free vibration of fgm timoshenko beams with through-width delamination

SCIENCE CHINA Physics, Mechanics & Astronomy

© Science China Press and Springer-Verlag Berlin Heidelberg 2014 phys.scichina.com link.springer.com

*Corresponding author (email: [email protected])

• Article • May 2014 Vol.57 No.5: 927–934

doi: 10.1007/s11433-013-5248-5

Free vibration of FGM Timoshenko beams with through-width delamination

LI ShiRong1* & FAN LiangLiang2

1 College of Civil Science and Engineering, Yangzhou University, Yangzhou 225009, China; 2 School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China

Received December 28, 2012; accepted June 7, 2013; published online March 19, 2014

Free vibration of functionally graded beams with a through-width delamination is investigated. It is assumed that the material property is varied in the thickness direction as power law functions and a single through-width delamination is located parallel to the beam axis. The beam is subdivided into three regions and four elements. Governing equations of the beam segments are derived based on the Timoshenko beam theory and the assumption of ‘constrained mode’. By using the differential quadrature element method to solve the eigenvalue problem of ordinary differential equations governing the free vibration, numerical re-sults for the natural frequencies of the beam are obtained. Natural frequencies of delaminated FGM beam with clamped ends are presented. Effects of parameters of the material gradients, the size and location of delamination on the natural frequency are examined in detail.

functionally graded materials beam, free vibration, natural frequency, differential quadrature method, delamination

PACS number(s): 02.60.Cb, 02.60.Jh, 02.60.Lj

Citation: Li S R, Fan L L. Free vibration of FGM Timoshenko beams with through-width delamination. Sci China-Phys Mech Astron, 2014, 57: 927934, doi: 10.1007/s11433-013-5248-5

1 Introduction

Functionally gradient materials (FGMs) are novel, micro-scopically inhomogeneous materials in which the mechani-cal properties vary smoothly and continuously from one surface to another. For example, by gradually varying the volume fraction of constituents rather than abruptly chang-ing it over the interface can combine the best properties of ceramics and metal, i.e., the high temperature resistance, high fracture toughness, high hardness and low thermal conductivity of ceramics, and high strength, ductility and thermal conductivity of metal. So, FGMs are good in engi-neering applications, such as high resistance to large tem-perature gradients, reduction of stress concentration and so on. Therefore, FGMs have applied extensively in situations

where large temperature gradients are encountered. The studies of the mechanical behaviors of FGM structures un-der the thermal and mechanical loadings have become a new research field in the solid mechanics [1–7]. However, for the reasons of manufacturing process, delaminations are typical defects of FGMs that may reduce the stiffness and strength of the structure and affect the vibration characteris-tics of the structure. It is important to evaluate the influ-ences of delaminations on the natural frequencies of a de-laminated FGM beam because the damage detection and structural health monitoring problems are also considered in the frameworks of vibration analysis.

There are plenty of analytical models and numerical analyses that have been reported for the vibration analysis of delaminated composite laminates. However, only two models, ‘free mode’ and ‘constrained mode’, are most commonly used in these previous investigations. Wang et al.

928 Li S R, et al. Sci China-Phys Mech Astron May (2014) Vol. 57 No. 5

[2] examined the influence of a through-width delamination on the free vibration of an isotropic Euler-Bernoulli beam by using a ‘free mode’. Their basic assumption of ‘free mode’ is that the delaminated layers have different trans-verse deformations and without touching each other during the vibration process. Studies by Mujumdar and Surya-narayan [3] pointed out that the ‘free mode’ is physically inadmissible where there are the off-midplane delamina-tions. To avoid this incompatibility, they assumed that the delaminated layers keep in contact with each other at all their interfaces during the vibration but are allowed to slide over each other [4]. This model is referred to as the ‘con-strained mode’ because the delaminated layers are ‘con-strained’ to have identical transverse deformations. Howev-er, a more profound study by Shen and Grady [5] found that the ‘constrained mode’ model failed to predict some ex-perimental phenomena such as the open mode shapes. Luo and Hanagud [6] investigated the dynamic responses of the delaminated beams based on Timoshenko beam theory and presented an approach of FEM. Their results show a good agreement with the experiments. A more comprehensive review on the analysis of delaminated beams can be found in Della and Shu’s review article [7].

