Optimal intensity measures for probabilistic seismic demand models of a cable-stayed bridge based on generalized linear regression models

Soil Dynamics and Earthquake Engineering 131 (2020) 106024

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Optimal intensity measures for probabilistic seismic demand models of a cable-stayed bridge based on generalized linear regression models Junjun Guo a, b, M. Shahria Alam b, *, Jingquan Wang c, Shuai Li c, b, Wancheng Yuan a, ** a

State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, 1239, Siping Road, Shanghai, 200092, PR China School of Engineering, The University of British Columbia, Kelowna, BC, V1V1V7, Canada c School of Civil Engineering, Southeast University, Nanjing, Jiangsu, 210096, PR China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Optimal intensity measures Probabilistic seismic demand models Generalized linear regression models Lasso regression Cable-stayed bridge

Seismic intensity measures (IMs) play an important role in predicting the seismic responses of structures sub­ jected to strong earthquakes. This paper proposes a general procedure to identify the optimal IMs for a long span cable-stayed bridge subjected to far-field and near-fault ground motions based on generalized linear regression models. Firstly, the generalized linear regression models, such as ordinary least squares (OLS), ridge regression and Lasso regression are presented. Secondly, the three dimensional numerical model of the bridge is generated in the OpenSees platform. Thirdly, 22 IMs are considered, and 160 ground motions from four site conditions are selected to excite the bridge in longitudinal and transverse directions separately. Then, the optimal IMs are determined by Lasso regression, which is an extended version of OLS, and the quadratic polynomial regression model is adopted to establish the probabilistic seismic demand models of the bridge. The numerical results reveal that peak ground velocity (PGV) can be selected as the optimal IM if only one IM is considered in the seismic demand models. However, PGV has a poor predictive ability for the seismic responses in the transverse direction. Hence, PGV, peak ground displacement (PGD), root-mean-square of velocity (VRMS), specific energy density (SED), velocity spectrum intensity (VSI) and Fajfar intensity (FI) are selected as the optimal IMs by Lasso regression, and they are correlated with velocity except for PGD. The identified six IMs together can significantly increase the fitting ability of the models.

1. Introduction Cable-stayed bridges have been massively constructed in different parts of the world in the past few decades because of the efficient use of materials, their aesthetic configurations, and economic construction. These bridges play an important role in the transportation networks of a country [1]. Hence, it is important to investigate the seismic responses of the bridges subjected to potential earthquakes. The previous studies mainly focus on the seismic isolation strategies [1,2], failure mechanism [3–5], spatial variation effects of ground motion [6,7] and reliability analyses [8] for cable-stayed bridges. As a key step in performance-based seismic design, the fragility curve is adopted to describe the conditional probability that the structural demand reaches or exceeds the defined limit states under given intensity measure (IM). Probabilistic seismic demand models (PSDMs) are

established to correlate peak seismic responses with IM. Thus, IM plays an important role in performance-based seismic design. An optimal IM should reflect the main characteristics of an earthquake and correlate well with the peak seismic responses of the considered structure. Researchers have done much work to investigate the optimal seismic IMs for various structures, such as buildings and highway bridges. Most studies focus on the optimal IM for far-field ground motions. Cordova et al. [9] developed a two-parameter IM that can account for period softening, and the compound IM reflects the acceleration spectral shape. Kramer et al. [10] proposed an IM named cumulative absolute velocity (CAV5), and found that excess pore pressure generation in potentially liquefiable soils is closely related to the proposed IM compared to the other IMs. Bradley and Cubrinovski [11] investigated the optimal IM for pile foundations that embedded in liquefiable and non-liquefiable soils. They found that the velocity related IMs have good ability in predicting

* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (J. Guo), [email protected] (M.S. Alam), [email protected] (J. Wang), [email protected] (S. Li), yuan@ tongji.edu.cn (W. Yuan). https://doi.org/10.1016/j.soildyn.2019.106024 Received 13 September 2019; Received in revised form 24 December 2019; Accepted 25 December 2019 Available online 10 January 2020 0267-7261/© 2019 Elsevier Ltd. All rights reserved.

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the pile responses, especially the velocity spectrum intensity (VSI) that represents the integral of the velocity spectrum over a wide period range. Padgett et al. [12] studied the optimal IM for multi-span simply supported steel girder bridge class, and found that peak ground accel­ eration (PGA) can be used from the aspects of efficiency, practicality, sufficiency and hazard computability. Shafieezadeh and Ramanathan [13] introduced the concept of fractional order intensity measures (IMs) and used a single frame integral concrete box-girder bridge class and a multispan continuous steel girder bridge class as case studies. The nu­ merical results indicate that the proposed IMs significantly improve ef­ ficiency and proficiency while maintaining practicality and sufficiency. Kostinakis and Athanatopoulou [14] investigated the correlation be­ tween a large number of IMs and the corresponding damages to midrise reinforced concrete buildings. The results reveal that the spectral ac­ celeration at the fundamental period of the structure presents the strongest correlation with the maximum and average interstorey drifts. Kostinakis et al. [15] reviewed various developed scalar structure-specific seismic IMs and the problems associated with their use in practice. Wang et al. [16] investigated the optimal seismic IMs for bridges in liquefied and laterally spreading ground, and found that velocity-related IMs are superior to acceleration and displacement ones. With more and more near-fault ground motions are recorded and simulated, researchers start to investigate the optimal IM for structures subjected to near-fault ground motions. Makris et al. [17] investigated whether peak ground velocity (PGV) is an optimal IM for near-fault ground motions. They found that the long period velocity pulses caused by local, distinguishable acceleration pulses are more detri­ mental to civil structures. Zhong et al. [18] studied the appropriate IM for a cable-stayed bridge subjected to far-field and near-fault ground motions, respectively. Their numerical results reveal that the peak ground velocity (PGV) appears to be the best IM for the bridge. The aforementioned research works are referred to as scale-based optimal IM selection. Although some researchers looked into different IMs, how­ ever, for scale-based IM, only one IM is selected to establish the PSDMs of structures. As a result, the selected IM may have a high correlation with some of the peak responses but may have poor prediction ability for the others. In order to overcome the disadvantages of scale-based IM, researchers have proposed vector-based IM that usually contains two parameters. Baker et al. [19,20] proposed IMs including two parame­ ters: spectral acceleration at the first-mode period of vibration together with a measure of spectral shape. The accuracy of the demand pre­ dictions is improved compared to the traditional scale-based IM. Baker et al. [21–23] considered a vector-based IM composed of spectral ac­ celeration and epsilon (defined as the difference between the spectral acceleration of a record and the mean of a ground motion from a pre­ diction equation at the given period) that has the ability to predict the structural response. Although the proposed vector-based IMs increase the accuracy of demand predictions, it should be noted that they are related to accel­ eration. It is expected that the proposed vector-based IMs work well for short period structures, such as low-rise buildings and small span bridges as such short period structures are sensitive to acceleration. For long period structures, such as long-span cable-stayed bridges, they are sensitive to velocity and displacement. In addition, there is limited work on the investigation of the optimal IMs for long-span cable-stayed bridges. Therefore, it is essential to study the appropriate IMs for longspan cable-stayed bridges. This paper aims at proposing a general procedure to establish the PSDMs of the example bridge based on generalized linear regression models. Basic concepts of the regression models are introduced briefly, followed by the description of the prototype bridge. Then, 160 ground motions are selected and 22 IMs are introduced. After that, Lasso regression is adopted to identify the important IMs for each engineering demand parameter of the bridge. Finally, quadratic polynomial regres­ sion is used to establish the PSDMs of the bridge.

2. Generalized linear regression models The linear regression expects the target values to be linearly corre­ lated to the input variables. However, it is almost impossible to find a straight line (or hyperplane) that goes through all the points of input variables. As an alternative, the linear regression model is established by minimizing the error between the observed and the predicted values. The model takes the form as below: yi ¼ by i ðω; xi Þ þ e

(1)

b y i ðω; xi Þ ¼ ω0 þ ω1 xi1 þ ::: þ ωn xin

(2)

where ω ¼ ½ω1 ; ω2 ; :::; ωn �T is the regression coefficients vector,

xi ¼ ½xi1 ; xi2 ; :::; xin �T is the ith input variables vector, yi is the ith observed response, b y i is the ith predicted response and e is the error term. 2.1. Ordinary least squares Ordinary least squares (OLS) regression fits the linear model in Eq. (2) with coefficients ω to minimize the residual sum of squares between

the observed responses y ¼ ½y1 ; y2 ; :::; ym �T and the predicted responses b y ¼ ½b y1; b y 2 ; :::; b y m �T [24]. Mathematically, it finds the minimum value using the following form: f ðωÞ ¼

�2 xTi ω

m X

yi

(3)

i¼1

The error between the ith predicted response and its corresponding regression response is illustrated in Fig. 1(a). The minimum value of the cost function fðωÞ can be determined by differentiating with respect to the regression coefficients vector ω. The following expressions demon­ strate the details of the differentiation: 2

m X � yi xTi ω xi1 2 6 7 6 i¼1 7 6 7 6 m X � 7 6 7 6 2 yi xTi ω xi2 7¼6 i¼1 7 6 7 6 ⋮ 7 6 ⋮ 7 6 6 4 ∂f ðωÞ 5 6 m X 4 � 2 yi xTi ω xin

3

2

∂f ðωÞ 6 ∂ω1 6 6 6 ∂f ðωÞ ∂f ðωÞ 6 ¼6 6 ∂ω2 ∂ω 6 ∂ωn

x11 6 x12 26 4 ⋮ x1n

x21 x22 ⋮ x2n

xT1

¼

6 7 6 xT 7 6 7 26 2 7 6 ⋮ 7 4 5 xTm

2 � T 3 xm1 6 y1 x1 ω� 6 y xT2 ω xm2 7 2 76 6 5 ⋮ 6 ⋮ 4 � xmn ym xTm ω

⋯ ⋯ ⋱ ⋯

3T 2

2

7 7 7 7 7 7 7 7 7 7 7 5

i¼1

2

¼

3

y1

� xT1 ω � xT2 ω

6 6 y 6 2 6 6 ⋮ 4 � ym xTm ω

3 7 7 7 7 7 5

3 7 7 7 7¼ 7 5

2XT ðy

XωÞ

(4)

Finally, the normal equation is obtained as below:

∂f ðωÞ ¼ ∂ω

2XT ðy

XωÞ ¼ 0

if the inverse matrix of XT X exists, then, � 1 ω ¼ XT X XT y

(5)

(6)

Estimating the regression coefficients ω based on Eq. (6) is called the normal equation method. Another commonly used method is the gradient descent method [25]. Fig. 1(b) demonstrates the basic concept of gradient descent by minimizing the function of y ¼ x2 . The derivative 2

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Fig. 1. Illustration of minimizing the cost function f ðωÞ: (a) error term, (b) gradient descent.

coefficients for ridge regression can be estimated as below:

of the function is denoted as f ðxÞ ¼ 2x. Then the variable x can be updated as below: 0

x¼x

∂f ðωÞ ¼ ∂ω

(7)

0

αf ðxÞ

�

where, α > 0 is called the learning rate. Eq. (7) makes the function 0 gradually approach to the minimum. For example, if f ðxÞ at current step is larger than zero, then, the function fðxÞ can be reduced by moving x leftward. After a certain number of iterations using Eq. (7), the function fðxÞ can converge to the minimum. For the minimization problem in Eq. (3), the regression coefficients can be updated by the following expression:

ωj ¼ ωj

m X ∂f ðωÞ yi α ¼ ωj þ 2α ωj i¼1

�

xTi ω xij

ðj ¼ 1; 2; :::; nÞ

ω ¼ XT X þ λI

α

∂f ðωÞ ¼ ω þ 2αXT ðy ω

ω¼ω

(8)

The accuracy and speed of gradient descent are influenced by the initial value of ω and the learning rate α. More details about the gradient descent can be found in [26]. It should be noted that the regression coefficients obtained from OLS rely on the independence of the model terms. When input variables are highly correlated, the matrix XT X becomes close to singular. As a result, the OLS regression becomes sensitive to the random errors in the observed responses, which leads to large variance. In order to address the problems of OLS regression, shrinkage methods are proposed by statisticians [27], and the shrinkage model is generated by adding a penalty term in the cost function of Eq. (3). The commonly used shrinkage methods are Ridge regression and Lasso regression [27,28], and will be described in the following sections.

f ðωÞ ¼

m X

yi i¼1

n X �2 xTi ω þ λ ω2i

Pn

2 i¼1 ωi ,

(12)

XT y

α

� ∂f ðωÞ ¼ ω þ 2α XT ðy ω

XωÞ

� λω

(13)

2.3. Lasso regression Mathematically, the cost function for Lasso regression to minimize is: f ðωÞ ¼

m X

yi i¼1

n X �2 xTi ω þ λ jω i j

(14)

i¼1

Solving the problem of Lasso regression is difficult due to the discontinuity of the derivative of the penalty term. As a result, the aforementioned normal equation and gradient descent methods are not suitable for Lasso regression. The commonly used techniques to fit Lasso models are the coordinate descent [30] and least angle regression (LARS) [31]. Coordinate descent is an optimal algorithm that succes­ sively minimizes along coordinate directions to find the minimum value of a function. Least angle regression (LARS) is an algorithm that provides a mean to determine which variables should be included in the model. The implementation of LARS is more complicated than that of the co­ ordinate descent. A more detailed introduction of LARS can be found in the reference [31]. It is recommended to implement LARS on some professional platforms, such as the scikit-learn module in Python lan­ guage [32]. The penalty terms for ridge regression and Lasso regression are different, which leads to different results. This phenomenon will be explained from the perspective of geometry. Mathematically, the cost

2.2. Ridge regression Ridge regression adds a penalty term λ for ridge regression can be written as:

1

When two or more coefficients are correlated in the regression model, very large positive and negative weights may exist in the esti­ mated coefficients vector by the OLS method. This problem can be avoided by using ridge regression that can impose constraints on the regression coefficient. Similar to ridge regression, there is another shrinkage method called Lasso regression [29] that forces some regres­ sion coefficients to be exactly zero, which makes it easier to interpret the data. Meanwhile, the important variables can be identified by Lasso regression.

(9)

XωÞ

(11)

XωÞ þ 2λω ¼ 0

where, I is the identity matrix. Similarly, for the gradient descent method, the regression coefficients can be updated as below:

Similarly, equation (8) can be expressed in the following matrix form:

ω¼ω

2XT ðy

and the cost function

(10)

i¼1

where, λ is a positive number that controls the amount of shrinkage, the larger the value of λ, the greater the amount of shrinkage. Based on the normal equation method mentioned in section 2.1, the regression 3

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functions for ridge regression and Lasso regression are equivalent to Eqs. (15) and (16), respectively. 8 m X �2 > > > yi xTi ω > f ðωÞ ¼ < i¼1 (15) n X > > > > ω2j � t :

counterweight is applied to the concrete box girder region, which can. Increase the vertical stiffness of the bridge. Auxiliary piers are set at the concrete box girder area to support the heavy dead loads. The yield strength and compressive strength of the steel and concrete girders are 370 MPa and 30.2 MPa, respectively. The cables are arranged in two inclined planes, and a semi-harp cable pattern is used because of its efficiency [33]. Intersecting cables are adopted at the center part of the main span to avoid its large vertical deformation due to live loads. The cable spaces in the concrete and steel girders are 6 m and 9 m, respec­ tively. The designed tensile strength of the cables is 1670 MPa, and the safety factor of the cables is three. The bearings at the auxiliary piers can move freely in the longitudinal and transverse directions of the bridge. Wind-resistant bearings and viscous dampers are installed between the pylons and girders in the transverse and longitudinal directions of the bridge, respectively. The height of the pylon 4 and 5 are 252 m and 242 m, respectively. The pylon is divided into upper, middle and lower parts by the upper and lower cross beams. The typical cross-sections are presented in Fig. 3.

i¼1

8 m X > > > f ðωÞ ¼ yi > < i¼1

�2 xTi ω (16)

n X > > > > jω i j � t : i¼1

Taking an example with consideration of two variables ðω1 ; ω2 Þ, the residual sum of squares is a quadratic function with respect to ω1 ; ω2 . This function represents a paraboloid in three-dimensional space. When projected to the ðω1 ; ω2 Þ plane, the paraboloid becomes a series of elliptical contours, as demonstrated in Fig. 2. The point at the center of contours is estimated by the OLS method. At this point, the parameter t tends to become infinite, which means that the penalty term has no ef­ fect on the cost function. The intersect points in Fig. 2(a) and (b) are the estimated solution for the ridge regression and Lasso regression, respectively. Compared to the ridge regression, most of the coefficients estimated by Lasso regression are forced to become zero since the intersection point is on the coordinate axis. It can be concluded that the parameter t trades off the accuracy and complexity of the model. Based on the aforementioned discussion, the OLS method may lead to model overfitting, i.e. with small training error, but large test error. However, shrinkage methods (ridge regression and Lasso regression) trade off the accuracy and interpretation of the model. In the current research, Lasso regression is adopted to build a better model to find out the important input variables.

3.2. Numerical model of the bridge The three-dimensional finite element model of the cable-stayed bridge generated in the OpenSees platform [34] is demonstrated in Fig. 4. The bridge deck is expected to remain elastic during seismic events, and linear-elastic beam-column elements are adopted to simu­ late the deck elements. The properties of the deck elements presented in Table 1 can be assigned to the corresponding deck elements. The in­ clined stay cables are assumed to remain elastic with consideration of the above mentioned large safety factor. The bending and compressive capacities of the stay cables are negligible, and large-displacement truss elements assigned with tension-only uniaxial material is used to simu­ late the mechanical behavior of the cables. The sag effect of stay cables due to its self-weight is considered by the equivalent modulus of elas­ ticity. The cable nodes are rigidly connected with the corresponding pylon nodes and deck nodes, respectively. Single sphere steel bearings are used for the bridge, and the wind-resistant bearings are installed in the transverse direction of pylons 4 and 5. These single sphere bearings are characterized by transferring friction force. Thus, the transition piers (1 and 9) and auxiliary piers (2, 3, 6, 7 and 8) are expected to be free from severe damages when subjected to strong ground motions, and elastic beam-column elements are adopted to model them. For viscous dampers, the damping coefficient and damping exponent are 3000 kN/(m/s) and 0.3, respectively. To simulate the behavior of such

3. Example bridge and its modeling 3.1. Description of the bridge A three-span high-speed railroad cable-stayed bridge is considered in the current research, and the lengths of the main and side spans are 672 m and 324 m, respectively. The hybrid girder is composed of concrete box girders and steel box girders, as presented in Fig. 3(a). The crosssectional properties of the girders are summarized in Table 1. The

Fig. 2. Geometric interpretation of (a) ridge regression, (b) Lasso regression. 4

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Fig. 3. Prototype bridge: (a) elevation, (b) details of pylons and auxiliary piers.

reduce computing time, only the middle part top 10 elements, bottom 10 elements, and the lower part top 10 elements, bottom 10 elements of the pylons are simulated with fiber-based sections, and the other elements of the pylons are modeled with elastic beam-column elements. The fiber sections are made up of unconfined, confined concrete and reinforcing steel. The concrete behavior is modeled with concrete01 material in OpenSees. This material uses Mander’s model [35] to define the stress-strain curves for confined and unconfined concrete. In the current study, the compressive strengths of the confined and unconfined con­ crete are 36.8 MPa and 30.2 MPa, respectively. Steel01 material in OpenSees is adopted to simulate the reinforcing steel, and the yield stress of the steel is 400 MPa. The foundations of the bridge are simu­ lated with zero-length elements containing three translational and three rotational linear elastic springs. Rayleigh damping is adopted, and the modal damping ratio is set to 0.03 as a cable-stayed bridge is

Table 1 Cross-sectional properties of the girder. Cross-section

Aðm2 Þ

Iz ðm4 Þ

Iy ðm4 Þ

Jðm4 Þ

Concrete box girder Steel box girder

54.5 3.2

8776 432

150 11.9

518 24.3

Note: A ¼ cross-section area; Iz ¼ moment of inertia about the global z-axis. Iy ¼ moment of inertia about the global y-axis; J ¼ torsion resistance.

dampers, the uniaxial material Viscous Damper is adopted. The pylons (4 and 5) play an import role for the bridge. They not only transfer vertical forces to the base but also resist against unbalanced bending moments. As a result, their mechanical behaviors are complicated and they may experience severe damages during seismic events. In order to 5

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Fig. 4. Schematic view of the numerical model.

characterized by small damping. Based on the modal analysis of the bridge, the first periods of the bridge in the longitudinal, transverse and vertical directions are 12.4 s, 4.9 s, and 4.1 s, respectively.

motion database are adopted as input ground motions for analyses. The 160 ground motions are approximately divided into 4 categories as suggested by Ref. [36]. The first 40 ground motions are selected such that their horizontal response spectra match the computed median and log standard deviation [37]. predicted for a magnitude 7 earthquake at a distance of 10 km, and the average shear velocity in the top 30 m (Vs30 ) of the site is assumed to be 760 m/s. The characteristics of the selected four sets of ground motions are demonstrated in Fig. (5). In this study, for each ground motion, randomly select one component from two horizontal components as the representation in the horizontal direction. The selected horizontal component together with the corresponding vertical component simultaneously excites the bridge. Fig. 5 presents the

4. Ground motions In the current research, the site of the cable-stayed bridge may experience mid to large-magnitude earthquakes at near to moderate distances. In addition, the site may be excited by pulse-like ground motions. Therefore, it is essential to investigate the optimal intensity measures for the bridge subjected to both far and near field ground motions. A total of 160 ground motions selected from PEER strong

Fig. 5. 3% geometric mean response spectra (horizontal and vertical components) in logarithmic scale. 6

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3% geometric mean response spectra (selected horizontal and vertical components) for each ground motion set. It can be observed that the pulse-like ground motions (Fig. 5(d)) are rich in long-period contents. A total of 22 intensity measures are taken into account, and detailed descriptions of them are summarized in Table 2. These intensity mea­ sures can approximately represent the characteristics of strong ground motions, such as amplitude (PGA, PGV, and PGD), frequency content (ASI and ASV) and duration [47].

the critical seismic responses are recorded. Here, the critical responses refer to longitudinal bearing displacement (Δb L ), longitudinal pylon curvatures (e.g., at lower-bottom (Lϕ1 ), lower-top (Lϕ2 ), middle-bottom (Lϕ3 ) and middle-top (Lϕ4 )) and transverse pylon curvatures (e.g., at lower-bottom (Tϕ1 ), lower-top (Tϕ2 ), middle-bottom (Tϕ3 ) and middletop (Tϕ4 )). The four critical sections of the pylons are specified in Fig. 3 (b). Establishing probabilistic seismic demand models (PSDMs) is an important step for performance-based seismic design. PSDMs presents the relationship between seismic intensity measure (IM) and peak seismic responses. Cornell et al. [48] proposed a power exponent model to establish the PSDMs, as expressed in Eq. (17):

5. Probabilistic seismic demand models In the current study, the selected horizontal (either in the longitu­ dinal or transverse direction of the bridge) and vertical components of each ground motion simultaneously excite the cable-stayed bridge, and

Sd ¼ aIMb

where, a and b are the regression coefficients. Taking natural logarithm on both sides of Eq. (17) yield:

Table 2 Intensity measures description.

lnðSd Þ ¼ ln a þ b lnðIMÞ

Intensity Measure

Description

Definition

Units

PGA

Peak ground acceleration Peak ground velocity Peak ground displacement Peak velocity and acceleration ratio

� � PGA ¼ max�u€g ðtÞ�

g

� � PGV ¼ max� u_ g ðtÞ�

cm/s

� � PGD ¼ max� ug ðtÞ�

cm

� � vmax =amax ¼ max ​ � u_ g ðtÞ�= � � � � max ug ðtÞ sffiffiffiffiffiffiffiffiZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tot 1 aRMS ¼ ½u€g ðtÞ�2 dt t sffiffitot ffiffiffiffiffiffiZffiffiffi0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tot 1 vRMS ¼ ½ u_ g ðtÞ�2 dt t tot 0 sffiffiffiffiffiffiffiffiZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tot 1 dRMS ¼ ½ug ðtÞ�2 dt t Z tottot 0 1 AI ¼ ½ u_ g ðtÞ�2 dt ttot 0 3 pffiffiffiffiffiffi Ic ¼ ðaRMS Þ2 ttot R tot SED ¼ 0 ½ u_ g ðtÞ�2 dt

s

PGV PGD vmax = amax aRMS vRMS dRMS AI Ic SED CAV ASI VSI SMA SMV EDA

Root-mean-square of acceleration Root-mean-square of velocity Root-mean-square of displacement Arias Intensity [38] Characteristic Intensity [39]

Specific Energy Density Cumulative Absolute Velocity [40] Acceleration Spectrum Intensity [41] Velocity Spectrum Intensity [41] Sustained maximum acceleration [42] Sustained maximum velocity [42] Effective Design Acceleration [40]

� R tot � CAD ¼ 0 � u€g ðtÞ�dt

m/s

cm2 =s cm/s

VSI ¼

R 2:5

Sv ðξ ¼ 0:05; TÞdT

cm

0:1

the third highest absolute value of acceleration in the time-history

g

the third highest absolute value of velocity in the time-history

cm/s

the peak acceleration value found after low pass filtering the input time history with a cut-off frequency of 9 Hz The acceleration level below which 95% of the total Arias intensity is contained P 2 C =fi Tm ¼ P i 2 Ci Fourier Ci amplitudes fi frequencies between 0.25 and 20 Hz

g

Tm

Mean Period [44]

Ia

Compound acc.related IM [45]

Ia ¼ PGA � td

FI

Fajfar intensity [46] Compound vel.related IM [45] Compound disp.related IM [45]

FI ¼ PGV � td

1=3

td ¼ tð95%AIÞ

1=4

Iv ¼ PGV

2=3

�

tð5%AIÞ

1=3 td

1=3

Id ¼ PGD � td

A smaller value of ζ implies higher proficiency of the IM. It also demonstrates that a more proficient IM has a lower logarithmic standard deviation (βD=IM ) and a higher slope (b). A sufficient IM is conditionally statistically independent of ground motion characteristics, such as fault to site distance (R) and earthquake magnitude (M). A sufficient IM in­ dicates that the regression line of the residual between the observed responses and the corresponding estimated responses with respect to R or M as flat as possible. The hazard computability refers to the efforts required to generate the probabilistic seismic hazard curve for the chosen IM. In general, hazard maps or curves are readily available for PGA, PGV, PGD and specific spectral quantities, such as spectral accel­ erations at 0.2 s (Sa02), and 1.0s (Sa10). In this study, the IMs and observed responses are transformed into natural logarithmic space. A scatter diagram of the longitudinal bearing displacements (Δb L ) with respect to the IMs in Table 2 indicates that a simple linear model is unreasonable to fit such data. A cubic polynomial is adopted to fit the data, as expressed in Eq. (20):

g3=2 � s1=2

g� s

A95 parameter [43]

Id

cm

Sa ðξ ¼ 0:05; TÞdT

0:1

A95

Iv

cm/s

R 0:5

(18)

Selecting an optimal IM is challenging due to the uncertainties of ground motions. Previous researches mainly focused on selecting the optimal IMs for small span bridges subjected to far-field ground motions. There is limited work on the optimal IMs for long-span bridges, such as cable-stayed bridges. Previous researchers [49] proposed the criteria, such as efficiency, practicality, proficiency, sufficiency and hazard computability, to select the optimal IMs. Efficiency means the estimated seismic demand having small varia­ tion, and it is usually reflected by the logarithmic standard deviation of the seismic demand (βD=IM ). Practicality refers to the dependency of the seismic demand on the IM, which is measured by the slope b in Eq. (18). The larger value of b, the more practical of the IM. Proficiency is a composite measure with respect to efficiency and practicality, which is defined as below: � ζ ¼ βD=IM b (19)

g

ASI ¼

(17)

g

lnðSd Þ ¼ ω0 þ ω1 lnðIMÞ þ ω2 ln2 ðIMÞ þ ω3 ln3 ðIMÞ

(20)

where, ω0 ; ω1 ; ω2 and ω3 are the regression coefficients. In order to use the ordinary least square (OLS) model to estimate the regression co­ efficients, the expression in Eq. (20) can be rewritten as below:

s

lnðSd Þ ¼ ω0 þ ω1 Z1 þ ω2 Z2 þ ω3 Z3

g� s1=3

(21)

where Z1 ¼ lnðIMÞ; Z2 ¼ ln2 ðIMÞ and Z3 ¼ ln3 ðIMÞ. Eq. (21) is called the generalized linear regression model, and the above-mentioned normal equation or gradient descent methods can be adopted to estimate the regression coefficients in Eq. (21). In regression analysis, the coefficient of determination (R2 ) is usually adopted to measure how well the regression curve fits the data [50]. The larger value of the R2 , the better

cm/s3/ 4

cm2/3/ s1/3 cm.s1/ 3

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fitting degree of the regression curve to the data. Fig. 6 demonstrates the regression curves based on Eq. (21) for the peak longitudinal bearing displacement (Δb L ) with respect to the first 16 IMs in Table 2. It can be observed that the PGV, PGD and VRMS fit the data better than the other IMs. Given the space limitation, the coefficients of determination (R2 ) for the considered critical responses with respect to the 22 IMs in Table 2 are summarized in Table 3. The R2 greater than or equal to 0.8 are marked in bold. It is found that the PGV and VRMS are better IMs compared to the other IMs. Taking the hazard computability into consideration, PGV can be selected as the optimal IM for the cable-stayed bridge subjected to both far-field and near-fault ground motions. Fig. 7 presents the probabilistic seismic demand models of the critical responses with respect to PGV. The R2 of the transverse pylon curvatures (Tϕ1 ;Tϕ2 ;Tϕ3 ;Tϕ4 ) are smaller than those of the longitudinal pylon curvatures. This phenomenon implies that only PGV cannot ac­ count for all the main characteristics of a ground motion. More IMs should be considered to improve the ability of the regression curves fitting to the data. However, the challenge is how to determine the appropriate IMs that can significantly improve the model prediction ability. Too many IMs increase the complexity of the model, and usually, it is difficult to estimate the regression coefficients due to the singularity

of the matrix. On the other hand, only one IM cannot provide good predictions for all the critical components. In order to trade off the ac­ curacy and complexity of the model, Lasso regression is adopted to identify the necessary IMs in the current study. The Lasso regression presented in section 2.3 establishes an appro­ priate model by adjusting the penalty coefficient (λ). In the current study, a total of 22 IMs in Table 2 are considered, and they are trans­ formed into natural logarithmic space. Additionally, performing the same transformation for the critical seismic responses. The data were standardized to give each IM equal importance regardless of the units. The standardization for each IM is conducted based on the following expression: � 0 (22) xi ¼ ðxi minxi Þ ðmaxxi minxi Þ ði ¼ 1; 2; :::; 22Þ where xi is the ith column of the transformed IM matrix (160 � 22), and maxxi ; minxi ; are the maximum and minimum values of xi . The same standardization is performed for the critical seismic re­ sponses (Δb L , Lϕ1 ; Lϕ2 ; Lϕ3 ; Lϕ4 and Tϕ1 ; Tϕ2 ; Tϕ3 ; Tϕ4 ). After stan­ dardization, the maximum and minimum values of each variable are scaled to zero and one, respectively. Then, the standardized data are used for the following Lasso regression. A 10-fold cross validation is

Fig. 6. Probabilistic seismic demand models for the longitudinal bearing displacement (Δb L ) with respect to the first 16 IMs described in Table 2. 8

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Table 3 The coefficients of determination (R2 ) for the critical responses with respect to the IMs. IMs

Δb

Lϕ1

Lϕ2

Lϕ3

Lϕ4

Tϕ1

Tϕ2

Tϕ3

Tϕ4

PGA PGV PGD vmax =amax

0.35 0.81 0.80 0.32

0.59 0.91 0.80 0.29

0.60 0.92 0.79 0.27

0.61 0.92 0.79 0.27

0.60 0.89 0.76 0.26

0.47 0.82 0.79 0.28

0.63 0.73 0.57 0.12

0.37 0.77 0.80 0.34

0.47 0.81 0.77 0.27

aRMS

0.35

0.60

0.59

0.60

0.59

0.50

0.63

0.40

0.48

vRMS

0.82

0.87

0.86

0.87

0.85

0.85

0.70

0.83

0.84

dRMS

0.75

0.74

0.72

0.72

0.70

0.75

0.52

0.77

0.73

AI Ic

0.48 0.45

0.73 0.70

0.73 0.70

0.74 0.71

0.71 0.69

0.64 0.61

0.71 0.70

0.54 0.51

0.61 0.59

SED CAV ASI VSI SMA SMV EDA A95 Tm

0.79 0.49 0.33 0.62 0.33 0.71 0.38 0.33 0.20

0.83 0.68 0.61 0.90 0.58 0.87 0.65 0.58 0.19

0.83 0.69 0.62 0.89 0.59 0.87 0.66 0.59 0.19

0.83 0.69 0.62 0.89 0.59 0.87 0.67 0.59 0.19

0.80 0.65 0.60 0.90 0.58 0.84 0.65 0.58 0.20

0.83 0.65 0.49 0.75 0.47 0.81 0.52 0.45 0.20

0.62 0.58 0.65 0.83 0.62 0.77 0.66 0.61 0.09

0.82 0.59 0.38 0.64 0.37 0.73 0.41 0.35 0.24

0.81 0.61 0.47 0.73 0.46 0.79 0.51 0.45 0.20

Ia

0.42

0.64

0.66

0.67

0.65

0.56

0.66

0.47

0.55

FI Iv

0.80 0.73

0.88 0.80

0.88 0.80

0.88 0.80

0.85 0.77

0.82 0.78

0.67 0.59

0.79 0.76

0.80 0.75

Id

0.73

0.72

0.71

0.71

0.69

0.74

0.49

0.76

0.71

L

Note: Δb L longitudinal bearing displacement (m); Lϕ1 ; Lϕ2 ; Lϕ3 ; Lϕ4 longitudinal pylon curvatures at the critical sections of lower-bottom, lower-top, middlebottom and middle-top, respectively, as can be seen in Fig. 3(b), and Tϕ1 ; Tϕ2 ; Tϕ3 ; Tϕ4 Transverse pylon curvatures at the critical sections in Fig. 3(b). Bold values indicate the R2 greater than or equal to 0.8.

Fig. 7. Probabilistic seismic demand models for the longitudinal bearing displacement (Δb L ), longitudinal pylon curvatures (Lϕ1 ;Lϕ2 ;Lϕ3 ;Lϕ4 ) and transverse pylon curvatures (Tϕ1 ; Tϕ2 ; Tϕ3 ; Tϕ4 ) with respect to PGV.

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adopted to find out the optimal model [51], and randomly taking 80% data as training data, and the left 20% as test data. This work can be implemented by programming in Python. Let us take the regression between the longitudinal bearing displacement (Δb L ) and the consid­ ered IMs as an example. For each model, 80% selected training data are put into Eq. (14), and the value of the penalty coefficient (λ) is changed from very small to very large values, as expressed in Eq. (23): λ ¼ eði�0:1

20Þ

ði ¼ 1; 2; :::; 300Þ

and finally do not change. The optimal penalty coefficient (λ) is selected based on the minimum value of the test errors among the ten models. For the current example, the optimal λ equals to 0.0082 (i ¼ 152). The regression coefficient traces for the best model with respect to the iter­ ation steps as demonstrated in the left part of Fig. 8 (a). At the beginning of the iteration, the regression coefficients are almost unchanged due to the small values of the penalty coefficient (λ) that has little effects on the regression model. With the increased value of λ, the regression co­ efficients gradually decrease until all of them become exactly zero. At the optimal step (i ¼ 152), only the coefficients of PGV, PGD, VRMS and SED are not zero. For the purpose of comparison, the nonzero co­ efficients are divided by their maximum value. It is found that the longitudinal bearing displacement (Δb L ) is highly correlated with PGD and PGV. The Lasso regression results for the lon­ gitudinal and transverse critical seismic responses are illustrated in Figs. 8 and 9, respectively. Finally, PGV, PGD, VRMS, SED, VSI and FI are selected as the optimal IMs in the current research. AI in Fig. 9 (b) is excluded from the optimal IMs based on two reasons. First, the relative coefficient of AI is much smaller than that of VSI as shown in Fig. 9 (b). Second, AI only exists in Fig. 9 (b). Based on the definition of the IMs, it is shown that the selected optimal IMs (PGV, PGD, VRMS, SED, VSI and FI) are correlated with the velocity except for PGD. This phenomenon can be explained from the aspect of response spectra (acceleration, ve­ locity and displacement spectra). Short periods are the acceleration

(23)

where, i is the iteration number. For each iteration, the coefficients (ω) are determined by Lasso regression, and they are recorded. After 300 iterations, the training and test errors of the model can be calculated at each iteration point. Here, the training and test errors are defined as the following expression: errorðωÞ ¼

m 1X yi m i¼1

�2 xTi ω

(24)

where, m is the sample size, xi is the standardized input vector for the ith sample and yi is the standardized observed response. For training error at each iteration step, m is the total number of the training data and ω is the regression coefficient vector at the current step. Fig. 8 (a) presents the training error and test error for the ten models at each iteration step. It is found that the model errors decrease to the minima, then increase,

Fig. 8. Regression coefficients traces and model errors for longitudinal critical seismic responses: (a) bearing displacement (Δb L ), (b) pylon curvature at lowerbottom (Lϕ1 ), (c) pylon curvature at lower-top (Lϕ2 ), (d) pylon curvature at middle-bottom (Lϕ3 ), (e) pylon curvature at middle-top (Lϕ4 ). 10

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Fig. 9. Regression coefficients traces and model errors for transverse critical seismic responses: (a) pylon curvature at lower-bottom (Tϕ1 ), (b) pylon curvature at lower-top (Tϕ2 ), (c) pylon curvature at middle-bottom (Tϕ3 ), (d) pylon curvature at middle-top (Tϕ4 ).

Fig. 10. Correlation analyses between PGV and the other optimal IMs: (a) PGD, (b)vRMS , (c) SED, (d) VSI and (e) FI. 11

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sensitive regions and mediate and long periods are the velocity and displacement sensitive regions, respectively. The fundamental periods of the cable-stayed bridge in longitudinal and transverse directions are 12.4 s and 4.9 s, respectively. In addition, the higher modes have sig­ nificant effects on the seismic responses of the cable-stayed bridge. Therefore, it is concluded that the velocity and displacement related IMs can be provided for the optimal IMs analysis of long-period structures, such as cable-stayed bridges. Moreover, it is expected that the velocity related IMs have good prediction ability for the seismic responses of cable-stayed bridges subjected to far-field or near-fault ground motions or both. Fig. 10 presents the correlation analyses between the PGV and other optimal IMs (PGD, VRMS, SED, VSI, and FI). It is found that the PGV is highly correlated with the other optimal IMs. As a result, the correla­ tions between these optimal IMs should be considered while establishing the probabilistic seismic demand models with them. Therefore, the quadratic polynomial expression in Eq. (25) is adapted to fit the critical seismic responses. yquad ¼ ω0 þ

n X i¼1

ωi xi þ

n n X X i¼1 j¼iþ1

ωij xi xj þ

n X

ωii x2i

can be observed that the coefficients of determination (R2 ) for the quadratic polynomial regression with 6 IMs are all improved compared to that with PGV as shown in Table 3. As mentioned above, the hazard curves can be generated for PGA, PGV, PGD and specific spectral quantities, such as spectral acceleration at 0.2 s (Sa02) and 1.0s (Sa10). However, for other IMs in Table 2, it is difficult to obtain their hazard curves. Nevertheless, it is possible to generate the hazard curves for unavailable IMs by establishing the re­ lationships with the available ones [Bradley 2012]. In addition, the main purpose of the current research is that proposing a general procedure to select the optimal IMs for structures subjected to strong ground motions. For the performance-based seismic design, these ground motions can be selected based on the seismic risk of the site. 6. Conclusions Optimal intensity measures for probabilistic seismic demand models of a long-span cable-stayed bridge subjected to the seismic recorders with both far-field and near-fault ground motions have been investi­ gated using generalized linear regression models. The three dimensional numerical model of the bridge is generated in the OpenSees platform. A total of 160 ground motions attributed to four site conditions are selected to excite the bridge. The Lasso regression is adopted to identify the import IMs for each critical seismic response in the longitudinal and transverse directions. The coefficient of determination (R2 ) is used to estimate the fitness of the proposed quadratic polynomial to the data. Several main conclusions are drawn in the followings:

(25)

i¼1

where, ω0 ; ωi ; ωij and ωii are the coefficients of the constant term, firstorder term, cross term and second-order term, respectively. In the cur­ rent study, a total of 6 IMs (n ¼ 6) are selected, and xi refers to the ith optimal IM (lnðIMi Þ). Fig. 11 demonstrates the fitness of the quadratic polynomial Eq. (25) to the critical seismic responses of the bridge in the longitudinal and transverse directions, and the regression coefficients for each critical response are presented in the Appendix (Table A1). It

Fig. 11. Accuracy of the quadratic polynomial regression fitting the longitudinal bearing displacement (Δb L ), longitudinal curvatures (Lϕ1 ; Lϕ2 ; Lϕ3 ; Lϕ4 ) and transverse curvatures (Tϕ1 ; Tϕ2 ; Tϕ3 ; Tϕ4 ) of the pylon. 12

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J. Guo et al.

