Effects of ground motion directionality on seismic behavior of skewed bridges considering SSI

Soil Dynamics and Earthquake Engineering 127 (2019) 105820

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Effects of ground motion directionality on seismic behavior of skewed bridges considering SSI

T

H.R. Noori, M.M. Memarpour , M. Yakhchalian, S. Soltanieh ⁎

Department of Civil Engineering, Faculty of Engineering and Technology, Imam Khomeini International University, Qazvin, 34148-96818, Iran

ARTICLE INFO

ABSTRACT

Keywords: Skewed bridges Ground motion directionality Soil-structure interaction Spatial distribution of demand Fragility curves

This paper investigates the quantitative effects of ground motion directionality on fragility curves of highway bridges. For this purpose, the influence of soil-structure interaction (SSI) is also included in finite-element modeling, and models of reinforced concrete bridges are generated assuming five different skew angles to account for geometric irregularity. A set of horizontal pairs of ground motions is imposed to the bridge models through nonlinear dynamic analyses. Twelve various angles of seismic incidence, ranging from 0° to 180°, are considered to account for ground motion directionality. Probabilistic seismic demand models are constructed for four structural components (i.e., column, deck at abutment seat, shear key, and elastomeric bearing), and fragility curves are developed. Additionally, the spatial distribution of column demand is investigated using a method giving the maximum resultant of column responses along the bridge horizontal axes. The results show that the most critical angle of seismic excitations does not necessarily coincide the orthogonal axes of the bridge. Response sensitivity of the bridge components to ground motion directionality effects varies from 30% to 50%. Furthermore, when the spatial distribution of column demand is considered, the results are up to 20% more conservative than those obtained along the bridge principal axes only. Fragility results also indicate that the response sensitivity of components to variation in the angle of seismic incidence and the response sensitivity of columns to the method of obtaining curvature ductility demand are more pronounced in SSI models compared to those in fixed-base models.

1. Introduction Heretofore, several studies have evaluated the sensitivity of structural response to ground motion directionality. Yet, considerable attention is required when including directionality effects in seismic risk assessment of bridges, as one of the important structures of a transportation network. In one of the earliest works concerning different characteristics of earthquake three-dimensional components, Penzien and Watabe [1] pointed out the consideration of imposing excitations along a variety of directions. They showed that the maximum acceleration intensity of a recorded three-component ground motion is directed along the epicenter, while the minimum acceleration intensity is pointed along the vertical orientation. To determine the most critical angle of seismic incidence, the critical values of structural demand should be compared among the different directions of imposing ground motions [2]. In a study conducted by Athanatopoulou [3], it was suggested that imposing ground motions with various incident angles may lead to increase in the recorded response of structure up to 80% in comparison to the case in which excitations are only imposed along the

⁎

principal axes of structure. Such a significant difference in structural response motivated researchers to investigate the directionality effects on irregular structures. Rigato and Medina [4] concluded that applying bi-directional ground motion with different angles of incidence can produce up to 60% difference in the maximum values of engineering demand parameters (EDPs) in asymmetrical buildings. In the context of ground motion directionality, bridge structures have also been in the center of attention due to different mechanisms of transferring lateral loads in orthogonal axes. Bisadi and Head [5] concluded that ground motion characteristics, in addition to structural dynamic properties, are determinative factors in nonlinear assessment of bridges subjected to various angles of seismic excitation. It has been generally observed that changing the incident angle of applied ground motions may cause an increase of demand when the structure is assumed to behave linearly [6–8]. Specifically, when nonlinearity effects are accounted for in bridge models [9], the increase of demand is even more significant than that for a linear model [5]. The results of the previous studies have confirmed that the response sensitivity of bridge structures to the incident angles of three-dimensional excitations depends on structural

Corresponding author. E-mail address: [email protected] (M.M. Memarpour).

https://doi.org/10.1016/j.soildyn.2019.105820 Received 6 March 2019; Received in revised form 16 July 2019; Accepted 13 August 2019 0267-7261/ © 2019 Elsevier Ltd. All rights reserved.

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properties and number of applied ground motion records [10]. By development of fragility curves as a tool for predicting seismic behavior of structures, a question arose that how the angle of seismic incidence can increase the probability of exceeding component demand from a certain level of capacity. In this regard, Banerjee and Shinozuka [11] concluded that ground motion directionality plays an important role in estimating damage probability of structural components, and the critical angle of seismic excitation does not coincide the bridge orthogonal axes. In another study, Torbol and Shinozuka [12] concluded that ground motion directionality can change the fragility median values from 22% to 66%. Fragility curves can be utilized to compare the effects of ground motion directionality at bridge system level or components level. However, these effects are more evident in fragility curves of individual components [13]. Many studies have focused on incident angles of bi-directional ground motions so far. Nevertheless, the importance of the applied approach for receiving component demands (e.g., column curvature ductility) corresponding to each angle of excitation has been rarely addressed. Usually, the critical demand is assumed to be obtained from the longitudinal or transverse direction of structures. In the case of buildings with regular plan, Emami and Halabian [14] indicated that the maximum value of EDPs such as inter-storey drift and curvature ductility is obtained along a horizontal angular distance from the orthogonal directions of structure. They also observed that the peak EDP values were increased from 8% to 39% when considering both the effects of ground motion directionality and spatial distribution of EDPs, simultaneously. Substantial role of soil-structure interaction (SSI) in seismic assessment of structures has been addressed by several researchers using different methods of modeling soil-foundation-bridge system. The approach used for modeling the supports of piers and the assumptions made on the characteristics of the soil profile can change the probability of damage to various components in skewed bridges [15]. Moreover, Kappos and Sextos [16] focused on the influence of pilefoundation stiffness on flexibility of bridges. They indicated that the foundation type, number of piles, and pile arrangement affect the ductility demand of piers. Mylonakis and Gazetas [17] compared the differences in ductility demand of bridge columns between fixed-base and flexible-base structures. It was specified that the interaction role is not always beneficial, and considering SSI does not necessarily lead to smaller response. Zhang et al. [18] developed two-dimensional finiteelement models of bridge classes simulating SSI effects. They concluded that bridge vulnerability is affected by structural properties and inelastic deformation of the underlying soil. Padgett et al. [19] investigated the sensitivity of fragility curves to random variables in modeling bridge elements and soil-foundation material parameters. They revealed that modeling the uncertainties associated with seismic performance assessment of bridges are mostly influenced by the undrained shear strength of site soil, structural damping ratio, and the gap between deck and abutment. Aygün et al. [20] utilized fragility curves for reliability estimation of bridges under soil failure situations. They modeled a three-dimensional finite-element model of bridge integrated with two-dimensional soil-foundation system using p-y method. The nonlinear p-y springs have been commonly utilized by researchers in recent years to develop soil-foundation-bridge models with the purpose of seismic assessments in interaction problems [21–23]. Importance of ground motion directionality has been acknowledged in previous studies. Seismic behavior and structural vulnerability assessment of skewed bridges have been the scope of several research works in the past [24–26]. However, because of uncertainties associated with propagation, reflection, and refraction of seismic waves, most of the prestigious bridge codes and guidelines only provide general specifications about directionality effects (e.g., Eurocode [27], Caltrans [2], and AASHTO guide specifications for LRFD seismic bridge design [28]). Furthermore, the concept of receiving the most critical response of structural components has been rarely examined in the

context of skewed bridges. In the present study, the significance of considering different angles of seismic incidence is investigated for component fragility curves of five simply-supported reinforced concrete girder bridges with skewed superstructure. This type of highway bridges is susceptible to severe damage induced by earthquake due to irregular geometric configuration [29,30]. Finite element models of the bridges are developed in OpenSees. Then, a set of 40 horizontal pairs of ground motions is employed for performing nonlinear dynamic analyses, and probabilistic seismic demand models (PSDMs) are generated to obtain the required information for fragility assessments. To estimate the critical demand of the bridge columns, the maximum resultant of the response vectors obtained from the bridge orthogonal axes are used as the EDP in the probabilistic analyses. Lastly, fragility curves are distinctly developed for four structural components including columns, deck at abutment seats, elastomeric bearings, and shear keys. In this research, two sets of bridge models (i.e., fixed-base condition and soilfoundation system) are considered to include SSI effects in damage probability estimations of the bridge components. The main goals in the present study are:

• Seismic vulnerability evaluation of skewed bridges including ground motion directionality effects and soil-structure interaction. • Monitoring the range of changes in fragility curves associated with • •

different angles of seismic incidence for each skew angle and dissimilar pier support modeling assumptions. Investigating the effects of including spatial distribution of column curvature ductility demand in fragility assessments. Investigating the response sensitivity of engineering demand parameters to variation in the angle of seismic incidence and the sensitivity of column demand to spatial distribution in SSI and fixed-base models.

2. Numerical modeling 2.1. Case studies Five two-span reinforced concrete bridges with five skew angles of 0°, 15°, 30°, 45°, and 60° are considered in this study. Span length of the bridges is 19 m, and total width of superstructure deck is 17.3 m. The superstructure is constituted by eight precast I-shaped AASHTO type IV girders covered by 0.2-meter thick concrete slab. The concrete girders are supported by seat-type abutments at the two ends of the deck and a three-column pier at the center of the longitudinal layout. Diameter of the columns is 1.2 m with center-to-center distance of 6.5 m. Moreover, the free height of the columns is 6.5 m. The columns are integrated by a cap beam, and superstructure girders are joined to the cap beam with pinned-type connection. The compressive strength of concrete is 28 MPa, and the minimum yield strength of steel reinforcement is 400 MPa. The cross sections of columns and cap beam are illustrated in Fig. 1, where details of longitudinal reinforcement can be found. Before conducting nonlinear analyses on the bridge models, response-spectrum analyses were initially performed, and bridge members such as deck, columns, and cap beams were designed according to the AASHTO LRFD bridge design specifications [31]. It was assumed that the bridges were located at a site with clayey soil in the Los Angeles area. Sectional properties such as dimensions and reinforcement ratios were designated with respect to the demands given by the response-spectrum analyses. Lastly, material and sectional assignments for individual elements were assumed to be identical for all the bridge models to make sure that the outcomes of nonlinear time-history analyses on finite-element models will be a function of the irregularity type of bridge; not dependent on variation in cross sectional detailing or geometric properties of the components. The central pier is supported by a pile group foundation consisting of 10-meter deep cast-in-drilled-hole (CIDH) piles with plan arrangement of 5 × 2. The diameter of piles is 1.0 m, and the center-to-center 2

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Fig. 1. Geometric Characteristics and the finite element model of bridge.

distances between the piles are 3.0 m. Longitudinal reinforcement detailing of piles is also shown in Fig. 1. The soil at the site is assumed as unsaturated soft clay with the undrained shear strength (SU) of 40 kPa and unit weight (γ) of 18 kN/m3. The abutment seat consists of exterior shear keys, elastomeric bearing pads, and a gap between deck and abutment backwall. Sacrificial shear keys are featured to encounter the transverse movements of the deck. These components are usually detailed to act as fuses breaking off during the design earthquake so that they protect abutment piles from excessive damage. Girders rest on 0.1 m thick elastomeric bearings at abutment, thus forces are horizontally transferred from deck to abutment by the sliding friction between girders and elastomeric pads.

materials is modeled using Concrete01, which is based on Kent-Park model [33]. The strain at maximum and crushing strengths for unconfined concrete is 0.003 and 0.006, respectively. These parameters are respectively 0.0011 and 0.005 for unconfined concrete. Besides, Steel01 is utilized to model concrete reinforcement. The grillage method recommended by Priestley et al. [34] is applied to model the superstructure. Since the deck is not expected to undergo the flexural yielding, it is divided into a grillage of elastic beam-column elements. This method prevents missing the effects of skewness on seismic demands. Each column in the pier is discretized into five displacementbased beam-column (dispBeamColumn) elements [35]. Rigid elements are utilized to model the portion of columns embedded in cap beam and pile cap [34]. Piles are modeled with 20 dispBeamColumn elements to incorporate nonlinearity along these components. Cross sections for columns, piles, and cap beam are defined using fiber sections. Rigid elements are also used to model the pile cap. They connect pile head nodes to those of columns.

2.2. Finite-element modeling Nonlinear finite-element models are developed using OpenSees platform [32]. The confined and unconfined behavior of concrete 3

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One of the computationally efficient methods of accounting for soilpile interaction is beam-on-nonlinear-Winkler foundation (BNWF) method. There are sophisticated techniques to include SSI effects in modeling and analysis of pile-supported structures with BNWF [36]. In this study, the lateral interaction between soil and pile is modeled using nonlinear p-y springs recommended by Boulanger et al. [37]. Dynamic p-y method characterizes the relationship between lateral resistance and deflection of soil material. To model the soil surrounding pile shafts, PySimple1, Tzsimple1, and QzSimple1 uniaxial materials in OpenSees are applied to respectively simulate soil lateral resistance, axial friction, and pile tip bearing. Lateral capacity of the trailing rows of the pile group is modified using the recommendations of FHWA [38] in order to consider the group efficiency of closely spaced pile shafts. A summary of the finite-element model and geometric characteristics of the superstructure, substructure, and foundation are indicated in Fig. 1. The constitutive components of abutment are shear keys, elastomeric bearings, pounding between deck and abutment backwall, backfill soil, and supporting piles. Since abutment behavior can influence the overall response of bridge, a sophisticated arrangement of nonlinear springs is defined as boundary conditions at the two ends of superstructure. To model the abutment components, the force-displacement relationships shown in Fig. 2 are assigned to zero-length elements. This method of simulating abutment behavior has been commonly used in some of the previous works (e.g. [39,40]). Elastomeric bearings are designed to meet AASHTO requirements. They are modeled using Steel01 material in OpenSees [40]. Exterior shear keys are modeled along the bridge transverse direction. Force-displacement relationship following the recommendations by Bozorgzadeh et al. [41] is considered to simulate nonlinear hysteretic behavior of these members. The pounding effect of deck-abutment is modeled using impact element. Analytical force-deformation relationship of this element has been proposed by Muthukumar [42]. Backfill soil is modeled using HyperbolicGap uniaxial material only along the longitudinal direction. This material is based on a nonlinear closed form hyperbolic force-deformation relationship developed by Shamsabadi [43] to model soilabutment-bridge interaction. Abutment piles contribute to transfer gravity and lateral loads in both the longitudinal and transverse

