Structural Concrete - Hassoun

12.6 Moment-Magnifier Design Method 429
12.6.2 Magnified Moments in Nonsway Frames
The effect of the slenderness ratio Kl u /r in a compression member of a braced frame may be ignored
if Kl u /r ≤ 34 − 12 M 1 /M 2 ≤ 40,asgiveninSection6.2.5.IfKl u /r is greater than 34 − 12 M 1 /M 2 ,
then the slenderness effect must be considered. The procedure for determining the magnification
factor δ ns in nonsway frames can be summarized as follows (ACI Code, Section 6.6.4):
1. Determine if the frame is braced against sidesway and find the unsupported length, l u ,and
the effective length factor, K (K may be assumed to be 1.0).
2. Calculate the member stiffness, EI, using the reasonably approximate equation
EI = 0.2E cI g + E s I se
1 + β dns
(12.10)
or the more simplified approximate equation
EI = 0.4E cI g
1 + β dns
(12.11)
where
EI = 0.25E c I g (forβ dns = 0.6) (12.12)
E c = 57, 000 √ f c
′
E s = 29 × 10 6 psi
I g = gross moment of inertia of the section about the axis considered, neglecting A st
I se = moment of inertia of the reinforcing steel
β dns =
maximum factored axial sustained load
maximum factored axial load
= 1.2D(sustained)
1.2D + 1.6L
Note: The above β dns is the ratio used to compute magnified moments in columns due to
sustained loads.
Equations 12.11 and 12.12 are less accurate than Eq.12.10. Moreover, Eq.12.12 is obtained
by assuming β d = 0.6 in Eq.12.11.
For improved accuracy EI can be approximated using suggested E and I values from
Eq.12.2 divided by 1 + β dns :
(
I = 0.80 + 25 A )(
st
1 − M )
u
A g P u h − 0.5P u
I
P g ≤ 0.875I g (12.12)
0
I need not be taken less than 0.35I g
where
A st = total area of longitudinal reinforcement (in. 2 )
P 0 = nominal axial strength at zero eccentricity (lb)
P u = factored axial force (+ve for compression) (lb)
M u = factored moment at section (lb.in.)
h = thickness of member (in.)
3. Determine the Euler buckling load, P c :
P c = π2 EI
(Kl u ) 2 (12.13)
Use the values of EI, K, andl u as calculated from steps 1 and 2.