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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
A mixed BEM-FEM formulation for layered soil-superstructure interaction 71 Similarly, the surface forces can be expressed as: g = g ξ + (23) i 1 + g jξ2 gkξ3 After transforming the Cartesian coordinates into homogeneous coordinates - REDDY (1984) - and substituting equations (22) and (23) for Eq. (21), one has: T ( g ξ + g ξ + g ξ )( ⋅ wξ + w ξ + w )dA = ∫ (24) e i 1 j 2 k 3 i 1 j 2 kξ3 A Moreover, minimising the portion of potential energy due to external loading ∫ 1 2, 3 A and keeping in mind that the integral f ( , ξ ξ ) dA ξ can be determined by: ∫ A ξ η1 1 ξ η2 2 ξ η3 3 dA = 2A η ! η ! η ! ( η + η + η + 2) ! 1 1 2 2 3 3 (25) the transverse nodal force vector is then given by: ⎧ Fi ⎫ ⎡2 ⎪ ⎪ A = ⋅ ⎢ ⎨F j ⎬ ⎢ 1 ⎪ ⎪ 12 ⎩ Fk ⎭ ⎢⎣ 1 1 2 1 1⎤ ⎧ g ⎪ 1 ⎥ ⎥ ⋅ ⎨g 2⎥ ⎪ ⎦ ⎩g i k ⎫ ⎪ ⎬ ⎪ ⎭ j (26) By following the same procedure in the other directions, the relation between the nodal and surface forces for the case of the Free Formulation and DKT element can be written as: ⎧Fi ⎪ ⎨F j ⎪ ⎩ Fk l l l ⎫ ⎧g ⎪ ⎪ ⎬ = [ Q ] ⋅ ⎨g ⎪ ⎪ ⎭ ⎩ g l i l j l k ⎫ ⎪ ⎬ ⎪ ⎭ (27) where l = 1,2, 3 represent the three orthogonal directions of the element and the matrix Q given by: Cadernos de Engenharia de Estruturas, São Carlos, v.9, n. 38, p. 63-82, 2007
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SUMÁRIO Análise experimental de p
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