Fig. 3. Comparison of the moment-curvature relationships of a RC column obtained by OpenSees and UCFyber.

directions. A tri-linear hysteretic model recommended by Choi [44] is used to capture the response of these components. Fig. 2 displays material behavior and element arrangement of the abutment components. To verify the analytical model of columns, a moment-curvature analysis of a reinforced concrete section is performed in OpenSees. The results of this analysis are compared with those obtained from the cross section analysis software of UCFyber [39]. Fig. 3 illustrates the comparison of the moment-curvature relationships obtained by OpenSees and UCFyber. It can be seen that there is a satisfactory agreement between the two curves. As previously mentioned in this section, p-y elements are utilized to model the soil surrounding the piles. For this purpose, API specifications [45] are applied to calculate p-y material parameters implemented in OpenSees such as Pult and Y50. In regard to the soil depth, a p-y spring is modeled at each 0.5 m length of the piles. Fig. 4a shows the cyclic behavior of a p-y element modeled at depth of 0.5 m from the ground level. Pult is 134 kPa for this element, and Y50 is 0.031 m. The backbone curve of the p-y element approximates to Matlock's recommended backbone for soft clay [46]. It should be noted

Fig. 2. Element arrangement and material behavior of abutment components. 4

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Fig. 4. Cyclic responses of (a) p-y element, (b) shear key, and (c) elastomeric bearing pad.

that Pult is variable depending on the location of p-y elements along the depth of piles and the plan configuration of the pile group. Fig. 4b and c show the cyclic behavior of the shear key element and elastomeric bearing, respectively. In order to model inherent viscous damping in time-history analyses, Rayleigh (mass and stiffness proportional) damping is considered as 5% of critical damping at the first two modes of vibration [35]. The stiffness proportional damping is dependent on the deformations within the bridge structure and is regularly updated at each time step of analysis. As mentioned previously, two sets of bridge models with different attitudes of pier support modeling are generated for fragility assessments in this study: the models with fixed base and the models considering SSI. In the models with SSI, ground motions are imposed to the fixed nodes of p-y springs, HyperbolicGap elements, and abutment piles, whereas uniform excitation is applied to the fixed-base models.

time t given the incident angle of θ. The demand values are recorded along the longitudinal (x) and transverse (y) axes of the bridge models. Furthermore, rotation angle of i indicates the orientation of response vector and varies from 0° to 360°. Hence, Dt , i is the resultant demand of Dt , x andDt , y in time t given the incident angle of θ at the angular distance of i from the longitudinal axis. Alternatively, the square root of sum of squares (SRSS) method can also be used. The resultant demand value in this method will be the same as that given by Eq. (1). Actually, the maximum value of Dt , i can be calculated using SRSS of Dt , x and Dt , y at each time step of the nonlinear time-history analyses. This way, the rotation angle of i can be directly estimated. Fig. 5 presents a conceptual illustration of the ground motion directionality and the demand spatial distribution considered in this study.

3. Methodology

It should be noticed that demand vectors cannot be projected onto different orientations, with an angle of i, for all the components. For example, EDPs such as inter-story drift, pile head displacement, and ductility demand of columns are considered as the parameters for which Dt , i can be defined. However, spatial distribution is not applicable for EDPs of some bridge components (e.g., shear key and elastomeric bearing deformations), since the responses of such components are captured along the local axis of the elements used in finite element simulation. Moreover, spatial distribution is not applicable for deck longitudinal displacement at abutment seat, because this EDP is defined only along one of the main principal axes of bridge.

Dt , i = Dt , x cos i + Dt , y sin i

3.1. Definition of the applied directionality method In this study, seismic response of four structural components is evaluated given various directions of imposing earthquake excitations. The analytical procedure defined to predict seismic fragility of bridge columns is not only limited to variability in the angle of seismic incidence, but it also contains the spatial variability of column EDP oriented along the principal and non-principal axes of structure. This concept was previously discussed by Emami and Halabian [14] for building structures. First of all, horizontal pairs of the selected ground motions are rotated to the Strike-Normal (SN) and Strike-Parallel (SP) components. To take into account the uncertainty associated with the angle of seismic excitations, the orthogonal components of ground motions are rotated counterclockwise [47] to produce twelve angles of seismic incidence, θ (i.e., 0° < θ < 180°), with increments of 15°. The angle of seismic incidence is referred to the angle between SN component and the longitudinal axis of bridge. It should be noted that ground motion horizontal components are simultaneously imposed to the bridge models, and the vertical component is ignored. After producing ground motion pairs corresponding to different angles of seismic incidence, they are imposed to the bridge models through nonlinear time-history analyses, and responses of the bridge components are recorded along the orthogonal principal axes of structure. Lastly, the maximum possible response of each bridge component is calculated based on the demand values received form nonlinear time-history analyses. Obviously, the peak response of the bridge columns should be estimated using a combination rule. In other words, the resultant value of the response vectors should be obtained using curvature ductility demands recorded along the principal axes. To this aim, the orthogonal response vectors are projected onto several new orientations in plane using Eq. (1). In this equation, Dt , x and Dt , y are the column demands at

Fig. 5. Schematic view of directionality concept. 5

(1)

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3.2. Fragility framework Fragility curves are widely known as one of the useful tools for performance assessment of structures and a common format to statistically conceptualize the vulnerability of structural components. Actually, a fragility curve expresses the conditional probabilities that the structural demands caused by different levels of earthquake intensity exceed the structural capacity defined by a damage state. In this research, the results obtained from fragility curves are utilized to evaluate the effects of ground motion directionality. General formulation of fragility analyses is described as follows:

Pf = P {D

(2)

C|IM }

where Pf indicates the probability that a specific structural demand (D) exceeds a selected structural capacity (C) conditioned on a ground motion intensity measure (IM). A probability distribution for demand as a function of IM is called probabilistic seismic demand model (PSDM). Based on Eq. (2), PSDMs are required to be combined with reliable capacity thresholds (representing damage states) to assess the fragility of any component in a bridge model. To obtain EDP values, nonlinear dynamic analyses are conducted on each model using a set of ground motions with associated IM values. Then, the relation between EDPs and IMs is traced in a scatter plot with logarithmic space for individual components given the various angles of seismic incidence. This method is termed as ‘cloud’ analysis [47]. Finally, a linear regression is fitted to the cloud of EDP-IM data to calculate the conditional mean of ln(EDP) as follows:

Fig. 6. Dispersion values obtained using the investigated IMs assuming column curvature ductility as EDP for fixed-base columns given different skew angles.

