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The exact stationary probability density for modal energy H can be obtained from solving the reduced
Fokker-Planck-Komogorov (FPK) equation associated with the averaged stochastic differential
equation (14). That is
p(H) = C,exp[-] A.*.
J A a iLJ>
ft- e (,
where C n is a normalization constant.
H t H,]
'
(16)
After obtaining the root mean square responses of structural deflection and internal forces, the control
effectiveness is evaluated using the following performance index
K = -
-ov
(17)
where subscripts u and c denote the uncontrolled and controlled structure; s represents the structural
response variables, which can be displacement, velocity, acceleration or internal force.
A total of twelve evaluation criteria have been defined to assess the control effectiveness and efficacy.
They include: maximum and average displacement reduction; maximum and average base force
reduction; maximum and average bending moment reduction; maximum displacement of AMDs;
maximum control force of AMDs; ratio of total control force to weight of bridge; etc.
CASE STUDY OF TING KAU BRIDGE
The Ting Kau Bridge (TKB), as shown in Figure 1, is a three-tower cable-stayed bridge with two main
spans of 448 m and 475 m respectively, and two side spans of 127 m each (Bergermann & Schlaich
1996). The bridge deck has been separated into two carriageways with three monoleg towers located
between. Eight longitudinal stabilizing cables with length up to 465 m have been used to strengthen
the slender central tower. Due to adopting these special design configurations, the TKB is one of the
most flexible cable-stayed bridges in the world.
For modal and seismic response analyses, a precise three-dimensional finite element model of the TKB
has been developed that contains over 15,000 degrees of freedom (DOFs). The huge number of DOFs
in a full structural model for cable-stayed bridges makes it difficult to create computationally effective
Fig. 1 Cable-stayed Ting Kau Bridge