Civil Engineering Reference
In-Depth Information
for other codes of practice, substituting suitable material factors for the steel and
concrete.
The ultimate stress in the steel is assumed to be 460/
γ
m
, where
γ
m
is the material
factor (
2.5
). With
γ
m
= 1.15, the ultimate stress to be used in the calculation is 400 MPa
(
k
st
in Figure 3.5 = 0.87).
The strength of concrete in a bending member is known to be less than its strength
as described by the compression test. The British code BS5400 defi nes this reduction
as a factor of 0.67 with respect to
σ
cu
. The concrete strength must be further factored
down by the material factor
γ
m
, which covers the risk that it may not be as strong as
specifi ed, which for concrete is equal to 1.5. Thus,
k
c1
is 0.67 / 1.5 = 0.45.
When a simplifi ed rectangular stress block is considered, as in this example, the
ultimate strength of the concrete should be further reduced to 0.4
σ
cu
, or 20 MPa for
the quality of concrete considered (
k
c
2
= 0.4).
A fi rst guess at the lever arm
l
a
1
may be made by assuming that the centroid of the
reinforcement is 100 mm from the bottom fi bre of the beam and that the depth of the
compressive stress block equals the depth of the top fl ange, Figure 3.6 (b).
Hence
l
a
1
= 1 - 0.1 - 0.075 = 0.825 m.
The force required in both the steel and concrete to provide the internal couple
would be
F
=
M
/
l
a
1
= 2 / 0.825 = 2.42 MN
The area of reinforcing steel
A
st
required to produce an ultimate force of 2.42 MN
is:
A
st
= 2.42 / 400 = 6.05 × 10
-3
m
2
, or 6,050 mm
2
.
This area may be provided by approximately 7.5 bars of 32 mm. As a fi rst try adopt
8 bars, yielding 6,434 mm
2
, arranged as shown in Figure 3.6 (c). As recommended in
a former French code, it is good practice to limit the size of bars to 10 per cent of the
minimum dimension of the member containing the reinforcement, 350 mm in this
case; this is respected by adopting 32 mm bars. It should also be noted that the true
size of the deformed bars normally used for reinforcement is about 10 per cent larger
than their nominal size, due to the protrusion of the ribs. The bars are spaced vertically
at twice their nominal diameter and it has been assumed that the cover to the 12 mm
stirrups is 35 mm. The centroid of the reinforcement, adopting these rules, is 98 mm
from the beam soffi t.
The depth of the rectangular stress block in the 1.1 m wide fl ange working at
20 MPa required to produce this force is 2.42/(1.1 × 20) = 0.110 m, less than the
thickness of the slab.
The lever arm may now be recalculated as 1.0 - 0.098 - 0.110 / 2 = 0.847 m. The
force required of the internal couple becomes
F
= 2 / 0.847 = 2.36 MN, which is
close enough to the fi rst guess. Thus the choice of reinforcement consisting of 8 bars
of 32 mm is adequate as a preliminary design.
