Detecting changes in the structural behaviour of a laboratory bridge model using the contact-point response of a passing vehicle

ABSTRACT

Drive-by bridge condition monitoring, using in-vehicle sensors to monitor bridges, represents a potential solution for network-scale monitoring of bridge structures. This paper presents a proof of concept for using the vehicle contact-point (CP) response for drive-by condition monitoring of bridges. An expression is presented which allows the vibration response at the point of contact between the tyre and the bridge surface to be inferred from the in-vehicle measurements. Following a simple numerical demonstration of the concept, laboratory tests are undertaken to verify that the CP response can be used to detect the fundamental frequency of the bridge. Results show that the CP response can be used to identify the bridge frequency with greater certainty than the signals measured directly on the vehicle. It is also shown, for two simulated damage cases, that changes in bridge frequency can be detected. The CP response is seen to be more sensitive to changes in bridge frequency than the measured signals. It is also observed that the detected frequency is sensitive to the vehicle speed and mass, which is an important consideration when combining results from multiple vehicle passages. Overall, the results verify that the CP response can be used to enhance drive-by bridge monitoring regimes.

KEYWORDS:

• Drive-by
• bridge
• structural health monitoring
• SHM
• laboratory experiments
• contact-point response

Introduction

As transport networks age and the infrastructure deteriorates over time, inspection and maintenance of the infrastructure components becomes increasingly challenging. Bridges are particularly important components and failure of a bridge, or temporary closures, can have significant impacts on the functionality of the network, while also posing a major safety risk to road users. Limited funding availability for infrastructure owners means that prioritisation of maintenance budgets towards specific structures becomes more and more challenging. For this reason, there is a clear need for quick and efficient techniques for ongoing monitoring and inspection of bridges.

Drive-by bridge monitoring approaches utilise in-vehicle sensors to monitor bridge condition based on measurements taken as the vehicle traverses the bridge. The drive-by concept was initially proposed by Yang et al. (2004) where it was theoretically demonstrated that the fundamental frequency of a bridge can be extracted from the vibration of the traversing vehicle. Given that bridge frequency is sensitive to damage, this represented a method of monitoring changes in bridge condition over time, without the need to install sensors on the bridge, or the need for manual visual inspections. Since the concept was initially proposed, there have been numerous developments in the use of drive-by bridge inspection techniques. Different damage mechanisms can affect bridge vibrations in different ways so there have been various efforts to examine how drive-by techniques can be used to identify these changes, e.g. for scour (Fitzgerald et al., 2019), cracking in the deck (Malekjafarian et al., 2018; McGetrick & Kim, 2013) or changes in boundary conditions (Cerda et al., 2014; Mei & Gül, 2019) amongst others. Drive-by techniques have also been used for different applications, including calculation of bridge damping (Keenahan et al., 2014), estimation of vehicle properties and investigation of pavement characteristics (McGetrick et al., 2013, 2017). More recently there have been a number of developments where researchers have explored the use of Machine Learning approaches to account for environmental or operational factors which may influence the measured bridge properties during any given vehicle passage (Corbally & Malekjafarian, 2022; Locke et al., 2020; Malekjafarian et al., 2019). The concept of drive-by bridge monitoring is a rapidly growing research area and the latest developments have been summarised in a number of recent state-of-the-art review papers (Malekjafarian et al., 2022; Shokravi et al., 2020; Wang et al., 2022).

One of the primary challenges of using drive-by techniques to measure the dynamic properties of a bridge relates to the fact that the contribution of the bridge vibration to the overall vehicle vibration is usually small, making it difficult to isolate the bridge-related components from the response. Yang et al. (2020) developed a formulation which allowed the response at the point of contact between the vehicle and the bridge to be derived from the measured response within the vehicle. The formulation was developed for a single degree of freedom, sprung mass, model. This model represented a specific trailer which was designed to be towed across a bridge. The contact-point (CP) response calculated from the measurements within the vehicle was then used to detect the bridge frequencies. Results from field testing of the approach were promising and demonstrated the ability to extract the bridge frequencies with little interference from the vehicle frequencies.

Since the original study by Yang et al., there have been a number of publications where the concept of the CP response has been investigated in more detail. Zhan and Au (2019) proposed a double-pass normalised CP response which is evaluated by combining the responses from two vehicles of different masses. They showed how it can remove the pavement effects and be used to estimate mode shapes and damage locations. Xu et al. (2021) extended the derivation of the CP response for a sprung mass by considering the effect of vehicle damping. They demonstrated that consideration of vehicle damping is not necessary as its effects on the CP response are minimal. Yang et al. (2021) then proposed to use variational mode decomposition with a band pass filter (VMD-BPF) to enhance bridge frequency identification when using the CP response, while Hashlamon et al. (2021) used the CP response of a stationary vehicle parked on a bridge to estimate the bridge response when it is traversed by another moving vehicle. However, in order to extend the approach to more commonly used vehicles, the vehicle suspension and vibration of the vehicle body need to be considered. This would allow the CP response to be evaluated using regular vehicles, without the need for a specialised trailer. The ability to use regular vehicles for evaluating the bridge frequency lends itself well to crowdsourcing of information about the bridge, for example, using a fleet of instrumented vehicles such as buses or trucks. It would also facilitate continual monitoring of the bridge frequency as vehicles repeatedly drive across the bridge. Only two studies have extended the formulation for the CP response to a more sophisticated vehicle model than the sprung-mass model. One of these studies, by Nayek and Narasimhan (2020), extended the use of the CP response to a quarter-car model, proposing an input-state estimation procedure employing a Gaussian process latent force model to estimate the CP response. The other study was by the authors of this paper, where a new, more simplistic expression was proposed (Corbally & Malekjafarian, 2021), again based on a quarter-car vehicle representation. This expression for the CP response allowed the vehicle suspension and body vibration to be taken into consideration and could be used to directly infer the CP response from the accelerations measured on a vehicle. The authors used numerical simulations to demonstrate that the CP response greatly outperforms the response measured directly on the vehicle for bridge frequency identification. It was also demonstrated that the proposed expression outperforms the equivalent sprung-mass models when considering the effects of the vehicle suspension system, with the sprung mass model unable to detect the bridge frequency when applied using signals from the regular vehicle model. This indicates that the approach could more generally be applied to regular vehicles rather than specifically designed test trailers where the dynamic behaviour is not significantly influenced by the suspension system.

