Probabilistic Engineering Mechanics 22 (2007) 100–111 www.elsevier.com/locate/probengmech The Green’s functions of a vertically inhomogeneous soil with a random dynamic shear modulus M. Schevenels ∗ , G. Lombaert, G. Degrande, D. Degrauwe, B. Schoors K.U. Leuven, Department of Civil Engineering, Kasteelpark Arenberg 40, B-3001 Leuven, Belgium Received 4 December 2005; received in revised form 30 August 2006; accepted 6 September 2006 Available online 9 October 2006 Abstract This paper deals with the study of the Green’s functions of a layered soil with random characteristics. The dynamic shear modulus of the soil is modelled as a non-Gaussian random process that varies in the vertical direction and is characterized by a marginal probability density function and a correlation function. The stochastic finite element method is applied to a hybrid thin layer — direct stiffness formulation in order to obtain the stochastic system equations, which are solved by means of a Monte Carlo simulation. The influence of the variations of the dynamic shear modulus on the Green’s functions is illustrated for different excitation frequencies and receiver positions. c  2006 Elsevier Ltd. All rights reserved. Keywords: Green’s functions; Stochastic finite element method; Non-Gaussian random process; Monte Carlo simulation 1. Introduction Vibrations induced by road, rail and subway traffic are a common source of discomfort to people. Numerical models have been developed for the prediction of traffic induced vibrations in the free field or in the built environment [1–3]. These models consist of a finite element formulation for the vehicles and the buildings and a boundary element formulation that accounts for the wave propagation in the soil. The latter is based on the Green’s functions of a horizontally layered halfspace. The prediction models have been validated for both road traffic and rail traffic [4]. For road traffic, the dominant frequency is about 10 Hz and a good correspondence of the predicted and the measured response is observed. For rail traffic, however, the dominant frequency is about 50 Hz and the validation reveals a discrepancy between the predicted and the measured response in the higher frequency range. Given the crucial role of the Green’s functions in the prediction model, the dynamic soil characteristics governing these functions are a possible source of the discrepancy. The most common techniques for the in situ measurement of the dynamic soil characteristics include the spectral analysis ∗ Corresponding author. Tel.: +32 16 32 16 82; fax: +32 16 32 19 88. E-mail address: mattias.schevenels@bwk.kuleuven.be (M. Schevenels). of surface waves (SASW) test, the seismic cone penetration test (SCPT), the crosshole test, and the downhole test. These techniques allow one to estimate the variation of the dynamic shear modulus and the material damping ratio with depth. The SASW test and the SCPT test are complementary: the SASW test has a better accuracy at shallow depth, while the SCPT test achieves a better accuracy at larger depth. Both techniques are based on the measurement of local averages of the dynamic soil characteristics, resulting in a limited resolution in the sense that small scale variations of the dynamic soil characteristics are not revealed. The cone tip resistance measured in a cone penetration test (CPT) is the result of averaging over a smaller spatial scale. Therefore, it is possible to derive additional information on the small scale variations of the dynamic soil characteristics if a relation with the cone tip resistance is assumed. In the present paper, the influence of the (small scale) variations of the dynamic shear modulus on the Green’s functions of a soil excited at the surface is examined. While the importance of the variation of the material damping ratio is also recognized, the influence is not accounted for in the present paper as the in situ measurement techniques required to assess this variation are still in an early stage of development. As it is practically impossible to formulate an accurate deterministic description of the soil profile that accounts for the small scale variations of the dynamic shear modulus, a probabilistic 0266-8920/$ - see front matter c  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.probengmech.2006.09.001