In structural optimization a performance function, such as the volume, weight, or the construction costs of the structure, is optimized subject to certain constraints for the structural responses, like displacements, stresses, loads at the foundation, etc.. Besides the analytical and numerical difficulties of solving mostly highly nonlinear problems of this type, a main problem is that the objective and constraint functions depend not only on the design variables \(x_1,...,x_n\), but also on some model parameters \(a_1,....,a_r\), like material parameters, external load factors, cost coefficients, etc., which are not known at the decision phase. Depending on the partial information about the unknown parameter vector a, several decision criteria are available. Main criteria are the minimization and/or the restriction of the expected total costs – or other structural performance variables – in case of a random parameter vector, or the minimization or restriction of the maximum total costs – or other performance variables – if the parameter vector \(a\) is known to vary in a given set of parameters. In the present book the information about the unknown parameter vector \(a\) is described by means of certain given constraints for \(a\), mostly linear or quadratic ones. Corresponding to the minimax decision rule, in the present work the worst case or worst value of the relevant performance/response variables is determined for several structural design problems. Important examples are:
[A] Static problems: maximum of certain displacement components with respect to variations of the parameter vector in a given ellipsoid; maximum normal thermoelastic stress under certain temperature distribution in a beam; minimizing the lowest eigenvalue of a stiffness matrix of a rigid-body motion with respect to certain coefficients.
[B] Buckling: finding the worst imperfection which minimizes the buckling load under constraint on the norm of the imperfection; minimizing the buckling load with respect to uncertainties in the stiffness matrix; minimizing the maximum load in a plane frame subject to uncertain location and imperfection parameters.
[C] Vibration: minimizing the lowest eigenvalue with respect to parameters describing uncertainty in the stiffness matrix of a dynamic structure; maximum response of the structure with respect to the earthquake excitation parameter vector lying in a given ellipsoid, worst critical velocity in aeroelasticity.
[D] FEM: finding maximum and minimum displacements with respect to uncertainties in the stiffness matrix \(K\) and in the applied loading \(P\).
In the monograph several mechanical models and mathematical/analytical tools are provided, and comparisons with other approaches, as e.g. the probabilistic one, are given. Moreover, many applications to the optimal structural design are presented. Since some of the criteria are based on worst case scenarios, nested or two-stage optimization problems have to be considered. The book contains many examples and a large number of references.