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A variable material property approach for solving elastic-plastic problems
0 ELSEVIER
PII:SO308-0161(96)00079-B
Int. J. Pres. Vcs. & Piprng 71 (1997) 285-29 1 1997 Elsevier Science Limited. All rights reserved Prmted in Northern Ireland 0308%Olhl/Y7/$17.oG
A variable material property approach for solving elastic-plastic problems H. Jahed, R. Sethuraman* Department
of Mechanical
Engineering,
University
& R. N. Dubey
of Waterloo,
Waterloo.
Ontario,
Canada N2L .?GI
(Received 20 October 1996,accepted 13 November 1996)
The linear elastic solution of a boundary value problem is usedas a basisto generate its inelastic solution. The method considersthe material parameters as field variables. Their distribution is obtained in an iterative manner using the projection method, the arc-length method, and Nueber’srule. The rate of convergenceis compared.The problem of a thick-walled cylinder is considered asan example.The method yields the analytical solution in the limiting caseif the cylinder is made of non-hardening Tresca material and is under internal pressure.The paper also presentssolutionsfor a Mises solid whosehardening is describedby the Ramberg-Osgood formula. 0 1997Elsevier ScienceLtd.
1 INTRODUCTION
yield criterion considered.
The idea of using elastic solutions for inelastic behavior is not new. Stowell (1950),’ Mendelson and Manson (1957),2 and Nueber (1961)3 based their plastic analysis on known elastic results. Some of these methods are still used widely in research works. More recently, the idea of using finite element elastic analysis for approximation of limit loads, was used by Jones and Dhalla (1986)4 for pressure vessels and piping design. This method has been modified by
2 PROBLEM
properties
variables. The spatial distributions
variables
are found
in an iterative
and hardening,
(2)
t*It-,
11, = 11,q,
where
~
r=r,+r,
(3) (4)
and nj is the unit outward normal to the surface. The solution to boundary value problems requires a
of these field
constitutive form:
The
method is used for elastic-plastic analysis of a pressurized tube. Different material behavior such as
non-hardening
=
U,jn,
as field
manner.
(1)
tions are prescribed as follows:
elastic solution of a boundary value problem. The
material
= OIL!
where Uij represents stress tensor, comma followed by a suffix denotes partial differentiation, and repeated suffix implies summation. Further, boundary condi-
shown by Mackenzie and Boyle (1993),’ Nadarajah et al. (1993),’ and Shi et al. (1993).’ This paper presents a method for developing a comprehensive inelastic solution on the basis of the the
is
DEFINITION
uij.j
method for calculation of a conservative limit load by invoking the lower bound limit load theorem was
treats
and Tresca,
Consider a body of volume R enclosed within surface r. The formulation of a boundary value problem for the body consists of traction and/or displacements prescribed on r and the stress equilibrium condition in R. The equilibrium condition in the absence of body forces is expressed in the form
Marriot (1988)’ and later by Seshadri (1991).6 With this method the effect of local inelasticity is simulated by iterative elastic finite element analysis in which highly loaded regions of the structure are systematically weakened by reduction of the local modulus of elasticity. The effectiveness and practicality of this
method
such as von Mises
relation
which can take the following
Ei,
as well as different
=fCa;,)
(5)
Total strain is assumed to be the sum of an elastic part E; and a plastic part F$
* On leave from Indian Institute of Technology, Madras600036, India.
Ejj = t-7,+ F;; 28.5
(6)
H. Jahed et al.
3%
where the stress-strain
Elastic strain is given by Hooke’s law F’
11
1+v = __ E
(J-
”
- ’ (Tkksij E
where v and E are the Poisson ratio and elastic modulus, respectively. The plastic part is given by Hencky’s total deformation equation e; = +s;,
(8)
where s;j =
is the deviatoric given by
-
(9) stress, and 6 is a scalar value function uij
equations are
~(Tkk~ij
(10)
(16)
ez3!+EP E
ff > u,,
with r0 being the initial yield stress. In the case of a hardening material, the uniaxial stress-strain equation is assumed to be described by the Ramberg-Osgood formula:
& u +a! a II -=Eo PO [ uo1
(17)
where so is the strain at initial yield, (Y is the yield offset, and n is the hardening exponent. The effective values of v,~ and EcR can be obtained in the form
The equivalent plastic strain, E& and the equivalent stress, a:(,, are defined as follows e$q = V$$$
and
ueq = G
(11) The value of $ is obtained from the uniaxial stress strain curve. With the help of (7), (8), and (9), eqn (6) may be written as
Since all quantities inside parentheses in eqn (12) are material dependent parameters one can put this equation in the following alternative form, J&f &ff Here, v,~ and Eetr, the effective Poisson ratio and effective Young’s modulus, depend on the final state of stress at a point. Hence, V,~ and E,@ can be thought of as field variables describing the material properties at each point, since the final state of stress at every point is unique. Clearly, for constant elastic values of veff and E,@ eqn (13) describes the elastic behavior of the body. In general, (13) is capable of describing nonlinear behavior. A comparison between eqns (12) and (13) shows that,
(14)
For an elastic-perfectly loading
plastic solid under uniaxial
1 1 --=-+~5s E
(18)
Their distribution the solution.
