Model-Free Method for Damage Localization of Grid Structure
Model-Free Method for Damage Localization of Grid Structure
Yang, Qiuwei;Wang, Chaojun;Li, Na;Luo, Shuai;Wang, Wei
2019-08-09 00:00:00
applied sciences Article Model-Free Method for Damage Localization of Grid Structure Qiuwei Yang, Chaojun Wang, Na Li * , Shuai Luo and Wei Wang * School of Civil Engineering, Shaoxing University, Shaoxing 312000, China * Correspondence: lina@usx.edu.cn (N.L.); wellswang@usx.edu.cn (W.W.); Tel.: +86-0575-8834-1503 (N.L.) Received: 18 July 2019; Accepted: 5 August 2019; Published: 9 August 2019 Abstract: A model-free damage identification method for grid structures based on displacement dierence is proposed. The inherent relationship between the displacement dierence and the position of structural damage was deduced in detail by the Sherman–Morrison–Woodbury formula, and the basic principle of damage localization of the grid structure was obtained. That is, except for the tensile and compressive deformations of the damaged elements, the deformations of other elements were small, and only rigid body displacements occurred before and after the structural damage. According to this rule, a method for identifying the position of the damage was proposed for the space grid structure by using the rate of change of length for each element. Taking a space grid structure with a large number of elements as an example, the elastic modulus reduction method was used to simulate the damage to the elements, and the static and dynamic test parameters were simulated respectively to obtain the dierence in displacement before and after the structural damage. The rate of change of length of each element was calculated based on the obtained displacement dierence, and data noise was added to the simulation. The results indicated that the element with the larger length change rate in the structure was the most likely to be damaged, and the damaged element can be accurately evaluated even in the presence of noise in data. Keywords: grid structure; damage identification; Sherman–Morrison–Woodbury formula; displacement dierence; change rate of element length 1. Introduction In construction, buildings with large spans are becoming increasingly common, and lightweight, economical, beautiful grid structures have become the first choice for roofs. While these grid structures remain in service, damage will accumulate with the long-term eects of loads, material aging, fatigue, accidental loads, and environmental corrosion. If the damaged elements in the structure cannot be identified in time, serious accidents may result. In recent decades, many scholars have comprehensively studied [1–3] the damage identification of large structures. For example, Yin et al. [4] studied the degree of element damage in the quadrangular pyramid structure by simulating the displacement-time curve of the unit node, indicating that the displacement relationship can reflect the damage to the grid structure. Wang et al. [5] can accurately determine the damage positions of grid element units using hammer strokes to obtain the structural time domain signals. Li [6] combined the static virtual deformation method and the sequence quadratic programming algorithm for damage identification to quickly and accurately identify damage positions. Ouyang et al. [7] identified damage by the cubic interpolation of bending deflection of the beam in the case of smaller stress redistributions, after assuming structural damages. Huang et al. [8,9] used a random damage index to quantify the damage of a random beam structure through the static test, while considering the model and measurement errors. Hosseinzadeh et al. [10] recognized damage identification as the process of optimizing the stiness to match the external force with the displacement and proposed a method for calculating the Appl. Sci. 2019, 9, 3252; doi:10.3390/app9163252 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 3252 2 of 12 static displacement using flexibility, as well the cuckoo and optimization algorithm to identify damages. The displacement influence line [11] and strain influence line [12] are also used to identify the damage, and Alexandre Kawano conjectured that the time to measure the structural displacement also aects the result [13]. Most of the above studies are based on the model correction technique, which is a common method for damage identification [14–16], using finite element models. Nevertheless, it is dicult to establish accurate finite element models for large structures. In order to avoid finite element modeling, Yang et al. [17] used the Sherman–Morrison–Woodbury (SMW) formula to discuss the theoretical basis of structural damage localization in detail. Additionally, they took a beam structure and rigid frame structure as examples to explain the application of this theoretical basis to achieve damage localization. However, the grid structure has not been specifically discussed. In view of this, the principle in literature [17] was applied to the grid structure to deduce the physical connection between the position of damage and variation in displacement of the grid structure in detail. It was concluded that the basic principle of damage localization of the grid structure before and after the structural damage were small and only the rigid body displacement occurred, except for tensile and compressive deformations of damaged elements. Based on this, the rate of length change of each element was used as the index to identify the damage position for the space grid structure. The proposed method can realize damage recognition without a model, thus avoiding the complicated finite element modeling process, and can be implemented by using static or dynamic test data. Taking the space grid structure with 71 elements as an example, the static and dynamic test parameters were simulated respectively. The length change rate of the element was calculated according to the obtained displacement dierence. At the same time, data noise was added to the simulation. The simulation results indicated that the element with the larger rate of length change is the most likely to appear damaged, and the damaged element can be accurately evaluated even if the data are noisy. 2. Basic Principle of Structural Damage Localization Based on Displacement Dierence There are many displacement-based damage identification methods [18–20], but none of them can explain the physical connection between the position of damage and the displacement. Generally, considering a structure with n degrees of freedom, the displacement under the action of static loads should satisfy the following equation: Ku = l (1) where K(n n) is the overall stiness matrix of the undamaged structure and u is the displacement vector under the static load vector l. Equation (1) can be expressed as: u = Fl (2) where F is the flexibility matrix of the undamaged structure, that is, F = K . Structural damage can result in a decrease in the stiness and increase in flexibility. The model matrix of the undamaged structure is related to the model matrix of the damaged structure as follows: F = F + DF (3) K = K DK (4) The displacement u produced after structural damage can be expressed as: u = F l (5) d d Therefore, the displacement change Du can be expressed as: Du = u u = DFl = [(K DK) K ] (6) d Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 13 Appl. Sci. 2019, 9, 3252 3 of 12 u = F l (5) d d In order to further develop Equation (6), the following SMW formula is used: Therefore, the displacement change ∆u can be expressed as: 1 1 T 1 1 T 1 T 1 (K + XY ) = K K X(I + Y K X) Y K (7) −1 −1 Δu = u −u = ΔFl = [(K − ΔK) − K ] (6) DK = DKI , using Equation (7) to simplify Equation (6) can obtain Equation (8): In order to further develop Equation (6), the following SMW formula is used: 1 1 1 Du = K DK(I K DK) K l (8) T −1 −1 −1 T −1 −1 T −1 (K + XY ) = K − K X (I +Y K X ) Y K (7) In order to express the relationship between the change in displacement and damages, Equation ∆K = ∆KI , using Equation (7) to simplify Equation (6) can obtain Equation (8): (8) can be rewritten as: Du = d (9) −1 −1 −1 −1 Δu = K ΔK(I − K ΔK) K l (8) d = K DK (10) In order to express the relationship between the change in displacement and damages, Equation 1 1 = (I K DK) K l (11) (8) can be rewritten as: The physical meaning of Equation (9) is very important. This equation indicates that displacement change Du of the structure before and after Δ damage u = δd was a linear combination of characteristic (9) displacements. According to Equation (10), the characteristic displacement di was obtained by applying −1 d = K ΔK (10) the non-zero column vector in DK as a static load to the structure, and these non-zero column vectors in DK can be defined as characteristic forces. The characteristic force and characteristic displacement −1 −1 −1 δ = (I − K ΔK) K l (11) are dierent for dierent structure types. The characteristic forces and displacements of the beam structure and rigid frame structure have been discussed in detail in literature [17]. In the following The physical meaning of Equation (9) is very important. This equation indicates that chapters, we will discuss the characteristic force and displacement of grid structures in detail and displacement change ∆u of the structure before and after damage was a linear combination of obtain the basic principle of the damage localization of the grid structure. characteristic displacements. According to Equation (10), the characteristic displacement di was obtained by applying the non-zero column vector in ∆K as a static load to the structure, and these 3. Damage Identification of Grid Structure non-zero column vectors in ∆K can be defined as characteristic forces. The characteristic force and 3.1. charCharacteristic acteristic disp For lace cem and enDisplacement t are differenof t for dif Grid Structur ferent structure ty e pes. The characteristic forces and displacements of the beam structure and rigid frame structure have been discussed in detail in Generally, the structure shown in Figure 1 was taken as an example to illustrate the characteristic literature [17]. In the following chapters, we will discuss the characteristic force and displacement of force and displacement of the grid structure, and the essential relationship between the displacement grid structures in detail and obtain the basic principle of the damage localization of the grid structure. dierence and the damage position was analyzed. The node displacement vector in the structure and the element stiness matrix in the local coordinate system are expressed as: 3. Damage Identification of Grid Structure u = [x y x y ] (12) 1 1 2 2 3.1. Characteristic Force and Displacement of Grid Structure 2 3 1 0 1 0 6 7 Generally, the structure shown in Figure 1 was taken as an example to illustrate the characteristic 6 7 6 7 6 7 6 7 EA 0 0 0 0 6 7 force and displacement of the grid struc e ture, and the essential relationship between the displacement 6 7 K = 6 7 (13) 6 7 6 7 L 1 0 1 0 6 7 difference and the damage position was analyzed. The node displacement vector in the structure and 6 7 4 5 0 0 0 0 the element stiffness matrix in the local coordinate system are expressed as: 5 6 7 12 3 4 X Figure Figure 1. 