2014, Vol. 35
The Chinese Meteorological Society
Article Information
- Shi-rong LI, Ze-qing WAN, Jing-hua ZHANG 2014.
- Free vibration of func ionally graded beams based on both classical and first-order shear deformation beam theories
- Appl. Math. Mech. -Engl. Ed., 35 (5) : 591–606
- http: //dx. doi. org/10.1007/s10483-014-1815-6
Article History
- Received 2013-02-04;
- in final form 2013-12-10
1 Introduction
Functionally graded materials (FGMs) are a class of composites that have variational smooth and continuous spatial properties which can alleviate the stress concentrations found in laminated composites. Typically,these materials consist of a mixture of ceramic and metal or a combination of different materials. The gradient compositional variation in the constituents from one surface to the other provides an elegant solution to the problem of higher transverse shear stresses that are induced when two similar materials with a large difference in the material properties are bonded. As a result,FGMs are considered to have enormous potential applications in modern technology and engineering,especially in the thermal environments where the thermal stress concentration can be minimized due to the high temperature gradient. Consequently,the studies devoted to understanding the static and dynamic behaviors of FGM beams,plates,and shells have being received more and more attention in recent years. However,comparing with the studies on the dynamic response on FGM plates and shells,the research attention to the vibrations of FGM beams is relatively limited.
Based on the Euler-Bernoulli beam theory,the vibration responses of FGM beams have been widely studied by different approaches[1, 2, 3, 4, 5]. Alshorbagy et al. [1] investigated the free vibration of an FGM beam by the finite element method. Due to the deficiency in the Euler-Bernoulli beam theory for the consideration of the shear effect,the numerical results showed that the slenderness of the beam had no influence on the frequencies and mode shapes. S¸im¸sek and Kocat¨urk[2] studied the dynamic response of an FGM simply supported beam under a concentrated moving harmonic load,in which the effects of the material homogeneity,the velocity of the moving harmonic load,and the excitation frequency on the dynamic responses of the beam were discussed. Pradhan and Murmu[3] numerically studied the thermo-mechanical vibration of a simply supported sandwich beam with two FGM surface layers,resting on an axially variable elastic foundation and subjected to a transversely non-uniform temperature increase by the differential quadrature method (DQM),and considered the temperature dependence of the material properties. By assuming that the material properties changed in the thickness direction,Yang and Chen[4] analyzed the free vibration and buckling of FGM beams with vertical edge cracks,and examined the effects of the parameters of the material gradient,location,and numbers of the cracks and the boundary conditions on the natural frequencies and buckling loads of the beam. Li et al. [5] analyzed a small vibration of post-buckled FGM beams with surface-bonded piezoelectric layers in thermal environment by a numerical shooting method based the exact geometrically non-linear theory for axially extensible beams.
In the framework of the first shear deformation theory or the Timoshenko beam theory, many researchers studied the static and dynamic characteristics of FGM beams by analytical and numerical methods[6, 7, 8, 9, 10, 11]. Sina et al. [6] used a new beam theory to analyze the free vibration of FGM beams. Their investigation indicated that a little difference in the natural frequency existed between the new theory and the traditional first-order shear deformation beam theory. By introducing a new unknown function to decouple the governing equations,Li[7] presented analytical solutions for the static bending and free vibration of FGM Timoshenko and Euler-Bernoulli beams. Huang and Li[8] also studied the free vibration of axially FGMs with non-uniform cross-sections by using the integration technique to transform the differential governing equations into the Fredholm integral equations. Xiang and Yang[9] studied the free and forced vibration of a laminated FGM beam with variable thickness under heat conduction by a DQM numerical technique. Ma and Lee[10] carried out a further investigation on the small free vibration of FGM beams with/without thermal post-buckling deformation subjected to an in-plane temperature increase by a shooting method. Some interesting behaviors were illustrated numerically in the thermal post-buckling equilibrium paths and the frequencytemperature relationship due to the temperature dependence of the material property and the shear deformation. More recently,Li and Batra[11] derived an analytical relation between the buckling loads of the FGM Timoshenko beams and the homogenous Euler-Bernoulli beams with clamped-clamped (C-C),simply supported-simply supported (S-S),and clamped-free (CF) ends for the arbitrary transverse inhomogeneity of the material properties. Unfortunately, such a relation does no exist for the clamped-supported (C-S) and supported-clamped (S-C) end constraints. For these end conditions,an algebraic eigenvalue problem is derived to determine the critical buckling load of the FGM Timoshenko beam which is similar to that for finding the critical buckling load of a homogeneous Euler-Bernoulli beam with the same end constraints.
