ケーソン式防波堤の外的安定に関する安全性指標と感度係数の関係

Reliability-based design optimization is much more computationally expensive than deterministic design optimization. To alleviate the computational demand, the First Order Reliability Method (FORM) is usually used in reliability-based design. Since FORM requires a nonlinear transformation from non-normal random variables to normal random variables, the nonlinearity of a constraint function may increase. As a result, the transformation may lead to a large error in reliability calculation. In order to improve accuracy, a new reliability-based design method with Saddlepoint Approximation is proposed in this work. The strategy of sequential optimization and reliability assessment is employed where the reliability analysis is decoupled from deterministic optimization. The accurate First Order Saddlepoint method is used for reliability analysis in the original random space without any transformation, and the chance of increasing nonlinearity of a constraint function is therefore eliminated. The overall reliability-based design is conducted in a sequence of cycles of deterministic optimization and reliability analysis. In each cycle, the percentile value of the constraint function corresponding to the required reliability is calculated with the Saddlepoint Approximation at the optimal point of the deterministic optimization. Then the reliability analysis results are used to formulate a new deterministic optimization model for the next cycle. The solution process converges within a few cycles. The demonstrative examples show that the proposed method is more accurate and efficient than the reliability-based design with FORM.

Reliability-based design of tunnelling problems and insights for Eurocode 7

The efficiency of the first-order reliability method (FORM) and the accuracy of Monte Carlo simulations (MCS) are coupled in probability-based designs of reinforced rock slopes, including a Hong Kong slope with exfoliation joints. Load–resistance duality is demonstrated and resolved automatically in a foundation on rock with a discontinuity plane. Other examples include the lengthy Hoek and Bray deterministic vectorial procedure for comprehensive pentahedral blocks with external load and bolt force, which is made efficient and more succinct before extending it to probability-based design via MCS-enhanced FORM. The FORM–MCS–FORM design procedure is proposed for cases with multiple failure modes. For cases with a dominant single failure mode, the time-saving importance sampling (IS) and the fast second-order reliability method (SORM) can be used in lieu of MCS. Two cases of 3D reinforced blocks (pentahedral and tetrahedral, respectively) with the possibility of multiple sliding modes are investigated. In the case of the reinforced pentahedral block, direct MCS shows that there is only one dominant failure mode, for which the efficient method of importance sampling at the FORM design point provides fast verification of the revised design. In the case of the reinforced tetrahedral block, there are multiple failure modes contributing to the total failure probability, for which the proposed MCS-enhanced FORM procedure is demonstrated to be essential. Comparisons are made between Excel MCS and MATLAB MCS.

A new approach for reliability-based optimization of water distribution networks is presented. The approach links a genetic algorithm (GA) as the optimization tool with the first-order reliability method (FORM) for estimating network capacity reliability. Network capacity reliability in this case study refers to the probability of meeting minimum allowable pressure constraints across the network under uncertain nodal demands and uncertain pipe roughness conditions. The critical node capacity reliability approximation for network capacity reliability is closely examined and new methods for estimating the critical nodal and overall network capacity reliability using FORM are presented. FORM approximates Monte Carlo simulation reliabilities accurately and efficiently. In addition, FORM can be used to automatically determine the critical node location and corresponding capacity reliability. Network capacity reliability approximations using FORM are improved by considering two failure modes. This research demonstrates the novel combination of a GA with FORM as an effective approach for reliability-based optimization of water distribution networks. Correlations between random variables are shown to significantly increase optimal network costs.

This article presents reliability analysis and reliability-based optimization of roadway minimum radius design based on vehicle dynamics, mainly focusing on exit ramps and interchanges. The performance functions are formulated as failure modes of vehicle rollover and sideslip. To accurately describe the failure modes, analytical models for rollover and sideslip are derived considering nonlinear characteristics of vehicle behaviour using the commercial software TruckSim. The probability of an accident is evaluated using the first-order reliability method and numerical studies are conducted using a single-unit truck model. To propose a practical application for the study, the reliability analysis for the minimum radius recommended by American Association of State Highway and Transportation Officials is conducted. The results show that, even if there are deviations from assumed design conditions of the current design guideline, the proposed design method can guarantee given target margins of safety against rollover and sideslip. Based on the reliability analysis, reliability-based design optimization is carried out and the results indicate new recommendations for minimum radii satisfying given target reliability levels.

