Summary of doctoral thesis in mechanical engineering and engineering mechanics: Dynamic of FGM beams with porosities in thermal environment under moving load

MINISTRY OF EDUCATION AND

VIETNAM ACADEMY OF

TRAINING

SCIENCE AND TECHNOLOGY

GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY
-----------------------------

BUI VAN TUYEN

DYNAMIC OF FGM BEAMS WITH POROSITIES IN
THERMAL ENVIRONMENT UNDER MOVING LOAD

Major: Engineering mechanics
code: 9520101

SUMMARY OF DOCTORAL THESIS
IN MECHANICAL ENGINEERING AND ENGINEERING
MECHANICS

Hanoi – 2018


The thesis has been completed at: Graduate University Science and
Technology – Vietnam Academy of Science and Technology.

Supervisors: 1. Assoc. Prof. Dr. Nguyen Dinh Kien
2. Dr. Tran Thanh Hai

Reviewer 1: Prof. Dr. Hoang Xuan Luong
Reviewer 2: Assoc. Prof. Dr. Tran Minh Tu
Reviewer 3: Assoc. Prof. Dr. Phan Bui Khoi

Thesis is defended at Graduate University Science and TechnologyVietnam Academy of Science and Technology at…o’clock...’, on
………. ,…….…, 2018

Hardcopy of the thesis be found at :
- Library of Graduate University Science and Technology
- Vietnam national library


1
INTRODUCTION
1. Relevance of the thesis topic
The effect of porosities which can be occurred within FGMs
during the process of sintering has been investigated recently [16, 17,
18, 19]. Due to FGM beams are often used in thermal environment,
investigation on the influence of temperature on free vibration of
FGM beams has been studied by several authors [20, 21]. To the
author’s best knowledge, to the date only Wang and Wu considered
forced vibration of FGM beam in thermal environment under moving
loads [22], in which dynamic response of FGM beam under a
harmonic moving load has been studied by Lagrange method. It
should be emphasized that in the authors [22] studied axially FGM
perfect beams (without porosities), and only the case of uniform
temperature is considered. Mathematically, the uniform temperature
distribution is a special case of nonlinear temperature field, and it is
relatively simple from computational point of view. Investigation on

vibration of FGM beams in thermal environment thus is important
from both the research and practical points of view.
2. Thesis objective
This thesis aims to develop finite element models for studying
vibration of FGM porous beams in thermal environment under
moving loads.
3. Scope
The thesis focuses on two-phase transverse FGM beams forming
from ceramic and metal. The external loads considered in the thesis
are the constant speed moving forces and moving harmonic forces.
4. Research methods
Both analytical method and finite element analysis are employed
in the thesis. The analytical method is used to derive equations of
motion for the beam, and the finite element method is then employed
to solve the governing equations and to determine the dynamic
characteristics of the beams.
5. Thesis organization


2
Beside Introduction section, the thesis contains 4 Chapters, a
Conclusion section and a list of publications relevant to the thesis.
References cited in the thesis are listed at the end of the thesis.
CHAPTER 1. OVERVIEW
1.1. FGM beams
Functionally graded materials (FGMs) can be considered as a
new type of composite material. These materials are often formed
from two or more constituent materials whose volume fraction varies
in one or more predefined spatial directions. FGMs overcome the
disadvantage problems such as delamination and stress concentration

which often seen in conventional composite materials. With such
advantages, FGMs have great potential in applications where the
operating conditions are severe, including spacecraft heat shields,
heat exchanger tubes, biomedical implants, flywheels, and plasma
facings for fusion reactors, etc. [24].
FGM beams, the structure considered in this thesis, are assumed
to be formed from two phases, metal and ceramic. The volume
fraction of constituents is considered to vary in a spatial direction,
namely in the beam thickness, by a power-law distribution as [3]
n

h
h
 z 1
Vc     ,   z  , Vc  Vm  1
(1.1)
2
2
h 2
where Vc, Vm are, respectively, the volume fraction of ceramic and
metal; z is the co-ordinate in the thickness direction, and n is material
grading index, defining the material distribution of the constituents.
In addition to the distribution (1.1), several authors also considered
the variation of the material properties in axial or both axial and
thickness directions.
1.2. Investigations on FGM beams
1.2.1. Mechanical behavior of FGM beams
The traditional analytical methods, especially Galerkin method,
are employed by researchers in studying mechanical behavior of
FGM beams [35-41]. Finite element method (FEM) is also widely

used to study the behaviour of FGM beams. Several finite element
beam models for analysis of FGM beams have been proposed in


