Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 1, pp. 39-52, Warsaw 2013

AN ANALYTICAL APPROACH TO LARGE AMPLITUDE VIBRATION AND

POST-BUCKLING OF FUNCTIONALLY GRADED BEAMS REST ON

NON-LINEAR ELASTIC FOUNDATION

Hessameddin Yaghoobi, Mohsen Torabi

Young Researchers Club, Central Tehran Branch, Islamic Azad University, Tehran, Iran

e-mail: yaghoobi.hessam@gmail.com; torabi.mech@gmail.com

In this paper, non-linear vibration and post-buckling analysis of beamsmade of functionally
gradedmaterials (FGMs) rest on a non-linear elastic foundation subjected to an axial force
are studied.BasedonEuler-Bernoullibeamtheoryandvon-Karmangeometric non-linearity,
the partial differential equation (PDE) of motion is derived.Then, this PDE problem is
simplified into an ordinary differential equation problem by using the Galerkin method.
Finally, the governing equation is analytically solved using the variational iterationmethod
(VIM). The results from the VIM solution are compared and shown to be in excellent
agreement with the available solutions from the open literature. Some new results for the
non-linear natural frequencies and buckling load of functionally graded (FG) beams, such
as effects of vibration amplitude, elastic coefficients of foundation, axial force, end supports
andmaterial inhomogenity are presented for future references.

Keywords: functionallygradedbeams,non-linearvibration,post-buckling,Galerkinmethod,
variational iterationmethod (VIM)

1. Introduction

Recently, a newclass of compositematerials knownas functionally gradedmaterials (FGMs) has
drawn considerable attention. Typically, FGMs aremade from amixture ofmetals and ceramics
and are further characterized by a smooth and continuous change of mechanical properties
from one surface to another. It has been reported that the weakness of the fiber reinforced
laminated composite materials, such as debonding, huge residual stress, locally largely plastic
deformations, etc., can be avoided or reduced in FGMs (Noda, 1991; Tanigawa, 1995). FGMs
were initially designed as thermal barrier materials for aerospace structures and fusion reactors
where extremelyhigh temperatureand large thermalgradient exist.With the increasingdemand,
FGMs have been widely used in general structures. Hence, many FGM structures have been
extensively studied, suchas functionally graded (FG)beams,plates, shells, etc. Furthermore, due
to huge application of beams in different fields such as civil, marine and aerospace engineering,
it is necessary to study their dynamic behavior at large amplitudes which is effectively non-
linear and therefore, is governed by non-linear equations. Non-linear free vibrationsand buckling
analysis of isotropic and composite beams have received a good amount of attention in the
literature (see References).

The Galerkin finite element method has been presented for studying non-linear vibrations
of beams describable in terms of the moderately large bending theory by Bhashyam and Pra-
thap (1980). Non-linear free vibration analysis of laminated composite beams was studied by
Kapania and Raciti (1989), using the perturbation method. Large amplitude free vibration of
an unsymmetrically laminated beam using von-Karman large deflection theory was studied by
Singh et al. (19991). Free non-linear and undamped vibration analysis of flexible non-prismatic
Euler-Bernoulli beams was investigated by Fertis and Afonta (1992). An analytical method for



40 H. Yaghoobi,M. Torabi

determining the vibration modes of geometrically non-linear beams under various edge condi-
tions was presented by Qaisi (1993). Nayfeh and Nayfeh (1995) obtained non-linear modes and
natural frequencies of a simply supported Euler-Bernoulli beam resting on an elastic founda-
tion with distributed quadratic and cubic non-linearities using the method of multiple scales
and the invariant manifold approach. The geometrically non-linear analysis of flexible sliding
beams based on the assumptions of Euler-Bernoulli beam theory was studied by Behdinan et
al. (1997a,b). Non-linear vibrations of an Euler-Bernoulli beam with a concentrated mass atta-
ched to it were investigated by Karlik et al. (1998). The beam carried its own weight as well
as other weights that were attached to the beam and participated in its vibrational motion.
Ganapathi et al. (1998) studied large amplitude free vibrations of cross-ply laminated straight
and curved beams using the spline element method. Patel et al. (1999) investigated non-linear
free flexural vibrations and post-buckling of laminated orthotropic beams resting on a class of a
two-parameter elastic foundation using a three-nodded shear flexible beam element. Azrar et al.
(1999) developed a semi-analytical approach to the non-linear dynamic response problem based
on Lagrange’s principle and the harmonic balancemethod. Hatsunaga (2001) presented natural
frequencies and buckling stresses of simply supported laminated composite beams taking into
account the effects of transverse shear and rotary inertia. A non-linearmodal analysis approach
based on the invariantmanifoldmethodwas utilized to obtain the non-linear normalmodes of a
clamped-clamped beam for large amplitude displacements by Xie et al. (2002). Guo and Zhong
(2004) investigated non-linear vibrations of thin beams based on sextic cardinal spline func-
tions, a spline-based differential quadrature method. Non-linear normal modes of vibration for
a hinged-hinged beam with fixed ends were evaluated, considering both the continuous system
and finite element models by Carlos et al. (2004).

