# An Exact Approach for Structural Damage Assessment

Authors: Q. W. Yang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

An exact approach is proposed for damage identification in statically determinate structures. The contribution of this study is twofold. Firstly, a rigorous disassembly formulation of structural global flexibility matrix is presented based on the matrix spectral decomposition, which can provide an exact relationship between the modifications of structural stiffness parameters and the associated flexibility matrix. Secondly, the static minimum-rank flexibility change is derived to obtain the exact flexibility change before and after damage. The proposed method is economical in computation and is simple to implement. For the statically determinate structures, the proposed method can exactly compute the elemental perturbed stiffness parameter only using a few of incomplete static displacement data. The efficiency of the proposed method is demonstrated by two statically determinate structures.


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## Body

## 1. Introduction

Detection, localization, and quantification of damage in a structure via techniques that examine changes in measured structural static/dynamic response have attracted much attention in recent years. The basic idea of this technique is that structural response parameters are functions of structural physical properties (mass, damping, and stiffness). Therefore, changes in the physical properties will cause changes in the response parameters. Recent surveys on the technical literature show that extensive efforts have been developed to find reliable and efficient numerical and experimental models to identify damage in structures [1–12].One important group of damage identification methods is the perturbation-based techniques [13–35]. These methods start with the derivatives of the response parameters to changes in material and physical parameters. These sensitivity coefficients are then used to calculate changes in the parameters that would force the analysis response parameters to match those measured in test. The common advantage of these methods is that they can detect structural damage by directly using the incomplete modal parameters without any eigenvector expansion or model reduction [24, 25]. However, the efficiency of perturbation analysis methods is limited because these methods are complicated and may be only suitable for small modifications of structural parameters [35]. For the cases with relatively large modifications of structural parameters, the first-order perturbation may be inaccurate. It has been pointed out that when the change of structural parameter is more than 15%, the second-order perturbation should be taken into account [36]. As an alternative, some researches [33–35] have used the iteration scheme to tackle the large damage case. It is anticipated that the computational cost of these existing sensitivity methods will be very expensive for large damage case, since a higher-order approximation should be performed or an iteration scheme must be used to estimate the perturbation parameters more precisely.To avoid the above shortcomings, an exact flexibility perturbation technique is developed in the paper for structural damage detection. The contribution of the present study is twofold. Firstly, a rigorous disassembly formulation of structural global flexibility matrix is presented based on the matrix spectral decomposition, which can provide an exact relationship between the modifications of structural stiffness parameters and the associated flexibility matrix. Secondly, a static minimum-rank formulation of structural flexibility change is presented to obtain the exact flexibility change before and after damage. The most significant advantage of the proposed method is that it is economical in computation and is simple to implement. Regardless of whether the damage is small or large, the proposed method can accurately compute the elemental perturbed stiffness parameter only using a few of incomplete static displacements without any higher-order approximation or iteration. In the following theoretical development, it is assumed that structural damages only reduce the system stiffness matrix and structural refined FEM has been developed before damage occurrence.


