Stresses in triangular plate with one edge under uniform loading

[roldy]

Homework Statement

A 45° triangular plate ABC built in at BC and loaded by a uniform pressure p₀ on the upper edge AB. The inclined edge AC is traction free. Find the stresses in the plate.

Homework Equations

Stress function:
[itex]\phi = r^2\left[A_1\cos(2\theta)+A_2+A_3\sin(2\theta)+A_4\theta\right][/itex]

The Attempt at a Solution

I know how to get the stresses in the plate. I'm just having trouble setting this problem up in terms of boundary conditions. I assumed that this is a punch on a rigid half-plane since one of the sides in fully built in. If my assumption is correct, then the boundary conditions below follow.

@ [itex]\theta = 0;\ U_r = 0,\ U_\theta = 0[/itex]
@ [itex]\theta = \pi/2;\ \sigma_{\theta r} = 0,\ \sigma_{\theta \theta} = -p_0[/itex]

Are these boundary conditions correct? Attached is an image of the plate.

 

[KataKoniK]

Your boundary conditions seem to be correct for the given problem. As you correctly mentioned, the plate can be treated as a punch on a rigid half-plane, with one side fully built in and the other side traction free.

At the fixed edge, the displacements in the radial and tangential directions are zero, as you have correctly stated. This is due to the fact that the plate is prevented from moving in these directions due to the built-in condition.

At the free edge, the stress in the tangential direction (σθθ) is equal to the applied pressure (p0), and the stress in the radial direction (σθr) is zero. This is due to the fact that the plate is free to deform in these directions, and there is no external force preventing it from doing so.

I hope this helps clarify the boundary conditions for the problem. Let me know if you have any further questions. Best of luck with your solution!