An analytical method and a semi-analytical method are proposed to analyze the dynamic thermo-elastic behavior of structures resting on a Pasternak foundation. The analytical method employs a finite Fourier integral transform and its inversion, as well as a Laplace transform and its numerical inversion. The semi-analytical method employs the state space method, the differential quadrature method (DQM), and the numerical inversion of the Laplace transform. To demonstrate the two methods, a simply supported Euler–Bernoulli beam of variable length is considered. The governing equations of the beam are derived using Hamilton's principle. A comparison between the results of analytical method and the results of semi-analytical method is carried out, and it is shown that the results of the two methods generally agree with each other, sometimes almost perfectly. A comparison of natural frequencies between the semi-analytical method and the experimental data from relevant literature shows good agreements between the two kinds of results, and the semi-analytical method is validated. Different numbers of sampling points along the axial direction are used to carry out convergence study. It is found that the semi-analytical method converges rapidly. The effects of different beam lengths and heights, thermal stress, and the spring and shear coefficients of the Pasternak medium are also investigated. The results obtained in this paper can serve as benchmark in further research.

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