# Residual Mode Vector-Based Structural Damage Identification with First-Order Modal Information

Authors: Shuai Luo; Zhenxin Zhuang; Wei Wang; Ping Jiang
Journal: Advances in Materials Science and Engineering
 (2021)

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## Abstract

Damage identification based on the change of dynamic properties is an issue worthy of attention in structure safety assessment, nevertheless, only a small number of discontinuous members in existing structure are damaged under service condition, and the most remaining members are in good condition. According to this feather, we developed an effective damage location and situation assessment algorithm based on residual mode vector with the first mode information of targeted structure, which utilized the quantitative relationship between first natural modes of global structure with the change of the element stiffness. Firstly, the element damage location is determined with exploitation of the sparseness of element stiffness matrices based on the discontinuity of damaged members. Then, according to the distribution characteristics of the corresponding residual mode vector, the nodal equilibrium equation about the damage parameter is established based on the residual mode vector, and the damage coefficients of structural elements are evaluated with the proposed equations. Two numerical examples are given to verify the proposed algorithm. The results showed that the proposed damage identification method is consistent with the preset damage. It can even accurately identify large-degree damages. The proposed algorithm only required the first-order modal information of the target structures and held few requirements of analysis resource; hence when compared with existing methods, it has obvious advantages for structural damage identification.


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## Body

## 1. Introduction

Structural damage identification technology shows its strong vitality in many engineering fields such as civil engineering, mechanical engineering, and aerospace engineering, which has benefited from the combination with modal measurement technology and finite-element analysis [1–3]. Various damage location and detection technologies have been developed based on structural vibration properties. The principle for vibration-based damage identification is that the damage-induced changes in the physical properties of element stiffness will cause detectable changes in system modal properties (natural frequencies and mode shapes).The investigation of mode eigen values for damage detection is common as the natural frequencies can be conveniently measured from just a few accessible points on the structure [4–7]. However, this method could not detect the subtle damage on structure. Damage detection methods have also been developed for the identification of damage directly based on measured mode shapes or mode shape curvatures [8–10]. A drawback of many mode shape-based methods is the complexity of having measurements from a large number of locations for higher order modal shapes [11–13]. For the purpose of fully utilized mode information, Kaouk and Zimmerman [14–16] defined the residual mode equation for damage detection based on mode information of structure, which can accomplish the dual tasks of determining the damage location and solving the damage situation effectively. This kind of damage detection method has received the widespread attention over the last decade [17–21]. There exists a common obstacle in the aforementioned detection methods, which is to obtain higher order of structural modal shapes. This requirement limits the practical application of the residual mode method. Recently, with the development of high-speed camera and sensitive sensor and vision recognition technology, the modal parameter identification technology of engineering structure has made remarkable progress [22–25]. We can effectively realize the accurate identification of the finite-order natural mode shapes (usually the first order) and natural frequencies (usually the first order) of the structural system. As such, more advanced analysis procedures are required to make the best use of first-order modal information from structural health monitoring.In this paper, the authors develop a damage detection method originally proposed by Kaouk et al. based on the residual mode equation with only the first order of structure mode information. The residual mode vector is developed based on the residual mode equation for structural element damage detection, in which the relationship between the stiffness matrix of damaged element and first-order mode information is derived from the residual mode equation. Then based on the relationship, the element damage location is determined with exploitation of the sparseness of element stiffness matrices. Finally, according to the distribution characteristics of residual mode vector, a new damage assessment algorithm is proposed.The presentation of this work is organized as follows: in Section2, the proposed algorithm based on the residual mode vector is developed. In Section 3, the process of structural damage identification using the established method is introduced numerically in detail with a simple supported beam. Moreover, the source of the accelerated formula is also discussed in detail. A two-story framework example is shown in Section 3.2 to show the feasibility and the superiority of the proposed method. The conclusions of this work are summarized in Section 4.


## 2. Problem Formulation

### 2.1. Fundamental Assumptions

In the following theoretical development, the following assumptions are made in damaged structures:(1)
Structural damages only reduce the system stiffness matrix, and structural refined numerical model has been developed before damage occurrence.(2)
It is assumed that the effect of damage on the mass properties of the structure is negligible, and we assume that the structure under consideration is undamped.(3)
The crack-induced structural member damage is not continuous in the finite-element model, this is reasonable in the previous study [26–29], and furthermore we could change the scale of element size in the numerical model.


