Transient Excitation of an Elastic Half Space by a Point Load Traveling on the Surface

Abstract

The propagation of transient waves in a homogeneous, isotropic, linearly elastic half-space excited by a traveling normal point load is investigated. The load is suddenly applied and then it moves rectilinearly at a constant speed along the free surface. The displacements are derived for the interior of the half-space and for all load speeds. Wave front expansions are obtained from the exact solution, in addition to results pertaining to the steady-state displacement field. The limit case of zero load speed is considered, yielding new results for Lamb's point load problem. INTRODUCTIONGround motions excited by moving surface forces arise, for example, from nuclear blasts and from shock waves generated by supersonic aircraft, and they interact with structures causing extensive damage. A mathematical problem of fundamental importance in these applications is that of an elastic half-space whose surface is excited by a normal point load which is suddenly applied and which subsequently moves rectilinearly at a constant speed.In recent years several solutions to this probl em have been given for the surface of the half-space. t Numbers in brackets designate References at end of paper. shown to form the steady-state displacement field when the load speed exceeds both of the body wave speeds. FORMULATION OF THE PROBLEMThe subject half-space problem is depicted in Fig. 1 The constants cd and cs, defined by c! = (X. + 2µ)/v and c; = µ/v, represent the dilatational and equivoluminal body wave speeds, respectively, where X. and µ are the Lam~ constants and v is the material density. The stresses T .. are related to the displacements by lJwhere tensor notation is employed.The boundary conditions at z = 0 take the formwhere o is the Dirac delta function. To represent quiescence at t = 0, the initial conditions appear asFinally, the potentials cf> and ljJ, and the space derivatives of the patentials, are required to vanish at infinity. FORMAL SOL UT IONA solution of the wave equations (1) that satisfies the initial conditions (6) and the boundedness condition at infinity can be computed Soos-n z+i(kx+vy) (1 7a)(1 7b)(1 7c)(1 7d) In view of the symmetry properties u (r,9,z,p) =u (r,-9,z,p) xa xa u (r,9,z,p) = -u (r,-9, z ,p)u (r,9,z,p) =u (r,-9,z,p), za za u. , and hence u., are only inverted for 0 :S 9 <'TT. Since u. has different Ja J Ja forms depending on the speed of the load relative to the body wave speeds, the inversion of each u. is separated into three cases. In particular, S(X)s(X)_ _E...c (mdz-iqr)where ~ is the position vector. uzd is converted into the Laplace transform of a known function by mapping (1 /cd)(mdz-iqr) into t through a contour integration in a complex q-plane. To this end, the singularities of the integrand of U:zd are branch points at -9-for t > twd, where Equation (26) defines one branch of a hyperbola with vertex 2 .!.2 r/p and asymptotes arg q = ±r/z. As shown in Fig. 2 (27) + -by a solid line labeled with qd and qd, this hyperbola is parametrically described by t as t varies from twd ...

An overview of a unified theory of dynamics of vehicle–pavement interaction under moving and stochastic load

“…Considerable effort has been devoted to describing pavement surface roughness [24][25][26]. The peak-to-valley measurements, the average deviation from a straight edge, and the cockpit acceleration are several distinct approaches that have been suggested for pavement surface characterization [27]. Practical limitations of these three approaches could be found in the study of Hsueh and Penzien [28].…”

“…Comparing (53) with (27), it is clear that the left side of (53) is just the PSD given by S AC ðXÞ. Since it has been proved that S AC ðXÞ ¼ gðXÞS nn ðXÞ where gðXÞ is an ordinary coherence function and S nn ðXÞ is one-dimensional PSD roughness.…”

Propagation of Transient Elastic Waves in Stratified Anisotropic Media

“…Meanwhile, many authors have used the method to solve a variety of specific problems. We mention the solution of Lamb's problem, where the elastic half space is considered (de Hoop 1961, Gakenheimer 1969. Some authors only used the two-dimensional version of the method (Achenbach 1973, p.298).…”

Analysis of ground motion due to moving surface loads induced by high-speed trains

“…The solution was written in terms of integrals that had to be numerically evaluated. Payton [5], and Gakenheimer and Miklowitz [6] obtained soil surface and internal displacements, respectively, when a point load is suddenly applied and then travels at constant speed over an elastic half-space surface. Fryba obtained in 1972 the stationary solution for this problem and Barros et al [8] for a layered half-space.…”