Effect of Random Material Properties on the Variability of Natural Frequency of Functionally Graded Beam

There has been considerable attention on functionally graded materials applied in most engineering aspects. However, the manufacturing of these materials with a fully specified material gradation profile is difficult, which causes them to have significant variability in structural and mechanical properties. As such, it is required that the randomness in material properties is properly handled for the accurate prediction of responses by structures, with stochastic analysis being a useful tool in this regard. However, most research work in this regard has been limited to static behavior of functionally graded material beams and plates, with few works on eigen-analysis of these structures.

In a recent paper published in KSCE Journal of Civil Engineering, Thuan Nguyen Van and Hyuk Chun Noh developed a finite element for a beam structure made of functionally graded material with varying width. This has been applied in the finite element stochastic analysis to acquire the beam’s natural frequency response variability within the Monte Carlo Simulation framework.

The authors modeled a functionally graded beam, whose width varied axially, with 2-node beam finite elements developed. It was assumed that along the beam’s mid-plane, the mass density and elasticity modulus varied randomly in the axial direction. Mathematical expressions were applied in order to model the variations as 1-dimensional univariate homogenous stochastic processes. The method of spectral representation was used to numerically generate 10,000 heterogenous random samples.

The research team observed that the effect that the correlation distance has on the random process characteristics is noticeable, such that when the distance tends towards infinity, the random variable state is attained, and is constant over the whole domain. The generated random samples were seen to fulfill the initially assigned statistical moments, as well as exhibiting homogeneity characteristics.

The authors tested the cogency of the modelled beam finite element by evaluating the coefficient of variation through varying the stochastic field standard deviation, and the size of the correlation distance. They observed that the variation of this parameter depends on the correlation distance value, such that it converges to values taken to be over 50% of the assumed stochastic field standard deviation, as this distance tends towards infinity. The variability of response was the same for all natural frequencies, and the boundary conditions were seen not to severely affect the coefficient of variation of random frequencies.

It was noted that the response coefficient of variation is a linear function of input randomness, and that mesh refinement has no effect on this. Further, the non-uniformity parameter was observed not to affect the response variability. Also, the parameter that characterizes the variation of material properties along the direction of the thickness, was observed not to affect the natural frequency coefficient of variation. In addition, the ratio of elastic modulus only affected the beam’s deterministic stiffness and not the stochastic responses.

Further tests by Nguyen and Noh showed that in negative perfect correlation condition for mass density and elastic modulus, the maximum coefficient of variation exceeded the stochastic process’s input standard deviation, but was approximately 75% when the parameters had no correlation. Although the system is linear, there was a nonlinear increase in the coefficient of variation of the response, with an increase in the coefficient of variation of stochastic processes.