Aprismatic Beams – A Mathematical Model and Application to a One Kilometre Arch Bridge

Design engineers, like all humans, are driven by Nash game theory to maximize return and hence simplify design. A determination of the optimal shape of beams to maximize strength and minimize costs has been an area of significant research since the 1970’s. However, real cost constraints in the market place usually see the selection of standard beams with invariant inertia tensor properties being used for most buildings throughout the world. The more challenging problem is the development of a beam of varying cross sectional area, this type of beam provides savings in terms of the quantity of steel and the mass of the ultimate building or bridge without degrading safety and can when manufactured in quantity to reduce costs. The purpose of the paper is to outline the mathematical development of aprismatic beams for everyday use in engineering to reduce material usage and hence human impact on the global environment. An example is provided using a 1 km arch bridge.