Structural Concrete - Hassoun

16.7 Introduction to Limit Design 579
strength of the structure can be increased as more sections reach their strength capacity. Although
the yielding of the reinforcement introduces large deflections, which should be avoided under service
loads, a statically indeterminate structure does not collapse when the reinforcement of the first
section yields. Furthermore, a large reserve of strength is present between the initial yielding and
the collapse of the structure.
In steel design, the term plastic design is used to indicate the change in the distribution of
moments in the structure as the steel fibers, at a critical section, are stressed to their yield strength.
The development of stresses along the depth of a steel section under increasing load is shown in
Fig. 16.20. Limit analysis of reinforced concrete developed as a result of earlier research on steel
structures and was based mainly on the investigations of Prager and Hodge [4], Beedle et al. [5],
and Baker et al. [6]. Baker [7] worked on the principles of limit design, whereas Cranston [8] tested
portal frames to investigate the rotation capacity of reinforced concrete plastic hinges. However,
more research work is needed before limit design can be adopted by the ACI Code.
16.7.3 Plastic Hinge Concept
The curvature φ of a member increases with the applied bending moment M. For an underreinforced
concrete beam, the typical moment–curvature and the load–deflection curves are shown
in Fig. 16.21. A balanced or an overreinforced concrete beam is not permitted by the ACI Code
because it fails by the crushing of concrete and shows a small curvature range at factored moment
(Fig. 16.22).
Figure 16.20
Distribution of yield stresses in a yielding steel rectangular section.
Figure 16.21
Yielding behavior of an underreinforced concrete beam.