Based on higher order shear deformation theories,studies on bending and vibration of FGM beams were performed[12, 13, 14, 15, 16]. Aydogdu and Tashkin[12] studied the free vibration behavior of a simply supported FGM beam based on the first,parabolic,and exponential shear deformation beam theories,respectively,in which natural frequencies were obtained by the Navier type solution method. S¸im¸sek[13, 14] investigated the dynamic responses of functionally graded beams by different beam theories,in which a system of equations of motion was derived by Lagrange’s equations. Sallai et al.[15] presented the analytical solution of static bending of FGM beams by the parabolic,exponential,and Aydogdu model shape functions of the shear strain distribution in the cross-section,in which power-law,sigmoid,and exponential functions of Young’s modulus in the thickness directions were considered. Mahi et al.[16] analyzed the free vibration of FGM beams with the temperature dependent material properties. The formulation was derived based on a unified higher order shear deformation theory. The effects of the initial thermal stress on the natural frequencies were also discussed.
The plane elasticity theory has also been used to analyze the static bending of FGM beams by some authors[17, 18, 19] . Sankar[17] investigated the deformations of simply supported FGM beams with Young’s modulus varying exponentially in the thickness direction and subjected to symmetrical sinusoidal transverse loads. Zhong and Yu[18] studied the static bending of a cantilever FGM beam with arbitrary through-the-thickness variations of the material properties. Ding et al.[19] derived a stress function for the anisotropic functionally graded beam in analytical forms and presented the bending solutions for the beams with different boundary conditions in the sense of elasticity for plane problems.
In this paper,the free vibration of FGM beams will be investigated based on both the classical and the first-order shear deformation beam theories. The equations of motion of the FGM beam will be derived by totally considering the axial,the transverse,the rotational,the axial-rotational coupling inertia forces,and the shear deformation. A system of complicated coupling ordinary differential equations with the boundary conditions corresponding to the harmonic response of the beam will be solved by a numerical shooting method. Meanwhile,an analytical transition between the natural frequency of an FGM beam and that of the related homogenous one will be derived on the basis of the Euler-Bernoulli beam theory. Furthermore, the availability of the transition equation to be extended to predict the natural frequencies for FGM Timoshenko beams will be discussed approximately. 2 Mathematical formulation 2.1 Equations of motion
Consider a straight and uniform cross-section beam made from FGMs with the length l,the width b,the depth h,and a rectangular cross section (see Fig. 1). It is assumed that the material properties vary continuously in the thickness direction arbitrarily. Based on the first-order shear deformation beam theory,the displacement field of the beam is given by
 |
| Fig. 1. Geometric of functionally graded beams. |
where x is the axial coordinate taken along the geometrical centroidal axis,and z is the lateral coordinate in the thickness direction; t is the time variable; u0 and w0 are the displacements of a point on the geometrical centroidal axis in the x- and z-directions,respectively; and ∅ is the angle of rotation of the cross section about the y-axis. Assume that infinitesimal deformation yields the strain-displacement relations
If the deformation of the FGM beam obeys Hooke’s law,the stress-strain relations are
where E is the elastic modulus,and ν is the Poisson’s ratio,which are assumed to be functions of z.
