Reinforced concrete - The Design of Prestressed Concrete Bridges - page 38

The bending moment applied to a beam is resisted by an internal couple. In a
reinforced concrete beam, the tensile component of this couple is provided by the
reinforcement, while the concrete above the neutral axis, for a sagging moment,
provides the compressive component, Figure 3.4.
The force F to be provided by both the concrete and the steel reinforcement is given
by the equation F = M / l a , where M = the bending moment applied at the section, and
l a is the lever arm of the internal couple. The lever arm of the internal couple is the
distance between the centroid of the tension reinforcement and the centroid of the
compression forces in the concrete.
Figure 3.4 (b) shows a typical stress diagram for a beam at working load, when the
steel stress is still within the elastic range. It may be seen that there is an area just below
the neutral axis where the concrete is in tension. Lower down the section, as the tensile
stress in the concrete would exceed its limiting value the concrete cracks, and the stress
in the concrete falls to zero. The contribution of the zone of uncracked concrete in
tension to the overall bending strength is conventionally ignored in calculation.
The accurate calculation of the lever arm at working load is complicated, as the
position of the neutral axis is not fi xed but depends on the magnitude of the bending
moment. For a beam that is rectangular in cross section, a rough approximation may
be made by assuming that the depth of the neutral axis n is 0.5 d e below the top fi bre
of the beam, where d e is the depth of the beam measured to the centroid of the tensile
reinforcing steel, and is known as the 'effective depth'.
Then, as the centroid of the triangular compressive force diagram is located at n /3
below the top fi bre,

n ≈ 0.5 d e and l a ≈ d e - 0.5 d e /3 ≈ 0.83 d e .

Figure 3.4 Beam in bending at working load