1. Grid Grid structure structure with with 11 11 elements. elements. e T (12) u = [x y x y ] 1 1 2 2 Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 13 1 0 −1 0 0 0 0 0 EA K = (13) −1 0 1 0 0 0 0 0 Taking the No. 3 element in Figure 1 as an example, the element stiffness matrix in the global coordinate system of the element and the corresponding ∆K are expressed as: Appl. Sci. 2019, 9, 3252 4 of 12 1 3 1 3 − − 4 4 4 4 Taking the No. 3 element in Figure 1 as an example, the element stiness matrix in the global 3 3 3 3 coordinate system of the element and the corresponding DK are expressed as: − − EA 4 4 4 4 K = p p (14) 2 3 1 3 1 3 L 6 1 3 1 73 6 7 6 4 4 4 4 7 p p 6 − − 7 6 7 6 3 3 3 3 7 6 7 EA 4 4 4 4 6 7 6 4 4 4 4 7 p p K = 6 7 (14) i 6 7 1 3 1 3 6 7 L 6 3 3 3 37 6 7 4 4 4 4 6 p p 7 6 − − 7 4 5 3 3 3 3 4 4 4 4 4 4 4 4 ΔK = α k (15) DK = k (15) i i i i wher Where e αand i and K Kar t are e the the damage coefficient and damage coecient and element elem sti ent ness stiffnes matrix s mat in r the ix in t global he g coor loba dinate l coor system dinate i t of the ith element, respectively, and E, A, and L are the elastic modulus, section area, and length of system of the ith element, respectively, and E, A, and L are the elastic modulus, section area, and element, length ofr espectively element, respect . The ifirst velycolumn . The firof st column o the non-zer f the non o vector-in zerDoK vector is applied in ∆as K is the applie characteristic d as the force to the No. 3 element, as shown in Figure 2: characteristic force to the No. 3 element, as shown in Figure 2: F = γ 2 4 γ 5 EA γα = F = γ Figure 2. Force of element 3 (Fr is resultant force). Figure 2. Force of element 3 (Fr is resultant force). It can be seen from Figure 2 that the element 3 was in a state of force equilibrium. The same It can be seen from Figure 2 that the element 3 was in a state of force equilibrium. The same result can be obtained by using the remaining non-zero column vectors as characteristic forces. The result can be obtained by using the remaining non-zero column vectors as characteristic forces. The above results are universal, namely: (1) The characteristic force was the balance force system. (2) The above results are universal, namely: (1) The characteristic force was the balance force system. (2) The characteristic force of an element aects only that element, without aecting other elements. Therefore, characteristic force of an element affects only that element, without affecting other elements. when the characteristic force is applied to the structure, the corresponding characteristic displacement Therefore, when the characteristic force is applied to the structure, the corresponding characteristic has the following characteristics: only the element corresponding to the characteristic force exhibited displacement has the following characteristics: only the element corresponding to the characteristic tensile and compressive deformations, while the other elements were free from the eect of forces force exhibited tensile and compressive deformations, while the other elements were free from the and only performed rigid body motions. Furthermore, the change in structural displacement change effect of forces and only performed rigid body motions. Furthermore, the change in structural due to damages was a linear combination of the characteristic displacements associated with the displacement change due to damages was a linear combination of the characteristic displacements damaged elements. As for the displacement dierence vector under any load, except for tensile associated with the damaged elements. As for the displacement difference vector under any load, or compressive deformations of damaged elements, the other elements only performed rigid body except for tensile or compressive deformations of damaged elements, the other elements only movements, without significant deformations. Therefore, the rate of length change of each element can performed rigid body movements, without significant deformations. Therefore, the rate of length be calculated according to the displacement dierence of the structure before and after the damage, and the element with the larger rate of length change was the position where the damage was most likely to occur. 