Directions of eccentricity of shells: the R-funicularity perspective

Some of the authors recently extended the definition of funicularity for continuous shells, introducing the concept of relaxed funicularity (R-funicularity or RF). Funicular shells are defined as shells whose static behavior is given by only local membrane actions. Extension to RF is needed as funicular shells can only attain pure membrane behavior for very specific boundary conditions (BCs) and bending—membrane stiffness ratios. The RF was developed to measure a shell’s shape quality for a given set of loads and BCs, including ’small’ moments effects. Quantification of RF is made by defining a generalized eccentricity (GE) measure and verifying that the GE fall inside some eccentricity limits. Aim of this work is to discuss the nature of the GE and its associated eigenvalue problem, that allows to calculate principal eccentricities (PEs), principal modulus eccentricities (PMEs) and their directions (PED and PMED). It is also shown how the directions of PEs and PMEs are related to the relative angle between the principal directions of membrane and bending internal actions. A proper graphical representation of the eccentricity directions is also proposed and applied with some examples.