performance and reliability prediction of bridges … and reliability prediction of bridges based on...
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Performance and reliability prediction of bridges based on inspec-tion and monitoring data using Bayesian dynamic models
*Da-Gang Lu1) and Xue-Ping Fan2)
1) School of Civil Engineering, Harbin Institute of Technology,
Harbin 150090, P.R. China 1)
2) School of Civil Engineering and Mechanics, Lanzhou University,
Lanzhou 730000, P.R. China 2)
ABSTRACT
The performance degradation of bridge structures is one of worldwide concerned life-cycle problems for civil infrastructure. Since the deterioration of structural perfor-mance is a time-variant process with large amount of aleatory randomness and epis-temic uncertainties, it is very important to successively predict/forecast structural per-formance to ensure serviceability, safety, and robustness. To integrate the inspection and/or monitoring data, Bayesian updating techniques are usually used to predict struc-tural performance and conditions. However, the traditional prediction functions for pro-cessing inspection and/or monitoring data are normally defined as static polynomial re-gression functions, which are difficult to realize online, dynamic, real-time, and even long-term performance prediction. In this paper, some Bayesian dynamic models re-cently proposed by the authors are reviewed, which have been applied to predict per-formance and reliability of bridges based on inspection and monitoring data. The gen-eral mathematical models and the Bayesian updating algorithms of dynamic linear model (DLM) are firstly introduced. And then, a linear growth model is built to predict the short-term variation trends of member resistance of concrete bridges based on in-spection data. To make use of structural health monitoring (SHM) data to predict long-term performance and reliability of bridges with the everyday monitored extreme stresses, a combinational DLM model of structural extreme stress is set up according to the weights of the prediction precision for the multiple DLMs. Using the combinational DLM model, the reliability indices can be predicted for the ultimate limit state function with further considerations of randomness in the monitored extreme stresses. To incor-porate both historical and real-time monitoring data in the forecasting of time-dependent reliability, a dynamic nonlinear model (DNM) is given, which is realized through Markov Chain Monte Carlo (MCMC) simulation. Based on the built model and real-time monitored stress data, the reliability of bridges can be predicted in near real-time. Some case studies are provided to illustrate the applications and feasibility of the Bayesian dynamic models introduced in this paper.
1)
Professor 2)
Lecturer
1. INTRODUCTION
Under the actions of long-term ambient environments, particularly chemical attack from deicing salts, environmental stressors, as well as continuously increasing traffic volumes, the physical quantities of engineering structures are subjected to changes in both time and space (Frangopol and Liu, 2007). Additionally, due to overuse, overload-ing, aging or damage, these changes may cause degradation of structural perfor-mance, and further have serious impacts on the remaining life of existing structures (Straub and Faber, 2006). The degradation processes of structural performance (e.g. resistance, reliability) are time-variant and irreversible (Enright and Frangopol, 1998). Therefore, it is crucial to real-timely predict the time-dependent performance and relia-bility for safety and serviceability assessment of structures.
Through non-destructive inspection (NDI) and/or structural health monitoring (SHM) for bridges, the basic condition data and health information of bridge structures, including strain, stress and deflection of some specified structural components or struc-tures, can be obtained.
Based on the inspection data of NDI, some traditional methods (for instance, the index method, the hyperbolic method, Asaoka method, the time series method, the grey prediction method, neural network method, among others), have been used for predic-tion and updating of bridge performance (Zhong et al, 2003; Lan et al, 2006; Zhang et al, 2002; Enright & Frangopol, 1999; Zhang & Ellingwood, 1998). However, these methods are all deterministic in nature, since they don’t consider the uncertainty of in-spection data.
The research on SHM systems of bridges has experienced two stages. During the first stage, both global and local sensor systems have been installed in many large-span bridges worldwide, and the transition system of wireless data, data acquisition technology, and system integration technology have been developed (Li et al, 2006; Li & Miu, 2003). For the second stage, a large number of studies have focused on modal parameter identification, damage detection, and model updating using the collected health monitoring data (Qin, 2000). As regard the prediction and assessment of bridge reliability, some achievements have been obtained, for instance, the reliability assess-ment of a long span truss bridge (Susoy and Frangopol, 2008); the performance predic-tion by use of the monitored extreme stress data (Frangopol et al, 2008). However, the research on real-timely predicting and assessing the time-dependent performance and reliability of structures using real-time SHM data of bridges has been still in the initial stage.
In this paper, three Bayesian dynamic models recently proposed by the authors are reviewed (Lu & Fan, 2012, 2013, 2014; Fan & Lu, 2013a,b,c, 2014a,b,c,d,e, 2015, 2016a,b,c; Liu, Lu & Fan 2014; Liu, Fan & Lu 2014), which have been applied to predict performance and reliability of bridges based on inspection and monitoring data. Some case studies are provided to illustrate the applications and feasibility of the Bayesian dynamic models introduced in this paper. 2. DYNAMIC LINEAR MODELS AND BAYESIAN UPDATING
2.1 General Formulation of Dynamic Linear Models (DLMs)
As a kind of state-space models, the dynamic linear model takes the following as-sumptions (West & Harrison, 1997; Petris et al, 2009):
a. State variables, observation errors and state errors satisfy normal distributions;
b. State variables (t, t = 1, 2,…,n) are a Markov chain;
c. Conditionally on (t), the observational variables (yt, t = 1, 2, … ,n) are inde-
pendent of each other and yt depends on t only. A dynamic linear model includes a state equation and an observation equation.
The state equation models the changes of the system with time and reflects inner dy-namic changes of the system and random disturbances, while the observation equation reflects how the measured data depends on the current state parameters of the system. The general form of a dynamic linear model is:
, ~ N 0,T
t t t t t ty F V , (1)
1 , ~ N 0, , ( 1,2, , )t t t t t tG W t T , (2)
1 1 1 1( | ) ~ N ,t t t tD m C , (3)
where yt is a r-dimensional observation vector; Ft is the known (nr) matrix which is the
regression matrix of the observation equation; Gt is a (nn) state transfer matrix; t is a n-dimensional state parameters vector, which are all normal random variables; vt is a r-
dimensional observation error vector; t is a n-dimensional state error vector, which are all zero-mean normal random variables; Dt-1 is the valid information set at time t-1.
For each time t, the dynamic model corresponds to a combination of four elements
{F, G, V, W}t = {Ft, Gt, Vt, Wt}, where Ft is the known (nr) matrix which is the regres-
sion matrix of observation equation; Gt is a (nn) state transfer matrix; Vt is a (rr) vari-
ance matrix which indicates the uncertainty of observation errors; Wt is a (nn) variance matrix which indicates the model uncertainty recursive from time t-1 to time t; Vt and Wt are respectively observation errors and state errors, which are sometimes called obser-vation noise and state noise, it is assumed that Vt and Wt are mutually statistically in-dependent.
Fig. 1 The modeling process of BDLMs
2.2 Bayesian Updating of Parameters of DLMs Based on the Bayesian updating algorithm, with Eqs.(1)-(3), the relationship be-
tween observation vectors and state parameters vectors is obtained as
1 1| ~ N , , | ~ N ,T
t t t t t t t t ty F V G W
. (4)
The modeling process of a Bayesian dynamic linear model can be divided into two key steps, as shown in Figure 1. The first step is the recursion of state parameters from time t-1 to time t, while the second one is the updating of the posteriori probability dis-tribution based on the priori probability distribution of state parameters and inspection data yt at time t. 3. PERFORMANCE PREDICTION OF BRIDGES BASED ON INSPECTION DATA
USING BAYESIAN LINEAR GROWTH MODEL
3.1 Prediction of Time-dependent Performance of Bridges Based on Inspection Data
The bending capacity of concrete bridges is time-variant; whose degradation is an irreversible process. For predicting structural bending capacity, the inspection data of bridge member resistance can be used.
