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

Abstract

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.

Contribution to the free vibration problem of a free‐free beam with large end masses

“…Lastly, in the further authors' work the investigations conducted in this paper will be extended to the cases of Timoshenko and Reddy‐Bickford beams (see e.g. []).…”

DTM and DQEM for free vibration of axially loaded and semi-rigid-connected Reddy-Bickford beam

“…The expression for bending rotation w ′ j ( z j , t ) is given by The shear force function Q j ( z j , t ) can be obtained as: Similarly, the bending moment function M j ( z j , t ) can be obtained as: The higher‐order moment function M hj ( z j , t ) can be obtained as: The details of analytical solution are not given in this study. These details can be found in a paper presented in Reference 8.…”

“…The cross‐sectional displacements of Reddy–Bickford beam theory are shown in (Figure 1(c)) and according to Reddy–Bickford beam theory, the displacements of the rectangular beam can be written as 1, 6, 7: where w 0 ( x, t ) is the lateral displacement of the beam neutral axis; ϕ( x, t ) represents the rotation of a normal to the axis of the beam; u ( x, z, t ) and w ( x, z, t ) are the axial and lateral displacements of the beam, respectively; x is the beam position; z is the distance from the beam neutral axis; t is time variable; h is the height of the beam and γ = 4/(3· h 2 ). Yesilce and Catal compared the free vibration analysis of Reddy–Bickford pile with the results of Timoshenko pile by using analytical method 8. The details of analytical solution can be found in this study.…”

Vertical vibration of a large diameter pile embedded in inhomogeneous soil based on the Rayleigh-Love rod theory

“…In light of this, much attention has been attracted to the study of the dynamic response of piles and various kinds of pile-soil dynamic interaction models have been put forward in the past decades, such as the Kelvin-Voigt model (Nogami and Konagai, 1988;Yesilce and Catal, 2008a;2008b;2008c;Ding et al, 2011;Wu et al, 2012;Li et al, 2015), plane-strain model (El Naggar and Novak, 1994;Han and Sabin, 1995;Militano and Rajapakse, 1999;Wang et al, 2013), and 3D axisymmetric model (Nogami and Novák, 1976;Zhou et al, 2009;Lü et al, 2014;. In the Kelvin-Voigt model, the interaction at the pile-soil interface is simulated by a linear spring and a dashpot connected in parallel, which is far from accurate when simulating the radial inhomogeneity of the soil as the parameters are assigned based on the experience.…”