(1) Based on the criteria of efficiency, practicality, proficiency, suf­ ficiency and hazard computability, PGV is more appropriate than other IMs for establishing the probabilistic seismic demand models for the cable-stayed bridge subjected to the selected ground motions. However, PGV has a poor predictive ability for the seismic responses in the transverse direction. This means that more than one IMs should be incorporated in the probabilistic seismic demand models. (2) The optimal IMs for each critical seismic response are identified by Lasso regression that trades off the accuracy and complexity of the models. PGV, PGD, VRMS, SED, VSI, and FI are selected as the optimal IMs, and they are correlated with velocity except for PGD. This phenomenon can be explained from the aspect of response spectra where short periods are in the acceleration sensitive regions, and long periods are in the velocity and displacement sensitive regions. The seismic responses of cablestayed bridge are mainly attributed to the contributions of long period modes. (3) The selected optimal IMs are used to establish the quadratic polynomial regression that can significantly improve the co­ efficients of determination (R2 ) of the models. In addition, the probabilistic seismic demand models of the bridge can be built based on the selected optimal IMs. (4) In general, the established Eq. (29) with coefficients in Table A1 can give good predictions for the ground motions. This predictive equation can be used to establish the probabilistic seismic de­ mand models (PSDMs).

sake of brevity. The selected optimal IMs (e.g, PGV, PGD, VRMS, SED, VSI and FI) are used to establish the quadratic polynomial regression, which can significantly improve the probabilistic seismic demand model. However, the proposed procedure of combining IMs may seem complicated for practical use. It is possible to further reduce the number of IMs for simplicity although this may affect the accuracy. Declaration of competing interest This is to conform that all the authors of this manuscript are aware of the submission and have agreed to submit the paper to Soil Dynamics and Earthquake Engineering. CRediT authorship contribution statement Junjun Guo: Conceptualization, Methodology, Software, Valida­ tion, Formal analysis, Investigation, Resources, Data curation, Writing original draft, Writing - review & editing. M. Shahria Alam: Method­ ology, Validation, Formal analysis, Investigation, Resources, Writing review & editing, Supervision, Project administration, Funding acqui­ sition. Jingquan Wang: Validation, Formal analysis, Investigation, Resources, Writing - review & editing, Supervision. Shuai Li: Valida­ tion, Formal analysis, Investigation, Resources, Writing - review & editing. Wancheng Yuan: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Resources, Writing - review & editing, Supervision, Project administration, Funding acquisition. Acknowledgment

The main purpose of the current research is to propose a general procedure to identify the important IMs for the cable-stayed bridge subjected to strong ground motions. This procedure will be suitable for analyzing the optimal IMs for other structures, such as continuous girder bridges. For performance-based seismic design, a series of ground mo­ tions can be generated based on the probabilistic seismic hazard analysis of the site, and the methods presented in the paper can be adopted to establish the probabilistic seismic demand models. The geometric and material uncertainties are not considered in the current study for the

This research was supported by the China Scholarship Council under Grant No. 201806260168; The Ministry of Science and Technology of China under Grant No. SLDRCE19-B-19; The National Natural Science Foundation of China under Grant No. 51778471, 51978512. The first author would like to acknowledge the Faculty of Applied Science, the University of British Columbia for their support during the UBC visit and complete the study.

Appendix

Table A.1 Quadratic polynomial regression coefficients EDP

Δb

L

Lϕ1 Lϕ2 Lϕ3 Lϕ4 Tϕ1 Tϕ2 Tϕ3 Tϕ4

ω0

ω1

ω2

ω3

ω4

ω5

ω6

ω11

ω22

ω33

ω44

ω55

ω66

ω21

ω31

ω32

ω41

ω42

ω43

ω51

ω52

ω53

ω54

ω61

ω62

ω63

ω64

ω65

-5.20 0.17 -15.17 -0.85 -14.77 0.66 -15.50 0.95 -10.53 0.32 -10.82 1.36 -8.22 2.59 -10.82 1.78 -8.88 3.71

-5.58 -0.58 -1.29 -0.65 0.87 -0.39 0.22 -0.39 7.95 -0.40 -1.25 0.02 -0.03 -0.19 4.15 0.07 2.07 -0.27

-2.43 0.09 -2.90 0.52 -1.82 -0.82 -1.91 -0.85 -3.96 1.59 -2.70 -0.05 -0.65 -1.47 -4.83 0.69 -5.81 -0.24

3.20 0.47 1.54 0.10 0.80 0.06 0.77 -0.01 1.44 0.33 3.51 -0.20 2.75 -0.37 5.21 -0.10 6.45 -0.55

-1.29 -0.63 -0.02 -0.24 -0.15 0.51 -0.02 0.63 2.88 0.23 -0.47 0.15 -1.27 1.03 -0.38 -0.31 -0.19 1.36

1.38 2.79 -1.83 -1.66 -0.86 -1.81 -0.63 -1.53 -3.89 -2.58 -0.89 0.23 -1.75 -0.56 0.57 -0.10 -2.70 -1.36

7.52 -0.37 5.79 0.08 2.45 0.17 2.94 0.19 -6.11 -0.31 3.30 0.80 2.02 0.49 -1.74 0.64 2.45 1.54

-1.47 -1.41 1.76 -0.07 0.81 -0.04 0.64 -0.20 4.64 -0.13 0.35 -1.07 -0.65 -0.81 2.83 -0.98 2.52 -2.17

-0.32 1.12 -0.21 -0.13 -0.19 -0.24 -0.18 -0.30 -0.54 -0.29 -0.05 0.37 0.22 0.03 -0.53 0.98 -0.09 0.68

0.84 0.70 0.74 -0.72 0.10 2.07 0.00 2.25 0.24 -9.37 0.35 0.51 -0.14 4.12 0.79 -5.86 0.29 -1.09

0.02 0.74 0.13 1.48 -0.20 0.65 -0.20 0.82 0.09 2.06 -0.16 1.77 -0.35 1.02 0.09 2.87 -0.52 4.18

-0.97 1.57 0.35 0.64 0.17 -1.47 0.10 -1.65 0.54 -0.82 0.00 -1.26 0.39 -3.52 -0.62 -1.61 0.15 -4.61

0.15 -1.10 -1.30 -0.69 -3.55 1.52 -3.79 1.59 4.59 -2.34 -1.01 0.49 -4.11 2.64 3.81 -1.59 -1.66 1.36

0.14 -1.84 -0.55 1.39 -0.12 1.78 -0.22 1.79 -0.50 2.91 -1.95 -0.81 -1.17 0.14 -1.96 -0.81 -4.04 -0.08

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References

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