For the degradation prediction of structural performances, many scholars have proposed a lot of techniques, such as the index method, the hyperbolic method, Asaoka method, the time series method, the grey prediction method, neural network method, and so on (Zhong et al, 2003; Lan et al, 2006; Zhang et al, 2002; Enright & Frangopol, 1999; Zhang & Ellingwood, 1998). However, there are some restrictions when applying these methods to analyze and predict the observation sequence data of structural performance. For example, successive regression analysis of the data model is static in nature, and it is hard to reflect the time-variant process of structural re-sistance. The time series method is only suited to predict the smooth random data se-quence, while the grey model is fit to make forecasting of data sequence with strong trend. Neural networks need a great number of data information, but the inspection da-ta of bridge structural performance is still limited. Therefore, there are still some large errors between the results yielded by these prediction methods and the real data. So these methods cannot reflect the degradation phenomenon of structural performance in nature. In addition, they are mostly off-line instead of real-timely forecasting and moni-toring.
In view of the limitations of the above prediction methods, the dynamic measure of structural performance is treated as a time series; then a Bayesian dynamic linear growth model was introduced by the authors (Lu & Fan, 2012).
3.2 Bayesian Linear Growth Model Considering the time-dependent characteristics of structural resistance degrada-
tion, a relative simple linear growing model of structural performance was adopted (Lu & Fan, 2012). It can predict the short-term trend of structural performance.
Suppose that Yt is the observation vector of the bending capacity of RC girders at time t, Xt is the mean value of RC girder bending capacity, vt is the observation error. Then the observation equation is
~ N(0, )t t t t tY X V . (5)
The degradation process of the flexural capacity generally is a very complex dy-namic process. With the effects of loads and other environment factors, the mean value
of the bending capacity is time-variant. Let t be the changed value of the state variable at time t, then the state equation is established as
1 1 ,1 ,1 ,1 ~ N(0, )t t t t t tX X W ， , (6)
1 ,2 ,2 ,2 ~ N(0, ) t t t t tW ， , (7)
where, ,1t and ,2t are the noises of state parameters, both of which are zero-mean
normal random variables.
If 1 0T
tF , 1 1
0 1tG
, t
t
t
X
, 2
2
0
0
xW
, then the DLM can be expressed as:
, ~ N 0,T
t t t t t ty F V , (8)
1 , ~ N 0, ,( 1,2 , )t t t t tG W t T . (9)
Based on Figure1, Eq. (4) and Bayesian updating technology, the recursively up-dating processes of Bayesian dynamic models can be realized.
3.3 The recursion and updating of Bayesian dynamic linear model for bridge re-sistance
Bayesian dynamic forecasting model is applicable to prediction of the future state parameters based on past data, which can be recursively updated with the well-known Kalman filter (West & Harrison, 1997). The recursively updating process is as follows.
(1) the posteriori distribution at time t-1
For mean 1tm and variance matrix 1tC , there is
1 1: 1 1 1( | ) ~ N ,t t t ty m C . (10)
(2) the priori distribution at time t
1: 1( | ) ~ N ,t t t ty a R , (11)
where, 1t t ta G m , 1
T
t t t t tR GC G W .
(3) the one-step forecasting distribution at time t
1: 1( | ) ~ N ,t t t ty y f Q , (12)
where, 1: 1E( | ) T
t t t t tf y y F a , 1: 1Var( | ) T
t t t t t t tQ y y F R F V .
(4) the posteriori distribution at time t
1:( | ) ~ N ,t t t ty m C , (13)
where, 1:E( | )t t t t t tm y a Ae , 1:Var( | ) T
t t t t t t tC y R A A Q , 1
t t t tA R FQ , t t te y f
(One-step prediction error). According to the definition of HPD region, the forecasting interval of the one-step
inspection data is 1.645 , 1.645t t t tf Q f Q
.
3.4 Model Monitoring of Bayesian Dynamic Linear Model The main idea of dealing with model monitoring is to use one or more alternative
models to compare and evaluate linear model performance. In the reference (Zhang et
al, 1922), model monitoring is achieved through Bayesian factor under the normal as-sumption. The main idea is firstly to build an alternative model, then to combine existing models for constructing Bayesian factors.
The adopted probability density function of the alternative model is as follows: 2 0.5 2 2
1 (2 ) exp( 0.5 / )tep k e k , (14)
where, 0.5( ) /te t t te y f Q is the standard prediction error.
The formula of Bayesian factor is
0 1
1 1
( | )( )
( | )
t t
t t
p y DH t
p y D
, (15)
where, 0 (|)p is the forecasting probability density function of the existing model.
In this paper, the adopted monitoring criteria (West & Harrison, 1997) is: if k=3 and H(t)<0.15, then the corresponding inspection data is abnormal, and need to be re-moved; otherwise, the inspection data is normal.
3.5 Case Study
Take the bending capacity mean value R of a RC bridge girder as the perfor-mance indicator. The BDLM of its bending capacity is built to predict the time-dependent performance of concrete bridges. Suppose that both R and the degradation speed S follow normal distributions. Assume that the resistance degradation occurred in the 12th year. The priori parameters of the initial R and S are:
T
0 0
168.03 1.146 0.000425102.71, 0.261 , ,
1.146 0.00782 0.000127tm C V
. (16)
The reinforced concrete girder is respectively inspected in the 12th, 15th, 17th, 22nd and 27th year. The inspection data sequence is shown in Tab. 1.
Tab. 1 Inspection data of RC girder’s bending capacity
T (year)
Inspection data of bending capacity (kN·m) Mean value Coefficient of Variation
12 85.5 0.1 15 105.6 0.1 15 68.2 0.1 17 86.7 0.2 22 92.6 0.1 27 100.1 0.1 27 56.8 0.1
The priori model and the observation data are plotted in Figure 2. The BDLM is
built according to Eqs. (5)-(7). The inspection data-based model updating curves are shown in Figures 3 and 4.
The updated effect of the model is basically in line with the beam flexural capacity trends. There is abnormal data respectively in the 15th year and the 27th year; the dis-turbance of the abnormal data on the model can be removed by discarding the related data, it can be concluded that firstly choosing a suitable performance index (e.g. mean value, standard variance) according to the actual state of bridge girders, and then de-
termining the priori information and the inspection data, a Bayesian dynamic linear model can be built to predict the future performance index.
Fig. 2 Comparison of the priori model and the inspection data
Fig. 3 The means of the predicted bending capacity in model updating
Fig. 4 The variance of the predicted bending capacity in model updating
4. RELIABILITY PREDICTION OF BRIDGES BASED ON MONITORING DATA US-
ING COMBINATIONAL BAYESIAN DYNAMIC LINEAR MODEL
4.1 Combinational Bayesian Dynamic Linear Model for Prediction of Monitored Ex-treme Stresses of Bridges
Suppose that there are ( 2,3, )n n DLMs for monitored extreme stress (Fan &
Lu 2013), the th ( 1,2, , )i i n DLM is
Observation equation:
, , , , , ~ N 0, t i t i t i t i ty V . (17)
State equation or system equation:
, , 1 , , , , ~ N 0, ( 1,2, , )i t i t i t i t i tW t T . (18)
Normal priori distribution of the initial priori information:
, 1 , 1 , 1 , 1( | ) ~ N , i t i t i t i tD m C . (19)
If the initial priori information obeys Log-normal distribution, then the information variable can be transferred into a quasi-normal distribution (Wang 2006)
0' 1.645 'x , 0' ( 1.645) / ( )g x , (20)
0 5 10 15 20 25 30 35 40 45 5050
60
70
80
90
100
110
Abnormal data
Abnormal data
t/year
mean
valu
e o
f b
en
din
g c
ap
acit
y/k
N.m
Inspection data
Priori forecasting model
0 5 10 15 20 25 30 35 40 45 5060
65
70
75
80
85
90
95
100
105
110
t/year
Mean
valu
eo
f b
en
din
g c
ap
acit
y/k
N.m
Priori forecasting curve
Forecasting curve after updating with the 12th year data
Forecasting curve after updating with the 15th year data
Forecasting curve after updating with the 17th year data
Forecasting curve after updating with the 22nd year data
Forecasting curve after updating with the 27th year data
0 5 10 15 20 25 30 35 40 45 5050
100
150
200
t/year
Vari
an
ce o
f b
en
din
g c
ap
acit
y/k
N.m
Priori forecasting curve
Forecasting curve after updating with the 12th year data
Forecasting curve after updating with the 15th year data
Forecasting curve after updating with the 17th year data
Forecasting curve after updating with the 22nd year data
Forecasting curve after updating with the 27th year data
in which ' is the mean value, and ' is the standard deviation, ( )g is the actual
probability density function of the sample information.
