Structural Damage Assessment under Varying Temperature Conditions V. Meruane1, W. Heylen2 1 Universidad de Chile, Department Mechanical Engineering Beauchef 850, Santiago, Chile email:
[email protected] 2
K.U.Leuven, Department Mechanical Engineering Celestijnenlaan 300 B, B-3001, Heverlee, Belgium email:
[email protected]
Abstract Modal parameters such as natural frequencies and mode shapes are sensitive indicators of structural damage. However, they are not only sensitive to damage, but also to the environmental conditions such as, humidity, wind and most important, temperature. For civil engineering structures, modal changes produced by environmental conditions can be equivalent or greater than the ones produced by damage. This paper proposes a damage detection method able to deal with temperature variations. The objective function correlates mode shapes and natural frequencies, and a Parallel Genetic Algorithm handles the inverse problem. The numerical model of the structure assumes that the elasticity modulus of the materials is temperature dependent. The algorithm updates the temperature and damage parameters together. Therefore, it is possible to distinguish between temperature effects and real damage events. Simulated data of a three span bridge and experimental data of the I-40 Bridge validate the proposed methodology. Results show that the proposed algorithm is able to assess the experimental damage despite of temperature variations.
1
Introduction
The assumption of most damage detection methods is that damage will modify the stiffness, mass or damping distribution of the structure, which in consequence will alter the measured vibration data. Modal parameters such as natural frequencies and mode shapes are sensitive indicators of structural damage. However, they are not only sensitive to damage, but also to the environmental conditions such as, humidity, wind and most important, temperature. For civil engineering structures, modal changes produced by environmental conditions can be equivalent or greater than the ones produced by damage. Thus, before comparing sets of modal parameters to detect damage, the effect of changing temperature on modal properties must be considered. Researchers have instrumented several structures with thermocouples, anemometers and humidity sensors. The information gathered by these sensors has been used to eliminate environmental influences from the structure's dynamic response. Nevertheless, this information has not been directly applied to damage detection methods. In model-based damage assessment, the most promising strategy is to include this information within the numerical model [1]. Hence, allowing the numerical model to predict the effects of damage combined with temperature variations. However, up to now, the effects of temperature have not been incorporated in any model-based damage assessment algorithm. According to Sohn [2], damage detection techniques will not be accepted in practical applications, unless environmental and operational conditions are explicitly considered. Researchers have investigated the influence of environmental factors with respect to the modal parameters for different bridge constructions. Farrar et al. [3] studied the modal parameters variation of the Alamosa Canyon bridge caused by environmental effects, service conditions and data reduction methods. They
monitored the bridge every two hours over a period of 24 hours. The results show that the most significant sources of modal variability are the thermal gradients across the bridge deck. The first natural frequency varies approximately 5% during the measured period. This frequency variation shows a clear relation to the temperature differentials across the deck. To confirm this relation, a second test was performed a year later. Once again, a clear correlation is observed. The frequencies of the first, second and third modes varied by approximately 4.7%, 6.6% and 5.0% respectively over the 24-hour period. Alampalli [4] conducted several tests over nine months on an abandoned steel-stringer bridge with a concrete deck. He tested the bridge under undamaged and simulated damaged conditions to estimate the sensitivity of modal parameters to environmental variability and to damage. Alampalli's results show that the changes in natural frequencies caused by the freezing of the supports are larger than the changes caused by damage. As a part of the SIMCES project, a series of tests were conducted to the Z24 Bridge in Switzerland. Environmental variables, as well as the bridge modal parameters were monitored during a year. Then, progressive damage test were carried out on the bridge. The paper of Krämer et al. [5] provides a description of the experimental setup, data acquisition, and damage introduction. Peeters and De Roeck [6] studied the relation between natural frequencies variations and the environmental