A new approach toward damage localization and quantification of structures under changing temperature condition:

William  Soo Lon Wah; Yung-Tsang Chen

doi:10.1177/1461348418793079

A new approach toward damage localization and quantification of structures under changing temperature condition:

Wah, William Soo Lon; Chen, Yung-Tsang 2018-08-27 00:00:00 Most damage detection methods developed in the literature cannot give the locations and extent of damages under the presence of varying temperature condition. This is because temperature condition changes the vibration properties of a structure, which are commonly analyzed for damage detection, and temperature gradient throughout the structure makes it difficult to create a baseline for the undamaged structure, as the baseline is generally constructed using features obtained under a wide range of temperature conditions. In this paper, a new insight on how to approach damage detection using only a single temperature condition to create the baseline is proposed. This approach solves the damage detection under changing temperature problem in two stages by first quantifying the change of stiffness of all the elements in a structure due to temperature and damage effects, followed by removing the temperature effect, a global effect, to give the actual damage locations and extent. Using single temperature condition allows new measure- ments to be compared to a benchmark so that local deviation can be obtained, thus making the damaged elements identifiable. The proposed approach is tested using a beam structure model and a shear building under different gradient temperature conditions, and the results demonstrate that the method successfully eliminates the change in elemental stiffness due to temperature effect and gives correct damage locations and extent. The approach can be implemented with other existing damage detection methods that did not consider the effect of temperature so that structures under varying temperature condition can be analyzed. Keywords Damage detection, damage localization, damage quantification, temperature condition, temperature gradient, environ- mental conditions Introduction Engineering structures in general are subjected to constant degradation due to the harsh environmental and operational conditions they are faced with. To maintain their safety and reliability, researchers have been devel- 1–4 oping damage detection methods to assess the health condition of the structures to allow reparation/replace- ment of structural defects at an early stage before catastrophic failure occurs. Based on the information given on the state of damages, the damage detection methods can be classiﬁed into four different levels as given by Rytter Level 1: Detect the presence of damages only in the structure. Level 2: Detect and give the locations of damages in the structure. Department of Civil Engineering, The University of Nottingham, Ningbo, People’s Republic of China Corresponding author: Yung-Tsang Chen, Department of Civil Engineering, The University of Nottingham Ningbo China, 199 Taikang East Road, Ningbo, 315100, People’s Republic of China. Email: yung-tsang.chen@nottingham.edu.cn Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www. creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). Soo Lon Wah and Chen 573 Level 3: Detect, locate, and give the extent of damages in the structure. Level 4: Detect, locate, and quantify damages, and give the level of safety of the structure depending on the damage case. 1,2,6 1,2,7 In the literature, several level 2 and level 3 damage detection methods have been proposed and applied to numerical and experimental structures. However, most of the proposed methods were veriﬁed by assuming that the environmental conditions the structures are faced with are constant. Yet, it is well known that the changing environmental conditions, the changing temperature condition in particular, have a major inﬂuence on the dynamic properties of civil engineering structures, which are commonly used as damaged sensitivity features for damage detection. As a result, false alerts may occur. Thus, to provide a robust damage detection method, which does not give false alerts, the effects of the changing environmental conditions, especially the changing temperature condition, should be taken into consideration while deriving the method. Moreover, the temper- ature condition usually has nonlinear effect on the dynamic properties of the structures, and temperature gradient throughout the structures usually exists, both of which also need to be considered in the development of the damage detection method. 12–14 To tackle this temperature condition issue, several approaches, such as regression analysis and the use of 11,14,15 multivariate statistical tools to process the damage sensitivity features, have been proposed in the literature; however, most of these approaches can only give the presence of damages in the structures. Among the proposed 16–18 approaches, limited methods can perform level 2 of damage detection, and only a few methods can perform 10,19 level 3 of damage detection, i.e. give the presence, locations, and extent of the damages altogether, yet most of the existing methods do not perform well under gradient temperature condition and under nonlinear effect of the changing temperature condition. The difﬁculties in the development of level 3 damage detection under changing temperature condition may be attributed to the unﬁt approach adopted by the researchers, as the methods proposed in the literature usually use the concept of creating the baseline of an undamaged structure using damage sensitivity features captured from a wide range of temperature conditions so as to cover all possible 8,20,21 scenarios the structure may encounter, and compare new measurements with the baseline to see whether they lie within the normal range. However, by doing so, a new measurement from the structure under a speciﬁc temperature distribution is only being compared to measurements from the undamaged structure under normal temperature conditions; it is not being compared to a benchmark that represents the undamaged structure at exactly the same temperature distribution. Therefore, it is generally difﬁcult to determine which part of the structure differs from the one of the undamaged structure. It is difﬁcult to compare a new measurement captured at a speciﬁc temperature condition to the same condition from the undamaged structure because, as mentioned previously, structures are subjected to gradient temperature and to nonlinear temperature effect. These make obtaining the exact same temperature condition from the undamaged structure to be used as baseline difﬁcult, because there is an inﬁnite amount of temperature variations throughout the structure, and this is the main reason why new observations are only being compared to a set of undamaged measurements to see whether they behave similarly. To tackle this problem, some methods in the literature tried to predict the temperature condition along with the damage locations based on the features gathered; however, gradient temperature distribution makes it difﬁcult to get reasonable results because gener- ally, only temperature at one or a few locations are measured throughout the whole structure, hence leading to errors in the temperature proﬁle. Moreover, constructing the baseline from a wide range of temperature con- ditions has the drawback that it may reduce the sensitivity of the methods to detect small damages if proper consideration is not taken into account; it is also time-consuming to gather all the necessary features for damage detection, hence retarding the implementation of the methods. This paper proposes a new insight on how to approach damage detection under varying temperature condition by adopting only one temperature condition for the baseline of the undamaged structure. This approach ﬁrst treats damage detection under changing temperature condition in the same way as one treats damage detection assuming constant temperature condition, followed by determining the change in stiffness of each element of the structure due to external effects. After the change in elemental stiffness has been quantiﬁed, temperature effect is removed from the extent coefﬁcients by analyzing the trend of the extent coefﬁcients along the structure to give damage locations and extent. In this paper, the damage detection method by Soo Lon Wah and Chen is further extended to take into account the effect of changing temperature with gradient temperature distribution. The proposed method uses frequency responses, and stiffness and degrees-of-freedom of each element to obtain an extent coefﬁcient for each element to represent the effects of both temperature and damage on the structure. After the extent coefﬁcients for all elements have been found, the temperature effect is then removed to give altogether the actual damage locations and damage extent calculated based on using one temperature as 574 Journal of Low Frequency Noise, Vibration and Active Control 39(3) baseline. To demonstrate the proposed approach, a beam structure model and a shear building are used as examples for damage detection and quantiﬁcation, and the results obtained show that by using only one tem- perature condition for the baseline, damage localization and quantiﬁcation under gradient temperature condition and nonlinear temperature effect can be achieved. This approach can be implemented easily with other existing damage detection methods to tackle level 3 damage detection under varying temperature condition. Damage detection methodology This section introduces the proposed approach for damage detection, localization, and quantiﬁcation of civil engi- neering structures subjected to varying temperature condition and to gradient temperature distribution. In this approach, damages in a structure are assumed to be represented by reductions in elemental stiffness of the members of the structure, and the locations of the damages are given by the element number each element is assigned to. The approach consists of two stages. In the ﬁrst stage, an extent coefﬁcient that is related to the contribution of the temperature and damage effects affecting the structure is assigned to each element present in the structure. In the second stage, the temperature effect is removed from the extent coefﬁcient so that the damage extent can be obtained through the use of a damage coefﬁcient. The damage coefﬁcient, if not equals to zero, gives the actual damage extent of an element, and the location of the damage can be identiﬁed by the element number accordingly. Stage 1: Extent coefficient assignment Consider a structure with n degrees-of-freedom subjected to a forced harmonic excitation. The mathematical model representing the motion of the structure can be given by the equation of motion jxt € _ MXðtÞþ CXðtÞþ KXðtÞ¼ fe (1) where M, C, and K are the n n mass, damping, and stiffness matrices of the structure, respectively; XðtÞ is the n 1 displacement response vector of the structure; f is the n 1 amplitude vector of the forced harmonic excitation; pﬃﬃﬃﬃﬃﬃﬃ j is the imaginary unit ð¼ 1Þ; x is the angular frequency of the forced harmonic excitation. For simplicity, the damping matrix C is neglected. Thus, equation (1) becomes jxt MXðtÞþ KXðtÞ¼ fe (2) Structural damage or structural change due to temperature variation here is referred to as a reduction/increase in the stiffness property of the structure with no change in the mass property. Hence, the equation of motion of the structure under the damage and temperature effects subjected to the same forced harmonic excitation can be given as jxt MX ðtÞþ K X ðtÞ¼ fe (3) c c c where K is the n n stiffness matrix of the structure under the damage and temperature effects; X ðtÞ is the n 1 displacement response vector of the structure under the damage and temperature effects. The stiffness matrix and the displacement response vector of the structure under the damage and temperature effects can be expanded and rewritten as K ¼ K dK (4a) X ¼ X dX (4b) c Soo Lon Wah and Chen 575 where dK represents the small change of the stiffness matrix due to damage and temperature effects; dX represents the small change of the displacement response vector due to damage and temperature effects. Substituting equation (4) into equation (3) and noting equation (2) leads to MdXðtÞþ KdXðtÞþ dKX ðtÞ¼ 0 (5) Since the input excitation is harmonic, the steady-state output will also be harmonic. Therefore, at steady state jxt XðtÞ¼ Ye (6a) jxt dXðtÞ¼ dYe (6b) jxt X ðtÞ¼ Y e (6c) c c In which Y, dY, and Y represent the n 1 steady-state vibration amplitudes of XðtÞ; dXðtÞ, and X ðtÞ, c c respectively. Substituting equation (6) into equation (5) leads to equation (7), which is only valid for a speciﬁc excitation frequency x. ðK x MÞdY þ dKY ¼ 0 (7) Rearranging yields dKY ¼ðx M KÞdY (8) In which the left-hand side of the equation has an unknown term dK which represents the change in global stiffness of the structure as a result of damage and temperature effects affecting the structure. This change is due to the reduction/increase in elemental stiffness of all the elements of the structure due to the temperature effect, and the reduction in elemental stiffness of all the damaged elements. It can be represented by TE dK ¼ c K (9) s s s ¼ 1 where TE is the total number of elements in the structure; K is the elemental stiffness matrix of the sth element at reference temperature; c is the extent coefﬁcient of the sth element, which represents the percentage reduction/increase in stiffness of the element after the reduction/increase in stiffness due to the combination of damage and temperature effects, when compared to the original stiffness of the member obtained at reference temperature. Substituting equation (9) into equation (8) yields TE c K Y ¼ðx M KÞdY (10) s s c s¼1 Rearranging results into TE c K Y ¼ðx M KÞdY (11) s s c s¼1 576 Journal of Low Frequency Noise, Vibration and Active Control 39(3) Equation (11) can be decomposed into n number of equations, with each being expressed as TE c K Y ¼ v (12) s s;p c p s¼1 where K is the pth row in the elemental stiffness matrix of the sth element; s;p v is the pth row of the n 1 vector v, where v ¼ðx M KÞdY. Expanding equation (12) yields c K Y þ c K Y þ þ c K Y ¼ v (13) 1 1;p c 2 2;p c TE TE;p c p Hence, similar to equation (13), the system of the n number of equations due to the decomposition of equation (11) can be expressed as c K Y þ c K Y þ þ c K Y ¼ v 1 1;1 c 2 2;1 c TE TE;1 c 1 c K Y þ c K Y þ þ c K Y ¼ v 1 1;2 c 2 2;2 c TE TE;2 c 2 (14) c K Y þ c K Y þ þ c K Y ¼ v 1 1;n c 2 2;n c TE TE;n c n The system of equation above, which is strictly valid for a speciﬁc excitation frequency x, can be solved simultaneously to obtain the extent coefﬁcient of each element present in the structure. In order to solve all unknown extent coefﬁcients, the amount of degrees-of-freedom in the structure must be equal to or greater than the number of elements present in the structure. However, this is not always the case. To tackle this problem, several sets of equations may be generated using different excitation frequencies ðx ; x ; .. .; x Þ, where the sub- 1 2 e script e represents the amount of excitation frequency used. Following this approach, a new set of equations can be formed by selecting TE equations from the generated sets of equations, followed by solving the new set of equations simultaneously to obtain the extent coefﬁcients. Stage 2: Removal of temperature effect for damage coefficient assignment After each element has been assigned with an extent coefﬁcient, the temperature effect should be removed to give the damage locations and extent of the damaged elements. The extent coefﬁcient c obtained for each element after the application of equation (14) can be decomposed into its contributing factors, which are the temperature effect and the damage effect as follows c ¼ c þ c (15) s t;s d;s where c is a temperature coefﬁcient representing the percentage reduction/increase in elemental stiffness of the sth t;s element due to the effect of the changing temperature from the reference baseline temperature used to calculate the extent coefﬁcient c ; c is a temporary damage coefﬁcient representing the percentage reduction in elemental stiffness of the sth d;s element from the stiffness obtained at reference temperature used to calculate the extent coefﬁcient c due to the effect of damage. The coefﬁcient c is temporary because the amount of reduction in stiffness used to calculate the percentage is d;s compared to the stiffness obtained at reference temperature; however, since the temperature of the structure being monitored has changed, to know the actual percentage reduction in stiffness, this amount of reduction should be compared to the stiffness at the current temperature condition (without the effect of damage), and not to the stiffness obtained at the reference temperature condition. This is because, for the same amount of percentage reduction in stiffness, different temperature condition will have different amount of reduction in stiffness. Thus, a new damage coefﬁcient, D , needs to be deﬁned and assigned to the elements to get the actual percentage s Soo Lon Wah and Chen 577 reduction in elemental stiffness due to damage effect. To be able to obtain the damage coefﬁcients of the elements to represent the percentage reduction in stiffness due to damage, the temperature coefﬁcient of each damaged element needs to be estimated ﬁrst so that the stiffness of the element at its current temperature condition (without the effect of damage) can be obtained and be used for comparison. This can be achieved by using the temperature coefﬁcients of the undamaged elements to estimate the temperature coefﬁcients of the damaged elements. Thus, the temperature coefﬁcient of the sth element may be obtained as follows c ¼ g þ g L þ g L (16) t;s s s 1 2 3 where g ; g , and g are undetermined coefﬁcients; 1 2 3 L is the distance between the sth element and a preset reference zero coordinate system. Equation (16) is a quadratic equation representing the variation of the Young’s modulus across the structure from the initial Young’s modulus used as reference and obtained at the reference temperature, due to the gradient temperature across the structure and to the nonlinear effect of the changing temperature condition on the vibra- tion properties and frequency responses of the structure. Equation (16) is in the form of a quadratic equation so as to take into account nonlinear gradient of temperature across the structure as well as nonlinear effect produced by temperature condition. Equation (16) can be obtained by ﬁrst plotting the extent coefﬁcients of the all the elements, followed by ﬁnding the equation of the line joining all the extent coefﬁcient values together. However, it should be noted that the damaged elements should have their damage effects contribution to the extent coefﬁcients removed before creating the equation of the line. To do this, the graph of the extent coefﬁcients of all the elements versus the elements’ number (ranked in order with the distance between them) should ﬁrst be plotted. Then, examining the graph, the damage locations can be obtained by the appearance of peaks in the graph. These peaks represent the contribution of damage to the extent coefﬁcients. Thus, by eliminating the peaks in the graph, the equation of the line can then be generated. The values of the coefﬁcients g ; g , and g 1 2 3 can thus be obtained by deﬁning the line. Substituting any value of L in the equation which represents the distance of the sth element to the reference zero coordinate, the temperature coefﬁcient for that element can then be estimated. It should be noted that a lower/higher order polynomial function can be adopted to best represent the extent coefﬁcient proﬁle. To give the percentage reduction of elemental stiffness of the sth element due to damage, a damage coefﬁcient D may be deﬁned. This damage coefﬁcient compares the amount of reduction in stiffness due to damage of each element to the actual stiffness of the element due to temperature effect only. The damage coefﬁcient D can be related to the temporary damage coefﬁcient c as follows d;s d;s D (17) 1 c t;s where D is the damage coefﬁcient of the sth element; ð1 c Þ represents the elemental stiffness of the sth element at its current temperature condition without the t;s effect of damage, in terms of a percentage of the stiffness obtained at the reference temperature used to calculate the extent coefﬁcient. Substituting equations (16) and (17) into equation (15) leads to 2 2 c ¼ðg þ g L þ g L Þþ D ð1 ðg þ g L þ g L ÞÞ (18) s s s s 1 2 3 s 1 2 3 s Rearranging equation (18), the damage coefﬁcient D for each element can be obtained as c ðg þ g L þ g L Þ s s 1 2 3 s D ¼ (19) 1 ðg þ g L þ g L Þ 1 2 3 From equation (19), the damage coefﬁcient, D , for the sth element can be obtained. Undamaged element will have D value of zero, while damaged element will have D value ranging from zero to one, with one representing s s 578 Journal of Low Frequency Noise, Vibration and Active Control 39(3) fully damaged. It should be noted that this way of calculating c for each element allows the proposed method to t;s perform for structures, which have nonlinear temperature gradient across the structure as well as structures whose vibration properties and frequency responses are affected nonlinearly by the changing temperature condition. Summary of the method Below are the procedures to follow to implement the proposed damaged detection method, and Figure 1 summa- rizes the procedures in the form of a ﬂowchart. 1. Apply a harmonic excitation force to the structure in its undamaged state to gather frequency responses of the structure for the baseline of the undamaged structure. If there are more elements than the number of degrees-of-freedom available, several harmonic excitation forces with different excitation frequencies should be applied to the structure one at a time. � The temperature condition along the structure can have a nonlinear proﬁle which is usually the case for real- life structures. Start Obtain the undamaged structure for the baseline Apply a harmonic excitation force(s) to the structure to gather the frequency responses of the structure Apply the same harmonic excitation force(s) to the structure to be monitored to gather its frequency responses Use the frequency responses gathered and apply Eq. (14) to calculate the extent coefficient of each element Plot the extent coefficients against their element number and examine whether peaks appear in the plot Yes No Is there any peak? Create the equation of the line joining all the extent coefficients (minus damage contribution) in the form of Eq. (16) Apply Eq. (19) to calculate the damage coefficient of each element to get the damage extent and locations End End Figure 1. Flowchart of the procedures to follow for damage detection. 1. Use damage detection methods (that did not account 2. Remove the temperature distribution trend for temperature) to determine the change in stiffness of to obtain the actual damage extent and locations each element due to temperature and damage effects Soo Lon Wah and Chen 579 2. Apply the same harmonic excitation force(s) to the structure when monitoring is required to gather new frequency responses of the structure. � It does not matter whether the temperature is constant or has a gradient distribution throughout the structure. 3. Use the frequency responses obtained from steps 1 and 2, and apply equation (14) to obtain the extent coef- ﬁcient c of each element. 4. Plot the extent coefﬁcients obtained against the elements’ number and analyze the graph to locate the dam- aged elements. � The damaged elements will appear as peaks in the plot. 5. Use extent coefﬁcients from all the undamaged and damaged elements (ignoring the contribution of the damage effect to the extent coefﬁcient) for the creation of the equation of the line (temperature effect trend) in the form of equation (16) so that the temperature coefﬁcient c of each element can be calculated. t;s � Equation (16) is in the form of a quadratic equation; however, a lower/higher order polynomial function can be used to best represent the trend of the temperature condition. 