Structural System Reliability Assessment and Updating Using Chain-Structure by menez

Architectural Engineering November 2015, Volume 3, Issue 3, PP.36-43

Structural System Reliability Assessment and Updating Using Chain-Structure BNs †

Qi’ang Wang , Ziyan Wu, Zongming Cai School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710129, China †

Email: qawang@mail.nwpu.edu.cn

Abstract An efficient computational framework for structural system reliability analysis and Updating based on Chain-Structure Bayesian networks (BNs) is present in the paper. The framework combines BNs and structural reliability methods (SRMs) for reliability assessment and updating. BNs have advantages in evaluating complex probabilistic dependence structures and reliability updating, while SRMs are employed to assess the conditional probability table. The improved branch-and-bound (B&B) method is integrated with BNs to simplify the whole network. In order to further reduce computational demand, failure (or survival) path events are introduced to create chain-structure BNs. Considering the correlations between failure modes, the system reliability is obtained through the Probability Network Estimation Technology (PNET). Finally, the reliability updating is carried out through BNs inference. Results show that computational efficiency is improved by the Chain-Structure BNs. System reliability problems with both continuous and discrete random variables can be better resolved by combining BNs and SRMs. This approach is also able to update system reliability when new information available. Keywords: System Reliability; Chain-Structure BNs; Improved Branch-and-bound Method; Failure Path Events; PNET

1 INTRODUCTION Structural reliability is commonly defined as the probability of a structure performing its purpose adequately for the period of time under the operating conditions encountered. Assessment of structural system reliability will be strongly influenced by the post-disaster information, especially when information about the system is uncertain and evolves in time. Some traditional structural reliability methods, e.g., FORM and SORM, can’t handle discrete random variables. The Bayesian Network (BN) is effective for discrete random variables analysis and information updating. Therefore, the two methods are combined in the paper for reliability probabilistic analysis and updating. Many researchers have tried to combine BN and reliability methods to solve structural system reliability problems. Doguc and Ramirez-Marquez[1] applied a K2 algorithm which constructs the Bayesian network model based on historical data. Langseth and Portinale[2] presented a BN framework for reliability analysis with its emphasis on establishing networks and probabilistic reasoning. Straub and Kiureghian[3] combined Bayesian networks (BNs) and structural reliability methods (SRMs) to create an enhanced Bayesian network (eBN) with little consideration of network structure simplification. Mahadevan et al.[4] applied the BNs to assess the structural system reliability analysis, with the incorporation of multiple failure sequences and correlation between component-level limit states. Bensi et al [5] proposed Chain-Structured BNs to improve the computational efficiency, but it was not used for system reliability analysis and updating. There is little consideration of the computing efficiency of conditional probability table (CPT) and BN reasoning [6] in previous literature. The BN can’t be well used in complex system. Also, there is still a lack of ideal solution for system reliability estimation and updating with both continuous and discrete variables. This paper presents an efficient computational framework for structural system reliability analysis based on ChainStructure BNs. The framework combines BNs and structural reliability methods (SRMs) for reliability assessment and updating. The standard BN structure that defines the system node as a child of its constituent components is built and simplified through improved Branch and Bound (B&B) method. The Chain-Structure BN is proposed to - 36 http://www.ivypub.org/AE

improve computational efficiency. The CPT of each failure path is calculated through structural reliability method. Taking into consideration the correlation between each failure paths, the system failure probability can be obtained through PNET method. Also, the system failure probability can be updated when new information is available. Further, the most vulnerable component in the system can be identified through BN backward inference.

2 TRADITIONAL STRUCTURAL RELIABILITY ANALYSIS 2.1 Safety Margins The limit state function describes the safety margin “M” between the capacity of the structure (R) and the load acting (S) on it. Safety margin “M” may be written as: M = R-S. Both R and S are random variables. M<0 represents a failure state since the load S exceeds the capacity R. Taking a statically indeterminate truss structure in elastic-plastic stage as an example, failure of one member may not lead to the failure of the whole structure. However, it will cause the redistribution of internal forces in other residual members [7]. Then, the next most potential failure element is determined. The procedure is repeated until the system fails. During the procedure, internal forces can be obtained at every level of residual members. The safety margin can be calculated through subtracting internal force from resistance. In the case of an ideal elastic-plastic material, system failure criteria can be represented as the failure of the last failure member. Therefore, for a certain failure mode, safety margin of the structural system is actually that of the final failure member.

