Dharmendra Kumar, Lalit Kumar Sahu

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Elastic Analysis of Thick Walled Pressure Vessel Consisting of
Functionally Graded Material Under Uniformly Varying Pressure
Load
Dharmendra Kumar *, Lalit Kumar Sahu**
*(Department of Mechanical Engineering), Christian College of Engg and Tech, Bhilai, Chhattisgarh, India
Email: kdks124@gmail.com
**(Department of Mechanical Engineering), Christian College of Engg and Tech, Bhilai, Chhattisgarh, India
Email: lalitmech08@gmail.com
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Abstract:
Functionally graded Materials (or also can say, Functionally Gradient materials) are characterized as an
anisotropic material whose physical properties varies continuously as the dimensions varies randomly or
strategically, to achieve the desired characteristic.Material modelling, geometric modelling and finite
element modelling is done for the truncated cone pressure vessel and then numerical problem is solved
using the finite element software ANSYS 18.1.The study is mainly focused on comprehensive overview of
the uniformly varying pressure loading, angular velocity and different material combination for
manufacturing of functionally graded materials; choosing the best material and range of uniformly varying
pressure and angular velocity for formulation of FGMs in this field are presented.
Keywords —Pressure vessel, functionally graded material, ANSYS 18.1, finite element method.
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I. INTRODUCTION
Functionally graded materials are a class of
advanced composites formed of two or more
constituent phases with a gradual and continuously
variable chemical composition, microstructure and
material properties. They were initially introduced
by a group of Japanese scientists for the purpose of
addressing the needs of aggressive environment of
thermal shock in the space shuttle in 1984 [1]. Naki
Tutuncu 2007 [5] obtained solutions for stresses
and displacements in functionally-graded
cylindrical vessels subjected to internal pressure
alone are obtained using the infinitesimal theory of
elasticity. The material is assumed to be isotropic
with constant Poisson’s ratio and exponentially-
varying elastic modulus through the thickness.Zewu
Wang et al. [21] in this paper investigation of the
optimization design of the material distribution
properties for an FGM hollow vessel subjected to
internal pressure. By constructing an exponential
function and determining the material properties,
the general analytical solution of the stresses of the
FGM pressure vessel was given based on the Euler-
Cauchy formula. And then, an optimization model
for obtaining the optimal material distribution of
FGM vessel was proposed coupling the general
finite element (FE) code.M. Z. Nejad et. al. [17]
Stresses and the displacements in thick-walled
spherical shells made of functionally graded
materials subjected to internal shells made of
functionally graded materials subjected to internal
and external pressure are developed by an analytical
method. The mechanical properties, except
Poisson’s ratio, are assumed to obey the
exponential variations throughout the thickness.
The displacement and stresses distributions were
RESEARCH ARTICLE OPEN ACCESS

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compared with the solutions of the finite element
method (FEM) and good agreement was found.
II. PROBLEM FORMULATION
In pressure vessels reduction of weight is an
important criteria retaining or even enhancing its
stress bearing capacity. As Exponential law is
selected in the base paper therefore to solve the
problem through APDL - ANSYS Parametric
Design Language ANSYS 18.1 exponential law is
chosen and considered the case of rotating truncated
cone pressure vessel with uniformly varying
pressure loading, angular velocity and different
material combination. Analysis for stresses and
deformation will be carried out for FGM truncated
conical shell with axially varying properties used
for pressure vessels.
In an FGM the variation of material properties is
usually attained by adjusting the volumefractions of
two or more compatible constituents, and the
material used is in the base paper assumed to
befunctionally graded in the radial direction. Thus,
variations in the material properties such as
Young’smodulus and Poisson’s ratio may be
arbitrary functions of the radial coordinate. In the
base paper Young’s modulus with exponentially-
varying propertiesand constant Poisson’s ratio is
considered as the variation of Poisson’s ratio ν is
small forengineering materials
Modulus of elasticity of FGM pressure vessel
follow the exponential law
𝐸( 𝑟) = 𝐸 𝑒 ( )
Where r = r´/R,r´is the radial radius, r is the
normalized radial radius, and α is the elastic
gradedfactor of the FGM. E is the modulus of
elasticity, and Eois the modulus of elasticity of the
external wall.
A. Validation
The dimensions of the pressure vessel as taken
in base paper are Ri = 74 mm, Ro = 80.082
mm, height = 400 mm and that of a spherical
head are (Ri = 74 mm, Ro = 77.542 mm). Two
primary materials are Al2O3 alumina for
ceramic and stainless steel for metal,
respectively. And Eo=200 GPa, ν = 0.3, and
exponent factor is α = 1.5. The internal
pressure is 12 MPa. The complete problem of
the base paper was solved in ANSYS APDL,
in this project also ANSYS APDL 18.1 is used.
