Reliability Sensitivity-based Correlation Coefficient Calculation in Structural Reliability Analysis

The correlation coefficients of random variables of mechanical structures are generally chosen with experience or even ignored,which cannot actually reflect the effects of parameter uncertainties on reliability.To discuss the selection problem of the correlation coefficients from the reliability-based sensitivity point of view,the theory principle of the problem is established based on the results of the reliability sensitivity,and the criterion of correlation among random variables is shown.The values of the correlation coefficients are obtained according to the proposed principle and the reliability sensitivity problem is discussed.Numerical studies have shown the following results：（1） If the sensitivity value of correlation coefficient ρ is less than（at what magnitude 0.000 01）,then the correlation could be ignored,which could simplify the procedure without introducing additional error.（2） However,as the difference between ρs,that is the most sensitive to the reliability,and ρR,that is with the smallest reliability,is less than 0.001,ρs is suggested to model the dependency of random variables.This could ensure the robust quality of system without the loss of safety requirement.（3） In the case of ｜Eabs｜ρ0.001 and also ｜Erel｜ρ0.001,ρR should be employed to quantify the correlation among random variables in order to ensure the accuracy of reliability analysis.Application of the proposed approach could provide a practical routine for mechanical design and manufactory to study the reliability and reliability-based sensitivity of basic design variables in mechanical reliability analysis and design.

Dynamic Reliability Analysis of Braced Frame Structures Considering Inter Story Correlation

Adding buckling restrained braces(BRB)of reinforced concrete frame structure can effectively improve the safety performance of the structure.The dynamic reliability analysis based on Poisson continuous process assumption and the first exceeding failure probability can be used to obtain the failure probability of the buckling restrained brace frame system under earthquake load,and the relationship between the failure probabilities of each floor of the structure is analyzed to obtain the frame system reliability interval of frame structure.The results show that the reliability of BRB frame structure is higher than that of pure frame structure,and the discrete failure probability is lower.

A Reliability Allocation Method of CNC Lathes Based on Copula Failure Correlation Model

The current research of reliability allocation of CNC lathes always treat CNC lathes as independent series systems. However, CNC lathes are complex systems in the actual situation. Failure correlation is rarely considered when reliabil?ity allocation is conducted. In this paper, drawbacks of reliability model based on failure independence assumption are illustrated, after which, reliability model of CNC lathes considering failure correlation of subsystems is established based on Copula theory, which is an improvement of traditional reliability model of series systems. As the failure time of CNC lathes often obeys Weibull or exponential distribution, Gumbel Copula is selected to build correlation model. After that, a reliability allocation method considering failure correlation is analyzed based on the model established before. Reliability goal is set first and then failure rates are allocated to subsystems according to the allocation vector through solving the correlation model. Reliability allocation is conducted for t = 1. A real case of a CNC lathe and a numerical case are presented together to illustrate the advantages of the reliability model established consider?ing failure correlation and the corresponding allocation method. It shows that the model accords to facts and real working condition more, and failure rates allocated to all the subsystems are increased to some extent. This research proposes a reliability allocation method which takes failure correlation among subsystems of CNC lathes into consid?eration, and costs for design and manufacture could be decreased.

Assessment of Dependent Performance Shaping Factors in SPAR-H Based on Pearson Correlation Coefficient

With the improvement of equipment reliability,human factors have become the most uncertain part in the system.The standardized Plant Analysis of Risk-Human Reliability Analysis(SPAR-H)method is a reliable method in the field of human reliability analysis(HRA)to evaluate human reliability and assess risk in large complex systems.However,the classical SPAR-H method does not consider the dependencies among performance shaping factors(PSFs),whichmay cause overestimation or underestimation of the risk of the actual situation.To address this issue,this paper proposes a new method to deal with the dependencies among PSFs in SPAR-H based on the Pearson correlation coefficient.First,the dependence between every two PSFs is measured by the Pearson correlation coefficient.Second,the weights of the PSFs are obtained by considering the total dependence degree.Finally,PSFs’multipliers are modified based on the weights of corresponding PSFs,and then used in the calculating of human error probability(HEP).A case study is used to illustrate the procedure and effectiveness of the proposed method.

Strength Reliability Analysis of Stiffened Cylindrical Shells Considering Failure Correlation

The stiffened cylindrical shell is commonly used for the pressure hull of submersibles and the legs of offshore platforms. There are various failure modes because of uncertainty with the structural size and material properties, uncertainty of the calculation model and machining errors. Correlations among failure modes must be considered with the structural reliability of stiffened cylindrical shells. However, the traditional method cannot consider the correlations effectively. The aim of this study is to present a method of reliability analysis for stiffened cylindrical shells which considers the correlations among failure modes. Firstly, the joint failure probability calculation formula of two related failure modes is derived through use of the 2D joint probability density function. Secondly, the full probability formula of the tandem structural system is given with consideration to the correlations among failure modes. At last, the accuracy of the system reliability calculation is verified through use of the Monte Carlo simulation. Result of the analysis shows the failure probability of stiffened cylindrical shells can be gained through adding the failure probability of each mode.

