Torsion:

Torsion is defined as, "Act of twisting or turning that produces stress (deformation) when one end is twisted in one direction and the other is held motionless or twisted in opposite direction".

• If one end is held motionless  ↠  couple moment (circle remain circle but rectangle become rhombus) due to torque we have angular deformation (radial lines remain straight). For example: Transmission shaft.
• Since couple (torque or angular force) applied on body  ↠  produces angular stress and strain.
• for power production  ↠  generator  +  turbine.

Torsional Equation:

Some important points are:

• Polar Moment of Inertia  ↠  it is used to predict an object's ability to resist torsion.
• Greater the polar MOI, the more torque is required to turn shaft by a certain angle.
• Greater the polar MOI, the smaller the shear stress required to produce a given torque.
• More polar MOI, more diameter of shaft which means less stress.

Finally the torsional equation is:

Types of Shaft:

There are two types of shaft which are discussed ahead:

1. Stepped Shaft  ↠  take sections depending on either torque, length, polar MOI or shear modulus is changing. Then apply Compatibility equation for Torsional Equation.
2. Concentric Shaft  ↠  in which shafts are parallel. Then apply Compatibility equation for Torsional Equation.

Direction of Torque:

• Using right-hand rule, curl fingers in the direction of angle of twist (which is the direction of torque) and thumb pointing outward (+ve direction).
• You can make your own convention.

Power Transmission:

Shafts and tubes (circular cross-section) used to deliver power. Power transmitted by shafts subjected to torque is:

Determination of Shear Modulus or Torsion Test:

Shear modulus or Modulus of Rigidity is defined as, the ratio of shear stress to shear strain. Shear modulus is a material constant and is determine using Torsion Machine.

Torsion machine consist of:

• Load dial  ↠  in which load is changing.
• Load Range Selector  ↠  can change load from here.
• Torque Strain Recorder  ↠  tell us shear strain.
• Two Heads  ↠  One is Fixed head and second is Turning head to twist shafts/rods.
• Motor  ↠  provide torque to twist specimen.
• Movable Unit on Rails  ↠  on which fixed head is mounted used to fit specimen.

Shear Modulus Calculations:

• Specimen of known diameter and length is mounted in the machine where known amount of torque is applied to the shaft.
• We put T, L, J, Ø in Torsion equation to find Modulus of Rigidity.
• ASTM A938-18  ↠  Standard test method for torsion test of wires.

Mohr's Circle:

It enables you to visualize the relationship between the normal and shear stresses acting on various inclined plane at a point in the stressed body.

• Mohr's circle gives double angle.
• If Mohr's circle is drawn under torsion, specimen only possesses shear stress and have zero normal stress. Consider bi-axial load (load applied along two axis).
• Max. tension condition is obtained by rotating specimen 45 deg actual and 90 deg in Mohr's circle.
• If couple moment at both end, max. shear stress is at center and break from there.
• Ductile Failure  ↠  Flat failure due to Torsion along the plane of Max. shear stress.
• Brittle Failure  ↠  due to torsion failure in brittle material occur at 45-degree and occur along the plane of max. tension. 

Statically Indeterminate System:

Consider a prismatic bar (constant cross-section) shaft fixed from both ends. 

Solid Non-Circular Shafts:

• Torque applied on non-circular solid shaft will bulge or wrap.
• Shaft that have non-circular cross-section are not axisymmetric (symmetric along any axis).

Ques: Does square, triangle or rectangle shafts are axisymmetric?

No, because they cannot be produced by revolving around an axis.

Ques: Does cone is axisymmetric?

Yes, because it can be produced by revolving around an axis.

Planar Symmetric:

It is defined as, "a mirror image across a plane (or symmetric along a plane like images or mirror)".

• It helps us to reduce geometry size (reduce number of unknowns ↠ simplify a problem).
• Conditions of symmetry is that the load as well as the supports are symmetric across the plane.
• Cube is 1/8th symmetric.
• Tensile specimen can be modeled as 1/8th symmetric.
• Follow table to find shear stress and angle of twist for Non-circular sections.

References:

• Material from Class Lectures + Book named Engineering Mechanics of Materials by R.C. Hibbeler (10th Edition) + my knowledge. 
• Pics and GIF from Google Images.  
• Videos from YouTube.