The explicit expressions for an elastic-plastic trapezoidal ring element are presented and implemented in NASTRAN computer program. The material is assumed to obey the von Mises' yield criterion, isotropic hardening rule and the Prandtl-Reuss flow relations. For the purpose of demonstration, two elastic-plastic problems are solved and compared with previous results. The first is a plane-strain tube under uniform internal pressure and the second, a finite-length tube loaded over part of its inner surface. A very good agreement was found in both test problems

Chen, P. C. T.

Ohara, G. P.

English

NASA Technical Reports Server

IMPLEMENTATION OF A TRAPEZOIDAL RING ELEMENT
IN NASTRAN FOR ELASTIC-PLASTIC ANALYSIS
P. C. T. Chen and G. P. O'Hara
U. So Army Armament Research and Development Command
Benet Weapons Laboratory, LCWSL
Watervliet Arsenal, Watervliet, NY 12189
SUMMARY
The explicit expressions for an elastic-plastic trapezoidal ring element are
presented and implemented in NASTRAN computer program. The material is assumed
to obey the von Mises' yield criterion, isotropic hardening rule and the Prandtl-
Reuss flow relations. For the purpose of demonstration, two elastic-plastic
problems are solved and compared with previous results. The first is a plane-
strain tube under uniform internal pressure and the second, a finite-length tube
loaded over part of its inner surface. A very good agreement has been found in
both test problems.
INTRODUCTION
In recent years finite element method has been widely used for solving
complex nonlinear problems and many large-scale general purpose computer programs
have been developed (ref. i). The MARC and ANSYS systems have found wide appli-
cations, yet they are quite expensive. The piecewise linear analysis option of
the NASTRAN program provides an algorithm for solving nonlinear problems in mater-
ial plasticity (ref. 2). The load is applied in increments such that the stiff-
ness properties can be assumed to be constant over each increment. However, the
usefulness of this option is quite limited because only a few elements have been
implemented. These include rod, tube, bar elements for one-dimensional problems
and plate elements for two-dimensional plane stress problems. This paper
describes the implementation of a trapezoidal ring element in NASTRAN for solving
elastic-plastic problems of rotational symmetry.
The theoretical basis of our implementation follows the approach first
developed by Swedlow (ref. 3). A unique relationship between the octahedral
stress and the plastic octahedral strain is assumed to exist. The material is
assumed to obey the Mises' yield criterion, isotropic hardening rule and the
Prandtl-Reuss flow relations. The explicit expressions for axisymmetric plas-
ticity are derived. The element stiffness matrix and the stress data recovery
routines for the trapezoidal ring element are developed. Seven new subroutines
are implemented and added to the NASTRAN code.
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For the purpose of demonstration, two elastic-plastic test problems are
solved. The first is an infinitely long tube under uniform internal pressure.
The NASTRAN results are in excellent agreement with an exact solution based on a
finite-difference approach (ref. 4). The second problem is a thick-walled
cylinder of finite length loaded over part of its inner surface. The NASTRAN
results are compared with those obtained by a two-dimensional code with the use
of quadrilateral ring elements (ref. 5). A good agreement between the two
results has also been achieved.
CONSTITUTIVE RELATIONS
Following the development by Swedlow (ref. I), the constitutive relations
to he used in our formulation for solving elastic-plastic problems of rotational
symmetry will be presented here. In the development, a unique relationship
between the octahedral stress and the plastic octahedral strain is assumed to
exist and the use of ideally plastic materials is excluded. The total strain
components (er, e@, ez and Yrz) are composed of the elastic, recoverable defor-
mations and the plastic portions (eP, e_, ePz, and yP ). The rates of plastic
flow, C_p, etc.), are inaependent o_ a time scale and are simply used for con-
venience instead of the incremental values. The definitions of the octahedral
stress TO and the octahedral plastic strain rate _P in the case of rotational
symmetry are : o
To = (i/3)[((_r-GO)2+ (cr@-Crz)2+ (_z'_r)2 + 6Trz2]i/2 (i)
•P r..P.P. (_P__P)2eo (i/3) 2 .'P 'P.2= Lter-e@) + . + [ez-er) + (3/2)(.rz)2] I/2cP (2)
where (_r' _@' _z' Trz) are the nonvanishing stress components.
A unique relationship between TO and e_ is assumed and there exists a
function, MT(To), such that
2MT(_o) : "{'olEPo (3)
