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)

160 Reinforced concrete beams-the serviceability limit states
where fc; is the concrete stress at a distance X; from the neutral axis and is a
compressive stress when X; is measured into the compression zone; M is the
bending moment acting on the section and Ic is the second moment of area
of the (cracked) equivalent section, defined in eqn (5.2-9) below. At any
distance from the neutral axis, the steel stress lsi is simply ac times that in
the adjacent concrete; therefore
M
/s; = acTx;
c
Specifically, we have, from Fig. 5.2-2,
M
fc = TX
c
M
fs = ac T ( d - X)
c
t's = ac ~ (x - d')
c
(5.2-5(b))
(5.2-6)
(5.2-7)
(5.2-8)
where fc is the maximum compressive stress in the concrete, fs is the stress
in the tension reinforcement and f~ that in the compression reinforcement.
Referring again to Fig. 5.2-2(b),
lc = ~bx 3 + acA~(x - d') 2 + acAs(d - x) 2
= ~bx 3 + ac(/bd(x - d') 2 + ac(Jbd(d - x?
from which (and the reader should verify this) we have
bd3 = H~r + ace(t- ~r +ace'(~- ~r (5.2-9)
where /c is the second moment of area of the (cracked) equivalent section
and the other symbols have the same meanings as in eqn (5.2-4). Equation
(5.2-9) is plotted in Fig. 5.2-4.
Referring to Fig. 5.2-1, if r is the radius of curvature of the beam at the
section under consideration, then the curvature 1/r is immediately
obtained from the strain diagram as
1 = ~c
r x
Substituting into eqn (5.2-6) and noting that Ec = fciEc, we have
1 M
r Eclc
(5.2-10)
(5.2-11)
which is the well-known curvature expression in structural mechanics.
The following worked example has useful application in the calculation of
crack widths (see Section 5.6).
Example 5.2-1
Figure 5.2-5(a) shows the cross-section of a simply supported beam having
a 10 m span and supporting a dead load gk of 24 kN/m and an imposed load