Page 279 - Marks Calculation for Machine Design

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Brown˙C06
STATIC DESIGN AND COLUMN BUCKLING
y
Weak axis 261
h x
Strong axis
b
FIGURE 6.19 Rectangular cross section.

The area (A) of this rectangular cross section is (bh), so the radius of gyration (k) for the
weak axis is given in Eq. (6.26) as

1
hb
3
I weak
12 1 2 b b
k = k weak = = = b = √ = √ (6.26)
A bh 12 12 2 3
If the area moment of inertia for the strong axis were used in Eq. (6.26), then the radius
of gyration (k strong ) would be too large by a factor of (h/b), where
h h b h
k strong = k weak = √ = √
b b 2 3 2 3

6.2.1 Euler Formula
For long slender columns where the slenderness ratio (L/k) is greater than a certain value,
for example, 130 for A36 steel or 70 for 6061-T6 aluminum, buckling of the column is
predicted if the calculated axial stress (σ axial ) is greater than the critical stress (σ cr ) given
in Eq. (6.27), called the Euler Buckling formula,
2
P cr C ends π E
σ cr = = (6.27)
A 2
L
k
where
P
σ axial = (6.28)
A
and
P = applied compressive axial force
A = cross-sectional area of column
P cr = critical compressive axial force on column
C ends = coefficient for type of connection at each end of column
E = modulus of elasticity of column material
There are two important points to make from Eq. (6.27). First, the only material property
in this equation is the modulus of elasticity (E), so the critical stress is the same for low-
strength steel as for high-strength steel. Second, as the length (L) of the column increases,