Laboratory investigation of a bridge scour monitoring method using decentralized modal analysis

Scour is a significant issue for bridges worldwide that influences the global stiffness of bridge structures and hence alters the dynamic behaviour of these systems. For the first time, this article presents a new approach to detect bridge scour at shallow pad foundations, using a decentralized modal analysis approach through re-deployable accelerometers to extract modal information. A numerical model of a bridge with four simply supported spans on piers is created to test the approach. Scour is modelled as a reduction in foundation stiffness under a given pier. A passing half-car vehicle model is simulated to excite the bridge in phases of measurement to obtain segments of the mode shape using output-only modal analysis. Two points of the bridge are used to obtain modal amplitudes in each phase, which are combined to estimate the global mode shape. A damage indicator is postulated based on fitting curves to the mode shapes, using maximum likelihood, which can locate scour damage. The root mean square difference between the healthy and scoured mode shape curves exhibits an almost linear increase with increasing foundation stiffness loss under scour. Experimental tests have been carried out on a scaled model bridge to validate the approach presented in this article.


Introduction
Bridges with foundations in water can be affected by the washing away of soil, a concept termed scour erosion. 1 This phenomenon is responsible for a majority of bridge failures worldwide [2][3][4] and causes significant economic and travel disruption. 5 Scour erosion causes a local reduction in soil elevation in the vicinity of bridge foundations, which affects the capacity and stiffness of the foundation system. 6 Losses in stiffness of the foundation system can lead to serviceability issues such as differential settlement, inclination of piers and abutments, and cracking in decks as well as partial or complete structural collapse. 7 The increased frequency of extreme weather events due to climate change is placing growing pressure on infrastructure networks worldwide, and many bridges on these networks are approaching the end of their design lives. There is an urgency among asset owners and operators to tackle these issues to ensure a sustainable future for these networks. 8 The occurrence of scour around bridge foundations is typically managed by discrete monitoring and inspections. Visual inspections by asset management agencies remain the most common approach, which generally involves divers inspecting the condition of foundations and ranking bridges against rating criteria related to the perceived severity of the scour problem. The labourintensive nature of these types of inspection has inspired the development of sensor-based systems that can remotely detect and monitor scour evolution around bridge foundations. These systems include radar devices that measure changes in material properties and hence the location of the soil surface, 9,10 installed devices that float out of the soil when scour reaches their level, 11,12 and driven rod systems including magnetic sliding collars 12 where physical sensors move along driven rods and rest on the riverbed within a scour hole. A full review of the performance of these types of sensor at detecting scour is available in Prendergast and Gavin. 13 In recent years, online direct monitoring of changes in the structural response as a result of scour evolution has gained traction among researchers. Scour results in a stiffness change at foundations, which represents a change in the boundary conditions of the structure, and hence manifests as changes in modal properties. 14,15 Several researchers have investigated changes in modal properties as a way to detect and monitor scour erosion. 16 The majority of research has focused on identifying changes in natural frequency to detect scour. Prendergast et al. 17 investigated the changes in natural frequency of a single pile affected by scour and proposed a Winkler-based spring-beam numerical model capable of tracking the frequency changes. The same authors later developed a vehicle-bridge-soil interaction model 18 and investigated if changes in natural frequency due to scour of the central pier of a twospan integral bridge could be determined from vibrations in the structure under passing vehicles. 19 This approach was later extended to locating scour-related foundation stiffness losses, that is, which pier or abutment is experiencing scour, 20 and investigating the influence of ambient temperature changes on the approach. 21 Ju 22 developed a three-dimensional (3D) numerical model to study how scour affects the natural frequency of a bridge in the presence of water-added mass, by incorporating soil-fluid-structure interaction. The presence of water lowers the natural frequencies of a structure compared to the case where there is no water, and the effect of scour is to reduce the natural frequencies. However, a non-smooth frequency change is observed due to non-uniform foundation elements and layered soils. Kong and Cai 23 studied the effect of scour on the response of a bridge with traversing vehicles, in the presence of wave loads. Results suggested that scour has a significant influence on the lower frequencies of a bridge pile. Since the entire bridge is affected by scour, it is suggested that the response of the deck near the supports or the response of passing vehicles can be used to detect scour presence. Bao et al. 24 studied frequency-based scour detection from the perspective of identifying the physical meaning of predominant natural frequency, optimal location of sensors and the effect of scour hole shape. 25 They conclude that natural frequency reduces with scour progression, but the frequency values differ depending on whether the scour hole is symmetrical or unsymmetrical, due to differences in the soil constraints. Chen et al. 26 identified various modal frequencies of a cable-stayed bridge, developed a numerical model of the system, optimized the model boundary conditions and used the model to estimate the scour condition at a pier based on measured and predicted frequencies.
In addition to frequency-based scour detection, several researchers have investigated the monitoring of other dynamic parameters to infer the presence of scour. Xiong et al. 27 studied four dynamic indicators for scour detection in cable-stayed bridges, namely, frequency, modal assurance criterion (MAC), mode shape curvature and flexibility-based deflection. They suggest that flexibility-based deflection can be a sensitive indicator of scour as long as an accurate reference numerical model of the bridge is available. Furthermore, the use of mode shape curvature is not deemed practical due to the high number of sensors required. Natural frequency, while sensitive and practical to measure, can only qualitatively indicate scour presence due to its sensitivity to other damage that may be present in the structure. Elsaid and Seracino 28 also investigated mode shape curvature and flexibility-based deflection as well as flexibility-based curvature for detecting scour on a scaled model of a coastal bridge. Each of these indicators showed promise for scour detection. However, this study simply modelled scour as an increase in pile effective length and omitted any soil-structure interaction. Foti and Sabia 29 analysed scour development on a fivespan bridge by monitoring the asymmetric dynamic behaviour of a scoured pier, using an array of accelerometers along the foundation mat. Scour was detectable by analysing the covariance of measured accelerations along the foundation, but was not quantifiable. Malekjafarian et al. 6 proposed a mode shapebased scour monitoring method for bridges with shallow foundations, based on measuring changes in the modal amplitude at a given pier relative to the mean of the amplitudes at remaining piers as detected from output-only modal analysis. Using sensors located at each pier, the mean-normalized mode shape exhibited sensitivity to scour-related foundation stiffness loss at given piers. The method was experimentally validated on a laboratory-scale bridge.
To address the shortcomings associated with frequency-based damage detection, modal estimation techniques using bridge accelerations have received increasing attention over the past two decades for effective health monitoring and parameter identification. [30][31][32][33] Obtaining mode shapes from modal testing 31 or from output-only modal methods 33 typically suffers the common drawback of requiring large numbers of sensors installed on a bridge, which can make these systems expensive and difficult to maintain. To overcome these challenges, decentralized and multi-setup approaches for modal analysis have been proposed for damage detection techniques, [34][35][36][37][38][39][40] that uses independent groups of data with overlapping nodes to capture local spatial information in stitchable segments, or reference-based multi-setup stochastic subspace identification algorithms. 41,42 These approaches have been successfully implemented and tested both in numerical as well as field environment. For example, Sim et al. 39 analysed different network topologies for the decentralized approaches and studied the effect of changing overlapping nodes on the accuracy of the estimated global mode shapes, and Dohler et al. 41 computed statistical uncertainties and covariance mode shapes derived using a multi-setup subspace system identification algorithm and applied the approach on actual bridge vibration data.
Indirect health monitoring methods have come to the fore recently, which aim to estimate bridge dynamic properties using responses measured from moving sensors such as instrumented vehicles, reducing the sensor burden. 43,44 Indirect monitoring using passing vehicles has shown promise at detecting changes in natural frequencies, [45][46][47] damping 48 and road surface roughness of bridges. 49,50 These methods rely on modal identification approaches such as short time-frequency domain decomposition (STFDD), which was used to estimate bridge mode shapes from multiple vehicle acceleration measurements in a recent study. 51,52 The method works by measuring segments of mode shapes from signals arising in a vehicle crossing a bridge, using frequency domain decomposition (FDD), 53 which are combined together using common points between neighbouring segments. 35,36,39 Despite showing promise, the indirect STFDD method is critically influenced by the vehicle velocity and has been shown to only be effective when the vehicle is travelling at unrealistically slow velocities.