The plastic modulus, MT(To), can be related to the slope, ET, of the effective
stress-strain (_-e) curve by
1 1 1
= (4)
3MT(T o) ET E
where E is the elastic modulus and
Et = o/e
= (3/_)zo
: _ eo + _/E . (5)
102
The material is assumed to obey the Mises yield criterion and the Prandtl-Reuss
flow rule. The matrix relationship for the plastic flow in the case of rota-
tional symmetry is
e = [c] (6)
EZ _Z
#rz d
[C] : _ 1 -_ -_ 0
-V 1 -V 0
-_ -V 1 0
0 0 0 2(i+_)
+ 6To2MT(To) St2 SrSo SrSz 2SrTrz
S@2 SoSz 2S@Trz
Sz2 2SzTrz
SYM 4Trz 2 _ (7)
where v is the Poisson's ratio,
Sr = (2Or-O@-Oz)/3 ,
So = (2o@-Oz-ar)/3 ,
Sz = (2Oz-Or-O@)/3 • (8)
For strain-hardening materials, MT (or ET) _ 0, we can obtain the inverse of
[C] numerically and this procedure is chosen in developing NASTRAN program. For
ideally-plastic materials, MT = ET = 0, matrix [C] does not exist and the NASTRAN
program fails. However, its inverse [C]-I still exists and the closed form has
been derived in Ref. 6. In the axisymmetric case, the explicit form is (ref. 7)
2G -
_ 2G -l-v SYM- - -- SYM-[D] i-2_ A Sr2
i-_ SrS@ SO2
v V l-v SrSz SoSz Sz2 (9)
1 2
_0 0 0 _(i-_)_ _SrTrz S Ttz SzTrz Trz _
103
where
A = 3To2(1 + M,I,/G)
and
1E/(I+_) (10)G =
If we want toremove the limitation that the use of ideally-plastic materials
is excluded, we have to use Eq. (9) instead of Eq. (7).
TRAPEZOIDAL RING ELEMENT
The incremental displacement field employed for the trapezoidal ring
element are
Au(r,z) = 81 + 82r + 83z + 84rz ,
Aw(r,z) = 85 + 86r + 87z + 88rz . (ii)
The transformation from grid point coordinates to generalized coordinates is
{8}= [rsq]{Aq} (12)
where
{Aq}T = [AUl, Aw I, Au2, Aw2, Au3, Aw3, Au4, Aw4] ,
{8}T = [81, 82, 83, 84, 85, 86, 87, 88] . (13)
the elements of the inverse of the transformation matrix [F_^] "I are theand
coefficients of the 8's in Equations (ii), evaluated at the COl_rs of the
element.
The stiffness matrix is formed in the same manner as that for the aniso-
tropic elastic element. The final form referred to grid point coordinates is
[K] = [rsq]T[K ][FSq],
where
[2]= 2_fr[B]T[D][B]dzdr . (14)
[D], the matrix of material coefficients, is defined by Equation (9). The [B]
matrix is the same as the elastic case, but now it expresses the incremental
strains in terms of generalized coordinates
{Ae} = [B]{8} • (15)
104
NASTRAN IMPLEMENTATION
The NASTRAN implementation for an elastic-plastic trapezoidal ring element
follows the steps outlined in section 6.8, "Adding a structural element," of
reference 8. Changes were required in the functional modules PLAI, PLA3, and
PLA4, which included the writing of several new subroutines. These new routines
could easily be modeled after the existing code for the linear portion of the
program. There are two major differences in the nonlinear subroutines. First
is the new code for the calculation of the material stiffness matrix and second
is that thermal stresses and element force calculations are eliminated in the
nonlinear code.
The changes in PLAI allows this module to identify the new element as a
member of the piecewise linear element set and properly initialize the non-
linear Element Summary and Element Connection Property Tables. Three element
stress recovery subroutines were added to PLA3: PSAPRG, a driver for stress
data recovery; PSTRRI and PSTRR2, phase I and II stress recovery routines.
Element stiffness calculations in PLA4 require four new subroutines: PKAPRG,
a driver for nonlinear trapezoidal ring elements in PLA4, PKTRRI and PKTRR2,
stress recovery routines which generate stresses for the computation of the
nonlinear material matrix; and PKTRAP, the stiffness matrix generation routine
for nonlinear trapezoidal ring elements.
The computer system available for this work is IBM 360 Model 44. In order
to add the new code into Link 13, it became necessary to add a new branch to the
overlay trees to contain the new elements.
NUMERICAL EXAMPLES
For the purpose of demonstration, two elastic-plastic problems of rota-
tional symmetry were solved and compared with other results (refs. 4 and 5).
The first is an infinitely long tube under uniform internal pressure and the
second, a thick-walled cylinder of finite length loaded over part of its inner
surface. The material _roperties for both problems are the same. The elastic
constants are E = 30x10 ° psi, _ = 0.3 and the effective stress-strain curve is
represented by three line segments connecting the four points in the (e-_) plane,
(_,_) = (0.0,0.0). (0.00S, iS0,000 psi), (0.055, 225,000 psi), (0.I, 225,000 psi).
EXAMPLE i. Consider an infinitely long tube subjected to uniform internal
pressure p. The plane strain condition is assumed. The tube of outside radius
2" and inside radius i" has been divided into 25 trapezoidal ring elements.
