Probabilistic seismic response and uncertainty analysis of continuous bridges under near-fault ground motions

Abstract

Performance-based seismic design can generate predictable structure damage result with given seismic hazard. However, there are multiple sources of uncertainties in the seismic design process that can affect desired performance predictability. This paper mainly focuses on the effects of near-fault pulse-like ground motions and the uncertainties in bridge modeling on the seismic demands of regular continuous highway bridges. By modeling a regular continuous bridge with OpenSees software, a series of nonlinear dynamic time-history analysis of the bridge at three different site conditions under near-fault pulse-like ground motions are carried out. The relationships between different Intensity Measure (IM) parameters and the Engineering Demand Parameter (EDP) are discussed. After selecting the peak ground acceleration as the most correlated IM parameter and the drift ratio of the bridge column as the EDP parameter, a probabilistic seismic demand model is developed for near-fault earthquake ground motions for 3 different site conditions. On this basis, the uncertainty analysis is conducted with the key sources of uncertainty during the finite element modeling. All the results are quantified by the “swing” base on the specific distribution range of each uncertainty parameter both in near-fault and far-fault cases. All the ground motions are selected from PEER database, while the bridge case study is a typical regular highway bridge designed in accordance with the Chinese Guidelines for Seismic Design of Highway Bridges. The results show that PGA is a proper IM parameter for setting up a linear probabilistic seismic demand model; damping ratio, pier diameter and concrete strength are the main uncertainty parameters during bridge modeling, which should be considered both in near-fault and far-fault ground motion cases.

Keywords

continuous bridge / probabilistic seismic demand model / Intensity Measure / near-fault / uncertainty

Introduction

The concept of performance-based seismic design (PBSD) [¹,²] has been widely accepted by the researchers in current earthquake engineering research. This general design philosophy could focus on comprehensive performance objectives such as the structure’s usability, safety, and economics [³]. On this basis, Pacific Earthquake Engineering Research (PEER) Centre developed a new-generation of performance-based earthquake engineering (PBEE) framework based on full probability theory, involving four generalized variables: the Intensity Measure (IM), the Engineering Demand Parameter (EDP), the Damage Measure (DM), and the Decision Variable (DV) [⁴–⁶]. Nevertheless, this seismic design procedure contains a large number of uncertainty parameters, such as seismic hazard, seismic demand, seismic capacity, and structure modeling. Once the performance level with given exceeding probability is ensured, the uncertainty in either seismic demand or capacity may significantly increase the realization cost [⁷]. Therefore, how to recognize and quantify the uncertainty parameter are important to push PBEE design theory into practical engineering application.

As for specific modeling uncertainty in engineering structures, previous studies have already recognized the importance of modeling parameter uncertainty and developed appropriate methodologies for data quantification [⁸–¹⁰]. However, most of the seismic response analyses were mainly based on far-fault earthquake ground motions.

Pan et al. [¹¹] studied 10 typical continuous bridge samples of New York State by Latin Hypercube sampling method and restricted pairing approach. Uncertainties associated with the material strength, bridge mass, friction coefficient of bearings, and expansion-joint gap size were considered in the analysis. The fragility curve was calculated for each bridge with 10 different selected far-fault records. The results showed that the bridge mass and friction coefficient of bearings have a significant effect on the seismic response.

Tubaldi et al. [¹²] concluded that the seismic response and vulnerability of self-consolidating concrete (SCC) bridges could exhibit dual load path result. A fully probabilistic approach was proposed to consider the input uncertainty for both record variability and the model parameters, incremental dynamic analysis (IDA) was used to propagate all pertinent sources of uncertainties to the seismic demand. The results showed that the response variability induced by model parameter uncertainty is significant.

Padgett and DesRoches [¹³] used gross geometries and far-fault ground motion variability to evaluate the modeling parameters which can affect the seismic response based on a retrofitted bridge sample. The results revealed the savings in simulation, computational effort, and the fragility estimation can be achieved through a preliminary screening of modeling parameters. However, the propagation of these potentially variable parameters tends to be dominated by the uncertainty in ground motion and structural geometry parameter.

Wang et al. [¹⁴] analyzed the uncertainty influence based on the fragility of bridges, where the uncertainty sources were concluded from recent collapsed bridge. The results showed that incorporating modeling uncertainties cannot only increase the dispersion in the fragility but also shift the median capacity of components to a smaller value. Since the key parameters can influence the design details in a uniform risk design framework, such impacts on the bridge fragility will further change the estimated risk level.

The above-cited studies were mainly focused on far-fault ground motions. Compared with far-fault ground motions, near-fault ground motions are considerably different as they are characterized by short duration, pulse-like ground motion (one or more pulses), significant vertical component and different representative IM parameters [¹⁵–¹⁷]. Therefore, the seismic response, and the probabilistic seismic demand model (PSDM) may be different.

In this paper, the effects of near-fault ground motions are applied to a regular continuous highway bridge to compare the IM parameters and establish the PSDM model for uncertainty parameter analysis; a specific distribution of uncertainty parameters concerning geometry, strength, mass and damping ratio are selected for the calculation both in near-fault and far-fault conditions, the results are presented as the absolute difference (“swing”) between the low and high values from each parameter distribution to draw the final tornado quantification diagram [¹⁸].

Bridge modeling and selected uncertainty range

Bridge modeling

This paper only considers regular highway bridges, whose seismic responses are basically controlled by fundamental vibration mode. The specific definition of regular highway bridges according to the Chinese Guidelines for Seismic Design of Highway Bridges [¹⁹] is reported in Table 1. Since the seismic response of a regular bridge is mainly controlled by fundamental vibration mode, it can be simplified as a single degree of freedom (SDOF) system, the pier and rubber bearings contribute to the stiffness while the girder contribute to the mass in the longitudinal direction. In the transversal direction, shear keys are placed on the bent cap and abutments to restrict the transversal displacement of superstructure as usually the case in China.

A typical regular highway bridge is designed first according to the Chinese Guidelines for Seismic Design of Highway Bridges [¹⁹], and adopted as the prototype bridge in this study. The bridge is a three-span continuous PC girder bridge with span arrangement of 3 m × 20 m and width of 13.5 m. The hollow-slab girder and double-column piers are adopted in the bridge, and elastomeric pad bearings are placed on the cap beams of the piers. Each column of the piers has a circular cross section with a diameter of 1.2 m, and a height of 12 m. C30-grade concrete with a compressive strength of 30 MPa is used in the piers and C50-grade concrete with a compressive strength of 50 MPa is used in the girder. HRB335 rebar (f_yk = 335 MPa) and HPB 300 rebar (f_yk = 300 MPa) are used as the longitudinal and transverse reinforcement of the column, respectively. The longitudinal reinforcement ratio is 1.0% and the transverse one is 0.4% according to the Chinese Guidelines [¹⁹]. Capacity design method is adopted in seismic design of the bridge, and the piers are designed as ductile elements, while other components are designed as capacity protected elements.

The 3-dimensional finite element (FE) model of the prototype bridge is built using OpenSees software [²⁰]. The aftershock data indicate that the bridge girder usually behaves elastically unless falling-off [²¹], so the hollow-slab girder is modeled using the elastic beam element. The bridge deck is simplified as lumped mass. Both the columns and the cap beams are modeled using nonlinear fiber beam-column element as shown in Fig. 1. The Kent-Scott-Park constitutive model is used for the concrete while the Giuffre-Menegotto-Pinto constitutive model is adopted for the reinforcement. Zero-length elastic spring elements are used to model the bearings between the girder and the cap beams. The abutments are assumed to follow a bilinear behavior after yielding, and the pile-soil effect is not considered in this study. The 3-dimensional FE model of the bridge is shown in Fig. 2. The numbers of total elements and nodes are 100 and 92, respectively.