Influence of rock property correlation on reliability analysis of rock slope stability: From property characterization to reliability analysis

Cohesion(c) and friction angle(φ) of rock are important parameters required for reliability analysis of rock slope stability. There is correlation between c and φ which affects results of reliability analysis of rock slope stability. However, the characterization of joint probability distribution of c and φ through which their correlation can be estimated requires a large amount of rock property data, which are often not available for most rock engineering projects. As a result, the correlation between c and φ is often ignored or simply assumed during reliability studies, which may lead to bias estimation of failure probability. In probabilistic rock slope stability analysis, the influence of ignoring or simply assuming the correlation of the rock strength parameters(i.e., c and φ) on the reliability of rock slopes has not been fully investigated. In this study, a Bayesian approach is developed to characterize the correlation between c and φ, and an expanded reliability-based design(RBD) approach is developed to assess the influence of correlation between c and φ on reliability of a rock slope. The Bayesian approach characterizes the sitespecific joint probability distribution of c and φ, and quantifies the correlation between c and φ using available limited data pairs of c and φ from a rock project. The expanded RBD approach uses the joint probability distribution of c and φ obtained through the Bayesian approach as inputs, to determine the reliability of a rock slope. The approach gives insight into the propagation of the correlation between c and φ through their joint probability into the reliability analysis, and their influence on the calculated reliability of the rock slope. The approaches may be applied in practice with little additional effort from a conventional analysis. The proposed approaches are illustrated using real c and φ data pairs obtained from laboratory tests of fractured rock at Forsmark, Sweden.

Reliability Analysis on Repairable System with Dual Input Closed-Loop Feedback Link Considering Shutdown Correlation Based on Goal Oriented Methodology

Goal oriented( GO) methodology is a kind of success oriented system reliability analysis method and has been used widely.The repairable system with dual input closed-loop feedback link( DICLFL) considering shutdown correlation didn't make reliability analysis accurately based on existing GO methodology. So, a reliability analysis method used to deal with DICLFL considering shutdown correlation is provided based on GO methodology.Firstly, a new operator, which is used to describe DICLFL considering shutdown correlation,whose number is 1,is created and named as Type 9C operator. And then,the formulas of type 9C operator are derived based on Markov process theory. Finally,the new method presented in this paper is adopted to conduct the reliability analysis of an electro-hydraulic servo speed control system. The analysis result is compared with those of Monte Carlo simulation and fault tree analysis( FTA). The comparison results show that this new reliability analysis method based on GO methodology is feasible and meaningful for reliability analysis of repairable systems with DICLFL considering shutdown correlation.Meantime,it will be useful for more other applications.

Reliability assessment and correlation analysis of evaluating orthodontic treatment outcome in Chinese patients

This study aimed to assess the reliability of experienced Chinese orthodontists in evaluating treatment outcome and to determine the correlations between three diagnostic information sources. Sixty-nine experienced Chinese orthodontic specialists each evaluated the outcome of orthodontic treatment of 108 Chinese patients. Three different information sources： study casts （SC）, lateral cephalometric X-ray images （LX） and facial photographs （PH） were generated at the end of treatment for 108 patients selected randomly from six orthodontic treatment centers throughout China. Six different assessments of treatment outcome were made by each orthodontist using data from the three information sources separately and in combination. Each assessment included both ranking and grading for each patient. The rankings of each of the 69 judges for the 108 patients were correlated with the rankings of each of the other judges yielding 13 873 Spearman rs values, ranging from -0.08 to ＋0.85. Of these, 90% were greater than 0.4, showing moderate-to-high consistency among the 69 orthodontists. In the combined evaluations, study casts were the most significant predictive component （R2=0.86, P〈0.000 1）, while the inclusion of lateral cephalometric films and facial photographs also contributed to a more comprehensive assessment （R2=0.96, P〈0.000 1）. Grading scores for SC＋LX and SC＋PH were highly significantly correlated with those for SC＋LX＋PH （r（SC＋LX）vs.（SC＋LX＋PH）=0.96, r（SC＋PH）vs.（SC＋LX＋PH）=0.97）, showing that either SC＋LX or SC＋PH is an excellent substitute for all three combined assessment,

Reliability Distribution of Numerical Control Lathe Based on Correlation Analysis

Combined Reliability distribution with correlation analysis,a new method has been proposed to make Reliability distribution where considering the elements about structure correlation and failure correlation of subsystems.Firstly,we make a sequence for subsystems by means of TOPSIS which comprehends the considerations of Reliability allocation,and introducing a Copula connecting function to set up a distribution model based on structure correlation,failure correlation and target correlation,and then acquiring reliability target area of all subsystems by Matlab.In this method,not only the traditional distribution considerations are concerned,but also correlation influences are involved,to achieve supplementing information and optimizing distribution.

