Multi-Phase Linear Regression

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4

Multi-Phase Linear Regression: A Novel Method for the Identification of BaseIsolated Buildings Using Seismic Response Data Chao Xu1, J. Geoffrey Chase2, Geoffrey W. Rodgers2, Cong Zhou2 School of Astronautics, Northwestern Polytechnical University, 710072 Xi’an, China 2 Department of civil and Mechanical Engineering, University of Canterbury, Private bag 4800, Christchurch, New Zealand Email: [email protected], [email protected], [email protected]; [email protected] 1

ABSTRACT: Base-isolation is an increasingly applied earthquake-resistant design technique in highly seismic areas. The in-situ identification of physical parameters of the isolation layer and monitoring of the isolation performance has become a critical issue. In this paper, a simplified system identification method is proposed for obtaining insight into the structural property of based-isolated buildings. A bilinear hysteresis model is chosen for modeling isolation bearings. Interstory restoring forcedeformation hysteresis loops are first generated from recorded seismic response. A two-step regression analysis based method is proposed to identify the linear/nonlinear physical parameters of the isolation layer. In the first step, hysteresis loops are split into many loading/unloading half-cycles according to zero velocity points. Multiple linear regression is applied to all half-cycles to yield equivalent linear system stiffness and damping. Both parameters clearly vary with half-cycle displacement increment. A threshold is used to separate linear and nonlinear half-cycles. For linear half cycles, linear regression analysis estimates viscous damping coefficient. For nonlinear half cycles, a multi-phase linear regression analysis method is applied to estimate nonlinear model parameters. A proof-of-concept numerical investigation shows the feasibility of the method. The method is effective and robust even of 5% noise level. KEY WORDS: Seismic isolation; System identification; Nonlinear regression; Hysteresis 1

INTRODUCTION

of a Masing criterion to transform a multi-valued hysteretic restoring force function into a single-valued function so that the ordinary optimization methods can be applied. The method was then extended to other base-isolated civil structures [6-9]. Furukawa et al. [10] proposed a least-squares output-error minimization method to identify a base-isolated building in Kobe City affected by the 1995 Hyogoken-Nambu earthquake. The isolation system was identified based on three models: a linear equivalent model, a bilinear model and a trilinear model. The results show that model parameters can be reasonably estimated and the tri-linear model best fit the recorded response. Ahn and Chen [11] proposed a nonlinear model-based system identification method for a three-span continuous base-isolated bridge. They used the MengottoPinto model to model hysteresis behavior of the lead-rubber bearings. Model parameters were pulled out by a two-stage optimization algorithm. Xie and Mita [12] presented a method to estimate the restoring force of an isolation layer using component mode synthesis (CMS). The amplitude-dependent equivalent system stiffness and damping coefficients were identified to characterize the nonlinear isolation system. Oliveto et al. [13] developed a time domain nonlinear system identification procedure to determine the physical parameters of the hybrid seismic isolation system of a base-isolated building. The method used a bilinear hysteretic model and a constant coulomb friction model to model the high damping rubber bearings and low friction sliding bearing respectively. Nonlinear least-squares optimization method was used to obtain modal parameters. The Covariance Matrix AdaptationEvolution Strategy (CMS-ES) algorithm was proposed for identification of nonlinear base isolation system from earthquake records in [14]. However, all of these methods are

Seismic isolation utilizes flexible elements, such as rubber bearing or sliding or rolling mechanisms, often coupled with energy absorbing dampers, to reduce structural response. The basic concept is to shift the fundamental natural period of a building to a lower value than the dominant frequency component of ground motion. Base isolation has been one of the most popular and powerful means of seismic protection. However, isolation performance is directly dependent on the physical behavior of the isolation devices. If isolation devices degrade or fail due to aging, temperature cycles or exceeding their design capacity during an extreme event, structural safety is no longer guaranteed. Therefore, in-situ identification and monitoring of the physical parameters of isolation system is an urgent need. A number of researchers have investigated the identification of base-isolated structures using seismic records. Some of the earliest works were presented after the 1994 Northbridge earthquake [1-4]. However, these studies are based on timevariant equivalent linear system assumptions and only structural modal parameters are identified to characterize the isolation performance. Identification of the nonlinear physical parameters of the isolation system is more attractive because these parameters are used to specify isolators in design and provide valuable parameters for future response prediction. In addition, identifying these parameters would also provide insight on the nonlinear cycles and lifetime of the devices. Limited works addressed nonlinear physical parameter identification of base-isolated structures are reviewed here. Tan and Huang [5] proposed an iterative trial-to-error optimization procedure to identify nonlinear physical parameters of isolators. The essence of the study is application 1