conditions for the Z24 Bridge. They showed that temperature variations are the most important cause of frequency changes. The authors identified a bilinear relation between frequency and temperature. Two lines with a knee situated at 0°C describe this relation. The investigation demonstrates that the asphalt layer on the deck is the main responsible for this bilinear behaviour. Although the asphalt layer does not play any role at warm temperatures, it adds significant stiffness to the bridge at cold temperatures. Rohrmann et al. [7] used experimental data from six years monitoring of the Westend Bridge in Berlin, and a modified thermomechanical model to investigate the influence of the temperature parameters with respect to the natural frequencies. They showed that the variations of the natural frequencies are linearly related to the measured temperature. For the simplest case, they described this relation as,
∆ω = a 0T0 + a1∇T
(1)
Where ∆ω is the change in frequency, T0 the mean temperature of the structure and ∇T the temperature gradient through the height or width of the cross section. The measured data determine the value of the parameters a0 and a1. He et al. [8] deployed a long-term monitoring system on the Voigt Bridge. They applied an automated system identification process to the bridge data from one year monitoring. The results show that the relative changes of the first three identified natural frequencies are of the order of 10%-20% during the monitored period. Ko et al. [9] investigated the correlation of natural frequencies with temperature for the cable-stayed Ting Kau Bridge. They continuously monitored the bridge during one year, and identified the natural frequencies for the first ten modes at intervals of one hour. The researchers trained, using the measured data, ten multi-layer perceptron neural networks, each representing a natural frequency. The networks input data are the temperatures, while natural frequencies are the output. The trained neural networks were able to predict correctly the natural frequencies given the temperatures for the one-year measured data. According to the authors, the well-defined effect of the temperature on the natural frequencies means that this effect can be eliminated or separated from the measurements, giving the potential to detect small structural damages. There are a few recent studies in which, by monitoring the features over a long period, it is possible to eliminate the effect of environmental conditions of the data. Cornwell et al. [10] applied a linear regression model to describe the variation of the natural frequencies of the Alamosa Canyon Bridge caused by changes in the environmental temperature. They used the regression model to compute 95% confidence intervals for the first natural frequency. Data from the second test validates the confidence intervals, in general the data falls within the intervals except for the largest temperature differential. The authors stated that quantification of the environmental and operational variability requires measurements over years at different weather conditions. Peeters and De Roeck [11] proposed a methodology to distinguish normal frequency variations from abnormal changes due to damage. The authors fitted an ARX model to the healthy data of the bridge. The bridge is assumed damaged when a natural frequency lies outside its confidence interval. They successfully verified this methodology with experimental data of the damaged Z-24 Bridge. Sohn et al. [12] presented a novelty algorithm that explicitly takes into account
varying environmental and operational conditions. They computed a novelty index and its threshold value through an auto-associative neural network. This neural network is trained using features extracted from the healthy system under a range of normal conditions. A simplified model of a computer hard disk serves as a test case; the results show that the algorithm is capable of detecting the presence of damage even under varying ambient conditions. Nevertheless, the authors remark the need of further investigations in the structure of the neural network, the sensitivity of the index under noisy environments and to establish what degree of changes in the novelty index are statistically significant. Another group of methods seeks to remove the variability due to environment without measuring the environmental factors. This can be achieved by extracting features which are strongly sensitive to damage, but not very sensitive to the environment variability. Yan et al. [13] proposed a damage detection algorithm that can be applied under varying environmental and operational conditions. The method is based upon principal component analysis (PCA) and novelty detection. It has the advantage that it does not require measurements of the environmental variables: it only needs the vibration characteristics. The authors