Detection, Localization, and Quantification of Damage in Structures via Artificial Neural Networks

Tis paper presents a structural health monitoring method based on artifcial neural networks (ANNs) capable of detecting, locating


Introduction
Te presence of damage in civil structures, whether caused by natural deterioration processes or extreme events, can lead to their collapse causing accidents.Terefore, it is necessary to continuously monitor structures aimed at the user's safety and structural reliability.In this context, structural health monitoring (SHM) has become relevant in engineering for its perennial surveillance and analysis of systems.Te term SHM is linked to vibration-based methods that involve observing the structure over time using the system's dynamic responses (displacements, velocities, and accelerations), with the structural condition being determined by damage-sensitive features.Tese properties include natural frequencies, frequency response function, mode shapes, mode shapes curvature, modal strain energy, and fexibility matrix (Ren et al. [1]).
As damage detection methods generally use the comparison of information with the values obtained in the healthy condition (reference scenario), a methodology has better predictive capabilities when using physical parameters in the assessment process, such as numerical models.When a fnite element (FE) model efectively represents the structure's dynamic properties, it can be adopted along with other techniques to identify damage.In this way, model-based and physics-informed methods have been deeply used in SHM employing machine learning (ML) algorithms, such as ANNs (Villalba and Laier [27]), support vector machines (Salkhordeh et al. [28]), and decision trees (Mariniello et al. [29]).
Te approaches using the pattern recognition aspect within the ML discipline can predict and classify damage scenarios by interpolating relationships between data.Damage detection methods that use neural networks depend on the relations between input and output, in which a damaged state is estimated according to a set of damagesensitive features that can be generated through a calibrated FE model or a hybrid approach (numerical and experimental data).For example, the frst fve natural frequencies of the structure or a set of wavelet coefcients can be used as input data to obtain damage location index as outputs (Farrar and Worden [30]).
According to Rytter [31], damage detection methods are classifed into four levels: (1) damage detection, (2) damage localization, (3) damage quantifcation, and (4) prediction of the remaining service life of the structure.Te methods available in the literature fall under levels 1 to 3, while level 4 is considered part of the structural assessment.However, there is still no widely accepted methodology for any structure or even any type of damage.
Generally, damage detection methods based on ANN do not achieve all Rytter's three levels (Hekmati Athar et al. [32]: level 1, Weinstein et al. [33]: levels 1 and 2, Xie et al. [34]: levels 1 and 2, Bisheh et al. [35]: level 1), or the authors propose multistage approaches to complete the three tasks (Malekjafarian et al. [36]: 2-stage method, Nick et al. [37]: 2stage method).Detection typically involves identifying whether damage is present, followed by separate steps to locate the damage and quantify its severity.Tis fragmented approach may not fully capitalize on the relationship among these tasks, potentially resulting in suboptimal overall performance.Tus, a signifcant gap remains in approaches that seamlessly integrate the three critical tasks in a single stage.
In this context, this paper presents a framework for detecting, locating, and quantifying damage in structures via ANNs using natural frequencies and mode shapes identifed from the responses to ambient excitation in a single stage.Te main contribution of this work is a methodology easily applied to diferent structures using numerical data through a physics-informed neural network.A supervised learning algorithm is employed with input and output factors to represent the structure's modal properties and damage states, respectively.
Unlike many papers in the literature that test damage detection methods only in numerical examples or simple experimental tests, this work also evaluated the presented framework in a real structure, the Z24 bridge, showing that it has potential for applications in real practical situations.Te proposed method is not only able to detect single and multiple damages but is also able to satisfactorily locate and quantify them, which is crucial to data-driven decisionmaking.
Tree main steps are used to illustrate the presented framework in diferent structural applications: (1) numerical simulation of damage scenarios in two structures under ambient vibrations with diferent noise levels; (2) validation of the methodology through experimental testing in a lab system; and (3) application of the methodology in the Z24 bridge, a real benchmark structure.Tus, this paper is organized as follows.Section 2 presents the vibration-based damage detection method via artifcial neural networks, Section 3 illustrates the numerical examples, Section 4 studies the experimental system, Section 5 presents the analysis of the Z24 bridge, and Section 6 presents the conclusions.

