Yes, it is certainly possible to derive the equation of motion for shell structures using first-order shear deformation theory. First-order shear deformation theory is an improvement over classical shell theory as accounting for transverse shear becomes increasingly important for thick shells or when analyzing higher-order modes - similar to what we find for beams. While the derivation is possible, it can be mathematically intensive. The specific form of the equations will depend on the shell geometry and coordinate system chosen. Will likely require numerical methods to find solution in most practical cases.

The problem is because of its complexity there is almost no where driven such equations in the way that we can see all possible steps. It might be a little long, but would it be possible for this with your beautiful style?