Detection of fatigue cracking in steel bridge girders: A support vector machine approach

archives of civil and mechanical engineering 17 (2017) 609–622

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Original Research Article

Detection of fatigue cracking in steel bridge girders: A support vector machine approach Hassene Hasni, Amir H. Alavi *, Pengcheng Jiao, Nizar Lajnef Department of Civil and Environmental Engineering, Michigan State University, East Lansing, MI 48823, USA

article info

abstract

Article history:

This study presents an artificial intelligence approach for the detection of distortion-induced

Received 15 June 2016

fatigue cracking of steel bridge girders based on the data provided by self-powered wireless

Accepted 19 November 2016

sensors. The sensors have a series of memory gates that can cumulatively record the duration

Available online 31 January 2017

of the applied strain. The gates are activated as soon as the electrical charge generated by piezoelectric strain transducer exceeds pre-defined thresholds. In the present study, the

Keywords:

distribution of the sensor output has been characterized by a Gaussian cumulative density

Support vector machine

function. For the analysis, extensive finite element simulations were carried out to obtain the

Fatigue cracking

structural response of an existing highway steel bridge girder (I-96/M-52) in Webberville,

Steel bridges

Michigan. Different damage states were defined by extending the lengths of the crack at the

Self-powered wireless sensors

web gaps from 10 mm to 100 mm. Damage indicator features were extracted for different data

Damage detection

acquisition nodes based on the sensor output distribution. Subsequently, support vector machine (SVM) classifiers were developed to fuse the clustered features and identify multiple damage states. The results indicate that the models have acceptable detection performance, specifically for cracks larger than 10 mm. The best classification performance was obtained using the information from a group of sensors located near the damage zone. Published by Elsevier Sp. z o.o. on behalf of Politechnika Wrocławska.

1.

Introduction

Multi-girder steel bridges are widely used throughout the highways in the United States. One of the main factors affecting the performance of these structures is the application of the repetitive loading over the steel-girder components. These load-carrying components deform under the live (traffic load) and the dead load of the structure. A typical steel girder bridge is composed of three main parts: girders, diaphragms, and stiffeners. The diaphragms are structural elements that provide resistance to the transverse traffic and wind loading.

The stiffeners connect the girder to the diaphragm. Over many years, inspections conducted on steel-girder bridges revealed that these structures are suffering from fatigue cracking under cyclic loading [1]. More specifically, low resistance to fatigue has been observed in structural members subjected to out-ofplane distortion. The phenomenon of out-of-plane distortion is impacted by a variety of factors such as thermal forces, traffic flow, differential deflection of the adjacent beams, etc. [2,3]. Fig. 1 displays the schematic formation of fatigue cracks in a steel girder caused by out-of-plane distortion. Fig. 1(a) displays an illustration of a steel bridge before deformation in a perspective view, and Fig. 1(b) shows the side view of the

* Corresponding author. E-mail addresses: [email protected], [email protected] (A.H. Alavi). http://dx.doi.org/10.1016/j.acme.2016.11.005 1644-9665/Published by Elsevier Sp. z o.o. on behalf of Politechnika Wrocławska.

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Fig. 1 – Schematic illustration of distortion-induced fatigue cracking: (a) bridge before deformation in a perspective view, (b) side view of the bridge in the initial stage, and (c) different types of fatigue cracks caused by out-of-plane distortion (D).

bridge in the initial stage. Fig. 1(c) schematically shows the cracks caused by out-of-plane distortions. It can be seen that the deformations of the girder web are caused by the differential displacement (d) between the two girders, which leads to the out-of-plane distortion (D). Such distortion eventually causes fatigue cracks on the girders in the form of horseshoe and horizontal cracks. Therefore, fatigue cracks usually occur at the girder web gap due to out-of-plane distortion. The distortion-induced fatigue cracks may occur as horizontal or horseshoe cracks at the top or bottom of the girder to stiffener connections (Fig. 1 (c)). More details on the forming mechanism of these cracks can be found in [4,5]. Different models have been developed to investigate the behavior of bridges [6], with particular focus on the retrofitting approaches to deal with this common type of structural damage [5,7]. However, the selection of an appropriate repair strategy is complicated and depends on many factors. On the other hand, significant cost of maintenance and retrofitting of stiffener-girder connections implies the necessity of detecting the damage progression at early stages to prevent severe damage to the bridge structures. The main goal of this study is to develop a new artificial intelligence (AI)-based approach for the detection of distortioninduced fatigue cracking of steel bridges based on the data provided by a newly developed self-powered wireless sensor. The sensor is empowered using piezoelectric transducers

through harvesting energy from the mechanical loading experienced by the structure. Different studies have been conducted to investigate the energy harvested from ambient excitation [8]. In order to calibrate the AI detection models, different finite element (FE) models of steel girders with complex geometry components were developed and the structural response of the girder was subsequently obtained. The fatigue life of the girder was determined based on the J-integral concept and Paris Law [9,10]. Several damage states were defined by extending the crack lengths. Different numbers of sensing locations were defined to monitor the strain changes due to damage progression. The sensing nodes were placed around the connection between the webs and the stiffeners to determine the optimal sensors configurations that maximize the detection performance of fatigue cracking. Thereafter, features representing the sensor output were extracted from the strain data at the sensing nodes. The obtained features were then fed into a support vector machine (SVM) classifier to identify multiple damage states.

2.

The self-powered wireless sensor

Wireless sensors are widely used as alternatives to the traditional wired sensors for structural health monitoring (SHM) [11,12]. A major limitation in the deployment of the

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conventional wireless sensors is finding a reliable and costeffective power supply. Many of these sensors use batteries that have a limited power lifetime and it needs to be replaced regularly. A viable solution to this power dilemma is to harvest energy from the ambient excitations [13,14]. In this context, piezoelectric transducers are widely used to convert environmental mechanical energy into an electrical energy [15,16]. In general, piezo-based self-powering can be categorized as: (1) harvesting electrical energy from the signals that are different from the signal being monitored or (2) from the signal being monitored [17]. Nearly all of the existing energy harvesting studies are focused on the first approach [18,19]. Based on the second approach, a new class of self-powered wireless sensors (SWS) has been recently developed and tested by the authors [20–23]. This type of sensor uses piezoelectric transducers to empower an array of ultra-low power floating gate computational circuits. SWS has a series of memory cells that cumulatively store the duration of strain events when the amplitude of the input signal, coming from the piezoelectric material, exceeds different thresholds. In fact, the sensor acts as a non-volatile memory for data storage, which optimize the need of the sensor in power. The data could be retrieved offline without the need to power the sensor. A radio frequency identification (RFID) scanner can be used to periodically read the data stored on-board the sensor [20,23]. One of the main advantages of this sensing system is the fact that it is ‘‘response-based’’. All the effects due to variations in load location, load magnitude, traffic wander, environmental effects such as temperature and moisture, material aging and degradation are aggregated in the strain response recorded by the sensor over time. This feature makes the sensor suitable for long-term SHM. Most of the other existing solutions evaluate the conditions of the system at a given instant. These methods present only a snapshot at the time where the measurements are taken. Thus, the obtained results are highly influenced by the environmental conditions. Since the developed SWS records each and every event at all time, it will aggregate all these short-term fluctuations. Thus, if long-term shifts are observed in the results, they are most probably correlated with condition degradation. An illustrative example of the level crossing cumulative time counting implemented by the gates with a constant injection rate can be found in [21,22]. Despite several advantages offered by this self-powered sensing technology, interpretation of the compressed data generated by such systems is a challenging task. With the purpose of solving this, previous research of our group revealed the notable efficiency of the SWS for continuous monitoring of infrastructural systems [20]. In addition, previous studies by Alavi et al. [21–24] on damage detection using the SWS did not take into account the number of cycles to failure, which is crucial in the case of fatigue failure. The present study proposes an SVM approach for the detection of fatigue cracking of steel bridges using the data provided by the SWS. The emphasis was placed on the distortion-induced fatigue cracks occurred at girder to stiffener connections. The entire damage detection procedure can be divided into the following major phases:

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Numerical simulation of the targeted structure;
Extraction of strain data and generation of the distribution histograms based on the SWS sensing mechanism;
Extraction of preliminary damage indicator features;
Fusion of data from a network of sensors to define more informative damage indicators;
Fatigue damage classification using an SVM approach.

