# Static and Dynamic Solutions of Functionally Graded Micro/Nanobeams under External Loads Using Non-Local Theory

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## Abstract

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## 1. Introduction

## 2. Modeling and Extraction of Governing Equation

#### 2.1. Formulation of the Problem

#### 2.2. Governing Differential Equations for Functionally Graded (FG) Beam

#### 2.3. Dimensionless Governing Equations in Different Boundary Conditions

## 3. Static Analysis

## 4. Buckling Analysis

## 5. Dynamic Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of a Euler–Bernoulli beam before (

**a**) and after (

**b**) bending in coordinate system.

**Figure 4.**Non-local effect on the tip displacement of the beam with one free end versus the variation of functional property under uniform load.

**Figure 5.**Variations of critical buckling load by non-local parameter in s-s beams for different values of power index of FGM.

**Figure 6.**Variations of critical buckling load by non-local parameter in beams under different boundary conditions (n = 0.5).

**Figure 8.**The variations of max deformation of cl-cl beams versus the normalized velocity of applied load for different power index at (

**a**) ta = 0, (

**b**) ta = 0.2 and (

**c**) ta = 0.4.

**Figure 9.**The variations of the max deformation of cl-cl beams versus the normalized velocity of applied load for different non-local parameter values at n = 2.

**Figure 10.**The variations of the max deformation of beams with one free end versus the normalized velocity of applied load for different power index at (

**a**) ta = 0, (

**b**) ta = 0.2 and (

**c**) ta = 0.4.

**Figure 11.**The variations of the greatest deformation of beams with one free end versus the velocity of applied load for different non-local parameter values at n = 0.

Materials | E (GPA) | ρ (kg/m^{3}) |
---|---|---|

Metal (Aluminum) | 70 | 3960 |

Ceramic (Aluminum oxide) | 390 | 2700 |

BCs | k |
---|---|

Simply supported-Simply supported (S-S) | π |

Simply supported-Clamped (S-C) | 4.4934 |

Clamped-Clamped (C-C) | 6.2832 |

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**MDPI and ACS Style**

Moheimani, R.; Dalir, H.
Static and Dynamic Solutions of Functionally Graded Micro/Nanobeams under External Loads Using Non-Local Theory. *Vibration* **2020**, *3*, 51-69.
https://doi.org/10.3390/vibration3020006

**AMA Style**

Moheimani R, Dalir H.
Static and Dynamic Solutions of Functionally Graded Micro/Nanobeams under External Loads Using Non-Local Theory. *Vibration*. 2020; 3(2):51-69.
https://doi.org/10.3390/vibration3020006

**Chicago/Turabian Style**

Moheimani, Reza, and Hamid Dalir.
2020. "Static and Dynamic Solutions of Functionally Graded Micro/Nanobeams under External Loads Using Non-Local Theory" *Vibration* 3, no. 2: 51-69.
https://doi.org/10.3390/vibration3020006