3 - Initial cone with outward wave propagation

Abstract

This chapter explains the model of initial cone with outward wave propagation in vibration analysis. The chapter is limited to those relationships that are required to calculate the dynamic-stiffness coefficients and the effective foundation input motion. As the analysis is performed in the frequency domain, the derivation is also performed for harmonic excitation in the elastic medium. The first of the two building blocks used to calculate the vibrations of an embedded foundation consists of a disk embedded in a layered half-space, modeled as a double cone, with each cone exhibiting outward wave propagation. Such a model is shown in the chapter for the vertical degree of freedom. As a special case, the disk is placed on the surface of the half-space that leads to a one-sided cone model. The chapter discusses the translational one-sided cone. The hypotheses and construction of the one-dimensional strength-of-materials approach using a tapered bar with the cross-section increasing in the direction of wave propagation—as in a cone—is addressed. The chapter derives the analogous relationships for the rotational one-sided cone and addresses the modifications necessary for the vertical and rocking degrees of freedom when Poisson's ratio approaches 1/2 yielding an infinite dilatational-wave velocity. The chapter presents the dynamic-stiffness coefficients of a (circular) foundation on the surface of a homogeneous half space that are compared with the results of a rigorous analysis.