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Figure 4. Optimal earthquake input and associated structural responses for case 1 (a) Fourier ampli-
tude of the ground acceleration (b) Normalized inelastic deformation (c) Hysteretic restoring force (d)
Dissipated energy
To investigate the influence of the damping
ratio on the computed worst earthquake load,
limited studies were carried out. The damping
ratio was changed, namely, 0.01, 0.03 and 0.05,
while all other parameters were kept unchanged.
The critical earthquake is computed by solving a
new optimization problem for each case. The ef-
fect of the change in η 0 was seen to be similar to
that due to α. In other words, the value of the
damping ratio was not seen to significantly influ-
ence the frequency content of the earthquake
acceleration. It was observed, however, that the
ductility ratio and the maximum inelastic defor-
mation of the structure decrease for higher damp-
ing ratios. Thus, the ductility ratio decreases to
2.43 when the damping ratio is taken as 0.05 while
the ductility ratio increases to 2.89 when the
damping ratio reduces to 0.01. It was also observed
that the inelastic structure with higher damping
ratio dissipates more energy through damping
compared with the same structure with lower
damping ratio (see Figure 8(b)). The damage
index also reduces when the damping ratio in-
creases.
To assess the structure safety, Eq. (5) was used
to estimate the damage index of the structure
subjected to the critical earthquake load. The ef-
fect of the parameter β on the damage index is
examined first. Based on experimental tests, it
was reported that β ranges between 0.05 and 0.20
with an average value of 0.15 as suggested by
Park et al, (1987). Figure 9(a) shows the influence
of β on the damage index. To study the effect of
the initial natural frequency of the structure on
the damage index, the structure stiffness was
varied while keeping all other parameters un-
changed and the critical earthquake was com-
puted for each case. Subsequently, the value of
DIPA was calculated for each case. In the nu-
merical calculations β was taken as 0.15 and
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