Closed form solutions of the Navier’s Equations for axisymmetric elasticity problems of the elastic half-space

Closed form solutions are derived in this paper for Navier’s equations for axisymmetric elastic half-space problems. They are solved assuming body forces are disregarded. The Boussinesq problem is considered. The displacements are used to obtain the stress fields. The shear stress-free boundary conditions on the boundary plane and the equilibrium of vertical stress and applied load are used to completely determine displacements and stresses. Other axisymmetric load problems considered are: (i) uniform (ii) conical (iii) inverted conical distributions. In each case, the Boussinesq solution is used as a Green function, yielding the vertical stress field as double integration problem. The vertical stress field for uniform load is obtained in terms of complete elliptic integrals of the second and third kinds. The vertical stress distribution under the center of a circular foundation under uniform load is obtained as a particularization of the solution for vertical stress at any point in the elastic half-space. The same result is derived by using the point load solution as an integral Kernel function. For conical distribution of load, the point load solution is used as a Green function, reducing the problem to double integration. The closed form expressions obtained for the vertical stress distributions under the center of the circular foundation for all the axisymmetrical load distributions considered are radially symmetrical functions; which agree with the symmetrical nature of the problem. The results obtained for all the load types considered are identical with previous results found in the literature.