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Non-ergodicity and PEER's framework formula

机译：非遍历性和PEER的框架公式

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摘要

A framework formula for performance-based earthquake engineering, advocated and used by researchers at the Pacific Earthquake Engineering Research (PEER) Center, is closely examined. The formula was originally intended for computing the mean annual rate of a performance measure exceeding a specified threshold. However, it has also been used for computing the probability that a performance measure will exceed a specified threshold during a given period of time. It is shown that the use of the formula to compute such probabilities could lead to errors when non-ergodic variables (aleatory or epistemic) are present. Assuming a Poisson model for the occurrence of earthquakes in time, an exact expression is derived for the probability distribution of the maximum of a performance measure over a given period of time, properly accounting for non-ergodic uncertainties. This result is used to assess the approximation involved in the PEER formula for computing probabilities. It is found that the PEER approximation of the probability has a negligible error for probabilities less than about 0.01. For larger probabilities, the error depends on the magnitude of non-ergodic uncertainties and the duration of time considered and can be as much as 20% for probabilities around 0.05 and 30% for probabilities around 0.10. The error is always on the conservative side.
机译：由太平洋地震工程研究（PEER）中心的研究人员倡导和使用的基于性能的地震工程的框架公式已被仔细研究。该公式最初是用于计算绩效指标超过指定阈值的年平均速率的。但是，它也已用于计算性能指标在给定时间段内超过指定阈值的概率。结果表明，当存在非遍历变量（偶然的或认知的）时，使用公式来计算此类概率可能会导致错误。假设一个及时发生地震的泊松模型，可以得出一个精确的表达式，用于计算给定时间段内最大性能指标的概率分布，并适当考虑了非遍历不确定性。该结果用于评估PEER公式中涉及的近似计算概率。发现对于小于约0.01的概率，概率的PEER近似具有可忽略的误差。对于较大的概率，误差取决于非遍历不确定性的大小和所考虑的时间长度，对于0.05左右的概率，误差可能高达20％；对于0.10左右的概率，误差可能高达30％。该错误总是保守的。

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