5.3 FGM beam having correlation multiple randomness

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

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.

Figure 13.

Figure 14.

72

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

Mechanics of Functionally Graded Materials and Structures

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

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

the effect of correlation between two random parameters of elastic modulus and mass density. To this aim, we consider three correlation cases: +1.0, 0.0, and 1.0. When the random processes for elastic modulus, f <sup>E</sup>, and mass density, f <sup>ρ</sup>, are exactly the same, the correlation coefficient (CC) is +1.0. If the values have negative amounts, then the correlation coefficient is �1.0. The zero-correlation (CC = 0.0)

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

As shown in Figure 15, the maximum COV exceeds the input standard deviation of the stochastic process in the case of negative perfect correlation, while it is about 75% in the case of no correlation. However, in case of positive perfect correlation (Figure 15c), the response COV of natural frequency of FGM beam is small enough to be ignored. These results can easily be understood because the stiffness and mass matrix of each elements in Eq. (18) increase or decrease with the same rate in the case of positive perfect correlation. In case of negative perfect correlation, the ratio of random parts in the stiffness and mass are relatively large since the random parts

In the case, when we take the correlated multiple random material properties into account, we also obtained the slight nonlinear pattern of variation of COV in

To evaluate response variability due to a single parameter of the random Young's modulus and multiple uncertain material properties, a formulation in the context of stochastic finite element solution is suggested for the natural frequency of FGM beam. In deriving the formula for the covariance of the response, modified power spectral density and correlation function are defined by using the general formula of random processes. The Monte Carlo simulation is performed employing the statistical preconditioning scheme as a random process generation technique. The local average method is employed instead of mid-point rule in Monte Carlo

In FGM beam natural frequency, the response COV for correlation between a random of single parameter and two uncertain parameters is observed. The coefficient of variation of natural frequency can only reach about 50% of the input standard deviation of the stochastic process in a single parameter of the random elastic modulus. However, the number of values is increased over 100% of the input standard deviation of the stochastic process in multiple uncertain material properties, when the correlation distance tends to infinity. The results showed that the COV of natural frequency of FGM beam in a single parameter of the random Young's modulus and multiple uncertain material properties achieved maximum

There is a very small difference between deterministic natural frequency and probabilistic natural frequency of FGM beam for the case of positive perfect correlation. Also, the COV of natural frequency does not depend on the number of elements, Young's modulus ratio, and the ratio of non-uniformity parameter of FGM beam. The importance of these parameters needs to be studied as a further

<sup>L</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>

x3

<sup>L</sup><sup>3</sup> ; N<sup>3</sup> <sup>¼</sup> <sup>x</sup> � <sup>x</sup>

L þ x2 L2 ; N<sup>4</sup> <sup>¼</sup> <sup>3</sup> <sup>x</sup><sup>2</sup>

<sup>L</sup><sup>2</sup> � <sup>2</sup>

x3 L3 :

Hermite shape functions of beam finite element:

means that stochastic processes of the two parameters are theoretically

have opposite sign, which makes the response variability large.

terms of COV of stochastic fields, as shown in Figure 16.

independent [13].

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

6. Conclusions

simulation.

work.

75

A. Appendix

N<sup>1</sup> ¼ x 1 � 2

x L þ x2 L2 ; N<sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>3</sup> <sup>x</sup><sup>2</sup>

variability for d about 1.0.

COV of natural frequency as a function of the standard deviation of stochastic process: (a) negative perfect correlation (CC = 1.0), (b) no correlation (CC = 0.0), and (c) positive perfect correlation (CC = +1.0).

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

amounts, then the correlation coefficient is �1.0. The zero-correlation (CC = 0.0) means that stochastic processes of the two parameters are theoretically independent [13].

As shown in Figure 15, the maximum COV exceeds the input standard deviation of the stochastic process in the case of negative perfect correlation, while it is about 75% in the case of no correlation. However, in case of positive perfect correlation (Figure 15c), the response COV of natural frequency of FGM beam is small enough to be ignored. These results can easily be understood because the stiffness and mass matrix of each elements in Eq. (18) increase or decrease with the same rate in the case of positive perfect correlation. In case of negative perfect correlation, the ratio of random parts in the stiffness and mass are relatively large since the random parts have opposite sign, which makes the response variability large.

In the case, when we take the correlated multiple random material properties into account, we also obtained the slight nonlinear pattern of variation of COV in terms of COV of stochastic fields, as shown in Figure 16.
