Using Full-Scale Observations on Footbridges to Estimate the Parameters Governing Human–Structure Interaction

Van Nimmen, K.; Van den Broeck, P.

doi:10.1061/(ASCE)BE.1943-5592.0001975

Open access

Technical Papers

Nov 9, 2022

Using Full-Scale Observations on Footbridges to Estimate the Parameters Governing Human–Structure Interaction

Authors: K. Van Nimmen https://orcid.org/0000-0002-8188-1297 [email protected] and P. Van den BroeckAuthor Affiliations

Publication: Journal of Bridge Engineering

Volume 28, Issue 1

https://doi.org/10.1061/(ASCE)BE.1943-5592.0001975

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Abstract

The further development and improvement of prediction models for crowd-induced vibrations of footbridges requires detailed information on representative operational loading data. In this paper, an inverse method is used to estimate the parameters that govern human–structure interaction from the resulting structural response. The parameters of interest concern the dynamic characteristics of a mass-spring-damper (MSD) system, applied to describe the mechanical interaction between the pedestrian and the structure. The dynamic characteristics of the MSD interaction model are estimated by minimizing the discrepancy between the observed and the simulated power spectral density of the structural response. The parameter estimation procedure assumes that the dynamic behavior of the empty structure, the average weight, and the distribution of step frequencies in the crowd are known. The proposed approach is verified using numerical simulations and the influence of modeling errors is investigated. The results show that as footbridges and the human body are by nature lightly (

≲

2%) and strongly (≈30%) damped, respectively, the structural response is most sensitive to small variations in the natural frequency of the MSD interaction model. The results, furthermore, show that the parameter estimation problem is mostly sensitive to errors related to the mean value of the distribution of step frequencies and the structural modes’ natural frequency and modal mass. The impact of the structural modeling errors is found to decrease as the impact of human–structure interaction increases. Next, the approach is applied to two real footbridges where the walking behavior and the structural response induced by high pedestrian densities are observed. The results show that an estimate of the natural frequency (≈3.0 Hz) and damping ratio (

\approx 34 %

) of the MSD interaction model is obtained that is in accordance with recent findings in the literature. These estimates are, however, for the first time ever, it is believed, based on full-scale observations involving high pedestrian densities.

Introduction

Footbridges are often conceived as slender and lightweight structures. In consequence, these structures can be highly sensitive to dynamic pedestrian excitation (Živanović et al. 2005). Where the static traffic load for footbridges is typically 5 kN/m² (≈6 persons/m²) (NBN EN 1991-2 ANB 2013), a vibration serviceability assessment is generally performed, starting from 15 persons on the bridge deck, up to 1.5 persons/m² (AFGC 2006; Heinemeyer et al. 2009). Dynamically speaking, pedestrian densities of 0.5 persons/m², and beyond, can be considered high (AFGC 2006; Heinemeyer et al. 2009). Currently, the response of these structures to human-induced loading is predicted based on equivalent load models, upscaled from single-person force measurements (AFGC 2006; Heinemeyer et al. 2009). Although the dynamic performance under high crowd densities is often imperative for the design, these loading conditions have virtually never been verified (Georgakis and Ingólfsson 2014). Concerns about these load models are further strengthened by the fact that human–structure interaction (HSI) phenomena are not well understood (Bruno and Venuti 2009). The further development and calibration of pedestrian excitation models requires in-field observations, for they are the only way to obtain detailed and accurate information on representative operational loading data (Georgakis and Ingólfsson 2014; Živanović 2012). As direct force measurements are, in this case, practically infeasible, inverse methods, where the input or unknown model parameters are estimated from the resulting vibration response and a dynamic model of the structure, constitute a promising alternative.

HSI may refer to two different phenomena. The first phenomenon, also known as active HSI (Van Nimmen et al. 2021a) or structure-to-human (S2H) interaction (Ahmadi et al. 2018), refers to the case where a pedestrian changes walking behavior as a result of the (un)consciously perceived vibrations of the supporting structure. As, in the vertical direction, these active HSI phenomena arguably only occur for unacceptably high structural vibration levels, whereby pedestrians automatically stop walking (Dang and Živanović 2016), they are not taken into account in this study. The second phenomenon, also known as passive HSI (Van Nimmen et al. 2021a) or human-to-structure (H2S) interaction (Ahmadi et al. 2018), describes the mechanical interaction between the human body and the supporting structure. In the vertical direction, this phenomenon has been shown to have a significant impact on the dynamic behavior of the coupled crowd–structure system (Pimentel and Waldron 1996; Brownjohn and Fu 2005; Živanović et al. 2010; Dong et al. 2011; Salyards and Hua 2015; Shahabpoor et al. 2016a) and, in turn, on the vibration serviceability assessment of footbridges (Shahabpoor et al. 2017; Van Nimmen et al. 2017a, 2021a). This HSI phenomenon is focused on in the following.

In the literature, different approaches have been adopted to model and identify HSI for walking persons. Inspired by developments in biomechanics, multibody (Maca and Valasek 2011) and inverted pendulum (IP) (Bocian et al. 2013; Dang and Živanović 2013; Qin et al. 2013, 2014) models have been introduced to simulate the vertical pedestrian excitation and interaction. Experimental validation of these models is lacking and difficult, owing to the large number of model parameters involved.

Recent developments increasingly make use of a mass-spring-damper (MSD) model to describe vertical HSI (Van Nimmen et al. 2017a; Shahabpoor et al. 2016a; Caprani and Ahmadi 2016; Venuti et al. 2016; Zang et al. 2016). This model owes its popularity to its (1) simplicity and (2) ability to describe the low-frequency (0–10 Hz) dynamic behavior of the human body (ISO 1981; Brownjohn 2001; Matsumoto and Griffin 2003; Agu and Kasperski 2011; Caprani and Ahmadi 2016; Shahabpoor et al. 2016b; Cappellini et al. 2016; Van Nimmen et al. 2017a). This model is composed of a sprung mass m_h1, an unsprung mass m_h0, a spring k_h1, and a damping element c_h1. The nominal mass of the person is given by m_h = m_h1 + m_h0 with μ_h1 = m_h1/m_h. Silva and Pimentel (2011) used the MSD model to define the relation between the amplitudes of the first three harmonics of the ground reaction forces (GRFs) induced by a pedestrian on a rigid surface and the acceleration levels registered on the pedestrian’s lower back. Acceleration data were collected for the lower backs of 33 persons walking on a rigid surface. The GRFs were simulated using a model given in the literature (Živanović et al. 2005). Considering a body mass of 70 kg and a pacing rate of 1.8 Hz, in that study, a natural frequency of 2.64 Hz and a damping ratio of 55% are reported for the MSD model.

Jimenez-Alonso and Saez (2014), Zang et al. (2016), and Zhang et al. (2015) reported on experiments performed on a 16-m-long double steel U-beam laboratory setup. The laboratory setup was excited by up to 10 pedestrians, together with a known shaker input force. Jimenez-Alonso and Saez (2014) fitted the MSD model parameters to minimize the mean square error between the experimentally identified and numerically predicted fundamental vertical natural frequency of the occupied footbridge. This approach resulted in a natural frequency of 2.75 Hz and a damping ratio of 47% for the MSD model. Zang et al. (2016) and Zhang et al. (2015) conclude that as the shaker frequency was closer to the natural frequency of the laboratory setup, the excitation of the pedestrians can be considered negligible. Using the known shaker input force, the frequency response function (FRF) of the primary system (footbridge) was derived from experiments. The parameters of the MSD model were then found by fitting the predicted FRF to the identified FRF. In this way, a natural frequency of 1.85 Hz and a damping ratio of 30% were found for the MSD model.

Shahabpoor et al. (2016a) used a laboratory setup consisting of an in situ cast post-tensioned concrete slab with a length of 11.2 mm and a width of 2 m. The experiments considered 2–15 pedestrians walking on the slab. The walking force of each person was set to be equal to the forces registered by an instrumented treadmill for the pedestrian walking on a rigid surface at a self-selected walking speed. In addition, the slab was excited by a known shaker input force. To identify the parameters of the MSD model, the error between the predicted and experimentally identified peak FRF magnitude and the modal parameters of the occupied structure (natural frequency, modal mass, and modal damping ratio) was minimized. In doing so, only a single mode of the concrete slab was taken into account. That study reports results obtained for the MSD model for natural frequencies in the range 2.75–3.00 Hz and damping ratios in the range 27.5%–0%.

So far, the calibration of the MSD interaction model involved laboratory-scale setups and a limited number of pedestrians. This work aims to estimate the dynamic characteristics of an MSD interaction model from full-scale observations involving high pedestrian densities. To this end, a parameter estimation procedure is proposed. The procedure is applied to two real footbridges, where the walking behavior and the structural response induced by high pedestrian densities are observed. The novelty and contribution of this work are in the formulation of the parameter estimation procedure specifically for the problem of HSI on footbridges, the analysis of its sensitivity to modeling errors, and the application to full-scale experimental observations.

The outline of this paper is as follows. In the next section, the parameter estimation procedure is presented and the impact of modeling errors is investigated. Then the procedure is applied to two real footbridges. The final section summarizes the conclusions.

Parameter Estimation Procedure

Mathematical Framework

This study uses the crowd–structure model introduced by Van Nimmen et al. (2017a). The dynamic behavior of the empty structure is represented by its modal parameters. Each pedestrian in the crowd is modeled as the superposition of an autonomous force term, corresponding to the forces induced on a rigid surface, and a linear MSD system used to simulate the vertical mechanical interaction with the supporting structure. This framework assumes that there is no active interaction between the pedestrians and the structure. In other words, it is assumed that the walking behavior (and thus, the autonomous force term) is not influenced by the structural vibrations perceived by the pedestrians.

The first-order continuous-time state-space equation of motion of the coupled crowd–structure model is

\dot{x} (t) = A_{c} x (t) + B_{c} S_{p} (t) p_{f} (t)

(1)

with x (t) = {[\begin{matrix} z^{⊤} (t) u_{h}^{⊤} (t) {\dot{z}}^{⊤} (t) {\dot{u}}_{h}^{⊤} (t) \end{matrix}]}^{⊤}

(2)

where

x (t) \in {Re}^{n_{s}}

is the state vector; n_s = 2 (n_m + n_h),

z (t) \in {Re}^{n_{m}}

, is the modal coordinate vector; n_m = number of modes;

u_{h} \in {Re}^{n_{h}}

is the vector of displacements of the MSD interaction models; n_h = number of pedestrians;

p_{f} (t) \in {Re}^{n_{h}}

is the force vector collecting the time history of the autonomous force term p_f,k(t) of each pedestrian k of the n_h pedestrians as rows; and

S_{p} (t) \in {Re}^{n_{dof} \times n_{h}}

is a selection matrix that transfers the forces to the corresponding n_dof degrees of freedom (DOFs) of the model of the structure. The autonomous force term p_f,k(t) is modeled using the probabilistic single-person force model developed by Živanović et al. (2007) and depends on the mass (m_h,k) and the step frequency (f_s,k) of the pedestrian. The arrival times of the pedestrians are assumed to follow a Poisson distribution (Helbing and Molnar 1995; Živanović 2012). The system matrices

A_{c} \in {Re}^{n_{s} \times n_{s}}

and

B_{c} \in {Re}^{n_{s} \times n_{dof}}

are defined as

A_{c} = [\begin{matrix} 0 I \\ - {\bar{M}}_{h s}^{- 1} {\bar{K}}_{h s} - {\bar{M}}_{h s}^{- 1} {\bar{C}}_{h s} \end{matrix}]

(3)

B_{c} = [\begin{matrix} 0 \\ T_{p} \end{matrix}]

(4)

where

{\bar{M}}_{h s}

,

{\bar{K}}_{h s}

, and

{\bar{C}}_{h s}

are the generalized mass, stiffness, and damping matrices of the coupled crowd–structure system, and T_p is the generalized input transformation matrix, as defined in Van Nimmen et al. (2017a).

The power spectral density (PSD) of the structural accelerations (

G_{\ddot{u}}

) can be derived from Monte Carlo simulations, given that the PSD is representative of the considered pedestrian density (see also the next section).

The mathematical framework is visualized in Fig. 1.

Parameter Estimation Problem

The objective is to identify the parameters that govern the effect of HSI on the resulting structural response: the dynamic characteristics of the linear mechanical systems representing the pedestrians in the crowd. The parameter estimation procedure assumes that the following inputs are known.

1.

The dynamic behavior of the empty structure (the structure without pedestrians). The natural frequencies and mode shapes can be predicted using a finite-element (FE) model and the corresponding modal damping ratios estimated based on the used materials and the construction type (AFGC 2006). A much more accurate representation of the dynamic behavior of the structure can be obtained from experimental vibration data using system identification techniques (Peeters and De Roeck 1999; Reynders and De Roeck 2008).

2.

The structural response induced by the pedestrians. This requires vibration measurements on the structure during operation. The duration of the measurements should be sufficiently long that the collected data can be considered representative of the involved pedestrian density. This condition can be verified by checking that the PSD of the structural accelerations has converged.

3.

The average weight of the pedestrians. This information can be easily collected for each participant using a scale on site.

4.

The distribution of step frequencies in the crowd. This can be derived from empirical relations (e.g., as a function of the pedestrian density Bruno and Venuti 2009) but is preferably based on additional information collected in situ (e.g., the body motion registered for each pedestrian) (Van Nimmen et al. 2021b).

Van Nimmen et al. (2017b) show that when a sufficiently long duration of the structural response induced by a certain pedestrian density is considered, the statistical effect of the remaining unknown stochastic parameters, i.e., the effect of the randomness in the autonomous term, on the PSD of the resulting structural response can be averaged out. The randomness of the autonomous force term results from the arrival times and the intraperson variabilities. As these parameters are much more difficult to identify reliably from experimental observations, it is a great advantage to be able to average out their statistical effect.

Next, an optimization problem is formulated, where the objective function measures the difference between the PSD of the experimentally observed (

{\tilde{G}}_{\ddot{u}}

) and numerically simulated (

G_{\ddot{u}}

) structural responses within a frequency range defined by a lower (ω_l [rad/s]) and upper bound (ω_u [rad/s]). For now, it is assumed that the structural response is observed in a single output location. The residuals are computed at discrete frequencies ω_k [rad/s]:

ω_{k} = ω_{l} + k Δ ω for k = 0, \dots, N

, where N + 1 = (ω_u − ω_l)/Δω + 1 = number of samples; and Δω = selected frequency resolution. The decision variables in this optimization problem are the natural frequency f_h1, the modal damping ratio ξ_h1, and the mass ratio μ_h1 of the selected linear mechanical system (see the introduction). It is shown by Van Nimmen et al. (2017a) that the impact of the interperson variability of the parameters of the human body models on the resulting structural response is low. Therefore, these parameters are assumed to be identical for the different persons on the structure and are subject to the following physical constraints: f_h1 > 0, ξ_h1 > 0, and 0 < μ_h1 < 1. The problem, subject to four inequality constraints, is formulated in standard form:

\begin{aligned} \underset{f_{h 1}, ξ_{h 1}, μ_{h 1}}{minimize} & g (f_{h 1}, ξ_{h 1}, μ_{h 1}) = \frac{∥ {\tilde{G}}_{\ddot{u}} - G_{\ddot{u}} (f_{h 1}, ξ_{h 1}, μ_{h 1}) ∥_{2}^{2}}{∥ {\tilde{G}}_{\ddot{u}} ∥_{2}^{2}} \\ subject to & f_{h 1} > 0 \\ ξ_{h 1} > 0 \\ μ_{h 1} > 0 \\ μ_{h 1} < 1 \end{aligned}

(5)

and can be categorized as a nonlinear programming problem, owing to the nature of the objective function (Nocedal and Wright 2006). The inequality constrained optimization problem can be solved using the lsqnonlin solver (MATLAB 2018) by means of the trust-region-reflective algorithm.

In the case where the structural response is registered at m_dof DOFs, the many objectives can be combined into a single-objective scalar function as follows:

\begin{aligned} \underset{f_{h 1}, ξ_{h 1}, μ_{h 1}}{minimize} & \sum_{l = 1}^{m_{dof}} γ_{l} g_{l} (f_{h 1}, ξ_{h 1}, μ_{h 1}) \\ with \sum_{l = 1}^{m_{dof}} γ_{l} = 1 \\ γ_{l} > 0 l = 1, \dots, m_{dof} \end{aligned}

(6)

where γ_l is the weight factor for the structural response registered at DOF l. If desired, the objective function in Eq. (6) can be further extended with a frequency weight factor γ_ω to increase or decrease the contribution of certain frequency ranges.

Given the low variation that is reported in the literature for the mass ratio μ_h1, it is reasonable to consider this parameter fixed and equal to μ_h1 = 0.95 (Matsumoto and Griffin 2003; Shahabpoor et al. 2016b). In this way, the numbers of parameters and inequality constraints in the optimization problem in Eq. (5) are reduced to two (f_h1, ξ_h1) and two (f_h1 > 0, ξ_h1 > 0), respectively.

Fig. 1 visualizes the applied mathematical framework, the parameter estimation problem (objective function), and the input from the experimental observations.

Numerical Verification Example

The parameter estimation procedure is illustrated and verified using a numerical example. The structure under consideration is a footbridge with a span L of 50 m and a deck width of 3 m. The structure is modeled as a simply supported Euler–Bernoulli beam for which only the fundamental vertical bending mode is considered, with realistic values of its modal parameters (AFGC 2006; Heinemeyer et al. 2009): a sinusoidal mode shape; a modal mass of 50 × 10³ kg; a modal damping ratio of 0.5%; and a natural frequency of 2 Hz. In this work, the modal mass refers to the proportion of the physical mass of the structure that is engaged in that mode (Brownjohn and Pavić 2007). It is noted that this is different from the unit-value modal mass that is conventionally related to mass-normalized mode shapes (Brownjohn and Pavić 2007). The lower the modal mass, the more sensitive that mode is to dynamic excitation.

A pedestrian density of 0.5 pedestrians/m² is considered, with a corresponding walking speed of 1.30 m/s, a mean step frequency of 1.89 Hz (Bruno and Venuti 2009), and a standard deviation of the step frequencies of 0.175 Hz, as for the case of spatially unrestricted traffic (AFGC 2006) (

N (1.89, 0.175)

). The phase shifts between the different pedestrians are uniformly distributed in [0, 2π] (Van Nimmen et al. 2020) and their arrival times are assumed to follow a Poisson distribution (Helbing and Molnar 1995; Živanović 2012). Fig. 2 presents the PSD of the pedestrian load. The latter is determined based on the procedure detailed in Van Nimmen et al. (2020), using the dynamic load factors for the main and subharmonics of the walking load defined in Živanović et al. (2007). The mechanical properties of the MSD interaction model are set to approximate the dynamic behavior of a person with one or two legs slightly bent [f_h1 = 3.0 Hz; ξ_h1 = 0.30; m_h1 = 70 kg; and μ_h1 = 0.95].

