A Crack Identification Method for Bridge Type Structures under Vehicular Load Using Wavelet Transform and Particle Swarm Optimization

Abstract

In this work a crack identification method is proposed for bridge type structures carrying moving vehicle. The bridge is modeled as an Euler-Bernoulli beam, and open cracks exist on several points of the beam. Half-car model is adopted for the vehicle. Coupled equations of the beam-vehicle system are solved using Newmark-Beta method, and the dynamic responses of the beam are obtained. Using these and the reference displacements, an objective function is derived. Crack locations and depths are determined by solving the optimization problem. To this end, a robust evolutionary algorithm, that is, the particle swarm optimization (PSO), is employed. To enhance the performance of the method, the measured displacements are denoised using multiresolution property of the discrete wavelet transform (DWT). It is observed that by the proposed method it is possible to determine small cracks with depth ratio 0.1 in spite of 5% noise interference.

1. Introduction

Structures under vehicular loads have significant applications such as railway tracks, bridges, and roadways. Moving load yields larger deflections and higher stresses than equivalent static loading; thus, dynamics of such structures has received significant attention [1, 2]. If the carrying structure includes crack-like defects, then the impact of moving load becomes more pronounced [3–6]. Various damage detection methods have been developed for such structures using the continuous wavelet transform (CWT) [7–11]. They are based on the fact that CWT coefficients of beam dynamic response demonstrate local peaks at crack locations, and magnitudes of these peaks are proportional to crack depths. The significant deficiency of these methods is that CWT coefficients become insensitive to crack at higher moving load speeds. The reason is that time-dependent structural response becomes smoother at higher vehicle speeds. In this case crack-induced singularities disappear, and CWT coefficients of the response signal fail to extract damage info.

Some of the methods in the area of structural damage detection are based on model updating [12]. The basic idea is to update mathematical or finite element model of the structure to match the calculated response to that measured from damaged structure, so that damage parameters are estimated. This is achieved solving an optimization problem. To this end metaheuristic methods are generally preferred, as they do not require gradient info and have less possibility of being trapped by local minima in comparison with the gradient-based methods. The particle swarm optimization (PSO) is one of these algorithms. PSO is a stochastic optimization technique inspired by natural flocking and swarm behavior of birds and insects [13]. It is known to have less parameters and rapid convergence compared with the genetic algorithms (GA) [14] and has been successfully employed in model-updating-based damage detection applications [15–18].

Bilello and Bergman [5] state that changes in the time response of the carrying beam due to damage are more perceptible in comparison to the changes in the natural frequencies. Motivated by this conclusion and the approach of Buezas et al. [19], Gökdağ [20] introduced a model-update-based damage detection approach for beam type structures subject to vehicular load. In this respect, time-dependent displacements from several points on a cracked beam are obtained, and an objective function is defined by subtracting these from the ones calculated by the mathematical model of the structure. Then, PSO algorithm is employed to minimize this objective function, so that crack parameters are determined. In the present work, the performance of this approach is improved employing the multiresolution property of the discrete wavelet transform (DWT). Also, it was demonstrated that the proposed method is efficient for small cracks and better than CWT coefficients method in [7] at higher vehicle speeds.