in the body is obtained
2.1 Determination
of veff and
as part of
E,=
In order to solve a boundary value problem utilizing a constitutive relation of type (13) one needs to know the complete spatial distribution of veff and Eeff. If this distribution is known a priori the solution of the boundary value problem can be constructed immediately. Conversely, one has to find their distribution as part of the solution. For this reason, the following procedure is considered for determination of veff and E eff* From a purely elastic analysis, material points with the same value of ueq are defined. Next, curves are drawn through these points. These curves in the case of a pressurized tube are circles (Fig. 1). All of the material points lying on a curve represent a single point on the stress-strain curve that describes the material behavior. Therefore these material points pose the same vcff and E,# in the real loading provided the shape of the curves remain unchanged. Suppose that ucq remains constant over a small thickness of d.s.
(W
P” ~1)
Fig. 1. Strip in a thick-walled cylinder: (a) pressurisedtube:
(b) an isolatedstrip.
Elastic-plastic
AS a consequence Y,~ and Eeti remain constant throughout this strip defined by ds. Next we isolate the strip and construct a boundary value problem for this strip which is subjected to traction t! and displacement u! on its boundary with the neighboring strip. Solution of this boundary value problem is now needed such that the behavior of points within the constant Y,~ and Erff strip can be ascertained. For the problem of thick-walled cylinders this boundary value problem is that of a cylinder under internal and external pressure (Fig. 1). In this manner, the original problem is discretized into several infinitesimal strips. Each strip consists of points which have the same material properties. Hence, the proper solution of each strip is equivalent to the elastic solution. To implement this procedure a scheme is needed for evaluation of effective material properties each strip. This is discussed in the next section. It should be mentioned that this method demands the elastic solution of each strip with constant properties. For the case studied here this elastic solution is the Lami: solution. 2.2 Scheme evaluation
of veff and EeB
The material constants vCffand Eeti are obtained in an iterative manner. This iterative procedure continues until the v~~-E,~ curve of each strip follows that of experimentally obtained uniaxial material behavior. The following three schemes may be used: (i) Projection
method
First, a pseudo-elastic solution is obtained under the plastic load. This identifies the state of stress at each point [point a on the extended elastic line in Fig. 2(a)]. Keeping the strain the same, a new value for stress is obtained by finding the corresponding point on the
problems
287
real uniaxial stress-strain curve [point a’ in Fig. 2(a)] Equations (16) or (18) are used to determine the new values of veff and E,*. A new solution is now pursued on new values of veff and E,* The same procedure is carried out until all of the points line up on the stress-strain curve. (ii) Arc-length
method
From the pseudo-elastic solution, the (T,~/u~~ - E,JE~ curve is constructed. A point on this curve represents the state of stress of a particle in the body [point a in Fig. 2(b)]. Next a radius r is defined as r = g/(a,,/~r,)~ A point is located on the curve which is at the same [point a’ in Fig. 2(b)]. New then obtained with the help (iii)
Neuber’s
real uniaxial stress-strain distance r from the origin values of veff and Eeff are of (16) and (18).