This paper adopts the expression previously developed by the authors and builds upon the numerical proof of concept. The concept is initially introduced through a simple numerical example before being more extensively tested using measurements from a laboratory-scale vehicle-bridge interaction model. Results from the laboratory testing show that the proposed model for evaluating the CP response from in-vehicle vibration measurements can successfully be used to identify the fundamental frequency of the bridge. In addition to this, changes in the fundamental frequency caused by simulating damage in the bridge can also be detected, demonstrating the benefits of adopting the CP response model as part of drive-by bridge monitoring regimes. It should be noted that external operational or environmental factors such as vehicle speed or temperature can also affect the measured frequency. Although these factors are not explicitly considered in this controlled laboratory study, it has been shown that machine learning techniques can effectively be used to remove these influences and enable damage-related changes in the frequency to be isolated (Corbally & Malekjafarian, 2022).

Drive-by bridge monitoring

Various techniques have been studied to try and monitor bridge condition from passing vehicles. The vast majority of these drive-by bridge monitoring techniques aim to detect the dynamic properties of a bridge based on vibration measurements taken on the vehicle. In this study, the fundamental frequency of the bridge is extracted from the vehicle measurements. This section uses a simple numerical example to demonstrate the proposed concept and highlights the advantage of using the CP response to detect the bridge frequency rather than trying to extract it directly from the measured signals on the vehicle.

Numerical modelling of vehicle-bridge interaction

Figure 1 illustrates a numerical modelling approach which adopts a quarter-car vehicle representation to model a vehicle traversing a finite element model of a bridge. The finite element model uses 20 no. beam elements, 0.75 m long, with 2-degrees of freedom per node, representing rotational and vertical displacements. The bridge model represents a 25 m long, single-span, simply supported concrete beam-and-slab bridge, as shown in Figure 2. This bridge is one of many similar bridges which are commonly found in many countries. The bridge properties used in the finite element model are summarised in Table 1, and Rayleigh damping of 2% was included to model structural damping effects in the bridge (Clough & Penzien, 2003). The roughness of the road surface was considered by simulating a “Class A” pavement profile, in accordance with ISO 8606 (ISO, 2016). It is noted that the Class A pavement, simulated in accordance with ISO 8606, represents the irregularities which may be experienced on a pavement considered to be in very good condition as expected on well-maintained highway. This pavement is included in the numerical simulation to ensure that the numerical model is closer to reality, and does not consider an idealised, but unrealistic, situation with a perfectly smooth interaction surface between the vehicle tyre and the bridge.

Figure 1. Numerical vehicle-bridge interaction model.

Figure 2. Cross section of bridge used in numerical simulation.

Table 1. Bridge model properties.

Property	Value
Length (L)	25 m
Young’s Modulus (E)	35 GPa
Cross Sectional Area (A)	6.95 m^2
Second Moment of Area (I)	1.156 m^4
Mass Per Unit Length (m)	17,384 kg/m

In order to consider the interaction between the vehicle and the bridge, a quarter-car model was used to represent the vibration of the vehicle. The quarter car model, as shown in Figures 1 and 3, consists of two degrees of freedom, with two lumped masses, representing the vertical motion of the (i) vehicle body and (ii) the axle (including the wheel). The two masses are connected using a spring and dashpot representing the stiffness and damping properties of the vehicle suspension. The axle degree of freedom is connected to the bridge using a spring to represent the tyre stiffness. While the quarter-car is a simplified representation of the actual behaviour of a vehicle, it has been widely adopted to represent the two primary modes of vibration of a vehicle, i.e. “bounce” of the vehicle body and axle “hop” (Jazar, 2008). The properties used in the quarter-car model and the natural frequencies of the model are included in Table 2. These properties are considered to be broadly representative of the majority of single axle truck suspensions in highway use. The parameter values are representative of a single axle air suspension. Cebon (1999).

Figure 3. Quarter-car vehicle model.

Table 2. Quarter-car properties.

Property	Value
Body Mass (Mv)	8,900 kg
Axle/Wheel Mass (Mw)	1,100 kg
Suspension Stiffness (kv)	2×10^6 N/m
Suspension Damping (cv)	40,000 Ns/m
Tyre Stiffness (kT)	3.5 × 10^6 N/m
First Frequency (body bounce)	1.9 Hz
Second Frequency (axle-hop)	11.3 Hz

As discussed previously by the authors (Corbally & Malekjafarian, 2021), the CP response will not only be sensitive to the bridge vibration, but will also be sensitive to the irregularities in the pavement surface as the wheel moves up and down over these irregularities. In order to more accurately consider the interaction between the tyre and the pavement surface, a rigid-disk model, as depicted in Figure 4(a), was adopted. This rigid-disk model was originally proposed by Chang et al. (2011) and more details of the modelling approach can be found in their paper. When applying the rigid-disk model, the wheel makes contact with only the higher points of the pavement and does not make contact with every single point in the irregularities of the pavement surface. Figure 4(a) shows the actual path of the wheel as it crosses over the pavement, with an exaggerated vertical scale for the pavement to demonstrate the concept. Figure 4(b) shows a portion of the simulated Class A pavement surface along with the “adjusted profile” which represents the passage of the wheel over the pavement. The thicker line represents the point of the tyre directly below the wheel centre, which is not always the point which is in contact with the pavement. From Figures 4(a) and 4(b) it can be seen that the wheel does not make contact with many of the lower points on the road surface, and this “adjusted profile” is used in the vehicle-bridge interaction model to account for the passage of the wheel over the pavement.