### 2.2. Theoretical Development

One basic approach to obtain a qualitative feel for the effect of the damage on an individual mode is to monitor the change in mode information between the damaged structures and the undamaged structures. With the above assumptions, considering an initial design, the first-mode eigen equation of ann-DOF finite-element model of the undamaged structure exists and is given as(1)λ1Mϕ1=Kϕ1,where λ1 and ϕ1 are the first-mode eigen value and corresponding eigenvector separately and M and K are the n × n analytical mass and stiffness matrices. In this paper, only the first-order modal information of the system will be used to identify the damage location and damage situation, and this is also the important motivation for conducting this research. Let ΔK be the exact perturbation matrices that reflect the nature of the structural damage. Thus, the exact perturbation matrices are sparse matrices with the nonzero elements reflecting the state of damage. The eigen equation for the damaged structure is then(2)λ1dMϕ1d=K−ΔKϕ1d,where λ1d and ϕ1d are the first-mode eigen value and corresponding eigenvector measured precisely, as the first-order modal parameters of structures are the most easily obtained dynamic information in experimental process. ΔK is the n × n analytical mass and stiffness matrices:(3)ΔK=∑mαmKm,where αm is a scalar denoting the damage extent corresponding to the m-th element and the ΔK matrices represent the effect of damage on the structural property matrices. Thus, the exact damage matrices are sparse matrices with the nonzero elements reflecting the state of damage. Now equation (2) can be rewritten in the dynamic residual form:(4)ΔKϕ1d=K−λ1dMϕ1d=D,where D is the defined residual mode vector related to the first mode about the damaged system, and it could be depicted as(5)D=d1,d2,⋯dnT,where superscript T means transpose, and the same below, the proposed residual mode vector is n × 1 column vector. Then the location of damage element could be easily processed based on the distribute regulation of the proposed residual mode vector. Sort the elements in residual mode vector in absolute descending order, and then one can easily find that there is a great deal of differences: The amplitude of di in the proposed residual mode vector corresponding to the damaged element is much larger than that of the undamaged element; even, the elements in the proposed residual mode vector corresponding to undamaged element tends to 0, which could be used as the basis of filtering in the analysis process.Obviously, equation (4) is composed of n equilibrium equations at nodes; based on the assumption that there is only one element damaged at each node, for the node location of the damage element, the stiffness of the independent damage element is strictly proportional to the right-hand side of equation (4), and its ratio is the damage coefficient to be calculated. Based on this principle, the damage situation could be evaluated quantitatively.In the end, this section summarizes the key steps as follows:Step 1:.
calculating the residual mode vector
For the target structure, calculate residual mode vector by substituting the experimental results of first-order modal informationλ1d and ϕ1d into equation (4).Step 2:.
locating the damage element in structure and filter
Arrange the elements in the residual mode vector in the descending order of their absolute values. Divide the previous element by the next, and the last damage element is where the result tends to infinity, so as to locate the damage element; then the element in residual mode vector corresponding to the undamaged unit after positioning is artificially set to 0 to realize filtering.Step 3:.
Verifying the results of location identification
List the elements in residual mode vector after filtering according to their corresponding structural elements. Divide the elements in residual mode vector corresponding to the symmetrical node in the element, and the result whose quotient is −1 is the damaged component, so as toverify the results of location identification in Step 3, and prepare a new list for identification of the damage situation quantitatively.Step 4:.
Evaluating the damage coefficient
Take out the stiffness matrix of the damage element after positioning, establish the balance equation on the node position one by one according to the damage element code, and complete the damage identification by evaluating the damage coefficient of the balance equation.
To conclude this section, the process of the proposed vibration measurements using only first-order modal information mainly involves two key influence factors: precise of first mode information and sparse of element stiffness matrix. More details are explained in the following numerical examples.


## 2.1. Fundamental Assumptions

In the following theoretical development, the following assumptions are made in damaged structures:(1)
Structural damages only reduce the system stiffness matrix, and structural refined numerical model has been developed before damage occurrence.(2)
It is assumed that the effect of damage on the mass properties of the structure is negligible, and we assume that the structure under consideration is undamped.(3)
The crack-induced structural member damage is not continuous in the finite-element model, this is reasonable in the previous study [26–29], and furthermore we could change the scale of element size in the numerical model.


