Analysis and modelling of the seismic behaviour of high ... - Ingegneria

2. DUCTILITY AND SEISMIC RESPONSE OF STRUCTURES
A
5% Demand
Diagram
D
Demand Point
Demand Diagram
Capacity Diagram
Figure 2.8. Determination of a displacement demand
The yield strength and yield displacement are denoted by fy and uy,
respectively. If the peak (maximum absolute) deformation of the inelastic
system is umax, the ductility factor is defined as
umax
µ = ( 2.8 )
u
y
For the same bi-linear system, the natural vibration period of the equivalent
linear system with stiffness equal to the secant stiffness ksec is
T = T
eq N
µ
1+
αµ −α
( 2.9 )
where TN is the natural vibration period of the system vibrating within its
linear elastic range (u < uy). Moreover, the most common method for
defining an equivalent viscous damping is to equate the energy dissipated
in a vibration cycle of the inelastic system and of an equivalent linear
system. Based on this concept, it can be shown that the equivalent viscous
damping ratio is (Chopra, 1995)
ξ
eq
1 ED
= ( 2.10 )
4π
E
S
where the energy dissipated in the inelastic system is given by the area ED
2
enclosed by the hysteresis loop and Es = ksecum /2 is the strain energy of
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