Geometry model was converted into FE model
by the discretization method that changes
continuous material variation into stepwise
variation.
Fig 1 Radial stress of cylindrical part of the FGM pressure vessel
Fig 2: Radial stress of spherical part of the FGM pressure vessel
III. RESULTS
The aim of this project is to make a comparative
study of the radial displacements, radial stresses,
circumferential stresses and von mises stresses
developed in atruncated cone pressure vessel made
of functionally graded material of different internal

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pressure, angular velocity and material subjected to
beta angle β = 10°.
Fig 3 Geometry modelling of half of the FGM pressure vessel with Beta 10°
Elastic analysis of rotating thick truncated conical
shells made of functionally graded materials with
beta angle 10° is carried out for four different cases
with constant internal pressure of 12 MPa,
uniformly varying pressure of 4-12 MPa, 4-24MPa,
and 4-36MPa.
Fig 4 Comparison of radial displacement for varying internal pressure of
truncated cone pressure vessel
Fig. 4 show the variation of radial displacement
along the axial direction for different internal
pressure of angle of beta 10°. The value of radial
displacement is nearly equal to zero at the top and
reaches maximum at the bottom, which confirms
the constrained boundary condition applied on the
model.From the comparison of radial displacements
for variable internal pressure, it is observed that
variable internal pressure has a considerable effect
on the radial displacement. As the internal pressure
increases radial displacement also increases.
Fig. 5 show the variation of the radial stress along
the axis for different internal pressure. The value of
radial stress doesn’t show a steep curve for the two
cases i.e. 12MPa constant and 4-12MPa but as the
difference in pressure increases radial stress is
maximum at the bottom and minimum at the top for
cases 4-24MPa and 4-36MPa ofinternal pressure,
which is due to fix-free boundary condition applied
on the truncated cone pressure vessel.The value of
maximum radial stress is for uniformly varying 4-
36 MPa.
Fig. 6 show the variation of the circumferential
stress along the axial direction for different internal
pressure. Circumferential stress induced in the
truncated cone pressure vesselare tensile and
compressive both in nature. Magnitude of the
compressive stress is higher than the tensile
stress.The value of maximum stress is for uniformly
varying 4-36 MPa.
Fig 5 Comparison of radial stress for varying internal pressure of truncated
cone pressure vessel
-5.00E-02
0.00E+00
5.00E-02
1.00E-01
1.50E-01
2.00E-01
0
40.155
80.31
120.47
160.62
200.78
240.93
281.09
321.24
361.4
401.55
Radialdeformation
Axial profile line
UX 4-24 MPa UX 4-12 MPa
UX 4-36 MPa UX 12 MPa const
-25
-20
-15
-10
-5
0
0 100 200 300 400 500
Radialstress
Axial profile line
SX 4-24 MPa SX 4-12 MPa
SX 4-36 MPa SX 12 MPa const

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Fig 6 Comparison of circumferential stress for varying internal pressure of
truncated cone pressure vessel
Fig. 7 show the variation of the von mises stress
along the axial direction of the shell for different
internal pressure. Von mises stress induced in the
disks is completely tensile in nature. It is maximum
at the bottom for internal pressure as pressure is
gradually increased from top to bottom. Maximum
stress is noticed for uniformly varying 4-36 MPa.
Fig 7 Comparison of von mises stress for varying internal pressure of
truncated cone pressure vessel
Fig 8 Comparison of displacement vector sum of 3 different material for
truncated cone pressure vessel
In this section different metals like ASTM A516,
ASTM SB2092014T6 commonly known as 2014
T6 alloy, ASTM B367Gr6 commonly known as Ti-
5Al-2.5Sn used for fabrication of pressure vessel is
studied under FGM. To avoid the complexity of
analyzing all the graphs only displacement is
considered in this case also strain is a dependent
variable on material therefore, by varying different
metal deformation or strain will change. A
comparative analysis will help to choose the best
material as per application from low pressure vessel
to high pressure vessel and also economical.Fig.
8shows the variation of the displacement vector
sumalong the axial direction for different internal
pressure. These are three commonly used material
for pressure vessel and it is found that ASTM A
516 is found to have a minimum deformation than
Ti 5Al 2.5 Sn followed by 2014 T6 Al alloy i.e. the
later ones can be used for low pressure vessels.