Selected uncertainty range

The ultimate objective of the study is to quantify the significance during the seismic response analysis. To quantify the importance of different parameters, different sources of uncertainties that can affect both the structural seismic performance and demand are selected as the main influence factors; the details are listed as damping ratio (x), volume mass (m), pier diameter (D), longitudinal reinforcement diameter (d), concrete cover thickness (c), concrete compressive strength (f_c), and reinforcement yield strength (f_y).

Sources of uncertainty affecting structural performance are often characterized as either aleatory or epistemic. Aleatory uncertainty refers to inherent randomness, or stems from the unpredictable nature of events, whereas epistemic uncertainty is that which is due to a lack of knowledge, and stems from incomplete data, ignorance, or modeling assumptions. According to the related research and design codes, the selected uncertainty parameters are assumed as following given distribution as follows [²²–²⁵].

Under a strong ground motion, bridge structures will step into nonlinear behavior characterized with evident deformation, among which the input energy is primarily dissipated by the hysteretic displacement response. During the bridge modeling, damping ratio (x) is defined to help consider the energy dissipation mechanism. According to the Chinese Guidelines [¹⁹], the suggested damping ratio value for concrete structures is set as 0.05; during the modeling in OpenSees software, the Rayleigh damping is used with the combination of the structure mass and stiffness; according to the research of Nielson [²⁶], the damping ratio of structures is assumed following normal distribution and the coefficient of variation (COV) is assumed as 30%.

In research of Chinese concrete volume mass difference and variability, the concrete volume mass (m) should follow a normal distribution [²⁷], as for the specification of Chinese General Code for Design of Highway Bridges and Culverts [²⁸], the suggested m value should range as 25–26 kN/m³. On this basis, the volume mass of the girder and pier are assumed to follow a normal distribution, the mean value is set as 25.5 kN/m³, and COV parameter of the normal distribution is set as 10% [²⁹].

Geometry uncertainty parameters can also affect the nonlinear characteristics of the bridge structure, thus the pier diameter (D), the longitudinal reinforcement diameter (d), and the concrete cover thickness (c) are selected for the geometry uncertainty analysis following the normal distribution [³⁰]. The mean values are selected from the original designed parameter, the COV of geometry parameters in the normal distribution are selected as 5%.

According to the Chinese Code for Design of Concrete Structures [³¹], the standard compressive strength of concrete C30 is fixed by the standard test method: after the cube specimens with side length of 150 mm are maintained for 28 days, the overall distribution of the compressive strength should be more than 95% as 30 MPa for the reliability; for the HRB335 longitude reinforcement, the standard yield strength value f_y and the mean yield strength value f_ym should satisfy the relationship as f_y = (1–1.645s_y) f_ym to ensure the 95% guaranteed rate for the design value, where the s_y is the adjust coefficient. On these basis, the concrete compressive strength and the reinforcement yielding strength are assumed as lognormal distribution, the COV parameter of lognormal distribution for concrete and reinforcement strength are selected as 20% and 10%, respectively [³²]. Figure 3 listed the selected uncertainty parameters and their related distribution detail; according to the distribution range and the COV parameter value, the low and high limit values are calculated based on the designed distribution setting as in Table 2.

Probabilistic seismic analysis for near-fault earthquake

After selecting the key sources of uncertainty parameters and their related distributions, a PSDM should be developed to offer the basis for quantifying the parameter significance [³²]. Usually, the PSDM model is not only a key aspect for implementing the PBEE framework, but also the analysis basis for uncertainty parameters quantification. Generally, a PSDM model can be defined by a mathematical expression that correlates the IM parameter to a specific EDP parameter, and it can predict the value of the demand at a specific intensity value. Under the same IM parameter level, the influence of each uncertainty parameter can be compared with the fitting correlation result of the related EDP value.