In recent years, Sapountzakis and Tsiatas (2007) investigated the flexural buckling of com-
posite Euler-Bernoulli beams of arbitrary cross sections. The resulting boundary-value problems
were solved using the boundary elementmethod. Aydogdu (2007) investigated the thermal buc-
kling of cross-ply laminatedbeamswithdifferentboundaryconditions.Nayfeh andEmam(2008)
obtained a closed-form solution for the post-buckling configurations of beams composed of iso-
tropic materials with various boundary conditions. Jun et al. (2008) investigated free vibration
and buckling behavior of axially loaded laminated composite beams having arbitrary lay-up
using the dynamic stiffness method taking into account the influence of axial forces, Poisson
effect, axial deformation, shear deformation and rotary inertia. Pirbodaghi et al. (2009) used
the first-order approximation of the homotopy analysis method to investigate non-linear free
vibrations of Euler-Bernoulli beam.Malekzadeh et al. (2009) studied non-linear free vibrations
of laminated composite thin beams rest on a non-linear elastic foundation (including shearing
layer) with elastically restrained against rotation edges by the differential quadrature approach.
Gupta et al. (2009) studied non-linear free vibrations of isotropic beams using a simple iterative
finite element formulation. An exact solution for the post-buckling of a symmetrically lamina-
ted composite beamwith fixed-fixed, fixed-hinged, and hinged-hinged boundary conditions was
presented by Emam and Nayfeh (2009). Gupta et al. (2010a,b) recently applied the concept of
coupled displacement field criteria to investigate the post-buckling behavior of isotropic (Gupta
et al., 2010b) and composite beams (Gupta et al., 2010a). Gunda et al. (2010) employed the
Rayleigh-Ritz method to study large amplitude vibrations of a laminated composite beamwith
symmetric and asymmetric lay – up orientations.

More recently, the large amplitude vibration and post-buckling analysis of FG beams ha-
ve attracted increasing research efforts. Ke et al. (2010) used the direct numerical integration
method together with Runge-Kutta technique to find the non-linear vibration response of FG
beams with different end supports. Simsek (2010) studied the non-linear forced vibration of
Timoshenko FG beams under action of moving harmonic load. Ma and Lee (2011) presented
a further discussion of non-linear mechanical behavior for FG beams under in-plane thermal



An analytical approach to large amplitude vibration... 41

loading. Fallah and Aghdam (2011) studied large amplitude free vibrations and post-buckling
of FG beams subjected to an axial force.

The pursuit of analytical solutions for the non-linear equation arising in vibration and post-
buckling analysis of FG beams is of intrinsic scientific interest. The primary purpose of the
present paper is to investigate an analytical solution for non-linear vibration and post-buckling
of beams made of FGMs rest on a non-linear elastic foundation subjected to an axial force.
Analytical expressions for non-linear natural frequencies and buckling load of FG beams are
determined using the variational iteration method (VIM) given by He (1999).

2. Basic idea of variational iteration method

To illustrate the basic concept of the technique, we consider the following general differential
equation

Lu+Nu= g(x) (2.1)

where L is a linear operator, N a non-linear operator, and g(x) is the forcing term. According
to the VIM, we can construct a correction functional as follows

un+1(x)=un(x)+

x∫

0

λ[Lun(t)+Nũn(t)−g(t)] dt (2.2)

where λ is aLagrangemultiplierwhich canbe identifiedoptimally via theVIM.Thesubscripts n
denote the n-th approximation, ũn is considered as a restricted variation, that is δũn =0; and
Eq. (2.2) is called acorrection functional.The solution to linearproblemscanbesolved ina single
iteration step due to the exact identification of the Lagrange multiplier. In this method, it is
required first to optimally determine the Lagrange multiplier λ. The successive approximation
un+1, n ­ 0 of the solution u will be readily obtained upon using the determined Lagrange
multiplier and any selective function u0, consequently, the solution is given by

u= lim
n→∞
un (2.3)

3. Problem statement

Consider a straight FGbeamof length L, width b and thickness h rests on an elastic non-linear
foundation and subjected to an axial force of magnitude P as shown in Fig. 1. The beam is

Fig. 1. Schematic of the FG beamwith a non-linear foundation



42 H. Yaghoobi,M. Torabi

supported on an elastic foundation with cubic non-linearity and shearing layer. In this study,
material properties are considered to vary in accordance with the rule of mixtures as

P =PMVM +PCVC (3.1)

where P and V are thematerial property and volume fraction, respectively, and the subscripts
M and C refer to themetal andceramic constituents, respectively. Simplepower lawdistribution
from puremetal at the bottom face (z=−h/2) to pure ceramic at the top face (z=+h/2) in
terms of volume fractions of the constituents is assumed as (Ke et al., 2010)

VC =
(2z+h
2h

)n
VM =1−VC (3.2)

where n is the volume fraction exponent. The value of n equal to zero represents a fully ceramic
beam.Themechanical and thermal properties of FGMs are determined fromthe volume fraction
of material constituents.We assume that the non-homogeneous material properties such as the
modulus of elasticity E, Poisson’s ratio ν andmass density ρ can bedetermined by substituting
Eq. (3.2) into Eq. (3.1) as

E(z)=EM +(EC −EM)
(2z+h
2h

)n
ν(z)= νM +(νC −νM)

(2z+h
2h

)n

ρ(z)= ρM +(ρC −ρM)
(2z+h
2h

)n (3.3)

The force and moment resultants per unit length, based on classical theory of beams in a
Cartesian coordinate system, can be written as (Emam and Nayfeh, 2009)

{
Nx
Mx

}
= b

[
A11 B11
B11 D11

]


u,x+

1

2
w2,x

w,xx




 (3.4)

in which w and u are the transverse and axial displacements of the beam along the z and x
directions, respectively. The stiffness coefficients A11,B11 and D11 are given as follows

(A11,B11,D11)=

h

2∫

−
h

2

E(z)

1−ν2(z)
(1,z,z2) dz (3.5)

After some mathematical simplifications (Emam and Nayfeh, 2009), the governing equation of
non-linear free vibration of an FG beam in terms of transverse displacement can be written as

I1w,tt+b
(
D11−

B211
A11

)
w,xxxx+

(
P−
bA11
2L

L∫

0

w2,x dx−
bB11
L
[w,x(L,t)−w,x(0, t)]

)
w,xx =Fw

(3.6)

in which the comma denotes the derivative with respect to x or t. Furthermore, I1 and Fw are
the inertia term and reaction of the elastic foundation on the beam, which are defined as

I1 =

h

2∫

−
h

2

ρ(z) dz Fw =−kLw−kNLw
3+kSw,xx (3.7)

where kL and kNL are linear and non-linear elastic foundation coefficients, respectively, and
kS is the coefficient of shear stiffness of the elastic foundation.



An analytical approach to large amplitude vibration... 43

For convenience, in the subsequent analysis we use the following non-dimensional variables

x=
x

L
w=
w

r
t= t

√√√√b
(
D11−

B2
11

A11

)

I1L
4

(3.8)

where r=
√
I/A is the radiusof gyration of the cross section.UsingEqs. (3.6) and (3.7) together

with the dimensionless variables defined in Eq. (3.8), the dimensionless form of the governing
equation becomes

w,tt+w,xxxx+

(
P−
1

2
Λ

1∫

0

w2,x dx−B[w,x(1, t)−w,x(0, t)]