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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

very complex and depend on many user control parameters to gain the best accuracy. In this paper, a novel and simplified method for in-situ identification of physical parameters of nonlinear baseisolated buildings is developed and validated. Nonlinear isolators are modeled by a bilinear hysteretic model. A twostep identification procedure based on linear/nonlinear regression analysis techniques is proposed. The proposed method is computationally simple and can be generalized to several forms of nonlinearity of the isolation system. The method is demonstrated numerically over different noise. 2 2.1

2.2

Figure 1 illustrates the general hysteretic response that might be of a structure subjected to a strong-motion earthquake. It can be seen that the total restoring force is path dependent. However, within a time segment where the velocity holds the same sign, the restoring force is a singlevalue function of the deformation. Hence, the whole response history can be sliced into many sub-cycles and sub-half cycles according to the points where the velocity is zero. For example, the data points between x4 and x6 formulate a loading half cycle and the data points between x6 and x8 formulate an unloading half cycle. Within each half cycle, the total restoring force is either monotonically increased or decreased with displacement, and can be approximated by regression curves as shown in Figure 1.

METHOD AND PROCEDURE Structural model

A base-isolated building can be separated into two structural systems: a superstructure and a base isolation system. The isolation system may consist of rubber isolators and additional viscous dampers to absorb energy. Since the response of the upper superstructure can be effectively reduced to an acceptable level by the isolation system, it is reasonable to make a rigid body assumption for the superstructure. Therefore, the base-isolated structure can be described by a simple single-degree-freedom dynamic model: mx cx f ( x, x )

mxg

(1)

where x , x and x are response acceleration, velocity and displacement relative to the fixed base; xg is the ground

Figure 1. General structural hysteretic loops

acceleration; c is the damping coefficient of dampers; f is the restoring force of the isolators, m is the mass of superstructure. With a priori known structural mass, Equation (1) can also be written as

m( xg x) cx f ( x, x )

Partition of hysteretic response histories

Generally, during an earthquake event, some half cycles behave linearly and the others nonlinearly. For linear cases, a linear regression model is applied. For nonlinear cases, a nonlinear regression model is needed. Therefore, the first step is to separate the linear and nonlinear half cycles.

(2)

2.3 According to experimental observation, the nonlinear behavior of isolators can be generally described by a bilinear hysteretic model [15]. The model parameters, initial stiffness ke, hardening stiffness kp and yielding deformation dy, clearly characterized nonlinear properties of the isolation system. The left side of Equation (2) can be obtained from measured accelerations. The right side of Equation (2) is an unknown function of state variables. However, if the state variables, displacement and velocity are known, the model parameters of the right side of Equation (2) can be estimated through regression analysis. It is indeed a problem of regression modeling, where the left side of Equation (2) is considered as a dependent variable and the state variable, displacement and velocity are considered as independent variables. It should be noted that the bilinear restoring force function is nonlinear, path dependent and not a single-valued function of state variables due to hysteresis. Standard regression analysis cannot be applied directly. To identify the model parameters, a two-step regression-analysis based identification is proposed.

The first step: separation of linear and nonlinear half cycles

Since any one of these half cycles is either linear or nonlinear, a linear regression analysis is implemented to all half cycles first. It is equal to make an equivalent linear system assumption to each half cycle. Thus, the regression model can be written:

m( xg x)

cl x k l x

(3)

where kl is the effective linear system stiffness and cl is the effective linear system damping. Equation (3) holds for every measurement time step. Let us consider every time step as a sample. When independent variables ( x and x ) and the dependent variable ( m( xg x) ) at any time step within a half cycle are obtained, the model parameters can be estimated from standard multiple linear regression procedures [16]. The identified effective stiffness varies over different subhalf cycles. Varying kl is a significant indicator of the nature of the dynamic system. It is reasonable that the hysteresis curve is linear when the half cycle displacement increment 2

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method can be used to solve the regression problem which yields the estimates of D and xb simultaneously [21].

