Damper Optimization for Long-Span Suspension Bridges - Design Optimization of Active and Passive Structural Control Systems - page 117

α

F CV = (8)

each mode. The sum of the effective must
equal the total mass of the bridge. Effective
mass participation of 80 to 90% of the total
mass in any given response direction can be
considered sufficient.
3. Time-History Analysis: For the earthquake
time-history analysis of bridge models, three
analysis tools are available. Among them,
the approach (a) is adopted universally.
a. Step-by-step integration in the time
domain
b. Superstition of normalized modal time
histories in the time domain
c. Evaluation of frequency-dependent
response contributions with transfor-
mation to and superposition in the time
domain.

where F is the force of damper, C is the damping
coefficient, V is the relevant velocity and α is the
exponent generally varying from 0.2 to 1.0. The
linear relationship between F and V occurs when
α equals 1.0, and is named the linear damper.
Because of the 90 0 phase between the damp and
elastic force, the linear damper would give no
contributions to internal structural forces.
The dampers can only be simulated by software
with nonlinearity damp element. The COMBIN14
element whose nodes have three DOFs in the
ANSYS software is employed, accounting for
the compression and tension in the axis direction
and no bending and torsion, as shown in Figure 3.

Seismic Input

Numerical integration schemes for the time
domain can have problems with accuracy or pe-
riod distortion as well as numerical stability when
the integration step ∆ t is not small enough. As
a general rule, numerical stability in condition-
ally stable explicit time integration schemes can
be achieved when the time step ∆ t is selected
such that:

Two levels of seismic waves (10% and 2% prob-
ability in 50 years respectively) are provided
by Jiangsu Provincial Institute of Earthquake
Engineering. These waves, located at different
points along the RSB, include data in all the three
directions (longitudinal, vertical and lateral direc-
tions). In this chapter, the group of five seismic
waves with the 2% probability of exceedance is
employed as the seismic acceleration input, and
the wave leading to the most adverse situation is
selected for the further analysis.
Analysis results have shown that the lateral
seismic motion has little influence on the girder
displacement of long-span cable-supported bridg-
es in along-bridge direction, and the opposite is
obtained as to the vertical seismic motion. Hence
longitudinal + vertical seismic input is taken into
account, the seismic waves corresponding to the
most adverse case is shown in Figure 4.
The travelling wave effect is obvious for the
RSB case because of its long span (Abdel-Ghaf-
far & Rubin, 1982; 1983a; 1983b; Abdel-Ghaffar,
2000) and thus should be considered when per-
forming time domain analysis. The design seismic

T n

∆ t = π (7)

where T n represents the period of the highest
significant mode of vibration.

Damper Simulation

Numerous dampers have been developed to release
seismic forces such as soft steel damper, friction
damper, magneto-rheological (MR) damper, vis-
cous fluid damper, etc. In this study, the relative
mature viscous fluid damper is applied. The force
of damper varying with the relative velocity can
be formed as: