# Buckling of Rectangular Composite Pipes under Torsion

^{1}

^{2}

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## Abstract

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## Featured Application

**Thin-walled rectangular composite pipes, because of their light weight, high stiffness, and strength, can be applied to ultra-lightweight airplanes, satellites, spacecraft, automobiles, and civil structures.**

## Abstract

_{r},(±45)

_{1-r}]

_{s}. The layups were assumed to be symmetric, and tension-bending, torsion-bending, and tension-shear coupling stiffnesses were ignored. To establish a simple design method, a closed-form polynomial equation for the buckling load factor was derived by minimizing the weighted residuals of the safe and non-safe side errors, which were obtained by comparing the derived numerical results with the polynomial equations. As a result, the errors of the polynomial equation for the buckling load factor were 4.95% for the non-safe side and 12.4% for the safe side. The errors are sufficiently good for preliminary design use and for parametric design studies and optimization.

## 1. Introduction

## 2. Numerical Calculation

#### 2.1. Energy Method

_{mn}as follows:

_{mn}is a common coefficient to determine amplitudes of w and v and Equation (5) is the condition of constant the plates’ edge angles. This assumption will be validated later. The total potential energy Π can be expressed in terms of the strain energy U and work of the external force W as

_{p}becomes

_{xx}, D

_{xy}, D

_{yy}, and D

_{ss}are

_{xs}and D

_{ss}are non-zero even in a symmetric layup; when the number of layups is large enough, the following assumption can be used.

_{cr}:

_{cr}= T/(2bht), i = 1, 2,…; and j = 1, 2,…. The buckling torsional stress is obtained by minimizing the total potential energy.

_{mn}, the determinant of the coefficient matrix for Φ

_{mn}should be zero. The coefficient matrix separates into one for which m + n is odd and one for which m + n is even. It is well known that the thin isotropic plate under shear load, in which m + n is even has a lower buckling load than the one in which m + n is odd except for long plate. Numerical calculations, however, both the case that m + n is even and m + n is odd, were conducted because orthotropic stiffness may lead to lower buckling load for the case that m + n is odd.

_{s}is defined as

#### 2.2. Case of a Constant Edge Angle

_{max}and n

_{max}. To check the accuracy relative to the number of terms, a numerical calculation was conducted on a quasi-isotropic layup of T300 CFRP. The material properties of the quasi-isotropic layup are listed in Table 1. m

_{max}and n

_{max}= 10, 20,…, 100, and b = h = 100 mm. The results are shown in Figure 4.

_{max}= n

_{max}= 20 is within 1% of m

_{max}= n

_{max}= 100 for l/b ≤ 6, and the buckling stress τ

_{cr}is constant for l/b > 6 because of effect of boundary constraints at x = 0 and x = l vanishing; thus, m

_{max}= n

_{max}= 20 for l/b ≤ 6 and buckling stress τ

_{cr}(l/b = 6) for l/b > 6 were used in the subsequent numerical calculations.

#### 2.3. Case of a Variable Edge Angle

_{cr}as follows:

_{s}was obtained by setting the determinant of the coefficient matrix for Β

_{mn}and H

_{mn}to zero. To check the accuracy relative to the number of terms of n and m, a similar numerical calculation as shown in Figure 4 was conducted, and the same conclusion was obtained for the terms of n and m.

## 3. Closed-Form Polynomial Equation

_{r},(±45)

_{1-r}]

_{s}layups. Here, r is the ratio of the 0/90 layer number to the total layer number. Because the [(0/90)

_{r},(±45)

_{1-r}]

_{s}layup is often used, it is thus chosen. The numerical analysis described above is hard to use for design purposes and would entail a huge calculation cost if it were to be used for optimum design of the whole system. Thus, it is considered that a simple approximate design equation would still be useful. Table 2 shows the material properties of unidirectional pre-preg tape, and Table 3 shows the layup sequence used in the numerical analysis. The total ply number was 32, and the binding stiffness was systematically controlled by varying r.

_{ij}t

^{3}/12 = D

_{ij}(i, j = x, y, s). Advanced stiffness properties like through thickness properties discussed in Refs. [19,20,21] are not used because of to make quick and convenient estimates. Figure 7, Figure 8 and Figure 9 show the buckling load factor k

_{s}for l/b, h/b, and r. The ranges of l/b, h/b, and r are 1 ≤ l/b ≤ 6, 0.7 ≤ h/b ≤ 1, and 0 ≤ r ≤ 1.

