Identification of the scour depth of continuous girder bridges based on model updating and improved genetic algorithm

Abstract

Scour is one of the primary reasons for the collapse of bridges, as it severely reduces the stability of their piers. This study developed a model updating method based on an improved genetic algorithm (GA) to identify scour depth, combined with a parallel computing technique to accelerate the model updating procedure. The proposed GA adopted a gradient-like calculation as the mutation operation, while the population in GA was initialized based on the exponential distribution. A finite element model of a continuous girder bridge, whose soil-structure interaction was simulated by winkle beam theory, was established to give a sensitivity analysis of the scour effect. A moving mass-spring model was used to simulate the vehicle-bridge interaction, and dynamic responses under various scour damages were generated to verify the method’s performance. The results of the numerical simulation indicated that the transverse vibration modes of a continuous girder bridge were sensitive to the scour effect, and the proposed method can identify the scour depth with an error of less than 0.31 m.

Keywords

Continuous girder bridge scour depth identification model updating genetic algorithm modal analysis

Introduction

Bridge safety has attracted significant attention since bridges are crucial infrastructures in transportation essential to the national economy. Many bridges suffer extreme natural or human-made hazards in their life cycle, such as floods, scours, earthquakes, hurricanes, and vehicle collisions (Akiyama et al., 2020; Do et al., 2019; Yanweerasak et al., 2018; Yilmaz et al., 2018). Scour is the major cause of bridge failures among these hazards, as this hydraulic action removes soil from their foundations, resulting in deterioration of bridge integrity. Severe scour damage can significantly decrease bridge piers’ stability and lead to the entire bridge’s collapse (Hughes et al., 2007). Investigating many cases in the United States, Wardhana and Hadipriono (2003) confirmed that floods and scours were the main cause of bridge failures during the service period. Similarly, Ye (2012) found scour to be a common defect in highway bridges in China, whose scour depths were usually more than 5 m. Nevertheless, the scour condition of a river- or sea-crossing bridge cannot be readily determined in real time due to difficulties in underwater inspections. Hence, it is critical to identify potential scour damage to underwater substructures, especially scour depths, to guarantee bridge safety.

Visual inspection is a fundamental measurement to detect the scour of bridges, which requires a diver to measure the scour depth (Avent and Alawady, 2005; Browne et al., 2010). It is evident that underwater manual operation is dangerous and time-consuming. Although many underwater inspection instruments have been developed to obtain scour conditions, such as sonar (Falco and Mele, 2002; Hayden and Puleo, 2011; Shen et al., 2018), radar (Forde et al., 1999), fiber-brag grating sensors (Kong et al., 2017; Lin et al., 2006), and electromagnetic sensors (Lin et al., 2017; Maroni et al., 2020), these underwater device-based methods cannot be applied during flooding since inspections and installing sensors are mostly dangerous and unattainable during these periods. Hence, these methods are not suitable for long-term and real-time monitoring of scour depths. In addition, the real scour depths cannot be measured by these methods in some cases because scour holes are filled by new soil with a weaker mechanical property than the original one (Foti and Sabia, 2011). As a result, the integrity of the whole bridge may decay without an apparent increase in scour depth.

In contrast, vibration-based methods can determine the scour depths of bridge piles though the backfill phenomenon has occurred. Since an increase in piers’ unsupported length can reduce the lateral stiffness of a bridge, the dynamic properties such as natural frequencies and mode shapes will change due to scour. Prendergast et al. (2013) undertook a laboratory experiment and a field test of a single pier, finding an apparent reduction in natural frequency with an increasing scour depth. Wang et al. (2014) compared the dynamic properties of a reinforcement bridge with different scour conditions, where scour has a significant impact on the first few transverse and longitude modes. Scozzese et al. (2019) investigated the modal property variation of an arch bridge under scouring action, and the results showed a sensitivity of the transverse mode shape to scour. It was concluded that scour exactly yields a reduction in natural frequencies and changes in mode shapes, especially for transverse modes.

The studies mentioned above indicated a prospective application of vibration-based methods in scour depth identification, which can continuously monitor the scour depth in real time. Chen et al. (2014) evaluated scour depth by matching sensitive natural frequencies calculated from the finite model to operational modal analysis. Elsaid and Seracino (2014) undertook a laboratory test to study the changes in dynamic properties induced by scouring, comparing indicators under separate scour conditions. The research recommended mode-shape curvature and flexibility-based deflections as indicators of scouring. Prendergast et al. (2016) analyzed the natural frequencies of piers under vehicle loading via a vehicle-bridge-interaction method. The changes in natural frequencies were used to detect scour since an apparent reduction in the first mode’s natural frequency was observed. Li et al. (2017) investigated scour depth based on time-varying natural frequencies of a cable-stayed bridge. Since environmental effects significantly impact natural frequencies, the scour effect on frequencies was separated from them by a nonlinear principle analysis. Bao et al. (2017) investigated key issues about the scour detection method based on the natural frequency spectrum, such as the influence of sensor location on scour monitoring. The result indicates that the location of the sensor installation should be close enough to the top of a pier due to the high amplitude of vibration. Recently, Fitzgerald et al. (2019) proposed a wavelet-based method in which the average wavelet coefficients of bogie accelerations from a passing train were calculated at the beginning. Then, a scour indicator based on the difference in average wavelet coefficients was developed to locate the scour. However, the most vibration-based methods to monitor scour depths mainly use natural frequencies as the objectives, without full use of modal information such as mode shapes. Estimating the scour depths by just a few natural frequencies sensitive to scour may be an ill-conditioned problem, resulting in the inaccuracy of scour depth identification.

