Civil Engineering Reference
In-Depth Information
in the compression zone, and the tensile reinforcements, however, are still in elastic range
until failure; therefore, the ultimate deflection is negligible in contrast to under-reinforced
beams.
Insuciently reinforced beams, though forbidden in accordance with the code, actually
existed due to artificial errors in construction. For instance, misplacement of the reinforce-
ments at the ends of cantilever beams will produce insuciently reinforced beams and then
give rise to abrupt failure.
The load-deflection curves for over-reinforced beams, under-reinforced beams and insu-
ciently-reinforced beams are respectively plotted in Fig. 3.1. It illustrates that ductility of
the member decreases with the increasing of reinforcement ratio
ρ
. Furthermore, turning
points are not visible before failure for over-reinforced beams and insuciently reinforced
beams characterized as brittle failure patterns, distinct from under-reinforced beams. In
addition, it can be seen from Fig. 3.1 that sectional resisting moment of
L
3
−
15
is larger
than that of
L
3
−
14
owing to higher strength of concrete.
Fig. 3.1
Diagram of load and deflection.
First of all, the original beam should be distinguished among different beam types before
retrofitting. Reinforcement ratio
ρ
of the under-reinforced beam lies between the maximum
ratio
ρ
max
and the minimum ratio
ρ
min
, while that of the insuciently reinforced beam is
smaller than
ρ
min
and the over-reinforced beam is greater than
ρ
max
.
According to codes, the maximum reinforcement ratio of tensile reinforcement can be
expressed as:
f
cm
f
y
ρ
max
=
ξ
b
(3
.
1)
where
ξ
b
is the coecient of characteristic compressive height in critical failure, which equals
to 0.61 for grade I steel, 0.55 for grade II steel;
f
cm
is the factored concrete flexural compres-
sive strength equals to 1
.
1
f
c
;
f
y
is the factored tensile strength of the tensile reinforcement.
In accordance with provisions of the code, the minimum ratio of longitudinal reinforcement
ρ
min
=0
.
15% when concrete grade is under C35.
The insuciently reinforced beam must be retrofitted using the adding reinforcement
method discussed in this section.
With regard to under-reinforced beam, the necessity for retrofit may be judged from crack
width, reinforcement stress and deflection of structural members. In general, crack width
is a linear function of reinforcement stress, which indicates that the wider the crack, the
higher the reinforcement stress. The calculation formula of reinforcement stress in service