Fig. 2. PSD of pedestrian load (G_F) for pedestrian density of 0.50 persons/m².

Fig. 3 presents the PSD of the structural response at midspan, predicted without and with HSI. This figure illustrates that, in both cases, the structural response is dominated by a single peak. For the case without HSI, this peak is centered around the natural frequency of the footbridge (2 Hz). When HSI is accounted for, the peak is centered around the effective natural frequency of the coupled crowd–structure system (Van Nimmen et al. 2017a), which is, in this case, located at 1.93 Hz.

Fig. 3. (a) PSD of structural response at midspan induced by pedestrian density d = 0.5 pedestrians/m²; and (b) enlargement of the dominant frequency range. Black = without HSI; gray = with HSI.

Optimal Solution

In this example, the PSD of the structural response at midspan (frequency range 0–10 Hz) is considered as the input for the objective function [Eq. (5)]. The objective function is now evaluated for a realistic range of the two decision variables (Fig. 4): the natural frequency 1 ≤ f_h1 ≤ 6 Hz and the modal damping ratio 0.1 ≤ ξ_h1 ≤ 0.8. This figure shows that the objective function, and thus the impact of HSI, is mainly sensitive to changes in the natural frequency f_h1 of the MSD interaction model. Fig. 4 illustrates a generally low sensitivity of the objective function to the modal damping ratio ξ_h1 of the MSD interaction model. The sensitivity to ξ_h1 (slightly) increases for increasing values of f_h1.

Fig. 4. Evaluation of objective function in terms of natural frequency f_h1 and modal damping ratio ξ_h1 of MSD interaction model. Solid circle = solution of optimization procedure.

For different feasible sets of initial values, the optimization procedure is found to successfully converge toward the true minimum: f_h1 = 3.0 Hz and ξ_h1 = 0.3.

Influence of Modeling Errors

To gain insight into the influence of the modeling errors on the solution of the parameter estimation problem, a parameter study was conducted to consider a reasonable range of footbridge parameters and pedestrian densities.

1.

Footbridge parameters. The previously introduced structure of a simply supported Euler–Bernoulli beam with a span of 50 m and a width of 3 m was used. Only the fundamental vertical bending mode with a sinusoidal mode shape was considered. The following ranges of modal parameters were considered: 1 ≤ f₁ ≤ 6 Hz in steps of 0.5 Hz, and two values of the structural damping ratio ξ₁ = [0.5, 2.0]%.

2.

Pedestrian parameters. Instead of considering different pedestrian densities, three crowd-to-structure mass ratios (μ_m) were considered:

μ_{m} = [2.5, 5.0, 10.0] %

. The crowd-to-structure mass ratio was defined as the ratio between the modal mass added by the crowd as a mass per unit area (m_1,add) [see also Van Nimmen et al. (2021a)] and the modal mass of the empty structure (m₁).

In addition to the physical constraints formulated in Eq. (5), upper bounds of 10 Hz and 100% were set for the natural frequency and the damping ratio of the MSD interaction model, respectively.

The modeling errors are related to the inputs that the parameter estimation procedure assumes to be known (Section “Parameter Estimation Procedure”). Given the high accuracy with which the structural response can be measured, and the ease with which the weight of the pedestrians can be determined, the impact of these related errors are not investigated here. The modeling errors (±) considered in this study are related to the following.

1.

The dynamic behavior of the empty structure characterized by its modal parameters. The parameter estimation procedure presented in this paper assumes that the dynamic behavior of the empty structure is known. When this procedure is applied to full-scale data involving pedestrian-induced vibrations, it is common practice that an operational or experimental modal analysis is performed on the empty footbridge. Given the uncertainty bounds on the modal parameters identified using state-of-the-art system identification techniques (Peeters and De Roeck 1999; Reynders and De Roeck 2008), it is reasonable to assume errors of 1% and 10% on the experimentally identified natural frequencies and modal damping ratios, respectively. The corresponding modal masses can be calculated using a (preferably calibrated) FE model of the structure. For the calculated modal mass, an error of 2% is considered reasonable. Less satisfactory results may be characterized by errors of 5%, 30%, and 5% on the natural frequency, modal damping ratio, and modal mass, respectively.

2.

The distribution of step frequencies in the crowd. Assuming that the distribution of step frequencies follows a Gaussian distribution, the impact of an error of 1% and 2.5% on the mean value and standard deviation of the step frequencies was investigated.

