# Archived Plane Stress Transformation

#### theBEAST

1. Homework Statement
Here is the problem with my attempt at the solution:

The magnitude of my answers are correct, HOWEVER I am getting the wrong signs. For the force balance in the x direction I get a negative P but for the force balance in the y direction I get a positive P. Does anyone know why?

I was thinking it was because I have the wrong direction for shear stress on my free body diagram... But I thought this wouldn't matter since the math will always work itself out?

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#### Chestermiller

Mentor
The stress tensor in the members is $\vec{\sigma}=\frac{P}{(0.05)(0.08)}\vec{i}_x\vec{i}_x=250 P\vec{i}_x\vec{i}_x$ Pa. The unit normal to the joint is $\cos {25}\vec{i}_x+\sin{25}i_y$. From the Cauchy stress relationship, the stress vector acting on the joint is $250 P\cos {25}\vec{i}_x$. The component of this stress vector normal to the joint is $250 P\cos^2 {25}$. The unit tangent to the joint is $\cos{25} \vec{i}y-\sin{25}\vec{i_x}$. The component of the stress vector tangent to the joint is $250 P\sin{25}\cos {25}$. So, for the joint not to fail,
$$250 P \cos^2 {25}< 800000$$and $$250 P\sin{25}\cos {25}<600000$$So,$$P<3896\ N$$and $$P<6266\ N$$So the critical load is 3896 N.

This result is basically the same as the result obtained by theBEAST, and thus confirms his answer.

"Plane Stress Transformation"

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