Nonlinear incremental analysis of loading and design sensitivity problems

In this paper, it is demonstrated that incremental problems of nonlinear potential and nonpotential systems, including sensitivity problems, can be uniformly treated within the analysis of a homogeneous set of equations. Regular and critical states are considered. It is shown that rank analysis of a rectangular matrix of a homogeneous set of incremental equations reveals all possible problems associated with singularity conditions. When considering nonlinear design modification problems, it is necessary to use derivatives of the secant and tangent stiffness matrices. A direct approach to differentiation of stiffness matrices on a finite element level in sensitivity problems is also presented. Simple illustrative examples are discussed.