Free vibration of semi-rigid connected Reddy-Bickford piles embedded in elastic soil

The literature on free vibration analysis of Bernoulli-Euler and Timoshenko piles embedded in elastic soil is plenty, but that of Reddy-Bickford piles partially embedded in elastic soil with/without axial force effect is fewer. The soil that the pile partially embedded in is idealized by Winkler model and is assumed to be two-layered. The pile part above the soil is called the first region and the parts embedded in the soil are called the second and the third region, respectively. It is assumed that the behaviour of the material is linear-elastic, that axial force along the pile length to be constant and the upper end of the pile that is semi-rigid supported against rotation is modelled by an elastic spring. The governing differential equations of motion of the rectangular pile in free vibration are derived using Hamilton’s principle and Winkler hypothesis. The terms are found directly from the solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. The models have six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments. Natural frequencies of the pile are calculated using transfer matrix and the secant method for non-trivial solution of linear homogeneous system of equations obtained due to values of axial forces acting on the pile, total and embedded lengths of the pile, the linear-elastic rotational restraining stiffness at the upper end of the pile and to the boundary conditions of the pile. Two different boundary conditions are considered in the study. For the first boundary condition, the pile’s end at the first region is semi-rigid connected and not restricted for horizontal displacement and the end at the third region is free and for the second boundary condition, the pile’s end at the first region is semi-rigid connected and restricted for horizontal displacement and the end at the third region is fixed supported. The calculated natural frequencies of semi-rigid connected Reddy-Bickford pile embedded in elastic soil are given in tables and compared with results of Timoshenko pile model.