Reliability, redundancy and risk as performance indicators of structural systems during their life-cycle

Engineering Structures 41 (2012) 34–49

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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct

Reliability, redundancy and risk as performance indicators of structural systems during their life-cycle Benjin Zhu, Dan M. Frangopol ⇑ Dept. of Civil and Environmental Engineering, ATLSS Engineering Research Center, Lehigh Univ., 117 ATLSS Dr., Bethlehem, PA 18015-4729, USA

a r t i c l e

i n f o

Article history: Received 5 July 2011 Revised 13 March 2012 Accepted 14 March 2012 Available online 21 April 2012 Keywords: Structural systems Performance indicators Time-dependent Deterioration Reliability Redundancy Risk Event-tree

a b s t r a c t Structural reliability, redundancy and risk as performance indicators are expected to change over time due to structural deterioration and time-variant loadings. The objective of this paper is to investigate the effects of (a) the deterioration in structural resistance, (b) the type of system modeling, and (c) the correlations among the failure modes of components on the time-dependent reliability, redundancy and risk of structural systems. A representative three-component system is used to demonstrate a general approach for studying these effects. This approach is then applied to an existing highway bridge in Colorado. The bridge is modeled using different types of systems with the consideration of two extreme correlation cases among the failure modes of the girders. An event-tree model is used to assess the direct, indirect, and total risk associated with the failure of component/system due to corrosion and traffic loads. The results reveal the importance of realistic system modeling and time effects in the quantification of reliability, redundancy and risk. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction It has been widely accepted that uncertainties exist in every aspect of the structural safety assessment process. Due to these uncertainties, probabilistic methods have been applied in structural engineering planning and design [4,27,7,3]. Probability-based system performance indicators [36], such as reliability index, redundancy index and risk, have been introduced and used in design, assessment, maintenance, monitoring and prediction of the performance of structural systems [14,6,29,10,30,17,18,24,25,31–34]. Structural systems deteriorate due to mechanical and environmental stressors after they are put in to service. In addition, the loadings acting on these structural systems may also vary with time. Therefore, the values of structural performance indicators are expected to change over time, as well. Structural systems can be generally modeled as series, parallel or series–parallel combination of potential failure modes. The definition of system failure depends on the system modeling type. Therefore, studying the influences of time and system modeling type on the performance indicators can provide a realistic description of structural performance over time and help improving the design and management of structural systems. The objective of this paper is to investigate the effects of resistance deterioration, the type of system modeling, and the ⇑ Corresponding author. E-mail addresses: [email protected] (B. Zhu), [email protected] (D.M. Frangopol). 0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2012.03.029

correlations among failure modes on the time-dependent reliability, redundancy and risk of structural systems. The analysis of these structural performance indicators taking into account the above factors is demonstrated using three-component systems. During the risk analysis process, an event-tree model is used to determine the direct and indirect consequences associated with the failure of component/system due to corrosion and traffic loads. The proposed approach is applied to an existing highway bridge in Colorado. 2. Time-dependent reliability In structural reliability theory, the safe condition is the one in which the failure of the investigated component/system does not occur. For a structural component with resistance r and load effect s, its performance function is:

g ¼rs

ð1Þ

The probability that this component fails is:

Pf ðcomponentÞ ¼ P½g < 0

ð2Þ

For a structural system with at least two failure modes, its failure probability is defined as the probability of violating any of the limit states that are defined by its failure modes:

Pf ðsystemÞ ¼ P½any g i < 0

ð3Þ

where gi is the system performance function with respect to failure mode i. Due to the usual assumption of Gaussian distribution of

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B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49

performance functions, the reliability index associated with the evaluated structural component/system is given by:

b ¼ U1 ðPf Þ

ð4Þ

where U is the standard normal cumulative distribution function. In many previous studies, loads and resistances are assumed to be time-independent random variables [20,28,38,39,22]. Accordingly, the probability of failure obtained from Eq. (3) is kept unchanged during the lifetime of a structure. However, in most practical cases, resistances and loads are time-dependent random variables. In general, the resistances deteriorate and loads increase over time. Therefore, Eqs. (1)–(4) including the time effects can be rewritten as:

gðtÞ ¼ rðtÞ  sðtÞ

ð5Þ

Pf ðcomponentÞ ðtÞ ¼ P½gðtÞ < 0

ð6Þ

Pf ðsystemÞ ðtÞ ¼ P½any g i ðtÞ < 0

ð7Þ

bðtÞ ¼ U1 ðPf ðtÞÞ

ð8Þ

where t = time. It should be noted that the probabilities of failure obtained from Eqs. (6) and (7) are instantaneous. They define the probability of failure at a point-in-time, rather than evaluate the probability within a specified time interval, which is known as cumulative probability of failure. Since the instantaneous probability of failure changes over time, it is time-dependent. In this paper, the ‘‘time-dependent’’ performance indicators (reliability index, redundancy index, and risk) are considered ‘‘point-in-time’’. 3. Time-dependent redundancy

is the consequence associated with the failure at time t. The time-dependent probability of component or system failure of a structure, Pf(t), can be obtained after identifying the component’s performance function or the system failure modes. Consequences caused by components or system failure consist of two parts: (a) direct consequences, CDIR(t), which are related to local components failure, and (b) indirect consequences, CIND(t), which are associated with subsequent system failure [5]. Direct consequences are considered proportional to the initial damage since they include only the commercial loss aspect (i.e., the cost required to replace the damaged component/system), while the indirect consequences are not proportional to the initial damage since they consist of several loss aspects, such as safety loss, commercial loss and environmental loss [21]. An event-tree model for a general case in which component i fails is shown in Fig. 1. In this figure, Fcomp,i represents the event that component i fails; Fsubsys|Fcomp,i and F subsys jF comp;i represent the events that the subsequent system fails and survives given the failure of component i, respectively. The subsequent system discussed herein is the system without component i. In branch b1, only direct consequence exists since only component i fails and the subsequent system survives. However, in branch b2, both direct and indirect consequences occur because the subsequent system fails after component i fails. Therefore, based on the classification of consequences (i.e. direct and indirect consequences), the risk at time t caused by the failure of component i can be split into direct risk RDIR,i(t) and indirect risk RIND,i(t), which are computed as:

RDIR;i ðtÞ ¼ Pf ;comp;i ðtÞ  C DIR;i ðtÞ

ð11Þ

RIND;i ðtÞ ¼ Pf ;comp;i ðtÞ  Pf ;subsysjcomp;i ðtÞ  C IND;i ðtÞ

ð12Þ

System redundancy has been defined as the availability of system warning before the occurrence of structural collapse [30]. Several studies have been performed in presenting measures of quantifying redundancy for structural design or assessment [14– 16,19]. However, no agreement has been reached on redundancy measures yet. In this paper, the time-dependent redundancy index provided in [16]

where Pf,comp,i(t) is the failure probability of component i at time t; Pf,subsys|comp,i(t) is the probability of subsequent system failure at time t given the failure of component i; CDIR,i(t) is the direct consequences at time t associated with the failure of component i (i.e. the cost to replace this component); and CIND,i(t) is the indirect consequences at time t caused by the failure of component i (i.e. the cost to rebuild the subsequent system, safety loss and environmental loss). Finally, the total risk caused by the failure of component i is:

RIðtÞ ¼ bs ðtÞ  bfc ðtÞ

RTOT;i ðtÞ ¼ RDIR;i ðtÞ þ RIND;i ðtÞ

ð9Þ

is used, where bs(t) is the system reliability index at time t and bfc(t) is the reliability index associated with the probability of the first component failure at time t. The larger the difference between these two reliability indices, the higher redundancy the system has. This difference can be interpreted as the availability of system warning before failure. However, redundancy as defined in Eq. (9) cannot be used as the only metric for assessing structural safety. For example, let’s consider two structures whose system reliability indices are 5.0 and 3.0, respectively, and the reliability indices of first component failure are 3.0 and 1.0, respectively. Obviously, the two systems have the same redundancy index (i.e. 2.0), but the first system is much more reliable than the second. Therefore, the redundancy index defined in Eq. (9) should be combined with the information on other performance indicators (such as system reliability and risk) to obtain a more complete assessment of time-variant structural performance.

ð13Þ

5. Three-component systems Most structures can be modeled as series, parallel or series– parallel systems. Three types of three-component systems shown in Fig. 2 are used herein to study the effects of resistance deterioration, system type and correlation among components’ failure modes on the time-dependent reliability, redundancy and risk. In order to study the time effects on the performance indicators of components and systems, a resistance deterioration model presented in [30] is used herein. It is assumed that the deterioration in resistance is mainly due to a continuous cross-section loss over time. The mean lRi(t) and standard deviation rRi(t) of the timedependent resistance of component i at time t are:

lRi ðtÞ ¼ ½1  DRi ðtÞt  Ai ð0Þ  ðlFy Þi rRi ðtÞ ¼ ½1 þ DRi ðtÞt  Ai ð0Þ  ðrFy Þi

ð14Þ ð15Þ

4. Time-dependent risk Risk has become an increasingly important performance indicator. It is defined as the combined effect of chances and consequences of some failure or disaster in a given context:

RðtÞ ¼ Pf ðtÞ  CðtÞ

ð10Þ

where R(t) is the risk caused by a failure in a given context at time t; Pf(t) is the probability of occurrence of the failure at time t; and C(t)

Consequences

Fsubsys |Fcomp,i Fcomp,i Fsubsys |Fcomp,i

Branches

CDIR,i

b1

CIND,i + CDIR,i

b2

Fig. 1. Event-tree risk model for component i failure.

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B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49

component 1

component 3

component 2

(a) component 1 component 1 component 2

component 3

component 3

component 2

(b)

(c)

Fig. 2. Three-component systems: (a) series system; (b) parallel system; and (c) series–parallel system.

where Ai(0) is the initial cross-sectional area of component i; DRi(t) is the deterioration rate of component i at time t; (lFy)i and (rFy)i are the mean and standard deviation of the material yield stress Fy of component i, respectively; and Fy is assumed to be lognormal and constant over time. The load effect is also assumed to be lognormal. The distribution parameters of yield stress and load effect of each component are listed in Table 1. The initial cross-sectional areas and deterioration rates of components are assumed to be deterministic. The deterioration rates of components 1 and 2 are constant during lifetime, while the deterioration rate of component 3 follows an exponential function so that it increases over time. With the parameters given in Table 1, the mean and standard deviation of the resistance of each component at time t can be obtained using the deterioration models in Eqs. (14) and (15). Two cases of correlation among the failure modes of components are considered: statistically independent case, i.e., q(gi, gj) = 0, and perfectly correlated case, i.e., q(gi, gj) = 1, where q is the correlation coefficient. It should be noted that perfect correlation doesn’t mean that all the components will fail together. This situation will occur only when the performance functions of all the components are identical. The service life of the system is assumed to be 50 years. Reliability, redundancy and risk analysis for the three types of systems and two extreme correlation cases are performed.

5.1. Reliability analysis The time-dependent reliability profiles of the three systems shown in Fig. 2 are calculated using RELSYS [12]. The results are

Table 1 Parameters of the three-component system. Parameters

Component 1

Component 2

Component 3

Initial cross-section area Ai(0) (cm2) Initial deterioration rate DRi(0) (per year) Deterioration rate DRi(t) at time t (per year) Yield stress Mean lFyi Fy (kN/cm2) Std. dev. rFyi Load (kN) Mean lQi Std. dev. rQi

2.5

2.0

4.8

0.015

0.004

0.005

DR1(0)

DR2(0)

DR3(0)  (1 + 0.025)t

11.0

6.5

10.0

2.0 5.0 0.5

1.0 4.5 0.45

2.0 9.5 0.95

plotted in Fig. 3. This figure shows that (i) in the case of perfect correlation, the reliability of series system is decided by the lowest component reliability during lifetime, i.e., min[b1; b2; b3], while the reliability of parallel system is determined by the highest component reliability during lifetime, i.e., max[b1; b2; b3]; (ii) for the series–parallel system with perfectly correlated failure modes, the system reliability can be obtained by comparing the reliability of series component and the reliability of the subsequent system consisting of two parallel components, i.e., min[b3; max(b1; b2)]; (iii) for the series system, the system reliability in the independent case is slightly lower than that associated with the perfectly correlated case; and (iv) for the parallel system, the system reliability in the independent case is much higher than that associated with the perfectly correlated case. Given a predefined reliability index threshold, the lifetime of a component/system is considered as the period of time during which the reliability index of the component/system is not lower than the threshold. It is computed by finding the time when the reliability index reaches its threshold. For different reliability index thresholds, the lifetimes of components and systems are listed in Table 2. It is found that (i) component 2 has the longest lifetime among three components due to its lowest deterioration rate; (ii) for the series and series–parallel system, correlations among failure modes of components have no significant effect on system lifetime; however, for the parallel system, the lifetime associated with the independent case is longer than that associated with the perfectly correlated case; and (iii) the lifetime of parallel system is much longer than the lifetimes associated with the other two types of systems regardless of the correlation cases. In order to find the effects of components combination on the reliability of series–parallel system, three types of component combinations are investigated herein, as shown in Fig. 4. The time-dependent reliability profiles associated with combinations II and III are plotted in Fig. 5. Figs. 3d and 5 show that, in the independent case, system reliability is usually controlled by the reliability of the series component; however, if the reliability of the series component decreases much more slowly than those of the other parallel components (i.e., in ‘‘Combination II’’), system reliability will be lower than the reliability of the series component. Table 3 shows the lifetime of the series–parallel system with different component combinations. For ‘‘Combination I’’ and ‘‘Combination III’’, the lifetimes in the independent and perfectly correlated cases are almost the same. However, for ‘‘Combination II’’, the difference in lifetime between the two correlation cases is significant. In the perfectly correlated case, the lifetimes of ‘‘Combination I’’ and ‘‘Combination II’’ are the same and ‘‘Combination III’’ has

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B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49

(b) 9.0

8.0

Component 1 (β1)

Reliability index, β

Reliability index, β

(a) 9.0 7.0 Component 3 (β3)

6.0

Component 2 (β2)

5.0 β target =4.0

4.0

β target =3.0

3.0

β target =2.0

2.0

β1

8.0

Series system

7.0

β3

6.0

β2

5.0 4.0

4.0

3.0

3.0

2.0

2.0

Independent Perfectly correlated

1.0

1.0

0.0

0.0 0

5

10

15

20

25

30

35

40

45

0

50

5

10

15

14.0 12.0

(d) 9.0

10.0

β1 Perfectly correlated

β2

8.0 6.0 4.0 3.0 2.0

4.0 2.0

β3

0.0 0

5

10

15

20

25

25

30

35

40

β1

8.0

Parallel system

Independent

20

30

35

40

45

50

Time, t (years)

