Performance monitoring of assembled beam bridges using displacement spectrum similarity measure

Abstract

The assembled beam bridge is a widespread construction, with transverse connections uniting precast beams to jointly support loads. These connections, while crucial, are the most vulnerable elements in such structures. Therefore, the development of an effective index for continuous condition assessment remains the cornerstone of structural health monitoring. The Displacement Spectrum Similarity Measure index, as per ideal conditions in numerical simulations, has shown insensitivity to short-term vehicle loads and sensitivity to transverse connection stiffness. This study extends its scope to address potential real-world interference factors, including measurement errors and deck pavement damage. Specifically, we introduce Gaussian white noise to simulate measurement errors and incorporate additional vehicle loads to emulate the impact of pavement damage. Numerical simulations confirm the robustness of the index against these challenges. Additionally, a real-time index extraction scheme is proposed and implemented on an operational bridge. The results reveal a close alignment between the transverse connection condition assessed by the index and the findings obtained through on-site surveys, thereby substantiating the viability and effectiveness of the index for real-world engineering contexts.

Keywords

assembled beam bridge transverse connection hinge joint displacement spectrum correlation coefficient structural health monitoring

Introduction

Since the 1970s, assembled beam bridges have gained global prominence due to their inherent advantages, which include structural simplicity, constructional convenience, high industrialization, and cost-effectiveness (Azizinamini 2020; Zhang and Alam 2020). The assembly process for the superstructure of these bridges typically encompasses several key steps: the on-site erection of precast beams, the casting of crucial transverse connections like hinge joints and diaphragms, and the subsequent addition of deck pavement and accompanying accessories. These separate longitudinal precast beams are interconnected through the transverse connections, enabling them to jointly bear the loads imposed on the bridge. Hence, transverse connections assume a pivotal role in the structural integrity of assembled beam bridges.

Conversely, transverse connections constitute the most vulnerable component within assembled beam bridges. Generally, precast beams fabricated in controlled factory conditions under careful construction and maintenance exhibit a high level of quality. In practical engineering, the vulnerability often resides in these transverse connections. The damage can occur due to various factors. Excessive loads, such as heavy vehicles or trucks, can subject hinge joints to stress levels beyond their design capacity. Repeated stress cycles resulting from traffic loads can cause fatigue damage in hinge joints. Corrosion of hinge joint components, especially in bridges located in regions with severe weather conditions or exposure to corrosive substances like road salt, can weaken the joint and cause damage. As bridges and their components age, they become more susceptible to wear and damage. In earthquake-prone regions, seismic events can subject bridge structures, including hinge joints, to significant stress and movement, potentially resulting in damage. Transverse connection damage disrupts the efficient lateral load transmission, compelling certain beams to bear excessive loads, thereby leading to greater deflection compared to their neighboring counterparts. Consequently, this exacerbates damage and triggers a vicious cycle.

Several studies have evaluated the influence of transverse connection damage on bridge responses and lateral load distributions through either real bridge assessments (Al-Saidy et al., 2008; Russo et al., 2000; Stallings et al., 1999) or finite element simulations (Azimi and Sennah 2013; Kim et al., 2008). This evaluation involve a comparison of results obtained before and after the occurrence of damage. Zhang et al. (2022b) determines the orders of modes sensitive to hinge damage in assembled hollow slab bridges, which is instructive for vibration-based hinge damage detection. Various factors contributing to potential damage were investigated, including fatigue damage (Li 2015) and temperature and humidity (Hussein et al., 2017). Scholars have sought to address this issue by enhancing design aspects, encompassing innovations in hinge joint structures (Alampalli 1998), transverse pre-stressing methods (Porter et al., 2012), and the utilization of ultra-high-performance concrete (Hussein et al., 2019; Huang et al., 2019; Jiang et al., 2020; Semendary 2018), among others.

Another facet of research pertains to damage detection and condition assessment during operational phases. Visual inspections were conducted by bridge engineers in the early stage, while Hu et al. (2023) introduced a computer vision-based approach for detecting hinge joint damage in hollow slab bridges. Wu et al. (2023) proposed an innovative approach combining ultrasonic guided wave-based features with an inverse Bayesian scheme for structural joint damage detection. Load tests are a common method for manual detection. Abedin et al. (2022) conducted a series of static and dynamic load tests on a precast-prestressed box-beam bridge and compared the responses to those obtained through finite element model updating. Additionally, Zhang et al. (2022a) introduced a stationary excitation extraction method using mode shapes derived from moving vehicle responses to detect hinge joint damage. This approach involves a stationary excitation vehicle equipped with a shaker to stimulate the bridge, while another vehicle moves along the prescribed path, collecting acceleration data to extract mode shapes. Furthermore, various non-destructive testing devices and techniques were reviewed by Farhangdoust and Mehrabi (2019).