This study presents a parallelized framework for the grid-and-radius-based Morgenstern–Price method to enable an efficient reliability analysis of unsaturated finite soil slopes under correlated and non-Gaussian conditions. To address the limitations of deterministic techniques and the significant computational demands associated with traditional Monte Carlo simulation, the framework incorporates surrogate modeling approaches, namely polynomial chaos expansion–enhanced Monte Carlo simulation (PCE-MCS) and Kriging-based Monte Carlo simulation (K-MCS). This research conducts a comprehensive evaluation of these surrogate-assisted methods with the gradient-based approaches for estimating the probability of failure (pf) under correlated and non-Gaussian distributions. The surrogate models exhibit exceptional accuracy (R2 > 99.7%, mean absolute error <0.6%) and drastically reduce computational cost compared with traditional methods. Parametric and variability studies show that increasing the mean stability number (0.005–0.065) and friction angle (20°–30°) reduces the pf from 19.19% to 0.005% and 14.9% to 0.01%, respectively, while higher variability increases pf. Similarly, increasing the air-entry value reduces pf from 29.8% to 0.01%, whereas pf rises to 15.1% and 21.8% with increases in soil–water characteristic curve parameters nf (0.1–3.25) and mf (0.25–1.65). A detailed comparison against gradient-based techniques such as the first-order reliability method (FORM) and the second-order reliability method (SORM) underscores the superiority of surrogate-based approaches. For instance, the FORM shows errors up to 92.59%, while SORM errors reach 62.75%, particularly in highly nonlinear or variable scenarios. In contrast, PCE-MCS and K-MCS errors remain generally below 10%. Furthermore, reliability-based design contour charts are developed using K-MCS. These charts align with the USACE reliability classification, enabling rapid and reliable assessment of unsaturated slope stability without the need for large-scale simulations. Therefore, by effectively bridging traditional geotechnical engineering and advanced surrogate techniques, the proposed approach enhances practical decision-making in unsaturated slope design and supports the advancement of reliability-based design optimization in geotechnical practice.

A novel reliability-based design optimization (RBDO) method using simulation-based techniques for reliability assessments and efficient optimization approach is presented in this paper. In RBDO, model-based reliability analysis needs to be performed to calculate the probability of not satisfying a reliability constraint and the gradient of this probability with respect to each design variable. Among model-based methods, the most widely used in RBDO is the first-order reliability method (FORM). However, FORM could be inaccurate for nonlinear problems and is not applicable for system reliability problems. This paper develops an efficient optimization methodology to perform RBDO using simulation-based techniques. By combining analytical and simulation-based reliability methods, accurate probability of failure and sensitivity information is obtained. The use of simulation also enables both component and system-level reliabilities to be included in RBDO formulation. Instead of using a traditional RBDO formulation in which optimization and reliability computations are nested, a sequential approach is developed to greatly reduce the computational cost. The efficiency of the proposed RBDO approach is enhanced by using a multi-modal adaptive importance sampling technique for simulation-based reliability assessment; and by treating the inactive reliability constraints properly in optimization. A vehicle side impact problem is used to demonstrate the capabilities of the proposed method.

This paper presents the development and demonstration of an efficient reliability-based tool for the reinforced earth retaining wall design in heavy haul railway. Here, two major internal failure modes, rupture and pullout, are the focus in the demonstration. The First-Order Reliability Method (FORM) is adopted to estimate the probability of failure for each failure mode, and is implemented in a spreadsheet. The reliability analysis is conducted with the consideration of the effect of uncertainty in the reinforcement length, horizontal and vertical spacing between steel strips, tensile strength of steel strips as well as the material property of the backfills. Using this design tool, a few candidate designs can be easily obtained by meeting the acceptable probability of failure and the final design is determined based on cost. The results obtained using FORM are also verified by comparing with those from Monte Carlo simulation. This design tool is shown to be simple to use in the retaining wall design of the railway engineering.

Efficient system reliability analysis illustrated for a retaining wall and a soil slope

The deterministic approach is commonly used for designing geosynthetic-reinforced soil retaining walls. This approach ignores the natural variability of the soil properties, which may lead to an uneconomic and/or unsafe design. This chapter presents a reliability-based design approach for geosynthetic-reinforced retaining walls with granular backfills by using the first-order reliability method. The results are also verified with conventional Monte Carlo simulation. Reinforced soil retaining walls are designed considering both internal and external stability requirements. The associated modes of failure are formulated, and the design steps are discussed in detail. The steps involved in the reliability-based design are explained through illustrative examples.

System reliability analysis of rock wedge stability considering correlated failure modes using sequential compounding method

This paper presents a reliability-based analysis of road vehicle accidents and the optimization of roadway radius and speed limit design based on vehicle dynamics, mainly focusing on windy environments. The performance functions are formulated as failure modes of vehicle rollover and sideslip and are defined on a finite set of basic variables with probabilistic characteristics, so-called random variables. The random variables are vehicle speed, steer angle, tire–road friction coefficient, road bank angle, and wind speed. The probability of accident was evaluated using the first-order reliability method (FORM) and numerical studies were conducted using a single-unit truck model. The analysis demonstrates that wind is a significant factor when assessing vehicle safety on roads, and probabilistic studies such as reliability-based design optimization (RBDO) are necessarily required to enhance vehicle safety in windy environments. Accordingly, design optimization of roadway radius and speed limit was conducted, and new designs were proposed satisfying the target reliability. This study suggests that probabilistic mechanics and theory can be of value for analysis and design of wind-related vehicle safety.

This study proposes a system reliability analysis of rack storage facilities subjected to forklift colliding events. The proposed system reliability analysis consists of two steps: the first step is to identify dominant failure modes that most contribute to the failure of the whole rack facilities, and the second step is to evaluate the system failure probability. In the first step, dominant failure modes are identified by using a simulation-based selective searching technique where the contribution of a failure mode to the system failure is roughly estimated based on the distance from the origin in the space of the random variables. In the second step, the multi-scale system reliability method is used to compute the system reliability where the first-order reliability method (FORM) is initially used to evaluate the component failure probability (failure probability of one member), and then the probabilities of the identified failure modes and their statistical dependence are evaluated, which is called as the lower-scale reliability analysis. Since the system failure probability is comprised of the probabilities of the failure modes, a higher-scale reliability analysis is performed again based on the results of the lower-scale analyses, and the system failure probability is finally evaluated. The illustrative example demonstrates the results of the system reliability analysis of the rack storage facilities subjected to forklift impact loadings. The numerical efficiency and accuracy of the approach are compared with the Monte Carlo simulations. The results show that the proposed two-step approach is able to provide accurate reliability assessment as well as significant saving of computational time. The results of the identified failure modes additionally let us know the most-critical members and their failure sequence under the complicated configuration of the member connections.

This work presents reliability analysis of Nano Engineered Reinforced Concrete (NERC) beam designed according to EN1992-1-1(2008) and its reference standard. In the reliability analysis, variability in load, geometric properties, Optimal Nano Engineered Concrete (ONEC) grade and other design variables were considered. The reliability analysis utilized FORM (First Order Reliability Method) with FORTRAN subroutine programs developed for the NERC beam failure modes at ultimate limit state. Results indicated that NERC beam behaves in a similar manner to conventional reinforced concrete beam on strength aspect. At design point, the NERC beam failed to meet the target safety index values provided by EN1990: 2002 (EC0) as a reference standard of EN1992-1-1 (2008) for all the failure modes considered. Furthermore, variations in span and reinforcement ratio have been found to be capable of changing the criticality of failure modes of NERC beam. Also, variation in characteristic yield strength of steel was found to severely affect NERC beam safety. This study recommends that unnecessary variation of span, reinforcement ratio and characteristic yield strength of steel should be avoided in NERC beams at project implementation stage so as to reduce the possibility of unforeseen structural failures. The work suggested that the safety margin provided by EN1992-1-1 (2008) and its reference standards for the design of singly reinforced concrete sections without shear reinforcement be improved. Finally, the study observed the need for incorporation of reliability-based design in Civil Engineering design standards.

Reliability-based design for soil tillage machines