Seismic fragilities for non-ductile reinforced concrete frames – Role of aleatoric and epistemic uncertainties
Introduction
The consequence-based risk management paradigm [1] developed as part of the research program in the Mid-America Earthquake (MAE) Center requires quantitative modeling and propagation of all sources of uncertainty within a probabilistic risk assessment framework. This paradigm also has supported the MAE Center’s recent effort to examine the potential impact of future earthquakes that may occur in the New Madrid Seismic Zone (NMSZ) on civil infrastructure in the Central and Eastern United States (CEUS). The NMSZ is capable of generating great earthquakes infrequently (the return period of magnitude seven events in the NMSZ is believed to be approximately 500 years [2]). Buildings in population centers such as Memphis, TN and St. Louis, MO in proximity to the NMSZ have not been designed with any consideration of seismic effects until recently. Accordingly, a major portion of the building inventory in that region is at risk, to an unknown degree.
As part of this MAE Center effort, seismic fragilities, which describe the probabilities of failure to meet performance levels as functions of earthquake ground motion intensity, of three-, six-, and nine-story gravity load designed (GLD) reinforced concrete (RC) frames1 that typify pre-1990 RC construction practices for low-, mid-, and high-rise RC frames in the CEUS were evaluated in a recently published study [3]. These seismic fragilities were developed using finite element-based non-linear time history analysis (NTHA), in which the structural parameters were set equal to their median values, as did the analyses reported in most previous studies (e.g. [4]). Thus, the seismic demands on these frames and their fragilities reflected solely the uncertainty in earthquake ground motion intensity.
The question naturally arises as to the extent to which uncertainties in structural properties might influence the seismic demand and, by extension, the fragilities and damage estimates for GLD RC frames. The question is especially pertinent in light of the known deficiencies and uncertainties in the performance of beam–column joints in such frames (e.g., weak column–strong beam design, lack of shear reinforcement and insufficient bar development length within joints) [5], [6]. To address this question, this paper examines the contribution of uncertainties in structural strength, stiffness, etc. to the overall seismic fragilities of the GLD RC frames studied previously [3]. The sensitivity of frame response statistics to the uncertainties in beam–column joint model parameters, as well as material and structural properties, is investigated at various levels of earthquake hazard for Memphis, TN, which is the largest population center (approximately 1 million in its metropolitan area) close to the NMSZ. In addition, confidence bounds on the fragilities are developed to provide a sense of their accuracy for decision- and policy-makers in prioritizing risk mitigation efforts and in developing post-earthquake response and recovery strategies [7].
Section snippets
Modeling beam–column joints in finite element analysis of gravity load designed reinforced concrete frames
RC frame structures constitute 16% of the buildings with appraised value greater than $5 million according to a survey of the building infrastructure in Shelby County, TN [8]. Such GLD RC frames in the CEUS traditionally have been designed using detailing provisions of ACI Standard 318 [9] for gravity load (or gravity plus wind load) effects only. These reinforcing details provide only a limited degree of ductility for frames in low-seismic regions, which often leads to so-called weak
Earthquake ground motions
Most seismic fragility analyses of frames in high-seismic regions performed previously (e.g. [11]) utilize natural strong ground motion records in the NTHA. Such records are unavailable for sites in the CEUS due to the infrequent nature of the earthquakes in that region. Accordingly, ensembles of synthetic earthquake ground motions that were developed specifically for the NMSZ under the auspices of the MAE Center by Wen and Wu [12] were utilized in the finite element models. The Wen–Wu ground
Fundamental modeling approach
A seismic fragility is defined herein as the probability of reaching stipulated damage states (performance levels) as a function of a specified measure of earthquake ground motion intensity. The fragility is described by the conditional probability that the structural capacity, C, fails to resist the structural demand, D, given the seismic intensity (hazard), SI, (the terms C, D, and SI in units that are suitable for seismic fragility assessment are defined subsequently) and is commonly modeled
Sensitivity of seismic demand to parameter uncertainties
Frame response statistics were determined using the earthquake ensembles identified in Fig. 3 for three levels of earthquake hazard for Memphis, TN: 10%, 5%, and 2% PE in 50 years. The values of Sa(T1)3 for these mean hazard events are listed in Table 34 for all three frames, as obtained from the seismic hazard
Treatment of uncertainty in fragility estimates
Seismic fragilities that incorporate sources of uncertainty considered above can be derived efficiently using Latin-hypercube sampling (LHS) [21] coupled to the finite element structural models. LHS provides a stratified sampling scheme rather than the purely random sampling, providing a more efficient means for covering the probability space than naïve Monte Carlo simulation. The sampling plan is given bywhere P is an N × K matrix, in which each of the K columns is a random permutation
Conclusions
The sensitivity of the response statistics of gravity load designed reinforced concrete frames to the uncertainties in material and structural properties and modeling parameters was investigated at various levels of earthquake hazard for Memphis, TN. Damping, concrete strength, and joint cracking strain were found to have the greatest impact on the response statistics. However, the uncertainty in ground motion dominated the overall uncertainty in structural response. This finding is consistent
Acknowledgements
This research was supported by the National Science Foundation under grant EEC-9701785 to the Mid-America Earthquake Center. This support is gratefully acknowledged. However, the views expressed are solely those of the authors and may not represent the position of either the MAE Center or the NSF. The authors would like to express their appreciation to Prof. Ross Corotis, former Editor of Structural Safety, for managing the review of this manuscript in accordance with journal policies for peer
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