大地工程系統動態可靠度之研究

行政院國家科學委員會專題研究計畫 成果報告

大地工程系統動態可靠度之研究 研究成果報告(精簡版)

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 95-2221-E-011-057-

執 行 期 間 ： 95 年 08 月 01 日至 96 年 07 月 31 日 執 行 單 位 ： 國立臺灣科技大學營建工程系

計 畫 主 持 人 ： 卿建業

計畫參與人員： 博士班研究生-兼任助理：謝宜宏、陳奕竹 碩士班研究生-兼任助理：張景復、陳宏彬

報 告 附 件 ： 出席國際會議研究心得報告及發表論文

處 理 方 式 ： 本計畫可公開查詢

中 華 民 國 96 年 10 月 03 日

Approximate Real-time Reliability Estimation of Nonlinear Dynamical Systems

with Monitoring Data

Jianye Ching¹

ABSTRACT

Updating reliability for nonlinear dynamical systems with high dimensional uncertainties based on monitoring data is a daunting task. A usual way of handling this is to first update the probability density function (PDF) of all uncertainties and then to update the system reliability based on the updated PDF. However, when uncertainties are high dimensional, e.g.: uncertain dynamical excitation, this is not feasible since updating high dimensional PDF is intractable unless the system is linear and all uncertainties are Gaussian.

For nonlinear dynamical systems, this research proposes a way of bypassing the updating step for the PDF of the uncertain variables but directly updating the reliability. Since the updating of the high dimensional PDF is bypassed, the resulting method is feasible for updating reliability of nonlinear dynamical systems with high dimensional uncertainties. Several examples are used to demonstrate and to verify the proposed approach.

Key words: Bayesian analysis, reliability update, monitoring, Subset Simulation

1 (Corresponding Author) Assistant Professor, Dept of Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan. Phone: 886-2-27301055. Email: jyching@gmail.com.

1. INTRODUCTION

Sensor techniques have been prospering in the recent decade. Many engineering systems, e.g. bridges, buildings, airplanes, etc., are instrumented nowadays for damage detection and health monitoring. An important question is how to update the reliability of these systems based on the monitoring data. However, this is, in general, a very challenging research topic.

An attempt was made by Beck and Au [10] who proposed adaptive Metropolis-Hastings algorithm to first update the probability density function (PDF) of the uncertain system parameters and then to update reliability. A limitation of this approach is that the dimension of uncertainties cannot be too high since updating high dimensional PDF is infeasible. Ching and Beck [11] proposed another method based on ISEE [12] to update reliability of linear dynamical systems. Their approach is not constrained in the uncertainty dimension, but it can be only applied to deterministic linear dynamical systems with Gaussian uncertainties.

Updating reliability of nonlinear dynamical systems remains a pending research problem.

In this research, it is shown that it is not necessary to first update the PDF of uncertain parameters before the reliability is updated when the data is a scalar, i.e. the system reliability can be directly updated. Since the updating of the PDF of uncertain parameters is bypassed, the approach is not limited to the number of uncertainties. To relax the constraint of scalar data, an approximate technique is further proposed to condense all information provided in

the monitoring data into a single scalar data. This scalar data plays a role similar to sufficient statistics, i.e. conditioning on the scalar data, it is assumed that the failure event is approximately independent of the monitoring data.

With the new approach, as long as the PDF of the uncertainties and the probabilistic model of the target dynamical system are given, the approximate functional relationship between the updated failure probability and the scalar data can be obtained prior to the monitoring process. This means in real applications, it is not necessary to conduct the new method in an online manner. Instead, the relationship can be calculated a priori so that the reliability update can be achieved right away once the monitoring data is obtained. It is expected that the new approach is useful for safety monitoring and maintenance of existing engineering systems.

The structure of the paper is as follows: In Section 2, the problem of updating reliability is formally defined. In Section 3, a simple procedure for reliability updating via Monte Carlo simulation (MCS) is presented, while a more efficient method based on SubSim is presented in Section 4. In Section 5, examples are used to demonstrate the new approach, and in Section 6, discussions and conclusion will be given.

2. PROBLEM DEFINITION

The goal of regular dynamical reliability analyses is to estimate failure probability given

the probability distribution of the uncertainties in the target dynamical system and the

probabilistic model

M

P F M , where

F

on these measurements, i.e. to compute

P F D M . A usual way of estimating this

probability is through the following equation:

     





1

| , | , | , 1 | ,

N

i i

P F D M P F M f D M d P F M

   N 



   

where

 contains uncertain parameters of interest, and  is drawn from

^f

^

^{D M}

Here it has been assumed that conditioning on

, D and the failure event are independent.

With this approach, the PDF of the uncertain parameters is first updated through stochastic simulation, and then the updated reliability is simply the average of the reliabilities for which the uncertain parameters are fixed at their sampled values.