Dynamic Reliability Evaluation and Life Prediction of Transmission System of Multi-Performance Degraded Wind Turbine

Wind power is a kind of important green energy.Thus,wind turbines have been widely utilized around the world.Wind turbines are composed of many important components.Among these components,the failure rate of the transmission system is relatively high in wind turbines.It is because the components are subjected to aerodynamic loads for a long time.In addition,its inertial load will result in fatigue fracture,wear and other problems.In this situation,wind turbines have to be repaired at a higher cost.Moreover,the traditional reliability methods are difficult to deal with the above challenges when performing the reliability analysis of the transmission system of wind turbines.To solve this problem,a stress-strength interference model based on performance degradation is introduced.Based on considering the strength degradation of each component,the improved Monte Carlomethod simulation based on the Back Propagation neural network is used to obtain the curve of system reliability over time.Finally,the Miner linear cumulative damage theory and the Carten-Dolan cumulative damage theory method are used to calculate the cumulative damage and fatigue life of the gear transmission system.

Adaptive Radial Based Sampling for Reliability and Reliability Sensitivity Analysis Involving Truncated Correlated Normal Variables

The reliability and reliability sensitivity （ RS ） models are presented for the engineering problem involving truncated correlated normal variables （CNV）, and in the case an adaptive radial based sampling is used to analyze the reliability and the RS. In the presented models, the truncated CNV is transformed to general CNV, and the value domains of the truncated CNV are treated as multiple failure modes, then the reliability and the RS with the truncated CNV are transformed to the general cases, on which an e^cient radial based sampling is used to analyze the trans- formed reliability and RS. An adaptive strategy is employed to search for the optimal radial in the sampling, by which the robustness of the method is improved. After the model concepts and the detailed implementation are given, several examples are presented to demonstrate the feasibility of the model and the efficiency of the solutions.

Jackknife based generalized resampling reliability approach for rock slopes and tunnels stability analyses with limited data:Theory and applications

An efficient resampling reliability approach was developed to consider the effect of statistical uncertainties in input properties arising due to insufficient data when estimating the reliability of rock slopes and tunnels.This approach considers the effect of uncertainties in both distribution parameters(mean and standard deviation)and types of input properties.Further,the approach was generalized to make it capable of analyzing complex problems with explicit/implicit performance functions(PFs),single/multiple PFs,and correlated/non-correlated input properties.It couples resampling statistical tool,i.e.jackknife,with advanced reliability tools like Latin hypercube sampling(LHS),Sobol’s global sensitivity,moving least square-response surface method(MLS-RSM),and Nataf’s transformation.The developed approach was demonstrated for four cases encompassing different types.Results were compared with a recently developed bootstrap-based resampling reliability approach.The results show that the approach is accurate and significantly efficient compared with the bootstrap-based approach.The proposed approach reflects the effect of statistical uncertainties of input properties by estimating distributions/confidence intervals of reliability index/probability of failure(s)instead of their fixed-point estimates.Further,sufficiently accurate results were obtained by considering uncertainties in distribution parameters only and ignoring those in distribution types.

Genetic Algorithms of Structural Fuzzy Reliability Index

In this paper the simple generation algorithms are improved. According to the geometric meaning of the structural reliability index, a method is proposed to deal with the variables in the standard normal space. With consideration of variable distribution, the correlation coefficient of the variables and its fuzzy reliability index, the feasibility and the reliability of the algorithms are proved with an example of structural reliability analysis and optimization.

Seismic reliability analysis of large electric power systems

Based on the De.Morgan laws and Boolean simplification, a recursive decomposition method is introduced in this paper to identify the main exclusive safe paths and failed paths of a network. The reliability or the reliability bound of a network can be conveniently expressed as the summation of the joint probabilities of these paths. Under the multivariate normal distribution assumption, a conditioned reliability index method is developed to evaluate joint probabilities of various exclusive safe paths and failed paths, and, finally, the seismic reliability or the reliability bound of an electric power system. Examples given in the paper show that the method is very simple and provides accurate results in the seismic reliability analysis.

Derivation of Reliability Index Vector Formula for Series System and Its Application

In this study, a reliability index vector formula is proposed for series system with two failure modes in term of the concept of reliability index vector and equivalent failure modes. Firstly, the reliability index vector is introduced to determine the correlation coefficient between two failure modes, and then, the reliability index vector of a series system can be obtained. Several numerical cases and an analysis on offshore platform are performed, and the results show that this scheme provided here has better computational accuracy, and its calculation process is simpler for the series systems reliability calculations compared with the other methods. Also this scheme is more convenient for the engineering applications.

