Page 289 - Marks Calculation for Machine Design

P1: Shibu
January 4, 2005
14:56
Brown.cls
Brown˙C06
STATIC DESIGN AND COLUMN BUCKLING
is given by Eq. (6.37) as
P cr S y 271
σ cr = = ec (6.37)
A
1 + 2
k
Notice that Eq. (6.37) does not contain the length (L) or the slenderness ratio (L/k),
so an artificial value of a transition slenderness ratio must be established. If the amount
of lateral deflection owing to bending from the axis of the compressive loading is to be
some percentage of the eccentricity (e), then if this percentage is 1 percent, the transition
slenderness ratio is given by Eq. (6.38) as

L E
= 0.282 (6.38)
k
transition σ cr
If the slenderness ratio is less than this transition value, then the column is short. How-
ever, if the slenderness ratio is greater than this transition value, then the secant formula
applies.
U.S. Customary SI/Metric
Example 4. Determine whether the column in Example 4. Determine whether the column in
Example 2 is short, where Example 2 is short, where
eccentricity ratio = 1 eccentricity ratio = 1.6
S y = 40 kpsi S y = 270 MPa
3
3
E = 10 ×10 kpsi E = 70 ×10 MPa
solution solution
Step 1. Using Eq. (6.37), calculate the critical Step 1. Using Eq. (6.37), calculate the critical
stress as stress as
P cr S y P cr S y
σ cr = = ec σ cr = = ec
A A
1 + 1 +
k 2 k 2
40 kpsi 40 kpsi 270 MPa 270 MPa
= = = 20 kpsi = = = 104 MPa
1 + (1) 2 1 + (1.6) 2.6
Step 2. Using the critical stress found in step 1 Step 2. Using the critical stress found in step 1
calculate the transition slenderness ratio using calculate the transition slenderness ratio using
Eq. (6.38). Eq. (6.38).
L E L E

= 0.282 = 0.282
k transition σ cr k transition σ cr
3 3
10 × 10 kpsi 70 × 10 MPa
= 0.282 = 0.282
20 kpsi 104 MPa
= 0.282 (50) = 14 = 0.282 (673) = 190
Step 3. As the transition slenderness ratio is Step 3. As the transition slenderness ratio is
less than the slenderness ratio from Example 2, greater than the slenderness ratio from Example
the column is not short. 2, the column is short.