Two simple integral equation methods are proposed for the analysis of a vertical loaded pile. For the first one it is assumed that the axisymmetrical loads formed by Mindlin’s horizontal point forces be distributed along the z-axis in [0,L] of the elastic half-space, and are composed with the Boussinesq’s point force. In the second one in addition to the above fictitious loads, Mindlin’s vertical forces are distributed along the z-axis in [0,L]. The former reduces the problem of a vertical loaded pile embedded in a half-space with the following boundary \(conditions:\)
z\(=0\), \(r\neq 0\), \(\sigma_ z=\tau_{rz}=0\); \(0\leq z\leq L\), \(U(e,z)=a-e\) \((e\to a);\)
P\(=-2\pi [a\int^{L}_{0}\tau_{rz}(a,z)dz+\int^{a}_{0}r\sigma_ z(r,L)dr]\), to a Fredholm integral equation of the first kind; the latter reduces the same problem but with different boundary conditions: \(0\leq z\leq L\), \(U(e,z)=a-e\), (e\(\to a)\); \(W(a,z)=const\), to two coupled Fredholm integral equations of the first kind. For a loaded rigid pile, the former is suitable for the cases which permit slides between the pile and the medium, but the latter for the cases without slides between the pile and the medium.
Comparing with the current methods of fictitious loads distributed on the actual boundary, the above two methods have the following advantages: The integral equation (or equations) obtained is one-dimensional and non- singular. The effect of initial stress may be taken into account. For the first method, there are additional advantages: the settlement function must not be prescribed and the three-dimensional stress state in compressible piles may be taken into account.
A theorem on error estimation for an approximate solution of the Fredholm integral equation of the first kind is presented, and numerical examples of two analytic methods for single loaded piles have been calculated by a DJS-21 computer.