Torsion

If a shaft is rigidly fixed at one end & twisted at the other end by a torque applied in a plane perpendicular to the longitudinal axis of the shaft, the shaft is said to be in the state of torsion.
Effects:
1) It imparts an angular displacement of one end cross-section with respect to the other end.
2) It sets up shearing stresses on any cross section of the shaft perpendicular to its axis.
Twisting Moment
Algebriac sum of the moment of the applied couples that lie to one side of the section under consideration.
Shear Stress due to torsion (τ)
Shear stress produced due to applied torque(τ) at a distance ‘r’ from the centre of the shaft is given by,
(τ) = T*(r/J)
Where, r = distance from centre of the shaft
J = polar moment of inertia

Polar Moment of Inertia (MOI)
The polar MOI of the section is its second moment of area about an axis perpendicular to its plane. If the axis passes through the centroid, the polar MOI is equal to the sum of the other two MOI passing through the centroid & in the plane but perpendicular to each other.

Shear Strain due to Torsion
The angular displacement of the one surface of the shaft from its original position due to the applied torque is called the shear strain at the surface.
Since, there is certain rotation, the unit of shear strain is radians.

Modulus of Rigidity (G) = Shear stress/Shear strain
ASSUMPTIONS
1) The plane section of the shaft normal to its axis remains plane after the torque is applied.
2) All diameter in section which were straight before torque was applied remains straight.
3) The twist along the length of the shaft is uniform throughout.
4) The material of the shaft is uniform throughout.
5) Maximum shear stress induced in the shaft due to applied torque does not exceed the elastic limit.

Relation between torsional stress, strain & angle of twist
τ/R = GӨ/L
putting; GӨ/L = k
If τ1 & τ2 are the shear stresses at radius R1 & R2 at any point,
τ/R = τ1/R1 = τ2/R2
the shear stress at any point in the cross section of the circular shaft is proportional to its distance from the axis of the shaft and is zero at the axis and maximum at the surface.
Relation between twisting couple & shear stress
T/J = τ/R

T/J = GӨ/L = τ/R Equation of Torsion

Torsional Rigidity
Torque which produces a twist of 1 radian in shaft of unit length.
Polar Modulus = J/R
Angle of Twist : When torque τ is applied on a circular shaft the line AB on the surgace of the shaft moves to the position ABI producing a shear strain φ & simultaneously radius OB moves through an angle Ө to the corresponding position OBI. This angle is called twisting angle.
Strength of a solid shaft
Maximum torque /power transmitted by a solid shaft
Maximum torque will be transmitted when maximum shear stress is produced at the top surface of the shaft of radius R.
T = τπD3 /16
Power transmitted through shaft
Let a shaft transmitt torque T at N rpm.
Angle turned per minute = 2πN radians ( 360 degree in 1 revolution)
Power transmitted P= average torque * angle turned per second
P= Tavg * 2πN/60 watts

Comparison between solid & Hollow Shafts
By strength
Consider two shafts made of same material equal in weight and length and same maximum shear stress
If the exrernal diameter is twice the internal diameter then, the torque transmitted by hollow shaft is 1.443 times more than torque transmitted by solid shaft. Hence, hollow shafts are preffered for heavy torques.
We have the formula, Th/Ts = (n2+1)/n√(n2-1)
Where, D/d = n (D=external diameter; d=internal diameter)
Th = Torque transmitted by hollow shaft
Ts = Torque transmitted by solid shaft

By weight
Assumed that both the shafts are made of same material & same length. Let, the applied torque to both the shafts are same. The maximum shear stress will also be same in both solid & hollow shafts.
Let, Wh = weight of hollow shaft
Ws = weight of solid shaft
So we have, Wh / Ws = [(n2-1)n2/3 ]/[ (n4-1)2/3 ]

Percentage saving in wt. of shaft
% saving in weight = [Ds2 – (D2-d2)/Ds2]*100
Where, Ds = diameter of solid shaft
D = external diameter
d = internal diameter

Replacing of shaft
When a solid shaft is to be replaced by a hollow shaft or vice-versa, then the power transmitted by the new shaft should be equal to the power transmitted by the shaft to be replaced.
Composite Shaft
When two shafts of same or different length, cross-sections or materials are connected together to form a single shaft is called a composite shaft.
i) Shaft is series
When a composite shaft connected in series is subjected to a torque, then the torque transmitted by each individual shaft is same. Torque applied at one end of composite shaft is equal to the resisting torque at the other end.
Total angle of twist at the fixed end or the resisting end of the shaft is the sum of the angles of twist of two shafts.
ii) Shafts in parallel
When the driving torque is applied at the junction of the two connected shafts, they are said to be connected in parallel. Resisting torque developed at both the ends, torque transmitted by each shaft is different but the angle of twist for the shaft is same.

Plastic Torsion of circular bar
When the shaft is applied with twisting moment & increased gradually , at some stage, the extreme fibres of the bar will reach the yield point in shear. This is the maximum possible elastic twisting moment that the bar can withstand and is denoted by T. Further increase in the value of twisting puts the interior fibres at the yield point with yielding progressing from outer fibres inward. The limiting case occurs when all fibres are stressed to the yield point. At this point, the twisting moment is fully plastic.
For circular solid bar subjected to torsion,
Tp = (4/3)*T