Reliability Copula Model for Wind Turbine Gearbox Based on Failure Correlation

On the basis of each gear's failure correlation, the reliability Copula model of a wind turbine gearbox is established and a 1.5 MW wind turbine gearbox is taken as the research object. Firstly, based on the dynamic reliability model of mechanical parts, each gear's life distribution function of a wind turbine gearbox is obtained.The life distribution function can be used as the marginal distributions of the system's joint distribution. Secondly,Copula function is introduced to describe the failure correlation between parts, and the appropriate Copula function is selected according to the shape characters of Copula probability density function. Finally, the wind turbine gearbox system is divided into three parts according to the failure correlation of each gear. The Sklar theorem and the thought of step by step analysis are used to obtain the reliability Copula model for a wind turbine gearbox based on failure correlation.

Reliability analysis of urban gas transmission and distribution system based on FMEA and correlation operator

为了改进城市的气体的安全管理，传播和分发系统，失败模式和效果分析(FMEA ) 被用来构造管道网络的可靠性分析系统。解决主观的问题和多专家决策的无常，关联操作员被介绍进风险优先级号码(逆波兰表示法) 的计算。用与重量分析和专家调查途径一起的 FMEA， FMEA 评估桌子被给，包括五个失败模式，风险优先级数字，失败原因和效果，以及校正行动。结果证明那个关联操作员能直接处理语言学术语并且确定风险的优先级。

Reliability sensitivity of wind power system considering correlation of forecast errors based on multivariate NSTPNT method

The impact of wind power forecast errors (WPFEs) on power system reliability can be quantified by a sensitivity model, which helps to determine the importance of different wind farms. However, the unknown distribution and correlation of WPFEs make it difficult to calculate the reliability sensitivity. The existing univariate non-standard third-order polynomial normal transformation (NSTPNT) expresses the reliability sensitivity of WPFEs by a normal random variable with explicit distribution, and is not suitable for multiple wind farms with correlated forecast errors. In this paper, the univariate NSTPNT method is extended to the multivariate by deriving the analytical expression of the correlation coefficients before and after the transformation, to establish the transformation between the WPFEs and a normal random vector (RV) with the specific correlation. A reliability sensitivity model to the WPFEs expressed to the normal RV is then proposed. The numerical results validate the accuracy of the proposed multivariate NSTPNT and the sensitivity model. The maximum relative error for using the sensitivity to approximate the change of reliability with distribution parameters of the WPFEs is less than 2.42%. The necessity of considering the correlation of WPFEs is analyzed. The maximum relative error of the sensitivity reaches 83% when the correlation is ignored.

A type of polarimetric parameter for evaluating the reliability of model-based decomposition result and its application

The reliability of the model-based decomposition result,which is seldom investigated,is a key factor in determining whether decomposition parameters can be effectively applied to polarimetric synthetic aperture radar(PolSAR)applications.In this paper,a type of polarimetric parameter for evaluating the reliability of model-based decomposition result is proposed.It originates from the degree of correlation between the scattering models.The main idea is whether the measured power can be assigned to the corresponding scattering model.The more the measured power that can be allocated to the corresponding scattering model,the more reliable the decomposition result.The C-band Radarsat-2,L-band UAVSAR,X-band TerraSAR-X,and L-band E-SAR PolSAR data are selected for experiments.The results show that the type of polarimetric parameter can effectively represent the reliability of the decomposition result.The reliabilities of surface and double-bounce scattering are high in the ocean and orthogonal buildings,respectively.For oriented buildings,the reliability of the decomposition result is lower owing to the overestimation of volume scattering.For forested areas,reliability is generally higher if volume scattering is not overestimated.Furthermore,the results show that the reliability of soil moisture retrieval is positively correlated with surface scattering reliability.

Correlation between abnormal trends in the spontaneous fields of tectonic plates and strong seismicities

Tectonic activities, electrical structures, and electromagnetic environments are major factors that affect the stability of spontaneous fields. The method of corre- lating regional synchronization contrasts （CRSC） can determine the reliability of multi-site data trends or shortimpending anomalies. From 2008 to 2013, there were three strong earthquake cluster periods in the North-South seismic belt that lasted for 8-12 months. By applying the CRSC method to analyze the spontaneous field Esp at 25 sites of the region in the past 6 years, it was discovered that for each strong earthquake cluster period, the Esp strength of credible anomalous trends was present at minimum 30% of the stations. In the southern section of the Tan-Lu fault zone, the Esp at four main geoelectric field stations showed significant anomalous trends after June 2015, which could be associated with the major earthquakes of the East China Sea waters （Ms 7.2） in November 2015 and Japan＇s Kyushu island （Ms 7.3） in April 2016.