6. Apply equation (19) to obtain the damage coefﬁcient D of each element to get the damage locations and extent. � The undamaged elements will have damage coefﬁcient of zero, while the damaged elements will have damage coefﬁcient ranging from zero to one, with one representing fully damaged. It is worth noting that stage 2 of the method can be implemented with other damage detection methods that did not account for temperature effect to eliminate the inﬂuence of varying temperature condition and allow the locations and extent of damages to be obtained, while the damage detection method proposed in stage 1 can be replaced by other damage detection methods that can give the amount of change of stiffness due to external factors from a baseline stiffness. To this end, the procedures given above can be modiﬁed accordingly; the ﬁrst three steps in the procedure can be replaced by other damage detection methods, while the last three steps can be incorporated in the selected damage detection method to eliminate the temperature effect and to locate and quantify damages. Numerical examples To demonstrate the proposed approach for damage detection, localization, and quantiﬁcation, a beam structure representing a model bridge and a shear building are used as case studies. The two structures are adopted since it is well known that temperature condition along long span bridges and tall buildings usually varies, hence, inﬂuenc- ing the dynamic properties of the structures. Beam structure The beam structure used for demonstration is shown in Figure 2. It is 15-m long, with one end (end A) being pin connected and the other end (end B) being free. At 10 m from the pin-connected end, the structure rests on a roller support. The structure is assumed to be made of steel material with a Young’s modulus of 200 GPa and density of 7850 kg=m . The cross-sectional area and the second moment of area of the beam are 2 4 assumed to be 0.08 m and 0.0006 m , respectively. The two-dimensional numerical model of the structure is built in MATLAB and it is discretized into 15 beam elements of 1-m long each. A harmonic excitation force with x ¼ 2000 rads/s is applied at the free end (B) pointing downward to generate the frequency responses of the structure. Force A 12 34 5 6 7 8 9 10 11 12 13 14 15 B 10 m 5 m Figure 2. Two-dimensional beam structure. 580 Journal of Low Frequency Noise, Vibration and Active Control 39(3) E = 0.002T - 0.359T + 203.07 -40 -20 0 20 40 60 80 Temperature (°C) Figure 3. Hypothetical variation of Young’s modulus with temperature. Table 1. Descriptions of undamaged and damaged cases subjected to nonlinear temperature gradient across the structure. Case no. Temperature at A ( C) Temperature at B ( C) Element number Damage extent (%) Undamaged 1 20 14 – – Undamaged 2 10 25 – – Damaged 1 10 2 8 20 Damaged 2 35 23 3 and 12 10 and 10 To simulate the effect of varying temperature condition, the Young’s modulus of the steel material is assumed to be temperature dependent as suggested by An et al. Since it was found in the literature that temperature 25,26 condition usually has nonlinear relationship with the vibration properties of civil engineering structures, the hypothetical variation of Young’s modulus with temperature is assumed to be nonlinear as can be seen in Figure 3. To consider the effect of temperature gradient across the structure, two undamaged cases and two damaged cases are used to demonstrate the proposed approach, and the descriptions of the cases are presented in Table 1. Figure 4 illustrates the temperature variation across the beam structure. In this beam structure, damage is assumed to be represented by a reduction in elemental stiffness of the structural components. Since generally structures are faced with gradient temperature, a nonlinear temperature distribution is assumed for the baseline of the undamaged structure. The temperature condition of the baseline of the undamaged struc- ture from which subsequent measurements can be compared with is given in Figure 5. Note that any temperature could have been used as baseline. After applying the proposed damage detection method, and the application of equation (14), the extent coefﬁcient c of each element for each case can be obtained. Since generally noise affects the performance of damage detection methods, to demonstrate the robustness of the second stage of the proposed approach to eliminate the effect of temperature conditions to give the damage locations and extent, 10% of noise is added to the extent coefﬁcients. The performance of the approach of eliminating the effect of temperature is dependent upon the extent coefﬁcients, which in turn are dependent upon the techniques used to obtain these coefﬁcients in stage 1. Therefore, it is the extent coefﬁcients that are polluted with noise, to demonstrate that the proposed approach can still be applied for cases where the extent coefﬁcients are subjected to errors. The polluted extent coefﬁcients are given by ^c ¼ c 1 þ < (20) s s where ^c is the polluted extent coefﬁcient; @ is the speciﬁed noise level added to the extent coefﬁcients and taken as 10 in this case study; < is a random number between 1 and 1. Plots of the extent coefﬁcients (polluted with noise) for the four cases under consideration are presented in Figure 6. It can be seen from Figure 6(a) and (b) that for the undamaged cases, no damage is present in the structure, since no consequent peak appears in the plots. Some ﬂuctuations in the ﬁrst plot can be seen in the Young's Modulus E (GPa) Soo Lon Wah and Chen 581 (a) (b) T = 0.04L + 0.40L + 10 s s T = -0.04L + 0.20L + 20 T = 25 °C s s B T = 20 °C T = 14 °C T = 10 °C T = 0 °C T = 0 °C A 12 34 5 6 7 8 9 10 11 12 13 14 15 B A 12 34 5 6 7 8 9 10 11 12 13 14 15 B L L s s 10 m 5 m 10 m 5 m (c) (d) T = -0.03L - 0.35L + 35 s s T = 35 °C T = 23 °C T = 0.02L + 0.50L - 10 s s T = 2 °C T = 0 °C T = 0 °C T = -10 °C 34 5 6 7 9 11 13 B 5 6 9 B A 12 8 10 12 14 15 A 12 34 7 8 10 11 12 13 14 15 L L s s 10 m 5 m 10 m 5 m Figure 4. Temperature variation across the beam structure for (a) undamaged case 1, (b) undamaged case 2, (c) damaged case 1, and (d) damaged case 2. T = -0.04L - 0.4L + 5 s s T = 5 °C T = 0 °C T = -10 °C A 12 34 5 6 7 8 9 10 11 12 13 14 15 B 10 m 5 m Figure 5. Temperature variation across the undamaged beam structure used as baseline. middle portion of the beam; however, no obvious peak that indicates an abrupt and large change in stiffness exists. Figure 6(c) and (d) give the plots of the two damaged cases. It can be seen from both plots that peaks appear to alert the presence of damages in the structure. The ﬁrst damaged case (Figure 6(c)) has a peak at element 8, which indicates that element 8 is damaged, while the second damaged case (Figure 6(d)) has two peaks at elements 3 and 12, respectively, suggesting that these two elements are damaged. By constructing equation (16) to determine the temperature coefﬁcient of each element, and by applying equation (19), a damage extent of 20% for element 8 can be calculated for the ﬁrst damaged case, while for the second damaged case, elements 3 and 12 are found to have a reduction in elemental stiffness of 10% and 11%, respectively. From the four plots shown in Figure 6, it can also be seen that the extent coefﬁcient values of the undamaged elements generally vary smoothly from element to element throughout the structure. This can be attributed to the fact that, for the undamaged element, the amount of change in stiffness is due fully to the deviation in temperature from the baseline temperature condition, and since temperature condition along a structure usually changes gradually from location to location, the change of stiffness of each element, which is represented by the extent coefﬁcient, will increase/decrease steadily along the structure. Only a signiﬁcant variation in the amount of change in stiffness from an element to the nearby elements will produce a sharp change in the extent coefﬁcient proﬁle. 582 Journal of Low Frequency Noise, Vibration and Active Control 39(3) (a) (b) 0.04 0.06 0.036 0.04 0.032 0.028 0.02 0.024 c = 0.00004L + 0.00055L + 0.02413 c = 0.00016L + 0.00107L + 0.00899 t,s s s t,s s s 0.02 0 12 34 56 789 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Element number Element number (c) (d) 0.24 0.18 c = 0.00010L + 0.00206L - 0.02783 c = 0.000009L + 0.00071L + 0.04139 t,s s s t,s s s 0.15 0.16 0.12 0.08 0.09 0.06 -0.08 0.03 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 12 34 56 789 10 11 12 13 14 15 Element number Element number Figure 6. Extent coefficient plot of beam structure for (a) undamaged case 1, (b) undamaged case 2, (c) damaged case 1, and (d) damaged case 2. As damage is a localized effect which produces a signiﬁcant decrease in stiffness of the damaged element, it will produce a sudden change in the extent coefﬁcient plot. Thus, any sharp change/peak in the plot can be attributed to the effect of damage. Noise affecting the measurements will also create a change in the extent coefﬁcient plot, however, these ﬂuctuations will be much less when compared to the effect of damage and no distinct peak will occur. Peaks at the damage locations will always exist because the extent coefﬁcient combines the amount of change of stiffness due to temperature and damage effects together. It can therefore be concluded that for a damaged element, the temperature contribution to the extent coefﬁcient will follow the same trend as other undamaged elements’ extent coefﬁcients, while the contribution of damage will be an addition to that trend. By observing the extent coefﬁcient plot, the temperature and damage effects can be distinguished, and the damage extent can be quantiﬁed through the peaks’ contribution. Shear building A shear building (Figure 7(a)) is also used as another example to demonstrate the applicability of the proposed approach for damage localization and quantiﬁcation, since buildings’ vibration properties are commonly affected 27,28 by the changing temperature condition they are faced with. The numerical model of the shear building is the same as the one used as case study by Roy and Ray-Chaudhuri. The shear building is 12-story high with uniform story mass and stiffness distribution throughout the height of the structure. The story mass and stiffness prop- 5 8 1 erties adopted in this paper are 1 10 kg and 2 10 Nm deﬁned at reference temperature of 20 C, respec- tively. A harmonic excitation force with angular frequency x ¼ 1000 rads/s is applied at the ﬁrst ﬂoor level to generate the frequency responses of the structure. To simulate a varying temperature condition, it is assumed that the stiffness of the structure varies nonlinearly with the temperature condition. The relationship between the temperature condition and the story level stiffness is given by equation (21). The story stiffness is assumed to vary nonlinearly with the temperature condition because Young’s modulus of structures generally varies with a changing temperature condition, which as a result, affects the stiffness of the structures. The temperature distribution across the undamaged structure used as baseline is given in Figure 7(b). Damage is simulated as a reduction in story-level stiffness, and the scenarios applied to the shear building are presented in Table 2. The temperature proﬁle along the structure for the different cases are shown in Figure 8, and it is assumed that the temperature condition at the top of the building is greater than that at the bottom since buildings are subjected to greater radiation at their roof level than at their base level. Similar to the beam case study, 10% of noise is added in this shear building case study. Extent coefficient Extent coefficient Extent coefficient Extent coefficient Soo Lon Wah and Chen 583 (a) (b) T = 13.12 °C top m12 m12 k12 k12 m11 m11 k11 k11 m10 m10 T = 0 °C k10 k10 m9 m9 k9 k9 m8 m8 k8 k8 m7 m7 k7 k7 m6 m6 k6 k6 m5 m5 k5 k5 m4 m4 k4 k4 m3 m3 k3 k3 m2 m2 k2 k2 m1 m1 Force k1 k1 T = 10 °C base Figure 7. (a) Two-dimensional shear building structure and (b) temperature variation across the undamaged shear building used as baseline. Table 2. Descriptions of undamaged and damaged cases subjected to nonlinear temperature gradient throughout the shear building. Case no. Temperature at base ( C) Temperature at top ( C) Story level Damage extent (%) Undamaged 1 20 29 – – Damaged 1 15 27 3, 6, and 9 5, 10, and 15 Damaged 2 30 33 6 10 Damaged 3 5 8 6 10 (a) (b) (c) (d) T = 29 °C T = 27 °C T = 33 °C T = 8 °C top top top top m12 m12 m12 m12 k12 k12 k12 k12 m11 m11 m11 m11 k11 m10 k11 k11 k11 m10 m10 m10 T = 0 °C T = 0 °C T = 0 °C T = 0 °C k10 k10 k10 k10 m9 m9 m9 m9 k9 k9 k9 k9 m8 m8 m8 m8 k8 m7 k8 m7 k8 m7 k8 m7 k7 k7 k7 k7 m6 m6 m6 m6 k6 k6 k6 k6 m5 m5 m5 m5 k5 k5 k5 k5 m4 m4 m4 m4 k4 k4 k4 k4 m3 m3 m3 m3 k3 k3 k3 k3 m2 m2 m2 m2 k2 k2 k2 k2 m1 m1 m1 m1 L L L L s s s s k1 k1 k1 k1 T = 20 °C T = 15 °C T = 30 °C T = 5 °C base base base base Figure 8. Temperature variation throughout the shear building structure for (a) undamaged case 1, (b) damaged case 1, (c) damaged case 2, and (d) damaged case 3. 