Elsevier

Engineering Structures

Volume 245, 15 October 2021, 112904
Engineering Structures

Probabilistic demand models and fragilities for reinforced concrete frame structures subject to mainshock-aftershock sequences

https://doi.org/10.1016/j.engstruct.2021.112904Get rights and content

Highlights

  • A bivariable function considering both mainshock and aftershock intensity measures is used to develop demand model.

  • A Bayesian approach is used to calibrate demand models.

  • The Gardoni bounds is used to quantify the variability in the fragility estimates.

  • The uncertainties in the seismic fragility estimates are quantified.

Abstract

This paper presents a formulation for developing seismic demand models and estimating fragilities for reinforced concrete (RC) structures under mainshocks (MSs) only and mainshock-aftershock (MS-AS) sequences. The demand model for the MS only is a function of the MS intensity, while the model for the MS-AS sequence is a function of two variables, one describing the MS and one describing the AS. A Bayesian method is used to calibrate the demand models. The demand models are then used to estimate the fragility of RC structures. Predictive fragility curves are developed to include the uncertainties in the model parameters estimated by Bayesian approach, and confidence bounds of the fragility curves are developed to separate the effects of the uncertainty in the model parameters from the other sources of uncertainties. The proposed formulation is illustrated using a typical five-story RC frame building. The results show that the fragility curves for MS only tend to significantly underestimate the fragility curves for MS-AS sequences. Also, the uncertainty in the fragilities for the MS-AS sequence is, as expected, larger than that considering only the MS.

Introduction

Strong earthquakes usually result in significant damages of structures, and many studies in the literature have examined the seismic responses and fragilities of structures under strong earthquakes. Examples include Huang et al. [1], Bai et al. [2], Ranjkesh et al. [3] and Xu and Gardoni [4]. A mainshock (MS) of an earthquake is often accompanied by several aftershocks (ASs). Since the intensities such as the magnitude and peak ground acceleration of ASs are usually smaller than the intensities of the corresponding MS, most structural design codes do not consider AS characteristics as important factors in the seismic design [5]. However, many earthquake events in history have shown that AS can bring significant damage to structures and infrastructure. For example, Han et al. [6] reported that a number of structures and infrastructures survived after the MS of the 1999 Chi-Chi earthquake, but collapsed after following ASs. After the MS (with magnitude of 8.0) of the 2008 Wenchuan earthquake, five ASs with magnitude greater than 6.0 were recorded during the next 20 days [7], which lead to thousands of mud-rock flows and landslides [8]. Five months after the MS of the 2010 Christchurch earthquake in New Zealand, an AS with magnitude 6.2 caused widespread structures and infrastructures damage across Christchurch [9], which brought the death of 185 people [10]. The 2016 Central Italy seismic sequence caused heavy damage and casualties in the local town and nearby villages and hamlets [11]. All these events show that it is important to consider the AS effects in the structural design and seismic analysis.

In recent years, more researchers started to examine the influence of mainshock-aftershock (MS-AS) sequence on structural response and fragility assessment. Kumar et al. [12] conducted one of the first studies that quantified the effects of cumulative seismic damage on fragilities. Kumar and Gardoni [13], [14] developed a probabilistic model to compute the degradation of structural properties due to seismic damage and developed fragility curves for aftershocks. Kumar et al. [12] and Kumar and Gardoni [13], [14] showed that seismic damage due to MSs significantly increases the fragilities for ASs. Argyroudis et al. [15] summarized the recent fragility assessment of critical transportation infrastructure and proposed a framework for the fragility assessment for transport systems of assets under multiple hazards. Goda [16] investigated nonlinear behaviors of Single-Degree-of-Freedom (SDOF) systems subject to MS-AS sequence and found that ASs result in an obvious increase of ductility demand. Raghunandan et al. [17] analyzed the collapse fragility of reinforced concrete (RC) buildings using scaled MS and AS, and concluded that the collapse fragility could significantly increase when MS-damaged structures experience AS. Nazari et al. [18] used the back-to-back incremental dynamic analysis (back-to-back IDA) approach [19], [20] in both MS and AS time history analysis to develop demand models for a wood frame building. Zhang et al. [21] studied the state-dependent fragility curves of wood-frame houses in Canada under MS-AS sequences. They found that the AS caused significant additional damage and increased 4.8% damage probability of the Red tag damage state (no access) compared with the MS. Silwal and Ozbulut [5] and Veismoradi et al. [22] performed simulations on structures with energy dissipation devices and found that the impact of AS on structural response could not be neglected and energy dissipation devices make structures have higher capacities to resist AS. Di and Malavisi [23] studied the impacts of masonry infills on bare RC frame structures for MS-AS sequence. They pointed out that the masonry infills provided additional capacity to resist MS and AS, but both structures with and without masonry infills suffered significant damage when MS residual drifts occurred. Jeon et al.[24] proposed a framework for aftershock fragility assessment. They used the IDA approach for MS to simulate different initial damages, and used a cloud method to compute the AS damage and AS fragility curves. Consistently with the findings of Kumar et al. [12] and Kumar and Gardoni [13], Jeon et al. [24] noted that if a structure suffered significant damage in MS, the probability of being in a higher damage state in AS increases. Yu et al. [25] examined the collapse capacity using a number of real and artificial MS-AS sequences through SDOF systems. The result shows that the AS could yield a reduction of structural capacity and an increase in the variability of the structural demand. Li et al. [26] investigated the collapse fragility of steel structures subjected to MS-AS sequence. They found that the collapse capacity might reduce significantly under AS when a high intensity MS attack the structure. Zhang et al. [27] studied the seismic risk of a high-rise building considering MS-AS sequence. They found that the deformation capacity of the lateral force-resisting system reduces under MS.

