Renewal theory-based life-cycle analysis of deteriorating engineering systems

Structural Safety 50 (2014) 94–102

Contents lists available at ScienceDirect

Structural Safety journal homepage: www.elsevier.com/locate/strusafe

Renewal theory-based life-cycle analysis of deteriorating engineering systems R. Kumar a,1, P. Gardoni b,⇑ a b

United States Catastrophe Risk Engineering, AIR Worldwide, 388 Market Street, San Francisco, CA 94111, United States Department of Civil and Environmental Engineering, University of Illinois, 3118 Newmark Civil Engineering Laboratory, 205 N. Mathews Ave., Urbana, IL 61801, United States

a r t i c l e

i n f o

Article history: Received 5 March 2013 Received in revised form 11 February 2014 Accepted 29 March 2014

Keywords: Life-cycle analysis Deterioration Stochastic modeling Life-cycle cost Time-dependent reliability

a b s t r a c t Engineering systems typically deteriorate due to regular use and exposure to harsh environment. Under such circumstances the owner of a system must take important decisions such as whether to repair, replace or abandon the system. Such decisions can affect the safety of, and the benefits to the users and the owner. Life-cycle analysis (LCA) provides a rational basis for such decision making process. In particular, LCA can provide helpful information on the performance of a system over its entire life-cycle, like its time-dependent reliability, the costs associated with its operation, and other quantities related to the service life of the system. This paper proposes a novel probabilistic formulation for LCA of deteriorating systems named Renewal Theory-based Life-cycle Analysis (RTLCA). The formulation includes equations to obtain important life-cycle variables such as the expected time lost in repairs, the reliability of the system and the cost of operation and failure. The proposed RTLCA formulation is based on renewal theory and proposes analytical solutions for the desired LCA variables using numerically solvable integral equations. As an illustration, the proposed RTLCA formulation is implemented to analyze the life-cycle of an example reinforced concrete (RC) bridge located in a seismic region. This analysis accounts for the accumulated seismic damage in the bridge columns caused by the earthquakes occurring during bridge’s life-cycle. The analysis results provide valuable insight into the importance of seismic damage in a bridge’s life-cycle performance and the strategies to operate a system in an optimal manner. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Engineering systems have to be operated in a strategic manner in order to maximize the safety of, and the benefits to the users and owners. Such operation strategies can be effectively devised only by conducting a life-cycle analysis (LCA) of the system. In LCA, the performance of a system, over its entire life-cycle, is studied in terms of a variety of performance measures such as reliability and its dependence on age of the system, the costs and benefits of operation a system taking into account the influence of repairs and maintenances. LCA must factor in the uncertainties in the operating conditions (e.g., environmental conditions, intensity and time of occurrence of loads) and, for deteriorating systems, it is extremely critical to model the process of deterioration and its effect on life-cycle performance of the system.

⇑ Corresponding author. Tel.: +1 2173335103. E-mail addresses: [email protected] (R. Kumar), gardoni@illinois. edu (P. Gardoni). 1 Tel.: +1 (415) 276 4103; fax: +1 (415) 912 3112. http://dx.doi.org/10.1016/j.strusafe.2014.03.012 0167-4730/Ó 2014 Elsevier Ltd. All rights reserved.

In the past, several researchers have developed models for LCA using the well known renewal theory. Renewal theory based models attract attention because they minimize the need for computationally expensive simulations and offer analytical equations to estimate the life-cycle performance measures for a system. However, renewal theory models are built on a well known assumption i.e., when a system is repaired its original properties are restored. Any form of repair is considered to be a complete renewal of the system and therefore partial repairs cannot be handled in renewal theory. Furthermore, all renewal cycles are (i.e. time period between two renewals) are assumed to be independent of each other. With these assumptions, researchers have time to time proposed various LCA models. Rackwitz [1] proposes equations to compute the expected values of benefit derived from operating a system, the cost of failures, the availability of the system and to optimize the design of system based on minimum failure rates. The work develops all the equations for infinite time horizon and for only one type of failure of the system (i.e. either service ability or ultimate failure). Streicher and Rackwitz [2], and Joanni and Rackwitz [3] compute the expected values of failure cost and

95

R. Kumar, P. Gardoni / Structural Safety 50 (2014) 94–102

2. Life-cycle of an engineering system Fig. 1 shows the various events in the life-cycle of an engineering system that is experiencing deterioration. The state of the system at a given time t is described in terms of the probability of ultimate failure Pf(t) of the system given that a load acts on the system at time t. Changes in Pf(t) occur in the form of discrete or continuous increments. Discrete increments are due to shocks that cause sudden changes in the system properties. Continuous increments in Pf(t) are due to a gradual deterioration of the system

TDi

TIi

t Ii

Repair

Tli

Ii (preventive) Lag

pa

TDi+1

TIi+1

Li t Li

Tli+1

t Ii+1

Replacement

1.0

down

in use

Lag

down

Ii+1 (essential)

in use

Pf (t)

benefits of a deteriorating system that is being inspected at regular time intervals followed by repairs. The probability of repair or no repair after inspection is computed based on either age or condition of the system. Similarly, Noortwijk [4], Noortwijk and Weidi [5] and Weidi et al. [6] proposed equations for expectation and variance of life-cycle costs for infinite time horizon. Noortwijk and Frangopol [7] compare the results from a renewal model and simulation based model. Several researchers have also developed methods that are not based on renewal theory. Typically, such methods are either purely based on monte-carlo simulation [8– 16] or are analytical methods based on some simplifying assumptions. Simulations based methods are widely applicable but can be computationally expensive. Analytical methods not based on renewal theory [17–24] are typically unable to model deterioration as a stochastic process and mostly a deterministic function of time is used to represent deterioration. Furthermore, it is difficult to consider both service ability and ultimate types of failures using such methods. This paper proposes a novel renewal theory based LCA model (RTLCA) for deteriorating systems. The proposed model provides equations to compute the instantaneous probability of being in service, the expected values and variances of availability, age, benefit, and costs of operation and failures of the system for a finite time horizon. The model accounts for both serviceability and ultimate failures. Although many of the mentioned concepts exist in the available renewal models, in the past they have been discussed primarily in the context of infinite time horizon which is not representative of an engineering system’s life-span. In this paper, we derive all the equations for finite time horizon. In our knowledge, the proposed concepts and treatment of instantaneous probability of being in service and age is novel in LCA. The repair durations are considered dependent on the level of damage and are not considered negligible as has often been found in existing models. The presented formulation is made general so that equations are not dependent on any specific method to model deterioration as long as certain renewal probabilities and density functions can be computed. An example is presented to illustrate the proposed RTLCA formulation where the life-cycle of a reinforced concrete (RC) bridge is analyzed accounting for the deterioration caused by earthquakes and corrosion. The occurrence of earthquakes is modeled using a time-dependent stochastic process accounting for both main shocks and aftershocks. The example considers delays in repair after earthquakes and the damage accumulated during this delay due to aftershocks. Corrosion is modeled as a function of time with a random initiation time. This paper is organized into six sections including this introduction. The second section describes the events typically observed in the life-cycle of an engineering system and introduces a few definitions used in the paper. The third section develops the equations for computing various LCA variables based on renewal theory. The fourth section briefly describes an existing stochastic deterioration model that can be used for RTLCA. The fifth section uses the stochastic deterioration model and the proposed RTLCA formulation to analyze the life-cycle of an example RC bridge. The sixth section presents the conclusions derived from this work.

Li+1

t Li+1

Time, t Fig. 1. Life-cycle of an engineering system.

properties due to phenomena like corrosion of steel, alkali-silica reactions, delayed-ettringite formation, creep, etc. Fig. 1 shows that an engineering system experiences alternating phases of being in use and in down-time. A system is said to be in use at time t if the system is functioning at that time. On the other hand, a system is said to be down or experiencing down-time if the system is either abandoned or removed from the service for repairs or replacement. In this paper, we call the start of a down-time as an intervention (I). The down-time of a system ends when the repair or replacement is complete and the system starts functioning again. In this paper, we call this event renewal (L). As mentioned earlier, interventions can be preventive or essential. Preventive interventions are typically made when a pre-determined safety related intervention criterion is met. Some examples of intervention criteria are: the exceedance of a threshold intensity of the applied load, a serviceability type failure such as exceedance of a threshold level for damage or Pf(t), and reaching a pre-determined time elapsed since previous renewal (like in the case of a scheduled maintenance). Fig. 1 shows that the ith intervention Ii that occurs at time tIi is preventive and is conducted because Pf(t) P pa. The figure also shows that Iiþ1 is an essential intervention and occurs because the system experiences an ultimate failure at time t Iiþ1 because of which Pf(t) jumps to 1.0. The corresponding renewal events Li and Li+1 occur at time tLi and t Liþ1 , respectively. In the figure, T Ii is the time interval between Li1 and Ii and T Di is the down-time following Ii. For some systems, deterioration does not progress during the down-time because the system is removed from service and it is immediately repaired. However, in some cases (as shown in the figure) the actual repair work may not begin immediately at t Ii and a lag period (T li following Ii and T liþ1 following Ii+1) may exist during which the deterioration process may continue. Generally, this is the time required for the mobilization of the required resources. For example, an infrastructure that has been closed due to damage from an earthquake is still exposed to aftershocks before the repairs or replacement might take place. In such cases, the lag period may significantly affect the LCA and hence must be considered.

3. Economic feasibility considerations for a system The costs incurred in the life-cycle of the system after its initial construction can be grouped into either cost of operation COp(t) or failure losses CL(t). The cost COp(t) is the total cost of repairs and replacement of the system following the serviceability and ultimate failures in order to operate the system up to time t. The

96

R. Kumar, P. Gardoni / Structural Safety 50 (2014) 94–102

P1

cost of operation is written as C Op ðtÞ ¼ i¼1 ½cOpi ectLi 1fiNL ðtÞg , where cOpi is the cost of repair or replacement occurring between events Ii and Li, c is the discount rate to compute the net present value (NPV) of the cost, NL(t) is the number of renewals in time t and 1{X} is the indicator function which is equal to 1 if statement X is true and 0 otherwise. The failure losses CL(t) is the sum of losses arising from injuries, deaths or damage to user’s property until time t. Such losses are observed only during ultimate failures. The loss CL(t) does not include the cost of replacing the system. P ct I i1 Therefore, C L ðtÞ ¼ 1 fiNI ðtÞg , where c Li is the loss correi¼1 ½cLi e sponding to Ii and NI(t) is the number of interventions in time t. The value of cLi is 0 if Ii is not due to an ultimate failure and it is positive otherwise. In addition to the described costs, another important economic consideration is benefit Q(t). The benefit Q(t) is the NPV of direct or indirect benefit derived from operating a system for time t (e.g., price of automobile fuel saved by commuters using a bridge). Using a formulation similar to Rackwitz [1], the operation of a system for time t is considered economically justified if

Q net ðt Þ ¼ Q ðt Þ  C Op ðtÞ  C L ðtÞ  C C > 0

ð1Þ

4. Proposed formulation for LCA In this study, we propose the RTLCA formulation which is based on renewal theory [25]. In renewal theory, a renewal process NE(t) is the number of occurrences in time t of an event E, called the renewal event. The time intervals between consecutive occurrences of E, also called the renewal times, are considered statistically independent and identically distributed (SIID). Therefore mathematically, N E ðtÞ ¼ maxfn : tEn  tg, where max {} is the maximum value function, En is the nth occurrence of E and tEn is the time of occurrence of En Now if T Ei are the renewal times, then P tEn ¼ ni¼1 T Ei . As mentioned earlier, fT Ei g is assumed to be a SIID sequence in renewal theory and based on this property, the renewal theory offers analytical solutions in the form of integral equations to compute several quantities that describe the renewal processes. In the RTLCA formulation, we propose to model the occurrences of Li as a renewal process assuming, as required in renewal theory, that the renewal times fT Li g are SIID, where fT Li g is given by fT Ii þ T Di g. This assumption requires that the events in the time interval ðtLi ; t Liþ1  are independent of the events in all other such intervals. This implies that the decisions associated to Li must be based only on the events occurring within the time interval ðt Li1 ; t Li . It also implies that the loading and deterioration process in the interval ðt Li ; t Liþ1  should be independent of the loading and deterioration process in other such intervals. Furthermore, we need to assume that the system is completely renewed after Li and its properties at t Li are identical to the original properties at t = 0. This simplification along with the above mentioned assumptions are required to ensure that fT Li g is a SIID sequence. Furthermore, in the RTLCA formulation, the event Li can be of type LR or LF hereafter written as events Li  LF and Li  LR respectively. For any Li, the events Li  LF and Li  LR occur with probabilities P(LR) = P(Li  LR) and P(LF) = P(Li  LF) independently of i. The type of renewal does not depend on the events that occur after tIi . For example, Li  LR even if there is a failure of the system in the lag period following Ii. It follows that P(LR) + P(LF) = 1, and

fT L ðsÞ ¼ fT L jLF ðsjLF ÞPðLF Þ þ fT L jLR ðsjLR ÞPðLR Þ

ð2Þ

where fT L ðsÞ is the probability density function (PDF) of T Li ; fT L jLR ðsjLR Þ is the PDF of T Li given that Li  LR and fT L jLF ðsjLF Þ is the PDF of T Li given that Li  LF . Similarly,

fT I ðsÞ ¼ fT I jLF ðsjLF ÞPðLF Þ þ fT I jLR ðsjLR ÞPðLR Þ

ð3Þ

where fT I ðsÞ is the PDF of T Ii ; fT I jLR ðsjLR Þ is the PDF of T Ii given that Li  LR and fT I jLF ðsjLF Þ is the PDF of T Ii given that Li  LF . In the following, we propose equations using renewal theory to estimate some important LCA variables. In order to implement the model, probabilities and the conditional PDFs mentioned in Eqs. (2) and (3) have to be computed that may require simulation of the actual events of one renewal (note that all renewals are SIID). However, since only one renewal needs to be simulated, this method is computationally more efficient than conducting Monte Carlo simulations of the entire life-cycle.

4.1. Computing the age In order to determine the level of deterioration in a system at time t, it is important to determine the age of the system or the time for which the system has been operating without any repairs. In renewal theory, age is defined as the time elapsed since the last renewal event [25]. In RTLCA model, we define age at time t as the time elapsed since the last renewal given that the system is in use at t and the age is zero if the system is down at t. Mathematically, age KðtÞ ¼ ðt  tLi Þ1fin use at tg , where i = NL(t). In the following, we propose an integral equation for computing the expectation E½KðtÞ. Conditioning on the events related to first renewal, we have

E½KðtÞ ¼

Z

1

0

Z sL   E Kðt ÞjT I1 ¼ sI ; T L1 ¼ sL fT I T L ðsI ; sL ÞdsI dsL

ð4Þ

0

where fT I T L ðsI ; sL Þ is the joint PDF for T I1 and T L1 and

  E KðtÞjT I1 ¼ sI ; T L1 ¼ sL ¼ t

t < sI

¼0

sI  t < sL

¼ E½Kðt  sL Þ

t  sL

ð5Þ

Therefore, we have

  E½KðtÞ ¼ t 1  F T I ðt Þ þ

Z

t

0

E½Kðt  sL ÞfT L ðsL ÞdsL

ð6Þ

This integral equation can be solved numerically for all t, by first discretizing t as 0, Dt, 2Dt,. . . and re-writing Eq. (6) using a summation in the place of the integral. Then, after some rearrangement of terms, Eq. (6) yields an algebraic equation to solve for E[K(kDt)] in terms of E[K((k  1)Dt)], E[K((k  2)Dt)],. . .,E[K(0)]. Using E[K(0)] = 0, E[K(kDt)] for all k can be computed starting from k = 1 and moving forward. In order to compute the variance of K(t), we need to compute the E[K(t)2] Following the steps shown in Eqs (4)–(6), we get

Z t h h i i   E KðtÞ2 ¼ t 2 1  F T I ðt Þ þ E Kðt  sL Þ2 fT L ðsL ÞdsL

ð7Þ

0

Now variance of K(t) is given by E[K(t)2]  E[K(t)]2. The function E[K(t)] can be used to estimate the level of deterioration in a system at any time t. For example, it can be shown for the RTLCA formulation that the instantaneous rate of ultimate failure mF(t) can be approximated as

mI ðtÞ 

fT I jLF ½E½KðtÞjLF PðLF Þ 1  F T I ½E½Kðt Þ

ð8Þ

ðtÞ represents the expected state at time t of a Furthermore, if x ½KðtÞ can be deteriorating system with no-repair strategy then x approximately considered as the expected state at t of the system subject to the repair strategy corresponding to K(t). This is useful ðtÞ from no-repair strategy can be used to estimate x ðtÞ because x for any repair strategy after computing K(t).

97

R. Kumar, P. Gardoni / Structural Safety 50 (2014) 94–102

4.2. Computing the Availability It is useful to estimate the time for which a system is available for use in its life-span. This is because down-times cause inconvenience to the users and loss of income to the owner. In literature, availability of a system has been defined as the fraction of the time for which the system is available in a particular time-span Rackwitz [1]. Following the same idea, we define availability of a system for the time interval [0, t] as A(t) = [TA(t)]/t, where Rt T A ðtÞ ¼ 0 XðsÞds and XðsÞ ¼ 1fin use at sg . In order to estimate A(t), we first compute PS(t) which is the probability that the system is in use at time t. This implies that the expectation of A(t), E[A(t)], Rt is given by 0 P S ðtÞdt. In the following, we propose an integral equation to compute PS(t) for the RTLCA formulation. Based on the first intervention I1 and the Total Probability Rule [26], we have

