A new damage quantification approach for shear-wall buildings using ambient vibration data

Abstract

This paper presents a new approach to estimate damage severity for shear-wall buildings using diagonal terms of a modal flexibility matrix estimated from dynamic properties. This study aims to provide a fundamental concept for quantifying the damage of realistic buildings by investigating an idealized shear-wall building. Numerical studies were performed on a 5-story shear-wall building model to validate the applicability of the presented approach, using two damage patterns. With the numerical simulations, the proposed approach accurately determined the damage ratio of the specimens. Experiments were also conducted on a 5-story shear-wall building model for which the system parameters were almost the same as those in numerical simulations. The estimated damage-quantification results from the experimental validations demonstrated that the performance of the presented method for shear-wall buildings was both suitable and accurate.

Keywords

damage identification / modal flexibility / damage quantification / shear-wall buildings

Introduction

Most civil infrastructure is exposed to severe external loads over time due to factors. Therefore, it is essential to acquire explicit data on the condition of such structures to avoid catastrophic failure, to increase economic efficiency, and to extend structural service life. To do all this, structural health monitoring (SHM) to estimate structural integrity, has been given considerable attention by the civil engineering research community to evaluate the status and remaining service life of structures. As a result of a lot of effort to develop or improve the SHM technique, novel methodologies using changes in dynamic parameters have been presented [ 1– 3]. Vibration-based damage assessment consists of: 1) examining the damage occurrence, 2) specifying the damaged locations, and 3) calculating the severity of damages in specific locations.

Some researchers developed member-level damage-quantification approaches for building structures based on finite element (FE) model updating approaches [ 4– 9], Kalman filter-based damage-quantification approaches [ 10, 11], and the neural network-based approach [ 12– 14]. Recently, a modal, flexibility-based damage quantification approach was presented [ 15– 18]. Reynders and De Roeck [ 15] proposed a damage-detection and quantification method using quasi-static flexibility. Koo et al. [ 16] used damage-induced inter-story deflections estimated using modal flexibilities to evaluate damage severity. Hosseinzadeh et al. [ 17] presented a new method called the “cuckoo optimization algorithm” for quantification of structural damage. They confirmed the efficiency and robustness of the proposed approach through numerical and experimental studies.

In this study, a novel damage quantification approach for shear-wall buildings, based on diagonal terms of modal flexibility matrix estimated from dynamic properties, is presented. The proposed approach has the crucial advantage, compared to conventional approaches, of using FE-model-updating based on a lot of optimization variables. This is possible because damage quantification based on this approach is only dependent on the damage parameterization estimated from vibration data. For damage quantification, the proposed approach only used measured signals to construct a modal flexibility matrix, without FE model updating.

First, the theory for the estimation of the modal flexibility matrix from vibration data, and for damage quantification by change in a diagonal of the flexibility matrix, was explained. Next, numerical simulations and experiments are conducted on a 5-story shear-wall building structure using two damage patterns.

Theoretical background

Estimation of the modal flexibility from ambient vibration data

The modal flexibility matrix G_m using m lower modes [ 19] can be expressed as

(1) {\mathbf{G}}_m={\mathbf{\Phi }}_m{\mathbf{\Lambda }}_m^{-\mathrm{1}}{\mathbf{\Phi }}_m^{\mathrm{{\rm T}}}={\displaystyle \mathbf{\sum }_{i=\mathrm{1}}^m{\displaystyle \frac{{\mathbf{\Phi }}_i{\mathbf{\Phi }}_i^{\mathrm{T}}}{{\omega }_i^{\mathrm{2}}}}},

where {\mathrm{\Lambda }}_m=\left[\backslash {\omega }_i^{\mathrm{2}}\backslash \right] for which {\omega }_i is the i^th structural natural frequency, i = 1, 2,…, m; {\mathrm{\Phi }}_m=\left\{{\phi }_{\mathrm{1}},{\phi }_{\mathrm{2}}...,{\phi }_m\right\} ; and {\phi }_i is the ith mass normalized mode shape which can be obtained by existing mass-normalization techniques [ 20, 21].