However, many demand models for MS-AS sequence available in the literature are functions of the MS or AS intensity alone. In addition, the AS demand models based on limited number of MS initial damage values cannot represent all MS damage scenarios, and the damage values may not exactly equal to real MS damage. Although the back-to-back IDA or other hybrid IDA methods are simple, they do not reflect the real damage caused by the MS-AS sequence. Besides, the IDA approach is computationally expensive as it requires a large number of time history analyses [17], [24]. The computational cost would be extremely heavy when each time history analysis involves multiple ASs.

Secondly, many studies simply consider the MS-AS sequence as one ground motion record in the analysis, rather than analyzing the impact of each shock separately (e.g., Hosseinpour and Abdelnaby [28]). Those studies cannot reflect how the damage condition and fragility change with the number and size of shock sequences. In addition, previous studies use the inter-story drift as the engineering demand parameter (EDP) when quantifying the demand or damage condition for MS-AS sequence. Some researchers [14], [29], [30] found that the drift-based EDP cannot reflect the accumulative damages under sequential shocks, and therefore, may not be a good measure for demand modeling or fragility analysis.

Thirdly, some works reviewed above-considered fragilities as deterministic values, which neglect the uncertainties in the estimates (i.e., the statistical uncertainty, see Gardoni et al.[31]). They did not account for the uncertainty inherent in the demand models and provide biased estimates of the demand values. The fragility curves were calculated by the mean or median value of the demand models and did not capture the uncertainties involved in demand estimation. Jalayer et al. [32] used Bayesian Cloud Analysis to consider the uncertainties related to structural modelling and ground motion. As a result, most of them did not provide a potential interval of the fragility curves and there is no more information on the seismic fragility of structures for engineers to make a better decision for mitigation or maintenance planning.

This paper proposes a new formulation for the modeling of the structural demand and fragility analysis considering MS-AS sequences that addresses the above issues. To consider the influence of MS and AS together, we use unscaled MS-AS ground motions in a time history analysis and propose a demand model for MS-AS sequence that contains variables from both MS and AS. We add AS intensity in the MS-AS demand model to consider the impact of each AS. To quantify the uncertainties in the seismic fragilities, we calibrate demand models using the Bayesian approach and then construct the Gardoni bounds (Gardoni et al. [31]) to quantify the variability in the fragilities due to the statistical uncertainty. We implement the proposed formulation in the analysis of a typical five-story RC building considering the Park-Ang damage index as the damage indicator to compute the structural damage for MS and each corresponding AS.

The paper is organized as follows: Section 2 proposes an approach to formulate the demand models for an MS only and an MS-AS sequence. In Section 3, based on the proposed demand models, we review how to construct a predictive estimate of fragility (Gardoni et al.[31]) and develop the Gardoni bounds. Finally, in 4 Time-history analysis and seismic demand modeling of a typical five-story RC building, 5 Fragility analysis of the example building, we use a typical five-story building to illustrate the proposed formulation and obtain the difference of fragility between MSs only and MS-AS sequences.

Section snippets

Proposed formulation for seismic demand models

Seismic demand is the response of a structure under earthquake loads. This section first briefly reviews the commonly used EDPs and intensity measures (IMs) for seismic demand estimation, and then discusses the selection of EDP and IMs for the MS-AS demand modeling. After that, we present how to develop probabilistic models for seismic demands, and how to calibrate the models using a Bayesian approach when data are available.

Assessment of structural fragility

This section described how to estimate the structural reliability of buildings using demand models developed with the formulation proposed in Section 2.

Following the conventional notation in structural reliability [46], [47], we write a limit state function g(IM,Θ) as[48]gIM,Θ=C-DIM,Θwhere IM=IMM for MS only and IM=[IMM,IMA] for MS-AS sequence; C is the capacity of the building; D(IM,Θ) is the demand on the building given by Eq.(4) – (7). The fragility of a structure can be written as [48], [49]

Time-history analysis and seismic demand modeling of a typical five-story RC building

In this section, we implement the proposed formulation for the analysis of a typical five-story RC frame building. The section begins with an introduction of the selected building structure and the finite element (FE) model. After that, we select a set of MS-AS sequences as the input ground motions and perform time-history analysis to obtain the seismic demands. Finally, we develop probabilistic demand models following Sections 2 using the data obtained from the time-history analysis.

