Mechanical Engineering Department, [616964]

S. E. Esfahani
Mechanical Engineering Department,
Islamic Azad University,
South Tehran Branch,
Tehran, Iran
Y. Kiani
M. Komijani
M. R. Eslami1
Professor and Fellow of the Academy of Sciences
ASME Fellow
e-mail: [anonimizat]
Mechanical Engineering Department,
Amirkabir University of Technology,
Tehran, IranVibration of a Temperature-
Dependent Thermally
Pre/Postbuckled FGM Beam Over
a Nonlinear Hardening Elastic
Foundation
Small amplitude vibrations of a functionally graded material beam under in-plane
thermal loading in the prebuckling and postbuckling regimes is studied in this paper. Thematerial properties of the FGM media are considered as function of both position and
temperature. A three parameters elastic foundation including the linear and nonlinear
Winkler springs along with the Pasternak shear layer is in contact with beam in deforma-
tion, which acts in tension as well as in compression. The solution is sought in two
regimes. The first one, a static phase with large amplitude response, and the second one,a dynamic regime near the static one with small amplitude. In both regimes, nonlinear
governing equations are discretized using the generalized differential quadrature (GDQ)
method and solved iteratively via the Newton–Raphson method. It is concluded that
depending on the type of boundary condition and loading type, free vibration of a beam
under in-plane thermal loading may reach zero at a certain temperature which indicatesthe existence of bifurcation type of instability. [DOI: 10.1115/1.4023975]
Keywords: small amplitude vibration, temperature dependency, postbuckling,
Timoshenko beam theory, generalized differential quadrature
1 Introduction
Thermal stability analysis of isotropic homogeneous beamlike
structures and the vibration analysis in thermal field of beamswith/without elastic foundation are conventional topics in struc-
tural mechanics. Li et al. [ 1] analyzed the buckling and post-
buckling behavior of elastic rods subjected to thermal load. They
achieved the results by solving the nonlinear equilibrium equa-
tions of the slender pinned-fixed Euler–Bernoulli beams viathe shooting method. Li et al. [ 2] employed the shooting
method for solving the equations related to buckling and post-
buckling behavior of fixed-fixed elastic beam subjected to trans-
versally nonuniform temperature loading. Li et al. [ 3] studied the
natural frequency of slender Euler beams in thermal field withvarious boundary conditions. Thermal stability analysis of the
Euler–Bernoulli beams resting on a two-parameters nonlinear
elastic foundation is studied by Song and Li [ 4]a n dL ia n dB a t r a
[5]. In these studies the ability of Winkler foundation on mode
alternation of buckling configuration of a pinned-fixed andpinned-pinned beams is examined. In all of the above-mentioned
works, material properties are c onsidered to be independent of
temperature.
The FGMs are a class of novel materials in which properties
vary continuously in a specific direction. In recent years, a largenumber of researches on the subject of thermal stability belongs to
the stability behavior of the FGM beams.
Bhangale and Ganesan [ 6] investigated the vibration and buck-
ling of an FGM sandwich beam with constrained viscoelastic
layer in thermal environment using the finite element method.They studied the influence of temperature on natural frequency
and loss factors for the mentioned beam. Vibration and buckling
of FGM thin-walled box columns in thermal environments is
investigated by Ramkumar and Ganesan [ 7]. Finite element
method is employed based on the classical laminate plate theory.
Material properties are considered to be temperature dependent.
Zhao et al. [ 8] derived the nonlinear differential equations of post-
buckling for FGM rod subjected to thermal load. They considered
rods with both ends pinned and used the shooting method to solve
the equations. Ke et al. [ 9] studied the postbuckling behavior of
functionally graded material beams including an edge crack
effect. They assumed von Karman’s strain-displacement relations
combined with the Timoshenko beam theory along with an open
edge crack. They applied the Ritz method to obtain the nonlinear
governing equations and used the Newton–Raphson method to
linearize the discreted equations. Anandrao et al. [ 10] investigated
the buckling and thermal post buckling behavior of uniform slen-
der FGM beams. The von Karman strain-displacement relation-
ship is used to obtain the equations. Single-term Ritz method and
finite elements method are applied to obtain the response of the
beam. Ma and Lee [ 11] studied the behavior of FGM beams with
simply supported edges subjected to in-plane thermal loading.
The shooting method has been applied to solve the equations. In
this study material properties are assumed to be temperature
dependent. Recently Ma and Lee [ 12] have investigated the non-
linear static responses of FGM beams subjected to in-plane ther-
mal loading. They obtained an exact closed-form solution for the
response of clamped-clamped and pinned-pinned beams under
uniform temperature rise loading.
Besides, Li et al. [ 13] studied the thermal stability and nonlin-
ear vibrations of a hybrid FGM Timoshenko beam, where both
edges are clamped. For an FGM Euler beam bonded with/without
1Corresponding author.
Manuscript received October 7, 2012; final manuscript received February 26,
2013; published online August 22, 2013. Assoc. Editor: Glaucio H. Paulino.
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piezoelectric layers, Wang et al. [ 14] studied the thermal buckling,
postbuckling, and nonlinear vibrations in both pre/postbuckling
phases based on an exact analytical approach. This study isconfined to clamped-clamped edge supports and properties are
considered to be temperature dependent. Pradhan and Murmu
analyzed the free vibration of FGM sandwich beam including
thickness variations in thermal field [ 15]. Based on the first order
shear deformation beam theory, Xiang and Yang [ 16] examined
the transverse heat conduction effects on small free vibrations of
symmetrically laminated FGM beams. Recently, Komijani et al.
[17,18] studied the nonlinear thermal stability and small ampli-
tude vibrations of a piezoelectric beam with graded properties
under thermoelectrical loadings. It is shown that since the proper-ties of the smart FGM beam are not distributed symmetrically
with respect to the beam midsurface, the linear bifurcation buck-
ling may not happen in this type of structure.
Stability analysis of the FGM beams that are in contact with an
elastic foundation are limited in number. Sahraee and Saidi [ 19]
applied the differential quadrature method and then analyzed the
buckling and vibration of a deep FGM beam-columns resting on
a Pasternak-type elastic foundation. Most recently, Fallah andAghdam [ 20,21] studied the nonlinear vibration and postbuckling
behavior of functionally graded material beams resting on a
nonlinear elastic foundation subjected to axial thermal [ 20]o r
mechanical [ 21] forces. A single mode Galerkin-based method is
adopted to deduce the critical buckling and postcritical state of thebeams. In this analysis, properties are assumed to be temperature
independent and the response of the structure is confined to its
first mode. However, as reported by Hetenyi [ 22], the Winkler
elastic foundation greatly affects the buckled shape of the beam,
and therefore confining the buckled shape of an in-contact beamsimilar to its contactless shape causes the overestimation of both
critical buckling temperature and postbuckling shape.
The problem of small amplitude vibration of beams under in-
plane thermal or mechanical loadings is investigated employing
various solution methods. Finite element formulation of Komijaniet al. [ 18], shooting method solution of Li et al. [ 13], variational
iteration method (VIM) solution of Fallah and Aghdam [ 21],
single-term Galerkin solution of Wang et al. [ 14], differential
quadrature solution of Pradhan and Murmu [ 15], and the exact
solution of Emam and Nayfeh [ 23] are some of the methods used
to solve the resulted governing equations. Literature review indi-
cated that there is no report investigating the nonlinear thermal
stability analysis of temperature-dependent FGM Timoshenkobeams supported on nonlinear hardening elastic foundations with
general boundary conditions. The beam is analyzed under two
types of thermal loads, namely uniform temperature rising and
heat conduction across the thickness. Various combinations of
clamped, simply supported, and roller (sliding support) are con-sidered as the edge supports of the structure. Properties of the
graded medium are distributed across the thickness based on the
power law model, where for each constituent they are functions
of temperature. The GDQ method is adopted to discretize the
equation. The effects of various involved parameters are exam-ined on the response of the structure.2 Governing Equations
Consider a beam made of ceramic-metal FGMs with rectangu-
lar cross section b/C2hand length Lresting on a hardening three-
parameters nonlinear elastic foundation, as shown in Fig. 1.
Thermomechanical properties are graded across the thickness.
Since the volume fraction of each phase gradually varies in the
gradation direction, the mechanical properties of FGMs vary
along this direction. Here we assume a continuous alterationof the volume fraction of FGM beam with ceramic/metal as con-
stituent materials. The ceramic volume fraction V
cis represented
as the following function across the thickness coordinate z
[13,24,25]:
Vc¼1
2țz
h/C18/C19k
;Vm¼1/C0Vc (1)
where kis the volume fraction exponent. The Voigt model is a
common rule to represent the effective properties of composite
materials [ 24,25]. According to this model, the effective proper-
ties of FGM beam Pcan be expressed as
Pðz;TȚ¼PmðTȚțPcmðTȚ1
2țz
h/C18/C19k
;PcmðTȚ¼PcðTȚ/C0PmðTȚ
(2)
where the subscripts mandcrefer to the metal and ceramic con-
stituents, respectively.
The effective mechanical properties, such as Young’s modulus
E, thermal expansion coefficient a, and thermal conductivity Kare
considered to follow the distribution law [ 2], except Poisson’s
ratio /C23, that is assumed to be constant across the thickness since it
varies in a small range [ 24,25].
The nonlinear strain-displacement relations, according to von
Karman’s assumption, may be written as [ 12,25]
ex¼@u
@xț1
2@w
@x/C18/C192
cxz¼@u
@zț@w
@x(3)
Here, exandcxzstand as the axial and shear components of strain
tensor, respectively. Furthermore, uandware the components of
the displacement vector in axial and transverse directions,respectively.
Based on the Timoshenko beam theory, displacements on a
generic point of the beam may be expressed in terms of midpoint
characteristics of the beam as [ 26]
uðx;z;tȚ¼u
0ðx;tȚțzuðx;tȚ
wðx;z;tȚ¼w0ðx;tȚ(4)
Fig. 1 Coordinate system and geometry of a FGM beam resting over a three-
parameters elastic foundation
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where w0andu0are displacements of the middle surface ( z¼0)
along zandxdirections, respectively, urepresents the rotation of
the cross section as a function of time t. Now, substituting Eqs. (4)
into(3)gives the strains at a generic point in terms of displace-
ment components of the middle surface
ex¼@u0
@xț1
2@w0
@x/C18/C192
țz@u
@x
cxz¼uț@w0
@x(5)
Considering an element whose temperature is raised from the
reference temperature T0to the temperature T, Hook’s law may be
written as [ 25,27]
rx¼EðzȚex/C0aðzȚðT/C0T0Ț ½/C138
rxz¼EðzȚ
2ð1ț/C23Țcxz(6)
where rxandrxzare the axial and transversal components of the
stress tensor. Equations (5)are substituted into Eqs. (6)to reach
rx¼EðzȚ@u0
@xț1
2@w0
@x/C18/C192
țz@u
@x/C0aðzȚðT/C0T0Ț"#
rxz¼EðzȚ
2ð1ț/C23Țuț@w0
@x/C18/C19(7)
Relation between the stress component and stress resultants,
within the framework of the Timoshenko beam theory, are [ 25,26]
Nx¼ðh
2
/C0h
2rxdz
Mx¼ðh
2
/C0h
2zrxdz
Qx¼ðh
2
/C0h
2Ksrxzdz(8)
where Nx,Mx, and Qxstand for the in-plane force, bending
moment, and shear force resultants, respectively. Besides, Ksis
called the shear correction factor, which depends on the geometry,
boundary conditions, and loading type. Determination of the shearcoefficient is not straightforward. Normally, K
s¼p2=12 is used
for a rectangular section [ 17,18,25]. Substituting rxandrxzfrom
Eqs. (7)into(8)and integrating with respect to the zcoordinate,
result in
Nx¼E1@u0
@xț1
2@w0
@x/C18/C192"#
țE2@u
@x/C0NT
Mx¼E2@u0
@xț1
2@w0
@x/C18/C192"#
țE3@u
@x/C0MT
Qxz¼KsE1
2ð1ț/C23Țuț@w0
@x/C18/C19(9)
where MTand NTare the thermal moment and thermal force
resultants and E1,E2, and E3are stretching, coupling stretching-
bending, and bending stiffnesses to be obtained by using Eq. (2)
asE1¼ðh
2
/C0h
2EðzȚdz¼hE mțEcm
kț1/C18/C19
E2¼ðh
2
/C0h
2zEðzȚdz¼h2Ecm1
kț2/C01
2kț2/C18/C19
E3¼ðh
2
/C0h
2z2EðzȚdz¼h31
12EmțEcm1
kț3/C01
kț2ț1
4kț4/C18/C19/C20/C21
NT¼ðh
2
/C0h
2EðzȚaðzȚðT/C0T0Țdz
MT¼ðh
2
/C0h
2zEðzȚaðzȚðT/C0T0Țdz (10)
The equations of motion of FGM beams can be derived by
applying the principle of virtual displacements
ðT
0dUsțdUf/C0dT/C0/C1
dt¼0 (11)
where the total virtual strain energy of the beam dUscan be
written as
dUs¼ðL
0ðh
2
/C0h
2ðb
0rxxdexxțKsrxzdcxz ðȚ dydzdx (12)
The virtual strain energy of the nonlinear elastic foundation dUfis
expressed as
dUf¼ðL
0ðb
0Kww0dw0țKg@w0
@xd@w0
@x/C18/C19
țKNLw3
0dw0/C20/C21
dydx
(13)
In which the linear Winkler stiffness, the shear layer stiffness, and
the nonlinear Winkler stiffness are indicated as Kw,Kg, and KNL,
respectively.
