acharya finalreport
DESCRIPTION
análisis femTRANSCRIPT
Creep Analysis of a Thick Walled Spherical Pressure Vessel
Considering Large Strain
by
Tushar Kanti Acharya
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________ Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute Hartford, Connecticut
April, 2012 (For Graduation May, 2012)
ii
© Copyright 2012
by
Tushar Kanti Acharya
All Rights Reserved
iii
CONTENTS
Creep Analysis of a Thick Walled Spherical Pressure Vessel Considering Large
Strain……………………………………………………………...…………………...…..i
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
LIST OF SYMBOLS (WITH UNITS) ............................................................................. vi
ABSTRACT ................................................................................................................... viii
1. Introduction .................................................................................................................. 1
2. Theory .......................................................................................................................... 3
2.1. Assumptions………………………………………………………...……3
2.2 Derivation of Creep Strain Rate Equation…………….………….….3
2.3 Stress and Strain Distribution…………………………………………….5
2.4 Norton's Creep Law………………………………………………………5
2.5 Equivalent Stress ……………………………………………………...7
2.6 Radial Stress r and Effective Strain Rate as Functions of Radius……..7
2.7 Small Strain Solution for Radial and Tangential Stresses………………..8
3. Results of Analytical Formulae ................................................................................. 11
4. Finite Element Analysis and Results ......................................................................... 15
4.1 Equations and Input for ANSYS Analysis……………………………...15
4.2 Coarse Mesh Results…………………………………………………….18
4.3 Fine Mesh Results……………………………………………………….23
4.4 Coarse Mesh vs. Fine Mesh……………………………………………..28
5. Discussion .................................................................................................................. 30
6. Conclusions ................................................................................................................ 31
7. References .................................................................................................................. 32
Appendix………………………………………………………………………………33
iv
LIST OF TABLES
Table Description Page
1 Data for Exact Solution 11
2 Data for Analysis 16
3 Coarse and Fine Mesh Comparison 17
4 von Mises Stress Comparison (Coarse Mesh,2300days) 19
5 von Mises Creep Strain Comparison (Coarse Mesh,2300days) 20
6 von Mises Stress Comparison (Coarse Mesh, 2300days) 21
7 von Mises Creep Strain Comparison (Fine Mesh, ε=1.03) 23
8 von Mises Stress Comparison (Fine Mesh,3000days) 24
9 von Mises Creep Strain Comparison (Fine Mesh, ε=0.6) 25
10 von Mises Stress Comparison (Fine Mesh,3000days) 26
11 Percentage Result Deviation of Finite Element model vs. Exact Solution 29
v
LIST OF FIGURES
Figure Description Page
1 Creep Rate vs. Strain, both at Internal Radius 11
2 Strain at Internal Radius vs. Time 12
3 Tangential Strain Rate ε't at various Radii of Spherical Vessel 13
4 Stress Distribution at Various Radii of Vessel for εa=0.0 14
5 Stress Distribution at Various Radii of Vessel for εa=0.62 14
6 A Hemispherical part, inner and outer radius are 20in and 100in 17
7 A Hemispherical meshing shown with SOLID187 18
8 Coarse Mesh, von Mises Creep strain at internal radius 19
9 Coarse Mesh, von Mises Creep Stresses at internal radius,
with ε=0.93 and Time=3000days 20
10 Course Mesh, von Mises Creep strain at internal radius
for Time=2300days 21
11 Coarse Mesh, von Mises Creep Stresses at internal radius,
with ε=0.56 and Time=2300days 22
12 Comparison between Coarse Mesh and Exact Solution Results 23
13 Fine Mesh, von Mises Creep strain at internal radius
for Time=3000days 24
14 Fine Mesh, von Mises Creep Stresses at internal radius,
with ε=1.0 and Time=3000days 25
15 Fine, von Mises Creep strain at internal radius
for Time=2300days 26
16 Fine Mesh, von Mises Creep Stresses at internal radius,
with ε=0.6 and Time=2300days 27
17 Comparison between Fine model and Exact Solution 28
18 Comparison between Coarse vs. Fine w.r.t.Exact Solution 29
vi
LIST OF SYMBOLS (WITH UNITS)
Functions
)'(f function of effective creep strain
Geometry:
a inner radius of sphere (inches)
b outer radius of sphere (inches)
r radius of sphere at random position (inches)
r’ an element at radius r with further very small displacement u
x,y,z coordinate axis
Mechanical Data
E Young’s Modulus (psi)
B,C1,C2,C3,n experimental material constants
c material constant (psi)
Pressure
p internal pressure of spherical vessel (psi)
Strain and Strain Rate
ε effective strain
ε’ effective strain rate
a effective strain at inner radius a
b effective strain at outer radius b
'a strain rate at inner radius a
'b strain rate at outer radius b
εr radial strain
εt tangential strain
Stress
effective stress (psi)
a effective stress at radius a (psi)
b effective stress at radius b (psi)
vii
r radial stress (psi)
t tangential stress (psi)
FH effective stress per Finnie and Heller (psi)
rFH radial stress per Finnie and Heller (psi)
tFH tangential stress per Finnie and Heller (psi)
Time
t time (seconds)
viii
ABSTRACT
In this report, the creep analysis of a thick walled spherical pressure vessel made of
a homogeneous and isotropic material and subjected to internal pressure is considered.
