Research on structural damage identification of bridge engineering based on dynamic parameters

Under the booming economic development in China, the traffic volume and the number of heavy and overweight vehicles crossing bridges are increasing, and many bridges show an accelerated trend of aging and functional degradation [1–3]. In order to ensure the safety and durability of bridges, avoid and reduce the significant loss of national and people’s property, it is very important to test the structural working condition and structural characteristic parameters [4–6].

After the completion of the bridge, due to the influence of climate and environmental factors, the structural materials will be corroded and gradually aging, long-term static and dynamic loading, so that its strength and stiffness with the increase of time and reduce [7–9]. This will not only affect the traveling safety, but also shorten the service life of the bridge [10]. Detecting and monitoring the health condition of bridge structures and evaluating their safety performance on this basis is an important part of the daily management of bridge operation. Bridge health monitoring has a very important role [11–12]. The basic connotation of bridge health monitoring is to monitor and evaluate the structural condition of the bridge, to trigger early warning signals for the bridge in special climate and traffic conditions or when the bridge operation condition is seriously abnormal, and to provide a basis and guidance for bridge maintenance and repair and management decision-making [13–14].

There are dynamic and static methods for structural damage detection in bridge engineering [15–16]. The advantages of dynamic damage diagnostic method over the traditional damage detection methods (visual and appearance inspection, unbroken or half-broken test, on-site loading test, etc.) are that the state of the structure can be grasped in general, the loading equipment required for the test is simple, the test speed is fast, it does not cause any new damage to the members, and the dynamic diagnostic test can form a stage-by-stage correspondence with the static force data [17–19]. Therefore, more and more scholars and engineers are devoted to the research of bridge dynamic damage diagnosis technology. The dynamic parameter damage identification method aims to solve the following problems in structural damage detection, i.e., whether there is damage in the structure, to determine the location of structural damage, the degree of structural damage, and the effect of structural damage on the performance of the structure (prediction of the remaining life of the structure) [20–21]. Typical dynamic parameter diagnostics involves comparing the observed changes in dynamic parameters with those of a baseline and using the most likely of these changes to determine the damage to the structure. Structural dynamic damage assessment can be divided into four steps, firstly, the selection of vibration observation signals, secondly, the extraction of characteristic quantities that are linked to the damage state, then er identifying the presence or absence of damage in the structure, and finally identifying the location, nature, and extent of damage [22–24].

In this paper, we first clarify the types of structural damage and model the types such as stiffness reduction. Then, according to the theory of structural damage identification based on frequency change and flexibility curvature, the structural damage identification method is studied. Combined with artificial neural network, the damage identification method based on RBF neural network is proposed. A finite element model of the bridge structure is established, the input and output vectors of the samples are calculated, and then the neural network samples are constructed, including the input and output vectors of the bridge structure to be recognized under various damage states (damage location and damage degree). Finally, the weights between neurons are adjusted, and the neural network is trained and tested. The method is applied to the damage recognition test of arch bridges and simply supported girder bridges to verify the effectiveness of the method.

2

Method

2.1

Damage modeling and crack generation in bridges

Before carrying out bridge structural damage identification, we need to clarify the type of structural damage. There are mainly the following types of bridge damage models: 1)

Decrease in stiffness

In the actual project there is a common structural quality did not occur any change, but the stiffness of the structural unit EI decline, this type of damage in the finite element software is very easy to numerical simulation, but the existence of neglected damage in the adjacent units of the impact.

2)

Open cracks

In reinforced concrete bridges, when damage occurs to the structure, it is often in the form of one or several open cracks.Abdek Wahzb et al. proposed to simulate the damage in the region of open cracks in reinforced concrete girders by varying the change in Young’s modulus of the material. The following proposed equation was presented: 1 $\begin{array}{l} E I (x) = E I [1 - α \cos^{2} (π / 2 {(| x - l_{c} | / β L / 2)}^{m})] \\ (l_{c} - β L / 2 < x < l_{c} + β L / 2) \end{array}$

Where, l_c is the distance from the center point of the damaged area of the structure to the left end support, L is the length of the whole beam. α, β, m is the parameter of the crack. α indicates the extension coefficient of the crack along the beam height direction, the value can express the degree of structural damage, the value range is 0~1. When α = 1, it means that the beam opening crack has become a penetrating seam, and the structural damage area has completely lost the bending capacity. At α = 0, it means that the beam is not cracked. β indicates that the longitudinal extension coefficient of the crack along the beam can describe the scope of the damage area, and its value ranges from 0 to 1. m represents the change coefficient of the material stiffness in the damage area, and it is used to describe the change rule of the stiffness in the center of the damage area of the structure on both sides. EI is the material bending stiffness of the beam in the intact state.

3)

Breathing cracks

When the bridge is subjected to fatigue loading, it will cause breathing cracks in the beam. Some cracks will occur under the action of the tension phenomenon, this crack is called breathing cracks. When there is a load effect, the crack will unfold. When the loading disappears, the cracks close again. This breathing crack behavior we can use in the load process of elastic stiffness from the tensile state to the compression state to simulate.

4)

Torsion spring model

When damage occurs to the structure, causing unequal turning angles of the nodes at both ends of the crack, a concentrated torsion spring can be used to simulate the damage.

Tada defines the torsion spring stiffness from the perspective of fracture mechanics theory as K_c = 1/C, where: 2 $C = \frac{2 H}{E I} {(δ / 1 - δ)}^{2} (5.93 - 19.69 δ + 37.14 δ^{2} - 35.84 δ^{3} + 13.12 δ^{4})$ where H is the height of the beam section and δ is the ratio of crack depth to beam height.