Differential quadrature method (DQM) is a simple and highly efficient numerical technique that was introduced by Bellman et al. [8] in the 1970s. It was primarily used for structural analysis by Bert et al. [9] in 1988. Since then, the method has been widely used for static and dynamic analy-sis of structures [10–13]. Differential quadrature element method (DQEM) was firstly proposed by Chen to solve the boundary value problems of differential equations in a complicated domain [14], in which a whole domain is di-vided into several elements and the DQ discretization is applied on these elements and both the boundary conditions and the transition conditions are discretized at correspond-ing grid points and then the boundary problem of differen-tial equations was approximated into a solvable algebraic equations. At the last decade DQEM has become an alterna-tive approach of structure analysis [15–17].

In this paper we present a free vibration analysis of de-laminated FGM beams. Timoshenko beam theory and the ‘constrained mode’ are used to establish the governing equations of the problem. By using DQEM to solve these equations numerical results of the natural frequency of the

beam are obtained. Geometrical effects (e.g. the length and position of the delaminated region) on the free vibration responses are discussed in detail.

2 Formulations

Suppose that a rectangular-section beam of length l and thickness h with a through-width delaminate defect is made of functionally graded materials with the material properties changing continuously from the top to the bottom. A sche-matic sketch of the geometric size and coordinate system of the beam is shown in Figure 1. The delaminated beam is divided into four beam elements: two integral elements and two delaminated elements (as shown in Figure 2). Each of them is analyzed by Timoshenko beam theory. Additional postulation is that the two delaminated layers have identical transverse deformations in the deformation so that our analysis is based on the ‘constrained mode’ [4]. As shown in Figure 2, li and hi are length and thickness of the ith beam segment and Pi, Qi and Mi are the axial force, shear force and bending moment of each beam segment, respectively. The clamped-clamped boundary condition is adopted in the numerical example but the analysis process is the same for other cases.

We assume that the material properties of the FGM beam are varied continuously from the upper surface to the lower one in the thickness direction. Furthermore, variations of the material volume fraction along the thickness of the beam was described by the power law [18,19],

L U L( ) 1 / 2 / , ( ) 1 ,

( / 2 / 2).

nV z z h V z V

h z h

(1)

Figure 1 A schematic sketch of the FGM beam with a single through- width delamination.

Figure 2 Representation of the divided four Timoshenko beam elements with resultant forces on their ends.

Li S R, et al. Sci China-Phys Mech Astron May (2014) Vol. 57 No. 5 929

in which subscripts L and U refer to the material of the lower and of the upper surfaces respectively, VL and VU are the volume fractions of the materials at the lower and upper surfaces, n is the power law index that should be greater than or equal to zero, especially, when 0n and n representing the beam made of pure lower and upper sur-face materials. Hence, the Young’s modulus E, Poisson ratio and the mass density can be obtained by using the mixture rule as follows:

L U L U( ) ( ) ( ) ,E z E E V z E (2a)

L U L U( ) ( ) ( ) ,z V z (2b)

L U L U( ) ( ) ( ) .z V z (2c)

2.1 Equations of motion

We analyze the four divided beam segments by using Ti-moshenko beam theory according to which the first order shear deformation is taken into consideration. Displace-ments in the axial and the transverse directions of each beam element in its local coordinate can be expressed by

( , , ) ( , ) ( , ),i i i i i i i iu x z t u x t z x t (3a)

( , , ) ( , ),i i i i iw x z t w x t (3b)

in which ix is the local longitudinal coordinate and t is the

time variable; iu and iw are the longitudinal and trans-

verse displacements of the beam axis; i is the rotational

angle of the cross section; subscripts 1,2,3,4i represent the four divided beam elements.

For the case of small deformation, the normal and shear strains in the beam elements can be obtained from the dis-placement of eq. (3) and presented as follows:

,i i ii i

i i i

u uz

x x x

(4a)

.i i ii i

i i i

w u w

x z x

(4b)

On the assumption of linearly elastic constitutive relations, we have the stress-strain relationships:

( ) ( ) ,i ii i

i i

uz E z z

x x

(5a)

( )

.2[1 ( )]

ii i

i

wE z

z x

(5b)

where i and i are the normal and shear stresses; ( )E z and

( )z as functions of the global lateral coordinate z, which have been given in eq. (2).

Integrals of eq. (5) over the cross sections for each beam

give the resultant forces and moments as follows:

d ,i

i ii i i i i

S

uP S A B

x x

(6)

d ,i

ii i i i i

S

wQ S D

x

(7)

d ,i

i ii i i i i

S

uM z S B C

x x

(8)

where iS is the cross section area of the ith beam element,

is the Timoshenko shear correction factor which de-pends on the shape of the cross section of the beam ( 5 / 6 for a rectangular cross section). iA , iB , iC

and iD are the stiffness coefficients calculated by

2

2

2{ , , , } ( ) 1, , , d .

