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68
Valério da Silva Almeida & João Batista de Paiva
2
b
{ P
2 2 2 2
} = [ K ] ⋅{
U } + [ K ] ⋅{
U }
(13)
bt
t
bb
b
After considering the conditions (8.1), (8.2) and (10) in Eq. (13), one has:
{ U
2
b
} = −[(
K
1
tt
+ K
2
bb
− 1 2 2
) ⋅ K ] ⋅{
U }
(14)
bt
t
and, by substituting (14) in (12), one obtains:
2
t
{ P
} =
2 2 1 2 1 2 2
[ K ] −[
K ] ⋅ ([ K ] + [ K ]) ⋅[
K ]] ⋅{
U }
tt
tb
tt
bb
− (15)
bt
t
Or
{ P
ˆ 2 2
} = [ K ] ⋅{
}
(16)
2
t U t
The above equation considers the influence of regions 1 and 2. Hence,
applying equation (9) and considering expressions (8.1) and (8.2) on the layers i and
i+1, one has:
i
t
{ P } =
i i
ˆ i 1 i −1
i i
[ K ] −[
K ] ⋅ ([ K ] + [ K ]) ⋅[
K ]] ⋅{
U }
tt
tb
− (17)
bb
bt
t
Thus, for the last
i = η layer, one has:
{ P
ˆ η η
} = [ K ] ⋅{
}
(18)
η
t U t
η
η
where P
t
and U
t
are the nodal parameters of the soil’s surface.
At this point, the influence of the nonhomogeneous soil is entirely expressed by
Eq. (18), which can be solved directly by applying the given loading conditions on the
surface or by coupling the superstructure using FEM or BEM.
4 SUPERSTRUCTURE COMPOSED OF LAMINAR ELEMENTS
To simulate the raft via the Finite Element Method (FEM), we superposed two
independent formulations, one to represent the membrane effect and the other the
plate effect. The finite element adopted is a combination of the triangular membrane
element with a rotational degree of freedom called Free Formulation, according to
Cadernos de Engenharia de Estruturas, São Carlos, v.9, n. 38, p. 63-82, 2007