The dispersion of EDP-IM values was calculated using Eq. (4) for primary components of the bridge models. The results show that PGV has the highest efficiency in most of the cases. However, the most efficient IM may vary for different components or skew models. As an example, Fig. 6 illustrates the dispersion values obtained using the investigated IMs assuming column curvature ductility as EDP for the fixed-base models considering θ = 0°. Since the vibration period of the bridge models fall into the velocity-sensitive region of the response spectrum, it would be a reasonable result that the component demands are strongly correlated with PGV. In recent studies, velocity-related IMs such as PGV have been suggested as the best IM of choice for probabilistic seismic demand modeling of bridges [21,25,50]. Periods of vibration modes are given in Table 1 for each bridge model. The mode shapes corresponding to the first and second modes of vibration are illustrated in Fig. 7. According to this figure, the natural vibration configuration of each of these two modes differs as the degree of bridge irregularity varies from skew angle of 0° to 60°. Additionally, values of the coefficients obtained for the regression lines fitted to EDP-PGV logarithmic plots are presented in Tables 2 and 3 for fixed-base and SSI models, respectively. In these tables, a and b are the intercept and slope coefficient of the regression line, and β is the dispersion. For instance, these plots are shown in Fig. 8 for primary components of the bridge model with skew angle of 0°.

(3)

1n(EDP ) = a + b1n(IM )

where a and b are the regression coefficients. The standard deviation of regression residuals can be obtained using the following equation:

St dev =

n i

(a + b 1n imi))2

(1n edpi n

(4)

2

where edpi and imi are EDP and IM values from ground motion record i, and n is the number of records. This logarithmic standard deviation is also termed ‘dispersion’ (β). Assuming a lognormal distribution for the conditional mean of EDP, the fragility curve of exceeding the EDP level of y is defined as:

P (D > C|IM ) = 1

1n y

(a + b 1n im)

3.4. Limit state models

(5)

where φ( ) is the standard normal cumulative distribution function, and P(D > C|IM) is the probability of exceeding the EDP level of y given IM = im.

To estimate the probability of damage as a function of IM, PSDMs should be convolved with capacity models. Definition of component limit states is one of the key issues in fragility formulations. Usually, damage states of individual components are identified with discrete levels of qualitative descriptions and their corresponding quantitative values. Bridge components are classified in two categories based on their role in vertical stability and load-carrying capacity of structure [40]: primary components (e.g., columns and abutment seat), and secondary components (e.g., shear keys and elastomeric bearings). Component damage thresholds (CDT) are defined at four different levels for primary components (CDT-0, CDT-1, CDT-2, and CDT-4). The selected CDTs, which are based on the Caltrans design and operational experience, are recommended by Ramanathan [40]. In the case of secondary components, CDTs are only applicable at two levels of damage (i.e., CDT-0 and CDT-1). The qualitative descriptions of bridge CDTs are presented in Table 4. Moreover, lognormal median and dispersion values for damage state distribution of the examined components are summarized in Table 5. In this research, the EDPs have been chosen according to investigations on the previous studies and the results obtained from these studies (see Ref. [40]). The component EDPs and the corresponding damage states (CDTs) are aligned with the prestigious codes and

3.3. Selection of appropriate IM Record-to-record variability is one of the uncertainties associated with the cloud analysis method which is due to different characteristics of the selected ground motions for nonlinear time-history analyses. The choice of an optimal IM can significantly reduce the degree of uncertainty in probabilistic assessments. ‘Efficiency’ of an IM candidate adopted for performance-based analyses is one of the factors of its optimality. An efficient IM reduces EDP variability with respect to the estimated median values [48]. In other words, lower dispersion of EDPIM results signifies a more efficient IM. For this research, the efficiency of four common IMs (i.e., peak ground acceleration (PGA), peak ground velocity (PGV), peak ground displacement (PGD), and spectral response at the first mode period of structure (Sa(T1)) was compared to find the IM yielding the best correlation with EDPs. The values of IM for horizontal components of a selected ground motion (IMGM1 and IMGM2) were obtained separately. Then, the geometric mean of IMGM1 and IMGM2 was taken as the resultant IM [49] for generating EDP-IM plots. 6

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Table 1 Periods of vibration modes of the bridge models. Bridge models Mode number

First Second Third Fourth

Skew angle 0°

Skew angle 15°

Skew angle 30°

Skew angle 45°

Skew angle 60°

SSI

Fixed base

SSI

Fixed base

SSI

Fixed base

SSI

Fixed base

SSI

Fixed base

0.72 0.20 0.16 0.14

0.59 0.18 0.16 0.14

0.73 0.21 0.15 0.14

0.60 0.18 0.15 0.14

0.73 0.20 0.16 0.15

0.60 0.18 0.16 0.15

0.74 0.20 0.20 0.15

0.60 0.20 0.18 0.15

0.74 0.29 0.2 0.14

0.60 0.27 0.17 0.15

Fig. 7. The first and second mode shapes of the bridge models (Skew angles = 0°, 15°, 30°, 45°, and 60°).

7

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Table 2 Properties of the regression lines representing PSDMs for fixed-base models. Skew angle

0° 15° 30° 45° 60°

EDP Column curvature ductility

Abutment unseating

Shear key deformation

Bearing deformation

a

b

β

a

b

β

a

b

β

a

b

β

−2.37 −2.42 −2.37 −210 −2.11

1.13 1.03 1.02 0.95 0.96

0.34 0.34 0.38 0.34 0.26

−0.34 −0.27 −0.38 −0.59 −0.15

0.77 0.73 0.74 0.76 0.6

0.30 0.33 0.30 0.28 0.23

0.24 0.24 −0.16 −0.02 −0.02

0.4 0.34 0.45 0.40 0.40

0.36 0.28 0.38 0.37 0.37

−0.94 −0.66 −0.7 −0.88 −0.35

0.85 0.76 0.76 0.79 0.68

0.47 0.46 0.44 0.44 0.43

operational experience [51]. Actually, they are the most suitable representatives of structural demand for use in seismic fragility assessment of bridges. Many researchers in recent years have utilized the same EDPs as those used in the present study to examine seismic behavior of reinforced concrete bridges [25,26]. These are the most important reasons behind the selection of column curvature ductility, deck displacement, shear key deformation, and bearing deformation as the bridge component demands for seismic assessment. The EDPs also match with the objectives perused in this research; e.g., investigating the effects of spatial distribution of column demand through the fragility framework.

approach of receiving structural response?’ Few Attempts have been made in the past to answer this question in regard to skewed bridges. In the present study, a large number of fragility curves are developed to study directionality effects on skewed bridges with fixed-base and SSI models. The fragility curves are obtained using logarithmic mean of EDP-IM plots for different components at various limit states. The curves describe the vulnerability of four bridge components by evaluation of EDPs such as column curvature ductility, deck relative displacement at abutment seat, shear key deformation, and bearing deformation. Fragility curves are generated for five bridge models with the skew angles of 0°, 15°, 30°, 45°, and 60°. Additionally, vulnerability of columns is investigated based on three non-identical approaches for imposing ground motions and receiving component response. The influence of spatial distribution of column curvature ductility demand is also evaluated through the fragility framework. The main purpose of this study is to evaluate the response sensitivity of the structural components in skewed bridges to ground motion directionality effects. Hence, it is noteworthy to point out that investigation of the system fragility is not in the scope of this study. Actually, individual fragilities of the bridge components are focused to make a quantitative comparison among the angles of seismic incidence. Furthermore, the role of SSI is included in sensitivity evaluation of the bridge components. It should be noted that the incident angle of 0° means that SN and SP components of ground motion are applied along the longitudinal and transverse directions of bridge, respectively.

