EVALUATION OF THE PROBABILITY OF FAILURE WITH UNCERTAIN DISTRIBUTION PARAMETERS

Efficient method for fully quantifying the uncertainty of failure probability

Response uncertainty evaluation and dynamic reliability analysis corresponding to classical stochastic dynamic analysis are usually restricted to the uncertainties of the excitation. The inclusion of the parameter uncertainties contained in structural properties and excitation characteristics has become an increasingly important problem in many areas of dynamics. In the present paper, a point estimate procedure is proposed for the evaluation of stochastic response uncertainty, and a response surface approach procedure in standard normal space is proposed for analysis of time-variant reliability analysis for hysteretic MDF structures having parameter uncertainties. Using the proposed procedures, the response uncertainties and time-variant reliability can be easily obtained by several repetitions of stochastic response analysis under given parameters without conducting sensitivity analysis, which is considered to be one of the primary difficulties associated with conventional methods. In the time-variant reliability analysis, the failure probability can be readily obtained by improving the accuracy of the first-order reliability method using the empirical second-order reliability index. The random variables are divided into two groups, those with CDF and those without CDF. The latter are included via the high-order moment standardization technique. A numerical example of a 15F hysteretic MDF structure that takes into account uncertainties of four structural parameters and three excitation characteristics is performed, based on which the proposed procedures are investigated and the effects of parameter uncertainties are discussed. Copyright © 1999 John Wiley & Sons, Ltd.

In this paper tunnel probabilistic structural analysis (TuPSA) was performed using the first order reliability method (FORM). In TuPSA, a tunnel performance function is defined according to the boundary between the structural stability and instability. Then the performance function is transformed from original space into the standard normal variable space to obtain the design point, reliability index, and also the probability of tunnel failure. In this method, it is possible to consider the design factors as the dependent or independent random parameters with arbitrary probability distributions. A software code is developed to perform the tunnel probabilistic structural analysis (TuPSA) using the FORM. For validation and verification of TuPSA, a typical tunnel example with random joints orientations as well as mechanical properties has been studied. The results of TuPSA were compared with those obtained from Monte-Carlo simulation. The results show, in spite of deterministic analysis which indicates that the rock blocks are stable, that TuPSA resulted in key-blocks failure with certain probabilities. Comparison between probabilistic and deterministic analyses results indicates that probabilistic results, including the design point and probability of failure, are more rational than deterministic factor of safety.

Heliostats represent the most important maintenance cost in the solar power tower plant. The aim of this work is to provide a failure probability analysis for heliostat design in order to minimize this maintenance cost. Based on mechanics of material study and wind aerodynamic analysis, a performance function, with five random variables, has been developed wherein the random variables are: wind speed, inside and outside pedestal diameters, pedestal yield stress and mirror mass. Four main methods have been proposed: first order reliability method, second order reliability method, Monte Carlo (MC) method and subset simulation (SS) method. The variation of failure probability with the variation of pedestal wall thickness and wind speed, for different outside diameters and heliostat azimuth and elevation angles, has been simulated. The results show that SS is more efficient and accurate for small failure probabilities; however, MC is more accurate for high failure probabilities.

Parameter estimation of the three-parameter Weibull distribution is an important problem in reliability analysis and statistical modeling. Random right-censored data are widely encountered in engineering practice. Conventional least squares (LS) methods usually construct the empirical cumulative distribution function (CDF) based on rank statistics. However, this empirical assumption cannot adequately capture the nonlinear variation in failure probability with time in the Weibull distribution. To address this limitation, an iterative conditional probability based on conditional failure probability (ICP-CDF) is proposed. The method uses the parameter estimates obtained from the conventional LS approach as initial values, adjusts the ranks of failure data according to conditional failure probabilities, and updates the empirical CDF accordingly. Within a unified least squares estimation framework, an ICP-CDF-LS parameter estimation method is developed, in which both the CDF and distribution parameters are updated iteratively. Simulation studies and case analyses demonstrate that, compared with the LS and MLE methods, the proposed approach achieves superior overall performance in terms of estimation accuracy and stability, making it more suitable for practical engineering applications.

In this work, the reliability analysis is employed to take into account the uncertainties in a structure. Reliability analysis is a tool to compute the probability of failure corresponding to a given failure mode. In this study, one of the most commonly used reliability analysis method namely first order reliability method is used to calculate the probability of failure. Since finding the most probable point (MPP) or design point is a constrained optimization problem, in contrast to all the previous studies based on the penalty function method or the preference of the feasible solutions technique, in this study one of the latest versions of the differential evolution metaheuristic algorithm named improved (μ+λ)-constrained differential evolution (ICDE) based on the multi-objective constraint-handling technique is utilized. The ICDE is very easy to implement because there is no need to the time-consuming task of fine tuning of the penalty parameters. Several test problems are used to verify the accuracy and efficiency of the ICDE. The statistical comparisons revealed that the performance of ICDE is better than or comparable with the other considered methods. Also, it shows acceptable convergence rate in the process of finding the design point. According to the results and easier implementation of ICDE, it can be expected that the proposed method would become a robust alternative to the penalty function based methods for the reliability assessment problems in the future works.

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.

The effects of varying distribution type of random variables and environmental, operational, and design random variables influenced by a shock wave caused from various origins on the failure probability are systematically investigated using the first order reliability method (FORM) for buried pipeline. It is found that the failure probability of the buried pipeline increases with faster P-wave velocity and slower S-wave velocity. The failure probability is estimated to be the largest for the Weibull distribution and the smallest for the lognormal distribution. A set of similar values of the failure probability for the normal distribution and lognormal distribution are noted. The normalized margin is defined and estimated. Furthermore, the normalized margin is used to predict the failure probability simply by using the fitting lines between the failure probability and the normalized margin.

Methodologies to calculate failure probability and to estimate the reliability of fatigue loaded structures are developed. The applicability of the methodologies is evaluated with the help of the fatigue crack growth models suggested by Paris and Walker. The probability theories such as the FORM (first order reliability method), the SORM (second order reliability method) and the MCS (Monte Carlo simulation) are utilized. It is found that the failure probability decreases with the increase of the design fatigue life and the applied minimum stress, the decrease of the initial edge crack size, the applied maximum stress and the slope of Paris equation. Furthermore, according to the sensitivity analysis of random variables, the slope of Pairs equation affects the failure probability dominantly among other random variables in the Paris and the Walker models.

A methodology for the assessment of the service life of liquid carrying metalic pipelines subjected to pitting corrosion is presented. The estimate of pipeline life is obtained from the loss of liquid through pit holes during transportation. The growth of corrosion pits is modelled by a two parameter exponential function having time dependency and decreasing rate of pit growth. Parameters which are related to corrosion, pipeline dimension and liquid flow are treated as probabilistic variables. The First Order Reliability method is used to estimate the reliability index, the probability of failure and the relative contribution of various uncertain parameters. Numerical results are given for a typical example pipeline. The results show that the probability of failure increases nonlinearly with time and that the contribution of pit hole size and pitting corrosion parameters are very significant in the long run.

The reliability of a simple turbomachinery model was calculated to demonstrate the application of a newly developing system integration tool, Probabilistic Design and Analysis Framework(PRODAF), along with efficient probabilistic methods using a response surface method. The model represents a system consisting of hypothetical turbine components. The parts include a blade, disk, and shaft with an applied angular velocity. All the components were modeled with the properties of the nickel alloy, Inconel 718. A response surface was calculated for the system of components to improve probabilistic computational efficiency. In addition, a fast probability integration method, Advanced First Order Reliability Method (AFORM), was used for the probabilistic analysis in order to provide an efficient analysis as possible. Geometric dimensions, the applied load, and material yield strength were varied for this study. The probability of failure was determined using the maximum first principal stress response and the material yield strength. A simple G function using the difference between strength and loading stress was used to determine failure limits. The probabilistic sensitivity of the failure response relative to the individual variables was determined also with material yield strength having the greatest influence. The model was recreated with every iteration of the probabilistic analysis in order to vary the geometry. As a result, the response surface method has a significant impact on improving computational efficiency and enabling reliability analysis with rapid turnaround.

