Introduction to Elasticity/Sample final 4

From Wikiversity
Jump to navigation Jump to search

Sample Final Exam Problem 4[edit | edit source]

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 | edit source]

The Prandtl stress function must satisfy the compatibility condition

Plugging in the stress function, we have,

or,

The torque for a simply connected section is given by

For the elliptical cross-section, we have

Therefore,

or,

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

Therefore,