Second-order Reliability Method Research Articles

Marginal and total exceedance probabilities of environmental contours

Various methods have been proposed for defining an environmental contour, based on different concepts of exceedance probability. In the inverse first-order reliability method (IFORM) and the direct sampling (DS) method, contours are defined in terms of exceedances within a region bounded by a hyperplane in either standard normal space or the original parameter space, corresponding to marginal exceedance probabilities under rotations of the coordinate system. In contrast, the more recent inverse second-order reliability method (ISORM) and highest density (HD) contours are defined in terms of an isodensity contour of the joint density function in either standard normal space or the original parameter space, where an exceedance is defined to be anywhere outside the contour. Contours defined in terms of the total probability outside the contour are significantly more conservative than contours defined in terms of marginal exceedance probabilities. In this work we study the relationship between the marginal exceedance probability of the maximum value of each variable along an environmental contour and the total probability outside the contour. The marginal exceedance probability of the contour maximum can be orders of magnitude lower than the total exceedance probability of the contour, with the differences increasing with the number of variables. For example, a 50-year ISORM contour for two variables at 3-h time steps, passes through points with marginal return periods of 635 years, and the marginal return periods increase to 10,950 years for contours of four variables. It is shown that the ratios of marginal to total exceedance probabilities for DS contours are similar to those for IFORM contours. However, the marginal exceedance probabilities of the maximum values of each variable along an HD contour are not in fixed relation to the contour exceedance probability, but depend on the shape of the joint density function. Examples are presented to illustrate the impact of the choice of contour on simple structural reliability problems for cases where the use of contours defined in terms of either marginal or total exceedance probabilities may be appropriate. The examples highlight that to choose an appropriate contour method, some understanding about the shape of a structure's failure surface is required.

Efficient prediction of wind and wave induced long-term extreme load effects of floating suspension bridges using artificial neural networks and support vector machines

Long-term extreme load effects are one of the primary concerns in the design of civil and offshore structures. Such load effects can be evaluated using the accurate but computationally demanding full long-term method or the more efficient but approximate first-order and second-order reliability methods. Monte Carlo based methods enhanced with machine learning algorithms offer efficient alternatives to the traditional methods. Therefore, artificial neural networks and support vector machines are used as surrogate models for the limit state function to speed up the prediction of long-term extreme load effects. A three-span suspension bridge with two floating pylons under combined wind and wave actions is used as a case study. The cumulative density functions of the long-term extreme values corresponding to a bending moment value due to vertical deflections at the critical position of the girder are calculated. It is then shown that the artificial neural network and support vector machine-based approaches require less computational effort and yield more accurate results than the first- and second-order reliability methods.

Conditional probability-based system reliability analysis for geotechnical problems

In geotechnical engineering, multiple failure modes exist, and any one can contribute to the failure of a system. System reliability is related to both the probability of failure of each failure mode and the correlations between them, which is obviously different from the reliability analysis of a single mode. This paper focuses on system reliability analysis for geotechnical engineering application, and proposes an efficient system reliability analysis method based on conditional probability, which is implemented in the ubiquitous Excel spreadsheet platform. By using the so-called common source random variables concept, the system reliability problem is converted to the calculation of a conditionally independent failure event. The procedure is illustrated for a circular rock tunnel with three failure modes, a tetrahedral rock wedge with four failure modes, and a slope with two and eight failure modes. In addition, comparisons are made between the proposed method with the bimodal bounds approach and the multivariate normal cumulative distribution function method. The study also illustrates the use of the second-order reliability method (SORM) to refine the reliability indices and improve the accuracy of the estimated system reliability in a case where limit state surfaces are not hyperplanes.

