The basic properties of some typical systems’ reliability in interval form

A proper mathematical representation of uncertainties is indispensable for reliability analysis of a practical engineering structural system. A general uncertainty analysis approach is probability bounds analysis (PBA), which propagates constraints on a distribution function through mathematical operations. The uncertainty about a probability distribution is represented by the set of cumulative distribution functions lying entirely within a pair of bounding distribution functions, which is called a P-box. Interval analysis as a special case of PBA is useful when there is no or less probabilistic information. It is common sense that great efforts must be paid to get enough probabilistic information used for probabilistic analysis of large and complex engineering structural systems. Even if there is no or less probabilistic information; the interval of possible values of probability of an event can be easily specified, such as the interval value of each element’s reliability of an engineering structural system.

This paper aims to introduce the concept of system reliability and its relationship to the reliability of its individual elements in an interval form. In terms of extension principle, interval arithmetic and possibility degree formula (PDF) for ranking interval numbers, basic properties of system reliability in interval form are investigated. The conclusion is that relationships between point reliability (point reliability used to describe a precise value of probability reliability is distinct with interval reliability) of some typical systems, such as series system, parallel system, series–parallel system, parallel–series system and r/n(G) system, etc., and point reliability of their individual elements are maintained in their interval forms. This is called quasi-consistency in this paper. A simple review of order relations of interval numbers, which will play an important role in interval reliability analysis, is given. The proposed quasi-consistency establishes the foundations for interval reliability analysis of a complex engineering structural system.     

Evaluation of multivariate normal integrals for general systems by sequential compounding

System reliability analysis often requires efficient and accurate evaluation of a multivariate normal integral. Despite recent advances in system reliability analysis methods, it is still a challenging task especially when the definition of the system event is complex; the system has a large number of components; and/or the component events have significant statistical dependence. This paper presents a new method developed for evaluating multivariate normal integrals defined for general system events including series, parallel, cut-set and link-set systems. The method compounds two components coupled by union or intersection sequentially until the system becomes a single compound event. Efficient numerical procedures are developed for obtaining the reliability index of the new compound event, and the correlation coefficients between the compound event and the remaining component events, at each step of the sequential compounding. The accuracy and efficiency of the proposed method, and its applicability to various types and sizes of multivariate normal integrals are demonstrated by numerical examples.     

Analysis of a combined cooling, heating, and power system model under different operating strategies with input and model data uncertainty

Combined cooling, heating, and power (CCHP) system models have been used by many researchers to compare their performance with conventional systems. However, decisions based on the results of computer simulations need to take into account the uncertainty of these results to get insight into the level of confidence in the predictions. This paper presents an analysis of a CCHP system model under different operating strategies with input and model data uncertainty. However, the uncertainties that underlie the variation in input parameters such as the thermal load, natural gas prices and electricity prices are not readily available. Additionally, engine performance uncertainty can be difficult to characterize because of the nonlinearity of engine efficiency curves. This paper presents practical and novel approaches to estimating the uncertainty in these and other input parameters. A case study using a small office building located in Atlanta, GA, is described to illustrate the importance of the use of uncertainty and sensitivity analysis in CCHP system performance predictions, and how the primary energy consumption, operational cost, and carbon dioxide emissions are affected by the uncertainty associated with the model input parameters.     

单桩承载力计算模式的不确定性分析

岩土工程分析模型大都存在不确定性,可靠性分析需要考虑模型不确定性的影响, 本文提出了真实反映模型不确定性的计算方法。文中还根据实测资料分析了单桩极限承载力计算模式的不确定性, 通过比较分析表明用本文方法表示模型的不确定性是适当的。     

On the treatment of finite element structures in stochastic linear dynamics using a mode-based meta-model

This contribution presents a meta-model of the structural modes, which can be used for efficient dynamic analysis of finite element structures in the presence of uncertainties in the loading and structural parameters. The main goal consists in cutting down the computational time required for the repeated number of analyses in order to assess the impact of the uncertainties. The meta-model allows for an efficient evaluation of the structural modes which are represented as a linear combination of the modes of a reference solution. The calibration of the meta-model is based on the modes of about 30-50 samples of the structure obtained with Monte Carlo simulation. The mode-based meta-model can always be advantageously employed when a limited number of modes are required for analysis, such as, e.g. for mode superposition analysis. The numerical examples show the utilization of the meta-model for various applications such as uncertainty analysis, reliability assessment and design optimization.     