2[1 ( )]

i

i

h

i i i i i i ih

A B C D E z z z zz

(9)

Here iz is the local transverse coordinate of the ith beam

initiated from the geometrically central plane. As shown in Figure 2, relations between the local and the global trans-verse coordinates are 1 4 ,z z z 2 2( ) 2z z h h , 3z

3( ) 2z h h .

For the free vibration, equations of motion for the beam segments can be formulated based on the Timoshenko beam theory [20] in their individual coordinate systems as fol-lows:

2 2

2 2

2

2

, ( , ) ,

,

i i i ii i i i

i i

i ii i

i

P u Q wI q x t I

x xt t

MQ J

x t

(10)

where iI and iJ are the inertial quantities related to the

mass density ( )z defined by

2

2

2( , ) ( )(1, )d ,

i

i

h

i i i ih

I J z z z

(11)

and ( , )i iq x t is the transverse loadings applied on the i th

beam element. On the assumption of the ‘constrained mode’ we have 2 2 3 3( , ) ( , )q x t q x t and 2 2 3 3( , ) ( , ).w x t w x t

Herein, we only study the free vibration responses so that we can let ( , ) 0iq x t . By substituting eqs. (6)–(8) into eq.

(10) one obtains the equations of motion in terms of the displacements as follows:

2 2 2

2 2 2,i i i

i i i

u uA B I

x x t

(12)

2 2

2 2( , ),i i i

i i i

w wD I q x t

xx t

(13)


2 2 2

2 2 2.i i i i

i i i i i

u wC B D J

xx x t

(14)

2.2 Harmonic response of the beam vibration

By considering harmonic responses of the beam vibration, solutions to the partial differential eqs. (12)–(14) may be found by using the variable separation approach in the form of

{ ( , ), ( , ), ( , )}

{ ( ), ( ), ( )}cos( ),

i i i

i i i

u x t w x t x t

u x w x x t

(15)

in which is the natural frequency of the beam and

( )iu x , ( )iw x and ( )i x are the corresponding shape functions.

The assumption of ‘constrained model’ gives the relation-ship of

2 3( ) ( ).w x w x (16)

For convenience in the following analysis, we introduce the non-dimensional transformation as follows:

i i iU u l , i i iW w h , i i ix l , i i i il h . (17)

By substituting eq. (15) into eqs. (12)–(14) and ignoring the axial inertial force and the transverse loadings, we can eliminate the time variable and transform the partial differ-ential equations into ordinary ones in dimensionless form as follows:

2 2

( )12 2

d d0,

d dii i

i i

Uc

( 1,2,3,4),i (18)

2

( ) 222

d d0,

ddii i

iii

Wc W

( 1, 4),i (19)

2 2( ) ( ) ( ) 23 4 52 2

d d d0,

dd d

( 1,2,3,4),

i i ii i ii i

ii i

U Wc c c

i

(20)

2

232 26 7 8 22

2 22

dd d0,

d dd

Wc c c W

(21)

where the dimensionless parameters are defined as:

2 2( ) ( ) ( )1 2 32

2 2( ) ( )4 5

6 2 2 3

2 3 2 37 8 2

2 3 2 2 3 2

, , ,

, ,

/ ( ),

, .( ) ( )

i i ii i i i i i

i i ii i

i ii i i i

i i

B h I l B lc c c

D C hAl

D l J lc c

C C

c D D D

D h I Ic c

D D h D D l

2.3 Boundary and continuity conditions

For the clamped-clamped ends of the beam, non-dimensional boundary conditions are expressed as follows:

1 1 1 4 4 4(0) (0) (0) (1) (1) (1) 0.W U W U (22)

At the interconnecting points of beam segments, we have the following continuity conditions:

1 2 2 4 2 2(1) (0), (0) (1),hW h W hW h W (23)

1 1 2 2 2 3 3 3

4 4 2 2 2 3 3 3

(1) / (0) / (0) / ,

(0) / (1) / (1) / ,

h l h l h l

h l h l h l

(24)

1 1 2 1 1 2 2

1 1 3 1 1 3 3

(1) (1) / (0)

(1) (1) / (0) 0,

l U z h l l U

l U z h l l U

(25)

