Criteria for uniqueness and extrema of moments in continuous end spans with differential settlement

This paper algebraically analyses an end span of a continuous beam which experiences differential settlement under a traveling point load or, separately, a uniformly distributed load (UDL). The beam has two distinct section flexural stiffnesses along the span, with a given ratio between the stiffnesses and with given support stiffnesses. The analysis outputs a novel result that, for the UDL or for any given location of the point load, there is a specific location of stiffness change along the span at which the moment diagram is unique irrespective of the ratio between section stiffnesses. This result punctures the common assumption that stiffening parts of a continuous beam always serves to attract higher moments to those parts. Then, differential calculus is applied to the algebraic analysis to reveal another novel result that, again for the UDL or for any given location of the point load, the moment diagram is an extremum when the point of stiffness change along the span matches a salient point along the moment diagram. For the point load, an important restriction on the location of stiffness change that triggers the moment extremum is identified. Also for the point load, the extremum criterion is further differentiated to arrive at an expression for the specific ratio between the two section stiffnesses which just ensures that any given allowable moment at the inner support is not exceeded as the load travels along the span. Design and research uses of these results are discussed. The paper illustrates how algebra, plots, calculus and physical reasoning may be combined to enhance understanding of mathematically complex systems.