3
recent years [59-64], for example the works by Alshorbagy et al. [25]
Mohanty el al. [66, 67], Gan and Nguyen [70, 71, 72]. Eltaher et al.
[73, 74] considered the physical neutral axis position in the
derivation of a finite element model for free vibration analysis of
macro/nano FGM beams. Jin and Wang [76] used the quadrature
method to derive stiffness and mass matrices for free vibration
analysis of FGM beams. Based on the first-order shear deformation
theory, Frikha et al. [77] developed a mixed finite element
formulation for bending analysis of FGM beams.
1.2.2. FGM beams with porosities
Porosities can lower the material stiffness, and as a result they
reduce the ability to resist external loads of structural components.
Wattanasakulpong and Ungbhakorn [18], Wattanasakulpong and
Chaikittiratana [19] proposed a simple model for free vibration
analysis of FGM porous beams, in which the porosity volume is
equally divided to both the ceramic and metal phases. The model has
been employed by Ebrahimi and Zia [79] to study nonlinear free
vibration of FGM Timoshenko beams. Chen et al. [16] proposed a
concept “porosity coefficient” in their study of bending and stability
of FGM porous beams. The model in [16] is then extended by the
authors to nonlinear vibration of sandwich beams with FGM porous
core [80], and free and forced vibration of FGM Timoshenko beams
with porosities [81]. Shafiei and Kazemi [82] studied stability of
nano/micro FGM porous beams with modification of the poroposity
model in [18, 19] by considering non-uniform distribution of

porosities in the beam cross sections. The non-uniform distribution
of porosity model has also been employed to study vibration of 2DFGM beams [83].
1.2.3. FGM beams in thermal environment
Chakraborty et al. [84] derived a finite element Timoshenko
beam model for studying wave propagation in sandwich beams with
FGM core in consideration of a uniform temperature rise. Based on
the finite element method, Bhangale and Ganesan [85] investigated
the effect of temperature on natural frequency and loss factor of
FGM sandwich beams with visco-elastic core. Ching and Yen [86]
presented a numerical solution for the themo-mechanical


4
deformation problem of FGM beams. The differential quadrature
method (DQM) has been employed by Xiang and Yang [87] in
studying vibration of non-uniform layer FGM Timoshenko beams
prestress by temperature. Pradhan and Murmu [88] studied free
vibration of FGM sandwich beams resting on an elastic foundation.
DQM was also employed by Malekzadeh [89], Malekzadeh et al.
[90] in free vibration analysis of FGM circular arches and FGM
curved beams in thermal environment. Esfahani et al [92] examined
the influence of elastic foundation support and temperature rise on
the nonlinear stability of FGM Timoshenko beams by the general
DQM. Mahi et al. [30] presented an analytical method to evaluate the
effect of temperature rise on natural frequencies of shear deformable
FGM beams. Wattanasakulpong et al. [21] constructed the governing
equations of thermo-mechanical stability and free vibration of FGM
beams. Ma and Lee [95] proposed an analytical solution for
nonlinear behavior of FGM beam under thermal loading. Analytical
method has also been employed by Eroglu in free vibration analysis

of FGM beams in thermal environment [96]. Trinh et al. [98]
presented an analytical method for vibration and stability analysis of
FGM beams under thermo-mechanical loads. With the aid of RungeKutta method, Kiani et al. [99] examined the effect of environmental
temperature on low velocity impact behavior of FGM beams.
Ghiasian t al. [100] studied static and dynamic stability of FGM
Euler-Bernoulli beams subjected to uniform temperature rise.
Ebrahimi et al. [17] derived equations of motion for studying free
vibration analysis of FGM Euler-Bernoulli beams with porosities in
thermal environment.
1.2.4. FGM beams under moving loads
Lagrange multiplayer method has been employed by Şimşek and
his co-workers in studying vibration of FGM beams excited by
moving loads [4, 5, 6, 8, 10, 11]. Yang et al. [104] studied vibration
of cracked beams under a moving load by assuming an exponential
variation of material properties in the thickness direction. The Ritz
and differential quadrature methods was used by Khalili et al. [105]
to investigate dynamic behavior of FGM beams subjected to a
moving mass. Rajabi et al. [7] employed Petrov–Galerkin method to
transfer a system of the fourth order differential equations of BFGM