4 2 5 8 k ¼ 1:2 10 T 9:2 10 T þ 2:136 10 (21) s;t where k is the stiffness of the sth story considering temperature effect in Nm ; s;t T is the temperature condition of the sth story in C. The plots of the extent coefﬁcients versus story’s level for the four cases are presented in Figure 9. It can be seen from Figure 9 that the cases are well classiﬁed; the ﬁrst plot (Figure 9(a)) indicates no damage is present, while the following three plots ((Figure 9(b)–(d)) show that the structure is damaged due to the appearance of peaks. It can T = 0.75L + 20 T = 0.05L + 0.40L + 15 s s T = 0.02L + 0.02L + 10 s s T = 0.02L + 0.01L + 30 s s T = 0.02L + 0.01L + 5 s s 584 Journal of Low Frequency Noise, Vibration and Active Control 39(3) (a) (b) (c) (d) 12 12 12 12 11 11 11 11 10 10 10 10 9 9 9 9 8 8 8 8 7 7 7 7 6 6 6 5 5 5 5 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 0.024 0.03 0.036 -0.05 0.05 0.15 0.25 0 0.1 0.2 -0.1 0 0.1 0.2 0.3 Extent coefficient Extent coefficient Extent coefficient Extent coefficient Figure 9. Extent coefficient plot of shear building for (a) undamaged case 1, (b) damaged case 1, (c) damaged case 2, and (d) damaged case 3. again be seen that the temperature effect produces a relatively smooth change of extent coefﬁcient from undam- aged story to undamaged story, while damage effect produces a sharp change in extent coefﬁcient of the damaged story from the nearby stories. This sharp change in extent coefﬁcient represents the extra contribution in decreas- ing the stiffness of the damaged story. Applying equation (19), it is found that the ﬁrst damaged case has story levels 3, 6, and 9 to have a reduction in stiffness by 5%, 10%, and 16%, respectively, the second damaged case has 9% reduction in the sixth story, while the last case has a reduction in stiffness of 10% in the sixth story. For this damage detection method, the temperature trend can be found through the quadratic function given by equation (16). For example, from Figure 9(a), it can be seen that the ﬁrst undamaged case has a nonlinear change in extent coefﬁcient. This case is analyzed in this paper to demonstrate that, depending on the proﬁle of extent coefﬁcient, the function given through equation (16) can be adjusted to suit the extent coefﬁcient proﬁle. Being able to adjust the function order allows the proposed approach to closely represent the temperature coef- ﬁcient of each element, thus decreasing the chance of obtaining the wrong damage scenario. For example, for the ﬁrst undamaged case (Figure 9(a)), a cubic function is adopted; therefore, a cubic part is added to the quadratic function (equation (16)). One potential difﬁculty of the proposed approach is how to quantify the extent of damages through the peaks, since different temperature condition will produce different peak contributions. For instance, the last two dam- aged cases have the same damaged element with the same damaged extent, but with different temperature con- dition. Quantifying the peak amount by subtracting the temperature coefﬁcient of that element from its extent coefﬁcient, it is found that the second damaged case (Figure 9(c)) has a value of 0.083, while the third damaged case (Figure 9(d)) has a value of 0.104. The difference in these two values is attributed to the different temperature conditions the two cases are at, since different temperature condition will produce different amount of change of stiffness for the same percentage of reduction in stiffness. This issue can be overcome by deriving relationships between the change in stiffness due to damage, and the stiffness of the elements at the current temperature condition. For example, for the proposed damage detection method, the damaged elements can be quantiﬁed through equation (19), which relates the peak contribution to the stiffness of the members at the current tem- perature condition. Discussion of the proposed approach The principle behind most conventional damage detection methods that did not take into account temperature effect is to compare new measurements to a single benchmark so as to determine which member has a change in stiffness from the original elemental stiffness, and to quantify this change for damage localization and quantiﬁ- cation. Since damage is the only effect considered in the methods that will cause change in stiffness, only the damaged elements will have a reduction in stiffness, however, most civil engineering structures are subjected to changing temperature condition, and both damage and temperature effects will change the stiffness of the elements. Therefore, obtaining the total change in stiffness due to both damage and temperature effects ﬁrst, followed by removing the temperature effect, will facilitate damage localization and quantiﬁcation. Story level 3 2 c = 0.00001L - 0.00019L + t,s s s 0.00166L + 0.02694 Story level c = 0.00005L + 0.00103L + 0.01435 t,s s s Story level c = -0.00002L - 0.00037L + 0.04349 t,s s s Story level c = -0.00001L + 0.00039L - 0.01900 t,s s s Soo Lon Wah and Chen 585 In the aspect of creating a baseline for damage detection, most approaches have difﬁculty in creating a baseline when encountering temperature gradient along the structures, which generally has a great inﬂuence on the vibra- tion properties. This condition is commonly found when analyzing long span bridges and tall buildings because ambient solar radiations are not evenly distributed along the structures. Limited methods were proposed in the literature to tackle gradient temperature condition along the structures for damage localization and quantiﬁca- tion. The reason behind may be attributed to the difﬁculty in creating a baseline of the undamaged structure which is commonly constructed using a wide range of temperature conditions, to include all possible conditions the structure may encounter, since there is an inﬁnite amount of temperature proﬁles that the structure can be faced with. Moreover, gradient temperature may overestimate/underestimate temperature effect on the vibration prop- erties, if the temperature proﬁle is not accounted for properly, hence leading to false alerts. To tackle this gradient temperature and temperature condition issue in general, drawing the extent coefﬁcient proﬁle of the structure due to temperature effect, seems to be the best solution since no predeﬁned solution for that proﬁle is required to be included in the baseline, and this extent coefﬁcient proﬁle can be obtained using the global trend of change in stiffness. To this end, this paper uses a quadratic function to represent the global change in elemental stiffness which can then be related to the temperature distribution along the structure. It is important to allow the proﬁle of extent coefﬁcient to be represented by a quadratic function or even higher order polynomial function because, if the proﬁle can only be represented by a linear function, the temperature coefﬁcient would be underestimated/overestimated, hence underestimating/overestimating the contribution of temperature in changing the stiffness of the elements, leading to false damage extent and locations. It should be noted that the quadratic function deﬁning the temperature distribution proﬁle along the structure (e.g., Figure 4) is different from the one describing the proﬁle of extent coefﬁcient (e.g., Figure 6). This difference is attributed to the nonlinear relationship between the Young’s modulus of the material and the temperature as well as the temper- ature proﬁle of the baseline structure, which were accounted for by the extent coefﬁcient plot, and not by the temperature distribution proﬁle along the structure. Using only one reference temperature as baseline is proposed in this paper, as it has the advantage that the damage detection method can give the damage locations and extent since the structure can be compared locally to the baseline. Moreover, the method can be implemented immediately without requiring months of measurements to create the baseline of the undamaged structure. It also reduces the risk of false alerts which may occur for structures that have temperature–Young’s modulus relationship to be nonlinear, since, if not all conditions are included in the baseline, false alerts may occur. The proposed approach eliminates the risk that small damages are not detected, which may occur if a range of temperature condition is used, because the small damages may not alter the dynamic properties signiﬁcantly enough to trigger the damage alert. Therefore, by using the proposed approach, the difﬁculty of using a range of temperature condition for the baseline of the undamaged structure for damage localization and quantiﬁcation is enhanced, as it can easily tackle any temperature proﬁle along the structure. This approach is appealing because it can be incorporated in most existing damage detection methods easily by ﬁrst using the damage detection methods to quantify the changes in stiffness of all the elements due to temperature and damage effects, followed by removing the temperature contribution which is a global effect by ﬁnding the temperature trend, to obtain the damage locations and extent. Conclusion A new insight on how to approach damage detection of civil engineering structures subjected to varying temper- ature and to gradient temperature conditions by using a single temperature proﬁle as the baseline is introduced in this paper. The approach uses the concept of ﬁrst quantifying the change in stiffness of all the elements due to the effects of temperature and damage, and then the temperature effect which has a global trend on the structure is removed, to give the damage locations and extent. The approach is implemented with a proposed damage detec- tion method that did not account for temperature effect, to be able to tackle varying temperature and gradient temperature conditions. The damage detection method with the proposed approach incorporated is tested for different undamaged and damaged cases using a beam structure and a shear building subjected to gradient temperature conditions. The results demonstrate that, by quantifying the change in stiffness of all the elements due to external factors, the trend of temperature effect along the structure can be obtained, as well as the damage locations. This temperature effect is found to be different from the damage effect, which is a localized effect, thus, the actual damage locations and extent can be obtained by eliminating the temperature effect. The proposed approach can be implemented with other damage detection methods that did not account for temperature effect to obtain the damage locations and extent under changing temperature condition. 586 Journal of Low Frequency Noise, Vibration and Active Control 39(3) Declaration of conflicting interests The author(s) declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following ﬁnancial support for the research, authorship, and/or publication of this article: This work was supported by the National Science Foundation of China (51708306,51605233), Zhejiang Qianjiang Talent Scheme (QJD1402009), and Faculty Inspiration Grant. References 1. Carden EP and Fanning P. Vibration based condition monitoring: a review. Struct Health Monit 2004; 3: 355–377. 2. Fan W and Qiao P. Vibration-based damage identiﬁcation methods: a review and comparative study. Struct Health Monit 2011; 10: 83–111. 3. Dominik I, Iwaniec M and Lech Ł. Low frequency damage analysis of electric pylon model by fuzzy logic application. J Low Freq Noise Vib Active Control 2013; 32: 239–251. 4. Abouel-Seoud SA. Fault detection enhancement in wind turbine planetary gearbox via stationary vibration waveform data. J Low Freq Noise Vib Active Control 2017. DOI: 10.1177/1461348417725950. 5. Rytter A. 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Abstract

