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A Limit of Economy of Material in Shell Structures

  • Author(s): Mitchell, Toby
  • Advisor(s): Govindjee, Sanjay
  • et al.

An engineering structure can be optimized to maximize its stiffness under a set

of applied loads, or (equivalently) to ensure all stresses in the structure are at a

limit value. Such an optimum can be refined by including more members in the

structure, enlarging the solution space and allowing an improved result. Continuing

this process, we obtain the limit of refinement of a sequence of optimal structures.

Originally studied by Michell, the limit consists of a continuum of infinitely dense

members, and is the most optimal structure possible for a given sequence of structural


In this work, we lift the work of Michell from the plane onto curved three-

dimensional shell structures. This is accomplished in two parts: first, we consider

optimal continuum shells in their own terms, then we consider a sequence of optimal

discrete grid shells and examine their convergence to the limit. The properties of the

continuum guide our search for a correct sequence; in particular, we expect the grid

shell to converge to the lines of principal stress of the continuum in the limit. Crucial

to our result is the concept of a net, a discrete collection of continuous curves that

retains the topological characteristics of a grid shell but allows us to use calculus to

reason about structures near the limit. The continuum shell problem allows us to

determine the outer geometry of the limit (the curvature), while the net allows us to

determine the inner geometry (the spacing of members along the surface).

We examine the properties of this limit, showing that the primal version of

Michell's dual problem is a membrane shell. Under the assumptions of a grid shell

composed of planar quadrilaterals, the limit has lines of principal stress aligned with

its lines of curvature. We derive numerical methods for solving for the shape of limit

shells under the assumption of isotropic stresses in the shell and loads available from

a potential.

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