A dynamic stiffness-based framework for harmonic input estimation and response reconstruction considering damage

doi:10.1007/s11709-022-0805-5

Front. Struct. Civ. Eng.

2022, Vol. 16

Issue (4) : 448-460 https://doi.org/10.1007/s11709-022-0805-5

RESEARCH ARTICLE

A dynamic stiffness-based framework for harmonic input estimation and response reconstruction considering damage

Yixian LI¹, Limin SUN^2,³(

), Wang ZHU⁴, Wei ZHANG⁵

¹. Department of Bridge Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, China
². State Key Laboratory of Disaster Reduction in Civil Engineering, Department of Bridge Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, China
³. Shanghai Qizhi Institute, Shanghai 200092, China
⁴. Sichuan Highway Planning, Survey, Design, and Research Institute Ltd., Chengdu 610000, China
⁵. Fujian Key Laboratory of Green Building Technology, Fujian Academy of Building Research, Fuzhou 350028, China

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Abstract

In structural health monitoring (SHM), the measurement is point-wise but structures are continuous. Thus, input estimation has become a hot research subject with which the full-field structural response can be calculated with a finite element model (FEM). This paper proposes a framework based on the dynamic stiffness theory, to estimate harmonic input, reconstruct responses, and to localize damages from seriously deficient measurements. To begin, Fourier transform converts the dynamic equilibrium equation to an equivalent static one in the frequency domain, which is under-determined since the dimension of measurement vector is far less than the FEM-node number. The principal component analysis has been adopted to “compress” the under-determined equation, and formed an over-determined equation to estimate the unknown input. Then, inverse Fourier transform converts the estimated input in the frequency domain to the time domain. Applying this to the FEM can reconstruct the target responses. If a structure is damaged, the estimated nodal force can localize the damage. To improve the damage-detection accuracy, a multi-measurement-based indicator has been proposed. Numerical simulations have validated that the proposed framework can capably estimate input and reconstruct multi-types of full-field responses, and the damage indicator can localize minor damages even with the existence of noise.

Keywords dynamic stiffness principal component analysis response reconstruction damage localization under-determined equation

Corresponding Author(s): Limin SUN

Just Accepted Date: 21 March 2022 Online First Date: 04 July 2022 Issue Date: 09 August 2022

Cite this article:

Yixian LI,Limin SUN,Wang ZHU, et al. A dynamic stiffness-based framework for harmonic input estimation and response reconstruction considering damage[J]. Front. Struct. Civ. Eng., 2022, 16(4): 448-460.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-022-0805-5
https://academic.hep.com.cn/fsce/EN/Y2022/V16/I4/448

Fig.1 The concept of the static equivalent damage load.

Fig.2 Layout of the FEM and sensor arrangement.

Fig.3 The first two modal shapes (f₁ = 9.458 Hz,

ξ

₁ = 1%; f₂ = 20.202 Hz,

ξ

₂ = 2%).

Fig.4 The estimated deflection at the bridge center.

Fig.5 Time and spatial distribution of equivalent nodal force (the real external harmonic load is at node No. 18).

Fig.6 Equivalent nodal force at 2.52 s.

Fig.7 Frequency spectrum of the nodal force at node No. 18.

Fig.8 Deflection of the bridge.

Fig.9 Time and spatial distribution of equivalent nodal force.

Fig.10 Frequency spectrum of the nodal forces.

Fig.11 The estimated deflection at node No. 23 (bridge center).

Fig.12 Deflection of the bridge at 6 s.

Fig.13 Strain response at the bridge center.

Fig.14 Strain distribution at 6 s.

Fig.15 The estimated nodal force with measurement noise.

Fig.16 Frequency spectrum of the nodal force at the bridge center.

Fig.17 Dynamic deflection history at the bridge center.

Fig.18 The loading and damage conditions.

Fig.19 The maps of (a)

p ¯

and (b)

Δ

Fig.20 The normalized

Δ

p under different input frequencies. (a) 7, 9, and 11 Hz; (b) 15, 20, and 25 Hz.

Fig.21 The normalized ?p with varied damage percentage.

Tab.1 The damage and loading conditions

Fig.22 The normalized

Δ

p of the three loading conditions.

