Structural Concrete - Hassoun

11.3 Load–Moment Interaction Diagram 359
a strain of 0.003, whereas the stress in the steel bars (on the tension side) is less than the yield
strength, f y (f s < f y ). In this case P n > P b and e < e b .
4. Balanced Condition (P b ). A balanced condition is reached when the compression strain in the
concrete reaches 0.003 and the strain in the tensile reinforcement reaches ε y = f y /E s simultaneously;
failure of concrete occurs at the same time as the steel yields. The moment that
accompanies this load is called the balanced moment, M b , and the relevant balanced eccentricity
is e b = M b /P b .
5. Tension Failure. This is the case of a small axial load with large eccentricity, that is, a large
moment. Before failure, tension occurs in a large portion of the section, causing the tension
steel bars to yield before actual crushing of the concrete. At failure, the strain in the tension
steel is greater than the yield strain, ε y , whereas the strain in the concrete reaches 0.003. The
range of this case extends from the balanced to the case of pure flexure (Fig. 11.3). When
tension controls, P n < P b and e > e b .
6. Pure Flexure. The section in this case is subjected to a bending moment, M n , whereas the axial
load is P n = 0. Failure occurs as in a beam subjected to bending moment only. The eccentricity
is assumed to be at infinity. Note that radial lines from the origin represent constant ratios of
M n /P n = e = eccentricity of the load P n from the plastic centroid.
ε c
0
ε
ε
ε
ε
ε
ε
ε
ε
ε
ε
e
ε
e = e b
ε
ε
0
(a)
Figure 11.3 (a) Load–moment strength interaction diagram showing ranges of cases
discussed in text and (b) column sections showing the location of P n
for different load
conditions.