First-order Reliability Method Research Articles

An improved approximate integral method for nonlinear reliability analysis

In order to evaluate the failure probability corresponding to the Limit State Function (LSF) in structural reliability, the First Order Reliability Method (FORM) linearizes the LSF and directly calculates the failure probability based on the Most Probable Point (MPP). But this method is unable to effectively handle nonlinear problems. The Second Order Reliability Method (SORM), on the other hand, utilizes the Hessian matrix to obtain curvature information at the MPP and computes the failure probability with Breitung's second-order approximation formula. However, SORM is unable to maintain satisfactory computational accuracy in highly nonlinear problems due to its lack of detailed analysis on the shape of the limit state surface. To address this issue, this paper proposes an Improved Approximation Integration Method (IAIM). The initial approximation of the limit state surface is based on the intersection region between the hypersphere and the n-dimensional paraboloid. Subsequently, a statistical analysis of the deviation between the aforementioned region and its projected area is conducted to construct a corresponding fitting function, which is then utilized for dimensionality reduction in the multidimensional integration of the approximation area. Finally, several examples are presented to demonstrate the accuracy and feasibility of the proposed IAIM.

Multi‐objective reliability‐based robust design for a rock tunnel support system using Pareto optimality

AbstractIn the context of rock material and modeling uncertainties, the optimization of rock tunnel support systems is often conducted by selecting the most cost‐effective solution among several feasible options that typically rely on the engineer's experience, potentially leading to overlooking the most optimal design. To improve such a limitation, this paper presents a multi‐objective reliability‐based robust design, considering the cost, safety, and design robustness systematically while maintaining the computational efficiency. In this framework, the uncertainty‐based reliability constrains is performed using the first‐order reliability method (FORM) and an improved Hasofer–Lind–Rackwits–Fiessler recursive algorithm (iHLRF‐x). The design robustness, in terms of sensitivity index (SI), is evaluated using the normalized gradient of the system response to the noise factors, which can be efficiently obtained from the output of FORM analysis. Then, the Pareto front revealing the tradeoff between multiple objectives can be directly generated using the proposed optimization framework. To illustrate the effectiveness of this procedure, a set of the optimal design combinations of the shotcrete thickness and installation position for the exampled rock tunnel are obtained, and new perspectives pertaining the success of the reliability‐based robust designs are provided.

Response Extremes of Floating Offshore Wind Turbine Based on Inverse Reliability and Environmental Contour Method

Floating structures are subject to complex marine conditions. To ensure their safety, reliability analysis needs to be conducted during the design phase. However, because of the complexity of traditional full long-term analysis, the environmental contour method (ECM) based on the inverse reliability method, which can combine accuracy and efficiency, is extensively used. Due to the unique environment in the South China Sea, the probabilistic characteristics of three-dimensional (3D) environmental parameters of wind, wave and current are investigated. The ECs of the target sea are established via the ECM based on both the inverse first-order reliability method (IFORM) and inverse second-order reliability method (ISORM). It is found that the sea state forecasted by ISORM is more extreme and may lead to a more conservative design than IFORM. Furthermore, the wind–wave–current combination coefficient matrixes developed using the 3D ECs are proposed for the design of FOWTs in the South China Sea. The validity and practicality of the contours and matrixes are tested by using a floating offshore wind turbine (FOWT) as a numerical example. Then, the short-term response of the structure under the combined wind, wave and current conditions is calculated, providing a theoretical reference for the design of FOWTs.

