# Assessment of the Lateral Vibration Serviceability Limit State of Slender Footbridges Including the Postlock-in Behaviour

^{*}

## Abstract

**:**

## 1. Introduction

- Prelock-in: observed in low pedestrian traffic density scenarios. The lateral vibrations are small and slightly perceptible by humans. In this case, the loads exerted on the footbridge are similar to those applied on a rigid surface (not a swaying surface).
- Lock-in: it is the instability point, in which the acceleration response builds up suddenly. The number of pedestrians on the footbridge reaches the “critical number” that causes acceleration amplitudes sufficient to trigger the pedestrian–structure interaction phenomenon.
- Postlock-in: the crowd density increases above the “critical number”. The growth of the structural response accelerates. The auto-induced component of the load must be included when evaluating the footbridge response to pedestrian action.
- Saturation: above acceleration levels of around 1–1.2 m/s
^{2}[21], some pedestrians feel uncomfortable and choose to change their gait or stop. The growth of the structural response slows down.

^{2}[12,22,30].

_{p}for the auto-induced in-phase component of the pedestrian force and q

_{p}for the out-of-phase component [31,32], thus considering the change of the modal parameters of the structure in a simplified way. Nevertheless, the spectral methods apply to a stationary and uniform pedestrian crowd sufficiently low to omit the human–human interaction. Moreover, the Hivoss guideline [12] incorporates a response spectra method based on Monte Carlo simulations, but it assumes that the mean step frequency of the pedestrian stream is in resonance with the footbridge dominant mode.

## 2. Method to Evaluate the Lateral Vibration of Footbridges Including the Postlock-in Behaviour

^{−1}), which defines completely the dynamic characteristics of the footbridge for the mode considered, and takes the well-known expression:

^{−1}), M the modal mass (kg) and C the modal damping (Nsm

^{−1}). For the particular case of lightly damped footbridges, H(f) presents a sharp peak at f = f

_{b}(the footbridge’s natural frequency) and tends to zero when moving away from this value. Hence, the structural response is amplified only in a very narrow frequency interval centred on f

_{b}.

_{p}(Hz). The step frequency f

_{p}, is a random variable with a normal distribution of probability P(f

_{p}) (mean μ = 0.86 Hz and standard deviation σ = 0.08 Hz [39,40]). Other factors that also influence the pedestrian action (such as the walking speed or the step length) are considered to be represented by the step frequency and the geometric relations between them [39,40,41], and thus are not taken into account for characterising the pedestrian flow. In this paper, the step frequency f

_{p}is taken in the interval (μ−3σ, μ+3σ), which represents a confidence level of 99.7%, which is adequate for the accuracy of this study, since it nearly covers the complete pedestrian´s excitation range without compromising the computational cost of the problem.

#### 2.1. Footbridge Response Due to the Auto-Induced Component of the Pedestrian Load

_{0}sin(2πf

_{b}t) (m), where y

_{0}(m) is the amplitude of the lateral movement and f

_{b}(Hz) is the frequency. Ingólfsson et al. measured both the treadmill motion and the lateral forces that were exerted by the participants. The auto-induced components, which were determined through the cross-covariance between the measured pedestrian force and the velocity of the treadmill, were expressed as proportional to the treadmill velocity in the following way:

_{p}(f

_{b}/f

_{p},y

_{0}) (Nsm

^{−1}) that depends on the frequency ratio f

_{b}/f

_{p}and on the amplitude of the movement y

_{0}is defined by fitting an exponential function to the data [17], and thus considering the large randomness of the measured load. Therefore, the feedback function α(f) (Nm

^{−1}) can be obtained from Equation (2) and expressed as follows:

_{0}and is expressed as follows:

_{p}obtained using the amplitude dependent stochastic model [17] at four different vibration amplitudes (4.5, 10, 19.4 and 31 mm), with the value obtained using Equation (4) [42]. It can be observed that the value of c

_{p}decreases as the amplitude increases, resulting in the second order polynomial (Equation (4)) being an upper bound of the experimental results.

_{p}(f

_{b}/f

_{p}) on the step frequency f

_{p}shows the inter-subject variability in the auto-induced loads. In a real case assessment, there is a crowd of pedestrians walking randomly with different frequencies f

_{p}. Hence, using the probability distribution function of the step frequency P(f

_{p}), and applying the superposition principle, the contribution of each possible value of f

_{p}included in the interval (μ−3σ, μ+3σ) in the auto-induced load exerted by one pedestrian is expressed through the coefficient c

_{p}(f

_{b}) (Nsm

^{−1}), as follows:

_{p}(f

_{b}) calculated through Equation (5). It can be seen that between 0.42 and 1.23 Hz, the resonant harmonic of the auto-induced pedestrian force is positive, limiting the footbridge natural frequency interval of possible instability.

_{b})| expressed in Newton (N), which can be computed by the product of the coefficient c

_{p}(f

_{b}) (Equation (5)) and the amplitude of the footbridge velocity (y

_{0}´ = 2π f

_{b}y

_{0}(m/s)), as follows:

_{d}(m)), the equivalent spectral value of the auto-induced resonant harmonic exerted by N pedestrians |F

_{N}(f

_{b})| in Newton (N) is expressed in modal coordinates, as follows:

_{0,N}is the modal displacement amplitude of the footbridge due to N pedestrians before the HSI develops. Consequently, the spectral value of the auto-induced displacement response, |Y

_{N}(f

_{b})|, is computed by the product of the spectral value of the auto-induced resonant harmonic exerted by N pedestrians, |F

_{N}(f

_{b})| (Equation (7)), and the resonant amplitude of the complex response function, |H(f

_{b})| (Equation (1) particularised in the natural frequency f

_{b}), resulting:

_{N}(f

_{b})|. Consequently, the amplitude of the resonant acceleration response due to the auto-induced component a

_{v,N}(m/s

^{2}) can be expressed in terms of amplification ratio to the modal acceleration amplitude a

_{0,N}(m/s

^{2}) without pedestrian–structure interaction, as follows:

_{v,N}/a

_{0,N}and the number of pedestrians N:

_{d}, L, f

_{b}, c

_{p}(f

_{b}) and |H(f

_{b})|, the non-dimensional constant G (-) is computed as follows:

#### 2.2. Footbridge Response without Bridge Motion

_{p}) (N

^{2}Hz

^{−1}) for the first five harmonics [42]. In this paper, the lateral step frequency f

_{p}is introduced as a variable, which means that Equation (12) represents the PSD of the load exerted by any pedestrian who walks with any frequency f

_{p}:

_{j}, B

_{j}and σ

_{j}are obtained from the experimental data fit. W = 700 N is the mean of people weight. The variance σ

_{j}

^{2}, which represents the energy content in the load around the j-harmonic, can be taken as either the data mean value, in order to represent the mean of the pedestrian load, or the 95% fractile (or that with 5% probability of exceedance), in order to represent the peak value of the pedestrian load. The values A

_{j}, B

_{j}and σ

_{j}are summarised in Table 1.

_{p}) with the step frequency f

_{p}shows the inter-subject variability in the loads. As done before, using the probability distribution function of the step frequency P(f

_{p}) and applying the superposition principle, the contribution of each possible value of f

_{p}included in the interval (μ−3σ, μ+3σ) of the load exerted by one pedestrian is expressed through the function SF(f) (N

^{2}), which can be expressed independently from the step frequency, as follows:

_{p}= 0.86 Hz (mean value of the step frequency) and 3f

_{p}= 2.58 Hz. The second, fourth and fifth harmonics are nearly zero as the even harmonics are small because of the asymmetric character of the pedestrian step (Table 1) and P(f

_{p}) tends to zero outside of the interval 0.62 Hz and 1.10 Hz.

