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Analysis of Statically Indeterminate Reactions and Deflections of Beams Using Model Formulas: A New Approach

Abstract 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 sup- ports, 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 im- posed using also the model formulas. This feature allows one to readily determine statically inde- terminate 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.

I. Introduction

There are different well-known methods for determining deflections of beams in mechanics of materials. These methods may include the following: 1 1 0 (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, and (e) conjugate beam method.

This paper extends an earlier study on method of segments11 by using singularity functions and model formulas. As a result, the proposed new approach allows a considerable reduction in the number of segments required in the study. This new approach makes available an effective method for mechanics educators and practitioners when it comes to determining reactions and deflections of beams. It is aimed at contributing to the enrichment of one’s learning experience and to provide a means for independent checking on solutions obtained by other methods.

The paper goes over the description of sign conventions and derives four model formulas for the slope and deflection of a beam segment having a constant flexural rigidity and carrying a variety of commonly applied loads. These formulas, derived using singularity functions, form the basis for a new approach to solving problems involving reactions and deflections of beams. In contrast to the method of segments,11 the proposed new approach does not have to divide a beam into multiple segments even if the beam has multiple concentrated loads or simple supports not at its

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Jong, I., & Rencis, J. (2007, June), Analysis Of Statically Indeterminate Reactions And Deflections Of Beams Using Model Formulas: A New Approach Paper presented at 2007 Annual Conference & Exposition, Honolulu, Hawaii. 10.18260/1-2--1471