' d is small and nonlinear when ' d is larger than the structural yield deformation. A rapid drop in kl at large displacement increment can be viewed as a good indicator of inelastic behavior during that half cycle. The plot of kl versus ' d can thus be used to indicate the potential inelastic subhalf cycle. In addition, the estimated effective linear damping coefficient, cl is the measure of system energy dissipation. System energy dissipation capacity will dramatically increase due to the added hysteretic damping when nonlinearity is present. The analysis of cl versus ' d can also be used as another indicator of inelastic half cycles. Thus, linear and nonlinear half cycles can be easily separated using a threshold.

2.4

According to the bilinear assumption, the estimate of E 21 is the hardening stiffness, kp, and the estimate of E11 gives additional estimates of initial stiffness, ke. The estimated breaking point is the turning point of elastic and inelastic response within the nonlinear half cycle. It is related to the yield displacement of the isolation system. For a loading nonlinear half cycle i, the yield displacement can be estimated: d yi

The second step: parameter identification

For all identified linear half cycles, the multiple linear regression process yields many estimates of viscous damping coefficient, c. The statistical mean of c will be considered as the ‘actual’ viscous damping coefficient of the target isolation system and be used as a known parameter at the next identification step. Therefore, Equation (2) can now be written:

m( xg x) cx

f ( x, x )

E 10 E 11 x ® ¯E 20 E 21 x

x1 d x d xb xb d x d x n

d yi

E 20 E 21 xb

xn xbi 2

(8)

where xn is the displacement at the instant of most recent unloading reversal. Through the overall method proposed, many estimates of system initial stiffness, hardening stiffness and yield displacement are obtained. The last output of each physical parameter is derived by statistically averaging these estimated values over the time history.

(4)

3

PROOF-OF-CONCEPT STUDY

The proof-of-concept study considered is a three-story reinforced concrete frame structure located in Kobe City, which was studied in [10]. The base isolation system consists of eight high damping rubber isolators. There are acceleration sensors at the foundation, first floor and top floor. The building is modeled as a single-degree-freedom system. The upper structure is considered as a rigid body with a mass of 817.8 ton. The base isolation system is modeled with a bilinear model with parameters: initial stiffness ke=70390kN/m, hardening stiffness kp=12500 kN/m and yield displacement dy=0.005m. A viscous damping coefficient c=600kN.s/m is considered to simulates structural viscous damping of the base isolation system and upper structure. The considered ground motion is a record of the 1995 Kobe earthquake with peak ground acceleration of 0.509g. Simulated structural response accelerations are calculated by solving the SDOF dynamic equation using the Newmark method. To implement the proposed identification method, structural displacement and velocity responses must be first estimated from acceleration response through integration. Numerical integration is sensitive to noise and subject to drift. However, the procedure can be improved if additional lowsampled displacement measurements, such as by GPS sensors, are available. The multi-rate Kalman filtering method for displacement and velocity reconstruction proposed [23] was applied to give displacement and velocity estimates. In this case study, the low-solution-measured displacement was taken at 20Hz and Acceleration data was taken at 200Hz. The effect

(5)

where D ^E10 E11 E 20 E 21` is the vector of regression coefficients; xb is the unknown breaking point which satisfies the linear constrain equations:

E10 E11 xb

(7)

where x1 is the displacement at the instant of most recent loading reversal. For an unloading nonlinear half cycle I, the yield displacement can be estimated:

For nonlinear half cycles, the restoring force at right side is a nonlinear function of displacement. With a bilinear assumption, the nonlinear restoring force can consist of two segments: the line segment with a slope of ke when displacement is below a critical value and the line segment with another slope kp when above. Therefore, data points in a nonlinear half cycle must be divided into two segments, and a different linearly parameterized polynomial identified via regression analysis for each segment. . The difficulty is associated with the breakpoint between two segments is unknown. This regression analysis is a special nonlinear regression problem, called multi-phase linear regression with unknown transition points. The problem has attracted much attention in mathematics and several solutions have been presented [17-20]. In the recent years, the multiphase linear regression technique has been applied in some engineering fields [21-22]. Here, its application is proposed for identifying nonlinear physical parameters of isolation system. For the identification of nonlinear half cycles, a two-phase linear regression is defined: E ( y x)

xbi x1 2

(6)

Equations (5) and (6) formulate a two-phase linear regression problem. Constrained nonlinear least squares 3

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of noise is investigated at different added levels. A separate white noise corresponding to different levels of signal-noiseratios (SNR) was added to the simulated noise-free acceleration and displacement measurements, respectively, to mimic a realistic situation over a range of possible sensor performance. All these responses and prior known mass data were used as inputs to the identification procedure.

parameters. In particular, the identified initial stiffness mean error is 7.88%, the identified viscous damping mean error is 0.2% and the identified yield displacement mean error is 8.0%. The identified initial stiffness mean error is a little larger, 11.8%. This large error is because the very small elastic deformation experienced by the isolation system due to the very low yield displacement.

