# Dynamic Pulse Buckling of Composite Stanchions in the Sub-Cargo Floor Area of a Civil Regional Aircraft

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Test Case Description

_{t}and X

_{c}are respectively the longitudinal tensile and compressive strengths, Y

_{t}and Y

_{c}are respectively the transversal tensile and compressive strengths, and S

_{c}is the shear strength. The stacking sequence of the laminate is [45; −45; 90; −45; 45; 0; 0; 0; 0; 0; 45; −45; 90; −45; 45].

## 3. Results and Discussion

#### 3.1. Mesh Convergence Analysis

#### 3.2. Validation of the Numerical Model

- Experimental Test T1: Compressive test aimed to determine the stiffness of the structure;
- Experimental Test T2: Compressive test up to the total failure.

#### 3.3. Dynamic Buckling Analysis

^{−3}strain) is related to the damage propagation, not to the post-buckling regime. In the explicit analysis, the elements reach the total failure state faster with respect to the implicit one, leading to an earlier and smaller loss in stiffness.

- Model S: This model has been discretized by using four-node linear shell (S4) elements [34]. The same in-plane element size of Model 3B has been considered; however, obviously, no division in the thickness direction has been employed. The equivalent properties of the entire laminate, reported in Table 6, have been used for each element.
- Model 1R: This model has been discretized by using eight-node linear solid elements with a reduced integration scheme (C3D8R) [34]. The same in-plane mesh discretization used for Model 3B and Model S has been considered. One element has been placed in the thickness direction, in order to avoid the use of the layered option. The equivalent mechanical properties of the entire laminate (Table 6) have been assigned to the elements.

- Applied displacement = 1.0 mm (below the critical displacement);
- Applied displacement = 1.53 mm (equal to the critical displacement);
- Applied displacement = 2.0 mm (above the critical displacement).

