Bending and Axial Loads - Unified Theory of Concrete Structures - page 109

shown in Figure 3.9(f), a distinct kink point B reflects the yield point ( ε s = ε y ). The moment and

the curvature at this point are the yield moment, M y , and the yield curvature, φ y , respectively.

Both the curves before the yield point, OB, and after the yield point, BC, are slightly curved,
reflecting the nonlinear stress- strain curve of concrete in Figure 3.9(e).
Point C (for ultimate) and point B (for yielding) in Figure 3.14(a) are calculated from cracked
sections. Before cracking, however, the member is much stiffer and should be calculated
according to the uncracked sections. This uncracked M - relationship is shown in Figure

3.14(b) at low loads and extended up to the point A, representing the cracking of concrete.
The cracking moment and the cracking curvature at this distinctive point are designated as
M cr and

φ

φ cr , respectively. If the four points, O, A, B and C are connected by straight lines, we
have the so-called trilinear M - φ

curve.
If the uncracked region of the M -

curve is neglected and if the stress-strain curve of
the concrete is assumed to be linear up to the load stage when the steel reaches the yield
point, then the curve OB becomes a straight line as shown in Figure 3.14(c). When points
B and C are also jointed by a straight line, the M -

φ

φ curve becomes bilinear. This bilinear

M - curve captures the basic force-deformation characteristic of bending action, and is

very useful in defining the ductility of reinforced concrete members, as explained in the next
section.
The two distinctive points B (at yield) and C (at ultimate) in the bilinear M -

φ

φ curve can be

calculated as follows:

Singly Reinforced Beams
The yield moment M y and the yield curvature φ y , represented by point B in Figure 3.14(c),

can be calculated by the linear bending theory given in Sections 3.1.1 and 3.1.2. Changing ε s

to ε y in the strain diagram of Figure 3.1(b) and f s to f y in the stress diagram of Figure 3.1(c),

Equations (3.3) and (3.6) give:

A s f y d 1

k
3

M y = − (3.120)

ε y
d (1

φ y = (3.121)

− k )

The coefficient k has been derived in Equation (3,16) by solving five equations, or in
Equation (3.23) using the transformed area concept:

( n

k = ρ ) 2 + 2 n ρ − n ρ (3.122)

The nominal moment M n and the ultimate curvature φ u , represented by the point C in the

M - curve (Figure 3.14c), can be calculated according to the nonlinear bending theory of

Sections 3.2.2.5 and 3.2.2.7 for ductile beams. Equations (3.88), (3.87) and (3.100) give M n
and

φ

φ u as follows:

A s f y d

a
2

M n = − (3.123)

φ u = ε u
c

( c = a /β 1 ) (3.124)