## 2.2. Theoretical Development

One basic approach to obtain a qualitative feel for the effect of the damage on an individual mode is to monitor the change in mode information between the damaged structures and the undamaged structures. With the above assumptions, considering an initial design, the first-mode eigen equation of ann-DOF finite-element model of the undamaged structure exists and is given as(1)λ1Mϕ1=Kϕ1,where λ1 and ϕ1 are the first-mode eigen value and corresponding eigenvector separately and M and K are the n × n analytical mass and stiffness matrices. In this paper, only the first-order modal information of the system will be used to identify the damage location and damage situation, and this is also the important motivation for conducting this research. Let ΔK be the exact perturbation matrices that reflect the nature of the structural damage. Thus, the exact perturbation matrices are sparse matrices with the nonzero elements reflecting the state of damage. The eigen equation for the damaged structure is then(2)λ1dMϕ1d=K−ΔKϕ1d,where λ1d and ϕ1d are the first-mode eigen value and corresponding eigenvector measured precisely, as the first-order modal parameters of structures are the most easily obtained dynamic information in experimental process. ΔK is the n × n analytical mass and stiffness matrices:(3)ΔK=∑mαmKm,where αm is a scalar denoting the damage extent corresponding to the m-th element and the ΔK matrices represent the effect of damage on the structural property matrices. Thus, the exact damage matrices are sparse matrices with the nonzero elements reflecting the state of damage. Now equation (2) can be rewritten in the dynamic residual form:(4)ΔKϕ1d=K−λ1dMϕ1d=D,where D is the defined residual mode vector related to the first mode about the damaged system, and it could be depicted as(5)D=d1,d2,⋯dnT,where superscript T means transpose, and the same below, the proposed residual mode vector is n × 1 column vector. Then the location of damage element could be easily processed based on the distribute regulation of the proposed residual mode vector. Sort the elements in residual mode vector in absolute descending order, and then one can easily find that there is a great deal of differences: The amplitude of di in the proposed residual mode vector corresponding to the damaged element is much larger than that of the undamaged element; even, the elements in the proposed residual mode vector corresponding to undamaged element tends to 0, which could be used as the basis of filtering in the analysis process.Obviously, equation (4) is composed of n equilibrium equations at nodes; based on the assumption that there is only one element damaged at each node, for the node location of the damage element, the stiffness of the independent damage element is strictly proportional to the right-hand side of equation (4), and its ratio is the damage coefficient to be calculated. Based on this principle, the damage situation could be evaluated quantitatively.In the end, this section summarizes the key steps as follows:Step 1:.
calculating the residual mode vector
For the target structure, calculate residual mode vector by substituting the experimental results of first-order modal informationλ1d and ϕ1d into equation (4).Step 2:.
locating the damage element in structure and filter
Arrange the elements in the residual mode vector in the descending order of their absolute values. Divide the previous element by the next, and the last damage element is where the result tends to infinity, so as to locate the damage element; then the element in residual mode vector corresponding to the undamaged unit after positioning is artificially set to 0 to realize filtering.Step 3:.
Verifying the results of location identification
List the elements in residual mode vector after filtering according to their corresponding structural elements. Divide the elements in residual mode vector corresponding to the symmetrical node in the element, and the result whose quotient is −1 is the damaged component, so as toverify the results of location identification in Step 3, and prepare a new list for identification of the damage situation quantitatively.Step 4:.
Evaluating the damage coefficient
Take out the stiffness matrix of the damage element after positioning, establish the balance equation on the node position one by one according to the damage element code, and complete the damage identification by evaluating the damage coefficient of the balance equation.
To conclude this section, the process of the proposed vibration measurements using only first-order modal information mainly involves two key influence factors: precise of first mode information and sparse of element stiffness matrix. More details are explained in the following numerical examples.