4 4 2 4 4 2 2

4 4 3 4 4 3 3

(0) (0) / (1)

(0) (0) / (1) 0.

l U h l l U

l U h l l U

(26)

Additionally, equilibrium conditions should be intro-duced at the interconnecting ends of beam segments, which are expressed by the resultant forces:

1 2 3 4 2 3(1) (0) (0), (0) (1) (1),P P P P P P (27)

1 2 3 4 2 3(1) (0) (0), (0) (1) (1),Q Q Q Q Q Q (28)

1 2 3 2 2 3 3(1) (0) (0) (0) (0),M M M P P (29)

4 2 3 2 2 3 3(0) (1) (1) (1) (1),M M M P P (30)

where 2 2( ) / 2h h and 3 3( ) / 2h h are the lat-

eral positions of mid-planes of beams 2 and 3 in the global coordinate, respectively. Expressions of the non-dimensional resultant forces in the above equations are given as follows:

2

d d d, ,

d d d

d d.

d d

i i i i i ii i i

i i i i i

i i i ii

i ii i

U C h h WP Q

B l

U D hM

C l

(31)

3 Numerical method

A numerical approach of DQEM which was first introduced by Chen (refer to sect. 1) is adopted to solve the ordinary differential eqs. (18)–(21) associated with boundary and continuity conditions (22)–(30). For this purpose, the beam is divided into three regions (two integral regions and one delaminated region) and each region is treated as a differen-tial quadrature element. Firstly, the physical domain of each element should be normalized and then divided into a set of discrete grid points. Here we use a non-uniform grid distri-


bution of cosine type as follows [14]:

1 ( 1)

1 cos ,2 1j

j

N

( 1,2, , ).j N (32)

Then the differential equations are approximated into alge-braic equations at each internal grid point through a DQ discretization procedure. The boundary conditions are also transformed into algebraic equations at the corresponding boundary points by the similar procedure. According to DQM, the first and second order derivatives of a func-tion ( )f at grid point i is expressed as [9]:

2

(1) (2)2

1 1

d ( ) d ( )( ), ( ).

d di i

N N

im m im mm mi i

f fA f A f

(33)

Here (1)mnA and (2)

mnA represent the weighting coefficients

given in eqs. (34) and (35) which can be calculated by using Lagrange polynomials as the test functions. Details of cal-culations of the weighting coefficients can be found in ref. [9].

(1)

1 1,

(1)

1

( ) ( ), ,

( , 1,2, , ),1

,( )

N N

ij i k j kk kk i j k j

N

iik i kk i

A j i

i j N

A

(34)

(2) (1) (1)

1

.N

ij ik kjk

A A A

(35)

By using those DQ rules, eq. (18) (as an example of the governing equations) and eq. (27) (as an example of the boundary conditions) are rewritten in a DQ form as follows:

(2) ( ) ( ) ( )

11

( ) 0,

( 2,3, 1; 1,2,3, 4),

xNi i i

km m mm

A U c

k N i

(36)

(1) (1) (1)1 1 2

1 1

(1) (2) (3) (2) (3)321 2 3 2 32 2

1 2 3

0.

x

x

x

N

N m m mm

N

m m m m mm

hA B U C

l

hhA B U B U C C

l l

(37)

Other equations could be easily transformed into the corresponding algebraic equations by a similar process as eqs. (18) and (27) are done. Details of those algebraic equa-tions will not be given here for saving space. Finally, the discrete algebraic equations are written as a matrix form of

2

11 11

,bbb bi

ib ii iiN N

UK K

K K UU

0 0

0 (38)

where K is a coefficient matrix and U is the eigenvector.

Subscript i stands for inner grid points and b does for the boundary grid points. By eliminating vector bU from eq.

(38) we obtain the standard eigenvalue problem of algebraic equations for the free vibration of the delaminated FGM beam as:

1 2[ ]{ } { } { }.ii ib bb bi i i K K K K U U 0 (39)

By solving the above equations, the natural frequencies as well as the mode shapes can be finally obtained.

4 Numerical results and discussions

In the following numerical computation, we consider the FGM beam with length l = 1.0 m and thickness h = 0.02 m as shown in Figure 1. The position of the delamination is determined by parameter 2h and 1 2 2,l l which

represent the lateral and the longitudinal locations of the central point of the delamination, respectively.

The beam is assumed to be functionally gradient compo-sites of metal (Ni) and ceramic (SiC). The material proper-ties of the two material constituents are given in Table 1.