Reliability index, β

(c) Reliability index, β

Time, t (years)

45

50

7.0

Series-parallel system

Independent

6.0

Perfectly correlated β2

5.0 4.0

4.0

3.0

3.0

2.0

2.0

β3

1.0 0.0 0

5

10

15

Time, t (years)

20

25

30

35

40

45

50

Time, t (years)

Fig. 3. Time-dependent reliability profiles of: (a) components; (b) series system; (c) parallel system; and (d) series–parallel system.

slightly shorter lifetime than the other two combinations. However, in the independent case, the effect of component combinations on system lifetime is significant and ‘‘Combination II’’ has the longest lifetime. Therefore, although the components of series–parallel system are the same, different component combinations could lead to different system lifetimes. To study the effects of failure modes correlation on the series– parallel system reliability, five correlation cases of ‘‘Combination I’’ are considered as listed in Table 4: (i) the failure modes of all components are statistically independent, denoted as ‘‘IN123’’; (ii) the failure modes of all components are perfectly correlated, denoted as ‘‘PC123’’; (iii) only the failure modes of components 1 and 2 are perfectly correlated, denoted as ‘‘IN13, IN23, PC12’’; (iv) only the failure modes of components 1 and 3 are perfectly correlated, denoted as ‘‘IN12, IN23, PC13’’; and (v) only the failure modes of components 2 and 3 are perfectly correlated, denoted as ‘‘IN12, IN13, PC23’’.

Table 2 Lifetime of components and systems (years). Components and systems Component 1 Component 2 Component 3 Series system

Parallel system

Series–parallel system

Independent Perfectly correlated Independent Perfectly correlated Independent Perfectly correlated

btarget = 2.0

btarget = 3.0

btarget = 4.0

36 94 37

27 62 31

20 35 25

35 36

27 27

20 20

104 94

69 62

51 35

37

31

25

37

31

25

component 1

component 1

component 2

component 3

component 3

component 2

(b)

(a) component 2 component 1

component 3

(c) Fig. 4. Three types of components combinations of series–parallel system: (a) combination I; (b) combination II; and (c) combination III.

B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49 9.0

7.0

5.2. Redundancy analysis

8.0

Series-parallel system Combination II

β1 β3

Independent

6.0

Perfectly correlated β2

5.0

4.0

4.0

3.0

3.0

2.0

2.0 1.0 0.0 0

5

10

15

20

25

30

35

40

45

50

Time, t (years)

Reliability index, β

(b)

9.0

5.3. Risk analysis

Independent

8.0

Series-parallel system Combination III

Perfectly correlated

7.0 6.0

β3 β2

5.0

4.0

4.0 3.0

3.0

2.0

2.0

β1

1.0 0.0 0

5

10

15

20

25

30

35

40

45

50

Time, t (years) Fig. 5. Time-dependent system reliability profiles of series–parallel systems: (a) combination II; and (b) combination III.

Table 3 Lifetime of series–parallel systems (years). Series–parallel systems

btarget = 2.0

btarget = 3.0

btarget = 4.0

Components combination I

37 37

31 31

25 25

46 37

39 31

32 25

36 36

27 27

20 20

Components combination II Components combination III

Independent Perfectly correlated Independent Perfectly correlated Independent Perfectly correlated

Table 4 Five correlation cases of series–parallel system. IN123

PC123

q(gi, gj) = 0.0

q(gi, gj) = 1.0

i, j = 1–3

i,j = 1–3

A series system has no redundancy since the system will fail if any component fails. Therefore, only parallel and series–parallel system are studied in this section. The time-dependent system redundancy profiles of the two systems associated with the two extreme correlation cases are plotted in Figs. 7–9. These figures show that (i) the system redundancy index can decrease or increase over time; (ii) for both systems, redundancy associated with the independent case is higher than that associated with the perfectly correlated case; and (iii) the difference of system redundancy between the two extreme correlation cases is more significant in the parallel system than in the series–parallel system.

This section presents the profiles of time-dependent direct risk, indirect risk and total risk caused by: (i) only one component failure; and (ii) system failure in two extreme correlation cases. For a three-component system, risk caused by component failure or system failure can be evaluated using an event-tree model. The eventtree model for risk analysis associated with only one component failure is shown in Fig. 10. In the three main branches, the event Ei Ej Ek indicates that component i fails while components j and k survive. For the independent (INDP) and perfectly correlated (PC) cases, the probabilities of the event Ei Ej Ek , respectively, are:

PðEi Ej Ek ÞINDP ¼ PðEi Þ  ½1  PðEj Þ  ½1  PðEk Þ

ð16Þ

PðEi Ej Ek ÞPC ¼ min½PðEi Þ;

ð17Þ

ð1  PðEj ÞÞ;

ð1  PðEk ÞÞ

where P(Ei), P(Ej) and P(Ek) are the failure probabilities of component i, j and k, respectively. In the following six branches, the events F i jEi Ej Ek and F i jEi Ej Ek denote that the damaged system (without component i) fails and survives, respectively. Therefore, the probabilities of occurrence of paths bi0 and bi1, respectively, are

(a)

8.0

Series-parallel system Combination I

7.0

Reliability index, β

(a) Reliability index, β

38

6.0

Detail A

5.0

4.0

4.0

3.0

3.0

2.0

2.0 1.0

IN13, IN23, PC12

IN12, IN23, PC13

IN12, IN13, PC23

q(g1, g2) = 1.0 q(g1, g3) = 0.0 q(g2, g3) = 0.0

q(g1, g3) = 1.0 q(g1, g2) = 0.0 q(g2, g3) = 0.0

q(g2, g3) = 1.0 q(g1, g2) = 0.0 q(g1, g3) = 0.0

0.0 0

10

15

20

25

30

35

40

45

50

Time, t (years)

(b) Reliability index, β

Note: IN = Independent failure modes; PC = Perfectly correlated failure modes.

The system reliability indices associated with the five different correlation cases are plotted in Fig. 6. The figure shows that (i) the effects of correlations among components’ failure modes on series– parallel system reliability are not significant; therefore, the system lifetimes associated with these five correlation cases will be the same; (ii) in the three cases where the two parallel components are independent, their system reliability indices are almost the same during the whole lifetime, and their reliability indices are the highest among these five cases; and (iii) the case in which only the two parallel components are perfectly correlated has the lowest system reliability.

5

7.0

Detail A

6.5

IN123

6.0

IN12,IN13,PC23 PC123

5.5

IN12,IN23,PC13

5.0 4.5

IN13,IN23,PC12 4.0 5

10

15

20

25

Time, t (years) Fig. 6. Time-dependent system reliability profiles of series–parallel system in different correlation cases.