5
beams under a moving oscillator to a system of second order
differential equations, and then solving the system by Runge-Kutta
method. Wang and Wu [22] employed Lagrange method to examine
the effect of uniform temperature rise on dynamic bahaviour of
Timoshenko beams formed from axially FGM. Taking the effect of
neutral axial position into account, Gan and Nguyen [106]
formulated a finite beam element for dynamic analysis of multi-span
FGM Timoshenko beams. FEM has also been employed by Gan and

his co-worker in studying axially FGM beams under moving loads
[26], and FGM beams with an immediate support transverse by
moving force [107].
1.3. Studies on FGM beams in Vietnam
Using an analytical m method, Nguyen et al. [111] studied the
static bending and vibration of axially loaded FGM Timoshenko
beam. Bending and vibration of FGM beams have also been
considered by Thai and Vo [112] by different higher-order beam
theories. Based on the third-order shear deformation theory, , Vo et
al. [113] derived equations of motion for FGM sandwich beams with
a homogeneous core, and then employed the finite element method
to compute the natural frequencies of the beams. Vo et al [34]
developed a finite element model for bending and free vibration
analysis of FGM beams. The vibration and diagnosis problems of
cracked beams have been considered by Huyen [114], Huyen and
Khiem [115], Khiem et al [116, 117] by using analytical methods.
Based on the co-rotational approach, Nguyen et al. [118, 119, 120]
derived the finite element models for large displacement analysis of
tapered FGM beams. The finite element method was also employed
by Nguyen and his co-workers to study the large deformation
analysis of FGM frames [121], FGM sandwich frames [33].
Recently, the effect of plastic deformation on stability behavior and
nonlinear bending of FGM beams has been studied by the finite
element method [122, 123, 124].
Vibration of FGM beams under moving loads has been
investigated by several authors in recent years. Trung [13] studied
vibration of FGM beams due to a moving mass or a moving
harmonic load by the finite element method. Ha et al. derived a new
finite element model for vibration analysis of multi-span FGM

6
beams under a harmonic load [14], non-uniform FGM beams under
multiple moving loads [15]. Nguyen et al. [133] used polynomials
derived by Kosmatka to derive a finite element formulation for
vibration analysis of non-uniform FGM beams subjected to a
variable speed moving load. The Kosmatka polynomials have also
been employed by Nguyen et al. [9] in derivation of stiffness and
mass matrices for analyzing 2-D FGM beams under a moving load.
1.4. Comments and study orientation
As can be seen from the above literature review, the
investigation on vibration of FGM beams under moving loads has
been considered by very few authors in recent years. In [12], Ha has
successfully derived finite element formulations for vibration
analysis of FGM beams due to moving loads, but the effect of
porosities and temperature has not been examined. The influence of
porosities and temperature has been considered by several authors,
but only on the free vibration problem. Rigidities and mass moments
of FGM beams are altered when considering the effect of porosities.
In addition, the beams are not only under thermal loading due to the
temperature rise, but the elastic moduli are decreased also. These
factors significantly influence dynamic behavior of the beams, and it
is necessary to take them into consideration. From the above reason,
this thesis investigates the vibration of FGM beams with porosities in
thermal environment under moving loads.
CHAPTER 2. FGM BEAMS IN THERMAL ENVIRONMENT

2.1. FGM beam under moving loads
Fig. 2.1 illustrates a FGM beam with length of L, rectangular
cross section with constant width b and height h. The beam is under

actions of nF loads, F1, F2, … FnF, moving from left to right with a
constant speed v. The beam is assumed to be formed from two
constituent materials, ceramic and metal, whose volume fraction
varies according to a power law as
n

 z 1
Vc     ,Vc  Vm  1
 h 2

(2.1)

in which Vc , Vm are the volume fraction of ceramic and metal,
respectively; z is a co-ordinate along the thickness direction, and n


7
(nonnegative) is a material index, defining the distribution of the
constituent materials.
z

MÆt c¾t ngang dÇm

y

F nF

F2

F1 gèm (Ec, Gc,  c)

x
h
kim lo¹i (Em, Gm,  m)

L

z,w
lç rçng

y

h
b

b

Fig.2.1. FGM beam with porosities under moving loads
2.2. Porosities of FGM beam
In the model in [18, 19], the porosity volume fraction V
(V<<1) is assumed equally divided into ceramic and metal phases.
When the beam is in thermal environment, the material properties of
the FGM beam are evaluated according to
n

z 1
P( z,T )  P (T )  Pm (T )  
h 2
 Pm (T )   Pc (T )  Pm (T ) V
2


 c


 

(2.3)

where Pc and Pm are, respectively, the property of ceramic and metal,
and they depend upon temperature T (K) of the environment; Vα is
the porosity volume fraction.
2.3. Temperature in FGM beam
Temperature distribution along the thickness direction of FGM beam
can be obtained by solving Fourier equation [91, 103]


d 
dT 
 ( z)   0

dz 
dz 

(2.4)

with boundary conditions T = Tc at z = h/2 and T = Tm at z = - h/2. In
Eq. (2.4), the thermal conductivity κ(z) is assumed to be temperatureindependent. Solving Eq. (2.4) gives the temperature distribution
along the beam thickness in the form

