Discretised Airy stress functions and body forces

Abstract

This paper extends polyhedral Airy stress functions to incorporate body forces. Stresses of an equilibrium state of a 2D structure can be represented by the sec- ond derivatives of a smooth Airy stress function and the integrals of body forces. In the absence of body forces, a smooth Airy stress function can be discretised into a polyhedron as the corresponding structure is discretised into a truss. The differ- ence in slope across a creases represents the axial force on the bar, while the zero curvatures of the planar faces represent zero stresses voids of the structure. When body forces are present, the zero-stress condition requires the discretised Airy stress function to curve with the integrals of these body forces. Meanwhile, the isotropic angles on the creases still indicate concentrated axial forces. This paper discretises the integrals of body forces into step-wise functions, and discretises the Airy stress function into quadric faces connected by curved creases. The proposed method could provide structural designers (e.g. architects, structural engineers) with a more intuitive way to perceive stress fields.