# Analytical Investigation on the Dynamic Behavior of Multi-Span Continuous Beams Supported on Soil with Finite Depth

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## Abstract

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## 1. Introduction

## 2. Basic Relationships

## 3. Analytical Solution of the Continuous Beam Vibration

## 4. Numerical Tests and Model Validation

#### 4.1. Simply Supported Beam

#### 4.2. Two-Span Continuous Beam

## 5. Results and Discussion

#### 5.1. Effect of Soil Thickness

#### 5.2. Effect of the Viscous Damping Coefficient of the Soil

#### 5.3. Effect of Subgrade Reaction Coefficient

#### 5.4. Effect of Span Ratio

## 6. Conclusions

- As the thickness of the soil involved in the movement increases, the continuous beam’s natural frequency will decrease, and the resonance amplitude increases significantly. When comparing the dynamic responses of different beams, the enhancement impact of finite depth soil motion on the response amplitude of the indirect excitation beam section is more significant than that of the direct excitation beam section.
- The viscous damping coefficient of the soil and the coefficient of subgrade reaction have a greater inhibition effect on the indirect influence span than on the direct excitation span. As the coefficient of the subgrade reaction increases, the system’s natural frequency increases, and the resonant response amplitude decreases.
- The adjustment of the span ratio has a significant influence on the dynamic response of the multi-span beam system. (1) Compared with other span ratios, the odd-order natural frequency is the largest and the even-order natural frequency is the smallest when $\epsilon =1.00$. (2) The smaller the span ratio is, the more pronounced the increasing effect of the resonant frequency is. (3) When the span ratio is determined, no matter whether the load acts on the long span or the short span, the response amplitude of the beam span indirectly affected by the load is certain.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Amplitude-frequency curves at each span’s mid-span of a two-span continuous beam: (

**a**) the first span; (

**b**) the second span.

**Figure 4.**Effect of soil thickness on a two-span continuous beam: (

**a**) natural frequency; (

**b**) the first span; (

**c**) H = 5 m.

**Figure 6.**Impact of viscous damping coefficient of the soil on displacement of a two-span continuous beam: (

**a**) the first span; (

**b**) inhibition effect of each span.

**Figure 7.**Impact of subgrade reaction coefficient on vibration response of a two-span continuous beam: (

**a**) natural frequency; (

**b**) mid-span of the first span.

**Figure 8.**The suppressing impact of subgrade reaction coefficient on displacement of a two-span continuous beam.

**Figure 9.**Modal effect of span ratio on a two-span continuous beam: (

**a**) 6.096 m + 6.096 m (ε = 1.00); (

**b**) 4.572 m + 7.62 m (ε = 0.60); (

**c**) 3.048 m + 9.144 m (ε = 0.33).

**Figure 11.**Effect of span ratio on response amplitude of a two-span continuous beam: (

**a**) mid-span of the first span; (

**b**) mid-span of the second span.

Physical Meaning | Symbol | Unit | Numerical Value |
---|---|---|---|

The jth span length of a continuous beam | ${l}_{j}$ | $\mathrm{m}$ | 6.096 |

The beam’s width | $b$ | $\mathrm{m}$ | 0.61 |

The beam’s height | $h$ | $\mathrm{m}$ | 0.305 |

The per unit length beam’s mass | $m$ | $\mathrm{kg}\cdot {\mathrm{m}}^{-1}$ | 447.08 |

Damping factor per unit length | $c$ | $\mathrm{kN}\cdot \mathrm{s}\cdot {\mathrm{m}}^{-2}$ | 1.0 |

Moment of inertia of the per unit length beam | $\gamma $ | $\mathrm{kg}\cdot \mathrm{m}$ | 3.466 |

Young’s modulus of the beam | $E$ | $\mathrm{MPa}$ | 2.482 × 104 |

Poisson ratio of the beam | ${\nu}_{0}$ | —— | 0.25 |

Soil mass per unit thickness of the per unit length beam | ${\rho}_{s}$ | $\mathrm{kg}\cdot {\mathrm{m}}^{-2}$ | 1037 |

Viscous damping coefficient of the soil per unit thickness of the per unit length beam | ${c}_{s}$ | $\mathrm{kN}\cdot \mathrm{s}\cdot {\mathrm{m}}^{-3}$ | 3.6 |

Modulus of subgrade reaction | ${k}_{f}$ | $\mathrm{MPa}$ | 16.55 |

Soil thickness | $H$ | $\mathrm{m}$ | 5 |

Natural Frequency (Hz) | (Timoshenko S., 1974) [10] | (Thambiratnam and Zhuge, 1996) [26] | (Friswell et al., 2007) [27] | The Present Study | |
---|---|---|---|---|---|

Euler-Bernoulli | Timoshenko | ||||

${\omega}_{1}$ | 32.9063 | 32.9033 | 32.898 | 32.8749 | 32.8289 |

${\omega}_{2}$ | 56.8135 | 56.8193 | 56.808 | 56.8040 | 56.1037 |

${\omega}_{3}$ | 112.908 | 111.961 | 111.900 | 111.9186 | 108.303 |

${\omega}_{4}$ | 193.760 | 193.8085 | 182.7608 |

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**MDPI and ACS Style**

Li, D.; Yang, H.; Ma, J.; Wang, J.; Wang, C.; Guo, Y.
Analytical Investigation on the Dynamic Behavior of Multi-Span Continuous Beams Supported on Soil with Finite Depth. *Coatings* **2024**, *14*, 864.
https://doi.org/10.3390/coatings14070864

**AMA Style**

Li D, Yang H, Ma J, Wang J, Wang C, Guo Y.
Analytical Investigation on the Dynamic Behavior of Multi-Span Continuous Beams Supported on Soil with Finite Depth. *Coatings*. 2024; 14(7):864.
https://doi.org/10.3390/coatings14070864

**Chicago/Turabian Style**

Li, Da, Hang Yang, Jianjun Ma, Ju Wang, Chaosheng Wang, and Ying Guo.
2024. "Analytical Investigation on the Dynamic Behavior of Multi-Span Continuous Beams Supported on Soil with Finite Depth" *Coatings* 14, no. 7: 864.
https://doi.org/10.3390/coatings14070864