Structural Concrete - Hassoun

16.12 Rotation of Plastic Hinges 587
Figure 16.27 (a) Plastic rotation from moment–curvature and moment gradient and (b)
development of plastic hinges in a reinforced concrete continuous beam.
occurs on both sides of the maximum moment section over a finite length. This length is called the
plastic hinge length, l p . The hinge length, l p , is a function of the effective depth d, and the distance
from the section of highest moment to the point of contraflexure (zero moment).
Referring to Fig. 16.27a, the length L p /2 represents the plastic hinge length on one side of the
center of support; M u and φ u indicate the factored moment and ultimate curvature at the critical
section, whereas M y and φ y indicate the moment and curvature at first yield. The plastic curvature
at the critical section φ p is equal to φ u − φ y and the rotation capacity is equal to φ p l p .
The estimated length of the plastic hinge was reported by many investigators. Baker [7]
assumed that the length of the plastic hinge is approximately equal to the effective depth d. Corley
[12] proposed the following expression for the equivalent length of the plastic hinge:
l p = 0.5d + 0.2 √ ( ) z
d
(16.11)
d
where z is the distance of the critical section to the point of contraflexure and d is the effective depth
of the section. Mattock [13] suggested a simpler form:
l p = 0.5d + 0.05z (16.12)
Tests [14] on reinforced concrete continuous beams showed that l p can be assumed equal to 1.06d.
They also showed that the length of the plastic hinge, in reinforced concrete continuous beams
containing hooked-end steel fibers, increases with the increase in the amount of the steel fibers and
the main reinforcing steel according to the following expression:
l p =(1.06 + 0.13ρρ s )d (16.13)