    PS ðt Þ ¼ P in use at t; T I1 > t þ P in use at t; T I1  t 

ð9Þ

Noting that the system is in use if t < T I1 , we have





P in use at t; T I1 > t ¼ P T I1 > t



The computation of the function cOp ðsÞ is essential to compute E[COp(t)]. In some special cases cOp ðsÞ can be easily obtained. For example if cOp1 is independent of T L1 , then cOp ðsÞ ¼ E½cOp1  For other cases, simulation of the events in the first renewal may be required. 4.4. Computing the failure losses In the following, we propose the integral equation to compute E½C L ðtÞ for the RTLCA formulation:

E½C L ðt Þ ¼ E½C L ðtÞjL1  LR PðLR Þ þ E½C L ðt ÞjL1  LF PðLF Þ where

E½C L ðt ÞjL1  LR  ¼





P in use at t; T I1  t ¼

1

0

  P in use at t; T I1  t T L1 ¼ s fT L ðsÞds 

Since, the system is not in use in the interval ðtI1 ; tL1 Þ and given the assumption that system is completely renewed at t Li , we have Pðin use at t; T I1  tjT L1 ¼ sÞ ¼ 0 t
As a result, we have

PS ðtÞ ¼ ½1  F T I ðtÞ þ

  E C L ðt ÞjL1  LR ; T L1 ¼ s ¼ 0

PS ðt  sÞfT L ðsÞds

0

ð13Þ

( "Z 2 # Z t

2 ) t 1  Var½AðtÞ ¼ 2 E X ðsÞds ps ðsÞds t 0 0

E

t

XðsÞdt

¼2

0

Z

t

0

2

Z 0

Z

E½C L ðt ÞjL1  LR  ¼

Z

t

0

t

Z

E½C L ðt ÞjL1  LF  ¼

Z 0

  E C L ðt ÞjL1  LF ; T L1 ¼ s ¼0 ¼ cL ecðsT DF Þ

t

E½C L ðt ÞjL1  LF  ¼ cL ecT DF

ð14Þ

ð16Þ

0

   cOp ðsÞ þ E C Op ðt  sÞ ecs fT L ðsÞds

0

Z

t

ecs fT L jLF ðsjLF Þds

E½C L ðt  sÞ ecs fT L jLF ðsjLF Þds

ð24Þ

Z

Z

tþT DF

t

ecs fT L jLF ðsjLF Þds

E½C L ðt  sÞ ecs fT L ðsÞds

ð25Þ

The variance of COp(t) and CL(t) is difficult to obtain analytically. Since simulations of the first renewal are required for computing cOp ðsÞ the simulation data can be used to compute the variances of cOp1 ; cL1 and cOp1 þ cL1 . The coefficient of variation of COp(t), CL(t) and C Op ðtÞ þ C L ðtÞ can be assumed to be equal to that of cOp1 ; cL1 and cOp1 þ cL1 respectively.

Rt Mathematically, benefit Q ðtÞ ¼ 0 qecs XðsÞds accounting for the discount rate. Now the expectation E[Q(t)] can be written as

E½Q ðtÞ ¼ q

Z

t

P S ðsÞecs ds

ð26Þ

0

where q is the benefit derived from having the system in use for a unit time. The upper bound of E[Q(t)2] can be estimated as follows:

Now writing cOp ðsÞ ¼ E½cOp1 jT L1 ¼ s, we have t

tþT DF

4.5. Computing the benefit

   E C Op ðt ÞT L1 ¼ s ¼ 0

Z

ð23Þ

ð15Þ

Based on the definition of COp(t) and assuming complete renewal of the system at tLi , we have

  E C Op ðt Þ ¼

Z

0

0

s

t
< s  T DF s  T DF  t  s st

where TDF is the down-time after an ultimate failure. It follows that

þ

E½C Op ðtÞjT L1 ¼ sfT L ðsÞds

ð22Þ

where

ps ðsÞP½XðsÞ ¼ 1jX ðsÞ ¼ 1dsds

1

0

ð21Þ

Combining Eqs. (18), (21), and (24), we have

The expected value of COp(t) can be estimated as follows:

Z

  E C L ðtÞjL1  LF ; T L1 ¼ s fT L jLF ðsjLF Þds

0

4.3. Computing the cost of operation

E½C Op ðtÞ ¼

E½C L ðt  sÞecs fT L jLR ðsjLR Þds

1

E½C L ðt Þ ¼ PðLF ÞcL ecT DF

ps ðsÞps ðs  sÞdsds

ð20Þ

ts

Similarly,

þ

t

s

ð19Þ

Therefore,

where it can be shown that

2 #

t
¼ cL ecðsT DF Þ þ E½C L ðt  sÞ ecs

t

where F T I ðtÞ is the cumulative distribution function (CDF) of T Ii The variance Var[A(t)] is given by

"Z

  E C L ðtÞjL1  LR ; T L1 ¼ s fT L jLR ðsjLR Þds

and

ð11Þ

Z

1

0

Now it follows that

Z

Z

¼ E½C L ðt  sÞ ecs ð10Þ

ð18Þ

ð17Þ

Z t Z t h i E Q ðt Þ2  2 ecs ps ðsÞ ecs ps ðs  sÞdsds 0

s

ð27Þ

98

R. Kumar, P. Gardoni / Structural Safety 50 (2014) 94–102

Using the upper bound in Eq. (27) the variance of Q(t) is given by Var[Q(t)] = E[Q(t)2]  E[Q(t)]2. 5. Modeling of the deterioration process As explained in earlier section, in order to use the proposed LCA model we need to know the conditional PDFs fT I ðsÞ, fT I jLF ðsjLF Þ; fT L jLF ðtjLF Þ and the probabilities P(LR) and P(LF) that are explained in Eqs. (2) and (3). These quantities depend on the process of deterioration, the process of loads (magnitude and time of occurrence) and the intervention criteria. In this section, we briefly discuss a Stochastic Semi-Analytical (SSA) formulation originally proposed by Kumar et al. [27] for modeling the deterioration processes. This deterioration model can be used to compute the mentioned distributions and probabilities based on a given intervention criteria. It must be noted that the proposed LCA model does not depend on any particular method of modeling the deterioration process and can be implemented as long as the mentioned probabilities and density functions can be computed. The capacity of a deteriorating system at time t, Ct = C[x(t)] is a stochastic process, where x(t) represents the properties of system at t. Similarly, the demand Dtn ¼ D½xðtn Þ; Stn  is a stochastic process, where Stn is the nth load since t = 0 and tn is the time of occurrence of the nth load. The number of loads to failure nF is given by nF ¼ minfn : g tn  0g, where g tn ¼ C tn  Dtn and t is the time instant immediately before t. Accounting for the deterioration process, we have

