First-order Reliability Method Research Articles

An Efficient Reliability Analysis Method Based on the Improved Radial Basis Function Neural Network

Abstract This paper develops an efficient reliability analysis method based on the improved radial basis function neural network (RBFNN) to increase the accuracy and efficiency of structural reliability analysis. To solve the problems of low computational accuracy and efficiency of the RBFNN, an improved RBFNN method is developed by transferring the sampling center of Latin hypercube sampling (LHS) from the mean values of random variables to the most probable point (MPP) in the sampling step. Then, the particle swarm optimization algorithm is adopted to optimize the shape parameters of RBFNN, and the RBFNN model is assessed by the cross-validation method for subsequent reliability analysis using Monte Carlo simulation (MCS). Four numerical examples are investigated to demonstrate the correctness and effectiveness of the proposed method. To compare the computational accuracy and efficiency of the proposed method, the traditional radial basis function method, hybrid radial basis neural network method, first-order reliability method (FORM), second-order reliability method (SORM), and MCS method are applied to solve each example. All the results demonstrate that the proposed method has higher accuracy and efficiency for structural reliability analysis. Importantly, one practical example of an industrial robot is provided here, which demonstrates that the developed method also has good applicability and effectiveness for complex engineering problems.

Efficient System Reliability Analysis for Layered Soil Slopes with Multiple Failure Modes Using Sequential Compounding Method

Evaluating the system reliability of layered soil slopes is a challenging issue because multiple failure modes may be included along the slip surfaces, which makes the overall failure probability greater than any individual slip surface. In this paper, an efficient system reliability analysis concerning the layered soil slopes is conducted based on the sequential compounding method (SCM) that has the ability to compound multiple failure events into an equivalent event sequentially. First, the first order reliability method (FORM) is employed to quantify initial reliability indices and correlation coefficients among these failure modes. Subsequently, the SCM is used to calculate the equivalent reliability indices and correlation coefficients until the multiple failure events are reduced to a compound event, and then the system reliability of the slope is obtained accordingly. The application of the approach to probabilistic evaluation of layered slopes is illustrated by two typical examples, and the correctness is verified by a Monte Carlo simulation (MCS). The results show that the SCM can deliver accurate system failure probability and greatly improve the computational efficiency compared with the MCS, which is an advantageous and promising strategy in evaluating the system reliability of layered soil slopes.

First-order reliability method to problems involving multimodal distributions

First-order reliability method to problems involving multimodal distributions

Influence of Limit State Function’s Form of Geotechnical Structures on Approximate Analytical Reliability Methods

Approximate analytical methods have been frequently used in geotechnical engineering to estimate the reliability of geotechnical structures due to their efficiency and simplicity. The main spirit of these methods is using the moments of the limit state function to estimate the reliability index. However, the moments are strongly dependent on the form of the limit state function, resulting in the fact that these methods are sensitive to the form of limit state functions. This study aims to systematically explore how various equivalent forms of limit state functions affect the performance of several commonly used approximate analytical methods, including the first-order second-moment method, the first-order reliability method, and the point estimation method. The applicable conditions of these methods are illustrated through five typical geotechnical examples. The results indicate that the estimated accuracy for the first-order second-moment method and the point estimation method is affected by the form of the limit state functions. Although the form of the limit state function does not affect the accuracy of the first-order reliability method, it affects computational efficiency. The limit state functions with an equivalent logarithmic form are almost always favorable for the investigated examples and are thus recommended in practice.

L-Moments-Based FORM Method for Structural Reliability Analysis Considering Correlated Input Random Variables

Leveraging the properties of the first three linear moments (L-moments), this study proposes an effective normal transformation for structural reliability analysis considering correlated input random variables, in which the admissible range of the initial correlation matrix when employing this transformation is also presented. Subsequently, a practical procedure for structural reliability analysis, grounded in the proposed transformation and first-order reliability method (FROM), is proposed, accommodating instances wherein the joint probability density functions (PDFs) or marginal PDFs of the relevant random variables remain unknown. In comparison to the technique premised on the first three central moments (C-moments), the proposed method, based on the first three L-moments, exhibits a more extensive applicability. Various practical scenarios showcase the method’s effectiveness and precision in calculating the structural reliability index, considering diverse distributions, numerous variables, and complex structures.

Influence of load partial factors adjustment on reliability design of RC frame structures in China

The partial factor method has been widely used in building design and the partial factors to ensure the safety of structures are specified in the adopted codes. The load partial factors in the design expressions have been increased in the lasted code in China, which leads to theoretically increment in reliability and a growth in the consumption of construction materials. However, the influence of load partial factors adjustment on design of building structures arises different points among scholars. Some believe that it has a great impact on the design, some think the influence is small. This makes designers have doubts in the safety of structures and investors are also confused about the cost. In order to illustrate the influence of load partial factor adjustment on safety level and material consumption of RC (Reinforced Concrete) frame structures, reliability analysis and material consumption analysis are performed using First Order Reliability Method (FORM). The approach is carried out according to the load partial factors in Chinese codes of (GB50153-2008) and (GB50068-2018), respectively. Then, the influence of load partial factors adjustment is demonstrated with a case design of RC frame structures with different load partial factors in codes. The results show that the partial factor has a noticeable influence on the reliability index. The adjustment of load partial factors in design leads to an increase of the reliability index, which is about 8–16%. The increase of material consumption used in RC structures is about 0.75–6.29%. And the case indicated that the adjustment of load partial factors mainly result in the increase of reinforcement consumption, while have little effect on the concrete consumption. This study provides an analytical and conclusive insight into the influence of load partial factor adjustment on safety level and material consumption, which is can be applied to a wide range of structures.

Comparative study of deterministic and probabilistic critical slip surfaces applied to slope stability using limit equilibrium methods and the First-Order Reliability Method

This work presents the validation of the Morgenstern-Price method implemented in the Risk Assessment applied to Slope Stability (RASS) computational program to carry out deterministic and probabilistic analyses of slope stability. Deterministic analyses, based on the factor of safety approach, are performed using limit equilibrium methods. The probabilistic ones, on the other hand, are carried out through the direct coupling of these methods to the First Order Reliability Method (FORM). Initially, two benchmark cases are presented for validation of the computational routine related to the Morgenstern-Price method. Next, two illustrative examples are presented, with the investigation of the critical surfaces defined by deterministic and probabilistic criteria, which correspond to the minimum factor of safety, the maximum probability of failure, and the maximum quantitative risk. In the set of stability analyses, it was verified that both the numerical responses and the geometry of the critical surfaces can vary depending on the choice of the limit equilibrium method and the criterion for identifying the critical surface. The different possibilities presented by the methodology used in this study define not only a critical surface, but a set of critical surfaces that can help in the engineering decision-making process and slope risk management, complementing the widely used purely deterministic analyses in geotechnics.

Robust reliability‐based design approach by inverse FORM with adaptive conjugate search algorithm

AbstractThe concept of reliability‐based design (RBD) of geotechnical works has seen great success in recent years, nevertheless, its application in real‐world engineering practice is still limited. This phenomenon could be mainly attributed to two aspects: the engineer's unfamiliarity of reliability algorithms even in their simplest forms, and the lack of direct connection between reliability algorithms and deterministic design tools. To promote the geotechnical RBD application, this study proposes an inverse first‐order reliability method (FORM) integrating adaptive conjugate direction search technique (RBD via conjugate FORM) to solve classical RBD problems where the number of (multiple) limit state functions is usually equal to the number of unknown design parameters. Notably, the RBD via conjugate FORM can readily back‐calculate those unknown design parameters subject to prescribed reliability index or probability of failure, by means of iterative evaluation of the parametric sensitivities. The proposed algorithm is first verified by several well‐documented examples. Comparative studies show that when the reliability problem to solve is highly nonlinear, the RBD results based on steepest descent search direction may fall in oscillation and non‐convergence, while the proposed RBD via conjugate FORM algorithm performs robustly and efficiently. It is then illustrated in combination with standalone numerical codes to conduct risk‐based optimization of unknown design parameters for two soil slope stability models. The first slope RBD model achieves designing the slope angle under specified reliability index, and the second slope RBD model is to optimize multiple design parameters of the stabilization pile, which can meet different slope reliability requirements under static and seismic loading conditions simultaneously.

Bayesian updating on resistance factors of H-Piles with axial load tests

In the Load and Resistance Factor Design (LRFD) of piles, several design codes recommend higher resistance factors if load tests are conducted. However, no information is provided on how these resistance factors are determined. In this paper, a probabilistic approach based on Bayes' theorem and the First Order Reliability Method (FORM) is proposed to calibrate resistance factors for different numbers of load tests and the corresponding test results. In addition, within-site variability, design methods, types of piles, and ground conditions can also be considered. The proposed approach applied to H-piles under axial load tests shows consistent results with current design codes. Results show that resistance factors are significantly increased even if only one positive test is observed among all the tests. For low variability sites, the differences of resistance factors between various design methods are significantly reduced if one or more tests are positive, while for high variability sites, the differences of resistance factors are only slightly decreased, indicating that design methods should be considered in the latter case. Most of the increase in resistance factors is achieved with a small number of tests. For β-Method used in clay sites, 80% of the increase in resistance factors is achieved with two, four and five consecutive positive tests are observed for low, medium and high variability sites, respectively.

Determination of critical aquifer subsidence points by DFFITS-COOK distance method using Importance sampling reliability analysis

The aim of this study is to investigate aquifer structure subsidence by groundwater level drawdown. In this study, a performance function matrix was developed in the first-order reliability method (FORM) by the importance sampling in order to investigate the regional resilience in a saturated aquifer. The study region is located in southwestern Iran. The changes in the layers of vertical structure destruction are an important factor affecting the plain's recharge capacity. Preparing the subsidence prediction layers due to the MODFLOW finite-difference model and criterion for its changes was related to five scenarios of recharge changes in the aquifer water level from 2008 to 2019 and also from 2019 to 2029. The results indicated that failure probability in Izeh aquifer varies from 0 to about 89 %. Considering structure type, most of the internal plain areas have high local resilience. For the aquifer boundary strips, the highest resilience is only in the southwestern part of aquifer. The high-risk border areas have a wide fitness on porous boundary strips as the main recharge sources. The reason for severe spatial changes in the failure probability of reliability was investigated by out-of-date analysis methods. The anomaly pattern was used in three methods of defining decision function to determine out-of-point under Robust Covariance, One-Class SVM and Isolation Forest. DFFITS and DFFITS methods (difference in fits) with COOK Distance were used to confirm the existence of 106 abnormal cells in the two-dimensional digit of baseline indices mean. Results showed that the most critical areas can be found in the central parts around the two lakes. The plain's permeable boundaries, the parts between the northwest and the center of the aquifer, and the northern aquifer's limited areas are other abnormal regions.

Reliability analysis of flexural provisions of FRP-RC beams and sensitivity analysis based on FORM

A reliability analysis was conducted based on the flexural strength recommendations provided by the ACI 440.1R-15 guide concerning the glass fiber-reinforced polymer-reinforced concrete (GFRP-RC) beams. A statistical investigation was carried out by compiling the dataset of 235 specimens from the technical literature on the model uncertainty of the beams to select the best probability distribution. As such, two failure modes of concrete crushing and GFRP rupture were examined. Reliability was analyzed using the first-order reliability method (FORM) via iHLRF algorithm. Importance sampling was also employed to determine the efficiency of the FORM. Findings reveal that the flexural requirements of ACI 440.1R-15 regarding the FRP-RC beams are conservative for both failure modes, particularly the GFRP rupture failure. Subsequently, importance vector α and sensitivity vectors δ and η were conducted to rank the effect of the most important random variables and their probability distribution parameters on the reliability index. Results from the importance vector α demonstrate that the random variable d is the most significant variable in both failure modes amongst the examined design variables. Thus, uncertainty contributed with d has the greatest effect on the reliability index. Moreover, based on the results from sensitivity vectors δ and η, varying the mean and standard deviation of the random variable ML leads to maximum change in total failure probability of the problem. Finally, strength reduction factors of 0.65 and 0.75 were respectively proposed through calibration so as to reach the target reliability index for GFRP rupture and concrete crushing.

Probability analysis of rock slope using FORM based on a nonlinear strength criterion

Various uncertainties are frequently encountered in the assessment of landslide stability. However, the uncertainty of parameters in the Mohr–Coulomb strength criterion is a major concern in the studies, neglecting the uncertainty related to the strength criterion itself that is used to describe the strength of geotechnical materials. To assess the reliability of landslides, a new probabilistic analysis method is proposed in this study. According to the traditional limit equilibrium framework, an improved method based on a nonlinear strength criterion is deduced to determine the safety factor of landslides. The local mean value of the random variables on the failure surface is estimated by the analytical method, avoiding random field simulation. The first-order reliability method (FORM) is applied to estimate the failure probability of landslides. An example is analyzed using the proposed probability analysis method, and its effectiveness is primarily verified. The results show that the traditional limit equilibrium method based on the linear Mohr–Coulomb strength criterion estimates the stability and failure probability of landslides inaccurately.

Effect of System Reliability on Design of Metal Building Roof Purlins

This paper provides a framework to incorporate structural system reliability effects in the design of roof purlins in a typical metal building. Today, every roof purlin is considered as a separate component, and the effect of spatial variation in the demand loads and potential redistribution and load sharing in the roof system capacity is ignored in design. Component reliability is established by first-order reliability methods implemented through load and resistance factor design. Based on recent work in load-bearing cold-formed steel framing systems, the load and resistance factor design framework is extended from components to systems through an additional resistance factor to account for system influence. An archetypical metal building is designed and selected for this study. Monte Carlo simulations of a segment of the metal building roof are performed with consideration of both randomness in the demands and capacity and employing geometric and material nonlinearity in the response model of the roof. The simulations indicate that the system effect in metal building roofs is beneficial, and increases in the design capacity when evaluated against demands may be justified. Sensitivity to the target reliability (allowed probability of failure), deflection limits, and modeling assumptions are observed and discussed. Preliminary factors to account for roof system reliability are provided.

Reliability Assessment Methodology of Blast Protective Steel Moment Resisting Frame Using NiTi SMA-based Connection

This paper proposes a methodology framework for the reliability assessment of smart steel Moment Resisting Frame structures (MRFs) equipped with Nickle Titanium Shape Memory Alloy (NiTi SMA) connections subjected to blast loading. The reliability assessment framework is formulated based on a two-step approach algorithm. In the 1st Step, the Monte Carlo Latin Hypercube Sampling Strategy simulation (MC-LHS) is adapted to generate the uncertain parameters sample points. Considering the numerical simulations, the 2nd Step employs simplified performance functions and the generated random outcomes from the 1st Step. The proposed reliability approach is verified against direct Monte Carlo Simulation (MCS) and First-Order Reliability Method (FORM). The performance functions are columns’ axial force and bending moments, rotation capacity at the connections, and Inter-Story Drift Ratio (ISDR). Throughout the development of the reliability assessment, the probabilistic models are parametrized on geometrical properties, material properties, vertical loads, model errors, and charge weights. The developed reliability framework is applied to a prototype 4-story smart MRF. The structural safety level is obtained in terms of the Reliability Index (β). The results show that the reliability framework provides an accurate and efficient structural collapse prediction of the MRFs equipped with NiTi SMA-based connections. Finally, sensitivity analysis is performed to indicate the sensitivity of building collapse to blast wave characteristics, material strength, vertical gravity loads, and column profile dimensions. The sensitivity analysis results also confirm the efficiency of the proposed reliability framework in observing the highly sensitive parameters, which is explosive charge weight.

REINFORCED CONCRETE BEAM AT ULTIMATE LIMIT STATE CONSIDERING VARIABILITY IN THE CONCRETE MIX DESIGN PARAMETERS

Variability in design parameters of Civil Engineering infrastructure is inevitable at implementation in most practical situations, the effect of which can best be determined at design stage using reliability concepts. This study used First Order Reliability Method (FORM) to determine the effect of variation in Nano Engineered Concrete (NEC) mix design parameters on structural safety of Nano Engineered Reinforced Concrete (NERC) beams at ultimate limit state using reliability based sensitivity analysis. The variability in NEC mix design parameters was incorporated into EN1992-1-1(2008) design formulations using predictive models of NEC characteristic compressive strength developed on the basis of experimental data with the aid of DataFit statistical package. FORTRAN based subroutines of the NERC beam performance functions were developed and used for the reliability sensitivity analysis at ultimate limit state. Results indicate that variability in NEC mix design parameters affect structural safety of NERC beams in accordance with how the mix design parameter contributes to characteristic compressive strength development. Moreover, increase in nanosilica dosage beyond optimal value was found to have negative effect on structural safety of NERC beams. The study found that compressive strength contributes to tensile strength of NERC beams through composite action. Furthermore, the study suggested that the effect of variability in concrete mix design parameters be incorporated in design formulations of Civil Engineering Codes of Practice to aid estimation of safety of structural elements to be designed and produced with NEC for improved safety, sustainability and resilience of Civil Engineering infrastructure.

Reliability-based assessment of ship hull girder ultimate strength

Abstract A reliability-based approach is presented to investigate the effects of structural and load uncertainties on the reliability estimation of ship hull girders. Structural uncertainties included randomness in material properties, geometric properties, initial geometric imperfections, and corrosion behavior. Load uncertainties included statistical uncertainties, model uncertainties, environmental uncertainties, and uncertainties related to nonlinearity. The hull girder ultimate strength was calculated using Smith’s method, and the probabilistic density function was evaluated by employing Monte Carlo simulations. In the load estimation, the still water bending moment and wave-induced bending moment were calculated using a simplified formula of the International Association of Classification Societies-Common Structural Rules code and then modified with load parameters. The reliability index was estimated using a first-order reliability method considering the operating time, the duration of the ship in the alternate hold loading condition, and the severity of the corrosion rate. As a result, sagging conditions dominated the collapse mode. The reliability indexes were obtained for the observed cases, and the viability of the ship was assessed accordingly.

Application of first-order reliability method with orthogonal plane sampling for high-dimensional series system reliability analysis

The increasing complexity of modern engineering systems has motivated a shift of research focus from component-level reliability to system reliability with interdependent components. There is a growing demand for efficient reliability methods to analyze high-dimensional systems that involve numerous dependent components and component variables. The first-order reliability method (FORM), which is widely used for component-level reliability analysis, becomes inaccurate for high-dimensional systems composed of numerous components, each with a nonlinear high-dimensional limit state function. By integrating the orthogonal plane sampling, this paper proposes an improved FORM-based method to tackle the curse of dimensionality for series systems. The idea is to construct secant hyperplanes using the orthogonal plane samples so as to reduce the FORM error for high-dimensional nonlinear limit state functions. The design points of secant hyperplanes are projected to high-dimensional system space using an efficient procedure based on the specified correlation matrix of variables. Finally, the series system reliability is computed as high-dimensional multi-normal integral, which is addressed by the equivalent component method. Four numerical examples are investigated to demonstrate the accuracy and efficiency of the proposed method. Results indicate that the proposed method is significantly more efficient than the Monte Carlo simulation and more accurate than the conventional FORM.

Toward a Comprehensive Pavement Reliability Analysis Approach

Reliability has been incorporated in pavement design tools to account for input variability influence on predicted performance. As they are not based on a probabilistic method of uncertainty propagation, the reliability analysis methodologies that are currently implemented in pavement performance tools lack rigor and robustness. This paper investigates the potential of three reliability analysis methodologies for pavement application: the Pavement ME reliability analysis methodology, Monte Carlo simulation (MCS), and the first-order reliability method (FORM). The MCS and FORM involve a response surface method for the generation of a second-order surrogate model. The investigation was performed using inputs and performance data from accelerated pavement testing structures. Inputs that were identified as significant were characterized as random variables and their associated variability was established using measured structural and material properties. Pavement performance with respect to rutting was predicted using the ERAPave performance prediction tool, while MCS was used to generate the actual variability of the distress. The reliability analysis results have shown that a comprehensive reliability analysis methodology is required that effectively captures input variabilities and the error associated with surrogate models.

Fire Safety of Plane Steel Truss according to System Reliability Analysis Combined with FORM Method: The Probabilistic Model and SYSREL Computation

This article focuses on the reliability analysis of the plane steel truss under fire conditions. The safety of the structure was estimated by system reliability analysis combined with First Order Reliability Method (FORM). The authors created the C++ code, which enables us to prepare the advanced probabilistic model for bearing capacity in the selected time of fire duration. Searching cut-sets for system analysis was performed in the C++ code, where stiffness matrix spectral analysis was employed. It was found that a probabilistic model has significant influence on the reliability indices. The research showed that depending on the probabilistic model, the sensitivity of the reliability index to individual variables is different.

System reliability and sensitivity analysis of lateral loaded pile considering soil’s spatial variability

ABSTRACT Probabilistic analysis has been widely used to assess the inherent uncertainty of variables in laterally loaded pile systems, but the calculation is still difficult and time-consuming. The present study presents an efficient probabilistic analysis framework for a laterally loaded pile system. The performance of the system is defined as the lateral deflection at the pile head and maximum bending moment of the pile shaft, corresponding to two failure modes. Within this framework, the spatial variability of the soil and the correlation between failure modes are considered by the random field theory and the First-Order Reliability Method, respectively. Moreover, the Sequential Compounding Method is used as an efficient tool to determine the system reliability indexes. The framework is confirmed by comparing the reliability indexes of failure modes and systems with those of the Monte Carlo Simulation Method. Furthermore, a parametric analysis and system sensitivity analysis are performed. The results show that the auto-correlation distance, allowable lateral displacement at the pile head, and allowable bending moment of the pile shaft have a great influence on reliability indexes of failure modes and system, and the major parameter of soil in affecting pile is the elastic modulus compared with the undrained shear strength.