8
z

h/ 2

dz
dz
T  Tm  (Tc  Tm ) 
/ 
 ( z )  h/ 2  ( z )
 h /2

(2.6)

It can be seen from above equation that if Tc = Tm then T = Tc = Tm. In
this case, the temperature is the same at every points of the beam,
and it is called the uniform temperature rise (UTR). In case Tc ≠ Tm,
the temperature is a nonlinear function of the z co-ordinate. The
temperature field thus is a nonlinear temperature rise (NLTR). In this
thesis, a temperature rise T for the NLTR is defined in accordance
with the works in [17, 21], that is T = Tc – Tm = Tc - T0 , with T0 =
300K is the referenced temperature.
2.4. Effect of temperature on material properties
Touloukian [130] shows that a property P of a material depends
on temperature by a nonlinear relation as
2
3
P  P0 ( P1T 1  1  PT
)
1  P2T  PT

3

(2.18)

in which P0, P-1, P1, P2 and P3 the temperature-dependent
coefficients.
Figs. 2.2 and 2.3 illustrate the influence of the porosity volume
fraction Vα and the temperature rise ΔT on Young’s modulus of a
FGM beam formed from stainless steel SUS304 and alumina for
various values of V and for ΔT = 500K. As can be seen from the
figures, Young’s modulus decreases clearly when the effect of
porosities is taken into account, for both the case of UTR and NLTR.
By comparing Fig. 2.2(b) Fig. 2.3, one can see that the effective
Young’s modulus E of the FGM beam decreases more significantly
for the case of UTR.
2.5. Governing equations
2.5.1. Displacement field
Axial and transverse displacements at any arbitrary point of the
beam are given by
u ( x, z , t )  u0 ( x, t )  z ( x, t )
w( x, z , t )  w0 ( x, t )

(2.24)


9
in which u0(x,t) and w0(x,t) are, respectively, the axial and
transverse displacements of a point on the mid-plane; θ(x,t) is the
cross-sectional rotation, and t is the time variable.
2.5.2. Strain and stress fields

The normal and shear strains resulted from Eq. (2.24) are as
follows

 xx  u, x  u0, x  z , x

(2.25)

 xz  u, z  w, x  w0, x  

The notation (..),x in the above equation is used to indicate the
derivative with respect to x and (..),z is the derivative with respect to z
variable.
Based on Hook’s law, the normal and shear stresses corresponding to
the strains in (2.25) are given by

 xx ( z, T )  E ( z, T ) xx  E ( z, T )  u0, x  z, x 

(2.26)

 xz ( z, T )   G( z, T ) xz   G( z, T )  w0, x   

in which E(z,T) and G(z,T) are, respectively, the effective Young’s
modulus and shear modulus, and ψ is the shear correction factor.
340

340

320

320

n=0.1

280

300
n=0.5
E (GPa)

E (GPa)

300

n=1

260
240

n=5

220

180
-0.5

-0.25

n=0.1
n=0.5

260

0
0.25
z/h
(a) T=0 K,V =0

n=1

240
220

n=10

200

280

n=5

200
0.5

180
-0.5

n=10
-0.25

0

0.25
z/h
(b) T=0 K,V =0.1

0.5

Fig.2.2. Influence of porosity volume fraction on the effective
Young’s modulus of FGM porous beam


10
280

280
n=0.1

260
240

200
n=5

180

140
-0.5

-0.25

n=0.5

220
n=1

200
180

n=10

160

n=0.1

240

n=1

220

E( GPa)

E (GPa)

260

n=0.5

0
0.25
z/h

(a) NLTR, T=500 K, V =0.1

n=5

160
0.5

140
-0.5

n=10
-0.25

0
0.25
z/h
(b) UTR, T=500 K, V =0.1

0.5

Fig.2.3. Influence of temperature on the effective Young’s
modulus of FGM porous beam for UTR and NLTR
2.5.3. Strain energy
The strain energy (U ) can be written in the form
L

U

2
1 

A11u0,2 x  2 A12u0, x, x  A22 ,2x  A33  w0, x     dx (2.27)



20

in which V in the beam volume, A is the cross-sectional area; A11,
A12, A22 and A33 are, respectively, the axial, axial-bending coupling,
bending and shear rigidities of the beam.
2.5.4. Strain energy due to initial thermal stress
Assuming the beam is free stress at the reference temperature T0
and it is subjected to thermal stress due to the temperature change.
The initial thermal stress resulted from a temperature T is given by
[30, 91]

 xxT   E ( z, T ) ( z, T )T

(2.29)

The strain energy caused by the initial thermal stress has the
form [17, 30]
L

UT  

1
1
E ( z, T ) ( z, T )Tw0,2 x dV   NT w0,2 x dx

2V

20

(2.30)

where NT is the axial force resultant due to the initial thermal stress.


11
2.5.5. Kinetic energy
The kinetic energy of the FGM with porosities has the form
 

L

1
 I11 (u0 2  w 0 2 )  2 I12 u0  I 22 2  dx
2 0 

(2.32)

in which I11, I12 and I22 are the mass moments.
2.5.6. Potential of external loads
The external loads considered in the present thesis are a single
moving force, a single moving harmonic force or multiple moving
forces with uniform amplitude. The potentials of these moving forces
can be written in the following form
nF

   Fi w0 ( xi , t ) ( xi  vti )

(2.35)

i 1

where nF is the number of loads; w0(xi,t) is the deflection under the
load Fi; xi is the abscissa measured from the left end of the beam to
the position of the load Fi, and ti is current time calculated from the
time when the load Fi enters the beam.
2.6. Effect of temperature on the beam rigidities
Fig. 2.5 illustrates the influence of the temperature rise ΔT on the
bending rigidity A22 of the beam with a porosity volume fraction V =
0.1 for two types of the temperature rise, UTR and LNTR.
1.5

T=0K

1.4

T=100K
T=200K

1.4

1.3

T=500K

1.3

1.2

UTR, V =0.1

1.1

1.2
1.1

1

1

0.9

0.9

0.8
0

(a)

2

4

n

6

8

10

T=0K
T=100K
T=200K

A22/A220

A22/A220

1.5

0.8
0
(b)

T=500K

NLTR, V =0.1

2

4

n

6

8

10

Fig.2.5. Influence of temperature rise on the bending rigidity of
FGM porous beam with i Vα = 0.1: (a) UTR, (b) NLTR


12
For a given value of the material index n, the rigidity steadily
decreases when increasing ΔT. The UTR has more significant
influence on the rigidity than that of the NLTR. Furthermore, the
decrease of the rigidity is more significant for the beam associated
with a higher index n.
2.7. Equations of motion
Equations of motion for the beam are derived from Hamilton’s
principle and they have the form
 I11u0  I12  A11u0, xx  A12, xx  0

nF

0  A33  w0, xx  , x   NT w0, xx   Fi ( xi  vti )
 I11w
i 1

 I   I u  A u  A    A  w     0
12
12 0, xx
22 , xx
33
0, x

 22

(2.44)

2.8. Euler-Bernoulli beam
Equations of motion for an FGM Euler-Bernoulli beam are as
follows.

 I u  I w
  A11u0, xx  A12 w0, xxx  0
 11 0 12 0, x
0  I12u0, x  I 22 w
0, xx  A22 w0, xxxx  A12 u0, xxx  NT w0, xx
 I11w

nF
   F  ( x  vt )
i
i
i

i 1

(2.48)

CHAPTER 3. FINITE ELEMENT MODELS AND
NUMRRICAL POROCEDURE
3.1. Nodal displacement vector
The vector of nodal displacements for a two-node genetic FGM
element (i, j) has six components as

d  {ui wi  i u j w j  j }T

(3.1)


13
in which the superscript ‘T’ is used to indicate the transpose of a
vector or a matrix. The displacements u0(x), w0(x) and rotation (x)
are interpolated from the nodal displacements as

u0  Nu d , w0  N wd ,   N d

(3.2)

where Nu, Nw and N are, respectively, the matrices of shape
functions for u0(x), w0(x) and (x). Various finite beam elements with
six d.o.f. as in Eq. (3.2) for vibration analysis of FGM beams have
been recently derived by using different shape functions.
3.2. Hierarchical shape functions
Finite element method based the hierarchical shape functions is
described in [132, 133]. The first-order interpolation for a function
f(x) on an interval [0,l ] is

f  N1 f1  N 2 f 2 

1   
2

f1 

1   
2

f2

(3.3)

x
With   2  1 is the natural co-ordinate. The second-order
l
interpolation for f(x) can be written in the form
f  N1 f1  N 2 f 2  N 3 f 3

(3.4)

in which N3 in the quadratic polynomial with the following form
N3  1   2

(3.6)

Similarly, by adding a third-order polynomial to the right-hand
side of (3.4), one gets the 3rd interpolation with N4 of the form



N4   1   2



(3.8)

The higher-order interpolation can be constructed in a similar
way as described above.
Using the shape functions N1, N2, N3, N4 mentioned above, the
axial displacement u0, transverse displacement w0 and rotation  are
interpolated as follows

u0  N1u1  N 2u2 ,   N11  N 2 2  N 33
w0  N1w1  N 2 w2  N3 w3  N4 w4

(3.9)


