Fast integration algorithms for time-dependent structural reliability analysis considering correlated random variables

The load-carrying capacity of reinforced concrete bridges can degrade over their service life through the initiation of deterioration mechanisms induced by fatigue, corrosion, cracks, and spalling. Analyses of residual capacity and time-dependent reliability of such bridges are significantly important information in the decision-making process of identifying the nature and timing of rehabilitation. It is noted in the literature that deterioration mechanisms such as chloride ingress, sulfate attack, and alkali-silica reaction have variable rates of deterioration governed by a large number of uncertain parameters. In such instances, a probabilistic approach provides a rational basis for estimating the uncertainties of residual capacity and reliability. In this paper, the authors present a probabilistic method to evaluate the time-dependent reliability and failure of concrete bridge elements. The probabilistic distribution of surface chloride concentration, diffusion coefficient, critical chloride concentration, and material variables were identified from literature. Monte Carlo simulation was employed for modeling the increases of live loads and the degradation of the component to obtain the time-dependent reliability curves for design service life. The methodology is further demonstrated by a pier column as an illustrative example. Results showed the reduction in resistance and the risk of failure periods. Further parametric studies showed that concrete cover depth and water-cement ratio have significant influence on the time-dependent probability of failure and reliability index. Overall, it has been shown that the methodology is generic and valuable information on residual capacity and time-dependent probability of failure can be applied to performance assessment and lifecycle for both new and existing structures.

Time-dependent reliability (failure probability) aims at measuring the probability of the normal (abnormal) operation for structure/mechanism within the given time interval. To analyze the maximum probable life time under a required time-dependent failure probability (TDFP) constraint, an inverse process corresponding to the time-dependent reliability is proposed by taking the randomness of the input variables into consideration. The proposed inverse process employs the monotonicity between the TDFP and the upper boundary of the given time interval which reflects the life time, and an adaptive single-loop sampling meta-model for the time-dependent limit state function is presented to estimate the TDFP at the given time interval flexibly. Since the TDFP is generally monotonic to the upper boundary of the given time interval, thus by adjusting the probable upper and lower boundaries of the time interval in which the corresponding TDFPs include the required TDFP constraint, the proposed approach can always search the maximum probable life time at the required TDFP by the dichotomy. By introducing the time variable as an input which is the same level as the input random variables and constructing the adaptive single-loop sampling meta-model for the time-dependent limit state function in a longer time interval with the TDFP bigger than the required TDFP, the TDFP in any subintervals of the time interval involved in the constructed meta-model can be estimated as a byproduct of the constructed meta-model without any additional actual limit state evaluations. Then the efficiency for analyzing the maximum probable life time is improved by the dichotomy and the unified meta-model of the time-dependent limit state function. Two examples are employed to illustrate the accuracy and the efficiency of the proposed approach.

This article proposes two strategies for time-dependent probabilistic fatigue analysis considering stochastic loadings and strength degradation based on the failure transformation and multi-dimensional kernel density estimation method. The time-dependent safety margin function is first established to describe the limit state of the time-dependent failure probability for mechatronics equipment with stochastic loadings and strength degradation. Considering the effective safety margin points and the corresponding number of the load cycles, two strategies for transforming the time-dependent failure probability calculation to the static reliability calculation are then proposed. Multi-dimensional kernel density estimation method is finally employed to build the probability density functions and the reliability is estimated based on the probability density functions. An engineering case of a filtering gear reducer is presented to validate the effectiveness of the proposed methods both in computational efficiency and accuracy.

Time-dependent system reliability is computed as the probability that the responses of a system do not exceed prescribed failure thresholds over a time duration of interest. In this work, an efficient time-dependent reliability analysis method is proposed for systems with bivariate responses which are general functions of random variables and stochastic processes. Analytical expressions are derived first for the single and joint upcrossing rates based on the first-order reliability method (FORM). Time-dependent system failure probability is then estimated with the computed single and joint upcrossing rates. The method can efficiently and accurately estimate different types of upcrossing rates for the systems with bivariate responses when FORM is applicable. In addition, the developed method is applicable to general problems with random variables, stationary, and nonstationary stochastic processes. As the general system reliability can be approximated with the results from reliability analyses for individual responses and bivariate responses, the proposed method can be extended to reliability analysis of general systems with more than two responses. Three examples, including a parallel system, a series system, and a hydrokinetic turbine blade application, are used to demonstrate the effectiveness of the proposed method.

This paper outlines an improved simulation method for a time-dependent reliability analysis of structural systems subjected to stochastic loads. Strengths of structural elements are assumed to degrade with time proportionally during the life cycle of the structural system, and stochastic load events are modeled as Poisson processes. The structural failure probabilities are estimated through the life cycle simulation, in which the directional simulation method is applied just to sample random load variables to execute each life cycle simulation efficiently. Numerical examples are presented and results are compared with those obtained by the crude Monte Carlo simulation to show the effectiveness of the proposed method. It is shown that the proposed method is effective for the assessment of the time-dependent failure probability of structural systems.

RWE npower own and operate a 12km long 8″ diameter natural gas pipeline that supplies natural gas from the National Transmission System (NTS) to a CHP unit. The pipeline has a nominal wall thickness of 6.35mm, is constructed from API 51 X42 grade steel and has a maximum operating pressure of 75barg. The pipeline was commissioned about 8 years ago and has been operating safely since that time. The pipeline was designed in accordance with BS 8010 Parts 1 and 2, with consideration given to the Institute of Gas Engineers Recommendations, IGE/TD/1, IGE/TD/9 and IGE/TD/12. One of the requirements of IGE/TD/1 is that the time interval between in-line inspections should not normally exceed 10 years. However, IGE/TD/1 Edition 4 allows the time interval to be exceeded if justification can be demonstrated using risk based techniques. For older pipelines it is often possible to determine the required time dependent failure probability based on the results of previous ILIs. This allows the time to next ILI to be determined. This time will depend on what has been found previously and values of other pipeline parameters. However, in the case of this pipeline there are no previous ILI data. In view of the above a probabilistic approach to External Corrosion Direct Assessment (ECDA) was adopted. The method makes use of the results from the above ground surveys rather than ILI data. However, in this instance, rather than being used to determine the time interval to the next above ground survey the method was used to determine the time interval to the next (first) ILI. The method is based on structural reliability analysis (SRA) which is used to determine the time dependent probability of failure based on available data. In view of the quantity and quality of the available above ground survey data it was possible to use the method to extend the time to the next ILL by several years.

Time-dependent reliability is the probability that a system will perform its intended function successfully for a specified time. Unless many and often unrealistic assumptions are made, the accuracy and efficiency of time-dependent reliability estimation are major issues which may limit its practicality. Monte Carlo simulation (MCS) is accurate and easy to use but it is computationally prohibitive for high dimensional, long duration, time-dependent (dynamic) systems with a low failure probability. This work addresses systems with random parameters excited by stochastic processes. Their response is calculated by time integrating a set of differential equations at discrete times. The limit state functions are therefore, explicit in time and depend on time-invariant random variables and time-dependent stochastic processes. We present an improved subset simulation with splitting approach by partitioning the original high dimensional random process into a series of correlated, short duration, low dimensional random processes. Subset simulation reduces the computational cost by introducing appropriate intermediate failure sub-domains to express the low failure probability as a product of larger conditional failure probabilities. Splitting is an efficient sampling method to estimate the conditional probabilities. The proposed subset simulation with splitting not only estimates the time-dependent probability of failure at a given time but also estimates the cumulative distribution function up to that time with approximately the same cost. A vibration example involving a vehicle on a stochastic road demonstrates the advantages of the proposed approach.

Plant-specific human reliability analysis (HRA) tailors human error probabilities (HEPs) to a facility's unique contexts. The diversity of digital human-machine interfaces and operational cultures necessitates methods grounded in empirical data rather than generic databases to ensure true plant-specificity. Advancing beyond prior fragmented data applications, this study establishes a systematic pipeline integrating the HuREX (Human Reliability data Extraction) database with the EMBRACE (EMpirical data-Based crew Reliability Assessment and Cognitive Error analysis) method. By sharing a unified task taxonomy, this framework directly translates plant-specific simulator data into coherent HEP estimates. In an application of this framework to the APR1400 plant, time-dependent failure probabilities were derived using Bayesian inference on 30 simulator performance records. Concurrently, nominal primitive error probabilities (NPEPs) were estimated via logistic regression from 44,585 task records. Due to current data scarcity, performance shaping factor (PSF) multipliers were determined through structured expert elicitation. This data-method integration provides a replicable foundation for generating context-sensitive HEPs in digitalized control rooms, highlighting both empirical strengths and current data limitations. • A plant-specific HRA approach is developed by integrating HuREX with EMBRACE. • HuREX data support the estimation of the time and cognitive failure probabilities. • The proposed approach is applied to estimate HEPs for the APR1400 plant. • Limitations such as reliance on expert judgment are discussed.

Maintenance, rehabilitation and replacement of existing bridges by using cost effective management are increasingly important topics in efforts to deal with aging and deteriorating infrastructure. Recently, engineers have shifted the emphasis from design and construction of new bridges to maintenance and rehabilitation of existing bridges. This paper uses the structural reliability method with a predefined performance function to assess the component performance of the main load carrying members of an inner span in a multi-span bridge. The performance of the components is a function of various random variables with certain probabilistic distributions. The analysis is extended using the same performance function to evaluate the reliability of the bay as subsystem and bridge as a system. System reliability in this case is defined as the probability that a system (bridge) performs its intended function under operating conditions for a specified period of time using second order reliability method (SORM) and Monte Carlo simulations. In addition, deterioration in the main carrying elements (components) due to environmental impacts is introduced and the sensitivity parameters for certain random variables are re-evaluated to determine the inspection and maintenance period of the bridge. The deterioration is imposed as a percentage reduction in certain random variables. The paper involves use of multi reliability analysis at component, sub-system and system levels. Results show that probability of failure (Pf) increases with decrease in distance of the bridge from source of deterioration. Reliability index (β) decreased with increase in the age of the bridge because of corrosion of reinforcing steel. Results also show that the rate of probability of failure for bridges exposed to deteriorating effects increased dramatically after 45-50 years.

