Reliability analysis of vacuum sewerage systems using the total probability theorem

2011 ◽  
pp. 2072-2079
Author(s):  
K Miszta-Kruk
Keyword(s):  
Reliability Analysis ◽  
Total Probability ◽  
Total Probability Theorem

Time-Dependent Reliability Analysis Using the Total Probability Theorem

2015 ◽  
Vol 137 (3) ◽  
Author(s):  
Zissimos P. Mourelatos ◽  
Monica Majcher ◽  
Vijitashwa Pandey ◽  
Igor Baseski
Keyword(s):  
Reliability Analysis ◽  
Limit State ◽  
Random Processes ◽  
Time Dependent ◽  
Normal Space ◽  
Total Probability ◽  
Conditional Probabilities ◽  
Reliability Method ◽  
Standard Normal ◽  
Total Probability Theorem

A new reliability analysis method is proposed for time-dependent problems with explicit in time limit-state functions of input random variables and input random processes using the total probability theorem and the concept of composite limit state. The input random processes are assumed Gaussian. They are expressed in terms of standard normal variables using a spectral decomposition method. The total probability theorem is employed to calculate the time-dependent probability of failure using time-dependent conditional probabilities which are computed accurately and efficiently in the standard normal space using the first-order reliability method (FORM) and a composite limit state of linear instantaneous limit states. If the dimensionality of the total probability theorem integral is small, we can easily calculate it using Gauss quadrature numerical integration. Otherwise, simple Monte Carlo simulation (MCS) or adaptive importance sampling are used based on a Kriging metamodel of the conditional probabilities. An example from the literature on the design of a hydrokinetic turbine blade under time-dependent river flow load demonstrates all developments.

Time-Dependent Reliability Analysis Using the Total Probability Theorem

2014 ◽  
Author(s):  
Zissimos P. Mourelatos ◽  
Monica Majcher ◽  
Vijitashwa Pandey ◽  
Igor Baseski
Keyword(s):  
Reliability Analysis ◽  
Conditional Probability ◽  
Limit State ◽  
Random Variables ◽  
Random Processes ◽  
Time Dependent ◽  
Normal Space ◽  
Total Probability ◽  
Standard Normal ◽  
Total Probability Theorem

A new reliability analysis method is proposed for time-dependent problems with limit-state functions of input random variables, input random processes and explicit in time using the total probability theorem and the concept of composite limit state. The input random processes are assumed Gaussian. They are expressed in terms of standard normal variables using a spectral decomposition method. The total probability theorem is employed to calculate the time-dependent probability of failure using a time-dependent conditional probability which is computed accurately and efficiently in the standard normal space using FORM and a composite limit state of linear instantaneous limit states. If the dimensionality of the total probability theorem integral (equal to the number of input random variables) is small, we can easily calculate it using Gauss quadrature numerical integration. Otherwise, simple Monte Carlo simulation or adaptive importance sampling is used based on a pre-built Kriging metamodel of the conditional probability. An example from the literature on the design of a hydrokinetic turbine blade under time-dependent river flow load demonstrates all developments.

Time-Dependent Reliability Analysis of Vibratory Systems With Random Parameters

2016 ◽  
Vol 138 (3) ◽  
Author(s):  
Zissimos P. Mourelatos ◽  
Monica Majcher ◽  
Vasileios Geroulas
Keyword(s):  
Large Scale ◽  
Probability Of Failure ◽  
Time Dependent ◽  
Total Probability ◽  
Conditional Probabilities ◽  
Random Parameters ◽  
Random Vibrations ◽  
Input Parameters ◽  
Total Probability Theorem ◽  
Dependent Probability

The field of random vibrations of large-scale systems with millions of degrees-of-freedom (DOF) is of significant importance in many engineering disciplines. In this paper, we propose a method to calculate the time-dependent reliability of linear vibratory systems with random parameters excited by nonstationary Gaussian processes. The approach combines principles of random vibrations, the total probability theorem, and recent advances in time-dependent reliability using an integral equation involving the upcrossing and joint upcrossing rates. A space-filling design, such as optimal symmetric Latin hypercube (OSLH) sampling, is first used to sample the input parameter space. For each design point, the corresponding conditional time-dependent probability of failure is calculated efficiently using random vibrations principles to obtain the statistics of the output process and an efficient numerical estimation of the upcrossing and joint upcrossing rates. A time-dependent metamodel is then created between the input parameters and the output conditional probabilities allowing us to estimate the conditional probabilities for any set of input parameters. The total probability theorem is finally applied to calculate the time-dependent probability of failure. The proposed method is demonstrated using a vibratory beam example.