3. Numerical analysis of distortion-induced fatigue cracking 3.1. Geometry, loading and boundary conditions of the steel girder For the numerical simulation carried out in this study, the highway steel bridge (I-96/M-52) in Webberville, Michigan, U.S. was selected. The steel girder under consideration was modeled using ABAQUS Version 6.12. Fig. 2 shows the geometry, loading and boundary conditions of the selected structure. Fig. 2(a) presents the geometry of the structure. In Fig. 2(a1), a 3-D model was created using SolidWorks. Fig. 2(a2), (a3), and (a4) show the end view, top view, and side view of the model respectively. It can be seen that the overall length of the girder was 7.62 m and the three beams were spaced by 1.93 m. The steel girders have the same cross section (W920  233) as the I-96/M-52 bridge. A web gap length of 25.4 mm was used for the simulations. Fig. 2(b) displays the boundary conditions of the model. The steel girders were modeled with a simply supported boundary conditions. The top flanges of the three beams were restrained with respect to translation in the z direction and rotation along x axis. The loading was applied in the form of vertical displacements to the cut edges of the lower flanges of the two exterior I-beams (Fig. 2(b)). The imposed displacements were 5 mm vertical displacement at the left outer girder and 15 mm at the right outer girder. More details of the dimensions and material properties of the stiffener and diaphragm plates are listed in Table 1. A linear elastic material was selected for the analysis.

3.2.

Numerical model

3.2.1.

Shell element-based FE model

The structure was modeled using shell elements in ABAQUS. Shell elements performs sufficient computation efficiency since they formulate thin structures with much less elements than 3-D solid elements. Quadrilateral shell elements with reduced integration (S4R) were used in this study. The total number of elements and nodes of the intact structure were 180,320 and 182,050, respectively. The simulation of the intact structure took about 2 min. The meshed structure is shown in Fig. 3(a) and an isometric drawing of the used S4R element is shown in Fig. 3(b). A static analysis was selected for the simulations. Besides, the analysis was done for small deformations. At the region of web gap, a finer mesh was used with an element size smaller than 2 mm such that the stress concentration zones can be accurately captured. The maximum stresses in the central girder were located around the

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Fig. 2 – Selected steel structure: (a) technical drawings of the structure, including (a1) a perspective view (a2) an end view, (a3) a top view, and (a4) a side view, and (b) assembly of the structure in ABAQUS with loading and boundary conditions. Table 1 – Geometry and material properties of the girder. Geometry (mm)

Material Overall model Young's modulus (GPa)

200

Poison's ratio

0.3

Length Spacing Web gap length

Stiffener 7620 1933 25.4

Length Width Thickness

connection stiffener to web. As it seen in Fig. 4, the maximum obtained principal strain of the intact structure was 1.103  103. The stiffeners and diaphragms were meshed using an element size of 25 mm  25 mm and the beams were meshed using a 2 mm  2 mm element size around the stiffener to web

Diaphragm 812 140 12.7

Length Width Thickness

1863 457 12.7

Beams W920  223 (W36  150)

connection. The rest of the central girder was meshed using 2 mm  76 mm quadrilateral element. Different simulations were performed to find the optimal mesh size that guarantees the numerical convergence of the solution. The results of the convergence analysis are shown in Table 2 and Fig. 5. It should be noted that the strain values were obtained for an element of

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Fig. 3 – Mesh details: (a) meshed structure, and (b) S4R element.

Fig. 4 – Maximum principal strain around the area stiffener to web connection of the central girder.

Table 2 – Results of the numerical convergence analysis. Element size h (mm) 200 150 100 50 25 10 5 3 2 1.5

Maximum principal strain (104) 1.74 2.37 4.34 4.71 5.9 8.29 9.19 9.82 11.03 11.01

the central girder web at the connection stiffener-web. As mentioned above, this element presents the highest deformation of the central girder. Fig. 5 shows that the element size of 2 mm  2 mm is sufficient to obtain satisfactory results. In particular, the error between the element sizes of 1.5 mm  1.5 mm and 2 mm  2 mm is only 0.18% (2 me difference). Therefore, the 2 mm S4R elements provide adequate accuracy of the numerical results.

