Map of a Bending Problem for Self-Similar Beams into the Fractal Continuum Using the Euler–Bernoulli Principle

Abstract

The bending of self-similar beams applying the Euler–Bernoulli principle is studied in this paper. A generalization of the standard Euler–Bernoulli beam equation in the FdH3 continuum using local fractional differential operators is obtained. The mapping of a bending problem for a self-similar beam into the corresponding problem for a fractal continuum is defined. Displacements, rotations, bending moments and shear forces as functions of fractal parameters of the beam are estimated, allowing the mechanical response for self-similar beams to be established. An example of the structural behavior of a cantilever beam with a load at the free end is considered to study the influence of fractality on the mechanical properties of beams.

Keywords

• self-similar beam
• Euler–Bernoulli bending principle
• fractal continuum
• Menger sponge
• Sierpinski carpet