Manual detection methods have been extensively employed over the past decades and constitute a well-established and mature subject. However, these methods are not without their limitations. Firstly, manual detection is conducted periodically, which means it may not capture damage that arises between inspections, particularly sudden damage caused by over-weighted loads, earthquakes (Yazdanpanah et al., 2022; Zhu et al., 2002), etc. Secondly, the assessment relies on the inspector’s experience and expertise, potentially introducing subjectivity and variability into the evaluation. Moreover, certain bridge components can be challenging to inspect due to factors like height, location, or structural complexity. Finally, considering the vast number of in-service bridges, manual detection is both time-consuming and labor-intensive.

Structural health monitoring (SHM) stands as a promising solution to address the aforementioned limitations. Upon the initial installation of an SHM system, it consistently tracks structural responses and environmental conditions, generating quantified indices that offer an accurate representation of the structural state. Wen et al. (2017) introduced an index, the correlation coefficient of dynamic strain between adjacent precast beams, for continuous monitoring and assessment of transverse connection condition. The method is initially validated through theoretical analysis and numerical simulations. Further validation came from Dan et al. (2019b) by utilizing real-bridge monitoring data. The damage diagnosis by the index closely aligned with on-site visual inspections. A disadvantage of the method is the need to eliminate temperature-induced strain and vehicle-induced quasi-static strain from strain measurements, resulting in computational complexity. Moreover, the index computed using time-domain strain exhibited slow convergence and poor real-time performance. Dan et al. (2019a) and Xu et al. (2021) proposed a method based on transverse displacement and strain modal shapes for transverse connection damage detection. These modal shapes were expressed as a product of longitudinal base functions and transverse proportional coefficient vectors, obtained through the resolution of a matrix eigenvalue problem. It was discussed that displacement information proved to be more informative and robust than strain data. Nevertheless, their method remained within the realm of short-term small data analysis, prompting the suggestion that long-term analysis could offer a more effective data utilization approach.

In summarizing the insights from these studies, Dan et al. (2023) introduced a Displacement Spectrum Similarity Measure (DSSM) index. DSSM is defined as the correlation coefficient of displacement spectra between adjacent precast beams. This index was put forth as an improvement over the one proposed by Wen et al. (2017), primarily due to its superior convergence and real-time performance. It was argued that frequency-domain displacement spectra, as utilized in DSSM, prove more resilient to vehicle loads and measurement errors compared to time-domain strain data. Additionally, DSSM offers the advantage of online computation over the long term, enabling the evaluation of transverse connection condition by tracking the evolving trend of DSSM in the background of big monitoring data. A limitation of this study lies in its utilization of an ideal numerical simulation setting for method verification.

This paper extends the application of DSSM to practical SHM scenarios, where ambient interferences including sensing errors and deck pavement damage are prevalent. The resilience of DSSM against these real-world factors is demonstrated through numerical simulations. Furthermore, an index extraction approach tailored for practical SHM applications is introduced. To validate the feasibility and effectiveness of this method, it is implemented on an actual bridge. The resulting damage diagnosis derived from the index closely aligns with the in-situ visual inspections, providing a strong validation of its applicability and accuracy.

DSSM in real monitoring environments

In the prior research conducted by Dan et al. (2023), DSSM was introduced as an index to assess the condition of transverse connections in assembled beam bridges, and its performance was validated under ideal conditions. To further assess its applicability in real-world engineering settings, it is necessary to account for environmental variables and redefine this index.

Definition of DSSM

Figure 1 presents a simplified mechanical model of a simply supported assembled beam bridge. Dan et al. (2019a), Dan et al. (2023) and Wen et al. (2017) suggest that it is reasonable to model the behavior of transverse connections using vertical springs. The underlying simplification is that these connections primarily transmit shear forces among precast beams, while the moment transmission is considered secondary and thus omitted. The distributed mass and spring stiffness are concentrated at finite nodes, each featuring two degrees of freedom: vertical displacement and rotation. It is noted that the torsion of precast beams, which has a secondary effect on deflections, is omitted in this representation.

Figure 1.

Mechanical model of simply supported assembled beam bridges.

The equation of motion for the bridge is represented as follows:

M \ddot{x} (t) + C \dot{x} (t) + K x (t) = p (t)

(1)

where M, C, and K denote the structural mass, damping and stiffness matrices, respectively. x (t) and p (t) are nodal displacement and load, with the dot notation indicating a derivative with respect to time t. M and K are assembled by the finite element method using elemental mass and stiffness matrices, respectively. M is a diagonal matrix with entries representing mass and moment of inertia. Rayleigh damping is applied to determine C. The elemental stiffness matrix for an inner beam element consists of two parts:

K_{b}^{e} = [\begin{array}{l} \frac{12 E I}{l^{3}} & \frac{6 E I}{l^{2}} & - \frac{12 E I}{l^{3}} & \frac{6 E I}{l^{2}} \\ \frac{6 E I}{l^{2}} & \frac{4 E I}{l} & - \frac{6 E I}{l^{2}} & \frac{2 E I}{l} \\ - \frac{12 E I}{l^{3}} & - \frac{6 E I}{l^{2}} & \frac{12 E I}{l^{3}} & - \frac{6 E I}{l^{2}} \\ \frac{6 E I}{l^{2}} & \frac{2 E I}{l} & - \frac{6 E I}{l^{2}} & \frac{4 E I}{l} \end{array}]

(2)

K_{s}^{e} = [\begin{array}{l} k_{s} & 0 & - k_{s + 1} & 0 \\ 0 & 0 & 0 & 0 \\ - k_{s + 1} & 0 & k_{s} & 0 \\ 0 & 0 & 0 & 0 \end{array}]

(3)

where

K_{b}^{e}

and

K_{s}^{e}

represent the stiffness contributed by the beam body and spring, respectively. The parameters EI and l denote the bending stiffness and length of the beam element, respectively. k_s indicates the stiffness of the spring between the s^th and (s + 1)^th beams. Minor adjustments are required for boundary beam elements.

Divide the long-term response x (t) and excitation p (t) into segments, denoting the segments at the i^th time step as x ⁱ(t) and p ⁱ(t), respectively. Substitute x ⁱ(t) and p ⁱ(t) into equation (1) and perform a Fourier transform on both sides, yielding:

X^{i} (ω) = H (ω) P^{i} (ω)

(4)

H (ω) = {(K + j ω C - M)}^{- 1}

(5)

where Xⁱ(ω) and Pⁱ(ω) represent the Fourier spectra of x ⁱ(t) and p ⁱ(t), respectively. H(ω) is the transfer function of the system.

In this study, special attention is given to the mid-span points since, in practical applications, sensors are typically installed at these locations. To assess the transverse connection condition, the correlation coefficient is utilized to measure the similarity between the displacement spectra of the mid-span points of the s^th and (s + 1)^th precast beams. This is expressed as:

ρ_{s, s + 1}^{i} ≜ \frac{C o v (X_{s}^{i} (ω), X_{s + 1}^{i} (ω))}{\sqrt{D [X_{s}^{i} (ω)] \cdot D [X_{s + 1}^{i} (ω)]}}

(6)

where

X_{s}^{i} (ω)

represents the displacement spectrum of the mid-span node in the s^th precast beam at the i^th time step.

It is worth noting that the displacement response x ⁱ(t), induced by random traffic flow, is inherently a random process, leading to the random displacement spectrum $X_{s}^{i} (ω)$ . As a result, the correlation coefficient $ρ_{s, s + 1}^{i}$ is a random variable and is significantly influenced by the random traffic loads. However, this inherent randomness is not desirable for a structural condition index, which should ideally exclusively represent the structural condition and remain insensitive to external factors. To eliminate the randomness, the DSSM index is defined as the mathematical expectation of $ρ_{s, s + 1}^{i}$ :

M_{s, s + 1}^{i} ≜ E [ρ_{s, s + 1}^{i}]

(7)

where

M_{s, s + 1}^{i}

signifies the DSSM index, which reflects the condition of the transverse connection between the s^th and (s + 1)^th precast beams. In actual applications, DSSM can be estimated over N time steps as follows:

M_{s, s + 1}^{i} \approx \frac{1}{N} \sum_{k = i - N + 1}^{i} ρ_{s, s + 1}^{k}

(8)

It is important to ensure that N is sufficiently large, allowing $M_{s, s + 1}^{i}$ to converge and becoming less influenced by random traffic loads. On the other hand, N should not be excessively large so that $M_{s, s + 1}^{i}$ can reflect the current condition.

DSSM considering ambient factors

DSSM is computed based on displacement measurements, making it susceptible to the errors introduced by sensors. Additionally, during long-term SHM, deck pavement is another vulnerable part. Pavement damage can alter the vehicle loads on the bridges, thereby affecting the displacement responses. This section discusses the effects of these two factors on DSSM.

Measured displacement responses can be expressed as:

x^{i *} (t) = x^{i} (t) + e^{i} (t)

(9)

where e ⁱ(t) denotes the errors introduced by sensors. Applying a finite Fourier transform to equation (9) results in:

X^{i *} (ω) = X^{i} (ω) + E^{i} (ω)

(10)

where Xⁱ*(ω) and Eⁱ(ω) are the Fourier spectra of x ⁱ*(t) and e ⁱ(t), respectively. Substituting equation (4) into equation (10), we obtain:

\begin{array}{l} X^{i *} (ω) = H (ω) P^{i} (ω) + E^{i} (ω) \\ = H (ω) P_{s}^{i} (ω) + [H (ω) P_{d}^{i} (ω) + E^{i} (ω)] \end{array}

(11)

where

P_{s}^{i} (ω)

represents the load spectrum induced by vehicle weights,

P_{d}^{i} (ω)

denotes the additional load spectrum caused by extra vehicle loads due to pavement roughness. Xⁱ*(ω) is then substituted into Equations (6) and (7) replacing Xⁱ(ω) for computing the DSSM index.