MaxEnt feature-based reliability model method for real-time detection of early chatter in high-speed milling

Real-time detection of early chatter is a vital strategy to improve machining quality and material removal rate in the high-speed milling processes. This paper proposes a maximum entropy (MaxEnt) feature-based reliability model method for real-time detection of early chatter based on multiple sampling per revolution (MSPR) technique and second-order reliability method (SORM). To enhance the detection reliability, the MSPR is used to acquire multiple sets of once-per-revolution sampled data (i.e., MSPR data) and to overcome the shortcoming of the once-per-revolution sampling. The proposed MaxEnt feature-based reliability model method solves the issue of the real-time detection of early chatter while ensuring its reliability. The failure hazard function (FHF) is estimated as a chatter indicator by using the SORM with the MaxEnt feature. The proposed method consists of five steps. First, set the prior parameters. Then collect data by using the MSPR technique. Next, calculate a set of the standard deviation of the data collected as a chatter feature and estimate the chatter indicator FHF by applying the SORM with the MaxEnt feature. Finally, implement the real-time detection of early chatter based on the estimated chatter indicator FHF and the threshold FHF0. The proposed method is applied to the high-speed milling process. Two examples prove that the proposed method can detect two kinds of early chatter: the early-stage of a severe chatter and the slightly intolerable chatter.

A most probable point method for probability distribution construction

With nonlinearity and uncertainty existing in engineering problems, it is important to predict the probability distribution of a response of an engineering model. The probability distribution is often constructed without sufficient accuracy due to a high computational cost. In this paper, a most probable point (MPP) method for the probability distribution construction is proposed. First, predictive models of the MPP components are established based on the Gaussian mixture distribution (GMD) and the inverse first-order reliability method. A mixture of first- and second-order reliability methods is then used to calculate discrete points of the cumulative distribution function (CDF). Finally, the CDF of the response is constructed by the GMD. A mathematical example and three engineering examples are used to verify the effectiveness of the proposed method.

Dimension Reduction Method-Based RBDO for Dependent Interval Variables

Reliability-based design optimization (RBDO) under epistemic uncertainty (i.e., imprecise probabilistic information), especially in the presence of dependency of input variables, is a challenging problem. In this paper, we propose a dimension reduction-based RBDO framework considering dependent interval variables, which is pursued in a purely probabilistic manner. Most probable point (MPP) based dimension reduction method (DRM) is used for reliability evaluation due to its ability to circumvent the shortcomings of poor approximation by first order reliability method (FORM) and pronounced computational complexity by second order reliability method (SORM). For modeling correlation of input variables, copula is used instead of true joint cumulative density function (CDF). A flexible Johnson family of distributions is used to handle the stochastic but poorly known epistemic uncertainty. In order to handle the uncertainty in correlation measures, arisen due to interval data, expert suggested bounds of correlation measures have been recommended. For the overall RBDO problem, a decoupled approach to optimization is explored. Two numerical examples — one mathematical problem and one engineering problem — have been solved to properly explicate the proposed RBDO process. It is demonstrated that correlations in input variables have significant impact on the optimal design solutions.

Probabilistic analysis of corroded pipeline under localized corrosion defects based on the intelligent inspection tool

A methodology is developed in this paper to estimate the time-dependent reliability of a pipeline in Algeria, which is in exploitation from 1981 and with internal and external localized corrosion defects detected in 2009. A probabilistic approach was applied to a pipeline made of API 5L X60 steel for the long-distance transportation of natural gas (55 km), which crossing several geographical reliefs in different country areas. The analysis was carried out by taking into consideration the potential stochastic variables altering drastically the reliability of the pipeline, i.e., dimensions of the localized corrosion defects, their locations, and distribution, and the corrosion rate. The correlations between the pipeline spatial corrosion defects distribution and their dimensions, obtained by using an intelligent inspection tool are used to analyze the actual in-service corrosion attack rate and for enhancement of the accuracy of the reliability assessment. The failure scenario was considered as the moment when the pipeline operating pressure exceeds the failure pressure defined analytically in accordance with different commonly used standards. The assessment of the reliability index of the corroded pipeline subjected to internal pressure and the failure probability was done. The detailed reliability analysis is carried out by using the second order reliability method (SORM) for the basic random variables with different probabilistic density within a nonlinear limit state function. The latter is based on the limit analysis of the failure pressure model for different standards and the numerical analysis using the finite element method and ANSYS software.