Also the kinetic energy dTis given by
dT¼ðL
0ðb
0/C18
I1@u0
@t@du0
@tțI2@u
@t@du0
@tțI2@u0
@t@du
@t
țI3@u
@t@du
@tțI1@w0
@t@dw0
@t/C19
dydx (14)
where I1,I2, and I3are constants to be derived by utilizing Eq. (2)
as
I1¼ðh
2
/C0h
2qðzȚdz¼hqmțqcm
kț1/C18/C19
I2¼ðh
2
/C0h
2zqðzȚdz¼h2qcm1
kț1/C01
2kț2/C18/C19
I3¼ðh
2
/C0h
2z2qðzȚdz¼h31
12qmțqcm1
kț1/C01
kț2ț1
4kț4/C18/C19/C20/C21
(15)
The motion equations of an in-contact FGM Timoshenko beam
are obtained according to the virtual work principle [ 26]. Integrat-
ing Eq. (11) by part, with the consideration of Eqs. (9)and(10),
result in the following equations of motion:
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du0:@Nx
@x¼I1@2u0
@t2țI2@2u
@t2
dw0:@Qxz
@xț@
@xNx@w0
@x/C18/C19
/C0Kww0țKg@2w0
@x2/C0KNLw3
0¼I1@2w0
@t2
du :@Mx
@x/C0Qxz¼I2@2u0
@t2țI3@2u
@t2(16)
Due to the integration process, the natural and essential boundary
conditions are obtained as
Nx¼0o r u0¼known
QxzțðKgțNxȚ@w0
@x¼0o r w0¼known
Mx¼0o r u¼known(17)
For generalizing the subsequent results, the following nondimen-
sional variables are introduced and are used in the text
n¼x
L;U/C3¼u
L;d¼h
L;W/C3¼w
h;j¼Ks
2ð1ț/C23Ț
e1¼E1
Eref
ch;e2¼E2
Eref
ch2;e3¼E3
Eref
ch3
K/C3
w¼KwL4
Eref
cI0;K/C3
g¼KgL2
Eref
cI0;K/C3
NL¼KNLh2L4
Eref
cI0
N/C3
x¼NxL2
Eref
cI0;Q/C3
xz¼QxzL2
Eref
cI0;M/C3
x¼MxL
Eref
cI0;
NT/C3¼NTL2
Eref
cI0;MT/C3¼MTL
Eref
cI0
x/C3¼xhffiffiffiffiffiffiffiffi
qref
c
Erefcs
;g¼t
hffiffiffiffiffiffiffiffi
Eref
c
qrefcs
;K1¼I1
hqref
c;
K2¼I2
h2qref
c;K3¼I3
h3qref
c(18)
where I0is the moment of inertia of the cross section and Eref
cand
qref
care the elasticity modulus and density of the ceramic constitu-
ent at reference temperature. Substitution of the above nondimen-sional parameters into Eqs. (16) and utilizing Eqs. (9)gives the
governing equations of the beam in dimensionless forms as
e
1@2U/C3
@n2țe1d2@W/C3
@n@2W/C3
@n2țde2@2u
@n2
¼K1
d2@2U/C3
@g2țK2
d@2u
@g2
je1@u
@nțd@2W/C3
@n2/C18/C19
ț/C26
e1@U/C3
@nț1
2d2@W/C3
@n/C18/C192"#
țde2@u
@n/C0NT/C3
12d2/C27
d@2W/C3
@n2ț/C20
e1@2U/C3
@n2țd2@W/C3
@n@2W/C3
@n2/C18/C19
țe2d@2u
@n2/C21
d@W/C3
@n/C01
12K/C3
wd3W/C3ț1
12K/C3
gd3@2W/C3
@n2
/C01
12d3K/C3
NLW/C33¼K1
d@2W/C3
@g2
e2d@2U/C3
@n2țd3@W/C3
@n@2W/C3
@n2/C18/C19
țd2e3@2u
@n2/C0je1uțd@W/C3
@n/C18/C19
¼K2
d@2U/C3
@g2țK3@2u
@g2(19)Five possible types of boundary conditions that are combina-
tions of the clamped, roller, and simply supported edges are con-
sidered. Mathematical expressions for these classes of edgesupports are
Clamped ðCȚ:U
/C3¼W/C3¼u¼0
Simply supported ðSȚ:U/C3¼W/C3¼M/C3
x¼0
Roller ðRȚ:U/C3¼u¼Q/C3
xzțðK/C3
gțN/C3
xȚddW/C3
dn¼0
(20)
where
N/C3
x¼12
d2e1dU/C3
dnț1
2d2dW/C3
dn/C18/C192"#
ț12
de2du
dn/C0NT/C3
M/C3
x¼12
de2dU/C3
dnț1
2d2dW/C3
dn/C18/C192"#
ț12e3du
dn/C0MT/C3
Q/C3
xz¼12
d2je1uțddW/C3
dn/C18/C19(21)
3 Solution Procedures
The solution of equation of motion (19) is divided into two
regimes. Part of time-independent solution that is related to ther-
mal postbuckling analysis with large magnitude and part of
dynamic solution for free vibration with small magnitude that istime dependent. Thus, the total solution of Eqs. (19)is
U
/C3ðn;gȚ¼U/C3
sðnȚțU/C3
dðn;gȚ
W/C3ðn;gȚ¼W/C3
sðnȚțW/C3
dðn;gȚ
uðn;gȚ¼usðnȚțudðn;gȚ(22)
Substituting Eqs. (22) into (19) and collecting the static parts
result in the following time-independent equations which describe
the nonlinear stability behavior of a beam under in-plane thermal
loading:
e1d2U/C3
s
dn2țe1d2dW/C3
s
dnd2W/C3
s
dn2țde2d2us
dn2¼0
je1dus
dnțdd2W/C3
s
dn2/C18/C19
ț(
e1dU/C3
s
dnț1
2d2dW/C3
s
dn/C18/C192"#
țde2dus
dn
/C0NT/C3
12d2)
dd2W/C3
s
dn2/C01
12K/C3
wd3W/C3
sț1
12K/C3
gd3d2W/C3
s
dn2
/C01
12d3K/C3
NLW/C33
s¼0
e2dd2U/C3
s
dn2țd3dW/C3
s
dnd2W/C3
s
dn2/C18/C19
/C0d2e3d2us
dn2/C0je1usțddW/C3
s
dn/C18/C19
¼0
(23)
and linearizing the remaining part about the static equilibrium
position with a small amplitude dynamic response reaches us to
e1@2U/C3
d
@n2țe1d2@W/C3
d
@n@2W/C3
s
@n2țe1d2@W/C3
s
@n@2W/C3
d
@n2țde2@2ud
@n2
¼K1
d2@2U/C3
d
@g2țK2
d@2ud
@g2
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je1@ud
@nțd@2W/C3
d
@n2/C18/C19
ț(
e1@U/C3
s
@nț1
2d2@W/C3
s
@n/C18/C192"#
țde2@us
@n
/C0NT/C3
12d2)
d@2W/C3
d
@n2țe1@U/C3
d
@nțd2@W/C3
s
@n@W/C3
d
@n/C18/C19
țe2d@ud
@n/C20/C21
/C2d@2W/C3
s
@n2ț/C20
e1@2U/C3
d
@n2țd2@2W/C3
s
@n2@W/C3
d
@nțd2@W/C3
s
@n@2W/C3
d
@n2/C18/C19
țe2d@2ud
@n2/C21
d@W/C3
s
@nțe1@2U/C3
s
@n2țe1d2@W/C3
s
@n@2W/C3
s
@n2țe2d@2us
@n2/C20/C21
/C2d@W/C3
d
@n/C01
12K/C3
wd3W/C3
dț1
12K/C3
gd3@2W/C3
d
@n2/C01
4d3K/C3
NLW/C32
sW/C3
d
¼K1
d@2W/C3
@g2
e2d@2U/C3
d
@n2țd3@W/C3
d
@n@2W/C3
s
@n2țd3@W/C3
s
@n@2W/C3
d
@n2/C18/C19
țd2e3@2ud
@n2/C0je1udțd@W/C3
d
@n/C18/C19
¼K2
d@2U/C3
d
@g2țK3@2ud
@g2
(24)
According to Eqs. (24), solution of Eqs. (23) should be clear at
the beginning. The analytical solution of Eqs. (23) is complicated
due to the strong nonlinearity and the included couplings in the
partial differential equations. Therefore, to solve the problemnumerically, the GDQ method is employed. The ability of the
GDQ method to handle the nonlinear stability problems is exhib-
ited by many authors [ 28]. A brief overview of the GDQ method
is presented in Appendix A.
