While the foregoing discussion has been for elastic response spectra, most structures are not expected or even designed to remain elastic under strong ground motions.

**Inelastic response spectra**

While the foregoing discussion has been for elastic response
spectra, most structures are not expected or even designed to remain elastic
under strong ground motions. Rather, structures are expected to enter the
inelastic region, and the extent to which they behave inelastically can be
defined by the ductility factor

assuming the simplest force
deformation relationship is chosen. Figure 17.33a shows the elastic perfectly
plastic (elasto-plastic) force deformation relation, *f _{s} *(

1Ductility factor and yield strength reduction factor

The yield strength reduction factor *R _{y}* is
defined as

where *f*_{0} and *u*_{0} are the
minimum yield strength and yield deflection required for the structure to
remain elastic during ground motion.

ductility factor = *u _{m}*/

The inelastic deformation ratio is defined as the ratio of
deflection of inelastic and the corresponding linear system related by *µ* and *R _{y}*.

2Equations of motion and controlled parameters

The governing equation of motion is

The same numerical procedures discussed in Chapter 7 can also
be applied here with the difference that the time instants must be detected
accurately enough when the system changes from elastic to yield branch.

For a given ground excitation u˙˙g (t ) , u(t) depends on three
system parameters ωn(Tn = 2π/ωn), ρ, and uy and the ductility factor *µ* depends on *ω*_{n}*ρ* and *R _{y}*

3Inelastic response shock spectrum

*Peak deformation and ductility demand*

The deformation response of an inelastic system is obtained
from its initial elastic vibration period *T _{n}* and damping
factor

Inelastic response spectra can be
calculated in the time domain by direct integration, analogous to elastic
response spectra but with structural stiffness as a nonlinear function of
displacements *K* = *K*(*u*). If elastic plastic behaviour is
assumed, then elastic response spectra on the basis that at high periods *T _{n}*
> 33 s (

An inelastic design spectrum is most commonly created directly
from the elastic design spectrum. Observe then the spectral velocity *S _{v}*,
spectral displacement

In the acceleration constant
region, the reduction factor *R _{y}* is attained by equating
elastic and inelastic strain energies. The resultant reduction factor is 2

Several researchers have proposed equations of variation of *R _{y}*
with

where the periods *T _{a}*,

*u***˙˙**_{g}* *= 1*g*
; *u***˙*** _{g}* = 122 cm/s,

Spectra such as those described
above provides the basis for safety evaluation of new and existing structures
which will be discussed in later chapters.

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Civil : Structural dynamics of earthquake engineering : Inelastic response spectra |

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