# Symbolic Parametric Representation of the Area and the Second Moments of Area of Periodic B-Spline Cross-Sections

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. State of the Art

## 2. Materials and Methods

#### 2.1. Moments of Area of a Periodic B-Spline

#### 2.2. Parametric Representation of the B-Spline of a Triangle

#### 2.3. Comparison to Polygon Cross-Sections

#### 2.4. Numerical Comparison Framework

#### 2.4.1. Spline Cross-Section with Valid Cross-Section Property

#### 2.4.2. Spline Cross-Section as Valid Jordan Curve

#### 2.4.3. Spline Cross-Section Numerically Compared to Polygon and Image Cross-Section

## 3. Results

#### 3.1. Moments of Area Parametrization of a Triangle Control Polygon

#### 3.2. Moments of Area Parametrization of a Quadrilateral Control polygon Area

#### 3.2.1. Moments of Area Parametrization of a Rectangle Control Polygon

#### 3.2.2. Moments of Area Parametrization of a Parallelogram Control Polygon

#### 3.2.3. Moments of Area Parametrization of a Trapeze Control Polygon

#### 3.3. Moments of Area Parametrization of a Symmetric Pentagon Control Polygon

#### 3.4. Moments of Area Parametrization of a Symmetric Hexagonal Control Polygon

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Gross, D.; Hauger, W.; Schröder, J.; Wall, W.A. (Eds.) Balkenbiegung. In Technische Mechanik 2: Elastostatik; Springer: Berlin/Heidelberg, Germany, 2017; pp. 81–165. [Google Scholar] [CrossRef]
- Bendsoe, M.P.; Sigmund, O. Topology Optimization: Theory, Methods, and Applications, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar] [CrossRef]
- Changizi, N.; Warn, G.P. Topology optimization of structural systems based on a nonlinear beam finite element model. Struct. Multidiscip. Optim.
**2020**, 62, 2669–2689. [Google Scholar] [CrossRef] - Denk, M.; Rother, K.; Gadzo, E.; Paetzold, K. Multi-Objective Topology Optimization of Frame Structures Using the Weighted Sum Method. In Proceedings of the Munich Symposium on Lightweight Design 2021; Springer: Berlin/Heidelberg, Germany, 2022; pp. 83–92. [Google Scholar] [CrossRef]
- Fredricson, H.; Johansen, T.; Klarbring, A.; Petersson, J. Topology optimization of frame structures with flexible joints. Struct. Multidiscip. Optim.
**2003**, 25, 199–214. [Google Scholar] [CrossRef] - Lim, J.; You, C.; Dayyani, I. Multi-objective topology optimization and structural analysis of periodic spaceframe structures. Mater. Des.
**2020**, 190, 108552. [Google Scholar] [CrossRef] - Denk, M.; Rother, K.; Paetzold, K. Multi-Objective Topology Optimization of Heat Conduction and Linear Elastostatic using Weighted Global Criteria Method. In DS 106: Proceedings of the 31st Symposium Design for X (DFX2020); The Design Society: Bamberg, Germany, 2020; pp. 91–100. [Google Scholar] [CrossRef]
- Stangl, T.; Wartzack, S. Feature based interpretation and reconstruction of structural topology optimization results. In DS 80-6 Proceedings of the 20th International Conference on Engineering Design (ICED 15) Vol 6: Design Methods and Tools—Part 2 Milan, Italy, 27–30.07.15; Weber, C., Husung, S., CantaMESsa, M., Cascini, G., Marjanovic, D., Graziosi, S., Eds.; Design Society: Glasgow, UK, 2015; Volume 6, pp. 235–245. [Google Scholar]
- Nana, A.; Cuillière, J.-C.; Francois, V. Automatic reconstruction of beam structures from 3D topology optimization results. Comput. Struct.