Reliability Model of Gear With Correlated Failure Modes Based on Joint Distribution

2014 ◽  
Author(s):  
Lai Yue-Hua ◽  
Dong Hai-Ping ◽  
Yi Xiao-Jian ◽  
Ding Juan ◽  
Lei Hua-Jin
Keyword(s):  
Joint Distribution ◽  
Failure Modes ◽  
Physical Models ◽  
Total Probability ◽  
Traditional Model ◽  
Reliability Model ◽  
Interference Theory ◽  
Calculation Results ◽  
Set Up ◽  
Total Probability Theorem

In this paper, a precise reliability model for gear with correlated failure modes is presented. Firstly, physical models of stress and strength are set up against two main failure modes of a gear respectively. Then, the corresponding performance functions to the two failure modes are obtained according to stress-strength interference theory regarding randomness of variables in physical models of stress and strength. Furthermore, joint distribution of the two performance functions is deduced by Total Probability Theorem considering the correlation of random variables. So, reliability of a gear with correlated failure modes can be computed based on the joint distribution. Finally, an example is given and in this example the precise reliability model based on joint distribution and the traditional reliability model without considering the correlation of failure modes, are respectively adopted to calculate reliability of a gear. The calculation results are compared with that by Monte Carlo simulation and the compared results show that the gear’s reliability obtained by considering correlation of failure modes based on joint distribution is more accurate than that by the traditional model without considering the correlation of failure modes.

Hazard characterization for alternative intensity measures using the total probability theorem

2021 ◽  
pp. 875529302110492
Author(s):  
Michael W Greenfield ◽  
Andrew J Makdisi
Keyword(s):  
Ground Motion ◽  
Hazard Analysis ◽  
Total Probability ◽  
Ground Acceleration ◽  
Intensity Measures ◽  
Hazard Curve ◽  
Cumulative Absolute Velocity ◽  
Liquefaction Hazard ◽  
Hazard Curves ◽  
Total Probability Theorem

Since their inception in the 1980s, simplified procedures for the analysis of liquefaction hazards have typically characterized seismic loading using a combination of peak ground acceleration and earthquake magnitude. However, more recent studies suggest that certain evolutionary intensity measures (IMs) such as Arias intensity or cumulative absolute velocity may be more efficient and sufficient predictors of liquefaction triggering and its consequences. Despite this advantage, widespread hazard characterizations for evolutionary IMs are not yet feasible due to a relatively incomplete representation of the ground motion models (GMMs) needed for probabilistic seismic hazard analysis (PSHA). Without widely available hazard curves for evolutionary IMs, current design codes often rely on spectral targets for ground motion selection and scaling, which are shown in this study to indirectly result in low precision of evolutionary IMs often associated with liquefaction hazards. This study presents a method to calculate hazard curves for arbitrary intensity measures, such as evolutionary IMs for liquefaction hazard analyses, without requiring an existing GMM. The method involves the conversion of a known IM hazard curve into an alternative IM hazard curve using the total probability theorem. The effectiveness of the method is illustrated by comparing hazard curves calculated using the total probability theorem to the results of a PSHA to demonstrate that the proposed method does not result in additional uncertainty under idealized conditions and provides a range of possible hazard values under most practical conditions. The total probability theorem method can be utilized by practitioners and researchers to select ground motion time series that target alternative IMs for liquefaction hazard analyses or other geotechnical applications. This method also allows researchers to investigate the efficiency, sufficiency, and predictability of new, alternative IMs without necessarily requiring GMMs.

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