3.2. Damage Localization of Grid Structure Based on Displacement Dierence Measurement of static displacements can be performed by many dierent methods. Most often used are traditional techniques, using precise leveling, dial gauge, or inductive gauge. Recently, optical systems for displacement measurement have been successfully applied in actual engineering, and good eects were achieved, such as photogrammetry, laser techniques, and visual techniques. More details about displacement measurement can be found in References [21–25]. Appl. Sci. 2019, 9, 3252 5 of 12 According to the measured displacement dierence before and after the damage of each node, the rate of length change of each element can be calculated. The calculation formula is derived as follows: The displacement dierence can be decomposed as the displacement dierence Du , Du , and Du x y z in the x, y, and z direction, respectively. The calculation formula of the length of each element in the damaged structure is expressed as: 2 2 2 L = (x x ) + (y y ) + (z z ) (16) 1 2 1 2 1 2 x = x + Du , y = y + Du , z = x + Du (17) 1 0 x 1 0 y 1 0 z where x , y , and z refer to original coordinates of the previous node of the element in the x, y, and z 0 0 0 directions, respectively, x , y , and z refer to original coordinates plus the coordinates of Du , Du , 1 1 1 x y and Du , respectively, and x , y , and z refer to original coordinates of the post node of the element z 2 2 2 plus coordinates of the displacement dierence Du , Du , and Du of the corresponding node. Using x y z the results obtained in Equation (16), the change in length of the element before and after damage is expressed as: DL = L L (18) where L refers to the original length of the structural element. According to Equation (18), the rate of length change of the element is expressed as: DL " = (19) The determination of the damage position can be made based on the computed change rate of element length. In summary, the damage localization of grid structure based on the displacement dierence has the following steps: (1) Obtain the displacement dierence Du through experiment. It is preferred to use a laser range finder to obtain the displacement dierence in the static test. If a static test cannot be performed, we can use the dynamic flexibility method to indirectly obtain the displacement dierence (multiply the virtual force vector by the dynamic flexibility dierence matrix before and after damage). Similarly, it is not necessary to establish a finite element model when the dynamic flexibility method is used. The approximate flexibility dierence matrix can be obtained by the following formula: DF = F F (20) F = ' ' (21) j=1 where F and F are the flexibility matrices before and after damage, respectively, m is the measured number of modes, and and ' are the th order eigenvalues and eigenvectors (mode shapes), j j j respectively. The flexibility matrix dierence obtained by the dynamic flexibility method was considered in the following equation to obtain the change in displacement: Du = DF l (22) where l is the virtual force vector. Theoretically, the virtual vector can be selected arbitrarily according to need. For example, l = [0 — 1 — 0] . The specific steps for the selection of the virtual force vector can be obtained from literature [1]; (2) The length of element after damage was calculated by Equation (16), and the change in length of the element before and after damage was calculated by Equation (18). Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 13 F = ϕ ϕ (21) j j j =1 where F and Fd are the flexibility matrices before and after damage, respectively, m is the measured number of modes, and λj and φj are the th order eigenvalues and eigenvectors (mode shapes), respectively. The flexibility matrix difference obtained by the dynamic flexibility method was considered in the following equation to obtain the change in displacement: Δu = ΔF ⋅l (22) where lv is the virtual force vector. Theoretically, the virtual vector can be selected arbitrarily according to need. For example,lv = [0 --- 1 --- 0] . The specific steps for the selection of the virtual force vector can be obtained from literature [1]; (2) The length of element after damage was calculated by Equation (16), and the change in length Appl. Sci. 2019, 9, 3252 6 of 12 of the element before and after damage was calculated by Equation (18). (3) The rate of change of length ε of the element (strain) is calculated by Equation (19), and the (3) The rate of change of length " of the element (strain) is calculated by Equation (19), and the element with the larger length change rate was identified as the position where the damage was element with the larger length change rate was identified as the position where the damage was most likely to occur. most likely to occur. 4. Cases 4. Cases The space grid structure in Figure 3 had a total of 71 elements. The main parameters were as The space grid structure in Figure 3 had a total of 71 elements. The main parameters were as 3 3 -4 2 follows: E = 200 GPa, ρ = 7.8 × 10 kg/m ; A = 3.14 × 10 m ; L = 3 m. The three translational degrees of 3 3 4 2 follows: E = 200 GPa, = 7.8 10 kg/m ; A = 3.14 10 m ; L = 3 m. The three translational degrees freedom were constrained at nodes 1, 7, 8, and 14, and a concentrated force of 20 kN in the negative of freedom were constrained at nodes 1, 7, 8, and 14, and a concentrated force of 20 kN in the negative Y-axis direction was applied at nodes 4 and 11, as shown in Figure 3a. In the example, the element Y-axis direction was applied at nodes 4 and 11, as shown in Figure 3a. In the example, the element damage was simulated by the reduction in elastic modulus. damage was simulated by the reduction in elastic modulus. L L L L L L 15 16 18 19 1 2 3 4 56 7 20KN (a) (b) Figure 3. (a). X-Y axis elevation structure and force; (b). Standard view of the space grid structure Figure 3. (a). X-Y axis elevation structure and force; (b). Standard view of the space grid structure of of element. element. Consider two cases of single damage and multiple damages: (1) simulate 20% stiness damage for element 5 between nodes 5 and 6; and (2) simulate 20% stiness damage for element 35 between nodes 1 and 15, and 20% stiness damage for element 40 between nodes 7 and 19, respectively. First, we simulated the static test. According to the obtained static displacement data, under a concentrated force, the rates of length change of 71 elements before and after the damages occurred were calculated by Equation (15). The results obtained from the calculation without considering the noise are shown in Figures 4 and 5. L Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 13 Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 13 Consider two cases of single damage and multiple damages: (1) simulate 20% stiffness damage Consider two cases of single damage and multiple damages: (1) simulate 20% stiffness damage for element 5 between nodes 5 and 6; and (2) simulate 20% stiffness damage for element 35 between for element 5 between nodes 5 and 6; and (2) simulate 20% stiffness damage for element 35 between nodes 1 and 15, and 20% stiffness damage for element 40 between nodes 7 and 19, respectively. First, nodes 1 and 15, and 20% stiffness damage for element 40 between nodes 7 and 19, respectively. First, we simulated the static test. According to the obtained static displacement data, under a concentrated we simulated the static test. According to the obtained static displacement data, under a concentrated force, the rates of length change of 71 elements before and after the damages occurred were calculated force, the rates of length change of 71 elements before and after the damages occurred were calculated by Equation (15). The results obtained from the calculation without considering the noise are shown by Equation (15). The results obtained from the calculation without considering the noise are shown in Figures 4 and 5. Appl. Sci. 2019, 9, 3252 7 of 12 in Figures 4 and 5. Figure 4. Rate of change of element length obtained from the static data (element 5 had damage with Figure 4. Rate of change of element length obtained from the static data (element 5 had damage with Figure 4. Rate of change of element length obtained from the static data (element 5 had damage with no noise). no noise). no noise). Figure 5. Rate of change of element length obtained from the static data (elements 35 and 40 were Figure 5. Rate of change of element length obtained from the static data (elements 35 and 40 were damaged with no noise). Figure 5. Rate of change of element length obtained from the static data (elements 35 and 40 were damaged with no noise). damaged with no noise). It can be seen from Figures 4 and 5 that when the data used were noise-free, the rate of change of It can be seen from Figures 4 and 5 that when the data used were noise-free, the rate of change length of the damaged element was abnormally increased, and the length change rates of the remaining It can be seen from Figures 4 and 5 that when the data used were noise-free, the rate of change of length of the damaged element was abnormally increased, and the length change rates of the elements were all small, indicating that the length change rate of the element was feasible as the basis of length of the damaged element was abnormally increased, and the length change rates of the remaining elements were all small, indicating that the length change rate of the element was feasible for damage localization. remaining elements were all small, indicating that the length change rate of the element was feasible as the basis for damage localization. In order to test the noise resistance of the proposed method, data noise of 5% was added to the as the basis for damage localization. static displacement data to simulate the real measurement condition, and the results of the calculation In order to test the noise resistance of the proposed method, data noise of 5% was added to the In order to test the noise resistance of the proposed method, data noise of 5% was added to the sta using tic di the spnoise-containing lacement data to si data mula arte the e shown realin mea Figur sure esm 6ent and condit 7. The ioformula n, and thfor e res adding ults of t data he ca noise lculat is ion as static displacement data to simulate the real measurement condition, and the results of the calculation follows: using the noise-containing data are shown in Figures 6 and 7. The formula for adding data noise is using the noise-containing data are shown in Figures 6 and 7. The formula for adding data noise is as follows: Du = Du (1 + nl uni f rnd[ 1, 1]) (23) (j) (j) as follows: where Du* denotes the jth component in the displacement change Du, Du* denotes the corresponding (j) Δu = Δu × (1 + nl ×unifrnd[−1,1]) (23) component contaminated with noise, ( j ) unifrnd ( j ) [ 1, 1] denotes a random number in the range of 1 and Δu = Δu × (1 + nl ×unifrnd[−1,1]) (23) ( j ) ( j ) 1, and nl denotes the noise level. As stated before, nl = 5% is used in this example. where ∆u* denotes the jth component in the displacement change ∆u, ∆u*(j) denotes the corresponding where ∆u* denotes the jth component in the displacement change ∆u, ∆u*(j) denotes the corresponding component contaminated with noise, unifrnd [-1, 1] denotes a random number in the range of −1 and component contaminated with noise, unifrnd [-1, 1] denotes a random number in the range of −1 and 1, and nl denotes the noise level. As stated before, nl = 5% is used in this example. 