I am analyzing a built-up C-channel for bending purposes (2 point loads of 1200 N each @ 30 cm symmetrically from the CL). The flanges are made of a different material than the web, they are also of a different thickness. The two materials have very different elastic and strength properties. Assume a perfect bond between the flanges and web.
So far i have used the "equivalent section" trick to convert the flanges' dimensions as if they were made of material 2 and to find the stresses in both the web and flanges, after this i ran an FEA on the beam and the stresses are very close to what the theory predicts.
The next and hardest step in the analysis is to figure out the deflection of the beam due to the bending load. While the "equivalent section" trick is helpful to find the maximum stresses, i'm not sure it can be used to find the actual deflection. Additionally, i need to figure out the shear center of the section and all formulas i have seen so far assume a uniform thickness in the web and flanges which is not my case. Furthermore, since the shear center will be located somewhere to the left of the section, there will be a resultant torque on the section which will induce a shear stress in both the flanges and web. While Roark's formulas for stress and strain provide guidance on how to calculate torsional shear stresses in C-channels, it generally assumes uniform material properties which is not the case of this channel. The materials are a ton weaker in shear than they are tension/compression so this is actually my main concern.
Can anybody recommend any resources, tools or tricks to solve this problem? I have access to my college's mechanics of materials books and even in the advanced edition i could't find anything that might help me. In theory i could simply use FEA software for this section but i will have nothing to verify the results against and I always like to at least have an analytical ballpark number to double check.
Any help would be greatly appreciated. Thank you fellas.