The rule proposed by Neuber (1961)” for calculation of the local elastic-plastic stress and strain is employed in this case. From the pseudo-elastic solution the total equivalent strain energy is calculated. Next, a point on the uniaxial material curve which has the same strain energy is identified [Fig. 2(c)]. Th’ 1s 1ocates the point f on the stress-strain curve in Fig. 2(c) such that the two areas abed and aefs are equal. The stress and strain values at f are then used for evaluation of v,, and E,+ 3 APPLICATION TO ELASTIC-PLASTIC ANALYSIS OF PRESSURIZED TUBES structures, such as piping in boilers in power plants. machin-
(W
(3
C
d
f
G
a
(19)
rule
Many load-bearing chemical industries,
(a)
+ (~eqI~dZ
E
t
e
b
w rule
1 0
Fig. 2. (a) Projection method; (b) arc-length method; (c) . . Nueber’s
5 0
method.
1
2
Fig. 3. First elastic solution compared to final
solution.
3 elastic-plastic
H. Jahed et al.
28X
where for plane stress c
I
= Cl+
ydl
-
2v,ff)plrf
-pzrt 2 r2 -
E eK
r:
(21) c = C1+ Ye) (PI -p&5$ 2 Eeff r; - rf
00 0
0.4
and
c = Cl+ v,ff)(l - 2~,ff)p~r5 -w-f I rz - rf Eelf
4.8
c2 = (1-t V,K)(PI J
Fig. 4. Comparison
of results with analytical solution and ABAQUS.
ery in aeronautical and aerospace engineering, and pressure shells in nuclear powered installation are pressure vessels, generally made of hardening materials, which undergo inelastic deformation. Therefore, a good understanding of their elasticplastic behavior is needed for analysis and design difficulties purposes. Owing to the mathematical arising from nonlinearity, a comprehensive analytical elastic-plastic analysis of such members is not attainable in general, Utilizing the method outlined above a comprehensive plastic solution of a thick-walled cylinder is obtained here. Constant geq strips for a thick cylinder form circular strips with traction conditions of the same type (Fig. 1). The elastic solution for each strip is given by the following well known equation: u=C,r+-
C2 r
for plane strain. Here rl, r2, ply and p2 are the inner radius, outer radius, internal pressure, and external pressure of each strip, respectively. Hence the inside and outside displacement of the strip can be related to its inside and outside pressure in the following form:
(23)
The components
Cl2 =
Von Mlses Hardanlng
of coefficient matrix C are as follows
(l; y&&
-2
cif =
2 (1
-
&f)
21
=
_
1
22
+
4-2
r: - r:
E,K
V,K
4
Eeff r: - r: !
Materials
-m+-
h-0
2
I
(24) c
c
(20)
p2)rX
rz2 - r:
E,K -1.6
(22)
W Fig. 5. Comparisonof resultswith ABAQUS.
Elastic-plastic
problems
289
---.-._
Hudmlng
-
.
P, P, - P, Pt
--.
Materials
/ao=l Iv,= 1.85 1ao=2. 15 luo=2.55 ~
(a)
P, Ino= I 85
(b) Fig. 6. Stress variation (internal pressure).
for plane stress, and
C,,=
4 NUMERICAL
2 r,r: --~ ECffri - rf (25)
2
rfr2 CT>,= -EeFfri - r:
for plane strain. After assembling all strips, equations of the form
[C’l{U = m
a system
of linear (26)
is obtained and solved. One of the schemes mentioned above is used to find the new values of v,~ and EClf.
RESULTS
DISCUSSION
Numerical results have been obtained for a cylinder with rJr, = 5, v = 0.3, and E/u0 = 1000. In the case of hardening material the uniaxial stress-strain curve is assumed to follow formula (17) with (Y= 3/7 and ?I = 5. Results for different internal and external load values for closed end and open end cylinders are presented in this section. Figure 3 shows how the first elastic solution is converged upon the final elasticplastic solution. It should be noted that different points shown on the stress-strain curve correspond to different material points throughout the wall thickness. A quantitative comparison of stress and strain distribution according to Tresca and von Mises yield criterion obtained from the present method with other analytical solutions (Prager and Hodge, 1951)“’ and a
(b)
(a) Fig. 7. Strain variation
AND
(internal
pressure).