Spherical vessels are used as pressure vessels in power plants and in petrochemical
industry. Creep analyses of thick-walled spherical pressure vessels subjected to internal
and or external pressure are important in solid mechanics and in engineering
applications. In engineering design predicting the behavior of creep over the long term is
important. The strains considered are assumed to be large which necessitates the use of
finite strain theory for evaluating the expressions for stresses, creep strains and strain
rates. The general theory developed by N.S. Bhatnagar and V.K. Arya (1973) has been
applied to the solution of a specific problem using Norton’s law of creep. The
infinitesimal (small) strain discussion by Finnie and Heller (1959) was also considered
for comparison. In addition, a finite element approach has been implemented using
ANSYS. The analysis and numerical example will aid the designers in the prediction of
correct creep strains, strain rates and stresses in cases where large creep deformations of
spherical pressure vessels are permissible.
1. Introduction
The problem of creep deformation of thick-walled spherical pressure vessels has
been studied previously (3). Most studies assume that the strains are small and that
deformations can be referred to the original dimensions of the vessels rather than the
instantaneous deformed values. While these assumptions are sufficiently accurate for
small strains, under longer time exposures the deformations may attain a value where
small strains can no longer be assumed. With this in mind an attempt has been made by
Bhatnagar and Arya (1) to solve the problem of creep deformation of thick-walled
spherical vessels under internal pressure considering large strains. This study is based on
a steady state law for creep together with the assumption of a homogeneous,
incompressible, isotropic material.
The finite strain theory, developed in (2), for the case of plastic flow of thick-walled
tubes with large strains, has been extended for application to the creep deformation of a
thick-walled spherical pressure vessel. In this project various expressions for stresses,
strains and creep rates, based both on infinitesimal and finite strain theories are
presented following the descriptions in (1) and (3).
The large amount of previous work has been concerned with the investigation of
stresses and strains in the wall of the vessel assuming the infinitesimal strain theory to be
valid. For the elastic deformations and the early stages of creep, the assumption that the
strains are infinitesimal yields values of stresses and strains which are in good agreement
with the experimental observations. But under longer time creep deformation conditions
the strains keep accumulating and may reach a value so large that the use of infinitesimal
strain theory for the evaluation of stresses and strains is no longer valid.
In finite element analysis, membrane forces and bending moments of the nodes
cannot be easily expressed as due to implicit functions of strain-rate because of the high
nonlinearity of stress-strain relation in creep. These appear to be the main reasons why
the published literature, so far, is directed mainly towards problems of simpler nature
such as hollow hemispherical pressurized vessel. Even in these cases, some
simplifications have been usually introduced.
2
In chapter 2 below the large formulation of the problem is first presented and
followed by the small strain formula. Chapter 3 then presents results obtained using the
above formulae. Chapter 4 contains the finite element model results.
3
2. Theory
These chapter presents the development of large and small strain theory of creep of
hollow spherical pressure vessel. The large strain formulation follows (1) and small
strain development follows (3).
Consider a spherical vessel of internal and external radii a and b, respectively,
subjected to internal pressure p.
2.1 Assumptions
To investigate the deformation behavior of the vessel at high temperatures, where
creep is dominant the following assumptions are made.
1. The material is homogeneous and isotropic.
2. The volume of the material remains constant (condition of incompressibility).
Creep has no effect on the density of the material as summation of creep rates in
the three principal directions is zero.
3. The ratios of the principal shear strain-rates to the principal shear stresses are equal.
4. The effective stress and effective strain (creep) rate , are related by
σ =f(ε’). [1]
2.2 Derivation of Creep Strain Rate Equation
By symmetry, the principal stresses in the two tangential directions are equal. We
shall denote the stresses and creep rates in radial and tangential directions by subscripts r
and t, respectively.
