A New Approach to Analyzing Reactions and Deflections of Beams: Formulation and Examples

Abstract

This paper is aimed at developing a new approach to analyzing statically indeterminate reactions at supports, as well as the slopes and deflections, of beams. The approach uses a set of four general formulas, derived using singularity functions. These formulas are expressed in terms of shear forces, bending moments, distributed loads, slopes, and deflections of a beam having a constant flexural rigidity and carrying typical loads. These loads include (a) a bending moment and a shear force at the left, as well as at the right, end of the beam; (b) a concentrated force, as well as a concentrated moment, somewhere on the beam; and (c) a uniformly, as well as a linearly varying, distributed force over a portion of the beam. The approach allows one to treat reactions at supports (even supports not at the ends of a beam) as concentrated forces or moments, where corresponding boundary conditions at the points of supports are to be imposed. This feature allows one to readily determine reactions at supports as well as slopes and deflections of beams. A beam needs to be divided into segments for study if it contains discontinuities in slope at hinge connections or different flexural rigidities in different segments. Several examples are included to illustrate the new approach.

Using A Single Equation To Account For All Loads On A Beam In The Method Of Double Integration: A Caveat

“…This is why these terms are trivial and may simply be omitted in the expression for the loading function q in Eq. (11). For further illustration of singularity functions, see Example 1.…”

“…There are several established methods for determining deflections of beams in mechanics of materials. They include the following: [1][2][3][4][5][6][7][8][9][10][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 general formulas. Naturally, there are advantages and disadvantages in using any of the above methods.…”

Deflection Of A Beam In Neutral Equilibrium À La Conjugate Beam Method: Use Of Support, Not Boundary, Conditions

“…Is it possible to ascertain the deflection of a loaded beam in neutral equilibrium? The answer is yes according to the conjugate beam method, [1][2][3][4] but a resounding no according to all other established methods, [3][4][5][6][7][8][9][10][11][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 Castigliano's theorem, and (e) method of segments. These other methods all expect a beam to have sufficient well-defined boundary conditions for use in seeking a unique solution for the deflection of the beam.…”