14
in which u1, u2, 1,…, w4 are the values of the variables at the nodes
and inside the element.
3.3. Displacement field with constraint
A beam element for vibration analysis of FGM structures can be
derived from the 9 d.o.f in Eq. (3.9) and illustrated in Fig.3.2.
However, the derived element is more efficient with lower number of
d.o.f. by constraining the shear strain to constant [133]. Using the
interpolation (3.9), one can write the shear strain (2.25) in the form
 6



4

1

1



 xz    w4   3   2   w3  1   2  
2
2 
 l

l

(3.10)

1
1

   w2  w1  2 w4   1   2  2 3  
2
l


For xz = const, one needs
6
4
1
1
 w4  3  0 , w3  1   2  0
l
l
2

2

(3.11)

From (3.11) one gets

l
l
1   2  , w4  3
8
6
Using (3.12), we can rewrite (3.9) as follows
w3 

u

1    u
2

1



1    u ,  1      1      1   2 
 3
2
1
2
2

1    w  1    w
w
2

(3.12)

1

2

2



2
l 1 2



8

 

1

2

 2  



l 1   2
6



(3.13)

3

The shear strain is now of the form

1
1
2
(3.14)
 w2  w1   1  2   3
l
2
3
The finite beam element model in this thesis is derived from the
displacement field (3.13) and the shear strain (3.14). The vector of
nodal displacements has only 7 d.o.f. now.

 xz 

3.4 Element stiffness matrix


15

The element stiffness matrix is derived from the strain energy of
an element, Ue, and it has the form
1
(3.18)
U e  dT kd
2
in which k is the element stiffness matrix with the following form

k  k uu  k u  k   k 
1

(3.19)

1

k uu 

2
2
NTu , A11Nu , d  , k u    NTu , A12 N , d 

l 1
l 1

k  

2
NT , A22 N , d  ,

l 1

1

(3.20)

1

l 2
2
k    ( NTw,  NT ) A33 ( N w,  N )d
2 1 l
l
The stiffness matrices in (3.20) have following explicit forms

k uu

l

1
2

l2
l
2
4

1

1


A 
 A33  2l l 2
 11 
, k  

l  1 1 
l 3
3

 1  l

2

2
l
l
4
 2

2l
3
l2
3
4 2
l
9
2l

3
2

l
3

1
l
2
2l

3


1


l
2

l 
2 

l2 
4 

l2 
3 

l
 
2
2 

l 
4 


16

k 

1 0
A 
16
 22  0
l 
3
 1 0


1

A
0  , k u   12
l


1

0 1
1 0

0 0


 0 1
 1 0

0
0
0
0
0

1
1 0 
0 0

0 1
1 0 
0

(3.21)

For the compactivity, the stiffness matrices in Eq. (3.21) should
be extended to (7x7) matrices by adding zero entries corresponding
the d.o.f. in the nodal load vector.
3.5. Element stiffness matrix due to initial thermal stress
The element stiffness matrix caused by the initial thermal stress
is obtained from the strain emery due to the temperature rise (2.30),
which written for an element is

1 T
d kT d

(3.22)
2l
with kT is the stiffness due to temperature rise with the following
explicit form
UTe 

0
0
1
0 
1

1 2
1 
0
l
0
0  l2 
12
12 


NT 
4 2
(3.24)
kT 
0
l
0
0 

0
l 
45

 1
0
0
1
0 


1 2 
 0  1 l2
0
0
l
12
12 

3.6. Element mass matrix
The element mass matrix is derived from the shape functions for
the displacement field. To this end, we write the kinetic energy
(2.32) in the following form
1
 e  d T md
2

(3.25)

17
in which m = muu + mww + mu + m is the mass matrix for the
element, with
1

muu 

1

l
l
NTu I11N u d  , m ww   NTw I11N w d 
2 1
2 1
1

mu

1

l
l
   NTu I12 N d  , m   NT I 22 N d 
2 1
2 1

(3.26)

are, respectively, the mass matrices stemming form the axial
displacement, transverse deflection, axial displacement-crosssectional rotation and the rotation of the cross section. Explicit forms

for these mass matrices are easily to obtain.
3.7. Element based on exact shape functions
The exact shape functions for the first-order shear deformable
beam element have been derived in the thesis of Ha [12]. The
expressions for the element stiffness and mass matrices using the
exact shape functions have similar forms as that in [12], except for
the difference of the rigidities and mass moments.
3.8. Element based on Euler-Bernoulli beam theory
The element stiffness matrix for a FGM Euler-Bernoulli beam
with porosities have the following forms

k  k uu  k uw  k ww

(3.33)

where
l

l

k uu   NTu , x A11N u , x dx , k uw   2  NTu , x A12 N w, xx dx
0

0

l

k ww   N

T

w, xx

(3.34)

