Building Design and Construction Handbook - Merritt - Ventech!

5.100 SECTION FIVE
FIGURE 5.80 Strain distribution is shown in (a) and stress distribution in
(b) for a cross section of a beam as it is loaded beyond the yield point, for
the idealized stress-strain relationship in Fig. 5.79b: stage (1) shows the condition
at the yield point of the outer surface; (2) after yielding starts; (3) at
ultimate load.
A circular section has an M p/M y ratio of about 1.7, while a diamond section has
a ratio of 2. The average wide-flange rolled-steel beam has a ratio of about 1.14.
Plastic Hinges. The relationship between moment and curvature in a beam can
be assumed to be similar to the stress-strain relationship in Fig. 5.80b. Curvature
� varies linearly with moment until M y � M p is reached, after which � increases
indefinitely at constant moment. That is, a plastic hinge forms.
Moment Redistribution. This ability of a ductile beam to form plastic hinges
enables a fixed-end or continuous beam to carry more load after M P occurs at a
section, because a redistribution of moments takes place. Consider, for example, a
uniformly loaded, fixed-end, prismatic beam. In the elastic range, the end moments
of M L � M R � WL/12, while the midspan moment M C is WL/24. The load when
the yield point is reached at the outer surfaces at the beam ends is W y � 12M y/L.
Under this load the moment capacity of the ends of the beam is nearly exhausted;
plastic hinges form there when the moment equals M P. As load is increased, the
ends then rotate under constant moment and the beam deflects like a simply sup-