39

7.0

3.5

8.0

Parallel system Perfect correlation case

3.0 2.5

6.0

βSystem 2.0

5.0 4.0

βFirst component

1.5

3.0

1.0

2.0 0.5

Redundancy Index

1.0 0.0 0

5

10

15

20

25

30

35

40

45

(a)

8.0

Redundancy index, RI

9.0

Redundancy index, RI

(a) Reliability index, β

B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49

7.0

0.0 50

4.0 3.0

1.0 0.0

Redundancy Index

6.0

5.0 4.5

4.0

4.0

βFirst component

3.5

0.0 0

5

10

15

20

25

30

35

40

45

3.0 50

15

20

25

30

35

40

45

50

1.4 1.2

Series-parallel system

1.0

Independent 0.8

Perfectly correlated

0.6 0.4 0.2 0.0

Time, t (years)

0

Fig. 7. Time-dependent system redundancy profiles of parallel system in: (a) perfect correlation case; and (b) independent case.

8.0

5

10

15

20

25

30

35

40

45

50

Time, t (years) Fig. 9. Time-dependent system redundancy profiles of two types of systems: (a) parallel system; and (b) series–parallel system.

6.0

1.2 1.0

βFirst component

5.0

0.8 4.0

βSystem

3.0

0.6 0.4

2.0

Redundancy Index

1.0 0

5

10

15

20

25

30

0.2 35

40

45

Redundancy index, RI

1.4

Series-parallel system Perfect correlation case

7.0

Reliability index, β

(b) Redundancy index, RI

5.5

βSystem

Redundancy index, RI

Reliability index, β

6.5 6.0

2.0

0.0 50

Time, t (years)

(b) 8.0

Series-parallel system Independent case

βSystem

1.2

6.0

1.0

βFirst component

5.0

0.8 4.0 0.6 3.0

Redundancy Index

2.0

0.4 0.2

1.0 0

5

10

15

20

25

30

35

40

45

Redundancy index, RI

1.4

7.0

Reliability index, β

10

7.0

8.0

0.0

5

Time, t (years)

10.0

0.0

Perfectly correlated

2.0

0

Parallel system Independent case

12.0

(a)

Independent

5.0

Time, t (years)

(b) 14.0

Parallel system

6.0

0.0 50

Time, t (years) Fig. 8. Time-dependent system redundancy profiles of series–parallel system in: (a) perfect correlation case; and (b) independent case.

  Pðbi0 Þ ¼ PðEi Ej Ek Þ  1  PðF i jEi Ej Ek Þ     Pðbi1 Þ ¼ P Ei Ej Ek  P F i jEi Ej Ek

ð18Þ ð19Þ

For the branches corresponding to the path bi0, only direct consequences exist since only component i fails. However, for the branches related to the path bi1, both direct and indirect

consequences are present since both component i and the damaged system (i.e., without component i) fail. Therefore, the direct and indirect risks caused by only component i failure are

  RDIR;i ¼ Pðbi0 Þ  C DIR;i þ Pðbi1 Þ  C DIR;i ¼ P Ei Ej Ek  C DIR;i     RIND;i ¼ Pðbi1 Þ  C IND;i ¼ P Ei Ej Ek  P F i jEi Ej Ek  C IND;i

ð20Þ ð21Þ

which are similar to those presented in Eqs. (11) and (12). In this paper, the direct consequences associated with each component failure and the indirect consequences associated with system failure are assumed to be $10,000 and $100,000 respectively. The risk threshold is assumed to be $102 and risks below this threshold are neglected. By using the event-tree model and the equations above, the direct and indirect risk caused by the failure of only one component associated with two extreme correlation cases in different systems are plotted in Figs. 11 and 12. These figures show that (i) even under the assumption that direct loss is the same for each component (i.e., $10,000), the direct risks caused by the failures of different components are still different due to different component failure probabilities; (ii) the direct risks caused by a certain component failure in different systems are the same since the direct risk is only related to the failed component itself and is independent of the system type; (iii) the correlations among the failure modes of components almost have no effect on the direct risks in the first 40 years due to the fact that the probability values obtained from Eqs. (16) and (17) are almost the same when the failure probabilities of three components are relatively low in the first 40 years; after that, the direct risks associated with perfectly correlated cases are slightly higher than those associated with the independent case; (iv) the indirect risks caused by the failures of different components are significantly different for the series and series–parallel system; however, in the parallel system, the indirect risks due to different components failure are slightly different; (v) the indirect risk caused by the failure of only one component in the series, parallel and series–parallel system

40

B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49

Consequences

F1|E1 E2 E3 F1|E1 E2 E3

E1 E2 E3

F2|E2 E1 E3

Only one component failure

E2E1 E3 F2|E2 E1 E3 F3|E3 E1 E2

E3 E1 E2

F3|E3 E1 E2

Branches

CDIR,1

b10

CIND,1 + CDIR,1

b11

CDIR,2

b20

CIND,2 + CDIR,2

b21

CDIR,3

b30

CIND,3+ CDIR,3

b31

Fig. 10. Event-tree risk model for only one component failure in a three-component system.

1.0E+04

Component 3

Direct risk ($)

1.0E+03 PC

1.0E+02

Component 1

INDP 1.0E+01 1.0E+00 1.0E-01

Component 2

1.0E-02 0

10

20

30

40

50

Time, t (years) Fig. 11. Profiles of direct risk due to the failure of only one component (INDP = Independent failure modes, PC = Perfectly correlated failure modes).

(a) 1.0E+05

Component 3

Series system 1.0E+03

Indirect risk ($)

Indirect risk ($)

(b) 1.0E+02

Component 3

1.0E+04

Component 1 PC

1.0E+02

first reaches the predefined risk threshold at the year t = 11, 33 and 17, respectively; this indicates that, compared to the other two systems, the indirect risk associated with only one component failure in the parallel system is the lowest; and (vi) in general, the indirect risks associated with the perfectly correlated case are slightly higher than those associated with the independent case in all systems. The total risks caused by only one component failure in different systems are shown in Fig. 13. It is noticed that (i) for the failure of different components in different systems, the total risks associated with perfectly correlated case are slightly higher than those associated with independent case; (ii) for only component 1 (or 2) failure, the total risks caused to the parallel and series–parallel system are almost the same, and they are both lower than the total risk associated with the series system; and (iii) for only component 3 failure, the total risks caused to the series system and series– parallel system are the same, and they are both higher than the total risk caused to the parallel system. Therefore, it can be

INDP 1.0E+01 1.0E+00

Component 2

1.0E+01 Component 2

Parallel system

PC

1.0E+00 Component 1 1.0E-01

1.0E-01

INDP

1.0E-02

1.0E-02 0

10

20

30

40

50

0

10

Time, t (years)

30

40

50

Time, t (years)

(c) 1.0E+05

Component 3

PC

1.0E+04

Indirect risk ($)

20

INDP

1.0E+03

Component 1

Series-parallel system

1.0E+02 1.0E+01

1.0E+00 1.0E-01

Component 2

1.0E-02 0

10

20

30

40

50

Time, t (years) Fig. 12. Time-dependent risk profiles of indirect risk due to the failure of only one component in: (a) series system; (b) parallel system; and (c) series–parallel system.

41

B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49

(a) 1.0E+05

(b) 1.0E+02

Series system

1.0E+04 1.0E+03

Total risk ($)

Total risk ($)

Series-parallel system PC

1.0E+02

Parallel system

INDP 1.0E+01 1.0E+00

Only component 2 fails 1.0E+00 PC Parallel system

INDP 1.0E-01

Only component 1 fails

1.0E-01

Series-parallel system Series system

1.0E+01

1.0E-02

1.0E-02 0

10

20

30

40

50

0

10

20

Time, t (years)

(c) 1.0E+05

40

50

Series system

1.0E+04

Total risk ($)

30

Time, t (years)

1.0E+03

Only component 3 fails

1.0E+02

Parallel system

1.0E+01

Series-parallel system

1.0E+00

PC INDP

1.0E-01 1.0E-02 0

10

20

30

40

50

Time, t (years) Fig. 13. Time-dependent total risk profiles of different systems due to the failure of (a) component 1; (b) component 2; and (c) component 3.

(a) 1.0E+05

(b) 1.0E+04 Independent case

1.0E+03 Series system

Only component 1

1.0E+02 Only component 3

1.0E+01

Independent case

1.0E+03

Total risk ($)

Total risk ($)

1.0E+04

1.0E+00 Only component 2

1.0E-01

Only component 1 1.0E+02 Only component 3

1.0E+01

Parallel system

1.0E+00

Only component 2

1.0E-01

1.0E-02

1.0E-02 0

5

10

15

20

25

30

35

40

45

0

50

5

10

15

Time, t (years)

20

25

30

35

40

45

50

Time, t (years)

(c) 1.0E+05 Series-parallel system

Total risk ($)

1.0E+04 1.0E+03

Independent case

1.0E+02

Only component 1 1.0E+01 Only component 3

1.0E+00 1.0E-01

Only component 2

1.0E-02 0

5

10

15

20

25

30

35

40

45

50

Time, t (years) Fig. 14. Time-dependent total risk profiles due to the failure of component and system in the independent case: (a) series system; (b) parallel system; and (c) series–parallel system.

concluded that the failure of the same component in different systems could cause different total risks to these systems. In general, if a component is part of a series system, and an identical component is part of a parallel system, the component failure will cause higher total risk to the series system than to the parallel system. The total risks caused by the failure of the system in the independent and perfectly correlated cases are plotted in Figs. 14 and

15, respectively. Fig. 14 shows that (i) for the series system, the total risk caused by system failure is slightly higher than that caused by only one component failure; while for the parallel system, the total risk caused by system failure is much lower than that caused by only one component failure since the probability of parallel system failure is much less than that associated with a component failure; and (ii) for the series–parallel system, the total risk caused

42

B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49

(b)

1.0E+05 Perfect correlation case

Total risk ($)

1.0E+04 1.0E+03

1.0E+04 Perfect correlation case

1.0E+03

Total risk ($)

(a)

Series system Only component 1

1.0E+02 Only component 3 1.0E+01 1.0E+00

Only component 3

1.0E+01

Only component 1

1.0E+00

Parallel system

Only component 2

1.0E-01

1.0E+02

1.0E-01

1.0E-02

Only component 2

1.0E-02 0

5

10

15

20

25

30

35

40

45

50

0

5

10

15

Time, t (years)

(c)

25

30

35

40

45

50

1.0E+05 Perfect correlation case

1.0E+04

Total risk ($)

20

Time, t (years)

Series-parallel system

1.0E+03 1.0E+02

Only component 1

1.0E+01 1.0E+00 Only component 3 1.0E-01

Only component 2

1.0E-02 0

5

10

15

20

25

30

35

40

45

50

Time, t (years) Fig. 15. Time-dependent total risk profiles due to the failure of component and system in the perfect correlation case: (a) series system; (b) parallel system; and (c) series– parallel system.

by system failure is determined by the total risk caused by the failure of component 3, which is in the series position of the series– parallel system. The results of the perfectly correlated case presented in Fig. 15 indicate that (i) for the series system, the total risk caused by system failure is determined by the highest total risk caused by only one component failure during lifetime; (ii) for the parallel system, the total risk caused by system failure is between the highest and lowest total risks caused by the failures of each component, which is different from the independent case; and (iii) for the series–parallel system, the total risk caused by system failure is also mainly controlled by component 3 during most of the lifetime, which is similar to the independent case. Therefore, the result of the comparison between the total risks of system failure and only one component failure depends on the system type and the correlations among the components’ failure modes. It is noted from Fig. 10 that, for the three-component model, the event-tree model has already six branches if only one component failure is considered. For a real structure with many members, using this event-tree model may lead to combinatorial explosion problem which threatens the assessment of direct/indirect risk of the structure. Since the combinatorial explosion problem usually arises in risk assessment and decision-making areas which are closely related to event-tree and fault-tree models, extensive research on this problem has been performed in the recent decades. Kirkwood [26] developed an algebraic approach to address the combinatorial explosion of decision tree scenarios. Later, Binary Decision Diagrams (BDD) was introduced and proved to be an efficient approach to perform event-tree and fault-tree analysis [2,23]. Therefore, considering the available computational resources, the application of the risk assessment event-tree model in this paper to real structures is feasible. 6. Case study: Colorado State Highway Bridge E-17-AH An existing bridge in Colorado is presented herein as a case study. Bridge E-17-AH is located on 40th Avenue (State Highway

33) between Madison and Garfield Streets in Denver, Colorado. The bridge has three simple spans of equal length (13.3 m) and a total length of 42.1 m as shown in Fig. 16 [11,13]. The deck consists of a 22.9 cm layer of reinforced concrete and a 7.6 cm surface layer of asphalt. The east–west bridge has two lanes of traffic in each direction with an average daily traffic of 8500 vehicles. The roadway width is 12.18 m with 1.51 m pedestrian sidewalks and handrailing on each side. The slab is supported by nine standard-rolled, compact, and non-composite steel girders as shown in Fig. 17 [11,13]. The girders are stiffened by end diaphragms and intermediate diaphragms at the third points. Each girder is supported at one end by a fixed bearing and an expansion bearing at the other end [11,13]. Since the main objective of this case study is to demonstrate the effects of corrosion, system modeling type, and the correlations among the failure modes of components on the system performance indicators rather than assess the bridge safety, the failure modes of substructure and deck are not taken into account and the bridge failure is modeled as the combination of only girders’ flexural failure. Three types of system failure modes are considered: (i) any girder failure causes the bridge failure; (ii) all girder failures cause the bridge failure; and (iii) any two adjacent girders failures cause the bridge failure. Considering the symmetry within the span, the system models can be simplified as shown in Fig. 18 (the girders are numbered in Fig. 17). The limit-state equations of girders 1–5 are listed as follows [11]:

gð1Þ ¼ Z e ðtÞF y cmfg  145:32kconc  37:3ksteel  M trke ðtÞDF e Ibeam ¼ 0 ð22Þ gð2Þ ¼ Z i ðtÞF y cmfg  244:08kconc  28:8kasph  31:7ksteel  Mtrki ðtÞDF ie Ibeam ¼ 0

ð23Þ

gð3; 4; 5Þ ¼ Z i ðtÞF y cmfg  197:65kconc  57:64kasph  31:7ksteel  Mtrki ðtÞDF i Ibeam ¼ 0

ð24Þ

43

B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49

East 42.1 m 13.3 m

13.3 m

13.3 m

Fig. 16. Elevation of Colorado State Highway Bridge E-17-AH.

