3 optimum-design

© 2013. Md. Tanjin Amin, Aminur Rahman & Dr. Abu Rayhan Md. Ali. This is a research/review paper, distributed under the terms
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Global Journal of Researches in Engineering
Mechanical andMechanicsEngineering
Volume 13 Issue 11 Version 1.0 Year 2013
Type: Double Blind Peer Reviewed International Research Journal
Publisher: Global Journals Inc. (USA)
Online ISSN:2249-4596Print ISSN:0975-5861
Optimum Design of Compound Pressure Vessel
By Md. Tanjin Amin, Aminur Rahman & Dr. Abu Rayhan Md. Ali
University of Engineering & Technology (BUET), Bangladesh
Abstract- Effect of compounding investigated theoretically by finite element modeling. The
variables are interference radius & shrinkage tolerance. Equivalent von Mises stress is used as
yield criterion to evaluate the optimum interference radius. For two different working pressure the
effect of the ratio of outer & inner radius (b/a=k) value on the optimum interference radius is also
noticed. The optimum interference radius solely depends on shrinkage tolerance. Furthermore,
percentage reduction of von Mises tolerance is compared for different working pressures &
different k values as well as for different materials.
Keywords: autofrettage, residual stress, von mises stress, working pressure, pressure vessel,
elasto-plastic junction.
GJRE-A Classification : FOR Code:
OptimumDesignofCompoundPressureVessel
Strictly as per the compliance and regulations of:
091399p

Md. Tanjin Amin α
, Aminur Rahman σ
& Dr. Abu Rayhan Md. Ali ρ
GlobalJournalofResearchesinEngineering()A
© 2013 Global Journals Inc. (US)
Vol
23
Year0132umeXIIIIssueXIVersionI
Abstract- Effect of compounding investigated theoretically by
finite element modeling. The variables are interference radius
& shrinkage tolerance. Equivalent von Mises stress is used as
yield criterion to evaluate the optimum interference radius. For
two different working pressure the effect of the ratio of outer &
inner radius (b/a=k) value on the optimum interference radius
is also noticed. The optimum interference radius solely
depends on shrinkage tolerance. Furthermore, percentage
reduction of von Mises tolerance is compared for different
working pressures & different k values as well as for different
materials.
Keywords: autofrettage, residual stress, von mises stress,
working pressure, pressure vessel, elasto-plastic junction.
I. Introduction
ompound cylinders are widely used in the field of
high pressure technology such as hydraulic
presses, forging presses, power plants, gas
storages, chemical and nuclear plants, military
applications etc. To enhance load bearing capacity and
life of multilayer pressure vessels, different processes
such as shrink fit and autofrettage are usually employed.
Shrink fit increases load capacity. Many researchers
have focused on methods to extend lifetimes of vessels.
Majzoobi et al. have proposed the optimization of bi-
metal compound cylinders and minimized the weight of
compound cylinder for a specific pressure [1]. The
variables were shrinkage radius and shrinkage
tolerance. Patil S. A. has introduced optimum design of
two layer compound cylinder and optimized
intermediate, outer diameter and shrinkage tolerance to
get minimum volume of two layer compound cylinders
[2]-[3]. Hamid Jahed et al. have investigated the
optimum design of a three-layered vessel for maximum
fatigue life expectancy under the combined effects of
autofrettage and shrink fit [4]. Miraje Ayub A. & Patil
Sunil A. have found minimum volume of three-layer
open type compound cylinder considering plane stress
hypothesis [5]. Yang Qiu-Ming et al. have presented a
simple and visual tool to calculate the residual stress
and describe the distribution of residual stress for both
the elastic-perfectly plastic model and the strain-
hardening mode [6].
II. Analytical Approach
As stated earlier, an elasto-plastic behavior has
been designated in this work.
Figure 1 : Bi-linear Stress Strain Curve
The model, shown in fig. 1 is described as follows:
(1)
In which, σ is the effective stress, σy is the initial
yield stress, Ep
is the slope of the strain hardening
segment of stress strain curve, and ε is the effective
strain.
a) Tangential & Radial Stress Pattern
To observe the stress pattern in a compound
cylinder, a sample cylinder with internal diameter a =
0.1m, interference radius c = 0.15m, and external radius
b = 0.20m has been considered. Material Properties of
this cylinder summarized in table I
Table 1 : Material Properties
An outer cylinder with the internal diameter
slightly smaller than the outer diameter of the main
cylinder is heated and fitted onto the main cylinder.
When the assembly cools down to room temperature a
compound cylinder isobtained. In this process the main
cylinder is subjected to an external pressure leading to a
compressive radial stress at the interface. The outer
cylinder or the jacket is subjected to an internal pressure
leading to a tensile circumferential stress at the inner
wall. Under this condition as the internal pressure
increases the compression in the inner cylinder is first
released and then only the cylinder begins to act in
tension.
C
 P
y E
Material σy(MPa) E(GPa) υ
Steel 800 207 0.29
Author α: Department of Mechanical Engineering, Bangladesh
University of Engineering & Technology (BUET), Dhaka-1000,
Bangladesh. e-mail: moshia.nir@gmail.com
Author σ: Department of Mechanical Engineering, Bangladesh
University of Engineering & Technology (BUET), Dhaka-1000,
Bangladesh. e-mail: aminur.buet07@gmail.com
Author ρ: Professor, Department of Mechanical Engineering,
Bangladesh University of Engineering & Technology (BUET), Dhaka-
1000, Bangladesh. e-mail: armali@me.buet.ac.bd

Figure 2 : Radial & Tangential Stress Distribution
through the Walls of a Compound Cylinder
Though outer cylinder is fitted on the top of the
inner cylinder, a compressive stress is developed in the
near bore region. Here in the fig. 2, compressive stress
is developed in the inner diameter while tensile stress
occurs at the outer portion. Here the initial stress is
compressive so if internal pressure is applied, then a
tensile stress will be developed & this tensile stress even
quite high, but the resultant stress not that high because
of this compressive stress.
Comparison of Stress between Single cylinder &
Compound Cylinder
By using Lame’s equation for thick-walled
cylinder, the stress pattern is obtained for both single
cylinder & compound cylinder. If these two cylinders
undergoes same internal pressure then the overall
stress pattern will change, which is shown in Fig. 3, here
the internal pressure is 250 MPa.
Figure 3 : Comparison of Stresses between Single &
Compound Cylinder
Because of the compressive tangential stress at
inner bore, the resultant tangential stress becomes
lower in the compound cylinder than the original
tangential stress developed at the single cylinder. Radial
stress doesn’t vary significantly after compounding.
Radial stress doesn’t vary significantly after
compounding. Maximum stress occurs at the junction
point of inner & outer cylinder rather than inner bore.
c) Comparison of Overall von Mises Stresses
According to the third strength theory, the
equivalent stress is as follows [12]-[13]:
(2)
Figure 4 : Comparison of von Mises stresses between
Single & Compound cylinder according to the third
strength theory
The equivalent stress based on the fourth
strength theory is as follows [12[-[13]:
(3)
GlobalJournalofResearchesinEngineeringVol
24
()AYear0132
rt
III
eq  
      222
2
1
tzzrrt
IV
eq  
Substituting (Lame formula)
Eq.(2), we obtain:
2/)( rtz  
umeXIIIIssueXIVersionI
b)
(4)
III
eqrt
IV
eq 
2
3
)(
2
3

Figure 5 : Comparison of von Mises stresses between
Single & Compound cylinder according to the third
strength theory

Eq. (3) s the relation between the third & the
fourth strength theory for a cylinder and also the relation
between the Tresca’s & Mises yield form.
As the fig. 4 & 5 suggest, maximum von Mises
stress starts to decreases from inner bore to junction of
two cylinders. Stress suddenly increases because of the
contact pressure between two mating surface. It again
decreases from junction point to outer portion.
i. Sample calculation
• In this case study, a=0.1m, b=0.2m, c=.15m,
working pressure, Pworking = 250MPa.
• From third strength theory, MVS = 666 MPa
• From fourth strength theory, MVS =577 MPa
• From these two theories, it is observed that MVS for
both is very similar. Indeed there is no significant
variation between these two methods.
• Fourth strength theory is considered for all
comparison.
III. Numerical Result
The compounding process may be simulated
by Finite Element Method, making use of elastic-plastic
analysis. It is possible to model the compound cylinder
by applying pressure to the inner surface of the model,
and then analyzing it for different interference radius &
different shrinkage tolerance.
Using a 3D axisymmetric element available in
ANSYS V11 SP1, a finite element mesh of a cylinder with
an inside radius 100 mm and outside radius of 200 mm
was generated. An internal pressure of 250 MPa was
applied. The stress distributions were evaluated in the
compound cylinder. The von Mises equivalent stress
was used in the subsequent analysis.
Structural Steel of flexible stiffness behavior has
been used. Compound cylinder with the dimension; a =
0.1 m, b = 0.2 m, an elastic plastic material’s model
with σ = 800 MPa; Modulus of elasticity E = 207 GPa; ν
= 0.29; were used for numerical modeling.
Here, the effects of following factors are
considered in the compounding process. The
considered factors are:
• Interference radius; 2. Value of k (b/a); 3. Different
material at inner cylinder.
a) Change of Interference Radius
The cylinders were subjected to different
internal pressures of 100, 200 and 300 MPa. From the
numerical simulation with ANSYS software, the curve of
the von Mises stress distribution was obtained for each
working pressure. From the curve, the values of
maximum von Mises stresses (MVS) were extracted.
This stresses were then plotted against interference
radius for each working pressure. The results are shown
in fig. 6.
Figure 6 : Variation of MVS verses interference radius at
different working pressure
It is observed that with the increase of
interference radius, the MVS decreases slightly.
Because at small interference radius, lower contact
pressure is developed at the junction of the two
cylinders. When the radius increases then the contact
pressure increases and there is much compressive
stress at the inner bore region resulting lower MVS. With
the increase of interference radius much stress is
developed at the outer cylinder; resulting a higher
overall stress distribution.
The value of MVS is calculated for different
interference radius for different working pressures.
Table 2 : Von Mises Stresses at Different Interference
Radius for Different Working Pressure
From the table II, it is observed that the
minimum values of MVS are at 280 mm & 300 mm. This
means in those radius tangential stresses at inner wall of
inner cylinder and inner wall of outer cylinder almost
same.
Vol
25
Year0132
Interference
Radius (mm)
Pressure
100 (MPa)
Pressure
200 (MPa)
Pressure
300 (MPa)
240 203 386 576
260 190 379 569
280 191 366 558
300 184 369 554
320 193 375 568
340 195 390 604
For a constant value of interference radius (280
mm) the percent reduction of MVS is calculated for
different working pressure.
Table 3 : Effect of Working Pressure at Maximum Von
Mises Stress
Working
Pressure
(MPa)
MVS of
single
cylinder
(MPa)
MVS of
compound
cylinder
(MPa)
% reduction
of MVS
100 233 191 18.03
200 467 366 21.63
300 701 558 20.40

