Analysis Of Statically Indeterminate Reactions And Deflections Of Beams Using Model Formulas: A New Approach

This paper is intended to share with educators and practitioners in mechanics a new approach that employs a set of four model formulas in analyzing statically indeterminate reactions at supports, as well as the slopes and deflections, of beams. The model formulas, in algebraic form, are derived using singularity functions. They are expressed in terms of (a) flexural rigidity of the beam; (b) slopes and deflections, as well as shear forces and bending moments, at both ends of the beam; and (c) applied loads on the beam. The types of applied loads include: (i) concentrated force and moment; (ii) uniformly distributed moment; and (iii) linearly varying distributed force. Thus, these model formulas are applicable to most problems encountered in the teaching and learning of mechanics of materials, as well as in practice. As a salient feature, this new approach allows one to treat reactions at supports, even not at the ends of a beam, simply as concentrated forces or moments, where corresponding boundary conditions at the points of supports are imposed using also the model formulas. This feature allows one to readily determine statically indeterminate reactions at supports, as well as slopes and deflections at any positions, of beams. A beam needs to be divided into segments for analysis only when it has discontinuity in slope or in flexural rigidity. Several examples are provided to illustrate this new approach.