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Box, F.; Kodio, O.; O’Kiely, D.; Cantelli, V.; Goriely, A.; Vella, D. Dynamic Buckling of an Elastic Ring in a Soap Film. Phys. Rev. Lett.
**2020**, 124, 198003. [Google Scholar] [CrossRef] - Kodio, O.; Goriely, A.; Vella, D. Dynamic buckling of an inextensible elastic ring: Linear and nonlinear analyses. Phys. Rev. E
**2020**, 101, 053002. [Google Scholar] [CrossRef] - Ali, A.Y.; Hasan, H.M. Nonlinear dynamic stability of an imperfect shear deformable orthotropic functionally graded material toroidal shell segments under the longitudinal constant velocity. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2019**, 233, 6827–6850. [Google Scholar] [CrossRef] - Amabili, M.; Païdoussis, M.P. Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction. Appl. Mech. Rev.
**2003**, 56, 349–381. [Google Scholar] [CrossRef] - Alijani, F.; Amabili, M. Non-linear vibrations of shells: A literature review from 2003 to 2013. Int. J. Non-Linear Mech.
**2014**, 58, 233–257. [Google Scholar] [CrossRef] [Green Version] - Kubiak, T. Static and dynamic buckling of thin-walled plate structures. In Static and Dynamic Buckling of Thin-Walled Plate Structures; Springer: London, UK, 2013. [Google Scholar]
- Bhimaraddi, A.; Chandrashekhara, K. Nonlinear vibrations of heated antisymmetric angle-ply laminated plates. Int. J. Solids Struct.
**1993**, 30, 1255–1268. [Google Scholar] [CrossRef] - Kumar, P.; Srinivas, J. Vibration, buckling and bending behavior of functionally graded multi-walled carbon nanotube reinforced polymer composite plates using the layer-wise formulation. Compos. Struct.
**2017**, 177, 158–170. [Google Scholar] [CrossRef] - Kumar, L.R.; Datta, P.K.; Prabhakara, D.L. Vibration and stability behavior of laminated composite curved panels with cutout under partial in-plane loads. Int. J. Struct. Stab. Dyn.
**2005**, 5, 75–94. [Google Scholar] [CrossRef] - Kolahchi, R.; Zhu, S.-P.; Keshtegar, B.; Trung, N.-T. Dynamic buckling optimization of laminated aircraft conical shells with hybrid nanocomposite martial. Aerosp. Sci. Technol.
**2020**, 98, 105656. [Google Scholar] [CrossRef] - Bigoni, D. Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability; Cambridge University Press: New York, NY, USA, 2012. [Google Scholar]
- Javani, M.; Kiani, Y.; Eslami, M.H. Dynamic snap-through of shallow spherical shells subjected to thermal shock. Int. J. Press. Vessel. Pip.
**2020**, 179, 104028. [Google Scholar] [CrossRef] - Shokravi, M.; Jalili, N. Thermal dynamic buckling of temperature-dependent sandwich nanocomposite quadrilateral microplates using visco-higher order nonlocal strain gradient theory. J. Therm. Stress.
**2019**, 42, 506–525. [Google Scholar] [CrossRef] - Zhang, J.; Chen, S.; Zheng, W. Dynamic buckling analysis of functionally graded material cylindrical shells under thermal shock. Contin. Mech. Thermodyn.
**2019**, 32, 1095–1108. [Google Scholar] [CrossRef] - Yasunaga, J.; Uematsu, Y. Dynamic buckling of cylindrical storage tanks under fluctuating wind loading. Thin-Walled Struct.
**2020**, 150, 106677. [Google Scholar] [CrossRef] - Kubiak, T. Estimation of dynamic buckling for composite columns with open cross-section. Comput. Struct.
**2011**, 89, 2001–2009. [Google Scholar] [CrossRef] - Zizicas, G.A. Dynamic buckling of thin elastic plates. Trans. ASME
**1952**, 74, 1257. [Google Scholar] - Budiansky, B.; Roth, R.S. Axisymmetric dynamic buckling of clamped shallow spherical shells. NASA TN
**1962**, 1510, 597–606. [Google Scholar] - Lindberg, H.E.; Florence, A.L. Dynamic Pulse Buckling; Springer: Dordrecht, The Netherlands, 1987. [Google Scholar]
- Labans, E.; Abramovich, H.; Bisagni, C. An experimental vibration-buckling investigation on classical and variable angle tow composite shells under axial compression. J. Sound Vib.
**2019**, 449, 315–329. [Google Scholar] [CrossRef] - Zaczynska, M.; Abramovich, H.; Bisagni, C. Parametric studies on the dynamic buckling phenomenon of a composite cylindrical shell under impulsive axial compression. J. Sound Vib.
**2020**, 482, 11546. [Google Scholar] [CrossRef] - Mondal, S.; Ramachandra, L. Nonlinear dynamic pulse buckling of imperfect laminated composite plate with delamination. Int. J. Solids Struct.
**2020**, 198, 170–182. [Google Scholar] [CrossRef] - Karthick, S.; Datta, P.K. Dynamic instability characteristics of thin plate like beam with internal damage subjected to follower load. Int. J. Struct. Stab. Dyn.
**2015**, 15, 1450048. [Google Scholar] [CrossRef] - Wang, Y.-H.; Han, Z.-J. Buckling study of composite plates subjected to impact loading. IOP Conf. Ser. Earth Environ. Sci.
**2019**, 267, 022022. [Google Scholar] - Wang, P.; Li, S.-Q.; Yu, G.-J.; Wu, G.-Y. Dynamic Crushing Behavior of Graded Hollow Cylindrical Shell under Axial Impact Loading. Chin. J. High Press. Phys.
**2017**, 31, 778–784. [Google Scholar] - Gui, Y.; Ma, J. Buckling of step cylindrical shells under axial impact load. J. Vib. Shock
**2019**, 38, 200–205, 228. [Google Scholar] - Gui, Y.; Xu, J.; Ma, J. Dynamic Buckling of a Cylindrical Shell with a General Boundary Condition under an Axial Impact. J. Appl. Mech. Tech. Phys.
**2019**, 60, 712–723. [Google Scholar] [CrossRef] - Zhang, Z.; Taheri, F. Numerical studies on dynamic pulse buckling of FRP composite laminated beams subject to an axial impact. Compos. Struct.
**2002**, 56, 269–277. [Google Scholar] [CrossRef] - Bažant, Z.P.; Cedolin, L. Stability of Structures; Dover Publication, Inc.: Mineola, NY, USA, 2010. [Google Scholar]
- Riccio, A.; Raimondo, A.; Di Caprio, F.; Fusco, M.; Sanitá, P. Experimental and numerical investigation on the crashworthiness of a composite fuselage sub-floor support system. Compos. Part B Eng.
**2018**, 150, 93–103. [Google Scholar] [CrossRef] - Di Caprio, F.; Ignarra, M.; Marulo, F.; Guida, M.; Lamboglia, A.; Gambino, B. Design of composite stanchions for the cargo subfloor structure of a civil aircraft. Procedia Eng.
**2016**, 167, 88–96. [Google Scholar] [CrossRef] [Green Version] - Guida, M.; Marulo, F.; Abrate, S. Advances in crash dynamics for aircraft safety. Prog. Aerosp. Sci.
**2018**, 98, 106–123. [Google Scholar] [CrossRef] - Marulo, F.; Guida, M.; Di Caprio, F.; Ignarra, M.; Lamboglia, A.; Gambino, B. Fuselage Crashworthiness Lower Lobe Dynamic Test. Procedia Eng.
**2016**, 167, 120–128. [Google Scholar] [CrossRef] - Smith, M. ABAQUS/Standard User’s Manual, Version 2019; Dassault Systèmes Simulia Corp: Providence, RI, USA, 2019. [Google Scholar]
- Riccio, A.; Saputo, S.; Sellitto, A.; Raimondo, A.; Ricchiuto, R. Numerical investigation of a stiffened panel subjected to low velocity impacts. Key Eng. Mater.
**2015**, 665, 277–280. [Google Scholar] [CrossRef] - Riccio, A.; Cristiano, R.; Saputo, S.; Sellitto, A. Numerical methodologies for simulating bird-strike on composite wings. Compos. Struct.
**2018**, 202, 590–602. [Google Scholar] [CrossRef] - Zucco, G.; Weaver, P.M. The role of symmetry in the post-buckling behaviour of structures. Proc. R Soc. A Math. Phys. Eng. Sci.
**2020**, 476, 20190609. [Google Scholar] [CrossRef] [PubMed]

**Figure 3.**One element in the thickness direction: Coarser mesh (Mesh 1A), intermediate mesh (Mesh 1B), and finer mesh (Mesh 1C).

