# Basic Assumptions for Strength Design of Columns

At maximum capacity, the load on the longitudinal reinforcement of a concentrically loaded concrete column can be taken as the steel area Ast times steel yield strength Æ’y. The load on the concrete can be taken as the concrete area in com pression times 85% of the compressive strength of the standard test cylinder. Æ’c The 15% reduction from full strength accounts, in part, for the difference in size and, in part, for the time effect in loading of the column. Capacity of a concentrically  loaded column then is the sum of the loads on the concrete and the steel.

The ACI 318 Building Code applies a strength-reduction factor   0.75 for members with spiral reinforcement and   0.70 for other members. For small axial loads where A(P  0.10Æ’A , g  gross area of column),  may be increased u c g proportionately to as high as 0.90. Capacity of columns with eccentric load or moment may be similarly determined, but with modifications. These modifications introduce the assumptions made for strength design for flexure and axial loads.
The basic assumptions for strength design of columns can be summarized as follows.

1. Strain of steel and concrete is proportional to distance from neutral axis (Fig. 9.50c).
2. Maximum usable compression strain of concrete is 0.003 in / in (Fig. 9.50c).
3. Stress, psi, in longitudinal reinforcing bars equals steel strain s times 29,000,000 for strains below yielding, and equals the steel yield strength Æ’y, tension or compression, for larger strains (Fig. 9.50Æ’).
4. Tensile strength of concrete is negligible.
5. Capacity of the concrete in compression, which is assumed at a maximum stress of , must be consistent with test results. A rectangular stress distribution 0.85Æ’c (Fig. 9.50d) may be used. Depth of the rectangle may be taken as a  1c, where c is the distance from the neutral axis to the extreme compression surface and 1  0.85 for Æ’c  4000 psi and 0.05 less for each 1000 psi that Æ’c exceeds
4000 psi, but 1 should not be taken less than 0.65.
In addition to these general assumptions, design must be based on equilibrium
and strain compatibility conditions. No essential difference develops in maximum
capacity between tied and spiral columns, but spiral-reinforced columns show far more toughness before failure. Tied-column failures have been relatively brittle and sudden, whereas spiral-reinforced columns that have failed have deformed a great deal and carried a high percentage of maximum load to a more gradual yielding failure. The difference in behavior is reflected in the higher value of  assigned to spiral-reinforced columns.

Additional design considerations are presented in Arts. 9.83 to 9.87. Following is an example of the application of the basic assumptions for strength design of columns.  