3.2.2. Comparison between shell and 3D solid element-based FE model In order to validate the accuracy of the shell model developed in the previous, a 3-D solid finite element model was

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Maximum Principale Strain(×10-4)

12

Table 3 – A comparison of the shell and 3D FEM models. Shell

10 4

Maximum principal strain (10 ) Computational time (min)

8 6 4 2 0 0.0

0.1

0.2

0.3 0.4 1/h (mm-1)

0.5

0.6

0.7

Fig. 5 – Variation of the maximum principal strain as a function of the element size.

developed and simulated using ABAQUS. The FE model that has exact the same geometry, loading, and boundary conditions as the shell model. The entire steel girder was meshed using eight-node brick elements with reduced integration (C3D8R). Fig. 6 shows the assembly of the structure, the refined portion of the central girder (at connection stiffener to web),

11.03 2

3D solid Error (%) 10.93 41

0.91

and the maximum principal strain at the connection between the stiffener and the web. Different simulations were conducted to obtain the optimal element size that guarantees the numerical convergence. The optimum was found to be 2 mm  2 mm  2 mm close to the stiffener to web connection as shown in Fig. 6(a). The stiffeners were meshed with an element size of 20 mm  20 mm  4 mm. The element size of all the other regions varied between 50 mm and 150 mm. A summary of the comparison between the shell model (S4R element-based) and the 3D solid model (C3D8R element-based) is illustrated by Table 3. The error of the maximum principal strain occurred at the elements located at the connection stiffener to web was found as 0.91%. However, the computational time of the 3D solid finite element model was up to 20.5 times higher than the shell model. Compared to the 3D solid model, the shell model delivered accurate results in a much less computational time, and therefore, the shell model was used for the rest of the simulations.

Fig. 6 – 3D finite element model (a) assembly and refined mesh portion, and (b) maximum principal strain around the connection stiffener to web.

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Fig. 7 – Mesh around the crack tip.

3.2.3.

Fatigue analysis of the steel girder

In this paper, the contour integral method was used to calculate the stress intensity factor. A predefined crack was placed 25.4 mm (1 inch) from the upper flange of the central girder, with an initial length of 10 mm. Damage states were defined by increasing the crack length from 10 mm to 100 mm. For each damage sate, the maximum principal strains and the J-integral values were extracted to determine the number of cycles needed to propagate the crack and to compute the sensor output. The optimal element size of the damaged structure was 0.2 mm around the crack tip vicinity and 1.8 mm in the contact area between the stiffener and the web of the central girder. Triangular elements were used to mesh the crack front. In order to perform a contour integral analysis under ABAQUS, three different entities should be defined: the crack front, the crack tip and the crack extension direction. The crack front is the forward part of the crack and it is useful to evaluate the first contour integral (Fig. 7). The crack tip is a point (for 2-D parts) to be selected from the assembly where the crack extension direction is defined. Thereafter, the crack extension direction is defined by selecting the points representing the start and the end of the crack. Triangular elements of type S3 were used to mesh the crack front region and the remaining part is meshed using quadrilateral elements (S4R) as indicated in Fig. 7. The triangular elements are recommended to mesh the crack front. However, this type of elements should not be used to mesh the contour integral region. Quadrilateral elements should be used instead [25]. A finer mesh around the crack tip was used to capture the high stress field at the crack front region. The J-integral is calculated based on elements surrounding the crack front. More contours should be requested in the analysis to check the accuracy of the results. In some cases, J-integral estimates might vary from one contour to the other. A strong or gradual variation of these estimates indicates an error in the contour integral definition or the mesh is not small enough to accurately calculate the J-integrals [25]. In linear elastic

problems, the first and second contours are usually inaccurate. In the present study, 8 contours were requested to ensure the accuracy of the J-integral calculations. The percent variation of J-integral values between the last two contours varies between 1% and 3% depending on the damage state. Therefore, the average of the last 2 contours was used for the analysis. In addition, the element dimensions were 14 mm  14 mm for the stiffeners and 18 mm  15 mm for the diaphragms. Computational time took about 3 min for each case. The maximum principal strains for two typical crack lengths are presented in Fig. 8. Thereafter, based on the J-integral estimates, the stress intensity factor was calculated with respect to the crack length. Then, Paris Law was used to estimate the number of loading cycles to propagate the crack by Da based on the following equation [9,10]: Da ¼ cDKn DN

(1)

where c and n are material constants, a is the crack length, N is the number of cycles and K is the stress intensity factor. These parameters were taken 2.40  102 and 3.3, respectively [26]. As seen in Fig. 9, a 100 mm crack will occur after about six million cycles. Accordingly, the life span of the girder was divided into 6 different stages. These periods represent the date/time of readings of the sensor output for the post-processing. Each stage consists of 1 million loading cycle.