First, we consider the case where a delamination with length l2 = 0.2 m is at the centre of the beam ( = 0.5 m) and located on the mid-plane (h2 = 0.01 m). The influences of volume fraction index n on the free vibration of FGM beam with/without delamination are studied by showing natural frequencies of 1–8th modes in Figure 3. From it we can see that the natural frequencies increase as the volume fraction index n increases, which is due to the increase of the vol-ume fraction of ceramic constituent in the beam. It is clear that natural frequencies of the delaminated beam are lower than these of the perfect beam due to the reduction of stiff-ness induced by the delamination.

In order to examine the relation between natural frequen-cies and the length of delamination, calculations are per-formed for the FGM beam with the delamination position parameters, = 0.5 m, h2 = 0.008 m. Characteristic curves of the natural frequencies for different orders versus the length of delamination are shown in Figure 4. From it we can see that the natural frequencies decrease in a multi-step manner with increase of the delemination on the whole be-cause increase in the length of the delamination leads to the decrease of the total bending stiffness of the whole beam. However, by checking these curves locally we found that the decrease is not monotone. There is an approximate pe-riodic relation between the decreasing ratio of frequency and the length of delamination. We also found that this

Table 1 Material properties of the metal (Ni) and the ceramic (SiC)

Material Young’ modulus

(GPa) Poisson ratio Mass density

(kg/m3) Upper surface SiC 427 0.17 3100

Lower surface Ni 206 0.3 8890


Figure 3 Natural frequencies of perfect and delaminated FGM beams changing with the power law index n. (l2 = 0.2 m, h2 = 0.01 m, = 0.5 m). (a) modes 1–4; (b) modes 5–8.

Figure 4 Natural frequencies versus the delamination length for perfect and delaminated FGM beams. (h2 = 0.008 m, = 0.5 m, n = 2.0). (a) modes 1–4; (b) modes 5–8.

periodic relation depends on the orders of vibration modes. The wave numbers increase with the increment of the vibra-tion modes. An approximate relationship can be abstracted from Figure 4 as follows:

m k 1, (40)

where m denotes the wave numbers and k is the order of vibration mode under consideration.

Numerical calculations are also carried out for the de-laminated FGM beam with fixed delamination length, l2 = 0.2 m, located in the mid-plane (h2 = 0.01 m) for a continu-ous variation of axial position parameter, . Curves of the natural frequencies versus parameter, , for both the perfect and delaminated beams are plotted in Figure 5, from which we can also find a periodic variation of the natural frequen-cies along with the longitudinal position of the delamination. The relation between the wave numbers of the curves and the orders of the modes still satisfy eq. (40). For each mode the frequency changes periodically, or vibrates harmoni-ously in the vicinity of a constant value which is the natural frequency for = 0.2 m, or 0.8 m. As we expected that all the curves are symmetric about the middle point, or = 0.5 m due to the symmetry of the geometry, the material proper-ties and the end supports of the whole beam about the centre.

Figure 6 shows two groups of numerical results of free

Figure 5 Natural frequencies of perfect and delaminated FGM beams versus parameter (h2 = 0.01 m, l2 = 0.2 m, n = 2.0). (a) modes 1–4; (b) modes 5–8.


Figure 6 Natural frequencies of perfect and delaminated FGM beams changing with the distance between the delamination and the mid-plane. (l2 = 0.2 m, = 0.5 m, n = 0). (a) n = 0 ; (b) n = 2.

vibration of delaminated FGM beams in which the index of volume fraction is given by n = 0 (degenerated into a ho-mogeneous case, pure metal) and n = 2, respectively. In the numerical computation, the delaminated layer is assumed to be at the centre of the beam ( = 0.5 m) but the lateral posi-tion (h2) to vary in order to examine its effects on the fre-quency. The length of the delamination is also specified as l2 = 0.2 m. In the homogeneous material case (n = 0) we can see from Figure 6(a) that the natural frequencies of the beam are reduced when the delamination approaches the mid-plane in the transverse direction. As a limit, when the delamination coincides with the mid- plane the natural fre-quency reaches the minimum for each vibration mode. A similar variation of the frequency can also be found for the FGM beam (n = 2) in Figure 6(b). However, the position of the delamination at which the natural frequency reaches its minimum dose not coincide with the geometrical mid-plane because the distribution of Young’s modulus of the FGM beam is not symmetric about it. Further analysis will show that the transverse location of the delamination at which the natural frequency reaches the minimum coincides with the physical mid-plane which is always near to the full ceramic surface.