A novel multi-fidelity modelling-based framework for reliability-based design optimisation of composite structures

A new multi-fidelity modelling-based probabilistic optimisation framework for composite structures is presented in this paper. The multi-fidelity formulation developed herein significantly reduces the required computational time, allowing for more design variables to be considered early in the design stage. Multi-fidelity models are created by the use of finite element models, surrogate models and response correction surfaces. The accuracy and computational efficiency of the proposed optimisation methodology are demonstrated in two engineering examples of composite structures: a reliability analysis, and a reliability-based design optimisation. In these two benchmark examples, each random design variable is assigned an expected level of uncertainty. Monte Carlo Simulation (MCS), the First-Order Reliability Method (FORM) and the Second-Order Reliability Method (SORM) are used within the multi-fidelity framework to calculate the probability of failure. The reliability optimisation is a multi-objective problem that finds the optimal front, which provides both the maximum linear buckling load and minimum mass. The results show that multi-fidelity models provide high levels of accuracy while reducing computation time drastically.

System Reliability Analysis With Second-Order Saddlepoint Approximation

AbstractThe second-order saddlepoint approximation (SOSPA) has been used for component reliability analysis for higher accuracy than the traditional second-order reliability method (SORM). This work extends the second-order saddlepoint approximation (SPA) to system reliability analysis. The joint distribution of all the component responses is approximated by a multivariate normal distribution. To maintain high accuracy of the approximation, the proposed method employs the second-order SPA to accurately generate the marginal distributions of the component responses; to simplify computations and achieve high efficiency, the proposed method estimates the covariance matrix of the multivariate normal distribution with the first-order approximation to the component responses. Examples demonstrate the high effectiveness of the second-order SPA method for system reliability analysis.

Key-block based analytical stability method for discontinuous rock slope subjected to toppling failure

Abstract This study introduces a simplified semi-distinct element algorithm for discontinuous rock slopes under toppling instability assessment based on block theory. The presented algorithm is developed to investigate block, flexural and block-flexural types of toppling failures which have been coded in the Python high-level programming language. In order to investigate toppling instabilities for different modes, the three modelling steps, namely, geometrical, behavioural and mechanical simulations were conducted to appoint high-order calculation trending loops. These loops were based on the analytical description of modified second-order reliability method with first-order efficiency to determine the factor of safety values for discontinuous rock slopes. Each type of toppling failure was described based on main assumptions that were modified by geometric qualifications such as the analytical procedure, single block slip, and the Voronoi diagram algorithm for the stability analysis of the three different failure types. According to the stability assessment results of the studied slopes which were verified by the distinct numerical method via UDEC software, it has been clarified that the results obtained from the algorithm and the numerical analysis were in an appropriate trend. The used algorithm is superior in pinpointing the critical sliding zones, safety factor estimation and progressive failure analyses as compared to the numerical analyses. Moreover, the block theory based algorithm has higher processing capability and speed with respect to the numerical method.

System Reliability Analysis With Autocorrelated Kriging Predictions

AbstractWhen limit-state functions are highly nonlinear, traditional reliability methods, such as the first-order and second-order reliability methods, are not accurate. Monte Carlo simulation (MCS), on the other hand, is accurate if a sufficient sample size is used but is computationally intensive. This research proposes a new system reliability method that combines MCS and the Kriging method with improved accuracy and efficiency. Accurate surrogate models are created for limit-state functions with minimal variance in the estimate of the system reliability, thereby producing high accuracy for the system reliability prediction. Instead of employing global optimization, this method uses MCS samples from which training points for the surrogate models are selected. By considering the autocorrelation of a surrogate model, this method captures the more accurate contribution of each MCS sample to the uncertainty in the estimate of the serial system reliability and therefore chooses training points efficiently. Good accuracy and efficiency are demonstrated by four examples.

