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)

Beams with reinforcement having a definite yield point 89
In the beam at the ultimate limit state of collapse the values of is and fs
must satisfy eqn (4.2-8); they must also satisfy the stress/strain curve for
the steel. Therefore the required value of fs can be determined graphically
by solving eqn ( 4.2-8) simultaneously with the stress/strain curve, as
illustrated in Fig. 4.2-2. The ultimate moment of resistance is then
Mu = AJ,(d - kzx)
= Asfs(1 - r_/ 2 /s )d (4.2-9)
ktfcu
from eqn (4.2-6) after simplification.
In eqn (4.2-9), the quantity [1 - (!kzfslkdcu]d is the lever arm for the
ultimate resistance moment. The ratio of the lever arm to the effective
depth d is sometimes referred to as the lever arm factor. Similarly, the ratio
xld is sometimes referred to as the neutral axis factor.
4.3 Beams with reinforcement having a definite yield
point
Figure 4.3-1(a) shows the cross-section of a beam with reinforcement
bars, such as mild steel or hot-rolled high yield steel, which have a definite
yield point [y. If the steel ratio (! ( = As! bd) is below a certain value to be
defined later, it will be found that as the bending moment is increased the
steel strain fs reaches the yield value fy while the concrete strain fc is still
below the ultimate value feu (Fig. 4.3-1(b)). Such a beam is said to be
under-reinforced; in an under-reinforced beam, the steel yields before the
concrete crushes in compression. Since the concrete does not crush (and
hence the beam does not collapse) until the extreme compression fibre
strain reaches Ecu, the beam will continue to resist the increasing applied
moment; this it does by an upward movement of the neutral axis, resulting
in a somewhat increased lever arm while the total compression force in the
concrete remains unchanged. At collapse, the strain distribution is as in
Fig. 4.3-1(c); since the steel has a definite yield point, the steel stress is
equal to the yield stress. The ultimate resistance moment of an underreinforced
section is therefore given by eqn ( 4.2-9) with fs replaced by /y:
( kz /y)
Mu = As/y 1 - (! kdcu d (4.3-1)
The failure of an under-reinforced beam is characterized by large steel
strains, and hence by extensive cracking of the concrete and by substantial
deflection. The ductility of such a beam provides ample warning of
impending failure; for this reason, and for economy, designers usually aim
at under-reinforcement.
If the steel ratio (! is above a certain value, the concrete strain will reach
the ultimate value feu (and hence the beam will fail) before the steel strain
reaches the yield value fy, and the strain distribution at collapse is as
shown in Fig. 4.3-2. Such a section is said to be over-reinforced. From eqn
(4.2-6)