Figure 4. Modelling interaction between the wheel and pavement: (a) Illustration of disk model and (b) Adjusted path of wheel over pavement (after Corbally & Malekjafarian, 2021).

The numerical analysis of the vehicle-bridge interaction model involved simulating a constant speed passage of the quarter-car model across the bridge and solving the equations of motion for the coupled vehicle-bridge system to find the dynamic response of the vehicle as it passed over the bridge. These simulated vibration measurements, could then be used as the inputs for the formulation for the CP response, and the frequencies of the signal could be analysed to try and identify the bridge frequency. It is noted that while simulations cannot truly represent the sources of measurement error which will occur in a real-world scenario, 3% random noise was added to the generated acceleration signals to account for measurement errors which will occur in practice. In reality, the measured signals may contain errors for various reasons including electrical noise, poor installation of sensors, vibrations caused by external sources such as other traffic, crossing expansion joints, or modes of vibration which are not captured by the quarter-car vehicle model (e.g. pitching or rolling motion of the vehicle).

Evaluating the contact-point response

In order to extend the approach proposed by Yang et al. (2020) to more commonly used vehicles, the vehicle suspension and vibration of the vehicle body need to be considered when evaluating the CP response. The quarter-car model allows these effects to be taken into account. As shown in Figure 3, $y_V$ and $y_W$ represent the displacement of the vehicle body and the wheel/axle respectively and $u_{cp}$ represents the deflection at the contact point between the wheel and the surface of the bridge. The remainder of the vehicle properties are as described in Table 2.

Equation (1) shows the formulation of the equations of motion for the quarter-car model, with dot notation used to indicate the time derivatives of variables (i.e. velocity and acceleration).

(1) $\left[\begin{array}{cc}{M}_v& 0\\ 0& M_W\end{array}\right]\left\{\begin{array}{c}{\ddot{y}}_V\\ {\ddot{y}}_W\end{array}\right\}+\left[\begin{array}{cc}{c}_V& -c_V\\ -c_V& c_V\end{array}\right]\left\{\begin{array}{c}{\dot{y}}_V\\ {\dot{y}}_W\end{array}\right\}+\left[\begin{array}{cc}{k}_V& -k_V\\ -k_V& k_V+k_T\end{array}\right]\left\{\begin{array}{c}{y}_V\\ y_W\end{array}\right\}=\left\{\begin{array}{c}0\\ k_T{u}_{cp}\end{array}\right\}$(1)

Utilising the axle/wheel equation from Equation (1) and re-arranging in terms of the contact-point acceleration, $u_{cp}$, the CP response can be represented by the formulation shown in Equation (2)(2) ${\ddot{u}}_{cp}=\frac{M_W}{k_T}\frac{d^2{\ddot{y}}_W}{d{t}^2}+\frac{c_V}{k_T}\left(\frac{d{\ddot{y}}_W}{dt}-\frac{d{\ddot{y}}_V}{dt}\right)+\frac{k_V}{k_T}({\ddot{y}}_W-{\ddot{y}}_V)+{\ddot{y}}_W$(2) . The full details of the derivation can be found in Corbally and Malekjafarian (2021).

(2) ${\ddot{u}}_{cp}=\frac{M_W}{k_T}\frac{d^2{\ddot{y}}_W}{d{t}^2}+\frac{c_V}{k_T}\left(\frac{d{\ddot{y}}_W}{dt}-\frac{d{\ddot{y}}_V}{dt}\right)+\frac{k_V}{k_T}({\ddot{y}}_W-{\ddot{y}}_V)+{\ddot{y}}_W$(2)

In Equation (2)(2) ${\ddot{u}}_{cp}=\frac{M_W}{k_T}\frac{d^2{\ddot{y}}_W}{d{t}^2}+\frac{c_V}{k_T}\left(\frac{d{\ddot{y}}_W}{dt}-\frac{d{\ddot{y}}_V}{dt}\right)+\frac{k_V}{k_T}({\ddot{y}}_W-{\ddot{y}}_V)+{\ddot{y}}_W$(2) the $\frac{d^{n}y}{Vd{t}^n}$ notation is used to represent the n^th time derivative of the measured acceleration signals on the vehicle. All other parameters are as defined above and in Table 2. This relationship between the measured accelerations in the vehicle, and the acceleration at the contact point, can be used to infer the CP response directly from in-vehicle measurements, once the properties of the vehicle are known. It should be noted that the vehicle properties required to evaluate the formulation in Equation (2)(2) ${\ddot{u}}_{cp}=\frac{M_W}{k_T}\frac{d^2{\ddot{y}}_W}{d{t}^2}+\frac{c_V}{k_T}\left(\frac{d{\ddot{y}}_W}{dt}-\frac{d{\ddot{y}}_V}{dt}\right)+\frac{k_V}{k_T}({\ddot{y}}_W-{\ddot{y}}_V)+{\ddot{y}}_W$(2) may not be readily available, however, the authors have previously demonstrated that the CP response is not particularly sensitive to errors in the values of these vehicle properties (Corbally & Malekjafarian, 2022). It was demonstrated that variations of 20% in each of the vehicle properties (i.e. ±10% from the actual value) caused no change in the detected value of the bridge frequency or in any of the frequencies present in the portion of the FFT from 0–12 Hz. The only changes were seen in the relative magnitude of the peaks, with all observed changes being less than 5%. This is because the CP response is primarily governed by the bridge vibrations rather than the vehicle vibrations. Most of the vehicle properties will have a small influence on the bridge vibrations, apart from the overall mass of the vehicle body and the vehicle speed, neither of which are required to evaluate the CP response using equation (2)(2) ${\ddot{u}}_{cp}=\frac{M_W}{k_T}\frac{d^2{\ddot{y}}_W}{d{t}^2}+\frac{c_V}{k_T}\left(\frac{d{\ddot{y}}_W}{dt}-\frac{d{\ddot{y}}_V}{dt}\right)+\frac{k_V}{k_T}({\ddot{y}}_W-{\ddot{y}}_V)+{\ddot{y}}_W$(2) . Errors or inaccuracies in the other vehicle parameters, therefore, have a negligible influence on the calculated CP response.

