1. Mechanical Engineering
Unit-5.Torsion in Shafts and Buckling of Axially Loaded Columns Lecture Number-6
Mr. M.A.Mohite
Mechanical Engineering
S.I.T., Lonavala

2. Rankine’s Formula (Empirical formula- both short and long columns)
(a= Rankine’s constant)

3. Values ‘a' for Rankine formula
Typical values of ‘a' for use in Rankine’s formulae are given below in table.
Material
Crushing Stress
MPa
Value of ‘a’
Pin Ends
Fixed Ends
Low carbon
steel
315
1/7500
1/30000
Cast Iron
540
1/1600
1/64000
Timber 35
1/3000
1/12000

4. Illustrative Example-1
A hollow cylindrical cast iron column is 4 m long with both ends fixed. Determine the minimum diameter of the column if it has to carry a safe load of 250 KN with a factor of safety of 5. Take the internal diameter as 0.8 times the external diameter. Take σC = 550 N /mm2 and. a = 1/1600 in Rankine's formula.
Solution-Given , L=4m=4000mm
End conditions = Both ends fixed
Effective length = Le=0.5L=2000mm
Safe load=250N
Factor of Safety=5
Ext.Dia =D, Internal Dia.=0.8D,Value of a=1/1600

5. Illustrative Example-2….Contd

6. Illustrative Example-2
Calculate the diameter of a piston rod for a cylinder of 1.5 m diameter in which the greatest difference of steam pressure on the two sides of the piston may be assumed to be 0.2 N/mm2.
The rod is made of mild steel and is secured to the piston by a tapered rod and nut and to the crosshead
by a cotter. Assume modulus of elasticity as 200 kN/mm2 and factor of safety as 8. The length of rod
may be assumed as 3 metres.
Solution. Given : D = 1.5 m = 1500 mm ; p = 0.2 N/mm2 ; E = 200 kN/mm2 = 200 × 103 N/mm2 ;l = 3 m = 3000 mm
We know that the load acting on the piston,
W =

7. Illustrative Example-2….cond

8. Illustrative Example-2….cond

9. Practice Example-1
Find the Euler’s crippling load for a hollow cylindrical steel column of 38 mm external diameter and 35 mm thick. The length of the column is 2.3 m and hinged at its both ends. Take E = 200 GN/m2. Also determine the crippling load by Rankine’s formula, using
σc = 320 MPa ; and a =1/ [Ans kN ; 17.4 kN]
Hint- 1. Calculate Area
2. Calculate Moment of Inertia
3. Calculate K=√I/A,
4. Caculate Euler’s Crippling Load using-
5. Crippling load by Rankine’s formula, -