From the table III, it is observed that the percent
reduction of MVS is higher at moderate & higher working
pressures.
b) Change of Shrinkage Tolerance
In the preceding case minimum MVS was found
in between 280 mm & 300 mm of interference radius.
Now the cylinder is subjected to a working pressure 200
MPa and interference radius is considered as 280 mm.
From the curve, the values of maximum von Mises
stress (MVS) are extracted. This stresses are then
plotted against the shrinkage tolerance. The results are
shown in fig. 8.
Figure 8 : Variation of MVS versus different shrinkage
tolerance
It is observed that the developed MVS starts to
decrease with the increase of the shrinkage radius. With
the increase of the shrinkage radius, the compressive
stress in the inner bore region increases & so that MVS
decreases. But after a certain point with the increase of
the shrinkage radius, MVS increases because then the
stress of the inner bore of the outer cylinder increases.
And also there is a close relation between the numerical
result & theoretical result shown in the fig. 8.
c) Values of k
• Constant inner diameter
For constant inner radius (a = 0.1m) of the
inner cylinder & constant interference radius, c = 280
mm cylinder with different k values (k=2, 2.5, 3) are
subjected to same working pressure.
Here the working pressure remains constant at
Pworking = 200 MPa
From numerical simulation, the value & the
position of MVS are extracted. This MVS is then plotted
against the diametral interference for each k value. The
results are shown in fig. 9
Figure 9 : Effect of k (b/a) on the diametral interference
(constant inner radius = 0.1m)
As in fig. 9, suggests for different values of k,
the minimum MVS obtains a higher value of thicker
cylinder at lower diametral interference.
For a constant working pressure (Pworking =
200 MPa), percent reduction of MVS is calculated for
different k values.
Table 4 : Effect Of K = (B/A) On Mvs At Constant Inner
Radius
From table IV, it is observed that, percent
reduction of MVS is higher for higher value of k. So, the
process is more beneficial with the increase of the
thickness of the cylinder.
26
()AYear0132
k
MVS of single
cylinder
(MPa)
MVS of
compound cylinder
(MPa)
% reduction of
MVS
2.0 467 366 21.63
2.5 414 315 23.91
3.0 392 286 27.04
• Constant outer diameter
In the preceding case the inner radius of the
inner cylinder was kept constant. Now by assuming the
outer radius (b= 0.2m) constant, cylinder with three k
values of 2.0, 2.5, 3.0m are considered. From the
numerical simulation, the curve of MVS distribution is
obtained for each value. The developed MVS are then
plotted against diametreal interference at fig. 10.
Figure 10 : Effect of k (b/a) on the diametral interference
(constant outer radius = 0.2m)

As in fig. 10, suggest for different values of k,
the minimum MVS obtains a higher value of thicker
cylinder at higher diametral interference.
For a constant working pressure (Pworking =
200 MPa), percent reduction of MVS is calculated for
different k values.
Table 5 : Effect Of K = (B/A) On Mvs at Constant Outer
Radius
From the table V, it is noticed that for same k
value, developed MVS is almost same though the value
of inner & outer radius have been changed. So the
optimum diametral interference depends on k value
only. If the inner & outer radius are changed keeping the
k value constant, then there will be no such change in
diametral interference.
From above two graphs it can be said that,
constant inner radius of inner cylinder is more effective
than other one. Because in inner radius constant
process the minimum von Mises stress found in lower
diametral interference.
d) Different Material as inner cylinder
Now inner cylinder of the compound cylinder is
replaced by cast iron cylinder and copper cylinder
respectively. Here the working pressure remains
constant at Pworking = 200 MPa From the numerical
simulations, the curve of von Mises stress distribution is
obtained for different material model. From the graph,
the value & the position of the maximum von Mises
stress are extracted and plotted against diametral
interference.
Figure 11 : Effect of different material on diametral
interference
From the fig. 11, it is observed that cast iron &
copper alloy shows minimum von mises stress at small
diametral interference. This means if material of low
tensile stress is compounded by a high tensile stress
material, then at lower diamteral interference they show
the desired effect.
Here cast iron & copper alloy are separately
shown in fig. 12.
Figure 12 : Effect of different material on diametral
interference (Cast iron & Copper alloy)
Vol
27
Year0132
k
MVS of
single cylinder
(MPa)
MVS of
compound cylinder
(MPa)
% reduction of
MVS
2.0 467 366 21.63
2.5 415 313 24.57
3.0 392 289 26.28
From fig. 12, it is noticed that, these two metals
jacketed by steel work better at small diametral
interference.
IV. Conclusion
The following decisions can be taken from the
investigations mentioned in this paper:
1. In compound cylinder, maximum von Mises stress
does not occur at smaller interference radius
instead of that, it occurs at midway section.
2. Maximum stress depends on the interference radius
as well as diametral interference.
3. The optimum diametral interference depends on k
value only rather than inner & outer radius.
4. The thicker the cylinder the more will be the
reduction in MVS.
5. Two element cylinder of different material reduces
diametral interference more than two element
cylinder of same material.

