Ultimate moment analysis is the basis for reinforced concrete design in most
countries. In the United States, the method for computing the ultimate moment
is prescribed by the American Concrete Institute (ACI) Code 318-08.
The codes of other countries are similar, di¤ering mainly in minor details.
Here we discuss a simplified description of the method of analysis based
upon ACI Code 318-08. For the sake of brevity, we limit our discussion
to beams made of ‘‘moderate’’ strength concrete and ‘‘medium-strength’’
reinforcing bars, materials that are commonly used in construction.
As shown in the stress-strain diagram in Fig. 9.5, moderate-strength
concrete typically has an ultimate compressive strength of about 4000 psi,
which corresponds to a strain of approximately 0.0012. However, experience
shows that the concrete stays intact well beyond this magnitude of strain,
and does not crush until the strain reaches 0.003.
The stress-strain diagram of the steel used in moderate-strength
reinforcement (grade 60 steel bars) is shown in Fig. 9.6. These bars are perfectly
elastic up to the yield stress of 60 ksi. When deformed beyond the
yield strain of approximately 0.002, some strain hardening takes place,
which is ignored in the analysis. In other words, the reinforcement is idealized
as an elastic, perfectly plastic material, as shown by the horizontal
dashed line in Fig. 9.6.
As mentioned previously, failure of a reinforced concrete beam occurs
when the compressive strain in the concrete reaches the crushing strain of
0.003. The type of failure depends upon the amount of reinforcement. If the