VIBRATION OF FUNCTIONALLY GRADED SANDWICH BEAMS EXCITED BY a MOVING h ARMONIC POINT LOAD DAO ĐỘNG của dầm SANDWICH có cơ TÍNH BIẾN THIÊN CHỊU KÍCH ĐỘNG của lực điều hòa DI ĐỘNG

Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV

VIBRATION OF FUNCTIONALLY GRADED SANDWICH BEAMS EXCITED
BY A MOVING H ARMONIC POINT LOAD
DAO ĐỘNG CỦA DẦM SANDWICH CÓ CƠ TÍNH BIẾN THIÊN CHỊU KÍCH
ĐỘNG CỦA LỰC ĐIỀU HÒA DI ĐỘNG
Van Tuyen BUI1a, Quang Huan NGUYEN2b, Thi Thom TRAN2b, Dinh Kien NGUYEN2b
1
ThuyLoi University, Hanoi, Vietnam
2
Institute of Mechanics,VAST, Hanoi, Vietnam
a
;
ABSTRACT
The vibration of functionally graded (FG) sandwich beams excited by a moving
harmonic point load is studied by the finite element method (FEM). The beams are assumed
to be formed from a homogeneous metallic soft core and two symmetrical FG layers. Based
on the first-order shear deformation beam theory, a finite beam element is formulated by
using the exact shape functions. The implicit Newmark method is employed in computing the
dynamic response of the beams. The numerical results show that the formulated element is
capable to access accurately the dynamic characteristics of the beam by using just several
elements. A parametric study is carried out to highlight the material distribution, the core
thickness to the beam height ratio and the loading parameters on the vibration characteristics.
Keywords: FG sandwich beam, moving load, vibration, dynamic response, FEM.
TÓM TẮT
Dao động của dầm sandwich có cơ tính biến thiên (FG) chịu kích động của lực điều hòa
di động được nghiên cứu bằng phương pháp phần tử hữu hạn (FEM). Dầm được giả định có
một lõi kim lọại và hai lớp ngoài FG, đối xứng qua mặt giữa dầm. Phần tử dầm dựa trên lý
thuyết biến dạng trượt bậc nhất được xây dựng trên cơ sở các hàm dạng chính xác. Đáp ứng
động lực học của dầm được tính bằng phương pháp tích phân trực tiếp Newmark. Kết quả số
chỉ ra rằng phần tử xây dựng trong bài báo có khả năng đánh giá chính xác các đặc trưng động

lực học của dầm chỉ bằng một vài phần tử. Ảnh hưởng của sự phân bố vật liệu, tỷ số giữa độ
dày của lõi và chiều cao dầm cũng như các tham số của lực di động tới các đặc trưng dao
động của dầm được khảo sát chi tiết.
Từ khóa: dầm sandwich FG, lực di động, dao động, đáp ứng động lực học, FEM.
1. INTRODUCTION
Functionally graded (FG) sandwich material is a new type of composite which is widely
used as structural material in recent years. This new composite has many advantages,
including the high strength-to-weight ratio, good thermal resistance and no delaminating
problem which often met in the conventional composites. Investigations on the vibration
analysis of FG sandwich beams have been extensively carried out recently. Mohanty et al. [1]
proposed a finite element procedure for static and dynamic stability analysis of FG sandwich
Timoshenko beams. Bui et al. [2] used the meshfree radial point interpolation method to study
the vibration response of a cantilever FG sandwich beam subjected to a time-dependent tip
load. Adopting the refined shear deformation theory, Vo et al. [3] investigated the free
vibration and buckling of FG sandwich beams. In [4], Vo et al. presented a finite element
model for the free vibration and buckling analyses of FG sandwich beams.

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Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV
Analysis of beams subjected to moving loads is a classical problem in structural
mechanics, and it has been a subject of investigation for a long time. This problem becomes a
hot topic in the field of structural mechanics since the date of invention of FG materials by
Japanese scientists in 1984. A combination of strong and light weight ceramics with
traditional ductile metals remarkably enhances the vibration characteristics of the structures.
The investigations on the dynamic response of FG beams [5-8] in recent years have shown
that the dynamic deflections of an FG metal-ceramic beam considerably reduces comparing to
that of the pure beam. In addition, an FG beam induced by a soft core may improve the
dynamic behavior of the structure when it subjected to moving loads.