The numerical results based on the NASTRAN program have been obtained. For this
problem, a new finite-difference approach (ref. 4) can be used to generate exact
solution and to assess the accuracy of the NASTRAN code. Some of the results
for the displacements and stresses are presented graphically in Figures 1 and 2.
Twenty-five load increments are used in NASTRAN as shown in Figure I. The
radial displacements at the inside as well as outside surface are shown as
functions of internal pressure. Figure 2 shows the distributions of radial,
105
tangential and axial stress components in a pressurized tube when half of the
tube is plastic. The pressure required to achieve this state is 0.7378 _o based
on NASTRAN code and 0.7356 o o based on the finite-difference solution (ref. 4).
Both codes indicate that the maximum tangential stress occurs at the elastic-
plastic interface. As demonstrated in Figures 1 and 2, the NASTRAN results are
in excellent agreement with those based on the finite-difference approach
(ref. 4).
EXAMPLE 2. Consider a two-dimensional elastic-plastic thick-walled cylin-
der problem as shown in Figure 3. The tube with inner radius (i"), outer
radius (2") and length (4") is loaded uniformly over a middle portion (2") of
the inner surface. The mesh generation and the loading for the half of the
undeformed structure is shown in the figure. This problem was solved in (ref. 5)
based on a scale loading approach. The first load factor is the upper limit of
the elastic solution and ten additional increments were needed until one of the
outside elements becomes yielded. The same load factors were used to obtain
the NASTRAN solution. Both programs indicate that the sequence in which the
elements become plastic is 1,5,9,2,13,6,10,3,7,14,11,17,4. Some steps will
cause mgre than one element to become plastic and those elements with effective
stress _ > 0.99 oo have been considered as yielded. The numerical results for
the radial displacement at the inside, ua (point i) and outside, ub (point 5)
as functions of internal pressure are shown in Figure 4. The stress components
at the centroid of one inside element (No. I) are shown in Figure 5. The effect
of loading history on the displacements and stresses can be seen from Figures 4
and 5. A comparison of the results between NASTRAN program and reference 5
indicates that very good agreement has been achieved.
CONCLUSION
An elastic-plastic trapezoidal ring element has been implemented in NASTRAN
computer program. Its application to elastic-plastic problems of rotational
symmetry has been demonstrated by solving two thick-walled tube problems. The
NASTRAN results for both problems are in excellent agreement with the other
results.
REFERENCES
i. W. Pilkey, K. Saczalski, and H. Schaeffer, "Structural Mechanics Computer
Programs, Surveys, Assessments and Availability," University Press of
Virginia, 1974.
2. "NASTRAN Theoretical Manual," NASA SP-221 (01), 1972.
3. J. L. Swedlow, "The Thickness Effect and Plastic Flow in Cracked Plates,"
ARL 65-216, Aerospace Research Laboratory, Office of Aerospace Research,
October 1965.
106
4. P. C. T. Chen, "A Finite-Difference Approach to Axisymmetric Plane Strain
Problems Beyond the Elastic Limit," To appear in Trans. 25th Conference of
Army Mathematicians, 1979.
5. P. C. T. Chen, "Numerical Solution of Gun Tube Problems in the Elastic-
Plastic Range," Proceedings of the 1977 Army Numerical Analysis and Computer
Conference, pp. 423-439.
6. Y. Yamada, N. Yoshimura, and T. Sakurai, "Plastic Stress-Strain Matrix and
Its Application for the Solution of Elastic-Plastic Problems by the Finite
Element Method," Int. J. Mech. Sci., Vol. i0, pp. 345-354, 1968.
7. P. C. T. Chen, "Elastic-Plastic Solution of a Two-Dimensional Tube Problem
by the Finite Element," Trans. Nineteenth Conference of Army Mathematicians,
pp. 763-784.
8. "NASTRAN Programmers' Manual," NASA SP-223 (01), 1972.
lOq
4-0.0 ~----'--------------------'-0 1.0 2·0 3.0
(E/<So )(U(A J Ub)/a
1.0
Ub
.8 Ua
PRESSURE
FACTOR .6
P /<5:
...... ·4
0
ex>
...... + .... + NASTRAN
-2 REF. 4
FIGUJ{ -1
_STRESSES
J.o
0.5
+ , _ NASTRAN
- 0.5 ' REF./-4.
FIGURE-2
109
41 4-2. 43 4z/. , LS'
2.9 30 _1 32
:36 _o
ZS 26 2.7 2B
i
21 22 2_ Z4
_26 _0
I'l IB 19 20
_Zl 25
"_ h,,
I_ 14 /5 16
I& __0
i iiiii i i
9 I0 II 12
p _ !1 ,, IS
,I,
Z s 6 7 B
l _ 6 7 8 .9 I0
1 2 3 4_
,,, __ I z, _ 4 5__.r
FIGURE-3
II0
u"
.J. ... + NASTRAN
REF- 5
.0 '--- ---L- ......a-.. -..I
-0 1-0 2-0 3·0
( E/~ ) ( Ua. J U b ) / Q
FIGUJ£ - 4
.2.
1.0
.8
PRESSURE
FACTOR -6
P /00
-4
STRESSES ZN EL../.
o'o/_
./4
- , REF. 5
* " +* NASTRANI
.2.
•0 4, ...... _I"
'0 "2 oZ_. "6 "_ 1,0
P/d"_
FIGUF_- 5
112


Implementation of a trapezoidal ring element in NASTRAN for elastic-plastic analysis

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