This paper presents a novel extreme value distribution that characterizes the joint tail behavior of the block maxima/minima of a bivariate stationary Gaussian process and the associated concomitants. Unlike classical multivariate extreme value models, which focus on component-wise maxima and may misrepresent failure mechanisms driven by non-simultaneous extremes, the proposed formulation directly captures the statistical structure relevant to reliability problems governed by interaction-type ultimate limit state functions. The derivation is based on a probabilistic composition of peak values and their concomitants and leads to a closed-form semi-analytical mixture model expressed solely in terms of the spectral moments of the underlying processes. The resulting probability distribution admits sampling, enabling its implementation in reliability analyses without resorting to computationally intensive time-domain simulations.
The importance of the probabilistic seismic hazard assessment for designing new and evaluating existing structures is well established. The seismic hazard could be affected by the near-fault directivity effects. Often, the effects are not considered in seismic hazard mapping in the context of seismic design code development; this is the case for developing the fifth-generation Chinese seismic hazard maps. In the present study, the assessment and mapping of the seismic hazard for the Chinese mainland by considering the near-fault directivity effects was carried out, and the differences between the maps with and without directivity effects were quantified. The analysis results for 31 selected major cities indicated that the current seismic design spectrum overestimates the uniform hazard spectra (UHS) for several typically considered exceedance probabilities and for the natural vibration period, Tn, greater than about 3 s. The shape of the seismic design coefficient curve differs substantially from the standardized UHS. Using the developed seismic hazard maps that incorporate the directivity effects, it was shown that the spatial pattern of the maps for the peak ground acceleration (PGA) and spectral acceleration (SA) at different Tn values are neither identical nor directly proportional. This implied that simply scaling a single standardized UHS by using the PGA value may not accurately represent the UHS for different sites. The quantification of the differences between the maps with and without directivity effects indicates that the consideration of directivity effects can increase SA at Tn = 2 s by approximately up to 8% to 24% for exceedance probabilities ranging from 63% to 2% in 50 years. The range of increase becomes approximately up to 11% to 42% if Tn is equal to 5 s is considered.
Baltimore bridge collapse highlights the need to reassess and manage the ship impact risk for bridges across navigable waterways. On the other hand, the bridge protection upgrades are costly. Data from automated identification systems (AIS) and past accidents at a given bridge site, can lead to accurate risk estimate thereby resulting in efficient risk management. However, an investment into the site-specific data requires a metric to justify the expected benefit from such data for ship impact risk management. The present study proposes a value of information framework to quantify the expected benefit (EVoI) from the site-specific data. Specifically, the potential cost saving using the vessel aberrancy and spatial distribution data is assessed. The proposed framework is implemented on an example bridge to evaluate the EVoI for a range of bridge collapse risk. Moreover, a parametric study is carried out to assess the influence of risk management efficiency, time to next assessment (decision horizon) and prior belief on the value of site-specific information. Based on the findings, it is concluded that, the site-specific information is valuable, except at the extreme cases of high and low risk. Further, the combined EVoI from vessel aberrancy and spatial distribution is sub-additive. Parametric study reveals the impact of risk management efficiency, and optimum decision horizons for low and high risk scenarios.
To accurately evaluate the fragility of bridge systems while accounting for correlations in the seismic demands of individual component, this study employs system reliability theory. The bridge is modeled as series–parallel system, and a system fragility assessment method based on a Vine Copula is proposed. In this approach, each component of the bridge system is treated as a variable within the Vine Copula, and the interdependence of seismic demands among each components is explicitly considered. Furthermore, the Copula function is used to establish a joint probability density function for assessing system fragility. A segment of the isolated continuous beam bridge of the Hong Kong–Zhuhai–Macao Bridge was selected as the case study. The proposed method was applied to analyze the correlations among 14 key components, including piers and bearings. The Akaike information criterion was employed to optimize the Copula function and determined an appropriate vine structure, after which system fragility is evaluated. Finally, the results were compared with those obtained using the conventional first-order boundary estimation method and the Monte Carlo method to validate the effectiveness and accuracy of the proposed approach.