In this article, a novel approach to bridge scour detection is proposed using a decentralized approach based on re-deployable accelerometers that is verified in a laboratory environment. The approach does not require a reference model for detecting and locating damage, improving on the limitations of previous damage detection methods. While not explicitly modelled, the sensors could be deployed using unmanned aerial vehicles (UAVs), which would land, collect data and move to the next point on a bridge, in a sequence of operations. This would enable data acquisition to be carried out over the full length of a bridge with limited sensor requirements. It is envisaged that a given structure would remain operational during data acquisition; therefore, forced accelerations due to passing vehicles are measured during each data acquisition phase, which has the added advantage of assessing the performance of the approach when natural vehicle-related variability is incorporated. A variety of modal estimation methods exist, and there is significant variability in the accuracy of resulting modal properties depending on which method is adopted. Due to its simplicity and ease of execution with reasonable results, the FDD technique is used in this article to extract modal information from the measured bridge accelerations. The trade-off with ease of execution lies with the possible introduction of errors in the estimated modal parameters. FDD works by extracting segments of the mode shape, which can subsequently be combined to form a global mode shape. This approach is tested using only two accelerometers in this article, which are considered as being deployed to various points along a bridge over time. Both numerical modelling and experimental testing are performed in the present work. A one-dimensional (1D) half-car vehicle model is simulated to cross a four-span bridge model 54,55 with three piers having sprung masses and representative foundation stiffness. 56 A class 'A' road profile is included on the bridge. 57 In each phase of simulated measurement, bridge accelerations are calculated at two points on the bridge (for a given number of vehicle passages) and a portion of the mode shape is estimated. The global mode shape for each case, along with a fitted curve and mode shape gradient, is used to detect the location of scour-related foundation stiffness loss, without the requirement of a prior knowledge of a healthy reference mode shape. The root mean square (RMS) difference is used to estimate the severity of the scour problem relative to the theoretical healthy mode shape. The proposed approach is validated using a laboratory-scaled bridge. The results from experimental tests demonstrate the effectiveness of the reference-free and RMS indicators at detecting the presence and severity of bridge scour. The method may prove useful for rapid postflood bridge evaluation on a network.

Bridge model
A multi-span single-lane bridge is simulated in this article as a 1D finite element (FE) model 55 using MATLAB. The bridge model comprises four 20-m long, simply supported spans, each modelled using 20 Euler-Bernoulli beam elements, 58 with 4 degrees of freedom (DOFs; see Figure 1). The beginning and end of the bridge rest on undeformable supports, modelled as pins and rollers, whereas the internal connections are modelled as hinged supports, resting on deformable piers. The presence of potential eccentricities at the internal supports due to the pin and roller arrangement is not considered; therefore, potential bending moments in the piers are not modelled. The piers are instead idealized to only enable vertical motion, which may introduce minor errors into the model. Three internal piers, modelled as single-DOF sprung masses (with stiffness, k pier,i and mass, m pier,i ), are assumed to rest on shallow pad foundations, modelled using a foundation spring k f,i (i = 1, 2, ..., N p ), where N p is the number of piers. The properties of the foundation springs are derived to correspond to a 4 m 3 2 m shallow pad foundation in contact with a uniform medium dense sand deposit, 44,59 using the expression in equation (1) where G is the soil shear modulus (kPa), v is the Poisson's ratio, L and B are the foundation length (m) and width (m), respectively. The soil shear modulus is calculated from the elastic modulus using the expres- where E = 100,000 kPa. 59 Equation (1) is semi-empirical, and more information on similar expressions is available in Pais and Kausel 61 and Mylonakis et al. 62 The geometric and material properties of the bridge and foundation are provided in Table 1. A class 'A' road surface profile is randomly generated 57 according to the ISO standard. 63 An approach length of 100 m is used to ensure the equilibrium of vehicle dynamic behaviour when it enters the bridge. The dynamic response of the bridge due to the timevarying moving forces is given by a system of equations at each time step where M b , C b and K b are the bridge mass, the damping and stiffness matrices, respectively, and € y b , _ y b and y b are the vectors of bridge accelerations, velocities and displacements for each DOF, respectively. f int is a vector representing the interaction forces between the bridge and the vehicle 55 as it traverses the bridge.

Vehicle model
A half-car model, shown in Figure 2, is employed to represent a two-axle vehicle. 54,64 The vehicle has 4 DOFs, namely, body mass translation (y s ), body pitch (u), and two axle mass translations (y u,i ). The axle masses are represented by m u,1 and m u,2 , respectively, and are connected to the body mass (m s ) through springs with linear stiffness (k s,i ) and viscous damping coefficients (C s,i ). The vehicle makes contact with the road surface though tyre springs of linear stiffness (k t,i ). 54 In this study, a series of vehicles is programmed to cross the bridge, and Table 2 shows the mechanical properties adopted for all vehicles. 48,54,64 The body mass and velocity are varied for each vehicle to simulate the expected variability in traffic loading. A weighin-motion (WIM) database from Illinois, provided by the US Federal Highway Administration's Long-Term Pavement Performance Program, 65,66 is used to develop a distribution of representative vehicle masses and velocities. A population of two-axle vehicles, with axle spacings varying between 5 and 7 m, is extracted from the database. For each vehicle run in this article, the body mass and vehicle velocity are chosen randomly from where M v , C v and K v are the mass, the damping and stiffness matrices of the vehicle, respectively, and € y v , _ y v and y v are the vectors of vehicle accelerations, velocities and displacements, respectively. The vector f v represents the time-varying interaction forces applied to the vehicle DOFs by the road profile and the bridge displacements. In modelling terms, the bridge and vehicle models are coupled, 55 and the global equation of motion is formed by combining equations (2) and (3) where M g and C g are the coupled mass and damping matrices, respectively. K g is the time-varying coupled stiffness matrix that depends on the bridge static displacements due to the vehicle. F is the system force matrix, and € u, _ u and u are the vectors of accelerations, velocities and displacements of the global system, respectively. The equation of the coupled system is solved in MATLAB using the Wilson-Theta integration scheme. 67 Decentralized approach to estimate bridge mode shape The decentralized concept of using re-deployable sensors to estimate mode shapes requires the scaling and stitching together of mode shape segments calculated at discrete sensor locations. 35,37,39 A pair of accelerometers are placed at discrete points along the bridge and calculate the modal response at their location due to vehicles passing along the bridge. These sensors are moved to various points over time (as could be undertaken using UAVs or otherwise). The modal response (mode shape amplitude) corresponding to a given frequency is estimated from the discrete accelerations measured at these locations using FDD. This is achieved using the singular value decomposition, which is applied to decompose the spectral density matrix, Gˆ(jv i ), for each frequency i of the response 53 where U i is the unitary matrix of singular vectors, S i is a diagonal matrix containing singular values, and H denotes the complex conjugate of the matrix. The singular vectors provide the mode shape amplitudes corresponding to a selected frequency. The concept is illustrated in Figure 3, where two accelerometers (Acc 1 and Acc 2) are moved along the bridge in eight phases to build the global mode shape(s). In each phase, a vector of mode shape amplitudes [u li , u ri ] corresponding to a segment of a mode shape is extracted. These segments are then rescaled and stitched together using a common location between neighbouring segments. One accelerometer remains fixed while the other is moved to the next location to maintain continuity. The global mode shape (; i c ) for the ith frequency, using r segments [c 1 , c 2 , ..., c r ], can be calculated by In the numerical analysis in this article, the nodal accelerations simulated in the vehicle-bridge interaction model are used to represent the 'accelerometers', deployed at various points. The bridge length is divided into eight segments (corresponding to sensor placement phases) -each having a length of 10 m (half a span length). The accelerometer locations are chosen as the supports and the midspan of each simply supported beam, as illustrated in Figure 3. For a given placement of two sensors (i.e. one phase), a single vehicle is modelled to cross the bridge, with the parameters chosen from the population of vehicles as described previously. A segment of a given mode shape is derived for each phase from the nodal accelerations generated due to the vehicle crossing, with 2 s of free vibration assumed after the vehicle departs the bridge. To include measurement error, random noise is added to the acceleration signals 48 with zero mean value and a standard deviation ranging between 3.5% and 8.5%, depending on the gross vehicle weights. These percentages are based on a sample of field measurements and engineering judgement to incorporate the variability of signalto-noise ratio with respect to the amplitude of the signal.
A demonstration of the decentralized approach for estimating mode shapes is shown in Figure 4. Figure   Table 2. Properties of the half-car model.

Vehicle property Notation Value
Axle masses m u,1 750 kg m u,2 1100 kg Tyre stiffnesses k t,1 1.75 3 10 6 N/m k t, 2 3.5 3 10 6 N/m Suspension stiffnesses k s,1 0.5 3 10 6 N/m k s, 2 1.0 3 10 6 N/m Suspension damping C s,1 1.0 3 10 4 N s/m C s,2 15 3 10 3 N s/m 4(a) shows the discrete mode shape amplitudes (connected by straight lines) for the first mode of the fourspan bridge, as developed by moving two accelerometers along the bridge model in phases. Also shown in this plot are the theoretical first mode shape amplitudes at the same points, as extracted from an eigenvalue analysis on the model mass and stiffness matrices. 68 Figure 4(b) shows the same information, but for the second mode shape of the bridge. The estimated and theoretical mode shapes agree well, with an MAC analysis giving values of 0.9985 and 0.9976 for the first and second modes, respectively. This suggests that the redeployable approach using single overlapping sensor location performs well and is not significantly affected by variation in vehicle properties.

Scour detection approach
In this article, the damage due to scour occurrence is considered as a loss in foundation stiffness in a shallow foundation at a given pier. While other damage mechanisms are not explicitly considered (that may or may not be present in the bridge structure), the assumption underlying scour damage is that this typically occurs reasonably quickly, for example, during a discrete flood event. This would lead to a change in the modal properties before and after such an event. Other damage -for example, corrosion or bearing damagetends to occur more slowly. However, it should be noted that if damage leads to a change in the modal properties, this can be used to trigger a manual inspection, which will identify the source of damage, and so is still of practical use for a bridge manager. The mechanism for scour-related foundation stiffness loss is based on the overburden and   strain-dependency of soil shear modulus, G. This is a depth-dependent parameter and typically varies nonlinearly with mean stress level, so a reduction in soil elevation (and overburden stress) due to scour lowers the G value at the formation level of the foundation. Recent experimental tests on laterally loaded piles have revealed how the strength and stiffness of soil depend on the shape of the scour hole, highlighting the sensitivity of these properties to the amount of overburden removed. 25 In addition, scour is considered in this article to undermine a shallow foundation, reducing the soil-foundation contact area, which increases the stress in the remaining soil. Increasing the stress leads to an increase in strain, according to the nonlinear stressstrain behaviour of soil, and G typically reduces with strain. 69 Therefore, the combined action of reducing the soil elevation and undermining the shallow foundation can lead to significant reductions in vertical foundation stiffness, in accordance with equation (1). Many previous works have focused on scour effects on piled bridges, where the mechanism for scour reduces the lateral stiffness and capacity of piled foundations. 19,20,28,29,70 In the present work, the focus is on bridges with shallow foundations, where the vertical stiffness is affected by scour undermining the foundation, a mechanism that has received much less attention in previous literature.
Two scour damage cases are considered in this work, 25% and 45% foundation stiffness reductions at two piers. Table 3 shows the assumed geometrical and material properties for these two scour cases. For the case of 25% foundation stiffness loss, this could correspond, for example, to a scour hole undermining a pad such that it reduces the soil-foundation contact area by '30%, with a corresponding reduction in G of '9%. 69 For the 45% foundation stiffness reduction, this could correspond, for example, to a reduction in soilfoundation contact area of '50% and a reduction in G of '20%.
The following text describes the scour detection approach investigated in this article. Mode shapes are derived from the re-deployed sensors for scour at the central pier and the right-hand side pier (refer to Figure 3) for both scour cases in Table 3, using the numerical model. To assess the repeatability of the analysis, a representative set of vehicle runs is used to derive healthy and damaged mode shapes. Fifty first mode shape estimates are developed for the healthy case, and for each of four damage cases. The damage cases consist of 25% and 45% foundation stiffness loss at each of two piers, the second (central) and the third (right) piers. Since eight vehicle runs are required to estimate a single mode shape using the re-deployable sensors, this equates to a total of 400 (=50 3 8) vehicle runs, with variable body masses and velocities, for the healthy case and for each scour case. Figure 5 presents the results of the analysis. Figure 5(a) shows the first mode shapes for the healthy case and scoured cases at the central pier, and Figure 5(b) shows the same information for scour at the third pier. The shaded region for each mode shape shows the mean value 6 one standard deviation at each location of the inferred mode shape, based on the 50 vehicle passes in each case. The vertical lines in Figure 5 show the locations of the piers, with dashed line denoting healthy and dotted line denoting scour locations. It can be observed that the reduction in foundation stiffness at the central pier, Figure 5(a), and at the right-hand side pier, Figure 5(b), reduces the mode shape amplitude at  those points, changing the overall shape of the first mode shape. It can be seen in Figure 5(a) and (b) that the shaded regions of the modal amplitudes do not overlap with each other, when plotting the mean value 6 one standard deviation of the measured data set. It should be acknowledged that if the entire population of the distribution were considered, some overlap would undoubtedly occur, which would mask the presence of scour. This is a wider issue with many damage detection approaches and has been studied in other works. 