# Experimental Validation of a Double-Deck Track-Bridge System under Railway Traffic

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## Abstract

**:**

^{®}software, as well as in situ dynamic characterization tests of the structure, namely ambient vibration tests, for the estimation of natural frequencies, modes shapes and damping coefficients, and a dynamic test under railway traffic, particularly for the passage of the Alfa Pendular train. The damping coefficients’ estimation was performed based on the Prony method, which proved effective in situations where the classical methods (e.g., decrement logarithm) tend to fail, particularly in the case of mode shapes with closed natural frequencies, as typically happens with the first vertical bending and torsion modes. The updating of the numerical model of the bridge was carried out using an iterative methodology based on a genetic algorithm, allowing an upgrade of the agreement between the numerical and experimental modal parameters. Particular attention was given to the characterization of the ballast degradation over the longitudinal joint between the two half-decks, given its influence in the global dynamic behavior of this type of double-deck bridges. Finally, the validation of the numerical model was performed by comparing the acceleration response of the structure under traffic actions, by means of numerical dynamic analyses considering vehicle-bridge interaction and including track irregularities, with the ones obtained by the dynamic test under traffic actions. The results of the calibrated numerical model showed a better agreement with the experimental results based on the accelerations evaluated in several measurement points located in both half-decks. In the validation process the vertical stiffness of the supports, as well as the degradation of the ballast located over the longitudinal joint between half-decks, was demonstrated to be relevant for the accuracy and effectiveness of the numerical models.

## 1. Introduction

- -
- Development of an advanced methodology capable of characterizing the degradation of the continuous ballast layer over bridges, particularly in the longitudinal joints between adjacent decks of double-deck bridges. In these specific zones, the ballast is subjected to cyclic movements induced by rail traffic, which can significantly reduce the interaction effect between adjacent decks. The two-step proposed methodology consists, first, of a model updating strategy based on experimental modal parameters and using a genetic algorithm, and second, of a validation strategy to evaluate the robustness of the calibrated model in adequately simulating the dynamic response of the bridge for the train passages. An important contribution of this work is the validation of the dynamic responses on the deck carrying the passing train, as well as on the neighbourhood half-deck, which represents a challenge in terms of the model accuracy and efficiency.
- -
- The accurate characterization of the level of degradation of the ballast over the joint between half-decks, due to the cyclic shear strain induced by traffic loads, involves the use of a dedicated shear modulus degradation curve, proposed by Ishibashi et al. [30]. This curve can realistically estimate the ballast shear modulus reduction under the high amplitude dynamic displacements that occur during the trains’ passage.
- -
- The application of the Prony method is used for the estimation of the modal damping coefficients [31]. Comparatively to the classical Logarithm Decrement (LD) method, this method is also based on the free vibration period after the train crossing of the bridge; however, it is more efficient in situations where coupled modes of vibration are present (i.e., modes of vibration with close or very close frequencies). The existence of coupled modes is quite common in short-medium span railway bridges where the natural frequencies of the fundamental torsion and bending modes are close or even merged. The accurate modal damping estimation is decisive for the characterization of the dynamic response of the bridge, particularly in resonances or near resonances scenarios.

## 2. Numerical Modelling

#### 2.1. Bridge Numerical Model

^{®}[32]. To better simulate the transition zone in the abutments, an extension of the track was also modeled. Moreover, different materials were used to model the ballast on the longitudinal and transversal joints to allow the study of the degradation of the track in these regions.

#### 2.2. Train Numerical Model

^{®}software [32]. Figure 5 illustrates the dynamic model of one of its cars, including the location of the suspensions and centres of gravity of the different components. In this figure, k, c, m and I represent stiffness, damping, mass and rotational inertia, respectively; a, b and h refer to the longitudinal, transversal and vertical distances, respectively; s represents the gauge and R

_{0}represents the nominal rolling radius. The subscripts cb, b and w refer to the carbody, bogie and wheelset, respectively. Concerning the suspensions, the subscripts 1 and 2 denote the primary and secondary ones, respectively, while the subscripts x, y and z designate the longitudinal, transversal and vertical directions, respectively. All the aforementioned parameters related to the BBN vehicle are presented in Table 2.

#### 2.3. Methodology of the Train-Bridge Dynamic Interaction

#### 2.3.1. Wheel-Rail Contact Formulation

#### 2.3.2. Dynamic Equations of the Train-Bridge Coupling

**a**, and contact forces,

**X**, as unknowns (Lagrange multipliers method) that can be mathematically described as:

**a**and Δ

**X**are the incremental nodal displacements and contact forces, respectively, that can be computed through the Newton-Raphson method. Regarding the superscripts, $t+\Delta t$ refers to the current time step, while $i$ and $i+1$ indicate the previous and current Newton iteration, respectively. The dynamic analysis is solved based on a direct integration scheme based on the α-method [44].

^{®}[45], imports the structural matrices of the bridge and train models developed in a FE package (in this case ANSYS

^{®}[32]), allowing the study of structures with any degree of complexity in an efficient way. Figure 8 shows the framework of the VSI tool and the outputs that can be obtained in the dynamic analyses carried out by it. A detailed description of this numerical tool can be consulted in [40].

## 3. Dynamic Tests

#### 3.1. Ambient Vibration Test

^{®}software [46]. In Figure 10, it is possible to observe peaks (red dots) in the first three curves of the average normalized singular values of the spectral density matrices of all test setups, these peaks corresponding to the four identified global vibration modes depicted in Figure 11 (where f is the average value of the natural frequency and ξ is the average value of the damping coefficient).

#### 3.2. Test under Railway Traffic

## 4. Model Updating

#### 4.1. Mode Pairing

#### 4.2. Sensitivity Analysis

_{c}) and density (ρ

_{c}) of the concrete, the modulus of deformability of the ballast in the longitudinal joint (E

_{bl}), the density of the ballast (ρ

_{b}) and the vertical stiffness of the supports (K

_{v}).

#### 4.3. Optimization

^{®}[45] and ANSYS

^{®}[32] (see Figure 17). A detailed description of the proposed methodology can be found in Ribeiro et al. [14].

_{v}) and density of the ballast (ρ

_{b}), present a tightness variation of values in the optimization stage. In turn, parameters that are not so sensitive to the responses, e.g., modulus of deformability (E

_{c}) and density (ρ

_{c}) of the concrete, present a slightly higher dispersion of values in the optimization stage. Regarding the ballast density (ρ

_{b}), it showed a variation of only 2.2% between optimizations, while the latter presented variations lower than 1.2%. Moreover, it is interesting to notice that the values obtained for the vertical stiffness of the supports are close to the lower limit (average ratio of 9.5%), which indicates a possible degradation of these bearing devices. The modulus of deformability of the concrete (E

_{c}) presented average values close to that adopted in the initial numerical model (average ratio of 50.1%), which indicates that the initial value estimation was satisfactory. Despite the variation of the parameter values between the optimization runs was noticeably low (11.6%), this was the largest variation among the analysed parameters. The modulus of deformability of the ballast in the longitudinal joint (E

_{bl}) showed close values between the first three optimizations and a greater variation in the GA4 optimization. The average ratio obtained was 49.0%, which may indicate a degradation of the longitudinal joint, since the parameter value on the initial model corresponds to its upper limit.

## 5. Model Validation

#### 5.1. Initial Considerations

_{i}and f

_{j}, guaranteeing that all the vibration modes within this interval become underdamped and the remaining ones outside it become overdamped. In this work, f

_{i}corresponds to the first identified vibration mode (f

_{i}= 10.10 Hz), while f

_{j}corresponds to the fourth one (f

_{j}= 35.58 Hz). Such consideration guarantees the correct representation of the damping coefficient of the main mode of the structure (first mode), while avoiding an overestimation of the damping in the following identified modes. In the frequency range between 10.10 Hz and 35.58 Hz, the dynamic response of the bridge is not influenced by the dynamic behaviour of the track. The track dynamics associated with the movements of the ballast layer, typically occurs in the frequency range between 80 Hz and 150 Hz [51]. Thus, for solving the train-track-bridge dynamic interaction problem the track behaves as a rigid layer, i.e., no relative movements between the rails and deck occur due to the track dynamic flexibility.

