The extension of classical isotropic plate and shell solutions and finite element formulations to cope with orthotropic/monoclinic laminated and shear deformable structures often involves very complex intermediate stages and final results within the derivations. This paper examines four case studies covering the use of symbolic computation to manage this complexity. These case studies comprise the derivation of a catalogue of solutions to orthotropic circular plate problems, the formulation of two axisymmetric shell finite elements (respectively using Flugge's shear-rigid shell assumptions and the shear-flexible assumptions of Soldatos) and the derivation of the eighth-order governing differential equation for a laminated monoclinic or orthotropic shell. The emphasis is placed upon the techniques required to achieve these derivations using symbolic computation, and the considerable effort involved in putting the results into publishable form is noted.