Free-interface modal synthesis method is applied to civil structure, and a substructure method is proposed by introducing the method into global sensitivity method. The substructure expression of the derivatives of eigenvalues and eigenvectors with respect to elemental parameters is obtained. The accuracy of the application of free-interface modal synthesis method is evaluated with different retained modes in substructure, and then the effectiveness of the proposed substructure sensitivity method is illustrated through an 11-storey building under both single- and multidamage cases. Both the damage locations and the extent can be effectively identified. By comparing it with the identical results of global sensitivity method, the proposed method can be faster in detecting the damage location and more stable under multidamage cases. Since this substructure sensitivity method only needs to update sensitivity matrix in the substructure with relative small number of DOFs, it may save much computation effort and become more efficient.
To ensure the safety of large-scale civil infrastructures, more and more health monitoring systems are adopted to evaluate health condition and also to prevent sudden failure due to accumulation of component damages [
Sensitivity-based method is believed to be an effective way in model updating and damage detection. It is based on the fact that a perturbation in parameters will affect the outputs of a structure, such as frequencies and mode shapes [
This paper attempted to apply free-interface modal synthesis method in civil infrastructure and to introduce the method into the traditional global sensitivity method and to establish a substructure sensitivity method. The derivatives of eigenvalue and eigenvector with respect to system parameters will be calculated with the substructure method. With only those factors within certain substructures having nonzero value, this method is supposed to require few computation efforts and higher efficiency in damage detection. Numerical example is conducted on an 11-storey frame building. In order to maintain the accuracy during damage detection, the accuracy of modal synthesis method will be first discussed and then the damage detection ability of the proposed method will be verified under several damage cases.
The idea of modal synthesis method is to first divide a structure into a series of substructures with no bound constraints attached. After acquiring the dynamic characteristics of every substructure, the system equation can be reconstructed by taking the displacement coordinate of each substructure into account, and finally the modal information can be derived from this equation. To facilitate the description, it is assumed that the structure is divided into only two substructures, named substructures
Similar to the whole structure, taking substructure
The eigenvalues and eigenvectors can be solved from the corresponding modal equation in frequency domain. And if only the first
The nodes of the substructure can be divided into two sets: those attached to the interface and those with only internal forces applied on. It is expressed as subscription
By normalizing using mass matrix, the dominate equation of
Assuming that only the first
By using (
In order to reconstruct the system equation, it is supposed that the number of the retained modes of each substructure should exceed the number of DOFs of the interface
It can be noticed from (
Matrix
Since the boundary condition is assumed to be free for substructures
By substituting the second coordinate transformation equation (
By solving (
The dynamic response sensitivity-based model updating method is adopted here with
Equation (
The basic modal equation expressed as equation (
By further applying modal truncation and sparse matrix storage in the above procedure, the system resource will be greatly saved during the construction of sensitivity matrix. Subscription
By using free-interface modal synthesis method mentioned above, the
Assume that
Considering that there are no duplicated frequencies, then the dimension of
In order to solve
Initially, it is assumed that the analytical finite element model is intact, while damage is simulated as a decreasing of elastic modulus in certain element. The damage detection iterative procedure aims to detect the damage location and extent through the convergence of eigenvalue and eigenvector.
Due to the mode truncation, there will be errors while using the modal synthesis method mentioned above. Since it is not expected too much errors caused by the uncertainty of the system parameters, the accuracy of the modal synthesis method is first examined by a simple example. As shown in Figure
Eleven-storey frame structure (unit: mm).
The retained modes of the two substructures begin with 6 and 12 in Group 1 and increase by 2 for the next calculation until all the modes are included. Based on the modal analysis, it is expected that there should be a convergent point for the desired accuracy along with the increasing retained modes of the substructures. In this example, the first 8 eigenvalues and eigenvectors are selected to be examined. In order to further discuss how many modes should be retained, the following convergence index
The convergence index
Convergent index
Group | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
Retained modes | ||||||||||
|
6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 |
|
12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | 30 |
|
||||||||||
|
26859 | 24728 | 24514 | 23767 | 23447 | 23285 | 23182 | 23060 | 22952 | 22711 |
|
86.2 | 53.1 | 31.4 | 13.6 | 6.9 | 4.4 | 5.3 | 4.7 | 10.6 | 3.7 |
Eigen errors on Group 4.
Modes | Expected frequency (Hz) | Analytical frequency (Hz) | Differences (%) | MAC (%) |
---|---|---|---|---|
1 | 1.035 | 1.052 | 1.6 | 100 |
2 | 3.361 | 3.44 | 2.4 | 99.4 |
3 | 4.359 | 4.512 | 3.5 | 99.5 |
4 | 6.349 | 6.477 | 2 | 96.4 |
5 | 7.057 | 7.356 | 4.2 | 95.9 |
6 | 10.147 | 10.546 | 3.9 | 95.5 |
7 | 11.536 | 11.719 | 1.6 | 97.6 |
8 | 14.253 | 14.709 | 3.2 | 97.1 |
Convergent index
The first 4 mode shapes after modal synthesis (Group 4).
Usually, the experimental data collected by applying damages on some elements and the differences between the analytical and the experimental responses are calculated in target function and are to be minimized during the iteration procedure. For the example discussed in this paper, only the first 8 frequencies and mode shapes are included in the target function expressed as
The whole structure is simply divided as 2 substructures. Meanwhile, the frequencies and mode shapes of the damaged model are first calculated from finite element model and assumed to be known as experimental data. And all the damages are assumed to be loss in the young’s modulus of some elements, and their damage extents are defined by SRF values:
Damage detection results using free-interface modal synthesis based substructure sensitivity method.
Case 1
Case 2
Case 3
In this session, in order to evaluate the detection effectiveness of the proposed method, more damage cases are chosen randomly, including single damage and multidamage cases. The damage detection results of the proposed substructure method are compared with those by traditional global sensitivity method. Three damage cases are listed in Table
Damaged Cases 4, 5, and 6.
Damage location | SRF | |
---|---|---|
Case 4 | Element 5 | −30% |
Case 5 | Element 3 and 8 | −30% and −20% |
Case 6 | Element 1, 2, and 7 | −30%, −50%, and −20% |
For Case 4, both the global and the proposed substructure sensitivity method can detect the damage location and damage extent as well. As compared in Figure
Frequencies and mode shapes using the proposed substructure method.
Mode | Frequency | MAC (%) | ||
---|---|---|---|---|
Experimental data (Hz) | Analytical data (Hz) | Difference (%) | ||
1 | 1.032 | 1.033 | 0.1 | 100 |
2 | 3.353 | 3.353 | 0 | 100 |
3 | 4.29 | 4.291 | 0.02 | 100 |
4 | 6.32 | 6.321 | 0.02 | 100 |
5 | 6.967 | 6.969 | 0.03 | 100 |
6 | 10.129 | 10.137 | 0.08 | 99.9 |
7 | 11.46 | 11.462 | 0.02 | 100 |
8 | 14.155 | 14.162 | 0.05 | 99.8 |
9 | 14.283 | 14.292 | 0.06 | 99.8 |
10 | 16.65 | 16.653 | 0.02 | 99.7 |
Damage detection results (Case 4).
Substructure sensitivity method
Global sensitivity method
Further examining the convergent procedure, the SRF values of the proposed substructure on iterations 1, 3, 4, and 5 are graphed in Figure
SRF values of substructure sensitivity method (iterations 1, 3, 4, and 5).
SRF values of global sensitivity method (iterations 1, 4, 9, and 10).
Convergent curves of Case 4.
Substructure method
Global method
Case 5 is a multidamage case with two elements assumed to have stiffness reduction of 30% and 20%, respectively. Figure
Damage detection results (Case 5).
Substructure sensitivity method
Global sensitivity method
Convergent curves of Case 5.
Substructure method
Global method
Figure
Damage detection results (Case 6).
Substructure sensitivity method
Global sensitivity method
A free-interface modal synthesis method is applied to civil structures, and, by introducing to the sensitivity method, a free-interface modal synthesis based substructure sensitivity method is proposed for damage detection. The derivatives of eigenvalue and eigenvector with respect to elemental parameters are calculated with substructure approach and herein facilitate the calculation burden of sensitivity matrix.
The application of modal synthesis method and the effectiveness of the proposed substructure damage detection method were verified through a frame structure. As the mode truncation may bring errors to the identification procedure, a criterion for retaining mode numbers of substructure was first proposed. Although there is still some slight false identification, overall, the proposed substructure method was able to identify the damage location as well as the damage extent correctly in both single- and multidamage scenarios. Compared to traditional global sensitivity method, the proposed substructure method has a better speed in detecting the damage location and is even more stable in multidamage cases. More importantly, different from global sensitivity method which takes the whole structure in calculating the sensitivity matrix in every step, the proposed substructure method will only need to calculate and update sensitivity matrix in certain substructure which contains the parameter. And the updating procedure is done within a relative small dimension. By doing so, the proposed substructure method will have fewer calculation and storage burden, and it will be an advantage on model updating efficiency.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful for financial support by National Natural Science Foundation of China (no. 51108089), Ph.D. Programs Foundation of Ministry of Education of China (no. 20113514120005), and National Natural Science Foundation of Fujian Province (no. 2011J05128).