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

x
B
P
Buckling About the Strong Axis
The column could buckle about its strong axis, resulting in the buckled shape shown
in which the column bends about its z axis and deflects in the x–y plane. For this
manner of buckling, the column base is fixed and its upper end is free. From Figure
16.7, the appropriate effective-length factor is K z = 2.0 and the effective length of the
column is (KL) z = (2.0)(9 m) = 18 m. The critical buckling load is therefore
9 m
P
cr
2
2 2 6 4
π EIz
π (200,000 N/mm )(128 × 10 mm )
= =
2
2
( KL)
[(2.0)(9 m)(1, 000 mm/m)]
z
= 779,821 N = 780 kN
y
z
A
B
P
The effective-slenderness ratio for buckling about the strong axis is
(2.0)(9 m)(1, 000 mm/m)
( KL/ r)
z = = 138.5
130 mm
Buckling About the Weak Axis
Alternatively, the column could buckle about its weak axis. In this case, the column
would bend about its y axis and deflection would occur in the x–z plane as
shown. For buckling about the weak axis, the column is fixed at A and pinned at
B. From Figure 16.7, the appropriate effective-length factor is K y = 0.7 and the
effective length of the column is (KL) y = (0.7)(9 m) = 6.3 m. The critical buckling
load about the weak axis is therefore
P
cr
2
π EI 2 2 6 4
y π (200,000 N/mm )(18.4 × 10 mm )
= =
2
2
( KL)
[(0.7)(9 m)(1, 000 mm/m)]
y
= 915,096 N = 915 kN
y
z
A
9 m
The effective-slenderness ratio for buckling about the weak axis is
(0.7)(9 m)(1, 000 mm/m)
( KL/ r)
y = = 127.8
49.3 mm
The critical load for the column is the smaller of the two load values:
Pcr = 780 kN
Ans.
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 (200,000 MPa)
= = = 102.9 MPa < 250 MPa O.K.
2
2
( KL/ r)
(138.5)
Since the critical buckling stress of 102.9 MPa is less than the 250 MPa yield stress of the
steel, the critical load calculations are valid.
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