Civil Engineering Reference
In-Depth Information
1
2
EF
E
=
δ
(2.12.2)
y
y
where
F
y
and
δ
y
are the action and deformation at fi rst yield, respectively. The total hysteretic
energy dissipated before failure
E
t,H
can be computed as follows:
N
=
∑
E
=
E
i
(2.12.3)
tH
,
,
H
i
1
where the summation is over all cycles
N
up to failure and
E
i
,
H
is the hysteretic energy dissipated
in the
i
th cycle.
In seismic design, high available ductility is essential to ensure plastic redistribution of actions among
components of lateral resisting systems, and to allow for large absorption and dissipation of earthquake
input energy. Ductile systems may withstand extensive structural damage without collapsing or endan-
gering life safety; this corresponds to the 'collapse prevention' limit state. Structural collapse is caused
by earthquakes, which may impose ductility demand
μ
d
that may exceed the available ductility
μ
a
of
the structural system. Imminent collapse occurs when
μ
d
>
μ
a
.
Several factors may lead to reduction of available ductility
μ
a
. These include primarily (i) strain rate
effects causing an increase in yield strength, (ii) reduction of energy absorption due to plastic deforma-
tions under alternating actions, (iii) overstrength leading to structures not yield when they were intended
to yield and (iv) tendency of some materials to exhibit brittle fracture. These factors may affect both
local and global ductility. The effects of material, section, member, connection and systems properties
on the structural ductility are discussed in the next section.
2.3.3.1 Factors Infl uencing Ductility
(i) Material Properties
The ductility of structural systems signifi cantly depends on the material response. Inelastic deformations
at the global level require that the material possesses high ductility. Concrete and masonry are brittle
materials. They exhibit sharp reductions of strength and stiffness after reaching the maximum resistance
in compression. Both materials possess low tensile resistance, which is followed by abrupt loss of
strength and stiffness. The material ductility
μ
ε
can be expressed as the ratio of the ultimate strain
ε
u
and the strain at yield
ε
y
, i.e.
μ
ε
=
ε
u
/
ε
y
. Consequently, the ductility
μ
ε
of concrete and masonry in
tension is equal to unity, while
μ
ε
is about 1.5-2.0 in compression. For concrete, the higher the grade,
the lower is the inelastic deformation capacity. Metals and wood exhibit much higher values of
μ
ε
.
Mild steel has average values of material ductility of 15-20 if ultimate strains
ε
u
are limited to the
incipient strain hardening
ε
sh
, i.e.
ε
u
=
ε
sh
as shown in Figure 2.34. Values of
μ
ε
in excess of 60- 80 can
be obtained by using the deformation at ultimate strength. Similarly, metal alloys, such as aluminium
and stainless steels, exhibit values of material ductility as high as 70-80. These alloys do not possess
clear yield points and a conventional 'proof stress', i.e. stress corresponding to 0.2% residual strain, is
utilized to defi ne the elastic threshold (e.g. Di Sarno and Elnashai, 2003, among others).
Steel reinforcement can be utilized in plain concrete and masonry to enhance their ductility. Steel-
confi ned concrete exhibits inelastic deformations 5-15 times higher than plain concrete (CEB, 1996 ).
Strain at maximum compressive strength is about 0.3-0.4% for almost all grades of concrete. Uncon-
fi ned concrete exhibits very limited ductility
μ
ε
in compression. Confi nement limits the post- peak
strength reduction, thus increasing the residual resistance. Ductility of concrete is signifi cantly enhanced
due to confi nement provided by transverse steel reinforcement. Experimental simulations have also