4.2 Combinational recursion of multiple Bayesian dynamic linear models The combinational recursively updating processes are: (1) The posteriori distribution at time t-1:
For mean , 1i tm and variance , 1i tC , there is
, 1 , 1 , 1 , 1( | ) ~ N , i t i t i t i tD m C . (21)
(2) The priori distribution at time t:
, , 1 , , ( | ) ~ N ,i t i t i t i tD a R , (22)
where , , 1i t i ta m , , , 1 , i t i t i tR C W .
(3) One-step prediction distribution at time t:
, , 1 , , ( | ) ~ N ,i t i t i t i ty D f Q , (23)
where , , ,1: 1 , E( | )i t i t i t i tf y y a , , , , 1: 1 , , Var( | )i t i t i t i t i tQ y y R V .
According to the definition of the highest posteriori density (HPD) region, the pre-diction interval of the inspection data with a 95% guarantee rate at time t is
, , , , 1.645 , 1.645i t i ti t i tf Q f Q
, (24)
where , , 1.645
i ti tf Q is the predicted lower limit data, and , , 1.645
i ti tf Q is the predicted
upper limit data. (4) The posteriori distribution at time t:
, , , , ( | ) ~ N ,i t i t i t i tD m C , (25)
where, , , , , i t i t i t i tm a A e ;
, , , i t i t i tC A V ; , , , /i t i t i tA R Q ;
, , , i t i t i te y f ; , i tA is the adap-
tive coefficient. (5) The prediction probability distribution based on the arithmetic mean at time t:
, , , , ,
1
1, N ,
s
m t i t i t m t m t
i
P N f Q f Qs
, (26)
where, , ,
1
1 s
m t i t
i
f fs
, , , 21
1 s
m t i t
i
Q Qs
.
(6) The combinational prediction probability distribution at time t:
, , , , , ,
1
, N ,s
c t i t i t i t c t c t
i
P k N f Q f Q
, (27)
where, , , ,
1
s
c t i t i t
i
f k f
, ,
,
1
, 1
1
, 1,2, ,i t
i t
i t s
i
Qk i s
Q
and ,
1
1s
i t
i
k
, ,
1 1
,
1i t
s
c t
i
Q Q
, ,
1
i tQ is
the prediction precision of the ith predicted model (the reciprocal of the variance), 1
, c tQ
is the forecasted precision of the combinational predicting model. (7) Comparison of the prediction accuracy between combinational predicted model
and the predicted model based on the arithmetic mean (Jiang et al, 2002):
,
1 1
, m tc tQ Q . (28)
4.3 Reliability Prediction of Bridges Based on the Combinational BDLMs A five span continuous steel plate girder bridge is taken for the example (Fran-
gopol, Strauss & Kim 2008) in this paper. More details about the aim and the results of the monitoring program are given in the reference (Mahmoud, Connor & Bowman 2005). The extreme stress at the beam bottom in the middle part of the third span from the whole bridge is monitored. The reliability index can be predicted by
p2 2 2 2( )
R S C M M
R S C M M
, (29)
where, M and M are respectively the mean and the standard deviation predicted
with the combinational BDLM; R and R are respectively the mean value and the
standard deviation of the resistance; S and S are respectively the mean value and
the standard deviation caused by the dead weight of steel; C and C are the mean
value and the standard deviation of the stress caused by the weight of concrete respec-
tively; M is a factor assigned to the data provided by the sensors.
4.4 Case Study For the case-study bridge described above, the everyday monitored extreme
stress data is shown in Figure 5, which follows normal or lognormal distribution. The predicted stress results including stress and reliability results are shown in Figures 6-12.
