Bending and Axial Loads - Unified Theory of Concrete Structures - page 121

0 and e d p . Inserting e At the limiting case of concentric loading, e = = = d p from

Equation (3.132) into Equation (3.151) gives

85 f c bh A s f y N n = 0 . + A s f y + (3.152)

Equation (3.152) is exactly the same as Equation (3.130), meaning that the approximation
made for the equivalent rectangular stress block is exactly correct for the limiting case of
concentric loading.
We can now summarize the method of analysis for a column section failing in com-
pression. First, assume c

h and go through the trial-and-error procedures of Equations
(3.146)-(3.149). If c is found to be less than or equal to h , the assumption is correct and we
have a solution. Second, if c is found to be greater than h , then the load N n can be calculated
directly from Equation (3.151) with the given eccentricity e .
Although the approximation of Equation (3.150) allows us to determine the load N n when

≤

c h , the inability to calculate the position of the neutral axis does not permit us to determine

the curvature

>

h , the concrete is uncracked
and the curvature is very small. The effect of such curvature would be small on the overall
deflection of the member. If the curvature is desired, however, the position of the neutral axis
can best be obtained from numerical integration of the stress-strain curve of concrete.

φ u = ε u / c in this range. Fortunately, when c >

3.3.5 Bending-Axial Load Interaction
The methods of analysis of eccentrically loaded column sections have been presented in
Sections 3.3.1-3.3.4. These analyses reveal an interesting trend. With an increase of the
eccentricity e , the behavior of a column section changes in the following manner:

e = 0 : axial load

e < e b : compression failure

e = e b : balanced condition

e > e b : tension failure

=∞

e : pure bending

This whole range of interaction from axial load to pure bending can be easily visualized by
a bending-axial load interaction curve, shown in Figure 3.23.
Figure 3.23 gives a diagram with the bending moment M as the abscissa and the load N as the
ordinate. The point on the abscissa denoted as M o represents pure bending, and the point on the
ordinate denoted as N o represents the concentric load. The point B, representing the balanced
condition, has a coordinate of M nb and N nb calculated from Equations (3.139)-(3.141) in
Section 3.3.2. In the region between the point M o and the balanced point B we have the
tension failure zone, which can be analyzed by the method in Section 3.3.3. The compression
failure zone, which is located between the balanced point B and the point N o can be analyzed
by the method in Section 3.3.4.
Three characteristics can be observed from the bending-axial load interaction curve in
Figure 3.23:

1. Any point on the interaction curve, such as point A, represents a pair of values, M n and
N n , which will cause the column section to fail. Points inside the curve are safe and points
outside the curve are unsafe.