2. Material and Method

2.1. Dynamic Response of the Vehicle-Beam System

Figure 1 illustrates the bridge-vehicle system. Euler-Bernoulli beam model is considered for the bridge, and half-car model is adopted for the vehicle moving with speed 𝑉. An open crack with depth ℎ₁ is located at 𝑧₁ on the beam. Surface unevenness of the beam is disregarded, and tires are assumed to be always in contact with the beam [9]. Linear deflections of the beam are considered. Longitudinal deformations and bending-longitudinal coupling are neglected. Under these assumptions the equations of motion for the vehicle-beam system can be derived as follows: 𝐼_𝑣0000𝑚_𝑣0000𝑚₁0000𝑚₂𝑑̈₁𝑑̈₂𝑑̈₃𝑑̈₄+𝑐₁𝐿²₁+𝑐₂𝐿²₂𝐿₂𝑐₂−𝐿₁𝑐₁𝐿₁𝑐₁−𝐿₂𝑐₂𝐿₂𝑐₂−𝐿₁𝑐₁𝑐₂+𝑐₁−𝑐₁−𝑐₂𝐿₁𝑐₁−𝑐₁𝑐₃+𝑐₁0−𝐿₂𝑐₂−𝑐₂0𝑐₂+𝑐₄𝑑̇₁𝑑̇₂𝑑̇₃𝑑̇₄+𝑘₁𝐿²₁+𝑘₂𝐿²₂𝐿₂𝑘₂−𝐿₁𝑘₁𝐿₁𝑘₁−𝐿₂𝑘₂𝐿₂𝑘₂−𝐿₁𝑘₁𝑘₂+𝑘₁−𝑘₁−𝑘₂𝐿₁𝑘₁−𝑘₁𝑘₃+𝑘₁0−𝐿₂𝑘₂−𝑘₂0𝑘₂+𝑘₄𝑑₁𝑑₂𝑑₃𝑑₄=00𝑘₃𝛿_I𝑢+𝑐₃𝛿_I𝑢̇𝑘₄𝛿_II𝑢+𝑐₄𝛿_II𝑢̇,𝜕𝜌𝐴²𝑦(𝑧,𝑡)𝜕𝑡²()+𝐶𝜕𝑦𝑧,𝑡𝜕𝜕𝑡+𝐸𝐼⁴𝑦(𝑧,𝑡)𝜕𝑥⁴=−𝑃_I−𝑃_II, (1)(2) where 𝐼_𝑣,𝑚_𝑣,𝑚₁,𝑚₂,𝐿₁,𝐿₂,𝑘₁,𝑘₂,𝑘₃,𝑐₁,𝑐₂,𝑐₃,𝑐₄ are the vehicle parameters shown in Figure 1 and 𝐶 is the damping of beam material. 𝑑_𝑖, 𝑖=1∶4, denote vehicle degrees of freedom, and 𝛿_I and 𝛿_II are the Dirac delta functions defined as follows: 𝛿_II=𝛿(𝑧−𝑉𝑡),𝛿_I=𝛿𝑧−𝑉𝑡−𝐿₁+𝐿₂, (3) where 𝑃_I and 𝑃_II are the interaction forces acting on the beam through the contact points I and II as follows:𝑃_I=𝑚₁+𝐿₂𝐿₁+𝐿₂𝑚_𝑣9.81+𝑘₃𝛿_I𝑢−𝑑₃+𝑐₃𝛿_I𝑢̇−𝑑̇₃,𝑃_II=𝑚₂+𝐿₁𝐿₁+𝐿₂𝑚_𝑣9.81+𝑘₄𝛿_II𝑢−𝑑₄+𝑐₄𝛿_II𝑢̇−𝑑̇₄, (4a)(4b)where 𝑢 is the vertical displacement at the tire contact point, that is, 𝑢=𝑦(𝑧,𝑡), and its derivative with respect to time is 𝑢̇=𝑑𝑦(𝑧,𝑡)/𝑑𝑡=𝑉𝜕𝑦(𝑧,𝑡)/𝜕𝑧+𝜕𝑦(𝑧,𝑡)/𝜕𝑡 and ^′=𝑑/𝑑𝑧. At the crack location vibration modes should satisfy the following compatibility equations [7, 11]:𝑌₁𝑧₁=𝑌₂𝑧₁,𝑌^′₁𝑧₁+𝐸𝐼𝑘𝑌₁^′′𝑧₁=𝑌^′₂𝑧₁,𝑌₁^′′𝑧₁=𝑌₂^′′𝑧₁,𝑌₁^′′′𝑧₁=𝑌₂^′′′𝑧₁, (5a)(5b)where 𝑌_𝑖(𝑧)=𝐶_𝑖1cos(𝜆𝑧)+𝐶_𝑖2sin(𝜆𝑧)+𝐶_𝑖3cosh(𝜆𝑧)+𝐶_𝑖4sinh(𝜆𝑧),𝑖=1,2,𝜆=𝜌𝐴𝜔²(𝐸𝐼)^0.25, (6) where 𝜆 and 𝜔 in (6) are eigenvalue and natural frequency parameters, respectively. 𝐶_𝑖𝑗 are the constants to be determined by solving the relevant eigenvalue problem. The factor “1/𝑘” in (5a) is called crack compliance. This is, for a rectangular beam with crack depth ratio of ℎ₁, defined as follows [3, 6, 7, 11]: 1𝑘=2ℎ𝐸𝐼ℎ₁1−ℎ₁²×5.93−19.69ℎ₁+37.14ℎ²₁−35.84ℎ³₁+13.12ℎ⁴₁,ℎ₁=ℎ₁ℎ. (7)

Figure 1 

Bridge-vehicle system.