Utilizing the GDQ method to discrete to the dimensionless gov-
erning equations (23), one may obtained discretized form of the
equations governing the pre/postbuckling equilibrium path of the
beam. Equations and the associated boundary conditions are given
in Appendix B.
The system of algebraic equations and associated boundary
conditions presented in Appendix Bmay be written in the form
½K
s/C1383N/C23NXsðȚ3N/C21¼FðȚ3N/C21 (25)
where ½Ks/C1383N/C23Nis the nonlinear stiffness matrix which depends
on both unknown variable vector XsðȚ3N/C21and temperature. It
must be noted that the right-hand side of Eq. (25)may be different
for each set of boundary conditions. The force matrix FðȚ3N/C21is
obtained through the thermally induced stress resultants and bend-ing moments for the simply supported boundary conditions at the
ends of the beam ( n¼0, 1) and vanishes when the beam is
clamped or roller at the ends (see the definition of thermal
moment in Eq. (21)). Thermal buckling, without consideration of
the magnitude of the temperature difference, occurs only when
FðȚ
3N/C21¼0. Otherwise, lateral deflection occurs when
FðȚ3N/C216¼0. The numerical algorithm to solve the postbuckling
behavior in each case is given by Liew et al. [ 29].
When the solution of static phase is accomplished, small free
vibration analysis is followed. The discrete form of the governing
equations along with the associated boundary conditions are given
in Appendix C. The presented equations are linear with respect to
the dynamic variables denoted by a subscript d. Solution of this
system is obtained as an eigenvalue problem. The eigenvalues of
the established system of equations present the nondimensional
frequency of the beam defined as x*.
4 Types of Thermal Loading
Two distinct types of thermal loadings are considered for the
beam. Uniform temperature rise and heat conduction across the
thickness are used widely in the literature and are expected for
design purposes.4.1 Uniform Temperature Rise (UTR). In the case of uni-
form temperature rise loading, the beam experiences a unifiedtemperature. Therefore, thermal force and moment resultants are
obtained based on Eq. (10)as
N
T¼hðT/C0T0ȚEmamțEcmamțEmacm
kț1țEcmacm
2kț1/C18/C19
MT¼h2k
2ðkț1ȚðT/C0T0ȚEmacmțEcmam
kț2țEcmacm
2kț1/C18/C19 (26)
4.2 Heat Conduction (HC). The case of heat conduction
across the thickness is considered. The one-dimensional heat con-duction equation with the prescribed temperature boundary condi-
tions on top and bottom surfaces of the beam are
Th
2/C18/C19
¼Tc;T/C0h
2/C18/C19
¼Tm
d
dzKðzȚdT
dz/C20/C21
¼0(27)
Solution of the above equation is obtained based on power series
solution as
T¼TmțTc/C0Tm
RXn
j¼01
kjț11
2țz
h/C18/C19kjț1
/C0Kcm
Km/C18/C19j"#
(28)
with
R¼Xn
j¼01
kjț1/C0Kcm
Km/C18/C19j"#
(29)
Sufficient terms of series expansion should be taken into account
to assure the convergence of the temperature profile of Eq. (28).
The above temperature distribution, when substituted into
Eq.(10), yields the thermal force and moment resultants as
NT¼hðTm/C0T0ȚAțðTc/C0TmȚS
R/C20/C21
MT¼h2ðTm/C0T0ȚB
2țðTc/C0TmȚ
RP/C0S
2/C18/C19 /C20/C21 (30)
with
A¼EmamțEcmamțEmacm
kț1țEcmacm
2kț1
B¼k
kț1/C18/C19EmacmțEcmam
kț2țEcmacm
2kț1/C18/C19
S¼Xn
j¼0/C201
kjț1/C0Kcm
Km/C18/C19j/C18Emam
kjț2țEmacmțEcmam
kjțkț2
țEcmacm
kjț2kț2/C19/C21
P¼Xn
j¼0/C201
kjț1/C0Kcm
Km/C18/C19j/C18Emam
kjț3țEmacmțEcmam
kjțkț3
țEcmacm
kjț2kț3/C19/C21(31)
5 Numerical Result and Discussions
In the present study, unless otherwise stated, a functionally
graded material beam made of SUS304 as metal constituent andSi
3N4as ceramic constituent is considered. Both materials are
considered to be temperature dependent, where each of their
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properties may be presented as a function of temperature as
follows [ 30]:
P¼P0P/C01T/C01ț1țP1TțP2T2țP3T3/C0/C1
(32)
where P/C01,P0,P1,P2, and P3are constants and unique to each
constituent. For the constituents of this study, these constants are
given in Table 1. The temperature independent case (TID),
describes the condition when properties are evaluated at
reference temperature T0¼300 K. The case TD, on the other
hand, represents the conditions when properties are calculated at
current temperature based on the Toloukian model described by
Eq.(32). For the sake of generality, in all of numerical examples,
the frequency parameter X¼ðffiffiffiffiffi
12p
=d2Țx/C3¼xL2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
hðqref
c=Iref
cȚp
is
presented.
5.1 Convergence Studies. To obtain the sufficient grid
points of the GDQ method, a critical buckling temperature differ-
ence of isotropic homogeneous beam with arbitrary boundary con-
ditions is presented in Table 2. As one may see, when N¼17, the
desired accuracy is reached since both N¼15 and N¼17 present
the same results. In this study, for all results, the number of gridpoints is chosen as N¼17.5.2 Comparison Studies. To assure the validity and accu-
racy of the presented method, two comparison studies are per-formed. At first, the frequency parameter of an isotropic
homogeneous beam with both edges simply supported are eval-
uated and compared with the results of Ying et al. [ 31] based on
an exact two-dimensional elasticity solution. Numerical results
are given in Table 3. Small differences between the results is
because of the difference between flexural beam theory and elas-
ticity theory. The relative difference in the cases is at most 0.5
percent. As expected, the natural frequency obtained by beam
theory is a little larger than the one obtained by the exact elasticity
solution.
In another comparison study, the natural frequency parameter
of a beam with S–SandS–C boundary conditions are obtained
as a function of temperature and presented in Figs. 2and 3,Table 1 Temperature-dependent coefficients for SUS304 and Si 3N4[30]
Material Properties P/C01 P0 P1 P2 P3
a(K/C01) 0 12.33 /C210/C068.086 /C210400
E(Pa) 0 201.04 /C21093.079 /C210/C04/C06.534 /C210/C070
SUS304 K (W m/C01K/C01) 0 15.379 /C01.264 /C210/C03/C02.092 /C210/C06/C07.223 /C210/C010
q(kg/m3) 0 8166 0 0 0
/C23 0 0.28 0 0 0
a(K/C01) 0 5.8723 /C210/C069.095 /C210/C0400
E(Pa) 0 348.43 /C2109/C03.07/C210/C042.16/C210/C07/C08.946 /C210/C011
Si3N4 K( Wm/C01K/C01) 13.723 /C01.032 /C210/C035.466 /C210/C07/C07.876 /C210/C011
q(kg/m 3) 0 2170 0 0 0
/C23 0 0.28 0 0 0
Table 2 Study on accuracy of natural frequency parameter Xfor various boundary conditions of isotropic homogeneous
(SUS304) Timoshenko beams subjected to UTR loading without contact conditions. d50.025 is considered.
Number of grid points C–C C–S C–R S–S S–R
N¼7 22.283296 15.369488 5.601878 9.868031 2.468595
N¼9 22.291178 15.389042 5.588070 9.864203 2.467205
N¼11 22.291178 15.388854 5.588169 9.864247 2.467177
N¼13 22.291099 15.388858 5.588168 9.864246 2.467132
N¼15 22.291100 15.388860 5.588167 9.864245 2.467109
N¼17 22.291100 15.388860 5.588167 9.864245 2.467109
Table 3 A comparison on natural frequency defined asffiffiffiffi
Xp
for
S–S isotropic homogeneous Timoshenko beams with d51/15
and various parameters of elastic foundation
ðK/C3
w;K/C3
gȚ Present Ying et al. [ 31]
(0, 0) 3.13538 3.13227
(0,p2) 3.73399 3.72775
(0, 5p2/2) 4.29734 4.28886
(102, 0) 3.74642 3.74012
(102p2) 4.14349 4.13558
(102,5p2/2) 4.58360 4.57410
(104, 0) 10.03292 9.99583
(104,p2) 10.05703 10.01971
(104,5p2/2) 10.09288 10.05520
Fig. 2 A comparison between the results of this study and
those reported by Li et al. [ 3] for fundamental frequency of S–S
isotropic homogeneous Euler–Bernoulli beams. Temperatureparameter is defined as s5(12=d
2)aDT.
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respectively. Since the formulation of Li et al. [ 3] is based on
Euler–Bernoulli beam theory, the correction shear factor is chosen
as an adequately large number. Beam is under UTR loading andproperties are TID. Results of this study and Li et al. [ 3] are coin-
cident which proves the efficiency of the present method.
5.3 Parametric Studies. As a benchmark study, the first
three frequencies of FGM beams with various boundary condi-
tions are presented in Tables 4to8. It is seen that for the constitu-
ents of this study, as the power law index increases, the natural
frequency of the system decreases. For each case of edge supports,an increase in Winkler or Pasternak constant of elastic foundation
results in higher natural frequency. This is due to the higherelastic stiffness of the beam when it is in-contact with foundation.
As expected, for a prescribed contact condition and power law
index, the C–C beam has the highest natural frequency and S–R
has the lowest one.
For a contactless beam, the fundamental frequency parameter
as a function of temperature rise is depicted in Figs. 4and5for
theS–SandC–C cases of boundary conditions, respectively. For
the case of a beam with both edges clamped, it is seen thatbefore a prescribed temperature, i.e., the bifurcation point tem-
perature, as temperature increases the frequency parameter
diminishes. This is due to the decrease in total stiffness of the
beam since geometrical stiffness diminishes as temperature rises.
Near the bifurcation point, frequency approaches zero. After thebifurcation point, an increase in temperature results in higher
frequency. This feature refers to the higher elastic stiffness of
the beam resulted from the von Karman nonlinearity [ 3]. It is
seen that temperature dependency of the constituents leads to
more accurate results, where with the assumption of constantmaterial properties, bifurcation points, are exaggerated. Besides,
in prebuckling range, with the assumption of temperature de-
pendency, predicted frequency is less than the one obtained withthe temperature-independent assumption. This is due to the less
elasticity modulus of the constituents in the TD case. A compar-
ison of Figs. 4and5reveals that the behavior of an FGM beam
with the S–Sboundary conditions is totally different from those
with the C–C boundary conditions. For the FGM beam with
both edges simply supported, frequency does not approach zero,
which somehow proves the nonexistence of bifurcation type
buckling. This is expected since a simply supported edge does
not handle the moment and the total bending moment is affected
by the temperature loading. Since the statement of bendingmoment is nonhomogeneous in terms of u,w, and u, the
resulted system of equations cannot be posed as an eigenvalue
problem and the load path of the beam within the studied range
is unique and stable. It should be mentioned that, however, the
load-deflection path of S–S beams is free of the bifurcation
point, but the same as the C–C case, frequency decreases up to a
definite temperature and then increases.
The influence of the elastic foundation on fundamental fre-
quency of an FGM beam for the C–C and S–S boundary
Fig. 3 A comparison between the results of this study and
those reported by Li et al. [ 3] for fundamental frequency of C–S
isotropic homogeneous Euler–Bernoulli beams. Temperatureparameter is defined as s5(12=d
2)aDT.
Table 4 The first three natural frequencies of lateral vibration for C–C FGM beams with d50.04, various power law indices, and
contact conditions
ðK/C3
w;K/C3
gȚ k¼0 k¼0.5 k¼1 k¼2 k¼5 k¼10 k¼1
First 22.1644 15.2640 13.3783 12.0136 10.9226 10.4046 9.5879
(0, 0) Second 60.4948 41.6768 36.5201 32.7792 29.7880 28.3764 26.1690
Third 117.1439 80.7439 70.7321 63.4475 57.6222 54.8942 50.6743
First 24.3190 16.9750 14.9679 13.5055 12.3396 11.8036 10.9998
(100, 0) Second 61.3204 42.3365 37.1344 33.3570 30.3383 28.9207 26.7208
Third 117.5749 81.0884 71.0531 63.7496 57.9100 55.1791 50.9632
First 26.7008 18.8431 16.6923 15.1163 13.8620 13.3008 12.4958
(100, 10) Second 64.9355 45.2091 39.8021 35.8607 32.7161 31.2685 29.0911
Third 121.6856 84.3659 74.1013 66.6141 60.6349 57.8726 53.6897
First 26.2978 18.5298 16.4042 14.8483 13.6099 13.0535 12.2501
(200, 0) Second 62.1350 42.9860 37.7386 33.9250 30.8787 29.4549 27.2615
Third 118.0043 81.4316 71.3726 64.0502 58.1965 55.4624 51.2504
First 28.5147 20.2545 17.9914 16.3270 15.0039 14.4215 13.6093
(200, 10) Second 65.7053 45.8179 40.3664 36.3896 33.2179 31.7633 29.5885
Third 122.1006 84.6958 74.4077 66.9019 60.9085 58.1428 53.9624
First 31.4968 22.5587 20.1065 18.2945 16.8558 16.2358 15.4036
(500, 0) Second 64.5172 44.8782 39.4959 35.5747 32.4460 31.0024 28.8228
Third 119.2833 82.4524 72.3227 64.9439 59.0474 56.3040 52.1027
First 33.2699 23.9955 21.4210 19.5138 18.0001 17.3547 16.5050
(500, 10) Second 67.9624 47.5976 42.0139 37.9322 34.6796 33.2033 31.0329
Third 123.3371 85.6777 75.3195 67.7579 61.7221 58.9461 54.7725
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conditions are depicted in Figs. 6and7, respectively. As previ-
ously discussed, the TD case results in more accurate conclusionsand therefore in the following discussion only this case is
addressed. It is seen that an increase in the Winkler or Pasternak
constants of elastic foundation results in higher stiffness and
therefore fundamental frequency and critical buckling temperature
are increased. For the case of C–C beams, the nonlinear coeffi-
cient of elastic foundation has no effect on frequency parameter
of the beam prior to buckling. This is expected since the prebuck-ling deformation of the beam is linear. In contrast, in the S–S
beams nonlinear coefficient of elastic foundation affects the fun-damental frequency with the initiation of temperature loading.