Numerical demonstration of the contact-point response

An illustrative numerical example of a low-speed vehicle passage is used here to demonstrate the concept. A vehicle passage at a speed of 3 ms⁻¹ across the bridge was simulated in MATLAB and the simulated vertical acceleration signals from the quarter car were used as inputs to Equation (2)(2) ${\ddot{u}}_{cp}=\frac{M_W}{k_T}\frac{d^2{\ddot{y}}_W}{d{t}^2}+\frac{c_V}{k_T}\left(\frac{d{\ddot{y}}_W}{dt}-\frac{d{\ddot{y}}_V}{dt}\right)+\frac{k_V}{k_T}({\ddot{y}}_W-{\ddot{y}}_V)+{\ddot{y}}_W$(2) to allow the CP response to be evaluated.

Figure 5 shows the frequency spectrum of both the axle response and the CP response after applying a Fast Fourier Transform (FFT) to the simulated signals. The first frequency of the bridge is also shown, along with the two vehicle frequencies. It is seen that the axle response has a distinct peak at the axle-hop frequency of the vehicle, and the bridge frequency is much less distinct. However, the CP response displays a clear peak at the bridge frequency and is not governed by the vehicle frequencies. This demonstrates the benefits of using the CP response for drive-by bridge frequency monitoring, albeit, in an idealised numerical model. The remaining sections of this paper aim to apply the approach using measured data from a laboratory-scale experiment to verify the CP response expression using real measured data.

Figure 5. FFT of axle response and CP response during 3 ms⁻¹ passage of quarter-car over bridge.

Laboratory-scale vehicle-bridge-interaction model

A scaled vehicle-bridge interaction laboratory model was used to test the application of the theory. The laboratory model was developed to replicate, at a reduced scale, the interaction between the vehicle and the bridge in a controlled environment. The following sections describe the bridge and vehicle models along with the direct modal testing of the bridge.

Bridge model

The bridge model, shown in the photographs in Figure 6 and the detailed sketches in Figure 7, consists of a 5 mm deep, 600 mm wide steel plate representing the bridge deck, with 6 no. steel angle beams (20 × 20 × 3 mm) bolted to the underside of the deck. The bridge deck spans 2 m between the supports, which were designed to allow simply-supported rotational behaviour at each end. The supports consist of a steel shaft which spans between two pillow-block bearing units and a 3D printed bearing shelf which surrounds the steel support shaft and provides a plinth to support the beams. The laboratory model was designed to represent the behaviour of a typical concrete beam and slab bridge under vehicular loading and is based on an existing bridge (shown in Figure 2). Due to the complexities of modelling the geometry of the bridge at a reduced scale, various scaling laws were employed to optimise the geometry of the cross section so that the span-deflection ratio under the load of a typical truck would remain constant, and the first natural frequency of the bridge would be in a similar range to that of the full scale bridge (the bridge length is scaled using a factor of 1:12.5). Timber approach spans were provided at each end of the bridge to allow the vehicle to accelerate/decelerate during each vehicle passage.

Figure 6. Bridge model (a) Fully Constructed and (b) During construction, showing support conditions and placement of beams.

Figure 7. Detailed sketches of laboratory bridge model (dimensions in mm).

In relation to the use of steel in the laboratory model, when replicating a concrete bridge, it is noted that it is possible to create a scaled model using a different material to the full scale bridge, however this needs to be accounted for in the scaling process. As discussed by Ramu et al. (2013), the primary objective when creating a model in a different material is to identify the parameters of interest, and to establish a dimensionless relationship which can be maintained between the full-scale and reduced scale structures. In the case of this study, the bending stiffness (EI) has been optimised so that the ratio between the span and the midspan static deflection under the load of a fully laden truck (representing 5% of the bridge mass) is maintained, and in this way, the difference in the Young’s Modulus between the two materials is accounted for, while still providing a realistic structural cross section which represents a beam and slab structure. In addition to this, the ratio of the vehicle mass to the overall mass of the bridge is also maintained at 5% and the vehicle speed is scaled using a constant relationship, defined in Equations 3(3) $\gamma =\frac{c}{2f_{b,1}L}$(3) . Despite the best efforts to model the most important aspects of the vehicle-bridge-interaction problem at a reduced scale, it is noted that as with any physical scale-models, there will be certain properties which will not have the same behaviour, in some part due to the scaling, and also due to the choice of material. In this case, the structural behaviour, and dynamic interaction between the vehicle and the bridge has been maintained between the two models, however, the damping levels in the steel structure are likely to be lower than that of concrete and the geometric shape of the cross section and beams is not the same.

Vehicle model

A remote-controlled vehicle was used for the experiments. The Axial SCX10™ III Jeep (Axial-Racing, 2022), a commercially available vehicle, has three speed settings and allows some of the vehicle components to be modified or replaced. The vehicle suspension system uses oil-filled shocks with single coilover springs and tuneable damping. For the purposes of testing, the outer plastic cover of the vehicle was removed, as shown in Figure 8, to allow wireless accelerometers to be attached to the vehicle. The sketches in Figure 9 show the specific locations of the accelerometers along with the relevant dimensions. Three accelerometers were installed, at the locations depicted in Figures 8 and 9, to allow vibrations to be measured on the vehicle body at the front and back of the vehicle, and also on the rear axle.