Fig.23 The averaged

Δ

Fig.24 The normalized-averaged

Δ

p with noise.

Fig.25 The averaged normalized

Δ

p with noise.

1	C Q Yang, D Yang, Y He, Z S Wu, Y F Xia, Y F Zhang. Moving load identification of small and medium-sized bridges based on distributed optical fiber sensing. International Journal of Structural Stability and Dynamics, 2016, 16( 4): 1640021 https://doi.org/10.1142/S0219455416400216
2	M Lydon, S E Taylor, D Robinson, A Mufti, E J O Brien. Recent developments in bridge weigh in motion (B-WIM). Journal of Civil Structural Health Monitoring, 2016, 6( 1): 69–81 https://doi.org/10.1007/s13349-015-0119-6
3	Y Yu, C S Cai, L Deng. State-of-the-art review on bridge weigh-in-motion technology. Advances in Structural Engineering, 2016, 19( 9): 1514–1530 https://doi.org/10.1177/1369433216655922
4	T Bao, S K Babanajad, T Taylor, F Ansari. Generalized method and monitoring technique for shear-strain-based bridge weigh-in-motion. Journal of Bridge Engineering, 2016, 21( 1): 04015029
5	A Lansdell, W Song, B Dixon. Development and testing of a bridge weigh-in-motion method considering nonconstant vehicle speed. Engineering Structures, 2017, 152 : 709–726 https://doi.org/10.1016/j.engstruct.2017.09.044
6	H Zhao, N Uddin, E J O’Brien, X Shao, P Zhu. Identification of vehicular axle weights with a bridge weigh-in-motion system considering transverse distribution of wheel loads. Journal of Bridge Engineering, 2014, 19( 3): 04013008 https://doi.org/10.1061/(ASCE)BE.1943-5592.0000533
7	T H T Chan, S S Law, T H Yung, X R Yuan. An interpretive method for moving force identification. Journal of Sound and Vibration, 1999, 219( 3): 503–524
8	L Yu, T H T Chan. Moving force identification based on the frequency–time domain method. Journal of Sound and Vibration, 2003, 261( 2): 329–349 https://doi.org/10.1016/S0022-460X(02)00991-4
9	S S Law, T H Chan, Q Zeng. Moving force identification: A time domain method. Journal of Sound and Vibration, 1997, 201( 1): 1–22 https://doi.org/10.1006/jsvi.1996.0774
10	A K Amiri, C Bucher. A procedure for in situ wind load reconstruction from structural response only based on field testing data. Journal of Wind Engineering and Industrial Aerodynamics, 2017, 167 : 75–86 https://doi.org/10.1016/j.jweia.2017.04.009
11	A Kazemi Amiri, C Bucher. Derivation of a new parametric impulse response matrix utilized for nodal wind load identification by response measurement. Journal of Sound and Vibration, 2015, 344 : 101–113 https://doi.org/10.1016/j.jsv.2014.12.027
12	S S Law, J Q Bu, X Q Zhu. Time-varying wind load identification from structural responses. Engineering Structures, 2005, 27( 10): 1586–1598 https://doi.org/10.1016/j.engstruct.2005.05.007
13	J S Hwang, A Kareem, H Kim. Wind load identification using wind tunnel test data by inverse analysis. Journal of Wind Engineering and Industrial Aerodynamics, 2011, 99( 1): 18–26 https://doi.org/10.1016/j.jweia.2010.10.004
14	L Zhi, Q S Li, M Fang, J Yi. Identification of wind loads on supertall buildings using Kalman filtering-based inverse method. Journal of Structural Engineering, 2017, 143( 4): 06016004 https://doi.org/10.1061/(ASCE)ST.1943-541X.0001691
15	L Zhi, M Fang, Q S Li. Estimation of wind loads on a tall building by an inverse method. Structural Control and Health Monitoring, 2017, 24( 4): e1908 https://doi.org/10.1002/stc.1908
16	Y Li, H Huang, W Zhang, L Sun. Structural full-field responses reconstruction by the SVD and pseudo-inverse operator-estimated force with two-degree multi-scale models. Engineering Structures, 2021, 249 : 112986 https://doi.org/10.1016/j.engstruct.2021.112986
17	S Gillijns, B de Moor. Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough. Automatica, 2007, 43( 5): 934–937 https://doi.org/10.1016/j.automatica.2006.11.016
