# Bimetallic Thin-Walled Box Beam Thermal Buckling Response

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Kinematics

#### 2.2. Constitutive Equations

#### 2.3. Finite Element Formulation

## 3. Results and Discussion

#### 3.1. Box Beam

#### 3.2. L-Frame

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 4.**Critical temperatures vs. displacement for a clamped–clamped beam ${L}_{1}=6\mathrm{m}$ with $\lambda =0.2$.

**Figure 5.**Critical temperatures vs. displacement for a clamped–clamped beam ${L}_{1}=6\mathrm{m}$ with $\lambda =0.5$.

**Figure 6.**Critical temperatures vs. displacement for a clamped–clamped beam ${L}_{1}=6\mathrm{m}$ with $\lambda =0.8$.

**Table 1.**Temperature dependent coefficients [19].

Material | Properties | P_{0} | P_{1} | P_{1} | P_{2} | P_{3} |
---|---|---|---|---|---|---|

Ti–6Al–4V | E (Pa) | 122.56 × 10^{9} | 0.0 | −4.586 × 10^{−4} | 0.0 | 0.0 |

α (1/K) | 7.5788 × 10^{−6} | 0.0 | 6.638 × 10^{−4} | –3.147 × 10^{−6} | 0.0 | |

SUS304 | E (Pa) | 201.04 × 10^{9} | 0.0 | 3.079 × 10^{−4} | –6.534 × 10^{−7} | 0.0 |

α (1/K) | 12.330 × 10^{−6} | 0.0 | 8.086 × 10^{−4} | 0.0 | 0.0 |

**Table 2.**Critical buckling temperatures of the beam ${L}_{1}=6\mathrm{m}$ for different boundary conditions and material thickness ratios.

λ | |||||||||
---|---|---|---|---|---|---|---|---|---|

BC | Mode | Method | 0 | 0.2 | 0.4 | 0.5 | 0.6 | 0.8 | 1 |

C-C | Y | Present | 128.87 | 135.26 | 145.83 | 153.64 | 164.09 | 199.19 | 284.51 |

Shell | 130.22 | 138.77 | 151.18 | 159.72 | 170.71 | 205.83 | 287.97 | ||

X | Present | 239.24 | 252.56 | 273.89 | 288.9 | 308.62 | 373.43 | 528.16 | |

Shell | 238.95 | 254.69 | 277.45 | 293.11 | 313.24 | 377.6 | 527.52 | ||

C-S | Y | Present | 66.58 | 69.91 | 75.39 | 79.44 | 84.84 | 102.97 | 146.98 |

Shell | 65.98 | 70.313 | 76.604 | 80.936 | 86.504 | 104.29 | 145.66 | ||

X | Present | 122.87 | 130.02 | 140.09 | 148.58 | 158.73 | 192.03 | 271.48 | |

Shell | 120.46 | 128.39 | 139.88 | 147.78 | 157.93 | 190.37 | 265.92 | ||

S-S | Y | Present | 32.72 | 34.368 | 37.07 | 39.06 | 41.72 | 50.62 | 72.24 |

Shell | 32.196 | 34.314 | 37.39 | 39.507 | 42.226 | 50.904 | 71.077 | ||

X | Present | 60.28 | 60.29 | 69.07 | 72.86 | 77.83 | 94.15 | 133.08 | |

Shell | 58.725 | 62.6 | 68.208 | 72.063 | 77.015 | 92.827 | 129.64 |

**Table 3.**Critical buckling temperatures of the beam ${L}_{2}=8\mathrm{m}$ for different boundary conditions and material thickness ratios.

