APPLICATION OF DIFFERENTIAL TRANSFORM IN FREE VIBRATION ANALYSIS OF TIMOSHENKO BEAMS RESTING ON TWO-PARAMETER ELASTIC FOUNDATION

Non-prismatic beams have received great attention from engineers due to their capability in optimizing the strength and weight of the structure. In recent years, many researchers have worked on engineering problems related to static and dynamic analysis of either Euler–Bernoulli [1–3] or Timoshenko [4,5] beams. For short thick beams and rotating machineries, the Timoshenko beam theory presents a more realistic model in comparison with the Euler–Bernoulli beam theory due to both the shear deformation and rotary inertia. When encountering problems such as beams on different types of elastic foundations, including buried pipelines, shallow foundations, and piles, understanding the static and dynamic response of Timoshenko beams on elastic foundations seems to be of great significance. To achieve this, a perception of the interaction between the soil and structural elements is required. Some researchers have investigated the effect of different soils on structural members. Mahmood and Ahmed [6] evaluated the sensitivity of concrete-reinforced beam structures to different behaviors of the soil and the interface layer when influenced by an earthquake excitement. Due to the complex behavior of different types of soils, it is difficult to obtain analytical solutions; therefore, simplified mechanical models have been proposed by several researchers, among which are one-parameter, two-parameter ,and three-parameter elastic foundations. First, Winkler [7] proposed a simple model with only one parameter, that is, the stiffness of linearly elastic and mutually independent vertical springs. Yankelevesky and Eisenberger [8] performed an exact analytical solution for a finite element beam-column resting on Winkler foundation leading to derivation of exact static stiffness matrix. Later, Yankelevsky