Free vibration of semi-rigid connected and partially embedded piles with the effects of the bending moment, axial and shear force

Natural circular frequencies of piles partially embedded in the soil whose upper ends are semi-rigid connected are calculated using the method of initial values, the transfer matrix, the small-displacement theory and the Winkler model. It is assumed that the behavior of the material is linear-elastic, and that the axial force along the pile length and the modulus of subgrade reaction to be constant. The semi-rigid connections of the upper ends of the piles are modeled by elastic rotational springs.The transfer matrices are obtained due to the boundary conditions of the piles. Natural circular frequencies and relative stiffness of the piles are calculated for a non-trivial solution of a linear homogeneous system of equations obtained due to the terms of the transfer matrices of the piles, the stiffnesses of the linear-elastic rotational springs and ratio of embedded length of the pile to total length of the pile; the results are presented in graphs.