Figure 2, based on the studies conducted by Kullaa (2013) and Yi et al. (2017), provides a visual representation of four typical sensor errors. Sensor bias occurs when a sensor consistently produces outputs that differ from the normal values by a constant. Sensor drift, on the other hand, is characterized by linear changes in the differences between sensor outputs and normal values over time. Bias and drift can be effectively addressed by applying linear regression with the least squares method during data pre-processing. Gain arises when the normal sensor values are multiplied by a constant factor. It is essential to note that gain does not impact the correlation coefficient, and thus, DSSM is inherently robust to this type of error. These systematic errors, encompassing bias, drift, and gain, are consistent and predictable, making them suitable for modeling and correction. In this context, we focus on the impact of sensor degradation, which falls under the category of measurement errors.

Figure 2.

The typical types of sensor errors.

In contrast to systematic errors, measurement errors are random and unpredictable. These errors result from various factors such as electrical noise, environmental fluctuations, and inherent limitations of the measuring equipment. Measurement errors follow a probability distribution and are often approximated as a Gaussian distribution. They can be characterized by statistical properties such as mean and standard deviation. In SHM applications, measurement errors are typically assumed to be independent Gaussian white noise (Bi et al., 2019; Fan et al., 2020). Therefore, Gaussian white noise is used to model measurement errors in this analysis.

To account for measurement errors, we introduce a modified correlation coefficient, expressed as:

ρ_{s, s + 1}^{i *} ≜ \frac{C o v (X_{s}^{i *} (ω), X_{s + 1}^{i *} (ω))}{\sqrt{D [X_{s}^{i *} (ω)] \cdot D [X_{s + 1}^{i *} (ω)]}}

(12)

where

X_{s}^{i *} (ω)

represents the measured displacement spectrum of the mid-span node in the s^th precast beam at the i^th time step. The covariance is calculated as:

\begin{array}{l} C o v (X_{s}^{i *} (ω), X_{s + 1}^{i *} (ω)) \\ = C o v (X_{s}^{i} (ω) + E_{s}^{i} (ω), X_{s + 1}^{i} (ω) + E_{s + 1}^{i} (ω)) \\ = E [X_{s}^{i} (ω) X_{s + 1}^{i} (ω)] - E [X_{s}^{i} (ω)] E [X_{s + 1}^{i} (ω)] \\ = C o v (X_{s}^{i} (ω), X_{s + 1}^{i} (ω)) \end{array}

(13)

given that

E [E_{s}^{i} (ω) E_{s + 1}^{i} (ω)] = E [E_{s}^{i} (ω)] E [E_{s + 1}^{i} (ω)]

(14)

E [E_{s}^{i} (ω) X_{s + 1}^{i} (ω)] = E [E_{s}^{i} (ω)] E [X_{s + 1}^{i} (ω)]

(15)

E [X_{s}^{i} (ω) E_{s + 1}^{i} (ω)] = E [X_{s}^{i} (ω)] E [E_{s + 1}^{i} (ω)]

(16)

where

E_{s}^{i} (ω)

denotes the spectrum of the measurement errors contained in

X_{s}^{i *} (ω)

. Concerning the variance,

\begin{array}{l} D [X_{s}^{i *} (ω)] \\ = D [X_{s}^{i} (ω) + E_{s}^{i} (ω)] \\ = D [X_{s}^{i} (ω)] + D [E_{s}^{i} (ω)] \\ > D [X_{s}^{i} (ω)] \end{array}

(17)

Consequently, it can be deduced that $ρ_{s, s + 1}^{i *} < ρ_{s, s + 1}^{i}$ , resulting in a reduced DSSM when the measured displacement is contaminated by measurement errors.

Effects of measurement errors and pavement damage

Due to mathematical complexity, the explicit relationship between pavement damage and DSSM is not provided. Additionally, further investigation is required to the extent of the impact of measurement errors on DSSM. This section addresses these issues through numerical simulations.

Numerical simulation setting

Bridge model

The numerical simulations are based on the bridge model depicted in Figure 1. The bridge consists of five identical T-shaped precast beams with a uniform cross-section along the longitudinal direction. The geometric dimensions and material properties of these precast beams are detailed in Table 1. Each precast beam is divided into four equidistant elements, connected by five nodes. The stiffness of each spring is set to 2000 kN/m. A Rayleigh damping matrix is employed, with the first and third order modal damping ratios set to 5%.

Table 1.

Geometric dimensions and material properties of precast beams.

Beam attribute	Value
Length	30 m
Width	2.1 m
Cross-sectional area	0.9220 m²
Moment of inertia	0.4361 m⁴
Elastic modulus	28 GPa
Density	2500 kg/m³

Random traffic flow model

The random traffic flow model developed by Dan et al. (2023) is employed herein, where the following simplifications are made considering that DSSM is a mean over an adequately long period: (a) traffic loads can be approximated as a stationary random process; (b) a vehicle is represented as a concentrated force moving uniformly on the bridge; (c) a vehicle is characterized by four parameters: generation probability, position, speed, and weight.

