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

where c is the outside radius of the tube (c = 130 mm/2 = 65 mm). Now, set Equation (b)
equal to the allowable compression stress determined in Equation (a), and solve for the
maximum eccentricity e:
⎡ 1 (65 mm) e ⎤
30.26 MPa = (40,000 N) ⎢
+
2 4⎥
⎣4,614.2 mm 8,053,246 mm ⎦
30.26 N/mm
40,000 N
2
1
65 mm
− = ⎡ e
2 4
4,614.2 mm ⎣ ⎢ ⎤
⎥
8,053,246 mm ⎦
∴ e = 66.9 mm
Ans.
max
(b) Maximum eccentricity based on the interaction method: To determine the maximum
eccentricity e, the interaction equation for axial and bending stresses is expressed as
P Pec ⎡ 1 ec ⎤
+ = P⎢
+ ⎥ = 1
A( σ ) I( σ ) ⎣ A ( σ ) I( σ ) ⎦
allow a allow b allow a allow b
(c)
From Equation (a), the allowable compression stress was found to be 30.26 MPa therefore,
(σ allow ) a = 30.26 MPa. The allowable bending stress was specified as (σ allow ) b = 150 MPa.
The maximum allowable eccentricity e max based on the interaction method can be
computed with these values, along with P = 40 kN:
⎡ 1 ec ⎤
P⎢
+ ⎥ = 1
⎣ A ( σ ) I( σ ) ⎦
⎡
1
(65 mm) e ⎤
(40,000 N) ⎢
+
⎥ = 1
2 2 4 2
⎣(4,614.2 mm )(30.26 N/mm ) (8,053,246 mm )(150 N/mm ) ⎦
allow
a
allow
b
⎡ ( 65 mm)
⎤ 1
1
⎢
e
4 2 ⎥ = −
2 2
⎣(8,053,246 mm )(150 N/mm ) ⎦ 40,000 N (4,614.2 mm )(30.26 N/mm )
∴ e = 332 mm
Ans.
max
Since the effective-slenderness ratio of the tube is relatively large, the allowable compression
stress computed in Equation (a) is relatively small. Because the allowablestress
method depends entirely on this allowable stress, the 66.9 mm maximum eccentricity
that it arrived at is very conservative. In the interaction method, only the axial
stress term (i.e., P/A) is directly affected by the small allowable compression stress.
The bending stress component, which is a significant portion of the total stress, is
divided by the 150 MPa allowable bending stress. Therefore, the maximum eccentricity
determined from the interaction method is much larger than the eccentricity found from
the allowable-stress method.
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