# Cross-Sectional Performance of Hollow Square Prisms with Rounded Edges

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## Abstract

**:**

## 1. Introduction

## 2. Modeling a Rounded Square with Filleted Corners

## 3. Cross-Sectional Performance

## 4. Numerical Conditions

## 5. Results: Improvement Ratio

## 6. Discussion

#### 6.1. Variable Range Estimation of $\eta $ and $\zeta $

#### 6.2. Effect of the Fillet at the Vertices

#### 6.3. Implication for the Mechanical Rigidity and Strength

#### 6.4. Versatility of the Theoretical Model

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**(

**a**) Circular sector defined by the angles $\alpha $ and $\beta $ and the radius $\rho $. (

**b**) Right-angle triangle with the base d and the apex $\varphi $.

## Appendix B

**Figure A2.**Diagram of the four domains in the first quadrant ($x>0$ and $y>0$) of the rounded square. Each domain is marked by a gray area. ${J}_{i}$$(i=1,\phantom{\rule{3.33333pt}{0ex}}2,\phantom{\rule{3.33333pt}{0ex}}3,\phantom{\rule{3.33333pt}{0ex}}4)$ represents the second moment of area of the ith domain with respect to the x axis. (

**a**) Domain 1; the gray area obtained by excluding the shaded right triangle from the vertically elongated sector. (

**b**) Domain 2; the gray area obtained by excluding the triangle from the long sector. (

**c**) Domain 3 (the gray small sector at the upper right) and Domain 4 (the gray kite-shaped square obtained by excluding the two shaded triangles from the right square).

Parameters | $\mathit{\rho}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | d | $\mathit{\varphi}$ |
---|---|---|---|---|---|

Domain 1 | $\frac{\ell}{2cos\theta}+h$ | $\theta $ | $\frac{\pi}{2}$ | $\frac{\ell}{2}(tan\theta -1)$ | $\frac{\pi}{2}-\theta$ |

Domain 2 | $\frac{\ell}{2cos\theta}+h$ | 0 | $\frac{\pi}{2}-\theta$ | $\frac{\ell}{2}\left(tan\theta -1\right)$ | $\frac{\pi}{2}-\theta$ |

Domain 3 | h | $\frac{\pi}{2}-\theta$ | $\theta $ | n/a | n/a |

Domain 4 | n/a | n/a | n/a | $\frac{\ell}{2}$ | $\frac{\pi}{2}-\theta$ |

## Appendix C

## References

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**Figure 1.**(

**a**) Photo of the cross-section of a square bamboo: Chimonobambusa quadrangularis (Franceschi) Makino. (

**b**) Approximate curve of the square bamboo’s cross-section. (

**c**) Square cross-section of the mint stem; Lamium album var. barbatum [12]. (

**d**) Triangular cross-section of the papyrus stem; Cyperus microiria [13].

**Figure 2.**A rounded square model with a four-fold symmetry. Schematic definitions of the three parameters, ℓ, $\theta $, and h are shown.

**Figure 3.**(

**a**) Geometric variation of a rounded square due to changes in the variables $\theta $ and ℓ under the condition where the enclosed area $A=\pi {a}_{0}^{2}$ is fixed. The constant ${a}_{0}$ serves as the unit of length. Dotted lines show the reference squares with side length ℓ. (

**b**) Dependence of the small sector’s radius h on $\theta $. The value of $\ell /{a}_{0}$ is varied with an interval of $0.2$.

**Figure 4.**Improvement ratios, $\eta $ and $\zeta $, for the case of ${\ell}_{\mathrm{out}}/{a}_{\mathrm{out}}=1.2$ and ${\ell}_{\mathrm{inn}}/{a}_{\mathrm{inn}}=0.8$. (

**a**) $\eta $ as a function of ${\theta}_{\mathrm{out}}$. (

**b**) $\zeta $ as a function of ${\theta}_{\mathrm{out}}$. Upper (black) and lower (gray) branches correspond to the results with respect to Axis-1 and Axis-2 depicted in Figure 1b, respectively.

**Figure 5.**Improvement ratios for the case of ${\ell}_{\mathrm{out}}/{a}_{\mathrm{out}}=0.8$ and ${\ell}_{\mathrm{inn}}/{a}_{\mathrm{inn}}=1.2$. (

**a**) $\eta $; (

**b**) $\zeta $.

**Figure 6.**(

**a**) A model of a triangular cross-section with rounded sides and filleted corners, mimicking the cross-section of a papyrus; see Figure 1. (

**b**) A model of a non-symmetric hollow square cross-section.

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**MDPI and ACS Style**

Shima, H.; Furukawa, N.; Kameyama, Y.; Inoue, A.; Sato, M.
Cross-Sectional Performance of Hollow Square Prisms with Rounded Edges. *Symmetry* **2020**, *12*, 996.
https://doi.org/10.3390/sym12060996

**AMA Style**

Shima H, Furukawa N, Kameyama Y, Inoue A, Sato M.
Cross-Sectional Performance of Hollow Square Prisms with Rounded Edges. *Symmetry*. 2020; 12(6):996.
https://doi.org/10.3390/sym12060996

**Chicago/Turabian Style**

Shima, Hiroyuki, Nao Furukawa, Yuhei Kameyama, Akio Inoue, and Motohiro Sato.
2020. "Cross-Sectional Performance of Hollow Square Prisms with Rounded Edges" *Symmetry* 12, no. 6: 996.
https://doi.org/10.3390/sym12060996