Ijciet 10 02_062

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International Journal of Civil Engineering and Technology (IJCIET)
Volume 10, Issue 02, February 2019, pp. 644-652, Article ID: IJCIET_10_02_062
Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=10&IType=02
ISSN Print: 0976-6308 and ISSN Online: 0976-6316
© IAEME Publication Scopus Indexed
RESPONSE OF FUNCTIONALLY GRADED
BEAMS OF DIFFERENT MATERIAL
COMBINATIONS UNDER THERMAL
ENVIRONMENT
N. Pradhan
Department of Civil Engineering, ITER, Siksha 'O' Anusandhan Deemed to be University,
Bhubaneswar, India
S. K. Sarangi *
Department of Mechanical Engineering, National Institute of Technology, Patna, India
* Corresponding author
ABSTRACT
In the present work, static and free vibration response of functionally graded beam
is investigated under thermal environment using Finite Element method (FEM). The
functionally graded (FG) material beam is considered to be composed of various
material combinations like metal/metal, metal/ceramic and ceramic/ceramic. The
material properties of the beam are assumed to be graded in the thickness direction
according to a simple power law distribution. The deflection and mode shapes of the
FG beams are obtained for Clamped-Free, Clamped-Clamped and Simply Supported
boundary conditions under both mechanical and thermal load. Numerical results are
obtained for the model in ANSYS software to show the influence of grading of materials,
material constituents, boundary conditions, volume fraction and temperature
conditions on the response of the FG beams.
Key Words: Beam, Functionally graded material, Thermal loading
Cite this Article: N. Pradhan and S. K. Sarangi, Response of Functionally Graded
Beams of Different Material Combinations under Thermal Environment, International
Journal of Civil Engineering and Technology, 10(02), 2019, pp. 644–652
http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=02
1. INTRODUCTION
Composite materials have been widely used in automobile industries, aerospace, structural
components and many other industrial applications due to their high specific strength, specific
stiffness, anti-corrosion ability etc. These composite materials have the limitations such as de-

N. Pradhan and S. K. Sarangi
lamination and de-bonding between matrix and fibre under extreme thermal loads due to
mismatch of mechanical properties. Also fiber and matrix having different coefficients of
thermal expansion develop residual stresses that may be present in it. To improve the strength
dispersion, weight and structural entireness, a new class of advanced composite material known
as Functionally Graded (FG) material was proposed having the ability to resist high
temperature gradient while maintaining structural integrity. FG materials are novel
microscopically inhomogeneous composite materials usually made from a mixture of ceramic
on the surface getting exposure to high temperature which provides thermal resistance while
the metallic constituent provides strength and toughness of the component.
The material properties vary continuously and smoothly from one surface to other by
gradually varying the volume fractions of constituents according to power law. Because of the
increased applications of the FG materials involving thermal structures in advanced aircraft
and aerospace engines, computer circuit boards, their characteristics under thermal
environment have attracted the attention of researchers in recent years [1]. Trun-kien Nguyen
et al [2] developed First Order Shear Deformation Functionally graded plate model. The
material properties of FG plate are assumed to be isotropic at each point and vary through the
thickness according to power law. Mohammad Talha et al [3] carried out free vibration and
static analysis of FG plates using higher order shear deformation theory. Numerical results are
presented for different thickness ratios, aspect ratios and volume fraction index with different
boundary conditions. X.L. Huang et al [4] presented the non linear vibration and dynamic
response of FG plates in thermal environments. They used higher order shear deformation plate
theory and von Karman-type equation that includes thermal effects. Alshorbagy et al. [5] used
Finite element method to study free vibration behavior of functionally graded beams with
material gradation axially and transversally along the thickness direction. Numerical results
were presented in both tabular and graphical forms to represent the effects of different material
distribution, slenderness ratios, and boundary conditions on the dynamic characteristics of the
beam. Mehta et al. [6] analyzed the static and dynamic behavior of functionally graded beam
by finite element method and for different material distribution and boundary conditions.