15.20 m 1.51 m

2.03 m

2

3

2.03 m

2.03 m

2.03 m

5

6

7

4

2.03 m 1.51 m

8

9

9.7 m

0.61 m

3.05 m

1.83 m

2.13 m 0.61 m

1

2.03 m

4.88 m

4.88 m

4.88 m

14.63 m Fig. 17. Cross section of Colorado State Highway Bridge E-17-AH.

where Ze and Zi are the plastic section modulus of girder 1 (G1) and girders 2–5 (G2–G5), respectively; Fy is the yield strength of steel girders; cmfg is the modeling uncertainty factor of steel girder; kasph ; kconc and ksteel are the weight uncertainty factors of asphalt, concrete and steel, respectively; DFe, DFi–e and DFi are the traffic load distribution factors of exterior (G1), exterior–interior (G2) and interior (G3, G4 and G5) girders, respectively; Mtrk-e and Mtrk-i

are the traffic load moment on girder 1 (G1) and girders 2–5 (G2– G5), respectively; and Ibeam is the impact factor of traffic load. The parameters of these random variables are listed in Table 5 [11]. In order to study the time effects on system reliability, redundancy and risk, the variations of girder capacity and live load over time need to be known. In this example, the live load model discussed in [11] is used herein to predict the time-dependent traffic

44

B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49

Girder 1

Girder 3

Girder 2

Girder 4

Girder 5

(a) Girder 1 Girder 2

Girder 1

Girder 2

Girder 3

Girder 4

Girder 2

Girder 3

Girder 4

Girder 5

Girder 3 Girder 4 Girder 5

(c)

(b) Fig. 18. Simplified system models for Bridge E-17-AH: (a) series system; (b) parallel system; and (c) series–parallel system.

volume and estimate the distribution type of the maximum traffic load moment and its associated distribution parameters. According to this model, the maximum traffic load moment follows a type I extreme value distribution and its parameters at the year t can be obtained as follows:

lM ðtÞ ¼ ruðtÞ þ l þ ðcr=aðtÞÞ pffiffiffi rM ðtÞ ¼ ðp= 6Þðr=aðtÞÞ

ð25Þ ð26Þ

where

l ¼ kMmax r ¼ dl

ð27Þ ð28Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðtÞ ¼ 2 lnð365ADTT tÞ uðtÞ ¼ aðtÞ 

ð29Þ

ln½lnð365ADTT tÞ þ lnð4pÞ 2aðtÞ

ð30Þ

in which, ADTT is the average daily truck traffic considered to be 850 trucks per day for this bridge [11]; k is the ratio between the traffic load moment and the HS-20 moment; d is the coefficient of variation; Mmax is the critical traffic load moment under the HS-20 truck load, which is equal to 351.2 kN m [11]; and c = 0.5772 (the Euler number). The values associated with the parameters k and d are 0.65 and 0.32, respectively [11]. The corrosion model used for the steel girders in this paper is based on Albrecht and Naeemi [1]. The average corrosion penetration c(t) at time t is:

cðtÞ ¼ ptq

ð31Þ

where p and q are the regression random variables based on the environment and type of steel. For exterior (G1) and interior–exte-

Table 5 Parameters of the random variables associated with the material properties and traffic load effects for the Bridge E-17-AH [11]. Variables

Random

Variables

Random

Fy (Mpa)

N[250, 30] N[1.11, 0.128] N[0.982, 0.122] N[1.14, 0.142] N[1.309, 0.163]

kasph kconc ksteel Ibeam –

N[1.0, 0.25] N[1.05, 0.105] N[1.03, 0.082] N[1.14, 0.114] –

cmfg DFe DFi-e DFi

Notes: N[l, r] = the random variable is normally distributed; the mean is l and the standard deviation is r.

rior (G2) girders, the means (coefficients of variation) associated with p and q are 80.2 (0.42) and 0.593 (0.4), respectively. The correlation coefficient between p and q is assumed to be 0.68. For interior girders (G3, G4, and G5), the two random variables are assumed to be uncorrelated and the means (coefficients of variation) of p and q are 34.0 (0.09) and 0.65 (0.1), respectively. Based on the parameters of random variables and the limitstate equations of girders, the time-dependent reliability indices of each girder are plotted in Fig. 19a. This figure shows that the exterior girder (G1) has the highest lifetime reliability while an interior girder (G3, G4 or G5) has the lowest reliability up to 60 years. After that time, the reliability of exterior–interior girder (G2) becomes the lowest due to its larger corrosion rate. The reliability indices of three types of systems considering three correlation cases among the failure modes of girders (independent case: q = 0; partially correlated case: q = 0.5; and perfectly correlated case: q = 1) are plotted in Fig. 19b–d. It can be seen that for the series system (see Fig. 19b), the reliability index in the case of perfectly correlated is determined by an interior girder (G3, G4 or G5) when t 6 60 years, and by the exterior–interior girder (G2) when t > 60 years. In the case of independence, the reliability index of the series system is smaller than that associated with the perfectly correlated case. For the parallel system and series–parallel system, the reliability index in the perfectly correlated case is decided by the exterior girder (G1) and an interior girder (G3, G4 or G5) during the entire lifetime, respectively. Contrary to the series system, the reliability indices associated with the independent case of parallel and series–parallel systems are much larger than those associated with the perfectly correlated case. Compared to the average value of the reliability indices associated with the two extreme cases, the reliability index in the partially correlated case is slightly lower in the series system but slightly higher in the parallel and series–parallel systems. The profiles of redundancy indices of the parallel and series– parallel systems associated with three correlation cases are shown in Fig. 20. This figure indicates that (i) the redundancy indices of both systems decrease slightly over time in all correlation cases; (ii) the redundancy index of series–parallel system associated with the perfectly correlated case is almost zero; (iii) for both systems, the redundancy index associated with the independent case is higher than that associated with the perfectly correlated case; and (iv) the redundancy indices associated with the partially correlated case in the two systems are both higher than the average values of the two extreme correlation cases.

45

(a) 5.5 Reliability index

5.0

(b)

5.5

Reliability index

B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49

4.5

G1

4.5 4.0 3.5 G2

3.0 2.5 2.0

G1

5.0 Series System

4.0

Perfectly correlated

3.5

G2

3.0

Partially correlated

2.0 1.5

G3, G4, G5

Independent

1.0

1.5 0

10

20

30

40

50

60

70

0

80

10

20

30

8

Independent

9

Parallel System

7 6

G1

Partially correlated

5 4

G2 Perfectly correlated

3 2

40

50

60

70

80

Time, t (years)

(d)

5.5

Reliability index

10

Reliability index

Time, t (years)

(c)

G3, G4, G5

2.5

4.5

Series-parallel System

5.0 G1

4.0

Independent

3.0 2.5 2.0

G2

1.5

G3, G4, G5

1

Partially correlated

3.5

G3, G4, G5

Perfectly correlated

1.0 0

10

20

30

40

50

60

70

80

0

10

20

Time, t (years)

30

40

50

60

70

80

Time, t (years)

Fig. 19. Time-dependent reliability profiles of Bridge E-17-AH: (a) girders; (b) series system; (c) parallel system; and (d) series–parallel system.