2.2

Dynamic properties of the structure

2.2.1

Solving for structural self-oscillation characteristics

When a structural system has N degree of freedom, its dynamic balance equation is: 3 $M y^{'} + C y^{'} + K y = f (t)$

In Eq. M,C,K are the mass, damping, and stiffness matrices, respectively, for the linear vibration regime of the structure. For the linear vibration system, each matrix can be represented by the n×n nd order real coefficient pair matrix. y^m,yⁿ,y are the acceleration, velocity, and displacement matrices of the structural system, and f(t) is the nth order matrix of the external excitation of the vibration system.

When the structural vibration system does not consider the effect of damping, equation (3) can be rewritten as: 4 $M y^{'} + K y = f (t)$

Let Eq. (4) be solved in the form: 5 $y = Y \sin (ω t + α)$

Eq. (5) where Y is the displacement magnitude vector, eliminating the common factor: 6 $\sin (ω t + α)$

Get: 7 $(K - ω^{2} M) Y = 0$

Equation (7) is the frequency equation of the multi-degree-of-freedom system. Let Y⁽⁰⁾ denote the dominant mode vector corresponding to frequency ω_i : Y^(i)T = (Y_1i,Y_2i,⋯Y_ni), and substitute ω_i and Y⁽ⁱ⁾ into (7): 8 $(K - ω^{2} M) Y^{(i)} = 0$

In Eq. i = 1,2,…n, N vector equations can be obtained, from which the N main vibrational shapes can be found Y^(l),Y⁽²⁾,…Y⁽ⁿ⁾. In Eq. (7), since the determinant is zero, only N–1 of the N algebraic equations in the above equation are independent, and thus only the relative value of each vibrational shape can be obtained, i.e., we can uniquely determine the shape of the main vibrational shape, but not the unique amplitude.

In order to be able to make the amplitudes of Y⁽⁰⁾ also have corresponding specific values, it is necessary to standardize the matrix of vibrational shapes, and the following two methods are common. One is to artificially specify that an element of the main vibration pattern Y⁽⁰⁾ is 1, thus obtaining the main vibration pattern. The second is to specify that each element of the main vibration pattern Y⁽ⁱ⁾ satisfies $Y_{i}^{T} M Y_{i} = I$ , and the vibration pattern vector Y⁽ⁱ⁾obtained by using the above two methods is the structurally normalized vibration pattern, which is also called the mass-normalized vibration pattern.

By using the above two methods, the normalized vibration vectors corresponding to the structure at any frequency can be obtained, and these N normalized vectors are formed into a square matrix, which is the structural vibration matrix, as in equation (9) below: 9 $Y = [Y_{1}, Y_{2}, \dots Y_{n}] = [\begin{matrix} Y_{11} & Y_{21} & \dots & Y_{n 1} \\ Y_{12} & Y_{22} & \dots & Y_{n 2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ Y_{1 n} & Y_{2 n} & \dots & Y_{n n} \end{matrix}]$

From the above analysis, we can conclude that the frequency of the structural vibration system and the corresponding vibration pattern depend only on the structural system’s own mass matrix M and stiffness matrix K, and have nothing to do with any external factors of the system, which is completely determined by the system’s own physical properties, and is an inherent property of the structural system.

2.2.2

Orthogonality of the principal vibration modes of the structure

For the same structural vibration system, the orthogonal nature of the principal vibration modes is satisfied between the inherent vibration modes corresponding to any two different frequencies. The equation (8) is obtained by replacing i with k and l, respectively: 10 $K Y^{(k)} = ω_{k}^{2} M Y^{(k)}$ 11 $K Y^{(l)} = ω_{l}^{2} M Y^{(l)}$

Multiplying both sides of Eq. by Y^(I)T and both sides by Y^(k)T yields: 12 $Y^{(l) T} K Y^{(k)} = ω_{k}^{2} Y^{(l) T} M Y^{(k)}$ 13 $Y^{(k) T} K Y^{(l)} = ω_{l}^{2} Y^{(k) T} M Y^{(l)}$

Since K = K^T,M = M^T, it is obtained by transposing both sides of Eq. (13): 14 $Y^{(l) T} K Y^{(k)} = ω_{l}^{2} Y^{(l) T} M Y^{(k)}$ obtained by subtracting equations (12) and (14): 15 $(ω_{k}^{2} - ω_{l}^{2}) Y^{(l) T} M Y^{(k)} = 0$

If ω_k ≠ ω_l, then we get: Y^(l)T MY^(k) = 0, which shows that for the structural mass matrix, for the same system of vibration, the principal modes of vibration corresponding to it at any two frequencies are orthogonal to each other. Similarly, Y^(I)T KY^(k) = 0 is obtained, which shows that for the structural stiffness matrix, the principal modes of vibration corresponding to the same system of vibration at any two frequencies are orthogonal to each other.

2.3

Structural damage identification based on frequency variation

Frequency is a physical parameter that relates the overall stiffness of a structure to its overall mass. Damage to a part of the structure will cause changes in the intrinsic frequency of the structure, but for civil engineering structures, the mass of the damage is generally not damaged, the intrinsic frequency change method of identification is not to take into account the effect of changes in the structure’s mass, and only take into account the structural stiffness reduction, the damping ratio increases the damage identification method, therefore, the change in intrinsic frequency is often due to the changes in the structure’s own stiffness caused by the comparison of the Before and after the damage can be recognized structural damage by the structural frequency change.

Generally, the deformation that occurs in the neutral axis of a member is called curvature, and beams are structures that mainly bear bending moments. Structural damage can change the curvature of the structure under external loads, so the curvature mode is also an important physical parameter for damage identification. The curvature mode is a special form of vibration characteristics of the bending structure, therefore, the sensitivity of its parameters to structural damage ensures the accuracy of the damage identification results.

Take ordinary beam structure as an example, according to the theory of material mechanics, when the beam bends under the action of external load, the basic formula of its bending deformation is: 16 $\frac{1}{γ} = \frac{M}{E I}$

Where: γ is the radius of curvature corresponding to the deformation of the structure at the position of the central axis under the action of external forces, $1 / γ$ is the curvature, EI is the bending stiffness, and M is the bending moment.

