Physically-based Dirac’s delta functions in the static analysis of multi-cracked Euler–Bernoulli and Timoshenko beams

Dirac’s delta functions enable simple and effective representations of point loads and singularities in a variety of structural problems, leading very often to elegant and otherwise unworkable closed-form solutions. This is the case of cracked beams under static loads, whose theoretical and practical significance has attracted in recent years the interest of many researchers. Nevertheless, analytical formulations currently available for this problem are not completely satisfactory, either in terms of computational efficiency, when the continuity conditions must be enforced with auxiliary equations, or in terms of physical consistency, when the singularities in the beam’s flexural rigidity are represented with Dirac’s delta functions having a questionable negative sign. These considerations motivate the present study, which offers a novel and physically-based modelling of slender Euler–Bernoulli beams and short Timoshenko beams with any number and severity of cracks, conducing in both cases to exact closed-form solutions. For validation purposes, a standard finite element code is used, along with two nascent deltas (uniform and Gaussian density functions) to describe a smeared increase in the bending flexibility around the abscissa of the crack.