_{1}norm between the vectors containing numerical (N) results and experimental (E) data and ${\Vert E\Vert}_{1}$ is the L

_{1}norm of the experimental data.

#### 5.2. Comparison between Numerical and Experimental Bridge Response before and after the Updating Process

#### 5.3. Influence of the Vertical Stiffness of the Bearing Supports

_{v}equal to 194.9 MN/m, which is close to the original lower bound adopted in the updating process. However, it was noticed that the numerical acceleration responses calculated on both tracks tend to match the corresponding experimental ones for lower levels of stiffness, which can be justified by a degradation of the elastomeric material present in the bearing supports, as well as by a reduction in the confinement provided by the steel sheets or by the possible non-linear behaviour of these devices. Based on this assumption, a modified model was obtained through a successive reduction in the vertical stiffness of the supports until reaching the value of 132 MN/m, which guaranteed the best compromise in terms of agreement between the numerical and experimental accelerations on both track sides. Figure 25 compares the numerical midspan accelerations obtained in the opposite (OT) and running tracks (RT) with those recorded experimentally by the accelerometers installed at these locations. A significant improvement in the agreement between the results is notorious, showing that the numerical model with the proposed modification captures with more accuracy the actual behaviour of the bridge. This conclusion is evident not only by observing the responses, but also by looking into the nMAE indicator, which decreased to 14.6% (22.5% in the previous model) and 22.6% (33.8% in the previous model) with respect to the results obtained in the OT and RT, respectively. Finally, in the frequency domain, a better match between the amplitudes of the peaks has also been achieved, especially in the main peak related with the first mode of vibration of the bridge around 10 Hz.

#### 5.4. Influence of the Degradation of the Longitudinal Joint

_{b2}, from 145 MPa to 76.25 MPa (see Figure 18), which demonstrates a possible degradation of the ballast over this region. However, this degradation tends to be more pronounced with the shear strain levels (distortion) that may occur in these joints due to the relative cyclic movements between adjacent half-decks caused by the train passages. These effects, which have been reported in the literature [8,53], may lead to modifications in the structural response. Hence, based on the numerically evaluated distortions of the ballast at the joint level, the degradation of the elasticity modulus of the ballast has been estimated to evaluate if it may influence the correspondence between numerical and experimental data.

_{0}, can be mathematically described by:

_{p}(see Ishibashi et al. [30] for details regarding the curve equation). Given the nature of the ballast material, a small confining pressure of 1.5 kPa and a null plasticity index (PI) has been adopted to trace the degradation curve presented in Figure 26b. The results show that the maximum level of shear strain due to the train passage is equal to $6.04\times {10}^{-4}$, which corresponds to a degradation of 16.6% of the shear (and elasticity modulus of the ballast over the joint). Therefore, it is acceptable to consider reductions of E

_{b2}in this order of magnitude in the numerical model (16.6% of 76.25 MPa equal to 12.66 MPa).

_{b2}= 12.66 MPa), an optimal value of 36 MPa has been obtained, which guaranteed a compromise in the agreement between the numerical and experimental data on both track sides. Figure 27 depicts the midspan numerical and experimental acceleration results relative to both the opposite (OT) and running tracks (RT) in the time and frequency domains. Note that, by degrading the longitudinal joint, the continuity between both half-decks decreases, leading to a slight decrease in the accelerations in the OT side and an increase in the RT side. The effect caused by this modification led to the lowest values of the nMAE indicator on both OT and RT sides, which took the values of 13.4% to 21.1%, respectively. In the frequency domain, the enhancements made in the numerical model described in Section 5.3 and Section 5.4 also led to notorious improvements, since the errors in the main peak’s amplitude related with the bridge’s first mode of vibration decreased from 46% and 54% in the OT and RT sides, respectively, to just 11% and 8%.