Reliability analysis of shallow foundation using the important-region-based method

ABSTRACT To improve the computational efficiency and accuracy is a constant pursuit for evaluating the stability of geotechnical structures built in spatially variable soils. This study proposes an important-region-based Karhunen – Loève expansion and first-order reliability method (KL-FORM-IR method) to perform reliability analysis for shallow foundation resting on spatially variable soil. The KL-FORM-IR method further improves the computational efficiency by ignoring the variability of the soil parameters outside the important region. An identification method for locating the important region of shallow foundation is first proposed, and the effect of the scale factor (K f), autocorrelation distance of soil properties, and foundation width (B) on the identification of important regions are investigated. For the purpose of validation, the proposed KL-FORM-IR method compared with the KL-FORM and KL-MCSM by performing reliability analyses based on the same numerical model. Conclusively, the KL-FORM-IR method can improve the computational efficiency and it contributes a new approach to the reliability analysis of shallow foundation resting on a spatially variable soil.

Efficient slope reliability and sensitivity analysis using quantile-based first-order second-moment method

This paper introduces a novel approach for parameter sensitivity evaluation and efficient slope reliability analysis based on quantile-based first-order second-moment method (QFOSM). The core principles of the QFOSM are elucidated geometrically from the perspective of expanding ellipsoids. Based on this geometric interpretation, the QFOSM is further extended to estimate sensitivity indices and assess the significance of various uncertain parameters involved in the slope system. The proposed method has the advantage of computational simplicity, akin to the conventional first-order second-moment method (FOSM), while providing estimation accuracy close to that of the first-order reliability method (FORM). Its performance is demonstrated with a numerical example and three slope examples. The results show that the proposed method can efficiently estimate the slope reliability and simultaneously evaluate the sensitivity of the uncertain parameters. The proposed method does not involve complex optimization or iteration required by the FORM. It can provide a valuable complement to the existing approximate reliability analysis methods, offering rapid sensitivity evaluation and slope reliability analysis.

The first-order time-variant reliability expansion method

Time-variant reliability problems are frequently encountered in engineering due to factors like material degradation or random loading modeled as random processes. The PHI2 method, which employs the First Order Reliability Method (FORM), is commonly used to solve such problems. However, it requires repeated searches for Most Probable Points (MPPs), making it computationally expensive. To improve efficiency with little sacrifice of accuracy, this study proposes a First Order Time-variant Reliability Expansion (FOTRE) method, which provides an efficient explicit formulation for MPP regarding time, in contrast to the expensive optimization approach of the PHI2 method. It requires only a single accurate search for the so-called “worst MPP” over the whole lifespan and offers the “adaptive accuracy of outcrossing rate”, which avoids the repeated search for MPPs ensuring computational accuracy. The inspiration behind the FOTRE method stems from the observation that the outcrossing rate tends to be small at time points with relatively large reliability indexes compared to the minimum reliability index βmin, which has a negligible impact on the subsequent structural failure probability over the entire lifespan. This innovative approach significantly improves the efficiency of solving time-variant reliability problems without compromising much of the numerical accuracy. The effectiveness and accuracy of the FOTRE method are demonstrated through several numerical examples.

Fast physically-based probabilistic modelling of rainfall-induced shallow landslide susceptibility at the regional scale considering geotechnical uncertainties and different hydrological conditions

The inherent uncertainty in hydro-geotechnical parameters presents a significant challenge for accurately predicting rainfall-triggered shallow landslides in mountainous regions. In this study, a novel probabilistic framework was developed and implemented in the “Py.GIS-FSLAM-FORM” software, designed to address the complexities associated with parameter uncertainty, correlation, and distribution. By combining the Fast Shallow Landslide Assessment Model (FSLAM) with the First-Order Reliability Method (FORM), we have enhanced the traditional probabilistic approach to create more accurate landslide susceptibility maps. This study emphasizes the uncertainly of geotechnical parameters and the critical influence of hydrological conditions on landslide susceptibility, especially focusing on the interaction between antecedent recharge (qa) and event rainfall (Pe). In our study area (Val d’Aran, Spain), the probabilistically based results revealed that areas of very high susceptibility are significantly affected by event rainfall, particularly on slopes of 30–40 degrees and aspects between 100 and 250 degrees. The variability in geotechnical parameters, especially the coefficient of variation (COV) in cohesion and friction angle, plays a crucial role in landslide susceptibility assessment, with increased COVs leading to greater landslide uncertainty. Additionally, cross-negative correlations and non-normal distributions of geotechnical parameters substantially influence the spatial distribution of landslides, notably when combining antecedent recharge with event rainfall. These results highlight the importance of incorporating parameter variability and hydrological conditions in susceptibility models to improve the precision of regional landslide forecasts. While the study was performed in Val d'Aran, its methodologies and conclusions are relevant to mountainous areas worldwide, offering insights for refining landslide prediction models and susceptibility assessments, contributing to global efforts in landslide disaster prevention.