To solve the coupled equations (1) and (2) vibration modes of the cracked beam are obtained first by solving the relevant eigenvalue problem (for details see [11]). Then, the series solution 𝑦(𝑧,𝑡)=𝐘(𝑧)^T𝐪(𝑡) is formed to determine dynamic response of the beam, where 𝐘(𝑧) is the vector of size 𝑁_𝑚×1 containing vibration modes of the cracked beam, 𝑁_𝑚 is the number of modes used, T means transpose, and 𝐪(𝑡) stands for the modal coordinates. Substituting 𝑦(𝑧,𝑡)=𝐘(𝑧)^T𝐪(𝑡) into (2), premultiplying the equation by 𝐘(𝑧), integrating from 0 to 𝐿, and finally combining with (1) lead to the coupled beam-vehicle differential equations [20, 21]: 𝐌_{𝑁𝑚×𝑁𝑚}𝟎_𝑁𝑚×4𝟎_4×𝑁𝑚𝐌_4×4𝐪̈𝐝̈+𝐂+𝐀₈𝐀₁₀−𝐀₆𝐂𝐪̇𝐝̇+𝐊+𝐀₇𝐀₉−𝐀₅𝐊𝐪𝐝=𝐅₁𝟎, (8) where𝐌=^𝐿₀𝜌𝐴𝐘𝐘^T𝑑𝑧,𝐂=^𝐿₀𝐶𝐘𝐘^T𝑑𝑧,𝐊=^𝐿₀𝐸𝐼𝐘𝐘^{′′′′T}𝑑𝑧, (9a)(9b)(9c) where 𝐌,𝐂, and 𝐊 are the mass, damping, and stiffness matrices in (1), that is, 𝐼𝐌=^𝑣0000𝑚_𝑣0000𝑚₁0000𝑚₂,𝑐𝐂=¹𝐿²₁+𝑐₂𝐿²₂𝐿₂𝑐₂−𝐿₁𝑐₁𝐿₁𝑐₁−𝐿₂𝑐₂𝐿₂𝑐₂−𝐿₁𝑐₁𝑐₂+𝑐₁−𝑐₁−𝑐₂𝐿₁𝑐₁−𝑐₁𝑐₃+𝑐₁0−𝐿₂𝑐₂−𝑐₂0𝑐₂+𝑐₄,𝑘𝐊=¹𝐿²₁+𝑘₂𝐿²₂𝐿₂𝑘₂−𝐿₁𝑘₁𝐿₁𝑘₁−𝐿₂𝑘₂𝐿₂𝑘₂−𝐿₁𝑘₁𝑘₂+𝑘₁−𝑘₁−𝑘₂𝐿₁𝑘₁−𝑘₁𝑘₃+𝑘₁0−𝐿₂𝑘₂−𝑘₂0𝑘₂+𝑘₄, (9d)(9e)(9f)where 𝐝 is the vector including the coordinates 𝑑_𝑖 in (1). Additionally,𝐀₁=𝐘(𝑧)^T𝑘₃+𝑉𝑐₃𝐘^′(𝑧)^T𝛿_I𝐀₂,=𝐘(𝑧)^T𝑐₃𝛿_I,𝐀₃=𝐘(𝑧)^T𝑘₄+𝑉𝑐₄𝐘^′(𝑧)^T𝛿_II𝐀₄,=𝐘(𝑧)^T𝑐₄𝛿_II,𝐀₅=𝟎_2×𝑁𝑚𝐀₁𝐀₂,𝐀₆=𝟎_2×𝑁𝑚𝐀₂𝐀₄𝐀₇,=𝐀₁+𝐀₃,𝐀₈=𝐀₂+𝐀₄,𝐀₉=00−𝑘₃−𝑘₄𝐀₁₀,=00−𝑐₃−𝑐₄,𝐀_𝑗=^𝐿₀𝐘(𝑧)𝐀_𝑗𝐅𝑑𝑧,𝑗=7,8,9,10,₁=^𝐿₀𝐘(𝑧)𝑊₁𝛿_I+𝑊₂𝛿_II𝑊𝑑𝑧,_I=𝑚₁+𝐿₂𝐿₁+𝐿₂𝑚_𝑣𝑊9.81,_II=𝑚₂+𝐿₁𝐿₁+𝐿₂𝑚_𝑣9.81. (10a)(10b)(10c)(10d)(10e)(10f)(10g)(10h)(10i)(10j)(10k)(10l)(10m)Newmark-Beta method (with 𝛽=1/6 and 𝛾=1/2 [22]) was employed in this work to solve this set of equations. For this purpose, computer codes were written in MATLAB environment. To check the reliability of the codes the midspan response of the beam subject to moving vehicle is obtained as shown in Figure 2, employing the same parameters in [21]. One can verify the agreement by comparing Figure 2 with the relevant figure in [21].

