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)

Design procedure for rectangular beams 111
Table 4.6-1 Lever-arm and neutral axis depth factors [14]
K= Mibd 2 f'" 0.05 0.06 O.o7 0.08 0.09 O.HXI 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 11.35 0.39 0.40 0.43 0.45 0.47 0.50
Comments
(a) The formula As= M/(0.87/yz) follows from Fig. 4.6-1(b ), where it is
seen that, by taking moments about the centroid of the concrete
stress block, M = (0.87/yAs)z.
(b) Table 4.6-1 is extracted from the I.Struct. E. Manual [14]. The leverarm
factors zld in the table are given by the BS 8110 formula:
~ = 0.5 + ~(0.25- 0~9)
where K = M!Ucubd 2 ). For the derivation of this formula, see
Example 4.6-10.
(c) Note that the zld and x/d values given in Table 4.6-1 forK= 0.156
are in fact equally valid forK> 0.156 (though this is not pointed out
Step3
in the I.Struct.E. Manual [14]). That is, forK 2:: 0.156, zld = 0.775
and xl d = 0.50. Study Example 4.6-11 for a full explanation.
If the design moment M > M u of Step 1:
(3a) Compression reinforcement is required and its area A~ is given by
A' M- Mu
s = 0.87/y(d - d')
(4.6-13)
where d' is the depth of the compression steel from the concrete
compression face (Fig. 4.6-2(a)).
(3b) If
d' > (1 - l:t__) use 700(1 - d')
X 800 ' X
in lieu of 0.87/y in the formula for A~.
(3c) The area of tension reinforcement As is calculated from
Comments
A Mu A'
s = 0.87/yz + s
where z = 0.775d (see Comment (c) below).
(4.6-14)
(a) The formula for A~ in Step (3a) was derived earlier as eqn (4.6-6).
(b) The formulae in Step (3b) were derived earlier in Example 4.6-2.
Example 4.6-2 shows that, where d' /d > (1 - fyl800), then the
compression reinforcement A~ does not reach the design stress 0.87/y
so that, in using the formula in Step (3a), a reduced stress f~ =
700(1 - d' lx) should be used.