**2017**, 189, 62–82. [Google Scholar] [CrossRef] - Tang, P.-S.; Chang, K.-H. Integration of topology and shape optimization for design of structural components. Struct. Multidiscip. Optim.
**2001**, 22, 65–82. [Google Scholar] [CrossRef] - Denk, M.; Klemens, R.; Paetzold, K. Beam-colored Sketch and Image-based 3D Continuous Wireframe Reconstruction with different Materials and Cross-Sections. In Entwerfen Entwickeln Erleben in Produktentwicklung und Design 2021; Stelzer, R., Krzywinski, J., Eds.; TUDpress: Dresden, Germany, 2021; pp. 345–354. [Google Scholar] [CrossRef]
- Denk, M.; Rother, K.; Paetzold, K. Fully Automated Subdivision Surface Parametrization for Topology Optimized Structures and Frame Structures Using Euclidean Distance Transformation and Homotopic Thinning. In Proceedings of the Munich Symposium on Lightweight Design 2020; Pfingstl, S., Horoschenkoff, A., Höfer, P., Zimmermann, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2021; pp. 18–27. [Google Scholar] [CrossRef]
- Amroune, A.; Cuillière, J.-C.; François, V. Automated Lofting-Based Reconstruction of CAD Models from 3D Topology Optimization Results. Comput. Aided Des.
**2022**, 145, 103183. [Google Scholar] [CrossRef] - Piegl, L.; Tiller, W. The NURBS Book. In Monographs in Visual Communications; Springer: Berlin/Heidelberg, Germany, 1995. [Google Scholar] [CrossRef]
- Denk, M. Curve Skeleton and Moments of Area Supported Beam Parametrization in Multi-Objective Compliance Structural Optimization. Ph.D. Dissertation, Technische Universität Dresden, Saxony, Germany, 2022. Available online: https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-822020 (accessed on 12 February 2023).
- Denk, M.; Rother, K.; Höfer, T.; Mehlstäubl, J.; Paetzold, K. Euclidian Distance Transformation, Main Axis Rotation and Noisy Dilitation Supported Cross-Section Classification with Convolutional Neural Networks. In Proceedings of the Design Society; Cambridge University Press: Cambridge, UK, 2021; Volume 1, pp. 1401–1410. [Google Scholar] [CrossRef]
- Denk, M.; Rother, K.; Neuhäusler, J.; Petroll, C.; Paetzold, K. Parametrization of Cross-Sections by CNN Classification and Moments of Area Regression for Frame Structures. In Proceedings of the Munich Symposium on Lightweight Design 2021, Online, 6 August 2022; Springer: Berlin/Heidelberg, Germany, 2022; pp. 93–103. [Google Scholar] [CrossRef]
- Azeem, M.; Jamil, M.K.; Shang, Y. Notes on the Localization of Generalized Hexagonal Cellular Networks. Mathematics
**2023**, 11, 844. [Google Scholar] [CrossRef] - Nadeem, M.F.; Azeem, M. The fault-tolerant beacon set of hexagonal Möbius ladder network. Math. Methods Appl. Sci.
**2023**. [Google Scholar] [CrossRef] - Overhauser, A.W. Analytic Definition of Curves and Surfaces by Parabolic Blending. arXiv
**2005**, arXiv:cs/0503054. [Google Scholar] - El-Abbasi, N.; Meguid, S.A.; Czekanski, A. On the modelling of smooth contact surfaces using cubic splines. Int. J. Numer. Methods Eng.
**2001**, 50, 953–967. [Google Scholar] [CrossRef] - Catmull, E.; Rom, R. A CLASS OF LOCAL INTERPOLATING SPLINES. In Computer Aided Geometric Design; Barnhill, R.E., Riesenfeld, R.F., Eds.; Academic Press: Cambridge, MA, USA, 1974; pp. 317–326. [Google Scholar] [CrossRef]
- Curry, H.B.; Schoenberg, I.J. On Pólya frequency functions IV: The fundamental spline functions and their limits. J. Anal. Math.