As the impact of modeling errors on the parameter estimation problem will always be case-specific, the objective of the analysis presented here is not to provide an exact quantification of the resulting estimation errors. The objective is to qualitatively assess in which conditions (i.e., mass ratio and modal parameters of a structural mode) it is feasible to get a reliable estimate of the MSD interaction model parameters, given the reasonably expected modeling errors.

Fig. 5 shows the ratio of the estimated and true values of the MSD interaction model parameters resulting from modeling errors on the structural modal parameters. When these ratios are close to unity, the impact of the modeling errors is low, and vice versa. When analyzing these results, it is important to keep in mind some key results from previous research involving the impact of HSI (Van Nimmen et al. 2017a). The impact of HSI on a structural vibration mode is mainly reflected in a (considerable) increase of the effective damping ratio of the coupled crowd–structure system in relation to the modal damping ratio of the empty structure. The impact of HSI increases as: (1) the frequency of the MSD interaction model is closer to the optimal tuned mass damper (TMD) frequency for accelerations (Asami et al. 2002) (i.e., for f_h1 smaller than but close to f₁, in the following expressed as

f_{h 1} ≲ f_{1}

); (2) the crowd-to-structure mass ratio increases; and (3) the damping ratio of the footbridge decreases. The impact of HSI is most sensitive to the frequency ratio f_h1/f₁, where the largest impact of HSI is found for f₁ larger than but close to f_h1, in the following expressed as

f_{1} ≳ 3

Hz with f_h1 = 3 Hz. The following observations can now be made from Fig. 5, for the considered range of modeling errors:

1.

The impact of the structural modeling errors decreases as the impact of HSI increases, thus as: (1)

f_{1} ≳ 3

Hz; (2) the crowd-to-structure mass ratio increases; and (3) the damping ratio of the footbridge decreases.

2.

The accuracy of the estimates of the natural frequency and the modal damping ratio generally go hand in hand: if the estimate of the natural frequency is close to its true value, so is the estimate of the damping ratio.

3.

As the natural frequency of the structural mode decreases from 2.5 Hz onward, the impact of the modeling errors increases rapidly, in particular for low mass ratios. This is because the impact of HSI decreases rapidly as f₁ decreases below 2.5 Hz. Conversely, as the natural frequency of the structural mode (f₁) increases from 3 Hz onward, the impact of these modeling errors increases slowly as the impact of HSI decreases slowly with increasing f₁.

Fig. 5. Impact of error on modal mass (black, ε_m = 2%; gray, ε_m = 5%); damping ratio (black, $ϵ_{ξ} = 10$ %; gray, $ϵ_{ξ} = 30$ %) and natural frequency (black, ε_f = 1%; gray, ε_f = 2%) of footbridge, on ratio of estimated and true value of natural frequency ( ${\hat{f}}_{h 1} / f_{h 1}$ ) and damping ratio ( ${\hat{ξ}}_{h 1} / ξ_{h 1}$ ) of the MSD interaction model, in terms of natural frequency of footbridge (f₁), for three damping ratios of the footbridge (open circle, ξ₁ = 0.5%; multiplication sign, ξ₁ = 1.0%; plus sign, ξ₁ = 5%) and three mass ratios ( $μ_{m} = {2.5 %, 5.0 %, 10.0 %}$ ). Solid circle: constraint is active for optimal solution.

In practical terms, it is advisable to minimize the influence of structural modes with a natural frequency below ≈2 Hz in the parameter estimation problem, because of the adverse effect on the accuracy of the obtained results, even for small structural modeling errors. This can, for example, be enforced by only considering the frequency range >2 Hz in the objective function. For modes with a natural frequency of 3 Hz and higher, the impact of the structural modeling errors increases as the natural frequency increases. This is particularly relevant for modeling errors related to the natural frequency. More generally, the higher the mass ratio, the more robust the obtained result is against structural modeling errors.

Fig. 6 shows the ratio of the estimated and true values of the MSD interaction model parameters resulting from modeling errors on the mean value and the standard deviation of the distribution of step frequencies. The following observations are made for the considered range of modeling errors.

1.

The impact of the modeling errors decreases for increasing mass ratio.

2.

The results are mainly sensitive to errors on the mean value of the distribution of step frequencies. The errors introduced by the standard deviation of the step frequencies stay (well) below 10% (except for the case where the mass ratio is lower than 5%).

3.

The impact of the error on the mean value of the distribution of step frequencies is as follows:

a.

On the estimated natural frequency of the MSD interaction model, for structural modes with a frequency equal to or higher than 2.5 Hz and

μ_{m} \geq 5 %

, the estimation error is limited to 15%. For structural modes with a frequency lower than 2.5 Hz, the estimation error can increase up to 200%.

b.

On the estimated damping ratio of the MSD interaction model, for structural modes with a frequency equal to or higher than 2.5 Hz, the estimation error is limited to 50% (

μ_{m} = 5 %

) and 25% (

μ_{m} = 10 %

). For structural modes with a frequency close to 1.5 Hz (and 4.5 Hz for

μ_{m} = 2.5 %

), the estimation error can increase up to 200% or more.