Damage Detection Method via Artificial Neural Networks
Te damage detection method via neural networks presented in this paper builds a model that provides a relationship between modal parameters (input data) and structural properties (output data).Tis relationship between the data of a system is determined through a training process inspired by the structure and operation of the human brain.In this way, artifcial neural networks try to reproduce human behaviors of learning, association, generalization, and abstraction.Te interconnected processing elements (artifcial neurons) perform simple operations and transmit their results to neighboring individuals.An artifcial neuron j has a set of inputs S i (dendrites) and an output S j (axon).Te connections (synapses) formed between neurons have numerical values representing the strength of each bond.Tese values are called synaptic weights (w i ) and are used to store knowledge.With the formation of the neural network model, diferent input data are applied to each neuron, and these values are weighted by the weight of each synapse, as shown in Figure 1.

Shock and Vibration
If the activity of a neuron reaches a certain value of the activation function f, the neuron propagates the received signal along the axon.Te sum of the m inputs weighted by the respective synaptic weights adding a bias term θ j defnes the activation state: Each ANN can assume a diferent architecture according to its neuron's organization.Te distribution of processing elements occurs in layers consisting of an input layer, an output layer, and one or more hidden layers.Tus, the number of existing layers and the number of elements in each form the architecture of a neural network.
Te learning process can be divided into two algorithms: supervised learning and unsupervised learning.Te learning algorithm is supervised when the training generates its results by labels of each set; i.e., there is a correct answer.However, when the input data are not linked with preestablished labels, the algorithm is unsupervised.While supervised learning applied in damage detection methods requires data from all possible damage scenarios, unsupervised learning only needs data from the system's normal condition (reference scenario).However, according to Farrar and Worden [30], it is not possible to diagnose damage scenarios beyond level 2 (location) in unsupervised learning.Terefore, in this work, supervised learning was used.
Te training dataset for neural networks was obtained through numerical models, with the input data linked to the structural condition by identifcation, location, or quantifcation labels (output data).Te database of each system was divided as follows: 70% for training, 20% for validation, and 10% for testing.
Te input factors (IFs) were based on natural frequencies or mode shapes to detect damage by the ANN model.Tese modal parameters can capture essential characteristics of a dynamic system using a few input features.Te resulting reduction in dimensionality can signifcantly improve computational efciency and model training speed, making it advantageous for practical applications and large-scale structures.In addition, the method employed normalized modal parameters because normalization enhances the consistency and comparability of the input data across various structural conditions.By transforming raw modal parameters into a standardized format, the model becomes highly efective in capturing the structural variations.
In this paper, the frst type of IF derives from calculating normalized natural frequencies.Multiple vibrational modes are employed to provide diverse information to the network, given that higher modes are more sensitive to damage.Each input factor j (IF j ) was computed through the value of each frequency j in the damaged condition (ω j d ) and the respective frequency j in the healthy condition (ω j h ), according to equation (2).However, the maximum number of vibrational modes in the input dataset depends on the conformity of the structure with the FE model and the performance of the system identifcation method.
Te second type of IF originates from normalized mode shapes, with the largest displacement value set to 1. Tis modal parameter provides spatial information about the structural system, leading to the adoption of only the frst vibrational mode.Moreover, only the degrees of freedom referring to the predominant displacements of the structure were considered, as most of the structural information resides in these points.
Te outputs were chosen to simulate the stifness reduction caused by the damage in each element or region i of a structural system.Terefore, an output factor (OF i ) was arbitrated ranging from 0 to 1, where 1 signifes no damage, and 0 means that the element loses its stifness completely.As the neural network model does not have a way to limit output values, depending on the dataset and the ANN training process and architecture, the stifness reduction factor obtained for each element can be slightly less than zero or greater than one.In this way, a process of trial and error was executed pursuing the best hyperparameters related to network structure for each system.
Figure 2 exemplifes possible output factors obtained from diferent input factors.Te system illustrated has three input factors (e.g., the frst three natural frequencies) and fve elements as output factors.Tere are two scenarios: (i) the healthy scenario and (ii) the damage scenario, where it is introduced 20% of stifness reduction in element 4 (OF 4 � 0.8).

