3. Finite element formulation for FGM beam and frequency analysis

#### 3.1 Finite elements

In case of four degree of freedom beam element, as shown in Figure 4, the transverse displacement function may be assumed as a cubic polynomial in x, and the corresponding shape functions are Hermite interpolation functions.

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

#### Figure 4. Beam element.

The width and mass per unit length of the element are

$$b\_{\epsilon}(\mathbf{x}) = b\_1 + \frac{b\_2 - b\_1}{L\_{\epsilon}} \mathbf{x},$$

$$m\_{\epsilon}(\mathbf{x}) = m b\_{\epsilon}(\mathbf{x}) = m \left(b\_1 + \frac{b\_2 - b\_1}{L\_{\epsilon}} \mathbf{x}\right),$$

where <sup>m</sup> <sup>¼</sup> <sup>Ð</sup> h=2 �h=2 ρð Þz dz and ρð Þz denote the mass density at z.

The nodal displacement vector of the element is

$$\{q\}\_{\epsilon} = \begin{Bmatrix} q\_1 \ q\_2 \ q\_3 \ q\_4 \end{Bmatrix}^T,$$

then, the displacement field is

$$w\_e = \langle \mathbf{N} \rangle \{ q \}\_{e^\*}$$

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 degree of freedom.

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

$$D\_{11} = \int\_{-h/2}^{h/2} \frac{E(z)}{1 - \nu^2} z^2 dz, 0$$

where ν is Poisson's ratio.

### 3.2 Application of Hamilton's principle

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

consideration, change of material constants and width depending on the

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

the corresponding shape functions are Hermite interpolation functions.

3. Finite element formulation for FGM beam and frequency analysis

In case of four degree of freedom beam element, as shown in Figure 4, the transverse displacement function may be assumed as a cubic polynomial in x, and

corresponding parameters are shown in Figure 3.

Mechanics of Functionally Graded Materials and Structures

3.1 Finite elements

Figure 3.

64

using Hamilton's principle. The strain energy expression Ue for bending is given as following:

4. Modeling of randomness

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

4.1 Mathematical expression

density can be written as

of ωu:

67

where ε<< 1:

In order to model the randomness in the material properties, the modulus of elasticity and mass density along the mid-plane are assumed to vary along its length

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

<sup>E</sup>0ð Þ¼ <sup>x</sup> <sup>E</sup><sup>0</sup> <sup>1</sup> <sup>þ</sup> <sup>f</sup> <sup>E</sup>ð Þ <sup>x</sup> � �,

h i,

An cos ωnt þ ϕ<sup>n</sup> ð Þ,

,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Sffð Þ ω<sup>n</sup> Δω

ρ0ð Þ¼ x ρ<sup>0</sup> 1 þ f <sup>ρ</sup>ð Þ x

where x is the coordinate along the axis of the FGM beam, E0, ρ<sup>0</sup> are the expected values of E<sup>0</sup> and ρ0, respectively, and f <sup>E</sup>ð Þ x , f <sup>ρ</sup>ð Þ x are one-dimensional

The numerical generation of sample functions of Gaussian zero-mean homogeneous stochastic processes, which describe the randomness in parameters of the structure, is accomplished using the spectral representation method. For a one-

stochastic process which are homogeneous with zero-mean values.

dimensional univariate (1D-1V) stochastic process, we have [11]

An ¼

Sffð Þ ω<sup>n</sup> dω<sup>n</sup> ¼ ð Þ 1 � ε

q

<sup>Δ</sup><sup>ω</sup> <sup>¼</sup> <sup>ω</sup><sup>u</sup> N ,

ω<sup>n</sup> ¼ nΔω, n ¼ 0, 1, 2, …, N � 1:

In Eq. (21), ω<sup>u</sup> denotes the upper cut-off frequency beyond which the power spectral density function Sffð Þ ω<sup>n</sup> may be assumed to be zero for either mathematical or physical reasons. The following criterion is usually used to estimate the value

∞ð

Sffð Þ ω<sup>n</sup> dωn,

<sup>ω</sup><sup>2</sup> ð Þ<sup>n</sup> , � <sup>∞</sup> <sup>&</sup>lt;ω<sup>n</sup> <sup>&</sup>lt; <sup>∞</sup>,

0

The uniform random phase angle ϕ<sup>n</sup> in Eq. lies in the range of 0½ � ; 2π . The power

here, σ<sup>f</sup> denotes the standard deviation of the stochastic process f xð Þ and d is the

In all examples, the coefficient of variation ð Þ COV of natural frequency, which is

defined as a ratio of the standard deviation of response to the absolute mean

f xð Þ¼ ffiffi 2 <sup>p</sup> <sup>∑</sup> N�1 n¼0

ωðu

0

spectral density function used in Eq. (20) is given as

1 ffiffiffi <sup>π</sup> <sup>p</sup> <sup>σ</sup><sup>f</sup> 2 d e �d<sup>2</sup>

correlation distance of the stochastic process along the x axis.

response, will be used to give the variability of the response.

Sffð Þ¼ ω<sup>n</sup>

of FGM beam in a random manner. We can model these variations as onedimensional univariate (1D-1V) homogeneous stochastic processes. The simple mathematical expressions for the randomly varying modulus of elasticity and mass

$$U\_{\epsilon} = \int\_{0}^{L\_{\epsilon}} \frac{b\_{\epsilon} D\_{11}}{2} \left(\frac{\partial^{2} w\_{\epsilon}}{\partial \mathbf{x}^{2}}\right)^{2} d\mathbf{x} \,\mathrm{L}$$

The kinematic energy Te for flexural vibration is

$$T\_{\epsilon} = \int\_{0}^{L\_{\epsilon}} \frac{mb\_{\epsilon}\dot{w}\_{\epsilon}^{2}}{2} d\infty.$$

where

$$K\_{\varepsilon} = \bigcap\_{\tilde{0}}^{L\_{\varepsilon}} b\_{\varepsilon} D\_{11} \langle \mathbf{N} \rangle^{T} \langle \mathbf{N} \rangle d\mathbf{x}.$$

$$M\_{\varepsilon} = \int\_{0}^{L\_{\varepsilon}} m\_{\varepsilon} b\_{\varepsilon} \langle \mathbf{N} \rangle^{T} \langle \mathbf{N} \rangle d\mathbf{x}.$$

Substituting Eq. (7) into Eqs. (9) and (10), the following can be obtained:

$$U\_{\epsilon} = \frac{1}{2} \{q\}\_{\epsilon}^{T} K\_{\epsilon} \{q\}\_{\epsilon^{\prime}}$$

$$T\_{\epsilon} = \frac{1}{2} \{\dot{q}\}\_{\epsilon}^{T} M\_{\epsilon} \{\dot{q}\}\_{\epsilon}.$$

The governing differential equations of motion and the related governing equation can be derived using Hamilton's principle

$$\delta \int\_{t\_1}^{t\_2} \left( \sum\_{1}^{N} U\_{\varepsilon} - \sum\_{1}^{N} T\_{\varepsilon} \right) dt = 0,$$

where N denotes the number of finite elements. Substituting Eqs. (13) and (14) into Eq. (15), the following can be obtained:

$$M\ddot{q} + \mathcal{K}q = 0,$$

here, q is the nodal displacement of the beam.

For simple harmonic vibration, we assume the displacements to be q xð Þ¼ ; <sup>t</sup> w xð Þe<sup>i</sup>ω<sup>t</sup> . Accordingly with Eq. (16), we can obtain the following:

$$(\mathcal{K} - a^2 \mathcal{M})\omega = 0,$$

where K is the assembled global stiffness matrix of Ke, the element stiffness matrix, which is given in detail in the Appendix. For Eq. (17) to be valid, i.e., to have nontrivial solution, the following needs to be satisfied:

$$\det\left(K - \alpha^2 \mathcal{M}\right) = \mathbf{0}.$$

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