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)

138 Reinforced concrete beams-the ultimate limit state
apply where the moment redistribution leads to a reduction in the bending
moment at a given section; where it leads to an increase in the bending
moment, no restriction is necessary.
In the design of continuous beams it is easy to overlook one important
point. Consider the fixed-end beam in Fig. 4.9-5(a). The elastic bending
moment diagram corresponding to the ultimate loading is shown by the
chain-dotted line in Fig. 4.9-5(b). The full line shows the redistributed
moment diagram for use in the design of the individual cross-sections for
the ultimate limit state. The dotted line is the elastic moment diagram
corresponding to the service loading (see Table 1.5-1 for the Yr factors for
service loads and for ultimate loads). Figure 4.9-5(b) shows that the
region ab, though under a sagging moment at the ultimate condition, is
under a hogging moment at the service condition. The ultimate load
condition requires no top reinforcement in the region ab, and consequently
wide cracks would develop there at the service condition. To guard against
such cracking, BS 8110 imposes the condition that
Mu <1: 0.7Me (4.9-4)
where Mu is the ultimate resistance moment provided at any section of the
member, and Me is the moment at that section obtained from an elastic
maximum-moments diagram covering all appropriate combinations of
ultimate loads.
Equation (4.9-4) is compatible with the above-mentioned 30% limit on
the moment reduction that is permitted in moment redistribution.
Returning to Section 4. 7, it is now clear why the design formulae and
procedure there are stated as valid for only up to 30% moment redistribution.
Note that the design formulae in Section 4.6 imply that xld may
reach 0.5. For xld to reach 0.5, eqn (4.9-3) states that Pb ~ 0.9; in other
words, there must be no more than 10% moment redistribution (see eqn
4.7-2). Hence the design formulae and procedure in Section 4.6 are valid
only for up to 10% moment redistribution.
Example 4.9-1
The span lengths of a three-span continuous beam ABCD are: exterior
spans AB and CD, 8 m each; interior span BC, 10m. The characteristic
Elastic (ultimate load)
(a)
(b)
Fig. 4.9-5