Fatigue correlation reliability evaluation of heavy-haul railway bridges

doi:10.1007/s10483-025-3269-8

摘要/Abstract

Abstract:

The fatigue of heavy-haul railway bridges is considered a key concern due to high stress levels and cyclic loading. The evaluation of fatigue reliability is required to include factor correlations. A major challenge is presented by the construction of the cumulative distribution function (CDF) and the description of correlations between random variables. In this study, the copula function is used to analyze the fatigue failure probability of the Shuohuang heavy-haul railway bridge. A C-vine copula (CVC)-based joint probability density function (JPDF) is derived with eight correlated parameters. To enhance efficiency in small failure probability calculations, the subset simulation and most probable point (MPP) Monte Carlo importance sampling are introduced based on the Rosenblatt transform and C-vine model. Comparisons with traditional Monte Carlo methods confirm that high accuracy and efficiency are achieved. The results show that when parameter correlations are ignored, failure probability is underestimated, increasing safety risks in bridge assessments.

Key words: heavy-haul railway bridge, fatigue correlation reliability, correlated random variable, C-vine copula (CVC), subset simulation method, Monte Carlo important sampling

中图分类号:

O346.2

. [J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(7): 1295-1314.

Mingyang ZHANG, Mengcheng CHEN, Wei FANG, Kaicheng XU, Hong HUANG. Fatigue correlation reliability evaluation of heavy-haul railway bridges[J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(7): 1295-1314.

图/表 16

Copula function family	Name	$C θ (u 1, u 2, u 3, ⋯, u n)$	$θ$	$τ$
Elliptical family	Gaussian	$Φ θ (Φ 1 − 1 (u 1), Φ 2 − 1 (u 2), Φ 3 − 1 (u 3), ⋯, Φ n − 1 (u n))$	$(− 1, 1)$	$2 π arcsin θ$
Elliptical family	$t$	$Φ θ (T m − 1 (u 1), T m − 1 (u 2), T m − 1 (u 3), ⋯, T m − 1 (u n))$	$(− 1, 1)$	$2 π arcsin θ$
Archimedean family	Gumbel	$exp (− (∑ i = 1 n (− ln (u i)) θ) 1 / θ)$	$θ ⩾ 0$	$(θ − 1) / θ$
	Clayton	$max ((∑ i = 1 n u i − θ − n + 1) − 1 / θ, 0)$	$[− 1, ∞) − {0}$	$θ / (θ + 2)$
	Frank	$− 1 θ ln (1 + ∏ i = 1 n (e − θ u i − 1) (e − θ − 1) n − 1)$	$(− ∞, ∞) − {0}$	$1 − 4 θ (1 − D 1 (θ))$

Copula function	Dependence parameter $θ^$	AIC value
Gaussian	$− 0.405 9$	$− 94.321 1$
Clayton	$1.45 × 10 − 6$	2.000 4
Gumbel	1	2.008 0
Frank	$− 4.252 4$	$− 43.294 7$

Variable	Mean	Standard deviation	Distribution function
$D c$	1	0.3	Lognormal
$C$	$1.46 × 1015$	$6.58 × 1014$	Lognormal
$m$	4.382 7	0	Deterministic
$S re1$	10.136	3.179	Gamma
$S re2$	11.474	3.551	Normal
$S re3$	11.501	3.572	Normal
$N 1$	441	57	Weibull
$N 2$	732	101	Weibull
$N 3$	1 355	220	Normal

Variable	$S re1$	$S re2$	$S re3$
$S re1$	1	0.2	0.3
$S re2$	0.2	1	0.4
$S re3$	0.3	0.4	1

Copula function	Dependence parameter $θ^$	AIC value
Gaussian	0.309 0	-47.471 5
Clayton	0.500 0	-30.986 4
Gumbel	1.250 0	-38.008 5
Frank	1.860 9	-43.294 7

Copula function	Dependence parameter $θ^$	AIC value
Gaussian	0.587 8	-93.281 9
Clayton	1.333 0	-2.231 1
Gumbel	1.666 7	-68.234 8
Frank	4.161 1	-86.862 9

Copula function	Dependence parameter $θ^$	AIC value
Gaussian	0.277 5	-14.754 7
Clayton	0.307 5	-9.383 8
Gumbel	1.152 8	-11.071 0
Frank	1.683 9	-13.232 3

Calculation method	$P f$	Sample number
MCS	$2.060 0 × 10 − 5$	$1 × 107$
CVC for function subset simulation method	$2.190 0 × 10 − 5$	$6 × 103$
MPP Monte Carlo importance sampling method	$2.074 8 × 10 − 5$	$1 × 104$

Calculation method	$P f$	Sample number
MCS	$2.460 0 × 10 − 5$	$1 × 107$
CVC for function subset simulation method	$2.659 0 × 10 − 5$	$6 × 103$
MPP Monte Carlo importance sampling method	$2.540 7 × 10 − 5$	$1 × 104$

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Year	Regular train	10 000-ton train	20 000-ton train	Year	Regular train	10 000-ton train	20 000-ton train
2018	21 870	15 877	9 010	2008	24 269	–	–
2017	26 535	15 612	5 774	2007	23 945	–	–
2016	30 081	14 987	1 532	2006	20 233	–	–
2015	19 830	13 753	–	2005	16 760	–	–
2014	34 026	11 592	–	2004	13 384	–	–
2013	34 053	11 261	–	2003	9 181	–	–
2012	27 573	11 151	–	2002	5 514	–	–
2011	22 882	4 838	–	2001	2 821	–	–
2010	22 568	2 450	–	2000	956	–	–
2009	26 878	–	–