Civil Engineering Reference
In-Depth Information
0
The residual force vector
{R}
can be considered as a measure of the
deviation from the equilibrium state.
To implement this iterative procedure, first of all, solve for the first
approximate displacements {
{
R
}=[
K
({
d
})]{
d
}+{P}
„
(9.53)
d
1
} using the initial stiffness [
K
o
] and the initial
total load [
P
o
]
{
d
l
}=[
K
o
]
-1
{
P
o
}
(9.54)
e
}.
Suppose that it is possible to express the nonlinear stress-strain
relationship of the material by
and work out the strain {
e
1
} from {
e
}=
[B]
{
d
})
If the tangential elasticity matrix [
D
o
] is used and an initial stress {
{
s
}=
f
({
e
(9.55)
s
01
} is
introduced
{
s
l
}=
[D
0
]
{
e
l
}-{
s
01
}
(9.56)
where {
{
s
01
}=[
D
0
] {
e
l
}-
f
({
e
1
})={
s
e1
}-{
s
1
}
(9.57)
where {
s
el
}=
[
D
0
]
{
e
1
} is the elastic stress. The excess initial stress {
s
01
}
e
l
} can then be obtained.
The initial stress in an element may be considered as the difference in
stress between the non-linear stress actually exists in the element due to the
deformation and the elastic stress.
The unbalanced residual forces on the structure are the assembly of all
the element residual forces given by
corresponding to {
(9.58)
Hence it is now possible to make an adjustment to the displacement as follows:
{
Dd
1
}=[
K
0
]
-1
{
R
1
}
(9.59)
(9.60)
Therefore {
Dd
1
}
The procedure is repeated until {
d
2
}={
d
1
}+{
Dd
n
} is sufficiently close to zero.
Here, in this case, a constant stiffness [
K
0
] has been employed in one
stage of iteration and the method is thus called the initial stiffness
method.
One distinctive advantage of this method is that the same stiffness matrix
is used at each step of iteration (
Figure 9.12
)
Once the stiffness matrix is
inverted, it only involves a small amount of computing effort in each
subsequent iteration step for determination of {
Dd
1
}. But the rate of
convergence is slow. Other methods with variable stiffness matrix
[K]
such
as the secant stiffness method and Newton-Raphson method may have a