Elastic analysis of rotating thick truncated conical
shells made of functionally graded materials with
beta angle 10° and constant internal pressure of 12
MPa is carried out for three different cases of
angular velocity of 1rad/sec, 2 rad/sec and 5 rad/sec.
Fig. 9 show the variation of the displacement vector
sumalong the radius for different internal pressure.
Three angular velocity used are 1 rad/sec, 2 rad/sec,
5 rad/sec for pressure vessel and it is found that all
the three angular velocities are exhibiting nearly
-50
0
50
100
150
200
250
300
350
400
450
500
Circumferentialstress
Axial profile line
SZ 4-24 MPa SZ 4-12 MPa
SZ 4-36 MPa SZ 12 MPa const
0
100
200
300
400
500
0 100 200 300 400 500
Vonmisesstress
Axial profile line
SEQV 4-24 MPa SEQV 4-12 MPa
SEQV 4-36 MPa SEQV 12 MPa const
0.00E+00
5.00E-02
1.00E-01
1.50E-01
2.00E-01
2.50E-01
0 2 4 6 8
USUM ASTM A516
USUM 2014 T6 Al alloy
USUM Ti 5al 2.5Sn

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same values thus angular velocity has very
negligible effect and can be used with all pressure
vessels.
Fig. 10 show the variation of the von mises stress
along the radius for different internal pressure. Von
mises stress induced in the disks is completely
tensile in nature. It is found that all the three
angular velocities are exhibiting nearly same values
thus angular velocity has very negligible effect can
be used with all pressure vessels.
Fig 9 Comparison of displacement vector sum of 3 different angular velocity
for truncated cone pressure vessel
Fig 10 Comparison of von mises stress of 3 different angular velocity
for truncated cone pressure vesselexample of a book in a series in [2]
IV. CONCLUSIONS
The results obtained may be concluded as:
1. As the internal pressure increases radial
stress, circumferential stress, von mises stress and
radial displacement also increases.
2. As the angular velocity increases
circumferential stress, von mises stress and radial
displacement remains same therefore angular
velocity has very negligible effect in a pressure
vessel.
3. Three different pressure vessel grade
material were analyzed and within the given
boundary conditions ASTM A516 is found to be
capable of withstanding the maximum pressure.
REFERENCES
[1] Ali Kattan, Reem A. Alrawi, “Pressure Vessel Design Using the
Dynamic Self-adaptive Harmony Search Algorithm SDIWC ISBN:
978-0-9891305-4-7 2014 pp. 11-16.
[2] M. Jabbari, A. Bahtui, M.R. Eslami, “Axisymmetric mechanical and
thermal stresses in thick short length FGM cylinders,” International
Journal of Pressure Vessels and Piping 86 2009 pp. 296–306.
[3] Abbas Vafaeesefat, “Optimization of composite pressure vessels with
metal liner by adaptive response surface method” Journal of
Mechanical Science and Technology 2011, pp. 2811-2816.
[4] Mukund Kavekar, Vinayak H.Khatawate, Gajendra V. Patil, “Weight
reduction of pressure vessel using FRP composite material”
International Journal of Mechanical Engineering and Technology
(IJMET), Volume 4, Issue 4, ISSN 0976-6359, pp. 300-310.
[5] Naki Tutuncu, “Stresses In thick-walled FGM cylinders with
exponentially-varying properties” Engineering Structures 29 2007 pp.
2032–2035.
[6] Hareram Lohar, Susenjit Sarkar, Samar Chandra Mondal, “Stress
analysis and burst pressure determination of two layer compound
pressure vessel” International Journal of Engineering Science and
Technology. ISSN: 0975-5462, Vol. 5 No.02 February 2013 pp. 349-
353.
[7] Deepak Thomas, Colin Johnson, Sarath Krishna,N.Sivaselva Kumar,
“Optimization Of Pressure Vessel Using A Composite Material”
International Journal of Advancements in Research & Technology,
Volume 3, Issue 3, March-2014 ISSN 2278-7763 pp. 91-96.
[8] K. Sahitya Raju, Dr. S. Srinivas Rao, “Design Optimisation Of A
Composite Cylindrical Pressure Vessel Using FEA” International
Journal of Scientific and Research Publications, Volume 5, Issue 12,
December 2015, ISSN 2250-3153 pp. 522-530.
[9] Shyam R Gupta, Ashish Desai, “Optimize nozzle location for
minimization of stress in pressure vessel” International Journal for
Innovative Research in Science and Technology, Volume ,1 Issue 1,
June 2014, ISSN 2349-6010.
[10] Shildip D. Urade, Dr. D. V. Bhope, Prof. S. D. Khamankar, “Stress
analysis of multilayer pressure vessel” International Journal of
Engineering and Technical Research ISSN: 2321-0869, Volume-2,
Issue-9, September 2014.