tested the algorithm performance with two application cases; a simulated bridge and an experimental wooden bridge model. A similar approach based on PCA and novelty detection is proposed by Hu et al. [14]. The authors illustrate the methodology with undamaged data of the new Coimbra footbridge under varying temperature conditions. As expected no damage is detected despite the temperature variations. Deraemaeker and Preumont [15] introduced a new feature derived from the concept of spatial filters. A spatial filter acts as a single sensor that is built from a linear combination of a large network of sensors. A modal filter is designed in a way that the single output mimics the behaviour of a single DOF system. The filter can be tuned to any mode in the frequency band of interest. The authors demonstrated that damage is characterized by the occurrence of spurious peaks in the modal filters at the resonance frequencies of the structure, whereas global environment changes do not modify significantly the output of the modal filter. This property makes modal filters an attractive feature for damage detection. Deraemaeker et al. [16] studied the problem of output-only damage detection under changing environmental conditions. The authors studied two indicators of damage: natural frequencies and mode shapes identified by an automated subspace identification methodology, and peak indicators extracted from the output of modal filters. They build a statistical model of the effect of environment using factor analysis. This model allows them to remove the effect of the environment from the ambient vibration data. Simulated data of a bridge with temperature variations serves as a test case. The results show that if the temperature effects are not removed, natural frequencies cannot be used to detect damage, whereas mode shapes and modal filters could be used. On the other hand, if temperature effects are removed, all features are able to distinguish between the damaged and undamaged case. The above damage detection methods provide an indication only about the presence of damage. These methods do not give information about the location and extent of the damage. To locate and quantify damage most users apply a two-step approach. First, they eliminate the effect of environmental conditions from the data using some of the techniques previously discussed. Second, damage is located and quantified using another damage detection algorithm. Kim et al. [17] proposed a methodology to detect, locate, and quantify damage under temperature varying conditions. However, this methodology is limited to beamlike structures. The authors constructed a statistical damage-warning model based on experiments of a model girder bridge. They formulated an empirical frequency correction using the relation between temperature and frequency ratios. Then, a frequency-based damage index method is used for damage location and quantification. Although the methodology can correctly detect the occurrence of damage, location and quantification accuracy decreases as the temperature gap between the undamaged and damaged models increases. To detect, locate and quantify structural damage accurately, model-based methods are needed. Hence, it is necessary to model directly the impact of the environment on the dynamical characteristics of a structure. The above investigations indicate that the most significant source of modal variability is temperature. Thus, to implement a model-based damage detection algorithm able to deal with environment variability, the numerical model must take into account the effect of temperature variations in the dynamic response of the structure. Pham [18] studied the effects of ambient temperature on the dynamic properties of a bridge using field experiments and numerical analysis of the Attridge Drive Overpass located in
Saskatoon, Saskatchewan. Pham concluded that the change of ambient temperature mainly affects the elastic modulus of the construction materials and therefore the stiffness of the entire bridge. The present study proposes a model-based damage detection method able to deal with temperature variations. This approach extends the damage detection algorithm developed in [19] to include temperature variations. The objective function correlates mode shapes and natural frequencies, and a Parallel Genetic Algorithm handles the inverse problem. The numerical model of the structure assumes that the elasticity modulus of the materials is temperature dependent. The algorithm updates the temperature and damage parameters together. Therefore, it is possible to distinguish between temperature effects and real damage events. Simulated data of a three span bridge and experimental data of the I-40 Bridge [20] validate the proposed methodology. Results show that the proposed algorithm is able to assess the experimental damage despite of temperature variations.
2
Formulation of the optimization problem
Damage is represented by an elemental stiffness reduction factor β i, defined as the ratio between the stiffness reduction to the initial stiffness. To consider the effect of varying temperature conditions, the temperature is included as a variable of the numerical model. Hence, the stiffness matrix depends on the stiffness reduction factors and the temperature:
K d ( β , T ) = ∑ (1 − β i ) K i (Ti )
(2)
i
Where Kd is the stiffness matrix of the damaged structure, Ki is the stiffness matrix of element i, β i is the ith stiffness reduction factor, and Ti is the temperature of element i. The value β i=0 indicates that the element is undamaged whereas 0< βi ≤1 implies partial or complete damage. The problem of detecting damage is a constrained nonlinear optimization problem, where the damage reduction factors βi are defined as updating parameters. The objective function correlates mode shapes and natural frequencies. To avoid the need of an accurate numerical model, the correlation between the numerical and experimental modes in the undamaged state is included in the objective function [21]. In this case, a perfect correlation between the numerical and experimental models is not required; hence initial differences between them are acceptable. The only requisite is that the numerical model must reproduce the same relative changes in the modal properties due to damage than the experimental model. Similar approaches were the initial correlation is incorporated into the objective function are presented by Friswell et al. [22], Hao and Xia [23] and Titurus et al. [24].The error in natural frequencies is represented by the ratio between the experimental and analytical eigenvalues,
ε λ ,i ({β }) =
λ A,i ({β }) λ A0,i ω A2 ,i ({β }) ω A2 0,i − = − 2 λ E ,i λ E 0 ,i ω E2 ,i ω E 0 ,i
(3)
The subscripts A and E refer to analytical and experimental respectively and the subscript 0 refers to the initial undamaged state. λi is the ith eigenvalue and ωi is the ith natural frequency. The difference between modes is represented by the Modal Assurance Criterion (MAC). MAC is a factor that expresses the correlation between two modes. A value of 0 shows no correlation whereas a value of 1 shows two completely correlated modes. The error is defined by,
ε MAC ,i ({β }) = (MAC (φ A,i ( β ), φ E ,i ) − MAC (φ A0,i , φ E 0,i ) )2
(4)
Where φA,i and φE,,i are the ith analytical and experimental mode shape. The subscript 0 refers to the initial undamaged modes. If the number of measured DOFs is lower than the numerical DOFs, a partial MAC is used. Hence, no mode shape expansion is needed. In equations (3) and (4), the goal is not to reach a
perfect match between the numerical and experimental parameters, but rather to reach the same correlation than in the undamaged case. This considers initial errors in the numerical model. The objective function is defined as the normalized sum of the errors plus a damage penalization term,
J ({β }) =
Fλ ({β }) FMAC ({β }) + + FD ({β }) Fλ , 0 FMAC , 0 Fλ ({β }) = ∑ ε λ ,i ({β })
(5)
i
FMAC ({β }) = ∑ ε MAC ,i ({β }) i
Fλ,0, and Fmac,0 refers to the initial values of the sums (β=0). FD is a damage penalization function. Damage penalization helps to avoid false damage detection caused by experimental noise or numerical errors [21,25]. Two damage penalization functions are used:
FD ,1 = γ 1 ∑ β i i
1 β i > 0 FD , 2 = γ 2 ∑ δ i , δ i = i 0 β i = 0
(6)
The first penalizes the total amount of damage. The second, on the other hand, penalizes the number of damage locations. Depending on the damage pattern expected, one can use the first function, the second, or a combination of both. The value of the constants γ1 and γ2 depend on the confidence in the numerical model and the experimental data. The optimization problem is defined as,
min J ({β }) subject to 0 ≤ β i ≤ 1
3
(7)
Damage detection algorithm
The optimization algorithm was developed in a previous work [19]. A parallel Genetic Algorithm (GA) programmed in Matlab and run in a cluster handles the optimization. The gene of each chromosome is the stiffness reduction factor of each element. Each chromosome represents one possible damage distribution. The algorithm employs a multiple population GA with five populations and a neighbourhood migration. The penalization function selected is the sum of FD,1 and FD,2 (see equation (6)) with γ1=γ2=γ. A normalized geometric selection is used as was recommended by Meruane and Heylen [25]. To ensure an effective search with an adequate balance between exploration and exploitation, each population works with a different crossover, being the following ones: arithmetic crossover [26], heuristic crossover [27], BLX-0.5 crossover [28], two point crossover [29] and uniform crossover [30]. In addition, each population applies both boundary and uniform mutations. Each population has a size of 40 individuals and the crossover and mutation probabilities are: pc=0.80 and pm=0.02 respectively. The migration interval is automatically adjusted. If a population has no improvement after a predefined number of generations, the GA stops and exchanges the individuals with their neighbours. This exchange of individuals is synchronous i.e., the algorithm waits until the five populations are ready to perform the migration. At each migration, each population sends its best individual, whereas its worse individual is replaced by the received individual. Before each migration, the best individuals from all populations are compared, if they are all the same the optimization is finished. Figure 1 illustrates this process.
Because the appropriate value for γ it is not known, its value is dynamically adjusted as shown in Figure 2. First the solution with γ=0 is computed, next the value of γ is increased by δ and the solution is recomputed. The algorithm repeats this process until it reaches a stable solution. The solution is defined stable if after three consecutive steps it remains the same. The value of δ used is 0.02.
The variables of the optimization problem are the stiffness reduction factors and the temperature. The numerical model assumes that the temperature changes only affect the material properties and produce thermal contraction and expansion. The boundary conditions of the structure are assumed not to change with temperature. Figure 3 shows the relations between Young's modulus of steel and concrete with temperature, these are the same curves used by Yan et al. [13]. The temperature dependence of Young's modulus is related to the atoms properties and it is very challenging to derive analytical relations [31]. Nevertheless, there are many experimental studies concerning the effect of temperature on Young´s modulus of concrete and steel [32-34].
Figure 1: Parallel optimization
Figure 2: Damage detection algorithm
Figure 3: Young modulus of a) steel and b) concrete versus temperature.
4 4.1
Application cases Numerical study
A numerical model of a three span bridge serves as an initial test case to study the performance of the damage detection algorithm. The primary purpose of the numerical study is to evaluate if the algorithm can distinguish between modal changes caused by damage and the ones due to temperature variations. The algorithm should be able to detect, locate, and quantify damage independently of the temperature variations. In addition, the algorithm should not detect damage in the presence of temperature variations if the structure is undamaged. The structure shown in Figure 4 is a three-span bridge similar to the one presented by Yan et al. [13]. The end spans are of equal length, 15m, and the middle span is 30m long. The bridge is made of two materials, namely steel and concrete, generic material properties are used. The numerical model assumes that the elasticity modulus of the materials is temperature dependent (Figure 3). The area of cross section and moment of inertia of the simulated bridge are 0.05 m2 and 1.66⋅10-4 m4, respectively. The numerical model is built in Matlab with 2D beam elements. The bridge is discretized by 32 beam elements and the motion is restricted to in plane vibrations. It is assumed that the first six bending modes are available and vertical displacements are measured in the 33 nodes. The algorithm considers the 32 elements of the numerical model possible damage locations. Thus, there are 32 updating parameters. To account for the experimental noise, the mode shapes are polluted by 1% random noise and the natural frequencies are polluted by 0.1% noise under the assumption that the measurement error in natural frequencies can be taken around 1/10th of that in mode shapes [35].
Figure 4: Numerical model of a three-span bridge Damage is introduced in the central region of the middle span as a stiffness reduction of 30% in element 16. The baseline condition is the undamaged bridge with a uniform temperature of 20°C. The bridge is subjected to variations of temperature as well as temperature gradients, as shown in Figure 4. At the left end of the bridge the temperature, T1, varies from 0 to 20°C, whereas at the right end, the temperature, T2, varies from 0 to 30°C. The temperature along the bridge varies linearly between these two ends. The following five cases are studied: 1. The bridge is undamaged and there is a uniform reduction of temperature to 0°C. 2. The bridge is undamaged and there is a gradient of temperatures from 0°C to 30°C. 3. The bridge is damaged and there is no temperature variation. 4. The bridge is damaged and there is a uniform reduction of temperature to 0°C. 5. The bridge is damaged and there is a gradient of temperatures from 0°C to 30°C.