g tn ¼ C 0 CC;tn  Y tn CD;tn

ð28Þ

where tn is the time of occurrence of the nth load, Y tn ¼ D½xð0Þ; Stn  are SIID and CC,t and CD,t represent the total effect of deterioration on capacity and demand at time t considering the events in timebt þ span [0,t]. By suitable rearrangement of terms, nF ¼ minfn : Y n b t ¼ D½xð0Þ; St =C½xð0Þ is the normalized demand W tn  1g, where Y n n with respect to the capacity of the undeteriorated system, and b t are SIID ranWt = [1  CC,t/CD,t]. As a result of these definitions, Y n dom variables and Wt is a stochastic process that captures the effect of deterioration on the system. The process Wt may consist of both shock and gradual deterioration process. Assuming that the process of gradual deterioration and shocks are mutually independent, Wt can be written as

Wt ¼

bt g 6.1. Stochastic model for ft n gandf Y n The process {tn} can be modeled using a stochastic model that can appropriately represent the time of occurrences of loads and b t g can be modeled by developing a CDF of Y bt . A the process f Y n

n

finite element (FE) model of the bridge is developed in OpenSees [28] in order to assess its dynamic properties. The details regarding the FE model is described in Kumar [30]. The described FE model is b t conditioned on the value used to compute F b ðyjSa Þ, the CDF of Y n Y jSa of pseudo-spectral acceleration Sa at the natural period of the bridge. This conditional CDF can be computed using the probabilistic deformation demand model developed by Gardoni et al. [31] and probabilistic capacity model developed by Gardoni et al. [32] and Choe et al. [33]. Now in order to compute the PDF of Sa, fSa ðsÞ, we use the regional seismic hazard curve for Los Angeles corresponding to the natural period of the bridge. The seismic hazard curve is obtained using OpenSHA [34]. The CDF F b ðyÞ is now Y R computed by performing the integration F b ðyjSa ÞfSa ðsÞds. For Y jSa this example, it is found that F b ðyÞ closely matches the CDF of a Y

L1

Z t n þ Rð t Þ

L2

Deck Column

Abutment

N ðt  Þ

X

CA. In this example, we assume that a bridge is repaired after the ith earthquake if the value of Pf ðtþ i Þ  pa , where pa is pre-determined acceptable probability of failure. Fig. 2 shows the example bridge modeled in OpenSees [28] with structural properties shown in Table 2. The bridge column is modeled as a nonlinear beam element with shown RC section properties and the bridge deck is modeled as rigid. The abutments and soil-structure interaction are modeled using springs with stiffness values provided in Huang et al. [29]. In this example, we consider the deterioration of the bridge caused by corrosion and seismic damage. We consider that the failure of the bridge can be caused by excessive lateral deformation of the bridge column (this is the typical failure mode for bridges designed according to the current seismic design specifications).

Hc

Abutment

ð29Þ

i¼1

where N(t) is the number of loads or shocks in the time interval [0,t], Z tn is the shock at tn and R(t) is the state of the gradual process at t. The application of the RTLCA requires 5 following steps: (i) Model the process of occurrence of loads by modeling the time of occurrence {tn} and fY tn g. (ii) Model the deterioration process by modeling shock process fZ tn g and gradual process R(t). (iii) Select of an criteria for preventive interventions. (iv) Simulate the event of first renewal using the modeled loading and detrioration process, and intervention criteria to compute the probabilities and density functions required for implementing the proposed RTLCA model. (v) Solve the proposed equations to compute the expectation and variance of the life-cycle performance variables. 6. Numerical example For the purpose of illustration of the RTLCA formulation, this section uses the proposed formulation to conduct a LCA of an example RC bridge. The bridge is a typical highway bridge with one single-column bent assumed to be located in Los Angeles,

Longitudinal steel

Ground level Pile

Dc Column Section

Fig. 2. Example RC bridge with one single-column bent.

Table 1 Structural properties of the example bridge. Parameter

Value

Shorter Span Long-to-Short span ratio Column height Column diameter Concrete cover Yield strength of longitudinal steel Compressive strength of concrete Longitudinal reinforcement ratio Volumetric ratio of transverse steel soil type abutment type Axial load ratio on bridge column Natural period of bridge

29.45 m 1.36 8.5 m 1.5 m 0.040 m 642.15 MPa 40.55 MPa 2.20% 0.90% D C 0.11 0.84 s

99

R. Kumar, P. Gardoni / Structural Safety 50 (2014) 94–102 Table 2 Probabilities and probability distributions for the renewal model. pa

P(LR)

fT L ðtÞ

0.1 0.2 0.3 0.4

0.916 0.850 0.774 0.714

5.089, 5.629, 6.086, 6.380,

fT I ðtÞ 8.044 7.630 7.368 7.253

4.873, 5.318, 5.696, 5.932,

8.252 7.890 7.656 7.563

Gamma distribution with parameters (0.678, 0.16). Using F b ðyÞ, Y the probability of failure of the system in as-built state is given by P f ð0Þ ¼ 1  F b ð1Þ ¼ 7:6716E  04. Y

