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)

158 Reinforced concrete beams-the serviceability limit states
• As •
Section
(a)
Bs
Strains
(b)
II
·~!L_·--·t
d-x
t
Fig. 5.2-1
!Acfc + Alf~ = Asfs
where Ac is the area of the concrete in compression and A~ and As are
respectively the area of the compression steel and that of the tension steel.
Using eqns (5.2-1) and (5.2-2) to express all stresses in terms of tc, we
have
which simplifies to
Ac(~) + (acA~)(x - d') = (aeAs)(d - x) (5.2-3)
Equation (5.2-3) states that the neutral axis of the cracked section passes
through the centroid of the transformed section or equivalent section,
obtained by replacing the areas A~ and As by their respective equivalent
concrete areas acA~ and aeAs (Fig. 5.2-2(b)). In the figure, the areas of
concrete displaced by the compression bars are indicated by voids;
however, in practice such voids are usually ignored* and the concrete
compression area Ac is taken as the nominal area bx, where b is the beam
width. On writing bx for Ac, e'bd for A~ and ebd for As, eqn (5.2-3)
becomes
!bx 2 + aee'bd(x - d') = acebd(d - x)
from which (and the reader should verify this) the neutral axis depth factor
xld is
j = -ac(e + e') + ~{ a~(e + e') 2 + 2ac( (! + ~ e')} (5.2-4)
• The voids can be allowed for by writing (ae - l)e' for aee' in eqns (5.2-4) and (5.2-9).