This study proposes a modal updating-based method to determine scour depth, fully using the modal parameters sensitive to the scour effect. In the following section, a soil-structure interaction method and simulation of the scour effect are introduced initially. A modal updating method based on an improved genetic algorithm (GA) is then detailed in Scour Depth Identification based on Genetic Algorithm section, and the framework of scour depth identification is illustrated at the same time. In Numerical Simulation of Continuous Girder Bridge section, a numerical simulation is established to find the sensitive modes of a continuous girder bridge, where the scour effect on the natural frequency and mode shape is discussed. Scour Depth Identification of Continuous Girder Bridge section gives a demonstration of the scour depth identification, verifying the feasibility of the proposed method. In the end, the key conclusions are remarked.

Scour effect on the boundary condition of a bridge

Soil-structure interaction of a bridge

Generally, all the degrees of freedoms (DOFs) at the bottom of each pier are strictly constrained when establishing a finite element (FE) model of a bridge. However, the soil-structure interaction of a bridge should be simulated exactly for a modal updating-based procedure to estimate the piers’ scour depth. The soil around the bridge’s piles is commonly modeled as a series of closely spaced springs, where stiffness coefficients are determined by the property and depth of the surrounding soil. Hence, the piles are regarded as Winkler beams supported by many springs in which the piles’ restraint reactions are proportional to their displacements.

In this study, the m-method recommended in the standard (JTG 3363-2019, 2019) has been utilized to simulate soil-structure interactions, where the soil is supposed to be an elastic linear continuous material with a small strain assumption. Although this method may not be suitable for seismic analysis due to the large deformation of the soil, linear modal analysis based on the assumption of small deformation can meet the requirement. According to the hypothesis of the m-method, the transverse resistance of the soil is proportional to transverse displacement

σ_{yz} = m_{z} z_{s} y_{z}

(1)

where $z_{s}$ is the embedded depth, $σ_{yz}$ is the transverse resistance of the soil, $m_{z}$ is the m value corresponding to the properties of the soil at the embedded depth, and $y_{z}$ is the transverse displacement at the embedded depth.

Then, the stiffness coefficient of the soil spring can be given by the following equation because the transverse resistance is positively correlated to the displacement

k_{s} = \frac{F_{y}}{y_{z}} = \frac{h_{sz} b_{c} σ_{yz}}{y_{z}} = h_{sz} b_{c} m_{z} z_{s}

(2)

where $h_{sz}$ is the thickness of the soil layer equal to the distance between the neighboring nodes in the FE model, and $b_{c}$ is the calculative width of a pile.

In the equation mentioned above, the calculated width $b_{c}$ is related to the pile layout and the dimensions of the pile section

b_{c} = c_{1} c_{2} (D_{p} + 1)

(3)

where $c_{1}$ is the interaction coefficient between piles that equals 1 for the layout of single-row piles, $c_{2}$ is the shape coefficient of the pile that equals 0.9 for a circular section, and $D_{p}$ is the diameter of the pile with a circular section.

Scour effect on the boundary condition

The scour effect changes the transverse support length of each pile, leading to a variation in boundary conditions. Therefore, the soil springs in the range of scour depth will be removed, as shown in Figure 1, to simulate the scour effect on the boundary condition. Simultaneously, the stiffness coefficient should be revised because the embedded depth of each soil layer has decreased. In addition to transverse elastic restraints, the vertical displacement at the bottom of each pile is restricted, indicating that the reduction of the bridge’s vertical stiffness is not considered in this study. In other words, it is hypothesized that the pile is always vertically supported by a soil layer without significant influence of the scour effect.

Figure 1.

Simulation of the scour effect.

Scour depth identification based on genetic algorithm

The framework of scour depth identification

This study proposed a model updating-based method to identify a continuous bridge’s scour depths, where those of the bridge’s FE model are optimized to match the analyzed modal parameters to the target modal parameters. The framework of the proposed scour depth identification method is illustrated in Figure 2. Notably, a GA considering a prior probability distribution of scour depth is chosen as the optimizer. Simultaneously, a parallel computing technique is applied to decrease the time cost of optimization because the operations of GA, such as fitness evaluation of individuals and random search, have excellent parallelism. In addition, target modal parameters for model updating can be obtained from the data-driven stochastic subspace identification (Data-SSI) algorithm (Peeters and Roeck, 1999).

