Fractal Continuum Calculus of Functions on Euler-Bernoulli Beam

Abstract

A new approach for solving the fractal Euler-Bernoulli beam equation is proposed. The mapping of fractal problems in non-differentiable fractals into the corresponding problems for the fractal continuum applying the fractal continuum calculus (FdH3-CC) is carried out. The fractal Euler-Bernoulli beam equation is derived as a generalization using FdH3-CC under analogous assumptions as in the ordinary calculus and then it is solved analytically. To validate the spatial distribution of self-similar beam response, three different classical beams with several fractal parameters are analysed. Some mechanical implications are discussed.

Keywords

• fractal continuum calculus
• Hausdorff dimension
• Euler-Bernoulli beam
• transversal displacement