Poisson white noise, as a typical non-Gaussian random excitation, has received considerable attention in both theoretical and practical aspects of random vibration analysis. This research is devoted to developing an effective explicit time-domain approach for the random vibration analysis of energy-dissipating structures equipped with nonlinear viscous or hysteretic dampers under nonstationary Poisson or filtered Poisson white noise excitations. The present approach is developed within the framework of equivalent linearization (EL), with the equivalent linear system derived without the conventional assumption of Gaussian response. The successive nonstationary non-Gaussian random vibration problems of a sequence of linearized systems involved in the iteration process of EL are solved using the explicit time-domain method (ETDM), where the first four orders of moments of dynamic responses are derived via explicit formulation using the cumulant and moment operation rules, while the complex fractional-order or higher-order moments involving the absolute values of the damper responses are estimated via explicit simulation using a sufficient number of samples. The effectiveness of the present EL-ETDM is validated through four numerical examples, including two oscillators equipped respectively with a nonlinear viscous damper and a nonlinear hysteretic damper under nonstationary Poisson white noise excitations, and two 5-degree-of-freedom structures installed respectively with nonlinear viscous dampers and nonlinear hysteretic dampers subjected to nonstationary filtered Poisson white noise excitations.
Concrete acceptance continues to rely almost exclusively on compressive strength, yet deterioration-driven failures under aggressive exposure, such as reinforcement corrosion and freeze–thaw attack, remain a leading cause of premature decommissioning, even for strength-compliant members. This mismatch points to a fundamental gap: in-service performance is governed not by any single metric but by the joint risk of non-compliance across coupled strength, durability and denseness indicators, whose contradictions the current pass/fail paradigm systematically overlooks. Addressing this gap, the present study develops a data-driven multi-indicator quality assessment workflow that chains point prediction, distribution-free uncertainty quantification and tail-dependent joint non-compliance modelling into an integrated assessment pipeline. Eight machine-learning models are benchmarked for simultaneous prediction of compressive strength, electrical resistivity and ultrasonic pulse velocity on 4420 specimens with synchronous measurements; the reported accuracy (R2 = 0.984–0.989 for compressive strength) strictly reflects in-domain interpolation, as leave-one-group-out cross-validation by mix design yields negative R2 for key indicators. Split conformal prediction then furnishes finite-sample prediction intervals whose coverage guarantees hold without distributional assumptions, while grouped analyses expose pronounced coverage heterogeneity across curing ages and aggregate types. A trivariate t-copula fitted to the prediction residuals reveals that the dependence among indicators is concentrated in the extreme tails: when compressive-strength residuals fall in the worst 5% region, the conditional probability of concurrent resistivity deterioration is roughly sevenfold higher than the independence baseline (an order-of-magnitude estimate given the small tail sample), a finding that reframes “strength-compliant yet durability-deficient” cases as a robustness deficit of the acceptance system itself. Joint non-compliance probabilities estimated via Monte Carlo simulation provide a graded quality-screening map that can inform acceptance decisions and life-cycle quality management workflows. The framework is modular: base model, conformal strategy and copula family are each independently replaceable. It is also, in principle, readily transferable to broader multi-indicator quality assessment problems in construction materials and infrastructure systems.
This work presents an advanced nonlinear modeling framework for Ball Track Nonlinear Energy Sinks (BT-NESs), where a heavy ball rolls along a generic, rubber-lined curved track. The device merges the mechanical simplicity of ball-type dynamic vibration absorbers with the broadband vibration mitigation of translational T-NESs to improve structural resilience. Owing to the non-holonomic rolling-without-slip constraint, the resulting formulation is substantially more complex than that of translational T-NESs, which explains the still limited development of the BT-NES literature. In particular, to the authors’ knowledge, no existing study has addressed the performance- and reliability-based seismic design optimization (PBSDO/RBDO) of BT-NESs, even for circular track configurations. The present work fills this gap and further advances the state of the art by introducing two key contributions beyond existing PBSDO frameworks for translational T-NESs: (i) an equivalent linear pushover-based surrogate model derived via static condensation, providing a consistent and computationally efficient alternative to uniform stiffness-degradation approaches; and (ii) a systematic comparison of two intensity measures—peak ground acceleration (PGA) and spectral acceleration at a fixed period of 1 s, Sa(T=1s)—within the probabilistic optimization loop. The proposed framework is applied to a medium-rise steel building in Chile. The BT-NES track shape is described by a set of rational functions covering a wide range of curvatures, and the objective is to minimize the expected life-cycle damage cost associated with slight, moderate, and extensive damage limit states under four spectrally distinct ground-motion types. Results show that the optimized BT-NES reduces the total expected repair cost by approximately 50% compared to the uncontrolled structure. The results further indicate that PGA-based assessments may significantly overestimate seismic risk for low-frequency-dominated excitations, although the optimal track shape remains largely unaffected.
