analysis. The third test problem IS an mtimte. ctrcular. cylindrical shell with an axially symmetrtc rmg load. There are both large hoop stresses and localized bending around the applied ring load. so thts problem WIII test the combined bending and membrane capabilitres of the CDCJM4 element. Because displacements
RF24
4PPLICATION
TO .AtiTO FRAiiE
with
CD""4
a * Cylxnder
Radius
t = Cylrnder
Th1chness
= Poisson's x = distance
-0.1
L
Bx - Distance
Fig. 5, Displacements
CRUSH
Since the mass of the auto frame ISa small II acuun tit the total vehicle mass. the crashworthiness of an auto can be studied from data extrapolated from quasrstatic crush of the frame. Lsmg this same argument.
from
Load
rut10 from
load
Point
from static analysis of ring loaded cylindrical shell.
2.0-
1.0
2.0
3.0 D~menslonless
4.0 Radial
5.0 D~aplacement
tJ.0 - N/WY
FIN. 6. Plastic collapse of ring loaded cylindrical shell
7,
5.0
Elastic-plasticanalysiswithNASTRAN crush tests can be made on isolated frame parts which is a convenient mode of testing in the laboratory. The first example of static frame crush is on the front-most part of the frame known as the “front horn structure”. A NASTRAN Rigid Format 6 plastic collapse analysis using CDUM3 beam elements and a one-half model of the structure is shown in Fig. 7. NASTRAN deformed plots have been superimposed to illustrate the progressive collapse under a longitudinal crush load. It is interesting to note from the
user elements
361
deformed plots the obvious formation of plastic hinges. Plastic hinges are not used here as analysis tools but arise simply as a natural consequence of highly locaked plastic flow. The second example exammes that part of an auto frame called the Worquebox~.It is just behind the front horn structure. Figure 8 shows a detailed shall model of one-half the strufzture. That part assumed plastic is indkated in Fig. 8 and was modeled with 402 CDUM4 shell elements. The elastic part was modeled with 140 CQUADZ elements and 48 CIRIA2 elements.
those static
Fig. 7. Plastic collapse of auto frame front horn structure
Fig. 8. Shell element model of auto frame torquebox NASTRAN
CDUM3
Bar Node1
AXMI
hrtusl
Crush
Load
0.8
l.0
1.2
1.4
1.6
Test
1.8
Data
2.0
Fig. 9. Torquebox plastic collapse.
2.2
2.4
2.6
2.8
3.0
361
L. A.
hRKlN
Figure 9 shovvs the load-deflection curve from a N QSTRAN plastic collapse analysis of this shell model and compares it with test data. Also plotted in Fig. 9 is the NASTRAN collapse analysis of a CDUM3 beam model of this same structure. It is seen the test data Indicates the actual structure is very flexible. This is due to the added deformation of the support and loading tixtures. Correcting this by shifting the test data to the left so the linear part matches the linear part of the calculated curve gives the “modified data” curve. Good agreement is seen between the shell model and the modified test data. The load-deflection curve m Fig. 9 for the CDUM3 beam model shows a definite collapse load only 5”” above test results. Houever. the overall structural behavior predicted by the CDLM3 model is much stiffer than test data. Lnderstandably, the beam model seems less able to predict the details of the local deformation and yielding than the shell model. In conclusion, thin-ualled beam and triangular shell elements have been added to the elastic-plastic capability of NASTRAN. These have been verified by correlation Rith known solutions. Both new elements are seen to have useful roles in practical, industrial applications such as auto frame crush calculations.
This work is viewed as a first major step in extending NASTRAN
capability.
Areas of interest
for future
uork include the metal forming problem, development of solid elements and alternative forms for the stressstrain relationship.
REFERENCES 1. The A.4STRRdA Programmer‘s
Manual, NASA W-223 Scientific and Technical Information Office NASA. Washington. D C (1973 ). 2. G. P Bazeley. Y K. Cheung. B X4 Irons and 0 C Zienkiewcz. Triangular elements in plate bendingconformmg and non-conforming soltmons. Proc .Air Force
Firsr Conj
\larrl.u .Clerhod> Srructural
Meclr
.
4FFDL-TR-6640. pp 547-576 Wright-Patterson 4 1. B., Ohro (1966). 3. P. G. Hodge. Jr. and T. Belytschko, Numertcal methods for the limit analysis of plates. J Appl. Me&. 35.796802 (1968). 4. S. Ttnwshenko and S. Womousky-Krteger, 2%wr o/ Plares and Shells. 2nd Edn. McGraw-Hill. New York (19591. 5. D. C. Drucker, Limit analysis of cylindrrcal shells under axially-symmetric loading Proc First Midwesrern Conf’ So/id Mech. (1953).