Figure 2.

The framework of scour depth identification.

Objective function of model updating

A critical point of a model updating method is the objective function in which the mean square error of natural frequencies is always used. However, the number of sensitive modes whose natural frequencies can be obtained from an ambient test or long-term monitoring is usually less than the number of scour depths to monitor. Hence, the mode shapes should be considered in the objective function to offer more modal information, reducing the degree of illness. Hence, the objective functions can be defined by the following equations

\underset{d_{s}}{mnimize} . J_{1} (d_{s}) = \frac{1}{N_{m}} \sum_{i = 1}^{N_{m}} {(1 - \frac{f_{a, i}}{f_{m, i}})}^{2}

(4)

\underset{d_{s}}{mnimize} . J_{2} (d_{s}) = \frac{1}{N_{m}} \sum_{i = 1}^{N_{m}} {(1 - MAC (φ_{a, i}, φ_{m, i}))}^{2}

(5)

where $J_{i} (d_{s}) (i = 1, 2)$ is the i-th objective function in this study, $d_{s =} {d_{s, 1}, d_{s, 2}, \dots, d_{s, N_{p}}}$ is the vector composed of scour depths, N_p is the number of piers, $N_{m}$ is the number of target modes, $f_{a, i}$ is the analyzed frequency of the i-th mode, $f_{m, i}$ is the target frequency of the i-th mode, $φ_{a, i}$ is the analyzed mode shape of the i-th mode, $φ_{m, i}$ is the target mode shape of the i-th mode, and $MAC$ is the modal assurance criteria (MAC) given by equation (6)

MAC (φ_{a, i}, φ_{m, i}) = \frac{{(φ_{a, i}^{T} φ_{m, i})}^{2}}{(φ_{a, i}^{T} φ_{a, i}) (φ_{m, i}^{T} φ_{m, i})}

(6)

Improved genetic algorithm

Fitness function

The GA is a widely used optimization algorithm referred to as the natural evolution mechanism, where the potential solutions are regarded as individuals in a population, and the parameters to be optimized are the genes of these individuals. For a classical GA procedure, the individuals are randomly generated at the beginning. The individuals are then evaluated by the predefined fitness function and elected for the new generation according to their fitness values. The optimal solution will be finally determined as the new generation’s individuals have better performance than those of the previous generation.

In this paper, each scour depth vector is regarded as an individual in the evolution procedure, and the components of these vectors are individuals’ genes. Then, the fitness function is defined by the combination of the reciprocals of the two objective functions mentioned before, and each objective is normalized to eliminate dimensional differences

Fitnes s_{i} = w_{1} \frac{min (J_{1} (d_{s, i}))}{J_{1} (d_{s, i})} + w_{2} \frac{min (J_{2} (d_{s, i}))}{J_{2} (d_{s, i})}

(7)

where $Fitnes s_{i}$ is the fitness value of the i-th individual, $d_{s, i}$ is the i-th scour depth vector, and $w_{1}$ and $w_{2}$ are the weight factors of the two objective functions that are utilized to balance the influence on the objective function since the reliabilities of frequencies and mode shapes are not the same. Specifically, $w_{1}$ and $w_{2}$ are set to 0.6 and 0.4, respectively.

Selection operation

As shown in Figure 3, an elite strategy in which the optimum individuals are directly reproduced to the environmental selection in the next generation is employed in the selection operation because the canonical GA in which the individuals are selected based on the probability may not converge to a global optimum solution (Holland 1992; Rudolph, 1994). In addition, the optimum solution will not be worse in the next generation, which can maintain the convergent speed. Particularly, the individuals, including the parents and offspring, are ranked by their fitness values before the selection operation. Then, those with large fitness values will be preserved for the next generation, in which the selection proportion $R_{s}$ is set to 10% in this paper. Therefore, the number of individuals to be directly reproduced is given by the following equation

N_{sele} = Round (R_{s} \cdot N_{indi})

(8)

where $N_{sele}$ is the number of individuals involved in the selection operation, $N_{indi}$ is the number of all the individuals, and $Round (•)$ is the rounding function.

Figure 3.

The elite strategy for producing the next generation.

Crossover operation

A crossover operation was executed between the selection and mutation operations in this study. Specifically, the rest of the individuals, except the individuals for direct reproduction and mutation, are all processed by a crossover operation. The proposed GA adopted a multiple-point crossover function, in which the pairs of individuals are randomly matched to produce offspring, and they are directly replaced by the offspring.

The genes for crossover are stochastically selected as shown in Figure 4, while the number of genes for crossover operation is given by

N_{gen, cr} = round (\frac{1}{2} N_{gen})

(9)

Figure 4.

Illustration of the crossover operation.

in which $N_{gen}$ is the number of genes that is equal to the number of components in the scour depth vector, and $N_{gen, cr}$ is the number of genes for the crossover operation.

