Keywords: boundary element methods, heterogeneous soil, soil-foundation interaction, method of successive rigidity, soil-foundation-system, finite medium.

In recent decades, soil-foundation interaction problems have been investigated by many researchers in which the supporting medium is supposed to have elastic behaviour. The models for elastic responses of soil have been developed along two distinct lines. In the first, the soil is represented as a mutually-independent spring system (Winkler's model), in which the disturbances caused in a certain region are not transmitted to another location outside it. In the second approach, the soil is assumed to be a continuous elastic medium, so that the effects of loading can also be observed away from the loading region. Many models have been developed taking into account the geometrical and mechanical properties of the continuum, such as the well-known Mindlin's solution to the problem of a homogeneous, semi-infinite, isotropic, linear elastic solid under static concentrated loading.

Some researchers have presented formulations for the soil-foundation interactions that occur when the supporting medium is assumed to be a Mindlin half space and these analyses can be divided into three groups: raft-soil , pile-soil and raft-pile-soil interaction problems. In the first group there are analytical solutions to the complete raft-soil problem, based on variational methods for particular geometry and loading of the raft [1].

In solutions to the second type of problem, exploring the interactions between pile groups and the soil, the elastic contribution of each pile to the global performance has been analysed classically, by using load-transfer curves (LTC) and a direct continuum approach [2]. The load-transfer curves have been extended to model the third type problem (raft-pile-soil), both for the rigid rafts [3] and the flexible caps.

In all articles discussed so far, the soil is assumed to be a homogeneous, isotropic, linear elastic solid which has an infinite stratum. However, in some cases the infinite thickness of a compressible medium can lead to overestimated values for the elastic displacements in soil-structure problems. In the literature, techniques proposed to model finite layer media fall basically into four distinct types. In the first, the contribution of each layer is substituted by an equivalent spring system. However, these approaches are strongly dependent on the model assumptions to evaluate the equivalent stiffness coefficient of the spring system. The second line of research has been developed from Burmister's model, which was initially proposed for a layered elastic solid under a concentrated vertical load acting on the free surface. In Chan et al. [4], the Burmister approach was extended to the concentrated forces acting within a layered half space. However, the governing partial differential equations (PDE) for these models lead to a cumbersome procedure to get approximated numerical solutions and there are no closed-form solutions to this fundamental elastic problem.

The third approach to a finite elastic stratum is the called finite layer technique. In general, this provides no representation of the shear stresses along the shaft at points within the layer domain, so that these stresses are adopted as equivalent concentrated forces acting at the nodal points located on the layer interfaces, and Bessel's functions are employed to evaluate the soil stiffness matrix.

The main objective of this article is to apply the BEM formulation to analysis the nonhomogeneous stratified half-space soil/pile interaction by using the so-called "sucessive stiffness method" developed by [5]. This alternative approach consists in treating each stratum as a sub-region and applying the boundary conditions between the common interfaces. So, it is assembled, in a ascending order, a matrix which will incorporate all the influences of the descending layers. In the present paper, it is improved the cited approach by considering the influence of the foundation in the final system, and the formulation is extended to three-dimensional problems analysis. The present numerical approach is compared with others formulations to some examples and the accuracy values found corroborate the formulation.

1

Zaman, M.M., Issa, A. and Kukreti, A.R. "Analysis of circular plate-elastic half-space interaction using an energy approach". Applied Math.l Modelling, 12, pp. 285-292, 1998. doi:10.1016/0307-904X(88)90036-4

2

Shen, W.Y., Chow,Y.K. and Yong, K.Y. "A variational approach for vertical deformation analysis pile groups". International Journal for Numerical and Analytical Methods in Geomechanics; 21 (11), pp.741-752, 1997. doi:10.1002/(SICI)1096-9853(199711)21:11<741::AID-NAG898>3.0.CO;2-D

3

Kuwabara, E. "An elastic analysis for piled raft foundations in a homogeneous soil". Soils and Foundations; 29(1), pp. 82-92, 1989.

4

Chan, K.S., Karasushi, P. and Lee, S.L. "Force at a point in the interior of a layered elastic half space". International Journal of Solids and Structures, 10, pp.1179-1199, 1974. doi:10.1016/0020-7683(74)90067-5

5

Maier, G.; Novati, G. "Boundary element elastic analysis by a sucessive stiffness method". Int. J. for Numerical and Anal. Methods in Geomechanics, v.11, p. 435-447, 1987. doi:10.1002/nag.1610110502

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