Figure 8. Vehicle model equipped with accelerometers.

Figure 9. Vehicle dimensions and accelerometer locations (a) Side elevation and (b) Plan view.

Direct testing of bridge modal properties

Before testing the CP response formulation, the bridge was tested directly to understand the dynamic behaviour of the structure. Initially, a finite element model of the bridge was developed to assess the expected modal behaviour of the bridge. Once the theoretical mode shapes and frequencies had been evaluated, 8 no. wireless accelerometers were placed on the surface of the bridge deck, at the locations depicted in Figure 10. The sensors positions were chosen to optimise the ability to extract the expected modes of vibration of the bridge, with sensors located at midspan, the two quarter-points and at the 1/8 span locations.

Figure 10. Direct modal testing of bridge (a) Sensor layout and impact locations and (b) Accelerometers on bridge deck during testing.

The natural frequencies and mode shapes of the bridge were evaluated using an output-only method, whereby the bridge was impacted with a mallet a total of 72 times. The bridge was impacted 9 times at each of the edge impact locations, and 15 times at each of the impact locations along the centreline of the bridge, as shown in Figure 10(a). These impact locations were chosen to ensure that all relevant mode shapes were excited. Signals from the 72 hammer impacts were then analysed in MATLAB and an algorithm was developed to automatically identify the natural frequencies and mode shapes of the bridge using Frequency Domain Decomposition (FDD). The results for the first four frequencies are depicted in Figure 11, where a surface was fitted between sensor locations to visualise the deflected shape of the bridge for each of the modes of vibration. It can be seen that the first and third mode shapes represent the first two global bending modes, with the second and fourth modes of vibration representing global torsion of the bridge deck. The natural frequencies and corresponding mode shapes were reasonably well aligned with the results from the FE model, with the exception of the second bending mode, which was 11.5% lower than that predicted by the FE model. It is not clear why such a large difference was observed for the second bending mode, and not for the other modes, however it is likely that this is a result of differences introduced during the construction and assembly of the laboratory model, meaning that it does not perfectly match the stiffness, geometry, fully composite behaviour and simply-supported boundary conditions which are used in the FE model. The results of hammer testing are compared to the FE modelling in Table 3, for the first four modes of vibration. It is noted, however, that for the purpose of the drive-by testing only the fundamental frequency of the bridge was of interest in this study. Measuring the dynamic properties of the bridge directly in this way, allowed the accuracy of the fundamental frequency detected from the in-vehicle measurements to be readily assessed for the drive-by testing.

Figure 11. Bridge mode shapes identified from testing (a) Mode 1, (b) Mode 2, (c) Mode 3 and (d) Mode 4.

Table 3. Dynamic properties obtained from hammer testing vs. finite element modelling.

Mode No.	Mode Type	Measured Frequency (Hz)	FE Model Frequency (Hz)	Difference
1	Bending	8.8	8.9	1.1 %
2	Torsional	15.3	15.4	0.6 %
3	Bending	29.9	33.8	11.5 %
4	Torsional	40.2	41.9	4.1 %

Drive-by identification of bridge fundamental frequency

The remote-controlled vehicle was driven over the bridge a number of times and the vibration data on the vehicle was collected during each passage. An algorithm was developed to automatically identify the point of entry to and exit from the bridge, based on peaks in the signals as it passed over a small gap at each end. The average vehicle speed during each passage was also automatically calculated from the signals. Figure 12 shows the acceleration signals measured on the vehicle during one crossing. The spikes in the signal as the axles enter and leave the bridge can clearly be seen. The lateral position of the vehicle, during each passage across the bridge, was not stringently controlled, so the lateral vehicle positions contained some variability, as would be experienced in practice.

Figure 12. Measured acceleration signals on vehicle during 0.7 ms⁻¹ passage.

In order to estimate the CP response, Equation (2)(2) ${\ddot{u}}_{cp}=\frac{M_W}{k_T}\frac{d^2{\ddot{y}}_W}{d{t}^2}+\frac{c_V}{k_T}\left(\frac{d{\ddot{y}}_W}{dt}-\frac{d{\ddot{y}}_V}{dt}\right)+\frac{k_V}{k_T}({\ddot{y}}_W-{\ddot{y}}_V)+{\ddot{y}}_W$(2) was used with the signals from the back axle and the body just above the back axle. The vehicle properties were estimated based on the datasheets for the vehicle, and where parameters were not known, reasonable estimates were used. The mass of the vehicle axle & wheels, $M_w$, was assumed to be 0.475 kg, with the suspension stiffness taken as $k_V=$ 455 Nm⁻¹, suspension damping $c_V$ = 20 Nsm⁻¹, and the tyre stiffness taken as $k_T=$ 2,500 Nm⁻¹. Exact measurements of the vehicle properties were not taken, to replicate a scenario where the vehicle properties are not known with exact certainty. As discussed previously, errors in estimating the vehicle properties should not have a significant impact on the calculated CP response. One of the primary advantages of using the proposed approach relates to the fact that only an estimate of the vehicle properties is required, as they will be difficult to measure in practice. The goal of the laboratory experiments presented in this paper is to verify that the bridge frequency can be detected using this method, despite the fact that the vehicle properties are not known with 100% accuracy. Figure 13 shows the CP response signal, evaluated using Equation (2)(2) ${\ddot{u}}_{cp}=\frac{M_W}{k_T}\frac{d^2{\ddot{y}}_W}{d{t}^2}+\frac{c_V}{k_T}\left(\frac{d{\ddot{y}}_W}{dt}-\frac{d{\ddot{y}}_V}{dt}\right)+\frac{k_V}{k_T}({\ddot{y}}_W-{\ddot{y}}_V)+{\ddot{y}}_W$(2) , corresponding to the same 0.7 ms⁻¹ vehicle passage shown in Figure 12. It can be seen that there are much higher frequency components in the signal, which corresponds to the findings of the numerical study presented by Corbally and Malekjafarian (2021). These higher frequency components tend to be associated with the roughness of the pavement surface. It is noted that in the laboratory, the steel surface of the bridge deck is reasonably smooth, compared to the expected road profile which would be experienced on a real bridge. However, given the geometry of the off-road tyres on the vehicle, there is a similar phenomenon at the contact point, whereby the CP response is composed of a combination of the bridge response and higher frequency vibrations caused by the roughness of the tyre and road surface. Although the relative contribution of the tyre-induced frequencies could not be easily quantified during testing, higher frequency vibrations induced by the interaction of the roughness of the tyres could be visually observed.

