Civil Engineering Reference
In-Depth Information
Once the force
F
is known, the maximum compressive stress in the concrete is
2
F
/
bn
, where
b
is the breadth of the beam, and the required area of reinforcing steel
A
st
=
F
/
σ
st
, where
σ
st
is the permissible working stress in the steel.
To summarise:
F
=
M
/
l
a
l
a
≈
0.83
d
e
σ
c
4
F
/
bd
e
A
st
=
F
/
≈
σ
st
For a more precise calculation, graphs and computer programs are available which
allow the position of the neutral axis, the working stress in the steel and the cross-
section area of the reinforcement to be calculated accurately.
Fortunately, most codes of practice now require reinforced concrete beams to be
designed in bending at the ULS, which is a much simpler calculation for the normal
case when the beam is under-reinforced. The steel reinforcement may be assumed to
be working at its yield stress,
σ
y
, reduced by a factor
k
st
that is defi ned by the code of
practice. The concrete compressive stress is assumed to be in the plastic range, with
a stress block typically of the shape shown in Figure 3.5
(a). The shape of this stress
block is derived directly from the factored stress/strain curve shown in Figure 3.2 (b).
The strength of concrete in compression for this purpose will be
k
c1
σ
cu
, where
σ
cu
is
the crushing strength of the concrete and
k
c1
is a reduction factor defi ned by the code
of practice (
3.4.3
). For preliminary calculations, this stress block may be simplifi ed to
a rectangular shape, working at a stress of
k
c2
σ
cu
as shown in Figure 3.5 (b), where
k
c2
is a factor further reducing the strength of the concrete to take account of this
simplifi cation.
The depth of the rectangular stress block that corresponds to an applied bending
moment must be calculated. Initially, this depth may be guessed, as may be the
arrangement of the tensile reinforcement, yielding a fi rst approximation to the lever
arm. The tensile force in the reinforcement and the compressive force in the concrete
corresponding to the guessed lever arm may then be calculated from
F
=
M
/
l
a
, where
M
is the applied bending moment at the ULS and
l
a
is the distance between the centre
of the tensile reinforcement and the centre of the guessed rectangular concrete stress
block. The depth of the compressive stress block required to provide this force, and
the number and size of the reinforcing bars may be then calculated, and a second
approximation made to their arrangement. A corrected lever arm is calculated, and a
revised
F
derived. Usually, convergence is achieved after very few repetitions of this
cycle, yielding a check on the adequacy of the width and depth of the beam, and a
good approximation to the steel area required to resist a known bending moment.
If the cross section is not rectangular, the centroid of the compressive force is
simply calculated by comparing the rectangular compressive stress block with the cross
section, Figure 3.5 (c).
