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)

92 Reinforced concrete beams-the ultimate limit state
The ultimate moment of resistance of a balanced section may be obtained
either from eqn (4.3-1) in which(} now satisfies eqn (4.3-6), or from eqn
(4.3-4) in which x now satisfies eqn (4.3-5).
As explained above, the immediate cause of failure of all three types of
beams is the crushing of the concrete when the compressive strain reaches the
ultimate value feu· However, in an under-reinforced beam, the failure is
initiated by the large strain increase in the tension reinforcement at yield.
For this reason, the failure of an under-reinforced beam is sometimes
referred to as a primary tension failure; that of an over-reinforced beam is
referred to as a primary compression failure.
4.4 Characteristics of some proposed stress blocks
(a) Hognestad et al
In the general flexural theory in Section 4.2 and in the more restricted
theory in Section 4.3, the properties of the concrete stress block have been
expressed in terms of the characteristic ratios k 1 and k 2 . Much research has
been carried out to study the characteristics of the stress block [1, 2, 5,
7-9). In particular, the tests by Hognestad et al. [7, 8) had a considerable
influence on American and, indirectly, British design thinking. Their
results are summarized in Fig. 4.4-1, in which the concrete cube strengths
feu have been obtained from the cylinder strengths f~ using a conversion
factor of 0.8. The figure shows that the ultimate strain feu varies with the
concrete strength; however, current American and British design codes
0·0036
0 -61---.P.oc:--t--"'o.--+-----t0·0032
ecu
0·4 0·0028
20 30 40 50 60
Cube strength feu ( N /mm2 )
Fig. 4.4-l Characteristics of Hognestad et al. 's stress block