analysis of pressure vessel using ANSYS software

1. Design and Analysis of Thick Pressure Vessels
Sumit S. Dharmarao1
1
PG Student, Department of Mechanical Engineering, Bramhadevdada Mane Institute Of Technology, Solapur,
Maharashtra
sumitdharmarao@gmail.com
Abstract- Thick walled cylinders are widely used in
chemical, petroleum, military industries as well as in
nuclear power plants. They are usually subjected to high
pressures. Currently ASME codes or empirical relations
are used for design and analysis of pressure vessels. Often
use of ASME leads to conservative approach in design of
thick pressure vessels. This increases weight of pressure
vessel. In order to optimize the design and to carry out
accurate analysis of thick pressure vessel, it is important
to use FEA based commercial tools. This not only reduces
weight of the vessel but also makes design procedure
faster. FEA based approach in design of thick pressure
vessel is not popular in the process industry. The process
equipment industry is just beginning to realize the
potential value of FEA for improved product design and a
faster development cycle. The design and analysis of thick
pressure vessel offers a good example of how FEA works
in practice. Until recently the primary analysis method
had been hand calculations and empirical curves.
Computer advances have made finite element analysis
(FEA) a practical tool in the study of pressure vessels,
especially in determining stresses in local areas. Having
tested three dimensional, symmetric and axi-symmetric
models, the preliminary conclusion is that finite element
analysis is an extremely powerful tool when employed
correctly. The analyst should be able to approximate the
solution using classical methodology (hand calculations) in
order to verify the solution. The key feature of the
paper is to determine the stress variations in the thick
walled pressure vessels using empirical as well as FEA
based approach.
Index Terms— pressure vessel, process equipment, cad,
FEA.
INTRODUCTION
A pressure vessel is a closed container designed to hold
gases or liquids at a pressure substantially different
from the ambient pressure. In an early effort to design a
tank capable of withstanding pressures up to 10,000 psi
(69 MPa), a 6-inch (150 mm) diameter tank was
developed in 1919 that was spirally-wound with two
layers of high tensile strength steel wire to prevent
sidewall rupture, and the end caps longitudinally
reinforced with lengthwise high-tensile rods. The
pressure differential is dangerous and many fatal
accidents have occurred in the history of their
development and operation. Consequently, their design,
manufacture and operation are regulated by engineering
authorities backed up by laws. Pressure vessels are used
in a variety of applications in both industry and the
private sector. They appear in these sectors as industrial
compressed air receivers and domestic hot water
storage tanks. Other examples of pressure vessels are
diving cylinder, recompression chamber, distillation
towers, autoclaves, and many other vessels in mining or
oil refineries and petrochemical plants, nuclear reactor
vessel, habitat of a space ship, habitat of a submarine,
pneumatic reservoir, hydraulic reservoir under pressure,
rail vehicle airbrake reservoir, road vehicle airbrake
reservoir and storage vessels for liquefied gases such
as ammonia, chlorine, propane, butane, and LPG.
Based on wall thickness pressure vessels can be
classified as:
• Thin Walled Pressure Vessels
• Thick Walled Pressure Vessels
A thick walled pressure vessel is the one that its wall is
10 times (or more) thicker than inside radius. Thick
walled pressure elements working in high temperatures
in power stations, chemical and petro chemical
industries are subjected to damage as a result of
mechanical loading, high temperature and corrosive
environment. Due to internal and external pressure,
high stress and strain exist in thick walled pressure
vessel.
In a Thick Walled Pressure Vessel;
• Yield strength is a continuous function of
radius.
• Radial stress is present, in addition to hoop
stress and longitudinal stress.
• The thick walled pressure vessel requires very
have high tensile strength.
• No Buckling effects which arise in thin walled
pressure vessels are countered with Pressure
vessels having thick walls.
• Failure points can be foreseen in thick pressure
vessels. Nevertheless, the failure would be
along the longitudinal direction.
STRESSES IN THICK PRESSURE VESSEL
Thick-wall theory is developed from the theory of
elasticity which yields the state of stress as a continuous
function of radius over the pressure vessel wall. The

2. state of stress is defined relative to a convenient
cylindrical coordinate system.
σt -Tangential Stress
σr - Radial Stress
σl -Longitudinal Stress
Typically in the design of thick pressure vessels, the
following equations (or theories) are mostly used. Use
of these equations depends on the type of material used
and the end construction.
• Lame’s equation
• Clavarino’s equation
• Birnie’s equation
• Barlow’s equation
A thick walled pressure vessel which is used in nuclear
plant and manufactured by L&T is selected for present
study. Various input properties of the thick walled
pressure vessel are as follows:
Internal radius, ri = 430 mm
External radius, ro = 500 mm
Internal pressure, pi = 2.943 MPa
External pressure, po = 0 MPa
Length = 2000 mm
End caps = Hemispherical end cap
TABLE I: PROPERTIES OF STEEL
Modulus of elasticity 200 GPa
Poisson’s ratio 0.33
Density 7850 Kg/m3
I. LAME’S EQUATION FOR THICK WALLED
CYLINDERS:
In cases where the material of the cylinder is being
Brittle (e.g. Cast Iron or Cast Steel), Lame’s stress
equations are used. It is based on the Maximum
Principle Stress Theory of failure, where maximum
principle stress is equated to the permissible stress for
the material.
Tangential Stress acting in a cylinder wall at radius r,
subjected to pressure:
Radial Stress acting in the cylinder at radius r, subjected
to pressure:
Longitudinal Stress acting in the cylinder at radius r:
Where,
Internal Pressure - pi
External Pressure - po
Internal Radius - ri
External Radius - ro
II. CLAVARINO’S EQUATION
Tangential Stress acting in the cylinder at any given
radius r:
Radial Stress acting in the cylinder at any given radius
r:
And
Where,
Constants - a, b
Internal Diameter - di
External Diameter - do
Poisson ratio -
Tables show the analytical calculations using Lame’s
and Clavarino’s equations