Bayesian analysis of uncertainty for structural engineering applications

There has been recent interest in differentiating aleatory and epistemic uncertainties within the structural engineering context. Aleatory uncertainty, which is related to the inherent physical randomness of a system, has substantially different effects on the analysis and design of structures as compared with epistemic uncertainty, which is knowledge based. Bayesian techniques provide powerful tools for integrating, in a rigorous manner, the two types of uncertainties. In a purely probabilistic viewpoint, the uncertainties merge, resulting in widened probability densities. From the viewpoint of design or experimentation, however, the two types of uncertainties have widely different effects. The purpose of this paper is to develop insight into these effects, using Bayesian-based analytical expressions for the aleatory and epistemic uncertainties. The paper goes beyond standard Bayesian conjugate distributions by incorporating the effects of model uncertainty, where the applicability of two or more analytical models are used to describe the structure of interest. The influence of multiple model uncertainties is explored for two problems: the Bayesian updating process as data is acquired, and the design of simple parallel systems.     

Modeling and analysis of uncertainties for risk-informed decisions in infrastructures engineering

Engineers deal with uncertainties in all their activities, and must often make decisions under conditions of uncertainty and risk. Infrastructures engineering is no exception—design codes are developed to ensure a desired level of safety and performance, or to ensure a specified operational life with a prescribed level of reliability; the required decisions must often be formulated without complete information and thus contain uncertainties. In considering uncertainties, it is important to recognize two broad types; namely, the aleatory type which is associated with natural randomness and the epistemic type which is associated with imperfect knowledge. Proposed here is a framework for the proper modeling and treatment of each type of uncertainty in the formulation of risk-informed engineering decisions. The concepts are illustrated with applications to bridges and offshore marine structures.     

PERMAS-RA/STRUREL system of programs for probabilistic reliability analysis

Programs for reliability analysis of structural, operational and other systems based on first- and second-order reliability concepts were made available as early as 1976 by the Technical University of Munich. In the meantime and since 1987 by RCP Consult GmbH (RCP) the programs have experienced many revisions, improvements and additional developments. The programs now cover the preparatory steps as well as the computational tasks in technical reliability, decision making under uncertainty and in statistical analysis. Important modules of strurel have also been embedded in the finite element program permas developed and maintained by INTES GmbH (INTES). The programs have been used in structural engineering and code making, in hydrology, operations research, financial planning and mathematical statistics and, in particular, in the nuclear power plant, offshore, ship, automotive and aerospace industry.     

Uncertainties in stormwater E. coli levels

Although water-quality monitoring programs have been widely used to identify and understand the level of pollution in urban stormwater systems, these data are often used without due consideration of the inherent uncertainties contained within these measurements. This study focuses on the uncertainties associated with the monitored levels of Escherichia coli, a common microbial indicator, in urban stormwater. Four sites located in Melbourne, Australia, were used to assess the uncertainty of six stormwater flow and E. coli variables: (1) discrete E. coli concentration, (2) stormwater flow rate, (3) stormwater event volume, (4) event mean concentration (EMC) of E. coli (i.e. a flow-weighted average of an event's E. coli concentrations), (5) E. coli load for each measured event, and (6) site mean E. coli concentration (SMC) (i.e. a volume-weighted average of the E. coli EMCs). Uncertainties of discrete E. coli samples were greater than 30%, while the uncertainty in stormwater flow measurements averaged greater than 97%, mainly due to the high uncertainties in measurements of very low flows. Propagation of these uncertainties, through their respective formulas, found that E. coli EMC uncertainties varied between 10% and 52% and that uncertainties relating to SMC estimates ranged from 35% to 55%. These results show the importance of considering uncertainty when using monitored data sets for any application, including those relating to stormwater management decisions. Suggestions are made about how to increase the accuracies of E. coli monitoring in urban stormwater and how to balance the different sources of uncertainties so that the overall combined uncertainties are minimised while keeping costs at a minimum.     