The displacement drift ratio (DR) at the top of pier is selected as the EDP parameter in realistic measures of the bridge damage, which is correlated with the deformation ability for the piers under different performance level. The proposed EDP can consider the influences of slenderness ratio, axial load ratio, reinforcement ratio, stirrup ratio, and other design parameters [³⁰]. In previous studies by the authors of this work, it was already found that in the far-fault domain, the peak ground acceleration (PGA) and DR at the top of pier follow a linear correlation relationship in natural logarithm coordinate system [³³]. However, due to the special features and characters of the near-fault earthquake, more IM parameters should be discussed to obtain the ideal PSDM model for near-fault earthquake.

Ground motion selection

The site categories of the selected records are classified based on the average shear wave velocity to a depth of 30m according to the Chinese guidelines for seismic design of highway bridges: type I (V_s30>510 m/s), type II (260 m/s<V_s30≤510 m/s) and type III (150 m/s<V_s30≤260 m/s) [³⁴]. All earthquake signals for near-fault and far-fault earthquake on 3 different site conditions are selected from the PEER ground motion database [³⁵]. For better consider the effects of the vertical earthquake component, all the records are collected and input from vertical and horizontal direction with its original property. Since the number of near-fault earthquake records with significant velocity is quite limited, site condition IV is not considered at this step of research. Figure 4 shows the magnitude (Mw) and the fault distance (R) distribution for the inputs of near-fault on 3 site conditions, a total of 75 earthquake records are selected as the database. Each event of the selected near-fault earthquake record is marked with the characters of a significant pulse in the velocity time-history, ratio of peak ground velocity (PGV) to PGAlarger than 0.15 and fault distance (R) within 20 km.

Near-fault PSDM

The chosen IMs are listed as PGA, PGV, PGD, the ratio of PGV/PGA, S_a, and predominant period, where the PGA is the PGD is the peak ground displacement, S_a is the spectral acceleration at the fundamental period with 5% damping, predominant period is the specific period when the maximum spectral energy is concentrated. The equations for the linear regression are also listed in Fig. 5, it shows that the PGA, PGV, PGD, and S_a have a positive fitting distribution with the bridge DR, whereas the PGV/PGAand predominant period show a negative fitting distribution. From the fitting result it can be concluded that PGAis the most correlated parameter for the bridge DRwith the maximum determination coefficient R and the minimum standard deviation P.

Therefore, a PSDM for the bridge sample can be established. By selecting the DRand PGAas the parameters for EDP and IM, respectively, the linear regression formula obtained in this section can be applied to estimate the DRvalue as Eq. (1), where parameters a and b represent the slope and intercept of the fitting line as demonstrated in Fig. 6.

(1)\ln (DR)=a\ln (\mathit{PGA})+b\mathrm{.}

Based on the analysis pre-settings, this PSDM model mainly focuses on the regular highway continuous bridges which are subjected to a near-fault earthquake with significant pulse at 3 different site conditions (Fig. 6). It can provide an estimation for the DRunder the near-fault pulse-like records and offer a suitable fitting model for the uncertainty parameter analysis.

Uncertainty analysis of the prototype bridge

Selected IM

Given a certain IM parameter value, the significance of different uncertainty can be quantified by the EDP result base on the established PSDM model. In this section, the uncertainty analysis between the near-fault and far-fault earthquake are compared both in elastic and nonlinear cases. By selecting the PGAas the main IM parameter, with the accordance of Chinese seismic design code for urban bridges [³⁶], the seismic design level can be divided into 2 levels by adjusting the PGAvalue: frequent earthquake evaluation (E1) and rare earthquake evaluation (E2). E1 corresponds to the earthquake with a Return Period of 475 years and E2 corresponds to the earthquake with a Return Period of 2500 years. The details of requirements are listed in Table 3. Levels of E1 and E2 evaluation are separately focused on the structure seismic response during the elastic and inelastic case. According to the Seismic Hazard Map of China, the uniform PGAvalue for E1 and E2 are, respectively, 0.05g and 0.20g. For providing enough fitting accuracy, a total of 90 earthquake records are selected from the PEER ground motion database considering both near-fault and far-fault records on 3 site conditions to ensure calculation accuracy.