Most damage detection methods developed in the literature cannot give the locations and extent of damages under the presence of varying temperature condition. This is because temperature condition changes the vibration properties of a structure, which are commonly analyzed for damage detection, and temperature gradient throughout the structure makes it difficult to create a baseline for the undamaged structure, as the baseline is generally constructed using features obtained under a wide range of temperature conditions. In this paper, a new insight on how to approach damage detection using only a single temperature condition to create the baseline is proposed. This approach solves the damage detection under changing temperature problem in two stages by first quantifying the change of stiffness of all the elements in a structure due to temperature and damage effects, followed by removing the temperature effect, a global effect, to give the actual damage locations and extent. Using single temperature condition allows new measure- ments to be compared to a benchmark so that local deviation can be obtained, thus making the damaged elements identifiable. The proposed approach is tested using a beam structure model and a shear building under different gradient temperature conditions, and the results demonstrate that the method successfully eliminates the change in elemental stiffness due to temperature effect and gives correct damage locations and extent. The approach can be implemented with other existing damage detection methods that did not consider the effect of temperature so that structures under varying temperature condition can be analyzed. Keywords Damage detection, damage localization, damage quantification, temperature condition, temperature gradient, environ- mental conditions Introduction Engineering structures in general are subjected to constant degradation due to the harsh environmental and operational conditions they are faced with. To maintain their safety and reliability, researchers have been devel- 1–4 oping damage detection methods to assess the health condition of the structures to allow reparation/replace- ment of structural defects at an early stage before catastrophic failure occurs. Based on the information given on the state of damages, the damage detection methods can be classiﬁed into four different levels as given by Rytter Level 1: Detect the presence of damages only in the structure. Level 2: Detect and give the locations of damages in the structure. Department of Civil Engineering, The University of Nottingham, Ningbo, People’s Republic of China Corresponding author: Yung-Tsang Chen, Department of Civil Engineering, The University of Nottingham Ningbo China, 199 Taikang East Road, Ningbo, 315100, People’s Republic of China. Email: yung-tsang.chen@nottingham.edu.cn Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www. creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). Soo Lon Wah and Chen 573 Level 3: Detect, locate, and give the extent of damages in the structure. Level 4: Detect, locate, and quantify damages, and give the level of safety of the structure depending on the damage case. 1,2,6 1,2,7 In the literature, several level 2 and level 3 damage detection methods have been proposed and applied to numerical and experimental structures. However, most of the proposed methods were veriﬁed by assuming that the environmental conditions the structures are faced with are constant. Yet, it is well known that the changing environmental conditions, the changing temperature condition in particular, have a major inﬂuence on the dynamic properties of civil engineering structures, which are commonly used as damaged sensitivity features for damage detection. As a result, false alerts may occur. Thus, to provide a robust damage detection method, which does not give false alerts, the effects of the changing environmental conditions, especially the changing temperature condition, should be taken into consideration while deriving the method. Moreover, the temper- ature condition usually has nonlinear effect on the dynamic properties of the structures, and temperature gradient throughout the structures usually exists, both of which also need to be considered in the development of the damage detection method. 12–14 To tackle this temperature condition issue, several approaches, such as regression analysis and the use of 11,14,15 multivariate statistical tools to process the damage sensitivity features, have been proposed in the literature; however, most of these approaches can only give the presence of damages in the structures. Among the proposed 16–18 approaches, limited methods can perform level 2 of damage detection, and only a few methods can perform 10,19 level 3 of damage detection, i.e. give the presence, locations, and extent of the damages altogether, yet most of the existing methods do not perform well under gradient temperature condition and under nonlinear effect of the changing temperature condition. The difﬁculties in the development of level 3 damage detection under changing temperature condition may be attributed to the unﬁt approach adopted by the researchers, as the methods proposed in the literature usually use the concept of creating the baseline of an undamaged structure using damage sensitivity features captured from a wide range of temperature conditions so as to cover all possible 8,20,21 scenarios the structure may encounter, and compare new measurements with the baseline to see whether they lie within the normal range. However, by doing so, a new measurement from the structure under a speciﬁc temperature distribution is only being compared to measurements from the undamaged structure under normal temperature conditions; it is not being compared to a benchmark that represents the undamaged structure at exactly the same temperature distribution. Therefore, it is generally difﬁcult to determine which part of the structure differs from the one of the undamaged structure. It is difﬁcult to compare a new measurement captured at a speciﬁc temperature condition to the same condition from the undamaged structure because, as mentioned previously, structures are subjected to gradient temperature and to nonlinear temperature effect. These make obtaining the exact same temperature condition from the undamaged structure to be used as baseline difﬁcult, because there is an inﬁnite amount of temperature variations throughout the structure, and this is the main reason why new observations are only being compared to a set of undamaged measurements to see whether they behave similarly. To tackle this problem, some methods in the literature tried to predict the temperature condition along with the damage locations based on the features gathered; however, gradient temperature distribution makes it difﬁcult to get reasonable results because gener- ally, only temperature at one or a few locations are measured throughout the whole structure, hence leading to errors in the temperature proﬁle. Moreover, constructing the baseline from a wide range of temperature con- ditions has the drawback that it may reduce the sensitivity of the methods to detect small damages if proper consideration is not taken into account; it is also time-consuming to gather all the necessary features for damage detection, hence retarding the implementation of the methods. This paper proposes a new insight on how to approach damage detection under varying temperature condition by adopting only one temperature condition for the baseline of the undamaged structure. This approach ﬁrst treats damage detection under changing temperature condition in the same way as one treats damage detection assuming constant temperature condition, followed by determining the change in stiffness of each element of the structure due to external effects. After the change in elemental stiffness has been quantiﬁed, temperature effect is removed from the extent coefﬁcients by analyzing the trend of the extent coefﬁcients along the structure to give damage locations and extent. In this paper, the damage detection method by Soo Lon Wah and Chen is further extended to take into account the effect of changing temperature with gradient temperature distribution. The proposed method uses frequency responses, and stiffness and degrees-of-freedom of each element to obtain an extent coefﬁcient for each element to represent the effects of both temperature and damage on the structure. After the extent coefﬁcients for all elements have been found, the temperature effect is then removed to give altogether the actual damage locations and damage extent calculated based on using one temperature as 574 Journal of Low Frequency Noise, Vibration and Active Control 39(3) baseline. To demonstrate the proposed approach, a beam structure model and a shear building are used as examples for damage detection and quantiﬁcation, and the results obtained show that by using only one tem- perature condition for the baseline, damage localization and quantiﬁcation under gradient temperature condition and nonlinear temperature effect can be achieved. This approach can be implemented easily with other existing damage detection methods to tackle level 3 damage detection under varying temperature condition. Damage detection methodology This section introduces the proposed approach for damage detection, localization, and quantiﬁcation of civil engi- neering structures subjected to varying temperature condition and to gradient temperature distribution. In this approach, damages in a structure are assumed to be represented by reductions in elemental stiffness of the members of the structure, and the locations of the damages are given by the element number each element is assigned to. The approach consists of two stages. In the ﬁrst stage, an extent coefﬁcient that is related to the contribution of the temperature and damage effects affecting the structure is assigned to each element present in the structure. In the second stage, the temperature effect is removed from the extent coefﬁcient so that the damage extent can be obtained through the use of a damage coefﬁcient. The damage coefﬁcient, if not equals to zero, gives the actual damage extent of an element, and the location of the damage can be identiﬁed by the element number accordingly. Stage 1: Extent coefficient assignment Consider a structure with n degrees-of-freedom subjected to a forced harmonic excitation. The mathematical model representing the motion of the structure can be given by the equation of motion jxt € _ MXðtÞþ CXðtÞþ KXðtÞ¼ fe (1) where M, C, and K are the n n mass, damping, and stiffness matrices of the structure, respectively; XðtÞ is the n 1 displacement response vector of the structure; f is the n 1 amplitude vector of the forced harmonic excitation; pﬃﬃﬃﬃﬃﬃﬃ j is the imaginary unit ð¼ 1Þ; x is the angular frequency of the forced harmonic excitation. For simplicity, the damping matrix C is neglected. Thus, equation (1) becomes jxt MXðtÞþ KXðtÞ¼ fe (2) Structural damage or structural change due to temperature variation here is referred to as a reduction/increase in the stiffness property of the structure with no change in the mass property. Hence, the equation of motion of the structure under the damage and temperature effects subjected to the same forced harmonic excitation can be given as jxt MX ðtÞþ K X ðtÞ¼ fe (3) c c c where K is the n n stiffness matrix of the structure under the damage and temperature effects; X ðtÞ is the n 1 displacement response vector of the structure under the damage and temperature effects. The stiffness matrix and the displacement response vector of the structure under the damage and temperature effects can be expanded and rewritten as K ¼ K dK (4a) X ¼ X dX (4b) c Soo Lon Wah and Chen 575 where dK represents the small change of the stiffness matrix due to damage and temperature effects; dX represents the small change of the displacement response vector due to damage and temperature effects. Substituting equation (4) into equation (3) and noting equation (2) leads to MdXðtÞþ KdXðtÞþ dKX ðtÞ¼ 0 (5) Since the input excitation is harmonic, the steady-state output will also be harmonic. Therefore, at steady state jxt XðtÞ¼ Ye (6a) jxt dXðtÞ¼ dYe (6b) jxt X ðtÞ¼ Y e (6c) c c In which Y, dY, and Y represent the n 1 steady-state vibration amplitudes of XðtÞ; dXðtÞ, and X ðtÞ, c c respectively. Substituting equation (6) into equation (5) leads to equation (7), which is only valid for a speciﬁc excitation frequency x. ðK x MÞdY þ dKY ¼ 0 (7) Rearranging yields dKY ¼ðx M KÞdY (8) In which the left-hand side of the equation