The curvature of the planar curve of the structure after bending occurs is given in the following equation: 17 $\frac{1}{γ (χ)} = \frac{y^{*}}{{(1 + y^{'})}^{3 / 2}} \approx y^{*}$

y is the displacement produced by the structure during the bending deformation, the initial conditions are irrelevant and are determined by the properties of the vibrating structure, which has two characteristics: superposition and orthogonality. The above equation shows that the deformation curvature of the structure is a second-order derivative function, which corresponds to each order of the displacement mode, the structure exists in the distribution of intrinsic curvature corresponding to the deformation curvature, and this distribution of intrinsic curvature with its counterpart is known as curvature mode vibration mode. Similarly, the curvature mode shapes have structural properties that are independent of the initial conditions and external forces applied to the structure.

Substituting Eq. (16) into Eq. (17) yields: 18 $y^{*} = \frac{M (χ)}{E I}$

According to the analysis of formula (18), the curvature is related to the bending stiffness of the structure, the greater the bending stiffness, the smaller the curvature, and the two are inversely proportional. When structural damage occurs locally, the local bending stiffness of the structure is reduced, so that the curvature increases at the damage, and the structural curvature changes at the damage location, using this feature can be used for damage identification of beam structures.

According to the mechanics of materials can be known: 19 $E I \frac{d^{2} y (x, t)}{d x^{2}} = M$

In the above equation, EI represents the flexural stiffness of the structure, y(x, t) represents the vibrational displacement of the structure at x at the moment of t, and M represents the bending moment of the structural section: 20 $\frac{d^{2}}{d x^{2}} (E I \frac{d^{2} y (x, t)}{d x^{2}}) = p (x)$

For the analysis of Eq. (20), p(x) represents the inertia force corresponding to the free vibration of the structure, and the solution equation for the inertia force p(x) is: 21 $p (x) = - ρ A \frac{\partial^{2} y (x, t)}{\partial t^{2}}$

In Eq. (21) A represents the cross-sectional area of the structure and ρ is the density of the material used in the structure. Substituting Eq. (21) into Eq. (20) yields the corresponding differential equation when the structure vibrates: 22 $\frac{\partial^{2}}{\partial x^{2}} [E I \frac{\partial^{2} y (x, t)}{\partial x^{2}}] + m (x) \frac{\partial^{2} y (x, t)}{\partial t^{2}} = 0$

In the above equation m(x) = ρA, is the mass size of the structure per unit length.

According to the theory of structural vibration modes, the solution of Eq. (22) can be expressed according to the principle of modal superposition as: 23 $y (x, t) = \sum_{r = 1}^{\infty} φ_{r} (x) q_{r} (t)$

In the above equation φ,(x),q_r (t) is the displacement mode shapes and displacement mode coordinates of the structural vibration, which are obtained from equation (18): 24 $δ (x) = \frac{\partial^{2} y (x, t)}{\partial x^{2}} = \sum_{r = 1}^{\infty} φ_{r}^{*} (x) q_{r} (t)$

The value of curvature function δ(x) reflects the rate of change of the slope of the structural bending vibration curve, which is proportional to the value of curvature mode vibration mode $φ_{r}^{*} (x)$ and should be a smooth curve. In the structural damage caused by local stiffness changes, the curvature function δ(x) is not a smooth curve, the curvature function at the damage will be a sudden change, so as to determine the location of the damage, which is also based on the curvature mode damage recognition technology of the core idea.

Curvature mode is the curvature value of displacement mode, its advantages are: the sensitivity to structural damage is higher than displacement mode, no need for structural damage before the modal information, suitable for bridge structural damage diagnosis: but curvature modal measurements need to be very close to the measurement point, the measurement is easy to be affected by the environment; therefore, the modal measurements and the modal separation technology is not complete enough at the present time.

2.4

Structural damage identification of bridge engineering based on RBF network

2.4.1

Basic theory of artificial neural networks

Artificial neural network is an information processing system that mimics the structure and function of the biological brain, of which the earliest proposed and most influential is the M-P model, which has been continuously improved by researchers and has become a widely used neuron model.

x_i is the input signal of the neuron, θ is the threshold value, f is the excitation function, ω_i is the connection weight between the higher-level neuron and the current-level neuron, and y is the output signal of the neuron.

The neuron processes the information according to the following steps: 1)

The information from the outside world is input to the neuron, and the input information is taken as a weighted value and subtracted from the threshold variable θ to obtain the net input of the neuron s, i.e: 25 $s = \sum_{i = 1}^{n} ω_{i} x_{i} - θ$

To unify the expressions, let ω₀ = –θ, x₀ = 1, rewrite the above equation as: 26 $s = \sum_{i = 0}^{n} ω_{i} x_{i}$

From the processing of the input signal by the neuron, it can be seen that the input of connection weight ω_i > 0 plays an excitation role for the summation, and the input of connection weight ω_i < 0 plays an inhibition role for the summation, so we call the first connection weight an excitation connection, and we call the second connection weight an inhibition connection.

2)

Perform some processing on the net input s to find the output value y, i.e.: 27 $y = f (s)$

Generally referred to as f(s) is the transfer function, the following are several common transfer functions, which satisfy the following relationships:

(1)

hard limit function 28 $f (s) = {\begin{array}{l} 1 & s \geq 0 \\ 0 & s < 0 \end{array}$

From the values of the hard limit function, we can see that this transfer function can be used to categorize data, which is achieved by assigning different values to each of the two different types of data.

(2)

Linear Functions 29 $f (s) = s$

The main role of linear functions is to realize function approximation.

(3)

Sigmoidal function

$f (s) = \frac{1}{1 + e^{- s}}$ function is very widely used, this transfer function can achieve a lot of functions, such as classification of the data, approximation of the function and optimization of the data, etc., and the adjustment of the neuron weights can be done using back propagation, so this transfer function is very important for neural networks.