)
w,xx+kLw+kNLw

3
−kSw,xx =0

(3.9)

where

P =
PL2

b
(
D11−

B211
A11

) Λ=
A11r

2

(
D11−

B211
A11

) B=
B11r

(
D11−

B211
A11

)

kL =
kLL

4

b
(
D11−

B211
A111

) kNL =
kNLr

2L4

b
(
D11−

B211
A111

) kS =
kSL

2

b
(
D11−

B211
A11

)

(3.10)

Assuming w(x,t) = V (t)φ(x) where V (t) is an unknown time-dependent function and φ(x) is
the first eigenmode of the beampresented inTable 1,whichmust satisfy the kinematic boundary
conditions. ApplyingGalerkin’smethod, the governing equation ofmotion is obtained as follows

V̈ (t)+(α1+Pαp+αkL +αkS)V (t)+α2V
2(t)+(αkNL +α3)V

3(t)= 0 (3.11)

where

α1 =

1∫
0

φ′′′′φ dx

1∫
0

φ2 dx

αp =

1∫
0

φ′′φdx

1∫
0

φ2 dx

α2 =−B[φ
′(1)−φ′(0)]αp

αkL = kL αkNL = kNL

1∫
0

φ4 dx

1∫
0

φ2 dx

αkS =−kSαp

α3 =−Λ
1∫
0

φ′
2
dx

(3.12)

The beam centroid is subjected to the following initial conditions

V (0)= a
dV (0)

dt
=0 (3.13)

where a denotes the non-dimensional maximum amplitude of oscillation. From Eq. (3.11), the
post-buckling load-deflection relation of the FG beam can be obtained as

PNL =−
α1+αkL +αkS +α2V +(αkNL +α3)V

2

αp
(3.14)

It should be noted that neglecting the contribution of V in Eq. (3.14), the linear buckling load
can be determined as

PL =−
α1+αkL +αkS

αp
(3.15)



44 H. Yaghoobi,M. Torabi

Table 1.Trial functions for a FG beamwith various boundary condition

Boundary condition φ(x) Value of q

Simply supported sin
qx

L
π

Clamped-clamped
(
cosh
qx

L
− cos

qx

L

)
−
coshq− cosq

sinhq− sinq

(
sinh
qx

L
− sin

qx

L

)
4.730041

Clamped-simply (
cosh
qx

L
−cos

qx

L

)
−
coshq− cosq

sinhq− sinq

(
sinh
qx

L
−sin

qx

L

)
3.926602

Supported

4. Implementation of VIM

Equation (3.11) can be simplified as

V̈ +β1V +β2V
2+β3V

3 =0 (4.1)

where β1 =α1+PαP +αKL +αKS, β2 =α2 and β3 =αKNL +α3. In order to solve Eq. (4.1)
using VIM, we construct a correction functional, as follows

Vn+1(t)=Vn(t)+

t∫

0

λ
[d2Vn(τ)
dτ2

+ω2Vn(τ)+β1Ṽn(τ)+β2Ṽ
2
n(τ)+β3Ṽ

3
n(τ)−ω

2Ṽn(τ)
]
dτ (4.2)

Its stationary conditions can be obtained as follows

λ′′(τ)
∣∣∣
τ=t
+ω2λ(τ)

∣∣∣
τ=t
=0 1−λ′(τ)

∣∣∣
τ=t
=0 λ(τ)

∣∣∣
τ=t
=0 (4.3)

Thus, the Lagrangian multiplier can therefore be identified as

λ=
1

ω
sin[ω(τ − t)] (4.4)

As a result, we obtain the following iteration formula

Vn+1(t)=Vn(t)+

t∫

0

1

ω
sin[ω(τ− t)]

·

[d2Vn(τ)
dτ2

+ω2Vn(τ)+β1Ṽn(τ)+β2Ṽ
2
n(τ)+β3Ṽ

3
n(τ)−ω

2Ṽn(τ)
]
dτ

(4.5)

From the initial conditions in Eq. (3.13), that we have it in the point t=0, an arbitrary initial
approximation can be obtained

V0(t)= acos(ωt) (4.6)

This initial approximation is a trial function, and it is used to obtain amore accurate approxi-
mate solution to Eq. (3.11). Here, ω is the non-linear frequency. Expanding the non-linear part,
we have

N[V0(t)] =β1acos(ωt)+β2[acos(ωt)]
2+β3[acos(ωt)]

