Civil Engineering Reference
In-Depth Information
from a detailed analysis of the structure. It follows that the supply should be equal or greater than the
demand for a safe seismic response in the inelastic range. Thus, the above discussion points towards
the necessity of studying the dynamic inelastic response of structures, in an attempt to quantify the
behaviour factors (
R
or
q
) necessary for the derivation of design forces from elastic forces. Finally,
inelastic spectra are generally derived by assuming elastic-plastic hysteretic models. However, SDOF
systems with different force-deformation relationships can also be employed, e.g. bilinear with harden-
ing and with stiffness and strength degradation, to more accurately represent the response of real
structures. In general, elastic-plastic non-degrading SDOF systems exhibit higher energy absorption
and dissipation than degrading systems. Therefore, estimates of force reduction factors based on the
former are sometimes unconservative (i.e. they overestimate
R
, hence underestimate the design force).
Use of
R
- factors based on elastic -plastic response should therefore be treated with caution, especially
for high levels of inelasticity.
The relationship between displacement ductility and ductility-dependent behaviour factor has been
the subject of considerable research. A few of the most frequently used relationships reported in the
technical literature are discussed below.
(i) Newmark and Hall ( 1982 )
The force reduction factor
R
μ
is defi ned as the ratio of the maximum elastic force to the yield force
required for limiting the maximum inelastic response to a displacement ductility
μ
. In this early study
R
μ
was parameterized as a function of
μ
(Newmark and Hall, 1982), as also discussed in Section 2.3.6 .
It was observed that in the long-period range, elastic and ductile systems with the same initial stiffness
reached almost the same displacement. As a consequence, the force reduction (or behaviour) factor can
be considered equal to the displacement ductility. This is referred to as the ' equal displacement ' region.
For short-period structures, the ductility is higher than the behaviour factor and the ' equal energy '
approach may be adopted to calculate force reduction. This approach is based on the observation that
the energy associated with the force corresponding to the maximum displacement reached by elastic
and inelastic systems is the same, as explained in Section 2.3.6 of Chapter 2. The proposed relationships
for behaviour factor are:
R
μ
=
1
when
T
<
0.05s
(3.22.1)
R
=
2
μ
−
1
when 0.12 s
<
T
<
0.5s
(3.22.2)
μ
R
=
μ
when
T
>
1.0 s
(3.22.3)
μ
while a linear interpolation is suggested for intermediate periods. The above is the fi rst and simplest
formulation used in practice. It has endured over the years due to its intuitive nature and has been con-
fi rmed by other studies, such as Uang (1991) and Whittaker
et al.
( 1999 ). Further refi nement though
is warranted since the force reduction factor may be a function of period within the regions defi ned by
equations (3.22.1) to (3.22.3) .
(ii) Krawinkler and Nassar ( 1992 )
A relationship was developed for the force reduction factor derived from the statistical analysis of 15
western USA ground motions with magnitude between 5.7 and 7.7 (Krawinkler and Nassar, 1992 ). The
records were obtained on alluvium and rock site, but the infl uence of site condition was not explicitly
studied. The infl uence of behaviour parameters, such as yield level and hardening coeffi cient
α
, was
taken into account. A 5% damping value was assumed. The equation derived is given as:
11
1
c
[
]
Rc
=
(
μ
−
)
+
(3.23.1)
μ
where:
