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Deflection of a Beam in Neutral Equilibrium à la Conjugate Beam Method: Use of Support, Not Boundary, Conditions

Abstract

Beams with flexural rigidity will deflect under loading. Is it possible to ascertain the deflection of a loaded beam in neutral equilibrium? The answer is yes according to the conjugate beam me- thod, but a resounding no according to all other established methods. The objective of this paper is to share with fellow engineering educators the insights, highlights, and several illustrative ex- amples for teaching the conjugate beam method. In particular, it is pointed out that (a) support conditions (or types), rather than boundary conditions, are what the conjugate beam method needs in finding solutions for deflections of loaded beams, (b) more support conditions than boundary conditions are usually known for beams in neutral equilibrium, and (c) the conjugate beam method often works better than other established methods in determining deflections of beams. It is demonstrated in this paper that the conjugate beam method does find the likely, or unique, deflection of a loaded beam in neutral equilibrium.

I. Introduction

All beams considered in this paper are elastic beams, which are longitudinal members subjected to transverse loads and are usually in static equilibrium. A beam is in neutral equilibrium if the force system acting on the beam is statically balanced and the potential energy of the beam in the neighborhood of its equilibrium configuration is constant.

Fig. 1 Actual beam Fig. 2 Conjugate beam The beam in Fig. 1 is in neutral equilibrium and will adopt a deflected shape. Is it possible to as- certain the deflection of a loaded beam in neutral equilibrium? The answer is yes according to the conjugate beam method,1– 4 but a resounding no according to all other established methods,3–12 such as (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 Castiglia- no’s theorem, and (e) method of segments. These other methods all expect a beam to have suffi- cient well-defined boundary conditions for use in seeking a unique solution for the deflection of the beam. The beam in Fig. 1 manifests only one known boundary condition (i.e., the deflection at the hinge support C is zero), which is simply insufficient to allow the other methods to settle on a unique solution. However, the conjugate beam method has no trouble with the beam in Fig. 1. This beam manifests three support conditions (i.e., free end at A, simple support at C, and free end at B), which are sufficient to allow a corresponding conjugate beam to be constructed as shown in Fig. 2. For now, the actual beam and the conjugate beam in Figs. 1 and 2 are used to

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Jong, I. (2008, June), Deflection Of A Beam In Neutral Equilibrium À La Conjugate Beam Method: Use Of Support, Not Boundary, Conditions Paper presented at 2008 Annual Conference & Exposition, Pittsburgh, Pennsylvania. 10.18260/1-2--4435