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TABLE 10.6
Procedure for Computer Implementation of the Wilson
θ
Method
Initial Step
1.
Select
t
, and let
θ
=
1
.
4
2.
Calculate the constants related to
t
2
c
0
=
6
/(θ
t
)
c
1
=
3
/θ
t
2
=
2
a
1
c
3
=
θ
t
/
2
c
4
=
c
0
/θ
c
5
=−
c
2
/θ
t
2
c
6
=
1
−
3
/θ
c
7
=
t
/
2
c
8
=
/
6
Initialize
V
0
,
V
0
, and
V
0
3.
4.
Form the effective stiffness matrix
K
=
+
+
K
c
0
M
c
1
C
Decompose (triangularize)
K
5.
=
LDL
T
At Each Step
c
2
V
n
+
2
V
n
)
+
1.
Calculate
P
n
+
θ
=
P
n
+
θ(
P
n
+
1
−
P
n
)
+
M
(
c
0
V
n
+
2
V
n
+
c
3
V
n
)
C
(
c
1
V
n
+
2.
Solve for the displacement
LDL
T
V
n
+
θ
=
P
n
+
θ
+
3.
Calculate displacements, velocities, and accelerations at
t
t
V
n
+
1
=
c
5
V
n
+
c
6
V
n
c
4
(
V
n
+
θ
−
V
n
)
+
V
n
+
1
=
V
n
+
V
n
+
1
+
V
n
)
c
7
(
V
n
+
V
n
+
1
+
2
V
n
)
V
n
+
1
=
V
n
+
t
c
8
(
where
V
n
+
1
V
n
+
1
V
n
+
1
V
n
V
n
V
n
X
n
+
1
=
X
n
=
2
t
2
1
−
βθ
/
3
−
1
/θ
−
βθ/
t
−
β/
2
A
=
t
(
1
−
1
/
2
θ
−
βθ
/
6
)
1
−
βθ/
t
−
β/
2
t
t
2
2
(
1
/
2
−
1
/
6
θ
−
βθ
/
18
)
t
(
1
−
βθ/
6
)
1
−
β/
6
and
θ
ω
−
1
3
6
t
2
+
θ
β
=
2
The characteristic equation corresponding to the eigenvalue problem of matrix
A
can be
obtained from Eq. (10.125). It can be seen that the Wilson
θ
method is unconditionally
stable if
θ
≥
1
.
37
.
In practice,
θ
=
1
.
4 is usually employed.
EXAMPLE 10.11 Transient Response of a Frame
The frame of Fig. 10.2 is chosen to illustrate the results of the use of different integration
schemes. The system mass and stiffness matrices
M
and
K
are given by Eq. (9) of Example
10.1 and Eq. (11) of Chapter 5, Example 5.5. The damping matrix
C
is assumed to be
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