Civil Engineering Reference
In-Depth Information
Equation (3.82) and express the compatibility equation in terms of the steel stress
f
s
and the
depth
a
. (2) Solve the new compatibility equation simultaneously with the force equilibrium
equation (3.80), to determine the unknowns
f
s
and
a
. (3) Insert the newly found unknowns,
f
s
and
a
, into the moment equilibrium equation (3.81), and calculate the moment
M
u
. (4) Calcu-
late the steel strain
ε
s
=
f
s
/
E
s
from the stress-strain relationship of steel. Knowing
ε
s
allows
us to calculate the ultimate curvature
φ
u
=
ε
s
/
(
d
−
c
) from the strain compatibility condition,
Figure 3.9(b).
Elaboration of the above procedures is not warranted, because over-reinforced beams are
seldom encountered. Over-reinforced beams are expected to fail in a brittle manner and
to exhibit small deflections. Consequently, the design of over-reinforced beams is either
prohibited or penalized by all the building and bridge codes.
3.2.2.4 Ductility of Under-reinforced Beams
In an under-reinforced beam, the steel bars are expected to yield before the crushing of
concrete. However, if the steel bars develop only a small plastic strain after yielding, the
resulting small deflection may not be adequate. To ensure the beam will develop a sufficiently
large deflection before collapse due to secondary crushing of concrete, the tensile steel bars
must be able to develop sufficiently large strains. This concept results in the ACI Code (ACI
318-08) specifying a
tensile steel strain limit of 0.005.
The ACI
tensile strain limit method
to ensure sufficient ductility of flexural beams is
illustrated in Figure 3.10. To implement this method, the Code defines a new effective depth
d
t
, defined as the distance from the extreme compression fiber to the centroid of the extreme
Figure 3.10
Strain limit methods for ductile beams
