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)

86 Reinforced concrete beams-the ultimate limit state
4.2 A general theory for ultimate flexural strengths
As a result of the extensive research work in the past several decades [ 1, 5],
the ultimate load behaviour of reinforced concrete beams is now quite well
understood. Current design methods in American and British codes are
based on the general theory described below. The following assumptions
are made:
(a) The strains in the concrete and the reinforcing steel are directly
proportional to the distances from the neutral axis, at which the strain
is zero.
(b) The ultimate limit state of collapse is reached when the concrete
strain at the extreme compression fibre reaches a specified value feu.
(c) At failure, the distribution of concrete compressive stresses is defined
by an idealized stress/strain curve.
(d) The tensile strength of the concrete is ignored.
(e) The stresses in the reinforcement are derived from the appropriate
stress/strain curve.
Figure 4.2-l(a) shows a beam cross-section having an area As of
longitudinal tension reinforcement and an area A~ of longitudinal compression
reinforcement; the distance d from the top face to the centroid of the
tension reinforcement is called the effective depth of the beam. Figure
4.2-l(b), in which x denotes the neutral axis depth, shows the strains
distributed in accordance with assumption (a). The assumption of linear
strain distribution, which implies that plane sections remain plane, is not
exactly correct but is justifiable for practical purposes. A critical review of
the research on this subject is given in Reference 6.
From assumption (b), the maximum concrete compressive strain has a
specified value feu at the instant of collapse. Therefore, the concrete strains
Cross-section Strains Stresses and forces
(a) (b) (c)
Fig. 4.2-1 Strain and stress distributions at failure