290
H. Jahed et al.
-
plme6o-M I
2.0 Fig.
R I R, I
30
I
4.0
I
5.0
8. Plane stress and plane strain stress distribution comparison.
finite element solution (Hibbit, 1994)” is given in Figs 4 and 5. The exact solution, given by Prager and Hodge (19X),‘” is matched by the present method. Also Figs 4 and 5 present a comparison of the present method solution to finite element elastic-plastic analysis using ABAQUS (Hibbitt, 1994).” The agreement between the two solutions is excellent. Additional results are given in Fig. 6. The results clearly show the effect of the material behavior on the stress distribution. As has been noticed by MacGregor et al., 1948’* and, more recently, by Bon and Haupt,13 the circumferential stress changes from tension to compression even for partially yielded cylinders [Fig. 6(b)]. However, for hardening materials the hoop stress remains tensile until the plastic zone is very large [Fig. 6(a)]. It is found in the case of ideal plasticity that the cylinder reaches the limit pressure very quickly and this pressure increases for hardening materials. Moreover, for hardening materials the limit
P;/cr,=l
--
Pi /u,=l.85
PihJo=1.4
R/Ri
0.0
I
I
I
3.0
2.0
1.o
Fig.
--..----
I
4.0
10. Variation
of material properties (hardening materials, von Mises yield condition).
pressure is reached at much higher strain values (Fig. 7). Figure 8 shows a comparison of plane stress and plane strain cases of a pressurized tube obeying the von Mises yield criterion. The figure indicates that predictions based on either plane assumption are fairly close. The behavior of an externally loaded cylinder has also been studied. The distribution of hoop and axial stress for various loadings are shown in Fig. 9. The spatial distribution of the material properties vetr and Eeff is given in Fig. 10. Several cases have been studied for non-hardening materials to compare the three schemes mentioned for obtaining convergent solution. The results show a higher convergence rate if Neuber’s rule is used. The number of iterations involved in these studies are tabulated in Table 1.
=d”o -1 2 -
_
. -
Von y1sss n-g _.-..-. __-..--
-
Paw., PO/s”.,.4
R I Rc
(a)
1 50
W
Fig. 9. (a) Hoop stress variation (external pressure); (b) axial stress variation (external pressure).
Elastic-plastic Table
1. Convergence
Load
comparison
Number of iterations Projection method
Arc-length method
Neuber’s rule method
37
14
5
67
47
6
92
71
7
12s
105
13
5 CONCLUSIONS
The idea of describing nonlinear behavior by linear analysis is implemented here. This is accomplished by simulating the behavior of the inelastic region through a collection of elements with pseudo-elastic response. The stress-strain response within each element is assumed linear even though each have different effective moduli. The difference in moduli allows each element to deform in a different manner as compared to a neighboring element. This results in an overall nonlinear behavior. In the case of a thick-walled cylinder, the region is divided into circular strips. The loading condition for each strip is identical and hence, the form of the
solution for each strip is the same as the solution given by the linear elastic analysis. Three schemes have been used in order to achieve rapid convergence of the iterative solution. The scheme based on the total strain energy provides the most rapid convergence.
Moreover, results obtained from the present method are in good agreement with other analytical and numerical results. The following comments summarize the main outcomes of this work: 1. A method is proposed to solve the inelastic problems using linear elastic analysis. 2. The present solution obtained for a thick cylinder made of Tresca materials matches the analytical
solution. 3. It was observed that for the case of non-hardening behavior Neuber’s rule for updating modulus gives
problems
291
faster convergence than the projection and arc-length method. 4. Results obtained for hardening materials are in excellent agreement with the elastic-plastic finite element analysis.
ACKNOWLEDGEMENTS
The first author wishes to thank the Ministry Higher Education of Iran for financial support.
