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)

124 Reinforced concrete beams-the ultimate limit state
Table 4.7-2 Lever-arm and neutral axis depth factors [14]
K= Mlbd2fcu 0.05 0.06 O.o? 0.08 0.09 0.100 0.104 0.110 0.119 0.130 0.132 0.140 0.144 0.150 0.156
zld 0.94 0.93 0.91 0.90 0.89 0.87 0.87 0.86 0.84 0.82 0.82 0.81 0.80 0.79 0.775
xld 0.13 0.16 0.19 0.22 0.25 0.29 0.30 0.32 0.35 0.39 0.40 0.43 0.45 0.47 0.50
30% 25% 20% 15% 0-10%
that, by taking moments about the centroid of the concrete block:
M = (0.87/yAs)z.
(b) Table 4.7-2 is extracted from the I.Struct.E. Manual [14]. The leverarm
factors zld are derived in Example 4.7-2.
(c)
In using Table 4.7-2, pay attention to the 'Limit of the table for
various % of moment redistribution' as stated at the bottom of the
table. Thus for 0-10% redistribution and K > 0.156, zl d and xl d are
still to be taken as 0.775 and 0.50 respectively (as though K = 0.156).
Similarly, for 15% redistribution and K > 0.144, zld and x/d are still
to be taken as 0.80 and 0.45 respectively. (Study Example 4.7-2
again, if necessary; note in particular the use of K and K' in the
formulae for zld.)
Step3
If the design moment M > Mu of Step 1:
(3a)
Compression reinforcement is required and its area A~ is given by
(see Comment (a) below)
A' M- Mu
s = 0.87/y(d - d')
where d' is the depth of the compression steel from the concrete
compression face.
(3b) If
(3c)
~ > ( 1 - ~), use f~ = 700 ( 1 - ~·)
(4.7-10)
in lieu of 0.87/y in the above formula for A~. See Comment (b)
below.
The area of the tension reinforcement As is given by (see
Comment (c) below)
A Mu A'
s = 0.87/yz + s
where z is taken from Table 4.7-2.
(4.7-11)
Comments
(a) The formula for A~ in Step (3a) was derived in Section 4.6 as eqn
( 4.6-6). Of course, Mu is now calculated from K' in Table 4. 7-1. See
also eqn (4.7-5).