Damage quantification by change in a diagonal of the flexibility matrix

For a shear-wall-building structure, the stiffness matrix can be estimated as

(2) \mathbf{K}=\left[\begin{array}{cccccc}{k}_{\mathrm{1}}+k_{\mathrm{2}}\hfill & -k_{\mathrm{2}}\hfill & \mathrm{0}\hfill & \cdots \hfill & \cdots \hfill & \mathrm{0}\hfill \\ -k_{\mathrm{2}}\hfill & k_{\mathrm{2}}+k_{\mathrm{3}}\hfill & -k_{\mathrm{3}}\hfill & \mathrm{0}\hfill & \cdots \hfill & \mathrm{0}\hfill \\ ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \mathrm{0}\hfill & \cdots \hfill & \cdots \hfill & -k_{n-\mathrm{1}}\hfill & k_{n-\mathrm{1}}+k_n\hfill & -k_n\hfill \\ \mathrm{0}\hfill & \mathrm{0}\hfill & \cdots \hfill & \cdots \hfill & -k_n\hfill & k_n\hfill \end{array}\right],

where n is the number of floors in the structure.

Then, the flexibility matrix G can be expressed as

(3) \mathbf{G}={\mathbf{K}}^{-\mathrm{1}}=\left[\begin{array}{ccccc}{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}\hfill & {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}\hfill & \cdots \hfill & {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}\hfill & {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}\hfill \\ {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}\hfill & {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\hfill & \cdots \hfill & {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\hfill & {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\hfill \\ ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}\hfill & {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\hfill & \cdots \hfill & {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{3}}}}\cdots +{\displaystyle \frac{\mathrm{1}}{k_{n-\mathrm{1}}}}\hfill & {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{3}}}}\cdots +{\displaystyle \frac{\mathrm{1}}{k_{n-\mathrm{1}}}}\hfill \\ {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}\hfill & {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\hfill & \cdots \hfill & {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{3}}}}\cdots +{\displaystyle \frac{\mathrm{1}}{k_{n-\mathrm{1}}}}\hfill & {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{3}}}}\cdots +{\displaystyle \frac{\mathrm{1}}{k_{n-\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_n}}\hfill \end{array}\right].

From Eq. (3), the diagonal of the flexibility matrix can be obtained as

(4) \mathbf{D}=\left[\backslash \mathbf{G}\backslash \right]=\left[\begin{array}{l}{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}\hfill \\ {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\hfill \\ ⋮\hfill \\ {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\cdots +{\displaystyle \frac{\mathrm{1}}{k_{n-\mathrm{1}}}}\hfill \\ {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\cdots +{\displaystyle \frac{\mathrm{1}}{k_{n-\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_n}}\hfill \end{array}\right].

Assume that damage has occurred at the m^th floor. Then, the diagonal of the flexibility matrix of the damaged structure can be expressed as

(5) {\mathbf{D}}^D=\left[\backslash {\mathrm{G}}^D\backslash \right]=\left[\begin{array}{l}{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}\hfill \\ ⋮\hfill \\ {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\cdots +{\displaystyle \frac{\mathrm{1}}{k_m^D}}\hfill \\ ⋮\hfill \\ {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\cdots +{\displaystyle \frac{\mathrm{1}}{k_m^D}}\cdots +{\displaystyle \frac{\mathrm{1}}{k_{n-\mathrm{1}}}}\hfill \\ {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\cdots +{\displaystyle \frac{\mathrm{1}}{k_m^D}}\cdots +{\displaystyle \frac{\mathrm{1}}{k_{n-\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_n}}\hfill \end{array}\right]=\left[\begin{array}{l}{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}\\ ⋮\\ {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\cdots +{\displaystyle \frac{\mathrm{1}}{(\mathrm{1}-\alpha )\times k_m^I}}\\ ⋮\\ {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\cdots +{\displaystyle \frac{\mathrm{1}}{(\mathrm{1}-\alpha )\times k_m^I}}\cdots +{\displaystyle \frac{\mathrm{1}}{k_{n-\mathrm{1}}}}\\ {\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\cdots +{\displaystyle \frac{\mathrm{1}}{(\mathrm{1}-\alpha )\times k_m^I}}\cdots +{\displaystyle \frac{\mathrm{1}}{k_{n-\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{k_n}}\end{array}\right],

where {\mathbf{D}}^D is the diagonal of the flexibility matrix of the damaged structure, k_m^I and k_m^D are the elementary stiffness of the intact and damaged structures at the mth floor, respectively, and \alpha is the damage severity, 0< \alpha <1 .