It may also be concluded from symmetry that the only displacement in this problem
is radial and is such that concentric spherical surfaces remain concentric and spherical
after deformation.
Considering the radial equilibrium in the deformed state, the mechanical
equilibrium equation is
4
)(2'
' rtr
dr
dr
[2]
The equation of compatibility is
1 tre
rr t
[3]
The effective stress σ and effective strain ε equations are
rt [4]
and
)(3
2rt [5]
The condition of incompressibility is
rt 2 [6]
The creep strain rate at any radius r as a function of given creep rate at the inner
radius a, is
'
)2/3(
)2/3(
)1()(1
)('
3
3
aa
a
er
a
er
a
[7]
Relation between creep strain rates and the creep strains is
'
)2/3(
)2/3(
1
1' aae
e
[8]
The equation for internal pressure p in terms of creep rate as
a
b
dp'
'
''3
2
[9]
Where
5
'
)2/3(3
)2/3(3
'
)1()(1
)(
aba
a
eb
a
eb
a
[10]
is the creep rate at the outer radius.
2.3 Stress and Strain Distribution
The effective stress can be obtained as a function of r from Eqn.1, provided the
effective strain a and creep rate'a , both at the inner radius a , are given.
The equation for radial stress r , at any radius r is given as
a
dpr
'
'
''3
2
[11]
And, the tangential stress at any radius r is given as
)'('
'3
2'
'
fdpa
t [12]
The creep rate t' , in the tangential direction, can then be obtained as
'
)2/3(
)2/3(
'
)1()(1
)(
2
13
3
ata
a
er
a
er
a
[13]
2.4 Norton’s Creep Law
Under secondary creep conditions, the effective stress and creep rate ' are
assumed to be related by Norton’s Law
6
n
c
B )(' [14]
Where B , n and c are experimental constants and c has the dimensions
of .
Eqn.14 can be rewritten as
)/1(
)/1(' n
nc
B
[15]
From Eqn.9, the equation for internal pressure p in terms of'a ,
'b , c , B and n
is
)(
3
2 )/1(')/1(')/1(
n
b
n
anc
B
np
[16]
From this and Eqn.10, the creep rate 'a at ar as a function of the strain
at ar , a is given as
nnnn
cn
nn
aa
a
eb
a
eb
a
n
Bp
]}
)1()(1
)({1[
2
3' /1
)2/3(
)2/3(
3
3
[17]
Note that the creep rate is function of the creep strain. Integrating Eqn.17 and
simplifying, the strain a can be obtained as a function of time t, given by
a
a
a
tn
Bp
eb
a
eb
a
nnc
n
nnnn
0
/1
)2/3(
)2/3(
2
3]}
)1()(1
)({1[
3
3
[18]
This expression can then be numerically integrated to compute the strain at the inner
radius for any given time.
7
2.5 Equivalent Stress
To find an alternate numerical solution for Equivalent stress at various radii of
vessel we use Eqn.7 in Eqn.15, to obtain
'/1
)2/3(
)2/3(
/1]
)1()(1
)([
3
3
an
nc
a
a
er
a
er
a
B
[19]
Inserting the expression for 'a from Eqn.17 in Eqn.19
]})1()(1
)({1[
])1()(1
)([
2
3
/1
)2/3(
)2/3(
/1
)2/3(
)2/3(
3
3
3
3
n
n
a
a
a
a
eb
a
eb
a
er
a
er
a
n
p
[20]
2.6 Radial Stress r and Effective Strain Rate as Functions of Radius
To find an expression for the radial stress at various radii of the vessel we insert
Eqn.15 in Eqn.11, to obtain
'
'
)1
(
)/1(''
3
2 a
dB
p n
n
nc
r
'
'/1
)/1(]'[
3
2an
nc n
Bp
[21]
8
However, since this
)/1()/1('
')/1( ']'[
' nn
an a
[22]
Using Eqn.7 in Eqn.22, we get
]})1()(1
)({1[']'[ )/1(
)2/3(3
)2/3(3
)/1('
)/1( ' nna
n
a
a
a
er
a
er
a
.
[23]
Combining Eqn.23, Eqn.17 and Eqn.21, we get
]
]})1()(1
)({1[
]})1()(1
)({1[
1[
/1
)2/3(3
)2/3(3
/1
)2/3(
)2/3(
3
3
n
n
r
a
a
a
a
eb
a
eb
a
er
a
er
a
p
[24]
Eqn.24 is a closed form expression of Eqn.11, which is used below in Fig-4 and 5 to
show the stress distribution at various radii.
2.7 Small Strain Solution for Radial and Tangential Stresses [3]
The expressions obtained for strain elastic and creep deformations are presented
below. The elastic stresses in a hollow sphere under internal pressure are given by the
Lame Equations
)2
1(3
3
33
3
r
b
ab
pat
[25]
9
)1(3
3
33
3
r
b
ab
par
[26]
Only two stresses need to be specified, since by symmetry the principal stresses in
the two tangential directions are equal. The condition of incompressibility
becomes drdur /'' and rut /'' , where u’ is the rate at which a radius r
is changing, we have
3'2'
r
Ctr [27]
In which C is a constant.
Now, the corresponding expressions for creep deformation are given.
Finnie & Heller [2] obtained theoretical expressions for creep stresses in spherical
vessel under small strain conditions assuming Norton’s Law. The equilibrium equation
per Finnie and Heller [2, Eqn7-35, p186] is
)(2 rFHtFHrFH
dr
dr
[28]
For a thick-walled creeping sphere under internal pressure prFH at ar
and 0rFH at br ; radial stress rFH [2, Eqn7-36, p187] is
]1)/[(
]1)/[(/3
/3
n
n
rFH ab
rbp [29]
so that
])(
[]1)/[(
)3
(3
/3
/3
n
n
n
nrFH
r
b
ab
p
ndr
d
[30]
Results obtained using Eqn.29 are shown in Fig-4 and Fig-5 for the assumption of
infinitesimal small strains for radial stresses.
Substituting Eqn.29 in Eqn.28, the tangential stress is obtained as
10
rr
tFH dr
dr 2 ]1)/[(
1]2
)23([)(
./3
/3
n
n
abn
n
r
b
p [31]
and the equivalent stress can be obtained as
rFHtFHFH [32]
Eqn.31 and Eqn.32 can be used to find the tangential stress in the spherical pressure
vessel for small strain conditions.
11
3. Results of Analytical Formulae
Using the formulation in chapter 2 above and the data considered by Bhatnagar and
Arya [1] and Rimrott [4] the creep deformation of a spherical vessel was investigated. The
following values of constants were taken.
Table-1: Data for Exact Solution
Descriptions Data
B 5.E-12 /day
σc 1000psi
n 6
b/a 5
p 40,000 psi
Fig-1 shows the effective creep rate a' at the internal radius a calculated in Excel
from Eqn.17 and plotted against the creep strain a at the internal radius. Note that the
creep strain rate increases with increasing creep strain.
Fig-1: Creep Rate Vs Strain, both at Internal Radius
12
Fig-2 shows the computed creep strain as a function of time obtained by numerical
integration of Eqn.18. The graph clearly indicates that the effective creep strain increases
with time. Moreover the figure also shows that the creep strain rate is not constant but
increases with time.
Fig-2: Strain at Internal Radius vs. Time
The tangential creep rate t' at various radii of the vessel was calculated from
Eqn.13 and Eqn.17, and is shown in Fig-3 for three values of the effective creep strain at
the inner radius a viz. 0.24, 0.56 and 0.76. From Fig-3 it is observed that for all radii,
the tangential creep strain rate increases with time. It is also observed that the tangential
creep strain rate decreases first quickly then slowly towards zero with distance from the
inner radius of sphere to the outer radius.
13
Fig-3: Tangential Strain Rate ε't at various Radii of Spherical Vessel
Fig-4 shows the distributions of radial tangential and equivalent stresses in the wall
of the vessel as obtained from Eqn.4, 20 and 24 at time zero. Also the radial, tangential
and equivalent stresses for small strain described by Finnie and Heller [2] are shown
(Eqn.29, 31 and 32). As expected, the results are very close since the computed strain
correction is negligible in this case.
The stress distribution in the wall of the vessel for 62.0a is shown in Fig-5.
Here stress distributions are shown following a considerable amount of deformation. The
radial, tangential and equivalent stresses for small strain described by Finnie and Heller
[2] are also plotted in Fig-5 (Eqns.29, 31 and 32). In this case, larger differences exist
between the predictions of small and large strain theories.
14
Fig-4: Stress Distribution at Various Radii of Vessel for εa=0.0
Fig-5: Stress Distribution at Various Radii of Vessel for εa=0.62
15
4. Finite Element Analysis and Results
In this project a finite element model was developed to compare with the results of
Bhatnagar and Arya. The finite element model was created using ANSYS Student
version 13 and the Norton Creep theory was applied. A hemisphere model was analyzed.
Although the analysis are made for a hemisphere and clamped to annular surface, the
method is quite general and other types of shells boundary conditions and geometries
can be treated similarly.
