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

Both tensile and compressive normal stresses will be produced by bending; however, the
compressive normal stress is the focus of our interest here.
AiSC Allowable-Stress Design Formulas
The AISC ASD formulas differentiate between short to intermediate-length columns, on
the one hand, and long columns, on the other, according to an effective-slenderness ratio
given by
E 29,000 ksi
4.71 4.71 133.681
σ = 36 ksi
=
Y
From Figure 16.7, the appropriate effective-length factor for a column fixed at its base
and free at its upper end is K y = K z = 2.0. The effective-slenderness ratios for buckling
about the strong and weak axes, respectively, of the W12 × 58 are therefore
KL z
r
z
(2.0)(11 ft)(12 in./ft)
KL y (2.0)(11 ft)(12 in./ft)
= = 50.0 and = = 105.2
5.28 in.
r 2.51 in.
y
The controlling effective-slenderness ratio for the column is 105.2. Since
KL/ ry
≤ 4.71 E/
σ Y, the column is considered to be an intermediate-length column and
the critical compression stress will be calculated from Equation (16.22). In this equation,
the elastic buckling stress σ e for the controlling effective-slenderness ratio is computed
from Equation (16.23):
σ
e
2
π E
=
⎛ KL
⎝ ⎜ ⎞
⎟
r ⎠
2
2
π (29,000 ksi)
= = 25.86 ksi
2
(105.2)
From Equation (16.22),
σ
cr
⎡ ⎛σ
⎞
Y ⎤ ⎡ ⎛
⎜ ⎟
36 ksi ⎞
⎜ ⎟⎤
= ⎢ ⎝σ
e
0.658
⎠⎥σ
= ⎢0.658 ⎝25.86 ksi ⎠
Y
⎥(36 ksi) = 20.10 ksi
⎢ ⎥
⎣ ⎦ ⎣⎢
⎦⎥
Finally, the allowable compression stress is determined from Equation (16.25):
σ
allow
σ cr 20.10 ksi
= = = 12.04 ksi
1.67 1.67
(a) Is the column safe for P = 25 kips, according to the allowable stress method? The
allowable stress method simply requires that the sum of the compressive axial stress and
the compressive bending stress be less than the allowable compression stress prescribed
by the pertinent AISC ASD column formula for centric loading. The sum of the compressive
axial stress and the compressive bending stress is
σ x = 1.47 ksi + 16.36 ksi = 17.83 ksi (C)
(d)
Since σ x is greater than the 12.04 ksi allowable compression stress, the column is not safe
for P = 25 kips, according to the allowable stress method. The ratio between the allowable
and actual stresses has the value
σ
σ
x
allow
17.83 ksi
= = 1.48 > 1 N.G.
12.04 ksi
(c)
Ans.
710