Structural Concrete - Hassoun

370 Chapter 11 Members in Compression and Bending
ε
ε
ε
Figure 11.9
ε ′ s ≥ ε y .
Strain diagram when compression controls. When ε s
c b
and
11.8 STRENGTH OF COLUMNS FOR COMPRESSION FAILURE
If the compressive applied force, P n , exceeds the balanced force, P b , or the eccentricity, e = M n /P n ,
is less than e b , compression failure is expected. In this case compression controls, and the strain in
the concrete will reach 0.003, whereas the strain in the steel is less than ε y (Fig. 11.9). A large part
of the column will be in compression. The neutral axis moves toward the tension steel, increasing
the compression area, and therefore the distance to the neutral axis c is greater than the balanced
c b (Fig. 11.9).
Because it is difficult to predict compression or tension failure whenever a section is given,
compression failure can be assumed when e < 2d/3, which should be checked later. The nominal
load strength, P n , can be calculated using the principles of statics. The analysis of column sections
for compression failure can be achieved using Eqs. 11.10 and Eqs. 11.11 given earlier and one of
the following solutions.
11.8.1 Trial Solution
This solution can be summarized as follows:
1. Calculate the distance to the neutral axis for a balanced section, c b :
( ) 87dt
c b =
(11.17)
87 + f y
where f y is in ksi.
2. Evaluate P n using equilibrium conditions:
P n = C c + C s − T (11.18)
3. Evaluate P n by taking moments about the tension steel, A s :
(
P n ⋅ e ′ = C c d − a )
+ C
2 s (d − d ′ ) (11.19)