Maxima and minima of moments in hyperstatic spans

Abstract

When a distributed load is applied to a hyperstatic (statically indeterminate) beam span of two distinct and given section flexural stiffnessses, maxima or minima of the critical moments within the span arise when a physically meaningful coincidence exists between the moment diagram and the flexural stiffness diagram for the span. In this paper, an insightful geometric argument is employed to identify that physically meaningful phenomenon for a span with both ends encastered. Symbolic analysis is then used to derive the coincidence for spans with rotational springs of any stiffness between zero (free rotation) and infinity (encastered) at one end or both ends. The vital role of implicit differentiation in enabling serendipitous discovery of the result is highlighted. Examples are used to show how the phenomenon and associated theory may be used to obviate iterative computations, thereby providing rapid and useful insight into the process of optimising fibre-reinforced polymer/steel plate layouts which create two section stiffnesses when connected to timber or steel beams. For reinforced concrete or steel–concrete composite spans it is shown that the phenomenon automatically occurs, and the effects on steel-to-concrete shear transfer of uncertainties in the flexural stiffness distribution are discussed again by recourse to the underlying theory.