Following Reasenberg and Jones [35], the occurrence of main shocks can be modeled as a homogeneous Poisson process [26], where the rate km ðMÞ of main shocks with magnitude Mm greater than or equal to M is given as follows:

km ðMÞ ¼ 10B1 B2 M

ð30Þ

where B1 and B2 are regional constants. It can be seen in Eq. (30) that the rate of all main shocks (i.e., Mm > 0) is given by km ð0Þ ¼ 10B1 . It can be derived from Eq. (30) that PðM m > MÞ ¼ 10B2 M . The instantaneous rate ka ðs; MÞ for aftershocks of magnitude Ma P M following a main shock of magnitude Mm is given by the modified Omori’s Law [36,37] as follows:

ka ðs; MÞ ¼

10B1 þB2 ðMm MÞ ðs þ cÞp

ð31Þ

where s is the time elapsed since the main shock, and c and p are regional constants. Rearranging the terms, the rate of all aftershocks given that Mm = M



km ð0Þ 1 ka ðs; 0Þ ¼ ½1  F Mm ðMÞ ðs þ cÞp

ð32Þ

where F Mm ðMÞ is the CDF for Mm. It is also found in past research [35–37] that the probability distribution of the magnitude of aftershocks is independent of the magnitude of main shocks and is indeed the same as that of main shocks. As a simplification, in this example, we assume that all earthquakes (main and aftershocks) at the bridge sites originate from a single point source. Now since the distribution of earthquake magnitudes is same for both main shocks b t also remains the same for and aftershocks, the distribution of Y n both main shocks and aftershocks. Based on Eq. (32), we assume that the time-dependent rate of aftershocks k0a ðsÞ following a main b t ¼ y is given as follows: shock with Y n

k0a ðsÞ ¼ h

km ð0Þ 1  F b ðyÞ

i



1 ðs þ cÞp

fT I jLF ðsjLF Þ

cOp ðsÞ

4.412, 4.896, 5.118, 5.390,

3.946, 4.391, 4.609, 4.858,

0.857 0.924 0.959 0.979

9.945 8.831 8.763 8.425

10.069 9.354 9.259 8.901

Kumar and Gardoni [38]. Secondly, we consider the effect of the seismic degradation on the static pushover properties of an RC column as modeled in Kumar and Gardoni [39]. Using these degradation models, Kumar [30] generates the data shown in Fig. 3 for b t for the bridge considered in the example. Based on Z tn and Y n the data, Kumar [30] estimated that the distribution of Z tn condib t is a Beta distribution with mean given tioning on the value of Y n b by 0:546 expð0:5= Y tn Þ and 0.95 probability interval centered on the mean shown in Fig. 2. In addition to seismic damage, we also include gradual deterioration caused by corrosion of reinforcing steel in the RC bridge columns. The effect corrosion can be modeled by computing the function R(t) in Eq. (29) using a detailed chloride diffusion equation and a capacity model for RC bridge columns. Choe et al. [40] shows the computation of deformation capacity of corrosion affected RC bridge columns. In this example, we assume that R(t) = 0 for t < tcorr and R(t) = [(t  tcorr)/20]1.2 and t P tcorr, where tcorr is a lognormal random variable with mean 50 years and variance 25 years representing the time of initiation of corrosion. 6.3. Intervention criteria, renewal time and cost of renewals As discussed earlier, there can be several criteria for interventions. In this example, we assume that a bridge is repaired after the ith earthquake if the value of P f ðt þ i Þ  a, where a is pre-determined acceptable probability of failure. As an example, we conduct the analysis for a = 0.1, 0.2, 0.3, 0.4. Generally, in the case of civil infrastructure systems, there is a time lag before repairs can be initiated. It is important to consider this time lag because earthquakes are usually followed by aftershocks which may cause further damage before the repairs. We assume, for this example, that the time lag is 3 months (0.25 years) to initiate repairs. We assume that the time to replace a bridge is 2 years and the time to repair a damaged bridge is given by a fraction of the 2 years proportional to the probability of failure at the time at which the repairs begins, i.e.,

1



Mean 0.95 Probability interval

0.9

ð33Þ

0.8

Y

Eq. (33) is exact if there exists a one-to-one mapping between b t and only one point source contribearthquake magnitude and Y n utes to the seismic hazard at the bridge site. 6.2. Modeling the deterioration process

fT L jLF ðtjLF Þ

0.7 0.6 Z tn

0.5 0.4

The failure of the bridge is taken as the event in which the deformation demand on the bridge column during an earthquake exceeds the available deformation capacity of the column. Both the deformation demand and capacity may be affected by structural deterioration of the bridge columns due to the experienced earthquakes. In this example, we consider two distinct seismic degradation phenomena in RC columns that affect the probability of failure of the columns. Firstly, we consider the effect of lowcycle fatigue in the longitudinal steel of RC columns caused by earthquakes using the seismic degradation model developed in

0.3 0.2 0.1 0

0

0.5

1

1.5

Yˆtn

2

2.5

3

Fig. 3. Modeling of shock deterioration process accounting seismic damage.

100

R. Kumar, P. Gardoni / Structural Safety 50 (2014) 94–102

Pf ðtIi þ 0:25Þ 2 years (where ðtIi þ 0:25Þ ¼ the time at which repairs begin accounting for the time lag.) We also assume the same proportionality also for cOpi , i.e., cOpi ¼ Pf ðt Ii þ 0:25Þ C C . In addition, the following values are considered for cF i , c and q: cF i ¼ 2:0C C , c = 0.04 year1 and q = 0.1CC year1. 6.4. Results and discussions Table 2 show the values of P(LR) and the parameters of Gamma distributions used to fit the distributions fT I ðtÞ, fT L ðtÞ, fT I jLF ðsjLF Þ, fT L jLF ðtjLF Þ. It is observed that P(LR) increases when a decreases. This implies that the system is more likely to have preventive interventions than essential interventions by decreasing the value of a. We also observe that expectations of T Ii and T Li decrease when a decreases. This implies that the frequency of intervention and renewal events increases by decreasing a. Similar effect is observed on the conditional renewal times T Li jF and T Ii jF . Table 2 also shows the value of cOp ðsÞ for all considered values of a. In general, cOp ðsÞ is a function of s which can be obtained by performing a statistical regression using the simulated cOp1 versus T Li data. In this example cOp1 is found to be independent of T Li . This is because cOp1 is a function of Pf(t) and the renewals are based on Pf(t) exceeding a constant value a. The results of the example bridge LCA show that for a long-term service life, it is more advantageous to have frequent repairs by lowering pa but for a short-term service life, it is not advantageous. This trend is observed for all the computed LCA variables which are individually discussed next. Fig. 4 shows Ps(t) plots for a values 0.1, 0.2, 0.3, 0.4. In the figure, the vertical axis represents Ps(t) and the horizontal axis represents t. It is observed that Ps(t) reaches convergence for the considered values of Ps(t) around t = 100 years. It is found that initially (i.e., for t < 20 years) Ps(t) is higher for higher values of a but in the long-term, Ps(t) is higher for smaller values of a. This is because smaller values of a necessitate more frequent interventions initially than the higher values of a. However, in the long-run with smaller a, bridges are less likely to have ultimate failures resulting in higher probability of being in use at a given time. Fig. 5 shows E[A(t)] plots for a values 0.1, 0.2, 0.3, 0.4. In the figure, the vertical axis represents E[A(t)] and the horizontal axis represents t. The plots imply that smaller values of a is beneficial in long-run because ultimate failures are avoided to a greater extent.