A22 N w, xx dx

0

with Nu is a matrix of linear shape functions, and Nw is a matrix of
cubic Hermite polynomials.
The element mass matrix has a similar form as that of the
stiffness matrix, m = muu + mww + mu + m , in which


18
l

l

m uu   NTu I11Nu dx, m ww   NTw I11N w dx
0

0

l

l

m u    N I N w, x dx, m   N
0

T
u 12

0

(3.36)

T
w, x 22

I N w, x dx

3.9. Discretized equations of motion
Ignoring damping effect of the beam, the equations of motion for
the FGM beam with porosities in thermal environment under moving
loads can be written in the context of the finite element analysis as

  (K  K )D  Fex
MD
B
T

(3.37)

 are, respectively, the vectors of structural nodal
in which D , D
displacements and accelerations; K B , K T , M , F ex are the stiffness
matrices due to the beam deformation and temperature rise, the mass
matrix and the nodal load vector of the structure, respectively.

3.10. Newmark method
Eq. (3.37) can be solved by the direct integration Newmark
method. The constant average acceleration method which ensures the
unconditional stability is employed in this thesis.
CHAPTER 4. NUMERICAL RESULTS AND DISCUSSION
4.1. Formulation verification
Table 4.4 compares the dimensionless deflection at the mid-span
for V = 0 and T = 0K of a FGM under a single moving load. At a
given value of the material index and the moving load speed, as can
be seen from Table 4.4, the maximum dimensionless deflections
obtained in the thesis are very closed to that Şimşek and Kocaturk [4]
using a semi-analytical method, and of Khalili et al. [105] using the
differential quadrature method.


19
Table 4.4. Comparison of maximum dimensionless mid-span
deflection (V = 0, T = 0)
Al2O3

n = 0.2

n = 0.5

n=1

n=2

(252 m/s)*

(222m/s)

(198m/s)

(179m/s)

(164m/s)

TBHi

0.9382

1.0307

1.1509

1.2569

1.3450

TBEx

0.9380

1.0402

1.1505

1.2566

1.3446

EBB

0.9329

1.0346

1.1445

1.2504

1.3377

Ref [4]

0.9328

1.0344

1.1444

1.2503

1.3376

Ref[105]

0.9317

1.0333

1.1429

1.2486

1.3359

Source

4.3. Dynamic response
4.3.1. Influence of temperature and porosities
Fig. 4.3 illustrates the relation between the dimensionless midspan deflection with the dimensionless travelling time for n = 0.5, Vα
= 0.1 and various temperature rise ΔT of the NLTR and two values
of the moving load speed, v = 30 m/s and v = 60 m/s. For both the
values of the moving load speed, the mid-span deflection is larger
when ∆T is higher for most the time the load on the beam.
1.5

1.5
(b) v = 60 m/s

1

w0L/2,t)/wst

w0(L/2,t)/wst

(a) v = 30 m/s

0.5

0
0






0.2

T=0
T = 50K
T = 100K
T = 150K
0.4
0.6
0.8
t/ T*

1

1

0.5

0





0

0.2

0.4

T=
T=
T=
T=

0
50K
100K
150K
0.6

0.8

1

t/ T*

Fig.4.3. Time history of the dimensionless mid-span deflection for
various temperature rise ∆T of NLTR (n=0.5, V = 0.1)

20
Fig. 4.5 shows the relation between the maximum dimensionless
mid-span deflection with the moving load speed v of the beam with n
= 1 for different values of ∆T and Vα in case of NLTR.
2.5

2.5

(b)

2

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

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

(a)

1.5
T=0
T=50K
T=100K
T=150K

1
0.8
0

100

200
v (m/s)

2

1.5
V=0
V=0.1
V=0.15
V=0.2

1
300

0.8
0

100

200

300

v (m/s)

Fig.4.5. Relation between the maximum dimensionless mid-span
deflection with moving load speed v for n = 1 and NLTR: (a)V=0.1,
T is variable, (b) T = 150K, V is variable.
From Fig. 4.5 one can see that the maximum dimensionless mid-span

deflection increases when increasing ∆T and Vα, regardless of the
moving load speed. In addition, as can be seen from Fig. 4.5(a), the
largest value of the mid-span deflection tends to be attained at a
lower moving load speed when ∆T is higher. The contrary is
occurred for the porosities as seen from Fig. 4.5(b), though it is not
significant as in Fig. 4.5(a).
The thickness distribution of the normal stress on the mid-span
section is depicted in Fig. 4.6 for the case of NLTR, and with v = 20
m/s, n = 0.5 and different values of ∆T and Vα. The maximum
amplitude of the normal stress in Fig. 4.6(a) is seen to be increased
when increasing ∆T.