3.5. Selection of ground motions In this research, horizontal pairs of forty unscaled ground motions are utilized to perform nonlinear time-history analyses. The horizontal response spectra of the selected ground motions match the median and logarithmic standard deviations predicted for an earthquake with strike-slip fault mechanism, moment magnitude of 7, and source-to-site distance of 10 km. The average shear wave velocity at the upper 30 m depth of soil layer (Vs30) is assumed to be 250 m/s. This standardized set of ground motion was proposed for use by Pacific Earthquake Engineering Research Center's Transportation Research Program (TSRP) with the purpose of structural and geotechnical analyses [52]. The list and characteristics of the selected ground motions are presented in Appendix A. Fig. 9 illustrates response spectra for the geometric mean of the selected horizontal components of ground motions compared to the target response spectra predicted by Boore and Atkinson [53] ground motion model.

4.1. Methods of receiving the component response Three distinct methods are utilized for the purpose of determining the maximum demand of the bridge components. In the first method (NN), the overall demand used for developing fragility curves is the maximum component response simply recorded along the principal orthogonal axes of bridge. In this method, the input ground motions are also imposed along the bridge principal axes. In other words, the SN and SP components of ground motions are imposed along the longitudinal and transverse directions of the bridge, respectively. The maximum demand of each bridge component is captured for each pair of ground motion records. Therefore, EDP values corresponding to the

4. Results and discussion In one hand, variation in incident angle of input motions can potentially result in significant difference in structural response. On the other hand, capturing the most critical demand of components has always been interesting for structural designers and bridge engineers. In this regard, a question arises that ‘how is the response sensitivity of the individual components to the ground motion directionality and the Table 3 Properties of the regression lines representing PSDMs for SSI models. Skew angle

0° 15° 30° 45° 60°

EDP Column curvature ductility

Abutment unseating

Shear key deformation

Bearing deformation

a

b

β

a

b

β

a

b

β

a

b

β

−4.99 −4.69 −4.23 −4.53 −4.96

1.63 1.52 1.35 1.4 1.48

0.75 0.78 0.84 0.79 0.79

−0.67 −0.67 −0.6 −0.68 −0.63

1.84 1.14 1.12 1.12 1.11

0.39 0.39 0.38 0.39 0.41

−3.88 −3.84 −3.62 −3.22 −2.38

1.13 1.12 1.06 0.91 0.99

0.93 0.89 0.9 0.91 0.92

−1.45 −1.6 −1.51 −1.51 −1.40

1.05 1.09 1.06 1.05 1.02

0.43 0.43 0.34 0.35 0.41

8

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Fig. 8. PSDMs for (a) columns with SSI, (b) deck unseating with SSI, (c) columns with fixed base, and (d) deck unseating with fixed base - skew angle of 0°.

PGV values of ground motions are obtained, and the fragility curves are developed. In the second method of receiving response, termed as DN, the horizontal pairs of the selected ground motions are imposed to the bridge models with twelve incident angles, θ. In this method, the SN component of each ground motion record makes an angle of θ with the bridge longitudinal axis. Nevertheless, the overall demand used in fragility assessments is the maximum component response recorded along the bridge principal axes. In the third method, referred as DD, the directionality of ground motions is considered as similar as that of the DN method, whereas the approach of receiving column response is different than that of the DN method. In the DD method, the column demands recorded along the bridge orthogonal axes in time step t (Dt , x and Dt , y ) are projected onto the radial direction corresponding to the angle of i (see Eq. (1)). The angular coordinate of i is the angle between the bridge longitudinal axis and a desired demand direction (see Fig. 5). This radial coordinate indicates the demand value recorded along a specific angle of i (0° < i < 360°) for an incident angle of θ in time step of t, Dt , i . Lastly, the maximum value of Dt , i is considered as the critical demand value applied in the PSDMs. The aforementioned method gives the highest value of Dt , i in each time step of nonlinear analysis as if the SRSS of Dt , x and Dt , y does. Using SRSS, the critical orientation of i can be directly calculated. A summary of description for the aforementioned attitudes toward receiving the peak component demand is represented in Table 6. Fragility curves of the five bridge

models with fixed-base and SSI considerations are developed for the three different approaches of NN, DN, and DD. 4.2. Effects of ground motion directionality In Fig. 10, fragility curves of four components are represented for the bridge model with skew angle of 0°. Each curve belongs to a specific angle of seismic incidence, so there are totally twelve fragility curves in each plot. The curves are developed for the DN method. Fragility curves related to the support modeling approaches (i.e., fixed base and SSI) are shown in separate plots. It is evident from the figures that the maximum discrepancy between the highest and the lowest bounds of the component fragility curves is variable depending on the bridge support condition. For example, the fragility curves associated with complete damage to the bridge columns (i.e., column failure) are illustrated in Figs. 10a and b for SSI and fixed base, respectively. The PGV value resulting in the failure probability of 50% ranges from 71 cm/s (θ = 15°) to 97 cm/s (θ = 90°). For the fixed-base model, this value varies between 65 cm/s (θ = 60°) to 83 cm/s (θ = 90°). In addition, vulnerability of a bridge component resulted from a particular incident angle can be quite different than that obtained from another one. It means that the critical angle of seismic excitations is not the same for all of the bridge components. It is also variable for dissimilar support modeling assumptions.

Table 4 Qualitative descriptions of bridge CDTs. The limit state sign

Primary

Secondary

CDT-0 CDT-1 CDT-2 CDT-3

Aesthetic damage Repairable minor functional damage Repairable major functional damage Component replacement

Aesthetic damage/Repairable minor functional damage Repairable major functional damage/Component replacement NA NA

Note: NA = Not Applicable.

