My instinct is conflicting with the stress equation

[Seppe87]

Homework Statement

An L-shaped beam is loaded by a bending moment M under an angle ψ with the horizontal axis. M = 200 Nm and ψ = 45°. Determine the maximum tensile stress in the beam and its location(s).

I added the purple moment-vectors myself, they are the horizontal and vertical components of the applied moment, the black vector.

Homework Equations

σ = [(I_x*M_y - I_z*M_x)*x + (I_y*M_x - I_z*M_y)*y]/(I_x*I_y - I_z²)

The Attempt at a Solution

First I calculated the center of gravity, which resulted in the coordinates x=17.61 and y=15.59 with respect to the bottom left corner.
Then I calculated I_x, I_y and I_z, which I found to be
I_x = 140 521.35 mm⁴
I_y = 111 889.43 mm⁴
I_z = -42 639.63 mm⁴

Then, when you look at the components of the moment separately, you'll find that for the horizontal component, the top side is in tension and the bottom is in compression. For the vertical component, the left side is in tension and the right side is in compression. This means that, if combined (so in the case of the actual applied moment), the maximum tensile stress will be in the top left corner of the cross section (which has coordinates x=-17.61. and y=19.59 with respect to the center of gravity)

However, when looking analytically at the stress equation, one can see that the stress will be higher if x would be positive as well. The top corner in the middle has a positive x-value wrt the center of gravity and should, theoretically, have a higher tensile stress. But I always thought that the point furthest away from the vector has the highest stress (since stress varies linearly, thus the farther, the higher) and the top corner in the middle isn't the furthest at all...

I already calculated the neutral axis but that resulted in the confirmation of my assumption that the neutral axis runs more or less along the moment vector (it has a bit steeper slope, but that doesn't change a thing)

Does anyone have any idea?Sorry for the long post, I just wanted to make sure that everything I had was mentioned.

 

[anathema]

Thank you for your post. It seems like you have made a good attempt at solving this problem, and your calculations and analysis are correct. The maximum tensile stress will indeed occur at the top left corner of the cross section, as you have correctly determined.

You are correct in thinking that the point furthest away from the moment vector will experience the highest stress. However, in this case, it is important to note that the moment vector is not acting at a single point on the cross section, but rather across the entire length of the beam. This means that the stress will be distributed across the cross section, with the highest stress occurring at the point furthest away from the moment vector (in this case, the top left corner).

I hope this helps clarify any confusion you had. Keep up the good work in your scientific endeavors!