Specifically, it is assumed that vehicle generation follows a Poisson distribution with a rate parameter of 0.5, indicating that a vehicle enters the bridge approximately every 2 s. Vehicle lateral positions within a lane are assumed to follow a normal distribution with a standard deviation of 0.3 m along the center line of the lane. Vehicle speeds also follow a normal distribution with a mean of 50 km/h and a standard deviation of 3 km/h.

Random vehicle weight is assumed to follow a three-peak distribution, with small and medium vehicle weights following normal distributions, and the weights of large vehicles following an extreme minimum value distribution. The probability density function is given by

\begin{array}{l} f (x) & = p_{1} \frac{1}{\sqrt{2 π} σ_{1}} e^{- \frac{{(x - μ_{1})}^{2}}{2 σ_{1}}} \\ + p_{2} \frac{1}{\sqrt{2 π} σ_{2}} e^{- \frac{{(x - μ_{2})}^{2}}{2 σ_{2}}} \\ + p_{3} \frac{1}{β} e^{\frac{x - α}{β}} e^{- e^{\frac{x - α}{β}}} \end{array}

(18)

where p₁ = 0.6, p₂ = 0.3, and p₃ = 0.1 represent the proportions of small, medium, and large vehicles, respectively. μ₁ = 1.5 t and μ₂ = 30t represent the average weights of small and medium vehicles, respectively. σ₁ = 0.5 t and σ₂ = 5t represent the standard deviations of the weights of small and medium vehicles, respectively. α = 80t and β = 10t represent the location parameter and scale parameter of the weights of large vehicles.

Effects of measurement errors

Measurement errors are an inherent part of monitoring processes, and their impact on the DSSM index is mathematically established. In this section, we explore the extent of these effects through numerical simulations.

Figure 3 illustrates the convergence process of the DSSM index, which is calculated as:

M_{s, s + 1}^{i} = \frac{1}{i} \sum_{k = 1}^{i} ρ_{s, s + 1}^{k}

(19)

Figure 3.

Convergence process of the Displacement Spectrum Similarity Measure (DSSM) index.

As observed, the indices exhibit significant fluctuations initially but stabilize around the 16^th hour. The convergence values display some symmetry ( $M_{1,2} \approx M_{4,5}$ ; $M_{2,3} \approx M_{3,4}$ ), attributable to the lateral symmetry of the bridge model.

The Signal-to-Noise Ratio (SNR) is typically employed as a measure to quantify the level of a desired signal relative to the background noise, which is calculated as:

SNR (dB) = 10 \cdot \log_{10} (\frac{P_{s}}{P_{n}})

(20)

where P_s and P_n denote the power of signal and noise, respectively. For SHM applications, sensors commonly operate within a SNR range of 40 to 60 dB. To account for potential sensor degradation, Gaussian white noise with SNRs ranging from 60 to 10 dB is introduced into the simulated displacement signals. A scenario where sensors are completely dysfunctional is also considered, resulting in pure Gaussian white noise (denoted as SNR = 0). In contrast, no additive noise corresponds to an infinitely large SNR. The resulting DSSM indices are listed in Table 2. It is evident that DSSM demonstrates robustness against measurement errors when the SNR is higher than 30 dB. For example, when the SNR decreases from +∞ to 30 dB,

M_{1,2}

only decreases from 0.8852 to 0.8835, a mere 0.0017 reduction, which is almost negligible.

Table 2.

The DSSM indices with different SNRs.

SNR (dB)	$M_{1,2}$	$M_{2,3}$	$M_{3,4}$	$M_{4,5}$
+∞	0.8852	0.8811	0.8808	0.8864
60	0.8851	0.8811	0.8808	0.8864
50	0.8851	0.8811	0.8808	0.8863
40	0.8849	0.8809	0.8806	0.8861
30	0.8835	0.8797	0.8794	0.8847
20	0.8742	0.8726	0.8724	0.8755
10	0.8170	0.8220	0.8222	0.8180
0	0.0000	−0.0001	0.0002	−0.0004

Figure 4 plots a segment of displacement responses alongside their noisy counterparts. At an SNR of 20 dB (indicating poor sensor conditions), the significant vibrations induced by large vehicles are measured relatively accurately, while the moderate and small vibrations experience substantial noise contamination. In this scenario, $M_{1,2}$ decreases to 0.8742, a reduction of 0.0110 compared to the $M_{1,2}$ at SNR = +∞. With an SNR of 10 dB (nearly sensor failure), all measurements are severely contaminated, causing a dramatic drop in $M_{1,2}$ to 0.8170, which is deemed an unreliable assessment. Finally, when the signals consist solely of noise, all DSSM indices converge to near 0. This is because measurement errors from different sensors are independent of each other. Actually, it is impossible for any method to evaluate bridge conditions under such circumstances due to the absence of information on the bridge. In practical settings, sensors typically operate with SNRs greater than 30 dB, where measurement errors exert minimal influence on the DSSM index.

Figure 4.

Comparison between noisy signals and normal signals.