Fig. 6. Impact of error on mean value (black, $ϵ_{μ_{f_{s}}} = 1$ %; gray, $ϵ_{μ_{f_{s}}} = 2.5$ %) and standard deviation (black, $ϵ_{σ_{f_{s}}} = 1$ %; gray, $ϵ_{σ_{f_{s}}} = 2.5$ %) of distribution of step frequencies on ratio of estimated and true value of natural frequency ( ${\hat{f}}_{h 1} / f_{h 1}$ ) and damping ratio ( ${\hat{ξ}}_{h 1} / ξ_{h 1}$ ) of MSD interaction model, in terms of natural frequency of footbridge (f₁), for three damping ratios of footbridge (open circle, ξ₁ = 0.5%; multiplication sign, ξ₁ = 1.0%; plus sign, ξ₁ = 5%) and three mass ratios ( $μ_{m} = {2.5 %, 5.0 %, 10.0 %}$ ). Solid circle: constraint is active for optimal solution.

For the considered range of modeling errors, it is shown that the results of the parameter estimation problem are mostly sensitive to errors related to the structural natural frequency and the mean value of the step frequencies. Furthermore, the impact of the structural modeling errors decreases as the impact of HSI increases. In other words, the impact of these modeling errors decreases as the sensitivity of the objective function to the decision variables increases.

The conclusions made here and in the numerical verification example (Section “Parameter Estimation Problem”), are case-specific and depend primarily on the dynamic behavior of the empty structure and the pedestrian density. For example, it can be expected that when a number of structural modes have a relevant contribution to the structural response, the optimization problem will be more robust against modeling errors: the insensitivity of a subset of structural modes to a decision variable can be countered by the sensitivity of another subset of structural modes. This is under the assumption that the structural modal parameters are estimated with a reasonable level of uncertainty and the natural frequencies are well spread over the interval 1–6 Hz. It is therefore advised for each case study (1) to investigate the duration of the experimental observations required to arrive at a reasonable level of convergence of the PSD of the structural response; and (2) to properly assess the impact of modeling errors at hand on the obtained optimal solution. Based on the insights gained in the latter, a greater weight can be assigned in the objective function [Eq. (6)] to specific frequency ranges and DOFs, such that the impact of modeling errors is minimized.

Case Study: Eeklo Footbridge

The case study and experimental data used in this section are fully documented in the Eeklo footbridge open access benchmark dataset (Van Nimmen et al. 2021b).

Structure

The Eeklo footbridge has three spans, a main central span of 42 m and two side spans of 27 m. The dynamic behavior of the footbridge is characterized by 14 modes, with a frequency up to 12 Hz (Fig. 7). An excellent agreement is found between the experimentally identified modal characteristics and the ones calculated by the calibrated FE model. For more information related to the footbridge and the model calibration, the reader is referred to Van Nimmen et al. (2014). The corresponding digital twin, describing the dynamic behavior of the footbridge, is part of the open access benchmark dataset (Van Nimmen et al. 2021b).

Fig. 7. Top and side views of first eight modes of Eeklo footbridge (compare with digital twin, calculated using ANSYS).

Full-Scale Observations

The bridge and pedestrian motion were registered simultaneously using wireless triaxial accelerometers on the deck and at the lower back of the pedestrians. Four data blocks were collected involving two pedestrian densities, 0.25 and 0.50 persons/m², representing a total of more than one hour’s worth of data for each pedestrian density. Van Nimmen et al. (2021b) show that it takes 1,000 s to limit the convergence error of the PSD of the structural response to less than 10%. These results indicate that the dataset collected for each pedestrian density, with a total length of more than 3,600 s, can be considered representative of the involved load case. The converged PSDs are shown in Fig. 8 for a pedestrian density of 0.50 persons/m². The PSD shows that the vertical structural accelerations are dominated by the contributions of the first (1.69 Hz) and second (2.97 Hz) modes of the footbridge. At the side spans, an important contribution of Modes 5 (5.68 Hz) and 8 (6.42 Hz) is also observed: two lightly damped vertical bending modes that have an antinode at the center of the side span. Similar observations are made for a pedestrian density of 0.25 persons/m².

Fig. 8. PSD of (black) vertical and (gray) lateral acceleration levels at center of (a) central; and (b) side span of Eeklo footbridge for pedestrian density of 0.50 persons/m².

Using the procedure detailed in Van Nimmen et al. (2018b), the step frequency of all pedestrians is reported in Van Nimmen et al. (2021b). The results show that the distribution of step frequencies can be well fitted by a Gaussian distribution. For pedestrian densities of 0.25 and 0.50 persons/m², the distribution of step frequencies is characterized by

N (1.77, 0.13)

Hz (χ² test, 94%) and

N (1.66, 0.19)

Hz (χ² test, 87%), respectively.

