Reinforced concrete - The Design of Prestressed Concrete Bridges - page 54

the wall, widen to a maximum, and fade out below the top. Reinforcement may then
be added only in the areas where the cracking is wider than is considered acceptable.
3.7 The ductility of reinforced concrete
3.7.1 General
Not only would a plain concrete beam have very little strength in bending, but as
concrete fails in tension in a brittle mode, there would be no warning of impending
collapse. These problems are overcome by reinforcing the concrete beam with ductile
steel bars. When over-stressed the steel yields, and before failing will undergo a degree
of strain hardening. Consequently, if a suitably reinforced concrete beam is subjected
to overloading, as the steel yields the defl ection of the beam will increase notably, and
it will crack grossly on the tension face. Furthermore, due to the strain hardening
of the steel reinforcement, the beam could sustain a small further increase in load
before collapsing. This ductile behaviour of reinforced concrete is a very important
characteristic, not only giving visible warning of overloading, but also making it very
tolerant of simplifi ed methods of analysis, and allowing practical arrangements of
reinforcement.
3.7.2 Minimum reinforcement
For the reinforced concrete to exhibit ductile behaviour, it must contain a minimum
percentage of reinforcement. This may be understood by considering an unreinforced
concrete beam of rectangular cross section. When subjected to a bending moment,
this beam will fail in a brittle mode when the tensile stress in the concrete reaches its
limiting value. We may call this limiting bending moment M ult,c .
If the beam were to be reinforced with less than a critical percentage of steel, the
moment of resistance of the reinforced section, assuming that the concrete is cracked
in tension and all the bending strength is provided by the steel reinforcement, M ult,st ,
would be less than that of the unreinforced beam ( M ult,st < M ult,c ). The beam would
still fail at M ult,c in a brittle mode, the steel instantly yielding and rupturing. If the
amount of reinforcement is greater than this critical value, when the moment reached
M ult,c the concrete would rupture, but the moment would be held by the reinforcement.
The critical percentage of high yield reinforcement for a bending member is generally
assumed to be about 0.15 per cent of the concrete area, placed close to the extreme
fi bre in tension. If mild steel is used, the percentage is increased in proportion to the
yield strength of the bars.
A similar consideration arises for members stressed in direct tension, where the
minimum percentage of high yield reinforcement required to give ductile behaviour is
widely accepted to be 0.6 per cent of the concrete area, distributed symmetrically in
the section.
3.7.3 Redistribution of bending moments
In continuous beams made of reinforced concrete, the bending moments over the
supports are generally greater than the moments in the span. When the loads are
increased from working values towards the ULS, it is convenient to allow hinges to