Effects of Hausdorff Dimension on the Static and Free Vibration Response of Beams with Koch Snowflake-like Cross Section

Abstract

In this manuscript, static and free vibration responses on Euler–Bernoulli beams with a Koch snowflake cross-section are studied. By applying the finite element method, the transversal displacement in static load condition, natural frequencies, and vibration modes are solved and validated using Matlab. For each case presented, the transversal displacement and natural frequency are analyzed as a Hausdorff dimension function. It is found that the maximum displacement increases as the Hausdorff dimension increases, with the relationship ymax=k0.79lndH+0.37, being k the iteration number of pre-fractal. The natural frequencies increase as ω∼M2.51, whereas the bending stiffness is expressed as EI=1165.4ln(dH+k). Numerical examples are given in order to discuss the mechanical implications.

Keywords

• Euler–Bernoulli beam
• Hausdorff dimension
• transversal displacement
• vibration modes