Civil Engineering Reference
In-Depth Information
point of view, equations which are not self-adjoint will always lead to unsymmetrical
matrices.
A second consequence of non-self-adjoint equations is that there is no energy formu-
lation equivalent to (2.129). It is clearly a benefit of the Galerkin approach that it can be
used for all types of equation and is not restricted to self-adjoint systems.
Equation (2.132) can be rendered self-adjoint by using the transformation
h
=
φ
exp
ux
2
c
x
exp
vy
2
c
y
(2.134)
but this is not recommended unless
u
and
v
are small compared with
c
x
and
c
y
, as shown
by Smith
et al
. (1973).
Equation (2.132) and the use of (2.134) are described in Chapter 8.
2.18 Further coupled equations: Biot consolidation
Thus far in this chapter, analyses of solids and fluids have been considered separately. How-
ever, Biot formulated the theory of coupled solid-fluid interaction which finds application
in soil mechanics (Smith and Hobbs, 1976). The soil skeleton is treated as a porous elastic
solid and the laminar pore fluid is coupled to the solid by the conditions of equilibrium
and continuity.
First, for two-dimensional equilibrium in the absence of body forces, the gradient of
effective stress from (2.50) must be augmented by the gradients of the fluid pressure
u
w
as follows:
∂σ
x
∂x
∂τ
xy
∂y
∂u
w
∂x
+
+
=
0
(2.135)
∂σ
y
∂y
∂τ
xy
∂x
∂u
w
∂y
+
+
=
0
where
σ
x
=
σ
x
−
u
w
, etc are “effective” stresses.
Assuming plane strain conditions and small strains, and following the usual sequence of
operations for a displacement method, the stress terms in equation (2.135) can be eliminated
in terms of displacements to give (Griffiths, 1994),
∂
E
(
1
−
ν
)
2
ν
)
2
2
2
u
∂x
−
∂
u
1
(
1
v
∂x∂y
∂
∂u
w
∂x
+
+
+
=
0
−
ν
)
+
ν
)(
1
2
ν
)
2
−
ν
)
2
∂y
2
(
1
(
1
−
2
(
1
(2.136)
E
(
1
−
ν
)
2
ν
)
2
2
2
1
∂
v
(
1
−
∂
v
∂x
u
∂x∂y
∂
∂u
w
∂y
+
+
+
=
0
−
ν
)
+
ν
)(
1
2
ν
)
2
−
ν
)
2
(
1
−
2
(
1
∂y
2
(
1
where
E
and
ν
are the effective elastic parameters.
