First-order Reliability Method Research Articles

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.

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.

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.

Generic fully coupled framework for reliability assessment of offshore wind turbines under typical limit states

Offshore wind turbines (OWTs) generally experience harsh marine environments with uncertainties in marine loads, soil properties, and material properties; therefore, the structural integrity assessment of OWTs is a challenging task. Most of the previously proposed methods for reliability assessment are time-consuming and neglect the coupling effects of aerodynamics, hydrodynamics, and OWT structural dynamics under complex environmental loads. In this study, a fully coupled framework for the reliability assessment of OWTs is proposed, in which a dynamic simulation tool in the time domain is combined with surrogate models. The established Kriging, multivariate regression, and support vector regression surrogate models are compared to determine the most feasible model. Furthermore, the influence of different solving methods, such as first-order reliability method (FORM), importance sampling (IS), and subset simulation (SS), on the estimation of structural reliability indexes are discussed. The superiority of the estimated reliability indexes in the safety assessment of OWTs is proved using the SS method based on the Kriging surrogate model. Subsequently, the potential failure modes of an OWT under different operational states are revealed, indicating that excessive tower vibrations must be eliminated for stable power production from OWTs. Furthermore, the SS method is found to be feasible as it ensures the estimation accuracy of the reliability indexes of the parked OWT based on Kriging and the other surrogate models. Finally, the significance of soil–structure interactions in the reliability index estimation is investigated, and the equivalent coupled spring boundary is proposed from the perspectives of structural safety and simulation accuracy.

Eurocode-compliant system-level reliability analyses of trussed portal frames under climatic loads

Eurocode 3 provides Geometrically and Materially Nonlinear Analysis with Imperfections-method (GMNIA) in which the entire structure can be designed in system-level. In GMNIA, reliability of structural system is verified by using a system safety factor which is obtained in Eurocode 3 based on numerical or experimental capacity test results. Unfortunately, such test results are scarcely published in the literature thus complicating the determination of the factor for a design engineer. In the so-called Direct Design Method, however, system safety factors are provided in advance for the design engineer by system-level reliability studies. This study determines the Eurocode-compliant GMNIA system safety factor for Warren truss portal frames by using the approach of the Direct Design Method. Advanced numerical models are utilized to perform Monte Carlo-simulations for entire structural systems. These simulations provide statistical distributions of system resistances which are employed in the First-Order Reliability Method to derive the system safety factors. Studied structures are made of S700 cold-formed hollow sections. Various system geometries, system configurations and load combinations consisting of snow and wind loads are investigated and a suitable system safety factor is proposed for the ultimate limit state design of Warren truss portal frames. The proposed system safety factor is applied in a practical comparison in which Warren system is designed both by the conventional Eurocode 3 method and GMNIA. This comparison reveals that GMNIA has a remarkable potential to offer reduced material consumption in design compared to the conventional method.

Relative Entropy Application to Study the Elastoplastic Behavior of S235JR Structural Steel.

The main issue in this work is to study the limit functions necessary for the reliability assessment of structural steel with the use of the relative entropy apparatus. This will be done using a few different mathematical theories relevant to this relative entropy, namely those proposed by Bhattacharyya, Kullback-Leibler, Jeffreys, and Hellinger. Probabilistic analysis in the presence of uncertainty in material characteristics will be delivered using three different numerical strategies-Monte Carlo simulation, the stochastic perturbation method, as well as the semi-analytical approach. All of these methods are based on the weighted least squares method approximations of the structural response functions versus the given uncertainty source, and they allow efficient determination of the first two probabilistic moments of the structural responses including stresses, displacements, and strains. The entire computational implementation will be delivered using the finite element method system ABAQUS and computer algebra program MAPLE, where relative entropies, as well as polynomial response functions, will be determined. This study demonstrates that the relative entropies may be efficiently used in reliability assessment close to the widely engaged first-order reliability method (FORM). The relative entropy concept enables us to study the probabilistic distance of any two distributions, so that structural resistance and extreme effort in elastoplastic behavior need not be restricted to Gaussian distributions.

A hybrid reliability assessment method for slab track with changing structural parameters

Changes in the dynamic properties of high-speed rail (HSR) slab track structures can have a great impact on train safety and ride quality. However, the dynamic response of wheel–rail systems, which is related to operational safety, has rarely been considered in existing service reliability assessments of track structures. To solve this problem, a hybrid method was developed, with the serviceability limit state (SLS) first defined with respect to the derailment coefficient (DC) and wheel unloading rate (WUR). In calculating the reliability index, the response surface method and the first-order reliability method were used to solve the implicit expression of wheel–rail force in the SLS equation. To reduce the computation cost, a surrogate model expressing the non-linear mapping of the wheel–rail interaction is proposed. The computation efficiency and accuracy of the hybrid method were compared with Monte Carlo simulation and a back-propagation neural network (BPNN). The computation time of the hybrid method was only 1/8.4 of the BPNN method, while the accuracy of the reliability index was 98% for the DC and 97% for the WUR. The hybrid method was used to assess the reliability of a typical slab track structure under changing stiffness and damping coefficients of the fasteners, cement asphalt mortar and foundation. The stiffness and damping of the fasteners had the most impact on wheel–rail dynamics and track reliability. This research provides new insights into reliability assessments for HSR slab track with respect to train operational safety.

Model for Wastage Allowance and Strength Properties of Pipe Piles Exposed to Marine Corrosion

The study’s objective is to analyze the mechanical properties of steel pipe piles as a part of a trestle bridge subjected to five years of natural marine corrosion degradation. Sixteen tensile specimens are extracted from the steel pipe piles in the splash, tidal, and immersion zones. The experimental tensile test results are used to establish regression equations defining the elastic modulus, yield strength, strain hardening index, and strength coefficient for the true stress–strain curves of the three regions. A non-linear time-dependent mathematical model is exploited to identify the corrosion degradation, using the data from one single corrosion degradation measurement campaign. The analysis indicates that the splash zone is experiencing the most severe corrosion degradation, and there are progressive losses in the mechanical properties of each zone as the corrosion degradation progresses. The established relationships of the mechanical properties, as a function of the ratio of corroded plate thickness to the as-built one, can be used as a fast-engineering approach to identify the mechanical properties of severely corroded piles. The corrosion degradation allowance is also defined using the first-order reliability method, accounting for existing uncertainties covered by the partial safety factors. By examining the impact of marine corrosion on the mechanical properties of marine structures and developing predictive models to assess the corrosion’s effect on material strength and corrosion allowance, the study aims to improve offshore structures’ safety, design, and maintenance.

Probabilidade de falha de pilares de concreto armado em situação de incêndio

Abstract The reliability of reinforced concrete (RC) columns in a fire situation is investigated in this paper by combining the General, Equilibrium, and Isotherm at 500 ° C Methods. The well-known Monte Carlo (MC) and the First Order Reliability Method (FORM) are used in a parametric study for the column’s safety. The columns, in a biaxial bending-compression situations, are analyzed by an algorithm that takes into consideration physical and geometric nonlinearities. The dead and live load actions, steel and concrete material strengths, section geometry, model error, and temperature are among the random variables taken into consideration. The findings indicate that increasing the cover can dramatically reduce the probability of failure.

Dimension reduction for constructing high-dimensional response distributions by accounting for unimportant and important variables

Probability distributions of responses have been widely used in structural analysis and design because of their complete statistical information. In practice, the dimensionality of input variables could easily reach hundreds or thousands, making it computationally expensive to obtain accurate distributions. In this paper, a generalized most probable point (MPP) method is developed to effectively build the response distributions of high-dimensional problems. First, a global index based on one-iteration MPPs is presented for dimension reduction, which is to divide the input variables into important and unimportant variables. After fixing the unimportant variables at their one-iteration MPP components, the MPP components of the important variables are obtained by performing the inverse first-order reliability method (FORM) in the reduced space. Predictive models of the all MPP components are then established to quickly estimate the MPPs of other cumulative distribution function (CDF) values. To accurately calculate CDF points of limit state functions with different shapes, a comprehensive uncertainty analysis method that accommodates the contributions of the important and unimportant variables is proposed by multiple combinations of FORM, second-order reliability method, and second-order saddlepoint approximation. Finally, the response distributions are generated based on Gaussian mixture distribution and all CDF points. The effectiveness of the proposed method is verified by a mathematical example and two engineering cases.

Reliability Analysis of Bearing Capacity of Soil of Combined Pile Raft Foundation (CPRF) of Historic Dharahara

When the raft foundation alone does not fulfil the design requirement, it may be possible to enhance the performance of the raft by economical addition of piles to transfer the heavy load to soft soil with a rather low total or differential settlement called a combined pile raft foundation (CPRF). The proposed twenty-one storey Dharahara tower at Sundhara, Kathmandu is being constructed with CPRF.This paper focuses on the reliability index, the probability of failure and reliability of Dharahara. Statistical analysis is carried out to determine the mean and variance of log-normally distributed geotechnical uncertainty parameters cohesion, unit weight, friction angle and the normally distributed loading on the foundation. First, the limit equilibrium performance function equation of bearing capacity is prepared using a First Order Reliability Method (FORM) and then solving the value of reliability index by coding on C Programming Language. Obtained value is validated with Euro Code-Basis of Structural Design. The results show that the reliability of the designed foundation system of Dharahara is 0.99999872% and the probability of failure is 0.00000128%. Sensitivity analysis is performed by varying uncertainty parameters and the result shows that friction angle is more sensitive than the other three parameters.

Probabilistic analysis of seasonal influence on the prediction of pile bearing capacity by CPT: A case study

In this study, we used a probabilistic approach to evaluate the probability of failure of small diameter bored piles in unsaturated sandy soils. The prediction of the bearing capacity was based on the Cone Penetration Test (CPT). We assessed the seasonal influence by analyzing CPT data from dry and wet seasons. The LCPC and Aoki-Velloso semiempirical methods were deterministically applied, and the Aoki-Velloso method was examined using reliability theory. The first-order reliability method (FORM) was used to evaluate the probability of failure and calculate the reliability index based on random variables obtained from CPT data considering the effect of season when predicting bearing capacity. A total of 8.0 m long bored piles with diameters of 0.25, 0.30, and 0.35 m were used for both deterministic and probabilistic analyses. We also evaluated the influence of load ratio variation on the reliability index and determined the pile length required to reach a target reliability index value. Reliability analyses considering CPT campaigns conducted in different seasons showed the effect of season on the prediction of bearing capacity. For example, an 8.0 m long pile with a 0.30 m diameter and 100 kN vertical load had a probability of failure of 38.7 % in the wet season and 0.01 % in the dry season. The reliability analyses provided insights into the influence of seasonal variability on in situ tests for pile bearing capacity, affecting the design of foundation elements.