Mutation operation

A mutation operation is employed to maintain diversity in the population, avoiding early convergence in a local minimum. In this study, the mutation probability is adaptive since the diversity gradually decreases, and the probability is given by the following equation

P_{mu, i}^{g} = 0.1 + 0.8 \cdot (1 - \frac{{Fitness}_{i}^{g}}{{Fitness}_{opt}^{g}})

(10)

where $P_{mu, i}^{g}$ is the mutation probability of the i-th individual in generation g, ${Fitness}_{i}^{g}$ is the fitness of the i-th individual in generation g, and ${Fitness}_{opt}^{g}$ is the fitness of the optimal individual in generation g.

The mutation operation in classical GA generates new individuals by a specific probability distribution, which has a poor convergence speed and cannot obtain the individuals with boundary values. For instance, pile foundations located on land have zero scour depth. If the lower bound is set to zero, the GA with the traditional mutation operation has no chance to produce suitable individuals. Therefore, a stochastic gradient-like mutation operation given by equations (11) and (12) are adopted to overcome the problems mentioned above

{\tilde{d}}_{s, i}^{g + 1} = {\tilde{d}}_{s, i}^{g} + ξ \otimes \frac{{\tilde{d}}_{s, opt}^{g} - d_{s, i}^{g}}{‖ d_{s, opt}^{g} - d_{s, i}^{g} ‖} Δ d

(11)

Δ d = Δ d_{0} [1 - \exp (Fitnes s_{opt} - {Fitness}_{i}^{g})]

(12)

where ${\tilde{d}}_{s, i}^{g + 1}$ is the scour depth vector of i-th individual in g+ 1 generation, ${Fitness}_{i}^{g}$ is the fitness value of i-th individual in g generation, $d_{s, opt}^{g}$ is the scour depth of optimum individual in g generation, $Fitnes s_{opt}$ is largest fitness in g generation, $Δ d$ is the iteration step length, $Δ d_{0}$ is an initial step length equal to 0.1, $ξ$ is a random vector with the components ranging from 0.9 to 1.1, the operation ⊗ is the elementwise multiplication, and the operation ‖•‖ is the modulus of the vector.

The components of the vector ${\tilde{d}}_{s, i}^{g + 1}$ may exceed the given range of scour depth after the mutation operation, and the result of equation (11) should be modified

d_{s, i k}^{g + 1} = {\begin{cases} {\bar{d}}_{s, l}, {\tilde{d}}_{s, i k}^{g + 1} > {\bar{d}}_{s, l} \\ {\tilde{d}}_{s, i k}^{g + 1}, {\underline{d}}_{s, l} \leq {\tilde{d}}_{s, i k}^{g + 1} \leq {\bar{d}}_{s, l} \\ {\underline{d}}_{s, l}, {\tilde{d}}_{s, i k}^{g + 1} < {\underline{d}}_{s, l} \end{cases}

(13)

where ${\tilde{d}}_{s, ik}^{g + 1}$ is the k-th component of scour depth vector ${\tilde{d}}_{s, i}^{g + 1}$ calculated by equation (11), $d_{s, ik}^{g + 1}$ is the modified value of ${\tilde{d}}_{s, ik}^{g + 1}$ , and ${\bar{d}}_{s, l}$ and ${\underline{d}}_{s, l}$ are the upper and lower bounds of the l-th scour depth, respectively.

However, the diversity of the population will decrease due to the gradient-like mutation operation. Thus, we set the probability of the i-th individual $P_{mu, i}^{dir}$ for selecting the mutation operation. When the random scalar $P_{mu, i}^{dir}$ is less than 0.5, the above-mentioned mutation operation is implemented; instead, another mutation given by the following equation is executed

d_{s, i}^{g + 1} = η \otimes d_{s, i}^{g}

(14)

where $η$ is the random vector with components from 0 to 2.

Initialization of individuals

It is known that the probability distribution of the scour depth is usually a uniform distribution in the classical GA, which is not suitable for scour depth identification. In particular, some piers may be located on the land where their scour depths are zero. Hence, the probability distribution is assumed to be an exponential distribution in which the scour depths in the FE models are initialized based on this probability distribution. It is based on the hypothesis that scour is usually caused by floods whose amplitudes can be described by an exponential distribution (Tubaldi et al., 2017) and that scour depth was positively correlated with their amplitudes. Thus, the prior probability distribution of scour depth is described by equations (15) and (16)

P (x < d_{s}) = 1 - \exp (- λ d_{s})

(15)

λ = - \frac{\ln (1 - α)}{[d_{s}]}

(16)

where $P (x < d_{s})$ is the prior probability distribution of scour depth, $d_{s}$ is the scour depth of a pile, $λ$ is a constant for calculating the prior probability distribution, $α$ is the user-defined confidence probability that is 90% in this paper, and $[d_{s}]$ is the allowed scour depth during a bridge’s operation.

