A Closed‐Form Solution for Earthquake Location in a Homogeneous Half‐Space Based on the Bancroft GPS Location Algorithm

A closed-form solution of transient response to suddenly applied loading distributed over a rectangular area on the surface of an elastic homogeneous half-space is developed for special purposes such as analysis of dynamic soil-structure interaction or contact problems. The solution is obtained using Laplace transform with respect to time and Fourier transform with respect to space. Inverse Laplace transform is implemented analytically. As extreme cases of rectangular loading, the solutions for a point force or finite line load can also be obtained. The advantages of this solution over most other solutions by numerical analyses are that the multiple integrations are reduced by one order, the singularity is removed from the integral kernel, and no additional discretization in the vicinity of the region of interest is required.

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Exact closed-form solutions for Lamb’s problem—II: a moving point load

Geophysical Journal International ◽

10.1093/gji/ggaa380 ◽

2020 ◽

Vol 223 (2) ◽

pp. 1446-1459

Author(s):

Xi Feng ◽

Haiming Zhang

Keyword(s):

Point Source ◽

Closed Form ◽

High Speed ◽

Moving Load ◽

Closed Form Solution ◽

Point Load ◽

Form Solution ◽

High Speed Rail ◽

Lamb’S Problem ◽

Homogeneous Half Space

SUMMARY In this paper, we report on an exact closed-form solution for the displacement in an elastic homogeneous half-space elicited by a downward vertical point source moving with constant velocity over the surface of the medium. The problem considered here is an extension to Lamb’s problem. Starting with the integral solutions of Bakker et al., we followed the method developed by Feng and Zhang, which focuses on the displacement triggered by a fixed point source observed on the free surface, to obtain the final solution in terms of elementary algebraic functions as well as elliptic integrals of the first, second and third kind. Our closed-form results agree perfectly with the numerical results of Bakker et al., which confirms the correctness of our formulae. The solution obtained in this paper may lay a solid foundation for further consideration of the response of an actual physical moving load, such as a high-speed rail train.

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Exact closed-form solutions for Lamb’s problem—III: the case for buried source and receiver

Geophysical Journal International ◽

10.1093/gji/ggaa485 ◽

2020 ◽

Vol 224 (1) ◽

pp. 517-532

Author(s):

Xi Feng ◽

Haiming Zhang

Keyword(s):

Closed Form ◽

Half Space ◽

Closed Form Solution ◽

Time History ◽

Natural Generalization ◽

Elastic Half Space ◽

Form Solution ◽

Elementary Functions ◽

Heaviside Step Function ◽

Lamb’S Problem

SUMMARY In this paper, we derive the exact closed-form solution for the displacement in the interior of an elastic half-space due to a buried point force with Heaviside step function time history. It is referred to as the tensor Green’s function for the elastic wave equation in a uniform half-space, also a natural generalization of the classical 3-D Lamb’s problem, for which previous solutions have been restricted to the cases of either the source or the receiver or both are located on the free surface. Starting from the complex integral solutions of Johnson, we follow the similar procedures presented by Feng and Zhang to obtain the closed-form expressions in terms of elementary functions as well as elliptic integrals. Numerical results obtained from our closed-form expressions agree perfectly with those of Johnson, which validates our explicit formulae conclusively.

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