_{00}, H

_{01},…, H

_{21}, R

_{1}and R

_{2}were determined by minimizing the following weighted residual and adjusting the weight w to make the non-safe side residual k

_{s+er}less than 5%.

_{s+er}was 4.77%, and k

_{s-er}was 11.9%.

_{cr}does not change in the region l/b > 6. Thus, the buckling load factor for the torsional buckling load of the orthotropic rectangular pipe can be described as follows:

_{max}= n

_{max}= 100 for the maximum k

_{s-er}error case (r = 0.5, b = l = 100 mm, t = 1 mm). The error is also plotted. The error in the region l/b > 6 is around −11%, which gives a safe and conservative side for the design. It is natural that constant gap appears in the region l/b > 6 because k

_{s}does not change in the region l/b > 6 and the constant k

_{s}’ at l/b = 6 is used. Note that the polynomial equation can be used within the ranges of 1 ≤ l/b ≤ 6, 0.7 ≤ h/b ≤ 1, and 0 ≤ r ≤ 1.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Fujita, K.; Nagai, H.; Oyama, A. A Parametric Study of Mars Airplane Concept for Science Mission on Mars. Trans. JSASS Aerosp. Technol. Jpn.
**2016**, 14, Pk_83–Pk_88. [Google Scholar] [CrossRef] - Nagai, K.; Oyama, A.; Mars Airplane, W.G. Mission Scenario of Mars Exploration by Airplane. In Proceedings of the 2013 Asia-Pacific International Symposium on Aerospace Technology, Takamatsu, Japan, 20–22 November 2013. [Google Scholar]
- Airoldi, A.; Bettini, P.; Boiocchi, M.; Sala, G. Composite Elements for Biomimetic Aerospace Structures with Progressive Shape Variation Capabilities. Adv. Technol. Innov.
**2016**, 1, 13–15. [Google Scholar] - Chang, C.Y. Segmented Compression Molding for Composite Manufacture. Proc. Eng. Technol. Innov.
**2017**, 5, 41–44. [Google Scholar] - Sahu, V.; Gayathri, V. Strength Studies of Dadri Fly Ash Modified with Lime Sludge–A Composite Material. Int. J. Eng. Technol. Innov.
**2014**, 4, 161–169. [Google Scholar] - Liu, C.M.; Chiang, M.S.; Chuang, W.C. Lean Transformation for Composite-Material Bonding Processes. Int. J. Eng. Technol. Innov.
**2012**, 2, 48–62. [Google Scholar] - Takano, A. Buckling Experiment on Anisotropic Long and Short Cylinders. Adv. Technol. Innov.
**2016**, 1, 25–27. [Google Scholar] - Ning, A.X.; Pellegrino, S. Experiments on Imperfection Insensitive Axially Loaded Cylindrical Shells. Int. J. Solids Struct.
**2017**, 115–116, 73–86. [Google Scholar] [CrossRef] - Wagner, H.N.R.; Petersen, E.; Khakimova, R.; Hühne, C. Buckling Analysis of an Imperfection-Insensitive Hybrid Composite Cylinder under axial Compression–Numerical Simulation, Destructive and Nondestructive Experimental Testing. Compos. Struct.
**2019**, 225, 111152. [Google Scholar] [CrossRef] - Podvornyi, A.V.; Semenyuk, N.P.; Trach, V.M. Stability of Inhomogeneous Cylindrical Shells under Distributed External Pressure in a Three-Dimensional Statement. Int. Appl. Mech.
**2018**, 53, 623–638. [Google Scholar] [CrossRef] - Weaver, P. On Optimisation of Long Anisotropic Flat Plates Subject to Shear Buckling Loads. In Proceedings of the 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, Palm Springs, CA, USA, 19–22 April 2004. [Google Scholar]
- Raju, G.; Wu, Z.; Weaver, P.M. Buckling and Postbuckling of Variable Angle Tow Composite Plates under In-Plane Shear Loading. Int. J. Solids Struct.
**2015**, 58, 270–287. [Google Scholar] [CrossRef] [Green Version] - Omidvari, A.; Hematiyan, M.R. Approximate Closed-form Formulae for Buckling Analysis of Rectangular Tubes under Torsion. Ije Trans. B: Appl.
**2015**, 28, 1226–1232. [Google Scholar] - Banks, W.M.; Rhodes, J. The Postbuckling Behaviour of Composite Box Sections. In Composite Structures; Marshall, I.H., Ed.; Springer: Berlin/Heidelberg, Germany, 1981; pp. 402–414. [Google Scholar]
- Loughlan, J. The buckling of composite stiffened box sections subjected to compression and bending. Compos. Struct.
**1996**, 35, 101–116. [Google Scholar] [CrossRef] - Vo, T.P.; Lee, J. Flexural–torsional buckling of thin-walled composite box beams. Thin-Walled Struct.
**2007**, 45, 790–798. [Google Scholar] [CrossRef] [Green Version] - Column Research Committee of Japan (Ed.) Handbook of Structural Stability; Corona Publishing Company: Tokyo, Japan, 1971; Part 4.
- Furusu, K. Studies on the Buckling of the Box Beam Composed of Thin Plates. Ph.D. Thesis, Seikei University, Tokyo, Japan, 2016. (In Japanese). [Google Scholar]
- Jones, R.M. Mechanics of Composite Materials; CRC Press: Boca Raton, FL, USA, 1998. [Google Scholar]
- Ng, Y.C. Deriving composite lamina properties from laminate properties using classical lamination theory and failure criteria. J. Compos. Mater.
**2005**, 39, 1295–1306. [Google Scholar] [CrossRef] - Kalkan, A.; Mecitoğlu, Z. A Method Based on Classical Lamination Theory to Calculate Stiffness Properties of Closed Composite Sections. J. Aeronaut. Space Technol.
**2017**, 10, 31–44. [Google Scholar]