This effect, however, is negligible.
The effects of various boundary conditions on frequency pa-
rameter of a beam subjected to uniform temperature rise loading
is depicted in Fig. 8. It is seen that responses of the C–C andC–R
beams are totally different from the other three types. In the C–C
and C–R cases, since edges are capable of supplying the extraTable 5 The first three natural frequencies of lateral vibration for C–R FGM beams with d50.04, various power law indices, and
contact conditions
ðK/C3
w;K/C3
gȚ k¼0 k¼0.5 k¼1 k¼2 k¼5 k¼10 k¼1
First 5.5801 3.8421 3.3678 3.0250 2.7510 2.6206 2.4138
(0, 0) Second 29.9847 20.6501 18.0983 16.2514 14.7755 14.0751 12.9708
Third 73.4283 50.5880 44.3260 39.7840 36.1545 34.4422 31.7637
First 11.4533 8.3591 7.5665 6.8689 6.3637 6.1566 5.9043
(100, 0) Second 31.6133 21.9477 19.3049 17.3850 15.8533 15.1402 14.0484
Third 74.1116 51.1342 44.8374 40.2626 36.6101 34.8930 32.2210
First 12.6982 9.2948 8.3558 7.6520 7.0945 6.8677 6.5960
(100, 10) Second 35.2964 24.8485 21.9884 19.8956 18.2300 17.4806 16.3948
Third 78.4420 54.5785 48.0346 43.2666 39.4646 37.7123 35.0695
First 15.2059 11.1799 10.0675 9.2311 8.5688 8.3030 7.9935
(200, 0) Second 33.1620 23.1728 20.4404 18.4490 16.8623 16.1350 15.0490
Third 74.7887 51.6747 45.3377 40.7355 37.0602 35.3380 32.6720
First 16.1642 11.8956 10.7157 9.8277 9.1247 8.8432 8.5171
(200, 10) Second 36.6900 25.9369 22.9917 20.8318 19.1140 18.3490 17.2599
Third 79.0821 55.0852 48.5044 43.7070 39.8825 38.1245 35.4842
First 23.0508 17.0391 15.3745 14.1176 13.1229 12.7298 12.2881
(500, 0) Second 37.4256 26.5104 23.5202 21.3250 19.5798 18.8065 17.7149
Third 76.7843 53.2632 46.8142 42.1224 38.3788 38.6408 33.9888
First 23.6939 17.5170 15.8064 14.5147 13.4924 13.0885 12.6350
(500, 10) Second 40.5846 28.9576 25.7683 23.4168 21.5493 20.7371 19.6277
Third 80.9718 56.5780 49.8872 45.0024 41.1106 39.3350 36.7003
Table 6 The first three natural frequencies of lateral vibration for S–S FGM beams with d50.04, various power law indices, and
contact conditions
ðK/C3
w;K/C3
gȚ k¼0 k¼0.5 k¼1 k¼2 k¼5 k¼10 k¼1
First 9.8558 6.8104 5.9860 5.3809 4.8791 4.6370 4.2634
(0, 0) Second 39.2590 27.0363 23.6933 21.2748 19.3439 18.4280 16.9827
Third 87.7174 60.4548 52.9828 47.5561 43.2060 41.1515 37.9449
First 14.0452 10.0771 8.9933 8.1864 7.5340 7.2501 6.8729
(100, 0) Second 40.5189 28.0417 24.6289 22.1541 20.1805 19.2551 17.8209
Third 88.2918 60.9140 53.4103 47.9581 43.5890 41.5306 38.3297
First 17.2073 12.4898 11.1955 10.2267 9.4492 9.1225 8.7132
(100, 10) Second 45.1507 31.7013 28.0189 25.3292 23.1901 22.2217 20.8026
Third 93.2395 64.8502 57.0669 51.3916 46.8536 44.7566 41.5912
First 17.2452 12.5052 11.2217 10.2509 9.4719 9.1447 8.7349
(200, 0) Second 41.7408 29.0138 25.5302 22.9999 20.9837 20.0481 18.6214
Third 88.8626 61.3698 53.8344 48.3568 43.9686 41.9062 38.7106
First 19.9054 14.5314 13.0532 11.9436 11.0563 10.6903 10.2460
(200, 10) Second 46.2504 32.5030 28.8143 26.0722 23.8923 22.9123 21.4923
Third 93.7801 65.2584 57.0464 51.7639 47.2070 45.1054 41.9426
First 24.4498 17.9503 16.1576 14.8079 13.7325 13.2940 12.7859
(500, 0) Second 45.2089 31.7408 28.0609 25.3685 23.2272 22.2589 20.8391
Third 90.5533 62.7171 55.0870 49.5337 45.0884 43.0061 39.8315
First 26.3935 19.4077 17.4793 16.0262 14.8697 14.4037 13.8622
(500, 10) Second 49.4028 35.0213 31.0787 28.1837 25.8849 24.8692 23.4400
Third 95.3837 66.5467 58.6391 52.8650 48.2517 46.1358 42.9792
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moment, the beam remains flat until a prescribed temperature in
which frequency approaches zero. After that, frequency increasesmonolithically as the beam deflects more. For three other cases,
however, the behavior is slightly different since the beam initially
starts to lateral deflection at the onset of thermal loading.