Fig. 5. Effect of a on the availability of the system.

Fig. 6. Effect of a on the age of the system.

Fig. 4. Effect of a on the values of Ps(t).

Figs. 6 and 7 show two measures that indicate the state of the bridge. Fig. 6 shows E[K(t)] versus t and Fig. 7 shows mF(t) versus t for the values of a considered earlier. Figs. 6 and 7 show that the example bridge is expected to deteriorate for the initial 25 years and then both E[K(t)] and mF(t) remain constant with small variation. It is also observed that the higher values of a result in greater deterioration of the bridge. However, there is a significant difference in the condition of a bridge as captured by E[K(t)] and mF(t). The values of mF(t) are more accurate indicator of the condition of the bridge because they indicate the amount of deterioration experienced since the last renewal while E[K(t)] indicates only the time elapsed since last renewal. Fig. 8 shows the relation between the expectation of the cost CTotal(t) = COp(t) + CL(t) and a. It is found that E[CTotal(t)] increases by increasing a. The rate of increase of E[CTotal(t)] decreases with time and E[CTotal(t)] is expected to eventually become constant. This is because the NPV of costs incurred after a sufficiently long time is small. This implies that the events after a sufficiently long period of time (t > 150 years in this example) have no relevance

R. Kumar, P. Gardoni / Structural Safety 50 (2014) 94–102

101

Fig. 7. Effect of a on the rate of ultimate failure for the system. Fig. 10. Coefficients of variation of K(t), A(t) and Q(t).

Fig. 8. Effect of a on the total expected total cost of operation and losses.

for decisions made at t = 0. Table 1 shows the values of coefficient of variation of ðcOp1 þ cL1 Þ for different values of a. These values, as mentioned earlier can also be considered as a reasonable estimate of coefficient of variation of CTotal(t). Fig. 9 shows the value of E[Qnet(t)]/CC with respect to t. At t = 0, E[Qnet(t)]/CC =  1 because the only cost incurred at t = 0 is the construction cost and there is no accumulated benefit. Gradually benefit accumulates and a breakeven (i.e., Qnet(t) = 0) is achieved around 12 years. Based on Fig. 9, it is found that it is economically advisable to lower the values of a. However, the figure does not imply that the benefits can be increased indefinitely by increasing the rate of interventions. This conclusion is correct only if interventions are conducted after an earthquake and hence the maximum rate of interventions can only be equal to the rate of earthquakes (i.e., repair after every earthquake.) Fig. 10 shows the plots of coefficients of variation of A(t), K(t) and Q(t), where the vertical axis represents coefficient of variation and horizontal axis represents time. It is found that the coefficients of variation are 0 at t = 0. This is because at t = 0 the system is in use and at that time there is no uncertainty about the state of the system. It is also found that initially the coefficients of variation increase with the increase in time and then converge to a particular value. 7. Conclusions

Fig. 9. Effect of a on the net benefit.

Life-cycle analysis (LCA) provides comprehensive information regarding the performance of an engineering system. In particular, LCA is important for making decisions regarding systems that are susceptible to deterioration during their service life. LCA models based on renewal theory are popular because they have an analytical approach that minimizes the need for computationally expensive simulations compared to purely simulations based methods. However, renewal theory based models have a well known limiting assumption i.e. at every repair restores the original state of the system. Renewal theory based models are interesting and are worth exploring because they provide considerable insight into the lifecycle of systems with relatively less computational effort than the approaches based purely on Monte Carlo simulations. This paper proposes a novel renewal theory based LCA model (RTLCA) for deteriorating systems. The proposed model provides equations to compute the instantaneous probability of being in

