This paper discusses characteristic features and inherent difficulties
pertaining to the lack of usual differentiability properties in probl
ems of sensitivity analysis and optimum structural design with respect
to multiple eigenvalues. Computational aspects are illustrated via a
number of examples. Based on a mathematical perturbation technique, a
general multiparameter framework is developed for computation of desig
n sensitivities of simple as well as multiple eigenvalues of complex s
tructures. The method is exemplified by computation of changes of simp
le and multiple natural transverse vibration frequencies subject to ch
anges of different design parameters of finite element modelled, stiff
ener reinforced thin elastic plates. Problems of optimization are form
ulated as the maximization of the smallest (simple or multiple) eigenv
alue subject to a global constraint of e.g. given total volume of mate
rial of the structure, and necessary optimality conditions are derived
for an arbitrary degree of multiplicity of the smallest eigenvalue. T
he necessary optimality conditions express (i) linear dependence of a
set of generalized gradient vectors of the multiple eigenvalue and the
gradient vector of the constraint, and (ii) positive semi-definitenes
s of a matrix of the coefficients of the linear combination. It is sho
wn in the paper that the optimality condition (i) can be directly appl
ied for the development of an efficient, iterative numerical method fo
r the optimization of structural eigenvalues of arbitrary multiplicity
, and that the satisfaction of the necessary optimality condition (ii)
can be readily checked when the method has converged. Application of
the method is illustrated by simple, multiparameter examples of optimi
zing single and biomodal buckling loads of columns on elastic foundati
ons.