## 3. Numerical Examples

### 3.1. Simply Supported Beam

A 12-element simply supported beam used in this example shown in Figure1 is employed to exercise the proposed method. The basic parameters of the beam are as follows: Young’s modulus E = 200 GPa, density ρ = 7.8 × 103 kg∕m3, length of each element L = 0.1 m, and cross-sectional area A = 2.5 × 10−3 m2. The finite-element model of the structure has 26 degrees of freedom.Figure 1
The finite element model of simply supported beam.Let the beam element stiffness matrix in local coordination can be represented as(6)Ke=EIL3126L−126L6L4L2−6L2L2−12−6L12−6L6L2L2−6L4L2.The corresponding element mass matrix is(7)Me=ρA420156L22L254L−13L22L24L313L2−3L354L13L2156L−22L2−13L−3L3−22L24L3.The local coordinate of each element is consistent with the global coordination; hence, the coordination transformation is unnecessary; the element stiffness matrix represented by all element node displacements in the global coordinate system is derived directly as equation (1). The deformation vector of each node in the overall coordinate system is set as(8)V=v1θ1v2θ2…v12θ12v13θ13T.Take element 3, for example, the deformation vector corresponding toK3e can be expressed as(9)V3e=v3θ3v4θ4T.According to the relationship of the deformation vector between local and global coordination, the element matrix under global coordination system is given by(10)V3e=SV,where the connective matrix S is 4 × 26, the element in row 1 and column 5 of S is 1; the element in row 2 and column 6 of S is 1; the element in row 3 and column 7 of S is 1; the element in row 4 and column 8 of S is 1; and the rest of the elements of S is 0. The connective matrix S can be expressed as(11)S=0…0000………00010000010001000…001000………00004×26.The elastic strain energy of element 3 can be derived based on the energy theorem as(12)U3=V3eTK3eV3e.Substitute equation (10) into equation (12), we can get(13)U3=VTS3TK3eS3,V=VTK3gV.Hence, the element stiffness matrix of element 3 under the global coordination can be expressed as(14)K3g=S3TK3eS3.The element stiffness matrix under the global coordination could be derived similarly; then the whole stiffness matrix of the beam is obtained as(15)K=∑i=1121−αiKig,where αi0≤αi<1 is taken as an unknown variable expressed as a damage coefficient of the corresponding element i, and for undamaged theoretical model, these damage coefficients are all zeros.With the same procedure, the whole mass matrixM could be derived; then the modal equilibrium equation of the undamaged structural system shown in equation (1) is obtained, and the damage identification of the structure will be studied with this equation. Obviously, for single element in the global system, its element stiffness matrix is sparse, which provides important clues for damage identification.For the damaged beam, preset the damage coefficient artificially asα2=0.15, α4=0.6, α8=0.13, and α10=0.3, and the rest elements are intact. Figure 2 shows the measured first eigenvector ϕ1d of the damaged system, with which the corresponding eigen value λ1d = 1033.5.Figure 2
The measured first modal eigenvector.Then the first-order eigen valueλ1d and eigenvector ϕ1d of the damaged beam are substituted into equation (4) to obtain the residual mode vector. Figure 3 shows the distribution diagram of residual mode vector obtained from equation (4). The vector is numbered according to the degrees of freedom (abscissa) of the element. Since each element is statically self-balanced according to mode shape, the residual mode vector corresponding to each element is symmetric about the x-axis; hence the sum is 0.Figure 3
The distribution diagram of residual mode vector.The first purpose in this analysis is to determine the location of the damage. Arrange the absolute value of the obtained residual mode vector from large to small in accordance with Step 2. It should be noted that for the absolute value of residual mode vector from large to small, the number of the corresponding abscissa in Figure3 of each ordered element of residual mode vector must be marked first before the sorting, which is important for the subsequent damage detection work. Then, according to Step 3, obtain the location map of damage element as shown in Figure 4. The abscissa is the number of degrees of freedom of the unit, the ordinate is the ratio, the peak position in the figure is 16, and each unit corresponds to four degrees of freedom. Therefore, it is clear that four damaged elements are present in the simply supported beam, which is consistent with the presetting.Figure 4
The location map of the damage element.The next step of this analysis is to verify the location of the structural damage. Referring to Step 4, the residual mode vector of the corresponding elements is arranged in Table1 according to the node code of each element, and the first variable of each element is divided by the third variable. The element whose quotient equals minus is the damage component, so as to verify the results of damage element location in the third step. Obviously, the results of damage location are consistent with the presetting, which verifies the results of Step 3 again.Table 1
Residual mode vector according to elements.
Element no.123456789101112Residual07431.3−7431.340591.4−40591.4743.4−743.400.0−15147.515147.50Force0−805.21548.4−18598.022657.1−2504.32578.600.0−5172.53657.80Vector7431.3−7431.340591.4−40591.4743.4−743.40.00−15147.515147.50.00Residual mode vector−805.21548.4−18598.022657.1−2504.32578.60.00−5172.53657.80.00The last step of this analysis is to determine the extent of structural damage with the damage location according to Step 4. Take out the stiffness matrix of the element after damage location, and calculateKmϕ1d in the left-hand side of equation (4), respectively, as shown in Figure 5. At the same time, take out the residual mode vector of the damage element located in Table 1, and express it in Figure 5. By comparison, for the given damage element, the elements in Dm are proportional to the product vector of the element stiffness matrix Km and the first-mode shape ϕ1d, and the ratio is the damage degree αm.Figure 5
Residual mode vector according to damage location.
(a)(b)(c)(d)(e)(f)(g)(h)As the located elements, 2, 4, 6, and 10 are discontinuous members in this study, and the following formula can be used to solve the damage coefficient:(16)αmKmϕ1d=Dm.Finally, the damage parameters obtained are listed in Table2. From the table, it can be easily seen that the results calculated by this method are accurate solutions with Equation (16) which reflects the advantage of the proposed algorithm.Table 2
The evaluated damage parameters.
Element no 2Element no 4Element no 6Element no 10K2ϕ1dD2α2K4ϕ1dD4α4K6ϕ1dD6α6K10ϕ1dD10α107431.249541.80.1540591.367652.20.6743.35718.20.13−15147.5−50491.60.3−805.2−5368.20.15−18598−30996.60.6−2504.2−19263.60.13−5172.5−17241.80.3−7431.2−49541.90.15−40591.4−67652.30.6−743.3−5718.20.1315147.450491.60.31548.310322.40.1522657.137761.80.62578.619835.40.133657.712192.60.3The results in this section indicate that only the first mode of the structure needs to be provided by the algorithm proposed in this paper. Compared with the commonly used identification methods, the experimental requirements were not strict. Therefore, the proposed method has obvious advantages for structural damage identification. This paper will continue to use an example of a frame structure to further verify the results.


### 3.2. Two-Story Frame Structure

The second example is a two-story frame structure as shown in Figure6. This frame was modeled with 24 equal elements of 0.2 m in length. Every node has 3 DOFs, an axial displacement, a transverse displacement, and a rotation. The properties of this structure are as follows: cross-sectional area A = 0.0336 m2; moment of inertia I = 1.6128 × 10−4 m4; Young’s modulus E = 200 GPa; and density ρ = 2500 kg/m3.Figure 6
Two-story frame structure.The stiffness matrix of frame element considering axial deformation can be expressed as(17)Kef=EAL100−100000000000000−100100000000000000+EIL30000000126L0−126L06L4L20−6L2L20001000−12−6L012−6L06L2L20−6L4L2.The corresponding element mass matrix is(18)Mef=ρAL6200100000000000000100200000000000000+ρAL420000000015622L054−13L022L4013L−3L210000005413L0156−22L0−13L−3L20−22L4L2.Taking element 3 as an example, suppose the element stiffness matrix in the local coordinate system isK3ef, and the displacement vector in the local coordinate system can be expressed as(19)V3ef=u3av3aθ3au4bv4bθ4bT.According to the energy theorem, the elastic strain energy of the third element is known to be(20)U3ef=V3efTK3efV3ef.The relationship between the deformation of element component 3 in the local coordination and the global coordination needs to transform in the numerical model as the local coordinate of each element is inconsistent with the global coordination, which is different from the simply supported beam example.The displacement vector in the global coordinate system can be expressed as(21)V3g=u3v3θ3u4v4θ4T.Then the mapping relationship between local coordinates and global coordinates can be found as [30](22)V3g=PV3ef,where the matrix P representing the mapping relationship is a 6 × 6 matrix as(23)P3=cosα3−sin0000sinα3cosα30000001000000cosα3−sinα30000sinα3cosα30000001,where a3 is the rotation angle of element 3 in the frame, and obviously the angle is π/2. By substituting equation (22) into equation (20), the following equations can be obtained by energy theorem as(24)U3ef=V3gTP3−1TK3efP3−1V3g.Thus, the stiffness matrix of element 3 under the global coordination can be derived as(25)K3eg=P3−1TK3efP3−1.Furthermore, the stiffness matrix of element 3 represented by the overall displacement vector of the frame structure in the global coordination is derived directly. The deformation vector of each node in the global coordination is set as(26)Vf=u1v1θ1…u24v24θ24T.As the simple supported beam model in example 1, the connective relationship between the element displacement vector and the overall displacement vector is given by(27)V3g=SVf,where the connective matrix S is 6 × 72. According to the corresponding relationship of Vf and V3g, the connective matrix S3 can be expressed as(28)S3=0…00…00…01000100010000000000…00…00…00…00…00…00000000001000100010…00…00…0,where the element in row 1 and column 7 of S3 is 1; the element in row 2 and column 8 of S3 is 1; the element in row 3 and column 9 of S3 is 1; the element in row 4 and column 10 of S3 is 1; the element in row 5 and column 11 of S3 is 1; the element in row 6 and column 12 of S3 is 1; and rest of the elements of S is zero. Substitute equation (27) into equation (24), the strain energy can be deduced as(29)U3e=VTS3TK3eg,S3V=VTK3gV.Substitute equation (25) into equation (29), then the stiffness matrix of element 3 under the global coordination can be expressed by the overall displacement vector as(30)K3g=S3TP3−1TK3efP3−1S3.The element stiffness matrix of the rest elements in global coordination can be obtained by the same procedure. Obviously, for each single element component, the stiffness matrix is a sparse matrix, and its sparse property is the key foundation of the proposed damage identification method.As the next step is establishing the whole stiffness matrix of the structure system in global matrix, the element stiffness matrix under the global coordination could be derived similarly, and the whole stiffness matrix of the frame structure can be expressed as(31)K=∑i=1241−αiKig,where αi0≤αi<1 is taken as an unknown variable expressed as the damage coefficient corresponding to element i, and for undamaged theoretical model these damage coefficients are all zeros. With the same procedure, the whole mass matrix M could be derived by the element mass matrix in equation (18), then the modal equilibrium equation of undamaged structural system shown in formula (1) is obtained, and the damage identification of the structure could be studied based on it. The stiffness of the single element in the global system is spare, whose nodal points are coupled. Thus, it would provide important clues for damage identification obviously.For the damaged frame structure, we could preset the damage coefficients artificially asα2=0.15, α4=0.3, α8=0.13, α10=0.6, α14=0.23, and α18=0.4, and the rest elements are intact. Figure 7 shows the measured first eigenvector ϕ1d of the damaged system, with which the corresponding eigen value λ1d = 58814.7. The same operation could be operated for damage identification of frame structures, which takes an example of the previous proposed beam.Figure 7
The measured first eigenvector of the frame structure.The first-order eigen valueλ1d and eigenvector ϕ1d of the damaged beam are substituted into equation (4) to obtain the residual mode vector. Take out the stiffness matrix of the element after damage location, and calculate the amount of Kmϕ1d in the left-hand side of equation (4), respectively, as shown in Figure 8(a) and take out the residual mode vector of the damage element and express it in Figure 8(b). By comparison, for the given damage element, the element value in residual mode vector is proportional to the product vector of the element stiffness matrix Km and the first mode shape ϕ1d, and the ratio is the damage degree αm.Figure 8
Component schematic diagram: (a) the amount ofKmϕ1d. (b) The residual mode vector of the damage element.
(a)(b)The damage coefficients can be evaluated by equation (16). Table 3 shows the amount of Kmϕ1d in the left-hand side of equation (16), and the residual mode vector corresponding to the located damage members of the frame structure. From the table, it can be seen that the results calculated by this method are accurately the same as those preset, which further proves the reliability of this proposed method.Table. 3
The evaluated damage parameters.
Element no 2Element no 4Element no 8Element no 10Element no 14Element no 18K2ϕ1dD2K4ϕ1dD4K8ϕ1dD8K10ϕ1dD10K14ϕ1dD14K18ϕ1dD183211025481653.735817201074516−440108−5721443327742599664321187.573873.12−2235587−8942355930163889524.4716689021500673836174498702.6406195324371722002432460559.426528651061146−993756−149063330389.799116.92−765979−99577.2−524497−314698419546.696495.72−681415−272566−3211025−481654−3581720−107451644010857214.04−4332774−2599664−321187−73873.12235587894234.8−5930163−889524−7166890−2150067−3836174−498703−4061953−2437172−2002432−460559−2652865−1061146351551.452732.71−1046734−3140201533214199317.8−342058−205235−19060.2−4383.84234297.893719.13α2=0.15α4=0.3α8=0.13α10=0.6α14=0.23α18=0.4According to the solution results of the two numerical examples, only the first-order mode of the structure needs to be measured by the method developed in this paper. Compared with the commonly used identification method, the experimental requirements are not strict, and the measured results are accurate solutions, so the solution results are better than the approximate methods such as sensitivity method, which is worth popularizing. Therefore this research develops a new strategy of structural damage identification based on first-order mode and opens up a new way for structural damage identification based on the dynamic test.


## 3.1. Simply Supported Beam

A 12-element simply supported beam used in this example shown in Figure1 is employed to exercise the proposed method. The basic parameters of the beam are as follows: Young’s modulus E = 200 GPa, density ρ = 7.8 × 103 kg∕m3, length of each element L = 0.1 m, and cross-sectional area A = 2.5 × 10−3 m2. The finite-element model of the structure has 26 degrees of freedom.Figure 1
The finite element model of simply supported beam.Let the beam element stiffness matrix in local coordination can be represented as(6)Ke=EIL3126L−126L6L4L2−6L2L2−12−6L12−6L6L2L2−6L4L2.The corresponding element mass matrix is(7)Me=ρA420156L22L254L−13L22L24L313L2−3L354L13L2156L−22L2−13L−3L3−22L24L3.The local coordinate of each element is consistent with the global coordination; hence, the coordination transformation is unnecessary; the element stiffness matrix represented by all element node displacements in the global coordinate system is derived directly as equation (1). The deformation vector of each node in the overall coordinate system is set as(8)V=v1θ1v2θ2…v12θ12v13θ13T.Take element 3, for example, the deformation vector corresponding toK3e can be expressed as(9)V3e=v3θ3v4θ4T.According to the relationship of the deformation vector between local and global coordination, the element matrix under global coordination system is given by(10)V3e=SV,where the connective matrix S is 4 × 26, the element in row 1 and column 5 of S is 1; the element in row 2 and column 6 of S is 1; the element in row 3 and column 7 of S is 1; the element in row 4 and column 8 of S is 1; and the rest of the elements of S is 0. The connective matrix S can be expressed as(11)S=0…0000………00010000010001000…001000………00004×26.The elastic strain energy of element 3 can be derived based on the energy theorem as(12)U3=V3eTK3eV3e.Substitute equation (10) into equation (12), we can get(13)U3=VTS3TK3eS3,V=VTK3gV.Hence, the element stiffness matrix of element 3 under the global coordination can be expressed as(14)K3g=S3TK3eS3.The element stiffness matrix under the global coordination could be derived similarly; then the whole stiffness matrix of the beam is obtained as(15)K=∑i=1121−αiKig,where αi0≤αi<1 is taken as an unknown variable expressed as a damage coefficient of the corresponding element i, and for undamaged theoretical model, these damage coefficients are all zeros.With the same procedure, the whole mass matrixM could be derived; then the modal equilibrium equation of the undamaged structural system shown in equation (1) is obtained, and the damage identification of the structure will be studied with this equation. Obviously, for single element in the global system, its element stiffness matrix is sparse, which provides important clues for damage identification.For the damaged beam, preset the damage coefficient artificially asα2=0.15, α4=0.6, α8=0.13, and α10=0.3, and the rest elements are intact. Figure 2 shows the measured first eigenvector ϕ1d of the damaged system, with which the corresponding eigen value λ1d = 1033.5.Figure 2
The measured first modal eigenvector.Then the first-order eigen valueλ1d and eigenvector ϕ1d of the damaged beam are substituted into equation (4) to obtain the residual mode vector. Figure 3 shows the distribution diagram of residual mode vector obtained from equation (4). The vector is numbered according to the degrees of freedom (abscissa) of the element. Since each element is statically self-balanced according to mode shape, the residual mode vector corresponding to each element is symmetric about the x-axis; hence the sum is 0.Figure 3
The distribution diagram of residual mode vector.The first purpose in this analysis is to determine the location of the damage. Arrange the absolute value of the obtained residual mode vector from large to small in accordance with Step 2. It should be noted that for the absolute value of residual mode vector from large to small, the number of the corresponding abscissa in Figure3 of each ordered element of residual mode vector must be marked first before the sorting, which is important for the subsequent damage detection work. Then, according to Step 3, obtain the location map of damage element as shown in Figure 4. The abscissa is the number of degrees of freedom of the unit, the ordinate is the ratio, the peak position in the figure is 16, and each unit corresponds to four degrees of freedom. Therefore, it is clear that four damaged elements are present in the simply supported beam, which is consistent with the presetting.Figure 4
The location map of the damage element.The next step of this analysis is to verify the location of the structural damage. Referring to Step 4, the residual mode vector of the corresponding elements is arranged in Table1 according to the node code of each element, and the first variable of each element is divided by the third variable. The element whose quotient equals minus is the damage component, so as to verify the results of damage element location in the third step. Obviously, the results of damage location are consistent with the presetting, which verifies the results of Step 3 again.Table 1
Residual mode vector according to elements.
Element no.123456789101112Residual07431.3−7431.340591.4−40591.4743.4−743.400.0−15147.515147.50Force0−805.21548.4−18598.022657.1−2504.32578.600.0−5172.53657.80Vector7431.3−7431.340591.4−40591.4743.4−743.40.00−15147.515147.50.00Residual mode vector−805.21548.4−18598.022657.1−2504.32578.60.00−5172.53657.80.00The last step of this analysis is to determine the extent of structural damage with the damage location according to Step 4. Take out the stiffness matrix of the element after damage location, and calculateKmϕ1d in the left-hand side of equation (4), respectively, as shown in Figure 5. At the same time, take out the residual mode vector of the damage element located in Table 1, and express it in Figure 5. By comparison, for the given damage element, the elements in Dm are proportional to the product vector of the element stiffness matrix Km and the first-mode shape ϕ1d, and the ratio is the damage degree αm.Figure 5
Residual mode vector according to damage location.
(a)(b)(c)(d)(e)(f)(g)(h)As the located elements, 2, 4, 6, and 10 are discontinuous members in this study, and the following formula can be used to solve the damage coefficient:(16)αmKmϕ1d=Dm.Finally, the damage parameters obtained are listed in Table2. From the table, it can be easily seen that the results calculated by this method are accurate solutions with Equation (16) which reflects the advantage of the proposed algorithm.Table 2
The evaluated damage parameters.
Element no 2Element no 4Element no 6Element no 10K2ϕ1dD2α2K4ϕ1dD4α4K6ϕ1dD6α6K10ϕ1dD10α107431.249541.80.1540591.367652.20.6743.35718.20.13−15147.5−50491.60.3−805.2−5368.20.15−18598−30996.60.6−2504.2−19263.60.13−5172.5−17241.80.3−7431.2−49541.90.15−40591.4−67652.30.6−743.3−5718.20.1315147.450491.60.31548.310322.40.1522657.137761.80.62578.619835.40.133657.712192.60.3The results in this section indicate that only the first mode of the structure needs to be provided by the algorithm proposed in this paper. Compared with the commonly used identification methods, the experimental requirements were not strict. Therefore, the proposed method has obvious advantages for structural damage identification. This paper will continue to use an example of a frame structure to further verify the results.


## 3.2. Two-Story Frame Structure

The second example is a two-story frame structure as shown in Figure6. This frame was modeled with 24 equal elements of 0.2 m in length. Every node has 3 DOFs, an axial displacement, a transverse displacement, and a rotation. The properties of this structure are as follows: cross-sectional area A = 0.0336 m2; moment of inertia I = 1.6128 × 10−4 m4; Young’s modulus E = 200 GPa; and density ρ = 2500 kg/m3.Figure 6
Two-story frame structure.The stiffness matrix of frame element considering axial deformation can be expressed as(17)Kef=EAL100−100000000000000−100100000000000000+EIL30000000126L0−126L06L4L20−6L2L20001000−12−6L012−6L06L2L20−6L4L2.The corresponding element mass matrix is(18)Mef=ρAL6200100000000000000100200000000000000+ρAL420000000015622L054−13L022L4013L−3L210000005413L0156−22L0−13L−3L20−22L4L2.Taking element 3 as an example, suppose the element stiffness matrix in the local coordinate system isK3ef, and the displacement vector in the local coordinate system can be expressed as(19)V3ef=u3av3aθ3au4bv4bθ4bT.According to the energy theorem, the elastic strain energy of the third element is known to be(20)U3ef=V3efTK3efV3ef.The relationship between the deformation of element component 3 in the local coordination and the global coordination needs to transform in the numerical model as the local coordinate of each element is inconsistent with the global coordination, which is different from the simply supported beam example.The displacement vector in the global coordinate system can be expressed as(21)V3g=u3v3θ3u4v4θ4T.Then the mapping relationship between local coordinates and global coordinates can be found as [30](22)V3g=PV3ef,where the matrix P representing the mapping relationship is a 6 × 6 matrix as(23)P3=cosα3−sin0000sinα3cosα30000001000000cosα3−sinα30000sinα3cosα30000001,where a3 is the rotation angle of element 3 in the frame, and obviously the angle is π/2. By substituting equation (22) into equation (20), the following equations can be obtained by energy theorem as(24)U3ef=V3gTP3−1TK3efP3−1V3g.Thus, the stiffness matrix of element 3 under the global coordination can be derived as(25)K3eg=P3−1TK3efP3−1.Furthermore, the stiffness matrix of element 3 represented by the overall displacement vector of the frame structure in the global coordination is derived directly. The deformation vector of each node in the global coordination is set as(26)Vf=u1v1θ1…u24v24θ24T.As the simple supported beam model in example 1, the connective relationship between the element displacement vector and the overall displacement vector is given by(27)V3g=SVf,where the connective matrix S is 6 × 72. According to the corresponding relationship of Vf and V3g, the connective matrix S3 can be expressed as(28)S3=0…00…00…01000100010000000000…00…00…00…00…00…00000000001000100010…00…00…0,where the element in row 1 and column 7 of S3 is 1; the element in row 2 and column 8 of S3 is 1; the element in row 3 and column 9 of S3 is 1; the element in row 4 and column 10 of S3 is 1; the element in row 5 and column 11 of S3 is 1; the element in row 6 and column 12 of S3 is 1; and rest of the elements of S is zero. Substitute equation (27) into equation (24), the strain energy can be deduced as(29)U3e=VTS3TK3eg,S3V=VTK3gV.Substitute equation (25) into equation (29), then the stiffness matrix of element 3 under the global coordination can be expressed by the overall displacement vector as(30)K3g=S3TP3−1TK3efP3−1S3.The element stiffness matrix of the rest elements in global coordination can be obtained by the same procedure. Obviously, for each single element component, the stiffness matrix is a sparse matrix, and its sparse property is the key foundation of the proposed damage identification method.As the next step is establishing the whole stiffness matrix of the structure system in global matrix, the element stiffness matrix under the global coordination could be derived similarly, and the whole stiffness matrix of the frame structure can be expressed as(31)K=∑i=1241−αiKig,where αi0≤αi<1 is taken as an unknown variable expressed as the damage coefficient corresponding to element i, and for undamaged theoretical model these damage coefficients are all zeros. With the same procedure, the whole mass matrix M could be derived by the element mass matrix in equation (18), then the modal equilibrium equation of undamaged structural system shown in formula (1) is obtained, and