Based on the plane elasticity theory,the equations of motion of the FGM beam are
Substituting Eq. (1) into Eqs. (4) and (5),performing integration over the cross section of
the beam,and using the traction boundary conditions 
and 
yield the
following equations of motion in terms of the resultant forces:
where q is the applied transverse force per length of the beam,and I0,I1,and I2 are the resultant inertial quantities defined by
in which A is the area of the cross section. The resultant forces and the bending moment are defined by
where κ is the shear deformation correction factor and is taken to be 5/6. Substituting Eqs.(2) and (3) into Eq. (10) gives the axial force FN,the shear force Fs,and the bending moment M in terms of the displacements as follows:
where A1,B1,C1,and D1 are the tension,tension-bending,shearing,and bending stiffness constants defined by
Substituting Eq. (11) into Eqs.(6)-(8) yields the equations of motion in terms of the displacements as follows:
2.2 Gradient of material properties
Herein,the material properties of the beam change only in the thickness direction. The Poisson’s ratio is assumed to be a constant for simplicity because it usually changes very little when it is compared with the Young’s modulus. The Young’s modulus and the mass density of the FGM beam can be written as
where Eb is the Young’s modulus,and ρb is the density of the reference homogenous material which is selected as the materials at the bottom surface of the FGM beam. They are defined by
fE(z) and fρ(z) are designed continuous dimensionless functions of the coordinate z (or stepwise continuous functions),which are expressed by
Substituting Eq.(16) into Eqs.(9) and (12),respectively,yields the stiffness coefficients and the inertia parameters as follows:
where the quantities with superscript stars are related to the reference homogenous material beams,whose definitions are given in Appendix A,and φi and φi (i = 0,1,2) are dimensionless coefficients totally standing for the inhomogeneity of the FGM beams which are calculated by the following integrations:
The material properties of the beam change as the power-law[1, 13, 15] functions of the coordinate z. The analytical expressions of functions fE(z),fρ(z) and the corresponding coefficients φi, φi(i = 0,1,2) are given,respectively,in Appendix A. For the reference homogenous beam,
3 Free vibration response of FGM beams
Assume that the vibration of beams is harmonic. Then,the dynamic response can be expressed as
where ω is the natural frequency,and u(x),w(x),and φ(x) are the shape functions. Substituting Eq. (21) into Eqs. (13)-(15),setting q = 0,and using some mathematical manipulations,we can finally arrive at a system of ordinary differential equations in dimensionless forms as follows:
in which the dimensionless quantities are defined as follows:
where Ω is the dimensionless natural frequency parameter; c is a constant which represents the inhomogeneity of the stiffness coefficients,and c = 1 corresponds to the homogeneous beam; and λij (i,j = 1,2,3) are constants representing the effects of the geometry,stiffness, inertia,and bending-tension coupling of the beam,and λ12 = λ21 = λ13 = 0 denotes for the reference homogenous beam. The analytical solutions of Eqs. (22)-(24) are difficult to be obtained due to that they are coupled with the three unknown functions U,W,and Φ. However, the numerical solutions of them can be found by some numerical techniques under specified boundary conditions. In order to write dimensionless boundary conditions for different end supports,the dimensionless resultant forces and bending moment corresponding to the mode shapes are given below :
where FN,Fs,and M are the amplitudes of the resultant forces FN,Fs and the bending moment M,respectively. In the numerical computation,the following four types of boundary conditions are considered.
(i) C-C
For the beam with two ends clamped,the boundary conditions can be written by
where the prime as superscript represents the derivatives with respect to ξ. The first equation in Eq. (10) is given by FN(1) = 0 using Eq. (26a).
(ii) S-S
For the beam with two ends simply supported,the boundary conditions are given by
The third equations in Eqs. (28a) and (28b) are derived from m(0) = m(1) = 0. By keeping in mind that
Eq. (28b) can be simplified as the uncoupled form of
(iii) C-F
The boundary conditions of the beam with C-F ends are as follows:
where the third equation in Eq. (29a) is given by fs(1) = 0.
(iv) C-S
The boundary conditions for C-S ends can be given by Eqs. (28a) and (27b) as follows:
From Eqs. (27b),(28a),and (30a),we can find that the axial displacement and the rotation angle are coupled in the boundary conditions due to φ1 ≠ 0. 4 Analogous transformation of frequency of FGM Euler-Bernoulli beams
By neglecting the axial,rotational,and tension-bending coupling inertia terms in Eqs. (13)- (15) and using the relation ϕ = ∂w/∂x ,the equation of motion of the free vibration for FGM Euler-Bernoulli beams in terms of the deflection can be obtained as follows:
Substituting the harmonic response of the transverse displacement in Eq. (21) into Eq. (31) and using the definition of the non-dimensional quantities yield the following non-dimensional ordinary differential equation:
where cρ is a dimensionless coefficient defined by
which integrates the non-homogeneity of an FGM beam both in the Young’s modulus and the mass density. Especially,for the reference homogenous beam with the material properties
it gives cρ ≡ 1. In this case,Eq. (32) reduces to
where W * and Ω * are the vibration mode shape function and the corresponding frequency of the reference homogenous beam,respectively. From the similarity between Eqs. (32) and (34), one can easily obtain the analogous transition relation between the natural frequencies of FGM beams and those of the corresponding homogenous beams as follows:
which shows that the natural frequency of an FGM beam is proportional to that of the corresponding homogenous beam with the same geometry and boundary conditions. The proportional coefficient depends on the variation of the material properties of the FGM beam or the functions fE(z) and ρE(z),and is determined by Eq. (33). Furthermore,it can easily show that the value of the coefficient in Eq. (35) will not be dependent on the boundary conditions. For the specified boundary conditions,the natural frequencies of a homogenous Euler-Bernoulli beam can be found even in the text books of dynamics of structures. Therefore,the solution of the free vibration of an FGM beam can be simplified as the calculation of the non-homogeneity parameter cρ.
