Geology Reference
In-Depth Information
2.1 Energy Dissipated by Inelastic
Structures
2009, 2010). Comprehensive reviews on these
aspects can be found in Moustafa (2011).
This chapter deals with the damage assess-
ment for inelastic structures under worst future
earthquakes using the critical excitations method.
The novelty of this research is in combining
damage indices, for the first time, with nonlinear
optimization and nonlinear time-history analysis
for assessing the structural performance under
possible future ground motions. The use of dam-
age indices provides a quantitative measure for
damage and necessary repair for the structure.
Bilinear and elastic-plastic force-displacement
relationships are taken to model the material
nonlinearity, and thus the present work is lim-
ited to non-deteriorating structures. Numerical
examples for one-storey and two-storey plane
frames without irregularities are provided. Future
practical applications of the proposed methodol-
ogy in seismic analysis and design of structures
are also discussed.
The energy dissipated by an
N
multi-degree-of-
freedom (MDOF) structure under the ground
acceleration
x
( )
can be computed by integrating
the equations of motion as follows (Zahrah &
Hall, 1984, Akiyama, 1985, Uang & Bertero,
1990, Takewaki, 2004, Kalkan & Kunnath, 2008):
t
N
1
2
=
∫
T
2
E t
( )
=
X MX
( )
τ
( )
τ τ
d
=
m x t
( )
K
i
i
i
1
0
(1a)
t
t
N
∑
∫
T
∫
E t
( )
=
X
( )
τ
CX
( )
τ τ
d
=
x
( )
τ
f
( )
τ τ
d
D
i
Di
i
=
1
0
0
(1b)
t
N
=
∫
E t
( )
=
x
τ
( )
f
( )
τ τ
d
−
E t
( )
(1c)
H
i
si
s
i
1
0
2. DAMAGE ASSESSMENT
OF INELASTIC STRUCTURES
UNDER EARTHQUAKE LOADS
where, M, C, are the mass and damping matrices
of the structure, respectively,
f
si
( )
is the
i
th hys-
teretic restoring force, X(
t
) is the structure dis-
placement vector and dot indicates differentiation
with respect to time. The quantities
E
K
(
t
),
E
D
(
t
),
E
s
(
t
) and
E
H
(
t
) represent the kinetic, damping,
strain and hysteretic energies, respectively
(Moustafa, 2009). For viscous damping models,
the damping energy reduces to
t
Damage assessment for structures is generally
based on the nonlinear response quantities under
earthquake loads (see Table 1). The bilinear and
the elastic-plastic models are shown in Figure 1.
The evaluation of the structural damage is usu-
ally carried out using damage indices which are
quantified in terms of the structure response and
the associated absorbed energy. Therefore, the
quantification of damage indices is carried out after
performing nonlinear time-history analysis for the
structure. The nonlinear time-history analysis for
the structure is performed by solving the equations
of motions using numerical integration schemes
(Moustafa, 2009, Hart & Wong 2000).
t
N
N
∑
=
∫
c x
( )
τ
x
( )
τ τ
d
.
ij
i
j
i
=
1
j
1
0
Note that equations (1) provide the relative
energy terms. Note also that, by the end of the
earthquake duration the kinetic and elastic strain
energies diminish. Thus, the earthquake input
energy to the structure is dissipated by hysteretic
and damping energies. The next section demon-
strates the use of the structure's response and the
hysteretic energy in developing damage indices.
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