Torsion

>>Torsion: If moment is applied in a plane perpendicular to the longitudinal axis of the beam (or) shaft, it will be subjected to Torsion

Ex: 

• Shaft Transmitting Torque or power.
• L beams
• Portico beams
• Curved beams
• Closed coiled springs.

>>Torsion Formula:

Where

T = Torque applied

θ = Twist of cross section

τs = Maximum shear stress due to torsion

R = Radius of shaft

L = Length of shaft

J = Polar moment of inertia

Assumptions:

• Plane normal sections of shaft remain plane after twisting.
• Torsion is uniform along the shaft
• Material of the shaft is homogeneous, and isotropic.
• Radii remain straight after torsion.
• Stress is proportional to strain i.e., all the stresses are with in elastic limit.

Note: 

• The stress setup at any point in a cross section is one of pure shear or simple shear.
• The longitudinal axis is neutral axis.
• The shear stress will vary linearly from zero at the centre to maximum at the outer surface  (any point on periphery)
  

>>Torsional Section Modulus:

Zp = J/R = Polar Moment of Inertia/ Radius of shaft

As the value of Torsional modulus increases. the Torsional strength increases. For Ex: A hollow circular shaft compared to that of a solid shaft of same area, will have more Torsional strength.

For a solid circular shaft,

For a hollow circular shaft,

D1 = Outer diameter

D2 = Inner diameter.

Torsional Rigidity:

CJ Unit : kg. cm2 or Nmm2

The torsional which produces unit twist per unit length.

Angle of Twist: θ = TL/CJ

>>Power Transmitted by a Shaft: In SI system: Power (P) is measured in watts (W)

>>Design of Shaft: To be safe against maximum permissible shear stress.

Diameter of shaft,

>>Composite Shafts : When two dissimilar shafts are connected together to form one shaft

the shaft is known as composite shaft.

Shafts in Series : If the driving torque applied at one end, and the resisting torque the other end, the shafts are said to have be connected in series.

For such shaft,

• both the parts carry some Torque i.e., T1 = T2
• Total angle of twist at fixed and is sum of separate angles of twist of two shafts.
  

Shaft in Parallel: If the Torque ‘T’ is applied at the junction of two shafts and resisting Torque at their remote ends, the shafts are said to be connected in parallel.

For such a case,

>>Combined bending and Torsion:

• Let a shaft be subjected to a bending moment of ‘M’ and twisting moment ‘T’ at a section.
  
• Equivalent Torque : It is the twisting moment, which acting along produce the maximum shear stress due to combined bending and Torsion.
  
• Equivalent Bending Moment : The bending moment to produce the maximum bending stress equal to greater principle stress
  

>>Comparison of Hollow and Solid Shafts:

• When the areas of solid and hollow sections are equal,
  
• when radius of solid shaft is equal to external radius of hollow shaft,
  
• The ratio of the weight of a hollow shaft and solid shaft of equally strong is
  

>>Strain energy due to torsion: