Civil Engineering Reference
In-Depth Information
3.13.2 Newmark or Crank-Nicolson method
If Rayleigh damping is assumed, a class of recurrence relations based on linear interpolation
in time can again be constructed, involving the scalar parameter
θ
which varies between
1/2 and 1 in the same way as was done for first order problems.
If using an assembly technique, the equations (3.118) are written at both the “0” and
“1” time stations,
+
f
k
[
K
m
]
)
d
U
d
t
[
M
m
]
d
2
U
d
t
2
[
K
m
]
{
U
}
0
+
(f
m
[
M
m
]
0
+
0
= {
F
}
0
(3.137)
+
f
k
[
K
m
]
)
d
U
d
t
[
M
m
]
d
2
U
d
t
2
[
K
m
]
{
}
1
+
(f
m
[
M
m
]
1
+
1
= {
F
}
1
U
and assuming linear interpolation in time,
}
0
+
t
(
1
−
θ)
d
U
d
t
0
+
θ
d
U
{
}
1
= {
U
U
d
t
1
(3.138)
d
U
d
t
d
U
d
t
0
+
t
(
1
−
θ)
d
2
U
d
t
2
0
+
θ
d
2
U
1
=
d
t
2
1
Rearrangement of these equations and elimination of acceleration terms leads to the fol-
lowing three recurrence relations,
f
m
+
[
M
m
]
+
(f
k
+
θt)
[
K
m
]
1
θt
{
}
1
U
f
m
+
[
M
m
]
1
θt
=
θt
{
F
}
1
+
(
1
−
θ)t
{
F
}
0
+
{
U
}
0
(3.139)
[
M
m
]
d
U
dt
1
θ
+
0
+
(f
k
−
(
1
−
θ)t)
[
K
m
]
{
U
}
0
d
U
d
t
d
U
d
t
1
θt
(
{
1
−
θ
θ
1
=
U
}
1
− {
U
}
0
)
−
(3.140)
0
d
2
U
d
t
2
d
U
d
t
d
U
d
t
d
2
U
d
t
2
1
θt
1
−
θ
θ
1
=
1
−
−
(3.141)
0
0
The algorithm requires initial conditions on displacements
{
U
}
0
and velocities
{
d
U
/
d
t
}
0
to
be provided in order to get started.
In the special case when
θ
=
1
/
4” method, which
is also the exact equivalent of the Crank-Nicolson method used in first order problems.
There are other variants of the Newmark type, but this is the most common.
The principal recurrence relation (3.139) is clearly similar to those which arose in first
order problems, for example (3.94). Although substantially more matrix-by-vector multipli-
cations are involved on the right hand side, together with matrix and vector additions, the
1
/
2 this method is Newmark's “
β
=