has an unknown term dK which represents the change in global stiffness of the structure as a result of damage and temperature effects affecting the structure. This change is due to the reduction/increase in elemental stiffness of all the elements of the structure due to the temperature effect, and the reduction in elemental stiffness of all the damaged elements. It can be represented by TE dK ¼ c K (9) s s s ¼ 1 where TE is the total number of elements in the structure; K is the elemental stiffness matrix of the sth element at reference temperature; c is the extent coefﬁcient of the sth element, which represents the percentage reduction/increase in stiffness of the element after the reduction/increase in stiffness due to the combination of damage and temperature effects, when compared to the original stiffness of the member obtained at reference temperature. Substituting equation (9) into equation (8) yields TE c K Y ¼ðx M KÞdY (10) s s c s¼1 Rearranging results into TE c K Y ¼ðx M KÞdY (11) s s c s¼1 576 Journal of Low Frequency Noise, Vibration and Active Control 39(3) Equation (11) can be decomposed into n number of equations, with each being expressed as TE c K Y ¼ v (12) s s;p c p s¼1 where K is the pth row in the elemental stiffness matrix of the sth element; s;p v is the pth row of the n 1 vector v, where v ¼ðx M KÞdY. Expanding equation (12) yields c K Y þ c K Y þ þ c K Y ¼ v (13) 1 1;p c 2 2;p c TE TE;p c p Hence, similar to equation (13), the system of the n number of equations due to the decomposition of equation (11) can be expressed as c K Y þ c K Y þ þ c K Y ¼ v 1 1;1 c 2 2;1 c TE TE;1 c 1 c K Y þ c K Y þ þ c K Y ¼ v 1 1;2 c 2 2;2 c TE TE;2 c 2 (14) c K Y þ c K Y þ þ c K Y ¼ v 1 1;n c 2 2;n c TE TE;n c n The system of equation above, which is strictly valid for a speciﬁc excitation frequency x, can be solved simultaneously to obtain the extent coefﬁcient of each element present in the structure. In order to solve all unknown extent coefﬁcients, the amount of degrees-of-freedom in the structure must be equal to or greater than the number of elements present in the structure. However, this is not always the case. To tackle this problem, several sets of equations may be generated using different excitation frequencies ðx ; x ; .. .; x Þ, where the sub- 1 2 e script e represents the amount of excitation frequency used. Following this approach, a new set of equations can be formed by selecting TE equations from the generated sets of equations, followed by solving the new set of equations simultaneously to obtain the extent coefﬁcients. Stage 2: Removal of temperature effect for damage coefficient assignment After each element has been assigned with an extent coefﬁcient, the temperature effect should be removed to give the damage locations and extent of the damaged elements. The extent coefﬁcient c obtained for each element after the application of equation (14) can be decomposed into its contributing factors, which are the temperature effect and the damage effect as follows c ¼ c þ c (15) s t;s d;s where c is a temperature coefﬁcient representing the percentage reduction/increase in elemental stiffness of the sth t;s element due to the effect of the changing temperature from the reference baseline temperature used to calculate the extent coefﬁcient c ; c is a temporary damage coefﬁcient representing the percentage reduction in elemental stiffness of the sth d;s element from the stiffness obtained at reference temperature used to calculate the extent coefﬁcient c due to the effect of damage. The coefﬁcient c is temporary because the amount of reduction in stiffness used to calculate the percentage is d;s compared to the stiffness obtained at reference temperature; however, since the temperature of the structure being monitored has changed, to know the actual percentage reduction in stiffness, this amount of reduction should be compared to the stiffness at the current temperature condition (without the effect of damage), and not to the stiffness obtained at the reference temperature condition. This is because, for the same amount of percentage reduction in stiffness, different temperature condition will have different amount of reduction in stiffness. Thus, a new damage coefﬁcient, D , needs to be deﬁned and assigned to the elements to get the actual percentage s Soo Lon Wah and Chen 577 reduction in elemental stiffness due to damage effect. To be able to obtain the damage coefﬁcients of the elements to represent the percentage reduction in stiffness due to damage, the temperature coefﬁcient of each damaged element needs to be estimated ﬁrst so that the stiffness of the element at its current temperature condition (without the effect of damage) can be obtained and be used for comparison. This can be achieved by using the temperature coefﬁcients of the undamaged elements to estimate the temperature coefﬁcients of the damaged elements. Thus, the temperature coefﬁcient of the sth element may be obtained as follows c ¼ g þ g L þ g L (16) t;s s s 1 2 3 where g ; g , and g are undetermined coefﬁcients; 1 2 3 L is the distance between the sth element and a preset reference zero coordinate system. Equation (16) is a quadratic equation representing the variation of the Young’s modulus across the structure from the initial Young’s modulus used as reference and obtained at the reference temperature, due to the gradient temperature across the structure and to the nonlinear effect of the changing temperature condition on the vibra- tion properties and frequency responses of the structure. Equation (16) is in the form of a quadratic equation so as to take into account nonlinear gradient of temperature across the structure as well as nonlinear effect produced by temperature condition. Equation (16) can be obtained by ﬁrst plotting the extent coefﬁcients of the all the elements, followed by ﬁnding the equation of the line joining all the extent coefﬁcient values together. However, it should be noted that the damaged elements should have their damage effects contribution to the extent coefﬁcients removed before creating the equation of the line. To do this, the graph of the extent coefﬁcients of all the elements versus the elements’ number (ranked in order with the distance between them) should ﬁrst be plotted. Then, examining the graph, the damage locations can be obtained by the appearance of peaks in the graph. These peaks represent the contribution of damage to the extent coefﬁcients. Thus, by eliminating the peaks in the graph, the equation of the line can then be generated. The values of the coefﬁcients g ; g , and g 1 2 3 can thus be obtained by deﬁning the line. Substituting any value of L in the equation which represents the distance of the sth element to the reference zero coordinate, the temperature coefﬁcient for that element can then be estimated. It should be noted that a lower/higher order polynomial function can be adopted to best represent the extent coefﬁcient proﬁle. To give the percentage reduction of elemental stiffness of the sth element due to damage, a damage coefﬁcient D may be deﬁned. This damage coefﬁcient compares the amount of reduction in stiffness due to damage of each element to the actual stiffness of the element due to temperature effect only. The damage coefﬁcient D can be related to the temporary damage coefﬁcient c as follows d;s d;s D (17) 1 c t;s where D is the damage coefﬁcient of the sth element; ð1 c Þ represents the elemental stiffness of the sth element at its current temperature condition without the t;s effect of damage, in terms of a percentage of the stiffness obtained at the reference temperature used to calculate the extent coefﬁcient. Substituting equations (16) and (17) into equation (15) leads to 2 2 c ¼ðg þ g L þ g L Þþ D ð1 ðg þ g L þ g L ÞÞ (18) s s s s 1 2 3 s 1 2 3 s Rearranging equation (18), the damage coefﬁcient D for each element can be obtained as c ðg þ g L þ g L Þ s s 1 2 3 s D ¼ (19) 1 ðg þ g L þ g L Þ 1 2 3 From equation (19), the damage coefﬁcient, D , for the sth element can be obtained. Undamaged element will have D value of zero, while damaged element will have D value ranging from zero to one, with one representing s s 578 Journal of Low Frequency Noise, Vibration and Active Control 39(3) fully damaged. It should be noted that this way of calculating c for each element allows the proposed method to t;s perform for structures, which have nonlinear temperature gradient across the structure as well as structures whose vibration properties and frequency responses are affected nonlinearly by the changing temperature condition. Summary of the method Below are the procedures to follow to implement the proposed damaged detection method, and Figure 1 summa- rizes the procedures in the form of a ﬂowchart. 1. Apply a harmonic excitation force to the structure in its undamaged state to gather frequency responses of the structure for the baseline of the undamaged structure. If there are more elements than the number of degrees-of-freedom available, several harmonic excitation forces with different excitation frequencies should be applied to the structure one at a time. � The temperature condition along the structure can have a nonlinear proﬁle which is usually the case for real- life structures. Start Obtain the undamaged structure for the baseline Apply a harmonic excitation force(s) to the structure to gather the frequency responses of the structure Apply the same harmonic excitation force(s) to the structure to be monitored to gather its frequency responses Use the frequency responses gathered and apply Eq. (14) to calculate the extent coefficient of each element Plot the extent coefficients against their element number and examine whether peaks appear in the plot Yes No Is there any peak? Create the equation of the line joining all the extent coefficients (minus damage contribution) in the form of Eq. (16) Apply Eq. (19) to calculate the damage coefficient of each element to get the damage extent and locations End End Figure 1. Flowchart of the procedures to follow for damage detection. 1. Use damage detection methods (that did not account 2. Remove the temperature distribution trend for temperature) to determine the change in stiffness of to obtain the actual damage extent and locations each element due to temperature and damage effects Soo Lon Wah and Chen 579 2. Apply the same harmonic excitation force(s) to the structure when monitoring is required to gather new frequency responses of the structure. � It does not matter whether the temperature is constant or has a gradient distribution throughout the structure. 3. Use the frequency responses obtained from steps 1 and 2, and apply equation (14) to obtain the extent coef- ﬁcient c of each element. 4. Plot the extent coefﬁcients obtained against the elements’ number and analyze the graph to locate the dam- aged elements. � The damaged elements will appear as peaks in the plot. 5. Use extent coefﬁcients from all the undamaged and damaged elements (ignoring the contribution of the damage effect to the extent coefﬁcient) for the creation of the equation of the line (temperature effect trend) in the form of equation (16) so that the temperature coefﬁcient c of each element can be calculated. t;s � Equation (16) is in the form of a quadratic equation; however, a lower/higher order polynomial function can be used to best represent the trend of the temperature condition. 6. Apply equation (19) to obtain the damage coefﬁcient D of each element to get the damage locations and extent. � The undamaged elements will have damage coefﬁcient of zero, while the damaged elements will have damage coefﬁcient ranging from zero to one, with one representing fully damaged. It is worth noting that stage 2 of the method can be implemented with other damage detection methods that did not account for temperature effect to eliminate the inﬂuence of varying temperature condition and allow the locations and extent of damages to be obtained, while the damage detection method proposed in stage 1 can be replaced by other damage detection methods that can give the amount of change of stiffness due to external factors from a baseline stiffness. To this end, the procedures given above can be modiﬁed accordingly; the ﬁrst three steps in the procedure can be replaced by other damage detection methods, while the last three steps can be incorporated in the selected damage detection method to eliminate the temperature effect and to locate and quantify damages. Numerical examples To demonstrate the proposed approach for damage detection, localization, and quantiﬁcation, a beam structure representing a model bridge and a shear building are used as case studies. The two structures are adopted since it is well known that temperature condition along long span bridges and tall buildings usually varies, hence, inﬂuenc- ing the dynamic properties of the structures. Beam structure The beam structure used for demonstration is shown in Figure 2. It is 15-m long, with one end (end A) being pin connected and the other end (end B) being free. At 10 m from the pin-connected end, the structure rests on a roller support. The structure is assumed to be made of steel material with a Young’s modulus of 200 GPa and density of 7850 kg=m . The cross-sectional area and the second moment of area of the beam are 2 4 assumed to be 0.08 m and 0.0006 m , respectively. The two-dimensional numerical model of the structure is built in MATLAB and it is discretized into 15 beam elements of 1-m long each. A harmonic excitation force with x ¼ 2000 rads/s is applied at the free end (B) pointing downward to generate the frequency responses of the structure. Force A 12 34 5 6 7 8 9 10 11 12 13 14 15 B 10 m 5 m Figure 2. Two-dimensional beam structure. 580 Journal of Low Frequency Noise, Vibration and Active Control 39(3) E = 0.002T - 0.359T + 203.07 -40 -20 0 20 40 60 80 Temperature (°C) Figure 3. Hypothetical variation of Young’s modulus with temperature. Table 1. Descriptions of undamaged and damaged cases subjected to nonlinear temperature gradient across the structure. Case no. Temperature at A ( C) Temperature at B ( C) Element number Damage extent (%) Undamaged 1 20 14 – – Undamaged 2 10 25 – – Damaged 1 10 2 8 20 Damaged 2 35 23 3 and 12 10 and 10 To simulate the effect of varying temperature condition, the Young’s modulus of the steel material is assumed to be temperature dependent as suggested by An et al. Since it was found in the literature that temperature 25,26 condition usually has nonlinear relationship with the vibration properties of civil engineering structures, the hypothetical variation of Young’s modulus with temperature is assumed to be nonlinear as can be seen in Figure 3. To consider the effect of temperature gradient across the structure, two undamaged cases and two damaged cases are used to demonstrate the proposed approach, and the descriptions of the cases are presented in Table 1. Figure 4 illustrates the temperature variation across the beam structure. In this beam structure, damage is assumed to be represented by a reduction in elemental stiffness of the structural components. Since generally structures are faced with gradient temperature, a nonlinear temperature distribution is assumed for the baseline of the undamaged structure. The temperature condition of the baseline of the undamaged struc- ture from which subsequent measurements can be compared with is given in Figure 5. Note that any temperature could have been used as baseline. After applying the proposed damage detection method, and the application of equation (14), the extent coefﬁcient c of each element for each case can be obtained. Since generally noise affects the performance of damage detection methods, to demonstrate the robustness of the second stage of the proposed approach to eliminate the effect of temperature conditions to give the damage locations and extent, 10% of noise is added to the extent coefﬁcients. The performance of the approach of eliminating the effect of temperature is dependent upon the extent coefﬁcients, which in turn are dependent upon the techniques used to obtain these coefﬁcients in stage 1. Therefore, it is the extent coefﬁcients that are polluted with noise, to demonstrate that the proposed approach can still be applied for cases where the extent coefﬁcients are subjected to errors. The polluted extent coefﬁcients are given by ^c ¼ c 1 þ < (20) s s where ^c is the polluted extent coefﬁcient; @ is the speciﬁed noise level added to the extent coefﬁcients and taken as 10 in this case study; < is a random number between 1 and 1. Plots of the extent coefﬁcients (polluted with noise) for the four cases under consideration are presented in Figure 6. It can be seen from Figure 6(a) and (b) that for the undamaged cases, no damage is present in the structure, since no consequent peak appears in the plots. Some ﬂuctuations in the ﬁrst plot can be seen in the Young's Modulus E (GPa) Soo Lon Wah and Chen 581 (a) (b) T = 0.04L + 0.40L + 10 s s T = -0.04L + 0.20L + 20 T = 25 °C s s B T = 20 °C T = 14 °C T = 10 °C T = 0 °C T = 0 °C A 12 34 5 6 7 8 9 10 11 12 13 14 15 B A 12 34 5 6 7 8 9 10 11 12 13 14 15 B L L s s 10 m 5 m 10 m 5 m (c) (d) T = -0.03L - 0.35L + 35 s s T = 35 °C T = 23 °C T = 0.02L + 0.50L - 10 s s T = 2 °C T = 0 °C T = 0 °C T = -10 °C 34 5 6 7 9 11 13 B 5 6 9 B A 12 8 10 12 14 15 A 12 34 7 8 10 11 12 13 14 15 L L s s 10 m 5 m 10 m 5 m Figure 4. Temperature variation across the beam structure for (a) undamaged case 1, (b) undamaged case 2, (c) damaged case 1, and (d) damaged case 2. T = -0.04L - 0.4L + 5 s s T = 5 °C T = 0 °C T = -10 °C A 12 34 5 6 7 8 9 10 11 12 13 14 15 B 10 m 5 m Figure 5. Temperature variation across the undamaged beam structure used as baseline. middle portion of the beam; however, no obvious peak that indicates an abrupt and large change in stiffness exists. Figure 6(c) and (d) give the plots of the two damaged cases. It can be seen from both plots that peaks appear to alert the presence of damages in the structure. The ﬁrst damaged case (Figure 6(c)) has a peak at element 8, which indicates that element 8 is damaged, while the second damaged case (Figure 6(d)) has two peaks at elements 3 and 12, respectively, suggesting that these two elements are damaged. By constructing equation (16) to determine the temperature coefﬁcient of each element, and by applying equation (19), a damage extent of 20% for element 8 can be calculated for the ﬁrst damaged case, while for the second damaged case, elements 3 and 12 are found to have a reduction in elemental stiffness of 10% and 11%, respectively. From the four plots shown in Figure 6, it can also be seen that the extent coefﬁcient values of the undamaged elements generally vary smoothly from element to element throughout the structure. This can be attributed to the fact that, for the undamaged element, the amount of change in stiffness is due fully to the deviation in temperature from the baseline temperature condition, and since temperature condition along a structure usually changes gradually from location to location, the change of stiffness of each element, which is represented by the extent coefﬁcient, will increase/decrease steadily along the structure. Only a signiﬁcant variation in the amount of change in stiffness from an element to the nearby elements will produce a sharp change in the extent coefﬁcient proﬁle. 582 Journal of Low Frequency Noise, Vibration and Active Control 39(3) (a) (b) 0.04 0.06 0.036 0.04 0.032 0.028 0.02 0.024 c = 0.00004L + 0.00055L + 0.02413 c = 0.00016L + 0.00107L + 0.00899 t,s s s t,s s s 0.02 0 12 34 56 789 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Element number Element number (c) (d) 0.24 0.18 c = 0.00010L + 0.00206L - 0.02783 c = 0.000009L + 0.00071L + 0.04139 t,s s s t,s s s 0.15 0.16 0.12 0.08 0.09 0.06 -0.08 0.03 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 12 34 56 789 10 11 12 13 14 15 Element number Element number Figure 6. Extent coefficient plot of beam structure for (a) undamaged case 1, (b) undamaged case 2, (c) damaged case 1, and (d) damaged case 2. As damage is a localized effect which produces a signiﬁcant decrease in stiffness of the damaged element, it will produce a sudden change in the extent coefﬁcient plot. Thus, any sharp change/peak in the plot can be attributed to the effect of damage. Noise affecting the measurements will also create a change in the extent coefﬁcient plot, however, these ﬂuctuations will be much less when compared to the effect of damage and no distinct peak will occur. Peaks at the damage locations will always exist because the extent coefﬁcient combines the amount of change of stiffness due to temperature and damage effects together. It can therefore be concluded that for a damaged element, the temperature contribution to the extent coefﬁcient will follow the same trend as other undamaged elements’ extent coefﬁcients, while the contribution of damage will be an addition to that trend. By observing the extent coefﬁcient plot, the temperature and damage effects can be distinguished, and the damage extent can be quantiﬁed through the peaks’ contribution. Shear building A shear building (Figure 7(a)) is also used as another example to demonstrate the applicability of the proposed approach for damage localization and quantiﬁcation, since buildings’ vibration properties are commonly affected 27,28 by the changing temperature condition they are faced with. The numerical model of the shear building is the same as the one used as case study by Roy and Ray-Chaudhuri. The shear building is 12-story high with uniform story mass and stiffness distribution throughout the height of the structure. The story mass and stiffness prop- 5 8 1 erties adopted in this paper are 1 10 kg and 2 10 Nm deﬁned at reference temperature of 20 C, respec- tively. A harmonic excitation force with angular frequency x ¼ 1000 rads/s is applied at the ﬁrst ﬂoor level to generate the frequency responses of the structure. To simulate a varying temperature condition, it is assumed that the stiffness of the structure varies nonlinearly with the temperature condition. The relationship between the temperature condition and the story level stiffness is given by equation (21). The story stiffness is assumed to vary nonlinearly with the temperature condition because Young’s modulus of structures generally varies with a changing temperature condition, which as a result, affects the stiffness of the structures. The temperature distribution across the undamaged structure used as baseline is given in Figure 7(b). Damage is simulated as a reduction in story-level stiffness, and the scenarios applied to the shear building are presented in Table 2. The temperature proﬁle along the structure for the different cases are shown in Figure 8, and it is assumed that the temperature condition at the top of the building is greater than that at the bottom since buildings are subjected to greater radiation at their roof level than at their base level. Similar to the beam case study, 10% of noise is added in this shear building case study. Extent coefficient Extent coefficient Extent coefficient Extent coefficient Soo Lon Wah and Chen 583 (a) (b) T = 13.12 °C top m12 m12 k12 k12 m11 m11 k11 k11 m10 m10 T = 0 °C k10 k10 m9 m9 k9 k9 m8 m8 k8 k8 m7 m7 k7 k7 m6 m6 k6 k6 m5 m5 k5 k5 m4 m4 k4 k4 m3 m3 k3 k3 m2 m2 k2 k2 m1 m1 Force k1 k1 T = 10 °C base Figure 7. (a) Two-dimensional shear building structure and (b) temperature variation across the undamaged shear building used as baseline. Table 2. Descriptions of undamaged and damaged cases subjected to nonlinear temperature gradient throughout the shear building. Case no. Temperature at base ( C) Temperature at top ( C) Story level Damage extent (%) Undamaged 1 20 29 – – Damaged 1 15 27 3, 6, and 9 5, 10, and 15 Damaged 2 30 33 6 10 Damaged 3 5 8 6 10 (a) (b) (c) (d) T = 29 °C T = 27 °C T = 33 °C T = 8 °C top top top top m12 m12 m12 m12 k12 k12 k12 k12 m11 m11 m11 m11 k11 m10 k11 k11 k11 m10 m10 m10 T = 0 °C T = 0 °C T = 0 °C T = 0 °C k10 k10 k10 k10 m9 m9 m9 m9 k9 k9 k9 k9 m8 m8 m8 m8 k8 m7 k8 m7 k8 m7 k8 m7 k7 k7 k7 k7 m6 m6 m6 m6 k6 k6 k6 k6 m5 m5 m5 m5 k5 k5 k5 k5 m4 m4 m4 m4 k4 k4 k4 k4 m3 m3 m3 m3 k3 k3 k3 k3 m2 m2 m2 m2 k2 k2 k2 k2 m1 m1 m1 m1 L L L L s s s s k1 k1 k1 k1 T = 20 °C T = 15 °C T = 30 °C T = 5 °C base base base base Figure 8. Temperature variation throughout the shear building structure for (a) undamaged case 1, (b) damaged case 1, (c) damaged case 2, and (d) damaged case 3. 