Hello, I am struggling with this problem. It is probably the easiest problem ever... What I did: The plane has 2 stress components. σ_{n} and σ_{s}. σ_{n} is a multiple of (l, m, k) vector. For σ_{s}, I made up a vector (a, b, c) which is orthogonal to (l, m, k). And I equated all vectors. I'm probably doing something wrong. Any help is appreciated!
1. The problem statement, all variables and given/known data 2. Relevant equations General plane formulas. 3. The attempt at a solution I thought that the plane has 2 stress components. σ_{n} and σ_{s}. σ_{n} is a multiple of (l, m, k) vector. For σ_{s}, I made up a vector (a, b, c) which is orthogonal to (l, m, k). And I equated all vectors. I'm probably doing something wrong. Any help is appreciated!
Hello, I am not an expert on elasticity but this really looks quite straightforward. Let's first find the stress vector T (I'm using T instead of σ to avoid confusion with the stress tensor). You will get it by multiplying the (diagonal) stress tensor by your normal vector as T=(σ_{1}l, σ_{2}m, σ_{3}n). It has two components as you wrote, T_{n} and T_{s}. The magnitude of T_{n} is simply the dot product of T and n and its direction is along n as you wrote. Vector T_{s} has to be the complement to the total stress vector. And for the second part - the shear stress will be maximum if vector T lies in your plane, e.g. the dot product of T and n is zero.
Whilst this thread properly belongs in the homework section, this needs comment. What is a stress vector?
We defined it similarly like the article on wiki does, so I won't rewrite it...Stress on Wikipedia Most likely there are other methods or other terminology, I'm not a native speaker so i can't tell the subtle differences that well, sorry about that.