基于事件树的天然气管道风险定量分析

提出了以事件树分析为主线,结合故障树分析的城镇天然气管道风险定量分析方法：将管道泄漏分为渗透泄漏、穿孔泄漏和开裂泄漏3种泄漏模式;以管道泄漏为故障树分析的顶事件,分析管道泄漏的原因和概率;以管道泄漏为事件树分析的初因事件,分析管道泄漏在各种情况下造成的后果序列;以故障树分析事件树中后续事件的概率,以初因事件的概率和后续事件的概率计算后果事件的概率;分别对各个后果事件进行后果危害程度分析和经济损失估计;累加每个后果事件的经济损失数学期望,得到整个管道系统的风险值。     

抗滑桩加固边坡的稳定可靠度分析

经常需要在试验资料不充分的条件下进行边坡工程设计,故如何从搜集到的各方面信息中统计得到工程设计所需参数具有重要的理论和工程意义。某边坡工程以实测两断层实验数据为基础,参照类似工程的实测资料分析结果,利用岩石分类确定断层的内摩擦角,利用反演分析和考虑残差的最小二乘拟合的方法确定断层的黏聚力;抗滑桩是横向受力体系,考虑抗滑桩4种失效模式的相关性,利用反分析的方法确定出抗滑桩的抗力统计参数;根据规范推荐的传递系数法确定出边坡滑动的功能函数,利用离散化降维解法求得边坡稳定的可靠度指标。分析结果表明:利用反演方法修正参数,减少了认识的不确定性,能够提高边坡工程设计的可靠性,在统计数据不足的情况下按可靠度进行边坡工程设计是一种可行方法;同时表明锚索抗滑桩加固边坡的效果是显著的。     

Probabilistic properties of thin-walled elements

Thin-walled structures are relatively complicated structural systems governed by many parameters. These include the dimensions, the material properties and imperfection measures. In principle all of these are not known with certainty at the design stage, and even after construction some degree of uncertainty remains.

The present paper reviews briefly how such uncertainties are considered in modern structural reliability theory to estimate nominal probabilities of failure. An example application will be given to illustrate the process of estimating the probability density function for the failure stress of stiffened plates subject to axial forces.     

Reliability of an engineering system under a strong earthquake with application to offshore platforms

This paper presents a method developed for the performance evaluation of an engineering system during a strong earthquake. The seismic reliability of the system is evaluated by considering the different possible failure modes, and seismic reliability of the components of the system. The method is applied to the oil production system at the Statfjord Field in the North Sea. The objective of the analysis is to document the reliability of oil production from the field in the event of an exceptionally large earthquake, and to investigate the effect of spatial variation of earthquake loading parameters on the computed system reliability. It is shown that, with a realistic model of the spatial variation of earthquake motion, there is very little correlation between the earthquake loading parameters at different platform locations. The main consequence of this lack of correlation is that the computed failure probability of the oil production system under the design earthquake is about one order of magnitude smaller than the probability of failure of individual platforms at the Statfjord Field.     

Component and system reliability analysis of nonlinear reinforced concrete grids with multiple failure modes

The bending moment redistribution is an inherent behavior in reinforced concrete grids, which increases the number of possible critical cross-sections susceptible to reach a limit state. Despite the fact that its influence could be important, the high number of failure cross-sections is not often considered in reinforced concrete reliability analyses. In this paper, a local approach of reliability analysis applied to grid structures is developed by taking into account the dominant failure modes. This approach is based on random sampling coupled with finite element analysis, through the use of a localized response surface technique. A mode selection strategy has been developed to capture the individual failure modes in order to classify their importance in the global system reliability. For large-scale systems, this procedure intends to reduce the global computational time in the reliability analysis. Numerical applications aim to show the effect of internal force redistribution, as well as the efficiency of the proposed approach and the interest of considering multiple failure modes.     

A PDEM-based perspective to engineering reliability: From structures to lifeline networks

Research of reliability of engineering structures has experienced a developing history for more than 90 years. However, the problem of how to resolve the global reliability of structural systems still remains open, especially the problem of the combinatorial explosion and the challenge of correlation between failure modes. Benefiting from the research of probability density evolution theory in recent years, the physics-based system reliability researches open a new way for bypassing this dilemma. The present paper introduces the theoretical foundation of probability density evolution method in view of a broad background, whereby a probability density evolution equation for probability dissipative system is deduced. In conjunction of physical equations and structural failure criteria, a general engineering reliability analysis frame is then presented. For illustrative purposes, several cases are studied which prove the value of the proposed engineering reliability analysis method.     

