Keywords: boundary element, finite element, BEM-FEM coupling, piles, pile groups, pile-soil interaction, dynamic impedances.

Foundations subject to strong static or dynamic loads and with high responsibility such as those of bridges, off-shore structures, support-walls, nuclear power plants or high buildings, are often solved using piles arranged in groups. Loads arising, for example, from the action of the wind, running machinery or sea waves are usually applied to a rigid cap connecting the top of the piles, so the action is distributed among them. Nevertheless, the different piles in a group will not generally support the same forces, unless the distance between them is big enough, which is not common. This interaction between the piles through the soil medium has been taken into account from early works. This way, the pile group impedances are usually strongly dependent on frequency and its behaviour varies with pile spacing, pile geometry, group size and soil and pile properties.

There are many publications related to this problem (see e.g. [1,2]). Two of the authors of the present work have developed a boundary element model for the computation of dynamic impedances of piles in elastic and poroelastic soils [3,4].

In this work a boundary element - finite element model is presented for the computation of time harmonic dynamic stiffness coeficients of piles embedded in an elastic halfspace. Piles are modelled using finite elements (FEM) as a beam according to the Bernoulli hypothesis, while the soil is modelled using boundary elements (BEM) as a continuum, semi-infinite, isotropic, homogeneous, linear, viscoelastic medium (the semi-infinite soil and radiation damping is easily represenseted by the BEM).

The dynamic model presented is based on previous static model developed by Matos Filho et al. [5], where it is assumed that the elastic soil is not disturbed by the piles and the tractions in the pile-soil interface are considered as a load applied within the half-space in the boundary integral representation of the soil.

In the present study, piles are modelled by the FEM as vertical beams according to the Bernoulli hypothesis, and are discretized using three-nodes elements with 13 degrees of freedom defined: two lateral displacements and a vertical displacement on each node, and two rotations on each one of the extreme nodes. Lateral displacements along the element are approximated by a set of fourth degree shape functions, while vertical displacements and tractions along the pile-soil interface are approximated by one of second degree. The sub-matrix that transforms nodal force components to equivalent nodal forces, and the stiffness and mass sub-matrices are defined.

The soil is modelled by the BEM as a linear homogeneous isotropic elastic un-bounded region. Generally, body forces are considered to be zero in elastodynamic problems. Nevertheless, in this case, the tractions within the soil along the pile-soil interface can be treated as loads applied within the half-space, as it is assumed that the soil continuity is not altered by the presence of the pile. The boundary element discretization of the half-space is made using nine-node quadrilateral or six-node triangular elements. The coupling BEM-FEM is imposed by compatibility and equilibrium conditions between the variables of the two methods along the pile shaft.

The main advantage of this model is the capacity of computing accurately stiffness coefficients with low computing times and low memory requirements in comparison to other methods that need to discretize the pile surface or volume. This way, pile groups with a big number of members can be analyzed without difficulty. Besides, once the surface (not necessarily flat) has been discretized, it has not to be changed to analyze different sets of piles, which can be modified easily. Other internal variables such as stress values along the pile can be obtained, and soil strata and rigid rocky beds can be easily taken into account. Furthermore, the model can be included into an existing BEM code by adding subroutines to obtain the mono-dimensional or surface integrals along the pile-soil interface, and modifying the system of equations in the way that is presented.

Several results are presented and compared to well known values taken from the literature, obtaining an excellent agreement.

1

M. Novak, "Piles under dynamic loads", 2nd Int Conf Recent Adv Geotech Earthquake Eng Soil Dyn, 21:145-62, 1992.

2

D.E. Beskos, "Boundary element methods in dynamic analysis: Part II (1986-1996)", Appl Mech Rev, 50:149-197, 1997. doi:10.1115/1.3101695

3

O. Maeso, J.J. Aznárez, "Numerical study of dynamic behavior of piles and pile groups in porous soils using the BEM." in 17 ASCE Engineering Mechanics Conference. EM2004. Newark, Delaware. USA, 2004.

4

O. Maeso, J.J. Aznárez, F. García, "Dynamic impedances of piles and groups of piles in saturated soils", Computer & Structures, 83, 769-782, 2005. doi:10.1016/j.compstruc.2004.10.015

5

R. Matos Filho, A.V. Mendonça, J.B. Paiva, "Static boundary element analysis of piles submitted to horizontal and vertical loads", Eng Anal Boundary Elem, 29:195-203, 2005. doi:10.1016/j.enganabound.2004.10.003

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