## 2. Theoretical Development

### 2.1. The Disassembly Formulation of Structural Flexibility Matrix Based on Spectral Decomposition

The developed theory begins with the disassembly of the (n×n) global stiffness matrix K, which can be obtained by the elemental eigenparameter decomposition technique [16, 23] as
(1)K=CPCT,
where the (n×N) matrix C is defined as the stiffness connectivity matrix representation of the connectivity between DOFs and the (N×N) diagonal matrix P has the elemental stiffness parameters pi (i=1~N) as its diagonal entries and n is the number of DOFs and N is the number of elemental stiffness parameters. The columns of matrix C physically embody the stiffness contribution to the global stiffness matrix in terms of the elemental stiffness parameters pi. The diagonal entries pi in the matrix P consist of the material and sectional properties of the element. The matrix C is independent of P and unchanged as damage occurs. A full description of the elemental eigenparameter decomposition technique can be seen in [16, 23]. According to (1), it is important to note that N≥n and the matrix C is of full rank (rank(Cn×N)=n) because K is of full rank (rank(Kn×n)=n). Assuming that αi (0≤αi≤1) is the ith elemental stiffness perturbation parameter, the value of αi is 0 if the ith element is undamaged and αi is 1 or less than 1 if the corresponding element is completely or partially damaged. Then the global stiffness matrix of the damaged structure can be assembled as
(2)Kd=CPdCT,
in which
(3)Pd=[p1(1-α1)p2(1-α2)⋱pN(1-αN)].The stiffness matrix perturbationΔK can be expressed as
(4)ΔK=K-Kd=CΔPCT,ΔP=diag(p1α1,p2α2,…,pNαN).Now we turn our attention to investigating the disassembly formulation of structural global flexibility matrix. For the statically determinate structures, as will be shown in Section3, n=N is valid. Then the disassemblies of the (n×n) global flexibility matrices F and Fd, for the undamaged and damaged structure, can be given as
(5)F=(C-1)TP-1C-1,(6)Fd=(C-1)TPd-1C-1.Apparently, (5) and (6) can be easily proved by F·K=Fd·Kd=In×n. Letting
(7)B=P-1,(8)Bd=Pd-1,(9)D=(C-1)T,
then (5) and (6) can be rewritten as
(10)F=DBDT,(11)Fd=DBdDT.From (10) and (11), it can be seen that the disassembly of the global flexibility matrix is similar in form to that of the stiffness matrix. Similarly, the matrix D is defined as the flexibility connectivity matrix and the diagonal entry bi in B is defined as the ith flexibility parameter (bi=1/pi). Subtracting (10) from (11), the flexibility matrix perturbation ΔF can be given as
(12)ΔF=DΔBDT
in which the diagonal matrix ΔB is
(13)ΔB=Bd-B=[b1(11-α1-1)b2(11-α2-1)⋱bN(11-αN-1)].Defineβi to be the ith elemental perturbed flexibility parameter that satisfies
(14)βi=11-αi-1,(15)ΔB=diag(b1β1,b2β2,…,bNβN).From (14), the perturbed stiffness parameter αi and the perturbed flexibility parameter βi are related as follows:
(16)βi=αi1-αi,
or
(17)αi=βi1+βi.According to the above discussion, if the perturbed stiffness parameterαi is given, we can obtain the exact ΔF by the course αi→(16)βi→(12)ΔF. Conversely, if ΔF is given, we can also obtain exact αi by ΔF→(12)βi→(17)αi.


### 2.2. The Static Minimum-Rank Formulation of Structural Flexibility Change

For the damage identification problem,ΔF can be firstly estimated by structural dynamic or static test, and then the stiffness perturbation parameters αi  (i=1~N) can be calculated by ΔF→(12)βi→(17)αi. To obtain the exact ΔF before and after damage, a static minimum-rank formulation of structural flexibility change is presented in this section by using a few of static displacement data. For an intact structure, the analytical static model can be expressed as
(18)Kuj=lj,
where uj is the displacement vector under the applied static load vector lj and j=1,…,m as it is assumed that only m static load vectors are used. Rewriting (18), one has
(19)uj=Flj.Similarly, the displacement vectorudj for the damaged structure can be obtained by
(20)udj=Fdlj.Therefore the change of the displacement vectorΔuj can be obtained as
(21)Δuj=ΔF·lj.Forj=1~m, (21) can be written in matrix form as
(22)ΔF·L=ΔU,
where
(23)ΔU=[Δu1,Δu2,…,Δum],L=[l1,l2,…,lm].Similar to the minimum-rank method [37, 38], the minimal rank solution of (22) for symmetric ΔF is given as
(24)ΔF=ΔU(ΔUTL)-1ΔUT.


### 2.3. Damage Detection

This section presents a step-by-step summary of the whole technique as follows.Step 1.
CalculateΔF by (24). In this step, the static displacement data of the damaged structure can be obtained by a static test on it, and the static parameters of the undamaged structure can be obtained by the analytical static model or through a static test on the intact structure.Step 2.
Compute the flexibility connectivity matrixD by (9) for the statically determinate structures.Step 3.
Compute the perturbed flexibility parametersβi (i=1~N) by (12).Step 4.
Calculate the perturbed stiffness parametersαi (i=1~N) using (17). Then structural damage can be assessed from these parameters.


## 2.1. The Disassembly Formulation of Structural Flexibility Matrix Based on Spectral Decomposition

The developed theory begins with the disassembly of the (n×n) global stiffness matrix K, which can be obtained by the elemental eigenparameter decomposition technique [16, 23] as
(1)K=CPCT,
where the (n×N) matrix C is defined as the stiffness connectivity matrix representation of the connectivity between DOFs and the (N×N) diagonal matrix P has the elemental stiffness parameters pi (i=1~N) as its diagonal entries and n is the number of DOFs and N is the number of elemental stiffness parameters. The columns of matrix C physically embody the stiffness contribution to the global stiffness matrix in terms of the elemental stiffness parameters pi. The diagonal entries pi in the matrix P consist of the material and sectional properties of the element. The matrix C is independent of P and unchanged as damage occurs. A full description of the elemental eigenparameter decomposition technique can be seen in [16, 23]. According to (1), it is important to note that N≥n and the matrix C is of full rank (rank(Cn×N)=n) because K is of full rank (rank(Kn×n)=n). Assuming that αi (0≤αi≤1) is the ith elemental stiffness perturbation parameter, the value of αi is 0 if the ith element is undamaged and αi is 1 or less than 1 if the corresponding element is completely or partially damaged. Then the global stiffness matrix of the damaged structure can be assembled as
(2)Kd=CPdCT,
in which
(3)Pd=[p1(1-α1)p2(1-α2)⋱pN(1-αN)].The stiffness matrix perturbationΔK can be expressed as
(4)ΔK=K-Kd=CΔPCT,ΔP=diag(p1α1,p2α2,…,pNαN).Now we turn our attention to investigating the disassembly formulation of structural global flexibility matrix. For the statically determinate structures, as will be shown in Section3, n=N is valid. Then the disassemblies of the (n×n) global flexibility matrices F and Fd, for the undamaged and damaged structure, can be given as
(5)F=(C-1)TP-1C-1,(6)Fd=(C-1)TPd-1C-1.Apparently, (5) and (6) can be easily proved by F·K=Fd·Kd=In×n. Letting
(7)B=P-1,(8)Bd=Pd-1,(9)D=(C-1)T,
then (5) and (6) can be rewritten as
(10)F=DBDT,(11)Fd=DBdDT.From (10) and (11), it can be seen that the disassembly of the global flexibility matrix is similar in form to that of the stiffness matrix. Similarly, the matrix D is defined as the flexibility connectivity matrix and the diagonal entry bi in B is defined as the ith flexibility parameter (bi=1/pi). Subtracting (10) from (11), the flexibility matrix perturbation ΔF can be given as
(12)ΔF=DΔBDT
in which the diagonal matrix ΔB is
(13)ΔB=Bd-B=[b1(11-α1-1)b2(11-α2-1)⋱bN(11-αN-1)].Defineβi to be the ith elemental perturbed flexibility parameter that satisfies
(14)βi=11-αi-1,(15)ΔB=diag(b1β1,b2β2,…,bNβN).From (14), the perturbed stiffness parameter αi and the perturbed flexibility parameter βi are related as follows:
(16)βi=αi1-αi,
or
(17)αi=βi1+βi.According to the above discussion, if the perturbed stiffness parameterαi is given, we can obtain the exact ΔF by the course αi→(16)βi→(12)ΔF. Conversely, if ΔF is given, we can also obtain exact αi by ΔF→(12)βi→(17)αi.


## 2.2. The Static Minimum-Rank Formulation of Structural Flexibility Change

For the damage identification problem,ΔF can be firstly estimated by structural dynamic or static test, and then the stiffness perturbation parameters αi  (i=1~N) can be calculated by ΔF→(12)βi→(17)αi. To obtain the exact ΔF before and after damage, a static minimum-rank formulation of structural flexibility change is presented in this section by using a few of static displacement data. For an intact structure, the analytical static model can be expressed as
(18)Kuj=lj,
where uj is the displacement vector under the applied static load vector lj and j=1,…,m as it is assumed that only m static load vectors are used. Rewriting (18), one has
(19)uj=Flj.Similarly, the displacement vectorudj for the damaged structure can be obtained by
(20)udj=Fdlj.Therefore the change of the displacement vectorΔuj can be obtained as
(21)Δuj=ΔF·lj.Forj=1~m, (21) can be written in matrix form as
(22)ΔF·L=ΔU,
where
(23)ΔU=[Δu1,Δu2,…,Δum],L=[l1,l2,…,lm].Similar to the minimum-rank method [37, 38], the minimal rank solution of (22) for symmetric ΔF is given as
(24)ΔF=ΔU(ΔUTL)-1ΔUT.


## 2.3. Damage Detection

This section presents a step-by-step summary of the whole technique as follows.Step 1.
CalculateΔF by (24). In this step, the static displacement data of the damaged structure can be obtained by a static test on it, and the static parameters of the undamaged structure can be obtained by the analytical static model or through a static test on the intact structure.Step 2.
Compute the flexibility connectivity matrixD by (9) for the statically determinate structures.Step 3.
Compute the perturbed flexibility parametersβi (i=1~N) by (12).Step 4.
Calculate the perturbed stiffness parametersαi (i=1~N) using (17). Then structural damage can be assessed from these parameters.


