Evaluate maximum shear stress:

Derive equation subjected to combined bending and torsion for finding out maximum shear stress and its equivalent twisting moment.                                                                                                

Sol.: In general we assume that shaft is subjected to torsion only but in actual practice because of weight of the pulley, couplings, pull in belts or ropes etc, shaft is subjected to bending. Therefore actually in shaft, both the shear stress because of torsion and direct stress because of bending are induced.
From the bending equation,
 
From torsion equation,
 
(i) The maximum principal stress at section
 
The bending moment corresponding to maximum principal stress is termed as equivalent bending moment, M_e . Therefore;
 
Where;

M_e = Equivalent bending Moment

M = Maximum Bending Moment from bending moment diagram

T = Torque transmitted

(ii) The maximum shear stress developed at section
 
The twisting moment corresponding to maximum shear stress on surface of shaft is called as equivalent twisting moment, T_e

. Therefore;
 
Where T_e = Equivalent Twisting moment
Finally;

Equivalent Bending moment
 
Equivalent Twisting moment
 
Maximum principal stress σ_max
 
Minimum Principal stress σ_min
 
Maximum shear stress ι_max