Shock and Vibration

Volume 2015 (2015), Article ID 193136, 12 pages

http://dx.doi.org/10.1155/2015/193136

## Nonlinear Regression Based Health Monitoring of Hysteretic Structures under Seismic Excitation

^{1}School of Astronautics, Northwestern Polytechnical University, Xi’an 710072, China^{2}Department of Mechanical Engineering, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand

Received 19 March 2015; Revised 25 June 2015; Accepted 29 June 2015

Academic Editor: Mickaël Lallart

Copyright © 2015 C. Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a health monitoring method using measured hysteretic responses. Acceleration and infrequently measured displacement are integrated using a multirate Kalman filtering method to generate restoring force-displacement hysteresis loops. A linear/nonlinear regression analysis based two-step method is proposed to identify nonlinear system parameters. First, hysteresis loops are divided into loading/unloading half cycles. Multiple linear regression analysis is applied to separate linear and nonlinear half cycles. Preyielding stiffness and viscous damping coefficient are obtained in this step and used as known parameters in the second step. Then, nonlinear regression analysis is applied to identified nonlinear half cycles to yield nonlinear system parameters and two damage indicators: cumulative plastic deformation and residual deformation. These values are closely related to structural status and repair costs. The feasibility of the method is demonstrated using a simulated shear-type structure with different levels of added measurement noise and a suite of ground motions. The results show that the proposed SHM method effectively and accurately identifies physical system parameters with up to 10% RMS added noise. The resulting damage indicators can robustly and clearly indicate structural condition over different earthquake events.

#### 1. Introduction

Whenever a strong motion earthquake occurs, buildings are expected to remain standing with various degrees of damage. Critical decisions must be made within a short period of time concerning whether the buildings are suitable for continued occupancy. Vibration-based structural health monitoring (SHM) has gained much interest and attention in the civil engineering community in recent years. It is recognised as a powerful tool to identify damage at its earliest stage and to determine the residual useful life of structures, especially for rapid evaluation after a major event [1].

Many vibration-based SHM methods for civil structures are based on identifying changes in modal characteristics [2–5]. However, only low frequency modes related to structural global deformation can be measured accurately, and these modal parameters are insensitive to localized damage in some cases and typically more applicable to structures where vibration response is highly linear [6]. Local diagnostic methods, such as impedance-based [7] and guided-wave based [8] methods, have been developed to improve sensitivity to local failure modes. However, they rely on close proximity to damage location and typically require many sensors distributed throughout a structure, which is currently impractical.

Advanced signal processing tools, such as wavelet analysis [9], empirical mode decomposition, and Hilbert transform [10], are also being proposed. These techniques offer the advantage of determining both the location and time of the damage. However, they cannot directly identify physical system parameters and quantify the level of nonlinear damage due to the absence of a physical system model. Therefore, a number of model-based system identification methods have been presented, including a range of time-domain filters to track time-variant model parameters [11–18]. However, only a few address nonlinear hysteresis and hysteresis-based damage indicators [19].

Hysteretic behaviour plays a critical role not only in seismic performance-based analysis and design [20, 21], but also in capturing the nonlinear yielding and energy absorption associated with damage [22]. A SHM method that captures hysteretic response would give more insight into structural nonlinearity and quantify the level of nonlinear damage.

Structural restoring force-displacement hysteresis loops can be constructed from measured responses [23–25]. Accelerometers are the most commonly used instruments in civil structures, and displacement and velocity have to be obtained from numerical integration. This procedure is fraught with major pitfalls due to the effects of noise, limiting accuracy of the hysteretic loops and damage detection methods based on hysteresis monitoring. However, recent advances in low-rate displacement sensors, such as GPS [26], enable sensor fusion methods that deliver accurate displacement, velocity, and acceleration. Several sensor fusion methods, such as the multirate Kalman filtering method [27], the cubic spline displacement correction method [28], the finite difference FIR filter method [29], and the finite element FIR filter method [30], have been proposed. These methods are expected to suppress measurement noise effectively and yield high quality hysteresis loops.

Structural damage indicators can be further extracted from constructed hysteresis loops. Secant stiffness was first calculated to determine the occurring of degradation and damage in [31]. System effective stiffness was extracted to describe the evolution of the structural stiffness in [32]. Evolution of hysteresis loop shape was considered as a rapid visual indicator of system degrading in [33]. Although these damage indicators can be used to indicate the occurrence of damage, they are largely qualitative. Damage indicators that can quantify structural damage and closely related to structural postevent safety and repair costs are urgently needed.

This research presents a simple and novel health monitoring method for hysteretic structures subjected to seismic excitation. A multirate Kalman filtering technique is applied to estimate high quality displacement and velocity from high-rate sampled acceleration and low-rate sampled displacement data. Hysteresis loops are constructed and a regression analysis based two-step method is proposed to identify preyielding, viscous damping coefficient, yielding displacement and postyielding stiffness, and resulting nonlinear damage indicators. The feasibility and robustness of the proposed method are illustrated for different noise levels over a suite of earthquake events.

#### 2. Construction of Hysteresis Loops

Toussi and Yao [23] first presented the idea of generating system hysteresis loops from recorded seismic response data. For this proof-of-concept study, it will be assumed that the structure in question can be adequately modelled as a single-degree-freedom (SDOF) system for simplicity and clarity. This situation is also true if the test structure responds primarily in a single mode and can be defined:where , , and are displacement, velocity, and acceleration related to the ground; is the total restoring force; is ground acceleration; and is the mass.

Rewriting (1) and including viscous damping restoring force yieldwhere is absolute acceleration; is viscous damping coefficient; and is stiffness restoring force. Assuming to be known* a priori* and to be measured, is consequently obtained. Dynamic displacement and velocity can be obtained from measured sensor data by integration and correction. Thus, hysteresis loops can be constructed by graphing the restoring force versus displacement with time as an implicit parameter.

Direct integration of measured acceleration to obtain velocity and displacement is sensitive to noise and can cause significant distortion of estimated displacement [34]. Data fusion of high-rate acceleration and low-rate displacement measurements can effectively suppress noise and yield good estimates of velocity and displacement. If high-rate acceleration and low-rate displacement measurements are available, estimation of displacement and velocity from the measurements can be modelled by a discrete dynamic system:where (3) is the system equation and (4) is the observation equation; is the measured acceleration and is the measured displacement. The state vector comprises the displacement and the velocity ; that is,Note that the subindex indicates a progression in time. is a matrix describing the system dynamics, is input matrix, and is design matrix, defined aswhere is the acceleration sampling interval. In (3) and (4), is a vector of acceleration measurement noise with distribution and is the vector of displacement measurement noise with distribution . Both are assumed to be Gaussian white noise processes with covariance and . Thus, and are given bywhere is the displacement sampling interval.

With (3) to (7), a discrete time multirate Kalman filter can be used to estimate the displacement and velocity at each acceleration sampling instant [27, 35].

#### 3. SHM Based on Regression Analysis of Hysteresis Loops

Many civil structures exhibit hysteresis when subject to severe cyclic loading. Figure 1 shows general hysteretic loops without considering system stiffness or strength degradation. A hysteretic cycle consists of a loading and an unloading half cycle. Any loading/unloading half cycle can be further divided into two nearly linear regimes: elastic and plastic, governed by , the preyielding stiffness, and , the postyielding stiffness, respectively. The elastic-plastic transition is generally smooth and gradual, but small. Omitting the transition process, the original half cycle can be represented by two line segments with different slopes, as shown in Figure 1, to capture the essential system dynamics.