## 3. Numerical Examples

To illustrate characteristics of the proposed damage detection algorithm, two statically determinate structures are presented. Example1 is a 23-bar plane truss structure and Example 2 is a cantilever beam with 12 elements.Example 1.
A 23-bar truss structure (shown in Figure1) is used in this example to verify the proposed method. The basic parameters of the structure are as follows: Young’s modulus E=200 GPa, density ρ=7.8×103Kg/m3, length of each bar L=1m, and cross-sectional area A=0.004m2. It is noted that this example is a statically determinated structure (N=n=23). Three load cases are shown in Table 1. The displacement data are contaminated with 3% random noise to simulate measurement error. Two damage scenarios were considered. The first one is a small damage case that elements 6, 12, and 17 are assumed to be damaged with stiffness losses of 15%, 15%, and 20%, respectively. The second scenario is a large damage case that elements 6, 12, and 17 have 60%, 70%, and 80% reductions in stiffness, respectively. Figure 2 shows the damage detection results for the small damage case and Figure 3 gives the identification results of the large damage case, respectively. For the small damage case, the calculated damage extents in Figure 2 without noise for elements 6, 12, and 17 are α6=0.1500, α12=0.1500, and α17=0.2000, which are exactly the assumed values (15%, 15%, and 20%). When 3% noise is introduced, the corresponding calculated damage extents are α6=0.1499, α12=0.1496, and α17=0.2017, which track the assumed values closely. For the large damage case, the calculated damage extents in Figure 3 without noise for elements 6, 12, and 17 are α6=0.6000, α12=0.7000, and α17=0.8000, which are exactly the assumed values (60%, 70%, and 80%). When 3% noise is introduced, the corresponding calculated damage extents are α6=0.5897, α12=0.7029, and α17=0.7888, which have 1.7%, 0.4%, and 1.4% relative errors as compared to the assumed values, respectively. From the above results, it is obvious that the stiffness perturbation parameter αi can be computed exactly by the proposed method for the error-free case regardless of whether the damage is small or large. When noise is considered, the predicted damage extents only have slight deviation from the true values because of the error in the measured data.
One of the main disadvantages of the proposed method is that the static loading conditions should be the same for the undamaged and damaged structures. In practice, it is very difficult to reproduce the same loading conditions in both cases. In this example, in order to investigate the effect of changes in the loading conditions on the damage detection, random errors of up to 7% in the static data of the original system are introduced to simulate the loading condition change to a certain extent. And then the damage detection result for the large damage case is shown in Figure4. It can be seen from Figure 4 that the prediction obtained is also reasonable.Table 1
Load cases for Example1.
Load case
Loads
1
F
1
=
10 KN; F2=0; F3=0.
2
F
1
=
10 KN; F2=10 KN; F3=0.
3
F
1
=
10 KN; F2=10 KN; F3=10 KN.Figure 1
23-bar truss structure with the static loads for Example1.Figure 2
Damage detection results when elements 6, 12, and 17 are damaged with stiffness losses of 15%, 15%, and 20% (Example1).Figure 3
Damage detection results when elements 6, 12, and 17 are damaged with stiffness losses of 60%, 70%, and 80% (Example1).Figure 4
Damage detection results considering the loading condition change simulated by adding 7% noise in the static data of the original system (Example1: large damage case).Example 2.
The second example is a cantilever beam as shown in Figure5. The basic parameters of the structure are as follows: Young’s modulus E=200 GPa, density ρ=7.8×103Kg/m3, moment of inertia I=1.0416×10-6m4, and cross-sectional area A=0.0025m2. The beam is modeled using 12 elements giving 24 DOFs (12 translational, 12 rotational) and the length of each element is L=0.1m. It is noted that this example is also a statically determinated structure (N=n=24). Four load cases are shown in Table 2. As before, the displacement data are contaminated with 3% random noise to simulate measurement error. Furthermore, only the transnational DOFs are used in the calculation of ΔF since it is difficult to measure the rotational DOFs. Two damage scenarios were considered. The first one is a small damage case that elements 4 and 9 are assumed to be damaged with stiffness losses both of 15%. The second scenario is a large damage case that elements 4 and 9 have 70% and 80% reductions in stiffness, respectively. Figure 6 shows the damage detection results for the small damage case and Figure 7 gives the identification results of the large damage case, respectively. For the small damage case, the calculated damage extents in Figure 6 without noise for elements 4 and 9 are α4=0.1500 and α9=0.1500, which are exactly the assumed values (15%, 15%). When 3% noise is introduced, the corresponding calculated damage extents are α4=0.1505 and α9=0.1490, which track the assumed values closely. For the large damage case, the calculated damage extents in Figure 7 without noise for elements 4 and 9 are α4=0.7000 and α9=0.8000, which are exactly the assumed values (70%, 80%). When 3% noise is introduced, the corresponding calculated damage extents are α4=0.7008 and α9=0.7987, respectively. It can be seen from Figures 6 and 7 that, only using the incomplete displacement data, precise results can be obtained by the proposed method for both small damage case and large damage case. It has been shown that the presented damage detection approach has remarkable advantage over the previous perturbation techniques in tackling the large damage case.