(4)

Gaussian function 30 $f (s) = e^{\frac{s^{2}}{δ^{2}}}$

The training process of neural networks can be categorized into two types: learning with a tutor and learning without a tutor.

2.4.2

Fundamentals of RBF Neural Networks

Radial basis function neural network is a three-layer forward network consisting of input, hidden and output layers. Figure 1 shows its network structure. It looks similar to the single hidden layer network in form, but they are completely different from each other. The input layer of the RBF neural network can realize the transformation of the data space through the transformation of the radial basis function, and the data from the hidden layer to the output layer is connected through the weights. Each of the input, hidden and output layers has a completely different role: the input layer consists of perceptual units, which are used to connect the network to the external environment; the role of the hidden layer units is to spatially transform the input data, so that the data can be categorized or regressed; and the role of the output layer is to provide a response to the activation pattern of the input.

Let the dimension of the input vector be m, the number of hidden units be n, and the dimension of the output vector be l. The mapping relationship of the RBF neural network consists of two parts:

The first part: the input data is spatially transformed by radial basis function. The j th hidden unit output is: 31 $h_{j} (x) = (‖ x - c_{j} ‖, σ_{j}) = e^{- \frac{{‖ x - c_{j} ‖}^{2}}{2 σ_{j}^{2}}}$

Eq: ϕ(∗)

- the radial basis function of the hidden unit;

||∗||

- denotes the number of paradigms, usually taken as the 2nd paradigm;

x

- n-dimensional input vector, i.e. x = [x₁x₂x₃……x_n]^T;

c_j

- the “center” vector of the jth nonlinear transformation unit;

σ_j

- the width of the j th nonlinear transformation unit.

The second part: the transformed implicit layer space data is weighted and summed, and the output is: 32 $f (x) = \sum_{j = 1}^{n} h_{j} (x) ω_{j}$

Eq: ω_j

- Weights between the jnd hidden unit and the output;

n

- the number of hidden units.

Assuming p sets of input and output samples x_p, y_p, define the objective function (sum of squares error): 33 $E (t) = \sum_{p} {‖ y_{p} - y_{p} (t) ‖}^{2} = \sum_{p} \sum_{k} {(y_{k p} - y_{k p} (t))}^{2} = \sum_{p} e_{p}^{2} (t)$

The goal of learning is to make E ≤ ε.

Eq: y_p(t)

- the actual output of the network after t weights adjustment for the p rd set of sample inputs.

ε

- the predetermined target error, ε > 0.

RBF neural network has more obvious difference with BP neural network, the input layer data of RBF neural network firstly Di through the radial basis function to realize the spatial transformation of data, which is the first learning stage. In addition, the data in the implicit layer space to the data weights in the output layer space is adjusted as the second learning stage.

2.4.3

Design of RBF neural network based damage recognition

The idea of using RBF neural network for structural damage identification is as follows: first of all, we have to choose the appropriate damage indicators, there are various types of damage indicators, different damage indicators can realize different functions, so we must reasonably choose the indicators with high identification efficiency, and the calculation of the damage indicators is generally obtained through the modal parameters of the structure; after determining and calculating the damage indicators, we need to establish a neural network The output parameters of the neural network should be determined according to the goal of damage recognition, such as determining the degree of damage, then the degree of damage as an output parameter; after the completion of the neural network training, when there is a new structural modal data input, the neural network will output the results of the recognition.

The basic steps for damage identification of bridge structures using RBF neural networks are shown in Figure 2. 1)

Prepare the calculation data. The computational data for damage indicators are generally related to the modal parameters of the structure, so we need to conduct field tests or establish a finite element model to obtain them.

2)

Calculate the input and output vectors of the sample. Before building a neural network, we first need to choose a suitable input vectors, and the calculation principle of input vectors will be explained in detail in the next section. The selection of output vectors should be based on the objectives of structural damage identification, which are generally: damage early warning, damage location identification, damage identification and prediction of the remaining service life of the structure.

3)

Construct neural network samples. Theoretically, the samples we select should include the entire sample space, i.e. Input and output vectors of the bridge structure to be recognized under various damage states (damage location and damage degree). In practice, if we utilize the entire sample space of the huge data without screening, it will not only greatly increase the sample collection and network training time, but also may not be able to achieve the desired results; similarly, in the other extreme case, it is even more unlikely that only a small number of training samples can achieve the recognition effect we want. Therefore, we should reasonably and appropriately choose the right amount of training samples.

4)

Train the neural network: The training process of RBF neural network is a process of constantly adjusting the weights between neurons, and the training of the neural network is completed when the error reaches the allowable range.

5)

Test the neural network. After the training of the neural network is completed, we have to evaluate the training results to test whether the neural network has met our functional requirements.

2.4.4

RBF Neural Network Design Functions

1)

Accurate design of RBF neural network

MATLAB Neural Network Toolbox provides function newrbe to accurately design the RBF neural network, which is called in the following form: 34 $n e t = n e w r b e (P, T, S P R E A D)$

Eq: P

- input parameters, expressed in the form of a matrix in MATLAB;

T

- output parameter, expressed in MATLAB in the form of a matrix;

SPREAD

- extension constant, the default value is 1;

net

- the return value of the function call, which can be exactly T when P is entered in the function.

The principle of function newrbe to realize the exact design is that the input parameter matrix is the same as the hidden layer matrix dimension, so it can achieve zero error.

2)

More efficient design of RBF neural network

Function newrb can be a good solution to the problem of network structure is too complex, its working principle is: the network in the beginning of the hidden layer neuron number of 0, the network calculates the neuron’s output whether to meet the error requirements, if not, then the hidden layer neuron number of iterations for 1, recalculate the neuron’s output whether to meet the error requirements, if the error requirements are met, then the network stops iterating, and if the network error does not meet the requirements, then the network iterates. Network error does not meet the requirements, then repeat the above steps until the error meets the requirements or the number of neurons in the hidden layer reaches the number of neurons in the input layer. The call form is as follows: 35 $n e t = n e w r b (P, T, G O A L, S P R E A D, M N, D F)$

Eq: GOAL

- target error;

MN

- maximum number of hidden layer neurons;

DF

- present frequency of the iterative process.