Trinch et al [7] studied vibration and buckling behaviors of Functionally Graded Beams in
thermal environment. Both mechanical and thermal loads are considered here. Effect of
boundary conditions, temperature distributions, material parameters and slenderness ratios on
the critical temperatures, critical buckling loads and natural frequencies are investigated under
thermal/mechanical loads. Praveen et al [8] investigated nonlinear transient thermo-elastic
analysis of ceramic-metal plates made of functionally graded material. Thermo-elastostatic and
thermo-elasto-dynamic response of plates subjected to pressure loading and thickness varying
temperature fields are examined. Numerical results for the deflection and stresses were
presented. Reddy et al [9] carried out the dynamic thermoelastic response of functionally
graded cylinders and plates using FSDT. HasanKurtaran [10] investigated large displacement
static and transient behavior of moderately thick deep functionally graded curved beams with
constant curvature using generalized differential quadrature method. Free vibration of two-
dimensional functionally graded structures by a meshfree boundary-domain integral equation
method was analyzed by Yang et al [11]. Akbas [12] studied free vibration characteristics of
an axially functionally graded cantilever beam subjected to temperature rising with
consideration of material-temperature dependent properties. Alexraj et al. [13] used Finite
Element Method (FEM) to study the static behavior of FG beams. Both mechanical and thermal
load is applied under mechanical and thermal load was to investigate the behavior of FGM
beam within the experience of Timoshenko beam theory. Calim [14] analyzed the transient
response of functionally graded Timoshenko beams having variation of material properties
along axial direction with variable cross-section.

Response of Functionally Graded Beams of Different Material Combinations under Thermal
Environment
In the present research, thermo-mechanical static and free vibration analysis of functionally
graded beams are presented for various material combinations (Metal-Metal, Metal-Ceramic
and Ceramic-Ceramic). Material properties of the beam vary through the thickness direction
based on power law. Uniform temperature is considered across the beam thickness and
effective material properties accordingly are calculated and used while developing the
simulation model. To develop the finite element model, ANSYS software is used and analysis
is carried out. Numerical results are obtained and presented for the FG beams considering
various boundary conditions and different values of power law index.
2. DEVELOPMENT OF MODEL
A beam made of functionally graded material having thickness h, width b, length L is
considered here. The effective material properties such as Young’s modulus E, poisson’s ratio
υ, Coefficient of thermal Expansion α of the functionally graded beam vary continuously along
the thickness direction (z axis direction) based on power-law which is given by [13].
= − + + = − + +
= −
ℎ
+
1
2
+
Where k is the power-law index which represents the material distribution through the
thickness of the beam, u and l denote upper and lower surface for the beam. Material properties
are calculated assuming a ten layered configuration for the proposed beam. Three different
material combinations such as metal/metal, metal/ceramic and ceramic/ceramic are considered
in this analysis. To carry out the analysis of the FG beam considering temperature effect, the
model is developed using ANSYS Mechanical APDL 15.0 software. Shell 281 (8 node)
element is considered to model the structure which is ideally suited for analyzing thin to
moderately thick structures and also it possesses the ability for modeling composite structures
with layered applications.
3. GEOMETRIC CONFIGURATION AND MATERIAL PROPERTIES
In this work, the width and thickness of the beam are taken as 0.1 m and 0.01m, respectively
while the length of the beam is considered as 1m for all analyses. The boundary conditions
considered for the analysis are Simply Supported (SS), Clamped-Clamped (CC) and Clamped-
Free (CF). The functionally graded beam considered here is modeled for a uniform temperature
T=500K. A unit pressure load is applied on the top surface. For metal/metal FG beam, the
materials taken are Steel at lower surface and Aluminium on upper surface. For metal/ceramic,
Steel is considered on lower and Alumina on upper surface whereas in ceramic/ceramic
material combination, Alumina on lower and Zirconia on upper side. The material properties
given in Table 1 are used for the analysis [7].
