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Title: Physically-based Dirac’s deltas functions in the static analysis of multi-cracked Euler-Bernoulli and Timoshenko beams
Authors: Palmeri, Alessandro
Cicirello, Alice
Keywords: Concentrated crack
Dirac’s delta function
Theory of distributions
Euler-Bernoulli beam
Timoshenko beam
Generalized functions
Issue Date: 2011
Publisher: © Elsevier
Citation: PALMERI, A. and CICIRELLO, A., 2011. Physically-based Dirac’s deltas functions in the static analysis of multi-cracked Euler-Bernoulli and Timoshenko beams. International Journal of Solids and Structures, 48 (14-15), pp. 2184-2195.
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 in non-trivial examples, 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.
Version: Accepted for publication
DOI: 10.1016/j.ijsolstr.2011.03.024
URI: https://dspace.lboro.ac.uk/2134/7692
Publisher Link: http://dx.doi.org/10.1016/j.ijsolstr.2011.03.024
ISSN: 0020-7683
Appears in Collections:Published Articles (Civil and Building Engineering)

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