Introduction to Elasticity/Sample final 4
Sample Final Exam Problem 4 
Consider the torsion of a prismatic bar having an elliptical cross-section as shown in the figure below.
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 .
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