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)

88 Reinforced concrete beams-the ultimate limit state
then obtained by taking moments about a convenient horizontal axis. For
e~ample, taking moments about the level of the tension reinforcement
gives
Mu = (kdcubx)(d - kzx) + A~f~(d - d') (4.2-4)
Or, by taking moments about the centroid of the concrete stress block,
Mu = A.f.(d - kzx) + Alf~(kzx - d') (4.2-5)
The Mu values from these equations are of course the same.
In Fig. 4.2-l(a) a rectangular section is shown, but the above theory is
of general validity, being equally applicable to the arbitrary cross-section
(provided it is symmetrical about a vertical axis).
Special case: As only
In the particular case of a singly reinforced beam, i.e. a beam with no
compression reinforcement, a graphical solution may conveniently be
used. Equation (4.2-3) now becomes
or
kdcubx = Asfs
J, = kdcub X
s As
From eqn (4.2-1),
Combining eqns (4.2-6) and (4.2-7),
(4.2-6)
(4.2-7)
J, = kdcu Ecu
s (2 Ecu + Es
where (2 is the steel ratio A.lbd.
(4.2-8)
Stress/strain curve
Fig. 4.2-2 Graphical solution for J. andEs at failure