This approach of updating reliability is feasible when the

 dimension is low. Beck

 dimension is high, e.g.: thousands, this approach

^f

^

^{D M}

technically challenging, and in many instances, practically infeasible. To circumvent this

difficulty, in Beck and Au (2002),

 only contains uncertain structural parameters, while the

^{P F}  ^{| }

^{, M} 

computed analytically based on the out-crossing rate method (Soong and Grigoriu 1993). The stationarity assumption may be reasonable for winds and ambient vibration, but not for strong motions and impacts, e.g.: earthquakes, which are obviously non-stationary by their nature. In order to remove the stationarity assumption, it seems necessary to augment the uncertain strong motions into

, but this, unfortunately, leads back to a high dimensional problem.

An exception occurs when the target dynamical system is linear and all uncertainties are

jointly Gaussian. In this case, it can be shown that drawing samples from

^f

^

^{D M}

easy since it is jointly Gaussian (Ching and Beck 2007). However, for nonlinear systems, the Gaussianity does not hold.

In this paper, a new approach of approximate reliability updating that is applicable to nonlinear dynamical systems with high dimensional uncertainties is proposed. The basic idea of the new approach is described as follows: If the monitoring data D can be transformed into a scalar index

 ( ) D

 ( ) D

 ( ) D

perspective of the failure event, it is argued that

^{| ,}   ^{| ( ), ,}   ^{| ( ),} 

P F D M

P F  D D M

P F  D M

where for the first equality, we have used the fact that conditioning on D,

 ( ) D

provide extra information with respect to D. According to Bayes’rule

     

 

   

   

 

( ) | , |

| ( ),

( ) |

( ) | , |

( ) | , | ( ) | ^C, 1 |

f D F M P F M

P F D M

f D M

f D F M P F M

f D F M P F M f D F M P F M

 





 



   

(3)

where

F

 is not involved, i.e. it is bypassed. Therefore, the dimension

 does not impose any limitation on the approach.

This new approach thus requires two components: (a) the search of such an approximate sufficient statistics

 ( ) D

^{P F}

^{ D M}

 ( ) D

^{( ) | ,} 

f  D F M

^f  ^{ ( ) | D F}

^{, M} 

P F M . In the next two sections, the detailed

^f

^

^{D F M}

^f  ^{ ( ) | D F}

^{, M} 

^{P F M}

Monte Carlo simulation (MCS) and Subset Simulation (SubSim) will be given. In a later section, a way of finding the approximate sufficient statistics

 ( ) D

will be proposed.

For notational simplicity, the symbol

M

 will

 ( ) D

M

3. ESTIMATION OF ^{P F}

^{ VIA MCS}

An approach of estimating

^{P F}

 based on MCS is presented in this section. This

^{P F}

 is small. However, the MCS approach is worth

^{P F}

 based on SubSim will be presented in the next section.

Let

R ( )  denotes the limit-state function that defines failure event F

R ( ) 1  

N

 samples  ^

^{: i  1... N} 

,

each sample corresponds to a monitoring data sample

D

( ) ( )

( )

i i



 D

R

R



 ^

^{: i  1... N} 

will be denoted by



According to the Law of Large Number, we have

 





1

1 1 ˆ

N i i

P F I R P F

N

   

where

^I

^f

^

^F

  ^|

f  F

F

N

N

R

 samples are distributed as ^f

^

^F

N

N

 samples are distributed as ^f   ^{ | F}

With these samples, the unknown PDFs

^f

^

^F

^f   ^{ | F}

using histograms. Let us denote the estimated histograms by

^f

^{ F}

^{f ˆ |}   ^{ F}

consequence,

     

   

 

ˆ | ˆ

| ˆ | ˆ ˆ |

1 ˆ

f F P F P F

f F P F f F P F

 

 

      

(5)

The performance of MCS in estimating

^{P F}

 is independent of the number of

^{P F}

 is small. The reason for this inefficiency is

P F . This is because the

R

R

R

f r . However, in order to estimate small

^{P F}

R

^f

^

^F

 ^|

f  F

^f

^

^F

^f

^

^F

region, failure samples in the tail of

^f

^

^F

4. ESTIMATION OF ^{P F}

 VIA SUBSIM

According to the previous discussion, the main drawback of MCS in estimating

 ^|

P F  stems from the fact that it is inefficient in propagating R

 | F

^{f r}

^f

^

^F

to propagate the samples into the tail in a more efficient manner. The SubSim procedure of propagating

