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70
Valério da Silva Almeida & João Batista de Paiva
6 BEM-FEM COUPLING
The BEM and FEM are coupled by representing the surface forces of the soil
given in Eq. (18) as reactions between the contact elements of the superstructure-soil
interface. Ideal friction conditions will also be admitted, without allowing for any relative
displacement between the two media. Provided the reactions of the FEM are
expressed by forces already integrated into the domain and transformed into
equivalent nodal forces in each element, the surface forces of the soil can be
expressed by nodal parameters distributed in the element. Hence, coupling requires
transforming the surface forces into equivalent concentrated forces or vice-versa.
In the present paper, the former transformation is given priority. This approach
is presented accordingly.
The approach developed up to this stage will be applied to the case of
transverse loading, although it can be extended to the other two directions. Figure 2
shows the transverse surface forces and the equivalent nodal forces.
Figure 2 - Equivalent surface and nodal forces
In Fig. 2, g i , g j and g k represent the surface forces at vertices i, j and k of a
generic element, while F
i, Fj
and F k
represent the equivalent nodal forces at the
respective nodes.
The work due to the external load in the element’s transverse direction can be
expressed as:
∫
T = g x , x ) ⋅ w(
x , x ) dA
e
A
( 1 2 1 2
(21)
where ( x , )
w and A are, respectively, the transversal displacement field in the
1 x 2
element domain and the area of the element. When the variation of the displacement
field is assumed to be linear, one has:
w = wξ
+ +
(22)
i 1 wjξ2
wkξ3
Cadernos de Engenharia de Estruturas, São Carlos, v.9, n. 38, p. 63-82, 2007