Keywords

1 Background of Bridge Screening by Drive-By Inspection

1.1 The Necessary of Digital Twin Bridge

Digital twin of infrastructure has a capacity to realize the data driven maintenance, the prediction of several future status and co-creation of new value with a different fields engineer. For example in Japan, the regular inspection for bridge can use drone or sensors instead of only direct visual method since 2022. The drone can capture the anomaly on surface with a detail by making 3D models and the sensors can detect the damage within the structure. 3D models with mechanical signal data (acceleration, temperature and etc.…) which is often called as Cyber-Physical System can estimate the equation of motion. With time going, the change can be extrapolated, and the several future can simulate. Data gained from digital models and real can be provide the explainability to the data analysis using AI such as DNN, RNN and Reinforced Learning. This explainability is shared with the civil engineer and data analyst, and can promote the innovation. Thus, Digital Twin of infrastructure can be “common language” for engineers. Especially, this study tries to construct the digital twin of bridges.

The number of bridge is often larger than the other infrastructures and the measurement should be cost-saver. Drone flight is often disturbed by weather conditions or regulations. For the detection of structural damage, sensors has a potential, however, the sensor putting on bridge directly may be cost-labor. The installation cost can be expensive with sensor increasing and the wireless sensors has a trade-off about the power capacity and the performance (contains sample rate, data sending and etc…). Although the sensor on bridge is ordinary at age of IoT, the sensing with ease are necessary in the dawn of construction and practical use of Digital Twin bridges. This study focuses on Vehicle Response Analysis (VRA) and the screening technology by them.

1.2 VRA

VRA is carried out by the data measured by sensors on the vehicle going over bridges. The data is analyzed based on Vehicle-Bridge Interaction (VBI) system and the change by bridge damage are detected. Generally, the damage index for bridge uses the natural frequency or the mode shape. Natural frequency (NF) is related to rigidity of the structure [1, 2]. The change in little damage are small and noisy, thus it should be detect by high accuracy accelerometer. Mode shape is considered to be more robust to noise than NF [3, 4]. However, bridge mode shape estimated from vehicle vibration is not accuracy. Thus, VRA is used for bridge screening.

1.3 Screening by Drive-by Inspection Technology

Drive-by bridge inspection is carried out by going vehicle with sensors. If bridge inspection is carried out by sensor data on vehicle only, the extreme high performance and huge data for supplement of the accuracy are necessary. On-going Monitoring can gather the data periodically. Screening aims the detection of data anomaly in wide area efficiently, and it doesn’t need many high accurate expensive sensors. The index of bridge screening must be sensitive and robust to noise, and mode shape is often used in numerical simulation. This study uses Spatial Singular Mode Angle (SSMA) [5]. SSMA is a damage index for bridge screening and previous studies propose practical use methods [6,7,8]. However, the generality to the bridge environment or mechanical parameter was not enough.

1.4 The Purpose and Contribution of This Study

This study carried out the experiment to 108 bridges under the management of local government in Japan, and the trend of their SSMA are verified. The result suggests that the bridge length – SSMA has a trend, and it shows the probability of detection without past data by direct visual inspection or measurement.

2 Vehicle-Bridge Interaction Theorem

The mathematical theorem of SSMA is shown. Calculation needs vibration and position at the front, rear axles of vehicle. The vibration should be obtained from mass points under the spring. The assumed vehicle (Half Car) model is shown in Fig. 1. This model has a rigid body as sprung-mass system, of which mass is \(m_{s}\), and of which inertia moment is \(I_{s}\). The point \({\text{G}}\) indicates the centre of gravity, and the distances from the point \({\text{G}}\) to the front and the rear axles are \(L_{1}\) and \(L_{2}\), respectively. In this Fig. 1, it is noted that \(L_{1}\) and \(L_{2}\) described as if as equal, however they are ordinary different because the engine often put near front wheel. The subscript i (= 1, 2) represents the front and rear axles. \(z_{si} \left( t \right)\) and \( z_{ui} \left( t \right)\) are the vertical displacements of the sprung-mass and the unsprung-mass. \(u_{i} \left( t \right)\) is the forced displacement input under the \(i\)-th axle. \(k_{si}\) and \(c_{si}\) are the spring stiffness and the damping of the spring-mass at the \(i\)-th axle. \(m_{ui}\), \(k_{ui}\) and \(c_{ui}\) are the mass, spring stiffness and damping of the unsprung-mass at the \(i\)-th axles, respectively. The equation of motion of the vehicle can be described by the following.

$$ {\mathbf{M}}_{{\text{V}}} \user2{\ddot{z}}\left( t \right) + {\mathbf{C}}_{{\text{V}}} \dot{\user2{z}}\left( t \right) + {\mathbf{K}}_{{\text{v}}} {\varvec{z}}\left( t \right) = {\mathbf{C}}_{{\text{P}}} \dot{\user2{u}}\left( t \right) + {\mathbf{K}}_{{\text{P}}} {\varvec{u}}\left( t \right) $$
(1)

respectively \(( \cdot )\) and \(( \cdot )\) denote the first-order and second-order time differentiation.

Fig. 1.
figure 1

Vehicle (Half Car) Model.

Full size image

Since the number of sensors are same with that of estimated mode shapes, when we set a sensor on each axle, only the first and second modes can be obtained. When we use only lower mode shapes, their variation can be explained only from two factors: the measurement environment and the structural change. The latter is, in other word, a damage. On the other hand, when we use more sensors, the main factor of variation becomes the ill condition problem, which means that the results depend only on noise, not on the status of the structure. On the other hand, the bridge displacement at position \(x \) and time \(t\) can be decomposed as follows:

$$ y\left( {x,t} \right) = \mathop \sum \limits_{k}^{{}} \phi_{k} \left( x \right)q_{k} \left( t \right) $$
(2)

\(\phi_{k} \left( x \right)\) is the \(k\)-th order mode shape, and \(q_{k} \left( t \right)\) is the \(k\)-th order basis coordinates. Substituting each axle position \(x_{i} \left( t \right)\) into Eqs. (2), the bridge displacement just under the \(i\)-th axle is shown below:

$$ y_{i} \left( t \right) = \mathop \sum \limits_{k}^{{}} \phi_{k} \left( {x_{i} \left( t \right)} \right)q_{k} \left( t \right) $$
(3)

Assuming that the road roughness at the position of \(x\) is \(R\left( x \right)\), the input component of the \(i\)-th axle of the vehicle at the time of \(t\) is shown below:

$$ r_{i} \left( t \right) = R\left( {x_{i} \left( t \right)} \right) $$
(4)

Then, the forced displacement inputs can be described by

$$ {\varvec{u}}\left( t \right) = {\varvec{y}}\left( t \right) + {\varvec{r}}\left( t \right) $$
(5)

On the other hand, assuming that

$$ {{\varvec{\Phi}}}\left( t \right) = \left[ {\begin{array}{*{20}c} {\phi_{1} \left( {x_{1} \left( t \right)} \right)} & {\phi_{2} \left( {x_{1} \left( t \right)} \right)} \\ {\phi_{1} \left( {x_{2} \left( t \right)} \right)} & {\phi_{2} \left( {x_{2} \left( t \right)} \right)} \\ \end{array} } \right] $$
(6)
$$ {\varvec{q}}\left( t \right) = \left\{ {\begin{array}{*{20}c} {q_{1} \left( t \right)} \\ {q_{2} \left( t \right)} \\ \end{array} } \right\} $$
(7)

Equations (5) can be rewritten in

$$ {\varvec{u}}\left( t \right) = {{\varvec{\Phi}}}\left( t \right){\varvec{q}}\left( t \right) + {\varvec{r}}\left( t \right) $$
(8)

Next, \(\phi_{k} \left( {x_{i} \left( t \right)} \right)\) can be discretized by interpolation as below:

$$ \phi_{k} \left( x \right) = \mathop \sum \limits_{{{\varvec{j}} = 1}}^{{\varvec{n}}} a_{jk} N_{j} \left( x \right) $$
(9)

When the base function \(N_{j} \left( x \right)\) is the Lagrangian function, the coefficient \(a_{kj}\) indicate the amplitude of \(k\)-th order mode shape at the discretized position \(x_{j}\). Figure 2 shows the concept of this interpolation.

Fig. 2.
figure 2

Concept of interpolation.

