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.