The dynamics of circular arches with multiple damage
Francesco CANNIZZARO, Ilaria FIORE, Annalisa GRECO, Salvatore CADDEMI, Ivo CALIO’download PDF
Abstract. The governing equation of the dynamics of the planar inextensible Euler-Bernoulli arch with multiple damage is tackled in this study by employing the distributional approach. Precisely, the presence of impairments is modelled via cracks that can be effectively embedded in the governing equation by means of the Dirac’s delta generalised functions. The governing equation is defined over a unique integration domain. The proposed integration strategy leads to closed form expressions of the displacement mode shapes maintaining the size of the problem as that of the undamaged arch regardless the number of cracks located along the span. The latter advantage avoids the enforcement of continuity conditions at the discontinuous sections. The proposed solution extends the integration procedure proposed in the static context to the vibration analysis and allows determining the modal characteristics of damaged circular arches. The versatility of the obtained closed form solution allows a straightforward execution of parametric analyses and is here adopted to evaluate the sensitivity of the eigenproperties of the multi-cracked circular arch to the change of meaningful geometric and mechanical parameters.
Closed form Solution, Circular Cracked Arch, Vibration Analysis
Published online 3/17/2022, 6 pages
Copyright © 2023 by the author(s)
Published under license by Materials Research Forum LLC., Millersville PA, USA
Citation: Francesco CANNIZZARO, Ilaria FIORE, Annalisa GRECO, Salvatore CADDEMI, Ivo CALIO’, The dynamics of circular arches with multiple damage, Materials Research Proceedings, Vol. 26, pp 411-416, 2023
The article was published as article 67 of the book Theoretical and Applied Mechanics
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
 P. Chidamparam and A. W. Leissa, “Vibration of planar curved beams, rings and arches,” Applied Mechanics Review, vol. 46, no. 9, pp. 467-483, 1993. https://doi.org/10.1115/1.3120374
 A. Greco and A. Pau, “Detection of a concentrated damage in a parabolic arch by measured static displacements,” Structural Engineering and Mechanics, vol. 39, no. 6, pp. 751-765, 2011. https://doi.org/10.12989/sem.2011.39.6.751
 U. Eroglu, G. Ruta and E. Tufekci, “Natural frequencies of parabolic arches with a single crack on opposite cross-section sides,” Journal of Vibration and Control, vol. 25, no. 7, pp. 1313-1325, 2019. https://doi.org/10.1177/1077546319825681
 A. D. Dimarogonas, “Vibration of cracked structures: a state of the art review,” Engineering Fracture Mechanics, vol. 55, no. 5, pp. 831-857, 1996. https://doi.org/10.1016/0013-7944(94)00175-8
 G. R. Irwin, “Relation of stresses near a crack to the crack extension force,” in 9th Congress of Applied Mechanics, Brussels, 1957.
 M. Krawczuk and W. Ostachowicz, “Natural vibrations of a clamped-clamped arch with an open transverse crack,” Journal of Vibration and Acoustics, Transactions of the ASME, vol. 119, no. 2, pp. 145-150, 1997. https://doi.org/10.1115/1.2889695
 I. Caliò, D. D’Urso and A. Greco, “The influence of damage on the eigen-properties of Timoshenko spatial arches,” Computers & Structures, vol. 190, pp. 13-24, 2017. https://doi.org/10.1016/j.compstruc.2017.04.012
 C. S. Huang, Y. P. Tseng, S. H. Chang and C. L. Hung, “Out-of-plane dynamic analysis of beams with arbitrarily varying curvature and cross-section by dynamic stiffness matrix method,” International Journal of Solids and Structures, vol. 37, pp. 495-513, 2000. https://doi.org/10.1016/S0020-7683(99)00017-7
 S. Caddemi, I. Caliò and F. Cannizzaro, “Closed-form solutions for stepped Timoshenko beams with internal singularities and along-axis external supports,” Archive of Applied Mechanics, vol. 83, no. 4, pp. 559-577, 2013. https://doi.org/10.1007/s00419-012-0704-7
 A. Palmeri and A. Cicirello, “Physically-based Dirac’s delta functions in the static analysis of multi-cracked Euler-Bernoulli and Timoshenko beams,” International Journal of Solids and Structures, vol. 48, no. 14-15, pp. 2184-2195, 2011. https://doi.org/10.1016/j.ijsolstr.2011.03.024
 F. Cannizzaro, A. Greco, S. Caddemi and I. Caliò, “Closed form solutions of a multi-cracked circular arch under static loads,” International Journal of Solids and Structures, vol. 121, pp. 191-200, 2017. https://doi.org/10.1016/j.ijsolstr.2017.05.026
 C. Bilello, Theoretical and experimental investigation on damaged beams under moving systems, University of Palermo: PhD Thesis, 2001.