Zhao-Hui Lu

The inverse transformation of the explicit fourth-moment standardization for structural reliability

In practical engineering, the probability distributions of some random variables are often unknown, and the only available information about these may be their statistical moments. To conduct structural reliability assessment without the exclusion of random variables with unknown probability distributions, an explicit fourth-moment standardization function has been proposed, and a single expression of its inverse transformation, that is, normal transformation, with limitations of inputting sets of the third and fourth moments (skewness and kurtosis) of random variables was derived. However, the clear definition of the complete expressions of the inverse transformation of fourth-moment standardization function under different combinations of skewness and kurtosis of random variables has not been provided yet. It is in this regard that four criteria are proposed to derive the complete inverse transformation of fourth-moment standardization function, and then the complete expressions of the inverse transformation are formulated. Through the numerical examples presented, the proposed complete expressions are found to be quite efficient for normal transformations and to be sufficiently accurate to include random variables with unknown probability distributions in structural reliability assessment.

Efficient evaluation of structural reliability under imperfect knowledge about probability distributions

Abstract We investigate the evaluation of structural reliability under imperfect knowledge about the probability distributions of random variables, with emphasis on the uncertainties of the distribution parameters. When these uncertainties are considered, the failure probability becomes a random variable that is referred to as the conditional failure probability. For the sake of transparency in communicating risk, it is necessary to determine not only the mean but also the quantile of the conditional failure probability. A novel method is proposed for estimating the quantile of the conditional failure probability by using the probability distribution of the corresponding conditional reliability index, in which a point-estimate method based on bivariate dimension-reduction integration is first suggested to compute the first three moments (i.e., mean, standard deviation and skewness) of the conditional reliability index. The probability distribution of the conditional reliability index is then approximated by a three-parameter square normal distribution. Numerical studies show that the computational efficiency of the proposed method was well above that of Monte Carlo simulations without loss of accuracy, and also show that neglecting parameter uncertainties will lead to the structural reliability being overestimated. The developed methodology provides a complete picture of structural reliability evaluation under imperfect knowledge about probability distributions.

A flexible distribution and its application in reliability engineering

Abstract Probability distributions of random variables are necessary for reliability evaluation. Generally, probability distributions are determined using one or two parameters evaluated from the mean and standard deviation of statistical data. However, these distributions are not sufficiently flexible to represent the skewness and kurtosis of data. This study therefore proposes a probability distribution based on the cubic normal transformation, whose parameters are determined using the skewness and kurtosis, as well as the mean and standard deviation of available data. This distribution is categorized into six different types based on different combinations of skewness and kurtosis. The boundaries of each type are identified, and the completeness of each type is proved. The cubic normal distribution is demonstrated to provide significant flexibility, and its applicable range covers a large area in the skewness–kurtosis plane, thus enabling it to approximate well-known distributions. The distribution is then applied in reliability engineering: simulating distributions of statistical data, calculating fourth-moment reliability index, finding optimal inspection intervals for condition-based maintenance system, and assessing the influence of input uncertainties on the whole output of a system. Several examples are presented to demonstrate the accuracy and efficacy of the distribution in the above-mentioned reliability engineering practices.

Empirical model of corrosion rate for steel reinforced concrete structures in chloride-laden environments

The corrosion rate of reinforcing steel is an important factor to determine the corrosion propagation of reinforced concrete structures in the chloride-laden environments. Since the corrosion rate of reinforcing steel is affected by several coupled parameters, the efficient prediction of which remains challenging. In this study, a total of 156 experimental data on corrosion rate from the literature were collected and compared. Seven empirical models for predicting the corrosion rate were reviewed and investigated using the collected experimental data. Based on the investigations, a new empirical model is proposed for predicting the corrosion rate in corrosion-affected reinforced concrete structures considering parameters including concrete resistivity, temperature, relative humidity, corrosion duration and concrete chloride content. The comparison between the experimental data and those predicted using the new empirical model demonstrates that the new model gives a good prediction of the corrosion rate. Furthermore, the uncertainty and probability characteristics of these empirical models are also investigated. It is found that the probability distributions of the model errors can be described as lognormal, normal, Weibull or Gumbel distributions. As a result, the new empirical model can provide an efficient prediction of the corrosion rate of reinforcing steel, and the model error analysis results can be utilized for reliability-based service life prediction of reinforced concrete structures under chloride-laden environments.

First passage probability assessment of stationary non-Gaussian process using the third-order polynomial transformation

In this article, an analytical moment-based procedure is developed for estimating the first passage probability of stationary non-Gaussian structural responses for practical applications. In the procedure, an improved explicit third-order polynomial transformation (fourth-moment Gaussian transformation) is proposed, and the coefficients of the third-order polynomial transformation are first determined by the first four moments (i.e. mean, standard deviation, skewness, and kurtosis) of the structural response. The inverse transformation (the equivalent Gaussian fractile) of the third-order polynomial transformation is then used to map the marginal distributions of a non-Gaussian response into the standard Gaussian distributions. Finally, the first passage probabilities can be calculated with the consideration of the effects of clumping crossings and initial conditions. The accuracy and efficiency of the proposed transformation are demonstrated through several numerical examples for both the “softening” responses (with wider tails than Gaussian distribution; for example, kurtosisu2009>u20093) and “hardening” responses (with narrower tails; for example, kurtosisu2009<u20093). It is found that the proposed method has better accuracy for estimating the first passage probabilities than the existing methods, which provides an efficient and rational tool for the first passage probability assessment of stationary non-Gaussian process.

Reliability-Based Assessment of Design Code Provisions for Circular CFT Stub Columns

This paper presents a probabilistic investigation of design code provisions such as AISC, DL/T and Eurocode 4 for circular CFT stub columns. The study is based on experimental results of 250 axially loaded circular CFT stub columns published in the literature. By comparing experimental results for ultimate loads with code-predicted column resistance, the error of resistance models is analyzed and it is found that it can be described using a normal random variable. Based on the investigations, reliability analyses were performed for the three design codes with/without considering the model error random variables. It is found that the model error is quite relevant for column safety, especially for the case of small load ratios. Consideration of model error resulted in increased reliability of stub columns designed according to AISC code, due to conservative column resistance predicted by the code. While for design codes of DL/T or Eurocode 4, consideration of model error produces a reduction of reliability indexes. Finally, the safety level of code provisions in terms of sufficient and uniform reliability criteria is also evaluated. Results show that the AISC design code provided very conservative results and the scatter of reliability indexes is the smallest. Eurocode 4 provided suitable results in terms of the sufficient criteria. The paper also shows that the reliability indices of columns designed following DL/T are slightly below the target reliability levels of ISO 2394.