By subtracting Eq. (4) from Eq. (5), the general equation of change in a diagonal of flexibility matrices due to damages, can be obtained as

(6) \mathrm{\Delta }\mathbf{D}={\mathbf{D}}^D-{\mathrm{D}}^I=\left[\begin{array}{l}\mathrm{0}\hfill \\ ⋮\hfill \\ \mathrm{0}\hfill \\ {\displaystyle \frac{\mathrm{1}}{\left(\mathrm{1}-\alpha \right)\times k_m^I}}-{\displaystyle \frac{\mathrm{1}}{k_m^I}}\hfill \\ {\displaystyle \frac{\mathrm{1}}{\left(\mathrm{1}-\alpha \right)\times k_m^I}}-{\displaystyle \frac{\mathrm{1}}{k_m^I}}\hfill \\ ⋮\hfill \\ {\displaystyle \frac{\mathrm{1}}{\left(\mathrm{1}-\alpha \right)\times k_m^I}}-{\displaystyle \frac{\mathrm{1}}{k_m^I}}\hfill \end{array}\right].

Finally, the damage ratio {\alpha }_m at the m^th floor can be simply expressed as follows:

(7) {\alpha }_m={\displaystyle \frac{k_m^I-k_m^D}{k_m^I}}={\displaystyle \frac{{\left[\mathrm{\Delta }\mathbf{D}\right]}_m^{\mathit{\text{'}}}}{{\left[{\mathbf{D}}^D\right]}_m^{\mathit{\text{'}}}}},

where {\left[\mathrm{\Delta }\mathbf{D}\right]}^{\mathit{\text{'}}}={\left[\begin{array}{l}\mathrm{0}\cdots \mathrm{0}{\displaystyle \frac{\mathrm{1}}{\left(\mathrm{1}-\alpha \right)\times k_m^I}}-{\displaystyle \frac{\mathrm{1}}{k_m^I}}\mathrm{0}\cdots \mathrm{0}\hfill \end{array}\right]}^{\mathrm{T}} is the differentiation of the \mathrm{\Delta }\mathbf{D} and {\left[{\mathbf{D}}^D\right]}^{\mathit{\text{'}}}={\left[\begin{array}{l}{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}}}{\displaystyle \frac{\mathrm{1}}{k_{\mathrm{2}}}}\cdots {\displaystyle \frac{\mathrm{1}}{\left(\mathrm{1}-{\alpha }_m\right)k_m^I}}\cdots {\displaystyle \frac{\mathrm{1}}{k_{n-\mathrm{1}}}}{\displaystyle \frac{\mathrm{1}}{k_n}}\hfill \end{array}\right]}^{\mathrm{T}} is the differentiation of the {\mathbf{D}}^D .

Numerical simulation

In this chapter, the efficiency of the presented damage-quantification technique is demonstrated using numerical simulation. In the example, the approach is applied to a 5-story shear-wall-building model with the model parameters shown in Table 1. Limited sensors (i.e., accelerometers) were assumed to be available on each floor and an additional concentration mass was introduced on each floor to consider sensor weight (i.e., each 0.4 kg). The following two damage patterns were considered (see Fig. 1) to confirm the efficiency of the presented approach in the described shear-wall-building model, and the bending stiffness of the columns were reduced to simulate damages with the same severity for all of the damage cases. In Damage Case 1 (DC1), the bending stiffness of Column 1 is reduced by 10%. In Damage Case 2 (DC2), the bending stiffness of Columns 1 and 3 is identically reduced by 10%.

Dynamic characteristics were estimated by eigenvalue analysis using MATLAB. Modal information changes due to progressive damage are depicted in Table 2, and intact mode shapes of the shear-wall building model are depicted in Fig. 2. The natural frequencies decreased about 0.517%–2.998%, while the modal assurance criterion (MAC) values were almost the same. Thus, the MAC values for the damaged mode shapes, based on intact mode shapes, were about 0.9983–0.9999. Since the number of obtainable modal information in full-scale structures is limited because of structural rigidity. For this reason, the lower three modes were used to construct each intact and damaged modal-flexibility matrix. Two-dimensional plots of the diagonal terms of modal-flexibility matrices and their changes due to progressive damage are depicted in Fig. 3. By applying the proposed approach using the estimated diagonal terms of the modal flexibility matrices, damage severity can be identified for each damage pattern. The final damage-quantification results generated by the proposed approach are shown in Fig. 4. It was found that the damage-quantification results agree well with the analytical damage severity of the numerical model.

