TORSION • Introduction, Round bar torsion, Non- uniform torsion • Relationship between Young’s Modulus, E, Poisson ratio, and modulus of rigidity • Power Transmission on round bar TORSION-INTRODUCTION • Occur when any shaft is subjected to a torque • This is true whether the shaft is rotating (e.g. drives shaft on engines, motors & turbines) or stationary (bolt/screw) • This torque makes the shaft twist and or end rotates relative to other inducing shear stress on any cross section • Failure might occur due to shear alone / Shear + stretching/bending ROUND BAR TORSION EQUATION The diagram show a shaft fixed at one end and twisted at the other end due to action of a torque T POLAR 2ND MOMENT OF AREA Solid shaft Hollow shaft NON-UNIFORM TORSION • Uniform/Pure torsion – torsion of prismatic bar subjected to torques acting only at the ends • Non-uniform torsion– the bar need not be prismatic and the applied torque may act anywhere along the axis of bar • Non-uniform torsion can be analysed by – Applying formula of pure torsion to finite segments of the bar then adding the results – Applying formula to differential elements of the bar and then integrating NON-UNIFORM TORSION • CASE 1: Bar consisting of prismatic segments with constant torque throughout each segment n n Ti Li i i 1 i 1 Gi ( I p )i • CASE 2: Bar with continuously varying cross sections and constant torque L L Tdx d GI p ( x ) 0 0 NON-UNIFORM TORSION • CASE 3: Bar with continuously varying cross sections and continuously varying torque L L T ( x ) dx d GI p ( x ) 0 0 NON-UNIFORM TORSION • Limitations – Analyses described valid for bar made of linearly elastic materials – Circular cross sections (Solid /hollow) – Stresses determined from the torsion formula valid in region of the bar away from stress concentrations (diameter changes abruptly/concentrated torque applied – For the case above, Angle of twist still valid – Changes in diameter is are small and gradually (angle of taper max 10o) RELATION BETWEEN E, AND G E G 2(1 ) MECHANICAL POWER TRANSMISSION BY SHAFT •Equation (2C) is the angular of equation (2A) •All 3 equations should be remembered n Ti Li i 1 Gi ( I p )i