Cones to model foundation vibrations: incompressible soil and axi-symmetric embedment of arbitrary shape

The recently streamlined strength-of-materials approach using cones to calculate vibrations of foundations embedded in layered half-spaces and full-spaces is applied to incompressible and nearly-incompressible soil and to axi-symmetric embedments of arbitrary shape. For incompressible soil the axial-wave velocity in the cones is limited to twice the shear-wave velocity and a trapped mass for the vertical motion and a trapped mass moment of inertia for the rocking motion moving as a rigid body with the under-most disk of an embedded foundation are introduced. In the case of a fully embedded foundation, a mass and a mass moment of inertia are also assigned to the upper-most disk. For an axi-symmetric embedment of arbitrary shape, the disks have varying radii. No modifications to the formulation are, however, required. For these two extensions the strength-of-materials approach using cones leads to the same sufficient engineering accuracy as is achieved in other more conventional cases. This is demonstrated in a vast study. Thus the same other advantages also apply: physical insight with conceptual clarity, simplicity and sufficient generality.