of
REFERENCES 1. Stowell, E. Z., Stress and strain concentration at a circular hole in an infinite plate. NACA TN 2073, 1950. 2. Mendelson, A. and Manson, S. S., Practical solution of plastic deformation in the elastic-plastic range. NACA TN 4088, 1957. 3. Neuber, H., Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law. Journal o,f Applied Mechanics, ASME, 1961,28(4), 554-550. 4. Dhalla, A. K. and Jones, G. L., ASME code classification of pipe stresses: a simplified elastic procedure. Znt. J. Pres. Ves. & Piping, 1986.26, 145. 5. Marriot, D. L., Evaluation of deformation or load control of stressunder inelastic conditions using elastic finite element stressanalysis.Proc. ASME PVP Conf. Vol. 136, 1988. 6. Seshadri, R., The generalized local stress strain (GLOSS) analysis--theory and applications. J. Pres. Ves. Tech., 1991,113,219. 7. Mackenzie, D. and Boyle, J. T., A method of estimating limit loads by iterative elastic analysis--I. Simple examples.Znt. J. Pres. Ves. & Piping, 1993,53, 77. 8. Nadarajah, C., Mackenzie, D. and Boyle. J. T., A method of estimating limit loads by iterative elastic analysis--II. Nozzle sphere intersections with internal pressureand radial load. Znt. J. Pres. Vex & Piping, 1993,53, 97. 9. Shi, J., Mackenzie, D. and Boyle, J. T., A method of estimating limit loads by iterative elastic analysis--III. Torispherical headsunder internal pressure.Int. J. Pres. Ves. & Piping, 1993,53, 121. 10. Prager, W. and Hodge, P. G., The Theory of Perfectly Plastic Solids. Wiley, New York, 1951. 11. Hibbitt. Karlsson and Sorensen. ABAQUS, User’s Manual, Ver. 5.5, 1995. 12. MacGregor, C. W., Coffin, L. F. and Fisher, J. C.. Partially plastic thick-walled tubes. J. Franklin Inst., 1948,245, 135. 13. Bonn, R. and Haupt, P.. Exact solution for large elastoplasticdeformation of a thick-walled tube under internal pressure. Znt. J. o,f Plasticity, 1995. 11(l), 99-118.
PII:SO308-0161(96)00079-B
Int. J. Pres. Vcs. & Piprng 71 (1997) 285-29 1 1997 Elsevier Science Limited. All rights reserved Prmted in Northern Ireland 0308%Olhl/Y7/$17.oG
A variable material property approach for solving elastic-plastic problems H. Jahed, R. Sethuraman* Department
of Mechanical
Engineering,
University
& R. N. Dubey
of Waterloo,
Waterloo.
Ontario,
Canada N2L .?GI
(Received 20 October 1996,accepted 13 November 1996)
The linear elastic solution of a boundary value problem is usedas a basisto generate its inelastic solution. The method considersthe material parameters as field variables. Their distribution is obtained in an iterative manner using the projection method, the arc-length method, and Nueber’srule. The rate of convergenceis compared.The problem of a thick-walled cylinder is considered asan example.The method yields the analytical solution in the limiting caseif the cylinder is made of non-hardening Tresca material and is under internal pressure.The paper also presentssolutionsfor a Mises solid whosehardening is describedby the Ramberg-Osgood formula. 0 1997Elsevier ScienceLtd.
1 INTRODUCTION
yield criterion considered.
The idea of using elastic solutions for inelastic behavior is not new. Stowell (1950),’ Mendelson and Manson (1957),2 and Nueber (1961)3 based their plastic analysis on known elastic results. Some of these methods are still used widely in research works. More recently, the idea of using finite element elastic analysis for approximation of limit loads, was used by Jones and Dhalla (1986)4 for pressure vessels and piping design. This method has been modified by
2 PROBLEM
properties
variables. The spatial distributions
variables
are found
in an iterative
and hardening,
(2)
t*It-,
11, = 11,q,
where
~
r=r,+r,
(3) (4)
and nj is the unit outward normal to the surface. The solution to boundary value problems requires a
of these field
constitutive form:
The
method is used for elastic-plastic analysis of a pressurized tube. Different material behavior such as
non-hardening
=
U,jn,
as field
manner.
(1)
tions are prescribed as follows:
elastic solution of a boundary value problem. The
material
= OIL!
where Uij represents stress tensor, comma followed by a suffix denotes partial differentiation, and repeated suffix implies summation. Further, boundary condi-
shown by Mackenzie and Boyle (1993),’ Nadarajah et al. (1993),’ and Shi et al. (1993).’ This paper presents a method for developing a comprehensive inelastic solution on the basis of the the
is
DEFINITION
uij.j
method for calculation of a conservative limit load by invoking the lower bound limit load theorem was
treats
and Tresca,
Consider a body of volume R enclosed within surface r. The formulation of a boundary value problem for the body consists of traction and/or displacements prescribed on r and the stress equilibrium condition in R. The equilibrium condition in the absence of body forces is expressed in the form
Marriot (1988)’ and later by Seshadri (1991).6 With this method the effect of local inelasticity is simulated by iterative elastic finite element analysis in which highly loaded regions of the structure are systematically weakened by reduction of the local modulus of elasticity. The effectiveness and practicality of this
method
such as von Mises
relation
which can take the following
Ei,
as well as different
=fCa;,)
(5)
Total strain is assumed to be the sum of an elastic part E; and a plastic part F$
* On leave from Indian Institute of Technology, Madras600036, India.