Full size image

When \(n = 2\) and \(x_{1} = L/3\) and \(x_{2} = 2L/3\), the Lagrangian function is

$$\begin{aligned} &N_{1} \left( x \right) = - \frac{3}{L}x + 2 \\ &N_{2} \left( x \right) = \frac{3}{L}x - 1 \end{aligned}$$
(10)
$$ \left[ {\begin{array}{*{20}c} {\phi_{1} \left( {x_{1} \left( t \right)} \right)} & {\phi_{2} \left( {x_{1} \left( t \right)} \right)} \\ {\phi_{1} \left( {x_{2} \left( t \right)} \right)} & {\phi_{2} \left( {x_{2} \left( t \right)} \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {N_{1} \left( {x_{1} \left( t \right)} \right)} & {N_{2} \left( {x_{1} \left( t \right)} \right)} \\ {N_{1} \left( {x_{2} \left( t \right)} \right)} & {N_{2} \left( {x_{2} \left( t \right)} \right)} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array} } \right] $$
(11)

By using Eqs. (10) and Eqs. (11), Eqs. (9) becomes

$$ {{\varvec{\Phi}}}\left( t \right) = {\mathbf{N}}\left( t \right){\mathbf{A}} $$
(12)

where the \(\left( {k, j} \right)\) component of the matrix \({ }{\mathbf{A}}\) is \(a_{kj}\). Assuming that the unsprung-mass parameters of the front and rear axles are same, which means that \(k_{u1} /m_{u1} = k_{u2} /m_{u2} = k_{u} /m_{u}\) and \(c_{u1} /m_{u1} = c_{u2} /m_{u2} = c_{u} /m_{u}\), the vertical acceleration vibrations of the unsprung-mass can be described by

$$ \user2{\ddot{z}}_{u} \left( t \right) = \left\{ {\begin{array}{*{20}c} {\ddot{z}_{u1} \left( t \right)} \\ {\ddot{z}_{u2} \left( t \right)} \\ \end{array} } \right\} = {\mathbf{N}}\left( t \right){\mathbf{A}}{\varvec{\sigma}}\left( t \right) + \overline{ \in }\left( t \right) $$
(13)

If the position of each axle \(x_{j} \left( t \right)\) are available, the interpolation matrix \({\mathbf{N}}\left( t \right)\) can be calculated. Since the unsprung-mass vibrations \(\user2{\ddot{z}}_{u} \left( t \right)\) and the interpolation matrix \({\mathbf{N}}\left( t \right)\) are known, we obtain

$$ {\mathbf{N}}^{ - 1} \left( t \right)\user2{\ddot{z}}_{u} \left( t \right) = {\mathbf{A}}{\varvec{\sigma}}\left( t \right) + \in \left( t \right) $$
(14)
$$ \in \left( t \right) = {\mathbf{N}}^{ - 1} \left( t \right)\overline{ \in }\left( t \right) $$
(15)

The left side of Eqs. (14) is the spatial correction of vehicle vibrations. Based on Eqs. (14), the mode shape \({\mathbf{A}}\) can be estimated by SVD (Singular Value Decomposition) of \({\mathbf{N}}^{ - 1} \left( t \right)\user2{\ddot{z}}_{u} \left( t \right)\). By SVD, the mode shape \({\mathbf{A}}\) and the bridge vibration component \({\varvec{\sigma}}\left( t \right) \) are calculated at the same time. The bridge components includes only information about the bridge vibration and unsprung-mass characteristics of the vehicle. Others are included in the error term \(\user2{ \in }\left( t \right)\): the vehicle responses: \({\varvec{z}}\left( t \right)\), \(\dot{\user2{z}}\left( t \right)\) and the road roughness: \({\varvec{r}}\left( t \right)\) and \(\dot{\user2{r}}\left( t \right)\). Since \({\mathbf{N}}^{ - 1} \left( t \right)\user2{\ddot{z}}_{u} \left( t \right)\) is time function, it can be described as data matrix \({\mathbf{D}} \in R^{2 \times T}\). \(T\) means the number of the measured data. The SVD of \({\mathbf{D}}\) is described by the product of an orthogonal matrix \({\mathbf{U}} \in R^{2 \times 2}\), a diagonal matrix \({{\varvec{\Sigma}}} \in R^{2 \times 2}\) and an orthogonal matrix \({\mathbf{V}} \in R^{T \times 2}\) (\({\mathbf{V}}^{{\text{T}}} {\mathbf{V}} = {\mathbf{I}}\): the unit matrix) as below:

