Using A Single Equation To Account For All Loads On A Beam In The Method Of Double Integration: A Caveat

When the method of double integration is applied to determine deflections of beams, one has the option of using a single equation containing singularity functions to effectively account for both concentrated and distributed loads on the entire beam without dividing the beam into multiple segments for integrations. This option is a right way and an effective approach to start the solution for the problem if the beam is a single piece of elastic body with constant flexural rigidity. However, this option becomes a wrong way and a misconception that will lead to a set of wrong answers if there exists in the beam (e.g., a combined beam) a discontinuity in slope or flexural rigidity. Unsuspecting beginners tend to miss the subtlety that a singularity function, like other functions, must have no discontinuity in slope if it is to be integrated or differentiated in its domain. Here, the domain lies along the beam. Since rudiments of singularity functions are a prerequisite background for sensible reading of this paper, they are included as a refresher. The purpose of this paper is to share with educators and practitioners in mechanics a caveat in analyzing hinge-connected beams – a pitfall into which beginners often tumble.