Example 1

Problem 2 (5 pts)

For a simply supported beam of length \(L\) and bending stiffness \(EI\), resting on a Winkler elastic foundation and loaded as shown below, determine the functions of deflection \(y(x)\), bending moment \(M(x)\), and shear force \(V(x)\) using a Fourier series representation.

Compute the corresponding Fourier coefficients and express all of the above functions as sums over \(k = 1,2,\ldots,K\). Also evaluate these functions at midspan \(x = L/2\) and for the lowest harmonic that yields a non‑zero contribution.

The governing differential equation of a beam on an elastic foundation is \( EI\,\frac{d^{4} y}{dx^{4}} + \kappa\, y(x) = q(x) \), where \(\kappa\) is the foundation stiffness.

Note: Evaluate definite integrals by first writing the antiderivative and then applying the limits of integration.

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Solution

Fourier approximation of the load and the response (deflection)

$$ q(x)=\sum_{k=1}^{K} q_{k} \sin\!\left(\frac{k \pi x}{L}\right),\qquad y(x)=\sum_{k=1}^{K} y_{k} \sin\!\left(\frac{k \pi x}{L}\right) $$

Load coefficients:

$$ q_{k}=\frac{2}{L} \int_{L/4}^{3L/4} q_{0} \sin\!\left(\frac{k \pi x}{L}\right)\,dx = -\frac{2 q_{0}}{k \pi}\Big[\cos\!\left(\frac{3 k \pi}{4}\right)-\cos\!\left(\frac{k \pi}{4}\right)\Big] $$

Deflection coefficients: substitute the Fourier expansions of \(q(x)\) and \(y(x)\) into the governing beam equation.

$$ EI\left(\frac{k^{4} \pi^{4}}{L^{4}}\right) y_{k} + \kappa\, y_{k} = q_{k},\qquad k=1,2,\ldots,K $$

Thus:

$$ y_{k} = \frac{q_{k}}{EI\left(\frac{k^{4} \pi^{4}}{L^{4}}\right) + \kappa} $$

Continuous function:

$$ y(x) = \sum_{k=1}^{K} y_{k} \sin\!\left(\frac{k \pi x}{L}\right) $$

Evaluate at \(x = L/2\) and \(k=1\).