Monte Carlo simulation study on the reliability of concrete-filled HSS beam-column design provisions

Revisions were recently proposed to the way in which concrete-filled hollow structural section (HSS) members are handled in CSA S16. These revisions were based on previous research, comparisons to experiments, and a first-order reliability method analysis of existing provisions for compression and flexural members. In this paper, this topic is further expanded by using Monte Carlo Simulations (MCS) to determine the inherent reliability of the previous and new design rules for concrete-filled rectangular hollow section (RHS) and circular hollow section (CHS) beam-columns. A representative set of concrete-filled RHS and CHS members with variations in concrete strength, wall slenderness, effective length, and loading eccentricity are analyzed. Using MCS, the reliability index (β+) of each is determined over a range of live-to-dead load (L/D) ratios. Inherent β+ values are compared to the code-specified target (i.e., β+ = 3.0 per Annex B of CSA S16) and the CSA S16:19 and AISC 360-16 provisions.

Reliability and sensitivity analysis of delamination growth of composite laminate structures using two efficient sampling methods

In recent years, composite structures have been used in a large number of applications in aerospace, machinery, marine, and civil engineering. However, there are inevitably many uncertainties in the whole life cycle of composite structures, which can easily lead to structural damage and failure. Therefore, it is important to analyze the reliability and sensitivity of composite structures. At present, most of the contributions use the first-order reliability method (FORM) and the second-order reliability method (SORM) to study the reliability of composite structures and compare them with the results of the Monte Carlo simulation. However, both methods have their limitations. FORM cannot guarantee the calculation accuracy for the highly nonlinear limit state equation, and the calculation efficiency of SORM is too low. Therefore, this paper proposes to use importance sampling (IS) and backpropagation neural network-based Monte Carlo (MC-BPNN) to study the reliability, sensitivity, and dispersion of delamination growth of composite laminates. The results show that compared with FORM and SORM, IS and MC-BPNN have higher calculation accuracy and efficiency and can more accurately evaluate the failure degree of composite structures and ensure their safe operation in the field of aerospace equipment. The universality of this method can make it being widely used in the reliability and sensitivity analysis of different composite materials as well as dispersion analysis.

Slope Stability and Reliability Analysis of Earth Embankment Constructed for the Doubling of Railway Track

Embankments of greater height pose threat to its stability and require geotechnical investigation and intervention. Conventionally, limit equilibrium approach is used to estimate the factor of safety of the slope to assess its stability. Over a period of time, with experience and engineering judgments, a factor of safety of 1.4 is deemed sufficient for Indian practice. In the approach, it is inherently assumed that different sources of uncertainty due to testing errors, model transformation, and inherent variability of the geo-material are taken care of with the use of single value of factor of safety. Still, a question remains, whether the suggested factor of safety is acceptable in a given environment of uncertainty or there is a need to further utilize the probabilistic approach. The present study demonstrates how combination of First Order Reliability Method (FORM), Finite Element Method (FEM) and Response Surface Method (RSM) can be useful in the probabilistic assessment of proposed construction of new earth embankment for the doubling of a railway track. It is further highlighted that conventional factor of safety approach when used in conjunction with probabilistic analysis brings rationality in decision making. Also, based on results of slope stability and reliability analysis, it is proposed to incorporate soil nailing to further improve the stability of the earth embankment.

