| For a thin plate or beam, if the deformation is on the order of the thickness and remain elastic, linear theory may not produce accurate results as it does not predict the in plane movement of the member. Therefore, a geometrically nonlinear, large deformation theory is required to account for the inconsistencies. This thesis discusses nonlinear bending and vibrations of simply-supported beams and plates. Theoretical results are compared with other well-known solutions. The effects of geometric nonlinearities are discussed. The equation of motion for plates with 'stress-free' and 'immovable' edges are derived using modal analysis in conjunction with the expansion theorem. Theoretical results are compared with a finite element simulation for plates. 'Immovable' edges are studied for beams. For large bending of beams with 'stress-free' edges, a theory by Conway is presented. A brief introduction to Duffing's equation and Gaussian curvature is presented and their relevance to nonlinear deformations are discussed. |