When assessing the risk of a bridge due to a hazard, the following factors need to be identified: the type of hazard, the probability of component/system failure given this hazard, and the associated failure consequences. In this example, the hazards are corrosion and traffic loads. The probability of a certain component or system failure due to these two hazards can be obtained from the reliability analysis above. Therefore, the only unknown factor is the consequence associated with the bridge failure. Quantification of the consequence caused by the failure of a bridge is a difficult task since it includes several aspects related to commercial, safety, and environmental losses [21]. Based on Rackwitz [35] and Stein et al. [37], the commercial and safety loss associated with the E-17-AH bridge failure are evaluated as follows:

2. Commercial loss: running cost CRunning Based on the length of the detour that users are forced to follow in the case of bridge failure, a general formula provided by Stein et al. [37] is:

C Running ¼ C Veh DAd

ð32Þ

where CVeh is the average running cost for vehicles ($/km); A is the average daily traffic (vehicles/day); D is the length of detour (km); and d is the duration of detour (days). The values of these parameters used for this bridge are presented in Table 7. 3. Commercial loss: time loss cost CTL Based on the time loss for users and goods traveling through the detour, a formula given by [37] is:

1. Commercial loss: rebuilding cost CReb Based on the repair costs of replacement options in Estes and Frangopol [13], the rebuilding costs of each item of the superstructure are listed in Table 6.

 

T Trk T Trk DAd þ C T v rk C TL ¼ C T v a Ocar 1  100 100 S

ð33Þ

where CTva is the value of time per adult ($/h); OCar is the average vehicle occupancy for cars; TTrk is the average daily truck traffic (%); CTvtk is the value of time for truck ($/h); and S is the average detour speed (km/h). The values of these parameters used for this bridge are also presented in Table 7.

7.0

Redundancy index

4. Safety loss cost CSL

Parallel, Independent

6.0 5.0

Based on the number of casualties and ICAFB (‘‘implied cost of averting a fatality for bridge engineering’’) in USA [35], the safety loss cost can be computed as:

Parallel, Partially correlated

4.0

Parallel Perfectly correlated

3.0

Series-parallel Partially correlated

Series-parallel Independent

2.0

Table 6 Rebuilding cost of each item in the superstructure of Bridge E-17-AH.

Series-parallel Perfectly correlated

1.0 0.0 0

10

20

30

40

50

60

70

80

Time, t (years) Fig. 20. Time-dependent redundancy profiles for different systems of Bridge E-17AH.

Rebuilding item

Notation

Cost ($)

Each girder Sidewalk, guard rails, the portion of slab under sidewalk (both sides) Slab (without the portion under sidewalk) Superstructure

CG CW

29,050 113,000

CS CSup

112,600 487,100

46

B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49

Table 7 Parameters for the consequences evaluation of Bridge E-17-AH. Definition and units of parameters

Notation

Value

Reference

Average daily traffic (ADT) (vehicles/day) Average daily truck traffic (ADTT) (%) Length of detour (km) Duration of detour (days) Average running costs for vehicles ($/km) Value of time per adult ($/h) Value of time for truck ($/h) Average vehicle occupancy for cars Average vehicle occupancy for trucks Safe following distance during driving (m) Average detour speed (km/h) Implied cost of averting a fatality for bridge engineering ($) Total bidge length (m)

A TTrk D d CVeh CTva CTvtk OCar OTrk DS S ICAFB L

8500 10 0.64 180 0.16 7.05 20.56 1.5 1.05 30 64 2.6  106 42.1

[11] [11] Based on the local transportation network [9] [37] [37] [37] [9] [9] [8] [37] [35] [11]

Table 8 Direct and indirect consequences caused by component and system failure of Bridge E-17-AH. Failure item

Direct consequence

Indirect consequence

Exterior girder (G1) Exterior–interior girder (G2) Interior girder (G3 or G4) Interior girder (G5) Superstructure

2CG + CW 2CG + CW + CS/3 2CG + 2CS/3 CG + CS/3 CSup

(CSup  2CG  CW) + CRunning + CTL + CSL (CSup  2CG  CW  CS/3) + CRunning + CTL + CSL (CSup  2CG  2CS/3) + CRunning + CTL + CSL (CSup  CG  CS/3) + CRunning + CTL + CSL CRunning + CTL + CSL

L þ1 DS



  T Trk T Trk OCar þ 1 OTrk ðICAFBÞ 100 100

ð34Þ

where L is the total bridge length (m); DS is the safe following distance during driving (m); and OTrk is the average vehicle occupancy for trucks. The values of these parameters used for this bridge are also listed in Table 7. For the failure of the bridge system, the direct consequence CDIR,S is determined by the rebuilding cost of the whole superstructure, and the indirect consequence CIND,S consists of the running cost, the time loss cost, and the safety loss cost. For the component (girder) failure, the direct consequence CDIR,C is the cost to replace the girder and its adjacent deck parts. Due to the symmetry consideration in system modeling, the failure of girder Gi (except G5) in Fig. 18 actually represents the failure of two girders: girder Gi and the other symmetrical girder. Therefore, the replacing cost of girder Gi (except G5) included in the direct consequence analysis is the double of CG, as listed in Table 8. For the girder G5, since it is located in the center line of the cross-section and has no symmetrical counterpart, its replacing cost is CG. Based on the assumption that the failure of a girder will cause the failure of its adjacent deck parts, the cost to replace these deck parts is included in the direct consequence evaluation. The failed deck parts associated with different girder failures are different. For the exterior girder (G1), its associated deck parts are defined as the sidewalk, guard rails and the portion of slab under the sidewalk; for the exterior–interior girder (G2), associated deck parts are the sidewalk, guard rails, the portion of slab under the sidewalk, and the portion of slab between G2 and G3; and for an interior girder (G3, G4 or G5), its associated deck parts are the portions of slab which are adjacent to the girder. The indirect consequence CIND,C includes the rebuilding cost of the damaged bridge system (without the girder and its affiliated deck parts), the running cost, the time loss cost, and the safety loss cost. The detailed items included in the direct and indirect consequences associated with the system or component failure are presented in Table 8. Considering an annual money discount rate rm of 2%, the future monetary value of the consequences CFV at the year t is given by:

C FV ¼ C PV ð1 þ rm Þt

ð35Þ

where CPV is the present monetary value of the consequences. Based on the time-dependent failure probabilities and the associated consequences, the time-dependent direct risk, indirect risk and total risk caused by component or system failure can be assessed using the risk event-tree model. The risk threshold in this example is defined as $102 and the risks below this limit are neglected. The direct risks caused by the failure of each girder are shown in Fig. 21. This figure shows that (i) the highest direct risk is caused by the failure of an interior girder (G3 or G4) in the first 40 years and then by the exterior–interior girder (G2) in the next 40 years; (ii) the lowest direct risk is caused by the failure of exterior girder (G1); and (iii) the effect of the correlations among the components’ failure modes on the direct risks can be neglected. The indirect risks caused by the failure of each girder are plotted in Fig. 22. The indirect risks of the parallel system in the independent case are not included in this figure since they are all less than the defined risk threshold ($102) during the lifetime. It can be found from Fig. 22 that: (i) in both series and series–parallel system, the indirect risks caused by the exterior girder (G1) failure are much lower than those caused by the failure of other girders;

1.0E+05

G2 1.0E+04

Direct Risk ($)

 C SL ¼

1.0E+03

G3, G4 1.0E+02

G5 PC

1.0E+01

INDP

1.0E+00

G1

1.0E-01 1.0E-02 0

10

20

30

40

50

60

70

80

Time, t (years) Fig. 21. Time-dependent direct risk profiles due to the failure of only one girder.

47

B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49

(a) 1.0E+07

(b) 1.0E+02

G3, G4, G5

Parallel system Perfect correlation case

Indirect risk ($)

Indirect risk ($)

1.0E+06

1.0E+05 Series system

PC

1.0E+04

INDP

G2

1.0E+03

1.0E+02

1.0E+01

G3, G4, G5 1.0E+00

1.0E-01

G1

G1

1.0E+01 1.0E+00 0

10

20

30

40

50

60

70

1.0E-02

80

0

10

20

Time, t (years)

(c) 1.0E+05

(d) 1.0E+05 Indirect risk ($)

1.0E+03 G5

1.0E+02

G2

Series-parallel System Independent case

1.0E+01 1.0E+00

G1

1.0E-01

40

50

60

70

80

G2

G4, G5

1.0E+04

G3

30

Time, t (years)

G4

1.0E+04

Indirect risk ($)

G2

G4, G5

1.0E+03

G3

G3

1.0E+02

G2

1.0E+01 G1

1.0E+00

Series-parallel system Perfect correlation case

1.0E-01 1.0E-02

1.0E-02

0

10

20

30

40

50

60

70

80

0

10

20

Time, t (years)

30

40

50

60

70

80

Time, t (years)

Fig. 22. Time-dependent indirect risk profiles due to the failure of only one girder in: (a) series system; (b) parallel system; (c) series–parallel system in independent case; and (d) series–parallel system in perfect correlation case.

(ii) in both correlation cases, the indirect risks caused by the failure of any interior girder (G3, G4 or G5) are the same in series system but slightly different in the series–parallel system; (iii) the correlations among components’ failure modes have almost no effect on the indirect risks in series system; however, for parallel and series–parallel system, the effect cannot be neglected; and (iv) the indirect risks caused by the failure of the same girder in different systems and different correlation cases are different.

(a) 1.0E+07

The total risks caused by the failure of each girder and each system in the independent case are plotted in Fig. 23. This figure shows that (i) in the series system, the total risk caused by system failure is higher than that caused by each girder failure; whereas in the series–parallel system, the system risk is slightly lower than the component risk in the first 18 years and higher than the component risk in the next 62 years; in the parallel system, the system risk is not shown in the figure since it is much less than the risk

(b) 1.0E+10

1.0E+06 Series system

G3, G4, G5

1.0E+04

Total risk ($)

Total risk ($)

1.0E+08

1.0E+05 G2

1.0E+03

G1

1.0E+02

1.0E+00

0

10

20

30

40

50

60

70

1.0E+04

G1 Parallel system Independent case

1.0E+02

Independent case

1.0E+01

G2 1.0E+06

1.0E+00

80

0

10

20

Time, t (years)

(c)

30

(d) 1.0E+05

1.0E+05 G3

G2

Total risk ($)

Total risk ($)

G5

G4

1.0E+02

Series-parallel system Detail B

1.0E+01 G1 1.0E+00

1.0E-02

10

20

30

40

50

Time, t (years)

70

80

G4

1.0E+04 G5

60

70

Detail B

G2 1.0E+03

0

60

G3

Independent case

1.0E-01

50

Series-parallel system

1.0E+04 1.0E+03

40

Time, t (years)

80

0

10

20

30

40

50

60

70

80

Time, t (years)

Fig. 23. Time-dependent total risk profiles due to the failure of component and system in independent case: (a) series system; (b) parallel system; (c) series–parallel system; and (d) detail B.

48

B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49

(a)

(b) 1.0E+05

1.0E+07 Series system 1.0E+05

G3, G4, G5

1.0E+04 Perfect correlation case

G2 1.0E+03

1.0E+02

0

10

20

30

40

50

G3, G4

1.0E+03 G5 1.0E+02

Parallel system

1.0E+01 G1

1.0E+00

G1

1.0E+01

1.0E+00

G2

1.0E+04

Total risk ($)

Total risk ($)

1.0E+06

Perfect correlation case

1.0E-01

60

70

1.0E-02

80

0

10

20

30

40

50

60

70

80

Time, t (years)

Time, t (years)

(c) 1.0E+07

Series-parallel system

1.0E+06

Total risk ($)

1.0E+05

G4, G5

1.0E+04

G3

1.0E+03

G2

1.0E+02 Perfect correlation case

1.0E+01 1.0E+00

G1

1.0E-01 1.0E-02

0

10

20

30

40

50

60

70

80

Time, t (years) Fig. 24. Time-dependent total risk profiles due to the failure of component and system in perfect correlation case: (a) series system; (b) parallel system; and (c) series–parallel system.

threshold $102; therefore, it can be concluded that the system risk is significantly lower than the component risk in the parallel system; and (ii) in both series and series–parallel system, the total risks caused by the failure of exterior girder (G1) are lower than those caused by the failure of other girders; however, the total risk associated with the exterior girder (G1) failure in the parallel system is much higher than that associated with an interior girder (G3, G4 or G5) failure (which are less than the risk threshold and not plotted in the figure). The total risks associated with the perfectly correlated case are presented in Fig. 24. It can be noticed that, (i) the total risk caused by the failure of the series system is determined by an interior girder (G3, G4 or G5) in the first 62 years and then by the exterior– interior girder (G2) in the next 18 years; (ii) compared with the total risks in the independent case (Fig. 23), the total risk associated with the failure of series system is slightly lower in the perfectly correlated case; however, for the parallel and series–parallel system, the total risks caused by system failure are much higher; and (iii) for the same bridge, different definitions of the system failure result in different total risks and the failure of the same girder may cause different total risks under different system failure definitions.

2.

3.

4.

7. Conclusions This paper investigated the effects of the structural resistance deterioration, the type of system modeling, and the correlation among the failure modes of components on the time-dependent reliability, redundancy, and risk of structural systems. A methodology for assessing lifetime reliability, redundancy and risk is presented and implemented using an existing highway bridge. The following conclusions are drawn: 1. The effect of correlation among the failure modes of components on the direct risk caused by a single component failure is independent of the type of system modeling. For the three

5.

different systems investigated, the direct risks in the perfectly correlated case are all slightly higher than those in the independent case. Similar conclusion is drawn for the indirect risk in series system. For the parallel system, the difference in the indirect risks between two extreme correlation cases is more significant than that in the series system. However, such a conclusion cannot be extended to the series–parallel system. The effect of deterioration of structural members will cause the degradation of reliability and the increase of risk over time. However, the tendency of redundancy changing with time is uncertain. It may decrease, remain the same, or even increase within the lifetime, depending on the effect of the deterioration on the resistance of all the system components. The event-tree model used in this paper considers the direct and indirect consequences caused by the component/system failure and can be used to assess the direct, indirect and total risk associated with the component/system failure for different systems. The direct risks caused by the failure of a certain component are independent of the system type. However, the corresponding indirect risks are influenced not only by the system type but also by the component position in the system. For a component in series position, the indirect risks caused by its failure in series and series–parallel systems are almost the same within the whole lifetime. However, for a component in parallel position, the indirect risks caused by its failure in parallel and series–parallel systems are almost the same in the first several decades and then the difference appears and increases as the deterioration of components gets worse. The total risks associated with the failures of different components in a system may be different. Failure of a component with higher lifetime reliability will cause lower total risk to the system. For the series or series–parallel system, the total risk caused by system failure is higher than that caused by component failure; however, for the parallel system, the opposite conclusion is drawn. Therefore, when assessing risk for series or

B. Zhu, D.M. Frangopol / Engineering Structures 41 (2012) 34–49

series–parallel system, the focus should be on the total risk caused by system failure; while for the parallel system, the attention should be focused on the total risk due to component failure.

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