102

R. Kumar, P. Gardoni / Structural Safety 50 (2014) 94–102

service, expected values and variances of availability, age, benefit, and costs of operation and failures of the system for a finite time horizon. The model is applicable to a wide variety of systems and operation strategies as long as certain renewal probabilities and density functions can be computed. An illustrative example is presented for the life-cycle analysis of a reinforced concrete (RC) bridge subject to deterioration caused by earthquakes and corrosion. In the example an operation strategy is analyzed where the bridge is repaired whenever the instantaneous probability of failure exceeds an acceptable limit. The results show that for a long-term service life, it may be more advantageous to have frequent repairs but for a short-term service life, it may not be advantageous or may even be disadvantageous. In other words it can be said that the desired positive effects of an operation strategy take some time to take effect. The proposed equations for finite time horizons allow the estimation of the time after which these effects can be observed. References [1] Rackwitz R. Optimization – the basis for code making and reliability verification. Struct Saf 2000;22(1):27–60. [2] Streicher H, Rackwitz R. Time-variant reliability-oriented structural optimization and a renewal model for life-cycle costing. Probabilist Eng Mech 2004;19(1–2):171–83. [3] Joanni A, Rackwitz R. Cost benefit optimization for maintained structures by a renewal model. Reliabil Eng Syst Saf 2008;93(3):489–99. [4] Van Noortwijk JM. Explicit formulas for the variance of discounted life-cycle cost. Reliabil Eng Syst Saf 2003;80(2):185–95. [5] Van Noortwijk JM, Van der Weide JAM. Applications to continuous-time processes of computational techniques for discrete-time renewal processes. Reliabil Eng Syst Saf 2008;93(12):1853–60. [6] Van der Weide JAM, Pandey MD, Van Noortwijk JM. Discounted cost model for condition-based maintenance optimization. Reliabil Eng Syst Saf 2010;95(3): 236–46. [7] Van Noortwijk JM, Frangopol DM. Two probabilistic life-cycle maintenance models for deteriorating civil infrastructures. Probabilist Eng Mech 2004; 19(4):345–59. [8] Mori Y, Ellingwood BR. Maintaining reliability of concrete structures, I: role of inspection/repair. J Struct Eng 1994;120(3):824–45. [9] Frangopol DM, Lin K-Y, Estes AC. Life-cycle cost design of deteriorating structures. J Struct Eng 1997;123(10):1390–401. [10] Neves LC, Frangopol DM, Cruz PS. Cost of life extension of deteriorating structures under reliability based maintenance. Comput Struct 2004;82(13– 14):1077–89. [11] Neves LC, Frangopol DM. Condition, safety and cost profiles for deteriorating structures with emphasis on bridges. Reliabil Eng Syst Saf 2005;89(2):189–98. [12] Kong S, Fragopol DM. Evaluation of expected life-cycle maintenance cost of deteriorating structures. J Struct Eng 2003;129(5):682–91. [13] Val DV, Stewart MG. Decision analysis of deteriorating structures. Reliabil Eng Syst Saf 2005;87(3):377–85. [14] Kumar R, Gardoni P, Sanchez-Silva M. Effect of cumulative seismic damage and corrosion on the life-cycle cost of reinforced concrete bridges. Earthquake Eng Struct Dyn 2009;38(7):887–905. [15] Kim S, Frangopol DM, Zhu B. Probabilistic optimum inspection/repair planning to extend lifetime of deteriorating structures. ASCE J Perform Construct Facilities 2011;25(6):534–44.

[16] Chun-Qing L, Mackie RI. A Risk-Cost optimized maintenance strategy for corrosion affected concrete structures. Comput Aided Civil Infrastruct Eng 2007;22(5):335–46. [17] Yang JN. Statistical estimation of service cracks and maintenance cost for aircraft structures. AIAA J. Aircraft 1976;13(12):929–37. [18] Mori Y, Ellingwood BR. Maintaining reliability of concrete structures, II: Optimum inspection/repair. J Struct Eng 1994;120(3):846–62. [19] Abaza K. Optimum flexible pavement life-cycle analysis model. J Transportat Eng 2002;128(6):542–9. [20] Deshpande VP, Damnjanovic ID, Gardoni P. Reliability based optimization models for scheduling pavement rehabilitation. Comput Aided Civil Infrastruct Eng 2010;25(4):227–37. [21] Val DV, Stewart MG, Melchers RE. Life-cycle performance of RC bridges: probabilistic approach. Comput Aided Civil Infrastruct Eng 2000;15(1):14–25. [22] Val DV. Effect of different limit states on life-cycle cost of RC structures in corrosive environment. J Infrastruct Syst 2005;11(4):231–40. [23] Wen YK, Kang YJ. Minimum building lifecycle cost design criteria. I: methodology. ASCE J Struct Eng 2001;127(3):330–7. [24] Wen YK, Kang YJ. Minimum building lifecycle cost design criteria. II: applications. ASCE J Struct Eng 2001;127(3):338–46. [25] Grimmett GR, Stirzaker DR. Probability and random processes. 3rd ed. New York: Oxford University Press; 2001. [26] Ang AH-S, Tang WH. Probability concepts in engineering: emphasis on applications in civil and environmental engineering. 2nd ed. New York: John Wiley and Sons; 2007. [27] Kumar R, Cline DBH, Gardoni P. A stochastic semi-analytical (SSA) approach to model deterioration in engineering systems. Struct Saf; 2012 [Submitted for publication]. [28] McKenna F, Fenves GL, Filippou FC, Mazzoni S. OpenSees: Open System for Earthquake Engineering Simulations. http://opensees.berkeley.edu/; 2008. [29] Huang Q, Gardoni P, Hurlebaus S. Probabilistic seismic demand models and fragility estimates for reinforced concrete highway bridges with one singlecolumn bent. ASCE J Eng Mech 2010;136(11):1340–53. [30] Kumar R. Stochastic Life-cycle analysis of deteriorating engineering systems and an application for seismic deterioration of RC highway bridges. Ph.D. Dissertation Texas A&M University; 2012. [31] Gardoni P, Mosalam KM, Der Kiureghian A. Probabilistic seismic demand models and fragility estimates for RC bridges. ASCE J Earthquake Eng 2003; 7(Special Issue 1):79–106. [32] Gardoni P, Der Kiureghian A, Mosalam KM. Probabilistic capacity models and fragility estimates for RC columns based on experimental observations. ASCE J Eng Mech 2002;128(10):1024–38. [33] Choe D, Gardoni P, Rosowsky D. Closed-form fragility estimates, parameter sensitivity and Bayesian updating for RC columns. ASCE J Eng Mech 2007; 133(7):833–43. [34] Field EH, Jordan TH, Cornell CA. OpenSHA: a developing community-modeling environment for seismic hazard analysis. Seismol Res Lett 2003;74(3):406–19. [35] Reasenberg PA, Jones LM. Earthquake hazard after and mainshock in California. Science 1989;243(4895):1173–6. [36] Omori F. On the aftershocks of earthquakes. J College Sci 1984;7(2):111–200. [37] Utsu T, Ogata Y, Matsu’ura RS. The centenary of the Omori formula for a decay law of aftershock activity. J Phys Earth 1995;43:1–33. [38] Kumar R, Gardoni P. Modeling structural degradation of RC bridge columns subject to earthquakes and their fragility estimates. ASCE J Struct Eng 2012; 138(1):42–51. [39] Kumar R, Gardoni P. Effect of seismic degradation on the reliability of reinforced concrete bridges. Eng Struct; 2012 [submitted for publication]. [40] Choe D, Gardoni P, Rosowsky D, Haukaas T. Probabilistic capacity models and fragility estimates for corroding reinforced concrete columns. Reliabil Eng Syst Saf 2008;93:383–93.