748
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
displacements for the nodes at the footing axes;
p
* = vector of
loads transferred to the footing from the structure (dead loads,
live loads on the floors, wind loads etc.).
2.1.3
The footing
Elongated footing may be supposed rigid in the transversal
direction. It gives the possibility to express the displacements of
the contact zone via the displacements of the footing giving the
following expression:
Aw w
ˆ
.
(3)
The loads on the footings transferred from the soil are summed
up according to the formula
pBp
ˆ
.
(4)
2.2
General system of equation
Eq. 1 and Eq. 3 give:
*
ˆˆ
wAw pC
,
(5)
while Eq. 2 and Eq. 4 give:
*
ˆ
p Kw pB
,
(6)
In further consideration system of Eq. 5 and Eq. 6 will be given
concrete expression.
3 SOIL MODELLED BY LINEARLY DEFORMABLE
HOMOGENEOUS HALF-SPACE
3.1
Model substantiation and Galerkin method
Small breadth of soil-structure contact zone leads to small depth
of deformable soil layer that allows considering the soil mass as
homogeneous continuum. Hence the computational domain may
be supposed homogeneous half-space; its linearity was
supposed earlier. Only normal loads on its surface are
considered; tangential loads are zero.
Earlier one of the authors obtained (Kholmyansky 2007) the
solution for three-dimensional problem about the system of
rigid punches on the half-space obtained with the boundary
element method and using Boussinesq solution.
Two specific variants of general numerical method of
weighted residuals where compared: collocation method and
Galerkin method (Finlayson 1972); the latter showed higher
accuracy and was chosen for further work.
Efficiency of that approach was illustrated by the fact that
the equilibrium of several hundreds of punches was considered
without difficulties. This paper continues to use that approach
for the discretization of Eq. 5.
3.2
Operator discretization
For discretization of the operator
C
, that describes the flexibility
of deformable foundation the simplest piecewise-constant basis
functions are chosen. The footing-soil contact zone
Ω
divided
into
n
boundary elements
Ω
j
.
ˆ
Each basis function corresponds to boundary element; the
function is unity for the points of the element and zero outside.
That makes the pressure field piecewise-constant and equal to
linear superposition of basis functions
N
j
:
n
j
j
j
Np p
1
ˆ
,
(7)
where
p
j
= specific normal load on
Ω
j
.
As a consequence Galerkin method provides instead of Eq. 5
its discrete form
*) (
) (
1
1
wa Aw a pc
m
j
j
n
j
j
ij
i
,
(8)
where
a
j
(
u
) = average
u
over the boundary element
Ω
j
;
m
j
—
area of
Ω
j
;
i
dxdy yx NCyxN
dxdy yx NCyxN NCN c
j
i
j
i
j
i
ij
) , )( ˆ )( ,(
) , )( ˆ )( ,(
ˆ,
;
(9)
[… , …] = scalar product of the two functions expressed by
integral of their product over
Ω
.
If
Ω
i
— a rectangle (
x
1
≤
x
≤
x
2
,
y
1
≤
y
≤
y
2
), then in Eq. 9
)
,
(
)
,
(
)
,
(
)
,
(
ˆ
1
1
2
1
1
2
2
2
y yx xF y yx xF
y yx xF y yx xF NC
i
;
(10)
F
(
a
,
b
) = displacement of the point of origin under the action of
unit load on the rectangle with the abscissas of its corner points
x
= 0 and
x
=
a
and with ordinates
y
= 0 and
y
=
b
:
a b
dxdy
y x
E
baF
0 0
2/12
2
)
(
1 ) ,(
;
(11)
E
,
ν
= deformation modulus and Poisson ratio of soil. The
formula obtained is a form of the well known method of
computation of half-space surface settlement by the
superposition of rectangular loads (see Terzhagi 1943). Another
form (Kholmyansky 2007) of well-known expression (see
Terzhagi 1943) for
F
(
a
,
b
) was obtained:
) /
Arsh(
) /
Arsh(
1 ) ,(
ba
b ab
a
E
baF
;
(12)
From this point on only uniform rectangular grids of boundary
elements are considered. Integration for the computation of
scalar product in Eq. 9 is performed numerically with the Gauss
2×2 cubature formula Гаусса.
The main difference of the described discrete method from
the well known Zhemochkin method (Zhemochkin and
Sinitsyn 1947) is the fact that the Galerkin approach is used
instead of collocation approach..
4 STRUCTURE MODELLED BY A BEAM
In case of narrow and stiff in transversal direction footing the
displacements of points in contact zone are determined by the
displacements on the footing axis; elements of matrix
A
are
units and zeros. Elements of matrix
B
are units and zeros too;
matrix
B
is transposed to
A
:
w
ˆ
T
AB
.
(13)
The structure model is supposed to be the Bernoulli-Euler beam.
For the discretization of the well known ordinary differential
equation of beam bending and computation of stiffness matrix
K
the method of real finite elements was applied
(Karamansky 1981).
