Ana Fernandez-Navamuel, Martin A. Díaz Viera, Matteo Croci
{"title":"A copula-based variational autoencoder for uncertainty quantification in inverse structural dynamics: Application to damage identification in an offshore wind turbine","authors":"Ana Fernandez-Navamuel, Martin A. Díaz Viera, Matteo Croci","doi":"10.1016/j.probengmech.2026.104054","DOIUrl":"10.1016/j.probengmech.2026.104054","url":null,"abstract":"<div><div>Identifying damage from limited sensor data is a challenging inverse problem due to solution multiplicity, which needs a probabilistic inference approach to quantify underlying uncertainty. Variational autoencoders (VAEs) are a powerful class of inference methods that have been proven effective for structural damage identification problems. However, all VAEs used in this context rely on simple Gaussian approximations, which cannot capture complex statistical dependencies among damage features. Gaussian mixture methods do not scale to high-dimensional problems: full-covariance models become computationally prohibitive, and diagonal-covariance models ignore correlations between damage features. To overcome these limitations, this work proposes a novel copula-based VAE architecture that enables the representation of highly correlated posterior distributions by decoupling the modeling of marginal and joint effects. We apply this methodology to the structural health monitoring of floating offshore wind turbine (FOWT) mooring systems, inferring the occurrence and severity of biofouling and anchoring damage from short-term platform rotation signals. We provide a comprehensive comparison of the copula and Gaussian mixture VAE approaches. Numerical experiments conducted on high-fidelity 2D synthetic datasets demonstrate that the copula VAE provides a computationally efficient solution that scales to higher dimensions while controlling parameter growth. Indeed, the proposed technique achieves the same performance with significantly fewer parameters than the Gaussian mixture alternatives, whose parameter counts grow prohibitively with the dimensionality of the latent space. Overall, these results suggest that copula-based VAEs provide a robust framework for uncertainty-aware damage detection in FOWT mooring systems.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"86 ","pages":"Article 104054"},"PeriodicalIF":4.1,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149200962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Na Meng, Rong Guo, Qian Wang, Jinzhong Ma, Junlin Li
{"title":"Probability-based quantification of high-risk regions for noise-induced tipping in panel structures","authors":"Na Meng, Rong Guo, Qian Wang, Jinzhong Ma, Junlin Li","doi":"10.1016/j.probengmech.2026.104050","DOIUrl":"10.1016/j.probengmech.2026.104050","url":null,"abstract":"<div><div>Panel structures operating in aerospace and transportation systems are frequently exposed to multiple interacting excitations, which may induce tipping from low-amplitude oscillations to catastrophic high-amplitude oscillations, thereby threatening structural safety and service life. To address this issue, this study investigates the tipping of panel structures under combined Gaussian colored noise and harmonic excitations. First, the deterministic panel model is shown to exhibit bistability, where low-amplitude and high-amplitude oscillation states coexist within specific ranges of excitation frequency and amplitude. Subsequently, the effects of noise intensity and correlation time on the tipping are analyzed. The results reveal that increasing the noise intensity or decreasing the correlation time significantly facilitates the occurrence of tipping and advances the onset of catastrophic high-amplitude oscillations. To quantify the risk associated with such transitions, a probability-based framework is proposed, and the concept of a high-risk region is introduced. This framework enables the identification of parameter regions in which noise-induced tipping is highly likely to occur. Furthermore, the dependence of the high-risk regions on both system parameters and noise characteristics is analyzed in detail. These findings may provide valuable insights for preventing noise-induced tipping in panel structures and improving their structural safety and reliability.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"86 ","pages":"Article 104050"},"PeriodicalIF":4.1,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149155369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Closed-form bias of mean-based degradation thresholds under joint bounded parameter uncertainty","authors":"Aaron Lim, Kyeong Ryong Kim, Kyo-Chan Koo","doi":"10.1016/j.probengmech.2026.104044","DOIUrl":"10.1016/j.probengmech.2026.104044","url":null,"abstract":"<div><div>Replacement thresholds for wearing components are routinely set by inserting mean parameter values into a multiplicative degradation law of Archard type and reading a critical time at the wear limit. When the wear coefficient and the load are known only as bounded intervals – the prevailing situation in industrial practice – this convention silently commits the operator to a threshold bias and an exceedance risk that have not been quantified, and that no single-parameter analysis can reveal. Building on the classical law of a product of independent uniform variables – for which no novelty is claimed – we derive a closed-form quantile function (elementary on an interior probability window, single-branch Lambert-<span><math><mi>W</mi></math></span> outside it) and, from it, the bias surface of the mean-based threshold over the square of floor ratios. The bias is strictly positive, vanishes identically whenever only one parameter is uncertain – a pure interaction effect of joint uncertainty – and on its elementary window depends only on the larger floor ratio. Recast on the probability axis, the mean rule is a quantile rule