Geotechnical uncertainty,modeling, and decision making

Modeling only constitutes one aspect of decision making. The prevailing limitation of applying modeling to practice is the absence of explicit consideration of uncertainties. This review paper covers uncertainty quantification (soil properties, stratification, and model performance) and uncertainty calculation with a focus on how it enhances the role of modeling in decision making (reliability analysis, reliability-based design, and inverse analysis). The key output from a reliability analysis is the probability of failure, where “failure” is defined as any condition that does not meet a performance criterion or a set of criteria. In contrast to the global factor of safety, the probability of failure respects both mechanics and statistics, is sensitive to data (thus opening one potential pathway to digital transformation), and it is meaningful for both system and component failures. Resilience engineering requires system level analysis. As such, geotechnical software can provide better decision support by computing the probability of failure/reliability index as one basic output in addition to stresses, strains, forces, and displacements. It is further shown that more critical non-classical failure mechanisms can emerge from spatially variable soils that can escape notice if the engineer were to restrict analysis to conventional homogeneous or layered soil profiles.     

The equivalent extreme-value event and evaluation of the structural system reliability

The idea of equivalent extreme-value event and accordingly a new approach to evaluate the structural system reliability are elaborated. For any type of compound random event as combination of a set of random events represented by inequalities, an equivalent extreme-value event is defined. Elaborated investigations show that correlative information among the component random events is inherent in the equivalent extreme-value event. Since the probability density function of the equivalent extreme value could be obtained through the probability density evolution method, the idea of equivalent extreme-value event leads to a new uniform approach to evaluate the structural system reliability for both static and dynamic problems. Particularly, the investigation points out that computation of the dynamic reliability essentially involves dealing with infinite-dimensional correlation information and that is why the widely-used out-crossing process theory could be only an approximate and somewhat empirical reliability evaluation rather than an exact approach. The proposed approach is discussed in detail on how to construct the equivalent extreme-value event and then implement the procedure numerically. Two examples, of which one deals with static problem comparing the results with exact solution, the other deals with a nonlinear frame structure subjected to stochastic ground motions, are illustrated to validate the proposed method. The investigations show that the proposed approach is of satisfactory accuracy and applicable to the structural reliability analysis of various structures.     

Improving project outcomes through operational reliability: A conceptual model

The work reported here is part of an on-going research effort concerned with developing a methodology to improve the outcomes of projects involving complex information systems. To facilitate the study, previous research concerned with high reliability organisations (HRO) is examined. An analysis of this literature is used to develop a conceptual model that explains how highly reliable operational capabilities are achieved and maintained. Cases of catastrophic failures are evaluated using this paradigm to illustrate its utility as an analytical research tool and its potential as a systemic framework for evaluating the operational capability of organisations prior to project initiation.     

Probabilistic risk assessment for evacuees in building fires

A probabilistic risk assessment method for evacuees in building fires is presented with consideration of some deterministic and stochastic factors. The time-dependent event tree technique is used to analyze probable fire scenarios based on operational reliability of fire protection systems. For a fire scenario, the time to untenable conditions is characterized by probability distribution with consideration of some uncertainties of design fire. Moreover, occupant pre-movement time, one of the most important proportions of evacuation time, is characterized by normal distribution to express its uncertainty. Based on calculated results of ASET and RSET for every fire scenarios, the expected number of casualties is obtained when untenable conditions occur. Moreover, according to some fire statistical data, expected risk to life (ERL) is calculated to express the risk severity. Finally, a case study for a supermarket building is presented to express the risk assessment method in detail.     

基于子集模拟的边坡可靠度分析方法研究

边坡稳定受到诸多不确定性因素的影响,比如土体性质的空间变异性以及地层情况的不确定性。这些不确定性因素的影响可以通过蒙特卡洛模拟(Monte Carlo Simulation, MCS)进行定量地分析。MCS概念简单并具有广泛的适用性。但是,在小概率失效区域内,MCS计算效率很低,需要庞大的随机样本量来保证一定的计算精度。本文提出了一种实用的边坡可靠度分析方法。通过采用一种高级的MCS方法(Subset Simulation, 子集模拟)来提高小概率区域内的计算效率以及计算精度,并以EXCEL的表单环境为平台,联合使用Visual Basic Application(VBA)编写计算程序。在该程序中,子集模拟、边坡稳定的确定性分析和不确定性分析分别由三个相对独立的计算模块实现。最后,本文以James Bay 土坝为例,简明地说明了所提出方法的有效性,并探索了临界滑动面的不确定性对边坡可靠度分析的影响。