## 6. Conclusions

- The ambient vibration test allowed the identification of four global modes, namely the first and second vertical bending modes and two torsion modes. These modes were also identified in the initial numerical model, although still with considerable differences in terms of frequency value (errors up to almost 5%).
- Experimental modal damping was identified using the free vibration records obtained in the tests under railway traffic. Given the difficulty in separating the contribution of the first bending and torsional modes for the free vibration (frequencies close to each other), the Prony method was used. By adopting this methodology to the records obtained during the tests under railway traffic, mean damping ratios of 6.69% and 4.72% were estimated for the first bending and torsional modes, respectively.
- Before performing the automatic optimization process, a sensitivity analysis was carried out, showing that 5 out of the 13 analysed parameters had a significant influence in the modal response, namely the modulus of elasticity of the concrete and ballast in the longitudinal joint, the density of concrete and ballast and the vertical stiffness of the bearing supports. Based on this outcome, an updating procedure based on a genetic algorithm was carried out to calibrate the numerical model. A significant reduction in the differences between numerical and experimental natural frequencies was achieved; more precisely, the average error of the frequencies decreased from 2.65% before updating, to 0.69% after updating. Regarding the MAC coefficients, they suffer an overall increase, reaching values close to 1.0 (between 0.968 and 0.997).
- Regarding the validation of the model, a good agreement between experimental and numerical results was observed, in particular with the model obtained after the updating. The improvement in the results was confirmed through the nMAE indicator, which reduced from 26.9% to 22.5% and from 43.2% to 33.8% regarding the responses obtained in the OT and RT sides, respectively. However, as confirmed with the results in the frequency domain, the amplitudes of the numerical and experimental responses were still considerably different, especially in the RT side. Therefore, to improve the results, and since the updating process was based on modal data obtained with ambient vibration measurements, two modifications in the FE model of the bridge were carried out with the objective of getting a closer match between experimental and numerical responses of the bridge under railway traffic.
- Since the vertical stiffness of the bearing supports proved to have a significant influence in the model updating process, an evaluation of its influence in the vertical response of the bridge was conducted. After performing successive changes to this parameter, a vertica1 stiffness of 132 MN/m was achieved, which guaranteed a compromise in terms of agreement between the numerical and experimental accelerations on both track sides. By doing so, the numerical time-histories significantly approached the experimental ones and the nMAE indicator suffered a significant reduction to 14.6% and 22.6% relative to the responses obtained in OT and RT sides, respectively. This enhancement was also notable in the frequency domain, since the amplitudes of the main peaks obtained in the numerical analyses increased, showing a better agreement with the measured data.
- Finally, the behaviour of the ballast located over the longitudinal joint was also analysed. The effects of a possible degradation of this material were evaluated based on the shear strain levels that occur in the joint due to the relative cyclic movements between the two adjacent half-decks. After evaluating the maximum plausible degradation of the ballast layer over the joint through a shear modulus degradation curve, a reduction in the modulus of elasticity of the ballast on this location was tested. An optimal value of 36 MPa was obtained, which guaranteed the best compromise in the agreement between numerical and experimental results. By doing so, the numerical results improved in relation to the experimental ones, leading to the lowest levels of the nMAE indicator on both sides, more specifically 13.4% to 21.1% with respect to the OT and RT sides, respectively.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Cascalheira bridge: (

**a**) global view; (

**b**) track view; (

**c**) cross-section (dimensions in cm).

**Figure 4.**Loading scheme of the Alfa Pendular train [14] (loads in kN).

**Figure 5.**Dynamic model scheme of a Alfa Pendular’s car (adapted from [38]): (

**a**) transversal view and (

**b**) lateral view.

**Figure 7.**Potential scenarios that arise from a valid solution of the nonlinear equations for contact search: (

**a**) actual contact and (

**b**) no contact.

**Figure 9.**Ambient vibration test: experimental setup including details of the accelerometer PCB 393B12 (dimensions in m).

**Figure 12.**Test under railway traffic: experimental setup including a detail of the accelerometer PCB 393A03 and highlighting the accelerometers located at midspan used to estimate damping (dimensions in m).

