**Application To Structural Mechanics**

The delta function can be used in structural mechanics to describe transient loads or point loads acting on structures. The governing equation of a simple mass–spring system excited by a sudden force impulse *I* at time *t* = 0 can be written

where *m* is the mass, ξ the deflection and *k* the spring constant.

As another example, the equation governing the static deflection of a slender beam is, according to Euler-Bernoulli theory,

where *EI* is the bending stiffness of the beam, *w* the deflection, *x* the spatial coordinate and *q*(*x*) the load distribution. If a beam is loaded by a point force *F* at *x* = *x*_{0}, the load distribution is written

As integration of the delta function results in the Heaviside step function, it follows that the static deflection of a slender beam subject to multiple point loads is described by a set of piecewise polynomials.

Also a point moment acting on a beam can be described by delta functions. Consider two opposing point forces *F* at a distance *d* apart. They then produce a moment *M* = *Fd* acting on the beam. Now, let the distance *d* approach the limit zero, while *M* is kept constant. The load distribution, assuming a clockwise moment acting at *x* = 0, is written

Point moments can thus be represented by the derivative of the delta function. Integration of the beam equation again results in piecewise polynomial deflection.

Read more about this topic: Dirac Delta Function

### Famous quotes containing the words application to, application, structural and/or mechanics:

“If you would be a favourite of your king, address yourself to his weaknesses. An *application to* his reason will seldom prove very successful.”

—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

“It is known that Whistler when asked how long it took him to paint one of his “nocturnes” answered: “All of my life.” With the same rigor he could have said that all of the centuries that preceded the moment when he painted were necessary. From that correct *application* of the law of causality it follows that the slightest event presupposes the inconceivable universe and, conversely, that the universe needs even the slightest of events.”

—Jorge Luis Borges (1899–1986)

“The reader uses his eyes as well as or instead of his ears and is in every way encouraged to take a more abstract view of the language he sees. The written or printed sentence lends itself to *structural* analysis as the spoken does not because the reader’s eye can play back and forth over the words, giving him time to divide the sentence into visually appreciated parts and to reflect on the grammatical function.”

—J. David Bolter (b. 1951)

“It is only the impossible that is possible for God. He has given over the possible to the *mechanics* of matter and the autonomy of his creatures.”

—Simone Weil (1909–1943)