71 Plotting the mean value 6 one standard deviation accounts for almost 70% of the total population. The change in the mode shape due to scour can potentially assist in locating scour damage. A damage indicator to locate scour is postulated based on fitting a polynomial curve to the estimated first mode shape amplitudes, 72 and identifying changes in these fitted curves between healthy and scoured conditions. To decide on the order of curve to be adopted, a maximum likelihood analysis 73 is carried out and the Akaike information criterion (AIC) 74 is calculated for several polynomials of increasing order. The AIC considers the trade-off between goodness of fit and complexity (order) of the polynomial used. Low-order polynomials and good fits both reduce the AIC value. The curve with the lowest AIC is considered as the most appropriate fit for a given set of modal amplitudes (given damage case in this article). AIC is calculated as twice the difference between the number of unknown variables and the maximum log-likelihood for each fit. 74 The number of unknown variables for each case equates to the degree of the curve fit, for example, a second-order polynomial has two unknowns and a third-order polynomial has three unknowns. Table 4 provides the values of likelihood and AIC for the fitted mode shapes from each scour condition and degree of polynomial fit, up to seventh order. It can be observed that the most appropriate degree of fit for each damage case is different. However, both third-and fourth-degree polynomials have good AIC scores for all cases. In order to allow comparison between cases, the same order of polynomial is desirable for all cases and the fourth order is chosen here for this purpose. Figure 6 presents the results of fitting fourth-degree polynomials to the modal amplitudes of Figure 5. Fourth-degree polynomials are also fitted to the envelope of the data, that is, to the mean value 6 one standard deviation of the estimated modal amplitudes. It can be seen, in Figure 6(a) in particular, that fitting polynomials removes much of the detail evident in Figure 5. However, the influence of damage is perhaps more obvious, with increased amplitude at scoured piers. The range of results indicates that 50 estimates of mode shapes should be enough to detect 25% stiffness loss at the central pier, and much less at the third pier. The maximum change in peak value occurs at the location of the scoured pier in each case, and the change in peak is larger for higher increases in foundation stiffness loss associated with scour. For scour at the central pier in Figure 6(a), the mean change in peak value for 25% and 45% foundation stiffness loss is 7.2% and 16.8%, respectively. For scour at the right-hand pier in Figure 6(b), the mean change in peak value for 25% and 45% foundation stiffness loss is 67.6% and 125.2%, respectively. This damage indicator provides an impartially objective metric, that is not dependent on any underlying parameters, as compared to using the mode shape amplitudes directly, and offers a way to locate scour without having prior knowledge of structural behaviour.
To quantify the severity of the scour damage (in terms of foundation stiffness loss), an indicator based on the RMS differences between the estimated mode shape amplitudes and the corresponding healthy mode shape is developed. The healthy mode shape values prior to any scour occurrence may be calculated from the physical measurements taken, or from an FE model of the undamaged bridge. The RMS difference is calculated as a square root of the sum of squared differences between the damaged and healthy mode shape amplitudes along the bridge where u Healthy and u Scoured are the amplitudes of healthy and scoured mode shapes, respectively, and m is the total number of points in the mode shape.
To investigate the effectiveness of the RMS differences at quantifying scour damage, the RMS differences between a healthy mode shape extracted from an eigenvalue analysis in the numerical model and the 50 mode shapes estimated for each damage condition are presented in Figure 7. The data are presented in terms of the mean value with an error bar (mean value 6 one standard deviation) showing the variability in the analysis. There is a near-linear variation in the RMS with percentage stiffness reduction with the R 2 value for the scour location at piers 2 and 3 being 0.993 and 0.998, respectively. It is noteworthy, however, that quantifying an actual extent of a scour hole affecting a given pier is still a challenge due to the nonlinear relationship between changes in soil G, and reduction in contact area should scour undermine the foundation. It should therefore be understood that the method adopted here can only infer an amount of foundation stiffness reduced due to scour occurrence. This is arguably a more useful metric than the actual scour hole magnitude as it is a direct measure of the structural distress due to scour.

Laboratory-scale model bridge
Using the decentralized approach to detect scourrelated foundation stiffness loss at bridge foundations is experimentally investigated in this section. A scaled   bridge model with four simply supported spans on three piers (see Figure 8) has been developed at the Kyoto University and is used to test the approach. 6,56 The model piers are supported on springs to simulate the behaviour of shallow foundations, and the abutments are considered as pinned. Bearing plates are used to create pinned and roller supports for each span. Four parallel springs are used at each pier to provide vertical stiffness (Figure 10), which can be reduced to simulate scour occurrence. Each spring has a stiffness of 49,000 N/m for the healthy case, providing a total stiffness of 196,000 N/m at each pier support. The spring stiffness was chosen so as to model an equivalent 4 m 3 2 m shallow foundation on a loose-to mediumdense uniform sand deposit. 6 The geometric and material properties of one of the bridge beams are shown in Table 5. Although the laboratory bridge is simplified relative to a real full-scale structure, it can provide a beneficial way for testing the scour damage detection approach postulated in this article in a preliminary sense. Results from this model should therefore be considered preliminary until full-scale validations are undertaken.