Experimental validation

In this section, experiments were performed on a 5-story shear-wall-building model. The structure was supported by bolt connections on the shaking table (see Fig. 5). Two damage patterns, the same used with numerical simulation, were described by replacement of the intact column with the damage column, as shown in Table 3.

To confirm the accuracy of the proposed damage-quantification approach, damage ratios estimated from two different experiments were compared to each other. To do this, first, damage ratios were evaluated using flexibility matrices composed of element stiffness obtained by static tests. Next, damage severities were determined by modal flexibility matrices calculated from dynamic properties using vibration measurements [ 16].

For the static push-over tests, the stiffness of target columns of the structure was directly estimated using inter-story displacements measured using laser-type displacement sensors and measured forces. In other words, the floor stiffness was simply calculated using the relationship among force, displacement, and stiffness. Flexibility matrices, as well as structural global-stiffness matrices, were constructed using the calculated floor stiffness. Last, damage severities were estimated by application of the presented damage quantification approach.

For the modal tests, vibration data was measured by accelerometers, which were attached on each floor, under the shaking table. Then, the dynamic properties extracted from vibration measurements were used to calculate modal flexibilities. Finally, damage ratios were identified by the estimated modal flexibilities using the proposed damage-quantification approach.

Static push-over tests

To carry out static push-over tests, a rod was connected to the test model’s top floor to deliver forces to the model, and a load cell was placed at the end of the rod. (see Fig. 5). Thus, horizontal forces were applied with identical values on all of the floors, based on the mechanics of materials theory. During movement of the shaking table with constant velocity, two laser-type displacement sensors, and the load cell, measured inter-story displacements and applied forces. A low pass filter was utilized to remove noises produced by small vibrations of the structure during movement to raw signals of the forces and displacements. The floor stiffness was estimated using the relationship between the forces and the displacements as shown in Figs. 6 and 7. Figure 8 shows two-dimensional plots of the diagonal terms of the flexibility matrices estimated by the inverse of the structural global stiffness matrix, and changes due to damage.

Modal tests

In this section, random vibration tests were performed under excitation by the shaking table, with random loads and five accelerometers (i.e., model name: PCB 393B12) installed on each floor. The sampling frequency (20 Hz) and anti-aliasing filter (i.e., 10 Hz) were applied to the raw measurement signals. The excitation occurred during 15 min for each measurement. This experiment was repeated five times for a pair of each intact and damaged case to reveal the unreliability of the modal characteristics and the damage severity estimated by proposed approach.

Time-domain measurements and their frequency-domain signals are depicted in Fig. 9. A stochastic subspace identification (SSI) method was used to extract the dynamic properties of the test model (see Fig. 10). The lower four modal information was used to calculate the modal flexibility matrices in the same manner as in the numerical simulations. Figures 11 and Table 4 show dynamic property changes (natural frequency and mode shape) under the described damages. It was shown that all of the natural frequencies clearly reduced due to damages (i.e., 0.43%–3.52%) while there were no remarkable changes of the MAC. The modal flexibility matrices were calculated using natural frequencies and mode shapes, and scaled by the known system-mass matrix. Figure 12 shows two-dimensional plots of the diagonal terms of the modal-flexibility matrices and their changes. Finally, Fig. 13 shows damage-quantification results of the static tests and the modal tests, with one sigma standard-deviation. Damage quantification provided by the static tests and the modal tests, agreed well with the analytically estimated damage ratio. Although the damaged column was originally designed to give a 10% EI reduction with respect to the intact case, there may have been fabrication errors by bolting forces.

Conclusions

This study presented a new damage-quantification approach for shear-wall buildings using diagonal terms of a modal flexibility matrix estimated from dynamic properties. This study was intended to provide a basic concept to solve the damage-quantification problem of more realistic multi-story buildings by investigating an idealized shear-wall building. The general equation for damage quantification of a shear-wall building based on a modal flexibility matrix was determined by analytical studies. Numerical simulations of a 5-story shear-wall-building model with two damage patterns were carried out. Damages were described by decreasing the bending stiffness (EI) of the columns with same severity (10%) for all of the damage cases. It was found that the presented approach accurately investigated the damage severity using the vibration data. Experiments were also conducted on a 5-story shear-wall-building model. To confirm the accuracy of the presented approach, two different experiments were conducted. First, damage severities were estimated by dynamic properties obtained from modal tests. Next, damage ratios were determined from static tests. Finally, damage-quantification results estimated by the static tests and the modal tests were compared. It was found that damage quantification in the two experiments agreed well with the analytically estimated damage ratio. The results emphasized the feasibility of the proposed approach. However, further studies are still required to expand the present concept to provide damage quantification of realistic building structures since this approach was limited to shear-wall buildings.

