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

ExAmpLE 16.9
40 kN
e
x
A 6061-T6 aluminum-alloy tube (outside diameter = 130 mm; wall thickness = 12.5 mm)
supports an axial load P = 40 kN, which is applied at an eccentricity e from the
centerline of the tube. The 2.25 m long tube is fixed at its base and free at its upper end.
Apply the Aluminum Association equations given in Section 16.5, and assume that the
allowable bending stress of the 6061-T6 alloy is 150 MPa. Determine the maximum
value of eccentricity e that may be used
(a) according to the allowable stress method.
(b) according to the interaction method.
y
2.25 m
Plan the Solution
Compute the section properties of the tube, and then use the Aluminum Association
equations to determine the allowable compression stress for the 2.25 m long fixed–free
column. Express both the allowable-stress and interaction methods in terms of P and e,
and solve for the allowable eccentricity e.
z
SolutioN
Section Properties
The inside diameter of the tube is d = 130 mm – 2(12.5 mm) = 105 mm.
The cross-sectional area of the tube is
π
A = [(130 mm) − (105 mm) ] = 4,614.2 mm
4
2 2 2
The moments of inertia about both the y and z centroidal axes are identical:
π
Iy
= Iz
= I = [(130 mm) − (105 mm) ] = 8,053,246 mm
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Similarly, the radii of gyration about both centroidal axes are the same:
4 4 4
4
8,053,246 mm
ry
= rz
= r = = 41.78 mm
2
4,614.2 mm
Allowable Compression Stress
From Figure 16.7, the effective-length factor for a fixed–free column is K = 2.0. Therefore,
the effective-slenderness ratio for the 2.25 m long 6061-T6 tube is
KL
r
(2.0)(2,250 mm)
= = 107.7
41.78 mm
Since this slenderness ratio is greater than 66, the allowable compression stress is determined
from Equation (16.24):
351,000 351,000
σ allow = MPa = = 30.26 MPa
2 2
( KL/ r)
(107.7)
(a)
(a) Maximum eccentricity based on the allowable-stress method: The axial and bending
stresses in the allowable-stress-method equation can be expressed as
P Pec ⎡ 1 ec⎤
σ x = + = P +
(b)
A I ⎣⎢ A I ⎦⎥
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