Free vibration of partially supported piles with the effects of bending moment, axial and shear force

The pile parts above the soil and embedded in the soil are called the first region and the second region, respectively. The fourth order partial differential equations of both regions for free vibration of partially supported pile subjected to bending moment, axial force and shear force are obtained using the small-displacement theory and the Winkler model. It is assumed that the behavior of the material is linear-elastic, and that axial force along the pile length and modulus of subgrade reaction for the second region to be constant. Shear effect is included in the partial differential equations by the second derivative of the elastic curve function with respect to shear deformation. Natural circular frequencies and relative stiffness of the pile are calculated for non-trivial solution of linear homogeneous system of equations obtained due to the values of axial forces acting on the pile, the shape factors, and the boundary conditions of the pile with both ends free and both ends simply supported; the results are presented in graphs.