In Figs. 9and10the effect of the heat conduction case of
thermal loading on the frequency parameter of contactless FGMbeams is depicted for the C–C andS–Scases of boundary con-
ditions, respectively. The TD case of properties for the heatconduction case developed by Shen [ 32] is studied herein. It is
seen that the behavior of the C–C beam in this case is also sim-
ilar to the case of UTR loading, where the behavior is of the
bifurcation type buckling. However, the behavior of a beam
with simply supported edges, even for the case of the reduction
of an FGM beam to a fully homogeneous one, is not of the
bifurcation type of instability. It is seen that for the constituentsof this study, critical buckling temperature decreases as the
power law index increases.Table 7 The first three natural frequencies of lateral vibration for C–S FGM beams with d50.04, various power law indices, and
contact conditions
ðK/C3
w;K/C3
gȚ k¼0 k¼0.5 k¼1 k¼2 k¼5 k¼10 k¼1
First 15.3433 10.5716 9.2703 8.3268 7.5680 7.2064 6.6372
(0, 0) Second 49.3767 34.0161 29.8129 26.7658 24.3268 23.1718 21.3595
Third 102.0257 70.3097 61.5994 55.2714 50.2126 47.8345 44.1345
First 18.3185 12.9204 11.4453 10.3637 9.4993 9.1104 8.5510
(100, 0) Second 50.3846 34.8210 30.5620 27.4702 24.9974 23.8351 22.0320
Third 102.5202 70.7051 61.9675 55.6178 50.5426 48.1610 44.4658
First 21.2027 15.1545 13.4973 12.2732 11.2979 10.8741 10.2990
(100, 10) Second 54.4712 37.0606 33.5672 30.2882 27.6720 26.4745 24.6921
Third 107.0224 74.2916 65.3016 58.7499 53.5112 51.1050 47.4443
First 20.8739 14.9035 13.2684 12.0614 11.0994 10.6802 10.1087
(200, 0) Second 51.3727 35.6076 31.2932 28.1570 25.6504 24.4804 22.6845
Third 103.0123 71.0982 62.3333 55.9621 50.8704 48.4854 44.7947
First 23.4457 16.8770 15.0742 13.7368 12.6729 12.2193 11.6247
(200, 10) Second 55.3864 38.7816 34.2343 30.9125 28.2633 27.0569 25.2761
Third 107.4940 74.6658 65.6490 59.0759 53.8309 51.4108 47.7527
First 27.1324 19.6887 17.6413 16.1148 14.9024 14.3968 13.7617
(500, 0) Second 54.2290 37.8696 33.3908 30.1236 27.5167 26.3216 24.5384
Third 104.4748 72.2648 63.4188 56.9824 51.8416 49.4456 45.7671
First 29.1572 21.2217 19.0366 17.4042 16.1085 15.5728 14.9108
(500, 10) Second 58.0456 40.8683 36.1617 32.7138 29.9673 28.7334 26.9522
Third 108.8962 75.7775 66.6803 60.0433 54.7495 52.3173 48.6660
Table 8 The first three natural frequencies of lateral vibration for S–R FGM beams with d50.04, various power law indices, and
contact conditions
ðK/C3
w;K/C3
gȚ k¼0 k¼0.5 k¼1 k¼2 k¼5 k¼10 k¼1
First 2.4666 1.7036 1.4972 1.3454 1.2107 1.1603 1.0672
(0, 0) Second 22.1371 15.2505 13.3709 12.0094 10.9170 10.3967 9.5762
Third 61.1498 42.1266 36.9185 33.1441 30.1256 28.6965 26.4524
First 10.3013 7.6167 6.8736 6.3121 5.8666 5.6904 5.4928
(100, 0) Second 24.2970 16.9649 14.9623 13.5026 12.3356 11.7978 10.9913
Third 61.9684 42.7799 37.5264 33.7157 30.6700 29.2354 26.9995
First 11.4367 8.4623 7.6388 7.0161 6.5224 6.3275 6.1101
(100, 10) Second 28.5142 20.2612 18.0017 16.3391 15.0151 14.4314 13.6179
Third 66.7966 46.6125 41.0838 37.0531 33.8395 32.3644 30.1564
First 14.3579 10.6359 9.6047 8.8245 8.2063 7.9633 7.6943
(200, 0) Second 26.2800 18.5220 16.4008 14.8472 13.6077 13.0497 12.2440
Third 62.7763 43.4242 38.1256 34.2788 31.2057 29.7650 27.5359
First 15.1931 11.2570 10.1663 9.3410 8.6872 8.4304 8.1465
(200, 10) Second 30.2217 21.5818 19.2140 17.4667 16.0705 15.4717 14.6477
Third 67.5469 47.2045 41.6318 37.5663 34.3257 32.8436 30.6376
First 22.4999 16.6869 15.0751 13.8549 12.8889 12.5106 12.0954
(500, 0) Second 31.4883 22.5574 20.1081 18.2973 16.8574 16.2363 15.4025
Third 65.1401 45.3021 39.8694 35.9152 32.7002 31.3001 29.0857
First 23.0419 17.0894 15.4390 14.1895 13.2003 12.8130 12.3880
(500, 10) Second 34.8457 25.1306 22.4617 20.4802 18.9066 18.2399 17.3746
Third 69.7491 48.9376 43.2344 39.0652 35.7448 34.2409 32.0377
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Fig. 6 Influences of three-parameters nonlinear elastic founda-
tionðK/C3
w;K/C3
g;K/C3
NLȚon the first mode frequency of linearly graded
C–C FGM beam with d50.04 subjected to UTR loading
Fig. 7 Influences of three-parameters nonlinear elastic founda-
tionðK/C3
w;K/C3
g;K/C3
NLȚon the first mode frequency of linearly graded
S–S FGM beam with d50.04 subjected to UTR loading
Fig. 4 Effect of temperature dependency and various power
law indices on the first mode frequency of S–S FGM beams with
d50.04 subjected to UTR loading
Fig. 5 Effect of temperature dependency and various power
law indices on the first mode frequency of C–C FGM beams
with d50.04 subjected to UTR loading
Fig. 8 Effect of various boundary conditions of linearly graded
FGM beam on the dimensionless frequency and deflection withd50.04 subjected to UTR loading
Fig. 9 Influences of various power law indices and tempera-
ture dependency on the first frequency of C–C FGM beam with
d50.04 subjected to HC loading
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6 Conclusion
To study the effects of boundary conditions, temperature de-
pendency, thermal loading type, and foundation coefficients on
small amplitude vibration of an FGM beam subjected to in-plane
thermal loading, the GDQ technique is adopted to discretized the
motion equations. Solution of the problem is posed into two
phases, one the static equilibrium phase and the other one is thesmall amplitude vibrations around the static response. In static
phase, the Newton–Raphson scheme is used to solve the resulting
system of equations, iteratively. The resulting response of the sys-
tem from static phase is imported into the small amplitude vibra-
tion equations. These equations are solved via a linear eigenvalueanalysis. As is shown, temperature dependency of the constitu-
ents, the Winkler and shear coefficients of elastic foundation,
greatly change the critical buckling temperature and frequenciesof the beam in both cases of thermal loading. The constant K
NL,
however, does not have any influence on DTcr. Besides, the behav-
ior of a beam greatly depends on type of thermal loading and
boundary conditions. The frequency parameter, also, inherits these
effects. It is found that for some cases of boundary conditions,fundamental frequency of the beam approaches zero around a pre-
scribed temperature which is the thermal bifurcation point. How-
ever, for the cases when the load-deflection path is not of the
primary-secondary one, frequency does not approach zero.
Appendix A
The derivatives of a function at a sample point can be approxi-
mated as a weighted linear summation of the value of the function
in the whole domain, which is the basic concept of the GDQ
method. The governing differential equations have been reducedto a set of algebraic equations by this approximation. The number
of algebraic equations depend on number of grid points. The mth
order derivative of the function f(x) with respect to xat a sample
point x
iis approximated by linear sum of all the functional values
at the whole grid points which is suggested by Bellman et al. [ 33]
as
dmfðxȚ
dxm/C12/C12/C12/C12
xi/C25XN
j¼1CðmȚ
ij/C1fðxjȚ (A1)
where Nis the number of grid points, xiis the location of grid
points, f(xj) is the function value at xj, and CðmȚ
ijare the weighting
coefficients corresponding to the mth order derivative. Quan andChang [ 34] suggested a Lagrangian interpolation polynomial to
overcome the numerical ill conditions in determining the weight-
ing coefficients CðmȚ
ij
fðxȚ¼XN
i¼1MðxȚ
ðx/C0xiȚMð1ȚðxiȚfðxiȚ (A2)
where
MðxȚ¼YN
j¼1ðx/C0xjȚ
Mð1ȚðxiȚ¼YN
j¼1ðxi/C0xjȚfor i¼1;2;3;…;N
by substituting Eq. (A2) into Eq. (A1) one may reach
Cð1Ț
ij¼XN
i¼1Mð1ȚðxiȚ
ðxj/C0xiȚMð1ȚðxjȚfor i;j¼1;2;3;…;Nand i6¼j
(A3)
Cð1Ț
ii¼/C0XN
j¼1;j6¼iCð1Ț
ij for i¼1;2;3;…;N (A4)
The coefficients of the first order weighting matrix can be gov-
erned by using Eqs. (A3) and(A4). Higher order coefficient matri-
ces can be expressed as follows:
Cð2Ț
ij¼XN
k¼1Cð1Ț
ikCð1Ț
kj for i;j¼1;2;3;…;N (A5)
Cð3Ț
ij¼XN
k¼1Cð1Ț
ikCð2Ț
kj for i;j¼1;2;3;…;N (A6)
Cð4Ț
ij¼XN
k¼1Cð1Ț
ikCð3Ț
kj for i;j¼1;2;3;…;N (A7)
Various types of grid distribution, which provide acceptable
results, have been introduced. However, in this article we use
normalized Chebyshev–Gauss–Lobatto grid points that are
xi¼1
21/C0cos pi/C01
N/C01/C18/C19/C20/C21
for i¼1;2;3;…;N (A8)
For more details about GDQ and method of distribution of grid
points, one may refer to [ 35,36].
