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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

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

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

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

Illustrative Example-2….Contd

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 =

Illustrative Example-2….cond

Illustrative Example-2….cond

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 =1/7500 [Ans. 17.25 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, -