in disguise whose implicit exceedance risk lies in the universal band <span><math><mrow><mo>[</mo><mn>40.3</mn><mtext>%</mtext><mo>,</mo><mn>50</mn><mtext>%</mtext><mo>)</mo></mrow></math></span> for every floor configuration; interval refinement shrinks the timing error but cannot repair the risk semantics. Positivity rests on a recent median–product inequality for positive symmetric factors, of which the bias surface is the exact quantification in the bounded-uniform case. The closed forms are benchmarked against a moment-matched lognormal, alternative bounded priors, and polynomial-chaos and maximum-entropy propagation; the risk conclusion survives every alternative, and any first-order propagation erases the bias entirely. The framework is demonstrated on coal-pulverizer roll tires, where the load-schedule floor ratio <span><math><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></math></span> gives the exact bias <span><math><mrow><mo>ln</mo><mn>3</mn><mo>−</mo><mn>1</mn></mrow></math></span>, an implicit risk of 45%, and a ten-percent risk policy that halves the mean-based replacement interval, with all conclusions stable over the admissible floor-ratio bracket <span><math><mrow><mo>[</mo><mn>0.25,0.40</mn><mo>]</mo></mrow></math></span>. The study contributes to degradation modeling an exact, assumption-transparent decision layer – quantile function, bias surface, and risk band – in which every statement depends only on dimensionless floor ratios and therefore transfers across equipment, materials, and units. For engineering uncertainty quantification more broadly, the closed forms supply a rare exact benchmark against which surrogate and moment-based propagation methods can be measured.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"86 ","pages":"Article 104044"},"PeriodicalIF":4.1,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149155311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Joint probability density function-driven simulation of multivariate non-Gaussian processes","authors":"Megan McCullough Hart, Liang Hu, Ahsan Kareem","doi":"10.1016/j.probengmech.2026.103957","DOIUrl":"10.1016/j.probengmech.2026.103957","url":null,"abstract":"<div><div>This study proposes a data-driven simulation method for multivariate non-Gaussian processes that reproduces not only spectral characteristics but also the joint probability density function (JPDF) of a given target time series. The method comprises three sequential steps that update the power spectral density (PSD), the marginal probabilistic density function (PDF), and the JPDF iteratively. The PSD correction step generates a time series that replicates the template’s Fourier amplitudes and phase differences, whose marginal PDF is then enforced via rank matching. More importantly, non-rigid point set registration based on the Coherent Point Drift (CPD) algorithm is used to align the marginal-PDF-corrected time series with the target’s JPDF. Repeating these steps iteratively yields simulations that jointly match the PSD, marginal CDF, and, importantly, the JPDF of the target time series. The effectiveness of the method is illustrated by a numerical example that simulates the wind pressure field on a low-rise building.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 103957"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148568035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Salvatore Russotto, Mario Di Paola, Antonina Pirrotta
{"title":"An innovative step-by-step stochastic linearization procedure for solving deterministic nonlinear differential equations","authors":"Salvatore Russotto, Mario Di Paola, Antonina Pirrotta","doi":"10.1016/j.probengmech.2026.103991","DOIUrl":"10.1016/j.probengmech.2026.103991","url":null,"abstract":"<div><div>Deterministic nonlinear differential equations are widely used in engineering to model the response of different mechanical systems. In most cases, exact analytical solutions are not available, and approximate techniques are therefore required. Deterministic linearization methods constitute a common class of approximate techniques. However, the development of methods that combine high accuracy with computational efficiency remains a challenging task.</div><div>In this paper, an innovative procedure based on a stochastic framework is proposed for solving deterministic nonlinear differential equations. The key idea is to replace the initial condition by a Gaussian random variable with small variance and mean equal to the deterministic initial value. This transformation turns the deterministic response into a stochastic process whose probability density function evolves according to the diffusion-free Fokker-Planck-Kolmogorov equation. Starting from this formulation, the evolution equation of the first-order statistics is derived, and an innovative numerical method, based on a step-by-step stochastic linearization, is introduced. The method is based on a subdivision of the mean value of the nonlinear term in different steps, so that within each step a closed-form linearized solution can be obtained together with the corresponding time instant. It is shown that, for sufficiently small variance of the initial condition, the mean value of the stochastic process coalesces with the solution of the original deterministic nonlinear equation.</div><div>The reliability of the proposed procedure is assessed through numerical applications to Kelvin-Voigt-type models with one and two nonlinear springs. The results demonstrate excellent agreement with both direct Runge-Kutta integration and quasi-linearization method, thereby confirming their accuracy and effectiveness. Additionally, it is worth highlighting the enhanced performance of the proposed procedure, as it strongly reduces computational time.