Internal Force and Moment Surfaces for Shells

Paper in proceeding, 2024

There are 4 fundamental 2nd order tensors in shell theory, firstly the stress and internal moment tensors which are in equilibrium with the applied loads and loading couples, and secondly the rate of membrane strain and the rate of bending tensors which are compatible with the velocity of the shell as it deforms. Shell theory is completed by the constitutive relationships which give the stress and moment in terms of the membrane strain and bending deformation. The fundamental tensors are associated with a number of vectors including the applied loads, loading couples, velocity and angular velocity. We show that the 4 fundamental tensors can all be linearly assembled from vectors using the gradient and also a 3rd order tensor containing the surface permutation tensor and the unit normal. Since a vector describes a surface in 3D space, all these quantities have a graphic significance which can be described using images and models - 3D graphic statics which is the application of differential geometry to statics. We demonstrate that if the moment is described in an appropriate way, using 2 vectors, then internal moments are in equilibrium with zero imposed loads and loading couples, although boundary moments and forces are required. Thus internal moments in shells are ‘redundant’ in the nomenclature of structural theory. For flat plates, one of the redundant moments has to be replaced by a non-redundant moment.