Abstract
Vibration-based method has been widely applied for damage identification of bridge. Natural frequency, mode shape, and their derivatives are sensitive parameters to damage. However, these parameters can be affected not only by the health of structure, but also by the changing temperature. It is essential to eliminate the influence of temperature in practice. Therefore, a fuzzy neural network-based damage assessment method is proposed in this paper. Uniform load surface curvature is used as damage indicator. Elasticity modulus of concrete is assumed to be temperature dependent in the numerical simulation of bridge model. Through selecting temperature and uniform load surface curvature as input variables of fuzzy neural network, the algorithm can distinguish the damage from temperature effect. Comparative analysis between fuzzy neural network and BP network illustrates the superiority of the proposed method.
1. Introduction
Bridge structure is playing significant role in modern transport system and economic development. With the rapid growth of traffic volume, the loads carried by bridges increase dramatically. External environment also exacerbates the deterioration of materials. Damage inevitably occurs in structures under the coupling effect of load and environment, which will lead to the deficiency of carrying capacity [1, 2]. Therefore, it is necessary to identify the structural damage and strengthen the bridge. The research on appropriate damage identification methods has received extensive attention [3, 4].
Vibration tests have been widely performed in bridge health monitoring. The dynamic characteristics such as eigenfrequencies, modal shapes, and damping ratios of structure contain effective information on bridge health status [5]. Vibration-based damage identification methods are proved feasible in laboratory and field testing. The theoretical background is that damage will modify the stiffness and mass of structure and then alter the modal data. Conversely, modal parameters can be regarded as damage indicators of structure [6]. However, these indicators are sensitive to not only damage, but also environmental conditions such as humidity, wind, and, most important, temperature [7, 8]. Wahab and de Roeck [9] conducted dynamic tests for a prestressed concrete bridge in spring and in winter and observed a change of 4%~5% in natural frequencies. Farrar et al. [10] found that the first eigenfrequency of Alamosa Canyon Bridge varies by approximately 5% during a 24 h time period. Zhou et al. [11] obtained 770 h modal frequencies and temperature data from the instrumented cable-stayed Ting Kau Bridge in Hongkong; the environmental variation accounts for changes in modal frequencies of 0.005 Hz and 0.018 Hz in absolute sense and 1.505% to 6.689% in relative sense for the first eight modes. Moser and Moaveni [12] presented results from a continuous monitoring system installed on Dowling Hall Footbridge. Significant variability in the identified natural frequencies is observed; these changes in natural frequencies are strongly correlated with temperature. Therefore, temperature effect must be effectively considered in practical application of damage identification.
Some researches have been conducted to solve this problem in recent years. One of the methods is to search the correlation between eigenfrequencies and corresponding temperatures [13–15]. Peeters and Roeck [13] adopted a Black-box model to describe the variations of eigenfrequencies as a function of temperature. Damage can be detected if eigenfrequencies of the new data exceed certain confidence intervals of the model. Sohn et al. [14] presented a linear adaptive model to discriminate the changes of natural frequencies due to temperature changes from those caused by structural damage or other environmental effects. Peeters et al. [15] used the ARX models to simulate the eigenfrequencies. If a new measured eigenfrequency lies outside the estimated confidence intervals, it is likely that the bridge is damaged. Z24 Bridge verified its feasibility. However, this kind of method possesses several drawbacks [16, 17]. Firstly, the optimal locations of temperature sensors may be difficult to determine. Secondly, the definition of environmental variables which affect the structural features is difficult. Moreover, it would be difficult for sensors to monitor environmental variables over a long time. Another group of methods can minimize the environmental effect without measuring temperatures. Yan et al. [16] proposed a principal-component-analysis- (PCA-) based method to distinguish between changes of modal data due to environmental variation and structural damage under linear or weakly nonlinear cases. In a companion paper [18], they conducted a further extension of the proposed method to handle nonlinear cases, which may be encountered in some complex structures. Sohn et al. [7] developed a novel detection technique which can take into account the environmental conditions of system in order to minimize false positive indications. Autoassociative neural networks are employed to discriminate system changes of interest such as structural damage from other variations. As pointed out by Meruane and Heylen [19], these methods cannot identify the damage location and severity. They proposed a damage detection method which is able to deal with temperature variations. The objective function correlates mode shapes and natural frequencies, and a parallel genetic algorithm handles the inverse problem.
