Vertical vibrations of a rigid circular body on a non‐homogeneous half‐space interrupted by a frictionless plane

Abstract

AbstractAn exact formulation is presented for the problem of a rigid circular body performing harmonic vibrations on an elastic half‐space whose shear modulus increases linearly with depth and is interrupted at some finite depth by a frictionless horizontal plane. The static case is derived in the limit of zero frequency vibrations while the known result for the uninterrupted half‐space is recovered in either extreme limit of the horizontal frictionless plane coinciding with the surface or when it is pushed down to an infinite depth.It is shown that the maximum effect of the interruption occurs when the frictionless plane is at a depth where the shear modulus is about 1·6 times the surface shear modulus. Furthermore, this maximum effect is equivalent to a reduction of about 5 per cent of the surface shear modulus or a reduction of about 2½ per cent in the natural frequency of the rigid body on an uninterrupted half‐space. The important conclusion, therefore, is that irrespective of the depth at which a half‐space isso interrupted, the surface shear modulus is still the dominant parameter and that both the increase in shear modulus with depth and the interruption are not only secondary but also opposing effects.