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The beam theory is used in the analysis and design of a wide range of structures, from buildings to bridges to the load-bearing bones of the human body. Beams resting on elastic foundation have wide application in many branches of engineering problems namely geotechnics, road, railroad and marine engineering and bio-mechanics. The foundation is very often a rather complex medium; e.g., a rubberlike fuel binder, snow, or granular soil. The key issue in the analysis is modelling the contact between the structural elements and the elastic bed. Since of interest here is the response of the foundation at the contact area and not the stresses or displacements inside the foundation material, In most cases the contact is presented by replacing the elastic foundation with simple models, usually spring elements. The most frequently used foundation model in the analysis of beam on elastic foundation problems is the Winkler foundation model. In the Winkler model, the elastic bed is modeled as uniformly distributed, mutually independent, and linear elastic vertical springs which produce distributed reactions in the direction of the deflection of the beam. However since the model does not take into account either continuity or cohesion of the bed, it may be considered as a rather crude representation of the elastic foundation. In order to find a physically close and mathematically simple foundation model, Pasternak proposed a so-called two-parameter foundation model with shear interactions. The first foundation parameter is the same as the Winkler foundation model and the second one is the stiffness of the shearing layer in the Pasternak foundation model.
Dynamic analysis is an important part of structural investigation and the results of free vibration analysis are useful in this context. Vibration problems of beams on elastic foundation occupy an important place in many fields of structural and foundation engineering.With the increase of thickness, existence of simplifying hypotheses in beam theories such as the ignorance of rotational inertial and transverse shear deformation in classic theory, application of determination coefficient in first-order shear theory and expression of one or few unknown functions based on other functions in higher-order shear theories is accompanied by reduction in accuracy of these theories. This represents the necessity of precise and analytical solutions for beam problems with the least number of simplifying hypotheses and for different thicknesses.
In the present study, the analytical solution for the problem of free vibration of homogeneous prismatic simply supported beam with rectangular solid sections and desired thickness resting on Pasternak elastic foundation is provided for completely isotropic behaviors under two-dimensional theory of elasticity and functions of displacement potentials. Characteristic equations of natural vibration are defined by solving one partial differential equations of fourth order through separation of variables and application of boundary conditions. The major characteristics of present study are lack of limitation of thickness and its validity for beams of low, medium and large thickness. To verify, the results of present study were compared with those of other studies. The results show that increases of foundation parameters is associated with an increased natural frequency, The intensity by increasing the ratio of thickness to length and in values larger than 0.2 and in the higher modes of vibration is reduced considerably.

Keywords: Natural Frequency, Free vibration, Deep Beam, elastic foundation, Potential Functions

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