Efficient Solution for Calculation of Upcrossing Rate of Nonstationary Gaussian Process

Firouzi, Afshin; Wang, Yang; Li, Chun‐Qing

doi:10.1061/(asce)em.1943-7889.0001420

Cited by 26 publications

(15 citation statements)

References 20 publications

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“…The failure time would then be the time elapsed before the load effect passes the barrier level (first passage time). This can be expressed as:

p_{f} ((), t) = p_{f} ((), 0) + ([], 1 - p_{f} ((), 0)) ([], 1 - e^{- \int_{0}^{t} ν ((), t) italicdt}),

where p f (0) usually equals 0 at t = 0 and can be determined using time‐independent reliability methods such as FORM (First Order Reliability Method) or SORM (Second Order Reliability Method) and ν ( t ) is the out‐crossing rate and can be calculated from the Rice formula as:

ν ((), t) = \int_{R}^{\infty} ((), \overset{Z}{true} - \overset{R}{true}) f_{Z \overset{Z}{true}} ((), R . \overset{Z}{true}) d \overset{Z}{true},

where

\overset{R}{true} ((), t)

is the slope of R ( t ) with respect to time,

\overset{Z}{true} ((), t)

is the time derivative of stochastic process Z ( t ) and

f_{Z \overset{Z}{true}}

is the joint probability function for Z ( t ) and

\overset{Z}{true} ((), t)

.…”

Section: Definition Of Problemmentioning

confidence: 99%

“…The analytical results are available for calculating the out‐crossing rate only in very specific cases. An accurate and simplified model of the Rice formula for calculating the out‐crossing rate of nonstationary Gaussian processes derived was by Firouzi et al as follows:

{ν^{+}}_{new} ((), t) = ω_{0} φ ({}, \frac{R (t) - μ_{z} (t)}{σ_{z} (t)}) normalΨ ({}, normalq ((), t)),

where ϕ is the standard normal probability density function and ω 0 is the cycle rate that is dependent on the correlation length and can be calculated as:

ω_{0} = \sqrt{{\ddot{ρ}}_{ZZ} ((), t . t)},

where

{\ddot{ρ}}_{ZZ}

is the second order derivation of the ACF of Z, which is calculated as follows:

{\ddot{ρ}}_{ZZ} ((), t . t) = \frac{2}{{normalθ}_{z}^{2}},

and Ψ{q( t )} is calculated as:

normalΨ ({}, normalq ((), t)) = φ ({}, normalq ((), t)) + ({}, normalq ((), t)) normalΦ ({}, normalq ((), t)),

where ϕ {q( t )} is the standard normal probability density function and Φ{q( t )} is the cumulative standard normal probability distribution evaluated at q(t), which is calculated as:

normalq ((), t) = ({}, \frac{R (t) - μ_{z} (t)}{σ_{z} (t)}) - ({}, \dot{R} (t) - true...)

…”

Section: Definition Of Problemmentioning

confidence: 99%

“…To apply this newly derived solution, the barrier level must be deterministic as indicated in the assumptions of Firouzi et al to solve the general Rice formula. In this study, AsM is used to relieve this limitation, which uses Monte‐Carlo simulation to generate realizations of random barrier level and subsequent integration of the results to obtain mean value of outcrossing rate, that is, E[ν(t)], of random barriers …”

Section: Definition Of Problemmentioning

confidence: 99%

“…where p f (0) usually equals 0 at t = 0 and can be determined using time-independent reliability methods such as FORM (First Order Reliability Method) or SORM (Second Order Reliability Method) and ν(t) is the out-crossing rate and can be calculated from the Rice formula as 13 :…”

Section: Time Dependent Reliability Analysismentioning

confidence: 99%

“…A new closed-form solution has been recently inspired by Firouzi et al for calculating the up-crossing of a nonstationary Gaussian process. 13 The present study uses the outcrossing of the remaining steel cables cross-sectional area from a variable level in a novel approach that is modeled as the first passage probability of a nonstationary Gaussian process in an asymptotic integration (AsM) framework. 14,15 It is shown that the proposed method can accurately predict remaining service-life of sleepers.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Time dependent reliability analysis of railway sleepers subjected to corrosion

Soltanian

Firouzi

Mohammadzadeh

2018

Structural Concrete

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Prestressed concrete sleepers, as a major component of railway infrastructure, are prone to many environmental deterioration or load induced damages during their service lives. Therefore, an accurate prediction of their remaining service life is of paramount importance for proper allocation of maintenance management budgets. In this study, the remaining cross sectional area of corrosion affected prestressing wires is modeled as a nonstationary Gaussian process while other time‐invariant uncertainties, that is, geometrical properties and load affects, are modeled as random variables. Hence, the timed‐dependent reliability of these assets are calculated using the first passage theory and the corresponding outcrossing rate of the stochastic process from the safe domain. The merit of this analytical method is its efficiency and accuracy in perdition of remaining service life. The applicability of this method is shown on a B70 PSC sleeper designed based on UIC‐713 code and it is found that pitting corrosion in chloride‐laden environment extremely diminishes the service life of these assets.

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“…The failure time would then be the time elapsed before the load effect passes the barrier level (first passage time). This can be expressed as:

p_{f} ((), t) = p_{f} ((), 0) + ([], 1 - p_{f} ((), 0)) ([], 1 - e^{- \int_{0}^{t} ν ((), t) italicdt}),

ν ((), t) = \int_{R}^{\infty} ((), \overset{Z}{true} - \overset{R}{true}) f_{Z \overset{Z}{true}} ((), R . \overset{Z}{true}) d \overset{Z}{true},

where

\overset{R}{true} ((), t)

is the slope of R ( t ) with respect to time,

\overset{Z}{true} ((), t)

is the time derivative of stochastic process Z ( t ) and

f_{Z \overset{Z}{true}}

is the joint probability function for Z ( t ) and

\overset{Z}{true} ((), t)

.…”

Section: Definition Of Problemmentioning

confidence: 99%

{ν^{+}}_{new} ((), t) = ω_{0} φ ({}, \frac{R (t) - μ_{z} (t)}{σ_{z} (t)}) normalΨ ({}, normalq ((), t)),

where ϕ is the standard normal probability density function and ω 0 is the cycle rate that is dependent on the correlation length and can be calculated as:

ω_{0} = \sqrt{{\ddot{ρ}}_{ZZ} ((), t . t)},

where

{\ddot{ρ}}_{ZZ}

is the second order derivation of the ACF of Z, which is calculated as follows:

{\ddot{ρ}}_{ZZ} ((), t . t) = \frac{2}{{normalθ}_{z}^{2}},

and Ψ{q( t )} is calculated as:

normalΨ ({}, normalq ((), t)) = φ ({}, normalq ((), t)) + ({}, normalq ((), t)) normalΦ ({}, normalq ((), t)),

where ϕ {q( t )} is the standard normal probability density function and Φ{q( t )} is the cumulative standard normal probability distribution evaluated at q(t), which is calculated as: