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Table 4. Response parameters for alternative constraint scenarios (α = 0.05, ζ = 0.03)
Case
x
max
(m)
μ
max
x
p
(m)
N
rv
*
DI
PA
Damage status
1
2
3
4
0.47
0.45
0.41
0.26
4.65
4.53
4.14
2.64
0.07
0.06
0.07
0.05
60
54
49
44
1.15
0.97
0.72
0.37
Total collapse
Damaged beyond repair
Damaged beyond repair
Repairable damage
*
N
rv
= number of yield reversals (see Table 1)
x
=
L
ε
/ cos
θ
=
0 0381
m.
is seen to be located in the stiff side of the initial
frequency of the inelastic structure.
y
y
Thus, brace 1 yields when
|
x
1
|
=
0 0381
.
4.2 Inelastic Two-Story
Frame Structure
m and brace 2 yields when
|
m. The objective function is taken as the weight-
ed damage indices in braces 1 and 2. In the nu-
merical analysis, the parameters of the Newmark
β-method were taken as
δ
x x
2
−
|
=
0 0381
.
1
A two-story braced building frame is considered
to demonstrate the formulation developed in this
chapter for MDOF inelastic structures (Moustafa
2009). The material behavior of the braces is
taken as bilinear (
k k
2
=
1 2
/ ;
α
=
1 6
/
and
the time step
∆
t
=
0 00.
s.
The results of this example are shown in Fig-
ures 10 and 11. In general, the feature observed
for the future earthquakes in the previous example
was also observed in this example. However, the
inelastic deformation and the associated damage
were seen to depend on the two vibration modes.
Thus, the maximum ductility ratio
μ
for case 1 is
4.34 while that produced from constraint case 4
is 2.27. Similarly, the maximum response reduces
from 0.15 m to 0.08 m when the constraints on
UBFAS and LBFAS are brought in. The earthquake
input energy to the inelastic system is mainly dis-
sipated by yielding and nonlinear damping of the
structure. The hysteretic and damping energies are
significantly higher than the recoverable strain and
kinetic energy. The kinetic and recoverable strain
energies are small and diminish near the end of the
ground shaking. The energy dissipated by yield-
ing is significantly higher than that dissipated by
damping. The weighted damage index for case 1
was about 0.96 implying total collapse while for
case 4 the damage index was about 0.35 implying
repairable damage.
It may be noted that the numerical illustrations
of the formulation developed in this paper were
demonstrated for simple structures with bilinear
=
γ
) as shown in Figure
1(a). The floor masses are taken as
m m
1
1
= = ×
.
Ns2/m, the cross-section-
al areas of the braces are
A A
1
1 75 10
5
2
−
4
=
=
6 45 10
.
×
2
m2, the Young's modulus =
2 59 10
11
.
×
N/m2, and
the strain hardening ratio = 0.10 (i.e., ratio of the
post-yield stiffness to the pre-yield stiffness).
When both braces are behaving elastically, the
stiffness matrix
K K
s
=
, if brace 1 yields
el
K K
s
=
1
, if brace 2 yields
K K
s
=
2
and if both
braces yield
K K
s
=
12
. These matrices are given
in Moustafa (2009). The structure is assumed to
start from rest. The first two natural frequencies
of the elastic structure were computed as
ω
1
=
.
rad/s. A Rayleigh
proportional damping
C M K
=
.
rad/s and
ω
2
6 18
16 18
=
a b
s
with a =
0.2683, b = 0.0027 is adopted. These values are
selected such that the damping ratio in the first
two modes is 0.03. This implies that the damping
forces in braces are nonlinear hysteretic functions
of the deformed shape of the structure. Let the
yield strain of the braces
ε
y
=
0 00.
for both
tension and compression. The braces will yield
at a relative displacement
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