(3) Network identification call function

Network identification call sim function: 36 $\tilde{T} = s i m (n e t, P)$

Eq: net

- the name of the network;

P

- input parameters, expressed as a matrix in MATLAB;

\tilde{T}

- output parameters, expressed as a matrix in MATLAB.

The sum-of-squares error E between the test sample output $\tilde{T}$ and the ideal output T is calculated using the sse function: 37 $E = s s e (T - \tilde{T}) = \sum_{i} {‖ T_{i} - {\tilde{T}}_{i} ‖}^{2} = \sum_{i} \sum_{j} {(T_{i j} - {\tilde{T}}_{i j})}^{2}$

E can be an important indicator of the overall recognition effectiveness of RBF neural networks.

2.5

Finite element modeling

2.5.1

Summary of works

In this paper, the RBF neural network damage identification case study is carried out in the background of a bridge, and the bridge data information comes from the Structural Engineering Test Center. The span structure of this bridge consists of 4 spans (Chongqing direction) simply supported girders + 1 hole 110m reinforced concrete slab arch + 2 spans (Chengdu direction) simply supported girders, and the deck system is simply supported girder-slab structure.

During the implementation of the safety assessment of the bridge structure, the finite element simulation of the bridge structure should adopt corresponding finite element simulation strategies for different analysis levels, analysis objectives and analysis methods. According to the principle of hierarchical depth and multilevel assessment, the safety condition assessment system of large span bridge structure should be divided into three levels: 1)

Damage early warning, the goal is comprehensive, instantaneous and visualization monitoring, and can quickly identify abnormalities and warn at the first time;

2)

Damage identification, the goal is to analyze and determine the nature of the anomalous information, and make a secondary warning of the structural damage site and its degree;

3)

Safety assessment, the goal is to assess the safety and reliability of the bridge structure on the basis of damage early warning and damage identification, and to provide a basis and guidance for the maintenance, repair and management decisions of the bridge structure.

Among them, the structural damage warning and damage identification method based on finite element model is mainly based on the overall condition monitoring technology based on vibration modal analysis and parameter identification, and discerns the state of the structure by analyzing the changes in the dynamic fingerprints related to the structural dynamics and the changes in the structural parameters. Therefore, the finite element simulation (simplified model) of bridge structure aiming at structural damage diagnosis (damage warning and damage identification) is aimed at accurately calculating the dynamic characteristics of the bridge structure such as self-oscillation frequency and vibration mode.

2.5.2

Finite element modeling

A large-scale finite element calculation program ANSYS is used to establish a three-dimensional finite element model for the structural analysis of the beam system of the Kouxi River mega arch bridge. The three-dimensional spatial structure of the structure is retained in the model, including the cross-section of the structure, connection between members, boundary and support conditions, etc., and the mass and stiffness of each member are described independently. The arch ribs, columns, transverse walls, transverse beams between columns and cover beams in the computational model are simulated using beam units, and the bridge deck is simulated using shell units and considered as a continuous bridge deck. The whole model is divided into 1,750 nodes and 1,390 cells. The coordinates of the spatial model are defined as x-direction along the bridge axis, y-axis vertically upward, and z-axis pointing to the side of the bridge side outward the origin is located at the center of the left arch foot section.

3

Results and Discussion

3.1

Analysis of arch bridge damage identification results

Using keras software, the sequence of building a sequential linear stack model is: convolutional layer 1 - overfitting-proof layer 1 - pooling layer 1 - convolutional layer 2 - overfitting-proof layer 2 - pooling layer 2 -flat layer -output layer. Where the activation function of the convolutional layer uses the relu function, the flat layer does not use the activation function, and the activation function of the output layer uses the softmax classification function in multiclassification. In this, the mechanism to prevent the overfitting layer is to randomly discard a specified percentage of weights, which increases the generalization ability of the deep network. In the selection of optimizer, stochastic gradient descent optimizer is used. When the loss value on the validation set rises for 3 consecutive cycles, it is considered that overfitting may have entered and training is stopped.

3.1.1

Inherent frequency

A finite element model of the arch bridge was established, and its first ten orders of frequency were extracted by using the subspace analysis method in the modal analysis of the finite element analysis software. The training data is 81 sets, and the test data is a combination set of 60 sets with 21 sets of damage degree of 75%, so the test data is 81 sets. By selecting the first ten order frequencies on the test set for detail, the specific input vectors are shown in Table 1.

Table 1.

First ten order frequencies of arch bridge

First order	Second order	Third order	Fourth order	Fifth order
0	2.45E-06	1.51E-05	4.04E-05	0.03247
9.45E-08	2.61E-06	1.54E-05	4.10E-05	0.03247
5.57E-07	2.51E-06	1.58E-05	4.18E-05	0.03247
0	2.58E-06	1.63E-05	4.28E-05	0.03246
5.78E-07	2.67E-06	1.72E-05	4.48E-05	0.03245
4.94E-07	2.45E-06	1.51E-05	4.01E-05	0.03247
4.10E-07	2.52E-06	1.65E-05	4.15E-05	0.03246
2.84E-07	2.71E-06	1.75E-05	4.31E-05	0.03246
0.03428	0.08173	0.08853	0.13198	0.23312
0.03428	0.08173	0.08848	0.13192	0.23312
0.03427	0.08172	0.08842	0.13183	0.23311
0.03426	0.08171	0.08834	0.13169	0.23309

From the table, it can be found that the arch bridge model of the first four orders of frequency values closer to 0, the reason is that the first four orders of deformation is mainly reflected in the horizontal direction of the displacement of the bridge abutment, so the fifth order frequency to the tenth order frequency is selected as the input term of the convolutional neural network.

