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

By Alessandro Palmeri, A. Cicirello. 2011

93 citations recorded by the source corpus.

Abstract 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.

Subfields: Civil and Structural Engineering, Mechanical Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/5dcee1f30c84ff9bf5d6e84d699fcad639257b2c

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In Civil and Structural Engineering

1. Ivo Caliò
2. Salvatore Caddemi
3. Antonino Morassi
4. Francesco Cannizzaro
5. Alessandro Palmeri
6. A. Cicirello

In Mechanical Engineering

1. Salvatore Caddemi
2. Ivo Caliò
3. Francesco Cannizzaro
4. Alessandro Palmeri
5. A. Cicirello

In Mechanics of Materials

1. Ivo Caliò
2. Salvatore Caddemi
3. Francesco Cannizzaro
4. Antonino Morassi
5. Alessandro Palmeri
6. A. Cicirello

Citations in the corpus

• Closed-form solutions for stepped Timoshenko beams with internal singularities and along-axis external supports (2013, background): “Validation of the results provided by the latter model has been provided in [15], where it has been shown how internal concentrated cracks can be alternatively modelled as concentrated flexural flexibility increments recovering the same solutions as in [11], and the Timoshenko model can also be trea”
• Multi-cracked Euler-Bernoulli beams: Mathematical modeling and exact solutions (2013, background): “This piecewise constant stiffness model corresponds to a family of nascent deltas in the bending flexibility considered in reference Palmeri and Cicirello (2011). The cantilever is subjected to a transversal load distribution p 1⁄4 pðxÞ 2 Lð 1;1Þ, where Lð 1;1Þ is the Lebesgue space of measurable an”