June 24, 2007
June 24, 2007
June 27, 2007
12.1535.1 - 12.1535.11
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 solu- tion 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 do- main. Here, the domain lies along the beam. Since rudiments of singularity functions are a pre- requisite background for sensible reading of this paper, they are included as a refresher. The pur- pose 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.
There are several established methods for determining deflections of beams in mechanics of ma- terials. They include the following:1-11 (a) method of double integration (with or without the use of singularity functions), (b) method of superposition, (c) method using moment-area theorems, (d) method using Castigliano’s theorem, (e) conjugate beam method, and ( f ) method using gen- eral formulas. Naturally, there are advantages and disadvantages in using any of the above meth- ods. By and large, the method of double integration is a frequently used method in determining slopes and deflections, as well as statically indeterminate reactions at supports, of beams. With- out use of singularity functions, the method of double integration has an advantage of needing a prerequisite in mathematics only up to simple calculus. However, it has the following drawback: it requires dividing a beam into multiple segments for separate integrations and studies whenever the beam carries concentrated forces or concentrated moments. This means that more constants of integration will be generated in the process of solution, and more boundary conditions will need to be identified and imposed to provide the needed number of equations for the solution.
In this paper, attention is focused on the method of double integration with the use of singularity functions. Mastery of the definition, integration, and differentiation of singularity functions, besides simple calculus, is a prerequisite for readers of this paper. For the benefit of a wider readership, a refresher on singularity functions is included in this paper. Readers, who are famil- iar with the sign conventions in mechanics of materials and the use of singularity functions, may skip the refresher on the rudiments in the early part (Sects. II and III) of this paper.
Jong, I. (2007, June), Using A Single Equation To Account For All Loads On A Beam In The Method Of Double Integration: A Caveat Paper presented at 2007 Annual Conference & Exposition, Honolulu, Hawaii. https://peer.asee.org/1809
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