5. Numerical example

The geometric dimensions of the example FGM beam are: h = 0.1 m, L = 1 m, and b<sup>0</sup> = 0.1 m. It is assumed that the material properties are E<sup>0</sup> = 70 GPa, ρ<sup>0</sup> = 2780 kg/m<sup>3</sup> , and ν = 0.33. E<sup>0</sup> denotes the Young's modulus at the mid-surface of the beam and E<sup>1</sup> at the top and bottom surfaces following Eq. (3) (Figure 6).

#### 5.1 Deterministic analysis results

The results in Figure 7 correspond to the prismatic homogeneous beam since the parameters in exponents, β and ψ, are all zero. The natural frequency is obtained in

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

Figure 6. FGM beam model.

COV <sup>¼</sup> Standard deviation

Standard deviation <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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

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

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 geometric dimensions of the example FGM beam are: h = 0.1 m, L = 1 m,

The results in Figure 7 correspond to the prismatic homogeneous beam since the parameters in exponents, β and ψ, are all zero. The natural frequency is obtained in

of the beam and E<sup>1</sup> at the top and bottom surfaces following Eq. (3) (Figure 6).

, and ν = 0.33. E<sup>0</sup> denotes the Young's modulus at the mid-surface

the processes employed to model the randomness in the system parameters.

and b<sup>0</sup> = 0.1 m. It is assumed that the material properties are E<sup>0</sup> = 70 GPa,

Varð Þ¼ <sup>ω</sup> <sup>E</sup> ð Þ <sup>ω</sup> � <sup>ω</sup> <sup>2</sup> h i

depending on the position in the domain of the structure.

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> � �.

Mechanics of Functionally Graded Materials and Structures

here,

Figure 5.

natural frequency.

4.2 Monte Carlo analysis

5. Numerical example

5.1 Deterministic analysis results

ρ<sup>0</sup> = 2780 kg/m<sup>3</sup>

68

j j Mean ,

varð Þ <sup>ω</sup> <sup>p</sup>

9 = ;:

Figure 7. The convergence between exact solution and finite element method.

the present study to compare with the results of analytical solution given by Hassen et al. [12].

The discrepancies between exact and finite element solutions for the frequencies for the first three modes are shown in Figure 8. The differences given in percentile tends to zero as the number of finite elements is increased, meaning the results are converging to exact solutions.

Figure 9 shows the first three normalized natural frequencies of uniform FGM beams for three cases of modulus ratio (E1=E0). The natural frequency increases as the ratio of Young's modulus increases from E<sup>1</sup> ¼ 0:2E<sup>0</sup> to E<sup>1</sup> ¼ 5E0.

#### 5.2 Variability of natural frequency due to randomness in elastic modulus

Figure 10 shows the COV of response versus the correlation distance (d) when the elastic modulus E<sup>0</sup> is random. The standard deviation of the random elastic modulus is denoted by σ<sup>f</sup> . In all cases, the COV of natural frequency shows similar trends, starting from small values for small correlation distance, up to large values

> for large correlation distance. In the FGM beam under consideration, the response variability, when the correlation distance tends to infinity, is obtained to be about

COV of natural frequency as a function of the correlation distance (d) for different standard deviation of

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

The relationship between COV of natural frequency and the COV of stochastic

The effect of mesh refinement on the COV of natural frequency is shown in Figure 12. The correlation distance log (d) is assumed to be from �3 to 1. As seen in Figure 12, the COV of natural frequency is not affected by the mesh refinement. Figure 13 shows the variation of the coefficient of variation (COV) depending on the non-uniformity parameter (ψ) of the FGM beam. As seen in the figure, the COV of response is not slightly affected by the non-uniformity parameter in par-

The overall features of the effect of Young's modulus ratio on COV of natural frequency are shown in Figure 14. The COV of natural frequency is not affected by the parameter β. These results can easily be understood because the standard deviation and the mean of natural frequency in Eq. (27) increase in the same rate.

process is shown in Figure 11. The standard deviation of stochastic process is changed from 0.0 to 0.25. As seen in Figure 11, the COV of response shows a

50% of the input standard deviation of the stochastic process.

slightly nonlinear pattern in all the cases of d = 0.01, 0.1, and 10.

COV of natural frequency as a function of the standard deviation of stochastic process f <sup>E</sup>ð Þ x .

ticular for large correlation distances.

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

Figure 10.

Figure 11.

71

stochastic process.

Figure 8. Error depending on mesh refinement (β ¼ 0, ψ ¼ 0).

Figure 9. First three normalized natural frequencies of FGM beams for different ratio of Young's modulus (ψ ¼ 0).

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

#### Figure 10.

the present study to compare with the results of analytical solution given by Hassen

The discrepancies between exact and finite element solutions for the frequencies for the first three modes are shown in Figure 8. The differences given in percentile tends to zero as the number of finite elements is increased, meaning the results are

Figure 9 shows the first three normalized natural frequencies of uniform FGM beams for three cases of modulus ratio (E1=E0). The natural frequency increases as

Figure 10 shows the COV of response versus the correlation distance (d) when the elastic modulus E<sup>0</sup> is random. The standard deviation of the random elastic modulus is denoted by σ<sup>f</sup> . In all cases, the COV of natural frequency shows similar trends, starting from small values for small correlation distance, up to large values

the ratio of Young's modulus increases from E<sup>1</sup> ¼ 0:2E<sup>0</sup> to E<sup>1</sup> ¼ 5E0.

Mechanics of Functionally Graded Materials and Structures

5.2 Variability of natural frequency due to randomness in elastic modulus

First three normalized natural frequencies of FGM beams for different ratio of Young's modulus (ψ ¼ 0).

et al. [12].

Figure 8.

Figure 9.

70

Error depending on mesh refinement (β ¼ 0, ψ ¼ 0).

converging to exact solutions.

COV of natural frequency as a function of the correlation distance (d) for different standard deviation of stochastic process.

for large correlation distance. In the FGM beam under consideration, the response variability, when the correlation distance tends to infinity, is obtained to be about 50% of the input standard deviation of the stochastic process.

The relationship between COV of natural frequency and the COV of stochastic process is shown in Figure 11. The standard deviation of stochastic process is changed from 0.0 to 0.25. As seen in Figure 11, the COV of response shows a slightly nonlinear pattern in all the cases of d = 0.01, 0.1, and 10.

The effect of mesh refinement on the COV of natural frequency is shown in Figure 12. The correlation distance log (d) is assumed to be from �3 to 1. As seen in Figure 12, the COV of natural frequency is not affected by the mesh refinement.

Figure 13 shows the variation of the coefficient of variation (COV) depending on the non-uniformity parameter (ψ) of the FGM beam. As seen in the figure, the COV of response is not slightly affected by the non-uniformity parameter in particular for large correlation distances.

The overall features of the effect of Young's modulus ratio on COV of natural frequency are shown in Figure 14. The COV of natural frequency is not affected by the parameter β. These results can easily be understood because the standard deviation and the mean of natural frequency in Eq. (27) increase in the same rate.

Figure 11. COV of natural frequency as a function of the standard deviation of stochastic process f <sup>E</sup>ð Þ x .

5.3 FGM beam having correlation multiple randomness

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

Figure 15.

73

It is natural to have preposition that not only the elastic modulus, but also the mass density of the material can have randomness. Therefore, we need to consider

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

Effect of the correlation between two random parameters: (a) negative perfect correlation (CC = 1.0), (b) no

correlation (CC = 0.0), and (c) positive perfect correlation (CC = +1.0).

Figure 12. Effect of mesh refinement on the COV of natural frequency (σ<sup>f</sup> ¼ 0:1).

Figure 13. Effect of non-uniformity parameter (ψ) on COV of natural frequency (Ne = 20, σ<sup>f</sup> = 0.15).

Figure 14. Effect of parameter β on the COV of natural frequency (Ne = 20, σ<sup>f</sup> = 0.2).

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