Adaptive importance sampling based neural network framework for reliability and sensitivity prediction for variable stiffness composite laminates with hybrid uncertainties

In this work, we propose to leverage the advantages of both the Artificial Neural Network (ANN) based Second Order Reliability Method (SORM) and Importance sampling to yield an Adaptive Importance Sampling based ANN, with specific application towards failure probability and sensitivity estimates of Variable Stiffness Composite Laminate (VSCL) plates, in the presence of multiple independent geometric and material uncertainties. The performance function for the case studies is defined based on the fundamental frequency of the VSCL plate. The accuracy in both the reliability estimates and sensitivity studies using the proposed method were found to be in close agreement with that obtained using the ANN based brute-force Monte Carlo Simulations (MCS) method, with a significant computational savings of 95%. Moreover, the importance of taking into account the randomness in ply thickness for failure probability estimates is also highlighted quantitatively under the sensitivity studies section.

A finite element model for estimating time-dependent reliability of a corroded pipeline elbow

PurposeThe aim of this paper is to investigate the failure probability in an irregular area in pipeline (elbow) over its lifetime. The reliability analysis is performed by using of an enhanced first-order reliability method / second-order reliability method (FORM/SORM) and Monte Carlo simulation methods: a numerical model of a corroded pipeline elbow was developed by using finite element method; also, an empirical mechanical behavior model has been proposed. A numerical case with high, moderate and low corrosion rates was conducted to calculate the deferent reliability indexes. The found results can be used in an application case for managing an irregular area in pipeline lifetime. Hence, it is necessary to ensure a rigorous inspection for this part of a pipeline to avoid human and environmental disasters.Design/methodology/approachThe present paper deals a methodology for estimating time-dependent reliability of a corroded pipeline elbow. Firstly, a numerical model of corroded elbow is proposed by using the finite element method. A mechanical behavior under the corrosion defect in time is studied, and an empirical model was also developed.FindingsThe result of this paper can be summarized as: a mechanical characterization of the material was carried out experimentally. A numerical model of a corroded pipeline elbow was developed by using the finite element method. An empirical mechanical behavior model has been developed. The reliability of a corroding pipe elbow can be significantly affected by corrosion and residual stress. A proportional relationship has been found between probability of failure and corrosion rate. The yield stress and pressure service have an important sensitivity factor.Originality/valueAiming to help Algerian gas and oil companies' decision makers, the present paper illustrates a methodology for estimating time-dependent reliability of a corroded pipeline elbow over its lifetime using numerical models by applying the finite element method. Firstly, a numerical model of a corroded pipe elbow was developed and coupled with an empirical mechanical behavior model, which is also proposed. A probabilistic is then developed to provide realistic corrosion parameters and time modeling, leading to the real impact on the lifetime of an elbow zone in pipeline. The reliability indexes and probability of failure for various corrosion rates with and without issued residual stress are computed using Monte Carlo simulation and FORM.

Reliability Assessment of Steel Lattice Tower Subjected to Random Wind Load by the Stochastic Finite-Element Method

This work concerns the stochastic finite-element method (SFEM) reliability analysis of a skeletal, very slender steel telecommunication tower calibrated with a full-scale pushover experiment carried out for the same tower. The basic SFEM implementation is completed with common application of the generalized stochastic perturbation technique (SPT) and of the response function method with polynomial basis of the statistically optimized order. Verification of the proposed stochastic technique is carried out for the first four probabilistic moments by a comparison with traditional crude Monte Carlo simulation and, alternatively, with semianalytical methodology using the same polynomial structural responses as SPT and the well-known probabilistic integral definitions. Characteristic mean wind speed is introduced here as the input random Gaussian variable, while the computational model is subjected to the wind profile linearly dependent upon this mean speed consistent with applicable European design codes. The SFEM analysis is carried out here including large deformations of the tower elements and dynamic excitations in each node induced by the given trial wind spectrum. Finally, second-order reliability method (SORM) analysis is provided on the basis of the resulting extreme normal forces in the tower legs and their counterparts taken directly from the full-scale pushover experiments. The obtained results show that reliability indices for these towers according to the engineering codes have larger magnitudes than these corresponding to the experimental statistics and SFEM.

SORM DG - an efficient SORM based on differential geometry

Abstract The first order reliability method (FORM) efficiently performs first order structural reliability analysis, but with limited accuracy. On the other hand, the traditional second order reliability method (SORM) was established to improve the results of FORM, but with a supplementary computational process. Proposed herein, is a new SORM, based on differential geometry making SORM more efficient without hindering accuracy. It can be used to perform the second order structural reliability analysis in engineering. A case in the geotechnical engineering field obtained from literature was solved aiming to demonstrate the ability of the analytical procedure via differential geometry. The advantages of the newly proposed approach whereby the reliability method by differential geometry (SORM DG) over the traditional SORM are discussed. The results show that the SORM DG optimizes the outcomes from FORM and achieves the accuracy of the traditional SORM, more efficiently.

Reliability analysis of metal‐composite adhesive joints under debonding modes I, II, and I/II using the results of experimental and FEM analyses

AbstractIn the present study, the experimental and finite element (FE) analyses are first carried out to investigate the deboning behavior of metal‐composite adhesive joints under modes of I and mode II loading. To conduct an FE on the debonding propagation, cohesive zone method (CZM), as well as maximum nominal stress and energy criteria, is applied. In the reliability analysis, to achieve the probability of debonding growth (PODG), limit state functions are formulated based on the energy release rate. To that end, the first‐order reliability method (FORM), the second‐order reliability method (SORM), and the Monte Carlo simulation (MCS) are used to calculate the PODG. The effect of initial debonding length on the PODG in all mentioned modes is investigated. Results obtained from reliability analysis reveal that the random variables including the initial debonding length, width, and thickness are the most sensitive variables to ascertain the PODG.

Current models for strength and dynamic design of underwater structures

As one of the key met-ocean environmental loads affecting underwater structural design, current is closely related to the failure process of underwater structures such as moorings and risers. In marine structural analysis, a current model depends mainly on its speed profile corresponding to the design criterion. The conventional model of the current speed extreme (CSE) profile ignores the interlayer correlation and renders the current profile model conservative. This study is focused on the analysis of a current profile model aimed at the strength and dynamic design of underwater structures with special consideration of the interlayer correlation based on the long-term prototype measured data in the “Liuhua” oil field in the South China Sea. For the current model related to strength design, an empirical orthogonal function (EOF) decomposition is first utilized for dimensional reduction. Then, the multi-dimensional extreme-value problem is converted into an inverse reliability problem. The inverse second-order reliability method (ISORM) is applied to update the reliability index for an optimal solution. The global energy for different multi-year return periods is treated as objective functions to determine the extreme-value profiles corresponding to strength design of underwater structures. For the current model related to dynamic design, a statistical classification method based on the clustering large application algorithm (CLARA) is performed to obtain a multi-profile current model by using a large long-term prototype monitoring current dataset. An average silhouette width is found and used to determine the final classification number. Finally, a multi-profile current model is established for the underwater structural dynamic design. The presented current models can be directly applied as the current loads for underwater structural design in the South China Sea. Moreover, besides the possibility to extend the proposed method to other ocean areas, it can also be used for developing models of wind, acoustic, and other field variables.