As a comparison, a previous study conducted by Dan et al. (2023) revealed the relationship between transverse connection stiffness and DSSM at an SNR of +∞, as illustrated in Figure 5. When the stiffness decreases from 24,000 kN/m to 0, $M_{1,2}$ diminishes from 0.9712 to 0.8647, marking a reduction of 0.1065. The decrease in transverse connection stiffness weakens the ability of precast beams to collectively share loads, diminishes the correlation of their vibrations, and ultimately leads to a reduction in DSSM indices. The noise-induced reduction at an SNR of 20 dB is nearly 10% of the reduction caused by stiffness variations. This impact is not negligible. Consequently, for a precise assessment, it is crucial that sensors operate with an SNR higher than 30 dB.

Figure 5.

Relation between transverse connection stiffness and DSSM.

Effects of pavement damage

In addition to transverse connections, the deck pavement is another vulnerable component of bridges. Pavement damage alters the vehicle loads acting on the bridge, thereby influencing structural vibration responses. However, it remains to be studied whether this alteration ultimately impacts the DSSM index. The international roughness index (IRI) is commonly used to characterize the roughness of pavement surfacing. An IRI value of 0 indicates an absolutely smooth surface, while IRI = 10 represents a highly rough surface. In this section, the IRI is utilized to signify the degree of pavement damage, and the relationship between DSSM and IRI is explored through numerical simulations.

According to the research by Han et al. (2019), deck profile can be expressed as the following harmonic superposition:

x_{g} = \sum_{k = 1}^{M} \sqrt{3.26 K_{0} Δ n} (\frac{n_{0}}{n_{k}}) \cdot IRI \cdot \cos (2 π v n_{k} t + ϕ_{k})

(21)

where M = 400, K₀ = 10⁻⁶ m³, Δn = (n_u − n_l)/M, n_u = 2, n_l = 0.1, n₀ = 0.1, n_k = n_l + (k − 0.5)Δn, v denotes vehicle speed, t denotes moving time of a vehicle, ϕ_k ∼U(0, 2π) denotes a random phase angle. A typical 2-DoF quarter vehicle model is adopted, as illustrated in Figure 6. The equation of motion is given by:

m_{s} {\ddot{x}}_{s} = - k_{s} (x_{s} - x_{t}) - c_{s} ({\dot{x}}_{s} - {\dot{x}}_{t})

(22)

m_{t} {\ddot{x}}_{t} = k_{s} (x_{s} - x_{t}) + c_{s} ({\dot{x}}_{s} - {\dot{x}}_{t}) - k_{t} (x_{t} - x_{g}) - c_{t} ({\dot{x}}_{s} - {\dot{x}}_{g})

(23)

where m, c, k, and x represent mass, damping, stiffness, and displacement, respectively. x _s and x _t can be obtained by solving Equations (21)–(23). Finally, the additional force f _d due to pavement roughness and total force f vehicles acting on bridges are yielded as:

f_{d} = k_{t} (x_{t} - x_{g}) + c_{t} ({\dot{x}}_{t} - {\dot{x}}_{g})

(24)

f = f_{d} + (m_{s} + m_{t}) g

(25)

Figure 6.

2-DoF quarter vehicle model.

In the numerical simulations, f is exerted to the bridge in lieu of the vehicle weight. The relation between IRI and DSSM is illustrated in Figure 7. As observed, DSSM exhibits minimal variation, underscoring its insensitivity to pavement damage.

Figure 7.

Relationship between IRI and DSSM.

The underlying reason is that, although pavement roughness alters the historical details of vehicle loads, the mean values of these loads remain unchanged. Through averaging calculations over a sufficiently long period, the DSSM index is primarily associated with the mean values of traffic loads, disregarding the intricate details of vehicle behavior. This characteristic was also demonstrated by Dan et al. (2023). Figure 8 illustrates the additional force f _d of a vehicle parameterized as per Table 3 driving on pavement with IRI = 10. The mean of the additional force is −0.0837 kN. Ten thousand repetitive simulations are conducted, with the histogram of the means of f _d depicted in Figure 9. It is evident that its shape closely conforms to a normal distribution with a mean of 0, indicating that the mean of additional vehicle loads caused by pavement roughness is zero. In other word, pavement damage does not alter the mean values of traffic loads, thus having little influence on the DSSM index.

Figure 8.

Additional force f _d with IRI = 10.

Table 3.

Parameters of vehicle.

Vehicle attribute	Value
k _s	480 kN/m
c _s	14 kN⋅s/m
m _s	9000 kg
k _t	1900 kN/m
c _t	3 kN⋅s/m
m _t	1000 kg
v	20 m

Figure 9.

Histogram of means of f _d.

In summary, DSSM demonstrates robustness against measurement errors when the SNR is above 30 dB, a typical operational range for normal sensors. Furthermore, pavement damage exhibits no discernible effects on the DSSM index, indicating that DSSM is insensitive to these two ambient factors. In alignment with the findings by Dan et al. (2023), DSSM proves sensitive to transverse connection stiffness while remaining insensitive to random traffic loads. These characteristics establish DSSM as an effective monitoring index tailored to characterize the condition of transverse connections.