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Table 5 Quantitative values of the bridge components limit states. Bridge component

Type

Column Abutment Seat

primary

Shear key Bearing

secondary

EDP

Unit

curvature ductility (μφ) displacement (δ) deformation deformation

Median values

– cm cm cm

dispersion

CDT-0

CDT-1

CDT-2

CDT-3

1 2.5 3.8 2.5

4 15.2 12.7 10.1

8 35.6 – –

12 53.3 – –

0.35 0.35 0.35 0.35

for different components of a bridge. The same plots are developed for the skewed bridge models. Comparing the fragility curves among the bridge models demonstrates that the geometric irregularity of bridge is another factor affecting vulnerability of components. In fact, the difference between the fragility curves is influenced by the variables such as the approach of support modeling, the angle of skewness, and the EDP considered in PSDMs. Fig. 11 presents fragility curves of four EDPs obtained using the three methods defined in Table 6. These curves are related to the bridge models with different angles of skewness considering SSI. In Fig. 11a, fragility curves associated with the curvature ductility demand of columns are presented for the bridge model with the skew angle of 60°. Each curve corresponds to one of the approaches of receiving response (NN, DN, and DD). As it was expected from the fragility curves related to the DD and DN methods, considering the spatial distribution of column curvature ductility demand leads to more conservative results. In other words, it is revealed that the probability of column failure in the examined limit state is higher in DD method rather than that in DN method. Similar trends can be found for the column fragility curves given all the bridge models with the examined skew angles. The curves related to the NN and DN methods are compared for primary and secondary components in Fig. 11. Generally, it can be concluded that including the effect of ground motion directionality increases the probability of exceeding structural demands from a predefined damage state at a specific seismic intensity. To obtain the fragility curves corresponding to the EDPs except the column curvature ductility, only the NN and DN methods are applied. As it was pointed out, projection of response vectors onto different orientations (DD method) cannot be applied for deck seating at abutments, shear keys, and elastomeric bearings. In addition to columns, vulnerability of the aforementioned components is sensitive to directionality effects. As it can be perceived from Figs. 11b to11d, the probability of damage to deck, shear keys, and elastomeric bearings increases at an angle of θ other than 0°. Besides, the order of vulnerability observed in the fragility curves related to NN and DN methods is similar to those shown in Fig. 11 for other limit states of the investigated components.

Fig. 9. Response spectra for the geometric mean of horizontal components compared to the target response spectra predicted by Boore and Atkinson [53] model.

The degree of response sensitivity of each component to ground motion directionality can be found from the range of differences between the upper- and lower-bound curves. This range of difference is small for some of the EDPs such as bearing deformation or column curvature ductility, whether the models are considered with fixed base or SSI. On the other hand, significant differences are observed between the fragility curves corresponding to deck unseating and shear key deformation. It can also be easily perceived that including SSI affects the probability of damage exceedance from a specific limit state. In regard to the effects of support modeling, when SSI effects are included in modeling and analysis, the probability of deck unseating reaches to 100% with lower PGVs in comparison to that when the bridge support is assumed to be fixed. In other words, deck at abutment seat is more vulnerable in the bridge model with SSI consideration since superstructure displacement is increased due to the movement of foundation and flexibility of pier base. Conversely, for every angle of seismic incidence, the probability of complete damage to columns (i.e., meeting CDT-3) is a larger value in the fixed-base model rather that in the model with SSI. To sum up, Fig. 10 shows that the critical angle of seismic incidence is dependent on the approach of support modeling and varies

4.3. Component vulnerability in different bridge models To interpret the results, the difference in median PGV values corresponding to each CDT is examined for the bridge models. Fig. 12 illustrates the PGV values associated with a failure probability of 50% (i.e., median PGV values) for CDT-3 of columns and CDT-1 of shear keys. This Figure compares the median PGV values obtained for the

Table 6 Methods of receiving component demands and direction of imposing ground motions. Method Description

Abbreviation

Direction of imposing ground motion

Approach of estimating response

Along the bridge orthogonal axes Along the angles of θ (0° < θ < 180°) Along the angles of θ (0° < θ < 180°)

Receiving maximum demand along the bridge orthogonal axes Receiving maximum demand along the bridge orthogonal axes Receiving maximum projected demand in plane along the critical angle of i

10

NN DN DD

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Fig. 10. Fragility curves of bridge components for 12 incident angles; Column failure (a) SSI (b) fixed base; deck unseating (c) SSI (d) fixed base; shear key break off (e) SSI (f) fixed base; bearing major functional damage (g) SSI (h) fixed base - skew angle of 0°.

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Fig. 11. Results of fragility analyses performed using the examined approaches of imposing ground motions and receiving response for (a) column curvature ductility demand - skew angle of 60°, (b) deck unseating - skew angle of 0°, (c) shear key break off - skew angle of 15°, and (d) bearing major functional damage - skew angle of 30°.

Fig. 12. Median PGV values corresponding to some CDTs obtained using the examined approaches of imposing ground motions and receiving response, for column curvature ductility at CDT-3 (a) fixed base and (b) SSI, and shear key deformation at CDT-1 (c) fixed base and (d) SSI.

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models with different skew angles and distinct support modeling assumptions. As it can be seen in Figs. 12a and b, the three applicable methods of imposing input motions and receiving the output responses are investigated using curvature ductility demand of columns as the EDP. According to Fig. 12b, for the models with SSI, median PGV values associated with CDT-3 are increased from skew angle of 0° to 60° when the NN method is used for fragility assessments. However, this increase is not of major significance when using the DN and DD methods. Fig. 12a shows the median PGV values corresponding to CDT-3 of columns for fixed-base models. Trends are almost similar to those for SSI models in Fig. 12b. However, an important change can be perceived by comparing Figs. 12a and b; there is an obvious difference between the median PGV values obtained from the NN method and those given by the two other methods. It is clear that as the skew angle is increased, this difference becomes greater in bridge models with SSI in comparison with fixed-base models. Therefore, it can be concluded that the effects of ground motion directionality are more highlighted in the models with SSI rather than those with fixed base. Based on the results, for the case in which the directionality effect of the input excitations is considered in analyses, including spatial distribution of column demands results in conservative fragility functions. In addition to columns, the median PGV values associated with the CDT of shear keys are estimated using the NN and DN methods. It is also found that considering SSI effects make the differences between the NN and DN methods greater in comparison with the case of fixed-base models. Figs. 12c and d attest the above-mentioned result for the bridge models with various angles of skewness.

Table 7 ΔDN-NN, ΔDD-DN, and ΔDD-NN percentage values for curvature ductility demand of columns given the examined skew angles, for models with fixed-base and SSI consideration. Skew angle

0° 15° 30° 45° 60° Average St. deviation

In the present study, quantitative indexes are used to compare the examined approaches of imposing input ground motions and receiving column response. To calculate these indexes, the ground motion intensity corresponding to the failure probability of 50% (PGV50), i.e., median PGV, is obtained for each approach of NN, DN, and DD. Then, the PGV50 values are used to calculate these indexes as follows:

=

50 50 (PGV NN PGV DN ) × 100 50 PGV NN

(6a)

DD DN

=

50 50 (PGV DN PGV DD ) × 100 50 PGV DN

(6b)

DD NN

=

50 50 (PGV NN PGV DD ) × 100 50 PGV NN

(6c)

SSI

ΔDN-NN (%)

ΔDD-DN (%)

ΔDD-NN (%)

ΔDN-NN (%)

ΔDD-DN (%)

ΔDD-NN (%)