To this end, first, it is necessary to develop a damage-quantification approach for bending dominant buildings. Next, by combining the two approaches, it is expected that damage-quantification approach for realistic buildings able to be developed.

References

[1] Doebling S W, Farrar C R, Prime M B, Shevitz D W. Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review. Technical Report, 1996

[2] Sohn H, Farrar C R, Hemez F M, Shunk D D, Stinemates D W, Nadler B R. Review of structural health monitoring literature: 1996–2001. Technical Report, 2004

[3] Fan W, Qiao P. Vibration-based damage identification methods: a review and comparative study. Structural Health Monitoring, 2011, 10(1): 83–111

[4] Fritzen C P, Jennewein D, Kiefer T. Damage detection based on model updating methods. Mechanical Systems and Signal Processing, 1998, 12(1): 163–186

[5] Teughels A, Maeck J, De Roeck G. Damage assessment by FE model updating using damage functions. Computers & Structures, 2002, 80(25): 1869–1879

[6] Jaishi B, Ren W X. Damage detection by finite element model updating using modal flexibility residual. Journal of Sound and Vibration, 2006, 290(1–2): 369–387

[7] Moaveni B, He X, Conte J P, Restrepo J I. Damage identification study of a seven-story full-scale building slice tested on the UCSD-NEES shake table. Structural Safety, 2010, 32(5): 347–356

[8] Moaveni B, Stavridis A, Lombaert G, Conte J P, Shing P B. Finite-element model updating for assessment of progressive damage in a 3-Story infilled RC frame. Journal of Structural Engineering, 2013, 139(10): 1665–1674

[9] Fu Y Z, Lu Z R, Liu J K. Damage identification in plates using finite element model updating in time domain. Journal of Sound and Vibration, 2013, 332(26): 7018–7032

[10] Ghee Koh C, Ming See L, Balendra T. Damage detection of buildings: numerical and experimental studies. Journal of Structural Engineering, 1995, 121(8): 1155–1160

[11] Yang J N, Lin S, Huang H, Zhou L. An adaptive extended Kalman filter for structural damage identification. Structural Control and Health Monitoring, 2006, 13(4): 849–867

[12] Rytter A, Kirkegaard P. Vibration Based Inspection Using Neural Networks Structural Damage Assessment Using Advanced Signal Processing Procedures (DAMAS) (Sheffield: University of Sheffield), 1997, 97–108

[13] Sahin M, Shenoi R A. Quantification and localization of damage in beam-like structures by using artificial neural networks with experimental validation. Engineering Structures, 2003, 25(14): 1785–1802

[14] Choo J F, Ha D H, Koh H M. A neural network-based damage detection algorithm using dynamic responses measured in civil structures. The Fifth International Joint Conference on INC, IMS and IDC, 2009

[15] Reynders E, De Roeck G. A local flexibility method for vibration-based damage localization and quantification. Journal of Sound and Vibration, 2010, 329(12): 2367–2383

[16] Koo K Y, Sung S H, Jung H J. Damage quantification of shear-wall buildings using deflections obtained by modal flexibility. Smart Materials and Structures, 2011, 20(4): 045010

[17] Zare Hosseinzadeh A, Bagheri A, Ghodrati Amiri G, Koo K Y. A flexibility-based method via the iterated improved reduction system and the cuckoo optimization algorithm for damage quantification with limited sensors. Smart Materials and Structures, 2014, 23(4): 045019

[18] Sung S H, Koo K Y, Jung H Y, Jung H J. Damage-induced deflection approach for damage localization and quantification of shear buildings: validation on a full-scale shear building. Smart Materials and Structures, 2012, 21(11): 115013

[19] Peeters B, DeRoeck G. Reference-based stochastic subspace identification for output-only modal analysis. Mechanical Systems and Signal Processing, 1999, 13(6): 855–878

[20] Parloo E, Verboven P, Guillaume P, Overmeire M V. Sensitivity-based operational mode shape normalization. Mechanical Systems and Signal Processing, 2002, 16: 757–767

[21] Bernal D. Mass scaling from known mass perturbations. Journal of Engineering Mechanics, 2004, 130(9): 1083–1088

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg