2. Theory formulation of FGM

An FGM is defined to be a material which has a continuous gradation throughthe-thickness (h). One side of the material is typically ceramic and the other side is typically metal. A mixture of the two materials composes the through-the-thickness characteristics. Let us consider a functionally graded beam shown in Figure 1. The parameters of the model FGM beam are as follows: L is the length of the beam, h is the thickness of the beam, and b is the width of the beam.

The elastic material properties vary through-the-beam thickness according to the volume fractions of the constituents using power law distribution (as shown in Figure 2).

The Stochastic Finite Element in the Natural Frequency of Functionally Graded Material Beams DOI: http://dx.doi.org/10.5772/intechopen.86013

Figure 1. Model of FGM beams.

response of structures with random material properties. In this direction, there is a reasonable body of recent research on the effect of uncertainties in material properties on the mechanical behavior of FGM. Investigators used stochastic simulation to study the effect of microstructural randomness on stress in FGM [2]. Ferrante et al. studied the effect of non-Gaussian porosity randomness on the response of functionally graded plate [3]. Yang et al. dealt with the stochastic bending response of moderately thick FGM plates [4, 5]. The effect of random material properties on post buckling response of FGM plate are presented in Lal

Mechanics of Functionally Graded Materials and Structures

However, the above mentioned literatures are for the static analysis. To the best of author's knowledge, few limited works have been done on the eigen analysis of FGM structures involving randomness in system parameters. Certain efforts have been made in the past to predict the dynamic behavior of structures with randomness. In most of the studies conducted, investigators dealt with the free vibration of functionally graded laminates with random material properties using first-order perturbation technique (FOPT) incorporating mixed type and semi-analytical approach to derive the standard eigenvalue problem [7]. Some of these papers presented the stochastic finite element method (SFEM) to investigate the natural frequency of functionally graded plates based on the higher-order shear deformation theory (HSDT) utilizing first-order reliability method and second-order reliability method [8]. In most cases, Jagtap et al. [9] examined the stochastic nonlinear free vibration response of FGM plate using HSDT with von-Karman kinematic nonlinearity via direct iterative stochastic finite element method. Shegokar et al. investigated the stochastic finite element nonlinear free vibration analysis of FGM beam with random material properties due to thermo-piezoelectric loadings [10]. The above mentioned literatures investigated the free vibration and nonlinear behavior of FGM beam and plate. The material properties, such as Young's modulus, shear modulus, and Poisson's ratio of FGM, are modeled as

In this chapter, the stochastic finite element solution is suggested to obtain the variability in the natural frequency of functionally graded material (FGM) beam. The elastic modulus and width of the FGM beam are assumed to vary in thickness and longitudinal directions following the exponential law. The uncertain material properties, such as modulus of elasticity and mass density of the FGM beam, are considered to be a one-dimensional homogeneous stochastic process. The stochastic finite element analysis of FGM beam is performed using the spectral representation method for the description of randomness in conjunction with Monte Carlo simulation (MCS). The response variability of natural frequency due to random elastic modulus and mass density in FGM beam is given. Furthermore, the effect of

An FGM is defined to be a material which has a continuous gradation throughthe-thickness (h). One side of the material is typically ceramic and the other side is typically metal. A mixture of the two materials composes the through-the-thickness characteristics. Let us consider a functionally graded beam shown in Figure 1. The parameters of the model FGM beam are as follows: L is the length of the beam, h is

The elastic material properties vary through-the-beam thickness according to the volume fractions of the constituents using power law distribution (as shown in

correlation between the two random parameters is observed as well.

the thickness of the beam, and b is the width of the beam.

et al. [6].

independent random variables.

2. Theory formulation of FGM

Figure 2).

62

#### Figure 2.

The variation of elasticity modulus along the corresponding directions.

$$E(z) = (E\_\circ - E\_m) \mathbf{g}(z) + Em$$

with

$$\mathbf{g}(\mathbf{z}) = \left(\frac{\mathbf{1}}{2} + \frac{\mathbf{z}}{h}\right)^p, \quad -\mathbf{0}.5h \le \mathbf{z} \le \mathbf{0}.5h, \quad \mathbf{0} \le p \le \infty$$

where E denotes the effective Young's modulus of elasticity, and Em and Ec represent the Young's modulus of metal and ceramic, respectively. The parameters g(z) and p represent the volume fraction of the metal and ceramic exponent, respectively.

In exponential law, for the material parameter of Young's modulus E, mass density ρ of the beam and the width of the beam b, with absolute values for z coordinate, which endows the symmetric characteristic to the beam with respect to mid-plane.

$$E(z) = E\_0 e^{\beta|x|}; \rho(z) = \rho\_0 e^{\beta|x|}; b(\varkappa) = b\_0 e^{\wp x}$$

In Eq. (3), E0, ρ<sup>0</sup> are the values of the Young's modulus and mass density at the mid-plane (z = 0) of the beam. The parameter β in the exponent characterizes the material property variation along the thickness direction. The width of the beam b varies according to the non-uniformity parameter ψ along the axis of the beam. In order to give the insight into the varying characteristics of the FGM beam under

The width and mass per unit length of the element are

The nodal displacement vector of the element is

where <sup>m</sup> <sup>¼</sup> <sup>Ð</sup>

Figure 4. Beam element.

degree of freedom.

beam

65

h=2

DOI: http://dx.doi.org/10.5772/intechopen.86013

�h=2

then, the displacement field is

where ν is Poisson's ratio.

3.2 Application of Hamilton's principle

beð Þ¼ x b<sup>1</sup> þ

með Þ¼ x mbeð Þ¼ x m b<sup>1</sup> þ

ρð Þz dz and ρð Þz denote the mass density at z.

The Stochastic Finite Element in the Natural Frequency of Functionally Graded Material Beams

f gq <sup>e</sup> ¼ q<sup>1</sup> q<sup>2</sup> q<sup>3</sup> q<sup>4</sup>

we ¼ h i N f gq <sup>e</sup>,

where h i¼ <sup>N</sup> h i <sup>N</sup><sup>1</sup> <sup>N</sup><sup>2</sup> <sup>N</sup><sup>3</sup> <sup>N</sup><sup>4</sup> , and Ni is Hermite shape function of <sup>i</sup>‐th

ð h=2

�h=2

Hamilton's principle may be a theoretical base for dynamical systems by its nature of integral form in time with Lagrangian density to account for continuous space. In this paper, the analysis of natural frequency of FGM beam is performed

D<sup>11</sup> ¼

In this case, the stiffness of the beam beD<sup>11</sup> is similar to EI of the homogeneous

E zð Þ <sup>1</sup> � <sup>ν</sup><sup>2</sup> <sup>z</sup><sup>2</sup>

dz,

� �<sup>T</sup>

b<sup>2</sup> � b<sup>1</sup> Le

x,

� �,

b<sup>2</sup> � b<sup>1</sup> Le

,

x

Figure 3. The variation of elasticity modulus, mass density, and width along the corresponding directions.

consideration, change of material constants and width depending on the corresponding parameters are shown in Figure 3.