18	H Fang, R A de Callafon. On the asymptotic stability of minimum-variance unbiased input and state estimation. Automatica, 2012, 48( 12): 3183–3186 https://doi.org/10.1016/j.automatica.2012.08.039
19	C S Hsieh. Extension of unbiased minimum-variance input and state estimation for systems with unknown inputs. Automatica, 2009, 45( 9): 2149–2153 https://doi.org/10.1016/j.automatica.2009.05.004
20	S Pan, D Xiao, S Xing, S S Law, P Du, Y Li. A general extended Kalman filter for simultaneous estimation of system and unknown inputs. Engineering Structures, 2016, 109 : 85–98 https://doi.org/10.1016/j.engstruct.2015.11.014
21	S Pan, H Su, H Wang, J Chu. The study of joint input and state estimation with Kalman filtering. Transactions of the Institute of Measurement and Control, 2011, 33( 8): 901–918
22	S Z Yong, M Zhu, E Frazzoli. A unified filter for simultaneous input and state estimation of linear discrete-time stochastic systems. Automatica, 2016, 63 : 321–329 https://doi.org/10.1016/j.automatica.2015.10.040
23	C S Hsieh, F C Chen. Optimal solution of the two-stage Kalman estimator. IEEE Transactions on Automatic Control, 1995, 44( 1): 194–199
24	Y Niu, C P Fritzen, H Jung, I Buethe, Y Q Ni, Y W Wang. Online simultaneous reconstruction of wind load and structural responses-theory and application to Canton Tower. Computer-Aided Civil and Infrastructure Engineering, 2015, 30( 8): 666–681 https://doi.org/10.1111/mice.12134
25	T S Nord, E M Lourens, O Øiseth, A Metrikine. Model-based force and state estimation in experimental ice-induced vibrations by means of Kalman filtering. Cold Regions Science and Technology, 2015, 111 : 13–26 https://doi.org/10.1016/j.coldregions.2014.12.003
26	C S Hsieh, F C Chen. Robust two-stage Kalman filters for systems with unknown. IEEE Transactions on Automatic Control, 2000, 45( 12): 2374–2378
27	C D Zhang, Y L Xu. Structural damage identification via response reconstruction under unknown excitation. Structural Control and Health Monitoring, 2017, 24( 8): e1953 https://doi.org/10.1002/stc.1953
28	L Zhi, Q S Li, M Fang. Identification of wind loads and estimation of structural responses of super-tall buildings by an inverse method. Computer-Aided Civil and Infrastructure Engineering, 2016, 31( 12): 966–982 https://doi.org/10.1111/mice.12241
29	W Zhang, L M Sun, S W Sun. Bridge-deflection estimation through inclinometer data considering structural damages. Journal of Bridge Engineering, 2017, 22( 2): 04016117 https://doi.org/10.1061/(ASCE)BE.1943-5592.0000979
30	L M Sun, W Zhang, S Nagarajaiah. Bridge real-time damage identification method using inclination and strain measurements in the presence of temperature variation. Journal of Bridge Engineering, 2019, 24( 2): 04018111 https://doi.org/10.1061/(ASCE)BE.1943-5592.0001325
31	Y Li, L Sun, W Zhang, S Nagarajaiah. Bridge damage detection from the equivalent damage load by multitype measurements. Structural Control and Health Monitoring, 2021, 28( 5): e2709 https://doi.org/10.1002/stc.2709
32	D Bernal. Load vectors for damage localization. Journal of Engineering Mechanics, 2002, 128( 1): 7–14 https://doi.org/10.1061/(ASCE)0733-9399(2002)128:1(7
33	D Bernal. Damage localization from the null space of changes in the transfer matrix. AIAA Journal, 2007, 45( 2): 374–381 https://doi.org/10.2514/1.25037
34	S Wold, K Esbensen, P Geladi. Principal component analysis. Chemometrics and Intelligent Laboratory Systems, 1987, 2( 1–3): 37–52 https://doi.org/10.1016/0169-7439(87)80084-9
35	I Jolliffe. Principal Component Analysis. 2nd ed. New York: Springer, 2010

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