In ANSYS the vessel was modeled as a hollow hemisphere (Fig-6) and SOLID187
elements were used for meshing (Fig-7). SOLID187 element is a higher order 3-D,
tetrahedral 10-node element. SOLID187 has a quadratic displacement behavior and is
well suited to modeling irregular meshes and non-linear problems.
The element is defined by 10 nodes having three degrees of freedom at each node:
translations in the nodal x, y, and z directions. The element has plasticity, hyper-
elasticity, creep, stress stiffening, large deflection, and large strain capabilities. It also
has a mixed formulation capability for simulating deformations of nearly incompressible
elasto-plastic materials, and fully incompressible hyper-elastic materials. [10] SOLID187
gives better approximation over Axi-symmetric element PLANE182; hence it is used for
the analysis.
4.1 Equations and Input for ANSYS Analysis
For Finite Element Analysis using the program ANSYS, Norton’s Law (Eqn.14
above) is written as (ANSYS Creep Model 10)
)exp()(' 31
2
T
CC C
nnc
B
)( [33]
Where nc
BC
1 , nC 2 and 03 C
The input data included in Table-2 were used in the calculations.
16
Table-2: Data for Analysis
Descriptions First Set Analysis Data Second Set Analysis Data
B 5.E-12 /day 5.E-12 /day
c 1000psi 1000psi
n 6 6
C1 (Eqn.45) 5.E-30 5.E-30
C2(Eqn.45) 6 6
C3(Eqn.45) 0 0
Internal Radius, a 20in 20in
Outer Radius, b 100in 100in
Internal Pressure, p 40,000 psi 40,000 psi
Young’s Modulous, E 5.5E6 psi 5.5E6 psi
Poisson’s Ratio 0.3 0.3
Number of Days to be
analyzed
3000 2300
In ANSYS the implicit Norton Creep model (Model 10) was selected in Material
modeling and considered half of a hollow sphere in analysis. A displacement DY=0 was
applied to the circular plane of the hemi-sphere, so that rigid body motion was avoided.
Pressure, p is applied to the inner surface of the sphere. The analysis was done for two
sets of data, 2300 days and 3000 days. The results are shown below.
In Fig-6 the hemispherical part is shown, which was considered for the analysis.
The hemisphere considered is a part of a thick spherical vessel under high internal
pressure of 40,000 psi. The inner radius is 20in and the outer radius of sphere is 100in. A
sample ANSYS input file is included in the Appendix.
17
Fig-6: A Hemispherical part, inner and outer radius are 20in and 100in
Fig-7 shows the meshed model. The increase in number of elements increases the
time required for solutions. In this paper results obtained using 4 and 5 subdivisions for
each main line in the geometry are compared. Table-3 shows the resulting discretization
in each case.
Table-3: Coarse and Fine Mesh Comparison
Mesh Type in Model Elements Nodes
4 Line Subdivision
(Coarse Mesh)
643 1144
5 Line Subdivision
(Fine Mesh)
1336 2223
18
Fig-7: A hemispherical meshing shown with SOLID187 elements
4.2 Coarse Mesh Results
Results obtained using ANSYS with the coarse mesh consisting of 643 elements are
now presented.
Fig-8 shows that the von Mises creep strain at the internal radius of the sphere
increases with time. The slope approximately matches that of the Exact Solution per
Bhatnagar and Arya’s. The creep strain at 3000days is 0.93 while per Exact Solution the
creep strain is 1.0 for 2984days. Fig-8 also shows that the creep strain rate increases with
time in agreement with the previous result (Fig.-1).
19
Fig-8: Coarse Mesh, von Mises Creep strain at internal radius
Fig-9 shows the computed von Mises creep stress in the hemisphere at 3000days.
The stress varies from 22743psi to 13493 from the inner to the outer radius. Per Exact
Solution for creep strain ε =0.93, the equivalent stress varies from 22802 psi to 12802
psi (see Table-4 and Fig-12 below).
Table-4: von Mises Stress Comparison (Coarse Mesh,2300days)
Radius )(exactm )(Ansysm %age Deviation
a = 20in 22802 22743 0.3
b = 100in 12802 13493 5.4
20
Fig-9: Coarse Mesh, von Mises Creep Stresses at internal radius,with ε=0.93 and
Time=3000days
Fig-10 shows the von Mises creep strain at the internal radius of the sphere as a
function of time for up to 2300days. Bhatnagar and Arya [1] discussed the results of their
model at effective creep strain ε =0.62. So for comparison we have included this time
line. It is observed that the effective creep strain increases with time. The creep strain at
2300days is 0.56 in ANSYS while per Exact Solution the creep strain is 0.56 for
2191days. The agreement is reasonable but ANSYS seems to under predict the strain
(Table-5).
