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)

A general theory for ultimate flexural strengths 87
at distanced and d' from the top of the beam can be obtained immediately
from the geometry of Fig. 4.2-1(b ). The relationship between the strain in
a reinforcement bar and that in the adjacent concrete depends on the bond
(see Chapter 6) between the concrete and the steel, but it is accurate
enough [1, 2] to assume here that they are equal. Therefore, the strains £ 8
in the tension reinforcement and c~ in the compression reinforcement are
given by the condition of compatibility as
d-x
Es = -X-Ecu (4.2-1)
X- d'
E~ = --X-Ecu (4.2-2)
Assumption (c) refers to an idealized stress distribution for the concrete
in compression, i.e. for the concrete above the neutral axis (shaded portion
in Fig. 4.2-1(a)). The stress distribution diagram (Fig. 4.2-1(c)) is
generally referred to as the stress block. A comparison of Fig. 4.2-1( c)
with Fig. 3.2-2(a) shows that the stress block in ultimate flexural strength
analysis is the stress/strain curve drawn with a horizontal axis for stress and
a vertical axis for strain. Since the beginning of this century, a large
number of ultimate strength theories have been proposed, but essentially
they differed only in the shape assumed for the stress block [5]. Hence if
the characteristics of the stress block are expressed in general terms,
ultimate strength equations can then be derived from the principles of
mechanics. The two relevant characteristics of the stress block (further
discussed in Section 4.4) are the ratio k 1 of the average compressive stress
to the characteristic concrete strength feu, and the ratio k2 of the depth of
the centroid of the stress block to the neutral axis depth. The forces on the
beam section can be expressed in terms of these characteristics:
concrete compression = ktfcubx
concrete tension = ignored (assumption (d))
reinforcement compression = A~f~
reinforcement tension = Asfs
where the steel tensile stress fs and the steel compressive stress f~ are
related to the strains Es and c~ by the respective stress/strain curves for the
reinforcement. From the condition of equilibrium,
ktfcubx = Asfs - A~f~ (4.2-3)
In eqns (4.2-1) to (4.2-3), the neutral axis depthx is in effect the only
unknown. For an arbitrary value of x, the steel strains £ 5 and c~ are given
by eqns (4.2-1) and (4.2-2), and the corresponding stressesfs andf~ by
the stress/strain curves. However, such a set of (x, fs, f~) values will not in
general satisfy eqn ( 4.2-3). In practice, a trial and error procedure is
usually adopted: a value of x is assumed, the steel strains (and hence
stresses) are then determined. If eqn (4.2-3) is not satisfied an adjustment
is made to x by inspection, and the procedure repeated (several times)
until eqn ( 4.2-3) is sufficiently closely satisfied. The ultimate flexural
strength M u (often called the ultimate moment of resistance) of the beam is