Bending and Axial Loads - Unified Theory of Concrete Structures - page 89

Figure 3.9 Singly reinforced rectangular sections at ultimate

The equilibrium condition can be derived from the distribution of stresses along the depth
of the cross-section, as given in Figure 3.9(c). Above the neutral axis is a compression stress
block sketched by a curve. The stresses in the compression stress block are related to the
strain in Figure 3.9(b) through the stress-strain relationship of the concrete. The stress-strain
relationship of concrete is assumed to be identical to the stress-strain curve obtained from the
compression test of a standard concrete cylinder, shown in Figure 3.9(e).
The compression stress block can be replaced statically by a resultant C in the equilibrium
equations. This resultant has a magnitude of C

k 1 f c bc , where k 1 is a coefficient representing
the ratio of the average stress of the compression stress block to the maximum stress of concrete
f c . The resultant C also has a location defined by a distance k 2 c , where k 2 is coefficient
representing the ratio of the depth of the resultant C from the extreme fiber to the depth of the
compression zone. These two coefficients k 1 and k 2 will be determined in Section 3.2.1.2.
Below the neutral axis in Figure 3.9(c), the small tensile stress of concrete adjacent to the
neutral axis is neglected. All the tensile stresses in the rebars are assumed to be concentrated
at the centroid of the reinforcing bars with a total area of A s . The stress in the rebar f s is
assumed to be related to the strain

=

ε s in Figure 3.9(b) through the stress-strain curve obtained
from the tension tests of bare reinforcing bars, Figure 3.9(f). For mild steel reinforcing bars,
the stress-strain curve is first linear, with a slope of E s up to the yielding strength f y , then
followed by a yield plateau, represented by a long horizontal straight line up to failure. Before
steel yielding, the total tensile force is denoted as T

= A s f s . After steel yielding, the total

tensile force is denoted as T = A s f y .