First, the damage detection algorithm is applied to the five cases, but temperature variations are not considered in the updating process. Thus, only the stiffness reduction factors are updated and the temperature remains constant at 20°C. This procedure allows determining the importance of including temperature variations in the damage detection process. Table 1 presents the results for the five cases. The results show that the algorithm detects damage in the first two cases even though the structure is undamaged. Since the algorithm does not consider temperature variations, it interprets them as damage. In the cases of damage with temperature variations, it is possible to locate the damage. However, quantification is not accurate (55% accurate). Case Element 16 16 16
N° 1 2 3 4 5
Simulated Damage % T1 °C 0.0 0.0 30.0 20.0 30.0 0.0 30.0 0.0
T2 °C 0.0 30.0 20.0 0.0 30.0
Element 17 21 16 16 16
Detected Damage % T1 °C 3.43 20 4.12 20 29.8 20 16.7 20 16.8 20
T2 °C 20 20 20 20 20
Table 1: Damage detected when the algorithm does not take into account temperature variations To achieve acceptable damage assessment predictions the temperature variations must be included in the damage detection process. The damage detection algorithm is applied to the five cases under study, the temperatures T1 and T2 are updated together with the stiffness reduction factors. Figure 5 shows the output of the damage detection algorithm for the different values of the damage penalization parameter γ. In general, it is observed that false damages are sequentially deleted by increasing the value of γ, until they are completely avoided and a stable solution is reached, once this condition is obtained the algorithm stops. The final damage and change of temperature detected are presented in table 2. Results show that the algorithm successfully distinguishes between damage and temperature variations. In the first two cases, the algorithm does not detect damage as expected, and accurately assess the temperature variations. In the damaged cases, the algorithm successfully locates the damage and quantifies it with an accuracy of 95%. However, in the presence of damage the predicted temperature variations are not accurate. This is explained by the fact that the modal properties, in this case, are more sensitive to damage than to temperature variations. To increase the algorithm accuracy it is recommended to measure the temperature in different locations in the structure. These temperature measurements can be used to restrict the temperature variations allowed during the implementation of the damage detection algorithm. Case N° 1 2 3 4 5
Element 16 16 16
Simulated Damage % T1 °C 0.0 0.0 30.0 20.0 30.0 0.0 30.0 0.0
T2 °C 0.0 30.0 20.0 0.0 30.0
Element 16 16 16
Detected Damage % T1 °C 0.0 0.0 31.3 20.3 29.7 0.0 28.5 1.3
Table 2: Final damage and temperature variations detected
T2 °C 0.0 30.8 11.8 11.0 21.0
Figure 5: Damage detection results for the simulated three span bridge, taking into account possible temperature variation
4.2
The I-40 Bridge
The algorithm is applied to the experimental data of the former I-40 Bridge over Rio Grande in New Mexico. Farrar et al. [20] tested this bridge to investigate the performance of modal parameters as indicators of structural damage. The experimental data obtained from the I-40 Bridge have become one the most studied data sets. Several independent damage detection methods have been applied to the experimental data from this test [20,36-39]. Figure 6 illustrates an elevation view of the portion of the bridge that was tested. It consists of three spans. The end spans are of equal length, 39.9m, and the middle span is 49.7m long. Figure 7 shows a cross section view. The bridge is made up of a concrete deck supported by two welded-steel plate girders and three steel stringers. Loads from the stringers are transferred to the plate girders by floor beams located at approximately 6.1m intervals. Cross bracing is provided between the floor beams. The portions of the plate girders over the piers had increased flange dimensions compared with the mid-span portions to resist the higher bending stresses at these locations. Forced vibration tests were conducted on the undamaged and damaged bridge. A hydraulic shaker excites the bridge with a uniform random signal between 2Hz and 12Hz, and 26 accelerometers measure the dynamic response as shown in Figure 8.
Figure 6: Elevation view of the portion of the I-40 Bridge that was tested
Figure 7: Geometry cross-section of the I-40 Bridge
Figure 8: Experimental setup of the I-40 Bridge
The introduced damage intends to simulate fatigue cracking that has been observed in plate girder bridges. Four levels of damage are introduced to the middle span of the north plate girder. These different levels of damage are introduced by making various torch cuts in the web and flange of the girder, Figure 9 illustrates them. The first level consist of a 0.61m long, 10mm wide cut through the web, in the second level this cut was extended to the bottom of the web. For the third level, the flange is cut halfway from both sides. At last, the bottom flange is completely cut to produce the fourth damage level. After each damage case, the structure is subjected to an experimental modal analysis.
