Civil Engineering Reference
In-Depth Information
9.6.2 Displacement Control Iteration Scheme
Loads can be applied to a structure using either of two different methods: under load control or
displacement control. To simulate the seismic behavior of a reinforced concrete structure sub-
jected to reversed cyclic loading, the entire load-displacement curve, including the ascending
branch, descending branch, and the hysteresis loops, can be obtained using the displacement
control scheme. On the other hand, the displacement control method also has advantages over
load control in the analysis procedure as described below.
1. Under load control it is impossible to indicate the behavior of the structure at a local limit
such as the temporary drop of force due to the initial concrete cracking. More importantly,
load control is incapable of producing the ultimate strength of the structure, or to trace
the behavior of the structure in the post-peak region. Under load control the tangent
stiffness matrix becomes nearly singular at the peak point of the load-displacement curve.
It was pointed out by Ayoub (1995) and Ayoub and Filippou (1998) that the failure of the
solution to converge is not an indication that the structure has reached its collapse point,
but rather a failure of the solution convergence. Under the displacement control, especially
the displacement control with arc length scheme (Batoz and Dhatt, 1979), it is possible to
obtain the behavior of the structure beyond the crack point and the maximum point and to
determine the entire response including ascending, descending and cyclic branches.
2. When there is no preference of load control or displacement control, the displacement
control method shows faster convergence and is more stable than the load control method.
This is observed in the nonlinear finite element analyses of reinforced concrete plane stress
structures in this topic.
Many researchers (e.g. Zienkiewicz, 1971; Haisler
et al.,
1977; Batoz and Dhatt, 1979)
have proposed the displacement control scheme to overcome the limits of the load control
method. Meanwhile, the arc length method has been developed to overcome the local and
global limit points in the nonlinear analysis, which treated the load factor as a variable. The
arc length method was originally proposed by Riks (1972) and was improved by Crisfield
(1981). A displacement control with an arc length scheme originally proposed by Batoz and
Dhatt (1979) is available.
9.6.3 Dynamic Analysis Iteration Scheme
9.6.3.1 Newmark's Method
Newmark's method (Newmark, 1959) and Wilson's method (Wilson
et al
., 1973) are two well-
known methods for dynamic analysis. These two methods have been implemented as a class
of objects in the OpenSees. The theoretical background
s
of Newmark's and Wilson's methods
are briefly summarized in this and the next section, respectively. The iterative procedures using
these two methods for nonlinear systems are described as well.
The explicit relationships of displacements, velocities, and accelerations from step
i
to step
i
+
1are:
u
i
+
1
=
u
i
+
(
t
)
u
i
,
(9.49)
t
)
2
u
i
+
1
=
u
i
+
(
t
)
u
i
+
0
.
5 (
u
i
,
(9.50)
