4. Modeling of randomness

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

∂2 we ∂x<sup>2</sup> � �<sup>2</sup>

mbew\_ <sup>2</sup> e 2

beD11h i <sup>N</sup> <sup>T</sup>h i <sup>N</sup> dx:

mebeh i <sup>N</sup> <sup>T</sup>h i <sup>N</sup> dx:

<sup>e</sup> Mef gq\_ <sup>e</sup>:

dt ¼ 0,

The governing differential equations of motion and the related governing equa-

dx:

dx:

beD<sup>11</sup> 2

> L ðe

> > 0

Ue ¼

The kinematic energy Te for flexural vibration is

Mechanics of Functionally Graded Materials and Structures

L ðe

0

Te ¼

L ðe

0

L ðe

0

Ue <sup>¼</sup> <sup>1</sup> 2 f gq T <sup>e</sup> Kef gq <sup>e</sup>,

Te <sup>¼</sup> <sup>1</sup> 2 f gq\_ <sup>T</sup>

> ∑ N 1

Ue � ∑ N 1 Te

Substituting Eqs. (13) and (14) into Eq. (15), the following can be obtained:

Mq€ þ Kq ¼ 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

<sup>M</sup> � � <sup>¼</sup> <sup>0</sup>:

. Accordingly with Eq. (16), we can obtain the following:

For simple harmonic vibration, we assume the displacements to be

<sup>K</sup> � <sup>ω</sup><sup>2</sup> <sup>M</sup> � �<sup>w</sup> <sup>¼</sup> <sup>0</sup>,

det <sup>K</sup> � <sup>ω</sup><sup>2</sup>

� �

tion can be derived using Hamilton's principle

δ ðt2

where N denotes the number of finite elements.

here, q is the nodal displacement of the beam.

have nontrivial solution, the following needs to be satisfied:

t1

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

Ke ¼

Me ¼

following:

where

q xð Þ¼ ; <sup>t</sup> w xð Þe<sup>i</sup>ω<sup>t</sup>

66

### 4.1 Mathematical expression

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 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 density can be written as

$$E\_0(\mathbf{x}) = \overline{E}\_0\left[\mathbf{1} + f\_E(\mathbf{x})\right],$$

$$\rho\_0(\mathbf{x}) = \overline{\rho}\_0\left[\mathbf{1} + f\_\rho(\mathbf{x})\right],$$

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 stochastic process which are homogeneous with zero-mean values.

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

$$f(\mathbf{x}) = \sqrt{2} \sum\_{n=0}^{N-1} A\_n \cos \left(\alpha\_n t + \phi\_n \right),$$

$$A\_n = \sqrt{2 \mathbf{S}\_{\mathcal{U}}(\alpha\_n) \Delta \alpha},$$

$$\Delta \alpha = \frac{\alpha\_n}{N},$$

$$\alpha\_n = n \Delta \alpha, n = 0, 1, 2, ..., N - 1.$$

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 of ωu:

$$\int\_0^{o\_n} \mathcal{S}\_{\overline{\mathcal{G}}}(o\_n) d o\_n = (1 - \varepsilon) \int\_0^{\infty} \mathcal{S}\_{\overline{\mathcal{G}}}(o\_n) d o\_n \mu$$

where ε<< 1:

The uniform random phase angle ϕ<sup>n</sup> in Eq. lies in the range of 0½ � ; 2π . The power spectral density function used in Eq. (20) is given as

$$S\_{\widetilde{\mathcal{Y}}}(o\_n) = \frac{1}{\sqrt{\pi}} \sigma\_{\widehat{f}} \,^2 d \, e^{\left(-d^{\widehat{\mathbb{L}}^2 o\_n^2}\right)}, \quad -\infty < o\_n < \infty,$$

here, σ<sup>f</sup> denotes the standard deviation of the stochastic process f xð Þ and d is the correlation distance of the stochastic process along the x axis.

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

Figure 6. FGM beam model.

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

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

Figure 7.

69

The convergence between exact solution and finite element method.

Figure 5. Example plot of random process used <sup>d</sup> <sup>¼</sup> <sup>0</sup>:01; <sup>0</sup>:1; <sup>1</sup>:0; <sup>σ</sup><sup>f</sup> <sup>¼</sup> <sup>0</sup>:<sup>1</sup> � �.

$$COV = \frac{\text{Standard deviation}}{|Mean|},$$

here,

$$\begin{aligned} \text{Standard deviation} &= \sqrt{\text{var}(a)} \\ \text{Var}(a) &= E\left[ \left( a - \overline{a} \right)^2 \right] \end{aligned} $$

and ω is the natural frequency of the FGM beam, and ω denotes the mean of the natural frequency.

#### 4.2 Monte Carlo analysis

In order to obtain the response variability in the natural frequency of the FGM beam, we employed the scheme of Monte Carlo simulation (MCS). As a matter of fact, the MCS corresponds to the deterministic analyses on a set of heterogeneous models of the given structure, in which the material properties have different values depending on the position in the domain of the structure.

The generation of heterogeneous random samples is accomplished by the aforementioned spectral representation scheme, and we use 10,000 samples for respective analyses. In particular, we adopt the local average scheme other than the midpoint rule in applying the MCS, with which better results can be obtained especially for the processes with small correlation distance. Figure 5 shows an example plot of the processes employed to model the randomness in the system parameters.