3
−ω2acos(ωt) (4.7)

Then

N[V0(t)] =
(
−aω2+aβ1+

3

4
β3a
3
)
cos(ωt)+

1

4
β3a
3cos(3ωt)

+
1

2
β2a
2cos(2ωt)+

1

2
β2a
2

(4.8)



An analytical approach to large amplitude vibration... 45

In order to ensure that no secular terms appear in the next iteration, the coefficient of cos(ωt)
must vanish. Therefore

ω=

√
β1+

3

4
β3a
2 (4.9)

Thus, the non-linear to the linear frequency ratio can be determined as

ωNL
ωL
=

√

1+
3

4

β3
β1
a2 (4.10)

Using the variational formula (4.5), we have

V1(t)= acos(ωt)+

t∫

0

1

ω
sinω(τ− t)

[d2acos(ωτ)
dτ2

+ω2acos(ωτ)

+
1

4
β3a
3cos(3ωτ)+

1

2
β2a
2cos(2ωτ)+

1

2
β2a
2
]
dτ

(4.11)

Then first-order approximate solution is obtained as:

V1(t)=
(
a+
β2a
2

3ω2
−
β3a
3

32ω2

)
cos(ωt)+

β2a
2

6ω2
cos(2ωt)+

β3a
3

32ω2
cos(3ωt)−

β2a
2

2ω2
(4.12)

Accordingly, inserting Eq. (4.12) into Eq. (3.14), the post-buckling load-deflection can be obta-
ined.

5. Results and discussion

In this Section, we present the results with the VIM, which was described in the previous
section for solving Eq. (3.11). To test the validity and accuracy of the method used in this
study, variations of the non-dimensional amplitude versus time obtained by the VIM and well-
establishedRunge-Kuttamethodaredisplayed inFig. 2.Thisfigure showsaverygoodagreement
between theVIM and numerical solution.

Fig. 2. Variation of the non-dimensional amplitude versus t obtained by VIM (solid line) and
NS (circle): (a) simply supported, (b) clamped-clamped, (c) clamped-simply supported; (kS =50,

kL =50, kNL =50, P =1, n=2, a=1)

Also, to demonstrate the accuracy of the mentioned method, the non-linear to linear fre-
quency ratio ωNL/ωL of the isotropic beams with simply supported (β1 = 97.4091, β2 = 0,
β3 = 24.3523) and clamped boundary conditions (β1 = 501.8177, β2 = 0, β3 = 76.3051) are



46 H. Yaghoobi,M. Torabi

Table 2.Comparison of the frequency ratio ωNL/ωL

a
SS CC

Present [29] [2] [28] [7] Present [29] [2] [28] [7]

1 1.0897 1.0897 1.0891 1.0897 1.0897 1.0554 1.0628 1.0221 1.0572 1.0552

2 1.3228 1.3229 1.3177 1.3228 1.3229 1.2067 1.2140 1.0856 1.2125 1.2056

3 1.6393 1.6394 1.6256 1.6393 1.6393 1.4235 1.3904 1.1831 1.4344 1.4214

4 2.0000 – – 1.9999 1.9999 1.6806 1.5635 1.3064 1.6171 1.6776

compared with those reported in the previous literature in Table 2. It is observed that the pre-
sent results agree very well with those given by Qaisi (1993), Azrar et al. (1999), Pirbodaghi et
al. (2009), Fallah and Aghdam (2011).

The material properties presented in Table 3 are applied to next verifications and to non-
linear analysis of FG beams. Table 4 shows the comparison of our solution with the published
data (Gunda et al., 2010; Ke et al., 2010; Fallah and Aghdam, 2011) for the frequency ratio
ωNL/ωL andbuckling load ratio PNL/PL for both clamped-clamped (CC) and simply supported
(SS) FGbeams.Asmentioned before, Eq. (4.1) contains a quadratic non-linearitymainly due to
the bending-stretching coupling. However, for CC beams either isotropic, composite or FG, the
quadratic term vanishes and, therefore, Eq. (4.1) is reduced to a Duffing equation. In this case,
the exact non-linear frequency ωexact can be determined as shown in Younesian et al. (2010).
Hence, in Table 5, the frequency ratio for CCFG beam is listed. A reasonably good agreement
with previous results for non-linear analysis of SS andCCFG beams can be observed in Tables
4 and 5.