2.2 Improved Branch and Bound method There are many different failure modes for the structural system with a high degree of redundancy. It is difficult to find all the failure modes and calculate failure probabilities, so Branch and Bound (B&B) method is employed to extract the dominant failure modes [8]. An improved B&B method was proposed by An Weiguang [7], which can improve the efficiency of identifying the dominant failure modes. Selecting failed elements in the failure path is called “branching”. According to criteria Pfl<ξPfs, secondary failure modes which have little or no effect on the system failure are ignored. Only dominant failure modes are retained. The procedure is known as “bounding”. The details of the improved B&B method are illustrated in the Fig. 1. Pfl is probability of occurrence of a failure mode; Pfs is the system failure probability. ξ is a constant number which is assigned to 2.2 in the paper. Set the initial state k=0, m=0, kk0, kz, ε , n = 0, Pfl=10-30.

Start

k = k+1

Select the first level potential failure element, only first i elements are retained. r1(1) has a maximum probability of failure.

Compute safety margin “M” of components.

Calculating of branching: arrange the failure elements in this failure level according to M.

P(r1(i+1))/P(r1(1))<δ1

Computation of bounding: Pfl<ξPfs

YES

k = k-1. Restore the total stiffness matrix and loading of the structures

End

m= kk0 + nkz or k=0

k = k+1. Restore the total stiffness matrix and loading of the structures NO

Form the new failure path

YES NO Any unselected failure element in this lever?

n=n+1 YES

NO | m=m+1

P’fs-Pfs |

≤ε

YES

FIG. 1 FLOW CHART OF IMPROVED B&B METHOD

2.3 PNET For a structural system with a high degree of redundancy, it is difficult or even impossible to calculate the probabilities of all the failure modes because of the huge number of failure modes and correlation between failure modes. Therefore, only dominant failure modes are considered and viewed as series system. Then, PNET is employed in the paper for system failure probability estimation. PNET method is a simple calculation method with - 37 http://www.ivypub.org/AE

high accuracy to analyze system reliability. The basic idea of PNET is: (1) the failure mode with highest probability is selected as representative mode. Correlation coefficients ρij between other failure modes with the representative mode are calculated. ρ0 is the demarcating correlation coefficient. ρij > ρ0 may be assumed to be perfectly correlated. ρij < ρ0 may be assumed to be statistically independent. On this basis, the possible failure modes can be divided into several groups, such that within each group the failure modes are highly correlated with a representative mode; the representative mode within each group is the failure mode having the highest probability of failure in the group. The representative modes between different groups are statistically independent. Thus, structural system reliability can be expressed as: m

(

Pfs = 1 − Π 1 − Pfj j∈1

)

(1)

Where m is the number of representative failure mode. Pfj is the probability of failure of the representative failure mode. The value of demarcating correlation is generally set to be 0.7 or 0.8. A value of 0.85 appears to be more appropriate for very important structure with fewer failure modes. Correlation coefficient between failure modes are need to be calculated in the PNET. The safety margins of the two failure modes are linear function of random variables X1, X2, ..., Xn, that means: M1 =

∑

; M2 =

ai X i

i =1

∑b X i

(2)

i =1

Where ai, bi are constants. Correlation coefficient between the two failure modes can be obtained by the following formula:

(

ρ[ M1 , M 2 ] = Cov[ M1 , M 2 ] σ M σ M n

)

(3)

2 2 = ∑ ∑ bi b j Cov[ X i ,X j ] . Cov[Xi, Where Cov[ M1 , M 2 ] = ∑ ∑ ai b j Cov[ X i ,X j ] , σ M = ∑∑ ai a j Cov[ X i , X j ] , σ M

=i 1 =j 1

i =1 j =1

=i 1 =j 1

Xj] is the covariance for two random variables Xi and Xj.

3 BN MODEL FOR STRUCTURAL SYSTEM RELIABILITY ASSESSMENT 3.1 Bayesian Network modeling A BN [9-11] is characterized by a directed acyclic graph consisting of a set of nodes representing random variables and a set of links representing probabilistic dependencies. The directed link will connect two nodes. The starting point is parent node and the end point is child node. Conditional probability table (CPT) describes the dependencies between child node and its parent node. R1

...

RN-1

RN P

...

N-1

FIG. 2 BNS OF A TRUSS STRUCTURE.

BNs construction can be generally divided into three steps: (1) Determine the number and the meanings of the variables in the interested domain. (2) Determine BNs topological structure based on dependent relationships between variables. (3) Learning the CPT of the given network structure. Suppose there is a truss structure with n bars. The bar number is 1, 2, …, N. The resistance of each bar is R1, R2, …, RN, respectively. The external load is denoted as P. S represents system state node. Fig. 2 shows the BN system for the truss structure. R1, R2, …, RN; and P in the Fig. 2 are continuous variables. 1, 2, …, N which are state nodes of each bar and can be obtained through - 38 http://www.ivypub.org/AE

traditional structural reliability method. 1, 2, …, N; and S are all binary variables with two states safe or failure.