For the tests involving a pedestrian density of 0.25 and 0.50 persons/m², the average body mass is 74.9 and 73.2 kg, respectively. The corresponding crowd-to-structure (modal) mass ratios (μ_m) are given in Table 1.

Table 1. Crowd-to-structure (modal) mass ratios (μ_j,m %) for Mode j of Eeklo footbridge for pedestrian density of 0.25 and 0.50 persons/m²

Mode	0.25 persons/m²	0.50 persons/m²
	μ_j,m (%)
Mode 1	0.3	0.7
Mode 2	5.3	10.5
Mode 3	1.3	2.7
Mode 4	0.2	0.4
Mode 5	5.6	11.2
Mode 6	1.6	3.3
Mode 7	1.2	2.4
Mode 8	5.7	11.4
Mode 9	1.2	2.4
Mode 10	0.9	1.7
Mode 11	6.5	13.0
Mode 12	0.5	1.1
Mode 13	1.1	2.2
Mode 14	0.3	0.6

Parameter Estimation

In a preliminary step, the digital twin of the Eeklo footbridge and an MSD interaction model with parameters corresponding to recent findings in the literature (natural frequency f_h1 = 3 Hz and modal damping ratio

ξ_{h 1} = 30 %

), were used to get a first idea of the impact of HSI for the Eeklo footbridge. Fig. 9 shows the FRF of the Eeklo footbridge, empty and with HSI for a pedestrian density of 0.50 persons/m². This figure shows that the impact of HSI is high for Modes 2 (f₂ = 2.97 Hz), 5 (f₅ = 5.68 Hz), and 8 (f₈ = 6.42 Hz). These modes are bending modes characterized by a low modal mass (and thus high crowd-to-structure modal mass ratio: ≈10% for a pedestrian density of 0.50 persons/m²) and a low damping ratio (0.2%–0.6%). The highest impact is found for Mode 2, as the natural frequency of the MSD interaction model is, in this case, close to the optimal TMD tuning frequency for accelerations (Asami et al. 2002). Fig. 9 furthermore shows that HSI has a negligible impact on the lateral structural response. The latter results from using a vertical MSD interaction model, as well as from the fact that the modes for which a high impact of HSI is found do not display significant lateral modal displacements.

Fig. 9. FRF for (black) vertical and (gray) lateral input and output acceleration at center of (a) central; and (b) side span of Eeklo footbridge. Dashed line = empty; solid line = with HSI, pedestrian density = 0.50 persons/m².

Next, the inputs for the parameter estimation procedure were set to match the conditions of the in-field observations involving pedestrian densities of 0.25 and 0.50 persons/m², respectively. The average body masses of the pedestrians were 74.9 and 73.2 kg, respectively. The distributions of step frequencies in the crowd were

f_{s} \approx N (1.77, 0.13)

Hz and

f_{s} = N (1.66, 0.19) Hz

, respectively. The digital twin of the Eeklo footbridge was used to describe the dynamic behavior of the empty footbridge. The structural response measured at the center of the main and the side span was used (close to the parapet). As the preliminary results show that HSI is most relevant for the vertical output, only the acceleration levels registered in the vertical direction were used in the parameter estimation problem. The PSD of the structural response was calculated, considering consecutive time windows of length T = 25 s, with 50% overlap. Given that the highest impact of HSI is expected for Modes 2, 5, and 8, the frequency range [2.5, 8.0] Hz is taken into account for the parameter estimation problem. Following the results of Section “Influence of Modeling Errors,” the high impact of HSI on the selected frequency range will limit the impact of modeling errors.

In Fig. 10, the objective function defined in Eq. (5) is evaluated for a realistic range of the two decision variables characterizing the MSD system: the natural frequency 1 ≤ f_h1 ≤ 6 Hz and the modal damping ratio 0.15 ≤ ξ_h1 ≤ 0.80. This figure shows that the objective function, and thus the impact of HSI, is mainly sensitive to changes in the natural frequency f_h1 of the MSD interaction model. These observations agree with the findings of the numerical verification example.

Fig. 10. Evaluation of objective function in terms of natural frequency f_h1 and damping ratio ξ_h1 of MSD interaction model, and (dashed lines) indicative uncertainty bounds on (solid circle) optimal solution, for pedestrian density of (a) 0.25 persons/m²; and (b) 0.50 persons/m².

For a pedestrian density of 0.25 persons/m² (0.50 persons/m²), an optimal solution of

f_{h 1}^{⋆} = 2.87

Hz,

ξ_{h 1}^{⋆} = 36

% (

f_{h 1}^{⋆} = 3.08

Hz,

ξ_{h 1}^{⋆} = 34

%) is found.