THE AIRY STRESS FUNCTION VECTOR IN 3 DIMENSIONS APPLIED TO FRAMES AND SHELLS AND ITS RELATIONSHIP WITH THE BELTRAMI-GÜNTHER TENSORS

The relationship between the geometry of structures and their structural action is perhaps best understood using stress functions. In 2 dimensions the force in a pin jointed truss member is a discontinuity in slope or 'fold' in the Airy stress function surface and a moment in a member is discontinuity in value or 'cliff' in the surface.We show that Airy stress function is actually only 1 component of a vector in 3 dimensions. Thus we can extend the ideas associated with the Airy stress function to fully 3-dimensional structures.The Beltrami stress functions in 3 dimensions were extended by Wilhelm Günther to apply to Cosserat materials and we show that the Günther stress function tensors can be derived without introducing the concept of stress. We explain why it is inevitable that we can have both the Maxwell and Morera versions of the Beltrami stress function tensor.We also develop Hermann Schaefer’s concept of a Krustenschale or ‘crust shell’ in which the forces and moments in a shell are discontinuities in the Günther stress function tensors.