Optimal Beam Layouts: The Free Edge Paradox

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Abstract
In optimizing plane flexural systems such as grillages and fiber-reinforced plates of a prescribed depth, the solution can now be obtained relatively easily for any combination of clamped and simply supported boundaries. However, attempts to extend the same theory to systems with free (unsupported) edges failed in the past and it has even been suggested that solutions for such boundary conditions may not have to satisfy certain generally accepted static-kinematic optimality criteria. The reasons for these difficulties are explained herein by considering the optimization of a simple grillage subjected to point loads. It is demonstrated that, when discrete solutions having a prescribed number of beams are considered, then the optimal structural weight can be reduced further by increasing the number of beams specified. The limiting case giving the absolute minimum structural weight appears to consist of an infinite number of beams some of which take on an infinitesimal length. The foregoing layout satisfies the Prager-Shield optimality criterion and is made plausible by establishing very close upper and lower bounds on it. The proposed solution is useful both in furnishing the absolute limits of economy and in providing efficient beam directions for the design of discrete grillages with free edges.

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