6. The size of a compound cylinder could be reduced
by significant amount with respect to its equivalent
single cylinder for the same working pressure.
7. The difference between a compound cylinder and
its equivalent single cylinder becomes more
significant at higher working pressures.
Appendix
Table : Nomenclature
V. Acknowledgment
The authors deeply acknowledge the financial &
technical support of the Department of Mechanical
Engineering, Bangladesh University of Engineering &
Technology, Dhaka, Bangladesh.
References Références Referencias
1. Majzoobi G.H. & Ghomi A., (2006) “Optimization of
compound pressure cylinders”, Journal of
Achievements in Materials and Manufacturing
Engineering, Vol. 15, Issue 1-2 March-April.
2. Patil Sunil A., (2005) “Optimum Design of
compound cylinder used for storing pressurized
fluid”, ASME International Mechanical Engineering
Congress and Exposition (Proceeding of IMECE05),
Nov 5-11,2005,Orlando, Florida USA.
3. Patil Sunil A., (2011) “Finite Element Analysis of
optimized compound cylinder”, Journal of
Mechanical Engineering Research, Vol. 3(1), Issue
March.
4. Jahed Hamid, Farshi Behrooz & Karimi Morvarid,
(2006) “Optimum Autofrettage & Shrink-Fit
Combination in Multi-Layer Cylinders”, Journal of
Pressure Vessel Technology, Transactions of the
ASME, pp. 196-200 , Vol. 128, MAY 2006.
5. Miraje Ayub A., Patil Sunil A.,(2011) “Minimization of
material volume of three layer compound cylinder
having same materials subjected to internal
pressure”, International Journal of Engineering,
Science and Technology , Vol. 3, No. 8, 2011, pp.
26-40.
6. Yang Qiu-Ming, Lee Young-Shin, Lee Eun-Yup, Kim
Jae-Hoon, Cha Ki-Up and Hong Suk-Kyun, (2009)
“A residual stress analysis program using a Matlab
GUI on an autofrettaged compound cylinder”,
Journal of Mechanical Science and Technology ,
Vol. 23, pp. 2913-2920.
7. A. Ghomi, Optimum Design of thick-walled pressure
cylinders (in Persian), MS.c final project, Bu-Ali SIna
University, Hammadan, Iran, 2005.
8. A. R. Ragab, S. E. Bayomi, Engineering Solid
Mechanics, Fundamentals and Applications. Boca
Raton CRC press, 1998.
9. Garret N. Vanderplaats, Numerical Optimization
Techniques for Engineering Design with
Applications, McGraw-Hill, New York, 1989.
10. Edgard Himmelblan, Optimization of Chemical
Processes, McGraw-Hill, 1989.
11. G. H. Majzoobi, G. H. Farrahi, M. K. Pipelzadeh and
A. Akbari, Experimental and Finite Element
Prediction of Bursting Pressure in Compound
Cylinders, International Journal of Pressure Vessels
and piping , vol. 81, pp 889-896, 2004.
12. G.J. Franklin, JLM. Morrison, Autofrettage of
cylinders: prediction of pressure, external expansion
curves and calculation of residual stresses
Proceeding of institute of Mechanical Engineers, vol.
174, pp. 947-74, 1960.
13. J.F. Harvey, Theory and design of pressure vessels,
New York: Van Nostrand Reinhold Company Ltd.,
1985.
28
()AYear0132
Symbol Meaning unit
a Inner radius (m)
b Outer radius (m)
c Interference radius (m)
σy Yield stress (MPa)
σθ Tangential stress (MPa)
σr Radial stress (MPa)
σz Longitudinal stress (MPa)

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