Shock and Vibration
Among the existing network topologies, it was chosen to work with the feedforward network.In this type of neural network, the fow of information processing occurs only towards the outputs; that is, there is no return of signals to the previous layers.Te networks were implemented through the feedforwardnet function of MATLAB [38].Te hyperparameters related to the training algorithm were fxed as 1000 epochs, 1 × 10 −7 stopping criterion based on the MSE goal, 10 maximum validation failures, and 1 × 10 −7 minimum performance gradient.In addition, the activation function for the hidden layers and the output layer were, respectively, hyperbolic tangent sigmoid transfer function and linear transfer function.
Figure 3 summarizes the framework for detecting damage in structures via ANNs.Initially, experimental and numerical responses to ambient excitation (acceleration signals) are used to identify the modal parameters through stochastic system identifcation, generating the numerical training data and experimental testing data.Next, the neural network is trained with the numerical dataset (input and output factors).Finally, the damage is detected, localized, and quantifed with the experimental dataset.

Numerical Examples
To test the efectiveness of the damage detection method presented, experimental tests of a cantilever beam and a 10bar plane truss were numerically simulated using matrix structural analysis.Tese structures were chosen to compare the results already obtained by other authors.
Te ambient vibration was simulated by white Gaussian noise.Ten, the system responses in accelerations were calculated by the Newmark method with an integration time-step of 0.005 s.Te presence of noise in the signals to reproduce experimental conditions was also simulated by the white Gaussian noise.It was considered sensors placed at all the structure's vertical nodes because of the data availability.Finally, the data-driven stochastic subspace identifcation (SSI-DATA) technique (Peeters [39]) was implemented to obtain the modal parameters for the damage detection methods.
3.1.Cantilever Beam.Te frst system analyzed was a metallic cantilever beam 750 mm long with a square box crosssection with an external dimension of 25.4 mm and a wall thickness equal to 1 mm.Te structure studied experimentally by Kaminski and Riera [40] was modeled with 25 Timoshenko beam elements, according to Figure 4. Te specifc weight, Young's modulus, Poisson's ratio, and Timoshenko shear factor are, respectively, 28 kN/m³, 68.6 GPa, 0.3, and 0.5.In addition, a concentrated mass of 18.2 g was added to all degrees of freedom to represent the presence of accelerometers in the experimental tests.To create the damping matrix, a damping ratio of 1% was considered in the frst and ffth vibration modes.
Te noise levels adopted in the acceleration signals were 3% and 5%.Te respective natural frequencies obtained by the fnite element model and through the stochastic system identifcation are in Tables 1 and 2.
Te frst two frequencies identifed by the SSI-DATA method, in both noise situations, are close to the original values of the FE model.However, the value of the third frequency presents an error of around 10% because the system responses did not have enough contribution from the highest frequencies under the action of the white Gaussian noise.Terefore, damage detection was performed only with data from the frst three vibration modes of the structure.
In training the neural networks, the frst mode shape was used as input data, considering only the vertical degrees of freedom.As output data, the 25 elements of the system were adopted.Te creation of the training set included single and multiple damage cases with two damaged elements.In these scenarios, the stifness reduction variations in each element were from 10% to 60% in 10% intervals.Tus, a total of 1800 cases of multiple damage (6 × (25!/(2! × 23!))) and 150 cases of single damage (6 × 25) were obtained, totaling 1950 datasets.
After some tests of network architecture, it has opted for only one hidden layer with 25 neurons.Te regression line is presented in Figure 5(a), comparing the model's predictions with the true labels of the dataset with a 10% tolerance limit of damage intensity, and Figure 5(b) indicates the mean squared error loss for the training and validation.Furthermore, Figures 6 and 7 show the results generated by the neural network for the three scenarios according to the applied noise levels.
Te method efciently identifed and located the damage in the cantilever beam case.Te quantifcation in the three scenarios was correct but with discrepancies up to 6.06%.4

Shock and Vibration
However, the damage values estimated by the ANNs were close to the exact values in both noise cases.Te ANN had a similar performance in damage detection compared to the results presented by Zeni [41] and Miguel et al. [15] with methods based on matrix updating.