1, and nl denotes the noise level. As stated before, nl = 5% is used in this example. Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 13 Appl. Sci. 2019, 9, 3252 8 of 12 Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 13 Figure 6. Length change rate of element obtained from the static data (element 5 was damaged with Figure an added 5% 6. Length noise) change . rate of element obtained from the static data (element 5 was damaged with an Figure 6. Length change rate of element obtained from the static data (element 5 was damaged with added 5% noise). an added 5% noise). Figure 7. Length change rate of element obtained from the static data (elements 35 and 40 were Figure 7. Length change rate of element obtained from the static data (elements 35 and 40 were damaged with an added 5% noise). damaged with an added 5% noise). Figure 7. Length change rate of element obtained from the static data (elements 35 and 40 were damaged with an added 5% noise). As can be seen from Figures 6 and 7, even if there was a noise interference of 5%, the damage of As can be seen from Figures 6 and 7, even if there was a noise interference of 5%, the damage of element 5 can be clearly judged from Figure 6, and the damage of elements 35 and 40 can be clearly element 5 can be clearly judged from Figure 6, and the damage of elements 35 and 40 can be clearly As can be seen from Figures 6 and 7, even if there was a noise interference of 5%, the damage of judged from Figure 7. This indicates that the proposed method had good anti-noise ability. judged from Figure 7. This indicates that the proposed method had good anti-noise ability. element 5 can be clearly judged from Figure 6, and the damage of elements 35 and 40 can be clearly As mentioned above, actual static loading may be dicult to perform on the actual engineering judged from Figure 7. This indicates that the proposed method had good anti-noise ability. structur As m e to ent obtain ioned ab the displacement ove, actual stat parame ic loadi ters. ng m Inasuch y be d aicase, fficult a t dynamic o perform test on t can he act be used ual eng to obtain ineerithe ng modal structure to obta parameters in the di and the spvirtual lacement para displacement meters. In parameters. such a caThat se, a dynam is, a method ic test can be used to obtain that uses the dynamic As mentioned above, actual static loading may be difficult to perform on the actual engineering flexibility the modalcombined parameters with and the vi virtual for rtua cel di wassp adopted lacement to pa obtai rameters. Tha n the displacement t is, a me data thod tha indirtectly uses the . The structure to obtain the displacement parameters. In such a case, a dynamic test can be used to obtain rdynam esults of ic the flexrate ibilit of y combi length change ned wiof th vi thert element ual force calculated was adopt by this ed t method o obtain ar t eh shown e displ inacemen Figurest dat 8–11a. the modal parameters and the virtual displacement parameters. That is, a method that uses the indirectly. The results of the rate of length change of the element calculated by this method are shown dynamic flexibility combined with virtual force was adopted to obtain the displacement data in Figures 8–11. indirectly. The results of the rate of length change of the element calculated by this method are shown in Figures 8–11. Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 13 Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 13 Appl. Sci. 2019, 9, 3252 9 of 12 Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 13 Figure 8. Length change rate of element obtained from the dynamic data (element 5 was damaged Figure 8. Length change rate of element obtained from the dynamic data (element 5 was damaged with no noise). Figure 8. Length change rate of element obtained from the dynamic data (element 5 was damaged with no noise). Figure 8. Length change rate of element obtained from the dynamic data (element 5 was damaged with no noise). with no noise). Figure 9. Length change rate of element obtained from the dynamic data (elements 35 and 40 were Figure 9. Length change rate of element obtained from the dynamic data (elements 35 and 40 were Figure 9. Length change rate of element obtained from the dynamic data (elements 35 and 40 were damaged with no noise). damaged with no noise). damaged with no noise). Figure 9. Length change rate of element obtained from the dynamic data (elements 35 and 40 were damaged with no noise). Figure 10. Length change rate of element obtained from the dynamic data (element 5 was damaged Figure 10. Length change rate of element obtained from the dynamic data (element 5 was damaged Figure 10. Length change rate of element obtained from the dynamic data (element 5 was damaged adding 5% noise). adding 5% noise). adding 5% noise). Figure 10. Length change rate of element obtained from the dynamic data (element 5 was damaged adding 5% noise). Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 13 Appl. Sci. 2019, 9, 3252 10 of 12 Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 13 Figure 11. Length change rate of element obtained from the dynamic data (elements 35 and 40 were Figure 11. Length change rate of element obtained from the dynamic data (elements 35 and 40 were damaged adding 5% noise). Figure 11. Length change rate of element obtained from the dynamic data (elements 35 and 40 were damaged adding 5% noise). damaged adding 5% noise). As seen in Figures 8 and 9, in the case of no noise interference, the length change rate of the As seen in Figures 8 and 9, in the case of no noise interference, the length change rate of the element obtained from the As seen in Figures 8 and dynam 9, in ic d the c ata can ase of no no also determi ise interference ne the dama , t ge posi he lention of the el gth change r em ate o ent i f th n a e element obtained from the dynamic data can also determine the damage position of the element in a relatively clear manner. When there was 5% noise interference, as seen in Figures 10 and 11, there element obtained from the dynamic data can also determine the damage position of the element in a relatively clear manner. When there was 5% noise interference, as seen in Figures 10 and 11, there relatively was a signi cle ficaant r m p ae nner. ak at t When there he position of t was h5% no e dama ise ged inelement, which also ind terference, as seen in Figure icated that the element s 10 and 11, there was a significant peak at the position of the damaged element, which also indicated that the element with the larger length change rate was most likely to sustain damage. Note that the results obtained was a significant peak at the position of the damaged element, which also indicated that the element with the larger length change rate was most likely to sustain damage. Note that the results obtained wi from the dyna th the larger l mi ength change ra c data were less a te wa ccurate tha s most likely n thos toe obt sustain ained f dam ro age. m t Note he stat that the resu ic data. This lts o is b be tained cause from the dynamic data were less accurate than those obtained from the static data. This is because the flexibility matrix obtained by the dynamic flexibility method was an approximation, and the from the dynamic data were less accurate than those obtained from the static data. This is because the flexibility matrix obtained by the dynamic flexibility method was an approximation, and the tdispla he flex cement ibility obta matr ined by ix obtaia npplying the ed by the dy virtual forc namic flexib e ba ilit sed on thi y method was an appr s was naturally l oximat ess aion ccura , atn ed tha thn e displacement obtained by applying the virtual force based on this was naturally less accurate than that that obtained from the direct static test. Therefore, in engineering practice, static conditions are displacement obtained by applying the virtual force based on this was naturally less accurate than obtained from the direct static test. Therefore, in engineering practice, static conditions are preferred to tpreferre hat obtd t aine od obt from ain tdi hspl e di acement rect stat par ic ta emet st. There ers inf or ore,der t in en o obt gine ain erin dg pr amaact ge loc ice, st alizat aticion condit resuio lts wit ns arh e obtain displacement parameters in order to obtain damage localization results with higher accuracy, higher accuracy, when the conditions permit. preferred to obtain displacement parameters in order to obtain damage localization results with when the conditions permit. higher accuracy, when the conditions permit. In order to study the sensitivity of the proposed method, Figures 12 and 13 present the results of In order to study the sensitivity of the proposed method, Figures 12 and 13 present the results the elemental length change rates when element 44 (between nodes 12 and 22) was damaged with of the elemental length change rates when element 44 (between nodes 12 and 22) was damaged with In order to study the sensitivity of the proposed method, Figures 12 and 13 present the results an added 5% noise. Specifically, Figure 12 gives the results when element 44 has 20%, 15%, and 10% of the e an added 5% noise. Sp lemental length ch ecifange r ically, Fi ates whe gure 12 giv n elemen es the r t 44e (b sults when e etween nod lement 44 has 20% es 12 and 22) was dam , 15%a , an ged d 10% with stiness reductions, and Figure 13 gives the results of 7.5% and 5% stiness damages. From Figure 12, stiffness reductions, and Figure 13 gives the results of 7.5% and 5% stiffness damages. From Figure an added 5% noise. Specifically, Figure 12 gives the results when element 44 has 20%, 15%, and 10% one can see that there was a significant peak at the damaged element 44, which indicated that the stiffness re 12, one can ductions, see that there wa and Figure s a signif 13 giv ice ant pea s the re ksults at the da of 7.maged element 4 5% and 5% stiffne 4,s which i s damages ndi.ca From ted tha Fig t the ure damage position can be determined in a relatively clear manner when the damage degree is more damage position can be determined in a relatively clear manner when the damage degree is more 12, one can see that there was a significant peak at the damaged element 44, which indicated that the than 10%. The results in Figure 13 are confused, and it was impossible to accurately identify the true da than mage posit 10%. The re ion ca sultn be determined s in Figure 13 arein coa rela nfused, tive and ly cle it wa ar m s impos anner w sibh le t en the d o accur aa m tealy ge id deg entriee fy tis more he true damage location, which indicated that the proposed method is ineective when the damage degree is damage location, which indicated that the proposed method is ineffective when