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 103991"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148568340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A probabilistic measure of SVE-to-RVE convergence","authors":"Martin Ostoja-Starzewski","doi":"10.1016/j.probengmech.2026.103979","DOIUrl":"10.1016/j.probengmech.2026.103979","url":null,"abstract":"<div><div>Heterogeneous materials such as composites, porous media, and polycrystalline solids exhibit effective constitutive responses that depend on both microstructure and boundary conditions when the observation domain is finite. Classical homogenization theory predicts that Dirichlet and Neumann boundary value problems yield identical effective tensors only in the representative volume element (RVE) limit, whereas for smaller domains the two responses generally differ.</div><div>In this work we introduce a probabilistic framework for characterizing this discrepancy. The effective tensor is modeled as a random element of the cone of symmetric positive-definite matrices and its probability distribution is derived from the principle of maximum entropy. The resulting model belongs to the matrix exponential family and, in an alternative maximum-entropy formulation, takes the form of a matrix-gamma-type distribution on the cone of symmetric positive definite (SPD) matrices.</div><div>Using this probabilistic description we define a coordinate-invariant anisotropy measure based on the Hellinger distance between the probability distributions associated with Dirichlet stiffness and the inverse Neumann tensor. The resulting expression admits a compact representation through a matrix Bessel kernel defined on the SPD manifold. In the scalar isotropic case the formula reduces to a closed form involving modified Bessel functions.</div><div>The analysis reveals a scaling law linking the anisotropy measure to the size of the computational domain. This relation leads to a statistical criterion for estimating representative volume element sizes based on the convergence of Dirichlet and Neumann response distributions.</div><div>The proposed framework connects ideas from homogenization theory, matrix-valued probability distributions, and information geometry, and provides a new perspective for analyzing finite-size effects and anisotropy in heterogeneous materials.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 103979"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148568396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zifeng Huang, Konstantin M. Zuev, Yong Xia, Michael Beer
{"title":"Upper generalization bounds for neural oscillators","authors":"Zifeng Huang, Konstantin M. Zuev, Yong Xia, Michael Beer","doi":"10.1016/j.probengmech.2026.103977","DOIUrl":"10.1016/j.probengmech.2026.103977","url":null,"abstract":"<div><div>Neural oscillators that originate from second-order ordinary differential equations (ODEs) have shown competitive performance in learning mappings between dynamic loads and responses of complex nonlinear structural systems. Despite this empirical success, theoretically quantifying the generalization capacities of their neural network architectures remains undeveloped. In this study, the neural oscillator consisting of a second-order ODE followed by a multilayer perceptron (MLP) is considered. Its upper probably approximately correct (PAC) generalization bound for approximating causal and uniformly continuous operators between continuous temporal function spaces and that for approximating the uniformly asymptotically incrementally stable second-order dynamical systems are derived by leveraging the Rademacher complexity framework. These bounds are further extended to the squared Wasserstein-1 distances between the probability measures of quantities of interest calculated from target causal operators and the corresponding learned neural oscillators. The theoretical results show that the estimation errors grow polynomially with respect to both MLP sizes and the time length, thereby avoiding the curse of parametric complexity. Furthermore, the derived error bounds demonstrate that constraining the Lipschitz constants of the MLPs via loss function regularization can improve the generalization ability of the neural oscillator. Numerical studies considering a Bouc–Wen nonlinear system under stochastic seismic excitation validate the theoretically predicted power laws of the estimation errors with respect to the sample size and time length, and confirm the effectiveness of constraining MLPs’ matrix and vector norms in enhancing the performance of the neural oscillator under limited training data.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 103977"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148568400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"In-sample versus ensemble optimisation of tuned mass dampers for structures with uncertainties","authors":"Sudip Chowdhury, Sondipon Adhikari","doi":"10.1016/j.probengmech.2026.103982","DOIUrl":"10.1016/j.probengmech.2026.103982","url":null,"abstract":"<div><div>Classical tuned mass damper (TMD) design assumes precisely known structural properties, but no systematic framework exists when the primary structure is uncertain due to material variability, construction tolerances, or environmental factors. This paper introduces a conceptual dichotomy in robust TMD design: in-sample optimisation, which determines parameters conditionally for individual structural realisations, versus ensemble optimisation, which seeks distributional robustness across the entire population of uncertain structures. This represents a fundamental departure from existing approaches, which treat uncertainty as a post-design consideration rather than embedding it within the optimisation formulation itself. A probabilistic framework is developed in which structural natural frequency is modelled as a random variable, and dynamic responses are characterised through closed-form analytical expressions for mean and variance, validated against Monte Carlo simulation. The in-sample approach applies fixed-point theory to each realisation and aggregates the resulting parameter distributions. The ensemble approach enforces equal-mean constraints on fixed-point frequencies across the uncertainty distribution and minimises response variance. Comparative analyses in both frequency and time domains demonstrate that while conventional deterministic TMDs remain sensitive to detuning, both proposed strategies effectively suppress resonance amplification. The ensemble design achieves superior performance, reducing maximum structural displacement by 21.50% relative to conventional TMD, compared to 19.32% for in-sample design. Beyond parameter optimisation, this work establishes in-sample versus ensemble as a transferable design philosophy for vibration control under uncertainty, with direct implications for adaptive versus mass-produced damper systems in engineering practice.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 103982"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148568401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Y.S. Zhao, Xuhui He, K. Shi, Zhengliang Xiang, Haiquan Jing
{"title":"A Bayesian-inference physics-informed extreme learning machine for suspension bridge buffeting analysis with uncertainty quantification","authors":"Y.S. Zhao, Xuhui He, K. Shi, Zhengliang Xiang, Haiquan Jing","doi":"10.1016/j.probengmech.2026.103993","DOIUrl":"10.1016/j.probengmech.2026.103993","url":null,"abstract":"<div><div>Long-span suspension bridges are highly susceptible to wind-induced buffeting, whereby efficient and reliable response analysis is indispensable for design evaluation and operational decision-making. However, classical analytical methods are not well suited for probabilistic inference and real-time prediction, and existing deep learning-based methods are hampered by inadequate physical interpretability and over-reliance on data-driven paradigms. To address such limitations, this study proposes a Bayesian-inference physics-informed extreme learning machine (BPIELM) framework for buffeting analysis of long-span suspension bridges. The proposed method integrates hybrid basis functions composed of Fourier and tanh functions, and embeds the governing buffeting dynamic equations into an extreme learning machine to achieve physics-informed modeling for buffeting time-history response prediction. A Bayesian extension is further adopted to derive the posterior distribution of the output weights, enabling uncertainty quantification of buffeting responses via posterior means and confidence intervals. The proposed framework is validated by comparing buffeting responses with finite element analysis results under different mean wind speeds, wind-speed noise levels, and turbulence intensities. The results show that the proposed method can accurately reproduce buffeting response time histories and yield reliable uncertainty intervals associated with model-parameter variability. Moreover, the method exhibits strong robustness under variable wind conditions and achieves substantially higher computational efficiency than conventional finite element analysis.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 103993"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148635365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mingjie Dai, Xiaohui Tan, Shanwei Liu, Zhihao Jiang, Zhitang Lu
{"title":"Effect of Karhunen-Loève Eigensolution accuracy on the reliability of foundation engineering","authors":"Mingjie Dai, Xiaohui Tan, Shanwei Liu, Zhihao Jiang, Zhitang Lu","doi":"10.1016/j.probengmech.2026.104007","DOIUrl":"10.1016/j.probengmech.2026.104007","url":null,"abstract":"<div><div>In the Karhunen–Loève (KL) series expansion framework for random field discretization, computational errors in common KL methods (e.g., the orthogonal series expansion (OSE) and Jacobi–Legendre–Galerkin (JLG) methods) arise not only from the truncation error due to the number of retained terms (<em>M</em>), but also from eigensolution accuracy related to the number of Gauss integration points (<em>N</em>1D GP) and basis function terms (<em>N</em><sup>1D</sup>). The OSE method with a full tensor-product construction and the JLG method with total-order truncation were employed for separable and nonseparable autocorrelation functions, respectively, thereby revealing the decoupled effects of the number of <em>N</em>1D GP and the number of <em>N</em><sup>1D</sup> on the eigenvalues and eigenfunctions. The results indicate that the global covariance error may not fully capture local reconstruction errors within the physical domain. For <em>N</em><sup>1D</sup> ≥ <em>M</em> + 2, <em>N</em>1D GP primarily influences eigenvalues (<em>λ</em>), while <em>N</em><sup>1D</sup> dictates fluctuations in eigenfunctions (<em>f</em>(<em>x</em>)). Accordingly, parameter selection criteria are established for one-dimensional and multi-dimensional random fields, respectively. Engineering case studies indicate that eigensolution accuracy affects the range of variation in load-settlement curves for shallow foundations and that the tilting failure mode of a pile group is particularly sensitive to it. Insufficient accuracy of the eigensolution causes drifts at the most unfavorable failure points and shifts in parameter sensitivity rankings within pile group systems. Furthermore, the convergence pattern of eigensolution errors directly maps onto the convergence behavior of the reliability assessment for pile group systems. This research provides a vital theoretical basis and accuracy parameter selection benchmarks for geotechnical reliability analysis within the KL framework.</div></div>","PeriodicalId":54583,"journal":{"name":"Probabilistic Engineering Mechanics","volume":"85 ","pages":"Article 104007"},"PeriodicalIF":4.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148737201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}