6. Conclusions

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

Mechanics of Functionally Graded Materials and Structures

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).

Figure 16.

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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 simulation.

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 variability for d about 1.0.

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 work.