Figure 13. Calculated CP response during 0.7 ms⁻¹ passage.

In order to estimate the fundamental bridge frequency from the drive-by measurements a series of passages of the vehicle across the bridge were performed. The vehicle has three speed settings, low, medium and high. The majority of the vehicle passages were at the low-speed setting, with a small number of crossings also performed at the higher speeds. Table 4 summarises the details of the vehicle crossings at each speed range, showing the mean vehicle speeds and the equivalent speed ranges, in kmh⁻¹ for the full-scale bridge upon which the laboratory model is based. The equivalent full-scale speeds of the vehicle crossings can be estimated using the geometric scaling of the bridge length (1:12.5), to work out a speed which would give the same duration for the vehicle crossing the bridge. However, this does not consider the scaling of vehicle-bridge interaction (VBI) which can more appropriately be scaled by maintaining a constant relationship between the driving frequency of the vehicle (which depends on the vehicle speed) and the fundamental frequency of the bridge. Equation (3) presents a formulation which can be used to maintain a relationship between the vehicle speed and bridge vibration (McGetrick et al., 2015):

(3) $\gamma =\frac{c}{2f_{b,1}L}$(3)

Table 4. Details of vehicle crossings for detection of bridge frequency.

Speed Setting	Speed Range (ms^−1)	Avg. Measured Speeds (ms^−1)	Equivalent Full-Scale Speed (kmh^−1)		No. of Crossings
			Geometric Scaling	VBI Scaling
Low	0–0.85	0.73	33	14	20
Medium	0.85–1.25	1.11	50	22	3
High	>1.25	1.52	68	30	3

where $\gamma$ is the dimensionless speed parameter, $c$ is the vehicle speed (ms⁻¹), $f_{b,1}$ is the fundamental frequency of the bridge (Hz) and $L$ is the length of the bridge span (m). The mean speeds for each set of vehicle crossings are converted to equivalent full-scale speeds using both approaches and shown in Table 4.

To estimate the bridge frequency from the drive-by measurements, the portion of the signals during each vehicle passage was extracted and the frequency spectrum of each signal was evaluated by applying a Fast Fourier Transform (FFT) to the data. The resulting FFTs for all vehicle passages in a given speed range were averaged, to give an overall frequency spectrum for each speed setting. The results were normalised before plotting and Figure 14 shows the FFTs for each speed setting. The measured bridge frequency, from direct testing, is also shown on the plots (8.8 Hz) for comparison, and it can be seen that the signals all contain a localised peak in the vicinity of the bridge frequency, however the peak is slightly shifted depending on the speed setting used. This shift is not unexpected as previous research has demonstrated that the measured frequency will be influenced by vehicle speed (Sitton et al., 2020). It should be noted that the medium and high-speed cases are only based on the average of 3 vehicle passages, however the low-speed case is based on the average of 20 passages, meaning that the reliability of the results at the medium and high-speed cases are likely to be lower, as discussed later. The limited vehicle passages at the higher speed settings was a result of limited accessibility to the laboratory during the COVID-19 pandemic, during the period when these experiments were carried out.

Figure 14. FFT plots of measured signals & calculated CP response for (a) Low, (b) Medium and (c) High speeds.

Examining the results from the low-speed setting in Figure 14(a), it can be seen that all of the measured signals contain multiple peaks, with the peaks associated with the bridge frequency slightly underpredicting it. Most importantly, it can be seen that the CP response exhibits a single distinct peak, just below the bridge frequency. This verifies the theory that the CP response is governed by the bridge frequency rather than the vehicle frequencies, making it a better indicator of bridge frequency. The accuracy of the predictions for the high-speed vehicle setting are closer to the actual bridge frequency, but due to the small number of vehicle passages at this speed, the results are less reliable, and the peak is less distinct. Despite this, it can be concluded that the CP response provides improved bridge frequency detection capabilities, but the influence of varying vehicle speed on the detected frequency should be considered if the measured frequency is to be used for condition monitoring.

