F. K. Kong MA, MSc, PhD, CEng, FICE, FIStructE, R. H. Evans CBE, DSc, D ès Sc, DTech, PhD, CEng, FICE, FIMechE, FIStructE (auth.)-Reinforced and Prestressed Concrete-Springer US (1987)

Moment redistribution-the fundamental concepts 133
4.9 Moment redistribution-the fundamental
concepts
Before discussing the ultimate load behaviour of reinforced concrete
continuous beams, we shall briefly refer to that of a continuous beam made
of an ideally elastic-plastic material, that is, a material having the
stress/strain relation in Fig. 4.9-l(a). An ideally elastic-plastic beam
section will have the moment/curvature characteristics in Fig. 4.9-l(b );
that is for a section of the beam subjected to an increasing moment M, the
curvature llr (where r is the radius of curvature) at that section increases
linearly with M until the value MP, called the plastic moment of resistance,
is reached; the curvature then increases indefinitely.
Figure 4.9-2(a) shows a two-span uniform beam made of such a
material, subjected to midspan loads Q; Figure 4.9-2(b) shows the elastic
bending moment diagram. Suppose the magnitude of Q is just large
enough for the moment at section C to reach the value Mp. Then, from
Fig. 4.9-l(b), it is seen that a further increase in the magnitude of Q, to Q'
say, will not increase the bending moment at C. A plastic hinge is said to
have developed at C, because after the moment at that section reaches Mp,
the beam behaves as though it is hinged there. Thus, under the increased
loads Q', the moment at B is
Ms = -f,.Ql + *(Q' -
Q)/
where 5Ql/32 is from Fig 4.9-2(b) and the increase of moment of
(Q' - Q)l/4 is the simple-beam bending moment corresponding to the
load increment (Q' - Q). Therefore, as Q is increased the moments at B
and D will eventually reach the value Mp (Fig. 4.9-2(c)) and the beam will
collapse in the mode in Fig. 4.9-2(d), where the beam is no longer a
structure but a mechanism; the collapse mode is often referred to as the
collapse mechanism. Let Qu be the value of Q at collapse. From Fig.
4.9-2(c),
MB
_ Qui_ Me
- 4 2
where now both M 8 and Me equal Mp; hence
fy
f
M
£
(a)
(b)
Fig. 4.9-l