Vehicle model
The model bridge is traversed by a scaled vehicle, comprising a two-axle tractor, towing a two-axle trailer, see Figure 9. The vehicle runs on tracks attached to the bridge beams. Both tractor and trailer have a main body consisting of a sprung steel plate. The tractor front and rear axles have two suspension springs (one for each wheel) of stiffness 1533 and 1753 N/m, respectively. The trailer has four suspension springs -each of stiffnesses is 8464 N/m. 56 An electronic controller is used to maintain a constant vehicle velocity while it traverses the bridge. 56 For the tests in this article, three vehicle velocities are considered, 1.14, 1.

Scour modelling and data acquisition
Seven accelerometers are deployed on the bridge: four at the midspans and three at the pier locations (an accelerometer at pier 2 is shown in Figure 10). To replicate a situation where only two sensors are available but re-deployable, data from only two sensors are used in any one phase of measurement. These sensors measure the vertical bridge accelerations due to the passing vehicle model. To model scour, four springs with stiffness 37,000 and 27,000 N/m are used to replace the 49,000 N/m springs at the pier support, giving stiffness reductions of 24.5% ('25%) and 44.9% ('45%), respectively. Four scour scenarios are investigated: 25% and 45% stiffness loss at pier 2, and 25% and 45% stiffness loss at pier 3. For each scour case considered, each of the nine vehicle cases traverses the bridge. Hence, there are nine healthy bridge cases and 9 3 4   (=36) damaged bridge cases. All 45 cases are repeated 30 times to test for repeatability.
To find a mode shape, the accelerations measured by two adjacent sensors are taken from the passage of one of the nine vehicle cases. For the chosen vehicle case, the results from one of the 30 runs are used in the procedure. The measured accelerations at the two sensor points are processed using FDD to obtain the modal amplitudes due to the given vehicle crossing at these sensor locations. One of the sensors is then considered to move to the next point on the bridge while the second sensor remains fixed, in keeping with the redeployable sensor approach. In reality, the sensors were not moved, but data were taken from one of the same sensors and one new one. A new set of accelerations is measured due to the passage of another vehicle (one of the 30 runs of one of the nine vehicles). Using FDD, the modal amplitudes at these two sensor locations are obtained. Considering the abutments as fixed and using seven locations of measurement, the process is repeated in six phases until the modal amplitudes are obtained along the entire bridge. The global mode shape is obtained by combining the mode shape segments in the same manner described in the numerical analysis previously, using the common sensor in each segment to scale the mode shape appropriately. It should be noted that due to the random nature in choosing the vehicle run to excite the bridge in the experimental tests, there are 60,480 possible ways in which the segments were generated using different vehicle combinations. This is calculated as permutations of nine possible vehicle cases in six phases to generate a mode shape (9P6). This is calculated assuming that each vehicle is run only once (instead of 30 times). The process is repeated for each of the four scour conditions, 25% and 45% at piers 2 and 3, respectively, to obtain damaged mode shapes. The results are described in the next section.

Analysis and results
The experimental mode shapes derived using the redeployable accelerometers are analysed in this section to ascertain their performance at detecting scour-related foundation stiffness loss at the piers. While sensors were considered as being re-deployed for the purpose of the analysis in this article, in reality, this was achieved using different sensors located at fixed points for each vehicle passage. In the actual tests, this means that data were acquired at several points along the bridge for each vehicle passage. It is therefore possible to compare the mode shape developed using re-deployed sensors with that measured directly from traditional fixed sensors along the bridge. 30 The first mode shape, derived using both approaches, is compared in Figure 11, where these are plotted as normalized to the greatest peak of the mode shape from the traditional method. The MAC value of the mode shape estimated using the redeployable sensors is 0.9989 as compared to the traditional mode shape, suggesting that the mode shape derived using re-deployable sensors and subsequent scaling and combining provides accurate results.
Since nine vehicles were used to obtain signals along the bridge for the healthy and damaged cases, a statistical analysis of the mode shapes derived using the redeployable sensors can be undertaken. Using the 60,480 possible combinations for a given mode shape, the mean first mode shape 6 one standard deviation, developed using the re-deployable sensor technique, is presented in Figure 12. Figure 12(a) shows the healthy case and scour at pier 2. Figure 12(b) shows the same information for the healthy mode shape and scour at pier 3. There is a significant difference in the amplitude of the measured mode shape at the point of scour. It is noteworthy that the modal amplitude is also affected at unscoured piers, but the changes at the scoured pier are much greater. At the scour location, the modal amplitude decreases with loss of foundation stiffness. At the unscoured  piers, the opposite effect is observed as the amplitude increases with loss of foundation stiffness. While these results are not as clean as those from the numerical study, they are consistent (see Figure 5). The method shows promise at enabling scour detection when experimental variability is present.