Figure 7 shows the magnitude (Mw) and the fault distance (R) distribution results for the uncertainty influence analysis, it consider both the near-fault and far-fault cases, the select criterion of the near-fault earthquake records remains similar as in Fig. 5, which ensure the significant pulse during the velocity time-history, a total of 90 earthquake records are selected as the analysis database.

Quantification of uncertainties

After the establishment of the PSDM model and IM level, the influence of different uncertainty parameters can be quantified. Among different global sensitivity analysis approaches, the Morris One-At-a-Time (MOAT) method is suitable for the linear regression result. Morris [³⁷] first presented the concept of elementary effects in 1991 as the derivatives over the space of parameters. This method is a simple but effective global method for screening a few important input parameters. The total effect and interaction effect for each parameter on the output can be described by the mean µ and standard deviation s.

During the uncertainty analysis, all parameters of the finite element model were first set equal to their respective mean values to carry the time-history analysis. Subsequently, each parameter was changed to the limit value separately and submitted to the original one for comparison. The absolute result difference of the low and high limit values from the parameter distribution is called the “swing” which can contribute to the tornado diagram as the final judgement [³⁸]. Based on the swing difference of uncertainty parameters at 3 different site conditions with near-fault and far-fault cases, the tornado diagrams are listed as follows where m is the mean value of 15 calculation results and the s is the standard deviation of the fitting results.

E1 Case

The quantification results of the uncertainties under E1 level at 3 different site conditions are present as Figs. 8–10. From the tornado distribution diagram, a few conclusions can be concluded: the distributions of m-swing act as the similar pattern with the s-swing; parameters of x, D, and f_c are the most effective uncertainty parameters for both near-fault and far-fault at 3 site conditions with a large swing value; when compare to the near-fault, the swing results of the far-fault are quite similar; the differences of the m-swing at 3 site conditions are not significant and the s-swing on site condition III correspond with the minimum value.

E2 Case

The quantification results of the uncertainties under E2 level on 3 site conditions are present as Figs. 11–13, from the tornado distribution diagram, it can be concluded that when compared with E1 level, the significance degree among the uncertainty parameters is more clear under the near-fault case; as to the near-fault earthquake, x is the most significant uncertainty parameter with the related distinctive swing difference; when compared with the far-fault ground motion, the near-fault correspond with smaller but more clear distinguish swing value; the uncertainty swing difference of far-fault among 3 site conditions is not significant.

Through the uncertainty quantification it can be concluded that: the parameter uncertainties of x, D, and f_c are all significant for both near-fault and far-fault earthquake at two seismic evaluation levels, however the near-fault earthquake show a better distinguish result for quantifying the influence degree of different uncertainties; the parameter uncertainties of m, d, c, and f_c have a weaker influence according to the quantification results.

Conclusions

The main purpose of this study is to establish a PSDM for regular highway continuous bridge under the near-fault ground motion and quantify the significance of different modeling uncertainty parameters during the seismic analysis. The results of this study demonstrate that:

1) PGAis a proper IM parameter for establishing the PSDM model for regular highway continuous bridges both in near-fault and far-fault earthquake cases.

2) In natural logarithm coordinate, the PSDM of PGA and the DRfollow a linear distribution relationship.

3) The damping ratio, pier diameter and the concrete strength are the significant uncertainty parameters for both near-fault and far-fault earthquake cases at two seismic design levels. Damping ratio acquired the maximum significant result under the near-fault earthquake with the E2 seismic design level.

4) Ongoing work is focused on more advanced stochastic modeling and uncertainty quantification methods to increase the result accuracy.

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