**1966**, 17, 71–107. [Google Scholar] [CrossRef] - Borisenko, V.V. Construction of Optimal Bézier Splines. J. Math. Sci.
**2019**, 237, 375–386. [Google Scholar] [CrossRef] - Clark, J.H. Parametric Curves, Surfaces and Volumes in Computer Graphics and Computer-Aided Geometric Design; Technical Report; Stanford University: Stanford, CA, USA, 1981. [Google Scholar] [CrossRef]
- Woodward, C.D. Cross-sectional design of B-spline surfaces. Comput. Graph.
**1987**, 11, 193–201. [Google Scholar] [CrossRef] - Pérez-Arribas, F.; Pérez-Fernández, R. A B-spline design model for propeller blades. Adv. Eng. Softw.
**2018**, 118, 35–44. [Google Scholar] [CrossRef] - Capobianco, G.; Eugster, S.R.; Winandy, T. Modeling planar pantographic sheets using a nonlinear Euler–Bernoulli beam element based on B-spline functions. PAMM
**2018**, 18, e201800220. [Google Scholar] [CrossRef] - Goel, A.; Anand, S. Design of Functionally Graded Lattice Structures using B-splines for Additive Manufacturing. Procedia Manuf.
**2019**, 34, 655–665. [Google Scholar] [CrossRef] - Weeger, O. Isogeometric sizing and shape optimization of 3D beams and lattice structures at large deformations. Struct. Multidiscip. Optim.
**2022**, 65, 43. [Google Scholar] [CrossRef] - Zhang, L.; Feih, S.; Daynes, S.; Wang, Y.; Wang, M.Y.; Wei, J.; Lu, W.F. Buckling optimization of Kagome lattice cores with free-form trusses. Mater. Des.
**2018**, 145, 144–155. [Google Scholar] [CrossRef] - Rozenthal, P.; Gattass, M. Geometrical properties in the B-spline representation of arbitrary domains. Commun. Appl. Numer. Methods
**1987**, 3, 345–349. [Google Scholar] [CrossRef] - Sheynin, S.; Tuzikov, A. Moment computation for objects with spline curve boundary. IEEE Trans. Pattern Anal. Mach. Intell.
**2003**, 25, 1317–1322. [Google Scholar] [CrossRef] - Sheynin, S.; Tuzikov, A. Area and Moment Computation for Objects with a Closed Spline Boundary. In Computer Analysis of Images and Patterns; Petkov, N., Westenberg, M.A., Eds.; Springer: Berlin/Heidelberg, Germany, 2003; pp. 33–40. [Google Scholar] [CrossRef]
- Soldea, O.; Elber, G.; Rivlin, E. Exact and efficient computation of moments of free-form surface and trivariate based geometry. Comput. Des.
**2002**, 34, 529–539. [Google Scholar] [CrossRef] - Huang, Z.; Cohen, F. Affine-invariant B-spline moments for curve matching. IEEE Trans. Image Process.
**1996**, 5, 1473–1480. [Google Scholar] [CrossRef] [PubMed] - Jacob, M.; Blu, T.; Unser, M. An exact method for computing the area moments of wavelet and spline curves. IEEE Trans. Pattern Anal. Mach. Intell.
**2001**, 23, 633–642. [Google Scholar] [CrossRef] - Flusser, J.; Suk, T. On the Calculation of Image Moments; Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic: Prague, Czechia, 1999. [Google Scholar]
- Soerjadi, R. On the Computation of the Moments of a Polygon, with Some Applications; Delft University of Technology: Delft, The Netherlands, 1968. [Google Scholar]
- Hally, D. Calculation of the Moments of Polygons; Technical Report ADA183444; Defense Technical Information Center: Fort Belvoir, VA, USA, 1987. [Google Scholar]
- Botsch, M.; Kobbelt, L.; Pauly, M.; Alliez, P.; Lévy, B. Polygon Mesh Processing; CRC Press: Boca Raton, FL, USA, 2010. [Google Scholar]