[11] Ms.T. R. Mangalware, Prof A. P. Shrotri, Mr. S. B. Khandagale,
“Design Optimization and Performance Analysis of Hybrid Composite
Overwrapped Pressure vessel: A Review”, International Journal of
7.50E-02
7.55E-02
7.60E-02
7.65E-02
7.70E-02
7.75E-02
7.80E-02
7.85E-02
0 2 4 6 8
Displacementvectorsum
Radial thickness
USUM 1 rad/sec USUM 2 rad/sec
USUM 5 rad/sec
150
155
160
165
170
175
0 2 4 6 8
VonmisesStress
Radial thickness
SEQV 1 rad/sec SEQV 2 rad/sec
SEQV 5 rad/sec

6. International Journal of Scientific Research and Engineering Development-– Volume 3 Issue 4, July-Aug 2020
Available at www.ijsred.com
ISSN : 2581-7175 ©IJSRED: All Rights are Reserved Page 135
Engineering Science and Management, Volume 2, Number 1, 2016pp.
64-66.
[12] Jignesh Dipakkumar Jani , Manoj Kumar Gupta , R. R. Trivedi, “Shape
Optimization of Externally Pressurized Thin-Walled End Dome
Closures with Constant Wall Thickness of Cylinder Vessel with Help
of Bezier Curve” International Journal of Engineering Science and
Innovative Technology, Volume 2, Issue 3, May 2013, ISSN: 2319-
5967 pp. 466-475.
[13] Nitant M. Tandel, Jigneshkumar M. Parmar, “A Review on Pressure
Vessel Design and Analysis” Volume 3, Issue 4, May 2013, ISSN -
2250-1991 pp. 119-121.
[14] Mutahir Ahmed, Rafi Ullah Khan and Saeed Badshah, Sakhi Jan,
“Finite Element Investigation of Geometry Effect on Pressure Vessel
under Combined Structural and Thermal Loads” International Journal
of Engineering and Advanced Technology Volume-4 Issue-2,
December 2014 ISSN: 2249 – 8958, pp. 118-124.
[15] H.L. Dai, Y.M. Fu, Z.M. Dong, “Exact solutions for functionally
graded pressure vessels in a uniform magnetic field” International
Journal of Solids and Structures, 2005, pp. 5570-5580.
[16] M. Shariyat, “Dynamic thermal buckling of suddenly heated
temperature-dependent FGM cylindrical shells, under combined axial
compression and external pressure” International Journal of Solids and
Structures, 2008, pp. 2598–2612.
[17] Mohammad Zamani Nejad, Majid Abedi, Mohammad Hassan Lotfian
and Mehdi Ghanna, “An exact solution for stresses and displacements
of pressurized FGM thick-walled spherical shells with exponential-
varying properties” Journal of Mechanical Science and Technology,
2012, pp. 4081-4087.
[18] A.M. Afsar, H. Sekine, “Optimum material distributions for prescribed
apparent fracture toughness in thick walled FGM circular pipes”
International Journal of Pressure vessels and Piping 2001, pp. 471-484.
[19] Tolga Akis, “Elastoplastic analysis of functionally graded spherical
pressure vessels” Computational Materials Science 2009 pp. 545–554.
[20] Y.Z. Chen *, X.Y. Lin, “Elastic analysis for thick cylinders and
spherical pressure vessels made of functionally graded materials”
Computational Materials Science, 2008, pp. 581–587.
[21] Zewu Wang, Shujuan Gao, Qian Zhang, Peiqi Liu, and Xiaolong Jiang,
“Distribution parameter optimization of an FGM pressure vessel” vol.
320 2011 pp. 404-409.
[22] A.R Ghasemi, A. Kazemian, M. Moradi, “Analytical and Numerical
Investigation of FGM Pressure Vessel Reinforced by Laminated
Composite Materials” Journal of Solid Mechanics Vol. 6, No. 1 2014
pp. 43-53.
[23] M. Zamani Nejad and G. H. Rahimi, “Deformations and stresses in
rotating FGM pressurized thick hollow cylinder under thermal load”
Scientific Research and Essay Vol.4 (3), March, 2009, ISSN 1992-
2248, pp. 131-140.
[24] M. Arefi and G. H. Rahimi, “Thermo elastic analysis of a functionally
graded cylinder under internal pressure using first order shear
deformation theory” Scientific Research and Essays Vol. 5(12), June,
2010, pp. 1442-1454.