Probabilistic framework for the assessment of the flexural design of concrete sleepers

An extensive study of the flexural performance of monoblock prestressed concrete sleepers in a light rail system was conducted as part of a research program funded by the Federal Transit Administration. Five consecutive sleepers deployed on the track were instrumented with strain gauges at their critical design cross-sections (center and rail seats) to obtain relevant flexural information during an uninterrupted period of 14 months. Results were compared with the projected design capacities obtained from the application of current design standards, resulting in glaring differences. The current design methodologies were deemed insufficient for the development of optimal design solutions for light rail applications. Furthermore, structural reliability analysis is employed to study the flexural capacity of the sleeper design. A capacity model based on the material and geometric properties of the sleeper design was developed. The demand model was derived from the field flexural data of over 27,000 train passes, fitting this information to predefined probability distributions. Four limit-state functions were defined to represent the typical flexural failure modes. The probability of failure was calculated using first-order reliability method, second-order reliability method, and Monte Carlo simulation. Ultimately, the analysis yielded consistent results for the three methods, showing largely low probability of failure at both design cross-sections under the studied demand level. In conclusion, the sleeper's capacity was higher than the existing field demands, indicating an overly conservative design approach.

Reliability Analysis of Reinforced Concrete Frame by Finite Element Method with Implicit Limit State Functions

Since the prediction of the seismic response of structures is highly uncertain, the need for the probabilistic approach is clear, especially for the estimation of critical seismic response parameters. Considering the uncertainties present in the material and geometric form of reinforced concrete (RC) structures, reliability analyses using the Finite Element Method (FEM) were performed in the context of Performance-Based Earthquake Engineering (PBEE). This study presented and compared the possibilities of nonlinear modelling of the reinforced concrete (RC) planar frame and its reliability analysis using different numerical methods, Mean-Value First-Order Second-Moment (MVFOSM), First-Order Reliability Method (FORM), Second-Order Reliability Method (SORM) and Monte Carlo simulation (MCS). The calibrated numerical models used were based on the previous experimental test of a planar RC frame subjected to cyclic horizontal load. Numerical models were upgraded by random variable (RV) parameters for reliability analysis purposes and, using implicit limit state function (LSF), pushover analyses were performed by controlling the horizontal inter-storey drift ratio (IDR). Reliability results were found to be sensitive to the reliability analysis method. The results of reliability analysis reveal that, in a nonlinear region, after exceeding the yield strength of the longitudinal reinforcement, the cross-sectional geometry parameters were of greater importance compared to the parameters of the material characteristics. The results also show that epistemic (knowledge-based) uncertainties significantly affected dispersion and on the median estimate parameter response. The MCS sampling method is recommended, but the First-Order Reliability Method (FORM) applied on a response model can be used with good accuracy. Reliability analysis using the FEM proved to be suitable for the direct implementation of geometric and material nonlinearities to cover epistemic (knowledge-based) uncertainties.

A multi-fidelity boundary element method for structural reliability analysis with higher-order sensitivities

A novel multi-fidelity modelling methodology for structural reliability analysis using the Boundary Element Method (BEM) with an Implicit Differentiation Method (IDM) is presented. Reliability analyses are conducted with methods such as Monte Carlo Simulation (MCS) and the First-Order Reliability Method (FORM). The higher-order sensitivities of the elastostatic Boundary Element Method equations with respect to changes in several geometric variables have been derived for the first time for use with the IDM for the purpose of conducting reliability analyses with the Second-Order Reliability Method (SORM), a more accurate alternative to FORM for problems with non-linear limit state functions. Multi-fidelity formulations involving the IDM have also been derived for the first time, making use of the metamodelling technique Kriging. The use of multi-fidelity modelling enables the creation of a model that has similar accuracy to a high-fidelity model, but with a computational cost similar to that of a low-fidelity model. By combining the accuracy of the IDM with the efficiency of multi-fidelity modelling the proposed methodology has the capability to be very effective when used for structural reliability analysis. The IDM is validated through a numerical example for which the analytical solution is known. A further two numerical examples featuring an I-beam section and a triangular support bracket with a large number of variables are also investigated. Results show that the employed multi-fidelity models were up to 6000 times faster in terms of CPU-time than the high-fidelity model, while also providing probabilities of failure that were up to 2225 times more accurate than the low-fidelity model. Overall, it has been shown that the use of the proposed IDM/multi-fidelity modelling methodology significantly improved the efficiency and accuracy of the above reliability analysis techniques when applied to complex problems involving a large number of random variables under high levels of uncertainty.