**Figure 19.**Correlation analysis between experimental and numerical modal parameters, before and after updating: (

**a**) errors of natural frequencies; (

**b**) MAC values.

**Figure 23.**Comparison between experimental and numerical vertical accelerations time-histories: (

**a**) OT and (

**b**) RT before updating; (

**c**) OT and (

**d**) RT after updating.

**Figure 24.**Comparison between experimental and numerical vertical accelerations in the frequency domain: (

**a**) OT and (

**b**) RT after updating.

**Figure 25.**Comparison between experimental and numerical vertical accelerations after updating and considering a modification in the vertical stiffness of the supports (K

_{v}= 132 MN/m): (

**a**) OT and (

**b**) RT in the time domain; (

**c**) OT and (

**d**) RT in the frequency domain.

**Figure 26.**Effects in the longitudinal joint due to the train passage: (

**a**) time-history of the shear strains and (

**b**) shear modulus degradation for ${\overline{\sigma}}_{0}=1.5\mathrm{kPa}$ and PI = 0.

**Figure 27.**Comparison between experimental and numerical vertical accelerations after updating and considering a modification in the vertical stiffness of the supports (K

_{v}= 132 MN/m) and in the modulus of elasticity of the ballast over the longitudinal joint (E

_{b2}= 36 MPa): (

**a**) OT and (

**b**) RT in the time domain; (

**c**) OT and (

**d**) RT in the frequency domain.

Parameters | Statistical Properties | Limits | Adopted Value | References | |||
---|---|---|---|---|---|---|---|

Distribution Type | Mean Value/Standard Deviation | Lower | Upper | ||||

E_{c} (GPa) | Modulus of elasticity of concrete | Normal | 37.4/3.74 | 29.9 | 44.9 | 37.4 | [14,34,35] |

ρ_{c} (kg/m^{3}) | Density of concrete | Normal | 2500/250 | 2000 | 3000 | 2500 | |

υ_{c} | Poisson ratio of concrete | - | -/- | 0.16 | 0.24 | 0.20 | |

E_{s} (GPa) | Modulus of elasticity of steel | Normal | 210/8.4 | 193.2 | 226.8 | 210.0 | [14] |

ρ_{s} (kg/m^{3}) | Density of steel | - | -/- | 7700 | 8000 | 7850 | |

υ_{s} | Poisson ratio of steel | - | -/- | 0.28 | 0.32 | 0.30 | |

E_{b} (MPa) | Modulus of elasticity of ballast | Uniform | 169/22.5 | 130 | 208 | 145 | [20,36,37] |

E_{bl} (MPa) | Modulus of elasticity of ballast (longitudinal joint) | Uniform | 79.8/37.7 | 14.5 | 145 | 145 | |

E_{bt} (MPa) | Modulus of elasticity of ballast (transversal joints) | Uniform | 79.8/37.7 | 14.5 | 145 | 145 | |

ρ_{b} (kg/m^{3}) | Density of ballast | Uniform | 1800/260 | 1350 | 2250 | 1800 | |

υ_{b} | Poisson ratio of ballast | - | -/- | 0.15 | 0.20 | 0.15 | |

K_{v} (MN/m) | Vertical stiffness of the support | Uniform | 355.5/111.7 | 162 | 549 | 324 | [33] |

K_{h} (kN/m) | Horizontal stiffness of the support | Uniform | 1730/476 | 905 | 2556 | 1809 |

**Table 2.**Parameters of the numerical model of the BBN vehicle (adapted from [39]).

Parameters | Unit | Value | |
---|---|---|---|

Car body | Car body mass | m_{cb} (kg) | 38,445 |

Car body roll moment of inertia | I_{cb,x} (kg·m^{2}) | 55,120 | |

Car body pitch moment of inertia | I_{cb,y} (kg·m^{2}) | 1,475,000 | |

Car body yaw moment of inertia | I_{cb,z} (kg·m^{2}) | 1,477,000 | |

Bogie | Bogie mass | m_{b} (kg) | 4858 |

Bogie roll moment of inertia | I_{b,x} (kg·m^{2}) | 2700 | |

Bogie pitch moment of inertia | I_{b,y} (kg·m^{2}) | 1931.5 | |

Bogie yaw moment of inertia | I_{b,z} (kg·m^{2}) | 3878.8 | |

Wheelset | Wheelset mass | m_{w} (kg) | 1711 |

Wheelset roll moment of inertia | I_{w,x} (kg·m^{2}) | 733.4 | |

Wheelset yaw moment of inertia | I_{w,z} (kg·m^{2}) | 733.4 | |

Primary suspension | Stiffness of the primary longitudinal suspension | k_{1,x} (N/m) | 4,498,100 |

Stiffness of the primary transversal suspension, | k_{1,y} (N/m) | 30,948,200 | |

Stiffness of the primary vertical suspension | k_{1,z} (N/m) | 1,652,820 | |

Damping of the primary vertical suspension | c_{1,z} (N·s/m) | 16,739 | |

Secondary suspension | Stiffness of the secondary longitudinal suspension | k_{2,x} (N/m) | 4,905,000 |

Stiffness of the secondary transversal suspension | k_{2,y} (N/m) | 2,500,000 | |

Stiffness of the secondary vertical suspension | k_{2,z} (N/m) | 734,832 | |

Damping of the secondary longitudinal suspension | c_{2,x} (N·s/m) | 400,000 | |

Damping of the secondary transversal suspension | c_{2,y} (N·s/m) | 17,500 | |

Damping of the secondary vertical suspension | c_{2,z} (N·s/m) | 35,000 | |

Longitudinal distance between bogies | a_{1} (m) | 19 | |

Longitudinal distance between wheelsets | a_{2} (m) | 2.7 | |

Transversal distance between vertical secondary suspensions | b_{1} (m) | 2.144 | |

Transversal distance between longitudinal secondary suspensions | b_{2} (m) | 2.846 | |

Transversal distance between primary suspensions | b_{3} (m) | 2.144 | |

Vertical distance between car body center and secondary suspension | h_{1} (m) | 0.936 | |

Vertical distance between bogie center and secondary suspension | h_{2} (m) | 0.142 | |

Vertical distance between bogie center and wheelset center | h_{3} (m) | 0.065 | |

Nominal rolling radius | R_{0} (m) | 0.43 | |

Gauge | S (m) | 1.67 |

Vehicle | Car Body (kg) | Bogie [×2] (kg) | Axle Average [×4] (kg) | Total Sum (kg) |
---|---|---|---|---|

BAS | 36,936 | 4858 | 1711 | 53,496 |

BBS | 37,752 | 4858 | 1711 | 54,312 |

RNB | 35,958 | 5204 | 1538 | 52,518 |

RNH | 37,548 | 5204 | 1538 | 54,108 |

BBN | 38,445 | 4858 | 1711 | 55,005 |

BAN | 37,345 | 4858 | 1711 | 53,905 |

Train Speed (km/h) | Mode 1 | Mode 2 | ||||
---|---|---|---|---|---|---|

f (Hz) | a_{max} (m/s^{2}) | ξ (%) | f (Hz) | a_{max} (m/s^{2}) | ξ (%) | |

110 | 9.21 | 0.0774 | 7.86 | 11.93 | 0.0233 | 4.64 |

135 | 9.57 | 0.0951 | 6.20 | 11.46 | 0.1803 | 4.76 |

140 | 9.55 | 0.0790 | 6.01 | 11.56 | 0.1899 | 4.77 |

$\overline{\xi}=$ 6.69% | $\overline{\xi}=$ 4.72% |

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## Share and Cite

**MDPI and ACS Style**

Saramago, G.; Montenegro, P.A.; Ribeiro, D.; Silva, A.; Santos, S.; Calçada, R.
Experimental Validation of a Double-Deck Track-Bridge System under Railway Traffic. *Sustainability* **2022**, *14*, 5794.
https://doi.org/10.3390/su14105794

**AMA Style**

Saramago G, Montenegro PA, Ribeiro D, Silva A, Santos S, Calçada R.
Experimental Validation of a Double-Deck Track-Bridge System under Railway Traffic. *Sustainability*. 2022; 14(10):5794.
https://doi.org/10.3390/su14105794

**Chicago/Turabian Style**

Saramago, Gabriel, Pedro Aires Montenegro, Diogo Ribeiro, Artur Silva, Sergio Santos, and Rui Calçada.
2022. "Experimental Validation of a Double-Deck Track-Bridge System under Railway Traffic" *Sustainability* 14, no. 10: 5794.
https://doi.org/10.3390/su14105794