Free vibration analysis of beams resting on Pasternak soil based on trigonometric shear deformation theory

Free vibration analysis of beams resting on elastic foundation is among the current research topics of interest. Although it is a common design approach to assume that the foundation where the beam rested on conforms to the Winkler hypothesis, the use of the Pasternak foundation model, which takes into account the shear interaction between the soil and the beam represents the structural behavior better. In addition to Pasternak foundation model, Touratier beam theory, which has a kinematically more extensive displacement function compared to polynomial shear deformation theories such as Timoshenko and Reddy-Bickford, is the mathematical model based on in this study. The results obtained from dynamic analyzes using finite elements method and similar may not coincide the results obtained by using analytical method, although a large number of nodes requiring long analysis periods are used. Dynamic stiffness method (DSM), differential quadrature element method (DQEM), generalized differential quadrature rule (GDQR), which give precise results in shorter periods with less nodes usage, are applied to Touratier beam. In addition, circular frequency values coincide with the analytical results are obtained by using variational iterative stiffness method (VISM) which has developed within the scope of this thesis. In this study, free vibration analysis of beams, which are resting on Pasternak foundation, subjected to an axial compressive force is performed. Equation of motions of a Touratier beam resting on Pasternak foundation that have uniform cross-section with different boundary conditions for single-span model and both uniform and non-uniform cross-section with semi-rigid end connections for multi-span model are solved by using DSM, DQEM, GDQR and VISM. Circular frequency values of beams are investigated by using Wittrick-Williams algorithm (WWA). The efficiency and reliability of the methods are presented in tables by comparing the values with those obtained from analytical method and elasticity theory. Transverse displacements, beam slopes, section rotations, shear forces and bending moments according to the position parameter are represented in figures.