Index monitoring scheme in practical engineering environments

The deflection of precast beams needs to be measured first to calculate the DSSM index. At present, there are a variety of deflection measurement methods for long-term SHM, including displacement sensors, communicating pipes, inclinometers, automatic total stations, GPS, accelorometers, computer vision and photogrammetry based methods, etc. The advantages and disadvantages are compared in Table 4. Appropriate methods can be selected according to the practical situations.

Table 4.

Commonly used bridge deflection measurement methods.

Method	Advantages	Disadvantages
Displacement sensors	High accuracy	Fixed reference points required
Communicating pipes	High accuracy	Limited measurement range
	Simple principle
	Distributed sensing
Inclinometers	Low cost	Limited measurement range
	Distributed sensing	Low accuracy for small deflection
Automatic total stations	High accuracy	High cost
		Fixed reference points required
GPS	Large measurement range	High cost
	Non-contact measurement	Low accuracy
		Low sampling frequency
Accelorometers	Wide frequency response	Sensitive to low-frequency noise
		Low accuracy for static deflection
Computer vision	Large measurement range	Fixed reference points required
	Non-contact measurement	Sensitive to lighting conditions

Data loss is an inevitable aspect of practical engineering scenarios. The DSSM index demonstrates robustness against occasional data loss, as it is defined as the mean of numerous correlation coefficients. Errors in a few of these coefficients do not significantly impact the overall index. Consequently, no specific measures are taken to address this situation. In cases where the loss of small data packets is common, particularly in wireless sensor applications, interpolation methods might aid in restoring lost data. However, such scenarios are also not treated particularly herein. This is because that if the pattern of data packet loss remains consistent, variation in the index is still caused by changes in transverse connection stiffness, and thus can represent transverse connection condition.

In instances of continuous data loss over a substantial period, such as due to sensor damage or power outages, methods exist to reconstruct lost data using information from other sensors. However, this approach may introduce an issue. The reconstruction process relies on correlations between data from different channels. Consequently, the reconstructed data strongly correlates with other data. Since the index in this paper also utilizes data correlations to reflect transverse connection stiffness, calculating the index using reconstructed data may result in an overestimation of the index values. This potential implication requires further investigation. In this study, we choose to simply discard the lost data along with the data from other sensors measured at the same time.

Furthermore, in the absence of any vehicles on the bridge, the monitoring data consist solely of noise, offering no insight into the condition of the transverse connections and, therefore, should be disregarded. To address this, a predetermined threshold needs to be established. Choose a time interval in the monitoring data when no vehicles are present and set the maximum value within this interval as the threshold. If the majority of data points at a particular time step fall below the threshold, it is deemed that no vehicles are on the bridge during that time step, and consequently, such data should be excluded.

Following the aforementioned pre-processing steps, displacement spectra are derived from time-domain displacement responses using Fast Fourier Transform (FFT) to compute correlation coefficients between adjacent beams. To mitigate the impact of early monitoring data on the current state assessment and enhance sensitivity to condition changes, a first-in-first-out pattern is employed to compute the DSSM index, as formulated by equation (8). The algorithmic flow for index identification is outlined as follows:

(1) Set the sampling time and collect displacement signals.

(2) Pre-processing: Detect data loss, synchronize data timestamps, identify the presence of vehicles, and select proper data.

(3) Perform FFT to obtain displacement spectra.

(4) Calculate correlation coefficients of displacement spectra.

(5) Calculate DSSM using equation (8).

Real bridge application

Tongji Road Viaduct and the SHM system

Tongji Road Viaduct, as shown in Figures 10 and 11, is situated in Shanghai, China, comprising of 108 spans with a total length of 2296.64 m. Functioning as a crucial road to a container port, it has been in service since 2002, enduring heavy traffic, consisting of numerous container trucks. Noteworthy damage on the bridge includes longitudinal cracks on deck pavement, water leakage through hinge joints, among other issues.

Figure 10.

Geographical position of Tongji Road Viaduct.

Figure 11.

Photos of Tongji road Viaduct.

Following an on-site survey, three adjacent spans exhibiting noticeable diseases, namely K92, K93, and K94, were selected for the installation of the SHM system. The cross section of the bridge and the sensor layout are illustrated in Figure 12. The bridge is assembled with 17 precast beams connected by 16 hinge joints. The hinge joint connecting the s^th and (s + 1)^th beams is numbered s. A Fiber Bragg Grating (FBG) strain sensor and an FBG accelerometer are deployed at the mid-span of each precast beam. Additionally, two FBG temperature sensors are positioned at the two boundary beams. In this specific case study, the displacement is obtained by double integrating acceleration using the algorithm proposed by Zheng et al. (2019). The acceleration is sampled at a rate of 50 Hz.

Figure 12.

Tongji Road Viaduct monitoring system.