Table-5: von Mises Creep Strain Comparison (Coarse Mesh,2300days)
Radius )(exact )(Ansys %age Deviation
a = 20in 0.6 0.56 6.7
21
Fig-10: Coarse Mesh, von Mises Creep strain at internal radius for Time=2300days
Fig-11 shows the computed von Mises creep stress in the hemisphere for a time
period of 2300days (and effective creep strain ε=0.56). Per analysis it is observed that
the stress varies from 19764psi to 10841 from inner to outer radius. Per Exact Solution
for creep strain ε =0.56 the equivalent stress varies from 20550 psi to 10550psi. This has
been plotted in Fig-12 (see also Table-6).
Table-6: von Mises Stress Comparison (Coarse Mesh, 2300days)
Radius )(exactm )(Ansysm %age Deviation
a = 20in 20550 19764 3.8
b = 100in 10550 10841 2.8
22
Fig-11: Coarse Mesh, von Mises Creep Stresses at internal radius,with ε=0.56 and
Time=2300days
Fig-12 plots the von Mises and Equivalent stresses at strains ε=0.93 and 0.56, as
functions of radius obtained from ANSYS and from the Exact solution respectively. The
Equivalent stress at ε=0.56 is aligned approximately with ANSYS calculated von Mises
stress. The same result is observed for ε=0.93. It is also shown that the ANSYS von
Mises stress values are slightly higher than the Exact stress. It seems that the ANSYS
values are more stringent than those obtained from the Exact Solution. So for design
purposes the ANSYS values are preferred.
23
Fig-12: Comparison between Coarse Mesh and Exact Solution Results
4.3 Fine Mesh Results
Fig-13 shows how the von Mises creep strain at the internal radius of sphere
increases with time for the Fine mesh model. The effective creep strain is 1.03 at
3000days. The creep strain is 1.03 for 3019 days per Exact Solution. The result is
slightly higher than the Coarse Mesh and Exact Solution results. This is an effect of the
finer mesh used (see Table-7). Table-7 shows von Mises creep strain deviation of
approx.0.60% at the creep strain ε=1.03.
Table-7: von Mises Creep Strain Comparison (Fine Mesh, ε=1.03)
ε )(exactDays )(AnsysDays %age Deviation
1.03 3019 3000 0.6
24
Fig-13: Fine Mesh, von Mises Creep strain at internal radius for Time=3000days
Fig-14 shows the von Mises creep stress in the hemisphere obtained from ANSYS
at time 3000 days (and effective creep strain ε=1.03 (approx)). Per analysis it is observed
that the von Mises stress varies from 24171 psi to 14777 psi from the inner to the outer
radius. Per Exact Solution for the creep strain ε =1.03 the equivalent stress varies from
23573 psi to 13573 psi. This has been plotted in Fig-7 and tabulated in Table-8. The
table shows that the ANSYS von Mises stress is more stringent with a 8.9% deviation at
the outer radius and a 2.5% deviation at the inner radius.
Table-8: von Mises Stress Comparison (Fine Mesh,3000days)
Radius )(exactm )(Ansysm %age Deviation
a = 20in 23573 24171 2.5
b = 100in 13573 14777 8.9
25
Fig-14: Fine Mesh, von Mises Creep Stresses at internal radius,with ε=1.0 and
Time=3000days
Fig-15 shows the von Mises creep strain at the internal radius of the sphere for time
period up to 2300 days. The creep strain at 2300 days is 0.6 in ANSYS while per Exact
Solution the creep strain is 0.6 for 2291days (Table-9). The ANSYS result seems to
under predict the Exact solution but it is better than the Coarse mesh results. Table-8
shows that the von Misses creep strain %age deviation is 0.4 compare to coarse mesh
with 0.6.
Table-9: von Mises Creep Strain Comparison (Fine Mesh, ε=0.6)
ε )(exactDays )(AnsysDays %age Deviation
0.6 2291 2300 0.4
26
Fig-15: Fine, von Mises Creep strain at internal radius for Time=2300days
Fig-16 shows the von Mises creep stress in the hemisphere for a time period of
2300days (and effective creep strain ε=0.6). Per analysis it is observed that the stress
varies from 20488psi to 11257 from inner to outer radius. Per Exact Solution for creep
strain ε =0.6, the equivalent stress varies from 20779 psi to 10779psi. This has been
plotted in Fig-17 (see also Table-10). The von Mises stresses in ANSYS have higher
deviation at the outer radius than at the inner radius.