In many cases, the flexibility is estimated in an approximate way by using modal parameters [39, 40] as
(25)ΔFmd=∑j=1m1λdjϕdjϕdjT-∑j=1m1λjϕjϕjT,
where ΔFmd is the dynamic flexibility change using the first m modes, m is the number of measured modes in modal survey, ϕj and ϕdj are the eigenvectors of undamaged and damaged structures, and λj and λdj are the corresponding eigenvalues, respectively. Apparently, the modal truncation in the computation of dynamic flexibility change will have an adverse effect on the damage detection. In this study, the eigenvalue decomposition of dynamic flexibility change is used to reduce the effects of modal truncation and measurement noise on the damage detection. Performing an eigenvalue decomposition of ΔFmd one can write
(26)ΔFmd=UΛUT,
where U=[u1,u2,…,un] is the eigenvector matrix consisting of the eigenvectors ui (i=1~n) and Λ is the eigenvalue matrix whose diagonal entries are the eigenvalues of ΔFmd, that is, diag(Λ)=(σ1,σ2,…,σn)T. In actual applications ΔFmd will have some negative eigenvalues. It is noted that ΔFmd obtained by a truncated set of modes using (26) is usually less than the exact change of structural flexibility. So one can have an intuitive guess that the negative eigenvalues of ΔFmd show the influence of the modal truncation in the computation of the dynamic flexibility change. Hence, all the negative eigenvalues should be set to zeros to reduce the errors caused by the modal truncation. On the other hand, when the damage is severe, the change in structural flexibility caused by the measurement noise is smaller than that caused by the damage. One can have another intuitive that the relatively smaller positive eigenvalues of ΔFmd can be seen as a product of the measurement errors and they will also be set to zero to reduce the influence of measurement errors. As a result, the filtered dynamic flexibility change can be obtained as
(27)ΔFfd=∑ir(uiσiuiT),
where r is the number of relatively larger entries in Λ, σ1,σ2,…,σr are the relatively larger eigenvalues of ΔFmd, and u1,u2,…,ur are the corresponding eigenvectors. In this example, only the first four modes are used to calculate the dynamic flexibility change by (26) and the modes are contaminated with 3% random noise to simulate measurement error. Figures 8 and 9 show the damage assessment results using the dynamic flexibility change for the small and large damage cases, respectively. It has been shown that the proposed method is applicable when the flexibility is estimated in an approximate way by using modal parameters.Table 2
Load cases for Example2.
Load case
Loads
1
F
1
=
10 KN; F2=0; F3=0; F4=0.
2
F
1
=
10 KN; F2=10 KN; F3=0; F4=0.
3
F
1
=
10 KN; F2=10 KN; F3=10 KN; F4=0.
4
F
1
=
10 KN; F2=10 KN; F3=10 KN; F4=10 KN.Figure 5
A cantilever beam with the static loads for Example2.Figure 6
Damage detection results when elements 4 and 9 are damaged with stiffness losses both of 15% (Example2).Figure 7
Damage detection results when elements 4 and 9 are damaged with stiffness losses of 70% and 80% (Example2).Figure 8
Damage detection results for the small damage case using the dynamic flexibility change (Example2).Figure 9
Damage detection results for the large damage case using the dynamic flexibility change (Example2).


## 4. Conclusions

An exact flexibility perturbation technique for damage identification in statically determinate structures has been developed in this study, which is based on matrix spectral decomposition and minimum-rank update theory. The most significant advantage of the proposed procedure is that it can obtain reliable extent of damage only by simple computation without any higher-order approximation or iteration, regardless of whether the damage is small or large. Two numerical examples are used to exercise this process and measurement noise is also simulated in damage detection. The results show the good efficiency and stability of the proposed method on the identification of damage on more than one element. It has been shown that the proposed procedure may be a promising method in structural damage identification. Future research on the technique can be carried out to tackle the damage detection problem of the statically indeterminate structures, to compare the method with other damage assessment techniques, and to demonstrate the procedure using experimentally measured data.



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