4

Table 1. Results of identified model parameters without noise

4.1

RESULTS AND DISCUSSION

Model parameters Initial stiffness [kN/m] Hardening stiffness[kN/m] Yield Displacement [mm] Viscous damping [kN.s/m]

Parameter identification without measurement noise

Figure 2 shows reconstructed hysteresis loops from recorded seismic response. It can be seen that the isolation system experience many nonlinear cycles during the earthquake. The elastic displacement is short because a very small yielding displacement of 5mm. Loading/unloading half cycles are identified according to zero velocity points from reconstructed hysteresis loops.

Mean 64842 13986 4.60 599

COV 0.18 0.36 0.28 1.08

True 70390 12500 5.00 600

Figure 2. Identified equivalent system parameters (a) stiffness (b) damping Figure 3 shows the identified equivalent linear system stiffness and damping parameters for each half cycle. Both are plotted versus half cycle displacement increment. It can be seen clearly that the equivalent linear system stiffness decreases and damping increases at larger half cycle displacement, which indicates the isolation system softens nonlinearly at larger displacement. Thus, each plot in Figure 3 can be divided into two regimes: one regime where both parameters show a nearly constant trend with displacement and the other regime where both parameters show a nonlinear variation with displacement. The former characterizes linear behavior, while the latter captures nonlinear behavior. From Figure 2, a threshold can be readily determined to separate nonlinear and linear regimes. Here, a threshold of 1.2mm was chosen. From identified linear half cycles, the estimate of viscous damping coefficient is obtained and used in the second identification step. Multi-phase linear regression is then applied to the identified nonlinear half cycles to estimate the other model parameters. The final identified model parameters are listed in Table 1, including the true simulated values. It can be seen from Table 1 that the proposed identification method gives good estimates of system initial model

Figure 3. Identified equivalent system parameters (a) stiffness (b) damping 4.2

Effect of threshold value chosen

The effect of the choice of threshold is investigated by varying its value. The identified model parameters for different threshold are listed in Table 2. It can be seen from Table 2 that the threshold value has an important effect on the identification results. If one choose a larger threshold, which means only the half cycles with very large displacement are considered nonlinear, the identification accuracy of hardening stiffness increases, but the accuracy of the other parameters decreases. If one choose a smaller threshold which means some linear half cycles will be consider nonlinear, the identification accuracy of hardening stiffness decreases obviously. However, in this situation, the identified viscous damping, initial stiffness and yield displacement suffer little change. In practice, several trials are needed to determine final reasonable threshold. Equally, one could identify each value using different thresholds to 4

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Table 3 shows that a lower estimate of structural mass will cause lower estimates of model parameters and the reverse is also true. The result occurs because the mass is proportional to the total restoring force. The variations of initial stiffness, hardening stiffness and viscous damping are nearly proportional to the variation of structural mass. It is noted that the coefficient of variation and the estimate of yield displacement seem unaffected by the variation of structural mass.

maximize accuracy given computational simplicity once half cycles are identified. In addition, one can also use a band of threshold instead of a single threshold as shown in Figure 3 to improve identification accuracy. Table 2. Results of identified model parameters without noise Model parameters (a) threshold = 0.8 mm, noise-free Initial stiffness [kN/m] Hardening stiffness[kN/m] Yield Displacement [mm] Viscous damping [kN.s/m] (b) threshold = 1.0 mm, noise-free Initial stiffness [kN/m] Hardening stiffness[kN/m] Yield Displacement [mm] Viscous damping [kN.s/m] (c) threshold = 1.2 mm, noise-free Initial stiffness [kN/m] Hardening stiffness[kN/m] Yield Displacement [mm] Viscous damping [kN.s/m] 4.3

Mean

COV

True

66206 19133 4.40 596

0.16 0.90 0.31 1.18

70390 12500 5.00 600

64902 14264 4.70 600

0.18 0.39 0.28 1.10

70390 12500 5.00 600

64842 13986 4.60 599

0.18 0.36 0.28 1.08

70390 12500 5.00 600

4.4

The parameter identification results from simulated responses with different levels of noise are listed in Table 4. At each noise level, the proposed identification procedure was run 100 times to give final statistical results. It can be seen from Table 4 that the parameter identification errors and coefficient of variation show a little increase as the noise level increases. However, the proposed identification method can give robust parameter estimates of initial stiffness, hardening stiffness and viscous damping even at 5% noise level. The identified stiffness damping parameter mean errors are all within 17% at this noise level. Table 4 Results of identified model parameters over different levels of noise

Effect of the estimate of structural mass

In practice, real structural mass cannot be determined exactly. The proposed identification method is tested over different mass inputs. The identification results are listed in Table 3. The threshold of 1.2mm was used here.

Model parameters (a) noise-free Initial stiffness [kN/m] Hardening stiffness[kN/m] Yield Displacement [mm] Viscous damping [kN.s/m] (b) 1% RMS noise Initial stiffness [kN/m] Hardening stiffness[kN/m] Yield Displacement [mm] Viscous damping [kN.s/m] (c) 5% RMS noise Initial stiffness [kN/m] Hardening stiffness[kN/m] Yield Displacement [mm] Viscous damping [kN.s/m] (d) 10% RMS noise Initial stiffness [kN/m] Hardening stiffness[kN/m] Yield Displacement [mm] Viscous damping [kN.s/m] (e) 20% RMS noise Initial stiffness [kN/m] Hardening stiffness[kN/m] Yield Displacement [mm] Viscous damping [kN.s/m]

Table 3. Results of identified model parameters with different mass estimates Model parameters Mean (a) mass = 80% of real mass, noise-free Initial stiffness [kN/m] 51874 Hardening stiffness[kN/m] 11189 Yield Displacement [mm] 4.60 Viscous damping [kN.s/m] 479 (b) mass = 90% of real mass, noise-free Initial stiffness [kN/m] 58358 Hardening stiffness[kN/m] 12587 Yield Displacement [mm] 4.60 Viscous damping [kN.s/m] 539 (c) mass = 100% of real mass, noise-free Initial stiffness [kN/m] 64842 Hardening stiffness[kN/m] 13986 Yield Displacement [mm] 4.60 Viscous damping [kN.s/m] 599 (d) mass = 110% of real mass, noise-free Initial stiffness [kN/m] 71327 Hardening stiffness[kN/m] 15385 Yield Displacement [mm] 4.60 Viscous damping [kN.s/m] 659 (e) mass = 120% of real mass, noise-free Initial stiffness [kN/m] 77811 Hardening stiffness[kN/m] 16783 Yield Displacement [mm] 4.60 Viscous damping [kN.s/m] 719

COV

True

0.18 0.36 0.29 1.08

70390 12500 5.00 600

0.18 0.36 0.29 1.08

70390 12500 5.00 600

0.18 0.36 0.28 1.08

70390 12500 5.00 600

0.18 0.36 0.29 1.08

70390 12500 5.00 600

0.18 0.36 0.29 1.08

70390 12500 5.00 600

Effect of noise level

5

Mean

COV

True

64842 13986 4.60 599

0.00 0.00 0.00 0.00

70390 12500 5.00 600

59977 14517 5.90 667

0.11 0.12 0.18 0.30

70390 12500 5.00 600

60683 14610 6.60 574

0.14 0.20 0.36 0.70

70390 12500 5.00 600

53698 17888 7.80 821

0.20 0.33 0.55 0.80

70390 12500 5.00 600

48030 20433 9.10 980

0.26 0.40 0.76 0.93

70390 12500 5.00 600

CONCLUSION

This paper develops a simple method for identification of nonlinear base isolation system. The method is based on a two-step regression analysis procedure. System parameters are identified over different half cycle intervals. Multiple-phase linear regression is used to identify nonlinear half cycles. 5 2603



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Feasibility of the proposed method has been demonstrated via a numerical example of a base-isolated structure subjected to 1995 Kobe earthquake excitation. The linear and nonlinear physical parameters of the isolation system can be extracted directly. The method is robust to noises even at the noise level of 5%. Overall, the proposed identification method is simple, direct and robust. The identification procedure is actually performed time segment by time segment, and requires no operator input except a threshold. This provide the potential application of the method in real-time or near real-time. Although the concept is proven focusing on structural systems that display bilinear hysteresis response, it can be easily extended to trilinear or degrading bilinear hysteretic systems. ACKNOWLEDGMENTS The authors are grateful to CSC and Northwestern Polytechnical University for the support of the project. REFERENCES [1]

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