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72
Valério da Silva Almeida & João Batista de Paiva
⎡2
A
[ Q ] = ⋅
⎢
⎢
1
12
⎢⎣
1
1
2
1
1⎤
1
⎥
⎥
2⎥⎦
(28)
By adding the contribution of all the finite elements and then minimising the
energy functional, one has:
[ K fem]
⋅ { U fem}
= { Ffem}
−[
Q ] ⋅{
Pr
}
(29)
where
U fem , F fem , P r ,
K fem and Q are, respectively, the vector of superstructure
displacements, the vector of equivalent nodal forces due to external loading, the
expanded vector of surface force due to soil reaction, the global stiffness matrix of the
shell structure and the expanded transformation matrix resulting from the contribution
of all the boundary elements.
Hence, by substituting Eq. (20) for Eq. (18) considering equilibrium and
compatibility conditions between the contact surfaces, one has:
[ K
fem
] ⋅{
U
fem
} = { F
fem
)
η η
(30)
} −[
Q ] ⋅[
K ] ⋅{
U }
t
Equation (30) can be simplified here, resulting in:
[ bem / fem bem / fem bem / fem
K ] ⋅ { U } = { F }
(31)
with:
[ Kbem
/ fem
] = [ K
fem
] + [ Q]
⋅[
Kˆ
η
]
(32)
7 NUMERICAL EXAMPLES
The algorithm developed herein allows for the independent analysis of both the
nonhomogeneous soil and pile and the analysis of the soil-superstructure as a set.
7.1 Finite layer with linear variation of modulus
This example presents an analysis of the soil considering it as a
nonhomogeneous, linear isotropic medium. For the specific case of a uniformly
Cadernos de Engenharia de Estruturas, São Carlos, v.9, n. 38, p. 63-82, 2007