[25] M.R. Eslami*, M.H. Babaei, R. Poultangari, “Thermal and mechanical
stresses in a functionally graded thick sphere” International Journal of
Pressure Vessels and Piping 2005 pp. 522–527.
[26] M. Jabbaria, S. Sohrabpourb, M.R. Eslami, “Mechanical and thermal
stresses in a functionally graded hollow cylinder due to radially
symmetric loads” International Journal of Pressure Vessels and Piping,
2002, pp. 493–497.
[27] M. Jabbari, A. H. Mohazzab, A. Bahtui, “One-Dimensional Moving
Heat Source in a Hollow FGM Cylinder”, Journal of Pressure Vessel
Technology 2009, Vol. 131 pp.021202-1 - 021202-7.
[28] A.T. Kalali, S. Hadidi-Moud, “A Semi-analytical Approach to Elastic-
plastic Stress Analysis of FGM Pressure Vessels” Journal of Solid
Mechanics Vol. 5, No. 1 (2013) pp. 63-73.
[29] X.-F. Li,X.-L. Peng, and Y.-A. Kang, “Pressurized Hollow Spherical
Vessels with Arbitrary Radial Nonhomogeneity” Vol. 47, No. 9,
September 2009, pp. 2262-2265.
[30] Peng-fei LIU, Ping XU, Shu-xin HAN, Jin-yang ZHENG, “Optimal
design of pressure vessel using an improved genetic algorithm” Journal
of Zhejiang University SCIENCE A ISSN 1862-1775, 2008 pp. 1264-
1269.
[31] Zeinab Mazarei, Mohammad Zamani Nejad and Amin Hadi, Thermo-
Elasto-Plastic Analysis of Thick-Walled Spherical Pressure Vessels
Made of Functionally Graded Materials, International Journal of
Applied Mechanics Vol. 8, No. 4 (2016) 1650054-1 1650054-25.
[32] Mohammad Rahim Nami, Hadi Eskandari, “Three-dimensional
investigations of stress intensity factors in a thermo-mechanically
loaded cracked FGM hollow cylinder” International Journal of
Pressure Vessels and Piping 89 (2012) 222-229.
[33] Mohammad Zamani Nejad a & Gholam Hosein Rahimi, “Elastic
analysis of FGM rotating cylindrical pressure vessels” Journal of the
Chinese Institute of Engineers Vol. 33, No. 4, (2010), pp. 525-530.
[34] Mohammad Zamani Nejad, Mehdi Jabbari, Mehdi Ghannad, “Elastic
analysis of FGM rotating thick truncated conical shells with axially
varying properties under non-uniform pressure loading” Composite
Structures 2014.
[35] Mohammad Zamani Nejad, Parisa Fatehi, “Exact elasto-plastic analysis
of rotating thick-walled cylindrical pressure vessels made of
functionally graded materials” International Journal of Engineering
Science, 2015 pp. 26–43.
[36] X.L. Peng, X.F. Li, “Thermoelastic analysis of a cylindrical vessel of
functionally graded materials” International Journal of Pressure
Vessels and Piping, 2010 pp. 203-210.
[37] Mojtaba Sadeghian , Hamid Ekhteraei Toussi, “Axisymmetric yielding
of functionally graded spherical vessel under thermo-mechanical
loading” Computational Materials Science, 2011, pp. 975–981.
[38] Hui-Shen Shen, Jie Yang, Sritawat Kitipornchai, “Post buckling of
internal pressure loaded FGM cylindrical shells surrounded by an
elastic medium” European Journal of Mechanics A/Solids 29 2010 pp.
448–460.
[39] L.H. You, J.J. Zhang, X.Y. You, “Elastic analysis of internally
pressurized thick-walled spherical pressure vessels of functionally
graded materials” International Journal of Pressure Vessels and Piping
2005 pp. 347–354.
[40] J.D. Claytona, J.J. Rencisb, 2000. Numerical integration in the
axisymmetric finite element formulation. Advances in Engineering
Software, 31: 137–141.
[41] Naki Tuntuncu, Murat Ozturk, “Exact solutions for stresses in
functionally graded pressure vessels” Composites Part B 2001, pp.
683-686.
[42] Azam Afshin, Mohammad Zamani Nejad, Kia Dastani, “Transient
thermoelastic analysis of FGM rotating thick cylindrical pressure
vessels under arbitrary boundary and initial conditions”, JCAMECH
Vol. 48, No. 1, June 2017, pp 15-26.
[43] Hong-Liang Dai, Yan-Ni Rao and Ting Dai, “A review of recent
researches on FGM cylindrical structures under coupled physical
interactions” Composite Structures, 2016.