Unit5 Torsion in Shafts and Buckling of Axially
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’ Low carbon steel 315 Pin Ends 1/7500 Cast Iron Timber 540 35 1/1600 1/3000 Fixed Ends 1/30000 1/64000 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 /mm 2 and. a = 1/1600 in Rankine's formula. Solution-Given , L=4 m=4000 mm • End conditions = Both ends fixed • Effective length = Le=0. 5 L=2000 mm • Safe load=250 N • Factor of Safety=5 • Ext. Dia =D, Internal Dia. =0. 8 D, 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/mm 2. 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 k. N/mm 2 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/mm 2 ; E = 200 k. N/mm 2 = 200 × 103 N/mm 2 ; 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/m 2. Also determine the crippling load by Rankine’s formula, using • σc = 320 MPa ; and a =1/7500 [Ans. 17. 25 k. N ; 17. 4 k. N] 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, -
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