Transcript Torsion - Tatiuc.edu.my

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