As for the numerical analysis in section 'Scour detection approach', a fourth-degree polynomial curve is fitted to the mode shape amplitudes and presented in Figure 13. Note that only the amplitudes at the points where sensors are deployed are used to generate the mode shapes in the figure. The purpose of this is to investigate the effectiveness of using a polynomial fit as a damage indicator to detect the location of scour. The fitted curves to the mode shapes of the healthy bridge differ significantly from those in the scoured bridge cases and follow the same trend as in the numerical analysis in Figure 6. This suggests that polynomial curve fitting to the modal amplitudes derived from redeployable sensors shows promise at enabling scour to be located on a bridge. The effect of the fitted curve is to smoothen the response obtained by direct analysis of the extracted modal amplitudes, enabling an easier assessment of damage presence.
The difference in RMS values between the mean estimated healthy mode shape and scoured mode shapes is calculated as a square root of the sum of squared differences between the damaged and healthy mode shape amplitudes along the bridge. The 60,480 mode shape combinations for each scour condition are analysed against a benchmark mean healthy mode shape (developed as the mean of the 60,480 healthy mode shape combinations). The results are shown in Figure 14. Error bars are shown on the plot to highlight the variability in the RMS values for each scour condition. A linear variation in RMS with foundation stiffness loss is evident from a trendline fitted to the data. The R 2 value in case of scour location at piers 2 and 3 is 0.992 and 0.983, respectively, suggesting a close fit. The results suggest that the RMS value may offer a way to quantify foundation stiffness loss due to scour relative   to a benchmark condition (pre-scour or otherwise). While the results are from scaled experimental testing, this is a promising find.

Conclusion
This article investigates an approach for detecting the loss in foundation stiffness of bridges resulting from scour erosion in a numerical and laboratory study using a decentralized modal analysis technique based on redeployable accelerometers. Bridge mode shapes contain valuable information relating to their structural condition and are sensitive to global changes, in particular, changes to boundary stiffness as can occur due to scour erosion. In this article, the mode shapes are estimated using a decentralized system of re-deployable accelerometers, whereby two accelerometers are considered to be placed sequentially at discrete points along a bridge. The modal amplitudes at sensor locations are extracted under operational conditions using output-only modal analysis with excitation from passing vehicles. The discrete mode shape segments developed along the bridge are scaled and combined together while ensuring that one sensor remains fixed as the other is moved. It is considered that the sensor re-deployment could be undertaken in the field using UAVs as part of a bridge assessment process.
Two damage indicators are proposed. The first is based on fitting polynomial curves to the discrete modal amplitudes to assist in locating scour damage, and the second is based on calculating RMS differences between healthy and damaged mode shapes to quantify the foundation stiffness loss due to scour. The approach is developed in this article and tested both numerically and using scaled experimental testing on a laboratoryscale bridge at the Kyoto University. The main findings of the study are as follows: 1. In the numerical and experimental analysis, the derived mode shapes from the re-deployable sensors compare well with those extracted from the theoretical eigenvalue analysis and the traditional output-only approach. This is significant as it demonstrates that the approach is unaffected by variability in the excitation, enabling the mode shape estimation by stitching together various mode shape segments. Furthermore, these mode shapes exhibit sensitivity to scour at two locations on a four-span representative bridge. 2. A fourth-degree polynomial fit to the modal amplitudes exhibits a reasonable change at the scour locations indicating its potential suitability as a scour damage indicator. It also has the benefit of smoothing the response of the mode shape while preserving the change at the damage location. 3. In the numerical and experimental analysis, the difference in RMS between the estimated healthy mode shapes and the scoured mode shapes is found to increase linearly with the reduction in scourrelated foundation stiffness.
The results in this article are an improvement on previous mode shape-based damage detection methods, which either rely on a high number of distributed sensors to calculate accurate modal data or on moving sensors such as vehicles, which tend to require very low velocities for accurate data acquisition. The analysis in this article suggests that two sensors sequentially moved along a bridge are capable of developing an estimate of the mode shape under given scour conditions with sufficient accuracy to enable the detection of scour.
Although the method was successfully tested using 1D numerical models and a simplified experimental arrangement, it is suggested that 3D numerical modelling and full-scale testing on a real bridge be undertaken before firm conclusions on the effectiveness of the approach be given.