Optimal Placement of Viscous Dampers for Seismic Building Design - Design Optimization of Active and Passive Structural Control Systems - page 37

but is limited to structures dominated by a single
mode response. The OCT approach was demon-
strated with passive linear viscous and viscoelastic
devices within a braced multistory building.
A good example of a gradient-based search
method is the optimum for minimum transfer func-
tions (Takewaki, 1997). This damper placement
technique takes as its objective the minimisation
of the sum of the interstory drifts of the transfer
function, evaluated at the structure's undamped
fundamental frequency. The method has since been
developed further for more complex structures,
multiple performance objectives, and optimal
sensitivity design to optimise total damping and
distribution (Takewaki, 2009). Since the damper
placement schemes are based on the dynamic
behaviour of the structure alone, the Takewaki
method is independent of the ground motion. The
method was demonstrated for two shear buildings
and assumed stationary ground motions. A pos-
sible weakness of the technique is its objective of
minimising the sum of a performance indicator
as opposed to the peak value, which is a more
appropriate damage indicator, and the exclusion
of design objectives in the method. The method's
status as an early benchmark method for damper
placement, its claimed independence from ground
motion characteristics, and the lack of verification
of the 1997 method for realistic building designs
and ground motions warrant further investigation.
The fully-stressed analysis/redesign proce-
dure is an analytical placement method that uses
engineering knowledge and a simple numerical
approach for damper placement (Levy and Lavan,
2006). Based on the principle of fully-stressed
design of truss members, the Lavan A/R method
uses a recurrence relationship to maximise ('fully-
stress') the dampers influence on the performance
parameter (e.g. drift or acceleration allowance) of
the building and minimise the total adding damping
necessary. The original procedure may be adapted
to constrain the total damping (Lavan and Levy,
2009). The Lavan method has been verified by
formal gradient-based optimisation and has been

applied to shear-frames, industrial frames (Levy
and Lavan, 2006), and 3D irregular frames (Lavan
and Levy, 2006). Levy and Lavan (2006) claim
the method achieves the optimal design, with a
desired uniform damage distribution, an inherent
consideration of performance based design objec-
tives, and efficiency based on realistic ground mo-
tion records and structures. The Lavan method is
shown to be more effective than an active control
method (Gluck et al., 1996) in terms of interstory
drifts for multiple structures and ground motions
(Lavan and Levy, 2009).

Techniques Chosen
for Further Study

To assess and illustrate the performance of a va-
riety of placement techniques, five methods were
applied to the retrofit of steel moment-resisting
frames under a range of seismic hazard levels. Two
are simple empirical rules, while the other three
attempt some form of optimisation. In each case
the key constraint is that the total added damping
is fixed at the same, constant value, enabling fair
comparisons of the schemes' performance.

1. Uniform: The total added damping C t is
uniformly distributed over the n storys, so
that the damping at floor i is simply:

C
n

C =

t

(1)

i

2. Stiffness-Proportional: The total added
damping C t is distributed over the n storys
in proportion to the stiffness k i at floor i :

k

C = C

i

(2)

i t

n

∑ 1

k

i

i =