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Session 2468

Effective Teaching and Learning of the Conjugate Beam Method: Synthesized Guiding Rules Ing-Chang Jong University of Arkansas

Abstract There are different established methods in Mechanics of Materials for determining deflections of beams. No matter which established method is used, one rightfully expects an identical solution to be obtained for the same problem. Well, not so fast! One will here see a puzzling scenario where a certain problem is amenable to solution only by the conjugate beam method, but not by any of the other methods at all. A loaded beam in equilibrium on a simple support is employed as an example of the puzzling scenario, solvable only by the conjugate beam method. The root cause of such a scenario lies in the fact that the conjugate beam method uses “support condi- tions” while all other methods use “boundary conditions” in the solutions. This paper contributes ten synthesized guiding rules for the conjugate beam method to effectively assist in its teaching and learning. Examples having different levels of complexity are included to illustrate the use of these rules. The solutions obtained by the conjugate beam method are checked and interpreted.

I. Introduction

Mechanics of Materials is either a required or an elective course in most undergraduate engi- neering curricula. Major established methods for determining deflections of beams, as taught in such a course, may include the following: 1-6 (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. The conjugate beam method was first derived, defined, and propounded for determining de- flections of beams in 1921 by Westergaard.1 It may well be called a “Westergaard method.” Readers interested in the development of this method are advised to refer to the original paper by Westergaard.1 Additionally, note that this method is one of the established methods for finding deflections of beams in the textbook by Timoshenko/MacCullough2 and that by Singer/Pytel.3 Nevertheless, this method is not easily found in most other textbooks.4,5,6

Solutions using the above methods (a) through (d ) all require that boundary conditions re- garding slopes or deflections at two or more different positions of a beam in equilibrium (e.g., zero or a specific slope, zero or a specific deflection, equal slopes, or equal deflections) be known. However, solutions using the above method (e) — conjugate beam method — require, instead, that support conditions regarding the types of support a beam has or the connections between the segments of the beam (e.g., fixed support, roller support, hinge support, internal

Proceedings of the 2004 American Society for Engineering Education Annual Conference & Exposition Copyright  2004, American Society for Engineering Education

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Jong, I. (2004, June), Effective Teaching And Learning Of The Conjugate Beam Method: Synthesized Guiding Rules Paper presented at 2004 Annual Conference, Salt Lake City, Utah. 10.18260/1-2--13636