Oil and gas industry is one of the most important industries contributing to the Malaysian economy. To extract hydrocarbons, various types of production platforms have been developed. Fixed jacket platform is the earliest type of production structure, widely installed in Malaysia’s shallow and intermediate waters. To date, more than 60% of these jacket platforms have operated exceeding their initial design life, thus making the re-evaluation and reassessment necessary for these platforms to continue to be put in service. In normal engineering practice, system reliability of a structure is evaluated as its safety parameter. This method is however, much complicated and time consuming. Assessing component's reliability can be an alternative approach to provide assurance about a structure’s condition in an early stage. Design codes such as the Working Stress Design (WSD) and the Load and Resistance Factor Design (LRFD) are well established for the component-level assessment. In reliability analysis, failure function, which consists of strength and load, is used to define the failure event. If the load acting exceeds the capacity of a structure, the structure will fail. Calculation of stress utilization ratio as given in the design codes is able to predict the reliability of a member and to estimate the extent to which a member is being utilised. The basic idea of this ratio is that if it is more than one, the member has failed and vice versa. Stress utilization ratio is a ratio of applied stress, which is the output reaction of environmental loadings acting on the structural member, to the design strength that comes from the member’s geometric and material properties. Adopting this ratio as the failure event, the reliability of each component is found. This study reviews and discusses the reliability for selected members of three Malaysian offshore jacket platforms. First Order Reliability Method (FORM) was used to generate reliability index and the probability of failure of the members. It was found that probability of failure is inversely related to reliability index for component reliability and variation in metocean values does not have much effect on the component reliability. High reliability indices indicate that component level reliability analysis is sufficient.

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.

Structural Reliability analysis is based on probability theory and a significant amount of literature exists on the subject and continues to be a subject of ongoing research. When the structural safety of a ship’s hull is considered, the ultimate hull girder strength should be evaluated. Existing literature on the subject often neglects the effect of temperature. This paper aims to determine the effects of temperature in ship structural reliability and to propose a procedure for analysing structures by taking into account temperature effects. The ultimate strength of the hull girder was calculated using a component approach, where the behaviour of the hull is evaluated based on the behaviour of the single structural components. A sample analysis for Tanker/FPSO structures is provided where the reliability analysis was carried out using a First and Second Order Reliability Method (FORM and SORM) analysis. The loading component was handled using extreme wave statistics and the Ferry Borges-Castanheta load combination method. Annual reliability indices and probabilities of failure were calculated for hogging and sagging conditions showing the effects of temperature along with Partial Safety Factors for all variables taken into account.

Fiber Reinforced Polymer (FRP) composites have been increasingly accepted in the construction industry as a promising substitute for steel. Bridge deck deterioration is one of the most common deficiencies in a bridge system. The use of FRP bars as reinforcement for concrete bridge decks provides a potential for increased service life, economic, and environmental benefits. This research presents the development of resistance models for concrete members (beams and slabs) reinforced with FRP bars (FRP-RC), for carrying out reliability analysis. The scope of this model is limited to the flexural behavior only; i.e. the failures are not shear failure and debonding. Probability of Failure,Pf , and Reliability Index, â, of FRP-RC sections are calculated using the developed resistance model. A wide range of design variables is covered in calibrating the flexural design of FRP-RC members, using First Order Reliability Method (FORM). This study results in the development of resistance models for FRP-RC bridge decks and girders which can also be used as resistance models for FRP-RC slabs and beams respectively. Also, the flexural reliability study on FRP-RC slabs and beams yielded parameters that affect the Probability of Failure,Pf , in terms of the Reliability Index, â. These results may be used to enhance the current recommendations for resistance factors.