Civil Engineering Reference
In-Depth Information
to sparsity, and “element-by-element” (mesh-free) algorithms can be developed using pcg
equation solution as shown in Chapters 9 and 12 .
3.12.2
Incremental load version
{
}
In (3.107),
is the total force applied and these equations are appropriate to linear systems.
Later in this topic we shall be concerned with non-linear systems similar to those described
in Chapter 6, in which it is desirable to apply loads incrementally and allow plastic stress
redistribution to equilibrate at each step.
If
f
is the change in load between successive times, the incremental form of the
first of (3.107) is
{
f
}
[
k
m
]
{
u
} +
[
c
]
{
u
w
} = {
f
}
(3.113)
where
are the resulting changes in displacement and excess pore pressure
respectively. Linear interpolation in time using the
θ
-method yields
{
u
}
and
{
u
w
}
} =
t
(
1
−
θ)
d
u
d
t
0
+
θ
d
u
{
u
(3.114)
d
t
1
and the second of (3.107) can be written at the two time levels to give expressions for
the derivatives which can then be eliminated to give the following incremental recurrence
equations (Sandhu and Wilson, 1969; Griffiths, 1994a; Hicks, 1995).
[
k
m
]
{
u
[
c
]
}
{
u
w
}
{
f
}
=
(3.115)
[
c
]
T
−
θt
[
k
c
]
t
[
k
c
]
{
u
w
}
0
The left hand side element matrix, again called [
k
e
], is formed from its constituent
matrices by subroutine
formke
and is symmetric. If using an assembly strategy, sub-
routine
fsparv
generates the global matrix. The right hand side vector consists of load
increments
u
w
}
0
. The fluid term is conveniently
computed without any need for assembly using an element-by-element product approach
(see Program 9.3). Subroutines
sparin
and
spabac
complete the solution for the incre-
mental displacements and excess pore pressures.
At each time step, all that remains is to update the dependent variables using
{
f
}
and fluid “loads” given by
t
[
k
c
]
{
{
u
}
1
= {
u
}
0
+ {
u
}
{
u
w
}
1
= {
u
w
}
0
+ {
u
w
}
(3.116)
3.13 Solution of second order time dependent problems
The basic second-order propagation type of equation was derived in Chapter 2 and at the
element level takes the form of (2.102), namely
[
m
m
]
d
2
u
d
t
2
[
k
m
]
{
u
} +
= {
f
(t)
}
(3.117)