3. TABLE II: ANALYTICAL CALCULATIONS USING LAME’S
EQUATION
Lame’s Equation
Radius
Radial
Stress
Tangential
Stress
Longitudinal
Stress
430.000 -2.9430 19.6607 8.3588
447.500 -2.0763 18.7940 8.3588
465.000 -1.3057 18.0234 8.3588
482.500 -0.6173 17.3350 8.3588
500.000 0.0000 16.7177 8.3588
TABLE III: ANALYTICAL CALCULATIONS USING
CLAVARINO’S EQUATION
Clavarino's Equation
Radius Radial Stress Tangential Stress
430.000 -4.0027 18.0359
447.500 -3.4393 16.9093
465.000 -2.9384 15.9074
482.500 -2.4910 15.0126
500.000 -2.0897 14.2100
Fig. shows radial stress, tangential stress and
longitudinal stress distribution along the thickness of
thick pressure vessel
Figure1: Radial stress distribution
Figure2: Tangential stress distribution
-4.5000
-4.0000
-3.5000
-3.0000
-2.5000
-2.0000
-1.5000
-1.0000
-0.5000
0.0000
400 450 500 550
Radialstress(MPa)
Radius(mm)
Lame's
equation
Clavarino's
equation
14.0
15.0
16.0
17.0
18.0
19.0
20.0
420 440 460 480 500 520
Tangentialstress(MPa)
Radius(mm)
Lame's
equation
Clavarino's
equation

4. Figure3: Longitudinal stress distribution
FE ANALYSIS USING ABAQUS
Pressure vessel analysis can be done using different
approaches as mentioned below:
• Axi-symmetric approach
• Analysis on a quarter section using cyclic
symmetry approach
• Analysis by drawing the complete vessel
In present study axi-symmetric approach is used. In this
analysis is carried out by drawing only a quarter section
of the entire vessel. This view gives a good
understanding of the application of loads on inside and
outside surfaces of the pressure vessel. Axi-symmetry
approach simplifies the model and also reduces the
computational time. This approach can be used if the
geometry is revolved about a particular axis. In Abaqus
axi-symmetry is used about Y-axis.
The thick walled pressure vessel data was analyzed
under different conditions. They are as follows
• Benchmarking using axi-symmetric analysis
without end caps
• Axi-symmetric analysis with hemispherical
end caps
• Axi-symmetric analysis of pressure vessel for
selected application
I. AXI-SYMMETRIC ANALYSIS OF PRESSURE
VESSEL FOR SELECTED APPLICATION
For this selected industrial data, FEA analysis is carried
out in Abaqus.
Axi-symmetry approach simplifies the model and also
reduces the computational time. This approach can be
used if the geometry is revolved about a particular axis.
In Abaqus axi-symmetry is used about Y-axis.
The boundary conditions are applied at two edges. The
first edge is the bottom edge, constrained to move only
along the radial direction and the other is at the end of
the hemispherical cap, constrained to move along the
longitudinal direction only. The details of the mesh are
as follows:
Mesh type: structured quad mesh
No. of nodes: 875
No. of Elements along Radial direction: 4
Element type: CAX4R – A four node axi-symmetric
quadrilateral element
Element shape: Rectangle element
(a) (b)
(c)
Fig 4: FE model of pressure vessel for selected application (a) Loads
and Boundary condition
(b) Meshed model (c) Zoomed view of mesh
0.000000
1.000000
2.000000
3.000000
4.000000
5.000000
6.000000
7.000000
8.000000
9.000000
420 440 460 480 500 520
Longitudinalstress(MPa)
Radius(mm)
Lame's
Equation

5. (a) (b)
(c) (d)
Fig 5: Thick pressure vessel shell with hemispherical end caps a)
Deformation b) Radial stress c) Tangential Stress d) Longitudinal
stress
(a)
(b)
(c)
Fig 6: Thick pressure vessel shell with hemispherical end caps
(Zoomed view) a) Radial stress (2D)
b) Tangential Stress (2D) c) Longitudinal stress (3D)
Tables show comparison of analytical and FE results.
From comparison it appears that axi-symmetric
approach produces fairly close results. Error in various
stress is very less when compared to theoretical values.
The error is small in longitudinal stress as well due to
considerable length of pressure vessel.
TABLE IV: FE RESULTS USING ABAQUS
FEA
Radius Radial
Stress
Tang.
stress
Long.stress
430.000 -2.50103 19.2187 8.3588
447.500 -2.09232 18.81 8.3588
465.000 -1.31938 18.0371 8.3588
482.500 -0.62916 17.3468 8.3588
500.000 -0.30317 17.0209 8.3588

6. CONCLUSIONS
In design of thick pressure vessels FEA tool can be
used effectively. Typically it helps the designer to
understand structural behavior of thick pressure vessel
in better fashion. Overall conclusions based on present
study are as below:
• Use of FEA tools in stress analysis of pressure
vessel has been demonstrated successfully.
• Axi-symmetric analysis approach simplifies
problem and reduces computational cost
drastically.
• From present study it appears that FE results are
in good agreement with Lame’s equation.
• Distribution and trend of stress within thick
pressure vessel along thickness is according to
theory.
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Edition
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CRC Press LLC, dated 2000
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Edition
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