Abstract
In general,there exist two types of deterioration scenarios:continuous deterioration[1]and sudden damage[2].The continuous deterioration usually refers to corrosion-induced deterioration which could cause cumulative damage during the structural service life.The sudden damage may endanger structural safety in a short period.Ignoring the relevant deterioration may overestimate the safety and service life of engineering structures.Thus,it is significant to establish a comprehensive framework for life-cycle assessment and systematically consider different deterioration scenarios.To date,reliabilityinformed assessment is one of the most popular approaches to consider the condition and safety evolution in life-cycle engineering comprehensively.
In this paper,a novel probability density function(PDF)-informed method(PDFM)is proposed to compute the system-level time-dependent reliability of deteriorating structures suffering from various deterioration scenarios.For a general deteriorating system,its performance function g can be written as
where t is the time parameter;Θ=[Θ1,Θ2,…,Θd]T is a d-elements random input vector;and b={bk,k=1,2,…,nd}is the random vector of nd critical time instants.Two types of deterioration modes are supposed:derivable deterioration and un-derivable deterioration.For derivable deterioration,the performance function G(Θ,t,b)is derivable to time t;and underivable deterioration refers to the continuous G(Θ,t,b)but discontinuous
(Θ,t,b)at the time instant b.
In this study,the PDFs of performance function under different types of deterioration modes are obtained to compute the time-dependent failure probability pf(t).For derivable deterioration,the PDF of performance function pYΘB(y,θ,b,t)can be solved by using existing algorithms of PDFM[3].For other deterioration scenarios,the“two-step translation method”is developed to calculate the pYΘB(y,θ,b,t).Applying the point evolution method[4],nsel selected points are acquired and the principal processes of“two-step translation method”are listed as following.
(1)For each representative pointθa,a=1,2,…,nsel,the critical time instants
can be identified and surrogate model Y(t)=G(θa,t,ba)is translated to
withΔg(t).
where
are the time instants just before and after ba,s;and
and
are G(θ,
,ba)and G(θ,
(2)Applying traditional PDFM then obtaining
t,b)is translated to pYΘB(y,θ,b,t)withΔg(t)at the inverse direction.
For illustrative purpose,a stochastic deteriorating system with one single sudden damage of random occurred time instant tdrop is assumed.
where both f0 and tdrop are Gauss random variable[f0～N(20,1)and tdrop～N(25,2)].
Figure 1 compares the differences between the two-step translation method and traditional PDFM by the PDFs after 23,25,and 27 years.It could be noticed that the PDFs obtained by the proposed method are smoother than traditional ones,which demonstrates the efficiency of the proposed method.
Figure 1 Comparison of PDFs at 23,25 and 27 years
Besides,supposing a simply supported beam with a cover thickness of 25 mm with a dimension of 6 000 mm×200 mm×500 mm which is subjected to uniform load.More detailed information refers to reference[5].Considering the ductile and brittle failure of an RC beam,the results of the PDF are presented in Figure 2.Figure 3 compares the reliability index obtained by PDFM and 1 million trials of Monte-Carlo simulation(MCS),which demonstrates the proposed method also suit for durability engineering.
Figure 2 PDF contour of RC beam
Figure 3 Comparison of time-dependent reliability index
Overall speaking,the PDFM based reliability analysis framework is developed and successfully applied in the general form of deterioration system.Numerical cases demonstrated that the proposed method could overcome the shortage of existing PDFM and maintain high computational accuracy comparing with MCS.