Numerical simulation of continuous girder bridge

Finite element model of a continuous girder bridge

The case study bridge is a three-span prestressed concrete girder bridge illustrated in Figure 5. This bridge’s layout is referenced to a municipal bridge located in China, but its dimensions were altered to simplify the case study. In this case, the total length of the bridge was 100 m, with spans of 30 m, 40 m, and 30 m. The height of the main girder was 1.8 m, while the width was 10.05 m. The piers of the bridge were identical in this study, with a height of 5 m and a plan size of 3 m × 1.5 m, excluding two parts of a half-circle with a radius of 1.5 m. The rock-socked piles at the different locations have the same diameter of 1 m and the center distance of 3.6 m. In addition, the length of each pile was set as 10 m in this study.

Figure 5.

Dimensions of the continuous prestressed concrete bridge: (a) bridge layout; (b) mid-section; (c) pile foundation layout (unit: cm).

A linear three-dimensional FE model of this continuous girder bridge was established by ANSYS software, and the FE model shown in Figure 6 was used to analyze the dynamic properties under separate scour conditions. The girder, piers, and piles were modeled by beam188 elements, and pile caps were simulated by MPC184 elements that are usually used to establish rigid constraints. Particularly, each MPC184 element connected the top nodes of the piles and the bottom node of the corresponding pier so that all the DOFs of these nodes were coupled. The deck was regarded as a series of lumped mass in which MASS21 elements were implemented, and their rotary inertias were not considered in this study. The element sizes of the girder and piers were set as 0.5 m since the BEAM188 element is a linear two-node element in 3 dimensions. Hence, the main girder was divided into 200 elements, while each pier contained 10 elements. In contrast, the element sizes of piles should be small enough to precisely simulate the scour effect in which each pile was divided into 100 elements with a size of 0.1 m.

Figure 6.

Finite element model of the prestressed concrete bridge.

The bearing at the top of each pier was modeled by three 1-dimensional COMBIN14 elements in which the stiffnesses of these springs along the -x, -y, and -z directions were 1 × 10⁹ N/m, 1 × 10¹² N/m, and 1 × 10⁹ N/m, respectively. In addition to these elements for simulating translational constraints, the torsional displacement of the girder at each bearing was coupled with the rotation at the top of the corresponding pier. In addition, the equivalent springs mentioned in Scour Effect on the Boundary Condition of a Bridge section were employed to model the soil-structure interaction in which the soil was hypothesized to be dense silty clay with an m value of 3 × 10⁷ N/m⁴. These equivalent springs were also modeled by 1-dimensional COMBIN14 elements in both the -x and -y directions in this FE model, which connected the pile nodes and fixed nodes. At the bottom of each pile, all the DOFs were completely restricted as the piles were supported by the rock.

Material properties are gathered in Table 1, whose values were determined according to the design code (JTG 3362-2018, 2018). For most concrete components in the FE model, their densities were assumed to be 2.5 × 10³ kg.m⁻³, but that of the deck was 2.4 × 10³ kg.m⁻³. The Young’s moduli of the girder, piers, and piles were set to 3.60 × 10⁴ MPa, 3.45 × 10⁴ MPa, and 3.25 × 10⁴ MPa, respectively, while the stiffness of the deck was ignored in this study.

Table 1.

Structural parameters of the continuous girder bridge FEM.

Structure component	Element type	Material	Young’s modulus (MPa)	Density (kg.m⁻³)
Main girder	Beam188	C50 concrete	3.60 × 10⁴	2.50 × 10³
Deck	Mass21	Asphalt concrete	—	2.40 × 10³
Piers	Beam188	C40 concrete	3.45 × 10⁴	2.50 × 10³
Piles	Beam188	C25 concrete	3.25 × 10⁴	2.50 × 10³

The Block Lanczos method was selected in ANSYS software to calculate theoretical modal parameters. Table 2 shows the result of modal analysis, and the mode shapes of the first eight modes are illustrated in Figure 7. According to related studies, the transverse stiffness of the whole bridge with soil-structure interactions was lower than that with fixed boundaries (Bao et al., 2019; Boulanger et al., 1999). Thus, the longitude and transverse vibration modes dominate the vibration with low frequency.

Table 2.

Modal Analysis of a continuous girder bridge based on the FEM.

Mode	Description of mode shape	Calculated frequency (Hz)
1	First longitude vibration mode	0.6185
2	First transverse vibration mode	1.8385
3	Second transverse vibration mode	2.5562
4	First vertical vibration mode	4.0157
5	Third transverse vibration mode	4.5624
6	Second vertical vibration mode	6.1507
7	Third vertical vibration mode	6.7664
8	Fourth transverse vibration mode	8.3358

Figure 7.

Calculated mode shape from FEM: (a)–(h) mode shape 1–8.

Sensitivity analysis of scouring effect

It is crucial to select modal parameters sensitive to the scour effect as the optimization objective because the insensitive modal parameters can yield severely ill conditions in model updating. Hence, a single factor sensitivity analysis was employed. In addition, only scour effects on piers P1 and P2 were analyzed, as the structure of the simulated bridge was symmetric. The relations between scour depths and natural frequencies are presented in Figure 8, and the variation ratio of natural frequencies with a scour depth of 1 m at piers P1 and P2 is presented in Figure 9.