The present work aims to study the vibration of an FG sandwich beam excited by a
moving harmonic load, which to the authors’ best knowledge has not been investigated so far.
The beam in this work is assumed to be formed from a homogeneous metallic soft core and
two symmetrical FG skin layers. Based on the first-order shear deformation beam theory, a
finite element beam formulation is derived and employed in computing the dynamic response
of the beam. A parametric study is carried out to highlight the effect of the material
distribution, the ratio of core thickness to beam height as well as the loading parameters on
the vibration characteristics of the beam.
2. MATHEMATICAL FORMULATION
Figure 1 shows a simply supported FG sandwich beam with length L, height h, width b,
core thickness h C in a Cartesian co-ordinate system (x, z). The beam is assumed to be
subjected to a harmonic load P=P 0 cos(Ωt), moving from left to right at a constant speed v.

Figure 1: FG sandwich beam under a moving harmonic load
The beam is assumed to be formed from a metallic soft core and two FG layers with the
volume fraction of the constituent materials follows a power-law function as follows
  2z + h
C

−
h
h
 C

=
Vc  0

  2 z − hC
 h−h
C







n





n

for

− h / 2 ≤ z ≤ − hC / 2

for − hC / 2 ≤ z ≤ hC / 2
for

(1)

hC / 2 ≤ z ≤ h / 2

and V m =1-V c . Here and afterwards, the subscripts ‘c’ and ‘m’ are used to indicate the
‘ceramic’ and ‘metal’, respectively. In Eq.(1), n is the material power-law index, defining the
variation of the constituent materials through the beam thickness. From Eq.(1) one can see
that the top and bottom surfaces of the beam are pure ceramic, and the core is full metal. The
effective property P(z) (such as Young’s modulus, shear modulus and mass density) can be

evaluated by Voigt model and having the form.

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Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV

P( z )

n

 2 z + hC 
( Pc − Pm ) 
 + Pm

 hC − h 

 Pm

n
 2 z − hC 

( Pc − Pm )  h − h  + Pm
C 



for

− h / 2 ≤ z ≤ − hC / 2

for − hC / 2 ≤ z ≤ hC / 2     

(2)

hC / 2 ≤ z ≤ h / 2

for

where P c and P m are the material properties of ceramic and metal, respectively. Based
on the first-order shear deformation beam theory, the displacements u 1 and u 3 in x and z
directions at any point of the beam are given by
u1 ( x=
, z , t ) u ( x, t ) − zθ ( x, t ),

(3)

u3 ( x, z , t ) = w( x, t ),

in which u(x,t) and w(x,t) respectively are the axial and transverse displacements of the
corresponding points on the mid-plane; θ(x,t) is the rotation of the cross section. The strains
and stresses based on Hook’s law resulted from Eq. (3) are as follows
εx =
u, x − zθ, x , γ xz =
w, x − θ

(4)

=
σ x E=

( z ) ε x , τ xz G ( z ) γ xz

where ε x , σ x , γ xz , τ xz are respectively the axial strain, axial stress, and the shear strain
and shear stress; G(z)=E(z)/[2(1+ν)] is the shear modulus. The Poisson’s ratio is assumed to
be constant in the present work.
From Eq. (4), the strain energy of the beam can be can written in the form
1 L
 A11u, x 2 + A22θ, x 2 + ψ A33 ( w, x − θ ) 2 dx
∫
0
2

=
U

(5)

In Eq. (5), ψ is the shear correction factor, equals to 5/6 for the rectangular section
herein; A 11 , A 22 , A 33 respectively are the axial, bending and shear rigidities, defined as
=
( A11 , A22 )

E ( z )(1, z )dA , A
∫=
∫ G ( z )dA
2

(6)

33

A

A

Using Eq. (2), one can write the rigidities in Eq. (6) in explicit forms as follows
Ec + nEm
n +1
(h − hC )( Ec − Em )  (h − hC ) 2 2h(h − hC )
h 2   bhC3 h 2 + hhC + hC2
=
−
+
+
A22

+
n+2
n + 1   12
4
3
 n+3
A=
bhC Em + (h − hC )
11

A=
bhC Gm + (h − hC )
33


 Em


(7)

Gc + nGm
n +1

Eq. (3) gives the kinetic energy of the beam in the form
=
T

with

1 L
 I11 (u 2 + w 2 ) + I 22θ 2  dx
∫
0
2

( I11 , I 22 ) = ∫ ρ ( z )(1, z 2 )dA

(8)
(9)

A

are the mass moments, and as the rigidities these mass moments can be computed
explicitly.