Figure 2 

Midspan deflection of beam (vehicle and beam parameters are from the work of Wu and Law [21]).

2.2. The Objective Function and Constrains

The aim is to correlate the response of the damaged beam to that calculated by the mathematical model of the structure. By this way damage configuration of the carrying structure will be determined. To achieve this, it is proposed to adjust crack sizes and locations by solving an optimization problem. The objective function of the problem is introduced as follows: 𝑓₁(𝐱)=^𝑁mp_𝑛=1^𝑇₀𝑦𝑧_𝑛,𝑡−𝑦𝑧_𝑛,𝑡max𝑦𝑧_𝑛,𝑡𝑑𝑡, (11) where 𝑁_mp is the number of measurement points on the beam, 𝑧_𝑛 is the location of 𝑛th measurement point on the beam, 𝑦 denotes the reference displacements (RDs) measured from damaged beam, 𝑦 stands for the corresponding displacements computed by the mathematical model of the structure, 𝑇 is the total time for the vehicle to move across the beam, and 𝐱 is the vector containing crack location and depth parameters, that is, 𝐱=ℎ₁,ℎ₂,…,ℎ_𝑁𝑐,𝑧₁,𝑧₂,…,𝑧_𝑁𝑐,ℎ_𝑖=ℎ_𝑖ℎ,𝑧_𝑖=𝑧_𝑖𝐿, (12) where ℎ is the height of beam cross-section and 𝑁_𝑐 denotes the number of cracks. As to the measurement points, it is better to choose them close to the midpoint, since maximum deflections occur around this point. Thus, five points on the beam were determined as {0.30.40.50.60.7}𝐿, that is, 𝑁_mp=5 in (11). Since number of cracks may not be known in advance, it is required to make initial assumption about their number. To this end, the beam span is divided into five equal intervals each of which is accompanied by a crack (see Figure 3). Thus, ten design parameters, that is, five crack locations and five depth ratios, should be determined to estimate damage condition of the structure. In general, if the beam span is divided into 𝑁 intervals, then the number of variables to be determined is 2𝑁. Ideally, the number of beam spans should be as large as possible. But this leads to more computational effort. Thus, based on experience, the beam is divided into five spans each of which is associated with a crack. With these explanations in mind, the optimization problem can be formulated as follows: min𝑓(𝐱)subjectto:0≤ℎ_𝑗<1,0.2𝑗−1≤𝑧_𝑗<0.2𝑗,𝑗=1,2,…,𝑁_𝑐. (13) Crack locations and depths are determined by solving (13). In this work the PSO algorithm is employed for this purpose, and its details are given in the next section.

Figure 3 

Five beam spans with cracks.

