|M.Sc Thesis||Department of Mechanical Engineering|
|Supervisor:||Prof. Altus Eli|
|Full Thesis text|
An analytical sensitivity (derivative) function for the buckling load is introduced for shape optimization. The derivative is derived by functional differentiation of the governing equation with respect to the shape function. Both the buckling load (P) and deflection (W) are considered as functionals of the shape function.
For beam-columns, an explicit analytical expression is obtained, depending on W up to its second spatial derivative. A finite element (FE) model is used to compute the derivative for any shape or boundary conditions, and an optimization using a generalization of the gradient projection method is performed. Results are compared with exact solutions and found to be more accurate than common FE optimization. An example of a clamped beam with two simple supports is studied, with special attention to the location of the middle support. The optimal area distribution is found to be composed of sections with the same shape as for the basic simply supported case.
The semi-analytical method is extended to the case of an axisymmetric circular plate. A FE model is developed and combined with the optimization algorithm. Stiffness optimization is conducted for a variety of boundary conditions, including a hole in the middle, where it is found that local stiffening at the hole edge is required to increase the buckling load.
The proposed method is further generalized for a more complex 2D plate. The optimization is combined with a commercial FE package. The results are verified by comparison between a plate strip and beam optimization. The 2D model is used to verify the axisymmetric plate model. For the case of simple square plates the optimization produces smooth geometries with higher improvement in the buckling load then common FE optimization. Amorphous shape plates with un-uniform meshes are studied.
The analytical gradient expression is independent of the FE model but requires its input. The optimization produces smooth geometries with fast convergence and high gains in the buckling load.