Structural Concrete - Hassoun

392 Chapter 11 Members in Compression and Bending
determine many unknowns, such as b, h, A s ,andA ′ s, within the ACI Code limitations. It is a
common practice to assume a column section first and then determine the amount of reinforcement
needed. If the designer needs to change the steel reinforcement calculated, then the cross section
may be adjusted accordingly. The following examples illustrate the design of columns.
11.13.1 Design of Columns for Compression Failure
For compression failure, it is preferable to use A s = A ′ s for rectangular sections. The eccentricity,
e, is equal to M u /P u . Based on the magnitude of e, two cases may develop.
1. When e is relatively very small (say, e ≤ 4 in.), a minimum eccentricity case may develop that
can be treated by using Eq. 10, as explained in the examples of Chapter 10. Alternatively, the
designer may proceed as in case 2. This loading case occurs in the design of the lower-floor
columns in a multistory building, where the moment, M u , develops from one floor system and
the load, P u , develops from all floor loads above the column section.
2. The compression failure zone represents the range from the axial to the balanced load, as
shown in Figs. 11.3 and 11.11. In this case, a cross section (bh) may be assumed and then the
steel reinforcement is calculated for the given P u and M u . The steps can be summarized as
follows:
a. Assume a square or rectangular section (bh); then determine d, d ′ ,ande = M u /P u .
b. Assuming A s = A ′ s, calculate A ′ s from Eq. 11.16 using the dimensions of the assumed
section, and φ = 0.65 for tied columns. Let A s = A ′ s and then choose adequate bars. Determine
the actual areas used for A s and A ′ s. Alternatively, use the ACI charts.
c. Check that ρ g =(A s + A ′ s)∕bh is less than or equal to 8% and greater or equal to 1%. If ρ g
is small, reduce the assumed section, but increase the section if less steel is required.
d. Check the adequacy of the final section by calculating φP n from statics; as explained in
the previous examples, φP n should be greater than or equal to P u .
e. Determine the necessary ties.
A simple approximate formula for determining the initial size of the column bh or the total
steel ratio ρ g is
P n = K c bh 2 or P u = φP n = φK c bh 2 (11.27)
where K c has the values shown in Table 11.2 and plotted in Fig. 11.19 for f y = 60 ksi and A s = A ′ s.
Units for K c are in lb/in. 3
The values of K c shown in Table 11.2 are approximate and easy to use because K c increases by
0.02 for each increase of 1 ksi in f ′ c. For the same section, as the eccentricity, e = M u /P u , increases,
P n decreases, and, consequently, K c decreases. Thus, K c values represent a load P n on the interaction
diagram between 0.8 P n0
and P b as shown in Fig. 11.3 or 11.11.
Table 11.2
Values of K c
(f y
= 60 ksi)
ρ g
(%) f ′ c = 4ksi f′ c = 5ksi f′ c = 6ksi
1 0.090 0.110 0.130
4 0.137 0.157 0.177
8 0.200 0.220 0.240
K c