2.3. The Particle Swarm Optimization Algorithm

PSO algorithm is initialized with a “swarm” composed of 𝑁 particles. Particles refer to the candidate points in the search space of the optimization problem. To obtain best solution each particle adjusts its trajectory toward its own previous best position and toward the previous best position of swarm. By this way, each particle moves in the search space with an adaptive velocity and stores the best position of the search space. Location (𝑥) and velocity (𝑣) of a particle are updated with the following equations [13, 14, 23]: 𝑣^𝑘+1_𝑖𝑗=𝑤𝑣^𝑘_𝑖𝑗+𝑐₁𝑅₁𝑝^𝑘_𝑖𝑗−𝑥^𝑘_𝑖𝑗+𝑐₂𝑅₂𝑝^𝑘_𝑔𝑗−𝑥^𝑘_𝑖𝑗,𝑥^𝑘+1_𝑖𝑗=𝑥^𝑘_𝑖𝑗+𝑣^𝑘+1_𝑖𝑗,𝑖=1,2,…,𝑁,𝑗=1,2,…,𝑚,𝑘=1,2,…,𝐾_max, (14) where 𝑘 is the iteration counter, 𝐾_max denotes the maximum number of iterations, 𝑚 is dimension of the problem, 𝑝^𝑘_𝑖𝑗 and 𝑝^𝑘_𝑔𝑗 are, respectively, the best positions of the 𝑖th particle and the swarm found until the 𝑘th iteration, 𝑅₁ and 𝑅₂∈𝑈(0,1), where 𝑈 means the uniform random distribution, and 𝑐₁ and 𝑐₂ are positive weighting constants called cognitive and social coefficients, respectively. These two constants regulate the relative velocity toward global and local best points. 𝑤 is the inertia coefficient employed to prevent swarm explosion. In this work the values of these parameters are selected as 𝑤=0.6, 𝑐₁=𝑐₂=1.7 [23]. Initial values of particles and their velocities were obtained drawing random numbers within the range of each dimension, that is, 𝑥¹_𝑖𝑗=𝑥_LB,𝑗+𝑅⋅𝑥_UB,𝑗−𝑥_LB,𝑗,𝑣¹_𝑖𝑗=𝑥_LB,𝑗+𝑅⋅𝑥_UB,𝑗−𝑥_LB,𝑗, (15) where 𝑥_LB,𝑗 and 𝑥_UB,𝑗 are, respectively, the lower and upper boundary values of the 𝑗th dimension and 𝑅∈𝑈(0,1). Besides, a particle moving beyond ranges was bounced back to the search space in the following way: 𝑥^𝑘_𝑖𝑗=𝑥_UB,𝑗−𝑅⋅𝑥^𝑘_𝑖𝑗−𝑥_UB,𝑗,if𝑥^𝑘_𝑖𝑗>𝑥_UB,𝑗,𝑥^𝑘_𝑖𝑗=𝑥_LB,𝑗+𝑅⋅𝑥_LB,𝑗−𝑥^𝑘_𝑖𝑗,if𝑥^𝑘_𝑖𝑗<𝑥_LB,𝑗. (16)

2.4. The Proposed Objective Function

In practice, the RDs in (11) should be measured from damaged structure. However, in this work they are produced by the mathematical model due to the lack of such experimental data. To simulate the effect of experimental noise arising from instrumentation, environmental conditions, numerical errors, and so forth, a sequence of random numbers are added to RDs in the following way [7, 9]: 𝑦(𝑧,𝑡)_noisy=𝑦(𝑧,𝑡)+𝑁_𝑝⋅𝐺⋅𝜎, (17) where 𝑦(𝑧,𝑡) is the calculated RD, which is free of noise, at point 𝑧 of the damaged beam. 𝑁_𝑝 is the noise percentage, 𝐺 is Gaussian distribution with zero mean and unit standard deviation, and 𝜎 is the standard deviation of 𝑦(𝑧,𝑡). If noisy data is employed in (11), then erroneous results may be obtained. Thus, the noise should be alleviated by a proper approach. To this end, multiresolution property of the DWT is exploited. By the DWT a time function 𝑓(𝑡) can be decomposed to an approximation function (𝐴_𝑀) at 𝑀th level and sum of details (𝐷_𝑚) up to that level as follows [24, 25]: 𝑓(𝑡)=𝐴_𝑀+^𝑀_𝑚=−∞𝐷_𝑚,𝐴_𝑀=^∞_𝑛=−∞𝑆_𝑀,𝑛𝜃_𝑀,𝑛(𝑡),𝐷_𝑚=^∞_𝑛=−∞𝑊_𝑚,𝑛𝜓_𝑚,𝑛(𝑡),𝑆_𝑚,𝑛=^∞_−∞𝑓(𝑡)𝜃_𝑚,𝑛(𝑡)𝑊𝑑𝑡_𝑚,𝑛=^∞_−∞𝑓(𝑡)𝜓_𝑚,𝑛(𝑡)𝜓𝑑𝑡,_𝑚,𝑛(𝑡)=2^−𝑚/2𝜓𝑡−𝑛2^𝑚2^𝑚, (18) where 𝑆_𝑚,𝑛 is the scale coefficient at 𝑚th level, 𝑊_𝑚,𝑛 is the wavelet coefficient, 𝜓 is the wavelet function, and 𝜃 is the scale function associated with 𝜓. 𝐴_𝑀 represents smooth part of the signal, whereas details include high-frequency components such as noise. If suitable wavelet and decomposition level (𝑀) are determined, then the approximation function of 𝑦 in (11) can be used instead of 𝑦 itself. In this case, the objective function in (11) can be revised as follows: 𝑓₂(𝐱)=^𝑁mp_𝑛=1^𝑇₀𝑦𝑧_𝑛,𝑡−𝑦_𝑀𝑧_𝑛,𝑡max𝑦_𝑀𝑧_𝑛,𝑡𝑑𝑡, (19) where 𝑦_𝑀 is the approximation function of 𝑦 at 𝑀th decomposition level.