The potential of the moving load P is simply given by
V=
− P0 cos(Ωt ) w( x, t ) δ ( x − vt )

where δ is the delta Diract function.
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Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV
Applying Hamilton’s principle to Eqs. (5), (8) and (10), one can obtain the equations of
motion for the beam in the form
I11u − A11u, xx =
0
 − ψ A33 ( −θ, x + w, xx=
I11w
) P0 cos(Ωt )δ ( x − vt )

(11)

I 22θ − A22θ, xx + ψ A33 ( −θ + w, x ) =
0

Eq. (11) has the same form as of homogeneous beams subjected to a moving harmonic
load, but the rigidities and mass moments are now defined by Eqs. (6) and (9), respectively. It
should be noted that due to the material property is symmetrically with respect to the midplane, the coupling axial stretch and bending terms are not appeared in the governing
equations as in case of FG beams.
3. FINITE ELEMENT FORMULATION
The finite element method is used herein in solving Eq. (11). To this end, the beam is

assumed being divided into a numbers of two-node beam elements with length of l. The
vector of nodal displacements for a generic element (i,j) has the following components
d = {ui

wi θi

wj θ j }

T

uj

(12)

where and hereafter a superscript ‘T’ denotes the transpose of a vector or a matrix. By
introducing the shape functions for the displacement field, we can write the displacements and
rotation inside the element as follows
=
u N=
N=
N wd
ud , θ
θd , w

(13)

where N u , N w , N θ are the matrices of the shape functions for u, w and θ, respectively.
The exact linear, quadratic and cubic polynomials previously deried by Nguyen et al. [7] by
solving the static equilibrium of a beam segment are employed herein to interpolate u, θ and
w, respectively. Using Eqs. (12) and (13), one can write the strain energy, kinetic energy and

the potential of the external load in term of the nodal displacement vector as follows
1 nELE T
1 nELE T
d kd
=
∑
∑ d ( k uu + kθθ + k γγ ) d
2 i 1=
2 i1
=
1 nELE  T  1 nELE  T
d ( m uu + m ww + mθθ )d
=
 T
∑ d md = 2 ∑
2 i 1=
=
i 1
=
U

(14)

−dT P0 cos(Ωt )NTw δ ( x − vt ) =
−dT f
V=

where nELE is the total number of the elements; k, m, f respectively are the element
stiffness, mass matrices and load vector of the element. The stiffness matrices k uu , k θθ , k γγ
and the mass matrices m uu , m ww , m θθ in Eq. (14) have the folloing forms

l

l

l

T
T
T
k uu =
∫ Nu , x A11Nu , x dx, kθθ =
∫ Nθ , x A22 Nθ , x dx, k γγ =−
∫ (N w , x Nθ ) ψ A33 (N w, x − Nθ )dx
0

0

l

=
m uu

0

l

N I N dx, m
I N dx, mθθ ∫ Nθ I
∫=
∫ N=

T
u 11

0

u

T
w, x 11

ww

(15)

l

T

w, x

0

22

Nθ dx

0

Using Eqs. (14) and (15) one can rewrite equations of motion for the beam in terms of
finite element analysis as follows.

 + KD =
MD
F

(16)

in which M, K and F respectively are the global mass, stiffness matrices and load
vector. These matrices and vector are obtained by assembling the element mass, stiffness and
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Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV
load vector m, k and f derived above in the standard way of the finite element analysis. Eq.
(16) can be solved by the direct integration Newmark method. Here, the average acceleration
method which ensures the unconditional stability [9] is employed.
4. NUMERICAL RESULTS
The derived element formulation has been implemented into a computer code and
employed in analysis of the FG sandwich beam subjected to a moving harmonic point load. A
simply supported beam composed of Aluminum (Al – metal phase) core and AluminumAlumina (Al-Al 2 O 3 ) FG layers is considered in this section. The material data for Aluminum
and Alumina are as follows: E m =70 GPa, ρ m = 2702 kg/m3 for Aluminum, and E c =390 GPa,
ρ c = 3960 kg/m3 for Alumina. The amplitude of the moving load is taken by P 0 =100kN. A
total of 500 steps are used for Newmark method in all the computations reported below.
Table 1 lists the fundamental frequency parameter μ of the FG sandwich beam for
various values of the core thickness to the beam height ratio h C /h and the material index n.
The frequency parameter in the Table is defined as follows