5 Conclusions

Free vibration response of a FGM Timoshenko beam with a through-width delamination, clamped at the two ends, is obtained by a numerical approach of differential quadrature element method. Numerical results of natural frequencies of the beam changing with parameters of the size and position of the delamination, and also with the material gradient pa-rameter are presented. The effects of the power law index of the material gradient, the length and the location of the de-lamination on the natural frequencies of the beam are ex-amined quantitatively in detail. For a fixed length and posi-tion of the delamination of the FGM beam, natural fre-quency increases with the increment in the value of index, n, due to the increase of the ceramic volume fraction. As we expected natural frequencies decrease while increasing the length of delamination. Natural frequency reaches the minimum when the delamination is longitudinally located at the beam center for a fixed transverse position. For a given axial position, the frequency decreases when the delamina-tion approaches the mid-plane of the beam. The numerical solution of this example shows that DQEM is an efficient way to analyze the vibration problem of delaminated FGM beams.

This work was supported by the National Natural Science Foundation of China (Grant No. 11272278).

1 Zou Y, Tong L, Steven G P. Vibration-based model-dependent dam-age delamination identification and health monitoring for composite structures—A review. J Sound Vib, 2000, 230: 357–378

2 Wang J T W, Liu Y Y, Gibby J A. Vibration of split beams. J Sound Vib, 1982, 84: 491–502

3 Mujumdar P M, Suryanarayan S. Flexural vibrations of beams with delaminations. J Sound Vib, 1988, 125: 441–461

4 Della C N, Shu D. Vibration of delaminated multilayer beams. Com-posites-Part B, 2006, 37: 227–236

5 Shen M H, Grady J E. Free vibrations of delaminated beams. AIAA J, 1992, 30: 1361–1370

6 Luo H, Hanagud S. Dynamics of delaminated beams. Int J Solids Struct, 2000, 37: 1501–1519

7 Della C N, Shu D. Vibration of delaminated composite laminates: A review. Appl Mech Rev, 2007, 60: 1–20

8 Bellman R E, Kashef B G, Cast G J. Differential quadrature: a tech-nique for the rapid solution of non-linear partial differential equations. J Comput Phys, 1972, 10: 40–52

9 Bert C W, Jang S K, Striz A G. Two new approximate methods for analyzing free vibration of structural components. AIAA J, 1988, 26: 612–618

10 Han J B, Liew K M. Axisymmetric free vibration of thick annular plates. Int J Mech Sci, 1999, 41: 1089–1109

11 Wang X, Tan M, Zhou Y. Buckling analyses of anisotropic plates and isotropic skew plates by the new version differential quadrature method. Thin-walled Struct, 2003, 41: 15–29

12 Karami G, Malekzadeh P, Mohebpour S R. DQM free vibration analysis of moderately thick symmetric laminated plates with elas-tically restrained edges. Composite Struct, 2006, 74: 115–125

13 Della C N, Shu D. Free vibration analysis of delaminated bimaterial


beams. Composite Struct, 2007, 80: 212–220 14 Zong Z. A variable order approach to improve differential quadrature

accuracy in dynamic analysis. J Sound Vib, 2003, 266: 307–323 15 Chen C N. The two-dimensional frame model of the differential

quadrature element method. Comput Struct, 1997, 62: 555–571 16 Han J B, Liew K M. Static analysis of Mindlin plates: the differential

quadrature element method (DQEM). Comput Methods Appl Mech Eng, 1999, 177: 51–75

17 Liu F L, Liew K M. Differential quadrature element method: A new appoch for free vibration analysis of polar Mindlin plates having dis-

continuities. Comput Methods Appl Mech Eng, 1999, 179: 407–423 18 Ma L S, Wang T J. Nonlinear bending and post-buckling of a func-

tionally graded circular plate under mechanical and thermal loadings. Int J Struct, 2003, 40: 3311–3330

19 Markworth A J, Romesh K S, Parks W P Jr. Modelling studies ap-plied to functionally graded materials. J Mater Sci, 1995, 30: 2183–2193

20 Li B, Tang Y Q, Chen L Q. Nonlinear free transverse vibrations of axially moving Timoshenko beams with two free ends. Sci Chi-na-Tech Sci, 2011, 54: 1966–1976

free vibration of fgm timoshenko beams with through-width delamination

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