After the training of the neural network is completed, some of its outputs are shown in Table 2. From the above table, it can be seen that the convolutional neural network with the intrinsic frequency of the arch bridge model as the input term has a good ability to recognize the damage degree of the structure. In the small damage degree, the neural network’s error determination for damage degree of 15% is 35% in individual cases, and the accuracy of damage degree discrimination is high in other working conditions.

Table 2.

Training results of arch bridge model

Degree of damage	Tag value	Actual output
0.25	0	0
0.35	1	1
0.45	2	2
0.55	3	3
0.65	4	4
0.25	0	0
0.35	1	2
0.45	2	2
0.55	3	3
0.65	4	4

Comparative analysis plots are drawn for all the output vectors and the training results are shown in Fig. 3.

From the comparison plot of predicted and actual output values in the training result analysis plot, it can be found that the two sets of data are basically on the same inclined plane. At the smaller degree of damage, there are some ups and downs; from the error distribution graph, we can get that the highest value of the ratio of the error to the standard value is 1, which indicates that when the judgment error is made, the difference from the accurate value is only one unit, and there is no situation in which the ratio is greater than 1, which indicates that the model neural network is more stable. Therefore, the analysis of all the output vectors can be synthesized to obtain that the final test set recognition accuracy is 78% with the frequency of the arch bridge model as the input term.

3.1.2

Frequency combinations

For the output layer, it is first labeled. It is divided into 5 categories according to the degree of damage. Divide the training set and test set. Since the first four order modes of the arch bridge finite element model are mainly manifested on the piers, the first four order frequency values are close to 0. In the frequency combination, the fifth order frequency to the tenth order frequency is also selected as the basis of the combination. The input items on the test set are shown in Table 3. a~f are fifth-order frequency - fourth-order frequency, sixth-order frequency difference, seventh-order frequency difference/sixth-order frequency difference, eighth-order frequency relative difference, ninth-order frequency, and tenth-order frequency - ninth-order frequency.

Table 3.

Input vectors of frequency combination test set

A	B	C	D	E	F
0.032431	0.807294	0.996617	0.946447	0.131961	0.10116
0.032428	0.807298	0.996617	9.597352	0.131895	0.101218
0.032423	0.807306	0.996616	9.603805	0.131794	0.101306
0.032416	0.807319	0.996615	9.612078	0.131618	0.101466
0.032432	0.807289	0.996616	9.619341	0.131951	0.101164
0.03243	0.80729	0.996615	9.59507	0.13188	0.101222
0.032427	0.807293	0.996614	9.599349	0.131781	0.101306
0.032422	0.807294	0.996612	9.60444	0.131621	0.101449
0.032432	0.807289	0.996617	9.607996	0.131934	0.101161
0.03243	0.80729	0.996616	9.590933	0.13185	0.101218

A total of 131 parameters were generated after the training of the neural network model was completed. During the training process of the model, its loss value and accuracy changes are shown in Fig. 4.

As can be seen from the decline curve of the loss value, when the number of training rounds reaches nearly 95 times, the loss ratio on the training set and the test set reaches the lowest. Since the accuracy requirement is satisfied, the iterative training is stopped. From the recognition accuracy graph, it can be seen that the accuracy rises faster in the first 10 iterations; when the number of training rounds reaches 50 times, the accuracy of the training set and the test set starts to rise and fall, indicating that it is about to enter the overfitting stage.

After the deep convolutional neural training is completed, the comparison of some of its predicted output values with the actual output values is shown in Table 4. As can be seen from the table comparing the labeled values with the actual output values, the damage recognition accuracy of this network is low when the damage degree is less than or equal to 30%; when the damage degree is greater than 30%, the recognition accuracy is high.

Table 4.

Label value and actual output value comparison

Degree of damage	Tag value	Actual output
0.25	0	1
0.35	1	2
0.45	2	3
0.55	3	3
0.65	4	4
0.25	0	0
0.35	1	2
0.45	2	2
0.55	3	3
0.65	4	4

All the output vectors were visualized and analyzed in the plotting software and the plots of the predicted values against the actual output values and the error distribution were obtained. The results are shown in Fig. 5.

From the training result analysis plot, it can be seen that when the predicted value is 0, some of the actual output values are 0, and the other part of the actual output value is 1. When the predicted value is 3, some of the recognized result values are 2. From the error distribution plot, it can be seen that the maximum value of the ratio of the error to the standard value in the test set is 1, which means that when the network predicts the wrong value, the proximity to the correct value is one level worse. Combining all the output vectors it can be learned that the damage recognition accuracy of the neural network is 71.6% when the arch bridge frequency combination is used as the input vector.

3.1.3

Curvature

For the curvature input vectors of the arch bridge, the same fifth order curvature values to the tenth order curvature values were selected as the basis for the analysis. The test set of curvature input terms for the finite element model of the arch bridge is shown in Table 5.

Table 5.

Test set of curvature input items

Fifth	Sixth	seventh	Eighth	Ninth	Tenth
5.63E-13	8.75E-09	-2.33E-08	-1.72E-08	1.92E-12	7.32E-11
6.38E-13	8.75E-09	-2.33E-08	-1.95E-08	1.92E-12	7.38E-11
7.29E-13	8.75E-09	-2.33E-08	-1.95E-08	1.95E-12	7.50E-11
8.37E-13	1.46E-08	-2.33E-08	-1.92E-08	1.95E-12	7.58E-11
6.15E-13	8.75E-09	-2.33E-08	-1.81E-08	1.87E-12	7.35E-11
7.50E-13	8.75E-09	-2.33E-08	-1.77E-08	1.92E-12	7.43E-11
9.12E-13	1.46E-08	-2.33E-08	-1.81E-08	1.90E-12	7.55E-11
1.16E-12	1.46E-08	-2.33E-08	-1.84E-08	1.95E-12	7.70E-11
5.51E-13	8.75E-09	-2.33E-08	-1.77E-08	1.84E-12	7.20E-11
6.10E-13	8.75E-09	-2.33E-08	-1.81E-08	1.84E-12	7.27E-11

After the training was completed, some of the training results were compared as shown in Table 6.

