# The Influence of Different Loads on the Footbridge Dynamic Parameters

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{i}the Fourier coefficient of the i-th harmonic.

- -
- A random gait, when the step frequency of passing pedestrians distributes under the probability curve and the phase angle of the first harmonic is random;
- -
- A synchronic gait when pedestrian gait coincides, that is, a uniform frequency and phase.

^{2}for the vertical and 0.20 m/s

^{2}for the horizontal direction, respectively. The maximum design acceleration limits for vibration frequencies are defined by other design requirements [47,48,49]. According to Eurocode [8,10], the control of these accelerations is not necessary, if the first vertical vibration frequency of the footbridge is higher than 5 Hz [50,51].

## 3. Results

^{2}(about 2/3 of the total magnitude of the acceleration component comprises up to 20 Hz) and the horizontal direction was up to 0.92 m/s

^{2}(about 1/3 of the total acceleration amplitudes to form component 20 Hz). Figure 9 illustrates two 2.3 and 5.2 Hz horizontal modes that run at a vertical excitation.

^{2}(Figure 9). Figure 9 shows six excitations (3.6, 6.2, 10.0, 13.79, 16.79 and 17.79 Hz) in the vertical direction.

## 4. Discussion

- arithmetic mean:$$\overline{x}=\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}{x}_{i},$$
_{i}—the measurement result. - standard deviation:$$\overline{{\sigma}_{x}=\sqrt{\frac{1}{n-1}{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}}},$$$$COV=\frac{{\sigma}_{x}}{\overline{x}},$$

## 5. Conclusions

^{2}. The comparative evaluation of the experimental results according to the ISO 10137 threshold revealed that pedestrians should not feel any discomfort caused by structural vibrations of the footbridge. The model developed using commercial FE package allows predicting the structural behavior. The research of the dynamic behavior of full operation of the footbridge, such as a faster response to human movement (jogging or running), are planned in the future.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Vibration measuring equipment: (

**a**) Measuring device “Brüel&Kjær”; (

**b**) Accelerometers 8344.

**Figure 5.**Loaded concrete slabs imitating crowd loading; (

**a**) View without concrete slabs, (

**b**) View of loaded footbridge.

**Figure 6.**Acceleration time histories and frequency spectra of acceleration amplitudes: The middle of the bridge (D3 point in Figure 4), a vertical acceleration signal with time and spectral frequency graphs after a group of people crossed the bridge; (

**a**) the interval of 0 to 64; (

**b**) the expanded signal from 35 to 40 s.

**Figure 7.**The middle of the bridge (D3 point in Figure 4) in the horizontal direction of the acceleration signal with time and spectral frequency graphs after a group of people crossed the bridge; (

**a**) the interval of 0 to 64; (

**b**) the expanded signal from 35 to 40 s.

**Figure 8.**The middle of the bridge (D3 point in Figure 4) in the vertical direction of the acceleration signal with time and spectral frequency graphs as a group of people runs on the bridge; (

**a**) the interval of 0 to 64; (

**b**) the expanded signal from 35 to 40 s.

**Figure 9.**The significance of the acceleration signal in the horizontal direction using time and spectral frequency graphs for the middle and the quarter of the footbridge (points D3 and D1 in Figure 4) as a group of people swings the bridge; (

**a**) the interval of 0 to 64; (

**b**) the expanded signal from 35 to 40 s.

**Figure 10.**Frequency spectra of acceleration amplitudes under impact excitation of unloaded (

**a**) and semi-loaded; (

**b**) bridge (1—midpoint, vertical; 2—quarter, vertical; 3—midpoint, horizontal).

**Figure 11.**The Fourier amplitude spectra (without a load and semi-loaded): (

**a**) vertical direction, respectively D1 (without load); (

**b**) vertical direction, respectively D1 (semi-loaded); (

**c**) vertical direction, respectively D3 (without load); (

**d**) vertical direction, respectively D3 (semi-loaded); (

**e**) vertical direction, respectively D5 (without load); (

**f**) vertical direction, respectively D5 (semi-loaded); (

**g**) horizontal direction, respectively D3 (without load); (

**h**) horizontal direction, respectively D3 (semi-loaded).

**Figure 12.**Detected modes using FDD (Frequency Domain Decomposition)—Operational Modal Analysis (OMA).

**Figure 13.**Results of the footbridge modal analysis: first 6 modes (

**a**–

**e**) and curves used primarily to calculate the damping ratio: a normalized correlation function for each mode and calculation of the damping ratio using the correlation function.

**Figure 15.**Mode shapes obtained from the FE model of the bridge (

**a**) 1st bending horizontal; (

**b**) 1st bending vertical; (

**c**) 1st torsional, (

**d**) 2nd bending vertical, (

**e**) 2nd torsional, (

**f**) 3rd bending vertical.

Top Chord | Bottom Chord | Vertical | Diagonal * (from 1 to 5 and from 20 to 24) | Diagonal * (from 6 to 19) | Deck Beam | Wind Bracing |
---|---|---|---|---|---|---|

RHS 300 × 200 × 12 | RHS 300 × 200 × 12 | SHS 200 × 200 × 10 | SHS 180 × 180 × 10 | SHS 180 × 180 × 5 | SHS 200 × 200 × 10 | RHS 200 × 100 × 5 |

**Table 2.**Experimental modes and corresponding natural frequencies after the impact excitation of the bridge and the OMA, along with statistical characteristics.

Experiment | Statistical Characteristics | Experimental Modes and Corresponding Natural Frequencies | |||||
---|---|---|---|---|---|---|---|

1 Mode | 2 Mode | 3 Mode | 4 Mode | 5 Mode | 6 Mode | ||

Impact excitation(1—without load) | $\overline{x}$, Hz | 2.28 | 3.56 | 5.16 | 6.20 | 9.98 | 13.79 |

${\sigma}_{x}$, Hz | 0.074 | 0.14 | 0.37 | 0.16 | 0 | 0.32 | |

COV, % | 3.25 | 3.93 | 7.17 | 2.55 | 0 | 2.32 | |

Impact excitation(2—semi-loaded) | $\overline{x}$, Hz | 2.02 | 2.88 | 4.52 | 5.52 | 7.70 | 12.50 |

${\sigma}_{x}$, Hz | 0.16 | 0.03 | 0.21 | 0.08 | 0.55 | 0.18 | |

COV, % | 7.92 | 1.04 | 4.65 | 1.45 | 7.14 | 1.44 | |

OMA(1—without load) | $\overline{x}$, Hz | 2.25 | 3.50 | 5.00 | 6.25 | 10.0 | 13.60 |

${\sigma}_{x}$, Hz | 0.074 | 0.14 | 0.37 | 0.16 | 0 | 0.45 | |

COV, % | 3.29 | 4.00 | 7.40 | 2.56 | 0 | 3.31 | |

Damping Ratio [%] | 1.713 | 1.077 | 0.905 | 0.621 | 0.440 | 0.521 | |

OMA(2—semi-loaded) | $\overline{x}$, Hz | 2.00 | 2.75 | 4.50 | 5.50 | 7.75 | 12.25 |

${\sigma}_{x}$, Hz | 0.16 | 0.03 | 0.21 | 0.08 | 0.55 | 0.24 | |

COV, % | 8.00 | 1.09 | 4.67 | 1.45 | 7.10 | 1.96 |

Mode | Frequencies (1—without Load; 2—Semi-Loaded) | Mode Type | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

f^{EXP,OMA}, Hz | f^{EXP impact}, Hz | f^{FEM}, Hz | Δ = f_{FE}/f^{EXP,OMA} | Δ = f_{FE}/f^{EXP impact} | |||||||

No | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | |

1 | 2.25 | 2.00 | 2.28 | 2.02 | 2.32 | 2.08 | 1.03 | 1.04 | 1.02 | 1.03 | 1st bending horizontal |

2 | 3.50 | 2.75 | 3.56 | 2.88 | 3.62 | 2.89 | 1.03 | 1.05 | 1.02 | 1.00 | 1st bending vertical |

3 | 5.00 | 4.50 | 5.16 | 4.52 | 5.01 | 4.72 | 1.00 | 1.05 | 0.97 | 1.04 | 1st torsional |

4 | 6.25 | 5.50 | 6.20 | 5.52 | 7.13 | 6.03 | 1.14 | 1.10 | 1.15 | 1.09 | 2nd bending vertical |

5 | 10.00 | 7.75 | 9.98 | 7.70 | 9.7 | 8.76 | 0.97 | 1.13 | 0.97 | 1.14 | 2nd torsional |

6 | 13.79 | 12.50 | 13.60 | 12.25 | 13.52 | 12.05 | 0.98 | 0.96 | 0.99 | 0,98 | 3rd bending vertical |

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**MDPI and ACS Style**

Kilikevičius, A.; Bačinskas, D.; Selech, J.; Matijošius, J.; Kilikevičienė, K.; Vainorius, D.; Ulbrich, D.; Romek, D.
The Influence of Different Loads on the Footbridge Dynamic Parameters. *Symmetry* **2020**, *12*, 657.
https://doi.org/10.3390/sym12040657

**AMA Style**

Kilikevičius A, Bačinskas D, Selech J, Matijošius J, Kilikevičienė K, Vainorius D, Ulbrich D, Romek D.
The Influence of Different Loads on the Footbridge Dynamic Parameters. *Symmetry*. 2020; 12(4):657.
https://doi.org/10.3390/sym12040657

**Chicago/Turabian Style**

Kilikevičius, Artūras, Darius Bačinskas, Jaroslaw Selech, Jonas Matijošius, Kristina Kilikevičienė, Darius Vainorius, Dariusz Ulbrich, and Dawid Romek.
2020. "The Influence of Different Loads on the Footbridge Dynamic Parameters" *Symmetry* 12, no. 4: 657.
https://doi.org/10.3390/sym12040657