Strength reduction factor based on probabilistic analysis for hybrid reinforced concrete beams

The American Concrete Institute (ACI) developed ACI-318 and ACI-440 codes, providing design guidelines for concrete beams subject to flexure and reinforced with steel or Fiber-Reinforced Polymer (FRP) bars, respectively. Although using a hybrid reinforcing system that combines steel bars and FRP bars has gained significant attention, ACI codes have no design guidelines for this concept. The main objective of this study is to determine the appropriate strength reduction factor for hybrid steel/FRP RC beams (i.e., concrete beams reinforced with a hybrid system of FRP and steel bars) by conducting a comprehensive probabilistic analysis over a wide range of hybrid steel/FRP RC beams dataset. This factor is a crucial component of the design standards outlined by the American Concrete Institute (ACI). The reliability index of the beams dataset under consideration is determined within the framework of reliability analysis using the First Order Reliability Method (FORM). Furthermore, a sectional analysis is employed as a mechanical model initially verified using an experimental dataset of 95 specimens. This model is utilized to forecast the flexural capacity of beams.The study's findings indicate a substantial correlation between the strength reduction factor and the reinforcement and live-to-dead load ratios. For the objective of conservative design, a strength reduction factor of 0.65 is suggested.

Sensitivity of fatigue reliability in wind turbines: effects of design turbulence and the Wöhler exponent

Abstract. Fatigue assessment of wind turbines involves three main sources of uncertainty: material resistance, load, and the damage accumulation model. Many studies focus on increasing the accuracy of fatigue load assessment to improve the fatigue reliability. Probabilistic modeling of the wind's turbulence standard deviation is an example of an approach used for this purpose. Editions 3 and 4 of the IEC standard for the design of wind energy generation systems (IEC 61400-1) suggest different probability distributions as alternatives for the representative turbulence in the normal turbulence model (NTM) of edition 1. There are debates on whether the suggested distributions provide conservative reliability levels, as the established design safety factors are calibrated based on the representative turbulence approach. The current study addresses the debate by comparing annual reliability based on different scenarios of NTM using a probabilistic approach. More importantly, it elaborates on the relative importance of load assessment accuracy in defining the fatigue reliability. Using the DTU 10 MW reference wind turbine and the first-order reliability method (FORM), we study the changes in the annual reliability level and its sensitivity to the three main random inputs. We perform the study considering the blade root flapwise and the tower base fore–aft moments, assuming different fatigue exponents in each load channel. The results show that integration over distributions of turbulence in each mean wind speed results in less conservative annual reliability levels than representative turbulence. The difference in the reliability levels varies according to turbulence distribution and the fatigue exponent. In the case of the tower base, the difference in the annual reliability index after 20 years can be up to 50 %. However, the model and material uncertainty have much higher effects on the reliability levels compared to load uncertainty. Knowledge about such differences in the reliability levels due to the choice of turbulence distribution is especially important, as it impacts the extent of lifetime extension through reliability reassessments.

Reliability analysis of fatigue crack growth in shallow shell structures using the Dual Boundary Element Method

This work presents a novel methodology for fatigue life reliability analysis using a newly developed Dual Boundary Element Method-based Implicit Differentiation Method (DBEM-IDM) in shallow shell structures. The proposed DBEM formulations evaluate stress intensity factors and fatigue life sensitivities in shallow shells considering geometrical variables and curvature. The IDM formulations, obtained through direct differentiation of the shallow shell DBEM formulations, uses the First-Order Reliability Method (FORM) to assess the reliability index and probability of failure of shallow shell structures with multiple cracks. Two numerical examples were simulated to illustrate the effectiveness of the proposed methodology. The first example examines a shallow shell with a centre hole and cracks subjected to various loads. The critical crack length required to ensure structural reliability and inspectability is determined. Fatigue reliability assessment is performed for the same example, employing limit state functions expressed in terms of fatigue life. FORM results are compared with the results of Monte Carlo Simulations (MCS), and provided a maximum difference of 3.67%. Parametric analyses are conducted to identify significant variables that affect reliability, and it was found that precise determination of design parameters is necessary to reduce the probability of failure, particularly for the Paris law constants. The second example, featuring more complex geometry, involved assessing fatigue reliability in a cylindrical shallow shell structure with multiple cracks. The results of this example indicate a 3.49% maximum difference between FORM and MCS. The results exhibit a low error compared to the MCS, and the inherent efficiency of the IDM-based FORM underscores its suitability for addressing uncertainty in reliability analysis.

