Sample Final Exam Problem 4 edit

Consider the torsion of a prismatic bar having an elliptical cross-section as shown in the figure below.

Torsion of bar with elliptical c.s.

The bar is subjected to equal and opposite torques   at the two ends which cause a twist per unit length of   in the bar.

The equation of the boundary of the cross section is


Since the Prandtl stress function   is zero on the boundary of the cross-section of a simply connected prismatic bar, we can choose the Prandtl stress function for the bar with an elliptic cross-section to be

  • (a) Determine the value of the constant  .
  • (b) Express the twist per unit length ( ) in terms of the applied torque ( ).
You will find the following results useful in evaluating the integral.
  • (c) What is the torsion constant of the section?
  • (d) Express the maximum shear stress in the bar in terms of  ,   and  .

Solution edit

The Prandtl stress function must satisfy the compatibility condition


Plugging in the stress function, we have,




The torque for a simply connected section is given by


For the elliptical cross-section, we have






The torsion constant   is given by


The stresses in the section are given by


The projected shear traction is


The maximum projected shear traction is at  . Hence,


To express the magnitude of the maximum shear stress in terms of  ,  , and  , we use