Structural Concrete - Hassoun

358 Chapter 11 Members in Compression and Bending
2. Forces in steel bars:
F s1 = A s1 f y = 4 × 60 = 240 K
F s2 = A s2 f y = 2 × 60 = 120 K
3. Take moments about A–A:
(952 × 10)+(240 × 2.5)+(120 × 17.5)
x = = 9.31 in.
952 + 240 + 120
Therefore, the plastic centroid lies at 9.31 in. from axis A–A.
4. If A s1 = A s2 (symmetrical section), then x = 10 in. from axis A–A.
11.2 DESIGN ASSUMPTIONS FOR COLUMNS
The design limitations for columns, according to the ACI Code, Section 22.2.2, are as follows:
1. Strains in concrete and steel are proportional to the distance from the neutral axis.
2. Equilibrium of forces and strain compatibility must be satisfied.
3. The maximum usable compressive strain in concrete is 0.003.
4. Strength of concrete in tension can be neglected.
5. The stress in the steel is f s = εE s ≤ f y .
6. The concrete stress block may be taken as a rectangular shape with concrete stress of 0.85 f ′ c
that extends from the extreme compressive fibers a distance a = β 1 c,wherec is the distance
to the neutral axis and β 1 is 0.85 when f ′ c ≤ 4000 psi (30 MPa); β 1 decreases by 0.05 for each
1000 psi above 4000 psi (0.008 per 1 MPa above 30 MPa) but is not less than 0.65. (Refer to
Fig. 3.6, Chapter 3.)
11.3 LOAD–MOMENT INTERACTION DIAGRAM
When a normal force is applied on a short reinforced concrete column, the following cases may
arise, according to the location of the normal force with respect to the plastic centroid. Refer to
Figs 1-3a and 11.3b:
1. Axial Compression (P 0 ). This is a theoretical case assuming that a large axial load is acting
at the plastic centroid; e = 0andM n = 0. Failure of the column occurs by crushing of the
concrete and yielding of steel bars. This is represented by P 0 on the curve of Fig. 11.3a.
2. Maximum Nominal Axial Load P n,max . This is the case of a normal force acting on the section
with minimum eccentricity. According to the ACI Code, P n,max = 0.80P 0 for tied columns
and 0.85P 0 for spirally reinforced columns, as explained in Chapter 10. In this case, failure
occurs by crushing of the concrete and the yielding of steel bars.
3. Compression Failure. This is the case of a large axial load acting at a small eccentricity. The
range of this case varies from a maximum value of P n = P n max to a minimum value of P n =
P b (balanced load). Failure occurs by crushing of the concrete on the compression side with