It is noticed that the relationship (35) is derived in the sense of the classical beam theory so that it will keep accurate only for the Euler-Bernoulli beams. Theoretically,for the Timoshenko beams,this kind of transformation relations will not exist. However,Eq. (35) can also be used to predict the natural frequency approximately by taking the value of Ω * as the frequency of a homogenous Timoshenko beam. Similarly,Eq. (35) may be extended to calculate the natural frequencies of thin FGM plates. 5 Numerical results and discussion
The shooting method is used to search for the numerical solution of the above mentioned two- point boundary value problem of the ordinary differential equations (22)-(24). The shooting method replaces a two-point boundary value problem by a sequence of initial-value problems. Thus,the unknown values of the functions at the initial point,ξ = 0, are estimated to start the computation. These are iterated upon with modified values obtained by the secant method until the prescribed boundary conditions at the final point x = 1 are satisfied. Herein,the fourth-order Runge-Kutta method with variable steps is used to integrate the initial problems, and the Newton-Raphson iteration method is applied to modify the unknown initial values to satisfy the boundary conditions at the terminal point. In the numerical computation,an relative error limit,Er = 10 −5,is taken to control the accuracy in the numerical integration and iteration. Details of this approach can be found in Refs. [5, 10, 20].
In the following computation,the material constituents of the FGM beam are considered to be composed by alumina and aluminum with the material properties as follows[6, 14]:
alumina: Et = 380 GPa,ρt = 3 800 kg/m3,νt = 0.23;
aluminum: Eb = 70 GPa,ρb = 2 700 kg/m3,νb = 0.23.
The reference homogenous beam is considered to be made of pure aluminium. First,we consider the effective material properties of the beam to be varied as the power-law functions shown in Eq.(A2). The values of the coefficients φi and φi(i = 0,1,2) for different values of the power-law index n are listed in Table 1. In Fig. 2,the curves of the parameters c and cρ versus the index n are plotted,from which it can be seen that the material inhomogeneity of the beam is more evident in the range of 0 ≤ n ≤ 3.
 |
 |
| Fig. 2. Coefficients c and cρ changing with index n for material properties to be varied by power-law functions. |
In order to show the accuracy of the present numerical method to predict the free vibra- tion response of the FGM Timoshenko beam,a comparison of the dimensionless fundamental frequencies obtained by the shooting method in the present paper with those in Refs. [6, 14] is given in Table 2. The non-dimensional frequency parameter λ in the literature is defined by
Inevitably,the present results show good agreement with those in Refs. [6, 14]. Therefore,the shooting method can be used to effectively solve the complicated ordinary differential equations (22)-(24) with various boundary conditions.
For some specified values of the slenderness parameter δ,Table 3,4,and 5 show the nondimensional fundamental frequencies of the FGM beams with different boundary conditions changing with the power-law index n. It needs to notice that,in these tables,for a specified boundary condition,the values in the first row are obtained from Eqs.(22)-(24) by the shooting method,and the values in the second row are estimated by Eq. (35),where Ω * is the dimensionless natural frequency of the homogenous Timoshenko beams taken from the value of the first row with n = 1011. From these tables,it can be seen that the difference between the frequencies solved by Eqs.(22)-(24) and those predicted by the analogous transition (33) is not much significant,and it becomes smaller as the values of δ decrease,which means that even for the Timoshenko FGM beam,we can use Eq.(35) to predict its natural frequency approximately if the frequency of the corresponding homogenous beam is known,instead of solving the complicated coupling differential equations (22)-(24) with the boundary conditions. For δ = 1/50,the results predicted by the two methods nearly coincide. However,theoretically, only for the Euler-Bernoulli FGM beam (δ = 0),Eq.(35) gives the exact transition for the natural frequency.
 |
 |
 |
Table 6-8 present the first three lower-order dimensionless frequencies of the FGM Timo- shenko beams for different values of the slenderness parameter δ and the power index n under the C-C boundary condition. In these tables,the values in the second row are also given by Eq.(35),where Ω * is the frequency of the homogenous Timoshenko beam. It can be found that for the frequencies corresponding to the second and third modes,Eq.(35) can also give a good approximation for the FGM Timoshenko beams. The biggest relative error between the frequencies obtained based on the first-order shear deformation beam theory by the shooting method and those predicted by Eq.(35) is not greater than 4%. Furthermore,it can be found that the effects of the shear deformation on the frequencies increase with the increase in the values of δ. Therefore,one can also use Eq.(35) to approximately evaluate the natural frequencies of the FGM beams from those of the reference homogenous beams in the sense of the Timoshenko beam theory.