Item | Value |
---|---|

Young’s modulus E_{x} = E_{y} [MPa] | 56,400 |

Shear modulus G_{xy} [MPa] | 21,500 |

Poisson’s ratio v_{x} = v_{y} [-] | 0.313 |

Item | Value |
---|---|

E_{L} [MPa] | 147,000 |

E_{T} [MPa] | 9800 |

G_{LT} [MPa] | 5096 |

v_{L} [-] | 0.32 |

Number of Layers | Ratio of 0/90 Layer | Bending Stiffness [N-mm^{2}] | ||||
---|---|---|---|---|---|---|

0/90 | ±45 | r | D_{xx} | D_{xy} | D_{yy} | D_{ss} |

0 | 32 | 0 | 3845 | 2996 | 3845 | 3158 |

4 | 28 | 0.125 | 4187 | 2654 | 4187 | 2816 |

8 | 24 | 0.25 | 4529 | 2313 | 4529 | 2474 |

12 | 20 | 0.375 | 4870 | 1971 | 4870 | 2133 |

16 | 16 | 0.5 | 5212 | 1630 | 5212 | 1791 |

20 | 12 | 0.625 | 5553 | 1288 | 5553 | 1450 |

24 | 8 | 0.75 | 5895 | 946.4 | 5895 | 1108 |

28 | 4 | 0.875 | 6237 | 604.7 | 6237 | 766.3 |

32 | 0 | 1 | 6578 | 263.1 | 6578 | 424.7 |

H_{00} | 1.1137 | H_{01} | 0.22213 |

H_{10} | −3.3798 | H_{11} | 6.0957 |

H_{20} | 11.382 | H_{21} | −7.4961 |

R_{1} | 1.0348 | R_{2} | 0.26278 |

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**MDPI and ACS Style**

Takano, A.; Mizukami, R.; Kitamura, R.
Buckling of Rectangular Composite Pipes under Torsion. *Appl. Sci.* **2021**, *11*, 1342.
https://doi.org/10.3390/app11031342

**AMA Style**

Takano A, Mizukami R, Kitamura R.
Buckling of Rectangular Composite Pipes under Torsion. *Applied Sciences*. 2021; 11(3):1342.
https://doi.org/10.3390/app11031342

**Chicago/Turabian Style**

Takano, Atsushi, Ryo Mizukami, and Ryuta Kitamura.
2021. "Buckling of Rectangular Composite Pipes under Torsion" *Applied Sciences* 11, no. 3: 1342.
https://doi.org/10.3390/app11031342