4.

Damage detection results and analysis

4.1.

Damage detection mechanism

The level crossing cumulative time counting implemented by the proposed SWS is schematically presented in Fig. 10. The main information that can be extracted from the sensors is the cumulative duration of strain events shown in Fig. 10. Based on previous studies [21–23], the sensor output can be expressed by the following Gaussian cumulative density function (CDF):

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Crack length (mm)

Fig. 8 – The FE results for (a) a = 20 mm (maximum principal strain = 2753 me), and (b) a = 70 mm (maximum principal strain = 3192 me).

   1 gm 1 þ erf pffiffiffi 2 s 2

120

CDFGaussian ðgÞ ¼

100

in which m, s, a and g refer to the mean of the strain distribution, standard deviation with respect to load and frequency, the total cumulative time of the applied strain measured by the entire gates and the gate number, respectively. For a better representation, the CDF can be transformed to a Probability Density Function (PDF) as indicated in Fig. 11.

80 60 40 20 0 0

2 4 6 Number of cycles (× 106)

Fig. 9 – Number of cycles vs. crack length.

8

(2)

For the analysis, 400 sensing nodes were placed in the horizontal and vertical directions on the upper half of the central girder around the stiffener to web contact area (20  20 with 20 mm spacing). These nodes represent the actual piezoelectric transducers (PZTs) that are attached to the structure. The average strain of the elements located inside a 10 mm diameter circle centered at the sensing node locations was used for the analysis. Fig. 12 displays the locations of the 400 data acquisition nodes on the structure. The geometry information is summarized in Table 4. Ten strain levels were defined for the girder as shown in Fig. 13.

Fig. 10 – Level crossing cumulative time counting implemented by SWS.

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Fig. 11 – Sensor output distribution (a) CDF fit, and (b) transformed PDF.

Fig. 12 – Locations of the sensing nodes.

4.2.

Information fusion

In order to evaluate the information provided by the sensing nodes, the m and s values for all of the 400 sensors for different damage scenarios were calculated. Then, a data fusion approach was proposed for the detection of damage progression in the girder. The basic idea was to use original parameters from the distribution (m and s), sensor location, and new features that include the effect of different groups of sensors. Subsequently, the damage state (D) was defined as follows: D ¼ function of ðm; s; xi ; yi ; Zi Þ

(3)

8 p p s s av > p > > Zs i ¼ i p > > > s STD > > > > > p p > > m mav > > > Zpmi ¼ i p > < m STD

Table 4 – Geometry of the sensing region, crack, and sensors. Geometry (mm) Sensing region (length  height  thickness) Crack length Distance between the centers of neighboring sensors Diameter of data acquisition nodes

where xi, horizontal distance of the sensing node from the sensing region center; yi, vertical distance of the sensing node from the sensing region center; m, mean of the PDF distribution; s, standard deviation of the PDF distribution; Zi, functions that include the information collected by the whole network of sensors. These functions are defined as follows:

400  400  12.7 10; . . .; 100 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}10 20 10

(4)

> p p > > s i mav > p > Z ¼ > p sm > i > s av > > > > > p p > > m mav > > : Zpmsi ¼ i p s av p

p

p

p

where mav ; mSTD ; s av ; s STD are, respectively, the average of m, standard deviation of m, average of s and the standard deviation of s of all of the sensors for a damage state p. The subscript i is the number of the sensor (the data acquisition node). The new defined features were inspired from the z-score function

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Strain Threshold Level (με)

600 500 400 300 200 100 0 1

2

3

4

5 6 7 Gate Number

8

9

10

Fig. 13 – Strain threshold levels.

