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

Critical Stress
The critical load equation [Equation (16.16)] is valid only if the stresses in the column
remain elastic; therefore, the critical buckling stress must be compared with the proportional
limit of the material. For structural steel, the proportional limit is essentially equal
to the yield stress.
The critical buckling stress will be computed with the use of the larger of the two
effective-slenderness ratios:
σ
cr
2
2
π E π (29,000 ksi)
= = = 16.83 ksi < 36ksi O.K.
2
2
( KL/ r)
(130.4)
Since the critical buckling stress of 16.83 ksi is less than the 36 ksi yield stress of the
steel, the critical load calculations are valid.
Allowable Column load
A factor of safety of 2.5 is required for this column. Therefore, the allowable axial load is
118.8 kips
Pallow = = 47.5 kips
Ans.
2.5
Lateral
bracing
z
y
C
x
A
P
B
17.5 ft
17.5 ft
Buckling about the weak axis.
ExAmpLE 16.4
A W310 × 60 structural steel shape (see Appendix B
for cross-sectional properties) is used as a column with
an actual length L = 9 m. The column is fixed at base A.
Lateral support is provided to the column, so deflection
in the x–z plane is restrained at the upper end; however,
the column is free to deflect in the x–y plane at B.
Determine the critical buckling load P cr of the column.
Assume that E = 200 GPa and σ Y = 250 MPa.
B
x
P
y
Weak
axis
Plan the Solution
Although the actual length of the column is 9 m, the
different end conditions with respect to the strong and
weak axes of the cross section cause markedly different
effective lengths for the two directions. Appropriate
effective-length factors based on the column end conditions
will be selected from Figure 16.7.
SolutioN
Section Properties
The following section properties can be obtained from Appendix B for the W310 × 60
structural steel shape:
y
z
A
9 m
z
Strong
axis
W310 × 60
6 4 6 4
I = 128 × 10 mm r = 130 mm I = 18.4 × 10 mm r = 49.3 mm
z z y y
The subscripts for these properties have been revised to correspond to the axes shown on
the cross section.
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