R

 | F

G

R

 | F

4.1 Procedure of SubSim

SubSim provides a more efficient way to propagate

G

^{f g .}

The benefit brought by SubSim is that the tail part of

^{f g}

^{P G}

^b

^b

G

In the following, the procedure of SubSim of estimating the tail part of

^{f g}

The first stage of SubSim is an MCS step: draw

N

G

^{f g . Note}

that these samples cannot effectively populate the tail of

f g . The next step is to choose a

threshold

b . Suppose there are

N

b ; therefore,

P G

b

 1

P G

b

N N and

1 1

(1:N)

G

f g G

b

N

N

b

f g G

b

N

N

N

f g G

b

being

^{f g}

G

b

N

N

N

f g G

b

Now the next threshold is chosen to be

b . Suppose there are

N

b ;

therefore,

P G

b G

b

P G

b G

b

N

N and

N

N

1 2

(1: ) ,

N

G

f g b

G b

while the other

N

N

b are distributed as

f g G

b

these

N

N

N

N

samples distributed as

f g G

b

have propagated to the desirable tail region of

f g . Later in the examples, the sample

number

N

N

For notational simplicity, the following notations will be used throughout the paper:

1

,

b b

f g P

 

 

  f g G

|  b

, P G  b

 

j j j j

b b b b

f

g P

 

  denote



 



f g b G b

P G b

G b

     

 ,

f

g

f g G

b

, 1 j j

P





P b   G b

bm

P

P G

b

Suppose the procedure is repeated for

m stages, the following samples will be

1

(1:N)

G

f

g ,

1

(1: ) ,

j j j

N

G

f

g

j  1,..., m  1

and ^{(1: )}

m

N

G

f

g . Also, the following probability estimates are available:

1 1

P

 N N

1| 1

j j

b b j

P

N

N

j  1,..., m  1

 

P G

b

      ^{ }

   

 

 

1 1

1 1

1 1

1

1

1 1 ,

1 1

( ) 1 ( ) 1

,

1 1 1 1

1 1

1 ( )

1 1

( ) 1

| | |

1 1

1

1 1

1

j j m

j

j j

m

m

b j j b b m b

j

N m N j

i i j n

b b b

i j j i n

m N

i j b

i j

i b i

P G b P G b G b P P G b b G b P P G b G b P

N N

I G b N I G b

N N N N N

I G b N

N N

I G b N













 

    



 



          

   

                 

 

     

 

 



   

 

  ^{ }

1 1

1 1

( ) ( )

,

1 1 1 1 1

1 1

j

j j m

N m N j N m

i n i j

b b b

j i n i j

N N

I G b I G b

N N







    

               

       

       

      

(6)

where we have used the fact that

1

 

1

, , , ,

1

1 1

j j j j j j

j

j n

b b b b b b b b b

n

N N

P P P P P

N N

  





 

         

 



 (7)

and that

 

 



1

1 1 1 1

m m m

m

j

b b b b b b

n

P P P P N



N



 

         

 



 (8)

Moreover,

^{f g}

   

  

1 1 1 1

1 1

1

, ,

1 1 1 1

,

1 1 1

ˆ ˆ 1 ˆ 1

j j j j m m

j j m

m

b b b b b b b b

j

j m

m

j n j

b b b b

j n j

f g f g P f g P f g P

N N N

f g N f g f g

N N N N

 







 

  

  

 

 

            

   



  

(9)

where

f ^ˆ

ˆ

j j

f

g

1

(1: ) ,

j j j

N

G



f g , the estimated ^{f g}

In SubSim, a popular choice is to pick the threshold values

 ^b

^{: j  1,..., m} 

all

P

N

N

N

N

b

1

(1: ) ,

j

j j

N b b

G

number of stages, i.e.

m , is usually chosen such that b

b P G

b G

b

 





1

10

P G b I G b

N







    

and

 ₁ ¹ , ₁ 

1

9 ˆ 9 ˆ 10 ˆ 10

10 10 ^{j j m}

m

j m

b b b b

j

f g f g f g f g



  



 



Since the SubSim samples propagate to the tail in a very efficient way, the

^{P G}

^b

estimator is more stable, and the estimated

^{f g}

The performance of SubSim is also independent of the degree of freedom of the system, system complexity, and the uncertainty dimension.

4.2 Estimating ^{P F}

 via SubSim

4.2.1 Estimation of ^{P F}

^{and f}   ^{ | F}

with SubSim

In the perspective of SubSim,

^{P F}

G

R

b

F

R  1

^f   ^{ | F}

 samples obtained in the j-th

1

(1: ) ,

j j j

N



 samples satisfying b

  R b

1

(1:N)



f

 ,

1

(1: ) ,

j j j

N



distributed as

f



j  1,..., m  1

m

N



f

 . Among

(1: )

m

N



bm

N

m

N



b

R

f

 . The other samples, denoted by 

R  1

distributed as

^f

^

^F

1 1

P

 N N

1| 1

j j

b b j

P N N

   (for

j  1,..., m  1

m bm

P F R

b

N N

adopted, one can verify that

 

  

1 1 1 1

1 1

1

, , ,1 ,1

1 1

,1

, ,1

1

|

9 ˆ 9 ˆ 10 ˆ 10

10 10

j j j j m m

m

j j m

m C

b b b b b b b b

j m

j m b

b b b b

j

f F f P f P f P

f f f N

N

   

  

 







  



  

     





where

f ^ˆ

 samples with histograms.

