Civil Engineering Reference
In-Depth Information
For the special case of
β
=
1
/
4and
γ
=
1
/
2, the method is identical to the Crank-Nicolson
θ
=
5 method described in Section 3.13.2.
Starting from the assembled form of (3.118), and using a “dot” notation to signify time
derivatives, we have
0
.
˙
¨
[
K
m
]
{
U
} +
[
C
m
]
{
U
}+
[
M
m
]
{
U
}={
F
}
(11.1)
˙
with known initial conditions on displacements and velocities,
{
U
}
0
and
{
U
}
0
.
˙
¨
Let
{
F
}
i
,
{
U
}
i
,
{
U
}
i
and
{
U
}
i
represent
conditions
at
time
t
=
it
,where
i
=
0
,
1
,
,...,
nstep
.
Assuming that [
M
2
], [
C
], [
K
],
β
,
γ
and
t
are constant, and that
{
F
}
i
is known for
m
m
m
˙
¨
all
i
, the following algorithm is used to obtain the values of
{
U
}
i
,
{
U
}
i
and
{
U
}
i
for all
i>
0.
1) Compute:
γ
βt
1
β(t)
[
K
]
=
+
[
C
m
+
[
K
m
]
]
2
[
M
m
]
2) Factorise [
K
] to facilitate step 8.
¨
˙
3) Solve the linear equations: [
M
]
{
U
}
0
= {
F
}
0
−
[
C
]
{
U
}
0
−
[
K
]
{
U
}
0
m
m
m
i
=
4) Set
0
5) Compute:
1
2
1
1
β(t)
1
βt
˙
¨
{
A
}
i
=
2
{
U
}
i
+
{
U
}
i
+
−
{
U
}
i
β
6) Compute:
1
1
γ
βt
{
−
γ
β
2
˙
t
¨
{
B
}
i
=
U
}
i
−
{
U
}
i
−
−
{
U
}
i
β
7) Compute:
F
1
= {
F
}
i
+
1
+
[
M
m
]
{
A
}
i
+
[
C
m
]
{
B
}
i
i
+
}
i
+
1
=
F
8) Solve the linear equations: [
K
]
{
U
i
+
1
9) Compute:
γ
βt
{
˙
{
U
}
i
+
1
=
U
}
i
+
1
− {
B
}
i
10) Compute:
1
β(t)
¨
{
U
}
i
+
1
=
2
{
U
}
i
+
1
− {
A
}
i
11) Increment
i
and repeat from step 5
Subroutine
beam mm
(see e.g. Program 4.6) forms the beam element consistent mass
matrix and subroutine
interp
takes the input load/time function data points and interpo-
lates linearly to give load/time function values at the resolution of the calculation time step.
The example and data shown in Figure 11.1 are of a cantilever of unit length modelled
with a single beam element, subjected to a tip loading given by a half-sine pulse with an