Figure 9: Damage scenarios introduced to the I-40 Bridge Processing of the bridge's experimental data reveals that the ambient temperature effect played a major role in the variation of the modal parameters. Table 3 summarizes the observed changes, with respect to the undamaged case, in the modal properties as a function of the damage level. Since the magnitude of the bridge's natural frequencies are proportional to its stiffness, the natural frequencies are expected to decrease with the progressive introduction of damage. However, the results presented in table 3 show that the frequencies magnitude increases for the first two damage levels. The increment of the natural frequencies magnitude is explained by a reduction of the ambient temperature. Mode N° 1 2 3 4 5 6
Case 1 MAC ∆ωE % 1.46 0.999 0.98 0.999 2.22 0.999 1.13 0.982 1.19 0.996 1.39 0.999
Case 2 MAC ∆ωE % 1.81 0.998 1.11 0.999 0.93 1.000 0.70 0.994 0.82 0.999 0.93 1.000
Case 3 MAC ∆ωE % -0.62 0.999 -0.36 0.999 -0.25 0.999 -0.60 0.985 -0.57 0.999 -0.84 0.999
Case 4 MAC ∆ωE % -7.87 0.860 -4.05 0.894 -0.03 0.998 -2.30 0.910 -0.30 0.997 -2.28 0.976
Table 3: Changes in experimental modes and frequencies after the introduction of damage, I-40 Bridge The numerical model is built in Matlab, with 4-node shell elements to model the concrete deck and the web of the plate girder, and three dimensional beam elements to model the flanges of the girder, stringers, floor beams and the concrete piers. Generic material properties are used. Table 4 shows the correlation between the numerical and experimental undamaged modes, where ωE and ωA are the experimental and numerical natural frequencies. The MAC values indicate the correlation between the experimental and numerical mode shapes. The maximum frequency difference is 2.45% and the minimum MAC value is 0.979.
Mode 1 2 3 4 5 6
MAC 0.997 0.992 0.994 0.979 0.982 0.981
ωE (Hz) 2.48 2.96 3.50 4.08 4.17 4.63
ωA (Hz) 2.48 3.02 3.58 4.18 4.14 4.70
∆ω (%) 0.00 2.03 2.29 2.45 0.72 1.51
Table 4: Initial correlation between numerical and experimental modes, I-40 Bridge 4.2.1
Damage detection
The numerical model assumes that the temperature of the whole structure is the same, thus the temperature varies uniformly along the structure. It should be noted that a uniform variation of temperature is a strong assumption since in most civil engineering structures, gradients of temperatures are reported [3]. Nevertheless, since we do not take into account any temperature measurements, a uniform variation is the best choice. The initial temperature of the undamaged bridge is set to 20°C. Therefore, in the damaged cases a uniform temperature change is imposed relative to this base condition of 20°C. The algorithm updates the temperature of the bridge together with the stiffness reduction factors. Thus, it is assumed that the algorithm can differentiate changes in the modal parameters produced by damage and/or temperature. The temperature is restricted to positive values, i.e. the relation between temperature and the natural frequencies is linear. If the temperature is lower than 0°C, the bridge properties would change due to the freezing of the asphalt layer [6]. Hence, to study cases with temperatures below freezing, the asphalt layer needs to be included in the numerical model, which is not the current case. Given the reduced number of response locations measured during the experimental tests, the number of possible damage locations must be restricted to a minimum. Otherwise, it is feasible to find combinations of damages that produce the same or a better correlation at the measured DOF than the real damage. Considering this, only the web elements of the plate girder are considered possible locations of damage. The web of the north and south plate girders are divided into 24 elements each, giving 48 stiffness reduction factors to be updated. Figure 10 illustrates the element numbering of the north web plate girder and the damage location. Damage is located at element 12 close to element 13 of the north plate girder, thus it is expected to detect damage in one of these two elements.
