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 91
Combining with eqn (4.3-2).,
or
ktfeu (x)2 (X) _ O
eEs d + feu d - feu - (4.3-3)
which may be solved for the neutral axis factor x/d (and hence for x) since
all the other quantities are known. The ultimate moment of resistance of
the over-reinforced beam may then be obtained by taking moments about
the tension reinforcement:
(4.3-4)
The failure of an over-reinforced beam is initiated by the crushing of the
concrete, while the steel strain is still relatively low. The failure is therefore
characterized by a small deflection and by the absence of extensive cracking
in the tension zone. The failure, often explosive, occurs with little warning.
A section is said to be balanced if the concrete strain reaches feu
simultaneously as the steel strain reaches fy; that is, if the strain
distribution at collapse is as shown in Fig. 4.3-3. The neutral axis depth
factor x/d of a balanced section has a unique value, which is in fact given by
eqn ( 4.2-7) with fs replaced by ey:
!d = f~ (for balanced section) (4.3-5)
feu ey
The steel ratio also has a unique value, given by eqn (4.2-8) with fs
replaced by /y and fs by ey:
e = k/eu feu (for balanced section) (4.3-6)
/y feu + ey
d-x
l_'---'
~
Fig. 4.3-3