Structural reliability analysis quantifies failure probabilities under multiple sources of uncertainty, yet computational expense of high-fidelity models renders direct Monte Carlo simulation impractical for engineering systems. Existing active-learning strategies frequently switch unstably between global exploration and boundary refinement, leading to redundant sampling and limited efficiency. This study develops active-learning Kriging framework that integrates a scale-free, coverage-aware acquisition rule with iterative reliability estimator. The acquisition combines three jointly normalized indicators-the distance to estimated limit-state surface, the predictive uncertainty, and the local sample spacing, to balance exploration and refinement while maintaining scale invariance. Joint normalization prevents dominance by any single indicator and suppresses clustering, while an annealed scheduling mechanism progressively shifts the search from global coverage to local boundary sharpening. Numerical benchmarks and engineering case studies show that the method produces accurate and smoothly convergent failure-probability estimates with substantially fewer true-model evaluations than representative active-learning criteria. The sampling trajectory preserves broad domain coverage, reduces run-to-run variability, and avoids premature focus on isolated failure regions. Run-time analysis further shows that the modest increase in per-iteration cost is outweighed in practice the significantly reduced number of iterations and leads to competitive wall-clock performance. These findings confirm that the proposed approach is both practical and robust for surrogate-assisted reliability assessment of complex engineering structures.
Ground-motion uncertainty and variability in structural parameters lead to highly dispersed seismic demands in bridge-mounted overhead catenary systems (OCS) of high-speed railways, which complicates quantitative safety assessment and risk-informed design decisions. This study proposes a principal-component-enhanced reliability-cost optimization framework to enable efficient estimation of low failure probabilities under limited nonlinear time-history analysis budgets. Multiple ground-motion intensity measures are compressed into a small set of principal components to represent loading scenarios and to reduce multicollinearity. A distributionparameterized XGBoost surrogate is trained to predict the parameters governing the residual-displacement distribution of OCS, and an additional Edgeworth-based tail correction is introduced to refine small-probability estimates when upper-tail accuracy becomes critical. The resulting failure probabilities are coupled with a consequence-based cost model, including material consumption, post-event repair, and service downtime, yielding a bi-objective trade-off between risk reduction and total cost. Group-wise cross-validation is adopted to prevent information leakage and to assess generalization across unseen scenario groups. A representative design example shows that the schemes recommended by the proposed framework reduce failure probability by about 30% with modest additional cost, demonstrating the framework's value for reliability-based decision support and seismic risk management of bridge-mounted OCS.
This paper presents a reliability-based design (RA-based) approach for the shear resistance of reinforced concrete (RC) beams strengthened with near-surface mounted (NSM) carbon fibre-reinforced polymer (CFRP) reinforcement. This approach integrates a semi-empirical model recently developed that couples the truss analogy and the simplified modified compression field theory within a double-loop iteration scheme. Despite the higher predictive performance of this model over the existing ones, it does not include a reliability framework to allow its use in direct design practice. Therefore, in the present work, this model is adapted to a design-oriented format by introducing characteristic material properties and partial factors in line with Eurocode philosophy and fib recommendations. The statistical variability of material, geometric, and loading parameters is characterised using established probabilistic models, while uncertainties in predicting the effective CFRP strain and the overall shear resistance are quantified through goodness-of-fit testing. Design Assisted by Testing (DAT) is employed to derive characteristic and design values of the effective CFRP strain, leading to a bond-related partial factor gamma(fb) for both NSM bars and laminates. The resistance partial factor gamma(R) is then calibrated for different target reliability indices using a comprehensive reliability analysis framework employing the First-Order Reliability Method (FORM) with Importance Sampling over an extensively sampled design space and multiple load ratios, enabling assessment across various reliability levels. Both constant and variable forms of gamma(R) are derived. The latter is expressed as a function of a combined transverse reinforcement parameter and shown to improve the uniformity of reliability levels across the design domain. The proposed RA-based design model is validated against an experimental database of 114 NSM-CFRP-strengthened beams and illustrated through a detailed design example. A freely available web-based design tool is also provided to facilitate practical application of the model.
Stochastic seismic response analysis and reliability assessment are generally computationally prohibitive, as they require repeated evaluations of large-scale computational models. Recent advances in neural operators have demonstrated strong promise for learning complex physical systems, but their application to stochastic structural dynamics remains limited. To address this gap, this study proposes an attention-enhanced Fourier neural operator (AttFNO)-assisted approach tailored for stochastic seismic response analysis and reliability assessment tasks. First, AttFNO is proposed by reformulating the conventional lifting stage as a linear attention-based fusion module, enabling the joint encoding of seismic ground motion excitation and structural parameters. Owing to explicitly learning the cross-time dependencies and the coupling of structural uncertainty and excitation, the attention-based module yields more accurate prediction of seismic response trajectories. Then, AttFNO is combined with probability density evolution method in one-stage and two-stage manners, so as to enable stochastic seismic response analysis and dynamic reliability assessment, respectively. The proposed method is validated on three typical engineering structures subjected to fully non-stationary stochastic ground motions, and the comparisons are made against several baseline neural operators. Results demonstrate that the proposed approach shows forecasting superiority regarding the seismic response trajectories and their evolutionary probability density functions across diverse engineering structures and stochastic ground motion models. Moreover, the failure probability estimated by the proposed method exhibits satisfactory accuracy while significantly reducing the computational time required by the state-of-the-art workflow.
Ballasted railway bridges are subject to dynamic excitation from passing trains, which can cause excessive vibrations in the ballast bed depending on the train speed, resulting in track instability. The Eurocode EN 1990 uses vertical deck acceleration as an indicator of safety, limited it to 3.5 m/s2 for ballasted bridges. Since experimental studies show that ballast instability occurs at about 7.0 m/s2, the normative limit seems to be arbitrarily based on a safety ratio of 2.0. The present paper examines the suitability of a lower safety ratio. The proposed methodology compares the physical acceleration limit with the design acceleration calculated at a critical speed corresponding to a failure probability of 10-4. An algorithm for the efficient assessment of critical speeds based on subset simulation is introduced, together with a parametric study for its optimization. A sensitivity analysis of the random variables of ballasted bridges allows the definition of two design scenarios in accordance with the Eurocode EN 1991-2. Results from the application of the methodology to four case study bridges show that design accelerations greater than the limit defined in the Eurocode can be found in ballasted bridges within the target probability of failure, suggesting that the safety ratio may be set lower than 2.0.
Typhoons pose substantial risks to power grid networks, particularly in industrial complexes where prolonged outages can result in severe economic and operational disruptions. Conventional typhoon risk assessments for power grids often rely on peak-wind or closest-approach hazard representations, which cannot capture the temporal evolution of wind direction and direction-dependent topographic amplification during storm passage. This study proposes a probabilistic risk-assessment framework for industrial-complex power grids based on a time-stepping, direction-dependent typhoon hazard model. The framework incorporates Monte Carlo simulation, fragility models for substations and transmission towers, recovery-time distributions, and site-specific grid and industrial information to estimate outage probabilities, durations, and cascading impacts on production disruptions. It is applied to the Gwangyang and Changwon industrial complexes in South Korea to examine how terrain, grid topology, and industrial characteristics jointly shape outage risk and economic loss. The results show that conventional hazard representations can underestimate expected economic loss, particularly for more frequent moderate-intensity typhoons, because they do not adequately capture the interacting effects of winddirection changes, topographic amplification, and network configuration. The case studies also show that outage risk and economic loss are governed not by hazard intensity alone, but by the combined effects of terrain, grid topology, and industrial characteristics. These findings highlight the importance of time-resolved, directiondependent hazard representation for more reliable typhoon risk assessment in industrial-complex power grids.
Risk assessment for spatially distributed infrastructure systems under tropical cyclone-induced strong winds requires models of the spatial correlation of wind speed residuals. This study characterizes this correlation by calculating residuals and semivariograms for sixteen tropical cyclones landfalling in the United States. We use the H*wind dataset as observational wind speeds and estimates from the HAZUS hurricane model as predicted wind speeds. To analyze the impact of modeling assumptions on the correlation structure, we considered two methods for wind speed residual decomposition and used classic and robust semivariogram estimators. We further considered three correlation model functional forms, along with weighted and non-weighted methods for fitting correlation models to the semivariogram estimates. The results show substantial variability among residual correlation structures across tropical cyclones, with varying sensitivity to modeling assumptions. Finally, we developed generalized mean and intensity-based correlation models to characterize the spatial correlation of wind speed residuals, both in terms of overall structure and the influence of storm intensity.
This study develops an advanced framework for system reliability assessment of engineering and environmental systems with multiple limit-state functions, based on an adaptive Kriging strategy termed SAK-PSO-HHs (System Adaptive Kriging with Particle Swarm Optimization and Hollow Hypersphere). Unlike conventional adaptive Kriging approaches that estimate only an aggregate system failure probability, the proposed method also derives component-level reliability indices, providing critical insight into dominant failure modes and localized sources of risk. The framework integrates Particle Swarm Optimization (PSO) to remove dependence on large sampling pools, reducing computational demand, while the Hollow Hypersphere (HHs) technique constrains the search domain by excluding known safe and failure regions. Iterative hypersphere refinement combined with a line-search procedure enables efficient and accurate estimation of both system- and component-level reliability indices. Benchmark comparisons with existing adaptive Kriging methods demonstrate that SAK-PSO-HHs achieves comparable accuracy with enhanced diagnostic capability. The framework is further applied to the Bie Creek watershed in eastern Taiwan—a representative site for Nature-based Solutions (NbS). The analysis identifies Cross-Section 46 as the critical component governing system reliability, consistent with historical engineering records. This integration of probabilistic reliability analysis with NbS-based watershed management establishes a robust foundation for risk-informed and adaptive river governance.
Radioactive waste disposals are engineered with concrete structures that must resist for very long times. Concrete structures are subject to chemical corrosion, which depends on the environment and is, thus, affected by climate change in the long term. We develop an enhanced Bayesian Network with imprecise probabilities to model the corrosion process while accounting for aleatoric and epistemic uncertainties. The modelling framework is applied to carbonation and chloride corrosion processes of concrete structures in radioactive waste disposals subject to climate change, considering the uncertainties associated with environmental parameters such as temperature, carbon dioxide concentration and relative humidity. The results demonstrate the potential of the enhanced Bayesian Network modelling framework proposed.
This study proposes a chloride ingress mechanisms-informed virtual reliability assessment (CIMI-VRA) framework for time-dependent safety evaluation of reinforced concrete structures subjected to chloride-induced corrosion. The framework integrates a high-fidelity and implementable reactive transport model that incorporates key physicochemical mechanisms to simulate long-term chloride ingress. To connect transport modeling with structural analysis in a physics-based manner, an electrochemical model is introduced to quantify corrosion propagation. Additionally, a virtual modeling technique using an extended support vector regression algorithm is employed to surrogate the physical model, enabling an efficient structural safety analysis workflow. Comparisons with experimental data confirm the effectiveness of the reactive transport model, while two illustrative examples of beam and column demonstrate that the virtual modeling technique can markedly accelerate structural reliability assessment with minimal loss in accuracy. This study further investigates the effects of various chloride ingress mechanisms on structural safety, filling a critical knowledge gap in the field. It is found that porosity evolution and particularly chloride binding considerably delay chloride ingress and the need for structural repairs. Neglecting or oversimplifying these mechanisms could lead to erroneous decision-making, underscoring the importance of this framework. Overall, this study integrates concrete material science with practical structural reliability analysis, enhancing fully physics-informed safety evaluations of infrastructure systems.
We consider the problem of time-variant structural reliability estimation of randomly excited nonlinear structural dynamical systems. The governing equations of motion are expressed as a set of Ito’s stochastic differential equations. The study focuses on probability of failure estimation using Monte Carlo simulation approach equipped with variance reduction technique based on Girsanov’s transformation. The first novel aspect of the study lies in formulation of a constrained stochastic optimal control problem whose solution leads to state-dependent suboptimal Girsanov’s controls. The optimal control problem is arrived at based on the principle of minimization of the sampling variance of Girsanov’s transformation-based failure probability estimator. The second novel aspect of the study involves formulating the derived optimal control problem within the framework of Markov decision process (MDP) and solving it using a reinforcement learning strategy. In this study, we employ the widely studied reinforcement learning algorithm, deep deterministic policy gradient (DDPG), to solve the derived optimal control problem. Numerical illustrations involving time-variant reliability estimation of linear/nonlinear, single/multiple dof systems subjected to non-stationary filtered Gaussian white noise excitations are presented. Results of the proposed procedure are compared those obtained by employing existing strategy for deriving state-independent Girsanov’s controls.
Evaluating rare events remains a challenge in structural and systems reliability analysis. While line sampling has emerged as a robust alternative to traditional Monte Carlo Simulation for rare event assessment, enhancements to conventional LS methodologies are actively sought. To address this, this study proposes a novel model-agnostic framework: Adaptive Surrogate Model-Assisted Radial-Based Line Sampling (AS-RBLS). The AS-RBLS approach offers three key advantages. Firstly, through the integration of LS with the radial-based partitioning mechanism, it effectively circumvents the limitation of algorithm efficiency resulting from an excessive number of samples falling into the safe domain during the execution of the line sampling algorithm. Second, a novel ‘Sequential Sorting’ algorithm is introduced. This non-intrusive algorithm, integrated with the adaptive surrogate model, efficiently determines the adaptive hypersphere radius and identifies important directions by performing merely two distance-sorting operations on candidate samples during surrogate model updating. Finally, the radial partitioning confines the failure domain exclusively outside the hypersphere, thereby enabling highly efficient short-range line searches along important directions solely within the outer hypersphere region. The proposed AS-RBLS method is systematically benchmarked against several existing approaches through five numerical case studies evaluating rare events. The results indicate that the proposed AS - RBLS notably enhances the efficiency of rare event estimation.