A sensitivity analysis was carried out to analyse the potential variability in the detected bridge frequency depending on how many vehicle passages were averaged to evaluate the frequency spectrum. Initially, the frequency spectrum was evaluated for all 20 low-speed vehicle passages. Then the number of vehicle passages used to evaluate the final frequency spectrum was increased incrementally from 1–20. Using this approach for the case of 1 vehicle passage means that there are 20 separate frequency spectra which could be used to estimate the bridge frequency, but for the case of 20 vehicle passages there is one single frequency spectrum, which is the average of all 20. For values in between 1 & 20, there are multiple possible combinations of vehicle passages which could be used. For example, for the case of 2 vehicle passages, there are 190 different combinations which could be used to calculate the average spectrum. The frequency spectrum was evaluated for all possible combinations of vehicle passages, and the peak value of the FFT was recorded for each combination. The peak value was considered to be the detected bridge frequency for that particular combination. Figure 15 shows the variation in the detected frequencies, showing the average value (solid blue line), the standard deviation bounds ($F_{\mu \pm \sigma }$) and the maximum/minimum ($F_{max/min}$). The value of the bridge frequency measured directly from the bridge while the vehicle was located at midspan is also included. This value was found to be 8.1 Hz and is depicted by the horizontal dashed black line. It is worth noting that this is 8% lower than the value which was found from direct testing of the bridge without the presence of the vehicle, and is also confirmed to be the frequency which is detected from the drive by testing. The results of the analysis show that the overall mean frequency detected from the CP-response was found to be 8.08 Hz, which almost exactly detects the frequency of the bridge with the vehicle at midspan. However, it can be seen from the graph, that the potential variability in the detected frequency can be quite high if only using a small number of vehicle passages. For example, if using a single vehicle passage, the error in the detected frequency could be as much as 7% away from 8.1 Hz. The use of 15 vehicle passages results in the errors being reduced to approximately 1% which suggests that at least 15 vehicle passages should be combined to ensure confidence in the detected frequency. It should also be noted that the influence of the vehicle mass cannot be neglected, and that if the mass of the vehicle changes, this should be accounted for when interpreting the detected frequency.

Figure 15. Variability in average value of detected bridge frequency depending on number of vehicle passages used.

Drive-by detection of bridge damage

To assess the ability of the CP response to detect changes in bridge frequency due to damage, the experiments were repeated, with two “simulated” damage conditions considered. The effect of damage on the bridge frequency was simulated by clamping masses to each side of the deck at midspan to artificially induce a change in the fundamental bridge frequency, as shown in Figure 16. The damage conditions and experimental details are summarised in Table 5 along with the frequency values measured directly from the bridge. It can be seen that the addition of lumped mass for the two damage cases resulted in a 2–8% reduction in frequency. To put this into context, for the concrete bridge upon which the experimental design is based (Figure 2), the same reduction in frequency would be experienced if a global loss of section of 5–20% was experienced across the width of the deck, at midspan. This represents significant damage, particularly considering it is unlikely that a similar level of damage would be experienced by all beams simultaneously. However, if all of the beams experienced delamination due to corrosion of the reinforcement along the whole length of the bridge, a similar reduction in frequency would be experienced with a loss of 12-60 mm of concrete, which would not be uncommon in a structure which has experienced advanced corrosion of the reinforcement. It is also noted that despite the extreme damage levels which would be required to achieve this reduction in frequency, the majority of research in the field of drive-by methods for bridge damage detection typically include more extreme damage conditions than those presented in this paper (e.g. Sarwar and Cantero (2021), Mei and Gül (2019).

Figure 16. Masses clamped at midspan of bridge to simulate damage.

Table 5. Details of drive-by damage detection experiments.

Damage Case	Bridge Freq. (Hz)	No. of Vehicle Crossings	Description
Healthy	8.8	20	Bridge in healthy condition
Dam1	8.6	20	1 kg each side @ midspan
Dam2	8.1	20	2 kg each side @ midspan

The experiments were carried out in the same way as the previous experiments, however this time, only the high-speed setting was used for the drive-by testing. Based on the three speed settings available for the remote-controlled vehicle, it was considered that the high-speed setting would be most suitable to represent speeds which could feasibly be used for full-scale drive-by bridge monitoring, with the lower speeds likely being too slow to facilitate drive-by monitoring under normal operational traffic conditions on most roads. It is worth noting that a significant portion of the existing literature has only been successful in determining the bridge properties, or identifying bridge damage, under low-speed driving conditions, as generally the longer duration signals allow improved detection of the bridge vibration (Malekjafarian et al., 2022; Wang et al., 2022). However, in order for the benefits of the drive-by approach to be realised, it must be capable of accurate bridge monitoring under real traffic speeds, which would not require closure of the bridge to other traffic.

Before each passage, the bridge was manually excited, by hitting it with a closed fist, to induce free vibration in the structure, and simulate a situation where other traffic is also crossing the bridge. The purpose of exciting the bridge in an uncontrolled manner, using a closed fist, was to replicate a scenario where the bridge would already be vibrating due to passing traffic, which is variable, and uncontrolled in nature. In free-flowing traffic conditions, for a 25 m bridge, it would be unlikely that that there would be additional heavy vehicles on the bridge at the exact same time as the monitoring vehicle, meaning that the effect of other vehicle masses on the bridge are not likely to be a major problem. However, vehicles driving ahead of the test vehicle, or driving in the opposite lane of the two-lane bridge, would tend to cause the bridge to be in free-vibration in many cases when the test vehicle reaches the bridge. Previous research has shown the beneficial effect of this external excitation in magnifying the contribution of the bridge frequency to the measured frequencies in the vehicle (Malekjafarian & OBrien, 2014; Yang et al., 2022; Yang et al., 2020). Figure 17 shows the frequency spectra for the axle response and the CP response for the healthy condition and the two damaged conditions. The actual bridge frequency, for each case, is also included in the plots, as represented by the three vertical lines. It is clear that the addition of mass at midspan has the effect of reducing the natural frequency of the bridge, and it can also be seen that the peak of the FFT plots, for both the axle and CP response give reasonably accurate measures of bridge frequency, albeit slightly under-predicted for each case. It is noted that the effect of the vehicle mass causes the observed frequencies to be slightly lower than when the bridge was directly tested without the vehicle on it. As discussed earlier, direct testing of the bridge with the vehicle located at midspan, showed that the observed frequencies from direct testing and from the drive-by testing were identical. This is an important consideration when combining results from vehicles of different masses, the effect of the vehicle mass needs to be taken into account when interpreting the results. Most importantly, it can clearly be seen that the peak shifts to the left as the level of simulated damage increases, providing confidence that this approach can be used to monitor changes in the bridge frequency. It can be seen once again that the CP response provides a more distinct peak at the bridge frequency than the axle response, although for the healthy case, it is not quite as distinct. It is also clear that manually exciting the bridge has increased the prominence of the peak for the axle response (a similar situation was observed for the two sensors on the vehicle body). This shows the benefits of using the CP response when bridge vibrations are lower, as seen in the previous section. In such cases, the vehicle frequencies tend to make it difficult to identify the peak associated with the bridge frequency, particularly as the exact frequency of a real bridge would not be known in advance, however, the CP response does not suffer from this effect and can be effectively used whether or not the bridge is already vibrating.

Figure 17. FFT Plots of (a) Axle Response and (b) Contact-Point Response, for the Healthy & 2 Damaged Cases (Directly Measured Bridge Frequencies Shown in Vertical Lines).

Finally, to assess the sensitivity of the CP response to damage, and compare it to the other signals, the peak values for the FFT plots for the two damaged cases were compared to the peak values identified for the healthy case. The percentage change in frequency from the healthy condition is shown in Figure 18. It is seen that the change in frequency is most distinct for the CP response, with the axle response being slightly less sensitive and the two body sensors (front and back) providing the exact same results, again, slightly less sensitive. Overall, it is seen that damage can be detected using any of the signals, however the CP response clearly demonstrates the advantage of being more sensitive to damage, but also allowing the bridge frequency to be more readily identifiable from the frequency spectrum.

Figure 18. Change in identified frequencies for both damage cases compared to those identified for the healthy case.

Discussion

This paper successfully demonstrates that the proposed equation to allow the CP response to be inferred from in-vehicle vibration measurements can successfully be applied to a real vehicle model and the bridge frequency can be more easily detected from the inferred CP response, compared to when the measured signals are used directly. The primary advantage of using the CP response relates to the fact that the peak associated with the bridge frequency is distinct and much more clearly distinguishable within the frequency spectrum, however it is also seen in Figure 18, that the changes in the peak of the CP response, due to the added mass, are more distinct than the changes seen from the sensors on the vehicle. The CP response shows changes of 5.9% and 8.2% for the two damage cases, whereas the sensors on the vehicle body show smaller changes of 3.5% & 7.1%. The sensor on the vehicle axle also shows a change of 7.1% for the more extreme damage case, with a change of 4.8% for the smaller damage case. The use of added mass on the bridge to simulate the effects of damage is of course a simplification, and cannot truly replicate the effects of damage. Despite this, the observed changes in bridge frequency, of 2–8%, do appear to align with some previous estimates in frequency change associated with damage. For example, Alampalli (1998) showed that bridge damage, in the form of sawcuts to the bottom flanges of bridge girders, caused changes of approx. 3–8% in the fundamental frequency of a bridge.

Given that temperature effects can cause significant changes in frequency, it may be difficult to accurately quantify changes related to damage. Peeters and De Roeck (2001) showed that, for a concrete bridge, changes in temperature over the course of a year accounted for variations of as much as 5–10% in the fundamental frequency of the bridge. It is also well documented, and demonstrated in this paper, that the speed and mass of the vehicle will affect the measured frequency when using drive-by techniques. Although this study does not explicitly consider these external factors, it has been shown in a separate study by the authors (Corbally & Malekjafarian, 2022) that machine learning techniques can be adopted to learn the influence of these environmental/operational factors and allow them to be removed, so that damage-related changes in the frequency spectrum can be isolated. In relation to the effects of measurement noise, in both the numerical and experimental results, it was seen that the measurement noise tended to be associated with high-frequency disturbances in the signals, and therefore did not have a detrimental effect on the bridge frequency detection in the 0–12 Hz range examined in this study.

A further limitation of this study relates to the simplifications required to develop a scaled model of the vehicle and bridge. Despite the fact that great care has been taken to consider scaling laws and to replicate the real-life scenario as accurately as possible, due to the challenges in scaling, there will always be inaccuracies in the model, and many other real-life effects may not be accurately captured. The irregularities in the road surface profile, and the resulting effect on the CP response, are not accurately modelled by the surface of the steel deck plate. In addition to this, the effect of additional traffic on the bridge, changes in road surface profile or in the vehicle properties over time may not be realistically modelled in the experiments. Full-scale testing on a real bridge would of course allow improved verification of the CP response formulation in capturing the bridge frequency, however, this also comes with its challenges. Most notably, when carrying out testing on an in-service bridge, it would not be possible to damage the bridge for the purposes of research experiments.

Conclusions

This paper presents a proof of concept for using the CP response of a passing vehicle to facilitate drive-by bridge condition monitoring. An expression previously developed by the authors, to allow the CP response to be inferred from the vehicle vibrations, is tested using a laboratory-scale vehicle-bridge interaction model. A numerical example is initially used to demonstrate that the bridge frequency is more easily identified using the CP response compared to the vibration response measured directly on the vehicle. A series of experiments are then carried out using the laboratory model and the bridge frequency is shown to be visible using the CP response. Experiments are also used to assess the ability of the CP response to monitor changes in bridge frequency which occur when damage is simulated in the bridge. The CP response is shown to be more sensitive to damage than any of the signals measured directly on the vehicle, further demonstrating the benefits of using it for bridge condition monitoring. These experimental results provide confidence in the proposed drive-by bridge monitoring method, however it is noted that the detected bridge frequency is sensitive to the vehicle speed and mass, which must be accounted for when combining information from different vehicles travelling at different speeds.

Disclosure statement

No potential conflict of interest was reported by the author(s).