4 2 5 8 k ¼ 1:2 10 T 9:2 10 T þ 2:136 10 (21) s;t where k is the stiffness of the sth story considering temperature effect in Nm ; s;t T is the temperature condition of the sth story in C. The plots of the extent coefﬁcients versus story’s level for the four cases are presented in Figure 9. It can be seen from Figure 9 that the cases are well classiﬁed; the ﬁrst plot (Figure 9(a)) indicates no damage is present, while the following three plots ((Figure 9(b)–(d)) show that the structure is damaged due to the appearance of peaks. It can T = 0.75L + 20 T = 0.05L + 0.40L + 15 s s T = 0.02L + 0.02L + 10 s s T = 0.02L + 0.01L + 30 s s T = 0.02L + 0.01L + 5 s s 584 Journal of Low Frequency Noise, Vibration and Active Control 39(3) (a) (b) (c) (d) 12 12 12 12 11 11 11 11 10 10 10 10 9 9 9 9 8 8 8 8 7 7 7 7 6 6 6 5 5 5 5 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 0.024 0.03 0.036 -0.05 0.05 0.15 0.25 0 0.1 0.2 -0.1 0 0.1 0.2 0.3 Extent coefficient Extent coefficient Extent coefficient Extent coefficient Figure 9. Extent coefficient plot of shear building for (a) undamaged case 1, (b) damaged case 1, (c) damaged case 2, and (d) damaged case 3. again be seen that the temperature effect produces a relatively smooth change of extent coefﬁcient from undam- aged story to undamaged story, while damage effect produces a sharp change in extent coefﬁcient of the damaged story from the nearby stories. This sharp change in extent coefﬁcient represents the extra contribution in decreas- ing the stiffness of the damaged story. Applying equation (19), it is found that the ﬁrst damaged case has story levels 3, 6, and 9 to have a reduction in stiffness by 5%, 10%, and 16%, respectively, the second damaged case has 9% reduction in the sixth story, while the last case has a reduction in stiffness of 10% in the sixth story. For this damage detection method, the temperature trend can be found through the quadratic function given by equation (16). For example, from Figure 9(a), it can be seen that the ﬁrst undamaged case has a nonlinear change in extent coefﬁcient. This case is analyzed in this paper to demonstrate that, depending on the proﬁle of extent coefﬁcient, the function given through equation (16) can be adjusted to suit the extent coefﬁcient proﬁle. Being able to adjust the function order allows the proposed approach to closely represent the temperature coef- ﬁcient of each element, thus decreasing the chance of obtaining the wrong damage scenario. For example, for the ﬁrst undamaged case (Figure 9(a)), a cubic function is adopted; therefore, a cubic part is added to the quadratic function (equation (16)). One potential difﬁculty of the proposed approach is how to quantify the extent of damages through the peaks, since different temperature condition will produce different peak contributions. For instance, the last two dam- aged cases have the same damaged element with the same damaged extent, but with different temperature con- dition. Quantifying the peak amount by subtracting the temperature coefﬁcient of that element from its extent coefﬁcient, it is found that the second damaged case (Figure 9(c)) has a value of 0.083, while the third damaged case (Figure 9(d)) has a value of 0.104. The difference in these two values is attributed to the different temperature conditions the two cases are at, since different temperature condition will produce different amount of change of stiffness for the same percentage of reduction in stiffness. This issue can be overcome by deriving relationships between the change in stiffness due to damage, and the stiffness of the elements at the current temperature condition. For example, for the proposed damage detection method, the damaged elements can be quantiﬁed through equation (19), which relates the peak contribution to the stiffness of the members at the current tem- perature condition. Discussion of the proposed approach The principle behind most conventional damage detection methods that did not take into account temperature effect is to compare new measurements to a single benchmark so as to determine which member has a change in stiffness from the original elemental stiffness, and to quantify this change for damage localization and quantiﬁ- cation. Since damage is the only effect considered in the methods that will cause change in stiffness, only the damaged elements will have a reduction in stiffness, however, most civil engineering structures are subjected to changing temperature condition, and both damage and temperature effects will change the stiffness of the elements. Therefore, obtaining the total change in stiffness due to both damage and temperature effects ﬁrst, followed by removing the temperature effect, will facilitate damage localization and quantiﬁcation. Story level 3 2 c = 0.00001L - 0.00019L + t,s s s 0.00166L + 0.02694 Story level c = 0.00005L + 0.00103L + 0.01435 t,s s s Story level c = -0.00002L - 0.00037L + 0.04349 t,s s s Story level c = -0.00001L + 0.00039L - 0.01900 t,s s s Soo Lon Wah and Chen 585 In the aspect of creating a baseline for damage detection, most approaches have difﬁculty in creating a baseline when encountering temperature gradient along the structures, which generally has a great inﬂuence on the vibra- tion properties. This condition is commonly found when analyzing long span bridges and tall buildings because ambient solar radiations are not evenly distributed along the structures. Limited methods were proposed in the literature to tackle gradient temperature condition along the structures for damage localization and quantiﬁca- tion. The reason behind may be attributed to the difﬁculty in creating a baseline of the undamaged structure which is commonly constructed using a wide range of temperature conditions, to include all possible conditions the structure may encounter, since there is an inﬁnite amount of temperature proﬁles that the structure can be faced with. Moreover, gradient temperature may overestimate/underestimate temperature effect on the vibration prop- erties, if the temperature proﬁle is not accounted for properly, hence leading to false alerts. To tackle this gradient temperature and temperature condition issue in general, drawing the extent coefﬁcient proﬁle of the structure due to temperature effect, seems to be the best solution since no predeﬁned solution for that proﬁle is required to be included in the baseline, and this extent coefﬁcient proﬁle can be obtained using the global trend of change in stiffness. To this end, this paper uses a quadratic function to represent the global change in elemental stiffness which can then be related to the temperature distribution along the structure. It is important to allow the proﬁle of extent coefﬁcient to be represented by a quadratic function or even higher order polynomial function because, if the proﬁle can only be represented by a linear function, the temperature coefﬁcient would be underestimated/overestimated, hence underestimating/overestimating the contribution of temperature in changing the stiffness of the elements, leading to false damage extent and locations. It should be noted that the quadratic function deﬁning the temperature distribution proﬁle along the structure (e.g., Figure 4) is different from the one describing the proﬁle of extent coefﬁcient (e.g., Figure 6). This difference is attributed to the nonlinear relationship between the Young’s modulus of the material and the temperature as well as the temper- ature proﬁle of the baseline structure, which were accounted for by the extent coefﬁcient plot, and not by the temperature distribution proﬁle along the structure. Using only one reference temperature as baseline is proposed in this paper, as it has the advantage that the damage detection method can give the damage locations and extent since the structure can be compared locally to the baseline. Moreover, the method can be implemented immediately without requiring months of measurements to create the baseline of the undamaged structure. It also reduces the risk of false alerts which may occur for structures that have temperature–Young’s modulus relationship to be nonlinear, since, if not all conditions are included in the baseline, false alerts may occur. The proposed approach eliminates the risk that small damages are not detected, which may occur if a range of temperature condition is used, because the small damages may not alter the dynamic properties signiﬁcantly enough to trigger the damage alert. Therefore, by using the proposed approach, the difﬁculty of using a range of temperature condition for the baseline of the undamaged structure for damage localization and quantiﬁcation is enhanced, as it can easily tackle any temperature proﬁle along the structure. This approach is appealing because it can be incorporated in most existing damage detection methods easily by ﬁrst using the damage detection methods to quantify the changes in stiffness of all the elements due to temperature and damage effects, followed by removing the temperature contribution which is a global effect by ﬁnding the temperature trend, to obtain the damage locations and extent. Conclusion A new insight on how to approach damage detection of civil engineering structures subjected to varying temper- ature and to gradient temperature conditions by using a single temperature proﬁle as the baseline is introduced in this paper. The approach uses the concept of ﬁrst quantifying the change in stiffness of all the elements due to the effects of temperature and damage, and then the temperature effect which has a global trend on the structure is removed, to give the damage locations and extent. The approach is implemented with a proposed damage detec- tion method that did not account for temperature effect, to be able to tackle varying temperature and gradient temperature conditions. The damage detection method with the proposed approach incorporated is tested for different undamaged and damaged cases using a beam structure and a shear building subjected to gradient temperature conditions. The results demonstrate that, by quantifying the change in stiffness of all the elements due to external factors, the trend of temperature effect along the structure can be obtained, as well as the damage locations. This temperature effect is found to be different from the damage effect, which is a localized effect, thus, the actual damage locations and extent can be obtained by eliminating the temperature effect. The proposed approach can be implemented with other damage detection methods that did not account for temperature effect to obtain the damage locations and extent under changing temperature condition. 586 Journal of Low Frequency Noise, Vibration and Active Control 39(3) Declaration of conflicting interests The author(s) declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following ﬁnancial support for the research, authorship, and/or publication of this article: This work was supported by the National Science Foundation of China (51708306,51605233), Zhejiang Qianjiang Talent Scheme (QJD1402009), and Faculty Inspiration Grant. References 1. Carden EP and Fanning P. Vibration based condition monitoring: a review. Struct Health Monit 2004; 3: 355–377. 2. Fan W and Qiao P. Vibration-based damage identiﬁcation methods: a review and comparative study. Struct Health Monit 2011; 10: 83–111. 3. Dominik I, Iwaniec M and Lech Ł. Low frequency damage analysis of electric pylon model by fuzzy logic application. J Low Freq Noise Vib Active Control 2013; 32: 239–251. 4. Abouel-Seoud SA. Fault detection enhancement in wind turbine planetary gearbox via stationary vibration waveform data. J Low Freq Noise Vib Active Control 2017. DOI: 10.1177/1461348417725950. 5. Rytter A. 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"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE

Published: Aug 27, 2018

Keywords: Damage detection; damage localization; damage quantification; temperature condition; temperature gradient; environmental conditions

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A new approach toward damage localization and quantification of structures under changing temperature condition:

A new approach toward damage localization and quantification of structures under changing temperature condition:

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A new approach toward damage localization and quantification of structures under changing temperature condition:

A new approach toward damage localization and quantification of structures under changing temperature condition:

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