**Figure 4.**Three elements in the thickness direction: Coarser mesh (Mesh 3A), intermediate mesh (Mesh 3B), and finer mesh (Mesh 3C).

**Figure 5.**Five elements in the thickness direction: Coarser mesh (Mesh 5A), intermediate mesh (Mesh 5B), and finer mesh (Mesh 5C).

**Figure 15.**Displacement, reaction, and energies as a function of time: 1.53 mm applied displacement.

ρ [g/cm^{3}] | th [mm] | E_{11} [MPa] | E_{22} [MPa] | G_{12} [MPa] | G_{13} [MPa] | G_{23} [MPa] | ν_{12} [-] | X_{t} [MPa] | X_{c} [MPa] | Y_{t} [MPa] | Y_{c} [MPa] | S_{c} [MPa] |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1.6 | 0.186 | 135,000 | 8430 | 4160 | 4160 | 3328 | 0.26 | 2257 | 800 | 75 | 171 | 85 |

Number of Elements in the Thickness | Element Size | Model Name | Number of Elements (Total) | Stiffness [kN/mm] | Computational Time |
---|---|---|---|---|---|

1 | 8 mm | Mesh 1A | 1056 | 56.220 | 10 s |

4 mm | Mesh 1B | 3572 | 56.317 | 15 s | |

2 mm | Mesh 1C | 12,920 | 56.340 | 21 s | |

3 | 8 mm | Mesh 3A | 3168 | 56.215 | 13 s |

4 mm | Mesh 3B | 10,716 | 56.305 | 18 s | |

2 mm | Mesh 3C | 38,760 | 56.327 | 33 s | |

5 | 8 mm | Mesh 5A | 5280 | 56.097 | 16 s |

4 mm | Mesh 5B | 17,860 | 56.180 | 21 s | |

2 mm | Mesh 5C | 64,600 | 56.192 | 52 s |

Stiffness | Failure Load | |
---|---|---|

Experimental | 54.8 kN/mm | 103.7 kN |

Numerical | 56.3 kN/mm | 106.0 kN |

Error | 2.6% | 2.2% |

Failure Displacement | Failure Load | |
---|---|---|

Implicit | 1.59 mm | 107.3 kN |

Explicit | 1.60 mm | 104.0 kN |

Error | 0.06% | 3.1% |

Model Name | Element Type | # of Elements in the Thickness | Laminate Type |
---|---|---|---|

Model 3B | Continuum Shell (SC8R) | 3 | Composite Layup |

Model S | Shell (S4) | 1 | Composite Layup |

Model 1R | 3D Stress (C3D8R) | 1 | Equivalent Laminate |

Model 3R | 3D Stress (C3D8R) | 3 | Equivalent Laminate |

th [mm] | E_{11} [MPa] | E_{22} [MPa] | E_{33} [MPa] | G_{12} [MPa] | G_{13} [MPa] | G_{23} [MPa] | ν_{12} [-] |
---|---|---|---|---|---|---|---|

2.79 | 60,267 | 37,845 | 8430 | 20,559 | 20,559 | 3328 | 0.43 |

Stacking Sequence | th [mm] | E_{11} [MPa] | E_{22} [MPa] | E_{33} [MPa] | G_{12} [MPa] | G_{13} [MPa] | G_{23} [MPa] | ν_{12} [-] | ν_{13} [-] | ν_{23} [-] |
---|---|---|---|---|---|---|---|---|---|---|

[0]_{5} | 0.93 | 135,000 | 8430 | 8430 | 4160 | 4160 | 3328 | 0.26 | 0.3 | 0.3 |

[−45, 45, 90, 45, −45] | 0.93 | 22,674 | 39,437 | 8430 | 28,759 | 28,759 | 3328 | 0.44 | 0.3 | 0.3 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sellitto, A.; Di Caprio, F.; Guida, M.; Saputo, S.; Riccio, A.
Dynamic Pulse Buckling of Composite Stanchions in the Sub-Cargo Floor Area of a Civil Regional Aircraft. *Materials* **2020**, *13*, 3594.
https://doi.org/10.3390/ma13163594

**AMA Style**

Sellitto A, Di Caprio F, Guida M, Saputo S, Riccio A.
Dynamic Pulse Buckling of Composite Stanchions in the Sub-Cargo Floor Area of a Civil Regional Aircraft. *Materials*. 2020; 13(16):3594.
https://doi.org/10.3390/ma13163594

**Chicago/Turabian Style**

Sellitto, Andrea, Francesco Di Caprio, Michele Guida, Salvatore Saputo, and Aniello Riccio.
2020. "Dynamic Pulse Buckling of Composite Stanchions in the Sub-Cargo Floor Area of a Civil Regional Aircraft" *Materials* 13, no. 16: 3594.
https://doi.org/10.3390/ma13163594