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This paper deals with a continuous beam resting on elastic support with elastic modulus derived from a random process. Governing equations of the stochastic higher-order finite element method of the free vibration of the continuous beam were derived from Hamilton's principle. The random process of elastic modulus was discretized by averaging random variables in each element. A solution for the stochastic eigenvalue problem for the free vibration of the continuous beam was obtained by using the perturbation technique, in conjunction with the finite element method. Spectral representation was used to generate a random process and employ the Monte Carlo simulation. A good agreement was obtained between the results of the first-order perturbation technique and the Monte Carlo simulation.

Based on the increase in tensile strength and toughness (43% and 30%, respectively), the feasibility of the hybrid nanofibers containing Single-walled carbon nanotubes and Nylon 66 on reinforcing cementitious materials has been clarified. The well-linking effect of nanofibers in the microstructure of hardened cement pastes has been found by scanning electron microscope (SEM) analysis and well-explained for the increase in mechanical strengths. Besides, field emission transmission electron microscope (FE-TEM) analysis has also been conducted to analyze the properties of the hybrid nanofibers.

This work presents the Monte Carlo simulation based on finite element analysis to investigate the natural frequencies of the beam on an elastic foundation with uncertain material. The elastic modulus is assumed as a homogeneous Gaussian random field. The random field is generated based on the spectral representation method. An iterative algorithm is employed using the finite element method to calculate natural frequencies with various the generated functions of elastic modulus. The effects of the random field on the response variability of natural frequencies and the effects of foundation coefficients are discussed in detail.

The free vibration of functionally graded material (FGM) beams on an elastic foundation and spring supports is investigated. Young\'s modulus, mass density and width of the beam are assumed to vary in thickness and axial directions respectively following the exponential law. The spring supports are also taken into account at both ends of the beam. An analytical formulation is suggested to obtain eigen solutions of the FGM beams. Numerical analyses, based on finite element method by using a beam finite element developed in this study, are performed in order to show the legitimacy of the analytical solutions. Some results for the natural frequencies of the FGM beams are given considering the effect of various structural parameters. It is also shown that the spring supports show the greatest effect on the natural frequencies of FGM beams.