Civil Engineering Reference
In-Depth Information
b
0.45
s
cu
n
Neutral
axis
d
1
0.87
s
Y
Total area of
reinforcement,
A
s
F
IGURE
12.10
Stress and
strain distributions in a
reinforced concrete beam
(a)
(b)
(c)
moment of resistance of a beam section is therefore based on the failure strength of
concrete in compression and the yield strength of the steel reinforcement in tension
modified by suitable factors of safety. Typical values are 1
.
5 for concrete (based on
its 28-day cube strength) and 1
.
15 for steel. However, failure of the concrete in com-
pression could occur suddenly in a reinforced concrete beam, whereas failure of the
steel by yielding would be gradual. It is therefore preferable that failure occurs in the
reinforcement rather than in the concrete. Thus, in design, the capacity of the concrete
is underestimated to ensure that the preferred form of failure occurs. A further factor
affecting the design stress for concrete stems from tests in which it has been found that
concrete subjected to compressive stress due to bending always fails before attaining
a compressive stress equal to the 28-day cube strength. The characteristic strength of
concrete in compression is therefore taken as two-thirds of the 28-day cube strength.
A typical design strength for concrete in compression is then
σ
cu
1
.
5
×
0
.
67
=
0
.
45
σ
cu
where
σ
cu
is the 28-day cube strength. The corresponding figure for steel is
σ
Y
1
.
15
=
0
.
87
σ
Y
In the ultimate load analysis of reinforced concrete beams it is assumed that plane
sections remain plane during bending and that there is no contribution to the bending
strength of the beam from the concrete in tension. From the first of these assumptions
we deduce that the strain varies linearly through the depth of the beam as shown
in Fig. 12.10(b). However, the stress diagram in the concrete is not linear but has
the rectangular-parabolic shape shown in Fig. 12.10(c). Design charts in Codes of
Practice are based on this stress distribution, but for direct calculation purposes a
reasonably accurate approximation can be made in which the rectangular-parabolic
stress distribution of Fig. 12.10(c) is replaced by an equivalent rectangular distribution
as shown in Fig. 12.11(b) in which the compressive stress in the concrete is assumed
to extend down to the mid-effective depth of the section at the maximum condition,
i.e. at the ultimate moment of resistance,
M
u
, of the section.
