Strain energy due to torsion

generator before application of torque

Consider a circular shaft of length L radius R, subjected to a torque T at one end (see Fig. 1.0). Under the action of torque one end of the shaft rotates with respect to the fixed end by an angle dφ. Hence the strain energy stored in the shaft is,

formula

Consider an elemental length ds of the shaft. Let the one end rotates by a small amount dφ  with respect to another end. Now the strain energy stored in the elemental length is,

formula

We know that

formula

where, G is the shear modulus of the shaft material and J is the polar moment of area. Substituting for  dφ from (3.0) in equation (2.0), we obtain

formula

Now, the total strain energy stored in the beam may be obtained by integrating the above equation.

formula

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