**Figure 1.**B-spline beam: Cross-section and centerline beam; Cartesian coordinates of control points; parametric coordinates of contour points.

**Figure 2.**Variation of the number of control points: B-spline segment; two segments; periodic B-spline; sharped one time repeated B-spline; polygon.

**Figure 3.**Parametrization of the control polygon: triangle; rectangle; parallelogram; quadrilateral; trapeze; symmetric pentagon; symmetric hexagon.

TSP | Tensor Product Spline | CSG | Constructive Solid Geometry |
---|---|---|---|

${C}^{1}$ | Tangential Continuity | $A$ | Zeroth Moment of Area (Cross-Sectional Area) |

${C}^{2}$ | Curvature Continuity | ${S}_{x},{S}_{y}$ | First Moment of Area |

$\mathit{T}$ | Monomial Basis | ${I}_{x},{I}_{y},{I}_{xy}$ | Second Moment of Area |

$\mathit{M}$ | Geometry Matrix | ${A}_{Img},{I}_{{x}_{Img}}$ ${I}_{{y}_{Img}},{I}_{x{y}_{Img}}$ | Moments of Area of the Binary Image |

${\mathit{C}}_{i}$ | Control Point of the Tensor Product Spline | ${A}_{Poly},{I}_{{x}_{Poly}}$ ${I}_{{y}_{Poly}},{I}_{x{y}_{Poly}}$ | Moments of Area of the Polygon |

$\mathit{P}\left(t\right)$ | Tensor Product Spline | ${A}_{yy}^{\left(0,0\right)},{I}_{yy}^{\left(0,0\right)}$ ${I}_{xy}^{\left(0,0\right)},{I}_{xx}^{\left(0,0\right)}$ | Moments of Area referred to the Origin of the Coordinate System |

${x}_{s},{y}_{s}$ | Center Points of the Cross-Section | ${A}_{TPS},{I}_{{x}_{TPS}}$ ${I}_{{y}_{TPS}},{I}_{x{y}_{TPS}}$ | Moments of Area of the Tensor Product Spline |

$c,b,h,p,a,d$ | Geometric Sizes | ${I}_{1},{I}_{2}$ | Principal Moments of Area |

$B$ | Binary Image | ${\u03f5}_{A},{\u03f5}_{{I}_{xy}},{\u03f5}_{{I}_{x}},{\u03f5}_{{I}_{y}}$ | Relative Errors |

$\sigma \xb2$ | Variance | $\overline{{\u03f5}_{[\dots ]}}$ | Mean Value of $[\dots ]$ |

Triangle | Rectangle | |
---|---|---|

$A$ | $c\frac{h}{2}$ | $bh$ |

${I}_{x}$ | $c\frac{{h}^{3}}{36}$ | $b\frac{{h}^{3}}{12}$ |

${I}_{y}$ | $ch\frac{{c}^{2}-cp+{p}^{2}}{36}$ | $\frac{{b}^{3}h}{12}$ |

${I}_{xy}$ | $c\frac{{h}^{2}\left(c-2p\right)}{72}$ | $0$ |

${x}_{s},{y}_{s}$ | $\left(\frac{c}{3}+\frac{p}{3},\frac{h}{3}\right)$ | $\left(\frac{b}{2},\frac{h}{2}\right)$ |

**Table 3.**Example of three cases of different control polygons as B-splines and alternative representations as six-line segment polygons along the curve and as 8 × 8 binary images.

Control Polygon | Case 1 $\mathbf{b}=1,\mathbf{p}=0.75,\mathbf{h}=1.5$ | Case 2 $\mathbf{b}=1,\mathbf{p}=2.0,\mathbf{h}=2.5$ | Case 3 $\mathbf{b}=1,\mathbf{p}=0.1,\mathbf{h}=0.5$ |
---|---|---|---|

Triangle | |||

Rectangle | |||

Parallelogram | |||

Trapeze | |||

Pentagon | |||

Hexagon |

Equation | Error [%] | ${\mathit{P}}_{100}$ | ${\mathit{P}}_{10}$ | ${\mathit{I}}_{128}$ | ${\mathit{I}}_{16}$ | |
---|---|---|---|---|---|---|

$A$ | $21c\frac{h}{80}$ | $\overline{{\u03f5}_{A}}$ | 0.007 | 0.879 | 1.922 | 11.951 |

${I}_{x}$ | $2833c\frac{{h}^{3}}{443520}$ | $\sigma \xb2\left({\u03f5}_{A}\right)$ | 0.000 | 0.000 | 0.168 | 4.990 |

${I}_{y}$ | $2833ch\frac{{c}^{2}-cp+{p}^{2}}{443520}$ | $\overline{{\u03f5}_{{I}_{x}}}$ | 0.015 | 1.776 | 4.299 | 19.075 |

${I}_{xy}$ | $2833c\frac{{h}^{2}\left(c-2p\right)}{887040}$ | $\sigma \xb2\left({\u03f5}_{{I}_{x}}\right)$ | 0.000 | 0.000 | 0.810 | 5.377 |

${x}_{s},{y}_{s}$ | $\left(\frac{c}{3}+\frac{p}{3},\frac{h}{3}\right)$ | $\overline{{\u03f5}_{{I}_{y}}}$ | 0.015 | 1.776 | 3.247 | 18.471 |

${I}_{x}{I}_{y}-{I}_{xy}^{2}$ | $8025889\frac{{c}^{4}{h}^{4}}{262279987200}$ | $\sigma \xb2\left({\u03f5}_{{I}_{y}}\right)$ | 0.000 | 0.000 | 0.291 | 5.914 |

$\overline{{\u03f5}_{{I}_{xy}}}$ | 0.015 | 1.776 | 5.046 | 25.759 | ||

$\sigma \xb2\left({\u03f5}_{{I}_{xy}}\right)$ | 0.000 | 0.000 | 1.575 | 6.264 |

Equation | Error [%] | ${\mathit{P}}_{100}$ | ${\mathit{P}}_{10}$ | ${\mathit{I}}_{128}$ | ${\mathit{I}}_{16}$ | |
---|---|---|---|---|---|---|

$A$ | $61b\frac{h}{90}$ | $\overline{{\u03f5}_{A}}$ | 0.004 | 0.505 | 2.674 | 13.488 |

${I}_{x}$ | $27371b\frac{{h}^{3}}{748440}$ | $\sigma \xb2\left({\u03f5}_{A}\right)$ | 0.000 | 0.000 | 1.110 | 5.877 |

${I}_{y}$ | $27371\frac{{b}^{3}h}{748440}$ | $\overline{{\u03f5}_{{I}_{x}}}$ | 0.008 | 1.009 | 3.858 | 16.736 |

${I}_{xy}$ | $0$ | $\sigma \xb2\left({\u03f5}_{{I}_{x}}\right)$ | 0.000 | 0.000 | 1.473 | 6.440 |

${x}_{s},{y}_{s}$ | $\left(\frac{b}{2},\frac{h}{2}\right)$ | $\overline{{\u03f5}_{{I}_{y}}}$ | 0.008 | 1.009 | 3.687 | 17.262 |

${I}_{x}{I}_{y}-{I}_{xy}^{2}$ | $749171641\frac{{b}^{4}{h}^{4}}{560162433600}$ | $\sigma \xb2\left({\u03f5}_{{I}_{y}}\right)$ | 0.000 | 0.000 | 1.248 | 5.963 |

Equation | Error [%] | ${\mathit{P}}_{100}$ | ${\mathit{P}}_{10}$ | ${\mathit{I}}_{128}$ | ${\mathit{I}}_{16}$ | ||||
---|---|---|---|---|---|---|---|---|---|

$A$ | $61a\frac{h}{90}$ | $\overline{{\u03f5}_{A}}$ | 0.004 | 0.505 | 2.218 | 12.118 | |||

${I}_{x}$ | $27371b\frac{{h}^{3}}{748440}$ | $\sigma \xb2\left({\u03f5}_{A}\right)$ | 0.000 | 0.000 | 1.075 | 4.609 | |||

${I}_{y}$ | $27371bh\frac{{b}^{2}+{p}^{2}}{748440}$ | $\overline{{\u03f5}_{{I}_{x}}}$ | 0.008 | 1.009 | 3.831 | 19.453 | |||

${I}_{xy}$ | $-27371b\frac{{h}^{2}p}{748440}$ | $\sigma \xb2\left({\u03f5}_{{I}_{x}}\right)$ | 0.000 | 0.000 | 1.489 | 5.294 | |||

${x}_{s},{y}_{s}$ | $\left(\frac{b}{2}+\frac{p}{2},\frac{h}{2}\right)$ | $\overline{{\u03f5}_{{I}_{y}}}$ | 0.008 | 1.009 | 3.463 | 20.586 | |||

${I}_{x}{I}_{y}-{I}_{xy}^{2}$ | $749171641\frac{{b}^{4}{h}^{4}}{560162433600}$ | $\sigma \xb2\left({\u03f5}_{{I}_{y}}\right)$ | 0.000 | 0.000 | 1.298 | 5.750 | |||

$\overline{{\u03f5}_{{I}_{xy}}}$ | 0.008 | 1.009 | 4.474 | 30.458 | |||||

$\sigma \xb2\left({\u03f5}_{{I}_{xy}}\right)$ | 0.000 | 0.000 | 1.920 | 9.431 |

Equation | Error [%] | ${\mathit{P}}_{100}$ | ${\mathit{P}}_{10}$ | ${\mathit{I}}_{128}$ | ${\mathit{I}}_{16}$ | |
---|---|---|---|---|---|---|

$A$ | $61h\frac{b+2p}{180}$ | $\overline{{\u03f5}_{A}}$ | 0.004 | 0.505 | 0.915 | 9.991 |

${I}_{x}$ | $\frac{{h}^{3}\left(4412605{b}^{2}+22420724bp+17650420{p}^{2}\right)}{273929040\left(b+2p\right)}$ | $\sigma \xb2\left({\u03f5}_{A}\right)$ | 0.000 | 0.000 | 0.013 | 3.937 |

${I}_{y}$ | $h\frac{17081{b}^{3}+75322{b}^{2}p+150644b{p}^{2}+136648{p}^{3}}{2993760}$ | $\overline{{\u03f5}_{{I}_{x}}}$ | 0.008 | 0.961 | 2.180 | 15.232 |

${I}_{xy}$ | $0$ | $\sigma \xb2\left({\u03f5}_{{I}_{x}}\right)$ | 0.000 | 0.000 | 0.132 | 4.967 |

${x}_{s},{y}_{s}$ | $\left(\frac{b}{2};h\frac{461b+1274p}{1098\left(b+2p\right)}\right)$ | $\overline{{\u03f5}_{{I}_{y}}}$ | 0.009 | 1.097 | 1.385 | 13.110 |

${I}_{x}{I}_{y}-{I}_{xy}^{2}$ | $\left(32\right)$ | $\sigma \xb2\left({\u03f5}_{{I}_{y}}\right)$ | 0.000 | 0.000 | 0.002 | 4.336 |

Equation | Error [%] | ${\mathit{P}}_{100}$ | ${\mathit{P}}_{10}$ | ${\mathit{I}}_{128}$ | ${\mathit{I}}_{16}$ | |
---|---|---|---|---|---|---|

$A$ | $h\frac{169b+218p}{288}$ | $\overline{{\u03f5}_{A}}$ | 0.003 | 0.332 | 2.438 | 14.852 |

${I}_{x}$ | $\frac{{h}^{3}\left(82305438169{b}^{2}+212310520756bp+120438027736{p}^{2}\right)}{20118067200\left(169b+218p\right)}$ | $\sigma \xb2\left({\u03f5}_{A}\right)$ | 0.000 | 0.000 | 1.298 | 6.766 |

${I}_{y}$ | $h\frac{1527254{b}^{3}+5667223{b}^{2}p+7684452b{p}^{2}+3712596{p}^{3}}{47900160}$ | $\overline{{\u03f5}_{{I}_{x}}}$ | 0.005 | 0.643 | 4.220 | 19.337 |

${I}_{xy}$ | $0$ | $\sigma \xb2\left({\u03f5}_{{I}_{x}}\right)$ | 0.000 | 0.000 | 1.454 | 7.039 |

${x}_{s},{y}_{s}$ | $\left(\frac{b}{2};h\frac{167189b+254368p}{2520\left(169b+218p\right)}\right)$ | $\overline{{\u03f5}_{{I}_{y}}}$ | 0.006 | 0.690 | 2.291 | 17.653 |

${I}_{x}{I}_{y}-{I}_{xy}^{2}$ | $\left(33\right)$ | $\sigma \xb2\left({\u03f5}_{{I}_{y}}\right)$ | 0.000 | 0.000 | 1.276 | 7.541 |

Equation | Error [%] | ${\mathit{P}}_{100}$ | ${\mathit{P}}_{10}$ | ${\mathit{I}}_{128}$ | ${\mathit{I}}_{16}$ | |
---|---|---|---|---|---|---|

$A$ | $301h\frac{b+p}{360}$ | $\overline{{\u03f5}_{A}}$ | 0.002 | 0.225 | 1.128 | 15.193 |

${I}_{x}$ | $\frac{{h}^{3}\left(1267299b+927031p\right)}{23950080}$ | $\sigma \xb2\left({\u03f5}_{A}\right)$ | 0.000 | 0.000 | 0.081 | 7.610 |

${I}_{y}$ | $h\frac{354311{b}^{3}+1131397{b}^{2}p+1267299b{p}^{2}+490213{p}^{3}}{5987520}$ | $\overline{{\u03f5}_{{I}_{x}}}$ | 0.003 | 0.398 | 2.405 | 20.943 |

${I}_{xy}$ | $0$ | $\sigma \xb2\left({\u03f5}_{{I}_{x}}\right)$ | 0.000 | 0.000 | 0.419 | 8.302 |

${x}_{s},{y}_{s}$ | $\left(\frac{b}{2},\frac{h}{2}\right)$ | $\overline{{\u03f5}_{{I}_{y}}}$ | 0.004 | 0.511 | 1.037 | 15.814 |

${I}_{x}{I}_{y}-{I}_{xy}^{2}$ | $\left(34\right)$ | $\sigma \xb2\left({\u03f5}_{{I}_{y}}\right)$ | 0.000 | 0.000 | 0.037 | 7.627 |

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**MDPI and ACS Style**

Denk, M.; Jäger, M.; Wartzack, S.
Symbolic Parametric Representation of the Area and the Second Moments of Area of Periodic B-Spline Cross-Sections. *Appl. Mech.* **2023**, *4*, 476-492.
https://doi.org/10.3390/applmech4020027

**AMA Style**

Denk M, Jäger M, Wartzack S.
Symbolic Parametric Representation of the Area and the Second Moments of Area of Periodic B-Spline Cross-Sections. *Applied Mechanics*. 2023; 4(2):476-492.
https://doi.org/10.3390/applmech4020027

**Chicago/Turabian Style**

Denk, Martin, Michael Jäger, and Sandro Wartzack.
2023. "Symbolic Parametric Representation of the Area and the Second Moments of Area of Periodic B-Spline Cross-Sections" *Applied Mechanics* 4, no. 2: 476-492.
https://doi.org/10.3390/applmech4020027