Assignment 4

3.20. Figure added. Statement modified.

A bar of rectangular cross-section is restrained in the x direction at the end faces, and in the y direction at the top and bottom faces, but is free in the z direction.  Use a semi-inverse approach to determine the stresses and strains in the bar for a temperature rise of T₁ degrees. Treat the problem as one of plane stress, and please follow these steps:

a) State on the figure the two boundary conditions on each of the four boundary faces of the bar.

b) Make the simplest assumptions about the solution for displacements and/or stresses that are consistent with the boundary conditions, then attempt to satisfy the governing differential equations and boundary conditions.

a) Plane Stress Boundary conditions shown on figure:

b) Simplest assumptions consistent with the displacement boundary conditions:

u = 0

v = 0

Check Governing Equations

1) Strain-displacement relations yield (e_x, e_y, g_xy) = 0

2) Stress-strain temperature relations of plane stress yield

s_x = s_y = -EaT₁/(1-n)

t_xy = 0

3) Assuming T₁ is constant, the D.E.'s of equilibrium are satisfied by constant stresses and zero body force.

Check the stress boundary conditions:

t_xy = 0  on all edges.

Conclusion: All governing equations and boundary conditions are satisfied. The solution is as obtained above.