$$ {\mathbf{D}} = {\mathbf{U\Sigma V}}^{{\text{T}}} $$
(16)

where \({\mathbf{U}}\) is the estimation of \({\mathbf{A}}\), and \({\mathbf{\Sigma V}}^{{\text{T}}}\) is the estimation of \({\varvec{\sigma}}\left( t \right)\) in the form of data matrix. In order for SVD of \({\mathbf{D}}\) to accurately estimate the bridge mode shape \({\mathbf{A}}\), the following conditions need to be satisfied: (a) \({\varvec{\sigma}}\left( t \right)\) is uncorrelated, (b) Error term \(\in \left( t \right)\) is white noise. The bridge vibration components \({\varvec{q}}\left( t \right)\) and \(\dot{\user2{q}}\left( t \right)\) are transient responses induced by the traffic loads, in this case. Thus, it is considered that the real values of \({\varvec{\sigma}}\left( t \right)\) does not satisfy the condition of a). While the SVD process gives the estimated bridge vibration components \({\mathbf{\Sigma V}}^{{\text{T}}}\), they are just uncorrelated signals near \({\varvec{\sigma}}\left( t \right)\). The error due to this affects on the estimated mode shape \({\mathbf{U}}\). This means that the estimation mode shape \({\mathbf{U}}\) and the succeeding index SSMA deviate slightly from the correct mode shape. This effect on SSMA, however, can be expected to be unchanging under the same measurement condition. Generally, a local damage on a bridge never influence the dynamic indices of the global system of the structure. Thus, it is expected that \({\mathbf{A}}\) remains unchanged even after the damage. However, because the local bridge responses are easily affected by the damage, the component \({\varvec{\sigma}}\left( t \right)\) changes. The estimation for it is \({\mathbf{\Sigma V}}^{{\text{T}}}\) and it cannot trace the transition. This error is included in the error of \({\mathbf{U}}\). This is the mechanism of SSMA to react a bridge’s local damage. The error term \(\user2{ \in }\left( t \right) \) does not include the bridge vibrations, but the vehicle vibrations \({\varvec{z}}\left( t \right)\), \(\dot{\user2{z}}\left( t \right)\) and the road profile \({\varvec{r}}\left( t \right)\), \(\dot{\user2{r}}\left( t \right)\). Because the influence of the damage on the bridge vibrations is a kind of pulse, the impact of the damage on the bridge vibration is not transmitted strongly to the vehicle. The error term may affect the result, but the effect is constant and can be ignored. It is noted that the property, which affects SSMA, is the running speed of the vehicle, because the conversion process in Eqs. (4) depends on time space. From the above consideration, although \({\mathbf{U}}\), the estimated bridge mode shape in Eqs. (16), is different from the actual value \({\mathbf{A}}\), it can be used as an evaluation index for the bridge health. It also suggests that SSMA is a possible indicator for bridge screening, because it is more sensitive than the actual mode shape.

3 Methodology of Experiment

3.1 Experiment Details

The experiment time is 8:00 AM to 5:30 PM on May, 2022 except with Sunday and total took 3 weeks. The traffic is not regulated thus the data can include the other vehicle vibration. The weather contains rain and thunder. Temperature range is 8.0 to 27.4 ℃ based on the data of Japan Meteorological Agency.

The experiment vehicle parameter and details are shown Table 1 and Fig. 3. Vehicle weight is 13.8 t including 6.8 t addition weight by steel plate. The edge of crane is directed to back of vehicle. Wheelbase is 4850[mm] and the velocity range is assumed as 20–60[km/h]. Notice the experiment bridge includes shorter than wheelbase (Table 2) 4 point coordinates which is entrance or exit of bridge based on GIS. Notice this 4 points coordinates has error to real position or GPS. GPS device uses mosaic-HAT CLAS and set above the front wheel axle. 3axis accelerometer are set on unsprung front and rear wheel axle (total: 2 sensors). The ADC has 23 bit/G resolution based on previous study [6,7,8]. Sampling rate is 300[Hz].