Table-10: von Mises Stress Comparison (Fine Mesh,3000days)
Radius )(exactm )(Ansysm %age Deviation
a = 20in 20779 20488 1.4
b = 100in 10779 11257 4.4
27
Fig-16: Fine Mesh, von Mises Creep Stresses at internal radius,with ε=0.6 and
Time=2300days
Fig-17 shows the Equivalent and von Mises stresses at ε=1.03 and 0.6, calculated
from the Exact Formula and from ANSYS respectively as functions of radius. The
equivalent stress obtained from Exact Formula at ε=0.6 agrees well with the ANSYS von
Mises stress. However, the ANSYS von Mises stress values are slightly higher than the
Exact Solution Equivalent stress values for ε=1.03. It seems that the ANSYS values are
more stringent than those obtained from Exact Solution. So for design purposes the
ANSYS values are preferred.
28
Fig-17: Comparison between Fine Mesh Model and Exact Solution
4.4 Coarse Mesh Vs. Fine Mesh
Fig-18 shows a summary of results comparing the Exact Equivalent stress at ε=1.0
and 0.6, the ANSYS von Mises stress at ε=1.03 and 0.6 using the fine mesh and the
ANSYS von Mises stresses at ε=0.93 and 0.56 using the coarse mesh model as functions
of the radius inside the sphere. The equivalent stress at ε=0.6 is aligned approximately
with the ANSYS von Mises stresses both for the coarse and fine mesh models. The
ANSYS result matches Bhatnagar and Arya’s results well up to the creep strain ε=0.6,
but for higher creep strain values the ANSYS von Mises results are higher than the Exact
values.
29
Fig-18: Comparison between Coarse vs. Fine Mesh and w.r.t. Exact Solution
In Table-11 we have summarized the deviations between the von Mises and
Effective stress results obtained from ANSYS and from the Exact solution. It is observed
that for the effective creep strain ε=0.6, the percentage deviation is +3.8% to -3.7%.
While for effective creep strain ε=1.0, the percentage deviation is +0.26% to -5.63%.
Table-11: Percentage Result Deviation of Finite Element model vs. Exact Solution
Description of Model Detail Maximum Deviation %
Minimum Deviation %
ANSYS Coarse Mesh 3000day vs. Exact Solution 0.26% -5.63%ANSYS Coarse Mesh 2300day vs. Exact Solution 3.82% -3.70%ANSYS Fine Mesh 3000day vs. Exact Solution -2.54% -9.57%ANSYS Fine Mesh 2300day vs. Exact Solution 1.40% -6.22%
In Table-11, the negative value indicates that the ANSYS results are higher than
those from the Exact Solution.
30
5. Discussion
From the analytical results for the large strain creep deformation of a thick walled
hollow spherical vessel shown in Fig-1, it is observed that the creep strain rate at the
inner radius 'a increases with strain. Fig-2 shows that the creep strain increases non-
linearly with time as the deformation under creep continues. Moreover the creep rate is
not constant but increases with time. In Fig-3 the tangential creep rate 't is found to
decrease rapidly with increasing radius for all the three values of a . The significant
results of the present investigations are observed in Fig-4 and Fig-5. The figures show a
negligibly small difference between the two stress distributions based on the
assumptions of finite and infinitesimal strains for 0a i.e. at the outset of the creep
process (i.e. 0a ), as it should be. However, after a considerable deformation under
creep (i.e. for 6.0a ) the difference between the radial and tangential stresses and for
the effective stress for the two assumptions, is quite large. It is to be expected that as the
strain at the inner boundary increases from its initial value, the radial, tangential and
effective stresses all increase continuously and the difference in the predictions from the
two theories goes on increasing.
The Finite Element model results show that the values of von Mises creep strain rate
and von Mises creep stress match those predicted by the Bhatnagar and Arya’s solution.
But when analyzing beyond the creep strain ε=0.6, the ANSYS predicted von Mises
stresses are higher than the Exact Solution values. The analysis results are not
significantly higher, but from design point of view they are noticeable. So for better
accuracy a designer should consider a finite element analysis for worst scenario cases.
31
6. Conclusions
In this work, a comparison is made between the Theory by Bhatnagar and Arya and
Finite Element Analysis for the creep analysis of isotropic and homogeneous thick-
walled spherical pressure vessels. Results show that the creep rate of the thick-walled
spherical vessel increases rapidly even though the creep rate of the same material when
subjected to constant true stress in simple tension is constant. This is an important effect
which is overlooked in the analyses making use of the infinitesimal strain theory.