This paper establishes a dynamic Bayesian network to model the growth of corrosion defects on energy pipelines. The integrated model characterizes the growth of defect depth by a homogeneous gamma process and considers the biases and random errors associated with the in-line inspection (ILI) tools. The distributions of the mean value and coefficient of variation of the annual growth of defect depth are learned from multiple ILI data using the parameter learning technique of Bayesian networks. With the same technique, the distributions of the biases and standard deviation of random errors associated with ILI tools are learned from ILI data and their corresponding field measurements. An example with real corrosion management data is used to illustrate the process of developing the model structure, learning model parameters and predicting the corrosion growth and time-dependent failure probability. The results indicate that the model can in general predict the growth of corrosion defects with reasonable accuracy and the ILI-reported and field-measured depth can be used to update the time-dependent failure probability in a near-real-time manner. In comparison with existing growth models, the graphical feature of Bayesian networks makes it more intuitive and transparent to users. The employment of parameter learning provides a semi-automated and convenient approach to elicit the probabilistic information from ILI and field measurement data. The above advantages will facilitate the application of the model in the practice of corrosion management in pipeline industry.

ABSTRACTIn the present study, reliability analysis of near surface disposal facility is performed, by assessing the probability of sequential failure of the multi barrier system using the contaminant transport model. The concentration and dose rate of the radionuclide evolve with time hence there is a need for time dependent reliability analysis. Due to the low values of expected probabilities of failure, an enhanced Monte Carlo (EMC) method and Subset simulation is employed. The Result of the analysis show that, the EMC method is useful to evaluate the probability of failure associated with the barrier system which has low probability of failure.

Due to the degradation, input loading and uncertainty in the design parameters usually involve random variables and random processes, reliability analysis for engineering design problems are usually time dependent. Many problems related to degradation have been treated as monotonic or statistically independent, therefore, the probability of failure only at the end of the lifetime of the structure are considered. To the issues of parameters with stochastic process, the outcrossing rate methods have been extensively developed to calculate the upper bound of time-dependent reliability. In these methods, the issue of proper choice of time interval is crucial and difficult. In this paper, a new method for time dependent reliability optimization based on the total probability theory and universal generating function is proposed. In the proposed method, firstly, Parameters with stochastic processes are discretized into some discrete random variables. Secondly, the discrete parameters are reformed into a new random process by the operation of the universal generating functions. Finally, based on the total probability theory, the probability of failure for each limit state function is analyzed using sequential optimization and time invariant reliability assessment method. Only the time invariant reliability method is needed in the proposed method, by conditioning the continuous random variables on the discrete random parameters. Numerical example is presented to demonstrate the performance of the proposed method.

In the past, residual factor R in strain-softening soil slopes has been included, either directly or indirectly, as a deterministic variable in both deterministic and probabilistic studies. This paper discusses the uncertainties associated with R and outlines a systematic approach for the reliability analysis of a natural slope in which shear strength parameters and pore pressure ratio are random variables, each assumed with a lognormal probability distribution. For the residual factor R, seven probability distribution options under the generalized beta-distribution system are considered. Slope reliability is computed based on the first order reliability method (FORM) and validated against Monte-Carlo simulation (MCS). Results obtained from two illustrative examples indicate that the probability of failure, with R as one of six random variables, can be orders of magnitude higher than that based on five random variables with R considered as a deterministic parameter. The magnitude of influence of R as a random variable is, however, highly dependent on its probability distribution, the left-skewed triangular distribution having the most significant influence in both the examples. Results of sensitivity analyses reveal that, for almost all of its assumed probability distributions, R is the most dominant among the six random variables. Effects of variation of some of the statistical and correlation properties of the other random variables, viz. the shear strength parameters and the pore pressure ratio, on the results of reliability analyses are also studied.

Time-dependent earthquake-fire coupling fragility analysis under limited prior knowledge: A perspective from type-2 fuzzy probability

Time-dependent structural systems (TDSSs) in engineering involve high dimensionality, nonlinearity, and complex uncertainties, complicating the reliability analysis compared to time-independent assessments. To address these challenges, this paper proposes an extremum-based back propagation neural network (BPNN) method for TDSS reliability analysis. The method adopts a double-loop structure. Specifically, the inner loop finds the minimum of the time-dependent performance function for a given realization of the random variables. This transformation converts the time-dependent problem into an equivalent time-invariant one. Then, the outer loop constructs a BPNN surrogate model to map the relationship between the random variables and the performance function minima. To improve computational efficiency, an adaptive sample selection strategy is integrated into the training process. This technique selects samples near the failure boundary to iteratively update the BPNN, ensuring high accuracy with a small training set. Once the stopping criterion is satisfied, the failure probability is estimated using Monte Carlo simulation (MCS). The trained BPNN model is used to rapidly predict the extremum for the large-scale sample pool. The proposed method is verified through three practical engineering cases: a four-bar mechanism, an aero-engine turbine disc, and a cantilever tube. Results show that the method remains accurate and efficient. The successful applications confirm the rationality and engineering applicability of the proposed model.