# Postbuckling and Free Vibration of Multilayer Imperfect Nanobeams under a Pre-Stress Load

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

#### 2.1. Kinematic Relations

#### 2.2. Equations of Motion

#### 2.3. Constitutive Equations

## 3. Static Analysis

#### 3.1. Buckling Load of Imperfect, Multilayer Nanobeams

#### 3.2. Postbuckling of Imperfect, Multilayer Nanobeams under Pre-Stress Loading

## 4. Free Vibrations

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Variation of the buckling amplitude with the axial pre-stress load of a perfect nanobeam with $\alpha =1$ and a variety of the nonlocal parameter, $\overline{\mu}.$

**Figure 4.**Variation of the static deflection with the pre-load at $a=1,\overline{\mu}=0.0$ and $\alpha =1$.

**Figure 5.**Variation of the static deflection with the pre-load at $a=1,\overline{\mu}=0.02$, and $\alpha =1$.

**Figure 6.**Variation of the static deflection with the pre-load at $a=1,\overline{\mu}=0.04$ and $\alpha =1$.

**Figure 7.**Variation of the static deflection with the pre-load at $a=1,\overline{\mu}=0.06$ and $\alpha =1$.

**Figure 8.**Variation of the static deflection with the pre-load at $a=1\mathrm{and}\alpha =3.434$. (

**a**) $\overline{\mu}=0.0;$(

**b**) $\overline{\mu}=0.02;$(

**c**) $\overline{\mu}=0.04;$(

**d**) $\overline{\mu}=0.06.$

**Figure 9.**Variation of the static deflection with the pre-load at $a=4$ and $\alpha =1$. (

**a**) $\overline{\mu}=0.0$; (

**b**) $\overline{\mu}=0.02;$(

**c**) $\overline{\mu}=0.04$; (

**d**) $\overline{\mu}=0.06$.

**Figure 10.**The variation of the first, third, and fifth natural frequencies with the applied axial load for $\alpha =1$ of the perfect nanobeam, $a=0$, and $\overline{\mu}=\left(0,0.02\right)$.

**Figure 11.**The variation of the first four natural frequencies with the applied axial load for $\alpha =1$ of the imperfect nanobeam, $a=1$, and $\overline{\mu}=\left(0,0.02\right)$. (

**a**) The first natural frequency; (

**b**) The second natural frequency; (

**c**) The third natural frequency; (

**d**) The fourth natural frequency.

**Table 1.**Nondimensional buckling load for a variety of initial curvatures and size parameter for $a=1.$

Nondimensional Imperfection Amplitude $\left(\mathit{a}\right)$ | Nondimensional Size Parameter $\left(\overline{\mathit{\mu}}=\mathit{\mu}/{\mathit{L}}^{2}\right)$ | |||||
---|---|---|---|---|---|---|

0 | 0.02 | 0.04 | 0.06 | 0.08 | 0.1 | |

0 | 39.4784 | 22.0603 | 15.3068 | 11.7192 | 9.4939 | 7.9789 |

1 | 66.6198 | 39.6802 | 28.5831 | 22.4276 | 18.4767 | 15.7086 |

2 | 76.6099 | 44.0773 | 30.4287 | 22.7649 | 17.8004 | 14.2962 |

3 | 78.8606 | 41.6370 | 25.8483 | 16.9195 | 11.1047 | 6.9832 |

4 | 74.6095 | 33.1988 | 15.4999 | 5.4418 | −1.1318 | −5.8044 |

**Table 2.**Non-dimensional buckling load for a variety of initial curvatures and size parameter for a $=2.$

Nondimensional Imperfection Amplitude $\left(\mathit{a}\right)$ | Nondimensional Size Parameter $\left(\overline{\mathit{\mu}}=\mathit{\mu}/{\mathit{L}}^{2}\right)$ | |||||
---|---|---|---|---|---|---|

0 | 0.02 | 0.04 | 0.06 | 0.08 | 0.1 | |

0 | 39.4784 | 22.0603 | 15.3068 | 11.7192 | 9.4939 | 7.9789 |

1 | 71.8484 | 42.4339 | 30.2078 | 23.3849 | 18.9855 | 15.8917 |

2 | 78.9568 | 42.4957 | 27.0549 | 18.3317 | 12.6551 | 8.6339 |

3 | 72.6622 | 30.2908 | 12.1536 | 1.8365 | −4.9113 | −9.7103 |

4 | 54.5237 | 6.8768 | −13.6671 | −25.4069 | −33.1108 | −38.6041 |

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**MDPI and ACS Style**

Emam, S.A.; Eltaher, M.A.; Khater, M.E.; Abdalla, W.S.
Postbuckling and Free Vibration of Multilayer Imperfect Nanobeams under a Pre-Stress Load. *Appl. Sci.* **2018**, *8*, 2238.
https://doi.org/10.3390/app8112238

**AMA Style**

Emam SA, Eltaher MA, Khater ME, Abdalla WS.
Postbuckling and Free Vibration of Multilayer Imperfect Nanobeams under a Pre-Stress Load. *Applied Sciences*. 2018; 8(11):2238.
https://doi.org/10.3390/app8112238

**Chicago/Turabian Style**

Emam, S. A., M. A. Eltaher, M. E. Khater, and W. S. Abdalla.
2018. "Postbuckling and Free Vibration of Multilayer Imperfect Nanobeams under a Pre-Stress Load" *Applied Sciences* 8, no. 11: 2238.
https://doi.org/10.3390/app8112238