3. Results and Discussion

3.1. Application

The following data [9] is employed for the numerical simulation: 𝐸=210 GPa, 𝜌=7860 kgm⁻³, 𝐿=50 m, 𝑏=1 m, ℎ=2 m, 𝑚_𝑣=12404 kg, 𝑚₁=𝑚₂=725.4 kg, 𝐼_𝑣=172160 kgm², 𝑘₁=1969034 Nm⁻¹, 𝑘₂=727812 Nm⁻¹, 𝑘₃=4735000 Nm⁻¹, 𝑘₄=1972900 Nm⁻¹, 𝑐₁=7182 Nsm⁻¹, 𝑐₂=2190 Nsm⁻¹, 𝑐₃=𝑐₄=0, and 𝐿₁=𝐿₂=3 m. The first five vibration modes of the beam, for which the natural frequencies are 1.9, 7.5, 16.9, 30, and 46.9 Hz, are considered. The sampling frequency of the simulation is 300 Hz which can capture the response of the first five vibration modes of the beam. Two percent modal damping is considered for each mode [7]. The damage scenarios in Table 1 are considered. In case 1 single crack with depth ratio of 0.1 is available at 25% of the beam length. In case 2 and case 3 multiple cracks exist on the beam. Note that the considered crack depths are small according to the relevant works [7–9]. Also, one may verify by the relevant works that the considered vehicle speeds are relatively high for these small cracks. In other words, the wavelet methods in the cited works cannot extract such small cracks through processing dynamic response recorded at higher speeds such as those in Table 1. However, it will be demonstrated that crack identification is possible for the cases in Table 1 by the method proposed in this work. The beam midspan deflections are compared in Figure 4. It is clear that deflections are relatively small and not bigger than 3 mm.

Table 1 

Damage scenarios.

(a)

(b)

(c)

Figure 4 

Comparison of beam midspan deflections for the cases in Table 1. U: undamaged, D: damaged.

For DWT a widely used wavelet, that is, Daubechies wavelet with four vanishing moments (db4) [25] shown in Figure 5, is employed. It is experienced that the approximation function at the fourth level (𝑀=4) is suitable since it is well correlated with noiseless data. This is demonstrated in Figure 6, considering Case 1 in Table 1. The midspan deflection of the beam is obtained for Case 1, and it is contaminated adding 5% Gaussian distribution. Then, the approximation (𝐴_𝑀) of this noisy data at the first level (𝑀=1) is obtained, and its comparison with the original data that is free of noise is illustrated in Figure 6(a). It is clear that 𝐴_𝑀 at this level is not suitable since it is not denoised sufficiently. This is true for those 𝐴_𝑀 at the levels 𝑀=2 and 3 (hence their graphs are not given). However, the 𝐴_𝑀 at 𝑀=4 is in well agreement with the noiseless data (see Figure 6(b)); thus, it can be employed as reference data in (19), that is, 𝑦_𝑀=4. At higher levels the 𝐴_𝑀 deviates from the noiseless data, since not only high-frequency components are removed but also some low-frequency parts that affect shape of the signal are removed. This observation is valid for other RDs on other points of the beam. Generally the suitable level depends on the length of reference data. That is, it may be higher when data is recorded at higher sampling frequencies.