Fragility analysis of the example building

This section develops fragility curves for the example five-story building using the demand models calibrated in Section 4. First, we discuss the capacities C in limit state function in Eq. (11). Next, we show the fragility curves considering the MS only and the MS-AS sequence, and discuss the differences in the results.

Bai et al. [61] proposed damage states, corresponding capacities (or limit states), and a probabilistic formulation to link limit states and damage sates. For this example,

Conclusions

This paper proposed a new formulation for the seismic demand modeling and fragility analysis of reinforced concrete structures considering the mainshock (MS) and mainshock-aftershock (MS-AS) sequence. Different from previous studies, we proposed to use spectral acceleration averaged over a certain period of range (Saavg) as the intensity measure (IM) in the demand modeling and fragility estimating, which capture Park-Ang damage index (DI).

For the MS demand model, we formulate the mainshock

CRediT authorship contribution statement

Zhou Zhou: Investigation, Writing - original draft, Software, Formal analysis, Visualization, Methodology. Hao Xu: Writing - original draft, Formal analysis, Visualization, Methodology. Paolo Gardoni: Writing - review & editing, Methodology, Formal analysis. Dagang Lu: Conceptualization, Supervision, Funding acquisition. Xiaohui Yu: Writing - review & editing, Conceptualization, Supervision, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported in part by the Youth Science Foundation of Heilongjiang, China (YQ2020E023), the National Science Foundation of China (51778198, 52078176), and the China Scholarship Council (Student number: 201806120239). Opinions and findings presented are those of the writers and do not necessarily reflect the views of the sponsor.

References (61)

  • C.H. Zhai et al.

    Inelastic displacement ratios for design of structures with constant damage performance

    Eng Struct

    (2013)
  • H. Xu et al.

    Probabilistic capacity and seismic demand models and fragility estimates for reinforced concrete buildings based on three-dimensional analyses

    Eng Struct

    (2016)
  • A. Furtado et al.

    Mainshock-aftershock damage assessment of infilled RC structures

    Eng Struct

    (2018)
  • W. Zhang et al.

    Experimental investigation and low-cycle fatigue behavior of I-shaped steel bracing members with gusset plate connections

    Thin-Walled Struct

    (2021)
  • Q. Huang et al.

    Probabilistic seismic demand models and fragility estimates for reinforced concrete highway bridges with one single-column bent

    J Eng Mech

    (2010)
  • R. Han et al.

    Seismic loss estimation with consideration of aftershock hazard and post-quake decisions

    ASCE-ASME J Risk Uncert Eng Syst A: Civil Eng

    (2016)
  • S. Hu et al.

    Stochastic procedure for the simulation of synthetic main shock-aftershock ground motion sequences

    Earthq Eng Struct Dyn

    (2018)
  • Z. Wang

    A preliminary report on the Great Wenchuan Earthquake

    Earthq Eng Eng Vibrat

    (2008)
  • Lin SL, Giovinazzi S, Pampanin S. Loss estimation in Christchurch CBD following recent earthquakes: validation and...
  • A. Sextos et al.

    Local site effects and incremental damage of buildings during the 2016 Central Italy earthquake sequence

    Earthquake Spectra

    (2018)
  • R. Kumar et al.

    Effect of cumulative seismic damage and corrosion on the life-cycle cost of reinforced concrete bridges

    Earthquake Eng Struct Dyn

    (2009)
  • R. Kumar et al.

    Modeling structural degradation of RC bridge columns subjected to earthquakes and their fragility estimates

    J Struct Eng

    (2012)
  • K. Goda

    Nonlinear response potential of mainshock–aftershock sequences from Japanese earthquakes

    Bull Seismol Soc Am

    (2012)
  • M. Raghunandan et al.

    Aftershock collapse vulnerability assessment of reinforced concrete frame structures

    Earthq Eng Struct Dyn

    (2015)
  • N. Nazari et al.

    Effect of mainshock-aftershock sequences on woodframe building damage fragilities

    J Perform Constr Facil

    (2015)
  • Ryu H, Luco N, Uma SR, Liel AB. Developing fragilities for mainshock-damaged structures through incremental dynamic...
  • Uma SR, Ryu H, Luco N, Liel AB, Raghunandan M. Comparison of main-shock and aftershock fragility curves developed for...
  • L. Zhang et al.

    Mainshock-aftershock state-dependent fragility curves: A case of wood-frame houses in British Columbia, Canada

    Earthq Eng Struct Dyn

    (2020)
  • F. Di Trapani et al.

    Seismic fragility assessment of infilled frames subject to mainshock/aftershock sequences using a double incremental dynamic analysis approach

    Bull Earthq Eng

    (2019)
  • J.S. Jeon et al.

    Framework of aftershock fragility assessment–case studies: older California reinforced concrete building frames

    Earthq Eng Struct Dyn

    (2015)
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