Figure 10: Element numbering of the north web plate girder and damage location The expected stiffness reduction factor in element 12 for the different damage cases are: 5%, 10%, 32% and 92%. These values were estimated using a detailed finite element model of the girder with the different cuts. The estimated stiffness reduction factors are the values that match the results of the simplified and detail models. The damage detection algorithm uses the first six modes of the bridge. Figure 11 shows the damage detection results for the different values of the damage penalization parameter γ. Table 5 presents the final damage and change of temperature detected; the numbers of the elements refer to the north plate girder since as expected no damage is detected in the south plate girder. With the exception of case 1, damage is correctly detected only at the middle of the north plate girder and the magnitude of the detected damage increases with an increment of the experimental damage. However, in the first case a different behaviour is observed; first the amount of damage detected is higher than in cases 2 and 3 and second false damage is detected in element 20. Damage is accurately quantified in cases
3 and 4, but in cases 1 and 2 the algorithm overestimates the amount of damage. The large reduction of temperature detected in cases 1 and 2 explains this behaviour. Most probably there are gradients of temperature instead of a uniform temperature. While a uniform temperature variation does not alter significantly the mode shapes, transversal gradients of temperature certainly do. This can explain the larger mode shape variation for mode 4 in case 1 compared to cases 2 and 3 (see Table 3). The experimental tests over the I-40 Bridge were performed from August 31 to September 11th, 1993. Farrar et al. [20] reported similar weather conditions during the measuring period. Temperatures range from morning lows of 12°C to afternoon highs of 35°C. The damage detection algorithm estimates a temperature variation of 30°C from case 1 to case 4, which is larger than the actual range of temperatures (a variation of 23°C). Nevertheless, it was shown in the simulated case (table 2) that temperatures predicted by the algorithm are not accurate. To improve the accuracy, the damage detection algorithm needs temperature measurements at different points of the structure.
Figure 11: Damage detection results for the I-40 Bridge. Left - Damage detected on the north plate girder, Right - Damage detected on the south plate girder
Case
Elements
Detected damage %
Expected damage in element 12 %
Change of temperature detected (°C)
1 2 3 4
12, 20 12 12 12, 13
32.67, 11.65 26.12 34.19 91.41, 91.87
5 10 32 92
-17.6 -9.67 4.26 12.16
Table 5: Damage and change of temperature detected in each case, I-40 Bridge
5
Summary and Discussions
This paper has proposed a model-based damage detection method able to deal with temperature variations. The approach extends the damage detection algorithm developed in [19] to incorporate temperature variations. The numerical model of the structure assumes that the elasticity modulus of the materials is temperature dependent. The algorithm updates the temperature and damage parameters together. Therefore, it is possible to distinguish between temperature effects and real damage events. Simulated data of a three span bridge and experimental data of the I-40 Bridge validate the algorithm. The numerical study demonstrates that the algorithm can distinguish between damage and temperature variations. Despite temperature variations in the undamaged cases the algorithm did detect that there was no damage. In the damaged cases, the algorithm successfully located and quantified the damage, even with temperature variations. In the case of the I-40 Bridge, the damage detected shows a good correspondence with the experimental damage in the four damaged cases. Despite the presence of experimental noise, modelling errors and varying temperature conditions, the algorithm correctly detects damage at the middle of the north plate girder. With the exception of the first case, the magnitude of the detected damage increases with an increment of the experimental damage. The results demonstrate that the algorithm is able to detect, locate, and quantify damage despite the temperature variations. However, before the algorithm can be used with confidence the following issues need to be addressed: •
In this study, gradients of temperature are only considered along the length of the bridge. Nevertheless, it has been shown that the most significant sources of modal variability are thermal gradients through the height or width of the bridge cross section [3,7]. In consequence, the damage assessment algorithm must be validated with gradients of temperatures along these directions.
•
It is crucial to measure the temperature at different locations in the structure. There must be enough measurement points to identify the bridge temperature and its gradients in the longitudinal, transversal, and vertical directions. The numerical model must include the effect of temperature changes and its gradients on the bridge dynamic properties.
•
The effect of temperature in the boundary conditions should be investigated and incorporated into the numerical model.
•
The numerical model should consider the nonlinear behaviour observed with temperatures below 0°C. The change in the young modulus with temperature must be considered for all materials in the structure. Some materials have an abrupt change of its properties with temperatures below freezing, as the case of the asphalt layer. These nonlinear changes should be studied and incorporated into the numerical model.
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