µ=

ω1 L2

ρm

h

Em

(17)

in which ω 1 is the fundamental frequency of the beam, and ρ m and E m are the mass
density and Young’s modulus of the core material, respectively.
The effect of the material index n and the core thickness to the beam height ratio h C /h on the
fundamental frequency of the beam is clearly seen from the Table. At a given value of the
h C /h, the fundamental frequency of the beam is smaller for the beam associated with a higher
index n. The effect of the h C /h ratio on the frequency is similar to that of the index n, and at a
given value of the index n, the frequency also decreases by raising the h C /h ratio. The
decrease in the fundamental frequency by raising the index n can be explain by the fact that,
as seen from Eq. (1), the beam associated with a higher index n contains more percentage of
metal. As a result, the rigidities of the beam, defined by Eq. (6), will be decreased, and this
leads to the lower frequency of the beam. The reduction in the fundamental frequency of the
beam by raising the h C /h ratio can be explained by the same reason as by raising the index n,
that is the rigidities of the beam reduced by raising the h C /h ratio.
It should be noted that the volume fraction of the constituent materials defined in this
paper is different from some works published before, e.g. the work by Vo et al. [3]. In Ref.
[3], the volume fraction of metal is defined first, and in this case, the volume fraction of metal
decreases by raising the index n. As a results, the frequency parameter in Ref. [3] increases by
increasing the index n. The authors have computed the frequency of the beam by using the
definition of the volume fraction in Ref. [3] (not shown herein), and a good agreement was
obtained.
Table 1: Fundamental frequency parameter μ of FG sandwich beam
h C /h

n
0
1/5
1/3
1/2
4/5
0.2

5.3926

5.4790

5.4842

5.3757

4.5456

0.5

5.3540

5.3631

5.3138

5.1468

4.3060

1

5.2395

5.1682

5.0675

4.8512

4.0342

2

4.9806

4.8351

4.6940

4.4499

3.7178

5

4.4054

4.2236

4.0774

3.8597

3.3343

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Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV
In Table 2, the maximum frequency parameter, max(f D ), of the beam is given for
various values of the the core thickness to the beam height h C /h ratio, and the material index
n. The efrequency factor f D is defined as follows
fD =

max( w( L / 2, t ))
w0

(18)

where w 0 is the deflection of simply supported homogeneous beam made of the core
material under a static load P 0 at the mid-span, that is
w0 =

P0 L3
48 Em I

(19)

The maximum deflection factor, as seen from Table 2 steadily increases when raising

the index n and the h C /h ratio. The increase in the maximum deflection factor can also be
explained by the reduction in the rigidities of the beam by increasing the index n and the core
thickness to beam height ratio.
Table 2: Maximum dynamic deflection factor max(f D ) of FG sandwich beam
h C /h
Index n
1/10
1/8
1/4
1/2
3/4
0.2

0.9731

0.9748

0.9879

1.0526

1.2277

0.5

1.0169

1.0200

1.0403

1.1180

1.2953

2

1.1752

1.1812

1.2152

1.3115

1.4690

3

1.2467

1.2531

1.2888

1.3836

1.5245

5

1.3462

1.3526

1.3874

1.4730

1.5875

Figure 2: Time-histories for mid-span deflection of FG sandwich beam at various values
of the index n, moving speed v, h c /h ratio and excitation frequency Ω.
In Figure 2, the time-histories for the mid-span deflection of the beam are depicted for
various values of the material index n, moving speed v, core thickness to beam height ratio
h C /h and excitation frequency Ω. In the figure, DT is the total time necessary for the load to
pass the beam. The following comments can be made from the figure.

• The dynamic response of the beam is governed by many parameters, including the
material distribution, core thickness to beam height ratio, moving speed and excitation
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Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV
frequency. These factors not only change the maximum amplitude of the dynamic deflection,
but the time at which the maximum deflection attains also.

• The dynamic deflection considerably increases when raising the moving speed. The
beam executes less vibration cycles when it subjects to a higher speed moving load than when
it subjects to a lower moving speed load. At a given value of the material index, the core

thickness to the beam height ratio and the moving speed, the dynamic deflection rapidly
increases when raising the excitation frequency towards to fundamental frequency of the
beam. Different from the moving speed, the number of vibration cycles which the beam
executes when it subjects to a higher frequency moving load is larger.

