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This study presents a Ritz-type analysis for obtaining the vibration frequencies and mode shapes of beams that have a discontinuity in slope. Such structures have a wide range of applications in engineering as they often represent structural idealizations of complex structures such as robotic arms, crack modeling and foldable wings of aircraft. In the present study, the beams have been modeled using Euler Bernoulli’s theory, and the discontinuity in slope is represented by a torsional spring connecting the two segments of the beam. The beams are discretized into subdomains based on the location of the discontinuity. Legendre polynomials are used to define the displacement variation over individual uniform domains. The continuity of displacement is applied at the interface of adjacent subdomains and is enforced into the global system of equations using a condensation procedure. This procedure eliminates the dependent Ritz constant obtained from the displacement continuity. This study focuses on obtaining the vibration frequencies and mode shapes of a simply-supported beam with a discontinuity in slope and compares the results with the analytical solution. The paper would be of interest to researchers involved with the structural health monitoring of beams with cracks, robotic arms, and vibrations of folding wings like the one being considered for 777-X.