 |
 |
 |
For the specific values
the effects of the power index on the first three dimensionless frequencies of the FGM beam with simply supported ends are presented in Table 9,in which the results are estimated by Eq.(35),the finite element method[5],and the variation method[2]. The values of Ω * in Eq.(35) are taken as the exact solutions of the frequency of the Euler-Bernoulli beam,and they are π2, 4π2,and 9π2,respectively. As we expect,good agreement is shown among the results obtained by the shooting method (the Timoshenko beam theory),the finite element method (the Euler-Bernoulli beam theory )[1],the Lagrange equations (the Euler-Bernoulli beam theory)[2],and the analogous transformation (the Euler-Bernoulli beam theory) when the beam is very slender. It is further proved that Eq. (35) is a simple and exact formulation for the prediction of the natural frequencies of FGM beams from those of the homogenous ones in the sense of the Euler-Bernoulli beam theory.
 |
Figure 3 shows a continuous variation of the dimensionless fundamental frequencies of the FGM Timoshenko beams for different boundary conditions versus the power index n for some specified values of l/h. The increase in the value of the power index leads to the increase in the dimensionless frequency,which is due to the fact that the increase in the value of n results in the increase in the volume of the metal,in other words,the decrease in the rigidity of the beam. Along with the increase in the values of l/h,the natural frequencies increase. Furthermore,the increment in the natural frequency due to the increase in l/h is the maximum for the C-C beam and the minimum for the C-F beam. Similar phenomena can also be found by examining the variation tendency of the frequencies with the boundary conditions in Table 1-4.
 |
| Fig. 3. Fundamental frequencies of FGM Timoshenko beams versus power exponent n for different boundary conditions. |
Finally,we consider that the material properties of the FGM beam vary the exponentiallaw functions as shown in Eq. (A5). Figure 4 illustrates the first two dimensionless frequencies of the FGM Timoshenko beams with different boundary conditions changing with the values of l/h = 1/δ. From the results,we can find that the effects of the slenderness ratio on the frequencies are more significant when l/h < 15 and that the frequencies tend to be constants when l/h > 20.
 |
| Fig. 4. First and second dimensionless frequencies of FGM Timoshenko beams versus l/h for exponent-law variation of material properties (see Eq. (A5)). |
The free vibration response of FGM beams is studied based on both the classical and the first-order shear deformation theories. The equations of motion for the functionally graded Timoshenko beams are derived by totally considering the first-order shear deformation and the longitudinal,transversal,rotational,and tension-bending coupling inertia forces by assuming the material properties of the beam to be changed arbitrarily in the thickness direction. By using the shooting method to solve the eigenvalue problem of ordinary differential equations in terms of the shape functions with different boundary conditions,the natural frequencies of the FGM beams are obtained numerically. The dimensionless frequencies are presented for the FGM beams with the material properties varying continuously in the thickness direction according to both power-law and exponent-law forms. The effects of the slenderness ratio,the material gradient parameters,and the boundary conditions on the frequencies are examined in detail. The numerical results show that the effects of the shear deformation on the frequencies tend to be more significant when the beams become “shorter” (or “thicker”) and have stronger constraints. Furthermore,for a given length-height ratio,the shear deformation effects are more evident for higher-mode frequencies than for lower-mode frequencies.
In the special case of the Euler-Bernoulli beam theory,a proportional transformation between the natural frequency of an FGM beam and that of a reference homogenous beam with the same geometry and boundary conditions is presented. As a result,the evaluation of the natural frequency of the FGM beam can be reduced to calculate the transformation factors if the frequency of the homogenous beam is known. This approach provides a simple and exact way to predict the natural frequencies of non-homogenous Euler-Bernoulli beams without solving the boundary value problem for differential equations. The numerical results also show that this analogous transition for natural frequencies can also be extended to approximately predicting the natural frequencies of FGM Timoshenko beams if the values of the homogenous Timoshenko beams are known.
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Appendix A1
The resultant stiffness and inertia coefficients of homogenous beams are given by
For the material properties of the FGM beam, the power-law functions fE(z) and fρ(z) are given by
where
The corresponding coefficients φi and φi are
in which α = rE − 1, and β = rρ − 1.
For the material properties changing as exponential functions, fE(z) andfρ(z) are given by
where
Similarly, one can arrive at the coefficients φi and φi (i = 0, 1, 2) as follows:
