First-order Reliability Research Articles

Probabilistic assessment of complex corrosion in pipelines considering River-Bottom Profile information

One of the important causes of failure and incidents in pipelines is corrosion. Due to the severe consequences and impact in several social, economic, and environmental areas, pipelines must be regularly inspected. The possibilities for pipeline assessment are standard equations and numerical simulations. While the first is simple and fast, it tends to be conservative and the second is expensive. The finite element method (FEM) has been successfully used to predict the failure pressure of pipelines under different conditions. Commonly assessment methods overlook the uncertainties for failure predictions, often relying on safety factors. However, it is imperative to consider uncertainties to ascertain the reliability and risk of corroded pipes. While the reliability analysis of pipelines with idealized defects has been extensively explored in the literature, studies considering complex corrosion profiles, are still demanding. This paper addresses this gap considering the traditional Monte Carlo (MC) and first-order reliability (FORM) methods. The assessment is based on the Effective Area Method (EAM) within the failure function. As well know, MC can be expensive due to the high number of function evaluations. To mitigate this, FORM is employed as it requires fewer function evaluations. However, function gradients are required as it relies on an iterative optimization algorithm. Particularly challenges arise when computing the gradient using EAM for burst pressure estimation. The EAM is an iterative procedure, and small perturbations can yield incorrect sensitivity values, causing problems for FORM convergence. To address this, we propose a novel procedure for efficiently and less conservatively obtaining the gradients required by FORM using EAM as a failure function. The modified FORM method achieved convergence within 15 iterations for all test cases. This alternative approach offers a faster, more precise and less conservative method for Level-2 probabilistic failure analysis of complex-shaped corrosion defects compared to traditional Level-1 methods.

Deterministic and reliability-based design of necessary support pressures for tunnel faces

This paper provides methods for the deterministic and reliability-based design of the support pressures necessary to prevent tunnel face collapse. The deterministic method is developed by extending the use of the unique load multiplier, which is embedded within OptumG2/G3 with the intention of determining the maximum load that can be supported by a system. Both two-dimensional and three-dimensional examples are presented to illustrate the applications. The obtained solutions are validated according to those derived from the existing methods. The reliability-based method is developed by incorporating the Response Surface Method and the advanced first-order second-moment reliability method into the bisection algorithm, which continuously updates the support pressure within previously determined brackets until the difference between the computed reliability index and the user-defined value is less than a specified tolerance. Two-dimensional reliability-based support pressure is compared and validated via Monte Carlo simulations, whereas the three-dimensional solution is compared with the relationship between the support pressure and the resulting reliability index provided in the existing literature. Finally, a parametric study is carried out to investigate the influences of factors on the required support pressure.

A dynamic inverse FORM method: Design contours for load combination problems

In static structural reliability problems involving a set of random variables, it is common to relate the failure probability to the event that a safety margin becomes negative. For dynamic responses we usually seek the mean out-crossing rate into the unsafe region. Standard FORM (First-Order Reliability) methods can be applied to both the static and dynamic problems. A challenge comes in the design context, when a target probability of failure or crossing rate is specified and a particular design variable (e.g., capacity) is sought. In the static case, inverse FORM methods have been suggested. The resulting “design contour” may be viewed, in the transformed space of standard Gaussian variables, as the locus of all possible FORM design points that can give the desired probability of failure value. These contours have been previously used for the static problem to determine “design contours” for various problems. We here develop analogous design contours for the dynamic problem, based on FORM principles, again defining the contour as a locus of all possible FORM design points. We apply these contours to study a practical load combination problem involving the vertical bending moment of a ship. Finally, we present some data on the accuracy of several methods of analysis, and the effect of modeling choices for the input variables. The FORM method is shown to be practical and quite accurate for a variety of situations.

Reliability Assessment of Tie-Down Cables for Cable-Stayed Bridges Subject to Negative Reactions: Case Study

Abstract This paper presents a reliability assessment for tie-down cables of cable-stayed bridges subject to negative reactions. The advanced first-order second-moment reliability method (AFOSM) is used to evaluate the reliability indexes. The strength of a tie-down cable and load parameters are selected as random variables. A Newton-type solution scheme with double iteration loops is used to solve the minimization problem defined in the AFOSM. Reactions at supports are calculated through the stiffness equation formed by a finite-element model, and the sensitivities of the reactions with respect to the random variables are obtained by the direct differentiation of the stiffness equation. The reliability indexes of the tie-down cables are evaluated for two cable-stayed bridges in service in Korea: the Second Jindo Grand Bridge and Incheon Grand Bridge. Three limit states on the pretension, required strength, and design strength are considered for the load combination of the Strength I limit state in LRFD b...

Innovative Probabilistic Methodology for Evaluating the Reliability of Discrete Levee Reaches Owing to Piping

AbstractTraditionally, levees are a popular measure widely adopted for flood control, accepted and trusted by populations living in floodplain areas. The presence of levees sometimes might even induce a false sense of safety in the population, influencing their decision to develop further in floodplains, because they feel safer. Thus, failures of levee systems are potentially devastating, as they might induce loss of human lives, damages to properties, and economic loss. This study proposes an innovative methodology to estimate the reliability of levee systems, accounting for different sources of uncertainty, and to divide and classify discrete levee reaches, according to different fragility classes. The reliability analysis is performed by evaluating the probability of failure, as a function of a certain failure mechanism, conditioned by a given hydraulic load. Fragility curves are determined using two different methods: Monte Carlo data generation and the so-called approximate, first-order reliability m...

Calibration of Side, Tip, and Total Resistance Factors for Load and Resistance Factor Design of Drilled Shafts

This paper presents a reliability-based analysis for the calibration of side, tip, and total resistance factors for axially loaded drilled shafts. Twenty-two drilled shafts that had been tested with the Osterberg cell (O-cell) method were collected from project archives at the Mississippi Department of Transportation and the Louisiana Department of Transportation and Development. This database was carefully selected to represent the typical subsurface soil conditions and design practice in Louisiana. The prediction of the load–settlement curves of the drilled shafts from soil borings was determined with the FHWA O'Neill and Reese design method and the SHAFT 6.0 computer program. The interpreted and predicted axial nominal resistances for the drilled shafts were determined with the FHWA 5% shaft diameter settlement criterion from predicted load–settlement curves. The measured nominal side, tip, and total axial resistances for drilled shafts were determined from the O-cell measurements. Statistical analyses were first performed to compare the predicted nominal axial resistances and the measured nominal resistances for the drilled shafts. In general, the selected design method underestimated the measured drilled shaft resistances. The first-order reliability and the Monte Carlo simulation methods were selected to determine the side, tip, and total resistance factors under the Strength I limit at the target reliability index of 3.0. The resulting calibrated side, tip, and total resistance factors were 0.3, 0.6, and 0.7, respectively.

Integrated probabilistic analysis of nuclear power plant building damage due to an aircraft crash

In order to ensure that nuclear power plant buildings are reliable and safe in case of external loading, it is very important to evaluate uncertainties associated with loads, material properties, geometrical parameters, boundaries and other parameters. Therefore, a probability-based analysis was developed as the integration of deterministic and probabilistic methods using existing state-of-the-art software. The subject of this paper is the integrated analysis of building failure due to impact by a commercial aircraft. The Monte Carlo Simulation, First-Order Reliability and the combined Monte Carlo Simulation and Response Surface methods were used for the probabilistic analyses. During an aircraft crash, the dynamic impact loading is uncertain. Therefore a relation expressed by the probability of failure of impacted wall and loading function was determined. With failure defined as concrete cracking and rebar rupture, the failure probabilities of the impacted wall were calculated as a function of the peak impact load. The integrated deterministic and probabilistic analysis approach was applied to the Ignalina Nuclear Power Plant in Lithuania. The conclusions from this analysis was that a through-the-wall crack in the concrete element of a plant wall may occur with a probability of 0.0266, but the failure probability of the reinforcement bars is very small, that is, near zero. Thus, no perforation of the impacted wall by structures of the aircraft should occur. The importance of performing a probabilistic analysis of crash events is shown by comparing results to those obtained by a mean value deterministic approach.

Stochastic finite-element modelling and optimization for net-shape forging of three-dimensional aero-engine blades

The stochastic nature of the forging process is a key influencing factor for achievable dimensional and shape accuracy of forged aerofoil blades for aero-engine applications. In this article, a finite-element-based stochastic modelling and optimization method is presented for net-shape forging of three-dimensional (3D) aerofoil blades. A combined Monte Carlo simulation and response surface method as well as the first-order second-moment reliability method are developed to quantify the stochastic characteristics of forging errors due to uncertainties of process parameters and forging conditions. A two-step optimization approach is presented to minimize systematic errors using a direct compensation method and to reduce random variations through a variance control procedure. A 3D blade forging case study is presented to evaluate the stochastic characteristics of the dimensional errors of forged aerofoil blades in comparison with measurement data and to demonstrate much improved forging accuracy using the proposed stochastic optimization method. The results from the case study also suggest that the stochastic-based optimization method may be easily applied to other metal-forming processes.

Probabilistic assessment of small earthfill dams

This paper focuses on a generic earthfill dam model and investigates, probabilistically, downstream and upstream slope stability, as significant limit states governing the long-term performance. Specific variables that define failure modes are identified and their probabilistic models defined. A parametric analysis is performed, using a simplified soil model, for different upstream and downstream slope gradients to evaluate notional reliabilities against slope failure. This is achieved using standard first-order second-moment reliability analysis. On the basis of these analyses, a methodology for sitespecific effects such as deterioration, maintenance patterns and the impact of climate change can be proposed.

Adaptive trust regions and response surfaces for reliability analysis

An efficient method for the reliability analysis of systems with non-linear limit states is presented in this paper. A trust-region-based adaptive response surface is used to search for the most probable point on the limit state. The adaptive response surface construction is aided by an optimal symmetric Latin hypercube sampling-based experimental design. First-order and second-order reliability estimates using the adaptive response surface are further improved with multi-modal adaptive importance sampling. Numerical examples including a vehicle side impact problem show that the proposed method has desirable efficiency and accuracy.

Probabilistic Slope Stability Analysis with Stochastic Soil Hydraulic Conductivity

The effects of stochastic hydraulic conductivity on the slope stability of an embankment dam are investigated using a combination of random field simulation, seepage analysis, and slope stability analysis. The hydraulic conductivity distribution is treated as a spatially stationary random field following a lognormal distri- bution. The turning band method is used to generate the spatial variability of the saturated hydraulic conductivity Ks in the domain. Different standard deviations of log hydraulic conductivity are investigated. For each sln Ks value of various realizations of hydraulic conductivity were generated and combined with a numerical s , ln Ks model to simulate water flow in an earth dam with variable Ks. The first-order second-moment reliability index b was employed to characterize the influence of the variability of Ks, and hence, pore-water pressures, on the stability of the downstream slope. A linear relationship between and the standard deviation of the factor sln Ks of safety sF was obtained from the simulation results. A relationship between b and , in which every 0.1 sln Ks increment of results in a decrease of 1.0 in b, is deduced based on the simulation results. Results of a sln Ks Shapiro-Wilk test for goodness of fit indicate that the factor of safety can be assumed to be normally or lognormally distributed when the saturated hydraulic conductivity follows a lognormal distribution and is sln Ks small (#0.5). When is large (>0.5), neither normal nor lognormal distributions provide a reasonable s ln Ks approximation of the factor of safety. Simulation results show that neither standard deviation nor coefficient of variation of the factor of safety is constant when only the variability of hydraulic conductivity is considered. While the results presented are directly applicable only to the particular earth dam geometry and boundary conditions studied, the methodology is general and may be extended to embankments with different boundary conditions. INTRODUCTION AND BACKGROUND Natural soils are highly variable and heterogeneous (Lamb 1966; Peck 1967; Vanmarcke 1977a; DeGroot and Baecher 1993). In practice, methods used for the analysis of slope sta- bility typically are deterministic, with soil properties charac- terized as constants for a given soil layer and each specified layer assumed to be homogeneous. The variability of soil properties presumably is taken into account indirectly for de- sign purposes by selecting a factor of safety, which is chosen based on experience and/or some design criteria (Griffiths and Fenton 1993; Duncan 1996). Many authors have pointed out the limitations of deterministic methods because they do not account explicitly for variability and uncertainty related to soil parameters (Vanmarcke 1977b; Oka and Wu 1990; Soulie et al. 1990; Kulhawy et al. 1991). Methods are needed to quan- tify uncertainties associated with soil properties and to include them in slope stability analysis (Anderson et al. 1982; Whit- tlestone et al. 1995). By incorporating uncertainties quantita- tively, reliability analysis can enhance engineering judgment and facilitate improved decision making (Li and Lumb 1987). Many water-retaining structures in North America are em- bankment dams (Fenton and Griffiths 1996) and the prediction of dam slope stability is of great importance. Hydraulic con- ductivity of the embankment soils is used to calculate all of the hydraulic parameters associated with dam performance, in- cluding seepage rates and amounts, pore-water pressures, and therefore, the state of effective stress. It follows that uncer- tainty associated with the spatial variability of hydraulic con- ductivity leads to uncertainty in estimating pore-water pres-

Reliability of Geometrically Nonlinear PR Frames

An efficient stochastic finite element method is proposed for the reliability analysis of nonlinear frame structures using the first-order second-moment reliability method. The assumed stress-field approach is used in computing nonlinear structural responses and the corresponding response gradients. Both geometric and material nonlinearities of structures are considered. The flexibility of a connection is represented by its moment-relative rotation relationship. Emphasis is placed on the comparison of the reliability indexes of linear structures, nonlinear structures, and nonlinear structures with flexible connections. Both the strength (combined axial compression and bending) and serviceability (deflection) limit states are considered. For stability-related structures, nonlinearity has a considerable influence on the structural reliability, and the reliability will be very different when the flexible connections are taken into consideration. The method is verified using the Monte Carlo simulation technique. The applicability, accuracy, efficiency, and robustness of the method are verified with the help of an example.

The stochastic finite element method in structural reliability

First-order reliability and finite element methods are used to develop a methodology for reliability analysis of structures with stochastically varying properties and subjected to random loads. Two methods for discretization of random fields are examined and the influence of the correlation length of random property or load fields on the reliability of example structures are investigated. It is found that the correlation length of load fields has significant influence on the reliability against displacement or stress limit states. The correlation length of property fields is significant for displacement limit states, but may not be significant for stress limit states. Examples studied include a fixed ended beam with stochastic rigidity and a plate with stochastic elasticity.

COMPUTER AIDED OPTIMUM STRUCTURAL DESIGN UNDER UNCERTAINTY

Abstract This paper is concerned with the incorporation of different forms of imprecision and uncertainty into the formulation and solution of optimum structural design problems. The statistical uncertainties inherent in material properties and structural loadings are modelled by a first-order second-moment reliability approach. Resulting constraint boundaries are then treated as “soft” boundary regions in which the non-statistical uncertainties, and quality aspects of design and construction are handled by fuzzy set operations.

Truncation Errors in First-Order Reliability

In a second moment format for a probabilistic structural design code, it is often necessary to estimate the mean and standard deviation of stress and strength, which in turn may be complicated functions of several random variables. In a broader sense, probabilistic analyses in many disciplines require knowledge of the first and second moments of complicated functions of random variables. This study attempts to describe, under special conditions, the errors introduced in the most widely used approximate method for making this calculation.