Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

Controlling slenderness ratio: Since r y < r z for this double-angle shape, the effectiveslenderness
ratio for y-axis buckling will be larger than the effective-slenderness ratio for
z-axis buckling. Therefore, buckling about the y centroidal axis will control for the compression
chord member considered here.
AiSC Allowable Stress Design Formulas
The AISC ASD formulas use an effective-slenderness ratio of
KL
r
= 4.71
to differentiate between short and intermediate-length columns, on the one hand, and long
columns, on the other. For σ Y = 36 ksi, this parameter is calculated as
E
σ
E 29,000 ksi
4.71 4.71 133.7
σ = 36 ksi
=
Y
(a) Allowable axial load P allow for KL = 8 ft: For an effective length KL = 8 ft, the controlling
effective-slenderness ratio for the double-angle compression chord member is
KL
r
KL
= = (8 ft)(12 in./ft)/0.885 in. = 108.5
r
y
Since KL/r y ≤ 133.7, the column is considered an intermediate-length column, so the
critical compression stress will be calculated with the use of Equation (16.22). The elastic
buckling stress for a slenderness ratio of 108.5 is
2
π E
σ e =
⎛ KL
⎝ ⎜ ⎞
⎟
r ⎠
2
Y
2
π (29,000 ksi)
= = 24.31 ksi
2
(108.5)
From Equation (16.22), the critical compression stress is
σ
cr
Y
36 ksi
= ⎡ σ ⎤
⎣0.658 e ⎦ σ = ⎡ ⎛ ⎞
⎜ ⎟⎤
σ ⎣0.658 ⎝24.31 ksi ⎠⎦ (36 ksi) = 19.37 ksi
Y
The allowable compression stress is determined from Equation (16.25):
σ
allow
σ cr 19.37 ksi
= = = 11.60 ksi
1.67 1.67
From this allowable stress, the allowable axial load for an effective length KL = 8 ft is
2
P = σ A = (11.60 ksi)(2.38 in. ) = 27.6 kips
Ans.
allow
allow
(b) Allowable axial load P allow for KL = 12 ft: For an effective length KL = 12 ft, the controlling
effective-slenderness ratio for the double-angle compression chord member is
KL
r
KL
= = (12ft)(12in./ft)/0.885 in. = 162.7
r
The elastic buckling stress for this slenderness ratio is
y
σ
e
2
π E
=
⎛ KL
⎝ ⎜ ⎞
⎟
r ⎠
2
2
π (29,000 ksi)
= = 10.81 ksi
2
(162.7)
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