Ejj = t-7,+ F;; 28.5
(6)
H. Jahed et al.
3%
where the stress-strain
Elastic strain is given by Hooke’s law F’
11
1+v = __ E
(J-
”
- ’ (Tkksij E
where v and E are the Poisson ratio and elastic modulus, respectively. The plastic part is given by Hencky’s total deformation equation e; = +s;,
(8)
where s;j =
is the deviatoric given by
-
(9) stress, and 6 is a scalar value function uij
equations are
~(Tkk~ij
(10)
(16)
ez3!+EP E
ff > u,,
with r0 being the initial yield stress. In the case of a hardening material, the uniaxial stress-strain equation is assumed to be described by the Ramberg-Osgood formula:
& u +a! a II -=Eo PO [ uo1
(17)
where so is the strain at initial yield, (Y is the yield offset, and n is the hardening exponent. The effective values of v,~ and EcR can be obtained in the form
The equivalent plastic strain, E& and the equivalent stress, a:(,, are defined as follows e$q = V$$$
and
ueq = G
(11) The value of $ is obtained from the uniaxial stress strain curve. With the help of (7), (8), and (9), eqn (6) may be written as
Since all quantities inside parentheses in eqn (12) are material dependent parameters one can put this equation in the following alternative form, J&f &ff Here, v,~ and Eetr, the effective Poisson ratio and effective Young’s modulus, depend on the final state of stress at a point. Hence, V,~ and E,@ can be thought of as field variables describing the material properties at each point, since the final state of stress at every point is unique. Clearly, for constant elastic values of veff and E,@ eqn (13) describes the elastic behavior of the body. In general, (13) is capable of describing nonlinear behavior. A comparison between eqns (12) and (13) shows that,
(14)
For an elastic-perfectly loading
plastic solid under uniaxial
1 1 --=-+~5s E
(18)
Their distribution the solution.
in the body is obtained
2.1 Determination
of veff and
as part of
E,=
In order to solve a boundary value problem utilizing a constitutive relation of type (13) one needs to know the complete spatial distribution of veff and Eeff. If this distribution is known a priori the solution of the boundary value problem can be constructed immediately. Conversely, one has to find their distribution as part of the solution. For this reason, the following procedure is considered for determination of veff and E eff* From a purely elastic analysis, material points with the same value of ueq are defined. Next, curves are drawn through these points. These curves in the case of a pressurized tube are circles (Fig. 1). All of the material points lying on a curve represent a single point on the stress-strain curve that describes the material behavior. Therefore these material points pose the same vcff and E,# in the real loading provided the shape of the curves remain unchanged. Suppose that ucq remains constant over a small thickness of d.s.
(W
P” ~1)
Fig. 1. Strip in a thick-walled cylinder: (a) pressurisedtube:
(b) an isolatedstrip.
Elastic-plastic
AS a consequence Y,~ and Eeti remain constant throughout this strip defined by ds. Next we isolate the strip and construct a boundary value problem for this strip which is subjected to traction t! and displacement u! on its boundary with the neighboring strip. Solution of this boundary value problem is now needed such that the behavior of points within the constant Y,~ and Erff strip can be ascertained. For the problem of thick-walled cylinders this boundary value problem is that of a cylinder under internal and external pressure (Fig. 1). In this manner, the original problem is discretized into several infinitesimal strips. Each strip consists of points which have the same material properties. Hence, the proper solution of each strip is equivalent to the elastic solution. To implement this procedure a scheme is needed for evaluation of effective material properties each strip. This is discussed in the next section. It should be mentioned that this method demands the elastic solution of each strip with constant properties. For the case studied here this elastic solution is the Lami: solution. 2.2 Scheme evaluation
of veff and EeB
The material constants vCffand Eeti are obtained in an iterative manner. This iterative procedure continues until the v~~-E,~ curve of each strip follows that of experimentally obtained uniaxial material behavior. The following three schemes may be used: (i) Projection
method
First, a pseudo-elastic solution is obtained under the plastic load. This identifies the state of stress at each point [point a on the extended elastic line in Fig. 2(a)]. Keeping the strain the same, a new value for stress is obtained by finding the corresponding point on the
problems
287
real uniaxial stress-strain curve [point a’ in Fig. 2(a)] Equations (16) or (18) are used to determine the new values of veff and E,*. A new solution is now pursued on new values of veff and E,* The same procedure is carried out until all of the points line up on the stress-strain curve. (ii) Arc-length
method
From the pseudo-elastic solution, the (T,~/u~~ - E,JE~ curve is constructed. A point on this curve represents the state of stress of a particle in the body [point a in Fig. 2(b)]. Next a radius r is defined as r = g/(a,,/~r,)~ A point is located on the curve which is at the same [point a’ in Fig. 2(b)]. New then obtained with the help (iii)
Neuber’s
real uniaxial stress-strain distance r from the origin values of veff and Eeff are of (16) and (18).