the damage identification of the structure could be studied based on it. The stiffness of the single element in the global system is spare, whose nodal points are coupled. Thus, it would provide important clues for damage identification obviously.For the damaged frame structure, we could preset the damage coefficients artificially asα2=0.15, α4=0.3, α8=0.13, α10=0.6, α14=0.23, and α18=0.4, and the rest elements are intact. Figure 7 shows the measured first eigenvector ϕ1d of the damaged system, with which the corresponding eigen value λ1d = 58814.7. The same operation could be operated for damage identification of frame structures, which takes an example of the previous proposed beam.Figure 7
The measured first eigenvector of the frame structure.The first-order eigen valueλ1d and eigenvector ϕ1d of the damaged beam are substituted into equation (4) to obtain the residual mode vector. Take out the stiffness matrix of the element after damage location, and calculate the amount of Kmϕ1d in the left-hand side of equation (4), respectively, as shown in Figure 8(a) and take out the residual mode vector of the damage element and express it in Figure 8(b). By comparison, for the given damage element, the element value in residual mode vector is proportional to the product vector of the element stiffness matrix Km and the first mode shape ϕ1d, and the ratio is the damage degree αm.Figure 8
Component schematic diagram: (a) the amount ofKmϕ1d. (b) The residual mode vector of the damage element.
(a)(b)The damage coefficients can be evaluated by equation (16). Table 3 shows the amount of Kmϕ1d in the left-hand side of equation (16), and the residual mode vector corresponding to the located damage members of the frame structure. From the table, it can be seen that the results calculated by this method are accurately the same as those preset, which further proves the reliability of this proposed method.Table. 3
The evaluated damage parameters.
Element no 2Element no 4Element no 8Element no 10Element no 14Element no 18K2ϕ1dD2K4ϕ1dD4K8ϕ1dD8K10ϕ1dD10K14ϕ1dD14K18ϕ1dD183211025481653.735817201074516−440108−5721443327742599664321187.573873.12−2235587−8942355930163889524.4716689021500673836174498702.6406195324371722002432460559.426528651061146−993756−149063330389.799116.92−765979−99577.2−524497−314698419546.696495.72−681415−272566−3211025−481654−3581720−107451644010857214.04−4332774−2599664−321187−73873.12235587894234.8−5930163−889524−7166890−2150067−3836174−498703−4061953−2437172−2002432−460559−2652865−1061146351551.452732.71−1046734−3140201533214199317.8−342058−205235−19060.2−4383.84234297.893719.13α2=0.15α4=0.3α8=0.13α10=0.6α14=0.23α18=0.4According to the solution results of the two numerical examples, only the first-order mode of the structure needs to be measured by the method developed in this paper. Compared with the commonly used identification method, the experimental requirements are not strict, and the measured results are accurate solutions, so the solution results are better than the approximate methods such as sensitivity method, which is worth popularizing. Therefore this research develops a new strategy of structural damage identification based on first-order mode and opens up a new way for structural damage identification based on the dynamic test.


## 4. Conclusions

This study presents an accurate algorithm for structural damage identification and location based on the measured first-order modal information of structure according to the sparse characteristics of the element stiffness matrix. The only location results required in determining the damage in the first-order modal equilibrium equation is the common residual mode vector whose nonzero elements are corresponding to the damaged member in the structure system. Finally based on the location results, the mechanical balance equation of the residual deformation of the located element is formulated for determining the damage extent. The proposed method has the advantages of low calculation resource requirement, fast positioning speed, and high recognition accuracy. As the higher order modal information could not be acquired sufficiently with the current technology, the process of computing the extent of structural damage using the proposed algorithm is shown to be computationally attractive and hence suitable for large-scale problems.



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*Source: 5526171-2021-07-03.xml*