Figure 8.

Calculated frequency versus scour depth: (a)–(h) modes 1–8.

Figure 9.

Variation ratio of natural frequencies (scour depth = 1 m).

Clearly, the scour had a minimal influence on the vertical vibration modes but a significant impact on the longitude and transverse vibration modes. It is inferred that the vertical vibration modes mainly depended on the properties of the girder that cannot be affected by scour as the piles were rock-socked. In other words, the natural frequencies of vertical vibration modes did not sharply decrease with scour condition development. In contrast, longitude and transverse vibration modes were sensitive to scour. Comparing scour effects at separate piers, the natural frequencies of the first longitude vibration mode changed in the same pattern, while those of transverse vibration modes deteriorated with different patterns. It can be found that scour led to a more considerable decline in the first longitude mode frequency than other modes. In addition, modes 2 and 8 were sensitive to the scour of P2, whereas modes 3 and 5 were sensitive to the scour of P1. The reason is that the transverse displacement of P2 was greater than that of P1 for mode shapes 2 and 8, while the conditions for mode shapes 3 and 5 were the opposite. Therefore, it is concluded that the scour effect has a significant impact on the natural frequencies of modes 1, 2, 3, 5, and 8.

Table 3 lists MAC values between mode shapes with and without the scour effect, and Figure 10 gathers the mode shapes of frequency-sensitive modes under different scour depths. The MAC value of mode 1 remained at 1.0 under various scour conditions, while that of mode 5 decreased within 0.002. Nevertheless, the MAC values of other modes decreased noticeably with increasing scour depth. This phenomenon is associated with the fact that the first longitude vibration mode of a continuous girder bridge is similar to the vibration mode of a single-degree-of-freedom (SDOF) system; thus, stiffness decline could not change its mode shape. However, the transverse vibration modes are different from the longitude vibration mode since the girder cannot be regarded as an SDOF system. Increasing scour depth significantly changes the relative stiffness between the piers, leading to an apparent change in the shapes of the transverse vibration modes. The declines in the MAC values of modes 2 and 3 induced by scour of P1 were more significant than those caused by scour of P2. In contrast, scour of P1 led to a lower descent in the MAC value of mode 8 than that of P2. Hence, the above results indicate that the scour effect apparently influenced the mode shapes of modes 2, 3 and 8 but not those of modes 1 and 5.

Table 3.

Modal assurance criteria between mode shapes with and without scour.

Mode	P1-1 m	P1-2 m	P1-3 m	P1-4 m	P1-5 m	P2-1 m	P2-2 m	P2-3 m	P2-4 m	P2-5 m
Mode 1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
Mode 2	0.993	0.965	0.909	0.832	0.748	0.998	0.992	0.986	0.980	0.974
Mode 3	0.994	0.973	0.932	0.875	0.987	0.999	0.996	0.993	0.990	0.987
Mode 5	0.998	0.991	0.982	0.971	0.959	1.000	1.000	0.999	0.999	0.998
Mode 8	0.958	0.810	0.638	0.518	0.438	0.924	0.820	0.744	0.687	0.640

Figure 10.

Mode shapes under different scour depths: (a)–(c) modes 1–3; (d) mode 5; (e) mode 8

Scour depth identification of continuous girder bridge

Transient analysis of dynamic responses

The numerical simulation, based on the FE mentioned above model, was used to verify the proposed method for identifying scour depth. Hence, full transient dynamic analysis in ANSYS software was employed to simulate the dynamic responses (e.g. acceleration) of the scoured bridge under vehicle loads, while Rayleigh damping (Liu and Gorman, 1995) with a structural damping ratio of 0.05 was used. In particular, a moving mass-spring model was adopted to model vehicle loads that the vehicle-bridge interaction can be considered. Figure 11 illustrates the vehicle-bridge vibration system, acceleration sensor layout, and road roughness. In the figure, $M_{ve}$ , $K_{ve}$ , and $v$ denote the mass, suspension stiffness, and velocity of the vehicle, respectively. We utilized the constraint equations in ANSYS software to establish the interaction between the vehicle and bridge, and the relationship among the different vertical displacements was described as

y_{ve, t} - c_{i} y_{i} - c_{i + 1} y_{i + 1} = r_{t}

(17)

c_{i} = \frac{z_{ve, t} - z_{i}}{z_{i + 1} - z_{i}}, c_{i + 1} = \frac{z_{i + 1} - z_{ve, t}}{z_{i + 1} - z_{i}}

(18)

Figure 11.

(a) Vehicle-bridge vibration system and acceleration sensor layout; (b) road roughness in the simulation. (c) power spectral density (PSD) of the road roughness.

where $y_{ve, t}$ is the vertical displacement of the vehicle at time t, $y_{i}$ and $y_{i + 1}$ are the vertical displacements of nodes i and i + 1, $r_{t}$ is the road roughness at the location of the vehicle at time t, $z_{ve, t}$ is the longitude location of the vehicle at time t, and $z_{i}$ and $z_{i + 1}$ are the longitude locations of nodes i and i + 1. The roughness presented in Figures 11(b) and (c) was randomly generated by the target power spectral density function of a Class A road (ISO 8608:2016, 2016).

Identification of single scour of the continuous girder bridge

We assumed that the scour depth at pier P2 was 2 m, those of the remaining piers were zero, and the suspension stiffness of the vehicle was 3 × 10⁵ N/m. Then, accelerations of the bridge under vehicle loads with separate masses of 25 t and 55 t and velocities of 80 km/h, 100 km/h, and 120 km/h were calculated based on the FE model. In addition, the sampling frequency was set as 100 Hz, and the total time was 10 s so that the modal identification algorithm needed sufficient data.

Figure 12 displaces acceleration responses extracted from the measured points A3, A7, and A11. Clearly, these acceleration responses have similar vibration modes, and the main difference among them is their maximum vibration amplitudes when the bridge is in a free vibration state. Comparing the responses, the vehicle with a velocity of 100 km/h leads to the largest vibration amplitude. In contrast, the vehicle with a velocity of 80 km/h causes the vibration with the lowest amplitude.

Figure 12.

Accelerations of the bridge under the different vehicle loads: (a)-(c) A3, A7, and A11.

We intercept the acceleration signals from 4 s to 10 s to obtain the free-vibration signals (if random vehicle loads are adopted, the excitations can be regarded as Gaussian noise, and it is not necessary to intercept the signals) and resample them to 10 Hz. Next, the Data-SSI algorithm was used to identify the target modal parameters of sensitive modes. In the modal identification algorithm, the range of the system order was from 2 to 100. Therefore, we obtained the modal parameters of the scoured bridge that are listed in Table 4. The identified frequencies were well matched to the theoretical frequencies, with relative errors of less than 3%. Hence, the result indicates that the SSI method captured the modal parameters of the sensitive modes from the dynamic response under the vehicle loads. Moreover, it is implied that the different velocities and masses of the vehicles may have a weak influence on the identification of the scour depths.

Table 4.

The results of modal identification under different vehicle loads.

Case	Vehicle parameters		Mode 2		Mode 3
Case	Mass (t)	Velocity (km/h)	DATA-SSI (Hz)	Relative error (%)	DATA-SSI (Hz)	Relative error (%)
S1	55	80	1.808	−0.1	2.691	+0.4
S2	55	100	1.809	+0.0	2.627	−2.0
S3	55	120	1.806	−0.2	2.698	+0.6
S4	25	80	1.799	−0.6	2.658	−0.9
S5	25	100	1.812	+0.2	2.682	+0.0
S6	25	120	1.762	−2.6	2.701	+0.7

The model updating-based method could be employed to determine the scour depths of the continuous girder bridge once its modal parameters were identified from the dynamic responses. The simulations in this paper were performed on a computer with 8 Intel Core i7-10700K CPUs and an RAM of 16.0 GB. In this procedure, the number of parallel networkers in MATLAB 2020R was set to 6, while the number of parallel processing steps in ANSYS software was 12. For the user-defined parameters in GA, the number of individuals in the population was 50, the convergence tolerance was 1 × 10⁻⁵, and the number of iteration steps was 100.

We take case S1 as an example, and the convergence processes of the objective function and scour depths are presented in Figure 13. According to the convergence diagram, the objective functions decreased sharply at the beginning of the optimization process. Then, the first objective function $J_{1} (d_{s})$ remained constant, while the second objective function $J_{2} (d_{s})$ increased rapidly. The difference in their trends is caused by the weight factors of the two objective functions, as we set them to 0.6 and 0.4, respectively. Thus, the optimization process was dominated by a decrease in the mean square error of the natural frequencies. Comparing the convergence of the identified scour depths, they held steady values after 12 iterations, indicating that the values tend to converge to the theoretical scour depths. Hence, the analysis result indicates that the proposed GA could quickly converge to a moderate ideal solution of scour depths, although some errors can be determined.

Figure 13.

The convergence process of genetic algorithm for case S1: (a)–(b) objective functions 1–2; (c)–(f) scour depths at piers P1–P4.

Table 5 gives a comparison between the identified scour depths under the various vehicle loads, and the errors of scour depths range from 0 m to 0.27 m. Evidently, there are small discrepancies between the absolute errors of the identified scour depths, and it seems that the errors are not related to the vehicle parameters.

Table 5.

Results of the identified scour depths identified under different vehicle loads.