Table 3.Material properties of the constituent materials of the FG beams

Properties of material SuS304 Si3N4

Young’s modulusE [GPa] 207.8 322.3

Poisson’s ratio ν [–] 0.3178 0.24

Material density ρ [kg/m3] 8166 2370

Table 4.Frequency ratio ωNL/ωL and buckling load ratio PNL/PL of the FG beam (kL =50,
kNL =10, kS =5, P =2, n=2)

a
ωNL/ωL PNL/PL

Present [10] [20] [7] Present [7]

SS 0 1.000 1.000 1.000 1.000 1.000 1.000

0.5 1.016 1.008 1.008 1.008 1.021 1.021

1 1.065 1.048 1.046 1.048 1.123 1.123

1.5 1.141 1.118 1.116 1.117 1.305 1.305

2 1.239 1.211 1.209 1.210 1.567 1.567

CC 0 1.000 1.000 1.000 1.000 1.000 1.000

0.5 1.015 1.015 1.015 1.015 1.038 1.038

1 1.058 1.058 1.058 1.058 1.152 1.152

1.5 1.126 1.126 1.126 1.126 1.343 1.343

2 1.215 1.215 1.215 1.215 1.609 1.609



An analytical approach to large amplitude vibration... 47

Table 5. Frequency ratio ωNL/ωL of the clamped FG beam

a
n=0.5 n=1 n=2

Present Exact Present Exact Present Exact

0 1.000 1.000 1.000 1.000 1.000 1.000

1 1.059 1.059 1.059 1.059 1.058 1.058

2 1.220 1.218 1.219 1.217 1.215 1.212

3 1.449 1.441 1.447 1.439 1.439 1.431

4 1.720 1.704 1.716 1.700 1.704 1.689

5 2.015 1.990 2.010 1.986 1.994 1.971

Fig. 3. Effect of the linear foundation stiffness on (a) SS FG, (b) CCFG and (c) CS FG beam for
frequency (solid line) and buckling load (dash line) ratios (kNL =50, kS =50, P =1, n=2)

After these verifications, we investigate the effects of foundation parameters, axial force,
vibration amplitude, end supports, such as SS, CC and clamped-simply supported (CS), and
material inhomogenity on thenon-linear freevibrationsandpost-bucklingbehavior ofFGbeams.
Figures 3a,b,c demonstrate effects of the linear foundation parameter for SS, CC and CS FG
beams. The curve marked kL = 0 represents the case when the beam just has the shearing
layer and non-linear foundation stiffness. It can be seen from these figures that all beams exhibit
typical hardening behavior, i.e., the non-linear frequency ratio increases as the linear foundation
parameter is decreased. Moreover, when the boundary conditions are CS, kL induces a higher
effect on the non-linear free vibrations and post-buckling behavior of FG beams.



48 H. Yaghoobi,M. Torabi

It can be observed from Figs. 4a,b,c that an increase in the value of the shearing layer
stiffness results in a decreasing hardening characteristic of the beam, i.e., a decrease in the rate
of kS causes an increase in the non-linear frequency and post-buckling strengthwith amplitude.
The top curves represent the case when the beam just has linear and non-linear foundation
stiffness. In addition, when the boundary conditions are CC, kS induces a lower effect on the
non-linear free vibrations and post-buckling behavior of FG beams. Nevertheless, an increase
in the value of the non-linear foundation parameter results in increasing non-linear frequency
and post-buckling strength with amplitude. This interesting behavior is shown in Figs. 5a,b,c.
Also, the effects of boundary conditions together with kNL on the non-linear free vibrations and
post-buckling ratio may be interpreted as shown in Fig. 3.

Fig. 4. Effect of the shearing layer stiffness on (a) SS FG, (b) CC FG and (c) CS FG beam for
frequency solid line) and buckling load (dash line) ratios (kNL =50, kS =50, P =1, n=2)

Moreover, the effect of the axial force on the non-linear natural frequency of SS, CC andCS
FGbeams is presented in Figs. 6a,b,c. Results in the figures reveal that as the value of the axial
load increases, the frequency ratio increases as well.