Appendix B
The governing equations and the associated equations for the
pre/postbuckling equilibrium states of the beam are
e1XN
j¼1Cð2Ț
ijU/C3
sjțe1d2XN
j¼1Cð1Ț
ijW/C3
sj !XN
j¼1Cð2Ț
ijW/C3
sj
țde2XN
j¼1Cð2Ț
ijusj¼0
Fig. 10 Effect of various power law indices and temperature
dependency on S–S FGM beams with d50.04 subjected to HC
loading
Journal of Applied Mechanics JANUARY 2014, Vol. 81 / 011004-11
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je1XN
j¼1Cð1Ț
ijusjțdXN
j¼1Cð2Ț
ijW/C3
sj !
ț(
e1"XN
j¼1Cð1Ț
ijU/C3
sj
ț1
2d2XN
j¼1Cð1Ț
ijW/C3
sj !2#
țde2XN
j¼1Cð1Ț
ijusj/C01
12d2XN
j¼1Cð0Ț
ijNT/C3
j)
/C2dXN
j¼1Cð2Ț
ijW/C3
sj/C01
12K/C3
wd3XN
j¼1Cð0Ț
ijW/C3
sjț1
12K/C3
gd3XN
j¼1Cð2Ț
ijW/C3
sj
/C01
12d3K/C3
NLXN
j¼1Cð0Ț
ijW/C3
sj !3
¼0
de2XN
j¼1Cð2Ț
ijU/C3
sjțd2XN
j¼1Cð1Ț
ijW/C3
sj !XN
j¼1Cð2Ț
ijW/C3
sj"#
țd2e3XN
j¼1Cð2Ț
ijusj
/C0je1XN
j¼1Cð0Ț
ijusjțdXN
j¼1Cð1Ț
ijW/C3
sj !
¼0i¼1;2;3;…;N (B1)
Here Cð0Ț
ijis the Kronecker dwhich is equal to one, when i¼j,
otherwise is equal to zero. Also, Cð1Ț
ijandCð2Ț
ijare the weighting
coefficient matrices of first and second order differentiations,
respectively. Besides subscript sindicates the static displacement.
The beam is divided into Ngrid points which indicate the number
of nodes in the ndirection.
The boundary conditions at edge points ( i¼1,N) may be writ-
ten also as:
For clamped end
U/C3
si¼W/C3
si¼usi¼0 (B2)
For simply supported edge
U/C3
si¼W/C3
si¼M/C3
x;si¼0 (B3)
For roller edge
U/C3
si¼usi¼Q/C3
xz;sițðKgțNx;siȚddW/C3
si
dx¼0 (B4)
Appendix C
The governing equations and the associated boundary condi-
tions for the small-scale vibrations of a beam in pre/postbucklingregimes are
e
1XN
j¼1Cð2Ț
ijU/C3
djțe1d2@2W/C3
si
@n2XN
j¼1Cð1Ț
ijW/C3
djțe1d2@W/C3
si
@nXN
j¼1Cð2Ț
ijW/C3
dj
țde2XN
j¼1Cð2Ț
ijudj¼@2
@g2K1
d2XN
j¼1Cð0Ț
ijU/C3
djțK2
dXN
j¼1Cð0Ț
ijudj !
je1XN
j¼1Cð1Ț
ijudjțdXN
j¼1Cð2Ț
ijW/C3
dj !
ț(
e1@U/C3
si
@nț1
2d2@W/C3
si
@n/C18/C192"#
țde2@usi
@n/C0d2
12NT/C3)
dXN
j¼1Cð2Ț
ijW/C3
djț"
e1 XN
j¼1Cð1Ț
ijU/C3
dj
țd2@W/C3
si
@nXN
j¼1Cð1Ț
ijW/C3
dj!
țe2dXN
j¼1Cð1Ț
ijudj#
d@2W/C3
si
@n2ț"
e1 XN
j¼1Cð2Ț
ijU/C3
djțd2@2W/C3
si
@n2XN
j¼1Cð1Ț
ijW/C3
dj
țd2@W/C3
si
@nXN
j¼1Cð2Ț
ijW/C3
dj!
țe2dXN
j¼1Cð2Ț
ijudj#
d@W/C3
si
@n
țe1@2U/C3
si
@n2țe1d2@W/C3
si
@n@2W/C3
si
@n2țe2d@2usi
@n2/C20/C21
dXN
j¼1Cð1Ț
ijW/C3
dj
/C01
12K/C3
wd3XN
j¼1Cð0Ț
ijW/C3
djț1
12K/C3
gd3XN
j¼1Cð2Ț
ijW/C3
dj
/C01
4d3K/C3
NLW/C32
siXN
j¼1Cð0Ț
ijW/C3
dj¼K1
d@2
@g2XN
j¼1Cð0Ț
ijW/C3
dj
/C0je1XN
j¼1Cð0Ț
ijudjțdXN
j¼1Cð1Ț
ijW/C3
dj !
țe2
dXN
j¼1Cð2Ț
ijU/C3
dj
țd3@2W/C3
si
@n2XN
j¼1Cð1Ț
ijW/C3
djțd3@W/C3
si
@nXN
j¼1Cð2Ț
ijW/C3
dj!
țd2e3XN
j¼1Cð2Ț
ijudj¼@2
@g2K2
dXN
j¼1Cð0Ț
ijU/C3
djțK3XN
j¼1Cð0Ț
ijudj !
(C1)
For the small amplitude free vibration analysis one may write
ð@2=@g2ȚhU/C3
dj;W/C3
dj;udji¼/C0 x/C32hU/C3
dj;W/C3
dj;udji.
The boundary conditions at edge points ( i¼1,N) may be writ-
ten also as:
For clamped end
U/C3
di¼W/C3
di¼udi¼0 (C2)
For simply supported edge
U/C3
di¼W/C3
di¼M/C3
x;di¼0 (C3)
For roller edge
U/C3
di¼udi¼Q/C3
xz;dițðKgțNx;siȚddW/C3
di
dnțNx;diddW/C3
si
dn¼0(C4)
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