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)

166 Reinforced concrete beams-the serviceability limit states
r-b
Fig. 5.2-8
(a) (b) (c) (d)
The lever arm of this force about the neutral axis is ~(h
- x); therefore
b(h - x) 3
moment due to concrete tension =! (d _ x) let (5.2-19)
It follows that when a moment Misapplied to the partially cracked section,
part of it ( eqn 5.2-19) is resisted by the concrete tension; the net moment
to be resisted by the concrete compression and by the forces in the
reinforcement is
1b(h-x) 3
Mnet = M - 1 (d _ x) fct (5.2-20)
Whence, in Fig. 5.2-S(c), the stresses are
fc = ~netX
c
(5.2-21)
(5.2-22)
where ae is the modular ratio £ 5 / Ec, and the second moment of area Ic is to
be calculated from eqn (5.2-9).
Similarly, the curvature produced by the moment Mnct is
(5.2-23)
where Mnct is obtained from eqn (5.2-20) and Ic is from eqn (5.2-9).
The following worked example has useful application in the calculation of
deflections (see Section 5.5).
Example 5.2-2
Figure 5.2-9 shows the cross-section of a beam, acted on by a load, part of
which is permanent. The bending moment M 1 due to the total load
is 48 kNm and the bending moment Mp due to the permanent load is