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Expanded Use of Discontinuity and Singularity Functions in Mechanics

Abstract

W. H. Macauley published Notes on the Deflection of Beams1 in 1919 introducing the use of discontinuity functions into the calculation of the deflection of beams. In particular, he introduced the singularity functions, the unit doublet to model a concentrated moment, the Dirac delta function to model a concentrated load and the Heaviside step function to start a uniform load at any point on the beam. Stephen H. Crandall and Norman C. Dahl incorporated these functions into their text on An Introduction to the Mechanics of Solids2 in 1959. The discontinuity functions allow writing a discontinuous function as a single expression instead of writing a series of expressions. The traditional approach requires that the different expressions be written for each region where a discontinuity appears, and when integrated, must be matched by evaluating the constants of integration. This is a mathematically laborious task that becomes more complex as the number of discontinuous functions increases. The Macauley functions are n ∀ x / a+n x ∝ a ∗ used to start a polynomial loading x / a ? ∀ at some point on the beam. These ∀ 0 x>a discontinuity functions appear in many, if not most, of the current mechanics of materials texts. There are two problems with the Macauley functions; first, they are very limited in the type of load functions that they model and second, for orders above n=1, they are difficult to stop if the region of application is only between a ∞ x ∞ b , where b is less than the length of the beam. The difficulty arises in introducing the negative of higher order polynomials at the point b. A method will be presented to analyze any continuous load function w(x) applied on the interval between a ∞ x ∞ b . Therefore, a single expression will be written for any beam loading. This expression will be integrated to determine the shear, moment, slope and deflection. Examples of different beam loadings are presented for a complete use of discontinuity functions. The use of discontinuity functions will be expanded to axial loadings, torsion of circular rods and particle dynamics.

Discontinuity Functions

The discontinuity functions were first introduced by the German mathematician A. Clebsch (1833-1872) in 18623. Walter D. Pilkey gives a complete history of Clebsch’s method in his 1964 article 4. He traced its long popularity in foreign countries and predicted it “will indubitably receive more attention in the future as the current trend is toward the engineering student becoming better equipped mathematically to cope with such analytical techniques.” Although the discontinuity functions, as defined by Macauley 1, appear in most mechanics of materials texts, they have not received the attention that Pilkey predicted. These functions were further developed by Oliver Heaviside (1850-1925), an English physicist and electrical engineer, and Paul A. M. Dirac (1902-1984), the 1933 Nobel Laureate in Physics.

There are two types of discontinuity functions; those that are singular at the point of discontinuity (singularity functions) and those that are not singular at the point of discontinuity.

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Soutas-Little, R. (2010, June), Expanded Use Of Discontinuity And Singularity Functions In Mechanics Paper presented at 2010 Annual Conference & Exposition, Louisville, Kentucky. 10.18260/1-2--16655