Retaining walls – reliability-based approach

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

Reliability-based design for soil tillage machines

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.

It is important to specify most dominant design parameters affecting safety of structures and to improve estimated accuracy of them in order to promote construction cost reduction through the reliability-based design. In this study, the dominant parameters for stability against sliding, overturning, and foundation failure of caisson-type breakwaters are specified by using safety index and sensitivity factor with FORM (First-Order Reliability Method). Moreover, correlations among failure modes in present design sections were described.

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.

Stochastic supply and fluctuating travel demand in transport systems leads to stochastic travel times and travel costs for travelers. This chapter will establish a dynamic road network design approach considering stochastic capacity and its influence on travelers’ choice behavior under uncertainty. This chapter will firstly work on the modeling of travelers’ departure time/route choice behavior under stochastic capacities. A reliability-based dynamic network design approach is proposed and formulated of which numbers of lanes on all the potential links are the design variables. A combined road network-oriented Genetic Algorithm and set evaluation algorithm is proposed to solve the dynamic network design problem. The proposed reliability-based dynamic network design approach is applied to a hypothetical network, and its solutions are compared to a corresponding static network design approach. It is concluded that the static network design approach may lead to inferior designs. In general static traffic assignment underestimates the overall total network travel time and total network travel costs. Dynamic network design approach appears to derive a fairly good allocation of road capacity over space and makes the best utilization of the network capacity over time.

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.

Fatigue cracks are an ongoing problem for aluminum high-speed vessels, and preventing fatigue cracks caused by wave loading is expected to be a significant challenge for future aluminum high-speed ferries and military vessels. To aid in this effort, a hot-spot fatigue design approach using first-order reliability methods (FORM) is constructed. Two different limit state functions are investigated, and the accuracy and consistency of the FORM method for the highly nonlinear fatigue limit state equations are evaluated through a comparison with Monte Carlo simulation results. The sensitivity of the resulting safety index to changes in the input variables, and their uncertainties, are presented graphically. The method is compared to existing design standards for four simple structural details.

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 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.

Reliability-based design (RBD) of tunnel concrete linings in weak rocks is challenging due to the uncertainty of geomechanical parameters and the ground-support interaction. This study introduces a transparent and efficient framework that integrates three key components: (1) implicit ground-support equilibrium analysis based on the Convergence-Confinement Method (CCM), (2) probabilistic evaluation of the safety factor ([Formula: see text]), reliability index ([Formula: see text]), and failure probability ([Formula: see text]) and (3) a data-driven surrogate model based on artificial intelligence for rapid parametric analysis. The uncertainties of the rock mass and concrete linings are treated through Monte Carlo sampling, and the outcomes are benchmarked against the First-Order Reliability Method (FORM) to verify accuracy and potential bias. The results are transformed into decision-making charts linking the thickness and compressive strength of concrete linings to target [Formula: see text] levels, thus providing risk-consistent objectives instead of a fixed safety factor criterion. Two types of linings are considered: conventional concrete (with an average uniaxial compressive strength of 20MPa) and fiber-reinforced reactive powder concrete (FRPC, with an average uniaxial compressive strength of 65MPa, developed by the research team). The findings demonstrate that increasing thickness and material quality significantly reduce [Formula: see text], achieving the reliability thresholds required for final tunnel support (2E-5). This effect is more critical at smaller thicknesses and with lower-quality materials. The use of FRPC also leads to a considerable reduction in [Formula: see text] at intermediate thicknesses, making it an efficient option when construction constraints limit thickness increase. The surrogate model successfully reproduces probabilistic trends with high consistency and slight conservatism. Ultimately, the integrated CCM-reliability-artificial intelligence framework bridges the gap between deterministic design and risk management, delivering economical and resilient tunnel lining designs in weak rocks.

In remotely operated vehicle (ROV), worm gearing is usually used to transfer movement and power between two staggered axes. In the design process of worm gearing, complex gear face and the exist of uncertainty make its design and modeling so difficult will affect its reliability. In this paper, a reliability-based design (RBD) method is proposed for worm gearing of remotely operated vehicle under two types uncertainties, and the first-order reliability method (FORM) is used to calculate the reliability of worm gearing. Finally, an engineering example is used to illustrate the effectiveness of the reliability design method for worm gearing of remotely operated vehicle.

This paper presents a simplified approach for reliability analysis of basal heave in a braced excavation considering the spatial variability of soil parameters, The first-order reliability method (FORM) with a variance reduction technique is employed to model the spatial variability in lieu of the conventional random field modeling (RFM). The proposed approach yields results that are comparable with those obtained using the conventional RFM approach that relies on Monte Carlo simulation. The proposed approach requires much less computational effort, is easy to use, and has potential as a practical tool for reliability-based design that has to deal with spatial variability of soils.