In this paper, a fuzzy neural network-based damage assessment method which can eliminate the temperature effect is proposed. Adaptive network-based fuzzy inference system (ANFIS) is a fuzzy inference system implemented in the framework of adaptive networks, which avoids the complexity and difficulty of traditional neural networks. Meanwhile, it can also overcome the shortcoming of poor learning ability for traditional fuzzy inference system [20]. Uniform load surface (ULS) is a derivative from modal flexibility, which is found to have much less truncation effect and is least sensitive to experimental error [21]. Therefore, ULS parameter is selected as damage indicator. Pham [22] pointed out that the changes of ambient temperature mainly affect the elastic modulus of the construction material and therefore the stiffness of the entire bridge. Shoukry et al. [23] obtained the relationship between temperature and elastic modulus of concrete. The numerical model of structure assumes that the elastic modulus of the materials is temperature dependent. Through considering temperature and ULS parameter in the calculating process of ANFIS, it can distinguish the damage from temperature effect. In order to verify its superiority, comparative analysis between fuzzy neural network and BP network is conducted.
2. Theoretical Background
2.1. Dynamics Background under Temperature Effect
According to existing research results, temperature alters the modal parameters through a complicated way. On one hand, temperature effect will affect mechanical properties of materials. On the other hand, the geometry is sensitive to temperature, which will change constraint conditions. A simply supported Euler-Bernoulli beam with uniform section shown in Figure 1 is considered for modal analysis under temperature effect.
It is assumed that mass and constraint conditions remain unchanged and temperature only affects the geometry and mechanical properties. The undamped flexural vibration frequency of order for beam structure can be calculated by [24–27] where and are the length and height of beam, respectively. is material density and is elastic modulus.
According to variational principles [28], it can be obtained that where represents an increment in the corresponding parameters.
Therefore, it can be furtherly derived that
Assuming that the thermal coefficient of linear expansion of the material is and the temperature coefficient of modulus is , it can be obtained that
Here we assume that the variation of modulus with temperature is linear for small changes in temperature. Consider the following:
It can be seen from (5) that /°C and /°C for concrete under 100°C, [25]. Therefore, modulus of concrete is the main factor altering the modal frequency.
In above calculation, constraint condition is not considered. The axial force exerted on both ends of the beam is where is the friction coefficient and is weight of the beam.
According to modal analysis, natural frequency of beam under axial force can be calculated by where is moment of inertia. Assuming that , (7) can be transformed into where , , so constraint condition has little influence on modal frequency.
Through above theoretical analysis, we can obtain that temperature effect alters modal frequency mainly by changing the elastic modulus of concrete. Shoukry et al. [23] got the relationship between temperature and modulus by laboratory tests which are listed in Table 1. Based on the research results, an ANFIS-based temperature effect elimination method is proposed.
2.2. Adaptive Network-Based Fuzzy Inference System (ANFIS)
Taking the fuzzy inference system with two inputs and one output, for example, the if-then rules are listed as follows [20]. Rule 1: if is and is , then . Rule 2: if is and is , then , where , are input variables for nodes, symbols , , , and are linguistic expressions, and are output variables and , , , , , are parameters.
The fuzzy inference mechanism of Sugeno model is shown in Figure 2, and the equivalent structure for ANFIS is shown in Figure 3. The same membership functions are adopted, and the output for th node at th layer is represented by .
Layer 1: every node in this layer has a corresponding node function. where symbols , are linguistic expressions (such as “small” or “large”). is membership degree for fuzzy set ; it specifies the degree to which the given inputs and satisfy the quantifier . is membership function which can be Bell, Sigmoid, or other related functions.
Layer 2: nodes in this layer are all fixed nodes represented by Π, and the output is the product of all inputs. Consider the following:
Output of each node represents the incentive intensity for one rule. Generally, the node function can be -Norms operators.
Layer 3: nodes in this layer are all fixed nodes represented by . The th node is used for getting the normalized incentive intensity. Consider the following:
Layer 4: nodes in this layer are adaptive nodes with corresponding function. Consider the following: where are normalized incentive intensity calculated by (11) and are parameters.
Layer 5: this layer is with only one node and labeled by , which is used to calculate the transferred message and acts as the overall output.
2.3. Uniform Load Surface (ULS)
The modal flexibility matrix can be calculated by (14) based on natural frequencies and mode shapes [21]. Consider the following: where is mass normalized mode shape vector, is natural frequency, is the order of modal data, and are node numbers, is the totally orders needed for calculation of modal flexibility matrix. It can be seen from (14) that the modal contribution to flexibility matrix decreases rapidly as the frequencies increase, so the flexibility matrix converges rapidly as the number of contributing lower modes increases. It reveals that an approximation of flexibility matrix can be obtained through several lower modes.
The deflection vector under uniform load which is called the uniform load surface [27] can be calculated by where
Uniform load surface curvature (ULSC) can be obtained by the second order differential of , as shown in the following equation: where is the element length.
The uniform load surface curvature difference (ULSCD) can be calculated by where , are the uniform load surface curvatures before and after damage, respectively.
Numerical simulation is conducted for a simply supported beam with rectangular cross-section in order to verify the effectiveness of ULS parameters. The length () is 9.0 m; the width () and height () of cross-section are 0.6 m and 1.0 m, respectively. The material is concrete with the compressive strength of 30 Mpa and density of 2500 kg/m3. Finite element model is constructed by ANSYS; it includes 15 elements and 16 nodes with element length of 0.6 m (Figure 4).
The damage severity of structure is represented by reduction in the element stiffness and it can be defined by where represents the damage severity of elements, is Young’s modulus of the bridge material, and the superscripts and represent undamaged and damaged elements, respectively.
Taking the damage identification of element 8 with damage severity of 0%, 10%, 20%, and 30% at temperature 30°C, for example, the ULS, ULSC, and ULSCD parameters can be calculated by (15)~(18); they are shown in Figures 5, 6, and 7.
As can be seen from Figures 5~7, ULS parameter can be used to identify the damage occurrence, while ULSC and ULSCD can localize the damage, and ULSCD possesses better effect.
3. Numerical Simulation for Damage Identification Based on ANFIS
3.1. Damage Identification Process and Characteristic Parameters
The specific calculation process for ANFIS-based damage identification is shown in Figure 8.
The normalized temperature and ULSCD parameters are selected as input variables of ANFIS, while damage severity of element is output one. Consider the following: where is normalized ULSCD vector and is normalized temperature vector.
Taking the damage identification of element 8, for example, its input variable can be represented by
Membership function for input variable is Gauss type; ANFIS can be initialized by fuzzy-C clustering. The ANFIS structure used in this paper is shown in Figure 9.
3.2. ANFIS-Based Damage Assessment under Temperature Effect
In order to determine the parameters of ANFIS, certain numbers of training samples are needed to realize the adjustment and optimization. 15 samples listed in Table 2 are selected for training and forming the ANFIS structure.
A hybrid learning algorithm based on back propagation and least squares is used for training, which can adjust the premise and conclusion parameters and produce if-then rule base automatically.
ANFIS can achieve adaptive adjustment of membership functions. The same initial function is adopted for input and output variables; it is shown in Figure 10.
The membership functions for input variables ( and ) after training are shown in Figures 11 and 12.
Test samples are constructed to verify the feasibility of ANFIS. The test samples and corresponding identification results are listed in Table 3.
As can be seen from Table 3, the proposed method in this paper can effectively identify the damage condition of bridge. It reveals that ANFIS can realize the adaptive identification and possesses favorable accuracy.
4. Comparative Analysis between ANFIS and BP Networks
4.1. Damage Identification Based on BP Networks under Temperature Effect
Changes of frequencies are used as damage identification parameters; it can be calculated by where and are natural frequencies before and after damage, respectively.
The input variable for BP neural networks is shown in the following equation:
Under the temperatures −20°C, −10°C, 0°C, 10°C, and 20°C, damage severity of 5%, 10%, 15%, and 20% for each element of 4, 6, and 8 is selected as training samples, while 7%, 12%, and 18% are test samples. The identification results are listed in Table 4.
The maximum relative error for BP neural networks-based damage assessment is 5.17%; the identification results are satisfactory.
4.2. Comparison of Identification Accuracy between ANFIS and BP
Comparative analysis is conducted in order to compare the superiority between BP-frequency and ANFIS-ULS-based methods. A similarity calculation formula is constructed to conduct evaluation considering the dimension difference between two methods. Consider the following: where and are the expected and actual outputs of damage identification, respectively. represents the distance between and . The bigger the is, the lower the relevance between and is. is the number of test samples.
According to (24) and Tables 3 and 4, the similarities are calculated for the identification results of BP and ANFIS. , , and . Therefore, the identification results of ANFIS are more relevant to the expected results. It reveals that ANFIS possesses more favorable accuracy.
5. Conclusions
Temperature effect can cause abnormal changes of modal parameters, which will lead to incorrect damage identification results. This paper presents an effective strategy for eliminating temperature effect in damage identification of bridge. ANFIS combines the advantages of neural networks and fuzzy inference system, which is used as damage identification algorithm. ULS, ULSC, and ULSCD are proved to localize damage locations accurately; ULSCD possesses more favorable effect. Therefore, temperature and ULSCD are treated as input variables of ANFIS for the damage assessment. In numerical simulation, elastic modulus of concrete is assumed to be temperature dependent; 15 samples are used for training and constructing ANFIS structure. Numerical simulation results reveal that the proposed method can effectively identify the damage condition of test samples. Comparative analysis is conducted for comparing the superiority between BP and ANFIS. A similarity calculation formula is constructed for evaluation, and the comparative analysis reveals that ANFIS results are more relevant to actual situation. It means that the proposed method in this paper possesses more favorable accuracy than BP network.
Considering the complexity of damage identification of bridge under temperature effect, the damage identification with more samples and damaged elements should be conducted in the future.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors express their appreciation for the financial support of National Natural Science Foundation of China under Grant nos. 51378236 and 51278222 and “985 Project” of Jilin University.