10-Bar Plane Truss.
Te second structure analyzed was a 10-bar plane truss (Figure 8) studied by Begambre and Laier [42] and Fadel Miguel et al. [14].All elements have a specifc mass of 7700 kg/m³, Young's modulus of 195 GPa, a moment of inertia of 3 × 10 −8 m 4 , and a cross-section of 4.2 × 10 −4 m 2 .Te damping ratio was assumed to be 1% in the 1 st and 3 rd vibration modes.
To compare with the results obtained by Begambre and Laier [42] and Fadel Miguel et al. [14], the scenario of 15% stifness reduction in bars 2 and 8 was analyzed.Both authors used a hybrid optimization approach: Begambre and Laier [42] used the PSO and Simplex algorithms, and Fadel Miguel et al. [14] employed a hybrid Nelder-Mead algorithm.Following these works, a noise of 3% was adopted in the acceleration signals of the healthy and damaged scenarios.
Table 3 presents the natural frequencies obtained by the fnite element model and the stochastic system identifcation.
In the neural network training, the input data were the frst mode shape (8 degrees of freedom), and the outputs were the structure bars.Te training set was created from the FE model with single damage cases and multiple damage cases with two damaged elements.In these scenarios, the variations of stifness reduction in each element were from 10% to 70% in 5% intervals.Tus, a total of 585 cases of multiple damage (13 × (10!/(2! × 8!))) and 130 cases of    Shock and Vibration single damage (13 × 10) were obtained, totaling 715 datasets.Te network architecture adopted was one hidden layer with 35 neurons.Te mean square error of the model during training and the regression line are presented in Figure 9.
Te results of the damage detection methodology are shown in Figure 10 and in Table 4 with the solutions obtained by Begambre and Laier [42] and Fadel Miguel et al. [14].Te framework used in this work correctly located and quantifed the damage scenario.

Experimental System
Te performance of the damage detection method was also verifed in a system through experimental tests of a steel cantilever beam.Te tests were carried out in the laboratory of the Applied Mechanics Group of the Federal University of Rio Grande do Sul (GMAP/UFRGS).Te steel cantilever beam is 420 mm long, 39.5 mm wide, and 1.2 mm thick (Figure 11(a)).Te specifc mass, Young's modulus, Poisson's ratio, and Timoshenko shear factor of the beam are 8193.9kg/m³, 210 GPa, 0.3, and 0.5, respectively.Tis system was modeled with 28 Timoshenko beam elements (Figure 11(b)) using matrix structural analysis.
Lateral cuts were made in some of the beam elements to represent possible damage.Te scenarios analyzed were as follows: (1) width reduction of element 13 to 33 mm; (2) width reduction of element 13 to 18.5 mm; and (3) width reduction of elements 8 and 13 to 17.5 mm and 18.5 mm, respectively.Te experimental test was repeated at each progressive damage step.

Dynamic Tests.
Te dynamic tests were executed in the healthy condition for later comparison with the damage scenarios by means of accelerometers and the pulse 12 channel Brüel and Kjaer type 3560 C acquisition system.As presented in Figure 12, the experimental tests were performed with three accelerometers, and Table 5 describes the properties of these devices.Te sensor placement sought efective coverage of the experimental beam's modal parameters, especially the mode shapes.Terefore, a measurement point was required at the beam's free end (maximum displacement in the frst mode shape) and two more measurement points were distributed equally in the beam's length.
To determine the modal parameters experimentally, small displacements were given at the beam, i.e., the system was in free vibration (Figure 13).At each stage, the dynamic tests were repeated to acquire the respective accelerograms.

System Identifcation.
Te three accelerograms obtained in each scenario were the input data for the SSI-DATA technique.Te frst fve natural frequencies were accurately determined, as shown in Table 6.Te highest diference found was 2.16% at the frst natural frequency.
Figure 14 exhibits the mode shapes identifed in the healthy scenario with the mode shapes of the numerical model.Te fourth and ffth mode shapes could not be determined due to the lack of acquisition points because only three accelerometers were used.However, the frst three modal shapes identifed have 99.99% correlations with the numerical model, as shown by the modal assurance criterion (MAC) (Abdel Wahab and De Roeck [6]) in Figure 15.

Damage Detection.
In the damage detection method based on neural networks for the case of the steel beam, the input data were the frst fve natural frequencies, and the outputs were the system's 28 elements.Te training data   16.
Figure 17 shows the results generated by the ANN for the three analyzed scenarios.Te proposed method presented efective damage localization results in the experimental tests, successfully identifying the damaged elements in all cases.In damage scenario 3, element 7 also displayed a fault state, possibly due to its proximity to the damaged region.
Te quantifcation results for damage detection showed some inaccuracies.However, the training did not include multiple damage cases because a larger database was impairing the network learning process, consequently affecting the generated results.