Table 1. Material properties.

Material E(GPa) ρ (Kg/m3
) α (1/K) υ
Steel (SUS304) 201.04 8166 12.33×10-6
0.32
Aluminium (Al) 70 2707 23.1×10-6
0.3
Alumina (Al2O3) 349.55 3800 6.8269×10-6
0.26
Zirconia (ZrO2) 244.27 3000 12.766×10-6
0.28
Variation of Young’s modulus (E) and density (ρ) along the thickness direction for three
different combinations of FG beams as metal/metal, metal/ceramic and ceramic/ceramic are
obtained and one such variation is presented in Fig 1.
Figure. 1. Variation of Young’s modulus (E) and density (ρ) through the thickness direction of
metal/metal FG beam
4. NUMERICAL RESULTS
This section divided mainly three sub sections such as static analysis, free vibration analysis
and transient response of Functionally Graded Beams.
4.1. Static analysis
First the grid convergence test is carried out and then numerical results are obtained. The
present ANSYS model is validated with the results available in literature [15] and the
comparison is presented in Table 2.
Table 2. Comparison of results obtained by present model with those available in literature [15]
Aspect ratio
Max. Deflection
Present Ferriera et al [15]
0.5 0.1215 0.1626
1 0.6886 0.7352
2 1.6820 1.9062
In this analysis, first the metal/metal FG beam is considered. Maximum transverse non-
dimensional deflection for various boundary conditions (CF, CC, and SS) under mechanical
and thermal loading are obtained. And presented in Table 3 for various power-law index. It is
observed that the maximum deformation decreases with increase in power-law exponent for all
boundary conditions (CF, CC, and SS) under mechanical load. The deflection increases with
increase in power index when k< 1 and decreases with increase in power index for k> 1 while
subjected to thermo-mechanical load for all boundary conditions.

Environment
Table 3. Maximum transverse non-dimensional deflection (w/h) for various power-law exponent
under both mechanical and thermo-mechanical load.
BCs Loading
Deflection in (m)
k=0 k=0.2 k=0.5 k=1 k=2 k=5 k=10
CF
T=0 0.212×10-4
0.141×10-4
0.131×10-4
0.122×10-4
0.112×10-4
0.101×10-4
0.944×10-5
T=500 0.01161 0.27064 0.2976 0.3133 0.28984 0.2010 0.1326
CC
T=0 0.439×10-6
0.293×10-6
0.272×10-6
0.252×10-6
0.233×10-6
0.209×10-6
0.196×10-6
T=500 0.000748 0.000894 0.00244 0.00256 0.002384 0.00168 0.00115
SS
T=0 0.224×10-5
0.140×10-5
0.128×10-5
0.118×10-5
0.111×10-5
0.103×10-5
0.976×10-6
T=500 0.000748 0.05121 0.05471 0.0572 0.03483 0.02209 0.01788
Next,the metal ceramic FGM beam made of steel (SUS304) and Alumina (Al2O3) is
considered. The top surface of the beam is assumed to be ceramic-rich, whereas the bottom
surface is metal rich andthe results obtained are presented in Table 4. It can be observed that
the deflection increases with increase in power exponent for all boundary conditions when
subjected to both mechanical and thermo-mechanical loading.But this smoothly decreases
when the beam deviates from ceramic to metal (k=5, 10) for all boundary conditions. The
deflected shapesfor the beams are also obtained and one for Clamped-Clamped boundary
condition and k=2 is shown in Fig 2.