[21,22]. As indicated by the equations above, the average and standard deviation of m and s include the effect of the whole group of sensors at the sensing location i.

4.3.

regime of austenitic stainless steel [28], non-destructive investigation of corrosion current density in steel reinforced concrete [29], prediction of the capacity of CCFT short columns [30], prediction of the pull-off adhesion of the concrete layers in floors [31,32], modeling of shear strength of RC deep beams [33], structural assessment and damage identification, prediction of the scour below submerged pipeline and scour depth downstream of sills [34,35], prediction of settlement of shallow foundations [36], maximum dry density and unconfined compressive strength of cement stabilized soil [37], factor of safety of soil nailed slopes [38], prediction of soil liquefaction susceptibility [39], etc. However, the proposed AI-based data fusion framework consists of the following main stages:
Structural simulation;
Information fusion in which features that are expected to characterize different properties of structures are extracted from a network of sensors; and
Fusion of the clustered features. The data fusion is performed using the AI classifier.

Damage detection using support vector machines

A damage detection process can be treated as a pattern recognition problem [21]. The solution is to use a classifier which can classify structures either as damaged or healthy. To this aim, an AI-based data fusion system was proposed for damage detection (Fig. 14). The AI techniques are considered as alternatives to existing traditional methods for tackling real world problems. They determine the model structure by automatically learning from data. AI has different well-known branches such as artificial neural network (ANN), fuzzy inference system (FIS), adaptive neuro-fuzzy system (ANFIS), and support vector machines (SVMs). In the last two decades, the AI methods have been widely used for tackling problems in civil engineering domains such as structural engineering, hydraulic engineering, geotechnical engineering, earthquake engineering, etc. In this context, some well-established studies are: calculation tensile strength and yield strength of dual phase steels [27], modeling true stress of dynamic strain aging

Fig. 14 – Data fusion flowchart.

Using the fitted m and s parameters obtained of a Gaussian distribution, different damage indicator features were obtained for a specified number of sensors. The defined features simultaneously fuse the information provided by array of scattered sensors. The damage indicator vectors were then used for the calibration of the classifier. Subsequently, a validation phase was performed to check the damage detection performance of the classifier. Among different AI techniques, SVMs have been widely used in the field of damage detection and structural identification [40,41]. Some of the main advantages of the SVM method are as follows [42]:
It has regularization parameters in order to avoid over fitting;
It uses the kernel trick which makes the user able to design the kernel via engineering approach;
The SVMs optimization techniques are based on a convex optimization to avoid local minima problems. SVM training always find a global minimum;
They provide an alternative solution when linear decisions hyper-plans are not sufficient to separate the classes by mapping the input data into a feature space resulting in a nonlinear classifier;
Other AI methods suffers from local minima, however the solution given by an SVM is always unique and global. Unlike ANN that they are based on empirical minimizations, the SVMs are less prone to overfitting because they use structural minimization. The main objective of SVM is to find an optimal hyperplane that maximize the margin between the nearest points of different classes as indicated in Fig. 15. SVM uses the kernel method to map the input feature to a higher dimensional space where the data becomes separable and then find an optimal hyperplane that maximize the margin between classes. This concept is called the 'kernel trick' as it illustrated in Fig. 16.

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formulation of the SVM method is defined by the flowing maximization problem: maxa 2 Rm WðaÞ ¼

m m X 1X ai  a a y y kðxi ; xj Þ 2 i;j¼1 i j i j i¼1

(5)

subject to the constraints of: 8 < 0m< ai < C X ai yi ¼ 0 :

and i ¼ 1 to m

(6)

i¼1

Fig. 15 – Margin and separating hyperplane illustration for the SVM method.

