Structural Concrete - Hassoun

428 Chapter 12 Slender Columns
4. If the factored column moments are zero or e = M u /P u < e min , the value of M 2 shall not be
taken less than M 2, min calculated using the minimum eccentricity given by ACI Code, Section
6.6.4.5.4:
e min = 0.6 + 0.03h (inch) (12.6)
M 2,min = P u (0.6 + 0.03h) (12.7)
where M 2, min is the minimum moment and e min is the minimum eccentricity. The moment M 2
shall be considered about each axis of the column separately. The value of K may be assumed
to be equal to 1.0 for a braced frame unless it is calculated on the basis of EI analysis.
5. It shall be permitted to consider compression members braced against sidesway when bracing
elements have a total stiffness, resisting lateral movement of that story, of at least 12 times
the gross stiffness of the columns within the story.
12.5.2 Sway Frames
In compression members not braced (sway) against sidesway, the effect of the slenderness ratio
may be neglected when
Kl u
≤ 22 (ACI Code, Section 6.2.5) (12.8)
r
12.6 MOMENT-MAGNIFIER DESIGN METHOD
12.6.1 Introduction
The first step in determining the design moments in a long column is to determine whether the frame
is braced or unbraced against sidesway. If lateral bracing elements, such as shear walls and shear
trusses, are provided or the columns have substantial lateral stiffness, then the lateral deflections
produced are relatively small and their effect on the column strength is substantially low. It can be
assumed (ACI Code, Section 6.6.4.4.1) that a story within a structure is nonsway if
∑
Pu Δ 0
Q = ≤ 0.05 (12.9)
V us l c
where ∑ P u and V us are the story total factored vertical load and horizontal story shear in the story
being evaluated, respectively, and Δ 0 is the first-order relative lateral deflection between the top
and bottom of the story due to V us . The length l c is that of the compression member in a frame,
measured from center to center of the joints in the frame.
In general, compression members may be subjected to lateral deflections that cause secondary
moments. If the secondary moment, M ′ , is added to the applied moment on the column, M a ,the
final moment is M = M a + M ′ . An approximate method for estimating the final moment M is to
multiply the applied moment M a by a factor called the magnifying moment factor δ, which must be
equal to or greater than 1.0, or M max = δM a and δ ≥ 1.0. The moment M a is obtained from the elastic
structural analysis using factored loads, and it is the maximum moment that acts on the column at
either end or within the column if transverse loadings are present.
If the P-Δ effect is taken into consideration, it becomes necessary to use a second-order analysis
to account for the nonlinear relationship between the load, lateral displacement, and the moment.
This is normally performed using computer programs. The ACI Code permits the use of first-order
analysis of columns. The ACI Code moment-magnifier design method is a simplified approach for
calculating the moment-magnifier factor in both braced and unbraced frames.