Response Evaluation - Fundamentals of Earthquake Engineering - page 230

stiffness of the initial elastic system. It was demonstrated that this assumption provides a reasonable
approximation of the damping (Anderson and Gurfi nkel, 1975). In addition, more refi ned formulations
of the damping matrix are not as important in inelastic systems as it is for their elastic counterparts. In
the inelastic range, the principal mechanism of energy dissipation is that due to irrecoverable deforma-
tions, as discussed in Section 2.3.3. The latter mechanism is accounted for by modelling the hysteretic
behaviour of the materials. Even though the role of hysteresis in damping is prevalent, the selection of
values of damping coeffi cients is of importance too, in both elastic and inelastic dynamic analyses
(Broderick et al ., 1994). Critical damping calibrated on target values in the lower modes can over- damp
the contribution of higher modes to the total response. This problem can be solved by adopting selec-
tively dissipative numerical integration schemes, e.g. the Hilber- Hughes - Taylor α - integration scheme
(Hilber et al ., 1977). By introducing intentional integration errors, observed as period elongation and
amplitude decay, spurious mode contributions can be eliminated, thus improving the overall quality of
the response calculations.
The application of time-stepping procedures to integrate the equations of motion of MDOF systems
requires controlled values of the time interval Δ t . Since higher modes are of short periods, small Δ t permits
accurate integration of higher modes. The higher modes are, however, poorly represented (from a fi nite
element discretization viewpoint) in the dynamic structural response; thus, it is not necessary to use a
time increment derived from the highest mode. Instead, a time increment suffi ciently small to integrate
the highest mode of interest should be utilized. For non-linear problems, where the reduction in stiffness
may lead to the sudden inclusion of higher modes of vibration in the integrated response, it is recom-
mended to employ the time integration algorithms developed by Hilber et al . (1977) , which require the
defi nition of three parameters, generally indicated as α , β and γ . Optimal solutions, in terms of accuracy,
analytical stability and numerical damping, are obtained for values of β = 0.25 · (1 − α ) 2 and γ = 0.5 −
α , with − 1/3 ≤ α ≤ 0. These algorithms, which exhibit low numerical damping for lower modes and high
damping for higher (generally poorly represented) modes, are implemented in several advanced or com-
mercial FE computer programs, e.g. the computer program Zeus-NL (Elnashai et al ., 2003 ).
It is noteworthy that modal and modal spectral analysis are primarily demand- oriented methods. In
other words, they normally provide estimates of the demand imposed by an earthquake on the structural
system investigated. They do not necessarily provide estimates of structural capacity, or ' supply ' . Only
response history analysis has the potential to be used in both ' demand ' and ' supply ' estimation. This
is explored further below.
4.6.1.3 Incremental Dynamic Analysis
Incremental dynamic analysis (IDA), also termed dynamic pushover, is an analysis method that can be
utilized to estimate structural capacity (or supply) under earthquake loading. It provides a continuous
picture of the system response, from elasticity to yielding and fi nally to collapse. The rationale behind
the IDA is derived by analogy with the incremental static analysis, or pushover, analysis which is dis-
cussed in Section 4.6.2.2. The concept of IDA is not new ( see , for example Bertero, 1977 ; Nassar and
Krawinkler, 1991). It has nevertheless more recently gained in popularity and wide use as a method to
estimate the global capacity of structural systems (Vamvatsikos and Cornell, 2004 ).
The method constitutes subjecting a structural model to one or more ground-motion records, each
scaled to multiple levels of intensity. Many dynamic analyses are undertaken and the response from
these analyses is plotted versus the record intensity level. The resulting curves, termed IDA curves,
give an indication of the system performance at all levels of excitation in a manner similar to the
load-displacement curve from static pushover. The steps for obtaining a single earthquake record IDA
are as follows:
(a) Defi ne a suitable earthquake record consistent with the design scenario, as discussed in Section
3.5 .