Mathematics

From the classical theory of linear elasticity, the principle of minimum complementary energy can be used to calculate compatible stress fields for static solids or structures. However, the minimum must only be sought among statically admissible stress fields, yielding a constrained problem. Based on the complementary energy principle, numerical methods have been developed both in the fields of analytical methods and stress-based finite element analysis (FEA). While analytical methods, such as power series approaches, are limited to problems with regular geometries and boundary conditions, the development of specific elements in stress-based FEA resulted in problems with large numbers of degrees of freedom, e.g., due to more than 20 degrees of freedom per element. The same holds usually true if accurate stress results have to be computed based on displacement-based FEA. In response, our approach involves a spline-based stress function that automatically ensures static admissibility and provides the solution, i.e., the stress field, by applying the principle of complementary energy. We show that the properties of splines allow for flexibility in terms of geometry and boundary conditions as well as efficiency in the number of degrees of freedom, yielding accurate approximations of the unknown stress field. We will present results for different types of 2D plane stress problems. First, we will use common benchmark problems, to which analytical methods can be applied, to validate our approach. Furthermore, we will consider composite beams with anisotropic material behavior as well as non-prismatic beams discussed in the literature, demonstrating the approach’s applicability to problems where obtaining accurate stress results remains a challenge for existing methods.