Therefore, the predictions based on the results from small or infinitesimal strain analyses
would be on unsafe side from a design point of view. The results from (1) have been
verified in ANSYS and good agreement has been found.
32
7. References
1. Bhatnagar, N.S. and Arya, V.K., Creep of Thick-walled spherical vessels under
internal pressure considering large strains, by. (Department of Mathematics,
University of Roorkee, Roorkee),1973.
2. MacGregor, C. W., Coffin, L. F., and Fisher, J. C. The plastic flow of thick-
walled tubes with large strains. J. appl. Phys., 19, 1948, p. 291.
3. Creep of engineering materials, Finnie, I., Heller, W.R., 1959. McGraw-Hill
Book Co., Inc., New York.
4. Johnson, A.E. and Khan, B., Creep of Metallic Thick-walled spherical vessels
subject to pressure and radial thermal gradient at elevated temperatures, Int. J.
Mech. Sci.,. Vol. 5, 1963, pp. 507-532.
5. Rimrott, F.P.J., Creep of thick-walled tubes under internal pressure considering
large strains, J.Appl.Mech. 26; Trans.ASME.Ser.E.81, 1959, p271.
6. You, L.H. and Ou, H., Steady-state creep analysis of thick-walled spherical
pressure vessel with varying creep properties, Journal of Pressure Vessel
Technology, ASME FEBRUARY 2008, Vol. 130. Issue 1, pp 014501-1-5.
7. Miller, G.K., Stresses in a spherical vessel undergoing creep and dimensional
changes, Int. J. Solids Structures Vol. 32, No. 14, 1995, pp. 2077-2093,
8. Arya, V.K., Debnath, K.K. and Bhatnagar, N.S., The spherical vessel with
anisotropic creep properties considering large strains, Int. J. Nonlinear
Mechanics, Vol. 15, 1980, pp. 185-193.
9. Nejad, M.Z., Hoseini, Z., Niknejad, A., and Ghannad, M., A New Analytical
Solution for Creep Stresses in Thick-walled Spherical Pressure Vessels, Journal
of Basic and Applied Scientific Research, 1 (11), 2011, p2162-2166,
10. Mechanical Behavior of Engineering Materials by Joseph Marin, Prentice Hall
Inc. 1962.
11. ANSYS Mechanical APDL Technical Manual
33
Appendix !* /PREP7 !* /NOPR KEYW,PR_SET,1 KEYW,PR_STRUC,1 KEYW,PR_THERM,0 KEYW,PR_FLUID,0 KEYW,PR_ELMAG,0 KEYW,MAGNOD,0 KEYW,MAGEDG,0 KEYW,MAGHFE,0 KEYW,MAGELC,0 KEYW,PR_MULTI,0 KEYW,PR_CFD,0 /GO !* /COM, /COM,Preferences for GUI filtering have been set to display: /COM, Structural !* !* ET,1,SOLID187 !* KEYOPT,1,6,0 !* MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,1,,5.5e6 MPDATA,PRXY,1,,0.3 TB,CREE,1,1,3,10 TBTEMP,0 TBDATA,,5e-30,6,0,,, SPHERE,100,20,0,180, /VIEW,1,,-1 /ANG,1 /REP,FAST !* LESIZE,ALL, , ,5, ,1, , ,1, MSHKEY,0 MSHAPE,1,3d CM,_Y,VOLU VSEL, , , , 1 CM,_Y1,VOLU CHKMSH,'VOLU' CMSEL,S,_Y !* VMESH,_Y1 !* CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* FLST,2,2,5,ORDE,2 FITEM,2,5
34
FITEM,2,-6 !* /GO DA,P51X,UY,0.0 FLST,2,2,5,ORDE,2 FITEM,2,3 FITEM,2,-4 /GO !* SFA,P51X,1,PRES,40000 TUNIF,723, /solu antype,static ! no inertia effects, so transient not needed ... nlgeom,on TUNIF,723, time,1.0e-5 deltim,1.0e-5,1.0e-8,1.0e-5 rate,off solve save time,10 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,20 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,30 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,40 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,50 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve
35
save time,100 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,200 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,300 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,400 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,500 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,600 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,700 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,800 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save
36
time,900 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,1000 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,1100 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,1200 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,1300 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,1400 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,1500 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,1600 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save
37
time,1700 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,1800 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,1900 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,2000 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,2100 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,2200 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,2300 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,2400 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save
38
time,2500 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,2600 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,2700 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,2800 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,2900 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save time,3000 deltim,2.0e2,1.0e-1,200 ! largest time step = 200 days ... rate,on,on nlgeom,on ! load drop matches cross section area decrease solve save fini