Furthermore, the influences of material inhomogenity in terms of the volume fraction expo-
nent, axial force and dimensionlessmaximum amplitude on the frequency ratio are presented in
Table 6. It is interesting to note from this table that the frequency ratio of the beam initially
increases and then decays by increasing in the value of n.



An analytical approach to large amplitude vibration... 49

Fig. 5. Effect of the non-linear foundation stiffness on (a) SS FG, (b) CC FG and (c) CS FG beam for
frequency solid line) and buckling load (dash line) ratios (kNL =50, kS =50, P =1, n=2)

Table 6. Effect of material inhomogenity on the frequency ratio of the FG beam (kL = 50,
kNL =50, kS =5)

a P
n

0 1 2 3 4 5

SS 0.5 0 1.02904918 1.02919972 1.02890989 1.02872178 1.02862044 1.02856965
5 1.03859244 1.03879152 1.03840822 1.03815943 1.03802539 1.03795822
10 1.05748102 1.05777493 1.05720904 1.05684167 1.05664372 1.05654453

CC 0.5 0 1.04011944 1.04035038 1.03990573 1.03961710 1.03946159 1.03938366
5 1.04796154 1.04823660 1.04770699 1.04736319 1.04717794 1.04708511
10 1.05961742 1.05995749 1.05930270 1.05887760 1.05864854 1.05853375

CS 0.5 0 1.04011944 1.04035038 1.03990573 1.03961710 1.03946159 1.03938366
5 1.04796154 1.04823660 1.04770699 1.04736319 1.04717794 1.04708511
10 1.05961742 1.05995749 1.05930270 1.05887760 1.05864854 1.05853375

6. Conclusion

Large amplitude vibration and post-buckling behavior of functionally graded beams rest on
non-linear elastic foundationwith simply supported, clamped-clamped and clamped-simply sup-
ported boundary conditions were investigated using the variational iteration method. This stu-
dy is within the framework of Euler-Bernoulli’s beam theory and von-Karman’s type of the
displacement-strain relationship. The accuracy of the method was investigated by comparing
the results with those available from the literature and well-established by the Runge-Kutta



50 H. Yaghoobi,M. Torabi

Fig. 6. Effect of the axial load on the frequency ratio of (a) SS FG, (b) CC FG and (c) CS FG beam
(kNL =50, kS =50, kL =50, n=2)

numerical method. The effects of foundation parameters, axial force, vibration amplitude, end
supports andmaterial inhomogeneity on non-linear dynamic behavior of FGbeamswere discus-
sed in detail. As a result, the influence of linear and shear layers of the foundation is to weaken
the non-linear behavior of the FGbeam,whereas the effect of the non-linear foundation stiffness
is to harden the beam response.

The presented expressions are convenient and efficient for the non-linear analysis of FG
beams. Finally, it has been attempted to show the capabilities and wide-range applications of
the VIM in solving such problems.

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Analityczne badania drgań o dużej amplitudzie i ugięcia po wyboczeniu belek z materiału

gradientowego spoczywających na nieliniowo sprężystym podłożu

Streszczenie

Wpracy przedstawiono analizę drgań nieliniowych i zjawisk następujących powyboczeniu w belkach
wykonanych z funkcjonalnych materiałów gradientowych (FGMs), spoczywających na nieliniowo sprę-
żystym podłożu i jednocześnie poddanych osiowemu ściskaniu. Na podstawie teorii Eulera-Bernoulliego
oraz przy uwzględnieniu geometrycznej nieliniowości von Karmana wyprowadzono cząstkowe równanie
różniczkowe ruchu takich układów. Równanie to sprowadzono do postaci różniczkowej zwyczajnej za po-
mocąmetody Galerkina. Na koniec, rozwiązano je analitycznie poprzez zastosowanie iteracyjnej metody
wariacyjnej (VIM), a uzyskane rozwiązanie porównano z innymi, już istniejącymi i znanymiw literaturze,
stwierdzając doskonałą zgodność.Otrzymano również nowe rezultatywpostaci określeniawpływuampli-
tudy drgań, sprężystości podłoża,wartości siły osiowej, rodzaju podparcia brzegóworaz niejednorodności
materiału na częstości własne i obciążenie krytyczne belek gradientowych.

Manuscript received December 2, 2011; accepted for print March 5, 2012