I have a beam with rectangular cross section, but rather an strange shape in the length wise direction as seen below:
So I have taken out the symmetry part of it:
And it actually has some nice radius's due to metal forming:
the beam is fixed at 0 and symmetric about 5. The Beam sees a Distributed load at s4. The distributed load is actually a gasket so when my beam deflect I actually need to iterate to make sure I reach a steady condition.
The reason why I would like analytic solution is for understanding of whats going on. The problem is a coupled problem, below the beam is a fluid flowing and also the beam has electric current flowing through it, so its all coupled and I would like to experiment with different materials.
So I made a function that represent my beam in Mathcad:
I have tried calculating the moment along the curve by saying that the axial force can be neglected and that the fixed(welded end) can rotate, it might not be 100% accurate but I assumed this will be close enough to the real world:
and the moment seems to make sense:
this also fits with FEM
The problem is that I can not figure out how to calculate the deflection:
Can you not simply take the derivative of the moment or something to get the deflection?
I have tried to solve this but it gives me completely wrong answer and not a nice symmetric deflection curve, I also need to guess the slope at the beginning of the beam and make sure it's the same as the end (6.03225deg). So I have the curve of the beam (inside/outside i.e outside -inside= thickness) as well as the moment and the Young's modulus and inertia of the cross section, so I should be able to calculate the deflection right?