The rule proposed by Neuber (1961)” for calculation of the local elastic-plastic stress and strain is employed in this case. From the pseudo-elastic solution the total equivalent strain energy is calculated. Next, a point on the uniaxial material curve which has the same strain energy is identified [Fig. 2(c)]. Th’ 1s 1ocates the point f on the stress-strain curve in Fig. 2(c) such that the two areas abed and aefs are equal. The stress and strain values at f are then used for evaluation of v,, and E,+ 3 APPLICATION TO ELASTIC-PLASTIC ANALYSIS OF PRESSURIZED TUBES structures, such as piping in boilers in power plants. machin-
(W
(3
C
d
f
G
a
(19)
rule
Many load-bearing chemical industries,
(a)
+ (~eqI~dZ
E
t
e
b
w rule
1 0
Fig. 2. (a) Projection method; (b) arc-length method; (c) . . Nueber’s
5 0
method.
1
2
Fig. 3. First elastic solution compared to final
solution.
3 elastic-plastic
H. Jahed et al.
28X
where for plane stress c
I
= Cl+
ydl
-
2v,ff)plrf
-pzrt 2 r2 -
E eK
r:
(21) c = C1+ Ye) (PI -p&5$ 2 Eeff r; - rf
00 0
0.4
and
c = Cl+ v,ff)(l - 2~,ff)p~r5 -w-f I rz - rf Eelf
4.8
c2 = (1-t V,K)(PI J
Fig. 4. Comparison
of results with analytical solution and ABAQUS.
ery in aeronautical and aerospace engineering, and pressure shells in nuclear powered installation are pressure vessels, generally made of hardening materials, which undergo inelastic deformation. Therefore, a good understanding of their elasticplastic behavior is needed for analysis and design difficulties purposes. Owing to the mathematical arising from nonlinearity, a comprehensive analytical elastic-plastic analysis of such members is not attainable in general, Utilizing the method outlined above a comprehensive plastic solution of a thick-walled cylinder is obtained here. Constant geq strips for a thick cylinder form circular strips with traction conditions of the same type (Fig. 1). The elastic solution for each strip is given by the following well known equation: u=C,r+-
C2 r
for plane strain. Here rl, r2, ply and p2 are the inner radius, outer radius, internal pressure, and external pressure of each strip, respectively. Hence the inside and outside displacement of the strip can be related to its inside and outside pressure in the following form:
(23)
The components
Cl2 =
Von Mlses Hardanlng
of coefficient matrix C are as follows
(l; y&&
-2
cif =
2 (1
-
&f)
21
=
_
1
22
+
4-2
r: - r:
E,K
V,K
4
Eeff r: - r: !
Materials
-m+-
h-0
2
I
(24) c
c
(20)
p2)rX
rz2 - r:
E,K -1.6
(22)
W Fig. 5. Comparisonof resultswith ABAQUS.
Elastic-plastic
problems
289
---.-._
Hudmlng
-
.
P, P, - P, Pt
--.
Materials
/ao=l Iv,= 1.85 1ao=2. 15 luo=2.55 ~
(a)
P, Ino= I 85
(b) Fig. 6. Stress variation (internal pressure).
for plane stress, and
C,,=
4 NUMERICAL
2 r,r: --~ ECffri - rf (25)
2
rfr2 CT>,= -EeFfri - r:
for plane strain. After assembling all strips, equations of the form
[C’l{U = m
a system
of linear (26)
is obtained and solved. One of the schemes mentioned above is used to find the new values of v,~ and EClf.
RESULTS
DISCUSSION
Numerical results have been obtained for a cylinder with rJr, = 5, v = 0.3, and E/u0 = 1000. In the case of hardening material the uniaxial stress-strain curve is assumed to follow formula (17) with (Y= 3/7 and ?I = 5. Results for different internal and external load values for closed end and open end cylinders are presented in this section. Figure 3 shows how the first elastic solution is converged upon the final elasticplastic solution. It should be noted that different points shown on the stress-strain curve correspond to different material points throughout the wall thickness. A quantitative comparison of stress and strain distribution according to Tresca and von Mises yield criterion obtained from the present method with other analytical solutions (Prager and Hodge, 1951)“’ and a
(b)
(a) Fig. 7. Strain variation
AND
(internal
pressure).
290
H. Jahed et al.
-
plme6o-M I
2.0 Fig.
R I R, I
30
I
4.0
I
5.0
8. Plane stress and plane strain stress distribution comparison.
finite element solution (Hibbit, 1994)” is given in Figs 4 and 5. The exact solution, given by Prager and Hodge (19X),‘” is matched by the present method. Also Figs 4 and 5 present a comparison of the present method solution to finite element elastic-plastic analysis using ABAQUS (Hibbitt, 1994).” The agreement between the two solutions is excellent. Additional results are given in Fig. 6. The results clearly show the effect of the material behavior on the stress distribution. As has been noticed by MacGregor et al., 1948’* and, more recently, by Bon and Haupt,13 the circumferential stress changes from tension to compression even for partially yielded cylinders [Fig. 6(b)]. However, for hardening materials the hoop stress remains tensile until the plastic zone is very large [Fig. 6(a)]. It is found in the case of ideal plasticity that the cylinder reaches the limit pressure very quickly and this pressure increases for hardening materials. Moreover, for hardening materials the limit
P;/cr,=l
--
Pi /u,=l.85
PihJo=1.4
R/Ri
0.0
I
I
I
3.0
2.0
1.o
Fig.
--..----
I
4.0
10. Variation
of material properties (hardening materials, von Mises yield condition).
pressure is reached at much higher strain values (Fig. 7). Figure 8 shows a comparison of plane stress and plane strain cases of a pressurized tube obeying the von Mises yield criterion. The figure indicates that predictions based on either plane assumption are fairly close. The behavior of an externally loaded cylinder has also been studied. The distribution of hoop and axial stress for various loadings are shown in Fig. 9. The spatial distribution of the material properties vetr and Eeff is given in Fig. 10. Several cases have been studied for non-hardening materials to compare the three schemes mentioned for obtaining convergent solution. The results show a higher convergence rate if Neuber’s rule is used. The number of iterations involved in these studies are tabulated in Table 1.
=d”o -1 2 -
_
. -
Von y1sss n-g _.-..-. __-..--
-
Paw., PO/s”.,.4
R I Rc
(a)
1 50
W
Fig. 9. (a) Hoop stress variation (external pressure); (b) axial stress variation (external pressure).
Elastic-plastic Table
1. Convergence
Load
comparison
Number of iterations Projection method
Arc-length method
Neuber’s rule method
37
14
5
67
47
6
92
71
7
12s
105
13
5 CONCLUSIONS
The idea of describing nonlinear behavior by linear analysis is implemented here. This is accomplished by simulating the behavior of the inelastic region through a collection of elements with pseudo-elastic response. The stress-strain response within each element is assumed linear even though each have different effective moduli. The difference in moduli allows each element to deform in a different manner as compared to a neighboring element. This results in an overall nonlinear behavior. In the case of a thick-walled cylinder, the region is divided into circular strips. The loading condition for each strip is identical and hence, the form of the
solution for each strip is the same as the solution given by the linear elastic analysis. Three schemes have been used in order to achieve rapid convergence of the iterative solution. The scheme based on the total strain energy provides the most rapid convergence.
Moreover, results obtained from the present method are in good agreement with other analytical and numerical results. The following comments summarize the main outcomes of this work: 1. A method is proposed to solve the inelastic problems using linear elastic analysis. 2. The present solution obtained for a thick cylinder made of Tresca materials matches the analytical
solution. 3. It was observed that for the case of non-hardening behavior Neuber’s rule for updating modulus gives
problems
291
faster convergence than the projection and arc-length method. 4. Results obtained for hardening materials are in excellent agreement with the elastic-plastic finite element analysis.
ACKNOWLEDGEMENTS
The first author wishes to thank the Ministry Higher Education of Iran for financial support.
of
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