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
n
i ipi
iin
ii IG
LT
11 )(
• CASE 1: Bar consisting of prismatic segments with constant torque throughout each segment
• CASE 2: Bar with continuously varying cross sections and constant torque
L
p
L
xGI
Tdxd
00 )(
NON-UNIFORM TORSION
• CASE 3: Bar with continuously varying cross sections and continuously varying torque
L
p
L
xGI
dxxTd
00 )(
)(
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
)1(2
EG
MECHANICAL POWER TRANSMISSION BY SHAFT
•Equation (2C) is the angular of equation (2A)
•All 3 equations should be remembered
n
i ipi
ii
IG
LT
1 )(