^{2}) is expressed in modal coordinates as follows:

^{2}) by using the following expression:

_{b}= 0.9 Hz, M = 165,880 kg, K = 5,304,000 Nm

^{−1}, C = 10,880 Nsm

^{−1}, L = 144 m and L

_{d}= 88 m. It can be observed that the structural response is amplified only in a very narrow interval centred in f

_{b}. Additionally, in most of the cases (footbridges with a natural frequency in the range 0.4–1.3 Hz), the dynamic response is only caused by the PSD of the first harmonic of the load.

_{0,mean}(m/s

^{2}) and maximum a

_{0,max}(m/s

^{2}) amplitude of the acceleration response without bridge motion due to one pedestrian uniformly distributed can be expressed by calculating the root mean square (RMS) value of the PSD response SY(f), as follows:

_{Ny}(Hz) is the Nyquist frequency. Thus, the probabilistic response is integrated over the frequency range to determine a single expected mean or maximum value. Equation (17) is used to calculate the maximum value of the acceleration response without bridge motion, while Equation (16) is used to calculate the mean value of the acceleration response considered to be perceived by pedestrians, in order to compare it with the threshold beyond which the HSI develops.

#### 2.3. Footbridge Response: General Formulation

- Prelock-in: the maximum acceleration response is calculated by using Equation (17).
- Lock-in: the criterion for the lateral lock-in is suggested to establish that the acceleration threshold beyond which HSI develops is 0.1–0.15 m/s
^{2}. When pedestrians perceive this acceleration level, they modify their gait and interact with the structure. The critical number of pedestrians is determined by comparing the mean value of the acceleration response calculated by using Equation (16), with 0.125 m/s^{2}. - Postlock-in: the maximum acceleration response is calculated by adding the response without bridge motion (Equation (17)) and that due to the auto-induced component of the load (Equation (10)). In Equation (10), the initial level of vibration that people perceive a
_{0,N}is obtained using Equation (16).

_{crit}and the maximum acceleration response a

_{N}(m/s

^{2}) is expressed as follows:

## 3. Numerical Applications

#### 3.1. Pedro e Inês Footbridge

^{2}that occurred with 145 people.

#### 3.2. Lardal Footbridge

^{2}for as few as 40 people. During excessive lateral vibrations (beyond 1 m/s

^{2}), it was observed that pedestrians accepted accelerations up to a given limit, beyond which some of them decided to stop, thereby limiting the accelerations (saturation level) [21].

#### 3.3. Numerical Response Evaluation

**.**The saturation level or acceleration value, beyond which some pedestrians feel uncomfortable and choose to stop, was assumed to be 1.2 m/s

^{2}, consistent with the recommendations of some authors [7,21] and the values recorded in the experiments on both footbridges. Equations (18–21) are represented in Figure 10 (the Pedro e Inês footbridge) and Figure 11 (the Lardal footbridge).

_{0}* (N) due to an idealised stream of N´ perfectly synchronised pedestrians, expressed in generalised coordinates; the parameter a

_{max}(m/s

^{2}), which controls the relation between the acceleration response and the number of pedestrians, also represented in Figure 10 and Figure 11; and the critical number of pedestrians obtained by using the two criteria of the Hivoss guideline. The relation between the acceleration response and the number of pedestrians is:

^{2}implies a high number of pedestrians that is beyond the limits of the full-scale crowd pedestrian tests (576 pedestrians in the case of Pedro e Inês and 218 pedestrians in Lardal), only the expression for pedestrian density lower than 1 p/m

^{2}was considered in this paper.

#### 3.4. Analysis of the Results

^{2}) gave a value of 167 pedestrians, which was far from the experimental result.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Full-scale crowd pedestrian tests on the Pedro e Inês footbridge [18]. Maximum lateral acceleration response vs. number of pedestrians. Stages of lateral vibration response.

**Figure 4.**The damping coefficient cp(fb) independent of the pedestrian step frequency f

_{p}obtained through Equation (5).

**Figure 5.**Experimental mean and 95% fractile power spectral density functions SF(f) of the pedestrian load exerted on static surface expressed independently of the pedestrian frequency.

