Closed form solutions of Euler–Bernoulli beams with singularities

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Multi-cracked Euler-Bernoulli beams: Mathematical modeling and exact solutions (2013, background): “Exact solutions, in static, dynamic and stability contexts, have been found by Caddemi and co-workers in a series of papers (Biondi and Caddemi, 2005; Biondi and Caddemi, 2007; Caddemi and Calió, 2009; Caddemi and Calió, 2008; Caddemi and Calió, 2012).”
Physically-based Dirac’s delta functions in the static analysis of multi-cracked Euler–Bernoulli and Timoshenko beams (2011, methodology): “This example has been suggested by the comment by Failla and Santini (2007) about a latent limitation of the rigidity modelling by Biondi and Caddemi (2005, 2007) when discontinuities of rotation and bending moment occur at the same location.”
Exact closed-form solution for the vibration modes of the Euler-Bernoulli beam with multiple open cracks (2009, methodology): “More precisely, the Euler–Bernoulli beam in the presence of a single crack concentrated at abscissa x0 along the axis can be modelled [46] using Dirac’s delta centred at x0 in the flexural stiffness E(x)I(x) of the beam (the Young modulus E(x) multiplied by the moment of inertia I(x) of the cross-se”
Closed-form solutions for stepped Timoshenko beams with internal singularities and along-axis external supports (2013, methodology): “The considered model is a generalisation of that proposed in [10,11] for a restricted range of singularities in the Euler–Bernoulli beam.”