30.0 35.4 29.7 31.3 31.7 31.6 2.3

2.0 1.8 1.9 8.8 11.0 5.1 4.4

31.4 36.6 31.1 37.3 39.2 35.1 3.6

31.5 41.6 52.5 51.9 47.6 45.0 8.7

2.0 0.0 3.5 10.9 19.7 7.2 8.1

32.9 41.6 54.2 57.1 57.9 48.7 11.0

50 50 where PGV50 NN, PGVDN, and PGVDD are the median PGV values corresponding to the NN, DN, and DD methods. Furthermore, ΔDN-NN is the relative difference in the median PGV values obtained from the DN and NN methods; ΔDD-DN is the relative difference in the median PGV values obtained from the DD and DN methods, and ΔDD-NN is the relative difference in the median PGV values obtained from the DD and NN methods. Since the DD method is only applied for the column EDP, ΔDDDN and ΔDD-NN cannot be determined for shear key, bearing, and deck at abutment seat. Fig. 13, as an example, provides an insight into the process defined via Eq. (6) for estimating Δ values. According to this figure, ΔDN-NN is 32%. In other words, when using the DN method, the 50% probability of complete damage to column is associated with a geometric mean PGV that is 32% lower than that of the NN method. From another point of view, in the case that the axes of horizontal ground motion pair coincide with the bridge principal orthogonal axes (NN), the median PGV value associated with CDT-3 of columns is 32% greater than the case that seismic excitation is imposed along different directions (DN). For the case that the angle of imposing seismic excitations varies and the spatial distribution is also considered (DD), the value of the index calculated by Eq. (6b) (ΔDD-DN) is 10%. ΔDD-NN gives the highest relative difference in median PGV values (ΔDD-NN = 39%). This is observed when the ground motion directionality and demand spatial distribution are both considered for one method (DD), and both of these effects are dismissed for another (NN). This quantitative comparison attests that the DD method is significantly more conservative attitude for obtaining curvature ductility demand. As it could be guessed, smaller earthquake intensity is required to cause the failure of a bridge column by using the DD method.

4.4. Effects of considering spatial distribution of column demand

DN NN

Fixed Base

Fig. 13. Differences of the median PGV values obtained using the three examined approaches of imposing ground motions and receiving response of columns.

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Fig. 14. Values of Δ computed for (a) column curvature ductility, (b) deck unseating, (c) shear key break off, and (d) bearing deformation at CDT-1 versus the skew angles of bridge.

Table 7 gives ΔDN-NN, ΔDD-DN, and ΔDD-NN percentage values of curvature ductility demand for the bridge models with different skew angles considering fixed base and SSI. This table shows the Δ values obtained using Eqs. (6a)-(6c). The effects of ground motion directionality and spatial distribution of column demand can be evaluated for various skew angles according to the results presented in Table 7. ΔDN-NN is 31.6% in average among the fixed-base bridge models. In other words, for the case that the seismic excitations are imposed only along the principal axes of bridge, the failure probability of 50% corresponds to an average geometric mean PGV that is 31.6% greater than the case that ground motion directionality is included (ΔDN-NN = 31.6%). However, this index is generally greater for the models with SSI (i.e., the average of ΔDN-NN is 45%). The average of ΔDD-DN values for fixed-base and SSI models are 5.1% and 7.2%, respectively. It means that the difference between the DD and DN methods is marginally greater in the models with SSI consideration. In total, it can be found that ΔDD-DN is significantly lower than ΔDN-NN and ΔDD-NN. The reason is that ΔDD-DN

only demonstrates the role of including spatial distribution of column demand, and the attitude of imposing ground motions is the same in both methods. To present a better comparison between the results achieved from the models with fixed base and SSI, Δ values for the four bridge components are plotted versus skew angles in Fig. 14. Fig. 14a indicates Δ values for the bridge columns. According to this figure, as the skew angle of bridge increases, ΔDD-DN also increases. It suggests that the influence of demand spatial distribution is highlighted in higher angles of skewness. Although ΔDD-DN values are nearly zero in the skew angles of 0° to 30° for both SSI and fixed-base models, this value is increased for the skew angles of 45° and 60°. Given the skew angle of 60°, ΔDD-DN rises up to 11% and 20%, for the bridge models with fixed-base and SSI, respectively. It can be concluded that the response of columns in skewed bridges is sensitive to the considerations of demand spatial distribution. In other words, given a specific value of PGV, the highest probability of complete damage to columns (i.e., μφ > 12) is given by a

Table 8 ΔDN-NN (%) values for bridge components. Skew angle

0° 15° 30° 45° 60° Average St. deviation

Fixed Base

SSI

Abutment unseating

Shear key deformation

Bearing deformation

Abutment unseating

Shear key deformation

Bearing deformation

22.6 15.5 19.7 −4.2 0.7 10.8 11.9

42.1 64.4 24.7 39.1 39.7 42.0 14.2

7.9 5.8 17.1 12.5 3.6 9.4 5.4

46.9 46.8 53.3 56.5 55.2 51.7 4.6

61.7 60.1 61.9 77.7 69.9 66.2 7.4

17.4 31.5 51.0 44.2 42.2 37.2 13.1

14

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response receiving orientation that is not along the longitudinal or transverse axis of bridge. Furthermore, with respect to Fig. 14a, it can also be found that among the fixed-base models the maximum percentage of ΔDN-NN appertains to the skew angle of 15°. However, changes of ΔDN-NN values are generally slight among different skew angles for fixed-base models (i.e., ΔDN-NN ranges between 30% and 35%). In bridge models with SSI, ΔDN-NN increases from 33% to 54% from the skew angle of 0° to 30°, and it remains at an almost steady value around 50% for the skew angles of 45° and 60°. Table 8 presents ΔDN-NN values for the EDPs of shear key, elastomeric bearing, and deck at abutment seat. It can be seen that considering SSI leads to increase in ΔDN-NN values obtained for each of these components. Moreover, Fig. 14b–d illustrate the changes in ΔDN-NN versus various skew angles for the bridge components. According to Fig. 14c, ΔDN-NN has a similar value for the skew angles of 0° to 30° in the bridge models with SSI. However, it rises up to a maximum value of 77.7% for the skew angle of 45°, and then it reduces from the skew angle of 45° to 60°. ΔDN-NN exhibits the greatest value for shear keys. Actually, shear keys are found as the bridge component exhibiting the highest sensitivity to ground motion directionality. When the bridge support is assumed to be fixed, ΔDN-NN increases to the maximum value of 64.4% for shear key in the abutment of the bridge model with skew angle of 15°. The role of shear key in abutments is to resist against the lateral forces transmitted from deck due to transverse movements. Therefore, shear keys are not considerably engaged under some angles of seismic incidence causing deck displacement be dominant along the bridge longitudinal axis. On the other side, when the stronger component of ground motion is oriented along the bridge transverse axis, deck displacement is dominant along the transverse direction, and thus there are larger shear key deformations. ΔDN-NN values for deck unseating and elastomeric bearings are also presented in Table 8, and their changes versus different skew angles are plotted in Figs. 14b and d for models with fixed-base and SSI consideration. In fixed-base models, there is an almost descending order for the variation of ΔDN-NN values corresponding to deck unseating from the skew angles of 0° to 60°. However, in the bridge models with SSI, ΔDNNN corresponding to deck unseating slightly increases by increasing the angle of skewness. For elastomeric bearings, the maximum variation of ΔDN-NN value among the bridge models with fixed base is less than 14% while this variation is about 34% among the models with SSI. ΔDN-NN rises up to 51% at the skew angle of 30° for elastomeric bearings in SSI models. It is noteworthy that median PGV values are significantly greater in the case of bridges with SSI assumption.

