Special Issue Article
Dynamic reliability prediction of bridge member based on Bayesian dynamic nonlinear model and monitored data
Advances in Mechanical Engineering 2016, Vol. 8(11) 1–10 Ó The Author(s) 2016 DOI: 10.1177/1687814016679313 aime.sagepub.com
Xueping Fan1,2 and Yuefei Liu1,2
Abstract Bridge monitoring systems provide a huge number of stress data used for reliability prediction. In this article, the dynamic measure of structural stress over time is considered as a time series, and considering the limitation of the existing Bayesian dynamic linear models only applied for short-term performance prediction, Bayesian dynamic nonlinear models are introduced. With the monitored stress data, the quadratic function is used to build the Bayesian dynamic nonlinear model. And two methods are proposed to handle with the built Bayesian dynamic nonlinear model and the corresponding probability recursion processes. One method is to transform the built Bayesian dynamic nonlinear model into Bayesian dynamic linear model with Taylor series expansion technique; then the corresponding probability recursion processes are completed based on the transformed Bayesian dynamic linear model. The other one is to directly handle with the built Bayesian dynamic nonlinear model and the corresponding probability recursion processes with Markov chain Monte Carlo simulation method. Based on the predicted stress information (means and variances) of the above two methods, first-order second moment method is adopted to predict the structural reliability indices. Finally, an actual engineering is provided to illustrate the application and feasibility of the above two methods. Keywords Bridges, monitored data, reliability prediction, Bayesian dynamic nonlinear models, Markov chain Monte Carlo simulation, first-order second moment method
Date received: 14 June 2016; accepted: 20 October 2016 Academic Editor: Jun Li
Introduction Bridges subjected to time-dependent loading and strength deterioration processes will experience the changes due to internal and external factors. Some of these changes would not only affect the serviceability and the ultimate capacity of structures1 but also make serious impacts on the remaining reliability of the existing bridges. Therefore, it is of great importance to know the dynamic reliability of the critical bridge members.2 Bridge monitoring systems provide a huge amount of the monitored data, such as stress, strain, and deflection. Proper handling of the continuously provided monitored data is one of the main difficulties in the
field of structural health monitoring (SHM).3,4 A sound number of studies about SHM information are mainly focused on the modal parameter identification, structural damage detection technology, data modeling, and so on.5–7 For research of the bridge reliability 1
Key Laboratory of Mechanics on Disaster and Environment in Western China, The Ministry of Education, Lanzhou University, Lanzhou, China 2 School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, China Corresponding author: Xueping Fan, Key Laboratory of Mechanics on Disaster and Environment in Western China, The Ministry of Education, Lanzhou University, 730000 Lanzhou, China. Email:
[email protected]
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).
2 prediction and assessment, there are some achievements, such as the reliability assessment of long span truss bridge,2 the reliability updating of a concrete bridge structure based on condition assessment with uncertainties,8 the structural performance prediction based on the monitoring data,9,10 reliability assessment of masonry arch bridges,11 structural real-time reliability prediction based on combinational Bayesian dynamic linear model (BDLM),12 and Bayesian forecasting of structural bending capacity of aging bridges based on dynamic linear model.13 However, based on the bridges’ health monitored data, the research on dynamically and reasonably predicting and assessing the structural reliability should be further studied. In this article, considering the limitation of BDLM, Bayesian dynamic nonlinear model (BDNM) is introduced to combine the monitored stress with bridge member’s reliability prediction. With monitored stress data and the quadratic function, the BDNM is first built, and then two methods are proposed to deal with the built BDNM and the corresponding probability recursion processes. One method is to transform the built BDNM into BDLM with Taylor series expansion technique and then the probability recursion processes are completed based on the transformed BDLM; the other one is to directly handle with the built BDNM and the probability recursion processes with Markov chain Monte Carlo (MCMC) simulation method. Based on the predicted stress information (means and variances) of the above two methods, the first-order second moment (FOSM) method is adopted to predict the structural reliability indices. Finally, an actual bridge is provided to illustrate the application and feasibility of the above two methods.
BDNM BDNM can incorporate all the useful monitored information into the model to update the prediction.14 They comprise a state equation, a monitored equation, and the priori information, where the state equation showing the changes in the system with time and reflecting the inner dynamic changes in the system and random disturbances is nonlinear. The observation equation expressing the relationship between the monitored data and the current state parameters of the system is linear. For the long-term prediction of the monitored data, the BDNMs have better prediction precision than BDLM which is mainly for short-term prediction of the monitored data, so the BDNMs are adopted to predict the performance data (stress and reliability indices) of the bridge in this article.
The built BDNM based on the quadratic function With long-term health monitoring data, the fitted quadratic function shown in equation (1) can be approximately
Advances in Mechanical Engineering and reasonably adopted to build the state equation. And then based on the built state equation and the monitored equation which is shown in equation (13), the BDNMs are built. State equations based on the quadratic function. For the long-term monitored data, the h(t) (fitted quadratic function) is commonly well-fitted to show the changing trend of the monitored data; therefore, the h(t) can be approximately and reasonably considered as the changing curves of the state variables, which are expressed as h(t) = at2 + bt + c
ð1Þ
ut = h(t) = at2 + bt + c
ð2Þ
ut + 1 = h(t + 1) = a(t + 1)2 + b(t + 1) + c
ð3Þ
where a, b, and c are all the regression coefficients of the monitored data. t is the time, the unit of which is day. ut is the state data at time t. Based on equation (2), the following equation can be obtained at2 + bt + c ut = 0
ð4Þ
The solutions of equation (4) are t21
=
b 6
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4a(c ut ) 2a
ð5Þ
Based on equation (3), the following equation can be obtained at2 + (2a + b)t + a + b + c ut + 1 = 0
ð6Þ
The solutions of equation (6) are
t21 =
b 2a 6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2a + b)2 4a(a + b + c ut + 1 ) 2a ð7Þ
The transformed state equation based on quadratic function. With equations (5) and (7), for t 0, considering the consistency and equality of the solutions, there are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4a(c ut ) = 2a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2a 6 (2a + b)2 4a(a + b + c ut + 1 )
b 6
2a
ð8Þ
After equation (8) is simplified, the transferred approximate state equation can be obtained as follows: If a.0, then the state equations are ut + 1 = ut + a +
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac + 4aut ,
t
b 2a
ð9Þ
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and ut + 1 = ut + a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac + 4aut ,
t.
b 2a
ð10Þ
t
b 2a
ð11Þ
If a\0, then the state equations are ut + 1 = ut + a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac + 4aut ,
where a, b, and c are constants; ut is the state parameter indicating the level of the monitored data at time t. For BDNM, vt + 1 and vt + 1 are, respectively, the monitored errors and state errors. It is assumed that error sequences vt + 1 and vt + 1 are internally independent, mutually independent, and independent of (ut jDt ). With equations (13)–(18), the updating relationship between the monitored data and state parameters can be obtained as
and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ut + 1 = ut + a + b2 4ac + 4aut ,
b t. 2a
ð12Þ
Equations (9)–(12) show that there exist two different cases for the approximate state equations, namely, equations (9) and (10) and equations (11) and (12). The case, which is more reasonable, accurate, and applicable, depends on the regressive coefficients (a, b, and c) and state parameter ut . The built BDNM. Based on section ‘‘The transformed state equation based on quadratic function,’’ the built BDNMs in this article are as follows: The monitored equation is yt + 1 = u t + 1 + n t + 1 ,
nt + 1 ;N½0, Vt + 1 , ð13Þ
t = 1, 2, . . . , T If a.0, then the state equations are ut + 1 = ut + a +
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac + 4aut + vt + 1 ,
vt + 1 ;N½0, Wt + 1 ,
b t 2a
ð14Þ
and ut + 1 = ut + a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac + 4aut + vt + 1 ,
vt + 1 ;N½0, Wt + 1 ,
t.
b 2a
ð15Þ
If a\0, then the state equations are ut + 1 = ut + a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac + 4aut + vt + 1 ,
vt + 1 ;N½0, Wt + 1 ,
t
b 2a
ð16Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac + 4aut + vt + 1 ,
vt + 1 ;N½0, Wt + 1 ,
t.
b 2a
It can be seen from equation (19) that the modeling processes of BDNM can be divided into two key steps. The first step is to obtain the priori probability distribution function (PDF) of ut + 1 at time t + 1 based on the state equation and the posteriori PDF of ut at time t; the second step is to obtain the posteriori PDF of ut + 1 at time t + 1 based on the priori PDF of state parameters at time t + 1 and monitored data yt + 1 at time t + 1. In this article, two methods are provided to deal with the built BDNM and the corresponding probability recursion processes. One method is to transform the built BDNM into BDLM with Taylor series expansion technique and the other one is directly to handle with the built BDNM and the probability recursion processes with MCMC simulation method.
Transformed BDLM and the corresponding probability recursion processes based on the built BDNM with Taylor series expansion technique In this section, the Taylor series expansion technique is adopted to transfer the built BDNM into BDLM. Usually, quadratic function of the monitored extreme stress data can better fit the changing trend of the monitored data for long-term prediction, and the trend data mean the state data of the monitored data. The fitted quadratic function can be transformed into linear state equation with Taylor series expansion technique: 1. Linearization of nonlinear state equations
ð17Þ
ut + 1 = E0 + Gt (ut mt ) + 0(ut mt )
ð20Þ
If a . 0, then
The initial state information (ut jDt );N½mt , Ct
ð19Þ
With Taylor series expansion technology, the nonlinear equations shown in equations (14)–(17) are transformed into approximate linear equation, namely
and ut + 1 = ut + a +
(yt + 1 jut + 1 );N½ut + 1 , Vt + 1 , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N½ut + a 6 b2 4ac + 4aut , Wt + 1
ð18Þ
E0 = mt + a +
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac + 4amt ,
t
b 2a
ð21Þ
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Advances in Mechanical Engineering
and E0 = mt + a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac + 4amt ,
t.
b 2a
ð22Þ
∂ut + 1 ∂gt (ut ) = = ut = mt ∂ut ∂ut ut = mt 2a 2a 1 + pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ut = mt = 1 + pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 2 b 4ac + 4aut b 4ac + 4amt b t ð23Þ 2a Gt =
The monitored equation is approximately similar to equation (13). The state equation ut + 1 = Gt ut + E0 Gt mt + vt + 1 ,
vt + 1 ;N½0, Wt + 1 ,
t = 1, 2, . . . , T
ð30Þ
The initial information (ut jDt );N½mt , Ct
ð31Þ
and ∂ut + 1 ∂gt (ut ) = = Gt = ut = mt ∂ut ∂ut ut = mt 2a 2a 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ut = mt = 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 2 b 4ac + 4aut b 4ac + 4amt b t. ð24Þ 2a If a \ 0, then pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E0 = mt + a b2 4ac + 4amt ,
b 2a
t
ð25Þ
3. The probability recursion processes of transformed BDLM BDLMs are applicable to the prediction of the future state parameters.14 With Bayesian method, the recursively updating processes of the transformed BDLM12,15 are as follows: (a) The posteriori distribution at time t For mean mt and variance matrix Ct, i = 1, 2, ., s, there is (ut jDt );N½mt , Ct
and E 0 = mt + a +
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac + 4amt ,
t.
b 2a
ð26Þ
ð27Þ
∂ut + 1 ∂gt (ut ) = ut = mt ∂ut ∂ut ut = mt 2a = 1 + pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ut = mt b2 4ac + 4aut 2a b = 1 + pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , t. 2 2a b 4ac + 4amt
ð28Þ
Gt =
2. Transformed BDLM The transformed BDLMs are as follows:
ð34Þ
where ft + 1 = E(yt + 1 jDt ) = at + 1 and Qt + 1 = var(yt + 1 jDt ) = Rt + 1 + Vt + 1 . According to the definition of highest posterior density (HPD) region,14 the predicted interval of the monitored data with a 95% confidential interval at time t + 1 is pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ½ft + 1 1:645 Qt + 1 , ft + 1 + 1:645 Qt + 1
where E0 and Gt are both constants; mt is the mean value of ut at time t. Furthermore, by combining equations (20)–(28), the transformed approximate linear state equation is
ð33Þ
where at + 1 = E0 and Rt + 1 = Gt + 1 Ct Gt0 + 1 + Wt + 1 . (c) One-step prediction distribution at time t + 1 (yt + 1 jDt );N½ft + 1 , Qt + 1
and
ut + 1 = Gt ut + E0 Gt mt + vt + 1
(b) The priori distribution at time t + 1 (ut + 1 jDt );N½at + 1 , Rt + 1
∂ut + 1 ∂gt (ut ) = ut = mt ∂ut ∂ut ut = mt 2a = 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ut = mt b2 4ac + 4aut 2a b = 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , t 2 2a b 4ac + 4amt
Gt =
ð32Þ
ð35Þ
pffiffiffiffiffiffiffiffiffiffiffi where ft + 1 1:645 Qt + 1 is the lower limit value pffiffiffiffiffiffiffiffiffiffiffi and ft + 1 + 1:645 Qt + 1 is the upper limit value. (d) The posteriori distribution at time t + 1 (ut + 1 jDt + 1 );N½mt + 1 , Ct + 1
ð36Þ
ð29Þ Ct + 1 = Rt + 1 where mt + 1 = at + 1 + At + 1 et + 1 , At + 1 Qt + 1 A0t + 1 , At + 1 = Rt + 1 =Qt + 1 , et + 1 = yt + 1 ft + 1 (one-step prediction error), At + 1 is the adaptive coefficient, and A0t + 1 is the transpose of At + 1 .
Fan and Liu
The probability recursion processes based on the built BDNM with MCMC simulation method The built BDNMs are shown in equations (13)–(18). With the monitored stress data, the probability function of the initial state parameters can be obtained with section ‘‘Determination of the main probability parameters of BDNM.’’ For example, the PDF of ut at time t is p(ut jDt ). With MCMC simulation method,16 a group of convergent samples from p(ut jDt ) can be (2) (3) (n) obtained as At = ½u(1) t , ut , ut , . . . , ut . At can be converted into the priori sample of ut + 1 jDt : (2) (3) (n) Bt + 1 = ½j(1) through t + 1 , jt + 1 , j t + 1 , . . . , j t + 1 Algorithm 1. Algorithm 1. 1. With MCMC simulation method, draw a group (2) (3) of convergent sample fv(1) t + 1 , vt + 1 , vt + 1 , . . . , v(n) t + 1 g from vt + 1 ;N ½0, Wt + 1 . (i) 2. For every sample u(i) t , 1 i n, let j t + 1 = gt + 1 (u(i) ) + v(i) t + 1 based on equation (14), (15), t (16), or (17), and then convergent sample Bt + 1 of priori probability distribution for state variable ut + 1 at time t + 1 can be obtained as (2) (3) (n) Bt + 1 = ½j(1) t + 1 , jt + 1 , j t + 1 , . . . , j t + 1 . For Bt + 1 , when monitored data yt + 1 at time t + 1 are obtained, with the MCMC simulation method, Bt + 1 can be converted into convergent sample At + 1 = (2) (3) (n) ½u(1) t + 1 , ut + 1 , ut + 1 , . . . , ut + 1 of posteriori PDF for state variable ut + 1 at time t + 1; the detailed steps are shown in Algorithm 2. Algorithm 2. For every sample j(i) t + 1 , 1 i n, 1. 2. 3.
4.
Let x0 = j(i) t + 1. Draw a sample z1 from f (yt + 1 jx0 ), namely, z1 ;f (yt + 1 jx0 ). x1 = z1 is accepted with the probability a(x0 , z1 ) = min (1, ((p(z1 jDt ))=(p(x0 jDt )))), or else z , if u r x1 = x0 . Namely, x1 = 1 , where x0 , if u r u follows uniform distribution: U[0, 1]. The same procedure is repeated M times, then let u(i) t + 1 = xM .
Steps (1)–(4) are repeated N times; convergent sam(2) (3) (n) ple At + 1 = ½u(1) t + 1 , ut + 1 , ut + 1 , . . . , ut + 1 of posteriori probability distribution for state variable ut + 1 at time t + 1 can be obtained. The distribution parameters of one-step prediction distribution are
5
E(yt + 2 jDt + 1 ) ’
M 1 X uk M k =1 t+1
D(yt + 2 jDt + 1 ) = D(At + 1 ) + Vt + 1 + Wt + 1
ð37Þ ð38Þ
where E(yt + 2 jDt + 1 ) is the mean value; D(yt + 2 jDt + 1 ) is the variance value. In Algorithm 2, according to the Metropolis– Hastings (M-H) algorithm of MCMC simulation method, f (yjx0 ) is adopted as N(x0 , s2 ), and s2 is approximately estimated with the variance of Bt + 1 . Commonly, p(ut + 1 jDt + 1 ) is unknown. By Bayesian theory, p(ut + 1 jDt + 1 )}p(yt + 1 jut + 1 , Dt )p(ut + 1 jDt ) can be obtained and ~p(ut + 1 jDt ) is constructed to approximately simulate p(ut + 1 jDt ); then through the monitored equation or state equation, the approximate PDF ~p(yt + 1 jut + 1 , Dt ) of p(yt + 1 jut + 1 , Dt ) is obtained; finally, ~p(ut + 1 jDt + 1 ) is approximately p(ut + 1 jDt + 1 ), namely, ~p(ut + 1 jDt + 1 )}~p(yt + 1 jut + 1 , Dt )~p(ut + 1 jDt ).
Determination of the main probability parameters of BDNM For the BDNM, the main probability parameters are Vt + 1, Wt + 1, mt, and Ct. The method of determining the main probability parameters is as follows. In this article, the interval period of model updating is 1 day. Vt + 1 is estimated with the variance of differences between the fitted trend data and monitored extreme stress data. According to the research,15,17,18 Wt + 1 can be solved with Wt + 1 =
Gt + 1 Ct G0 t + 1 + Ct d
ð39Þ
where Gt0 + 1 is the transpose of Gt + 1 . d is the discount factor, which often falls between 0.95 and 0.98. mt is the mean value of the state variable at time t, and Ct is the variance of the state variable at time t. They can be estimated with the smoothly processed or resampled monitored data at time t and before time t. If the initial state data follow the lognormal distribution, then the state data can be transformed into a quasi-normal distribution17 with equations (40) and (41); the distribution parameters are, respectively, m0 (mean value) and s0 (standard deviation) f( 1:645) g(x0 )
ð40Þ
m0 = x0 + 1:645s0
ð41Þ
s0 =
where g( ) is the actual fitted PDF of the sample data (lognormal probability density function), and the actual PDF (lognormal PDF) is G( ) and G(x0) = 0.05.
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Reliability prediction based on the built BDNM and FOSM
equations (43) and (44) and the built BDNM, the predicted reliability index bBDNMs is
FOSM 19,20
In this article, the FOSM method is adopted to predict the reliability indices. Suppose there are random variables R (generalized resistance) and S (generalized load effects including dead load effects and live load effects) which are internally independent and mutually independent, the mean value and standard variance of which are, respectively, as follows: mR , sR and mS , sS . The limit state function is g(R, S) = R S
ð42Þ
With FOSM method, the computation formula of the reliability indices can be obtained with mR mS b = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2R + s2S
ð43Þ
Reliability prediction formulas, respectively, based on the transformed BDLM and the built BDNM with FOSM method The limit state function of the second lateral span beam of some bridges is g(R, S, C, M) = R S C g M M
ð44Þ
where R is the steel yield strength, S is the stress caused by the dead weight of steel, C is the stress caused by the dead weight of the concrete, M is the monitored extreme stress predicted with the transformed BDLM or the built BDNM, and gM is a factor assigned to the data provided by the sensors. Prediction formula of reliability indices based on the transformed BDLM with FOSM method. With FOSM method, based on equations (43) and (44) and the transformed BDLM, the predicted reliability index bBDLMs is mR mS mC g M 3 mBDLMs bBDLMs = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2R + s2S + s2C + (gM 3 sBDLMs )2
ð45Þ
where mBDLMs and sBDLMs are the mean and standard deviation of M, respectively, which can be obtained with equation (34). mR and sR are the mean and standard deviation of R, respectively. mS and sS are the mean and standard deviation of S, respectively. mC and sC are the mean standard deviation of C, respectively. g M is a factor assigned to the data provided by the sensors. Reliability prediction formula based on the built BDNM with FOSM method. With FOSM method, based on
mR mS mC g M 3 mBDNMs ffi bBDNMs = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2R + s2S + s2C + (gM 3 sBDNMs )2
ð46Þ
where mBDNMs and sBDNMs are the mean and standard deviation of M, respectively, which can be obtained with equations (37) and (38). mR and sR are the mean and standard deviation of R, respectively. mS and sS are the mean and standard deviation of S, respectively. mC and sC are the mean and standard deviation of C, respectively. g M is a factor assigned to the data provided by the sensors.
Application to an existing bridge The I-39 Northbound Bridge, described in Frangopol and colleagues9,10 in detail, was built in 1961. It is a five-span continuous steel plate girder bridge. The total length of the bridge is 188.81 m. The explicit details about the aim and the results of the monitoring program for the whole bridge are given in Frangopol and colleagues.9,10 The extreme stress data at the beam bottom in the middle part of the second lateral span from the whole bridge are monitored for 83 days, and the corresponding limit state function is shown in equation (44). The monitored data displayed the variability of the stresses caused by traffic, temperature, shrinkage, creep, and structural changes. The stresses from the dead weight of the steel structure and the concrete deck are not included in the measured data. The day-by-day monitored extreme stress data are shown in Figures 2, 4, 6, and 7. In this example, with equations (13)–(46), the trend data of the monitored data are directly fitted as mt = 0:001147t2 + 0:05514t + 24:93
ð47Þ
where mt is the approximate state value at time t. To obtain the distribution parameters of the initial state information, the monitored stress data of the 83 days are smoothly processed or resampled, and then the initial information (the resampled data) of the monitored data is approximately solved. Through Kolmogorov–Smirnov (K-S) test for the initial information, the initial priori PDF is the lognormal probability density distribution or normal probability density distribution as shown in Figure 1.
BDNM based on the monitored data With equations (13)–(18), the built BDNM is as follows. The monitored equation is yt + 1 = ut + 1 + nt + 1 ,
nt + 1 ;N½0, V ,
t = 1, 2, . . . , T ð48Þ
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Figure 1. The initial priori probability density functions and the initial information.
Figure 3. Prediction precision based on the four cases of the built BDNM.
d = 0:98, Wt = Ct1 + Ct1 =d according to the actual engineering experience of the authors. LN½ is the lognormal density function, and Ct can be obtained with equation (51). Equation (51) shows that the initial information follows the normal distribution or lognormal distribution. So the following four cases are discussed to predict the monitored stress data:
Figure 2. Monitored and predicted data based on the four cases of the built BDNM.
The state equations are ut + 1 = ut 0:001147 + + vt + 1 ,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:1174 0:0046ut
vt + 1 ;N½0, Wt + 1 ,
t 24
ð49Þ
and ut + 1 = ut 0:001147 + vt + 1 ,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:1174 0:0046ut
vt + 1 ;N½0, Wt + 1 ,
t.24
ð50Þ
The initial state information mt jDt ;N½24:5052, 4:66352 or LN½3:1811, 0:18862 ð51Þ where yt + 1 are the monitored extreme stress data at time t + 1. mt + 1 is the state value of the monitored extreme stress at time t + 1. nt + 1 is the observational error. vt + 1 is the state error, which means the uncertainty of state variable. V = 29.04 which can be approximately obtained with the monitored stress data, and
Case 1. The initial state information follows the normal distribution, and then the BDNMs are built based on the normal distribution to predict the monitored extreme stresses with MCMC simulation method described in section ‘‘The probability recursion processes based on the built BDNM with MCMC simulation method.’’ Case 2. The initial state information follows the lognormal distribution; first, the lognormal distribution must be transformed into a quasi-normal distribution,17 and then the BDNMs are built based on the quasi-normal distribution to predict the monitored extreme stresses with MCMC simulation method described in section ‘‘The probability recursion processes based on the built BDNM with MCMC simulation method.’’ Case 3. The arithmetic mean of the one-step prediction mean values, respectively, obtained with case 1 and case 2 is considered as the predicted extreme stresses of the third case. Case 4. The fourth case is to build combinatorial BDNM with BDNM obtained with case 1 and case 2; the modeling processes of combinatorial BDNM are described in Jiang et al.21 in detail. From Figure 2, it is noticed that the predicted stress data of the four cases all fit the changing rules of the monitored extreme data, but as far as the prediction precisions of the four cases shown in Figure 3 are
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Advances in Mechanical Engineering
Figure 4. Predicted extreme stresses based on the four cases of the transformed BDLM.
concerned, the combinatorial prediction precision is the best. So the combinatorial prediction model of the monitored stresses is adopted to predict the structural reliability indices.
BDLM based on the monitored data Based on section ‘‘Transformed BDLM and the corresponding probability recursion processes based on the built BDNM with Taylor series expansion technique’’ and equations (48)–(51), the built approximate BDLMs are as follows: The monitored equation yt + 1 = m t + 1 + n t + 1 ,
nt + 1 ;N½0, 29:04
ð52Þ
The transformed approximately linear state equation mt + 1 = 0:967mt + 0:88 + vt + 1 , vt + 1 ;N½0, Wt + 1 ,
t 24
ð53Þ
and mt + 1 = 1:033mt 0:88 + vt + 1 , vt + 1 ;N½0, Wt + 1 ,
t.24
ð54Þ
The initial state information
Figure 5. Prediction precision based on the four cases of the transformed BDLM.
distribution. So the following four cases are discussed to predict the monitored extreme stresses: Case 1. The initial state information follows the normal distribution, and then the transformed BDLMs are built based on the normal distribution to predict the stress data. Case 2. The initial state information follows the lognormal distribution; first, the lognormal distribution must be transformed into a quasi-normal distribution,17,18 and then the BDLM is built based on the quasi-normal distribution to predict the stress data. Case 3. The arithmetic mean of the one-step prediction mean values, respectively, obtained with case 1 and case 2 is considered as the predicted extreme stresses of the third case. Case 4. The fourth case is to build combinatorial BDLM with BDLM obtained with case 1 and case 2; the modeling processes of combinatorial BDLM are described in Jiang et al.21 and Liu et al.22 From Figure 4, it is noticed that the predicted stress data of the four cases all fit the changing rules of the monitored stress data, but as far as the prediction precisions of the four cases shown in Figure 5 are concerned, the combinatorial prediction precision is the best. So the combinatorial prediction model (case 4) of the monitored extreme stress is adopted to predict the structural reliability indices.
mt jDt ;N½24:5595, 4:66502 or LN½3:1811, 0:18862 ð55Þ where d = 0:98, Wt + 1 = Ct + Ct =d according to the actual engineering experience of the authors. LN½ is the lognormal density function, and Ct can be obtained with equation (55). Equation (55) shows that the initial state information follows the normal distribution or lognormal
Reliability prediction based on the BDNM (case 4) and transformed BDLM (case 4) In Figures 3 and 5, we can know that the combinatorial prediction model is the best. So the combinatorial prediction model of the stresses is adopted to predict the structural reliability indices. The predicted results, which are shown in Figures 6–8, can show the changing
Fan and Liu
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Figure 6. Reliability indices based on case 4 of BDNM.
Figure 8. Comparison of reliability indices obtained with the built BDNM (case 4) and the transformed BDLM (case 4).
stresses predicted with the transformed BDLM (case 4) and the built BDNM (case 4).
Conclusion
Figure 7. Reliability indices based on case 4 of the transformed BDLM.
trends and the ranges of the monitored reliability indices. The design specifications are as follows.9 mR = 380 MPa and sR = 380 3 0:07 = 26:6 MPa, mS = 116:3 MPa and sS = 116:3 3 0:04 = 4:65 MPa, and mC = 108:8 MPa and sC = 108:8 3 0:04 = 4:35 MPa of the stresses of the structural dead weight yield mR mS mC g M 3 mM bp = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2R + s2S + s2C + (g M 3 sM )2 380 116:3 108:8 1:15mM = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 26:62 + 4:652 + 4:352 + (1:15 3 sM )2
The article first builds the BDNM based on the quadratic function and the monitored data, and then two methods are proposed to handle with the probability recursion processes; finally, based on the predicted stress data, the structural reliability indices are predicted with FOSM method. The following conclusions can be reached: from the prediction results shown in Figures 2–8, the prediction value and predicted range of the stress and reliability indices all fit the changing rules of the corresponding performances with the proposed two methods in this article, and the prediction precisions are both better and better with the updating of the monitored data. Acknowledgements The authors would like to thank the editor and the anonymous reviewers for their constructive comments and valuable suggestions to improve the quality of the article.
Declaration of conflicting interests ð56Þ
155 1:15 3 mM ffi = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 27:3512 + (1:15 3 sM )2 assigned to the bottom of the girder in the middle of the second lateral span which is shown in Frangopol and colleagues,9,10 where mM and sM are the mean and standard deviation, respectively, of the monitored extreme
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (project no. 51608243), the Natural Science Foundation of Gansu Province of China (project no. 1606RJYA246), and the Fundamental Research
10 Funds for the Central Universities (lzujbky-2015-300, lzujbky2015-301).
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