Area moment of inertia of a “flower-shaped” cross section and generalization for N-fold rotational symmetry and multipole moments

Motivated by three-point bending tests with a nanoindenter and the necessary analysis, the area moment of inertia of tiny bending specimens, with a special “flower-shaped” cross section, is calculated analytically. If such cross sections have a four-fold rotational symmetry, it can be shown that a rotation of the cross section by any angle, with respect to the reference plane, does not change the result. By demonstrating that this is valid also for N-fold rotational symmetry with N ≥ 3 and by generalization for multipole moments of any order, some interesting equations and principles are found. (Originally, the chapters 1–4 were an internal communication in the Institute of Advanced Ceramics, TUHH. Later on, the chapters 5–7 and the appendices were added, and the introduction was extended. Although the calculations are standard, they are perhaps also of interest to others. In order not to lose track of the idea and the calculations, they are published in this format.)

Schlagworte

bending test

area moment of inertia

rotational symmetry

second moment of area

central binomial coefficient

trigonometric identity

DDC Class

600: Technik

620: Ingenieurwissenschaften

More Funding Information

Deutsche Forschungsgemeinschaft (DFG)

Lizenz

https://creativecommons.org/licenses/by-nc-sa/4.0/

Name

TUHH-Moment-of-Inertia-2020-Jelitto.pdf

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1.04 MB

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Adobe PDF

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Area moment of inertia of a “flower-shaped” cross section and generalization for N-fold rotational symmetry and multipole moments