Case	Vehicle parameters		Scour depth (m)/Absolute error (m)
Case	Mass (t)	Velocity (km/h)	Pier 1	Pier 2	Pier 3	Pier 4
S1	55	80	0.04/+0.04	1.73/−0.27	0.00/+0.00	0.00/+0.00
S2	55	100	0.00/+0.00	1.99/−0.01	0.04/+0.04	0.00/+0.00
S3	55	120	0.00/+0.00	0.03/+0.03	1.98/−0.02	0.05/+0.05
S4	25	80	0.00/+0.00	1.79/−0.21	0.16/+0.16	0.04/−0.04
S5	25	100	0.00/+0.00	1.99/−0.01	0.04/+0.04	0.00/+0.00
S6	25	120	0.00/+0.00	1.78/−0.22	0.23/+0.23	0.00/+0.00

The error of the identification of scour depths is mainly induced by the characteristics of the GA. Essentially, the proposed GA is a heuristic optimization algorithm that seeks the optimal solution in the solution space based on a random searching strategy. Although the selection, crossover and mutation operations were implemented, the results of the optimization process still contain randomness. Hence, the optimized solution of scour depths did not match the given scour condition well in the end.

Identification of multiple scours of the continues girder bridge

We used the proposed method to identify multiple scours of the same continuous girder bridge for further verification. In this section, the vehicle mass was set as 55 t, while the velocity was 100 km/h. Four cases of multiple scours are listed in Table 6. In these cases, the scour depth of each scoured pier was assumed to be 1.0 m. Table 7 presents the results of the identification of multiple scours. The errors of the identification of multiple scours range from 0 m to 0.31 m. In contrast to the identification of the single scour case, the errors of the multiple-scour cases are moderately larger. The reason is that a single scour can induce an apparent change in the shapes of the sensitive modes. In contrast, multiple scours may lead to a slight variation in the relative stiffness between the piers, which means that the mode shapes will not change significantly. Hence, it may be difficult for the proposed method to identify multiple scour depths with great accuracy when the number of scoured piers increase.

Table 6.

Scour depths of multiple scour cases.

Case	Scour depth (m)
Case	Pier 1	Pier 2	Pier 3	Pier 4
M1	1.00	1.00	0.00	0.00
M2	0.00	1.00	1.00	0.00
M3	1.00	1.00	1.00	0.00
M4	1.00	1.00	1.00	1.00

Table 7.

The results of the identification of multiple scours.

Case	The results of the proposed method (m)/Absolute errors (m)
Case	Pier 1	Pier 2	Pier 3	Pier 4
M1	0.76/−0.24	0.83/−0.17	0.31/+0.31	0.14/+0.14
M2	0.00/+0.00	0.81/−0.19	1.06/+0.06	0.00/+0.00
M3	0.87/−0.13	1.24/+0.24	0.74/−0.26	0.14/−0.14
M4	1.21/+0.21	1.19/+0.19	0.71/−0.29	0.84/−0.16

Conclusions

This study focuses on the scour depth identification problem of continuous girder bridges, which is of great significance to ensure their safety. Hence, the scour effect on the modal parameters of a continuous girder bridge was detailed by a sensitivity analysis based on the FE model in which the soil-structure interaction was considered by the Winkle beam method. Then, the dynamic responses of the numerical model under the different vehicle loads (simulated by a moving mass-spring model) were extracted to offer the data to verify the feasibility of the proposed method. The main conclusions are the followings:

The natural frequencies of the transverse and longitude vibration modes were sensitive to the scour effect, especially the first longitude vibration mode. However, the shape of the first longitude vibration mode was not influenced by scour because this vibration mode was similar to the SDOF system, in which stiffness did not apparently change the mode shape. In contrast, the scour effect can significantly change the relative stiffness of the piers, leading to distinguished variations in the transverse vibration modes.

The SSI method could capture the modal parameters of the transverse vibration modes sensitive to the scour effect, in which the errors of frequency identification under different vehicle loads were less than 2%. It is implied that the velocity and mass of the vehicle will have a weak influence on the identification of the scour depth if the modal identification algorithm can successfully extract the modal parameters.

The model updating method based on the genetic algorithm was adopted to identify the scour depth of a continuous girder bridge. Compared to traditional vibration-based approaches, this method adds MACs to the objective function, considering the influence of mode shapes. The numerical simulation results show that the proposed method can determine scour depths with errors of less than 0.31 m, confirming the feasibility of this method.

The model updating method can be employed to assess the scour depth of a continuous girder bridge by its dynamic properties, such as natural frequencies and mode shapes, which is low-cost and convenient. However, the calculation efficiency of this method needs to be improved in future exploration, although the parallel technique has been used. The surrogate model is an alternative to decrease the time cost of optimization in model updating, but the simple description of mode shapes may be the most difficult issue due to the complex relationship between scour depths and mode shapes. Therefore, further studies will focus on surrogate models of modal shapes in which efficiency and accuracy should be balanced.

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: The research was financially supported by the National Key Research and Development Program of China (2020YFC151190), National Natural Science Foundation of China (Grant 51838004), Fundamental Research Funds for the Central University, and Postgraduate Research and Practice Innovation Program of Jiangsu Province, China (No. KYCX170120).

ORCID iDs

Shitong Hou

Gang Wu

Data availability statement

Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request, including the experimental data and simulation results presented in this paper.

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