The SVM classification process can be defined as follows: Find a decision function f that maps inputs x into labels y, where (x,y) are the input-output pairs. By identifying the function f, a new label xn could be classified as yn based on the sign of the discriminant function f(xn). The mathematical

2

The set (xi,yi) are the input pairs (data, class label), m is the number of training samples, and the a is a representative vector of the relative importance of training sample i in the classification of a new sample. C is the regularization parameter (C > 0) and k is the kernel function. Finally, in order to classify a new data xn, the discriminant function f is defined as follows: f ðxn Þ ¼

ai yi kðxn ; xi Þ þ b

(7)

i¼1:m

Class 1 Class 2

1.5

X

5 Class 1 Class 2 Seperating hyperplane

4

z = x 2 + y2

1

y

0.5 0 -0.5

3

2

1

-1 -1.5

0 5

r = 1.2

0

-2 -2

-1

0 x

1

2

y

-5

-4

0

-2

2

x

Fig. 16 – The mapping process using the transformation function: z ¼ x2 þ y2 : (a) non-linearly separable 2D data, and (b) linearly separable 3D data.

Fig. 17 – The SVM performance on the: (a) testing data, and (b) validation data.

4

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Fig. 18 – Set of sensors with the highest detection rate.

Typical SVM kernels are: linear kernel, polynomial, radial basis function (RBF) and sigmoid [40,41]. For the current study, the RBF kernel was used. The RBF kernel function is given by the following expression:

kðx; yÞ ¼ exp 

jjxyjj2 2d2

! (8)

where d is the RBF parameter to be specified along with the SVM parameter C. Detection rate was used to evaluate the classifier performance as follow:

Detection ¼

Number of data correctly classified Total number of data

(9)

The total number of data points was 2400 (400 sensing nodes  6 damage states). The labels to be classified were 6 damage states (D1, . . ., D6) each represented the girder condition after 1 million cycles. The data was divided into three subsets: 70% for training, 15% for testing and 15% for validation. The SVM algorithm was run for different sensor configurations by adding the number of sensors from 1 (single sensor) to 400 (the entire sensor network). The optimal

Fig. 19 – Confusion matrixes for the best sensor configuration.

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parameters d and C were found through an extensive searching algorithm. Fig. 17 displays the results for the testing and validation sets for different combination of the input parameters. The results indicate that the new Z features have good performance for both of the testing and validation sets. Using only the individual sensing features m and s results in a very low performance (around 20–33%). An interesting observation from Fig. 17 is that both curves present the highest values for a number of sensors below 30. This indicates that the damage is located in the upper area of the web where the first 30 sensors are located. The best detection rate for the training, testing and validation was obtained using Sensing nodes 4–16 as shown in Fig. 18. The optimal value of the d and C parameters were 0.9, 100, respectively. The best performance of the classifier was: - Training = 83% - Testing = 82% - Validation = 85% Fig. 19 displays the confusion matrixes for the best sensor configuration. As seen in Fig. 19, class 2 is misclassified as class 1 in all cases. This indicates that there is not important variation in strain amplitude between 10 mm and 20 mm crack lengths. Therefore, the classifier cannot differentiate between the two classes. However, for the other damage states, the detection performance is satisfactory.

5.

Conclusions

Many older steel girder bridges are affected by the distortioninduced fatigue cracking. In this study, an SVM-based procedure has been proposed for the detection of fatigue cracking in steel girders based on the data of the SWS. The sensor operates by harvesting the energy from the host structure. It can record the cumulative durations of the applied strain signal at predefined threshold levels. However, the sensor output is compressed into a single histogram. This makes the interpretation of the data very challenging. Moreover, detection of the fatigue cracking stages is a complicated task due to notable changes of the strain patterns during crack propagation. In order to evaluate the performance of the sensor, a numerical study was carried out on a typical girder. The damage detection process was divided into different phases: First, the data was collected from the FE strains and the features were extracted to define individual damage indicators (PDF parameters of m and s). Thereafter, a data fusion model was defined based on the previously extracted features and the sensors group effect. Finally, the new features were inputted to SVM classifiers. The results indicate that the classification performance was increased using the data fusion model. It was observed that the SVM models can accurately classify most of the damage stages, specifically for cracks larger than 10 mm. Besides, tracking the performance of the SVM models gives an insight into the damage location. Although the FE simulation results still remain the standard for most of the elasticity problems, verification of the proposed approach with laboratory or

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full-scale experiments would be an interesting topic for future study.

Acknowledgment The presented work is supported by a research grant from the Federal Highway Administration (FHWA) (DTFH61-13-C00015). Funding The U.S. Federal Highway Administration (FHWA) (DTFH6113-C-00015).

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