Table 4. Maximum transverse non-dimensional (w/h) deflection for various power-law exponent
BCs Loading
Deflection (m)
k= 0 k=0.2 k=0.5 k=1 k=2 k =5 k=10
CF
T=0 0.425×10-5
0.501×10-5
0.529×10-5
0.557×10-5
0.579×10-5
0.599×10-5
0.614×10-5
T=500 0.00343 0.09658 0.1345 0.1600 0.16618 0.1537 0.12278
CC
T=0 0.882×10-7
0.104×10-6
0.110×10-6
0.116×10-6
0.120×10-6
0.124×10-6
0.127×10-6
T=500 0.000218 0.000766 0.0010 0.00123 0.00127 0.00119 0.000976
SS
T=0 0.448×10-6
0.523×10-6
0.548×10-6
0.572×10-6
0.593×10-6
0.618×10-6
0.638×10-6
T=500 0.000816 0.004176 0.00447 0.00468 0.00526 0.00518 0.00152
(a) (b)
Figure 2. Deflected shape of clamped-clamped beamfor k=2 (a) under mechanical loading
(b) Under thermo-mechanical loading
Now, considering ceramic/ceramicFG beammade of Alumina (Al2O3) and Zirconia (ZrO2),
results are obtained and presented in Table 5.It is observed that the deflection decreases with

increase in power-index for all boundary conditions while applying mechanical load only
whereas the deflection increases with increase in power-index in case of thermo-mechanical
load for all boundary conditions. The deflected shape of SS beam for k=1 under mechanical
and thermo-mechanical loading is shown in Fig 3.
Table 5. Maximum transverse non-dimensional (w/h) deflection for various power-law exponent
BCs Loading
Deflection (m)
k= 0 k=0.2 k=0.5 k=1 k=2 k =5 k=10
CF
T=0 0.608×10-5
0.536×10-5
0.520×10-5
0.507×10-5
0.495×10-5
0.481×10-5
0.470×10-5
T=500 0.00641 0.1294 0.1558 0.1727 0.1683 0.1891 0.19578
CC
T=0 0.126×10-6
0.111×10-6
0.108×10-6
0.105×10-6
0.103×10-6
0.0998×10-6
0.097×10-6
T=500 0.000404 0.000997 0.0017 0.00128 0.00135 0.00571 0.00734
SS
T=0 0.641×10-6
0.561×10-6
0.543×10-6
0.529×10-6
0.517×10-6
0.503×10-6
0.493×10-6
T=500 0.000404 0.01065 0.01397 0.0175 0.0192 0.02602 0.0296
(a) (b)
Figure 3. Deflected shape of simply supported beam (k=1) under
(a) Mechanical loading (b) thermo-mechanical loading
4.2. Free vibration analysis
In this section, results of free vibration analysis for the functionally graded beam are presented
considering three different material combinations similar to the cases presented in the previous
section. First three non-dimensional frequencies for FG beams are calculated and presented in
Tables 6-8. Effects of non-negative power-law exponent and boundary conditions on the
frequencies are presented. It is observed that when the power exponent starts to deviate from
zero and it grows up, all the non-dimensional frequency reduces for all boundary conditions
(CF, CC and SS). It is also seen from the results that the FG beam consists of metal/ceramic
combination gives better performance for free vibration response. The mode shapes for the
beam are also obtained (Fig.4).
