Fractal Continuum Mapping Applied to Timoshenko Beams
Abstract
:1. Introduction
2. Terminology and Notations
2.1. Fractal Continuum Calculus, -CC
- -
- The fractional norm is given by , where , and the mapping of the fractional coordinates in the fractal continuum to the integer coordinates in the embedding Euclidean space is defined as
- -
- The distance between two points is given as , where .
- -
- The fractal gradient operator is , where and .
- -
- The divergence operator is .
- -
- The Laplacian is .
2.2. Fractal Beams
2.2.1. Carpinteri’s Beam
2.2.2. Balankin’s Beam
3. Fractal Timoshenko Beam Equations
Timoshenko Beam Equations in -CC
4. Transversal Displacement on Fractal Beams
- I
- Simply supported fractal beam
- II
- Cantilever fractal beam
Discussion of Results
5. Conclusions
- i
- The stiffness of the self-similar beam is governed by the parameters , , , and the iteration number of the pre-fractal.
- ii
- The flexural rigidity is affected by the geometry and fractal topology of the beam, such that when increases, decreases. However, Carpinteri’s beam is stiffer than Balankin’s beam, as shown in Figure 4a,c.
- iii
- The fractal mass of each iteration and the boundary conditions of beams, as well as the length scales of similarity, are linked with the scale effect of fractal beams.
- iv
- Using the generalized Timoshenko beam formulation is possible to obtain the exact solutions of a complex scale-invariant beam through the parameter , which includes its topology and fractal geometry.
- v
- The introduced solution coincides with the classical Timoshenko beam theory when .
- vi
- A devil’s staircase-like structural response is presented in Carpinteri’s beams.
- vii
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Beam | Dimension | ||||
---|---|---|---|---|---|
3 | |||||
Carpinteri’s | 1 | ||||
1 | |||||
3 | |||||
Balankin’s | 2 | ||||
1 |
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Samayoa, D.; Alcántara, A.; Mollinedo, H.; Barrera-Lao, F.J.; Torres-SanMiguel, C.R. Fractal Continuum Mapping Applied to Timoshenko Beams. Mathematics 2023, 11, 3492. https://doi.org/10.3390/math11163492
Samayoa D, Alcántara A, Mollinedo H, Barrera-Lao FJ, Torres-SanMiguel CR. Fractal Continuum Mapping Applied to Timoshenko Beams. Mathematics. 2023; 11(16):3492. https://doi.org/10.3390/math11163492
Chicago/Turabian StyleSamayoa, Didier, Alexandro Alcántara, Helvio Mollinedo, Francisco Javier Barrera-Lao, and Christopher René Torres-SanMiguel. 2023. "Fractal Continuum Mapping Applied to Timoshenko Beams" Mathematics 11, no. 16: 3492. https://doi.org/10.3390/math11163492