Z24 Bridge
Finally, the proposed damage detection method was applied and verifed on a real civil structure used as a benchmark in the scientifc community, the Z24 bridge (Switzerland).In this work, a multiple damage case caused by the settlement of one of the system's piers was analyzed.
Te Z24 bridge was selected by the Brite-Euram research project BE-3175, SIMCES (System Identifcation to Monitor Civil Engineering Structures), as a study object to develop a methodology for structural integrity monitoring.Te work     [43]).Tis structure was in the Canton of Bern in Switzerland, connecting Koppigen and Utzenstorf, and overpassed Highway A1, which connected the cities of Bern and Zurich.Te bridge had three spans and two lanes and was about 60 m long (Figure 18).Te two central piers were clamped into the girders, while the two triplets of columns at both ends supported the bridge at the endpoints.All supports were rotated regarding the longitudinal axis of the structure forming a slightly skew bridge (Peeters and De Roeck [44]).Te girder was a two-box cell with posttensioned concrete (Figure 19).In total, 17 progressive damage tests (PDTs) were performed, which are described in detail by Krämer [45], Maeck and De Roeck [46], and Reynders and De Roeck [47].Te PDTs were selected according to the occurrence frequency, based on the number of citations in the literature and the experience of Swiss bridge owners (Roeck [43]).Te frst set of PDTs considered the foundation settlement of the Koppigen pier.Te settlement scenarios were simulated by lowering the pier (x � 44 m), which caused several cracks in the girder, as illustrated in Figure 20.Other failure modes of bridge piers and their impact on structures are presented by Govahi et al. [48] and Liu et al. [49], for instance.
In this work, PDT 2 was adopted as the healthy scenario, and the 95 mm settlement of the column (PDT 6) as the damaged scenario.

System Identifcation.
Te modal parameters of the Z24 bridge were identifed from the data of the ambient vibration tests, which included nine setups of 15 accelerometers on the deck (2 triaxial, 3 biaxial, and 10 uniaxial), 2 triaxial accelerometers on one of the piers, and 3 reference accelerometers (1 triaxial and 2 uniaxial), totaling 33 records.Te sampling frequency and acquisition time were 100 Hz and 655.36 s, respectively.Table 7 presents the identifed natural frequencies by the SSI-DATA technique for the healthy and damaged scenarios.[50] and Reynders et al. [51] developed the FE model used in this work employing ANSYS APDL [52] with beam elements (BEAM4 and BEAM 44, 6 degrees of freedom per node).Te authors modeled the girder with 82 beam elements and the piers, columns, and abutments with 44 beam elements, according to Figure 21.Mass elements were added to represent the cross girders and foundations, considering concentrated translational mass and rotary inertial components.

Shock and Vibration
Initially, Young's modulus E 0 = 37.5 GPa and shear modulus G 0 = 20 GPa were considered.To account for the infuence of the soil on the system, spring elements were included around the pillars and at the base of the abutments.Te soil stifness parameters adopted for the springs were K y,p = 180 × 10 6 N/m³ and K h,p = 210 × 10 6 N/m³ (under the piers, at (x) = 14 and 44 m); K v,c = K h,c = 100 × 10 6 N/m³ (under the columns, at (x) = 0 and 58 m); K v,a = 180 × 10 6 N/m³ and K h,a = 200 × 10 6 N/m³ (at the abutments); K v,ac = K h,ac = 100 × 10 6 N/m³ (around the columns).
Tis numerical model had the values of Young's modulus and shear modulus of the bridge girder and soil stifness parameters updated according to the six natural frequencies identifed in the healthy scenario presented in Table 7. Tis was carried out through a minimization problem using the objective function: where  ω j is the identifed natural frequency of the jth mode and ω j is the natural frequency j of the numerical model as a function of the variable ∅.In this process, 36 variables were updated: 17 Young's moduli and 17 shear moduli, with the moduli of intermediate elements interpolated from the main elements; the vertical soil stifness under the piers K v,p and the horizontal stifness under the abutments K h,a .
Te diference between the frst six experimental frequencies and the updated numerical frequencies was less than 1%, as shown in Table 8.Te updated soil stifness parameters were K v,p � 147.5 × 10 6 N/m³ and K h,a � 146.4 × 10 6 N/m³, and the updated bending stifness (EIy) and torsional stifness (GIx) parameters in comparison with initial model values are presented in Figure 22.

Damage Detection.
Nine regions of the bridge girder were established (Figure 23) in the updated FE model, aiming to reduce the stifness reduction factors in the damage detection method.Te regions were defned to represent the areas of infuence of the piers and abutments in addition to girder partitions.
Te damage detection method used the frst six natural frequencies as input data and the girder regions as outputs.
Te training data included single damage cases of the fnite element model.In these scenarios, the stifness reduction variations in each region ranged from 5% to 80% in 5% intervals, totaling 144 datasets (16 × 9).After some In the case of the Z24 bridge, the damage detection method was able to locate the main damaged region (region 7), where the pier settlement and the cracks in the adjacent regions of the girder occurred.Secondary damage regions from cracking in neighboring areas of the pier (regions 6 and 8) were not as easily located, with only region 6 identifed.It was not possible to verify the damage levels identifed because there was no real quantitative response.
Te results showcased in Figure 25 for damage detection and localization demonstrate a superior performance on a global level compared to prior studies.For instance, in the work by Sony et al. [53], the absence of quantitative information related to physics properties within the damage index hindered the ability to infer damage without undertaking a global analysis of each structural component   It must be emphasized that the neural network needs data from the same domain in the training and test dataset to achieve the necessary accuracy.Tis work used model updating to align the numerical domain and the experimental domain.Tis process fne-tunes the numerical model using experimental data to reduce their discrepancy.
Another approach to address the divergence between different domains is data preprocessing.Te efects of mismatched modal parameters can be minimized by aligning the statistical properties of diferent datasets, such as mean, variance, and distribution.Alternatively, domain adaptation techniques, a subcategory of transfer learning, can be employed.Tis approach enables the model to generalize from one domain to another (e.g., numerical simulations to experimental systems) by learning their diferences and similarities.Tis can reduce the strict requirement for domain alignment and alleviate computational problems in large-scale structures.Shock and Vibration

Conclusions
Tis paper presented a method for damage detection via artifcial neural networks in a single stage using numerical training data.Te methodology was applied, compared, and evaluated through numerical simulations (cantilever beam and 10-bar truss) and experimental tests (steel beam and Z24 bridge).In this way, it was possible to evaluate the method's performance to identify, locate, and quantify the scenarios of single and multiple damages.Te results obtained allow drawing the following conclusions: (1) Te ANNs showed promising results in locating and quantifying damage and can be used for practical purposes.In the simulation cases, the method presented similar results to those found by other authors in the literature who used damage detection methods based on matrix updating.Moreover, even with noise in the acceleration signals and changes in the frst vibration mode of 0.015% (cantilever beam case), the framework could perform the three damage tasks satisfactorily.(2) In the experimental lab structure, the methodology proved to be efective in detecting, locating, and quantifying single and multiple damage scenarios with a mean diference of 0.47% and 2.26% between the frequencies in the healthy and damaged conditions, respectively.(3) In the study of the Z24 bridge, it was possible to verify the method's efectiveness in detecting and locating damage, proving that the methodology can be used in real civil structures.Additionally, even with an evaluation based only on natural frequencies, the method could estimate the damage caused by the pier settlement, despite a mean diference of 4.08% between the frequencies in the healthy and damaged conditions.
Future work can be carried out investigating illconditioning problems that can happen with larger numbers of elements as output factors.As was observed in the Z24 bridge, adopting a few input factors required fewer regions as outputs.Terefore, structures modeled with more elements should be investigated with a diferent approach.Furthermore, the framework should be extended to accommodate environmental and operational conditions, such as temperature variations and include transfer learning to avoid problems related to mismatched modal parameters.

Figure 2 :
Figure 2: Examples of output factors for a healthy scenario and a damage scenario.

Figure 12 :
Figure 12: Experimental beam and acquisition system.

Figure 21 :
Figure 21: FE model of the Z24 bridge.

Table 1 :
Numerical natural frequencies of the cantilever beam (Hz).

Table 2 :
Identifed natural frequencies of the cantilever beam (Hz).

Table 4 :
Results of the damage detection for the 10-bar truss structure.

Table 6 :
Natural frequencies of the experimental beam (Hz).

Table 7 :
Identifed natural frequencies of the Z24 bridge.

Table 8 :
Updated numerical frequencies of the Z24 bridge.