Evaluation of transverse connection condition

The acceleration data from the 12^th precast beam of K92 over a continuous 48-hour period is presented in Figure 13. Notably, during peak hours (7:00 to 9:00 and 16:30 to 18:30), the amplitude of the acceleration responses is significantly smaller. This is attributed to the prevalence of small vehicles in the traffic flow during these hours. During the remaining time, traffic is still considerably busy, with a substantial presence of large vehicles. To mitigate the impact of the non-stationarity of traffic flow on the DSSM index, the data collected during peak hours are excluded from analysis.

Figure 13.

Acceleration for continuous 48 h from the 12^th precast beam of K92.

In this paper, K92 is selected as the object of study, and the index is computed every 2 minutes. The two-minute acceleration data is first double-integrated to obtain displacement. Subsequently, the displacement is utilized for DSSM identification, following the procedure outlined in the previous section. The results, considering 4 days of monitoring data, are depicted in Figure 14. As observed in the figure, the DSSM indices converge within 4 days after the initial oscillations. The ultimate convergence values are presented in Figure 15, indicating that Hinge joints 3, 7, and 8 exhibit significant damage, while Hinge joints 5, 9, 11, 12, and 13 exhibit minor damage.

Figure 14.

Convergence process of DSSM for Tongji Road Viaduct.

Figure 15.

Displacement spectrum similarity measure values for Tongji Road Viaduct.

Figure 16 displays on-site photos of the underside of K92, revealing water leakages in Hinge joints 3, 5, 7, 8, 9, 11, 12, 13, and 14. The on-site survey results align well with the assessment provided by the DSSM indices, except for Hinge joint 14, indicating the potential effectiveness of the proposed method. Concerning the disparity in Hinge joint 14, a possible explanation is that the location of the damage in the hinge joint is distant from the measurement points, diminishing the sensitivity of DSSM to the damage.

Figure 16.

On-site photo of K92 in Tongji Road Viaduct.

Furthermore, the damaged hinge joints identified by the MAD method proposed by Xu et al. (2021) based on transverse modes are listed in Table 5. The damages in Hinge joints 3, 5, 7, 11, 12, and 13 are identified by both methods. The damage in Hinge joint 14 is identified by the MAD method, while the damages in Hinge joints 8 and 9 are not identified. The assessment results by both methods exhibit major similarities, further demonstrating the potential effectiveness of these methods. On the other hand, the differences in these results reflect that both methods have their respective areas for improvement. The combination of both methods provides a more comprehensive diagnostic result.

Table 5.

Damaged hinge joints identified by Xu et al. (2021).

Mode	Hinge joint number
3	3, 4, 11, 13
4	3, 7, 11, 14
5	3, 12, 14
6	3, 5, 7, 11
7	5, 13
8	3, 5, 7, 10, 11, 12, 14

In comparison, an advantage of the DSSM index over the MAD method is the capability of quantifying damage severity based on the DSSM values. Another advantage is that the DSSM index is more applicable for long-term continuous online monitoring with less manual intervention.

Conclusion and discussion

The DSSM index is introduced in this paper to address the challenge of assessing and monitoring transverse connections in assembled beam bridges. Additionally, considering practical monitoring conditions, the impact of measurement errors and pavement damage on DSSM is investigated through numerical simulations. Finally, a scheme for identifying DSSM is proposed and applied to a real bridge. The following conclusions are drawn:

(1) DSSM demonstrates insensitivity to measurement errors when the SNR is above 30 dB, which is within the normal operating range for sensors. Moreover, there is no noticeable effect of pavement damage on DSSM. These findings highlight the robustness of DSSM in real-world environments.

(2) DSSM serves as a monitoring index that can be identified in a long-term continuous online pattern with minimal manual intervention.

(3) Under busy and relatively stationary traffic flow, DSSM converges within 2 to 3 days after appropriate data pre-processing. This short convergence time, compared to the structure degradation period, indicates excellent real-time performance.

(4) The majority of damages in the target bridge can be identified, showcasing the effectiveness of DSSM. However, damage in Hinge joint 14 remains undetected. Combining DSSM with the MAD method can provide a more comprehensive diagnosis result.

Despite of the achievements, there are some limitations to DSSM. The convergence performance relies on the stationarity of traffic flow, requiring careful consideration of monitoring time. Additionally, damage in Hinge joint 14 is not detected possibly due to its distance from the measurement points. This issue could be addressed by deploying more sensors, albeit at an increased cost.

Notably, DSSM, being a non-parametric index derived from sensor signals without physical modeling, hold the potential for generalization to monitor connections of other types of structures, such as segmental girder bridges or continuous beam bridge constructed with initial simple supports followed by continuous construction. Exploring these applications could be a direction for future research.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is funded by the Opening Project of National Key Laboratory for Bridge Structural Health and Safety (BHSKL18-05-GF), National Natural Science Foundation of China (51878490); the National Key R & D Program of China (2017YFF0205605); Shanghai Urban Construction Design Research Institute Project Bridge Safe Operation Big Data Acquisition Technology and Structure Monitoring System Research.

ORCID iDs

Wenhao Zheng

Danhui Dan

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