The reliability-based structural optimization formulated using the advanced first order second moment (AFOSM) method contains a suboptimization problem. In this formulation, the numerical efficiency has been the main obstacle for a practical use. This corresponds to the process of obtaining the most probable failure point of a failure constraint, and requires solutions of the random state equation with random parameters. Excessive computational work is of main concern especially when they appear in the stiffness matrix.
This thesis presents a numerically efficient method to reduce the computational effort by using the Neumann expansion technique to deal with the random state equation in the suboptimization process. Through numerical implementation and application to various test problems, performance characteristics are studied and compared with the previous approach which uses full FEM solutions of original random state equations.
Model problems include truss and beam structures with uncertainties in design variables, material properties and loads under various distributions, both normal and non-normal. Reliability constraints in the stresses, displacements and eigenvalues as the functional requirements of the structures are considered.
The number of terms in the Neumann expansion can be the deciding factor for computational efficiency. It is seen, however, that the present method with a few terms can give fairly accurate solutions comparable to those obtained by using the full FEM analysis and reduce computational time, especially for complex structures. Computational efficiency is also dependent on the number of random variables appearing in the stiffness matrix as compared with the number of state variables. Other numerical strategies can be possible in selecting the number of Neumann expansion terms. A useful way is to utilize the Neumann expansion method at the beginning iterations, while the full analyses with the random state equations is applied only when more exact solution is required near the solution of the optimization problem. As a major disadvantage this will make the optimization process more complicated from a user's point of view. Unless the variance of random variables in the stiffness matrix is relatively large, the procedure can be applied routinely without any specific prior experience.
Several non-normal distributions are shown practically well treated by using equivalent normal distributions. The resulting optimal structures, however, can be conservative or nonconservative compared with the normal distribution case, depending on the kinds of distributions and active constraints. It is concluded that the approximate solution by the Neumann expansion can make the reliability-based optimal design problem effectively handled although this requires additional user input when high accuracy is necessary.
