Torsion of a Circular Bar (Statically Indeterminate)

 

Example:  Bar  ABCD  shown below is clamped at both ends and subjected to applied torques
T and 2T.    The bar has cross-section with polar moment of inertia, J, and shearing modulus of

elasticity, G.  Find the restraint torques at each end and the maximum shear stress in the bar.

                                                             
                                    

 

Strategy:  Use a free body diagram to identify the support torques at A and at D.  Write
the equation of equilibrium.   So more information is needed since there are two unknown

support torques.  Next apply the torque-rotation relation given by  ∆i = TiLi/JiGi .  Apply
the geometrical relation that the net rotation is zero.  Solve for the unknown torques.  Finally

determine the torque in each section of the bar.  Use it to calculate the shear stress in each

section of the bar.

 

The FBD is as follows:

                                  

 

          →ΣTx = 0    – TA – T  + 2T  +  TD  = 0     which yields:         TA  –  TD  =  T       (1)

 

Let   ∆  denote rotation.  Now the sum of the relative rotations must be zero as

follows:                        ∆B/A  +   ∆C/B  +   ∆D/C  =  0

 

The axial forces in each section are  T1,  T2,  and  T3  as depicted in the figure below.

 

                                       

 

For A to B:  T1  =  TA,        For B to C:  T2  =  TA + T,       For C to D:  T3  =  TA - T

 

 

        Click here to continue with this example.

 

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