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A mixed BEM-FEM formulation for layered soil-superstructure interaction
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Bergan & Felippa (1985) and the triangular plate element called DKT (Discrete
Kirchhoff Theory) described by Batoz (1980). This flat shell element is formulated by
generating a stiffness matrix for a plane triangle in a three-dimensional space. The
resulting system for the plate-membrane combination is represented by:
K
nodal
( FF DKT ) ⋅u
= F
( FF + DKT ) ( FF + DKT )
+
(19)
where the displacement parameters of element “e” are expressed for nodes i,j and k of
the vertices by:
T ⎧
{ u e}
= ⎨(
u v θ x w θ x θ x ) i (
⋅ ⋅
) j (
⋅ ⋅ )
3 1 2
⎩
k
⎫
⎬
⎭
(20)
where u, v e w are the displacements and θ , x
θ
1 x
e θ
2 x
are the rotations obtained in the
3
directions 1, 2 and 3, respectively, for each vertex node.
5 SUPERSTRUCTURE COMPOSED OF 3D BUILDINGS SUBJECTED TO
VERTICAL AND HORIZONTAL FORCES
The buildings are modeled using three-dimensional finite bar elements to
represent the linear elements of beams and columns, without considering the torsion
effect on them. The building’s slabs are considered as diaphragms with infinitely stiff
horizontal planes, for which reason the horizontal displacements on each floor
( u, v and θ x 3
) are the same. Therefore, all the influences of the columns and beams
on each floor are transferred to a single master node, which is the floor-type torsion
centre. This building model considerably reduces the degrees of freedom of the final
system.
The vertical forces are applied to the beams, and can be pointwise or
distributed, and the horizontal forces caused by the effect of wind, horizontal and
torsional forces, should be prescribed pointwise at the center of torsion for each floor.
Thus, the columns of the building’s ground floor are coupled with the raft’s shell
elements, forming a stiffness matrix that contains the influence of the raft-building set.
The influence of the soil must then be coupled to this system so that the soil-raftbuilding
mechanism can be considered.
Cadernos de Engenharia de Estruturas, São Carlos, v.9, n. 38, p. 63-82, 2007