Geology Reference
In-Depth Information
Figure 2. Flowchart for deriving critical earth-
quake loads
parameters are changed later to study their influ-
ence on the estimated worst earthquake loads and
the associated damage. The yield displacement is
taken as 0.10 m and the structure is taken to start
from rest. The objective function is adopted as
the Park and Ang damage index DIPA given by
Eq (5). The parameters of the Newmark β-method
are taken as
δ
=
1 2
/ ;
α
=
1 6
/
and
∆
t
=
0 005
.
s.
4.1.2 Quantification of Constraints
A set of 20 earthquake records is used to quan-
tify the constraint bounds E, M1, M2, M3,
M
4
( )
ω
and
M
5
(
ω
(COSMOS, 2005). Table 2 provides
information on these records. Based on numerical
analysis of these records, the constraints were
computed as E = 4.17 m/s1.5, M1 = 4.63 m/s2
(0.47 g), M2 = 0.60 m/s and M3 = 0.15 m and the
average dominant frequency was about 1.65 Hz.
The envelope parameters were taken as A0 = 2.17,
α
1
= 0.13, and
α
2
= 0.50. The convergence limits
ε ε
1
,
were taken as 10-6 and the convergence
criterion on the secant stiffness is taken as 10-3
N/m. The frequency content for
x t
g
( )
is taken as
(0.1-25) Hz. Additionally, in distributing the
frequenciesωi
ω
i
,
i
=
1 2
,
, ...,
N
in the interval
f
0 1 25
, it was found advantageous to select
some of these
ω
i
to coincide with the natural
frequency of the elastic structure and also to place
relatively more points within the modal half-
power bandwidth.
The constraint scenarios considered in deriv-
ing the worst earthquake inputs are listed in Table
3. The constrained nonlinear optimization problem
is solved using the sequential quadratic optimiza-
tion algorithm 'fmincon' of the Matlab optimiza-
tion toolbox (Caleman et al, 1999). In the nu-
merical calculations, alternative initial starting
solutions, within the feasible region, were exam-
ined and were found to yield the same optimal
solution. To select the number of frequency terms
N
f
a parametric study was carried out and
N
f
=
( . ,
)
4. NUMERICAL EXAMPLES
4.1 Bilinear Inelastic Frame Structure
A SDOF frame structure with mass 9
×
103 kg,
initial stiffness k1 = 1.49
×
105 N/m and viscous
damping of 0.03 damping ratio is considered
(initial natural frequency = 4.07 rad/s). The strain
hardening ratio is taken equal to 0.05. These
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