4.2.1 Estimation of ^f

^

^F

In principle, one can estimate

^f

^

^F



^f

^

^F

However, these samples mostly cluster around the main support region of

^f

^

^F

discussed as above,

 | F

^f

^

^F

accurately at the tail region. This can be achieved with another round of SubSim, described as follows.

Firstly, the _,1

bm

N  N



bm

N

are also distributed as

^f

^

^F



N



^f

^

^F

^f

^

^F

N

Secondly, the SubSim procedure is implemented to propagate the

 | F

tail of

^f

^

^F

b so that there are

N

 | F

b ;

P

 b F

P

 b F

N N and

These

N samples, denoted by

1

(1: )

| N



f

 

b F

f



F

N

N

b

f

 

b F

N

N

N

1 |

f

 F

R  1

 . b

Together with the original

N

N

N

| 1, 

f   b F

Then the next threshold

b is chosen. There are

N

1 2

(1: ) , |

N b b F



than

b , so they are distributed as

f



b

 b F

f



F

m times. Finally, the following samples will be obtained:

1

(1: )

| N



1 |

f

 F

1

(1: ) , |

j j j

N b b F



j j

f

 F

j  1,..., m  1

m

N



f



F

f  

b F

P

P



b F

N N

 

1| , 1

| ,

j j

b b F j j j

P

 P   b

  b F  N

N

j  1,..., m  1

 ^|

f  F

  ₁  ¹ , ₁   

1

9 ˆ 9 ˆ ˆ

| | | 10 | 10

10 10 ^{j j m}

m

j m

b b b b

j

f  F f  F

f

 F

f  F



 



where

f ^ˆ

ˆ

|

j j

f

 F

1

(1: ) , |

j j j

N b b F



In summary, with two rounds of SubSim,

^{P F ,}

^f

^

^F

^f   ^{ | F}

effectively estimated. Together with (5),

^{P F}

 estimate can be obtained. Please note that

Besides, with the proposed approach, the functional relationship between the updated failure probability and the scalar data

 can be obtained prior to the monitoring process as

5. SELECTION OF SUFFICIENT STATISTICS

6. EXAMPLES

Three examples are selected to demonstrate the use of the new method. The first example is an infinite slope, where the monitoring value is the height of the water table, and the failure is defined as the sliding of the slope. The second example is a ten-story building subjected earthquake motions, where the monitoring value is the maximum acceleration on

the roof, and the failure is defined as the exceedance of the maximum inter-story drift over a prescribed threshold. The third example is a real case study, a deep excavation problem, where the monitoring value is the maximum diaphragm wall deformation, and the failure is defined as the exceedance of the maximum ground settlement over a prescribed threshold.

There are some common features for the three examples: (a) the uncertainties are so significant that the failure probability is quite large when no monitoring data is available; (b) the physical quantities that directly define failure cannot be monitored easily, but the monitored value contains certain information about those quantities, so the knowledge of the monitoring value is helpful in reducing the uncertainties associated with the defined failure.

6.1 Infinite slope

In this hypothetical example, let us consider the infinite slope in Figure 2, where

H

 is the slope angle, h



 is unit weight of the saturated soil. The failure event is

       

    ²  

sin cos cos tan

sat

sat

h H h

R Z h H h

   

    

   

 

       

The height of the water table is uncertain and is uniformly distributed over the [0,

H

h   H U

U

Z



H

 . It is assumed that the uncertain U

variables



H

 are Gaussian with mean equal to   17 kN m /

,10 ,19 m kN m /

, 35   

  1.5 kN m /

, 3 ,1.5 m kN m /

, 3   

 is

20

 is the height of the water table h

For this case study, the prior failure probability, i.e. the failure probability without data

 

P F , is roughly 50%, indicating that the uncertainties are significant. Therefore, it is

desirable to update the failure probability with the monitoring data so that the uncertainties

can be reduced. Both MCS and SubSim are employed to estimate

^{P F}

 , but only the

^f

^

^F

^f   ^{ | F}

^{P F}

 is obtained.

The left-hand-side plot in Figure 4 shows the analysis result of the proposed approach with a single SubSim run with stage sample number

N

runs. Figure 5 shows the comparison of the variabilities in the

^{P F}

 estimates made by

To examine the analysis results, let us consider the following verifying procedure: