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 139
dead load Gk (inclusive of self-weight) is 36 kN/m and the characteristic
imposed load Qk is 45 kN/m. Draw the bending moment envelope (or the
maximum-moments diagram) for the ultimate condition. It is desired to
take advantage of moment redistributions to equalize as far as possible the
bending moments at the various critical sections. (See comments at the end
on the loading arrangements.)
SOLUTION
Stepl
With reference to Table 1.5-1 and to the description of its use in Section
1.5, the design loads are
Step2
1.4Gk + 1.6Qk = (1.4)(36) + (1.6)(45) = 122.4 kN/m
l.OGk = (1.0)(36) = 36 kN/m
There are three loading cases to consider (Fig. 4.9-6): Case 1 for
maximum hogging moment (designated negative here) at B, Case 2 for
maximum sagging moment at span BC and Case 3 for maximum sagging
moment at spans AB and CD. (Strictly, there should be a Case 4, which
is a mirror reflection of Case 1, but the reader should study the solution
here and satisfy himself that that case is in fact covered.) See also the
comments at the end of the solution.
Step3
Consider Case 1. The reader should verify that the elastic bending
moment diagram is that of the chain-dotted line in Fig. 4.9-6. (Hint:
The support moments, -1097 and -667 kNm, may be determined by
one of the methods in Reference 12, Chapters 4 and 6. The moment
506 kNm is obtained by considering AB as a simple beam supporting a
uniformly distributed load of 1.4Gk + 1.6Qk = 122.4 kN/m and acted on
by a hogging moment of 1097 kNm at B. Similarly for the moment
648 kNm.)
Step4
Consider moment redistribution for Case 1. Equation (4.9-4) permits
the hogging moment at B to be reduced, numerically, to 70% of the
elastic value, provided of course that xl d of the section as finally
designed does not exceed 0.3 (see eqn 4.9-3):
70% of ( -1097 kNm) = -768 kNm
The redistributed moment M 8 is -768 kNm; adjust Me to the same
value of -768 kNm. The reader should now verify that the redistributed
moment diagram is the full line. (Hint: The moments 632 kNm in AB
and 762 kNm in BC may be obtained using the hint in Step 3.)
StepS
Consider, for Case 1, the condition Mu <t 0.7Me imposed by eqn
(4.9-4). Strictly speaking, it is necessary to draw the complete moment
diagram with ordinates equal to 70% of the chain-dotted curve. However,
by inspection, only the portions represented by the dotted lines