21
0.5

0.5

(b)

(a)
0.25
z/h

z/h

0.25

0

-0.25

-0.25

T=0
T=50K
T=150K

-0.5
-50 -40

-20

*

0

0

20

40

xx

-0.5
-50 -40

V=0
V=0.1

V=0.2

-20

*

0

20

40

xx

Fig.4.6. Thickness distribution of the normal stress of mid-span
section: (a) V = 0.1, ∆T is variable, (b) ∆T = 100K, Vα is variable.
4.3.2. Influence of excitation frequency
Fig. 4.8 shows the time histories for the dimensionless mid-span
deflection of the FGM beam under a moving harmonic load for
various values of the excitation frequency , and for n = 1, Vα = 0.1,
v = 30 m/s.
2

 = 10 rad/s
 = 15 rad/s
 = 20 rad/s

w0(L/2,t)/wst

w0(L/2,t)/wst

1

0

-1

6
4
2
0
-2

(a)
-2

0

(b)
0.2

0.4
t/ T*

0.6

0.8

1

-4

0

0.2

0.4

0.6

0.8

1

t/ T*

Fig.4.8. Time histories for dimensionless mid-span deflection for
various values of excitation frequency: (a) NLTR, (b) UTR (n = 1,
Vα=0.1, v = 30 m/s)
For both the type of the temperature, the excitation frequency
significantly alters the mid-span deflection of the beam. Especially,
for the case of UTR, the mid-span deflection corresponding to  =


22
20 rad/s is much higher than that of the cases  = 10 rad/s and  =
15 rad/s. This sharp increase of the deflection might be caused by the
resonant phenomenon which happens when the excitation frequency
is near the fundamental frequency of the beam.
4.3.3. Influence of number of moving loads

Fig. 4.9 illustrates the influence of the moving load number nF
and the distance between the loads d on the time histories for the
dimensionless mid-span deflection for n = 3, Vα = 0.1, v = 30 m/s and
∆T = 100K. In Fig. 4.9, ∆T* is the total time, necessary for a single
load to cross the beam. The beam executes more vibration cycles and
the mid-span deflection is higher when it is subjected to more
number of moving loads. In addition, the time at which the mid-span
deflection attains a maximum value tends to be increased when the
beam is under more moving. The mid-span deflect is clearly
decreased when increasing the distance between the loads.
4

2
1

2
1
0

0
-1
0

d=L/8
d=L/5
d=L/3
d=L/2

(b)

3

w(L/2,t)/wst

3

w(L/2,t)/wst

4

nF=1
nF=2
nF=3
nF=4

(a)

0.5

1

t/T*

1.5

2

-1
0

0.5

1

1.5

2

t/T*

Fig.4.9. Influence of the load number and distance between the
loads on the time histories for dimensionless mid-span deflection for
n = 3, Vα= 0.1, v = 30 m/s, ∆T = 100K: (a) d = L/4 and nF is different,
(b) nF = 3 and d is different.


23
6

6

(a)

5

4
3
T=0
T=50K
T=100K

T=150K

2
1
0

100

200
v (m/s)

300

400

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

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

5

(b)

4
3
d=L/8
d=L/5
d=L/4
d=L/2

2
1
0

100

200
v (m/s)

300

400

Fig.4.10. Relation between maximum dimensionless mid-span
deflection with moving load speed for nF = 3, n = 1 and Vα = 0.1: (a)
d = L/4 and ∆T is variable, (b) ∆T = 100K and d is variable.
The dynamic behavior of the FGM beam in thermal environment
under multiple moving loads can be observed from Fig. 4.10. As in
case of the single moving load, the maximum mid-span deflection
increases by increasing ∆T. When the distance between the moving
loads is large enough, the peak of the curves in Fig. 4.10 becomes
abnormal. For example, the curve corresponding to d = L/2 in Fig.
4.10(b) attains the maximum value at a relative low moving load
speed.
CONCLUSIONS
The main conclusions of the thesis can be summarized as follows
1) The influence of porosities and temperature rise on vibration of
transversely FGM beams with power-law distribution in the
thickness direction has been studied. The material properties of
the beams are assumed to vary in the thickness direction by a

power-law function, and they are considered to be temperaturedependent. The effect of two types of temperature distribution,
UTR and NLTR, on the fundamental frequency has been
examined in detail.
2) The finite element models for dynamic analysis of FGM beams
with porosities in thermal environment have been developed. The
new model TBHi based on hierarchical shape functions and