Table 6. Firstthreedimensionless frequency parameters (λ1,λ2, λ3)for metal/metal FG beam for various
power-law index

Environment
Frequency BCs k= 0 k=0.2 k=0.5 k=1 k=2 k =5 k=10
λ1
CF 3.694283 3.679766 3.506782 3.373667 3.287526 3.267877 3.203754
CC 23.66334 23.5151 22.41246 21.56377 21.01464 20.88964 20.05455
SS 10.81631 10.61324 10.20052 9.829465 9.505339 9.294811 9.215425
λ2
CF 23.69755 23.04361 21.96158 21.12868 20.59008 20.46683 20.12867
CC 64.98998 64.75052 61.71979 59.39083 57.88204 57.53555 57.19169
SS 41.49945 41.11743 39.1854 37.69899 36.7367 36.5174 36.10687
λ3
CF 35.86871 35.44195 35.30291 35.19853 35.12002 35.0573 35.03054
CC 127.0054 126.8694 120.9527 116.3957 113.4483 112.7772 111.8588
SS 93.23757 92.96566 88.69808 85.36033 83.11031 82.4568 81.02698
Table 7. First three dimensionless frequency parameters (λ1 ,λ2, λ3 ) for metal/ceramic FG beam for
various power-law index
λ1
CF 6.829879 5.527676 5.146971 4.754424 4.4399 4.175033 4.040646
CC 43.62403 35.30993 32.87614 30.36647 28.357 26.666 25.80503
SS 19.07822 15.54574 14.50494 13.45274 12.583 11.78957 11.36851
λ2
CF 42.7749 34.6187 32.23447 29.77436 27.8059 26.14889 25.30459
CC 120.1413 97.24201 90.54458 83.62786 78.0927 73.43041 71.05759
SS 76.36465 61.80313 57.54432 53.15393 49.6407 46.68458 45.1758
λ3
CF 67.61019 56.87326 52.32059 47.8074 43.7819 40.08234 38.40996
CC 235.4142 190.5585 177.4269 163.8741 153.0187 143.8696 139.2204
SS 171.9926 139.3081 129.7423 119.8913 111.9833 105.2596 101.821
Table 8. First three dimensionless frequency parameters (λ1 ,λ2, λ3 ) for ceramic/ceramic FG beam for
various power-law index
λ1
CF 3.523844 3.440032 3.432089 3.4187 3.403723 3.4010 3.313254
CC 22.22295 21.96365 21.91395 21.82908 21.73422 21.71811 21.29572
SS 9.987337 9.642029 9.634313 9.60572 9.55988 9.532875 9.254888
λ2
CF 21.81629 21.5436 21.49436 21.41062 21.31735 21.30124 21.17749
CC 61.44809 60.48592 60.34976 60.1183 59.8596 59.81648 59.02979
SS 39.17113 38.47144 38.38294 38.23317 38.06524 38.03801 37.17416
λ3
CF 33.91393 33.48352 33.18988 33.08489 32.85947 32.62044 32.13844
CC 119.527 118.5119 118.251 117.7994 117.2978 117.2162 116.6337
SS 87.71221 86.6623 86.48075 86.15624 85.77954 85.69558 84.99058

Figure. 4. First and third mode shapes of metal/metal FG beam for CC boundary conditions with
power index k=2
5. CONCLUSIONS
Static and dynamic Responseof Functionally Graded beam with different material
combinations such as metal/metal, metal/ceramic and ceramic/ceramic and different boundary
conditions are presented. The mechanical and thermal properties of FG beam are assumed to
vary along thickness direction based on power-law distribution. A simulation model is
developed using ANSYS Mechanical APDL 15.0 software. Effect of material combination,
power-index and boundary conditions are studied. It is observed from the results that the
material combination of metal/ceramic gives better performance in static deflection, free
vibration and transient response compared to other material combinations of FG beam. Also
power-law index k and boundary conditions have significant effects on static characteristics,
free and transient vibrations of functionally graded beams.
REFERENCES
[1] Birman, V. and Byrd, L.W., 2007, “Modeling and analysis of functionally graded materials
and structures”, Applied mechanics Reviews, Vol. 60(5), pp. 195-216.
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for functionally graded material”, Composite structures, Vol. 83, pp. 25-36.
[3] Talha, M. and Singh, B.N., 2010, “Static response and free vibration analysis of FGM plates
using higher order shear deformation theory”, Applied mathematical modeling, Vol. 34,
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[4] Huang, X.L. and Shen, H.S., 2004, “Nonlinear vibration and dynamic response of FGP in
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[6] Mehta, R. and Balaji, P.S., 2013, “Static and dynamic analysis of functionally graded
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Environment
[9] Reddy, J.N. and Chin, C.D., 1998, “Thermoelastical analysis of functionally graded
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[15] Ferreira, A.J.M., Batra, R.C., Roque, C.M.C., Quin, L.F. and Martins, P.A.L.S., 2005,
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