**Figure 6.**The functions SX(f) (L = 144 m and Ld = 87.5 m) and H(f)

^{2}(f

_{b}= 0.9 Hz, M = 165,880 kg, K = 5,304,000 Nm

^{−1}and C = 10,880 Nsm

^{−1}[18]) are shown overlapped to represent Equation (15).

**Figure 8.**The Pedro e Inês footbridge. (Picture after https://structurae.net/en/structures/pedro-and-ines-bridge.).

**Figure 9.**The Lardal footbridge. (Picture after http://broer-vestfold.blogspot.com/2011/08/bro-v-kjrra-fossepark-i-lardal.html.).

j = 1 | j = 2 | j = 3 | j = 4 | j = 5 | |
---|---|---|---|---|---|

A_{j} | 0.900 | 0.020 | 0.774 | 0.0258 | 0.612 |

B_{j} | 0.043 | 0.031 | 0.026 | 0.064 | 0.026 |

σ_{j}/W (mean value) | 0.035 | 0.005 | 0.018 | 0.004 | 0.008 |

σ_{j}/W (95% fractile) | 0.054 | 0.008 | 0.025 | 0.006 | 0.0012 |

Footbridge | f_{b} (Hz) | M (kg) | ξ (-) | L (m) | L_{d} (m) | b (m) |
---|---|---|---|---|---|---|

Pedro e Inês | 0.91 | 165,880 | 5.80 × 10^{−3} | 144.00 | 88.00 | 4.00 |

Lardal | 0.83 | 18,000 | 2.50 × 10^{−2} | 91.00 | 80.00 | 2.40 |

Footbridge | |H(f_{b})| (mN^{−1}) | c_{p}(f_{b}) (Nsm^{−1}) | G (-) | a_{0,mean} (m/s^{2}) | a_{0,max} (m/s^{2}) | N_{crit} (-) | N_{sat} (-) |
---|---|---|---|---|---|---|---|

Pedro e Inês | 1.59 × 10^{−5} | 170.09 | 3.20 × 10^{−2} | 1.64 × 10^{−3} | 2.53 × 10^{−3} | 75 | 129 |

Lardal | 4.09 × 10^{−5} | 177.36 | 5.5 × 10^{−2} | 9.52 × 10^{−3} | 15.00 × 10^{−3} | 13 | 36 |

Footbridge | |H(f_{b})| (mN^{−1}) | F_{0}* (N) | a_{ma} (m/s^{2}) | N_{crit} (Arup´s) (-) | N_{crit} (acc.) (-) |
---|---|---|---|---|---|

Pedro e Inês | 1.59 × 10^{−5} | 11.20 | 1.20 × 10^{−2} | 73 | 167 |

Lardal | 4.09 × 10^{−5} | 33.45 | 7.40 × 10^{−2} | 31 | 4 |

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**MDPI and ACS Style**

Cuevas, R.G.; Jiménez-Alonso, J.F.; Martínez, F.; Díaz, I.M.
Assessment of the Lateral Vibration Serviceability Limit State of Slender Footbridges Including the Postlock-in Behaviour. *Appl. Sci.* **2020**, *10*, 967.
https://doi.org/10.3390/app10030967

**AMA Style**

Cuevas RG, Jiménez-Alonso JF, Martínez F, Díaz IM.
Assessment of the Lateral Vibration Serviceability Limit State of Slender Footbridges Including the Postlock-in Behaviour. *Applied Sciences*. 2020; 10(3):967.
https://doi.org/10.3390/app10030967

**Chicago/Turabian Style**

Cuevas, Rocío G., Javier F. Jiménez-Alonso, Francisco Martínez, and Iván M. Díaz.
2020. "Assessment of the Lateral Vibration Serviceability Limit State of Slender Footbridges Including the Postlock-in Behaviour" *Applied Sciences* 10, no. 3: 967.
https://doi.org/10.3390/app10030967