Civil Engineering Reference
In-Depth Information
6
6
δ
u
i
=
t
)
2
δ
u
i
−
t
)
u
i
−
3
u
i
.
(9.63)
(
δ
(
δ
3
δ
1
2
δ
δ
u
i
=
t
δ
u
i
−
3
u
i
−
t u
i
(9.64)
3
θ
6
k
i
=
k
i
+
t
c
+
t
)
2
m
(9.65)
(
θ
6
θ
3
c
u
i
+
3
m
u
i
+
θ
tc
δ
p
i
=
θ
p
i
+
t
m
+
(9.66)
2
A summary of iterative procedure of Wilson's method by the use of the modified
Newton-Raphson method is described as follows:
1. Calculate
p
i
based on Equation (9.66)
2. Determine the tangent stiffness
k
i
3. Calculate
k
i
based on Equation (9.65)
4. Solve
k
i
δ
δ
u
i
=
δ
p
i
and obtain trial
δ
u
i
,trial
u
i
=
u
i
+
δ
u
i
R
=
δ
5. Determine the resisting force
δ
f
and unbalanced force
δ
R
−
δ
f
6. Check convergence criterion; if yes, go to step 7; if no, go to step 4
7. Determine
1
δ
u
i
based on Equations (9.63) and
u
i
=
θ
δ
u
i
for
δ
t
=
θ
t
t
)
u
i
+
2
(
t
)
2
2
(
t
)
2
6
u
i
=
(
u
i
;
u
i
=
(
t
)
u
i
+
u
i
+
u
i
8.
9.
u
i
+
1
=
u
i
+
u
i
;
u
i
+
1
=
u
i
+
u
i
;
u
i
+
1
=
u
i
+
u
i
10. Proceed to the next time step
If
θ
=
1, Wilson's method will be the same as the linear acceleration method, which is
stable if
t
<
0
.
551
T
n
.If
θ
≥
1
.
37, Wilson's method becomes unconditionally stable.
9.7 Nonlinear Finite Element Program SCS
This chapter has presented the theoretical background of integrators for static and dynamic
analyses including the load control method, the displacement control method, Newmark's and
Wilson's methods. The integrators are incorporated with algorithms for nonlinear analysis.
These integrators and algorithms have already been implemented as classes of objects in the
OpenSees (Fenves, 2005), which makes it capable of performing nonlinear finite element
analysis. To perform analysis on reinforced concrete plane stress structures such as shear
walls, OpenSees is modified to extend its capability.
1. Appropriate uniaxial materials of steel and concrete are added to OpenSees for rein-
forced/prestressed concrete plane stress structures because of the following reasons. In
OpenSees the available uniaxial modules for steel and concrete are Steel01 and Concrete01.
The features of Steel01 and Concrete01 have been introduced in Section 9.1. Steel01 does
not consider the smeared yield strain and stress of embedded steel, and the unloading and
reloading paths do not take into account of the Bauschinger effect. Concrete01 does not