the damage degree than 10%. The results in Figure 13 are confused, and it was impossible to accurately identify the true less than 10%. Obviously, the cause of misjudgment lies in the adverse eect of data noise, because the is less than 10%. Obviously, the cause of misjudgment lies in the adverse effect of data noise, because damage location, which indicated that the proposed method is ineffective when the damage degree displacement changes caused by the damage are often submerged by the data noise when the damage the displacement changes caused by the damage are often submerged by the data noise when the is less than 10%. Obviously, the cause of misjudgment lies in the adverse effect of data noise, because degree is very small. For this case, 10% stiness reduction is the minimum detectable damage. the displace damage degr ment change ee is very sma s caused ll. For th by the dam is case, 10% a st ge iffne are often sub ss reduction mierged by th s the minime data noise um detectable when the damage. damage degree is very small. For this case, 10% stiffness reduction is the minimum detectable damage. Figure 12. Length change rate of element obtained from the static data with an added 5% noise (element Figure 12. Length change rate of element obtained from the static data with an added 5% noise 44 has 20%, 15%, and 10% stiness reductions). (element 44 has 20%, 15%, and 10% stiffness reductions). Figure 12. Length change rate of element obtained from the static data with an added 5% noise (element 44 has 20%, 15%, and 10% stiffness reductions). Appl. Sci. 2019, 9, 3252 11 of 12 Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 13 Figure 13. Length change rate of element obtained from the static data with an added 5% noise (element Figure 13. Length change rate of element obtained from the static data with an added 5% noise 44 has 7.5% and 5% stiness reductions). (element 44 has 7.5% and 5% stiffness reductions). 5. Conclusions 5. Conclusions We deduced the relationship between the characteristic force and characteristic displacement of the We deduced the relationship between the characteristic force and characteristic displacement of grid structure. The SMW formula was used to deduce the physical principle of damage identification the grid structure. The SMW formula was used to deduce the physical principle of damage by the displacement dierence in detail. Based on this principle, the rate of length change of the identification by the displacement difference in detail. Based on this principle, the rate of length element was proposed to implement the identification of damage position in the grid structure. A space change of the element was proposed to implement the identification of damage position in the grid grid structure was taken as an example. A numerical example was used to verify the result. The result structure. A space grid structure was taken as an example. A numerical example was used to verify indicated that the element with larger length change rate was the most likely position where damage the result. The result indicated that the element with larger length change rate was the most likely occurred. The proposed method has certain noise resistance and can obtain better identification eects position where damage occurred. The proposed method has certain noise resistance and can obtain in the case of single or multiple element damage. The biggest advantage of the proposed method is better identification effects in the case of single or multiple element damage. The biggest advantage that it is not necessary to establish a finite element model. In the specific application of the project, of the proposed method is that it is not necessary to establish a finite element model. In the specific the displacement dierence can be calculated only by a static test, such as using a laser range finder application of the project, the displacement difference can be calculated only by a static test, such as to obtain the displacement data of each node in order to calculate the displacement dierence. If the using a laser range finder to obtain the displacement data of each node in order to calculate the static displacement data cannot be obtained directly, it can be obtained from the parameters obtained displacement difference. If the static displacement data cannot be obtained directly, it can be obtained by dynamic testing. Therefore, the proposed method has good prospects for application. from the parameters obtained by dynamic testing. Therefore, the proposed method has good prospects for application. Author Contributions: Investigation, W.W.; methodology, Q.Y.; software, S.L.; writing—original draft, C.W.; writing—review and editing, N.L. Author Contributions: Investigation, W.W.; methodology, Q.Y.; software, S.L.; writing—original draft, C.W.; Funding: This research was funded by the National Natural Science Foundation of China (Grant numbers [11202138, writing—review and editing, N.L. 41772311]) and the Zhejiang Provincial Natural Science Foundation of China (Grant number [LY17E080016]). Conflicts of Interest: The authors declare no conflict of interest. Funding: This research was funded by the National Natural Science Foundation of China (Grant numbers [11202138, 41772311]) and the Zhejiang Provincial Natural Science Foundation of China (Grant number [LY17E080016] References ). 1. 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