Structural Concrete - Hassoun

580 Chapter 16 Continuous Beams and Frames
Figure 16.22
Yielding behavior of an overreinforced concrete beam.
Figure 16.23
Idealized moment–curvature behavior of reinforced concrete beams.
The significant part of the moment–curvature curve in Fig. 16.21 is that between B and C,
in which M u remains substantially, constant for a wide range of values of φ. In limit analysis, the
moment–curvature curve can be assumed to be of the idealized form shown in Fig. 16.23, where
the curvature, φ, between B and C is assumed to be constant, forming a plastic hinge. Because
concrete is a brittle material, there is usually considered to be a limit at which the member fails
completely at maximum curvature at C.
Cranston [8] reported that in normally designed reinforced concrete frames, ample rotation
capacity is available, and the maximum curvature at point C will not be reached until the failure or
collapse of the frame. Therefore, when the member carries a moment equal to its factored moment,
M u , the curvature continues to increase between B and C without a change in the moment, producing
a plastic hinge. The increase in curvature allows other parts of the statically indeterminate structure
to carry additional loading.
16.8 THE COLLAPSE MECHANISM
In limit design, the moment strength of a reinforced concrete member is reached when it is on the
verge of collapse. The member collapses when there are sufficient numbers of plastic hinges to