Figure 5 

The db4 wavelet (dashed) and its scale function (solid).

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Figure 6 

Comparison of 𝐴𝑀 at various levels with the original noiseless response of beam midspan.

After some trials the swarm size and maximum iteration number are determined as 30 and 100, respectively. Then the optimization problem is solved using the objective functions 𝑓₁ and 𝑓₂ in (11) and (19), and damage detection results are given in Figures 7 and 8, respectively. As the PSO algorithm is stochastic, the mean of 30 runs [16, 18] of the algorithm is displayed in the figures. Ideally, only the crack depth ratios at the true damage locations are expected to be nonzero. However, this is not sufficiently satisfied by the objective function 𝑓₁ in (11), the reason of which is the noise in RDs. For example, for Case 1 it seems there is a crack with depth ratio about 0.16 around the 0.07𝐿 on the beam (see Figure 7) though the only crack in this case is at 0.25𝐿 of the beam length and with depth ratio of 0.1. On the other hand, the standard deviation (SD) of crack depth ratio at true crack location is low compared with the value of relevant parameter. However, SD at a misleading crack point is high compared to the relevant crack depth ratio, which indicates that this is not real damage. Generally, SD at points close to beam ends is higher. This is because the curvature of vibration modes around the beam ends is small for simply supported beam [26].

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Figure 7 

Damage detection results (the objective function in (11) is employed).

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Figure 8 

Damage detection results (the objective function in (19) is employed).

Thus cracks close to beam ends do not have much effect on the dynamic response of the beam. According to Figure 8, all real crack points are predicted more accurately, and SD values are generally lower when compared to Figure 7. This implies that the proposed method produces more reliable results.

3.2. Comparison with CWT Coefficients Approach

Zhu and Law [7] demonstrated that magnitudes of CWT coefficients of beam dynamic response become large when moving load passes over crack point. The reason is that crack induces localized singularities in the structural response, and the CWT coefficients of the response are sensitive to these singularities. However, the deficiency of this approach is that CWT coefficients become insensitive to damage with increasing moving load speed. Hester and González [8] report that it is possible to detect crack depth ratios of 0.2 and 0.3 at the vehicle speed 7 m/s by their approach. However, their method fails to detect depth ratio of 0.1 at the same speed. Nguyen and Tran [9] state that cracks as small as 10% of beam height can be detected when the vehicle is moving at low velocity such as 2 m/s. The authors conclude that these small cracks are very difficult to be revealed at high vehicle speeds. Thus, one may verify that in their work the authors consider crack depth ratio of 0.5 to detect crack at the vehicle speed of 30 m/s. From the relevant literature one can conclude that crack depth ratio of 0.1 cannot be detected at a vehicle speed such as 20 or 30 m/s by the wavelet transform methods. In order to prove this, Case 2 in Table 1 is considered. Midspan deflection and its CWT coefficients are demonstrated in Figure 9. The scale interval is determined as in [8] using the relation 𝐹_𝑎=𝐹_𝑐/(𝑎Δ), where 𝐹_𝑐 denotes the centre frequency of the Mexican Hat wavelet (0.25 Hz), 𝑎 is the scale, Δ is the sampling period, and 𝐹_𝑎 is the frequency corresponding to the scale 𝑎. The maximum scale corresponds to the first natural frequency of the healthy beam, that is, 1.9 Hz. At a singularity point the CWT coefficients form a cone of influence [27]; that is, absolute values of the CWT coefficients become extreme and the region affected by the singularity increases along with scale. However, this variation is not observed in Figure 9; hence, one can conclude that the CWT coefficients fail to extract damage signature for the considered case.

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Figure 9 

The midspan deflection for Case 2 in Table 1 and its CWT coefficients.

4. Conclusions

In this work a damage detection method is proposed for bridge type structures under vehicular load. An objective function is defined by the time-dependent deflections on several points of damaged beam and the mathematical model of the structure. The reference data is denoised using the multiresolution property of the DWT. Then, the optimization problem is solved to determine crack locations and depths. It was observed that the proposed approach achieves identification of small cracks with depth ratio of 0.1 in spite of 5% noise interference and is superior than the CWT coefficients method. The present study should be verified by experimental data. This issue is planned to be dealt with in later studies.