Fig. 3.
figure 3

The position of sensor on vehicle

Full size image
Table 1. Parameter of vehicle
Full size table
Table 2. The experiment bridges.
Full size table

The experiment is carried out on 108 bridges (Table 2). L is bridge length. The length range are 1.9–330[m]. W is bridge width and Span is the number of span. Type is bridge type of the structure and material. Especially, RCAr: RC Arch, RCRa: RC Ramen, Cal: Calvert. RC bridge is 23 contains 1 arch and 4 ramen, PC bridge is 44, Steel bridge is 24, and Calvert is 16. Notice this categories is not strict. All bridges doesn't have a fatal damage in regular inspection just before, and this study defined as almost healthy. No. 45 is measured in previous experiments with different GPS accuracy [6,7,8]. In this experiment, No. 45 are measured over 100 times. No. 28, 33, 57, 74, 82, 89, 96 has only 1 data because of going through only or measurement failure. The others has 2 or more data. Multi-span bridges has 32. Although the original SSMA assumes to be applied to single span bridge data, this study verifies the data-driven method which calculate their SSMA as a single span.

3.2 Data Verification

The acceleration data and the FFT result on No. 45 is shown Fig. 4. In Fig. 4a, blue line is front, orange is rear. X axis is time[s], Y axis is acceleration [g]. No. 45 has a step by joint at entrance and exit, and the impact peaks appear. The peaks of rear (orange) delay from front. The amplitude of rear around the middle of bridge is larger than front ones. This is assumed to be occurred by front is near center of gravity. In Fig. 4b, also blue line is front, orange is rear. X axis is frequency [Hz], Y axis is power spectrum density [\({\text{m}}^{2} /{\text{s}}^{3}\)]. Y axis neglects under 5 Hz because the effect of electrical trend is considered. The front has peaks around 13 and 25 Hz, and the rear has peaks 8 and 30 Hz. Considering the amplitude of rear over around the middle of bridge is larger, it is expected that rear is affected by bridge vibration, and front is by engine vibration.

Fig. 4.
figure 4

The acceleration of vehicle (a) and the FFT result (b).

Full size image

4 Analysis and Discussion

Previous study suggested the relation of SSMA and bridge length [7]. The length - SSMA for all experiment bridges is shown Figs. 5 and 6. X axis is bridge length[m]. Y axis is SSMA[deg.]. In Fig. 5, the trend between length – SSMA are suggested, especially, under 40 m. Figure 6 shows length – SSMA under 40 m (92 bridges). The bridges under 40 m are almost a single span. Some bridge has a large variance. This reason is suggested that they are affected by the environment of location (near the intersection etc.…) in previous study [7, 8]. Additionally, this study uses 4 point coordinates defined from GIS only which can be wrong, and the effect of error may increase in case of short span bridges. As the other reason, the bridge type is considered. For example, calvert bridges can be considered that calvert bridges may be interacted with the ground around bridge base and their behavior of vibration are different with others.

Fig. 5.
figure 5

The length – SSMA with all length range.

Full size image
Fig. 6.
figure 6

The length – SSMA under 40m of bridge length.

Full size image

The trend curve is appeared to be bi-linear. The area in 0 to around 10 m increase and over 10 m is flat. This result may occurred by the relation of wheelbase and bridge length or calculation length. Because the calculation length is related to bridge length under the sample rate fixed, and the velocity has often a distribution curve such as a normal distribution, SSMA may have a trend corresponded to these conditions. On the other hands, this trend suggests that the screening method by SSMA can detect a damaged bridge without past data. The damage detection method in previous study using natural frequency or mode shape [1,2,3,4] ordinary need to compare the status of past data on health to latest. This proposal method may detect damage by using the distance between SSMA gained on unknown bridge and the trend curve. However, it should be considered that the trend curve can depend on the vehicle parameter, for example, weight.

5 Conclusion and Future Works

This study carried out the experiment to 108 bridges, and their SSMA trend is verified. The findings is as flow:

  1. 1.

    The SSMA shows that the length – SSMA has a trend appeared in previous study.

  2. 2.

    Some large variance of SSMA may be occurred with the location, position error or bridge type.

  3. 3.

    This trend suggests that the screening method by SSMA can detect a damaged bridge without past data because this method can use the distance between SSMA gained on unknown bridge and the trend curve instead of past data.

In future works, the analysis for bridge respectively is verified and it reveals the exclusion criteria such as the location near intersection, position error correction by GPS measurement on actual 4 point coordinates, and the effect from bridge types or vehicle weight. Additionally, this method can use DNN or RNN for improvement of detection accuracy or position error, and it validates how these method can apply.