Dynamic analysis of systems with distributed mass by using high-order shear deformation theory, differential quadrature (DQM) and differential transformation (DTM) methods

In this study, the mathematical model of Reddy-Bickford beam theory, one of the high order shear deformation theories, is used.One of the important subjects that is interested by many researchers and that protects its currency in structural and earthquake engineering is free vibration analysis of beams on elastic foundation. Dynamic analysis of beams on elastic foundation is investigated by different numerical methods in literature. These studies mostly include the solutions made by Finite Difference, Finite Elements, Boundary Elements and Perturbation Techniques that numerical or many scaled perturbation methods are used. High capacity computers are needed for problem solution since the most accurate conclusion can be obtained by using a lot of nodes or by making a lot of iterations in these numerical approximation methods. Differential Quadrature Element Method (DQEM) and Differential Transformation Method (DTM), which are developed as a result of the studies made for reducing the computer capacity and solution time problems in calculations, are the methods used in this study.In this study, equation of motions for Reddy-Bickford beam on Winkler elastic foundation that have uniform cross-section with different boundary conditions for single-span model and nonuniform cross-section with semi-rigid end connections for two-span model are solved by using Differential Quadrature Element Method (DQEM) and Differential Transformation Method (DTM), circular frequencies for the first three modes of the model obtained using these methods are compared with the ones obtained by analytic method and effectiveness and reliability of the methods are exposed. For this purpose, calculation algorithms and computer programs of DQEM and DTM are prepared and numerical results are obtained by these programs.