Keywords: stochastic analysis, thin-layer method, inhomogeneous half-space, uncertainty, soil-structure interaction, wave propagation.

Wave propagation in a layered half-space has many applications in seismology and civil engineering. Exact stiffness matrices and their approximations for a layered half-space were previously presented [1]. The method based on the discrete stiffness matrices is often referred to as a "thin-layer method". A spectral decomposition for solutions to the method was developed [2]. Since the method leads to rigorous and effective numerical models, it has been applied to wave-propagation problems in various layered systems [3]. In the conventional thin-layer method, properties of layered media have been assumed deterministic. However, all natural and man-made systems have intrinsic randomness in material properties. Therefore, a stochastic enhancement of the thin-layer method is highly desirable.

In this paper, a "stochastic thin-layer method" is developed for the analysis of wave propagation in an inhomogeneous half-space in antiplane shear. The shear modulus is assumed uncertain and characterised by a random field with vertically varying statistical properties. Then, it can be expanded using the Hermite polynomial chaos of a zero-mean and unit-variance Gaussian field with a correlation function [4]. Using the Karhunen-Loève expansion, the Gaussian field can be discretized into independent standard normal random variables [5].

The inhomogeneous half-space is represented by thin layers. The layers include not only ordinary layers but also continued-fraction absorbing boundary conditions for the infinite extent of the half-space [6]. Applying the Galerkin method not only in the spatial domain but also in the stochastic domains, a stochastic thin-layer method for an inhomogeneous half-space in antiplane shear is presented.

Using the stochastic methods, dynamic responses of an inhomogeneous half-space subjected to a transverse line load on its surface are obtained and verified by comparison with Monte Carlo simulations. The stochastic methods are found to provide accurate probabilistic treatment of half-space dynamics.

1

E. Kausel, J.M. Roësset, "Stiffness matrices for layered soils", Bulletin of the Seismological Society of America, 71, 1743-1761, 1981.

2

E. Kausel, "An explicit solution for the Green functions for dynamic loads in layered media", Research Report R81-13, Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1981.

3

E. Kausel, "The Thin-Layer Method in Seismology and Earthquake Engineering", in E. Kausel, G. Manolis, (Editors), "Wave Motion in Earthquake Engineering", WIT Press, 193-213, 2000.

4

S. Sakamoto, R. Ghanem, "Simulation of multi-dimensional non-gaussian non-stationary random fields", Probabilistic Engineering Mechanics, 17, 167-176, 2002. doi:10.1016/S0266-8920(01)00037-6

5

P.D. Spanos, R. Ghanem, "Stochastic finite element expansion for random media", Journal of Engineering Mechanics, 115, 1035-1053, 1989. doi:10.1061/(ASCE)0733-9399(1989)115:5(1035)

6

J.H. Lee, J.L. Tassoulas, "Consistent transmitting boundary with continued-fraction absorbing boundary conditions for analysis of soil-structure interaction in a layered half-space", Comput. Methods Appl. Mech. Engrg., 200, 1509-1525, 2011. doi:10.1016/j.cma.2011.01.004

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