Table 6.

Label value and actual output value comparison

Degree of damage	Tag value	Actual output
0.25	0	0
0.35	1	1
0.45	2	2
0.55	3	3
0.65	4	4
0.25	0	0
0.35	1	1
0.45	2	2
0.55	3	3
0.65	4	4

The graph of all the input vectors analyzed is shown in Figure 6. As can be seen from the table comparing the labeled values with the actual output values, the two are completely consistent within a certain ten sets of data selected randomly. Therefore, it can be preliminarily concluded that the accuracy of damage identification for this condition is high. The plot of all the output values against the predicted values shows that the two are basically on a complete slope. From the error distribution plot, it can be seen that there are individual data with an error to standard value ratio of 1, indicating that when the recognition error is made, it is only one place away from the accurate value. Combining all the output vectors gives a recognition accuracy of 76.5% for the deep convolutional network with the curvature of the arch bridge as the input term.

3.1.4

Curvature frequency combinations

Distinguish the input term from the curvature-frequency combination of the arch bridge finite element model in Chapter II. In this chapter, the combination of the fifth order curvature to the seventh order curvature and the fifth order frequency to the seventh order frequency is used as the input term of the neural network. Some of the input vectors for its test set data are shown in Table 7.

Table 7.

Curvature frequency combination test set input

Fifth curvature	Sixth curvature	Seventhr curvature	Fifth frequency	Sixth frequency	Seventh frequency
5.63E-13	8.75E-09	-2.33E-08	0.03247178	0.03428187	0.08173032
6.38E-13	8.75E-09	-2.33E-08	0.03246915	0.0342769	0.08172591
7.29E-13	8.75E-09	-2.33E-08	0.03246527	0.03426931	0.08171919
8.37E-13	1.46E-08	-2.33E-08	0.03245865	0.0342561	0.08170754
6.15E-13	8.75E-09	-2.33E-08	0.03247241	0.0342852	0.08173421
7.50E-13	8.75E-09	-2.33E-08	0.03247052	0.03428417	0.0817341
9.12E-13	1.46E-08	-2.33E-08	0.03246779	0.03428294	0.081734
1.16E-12	1.46E-08	-2.33E-08	0.03246338	0.03428143	0.08173379

When the neural network training is completed, some of the training results are shown in Table 8. From the above training results table, it can be seen that when the damage degree is less than 45%, there are some recognition errors, not only when the damage degree is small the recognition error rate is higher. However, when the damage degree is greater than or equal to 45%, the recognition accuracy is higher. Therefore, it can be judged that the neural network model for this input condition is only suitable for predicting structures with large damage values.

Table 8.

Training results

Degree of damage	Tag value	Actual output
0.25	0	1
0.35	1	1
0.45	2	2
0.55	3	2
0.65	4	4
0.25	0	0
0.35	1	1
0.45	2	2
0.55	3	3
0.65	4	4

The comparative analysis plots of all output vectors are plotted as shown in Fig. 7. From the training result analysis plot, it can be learned that the predicted values and the actual output values have vectors with recognition errors within the range of the five predicted recognition categories, but the whole still forms a skewed plane. From the ratio plot of error to standard value, it can be seen that the highest ratio is 1, indicating that the predicted results have some reference value. A comprehensive analysis shows that the identification accuracy of the arch bridge finite element model with curvature frequency as an input term is 81.5%.

3.2

Corrected damage identification for simply supported girder bridges

3.2.1

Finite element model correction

Before the model correction of the bridge, it is necessary to do sensitivity analysis of the dynamic parameters of the structure, ignore the parameters with low sensitivity, and select the dynamic parameters with high sensitivity to correct the model. The 5% range of the dynamic parameters is defined to avoid excessive modification of the dynamic parameters, which may cause large changes in the dynamic characteristics of the structure. Through the analysis of the parameters in this chapter,the height of the T-beam flange plate, the modulus of elasticity of the structural material and the mass density parameters are selected to analyze the effect of these parameters on the fixed frequency of the structure, and the results of the sensitivity analysis are shown in Figure 8.

From the above figure, it can be seen that the fixed frequency of the structure is affected by the height of the T-beam flange plate on the fixed frequency of the structure is very small, the sensitivity is very low and negligible, and affected by the modulus of elasticity of the structural material and the mass of the structure compared with the height of the T-beam flange plate has a higher sensitivity, which is suitable to be used as a correction parameter for the dynamic characteristics of the structure, in the process of correction, because there is no large difference in the modulus of elasticity of the structural material and the mass of the sensitivity, therefore, the weights are all are chosen as 1.

The model correction effects are shown in Tables 9 to 11. Table 9, Table 10 and Table 11 show the calculated and measured frequencies before and after correction, and the parameter correction values, respectively.

Table 9.

Fixed and measured frequency

Test span	Exponent	Vibration frequency (Hz)		Frequency error (Fc-Fe/Fe/Fe)/Fe	Damping ratio(%)
Test span	Exponent	Measured Fe	Computed Fc	Frequency error (Fc-Fe/Fe/Fe)/Fe	Damping ratio(%)
Span 3	1	3.759	3.427581	8.82%	3.268
Span 3	2	12.668	12.066219	4.75%	3.268

Table 10.

Corrected and measured frequency

Test span	Exponent	Vibration frequency (Hz)		Frequency error (Fc-Fe/Fe/Fe)/Fe	Damping ratio(%)
Test span	Exponent	Measured Fe	Computed Fc	Frequency error (Fc-Fe/Fe/Fe)/Fe	Damping ratio(%)
Span 3	1	3.759	3.569	5.05%	3.268
Span 3	2	12.668	12.531	1.08%	3.268

Table 11.

Parameter correction

Bridge model	Elastic modulus E (MPa)	Density p (kg/m³)
Prefix	3.54*10⁴	2558
Corrected	3.87*10⁴	2506

From the tables, it can be seen that the model correction effect is obvious, and the calculated values of the modal information of the bridge after correction are more consistent with the measured values of the modal test, which can pave the way for the later study of bridge damage identification.

3.2.2

Results of damage identification analysis

Finite element analysis is performed on the modified bridge model to extract the healthy state bridge modal data and calculate the vibration curvature modes of each node. The structural damage is simulated by reducing the elastic modulus, and in order to test the damage identification effect of the bridge, the following several damage working conditions are set, and the specific working condition settings are shown in Table 12.

Table 12.

Bridge damage condition

Operating condition	Degree of damage	Damage position
1	15	Unit 14
2	30	Unit 14
3	45	Unit 14
4	15	Unit 6, Unit23
5	15, 30	Unit 6, Unit23
6	15, 30, 45	Unit 6, 14, 23

Based on the working conditions of the model for modal analysis, displacement curvature mode is actually the second-order differential of the relative displacement of the vibration mode, you can get the relative rate of change of the curvature mode of each node of the structure through the central difference theory, through the formula (3.3) to calculate the relative rate of change of the curvature mode of each node, the use of Matlab to process the data, the finishing results are shown in Figure 9.

It can be seen from the above figure that there is an obvious mutation in the damage position linearly, and the mutation amplitude is proportional to the degree of damage. It is easy to find out the damage unit location of the structure from the line graph, and the effect is more satisfactory.

In order to do quantitative analysis of the damage degree of the damage unit, the relative rate of change of the curvature mode of each node in different damage conditions is used as a training sample to train and learn the neural network, and Table 12 is used as an output sample to test the effect of damage recognition. The established neural network model input vector dimension is 6, so the output vector dimension is 3, according to the empirical formula and trial calculation to determine the hidden layer vector dimension is 6 best. Therefore, the neural network here choose 6-6-3 network structure. The target error of the neural network is set to 0.0001, and the training function is selected as TRA function, which has higher precision and faster convergence. The learning rate is 0.05 and the number of iterations is set to 2000. The output results after training are shown in Table 13 below.

Table 13.

Neural network training results

Operating condition	Damage position	Target output	Training output
1	6, 14, 23	0, 0.15, 0	0.0019, 0.1566, 0.0021
2	6, 14, 23	0, 0.30, 0	0.0033, 0.3083, 0.0028
3	6, 14, 23	0, 0.45, 0	0.0028, 0.4574, 0.0026
4	6, 14, 23	0.15, 0, 0.15	0.1557, 0.0016, 0.1562
5	6, 14, 23	0.15, 0, 0.3	0.1533, 0.0036, 0.3061
6	6, 14, 23	0.3, 0.3, 0.45	0.3022, 0.3068, 0.4571

As can be seen from the table, in the structure of the undamaged unit, the output value of the neural network is basically zero, which can realize the correct judgment of the undamaged unit.

The identification of the structural damage location is also basically correct, in order to better see the quantitative identification effect on the damage degree of structural damage units, the relative error between the target output value and the training output value is derived, and the calculation results are shown in Tables 14 to 16, where Table 14, Table 15 and Table 16 are the output errors of damage conditions of Unit 6, Unit 14 and Unit 23 respectively.

Table 14.

Unit 6 damage operating condition output error

Damage operating condition	Target output	Training output	Error	Error ratio
4	0.15	0.1557	0.0057	3.80%
5	0.15	0.1533	0.0033	2.20%
6	0.3	0.3022	0.0022	0.73%

Table 15.

Unit 6 damage operating condition output error

Damage operating condition	Target output	Training output	Error	Error ratio
1	0.15	0.1566	0.0066	4.40%
2	0.3	0.3083	0.0083	2.77%
3	0.45	0.4574	0.0074	1.64%
6	0.3	0.3068	0.0068	2.27%

Table 15.

Unit 6 damage operating condition output error

Damage operating condition	Target output	Training output	Error	Error ratio
4	0.15	0.1562	0.0062	4.13%
5	0.3	0.3061	0.0061	2.03%
6	0.45	0.4571	0.0071	1.58%

From the table, it can be seen that the neural network designed in this paper can quantitatively analyze the damage degree of the damage unit of each condition, and the relative error is within 5%. The main reason for the error is that the correction accuracy of the finite element model needs to be improved, and the training samples are fewer, but the effect is more satisfactory, which verifies the feasibility of the method in practical engineering applications.

4

Conclusion

In this paper, based on the dynamic characteristics of bridge structures, RBF neural networks are utilized to effectively identify the location and degree of damage occurring in bridge structures. It is found that, for arch bridges, the accuracy of damage identification with intrinsic frequency as input is 78%, with frequency combination as input is 71.6%, and with curvature and curvature-frequency combination as input are 76.5% and 81.5%, respectively. Using the sensitivity method to correct the finite element model, the error of the bridge structural dynamic characteristic parameters can be reduced from 8.82% to 5.05%, and more accurate identification results can be obtained. Comprehensively, the damage identification method based on RBF neural network in this paper has good performance and meets the design expectation.

Language:: English

Publication timeframe:: 1 times per year
Journal Subjects:: Life Sciences, Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics, Physics, other

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Research on structural damage identification of bridge engineering based on dynamic parameters

Qianxue Liang

Published Online: Mar 19, 2025

Received: Oct 24, 2024

Accepted: Feb 07, 2025

DOI: https://doi.org/10.2478/amns-2025-0483

KeywordsEngineering structural damage, Curvature modulus, Artificial neural network, Finite element modeling

© 2025 Qianxue Liang, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Engineering structural damage, Curvature modulus, Artificial neural network, Finite element modeling