Probabilistically modelled geometrical imperfections for reliability analysis of vertically loaded steel frames

The article explores the stochastic modelling of 3D steel frames under static vertical load, with a main focus on random initial geometrical imperfections. The random input parameters for initial geometrical imperfections are derived from the tolerance criteria in accordance with European standard. In order to derive statistical parameters from the corresponding standard tolerances, two methods noted as #RSS (random storey sway) and #RSP (random storey position) have been utilized and compared. Both of these methods are used along with the first-order reliability method (FORM) and the advanced finite element method (FEM) – geometrically and materially nonlinear imperfect analysis (GMNIA) to numerically analyse several geometries of steel frame structures. Ultimate resistances of these structures are monitored and basic sensitivity studies are conducted. The results are also compared with the deterministic approach of the European standard. The stochastic approach of the FORM to estimate the ultimate resistance of steel frame structures serves as a verification tool of the classic, in engineering praxis widely used, deterministic approach. This study provides useful provisions for the advanced numerical analyses of steel frames of various geometries. Additionally, the estimations of ultimate resistance by the deterministic European standard approach is verified for selected frame geometries.

Approximate Integral Method for Nonlinear Reliability Analysis

In the realm of reliability analysis methods, the First-Order Reliability Method (FORM) exhibits excellent computational accuracy and efficiency in linear problems. However, it fails to deliver satisfactory performance in nonlinear ones. Therefore, this paper proposes an Approximate Integral Method (AIM) to calculate the failure probability of nonlinear problems. Firstly, based on the Most Probable Point (MPP) of failure and the reliability index β obtained from the FORM, the Limit State Function (LSF) can be equivalent to an Approximate Parabola (AP) which divides the hypersphere space into feasible and failure domains. Secondly, through the ratio of the approximate region occupied by a parabolic curve to the entire hypersphere region, the failure probability can be calculated by integration. To avoid the computational complexity in the parabolic approximate area due to high dimensionality, this paper employs a hyper-rectangle, constructed from chord lengths corresponding to different curvatures, as a substitute for the parabolic approximate area. Additionally, a function is utilized to adjust this substitution, ensuring accuracy in the calculation. Finally, compared with the calculated result of the Monte Carlo simulation (MCS) and the FORM, the feasibility of this method can be demonstrated through five numerical examples.

Probabilistic Design Methods for Gust-Based Loads on Wind Turbines

The IEC 61400-1 standard specifies design load cases (DLCs) to be considered in the design of wind turbine structures. Specifically, DLC 2.3 considers the occurrence of a gust while the turbine shuts down due to an electrical fault. Originally, this load case used a deterministic wind event called the extreme operating gust (EOG), but the standard now also includes an approach for calculating the extreme response based on stochastic simulations with turbulent wind. This study presents and compares existing approaches with novel probabilistic design approaches for DLC 2.3 based on simulations with turbulent wind. First, a semiprobabilistic approach is proposed, where the inverse first-order reliability method (iFORM) is used for the extrapolation of the response for electrical faults occurring at a given rate. Next, three probabilistic approaches are formulated for the calculation of the reliability index, which differs in how the aggregation is performed over wind conditions and whether faults are modeled using a Poisson distribution or just by the rate. An example illustrates the methods considering the tower fore-aft bending moment at the tower base and shows that the approach based on iFORM can lead to reductions in material usage compared to the existing methods. For reliability assessment, the probabilistic approach using the Poisson process is needed for high failure rates, and the reliabilities obtained for designs using all semiprobabilistic methods are above the target level, indicating that further reductions may be obtained via the use of probabilistic design methods.

Improved FORM and SORM Based on Improved Modified Symmetric Rank 1 Algorithm and Adaptive Kriging Model

Abstract The first-order reliability method (FORM) is simple and efficient for solving structural reliability problems but may have large errors and converge slowly or even result in divergence when dealing with strongly nonlinear performance functions. For this case, the existing second-order reliability method (SORM) achieves higher computational accuracy but with a consequent decrease in efficiency. To achieve a better balance between accuracy and efficiency, this paper proposes an improved FORM and an improved SORM. First, an improved modified symmetric rank 1 (IMSR1) algorithm, in which the line search strategy for step length is unnecessary, is proposed for iterations of the FORM, and an adaptive Kriging model with a rational update criterion is presented to improve the efficiency of the FORM. Then, an improved FORM with high efficiency and good convergence is proposed. Second, due to the good precision of the adaptive Kriging model at the final design point, the Hessian matrix is available easily without additional computational effort, and an improved SORM with the same efficiency as the improved FORM is presented naturally. Finally, the accuracy, efficiency, and convergence of the proposed methods are verified by numerical and engineering examples.

Resistance factors for reliability based-design of geogrid reinforced soil foundations against ultimate and serviceability limit state

In the recent decades, the use of geosynthetics to improve the overall soil’s bearing capacity and settlement behavior under shallow foundations has become a major topic. Despite several studies that have been carried out on Reinforced Soil Foundations (RSF), no design specifications have been set to design RSF against the Ultimate and Serviceability Limit State (ULS and SLS) in any of the established codes. This paper presents the development of resistance factors based on the Load and Resistance Factor design (LRFD) approach against the ULS and SLS for strip footing resting on C−ϕ soils. The resistance factors are developed using a spreadsheet-based reliability model based on the First Order Reliability Method (FORM) to be consistent with the ASCE-7 load factors. The resistance factors are calibrated at a targeted probability failure (Pf) of 0.13499% and 0.00317% for ULS and 2.27%, for SLS. A Finite Element Model (FEM) is developed to validate three analytical solutions to establish the reliability model. The FEM results showed an agreement with the analytical solutions. For ULS the developed resistance factors at Pf= 0.13499% ranged from 0.33 to 0.9 assuming lognormally distributed variables and 0.25 to 0.89 assuming normally distributed variables. The resistance factors against SLS at Pf= 2.27%. ranged from 0.39 to 0.85 considering lognormally distributed variables and 0.35 to 0.83 considering normally distributed variables.

Reliability based assessment of reinforced concrete columns under eccenric loads using refined first-order reliability method

Reliability based assessment of reinforced concrete columns under eccenric loads using refined first-order reliability method

Active Kriging-based conjugate first-order reliability method for highly efficient structural reliability analysis using resample strategy

Efficient structural reliability analysis method is crucial to solving reliability analysis of complex structural problems. High-computational cost and low-failure probability problems greatly limit the efficiency in structural reliability analysis problems, causing the safety and reliability of the structure to be questioned. In this work, a highly efficient structural reliability analysis method coupling active Kriging algorithm with conjugate first order reliability method (AK-CFORM) is proposed. Specifically, the resample strategy is considered to reduce the number of samples evaluated in each active learning process; the uniform sampling is used to better balance global and local optimal problems; the conjugate map is used to improve the robustness of analytical first order reliability method; and the approximate numerical differential formula is proposed to solve the problems of non-convergence when solving the gradient of the Kriging surrogate model. Finally, three numerical cases and four engineering cases are used to illustrate the effectiveness and robustness of the proposed method. The results show that the proposed AK-CFORM has greater advantages in the number of calling system response and surrogate model with robust and accurate performance.