Figure 3: Relation between the deflection parameter and the moving speed of FG
sandwich beam with various values of index n and h c /h ratio
In order to examine the effect of the material distribution and the core thickness on the
dynamic response of the beam, the relation between the deflection parameter f D and the
moving speed v of the FG sandwich beam under a moving point load (Ω=0) is depicted in
Figure 3 for different values of the index n and the core thickness to beam height ratio h C /h.
The following comments can be drawn from Figure 3

• At a given value of the index n and the core thickness to beam height ratio, the curve
represented the relation between the deflection factor and the moving speed of the FG
sandwich beam is similar to that of homogeneous beams [10]. When the moving speed is
larger than a certain value, the parameter f D steadily increases and it reaches a peak value
before descending. For the lower values of the moving speed, the parameter f D in Figure 3
both increases and decreases with increasing v. This phenomenon is associated with the
oscillations as seen from the time-histories depicted in Figure 2 as explained by Olsson in
Ref. [10].

• Regardless of the moving speed, the deflection factor f D increases when raising the
material index n or the core thickness to the beam height. This phenomenon, as explained
above, due to the decrease of the beam rigidities when raising the index n and the h C /h ratio.
5. CONCLUSIONS
The paper investigated the vibration of FG sandwich beam excited by a moving harmonic
point load by using the finite element method. The beam is assumed to be formed from a
homogeneous metallic soft core and two symmetrical FG layer. A beam element based on the
first-order shear deformation beam theory was formulated and employed in the investigation.

The direct integration Newmark method has been used in computing the dynamic response of
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Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV
the beam. The numerical results have shown that the vibration characteristics of the beam,
including the fundamental frequency and dynamic deflection factor, are strongly affected by
the material distribution, the core thickness to the beam height ratio, the speed and frequency
of the moving force. The dynamic deflection factor increases by raising the index n and the
core thickness to beam height ratio. The moving speed and the excitation frequency not only
alters the amplitude of the dynamic deflection but it also changes the vibration cycles which
the beam executes.
REFERENCES
[1] Mohanty, S.C., Dash R.R., & Rout, T., Static and dynamic stability analysis of a
functionally graded Timoshenko beam. International Journal of Structural Stability and
Dynamics, 2012, Vol. 12(4), DOI: 10.1142/S0219455412500253.
[2] Bui, T.Q., Khosravifard, A., Zhang, Ch., Hematiyan, M.R., & Golub, M.V., Dynamic
analysis of sandwich beams with functionally graded core using a truly meshfree radial
point interpolation method, Engineering Structures, 2013, Vol. 47, p 90-104.
[3] Vo, T.P., Thai, H.T., Nguyen, T.K., Maheri, A., & Lee, J., Finite element model for
vibration and buckling of functionally graded sandwich beams based on a refined shear
deformation theory, Engineering Structures, 2014, Vol. 64, p. 12-22.
[4] Vo, T.P., Thai, H.T., Nguyen, T.K., Maheri, A., Inam, F., & Lee, J., A quasi-3D theory
for vibration and buckling of functionally graded sandwich beams, Composite Structures,
2015, Vol. 119, p. 1-12.
[5] Şimşek, M., & Kocatürk, T., Free and forced vibration of a functionally graded beam
subjected to a concentrated moving harmonic load, Composite Structures, 2009, Vol. 90
(4), p. 465–473.
[6] Şimşek, M., Vibration analysis of a functionally graded beam under a moving mass by
using different beam theories, Composite Structures, 2010, Vol. 92 (4), p. 904-917.

[7] Nguyen, D.K., Gan, B.S., & Le, T.H., Dynamic response of non-uniform functionally
graded beams subjected to a variable speed moving load, Journal of Computational
Science and Technology, JSME, 2013, Vol. 7(1), p. 12-27.
[8] Le, T.H., Gan, B.S., Trinh, T.H., & Nguyen, D.K., Finite element analysis of multi-span
functionally graded beams under a moving harmonic load. Mechanical Engineering
Journal, Bulletin of the JSME, 2014, Vol. 1(3), p. 1-13.
[9] M. Géradin, D. Rixen, Mechanical Vibrations. Theory and Application to Structural
Dynamics, Second edition, John Willey & Sons, Chichester, 1997.
[11]M. Olsson, On the fundamental moving load problems, Journal of Sounds and Vibration,
1991, Vol. 145(2), p. 299-307.
AUTHORS’ INFORMATION
1.

Bui Van Tuyen (e-mail: ) is a Lecture at the ThuyLoi University.
His current research is finite element modeling of FG structures subjected to moving
loads.

2.

Nguyen Quang Huan (e-mail: nqhuan@.mail.ac.vn), Tran Thi Thom
(ttthom@.mail.ac.vn) and Nguyen Dinh Kien (ndkien@.mail.ac.vn) are research
members at the Institute of Mechanics, VAST. Their interested topic is development of
finite element formulations for analysis of solids and structures.

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