were developed for four bridge components (i.e., two primary and two secondary components). Based on the results, three methods were considered for the fragility assessments. In the first method (NN method), ground motions were imposed only along the bridge orthogonal axes. The maximum of the demand recorded along the longitudinal and transverse axes of bridge was also considered as the critical component demand. In the second method (DN method), the maximum value of the component response recorded along the longitudinal and transverse axes of bridge was captured for different angles of seismic incidence. In terms of receiving the critical demand of columns, another method was also taken into consideration (DD method); the method by which the maximum resultant demands could be resulted from an orientation other than the principal axes of bridge. Consequently, fragility curves were developed for three distinct approaches of imposing excitations and receiving component response. Furthermore, SSI effects were accounted for in three-dimensional finite-element models of bridges. To this aim, a model of pile group foundation and the surrounding soil was considered using the BNWF (dynamic p-y) method. This method was applied to provide the possibility of comparing fragility results obtained from SSI models with those obtained from the equivalent fixed-base models. Altogether, more than 4800 nonlinear dynamic analyses were conducted, and the outputs were processed. The most highlighted results are summarized as the followings:

• • •

•

5. Conclusions

•

In this paper, seismic vulnerability of skewed bridges was investigated under the influence of ground motion directionality. In this regard, five different skew angles were considered to generate the bridge models. For the phase of analyses, a ground motion record set was selected, and the horizontal pairs of the records were mathematically rotated along different directions to produce twelve angles of seismic incidence. Then, nonlinear dynamic analysis was performed given each incident angle, and time-history responses of individual components were recorded along the principal orthogonal axes. PSDMs

15

Including the directionality effects of the horizontal components of ground motion increases the probability of exceeding a predefined limit state for a structural component demand. The critical angle of seismic incidence varies with the change of EDPs, skew angles, and the pier support modeling assumptions. The maximum curvature ductility demand of the bridge columns is received along an orientation other than the principal orthogonal axes of bridge. The difference of median PGV values between the method that considers the spatial distribution of demand and the method in which the maximum demand is received along the bridge principal axes rises as the skew angle is increased. For the case that the highest demand value is measured along either the longitudinal or transverse axis of bridge, the median PGV associated with complete damage to columns is up to 20% higher than that for the case in which the receiving response is estimated using the demand spatial distribution method applied in this paper. The differences of median PGV values due to the change in the angle of seismic incidence and the approach of receiving response are higher for the models with SSI when compared with those for fixed-base models. It means that directionality effects are more pronounced when SSI effects are included in modeling and analysis. Vulnerability of bridge components such as deck at abutment, shear keys, and elastomeric bearings is also increased when ground excitations are imposed along directions other than longitudinal and transverse directions of bridge. In other words, when ground motions strike the bridge with an incident angle of θ (θ ≠ 0°), the damage to the above-mentioned components is up to 77% more severe than the case in which the incident angle of ground motions is 0°. Moreover, shear key is found to be the most sensitive component to directionality effects of ground motion.

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Appendix A Table A.1

Characteristics of the selected ground motions Record No.

Earthquake Name

Year

Station

Magnitude

Hypocentral Distance

Closest Distance

Vs30 (m/s)

Fault Normal Orientation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Mammoth Lakes-01 Chi-Chi, Taiwan Cape Mendocino Imperial Valley-06 Kocaeli, Turkey Imperial Valley-06 Chi-Chi, Taiwan Chi-Chi, Taiwan Kocaeli, Turkey Trinidad Spitak, Armenia Loma Prieta Chi-Chi, Taiwan Victoria, Mexico Loma Prieta Chalfant Valley-02 Chi-Chi, Taiwan Denali, Alaska Imperial Valley-06 Big Bear-01 Landers Northridge-01 San Fernando N. Palm Springs Loma Prieta Chi-Chi, Taiwan Chi-Chi, Taiwan Imperial Valley-06 Chi-Chi, Taiwan Duzce, Turkey Chi-Chi, Taiwan Loma Prieta Imperial Valley-02 Chi-Chi, Taiwan-03 Northridge-01 Chi-Chi, Taiwan-03 Loma Prieta Loma Prieta Chi-Chi, Taiwan Chi-Chi, Taiwan-06

1980 1999 1992 1979 1999 1979 1999 1999 1999 1980 1988 1989 1999 1980 1989 1986 1999 2002 1979 1992 1992 1994 1971 1986 1989 1999 1999 1979 1999 1999 1999 1989 1940 1999 1994 1999 1989 1989 1999 1999

Long Valley Dam (Upr L Abut) CHY036 Rio Dell Overpass – FF Delta Yarimca Calipatria Fire Station CHY034 NST Duzce Rio Dell Overpass, E Ground Gukasian Gilroy Array #4 TCU060 Chihuahua Fremont - Emerson Court Zack Brothers Ranch TCU118 TAPS Pump Station #10 El Centro Array #4 San Bernardino Yermo Fire Station Sylmar - Converter Sta LA - Hollywood Stor FF Morongo Valley Hollister - South Pine TCU055 CHY025 Brawley Airport CHY088 Duzce TCU061 Saratoga - Aloha Ave El Centro Array #9 TCU123 Jensen Filter Plant CHY104 Salinas - John & Work Coyote Lake Dam (Downst) CHY008 TCU141

6.06 7.62 7.01 6.53 7.51 6.53 7.62 7.62 7.51 7.2 6.77 6.93 7.62 6.33 6.93 6.19 7.62 7.9 6.53 6.46 7.28 6.69 6.61 6.06 6.93 7.62 7.62 6.53 7.62 7.14 7.62 6.93 6.95 6.2 6.69 6.2 6.93 6.93 7.62 6.3

15.52 44.74 24.55 35.17 25.07 58 46.82 89.2 99.52 78.22 36.68 36.79 46.07 38.29 57.86 17.47 44.49 84.89 28.9 47.33 86.28 21.87 41.57 12.66 51.31 36.74 33.13 44.29 69.24 14.09 42.81 32.35 15.69 39.64 21.78 42.95 49.58 35.49 69.73 57.97

15.46 16.06 14.33 22.03 4.83 24.6 14.82 38.43 15.37 – – 14.34 8.53 18.96 39.85 7.58 26.84 2.74 7.05 – 23.62 5.35 22.77 12.07 27.93 6.36 19.09 10.42 37.48 6.58 17.19 8.5 6.09 31.79 5.43 35.05 32.78 20.8 40.44 45.72

345.4 233.1 311.8 274.5 297 205.8 378.8 375.3 276 311.8 274.5 221.8 272.6 274.5 284.8 271.4 215 329.4 208.9 271.4 353.6 251.2 316.5 345.4 370.8 272.6 277.5 208.7 272.6 276 272.6 370.8 213.4 272.6 373.1 223.2 271.4 295 210.7 215

282 292 260 233 180 233 292 306 163 319 212 38 278 228 38 58 271 199 233 323 225 32 195 197 38 283 271 233 292 172 272 38 233 270 32 270 38 38 292 275

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