λ | |||||||||
---|---|---|---|---|---|---|---|---|---|

BC | Mode | Method | 0 | 0.2 | 0.4 | 0.5 | 0.6 | 0.8 | 1 |

C-C | Y | Present | 73.17 | 76.83 | 82.85 | 87.3 | 93.24 | 113.16 | 161.54 |

Shell | 73.542 | 78.368 | 85.375 | 90.2 | 96.405 | 116.24 | 162.35 | ||

X | Present | 135.52 | 142.96 | 154.88 | 163.38 | 174.53 | 211.15 | 298.53 | |

Shell | 135.02 | 143.91 | 156.78 | 165.62 | 177 | 213.37 | 298.08 | ||

C-S | Y | Present | 37.63 | 39.52 | 42.63 | 44.91 | 47.97 | 58.21 | 83.07 |

Shell | 37.323 | 39.774 | 43.33 | 45.783 | 48.933 | 58.997 | 82.396 | ||

X | Present | 69.35 | 73.33 | 79.45 | 83.81 | 89.53 | 108.31 | 153.09 | |

Shell | 68.256 | 72.754 | 79.261 | 83.736 | 89.487 | 107.87 | 150.68 | ||

S-S | Y | Present | 18.45 | 19.38 | 20.9 | 22.03 | 23.53 | 28.55 | 40.73 |

Shell | 18.211 | 19.409 | 21.149 | 22.346 | 23.885 | 28.793 | 40.204 | ||

X | Present | 33.95 | 35.9 | 38.9 | 41.04 | 43.84 | 53.03 | 74.95 | |

Shell | 33.278 | 35.473 | 38.651 | 40.836 | 43.642 | 52.603 | 73.466 |

**Table 4.**Critical buckling temperatures of the beam ${L}_{3}=10\mathrm{m}$ for different boundary conditions and material thickness ratios.

λ | |||||||||
---|---|---|---|---|---|---|---|---|---|

BC | Mode | Method | 0 | 0.2 | 0.4 | 0.5 | 0.6 | 0.8 | 1 |

C-C | Y | Present | 47.03 | 49.39 | 53.27 | 56.13 | 59.95 | 72.75 | 103.83 |

Shell | 47.158 | 50.252 | 54.745 | 57.839 | 61.818 | 74.536 | 104.11 | ||

X | Present | 86.74 | 91.72 | 99.37 | 104.82 | 111.98 | 135.47 | 191.49 | |

Shell | 86.605 | 92.308 | 100.56 | 106.23 | 113.53 | 136.86 | 191.19 | ||

C-S | Y | Present | 24.14 | 25.35 | 27.35 | 28.81 | 30.77 | 37.34 | 53.28 |

Shell | 23.965 | 25.538 | 27.823 | 29.397 | 31.419 | 37.881 | 52.906 | ||

X | Present | 44.43 | 46.99 | 50.91 | 53.71 | 57.38 | 69.4 | 98.09 | |

Shell | 43.872 | 46.763 | 50.945 | 53.822 | 57.518 | 69.335 | 96.854 | ||

S-S | Y | Present | 11.82 | 12.42 | 13.39 | 14.11 | 15.07 | 18.29 | 26.09 |

Shell | 11.695 | 12.464 | 13.582 | 14.35 | 15.338 | 18.491 | 25.818 | ||

X | Present | 21.74 | 22.99 | 24.91 | 26.28 | 28.08 | 33.96 | 47.99 | |

Shell | 21.396 | 22.807 | 24.85 | 26.255 | 28.059 | 33.82 | 47.234 |

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**MDPI and ACS Style**

Simonetti, S.K.; Turkalj, G.; Banić, D.; Lanc, D.
Bimetallic Thin-Walled Box Beam Thermal Buckling Response. *Materials* **2022**, *15*, 7537.
https://doi.org/10.3390/ma15217537

**AMA Style**

Simonetti SK, Turkalj G, Banić D, Lanc D.
Bimetallic Thin-Walled Box Beam Thermal Buckling Response. *Materials*. 2022; 15(21):7537.
https://doi.org/10.3390/ma15217537

**Chicago/Turabian Style**

Simonetti, Sandra Kvaternik, Goran Turkalj, Damjan Banić, and Domagoj Lanc.
2022. "Bimetallic Thin-Walled Box Beam Thermal Buckling Response" *Materials* 15, no. 21: 7537.
https://doi.org/10.3390/ma15217537