Dirac Delta Function - Application To Structural Mechanics

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 = x0, 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

\begin{align} q(x) &= \lim_{d \to 0} \Big( F \delta(x) - F \delta(x-d) \Big) \\ &= \lim_{d \to 0} \left( \frac{M}{d} \delta(x) - \frac{M}{d} \delta(x-d) \right) \\ &= M \lim_{d \to 0} \frac{\delta(x) - \delta(x - d)}{d}\\ &= M \delta'(x). \end{align}

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

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