| Abstract | The behavior of an axially loaded rigid cylindrical body embedded
in, and bonded to, an elastic half space is studied. The characteristics of this problem are governed by two parameters, the Poisson's ratio u of the half space and the Length to diameter ratio of the cylindrical body. By varying the latter, the results can be used to approximate the conditions encountered in rigid piles or caissons as well as cir- cular footings. The formulation leads to a mixed boundary value problem with res- pect to stress and displacement, which is intractable for exact solution. In the proposed approximate solution, the domain of the rigid body and the rest of the half-space which is the domain of interest will be treated together as one domain, and a field of distributed forces will be applied on the boundary of the rigid body. The latter is divided into a suitable number of cylindrical zones on the side and annular zones on the base subjected to radial and vertical distributed forces. The explicit solutions of the four cases of radial and vertical forces distributed over cylindrical and annular zones in the interior of an elastic halfspace are derived in this study. The intensities of the distributed forces acting on the cylindrical
and annular zones are so determined as to satisfy the prescribed conditions of both vertical and radial displacements at the centers of the zones. This leads to a set of simultaneous linear equations in which the coefficient matrix consists of the flexibility influence coefficients of the system of distributed forces. For large values of length to diameter ratio, which correspond to the case of a pile or caisson, the area of the base is very small in comparison with the area of the side and, consequently, the flexibility influence coefficients associated with forces in the two areas are not of the same order of magnitude. As e result the coefficient matrix is ill—conditioned and the solution by direct matrix inversion is not feasible. An incremental iterative Scheme is proposed to avoid this difficulty. The cylindrical elements on the side and the annular elements on the base are grouped into two systems. In the first system, the cylindrical boundary is divided into a suitable number of side elements as before while the base is divided into only one or two elements. The second system consists only of a suitable number of base elements as before. The two resulting sets of simultaneous linear equations, which incorporate the interactive effects between the two systems, are then inverted alternately and the solution is obtained by iterations between the two systems. This iteration scheme converges in three cycles with an accuracy of 0.1%. The results of parametric study of piles and caissons of practical length to diameter ratios are presented graphically to illustrate the stress distribution and load transfer behavior. The total load expressed in terms of the vertical displacement is also plotted against arguments of length to diameter ratio. The displacements at locations between adjacent nodes were computed and found to check satisfactorily with the prescribed boundary conditions. In order to check the equilibrium of vertical forces, the shearing stresses on the side and the normal stresses at the base are integrated numerically and the result agrees with the total vertical load within 17. accuracy, Finally, for the limiting case of rigid circular footings, the solutions agree with the exact solution within 0.5% accuracy. It is observed in this investigation that in piles and caissons of practical length to diameter ratios, high stress concentration occurs in the vicinity of the edge of the base as expected. For given prescribed displacement, the magnitudes of the shearing and normal stresses increase with increasing Poisson's ratio. For practical purposes, the applied load is linearly transferred along the depth and the proportion of the load carried on the base is less than 10%, being greater for higher values of Poisson's ratio.
The solutions of the four fundamental cases of cylindrical and
annular forces acting in the interior of a half-space derived in this
study, together with the proposed incremental iterative scheme, can be adopted in the treatment of other axisymmetric foundation problems such as elastic piles, caissons and circular footings where the deformations of the substructures are incorporated in the solutions. Also stepped taper piles and anchor blocks can be treated without fundamental difficulty. |