Fig. 5 Monitoring extreme stress Fig. 6 Predicted data of normal distribution
0 5 10 15 20 25 30 35 40-10
-5
0
5
10
15
20
25
30
35
40
T/day
Str
ess/M
Pa
Real-time monitoring extreme stress
Stress with elimination of trend data
Trend extreme stress
0 5 10 15 20 25 30 35 4010
15
20
25
30
35
40
T/day
Str
ess/M
Pa
Monitoring extreme stress
One-step forecasting data
Forecasting upper limit data
Forecasting lower limit data
0 5 10 15 20 25 30 35 4010
15
20
25
30
35
40
45
50
55
60
T/day
Str
ess/M
Pa
One-step forecasting monitoring extreme stress
Real-time monitoring extreme stress
Forecasting the upper limit ofmonitoring extreme stress
Forecasting the lower limit of monitoring extreme stress
0 5 10 15 20 25 30 35 4015
20
25
30
35
40
45
T/day
Str
ess/M
Pa
Monitoring extreme stress
One-step forecasting data
Forecasting upper limit data
Forecasting lower limit data
Fig. 7 Predicting results of lognormal distribution
Fig. 8 the arithmetic mean of the two distributions
Fig. 9 the combinational forecasting
model Fig. 10 the predicting stresses of the four cases
Fig. 11 Forecasting precision Fig. 12 predicted reliability indices
As seen from the forecasting precision of the above four cases, which are shown in
Figure 11, the forecasting precision of the combinational model is the best. So the combinational forecast model of the extreme stress is adopted to forecast structural re-liability indices with Eq. (29), the forecasting results are shown in Figure 12. 5. RELIABILITY PREDICTION OF BRIDGES BASED ON MONITORING DATA
USING BAYESIAN DYNAMIC NONLINEAR MODEL AND MCMC Due to the limitation of BDLM, the BDLM hasn’t been widely used in the field of
structural health monitoring. Therefore in consideration of the limitations of Bayesian dynamic linear model, a more reasonable Bayesian nonlinear dynamic model is intro-duced and established based on the monitored stress data of bridges. The correspond-ing probabilistic recursive process of the built nonlinear Bayesian dynamic model is completed through Markov Chain Monte Carlo (MCMC) simulation. The Bayesian dy-namic nonlinear models can increase the prediction precision of monitored data.
0 5 10 15 20 25 30 35 4015
20
25
30
35
40
45
T/day
Str
ess/M
Pa
Monitoring extreme stress
One-step forecasting data
Forecasting upper limit data
Forecasting lower limit data
0 5 10 15 20 25 30 35 4015
20
25
30
35
40
45
T/day
Str
ess/M
Pa
Monitoring extreme stress
Forecasting data of normal dictribution
Forecasting data of lognormal dictribution
Forecasting data based on the weights
Forecasting data based on the arithmetic mean
0 5 10 15 20 25 30 35 4010
20
30
40
50
60
70
80
T/day
Fore
casting v
ariance
Normal distribution
lognormal distribution
Forecasting based on the weights of the two distributions
Forecasting based on the arithmetic mean of the two distributions
0 5 10 15 20 25 30 35 403.8
4
4.2
4.4
4.6
4.8
5
T/day
Relia
bili
ty indic
es
Monitoring data
Forecasting data
Forecasting lower limit data
Forecasting upper limit data
5.1 Assumptions of Bayesian Dynamic Nonlinear Model
(1) The state variable ( t ) is a Markov Chain.
(2) Conditionally on ( t ), the observational variables ( 'ty s ) are independent and
ty depends on t only.
(3) The state equation is nonlinear, and the observation equation is linear. 5.2 Bayesian Dynamic Nonlinear Model for Prediction of Monitored Extreme
Stresses of Bridges A Bayesian dynamic nonlinear model introduced in the reference(Fan & Lu, 2014)
is Observation equation:
, ~ N 0, , ( 1,2, , )t t t t ty V t T . (30)
State equation:
1( ) , ~ N 0, , ( 1,2, , )t t t t t tg W t T . (31)
Initial information:
1 1 1 1( | ) ~ N ,t t t tm C D , (32)
where, t is the state parameter indicating the level of the observational data at time t;
1( )t tg is nonlinear.
With Eqs. (29)-(31), the updating relationship between observational data and state parameters can be obtained as:
1| ~ N , , N ( ), t t t t t ty V g W . (33)
5.3 MCMC Simulation of Bayesian Dynamic Nonlinear Models Based on the health monitoring data of bridges, the probability function of the initial
state parameters can be statistically obtained. For example, the density function of 1t
at time t-1 is 1 1( | )t tD .
For obtaining the simulation samples of state values at different time, in this paper,
with the MCMC simulation method, a group of convergent samples from 1 1( | )t tD
can be obtained as (1) (2) (3) ( )
1 1 1 1 1A , , , , n
t t t t t . 1A t can be converted into the priori
samples of 1|t tD : (1) (2) (3) ( )
tB , , , , n
t t t t through the algorithm 1.
Algorithm 1 (1) With MCMC simulation, draw a group of convergent sample
(1) (2) (3) ( )
1 1 1 1, , , n from ~ 0, t tW ;
(2) for every sample ( )
1
i
t , 1 i n , let 1
( ) ( ) ( )
1( )t t
i i i
tg
, then the convergent
sample tB of priori probability distribution for state variable t at time t can be ob-
tained as (1) (2) (3) ( ), , , , n
t t t t t B .
For tB , when the observational data ty at time t is obtained, with the MCMC
simulation method, the tB can be converted into the convergent sample
(1) (2) (3) ( ), , , , n
t t t t t A of posteriori probability distribution for state variable t at
time t, the detailed steps are shown in Algorithm 2 or Algorithm 3. Algorithm 2
For every sample ( )i
t ,1 i n ,
(1) Let ( )
0
i
tx ;
(2) draw a sample 1z from 0( | )tf y x , namely 1 0~ ( | )tz f y x ;
(3) 1 1x z is accepted with the probability 0 1 1 0( , ) max 1, ( | ) / ( | )t ta x z p z D p x D , or else
1 0x x ;
(4) The same procedure is repeated M times, then let ( )
t
i
Mx .
The steps (1)-(4) are repeated N times, then the convergent sample (1) (2) (3) ( ), , , , n
t t t t t A of the posteriori probability distribution for state variable t
at time t can be obtained. Algorithm 3
For (1) (2) (3) ( ), , , , n
t t t t t B ,
(1) Let 0 mean( )tx B , where mean( )tB is the mean value of tB ;
(2) draw a sample 1z from 0( | )tf y x , namely 1 0~ ( | )tz f y x ;
(3) 1 1x z is accepted with the probability 0 1 1 0( , ) max 1, ( | ) / ( | )t ta x z p z D p x D , or else
1 0x x ;
(4) The steps (1)-(3) are repeated N times, then the convergent sample (1) (2) (3) ( ), , , , n
t t t t t A of posteriori probability distribution for state variable t at
time t can be obtained. For Algorithm 2 or Algorithm 3, the distribution parameters (mean value and vari-
ance) of one-step prediction distribution are shown as
1 1
1
1( | ) ( )
Mk
t t t t
k
E y D fM
, (34)
1( | )= ( )+ +t t t t tD y D D A V W , (35)
where, 1( | )t tE y D is the one-step prediction mean value; 1( | )t tD y D is the one-step
prediction variance value in consideration of the observational error or state error. In Algorithm 2 or Algorithm 3, according to the M-H algorithm of MCMC simulation
method, 0( | )f y x is adopted as 2
0N( , )x , 2 is approximately estimated with vari-
ance of tB , commonly ( | )t tp D is unknown, by Bayesian theory,
1 1( | ) ( | , ) ( | )t t t t t t tp D p y D p D can be obtained, 1( | )t tp D
is constructed to approximately
simulate the 1( | )t tp D , then through the observational equation and state equation,
the approximate probability distribution function1( | , )t t tp y D
of 1( | , )t t tp y D
can be ob-
tained. Finally, ( | )t tp D is approximately ( | )t tp D , namely 1 1( | ) ( | , ) ( | )t t t t t t tp D p y D p D .
5.4 Reliability Prediction of Bridges Based on the BDNM and MCMC
In this paper, the deflection limit value [ω] is taken as the general bridge re-sistance, while the monitored deflection value f(t) is treated as the general load effect, so the limit state function is
[ ] ( )Z f t , (36)
where, f(t) is the monitored deflection value.
The first-order reliability index p is
[ ] ( )
2 2
[ ] ( )
f t
p
f t
, (37)
where, ( )f t and 2
( )f t can be obtained with the BDNM and MCMC simulation method.
5.5 Case Study A concrete filled steel-tube (CFST) arch bridge has three spans and a flying swal-
low type, the total span of the bridge is 740 m, the main bridge is 260 m, the bridge spans include 51m+158m+51m. The schematic diagram is shown in Figure 13, the everyday extreme deflection data at the beam bottom in the middle part of the 158m span from the whole bridge is monitored for 30 days. The monitored data is shown in Figure 14.
Fig. 13 The schematic diagram of the bridge Fig. 14 Monitored everyday deflection
data
The monitored data is resampled, and with the resampled data, the parameters of
-1 1( | )t tD ,
tW and tV are estimated. Through estimation, the state variable, observa-
tional error and state error all follow normal distribution (see Figure 15). Therefore, the built dynamic models are as following:
Observational equation , ~ N 0, t t t t ty V
State equation： 1 10.01596 ~ N 0, ( 1,2 , ) = /t t t t t t tW t T W C ， ，
Initial information： 1 1 1 1( | ) ~ N , t t t tD m C
where, 0.01596 is the changing rate of state data, is discount factor, which is adopt-
ed between 0.96~0.99 according to the engineering experience of the authors. From the above dynamic model, it can be known that, the model is dynamic linear
model. For illustrating the application and the flexibility of the built BDNM, the predicted
0 5 10 15 20 25 3042
43
44
45
46
47
48
49
50
Time/day
Deflection/m
m
Monitored everyday extreme deflection data
deflection data is shown in Figure 16. Figure17 shows that the prediction precision of BDNM is better and better.
Figure 18 shows that the reliability index solved with specification is 7.2 which is relatively conservative; the reliability index obtained by the kernel density estimation (Zhao, 2012) is 4.75; two kinds of Bayesian dynamic models considered both random-ness and uncertainty of monitored data, so the predicted reliability indices are smaller than the deterministic reliability indices. But the changing trend of the reliability indices by BDNM is in line with the changing trend of the reliability indices by BDLM (Fan & Lu, 2013). So the BDNM is more reasonable to predict structural reliability indices.
Fig. 15 PDF of monitored data Fig. 16 Predicted deflection value by BDNM
Fig. 17 Prediction precision of BDNM Fig. 18 Reliability indices of BDNM
6. CONCLUSIONS AND FUTURE WORKS
(1) Bayesian dynamic linear model can characterize the time-dependent process of the degradation of structural bending capacity (resistance), and overcome the defects of the traditional static forecasting models. According to the posteriori distribution of structural performance parameters, the structural performance can be forecasted on-line with time.
(2) The monitoring extreme stress-based combinatorial Bayesian dynamic linear model is firstly built, compared with single Bayesian dynamic linear and the forecast model based on the arithmetic mean of the two single Bayesian dynamic linear models,
42 43 44 45 46 47 48 49 500
0.05
0.1
0.15
0.2
0.25
Data
Density
Monitored deflection data
Normal distribution
0 5 10 15 20 25 3040
42
44
46
48
50
52
Time/day
Deflection/m
m
Monitored deflection data
One-step predicted deflection data
One-step predicted deflection upper data
One-step predicted deflection lower data
0 5 10 15 20 25 300.148
0.15
0.152
0.154
0.156
0.158
0.16
0.162
0.164
0.166
Time/day
pre
dic
tion p
recis
ion
Prediction precision of deflection by BDNM
0 5 10 15 20 25 300
1
2
3
4
5
6
7
Time/day
Relia
bili
ty indic
es
Predicted reliability indices by DLM
Predicted reliability indices by Specification
Predicted reliability indices by kernel density estimation
Deterministic reliability indices
Predicted reliability indices by BDNM
predicting extreme stresses are almost the same, but as far as the forecast precision is concerned, the combinational forecast model has the best prediction precision.
(3) Based on the combinational BDLM and BDNM of monitored extreme stresses, structural reliability indices are predicted. This paper considered the randomness and uncertainty of monitored data, so the predicted reliability indices are smaller. But the predicted smaller reliability indices may better reflect the actual state of the bridge.
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