Page 167 - Dynamic Loading and Design of Structures

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1999), for two site conditions (‘rock’ and ‘alluvium’), and four ductility factors: 1 (elastic
behaviour), 2 (low ductility), 3.5 (medium ductility) and 5 (high ductility). Note that the shape
of inelastic spectra is generally different from that of the corresponding elastic spectra; they
are much smoother than the latter, and smoothness increases with the ductility level. For
µ≥3.5 the strength requirement decreases monotonically with the period, regardless of soil
conditions. Inelastic behaviour appears to be more effective in reducing the maximum elastic
acceleration in the case of motions recorded on rock, but in all cases elastic force reduction is
very significant in the medium and long period range. Also of practical significance is the
observation that for µ≥3.5 inelastic strength demands are just slightly influenced by the
ductility level, for both rock and alluvium; the implication of this is that for relatively small
changes in the strength of medium and high ductility structures, the increase in the required
ductility is significant.

Design spectra
Seismic codes still rely upon the concept of inelastic spectrum for specifying design actions
(forces) to be used for elastic modal analysis of structures which are expected to respond
inelastically to the design earthquake. This is a rather crude approximation and errors tend to
increase as the level of inelasticity (or target ductility µ) and the fundamental natural period
(or the number of storeys) increase (Anagnostopoulos et al., 1978; Krawinkler and Nassar,
1992).
Since for design purposes several ground motions with different characteristics have to be
taken into account, an average inelastic response spectrum has to be used, and this would
generally involve considerable work. Hence, several attempts have been made to construct
(inelastic) design spectra directly from the corresponding elastic spectra, by appropriate
modification of the latter. The typical way to do this is to divide the ordinates of the elastic
response spectrum by a factor which depends on the type of inelastic behaviour (e.g.
elastoplastic, stiffness degrading, etc.) and the damping (typically 5 per cent is used for the
design spectra, as mentioned in Section 4.3.2), in addition to the period; i.e. for a given
hysteretic behaviour and damping ratio

(4.19)

where the subscripts ‘el’ and ‘in’ refer to the ordinates of the elastic and inelastic response
spectrum, respectively. Note that in eqn (4.19) T is the fundamental period of the structure
before yielding, often referred to as the elastic period. This period is not the effective or the
predominant period of the inelastically responding structure (particularly when the plastic
deformations are significant), hence it should not be forgotten that plotting Sin as a function of
the initial T is merely a convention.