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IASS-SLTE Symposium 2014: Shells, Membranes and Spatial Structures: Footprints


IASS Symposium 2014

SESSION: General 9: Computational Tools, Shape Finding and Optimization

A limit of economy of material in shell structures

< Table of Contents for General 9: Computational Tools, Shape Finding and Optimization
  • Proceedings Name: IASS-SLTE Symposium 2014: Shells, Membranes and Spatial Structures: Footprints
  • ISSN: (Electronic Version) 2518-6582
  • Session: General 9: Computational Tools, Shape Finding and Optimization
  • Title: A limit of economy of material in shell structures
  • Author(s): Toby MITCHELL
  • Keywords: Michell trusses, grid shells, isogeometric analysis, form-finding, differential geometry, optimization
Abstract
The limit problem of structural optimization, derived by Michell in 1904, describes the most optimal truss possible as an infinitely fine net of members that solves the dual problem of truss optimization. To generalize this limit problem to 3D shell structures, we derive its primal version by connecting forces in truss members to stresses in a membrane shell. Under the assumption that the truss is a grid of planar quadrilaterals, we show that the limit is a membrane shell with stresses aligned with its lines of principal curvature. Specialization to the case of isotropic membrane stresses and conservative loads yields a tight connection between the loads and the required shell shape, and a requirement that each truss member do the same work as every other leads to a theorem governing the spacing of members along the shape. We develop numerical methods to solve for optimal limit shell shapes, and conclude with a discussion of further work required to solve for grid shells proper.

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