3.2 Efficient Chain-Structure BN modeling Assuming all the variables are binary, Fig. 3 compares the computational demands, measured in terms of the total clique sizes, for converging and Chain-Structure BN structure. As the number of components increases, the computational demand of the BNs with converging structure increases exponentially, while that of the BN with the Chain-Structure structure increases linearly. Thus, it is critical to transfer converging BNs to Chain-Structure BNs for complex structures with a large amount of components, as shown in the Fig. 3.

Memory Demond

Converging Chain-like

250 200 150 100 50 0

Number of Components

FIG. 3 COMPUTATIONAL DEMANDS OF SYSTEM BNS

Fig. 4(a) shows the BN model with converging structure for a series system. Survival path event (SPE) is introduced into BN as an intermediate node denoted as Es,i,. It describes the state of the variables up to that event. The subscript i indicates that the particular SPE is associated with component i. The converging BN structure can be transferred to be a Chain-Structure BN structure by using SPEs, shown in the Fig. 4(b). For a series system, Es,N is in the survival state only if both Es,N-1 and CN are in the survival state. The probability distribution of Es,i is:

{

{Es ,i −1 = 1} ∩ {Ci = 1} E s ,i = 1 0 elsewhere

(4)

For a parallel system, failure path event Ef,i is introduced as intermediate node. Switching Es,i in the Fig. 4(b) into Ef,i. Then, the BNs obtained represents the parallel system:

0 {E f ,i −1 = 1} ∩ {Ci = 1} E f ,i =  elsewhere 1 C1

...

Ssys

(a) BNS WITH CONVERGING STRUCTURE

(5)

...

Es,1

Es,2

...

Es,i

...

Es,N

Ssys

(b) CHAIN-STRUCTURE BNS

FIG. 4 CONVERGING AND CHAIN-STRUCTURE BNS

4 CASE STUDY A truss structure with 10 bars was used as an example to illustrate the proposed method, shown in the Fig. 5. The degree of redundancy of the truss is 2. Element materials, the dimensions and load case are shown in Table 1. The distributions for material strength and load are assumed to be normal. The buckling is not considered for the compression element. The values of δ1, δ and ε, described in the Fig. 1, are 2.2, 2.2 and 1.0×10-8, respectively. Both kk0 and kz are 2. Firstly, the standard BNs was built, as shown in Fig. 6. P represents the load value. Ri (i = 1,2, ..., 10) is the allowance stress value for the respective bar member. Nodes from 1 to 10 indicate state of each bar. S represents system state. All bars and the system have binary state. “0” denotes the safe state and “1” denotes the failed state. - 39 http://www.ivypub.org/AE

TABLE 1 PARAMETERS OF MATERIAL, SIZE AND LOAD Allowable tensile stress Mean Allowable compressive stress Cross section /(m2) value/(MPa) mean value/(MPa) 1~6 200 100 0.01 7~10 50 25 0.01 The coefficient of variation ratio for stress CVR=0.1; Ei=2.06×105MPa(i=1,2,…,10); Load mean value P=2.06×105N; The coefficient of variation ratio for load CVR=0.2; The stress for each component is independent. Element number

Failure domain Fi for each bar can be obtained through structural force analysis. 1: M1 = F1 = R1 − 2.248P ≤ 0 1: M 3 = F3 = R3 − 0.713P ≤ 0 1: M 5 = F5 = R5 − 0.475P ≤ 0 1: M 7 = F7 = R7 − 0.855P ≤ 0 1: M 9 = F9 = R9 − 0.904 P ≤ 0

; F2 = 1: M 2 = R2 − 0.787 P ≤ 0 ; F4 = 1: M 4 = R4 − 2.252 P ≤ 0 ; F6 = 1: M 6 = R6 − 0.024 P ≤ 0 ; F8 = 1: M 8 = R8 − 0.945P ≤ 0 ; F10 = 1: M10 =− R10 0.898P ≤ 0

(6)

Where R1, R2, R6, R7, R9 are allowable tensile stresses, and R3, R4, R5, R8, R10 are allowable compressive stresses. 5

P 8

...

R10

...

3 6 10

4 P

FIG. 5 THE 10-BAR TRUSS STRUCTURE

FIG. 6 STANDARD BNS OF THE TRUSS STRUCTURE

For the standard BNs, there will be 210=1024 elements in the CPT. In order to reduce the amount of computation, the improved B&B method is employed to select the dominant failure modes. Secondary failure modes and nodes are deleted from the BNs. There are total 8 failure path for the truss structure, as shown in Fig. 7.

R10

FIG. 7 EIGHT DOMINANT FAILURE MODES

FIG. 8 THE SIMPLIED BNS

The eight major failure paths do not contain nodes 2, 5 and 6, which means that those nodes have very low failure probability. Therefore, the joint probability associated with nodes 2, 5 and 6, may be neglected. Then, the standard BNs can be simplified, as shown in the Fig. 8. For the BNs shown in Fig. 8, the reliability for each bar is analyzed by using first order second moment (FOSM) method. The effect of nodes P and Ri (i=1, 2, ..., 10) on each bar element can be represented by a conditional probability table, known as the first-level CPT. The remaining BN model will not consider nodes P and Ri. Firstlevel conditional probability table is shown in the Table 2. - 40 http://www.ivypub.org/AE

The eight dominant failure modes are introduced into BNs, and a new BNs can be obtained, as shown in Fig. 9. In this figure, all the local BNs for each failure modes have converging structure. In order to reduce calculation, failure path event Ejf,i is introduced into BNs. Where i represents the element node, j represents the number of failure mode (j = 1,2, …, 8). The local BN for the failure mode is revised to be Chain-Structure BNs. The CPT of the ChainStructure BNs is calculated, which is called the second-level CPT, shown in the Table 3 (b). In the Table:

P( F2= 1)= P( E 2f ,4= 1)

(7)

Thus, the failure probability of the second failure mode is 1.168×10-4. Second-lever CPTs of other failure mode are shown in Table 3 (a~h). Similarly, the failure probabilities of other failure mode can be obtained (Table 3). After all the failure path events are introduced into the BN, the final chain-structure BN is shown in the Fig. 10. 1

TABLE 2 THE FIRST-LEVEL CPT Bar number i

Probability of failure Pf

1.549×10-12

2.039×10-16

4.249×10-5

5.944×10-8

1.159×10-1

2.288×10-7

7.270×10

-2

FIG. 9 BNS WITH DOMINANT FAILURE MODES 10

E8f,8

9 E4f,8

E5f,8

4 E7f,8

E2f,10

E8f,10

E6f,8

E4f,10

E5f,10

E3f,10

E7f,4

E2f,8

E1f,10

E8f,3

E6f,7

E4f,7

E5f,9

E3f,9

E7f,1

E2f,4

E1f,4

Ssys

FIG. 10 THE FINAL CHAIN-STRUCTURE BN FOR THE TRUSS STRUCTURE TABLE 3 THE SECOND-LEVEL CPT OF 8 DOMINANT FAILURE MODES (a) FAILURE MODE 1 Element i 10 4 P(E1f,i) 7.270×10-2 1.168×10-4 (c) FAILURE MODE 3 Element i 10 9 P(E3f,i) 7.270×10-2 2.321×10-5 (e) FAILURE MODE 5 Element i 8 10 9 P(E5f,i) 1.159×10-1 7.240×10-2 2.321×10-5 (g) FAILURE MODE 7 Element i 8 4 1 P(E7f,i) 1.159×10-1 4.059×10-5 4.780×10-13

(b) FAILURE MODE 2 10 8 4 7.270×10-2 1.123×10-1 1.168×10-4 (d) FAILURE MODE 4 Element i 8 10 7 P(E4f,i) 1.159×10-1 7.240×10-2 2.321×10-5 (f) FAILURE MODE 6 Element i 8 7 P(E6f,i) 1.159×10-1 2.321×10-5 (h) FAILURE MODE 8 Element i 8 10 3 P(E8f,i) 1.159×10-1 7.240×10-2 2.525×10-14 Element i P(E2f,i)

After all the dominant failure mode probability are obtained, the system failure probability for the truss structure can be calculated through PNET. Firstly, the safety margins of all the eight dominant failure modes are listed in the Table 4. The correlation coefficient between the failure modes can be calculated using Equations 3: - 41 http://www.ivypub.org/AE

ρ= [ M1 , M 2 ] ρ= [ M 3 , M 5 ] ρ= [M 4 , M 6 ] 1

(8)

According PNET method, 8 dominant failure modes are divided into 5 groups. System failure probability can be calculated through Equation 1.

Pfs = 1 − (1 − 1.168 × 10 −4 ) × (1 − 2.321× 10 −5 ) 2 × (1 − 4.780 × 10 −13 ) × (1 − 2.525 × 10 −14 ) = 1.632 × 10 −4

(9)

Therefore, the system reliability index is estimated:

β = −Φ −1 ( Pfs ) = 3.593

(10)

TABLE 4 SAFETY MARGIN OF 8 DOMINANT FAILURE MODES Failure mode 1 2 3 4 5 6 7 8

Safety margin M 1 = R4 − 3P + 0.83R10 M 2 = R4 − 3P + 0.83R10 M 3 =R9 − 1.8P + R10

M 4 =R7 − 1.8P + R8 M 5 =R9 − 1.8P + R10 M 6 =R7 − 1.8P + R8 M 4 =R7 − 1.8P + R8

M 8 =R3 − 1.5P + 0.83R8

Through BN forward and backward reasoning, the system reliability can be updated when new information available, which is a main advantage compared with traditional structural reliability method. For instance, the 4th bar is destroyed, then the structural system failure probability may be updated using BN forward inference:

P( S= 1 F4= 1)= P( F4= 1, S= 1) P( F4= 1)= 0.417

(11)

BN allows the joint probability distribution over the system components and diagnostic observations to be expressed in compact form. The use of such a model along with graph-theoretic algorithms for probabilistic inference makes it possible to compute the probability of a component defect given the outcomes of diagnostic observations. Further, the most vulnerable components in the structure system can be identified. For the truss structure, to assume that the structural system is in the failure state, then the failure probability of all bars can be calculated. The 10th bar has the maximum failure probability. Therefore, the 10th bar is the most vulnerable component in the system. Maintenance or retrofit can be taken for the component to reinforce the whole system. P(10= 1 S sys= 1)= P(10= 1, S sys =1 ) P( S sys= 1)= 0.9355

(12)

Through BN forward inference, the system reliability can be evaluated and updated when new information available, which is a main advantage compared with traditional structural reliability method. In addition, the most vulnerable component in the system can be identified through BN backward inference, which is the critical information for system reinforcement.

5 CONCLUSIONS The paper develops a new exploration for system reliability evaluation by combining the BN and traditional structural reliability method. Structural reliability methods are used to compute the CPTs of the BN. BN inference is used for system reliability calculation and updating. The correlation between dominant failure modes is taken into account. A 10 bar truss structure is used as an example to illustrate the method. Firstly, the standard BN is constructed for the system. The improved B&B method is employed to select the dominant failure modes. Then, standard BN is simplified by removing secondary failure mode, and intermediate failure path event nodes are introduced to create a Chain-Structure BN. Finally, the system failure probability is estimated through PNET. Results show that Bayesian network reliability method combined with the traditional structural methods could better solve the information updating problem and deal with both continuous and discrete random variables. The BN structure is simplified through the improved (B&B) method. The computation efficiency is greatly improved by - 42 http://www.ivypub.org/AE

converting standard BNs to be Chain-Structure BNs. For further research, failure and survival path events can be combined to further simplify the network structure. In addition, the system reliability can be updated through BN forward inference. Using backward inference, the most vulnerable component in the system which can be identified the algorithm of this paper is also able to be combined with the structural health monitoring system to update realtime reliability.

ACKNOWLEDGMENT The research was supported by the National Natural Science Foundation of China under Award Number 51278420 and the Doctorate Foundation of Northwestern Polytechnical University under the Grant Number CX201408. The opinions, findings, and conclusions stated herein are those of the authors and do not necessarily reflect those of the sponsors. The authors are grateful to the editors and colleagues whose helpful comments improved the paper.

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AUTHORS 1

Qiâ&#x20AC;&#x2122;ang Wang (1986-), male, Master,

Ph.D

candidate,

mainly

engaged

Ziyan Wu (1962-), female, Ph.D., Professor, Doctoral

supervisor, mainly engaged the research of structural health

structural system reliability analysis and

monitoring

structure health monitoring, earned his

Northwestern Polytechnical University in 2006, acquired PhD.

master degree of structural engineering in

Email: zywu@nwpu.edu.cn

Northwestern Polytechnical University,

Xiâ&#x20AC;&#x2122;an, China. Email: qawang@mail.nwpu.edu.cn

and

reliability

assessment,

graduated

from

Zongming Cai (1992-), male, Bachelor, Master candidate,

mainly focused on performance-based design and fragility analysis, graduated from Northwestern Polytechnical University in civil engineering, acquired Bachelor Degree. Email: zmcai@mail.nwpu.edu.cn

- 43 http://www.ivypub.org/AE