Structural Concrete - Hassoun

430 Chapter 12 Slender Columns
4. Calculate the value of the factor C m to be used in the equation of the moment-magnifier factor.
For braced members without transverse loads,
C m = 0.6 + 0.4M 1
(12.14)
M 2
where M 1 /M 2 is positive if the column is bent in single curvature and negative if the member
is bent in double curvature. For members in which M 2, min = P u (0.6 + 0.03h) exceeds M 2 ,the
value of C m in Eq.12.14 shall either be taken equal to 1, or shall be based on the ratio of
computed end moments, M 1 /M 2 .
5. Calculate the moment magnifier factor δ ns :
C
δ ns =
m
≥ 1.0 (12.15)
1 −(P u ∕0.75P c )
where P u is the applied factored load and P c and C m are as calculated previously.
6. Design the compression member using the axial factored load, P u , from the conventional
frame analysis and a magnified moment, M c , computed as follows:
M c = δ ns M 2 (12.16)
where M 2 is the larger factored end moment due to loads that result in no sidesway and should
be ≥ M 2,min = P u (0.6 + 0.03h). For frames braced against sidesway, the sway factor is δ s = 0.
In nonsway frames, the lateral deflection is expected to be less than or equal to H/1500, where
H is the total height of the frame.
12.6.3 Magnified Moments in Sway Frames
The effect of slenderness may be ignored in sway (unbraced) frames when Kl u /r < 22. The procedure
for determining the magnification factor, δ s , in sway (unbraced) frames may be summarized
as follows (ACI Code, Section 6.6.4.6):
1. Determine if the frame is unbraced against sidesway and find the unsupported length l u and
K, which can be obtained from the alignment charts (Fig. 12.3).
2–4. Calculate EI, P c ,andC m as given by Eqs. 12.2 and Eqs. 12.10 through 12.14. Note that the
term β ds is used instead of β dns to calculate I and is defined as the ratio of maximum factored
sustained shear within a story to the total factored shear in that story.
5. Calculate the moment-magnifier factor, δ s using one of the following methods:
a. Magnifier method:
1
δ s =
1 − ( ∑
Pu ∕0.75 ∑ ) ≥ 1.0 (12.17)
P c
where δ s ≤2.5 and ∑ P u is the summation for all the factored vertical loads in a story and
∑
Pc is the summation for all sway-resisting columns in a story. Also,
δ s M s =
M s
1 − ( ∑
Pu ∕0.75 ∑ P c
) ≥ M s (12.18)
where M s is the factored end moment due to loads causing appreciable sway.
b. Approximate second-order analysis:
δ s = 1
1 − Q ≥ 1 or δ sM s = M s
1 − Q ≥ M s (12.19)