ARTICLE IN PRESS
Reliability Engineering and System Safety 92 (2007) 1651–1658 www.elsevier.com/locate/ress
Gamma processes and peaks-over-threshold distributions for time-dependent reliability
J.M. van Noortwijka,b,Ã, J.A.M. van der Weideb, M.J. Kallena,b, M.D. Pandeyc
aHKV Consultants, P.O. Box 2120, NL-8203 AC Lelystad, The Netherlands bFaculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, P.O. Box 5031, NL-2600 GA Delft, The Netherlands cDepartment of Civil Engineering, University of Waterloo, Waterloo, Ont., Canada N2L 3G1
Received in revised form 16 August 2006; accepted 2 November 2006 Available online 21 December 2006
Abstract
In the evaluation of structural reliability, a failure is defined as the event in which stress exceeds a resistance that is liable to deterioration. This paper presents a method to combine the two stochastic processes of deteriorating resistance and fluctuating load for computing the time-dependent reliability of a structural component. The deterioration process is modelled as a gamma process, which is a stochastic process with independent non-negative increments having a gamma distribution with identical scale parameter. The stochastic process of loads is generated by a Poisson process. The variability of the random loads is modelled by a peaks-over-threshold distribution (such as the generalised Pareto distribution). These stochastic processes of deterioration and load are combined to evaluate the time-dependent reliability. r 2006 Elsevier Ltd. All rights reserved.
Keywords: Deterioration; Load; Gamma process; Peaks-over-threshold distribution; Poisson process; Kac functional equation
1. Introduction consider loads that are larger than a certain threshold rather than the loads themselves, because the Poisson In structural engineering, a distinction can often be made assumption is then better justified [2,3]. In order to between a structure’s resistance and its applied stress. determine the probability distribution of extreme loads A failure may then be defined as the event in which—due (such as water levels, discharges, waves, winds, tempera- to deterioration—the resistance drops below the stress. tures and rainfall), we use the peaks-over-threshold This paper proposes a method to combine the two method. Exceedances over the threshold are typically stochastic processes of deteriorating resistance and fluctu- assumed to have a generalised Pareto distribution [4,5]. ating load for computing the time-dependent reliability of a Given the above definitions of the stochastic processes of structural system. deterioration and load, they are combined to compute the As deterioration is generally uncertain and non-decreas- time-dependent reliability. This method extends the ap- ing, it can best be regarded as a gamma process (see, e.g. proach of Mori and Ellingwood [6], who modelled the [1]). The gamma process is a stochastic process with uncertainty in the deterioration process through a prob- independent non-negative increments having a gamma ability distribution of the damage growth rate rather than distribution with identical scale parameter. The loads occur assuming it as a stochastic process. at random times according to a Poisson process. We The paper is organised as follows. The stochastic processes of deteriorating resistance and fluctuating load are presented in Sections 2 and 3, respectively. These two stochastic processes are combined for the purpose of time- ÃCorresponding author. HKV Consultants, P.O. Box 2120, NL-8203 AC Lelystad, The Netherlands. Tel.: +31 320 294203; dependent reliability analysis in Section 4. The proposed fax: +31 320 253901. method is applied to a dike reliability problem in Section 5 E-mail address: [email protected] (J.M. van Noortwijk). and conclusions are formulated in Section 6.
0951-8320/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2006.11.003 ARTICLE IN PRESS 1652 J.M. van Noortwijk et al. / Reliability Engineering and System Safety 92 (2007) 1651–1658
2. Stochastic process of deterioration Let XðtÞ denote the deterioration at time t, tX0, and let the probability density function of XðtÞ, in conformity with Deterioration can be regarded as a time-dependent the definition of the gamma process, be given by stochastic process fXðtÞ; tX0g where XðtÞ is a random f ðxÞ¼GaðxjvðtÞ; uÞ (1) quantity for all tX0. At first glance, it seems possible to XðtÞ represent the uncertainty in a deterioration process by the with normal distribution. This probability distribution has been vðtÞ vðtÞ used for modelling the exchange value of shares and the EðXðtÞÞ ¼ ; VarðXðtÞÞ ¼ . (2) u u2 movement of small particles in fluids and air. A character- istic feature of this model, also denoted by the Brownian A component is said to fail when its deteriorating motion with drift (see, e.g. [7, Chapter 7]), is that a resistance, denoted by RðtÞ¼r0 XðtÞ, drops below the structure’s resistance alternately increases and decreases, stress s (see Fig. 1). For the time being, we assume both the like the exchange value of a share. For this reason, the initial resistance r0 and the stress s to be deterministic. Let Brownian motion is inadequate in modelling deterioration the time at which failure occurs be denoted by the lifetime which proceeds in one direction. For example, a dike of T. Due to the gamma distributed deterioration, Eq. (1), the which the height is subject to a Brownian deterioration can, lifetime distribution can then be written as according to the model, spontaneously rise up, which of FðtÞ¼ PrfTptg¼PrfXðtÞXr0 sg course cannot occur in practice. In order for the stochastic Z 1 GðvðtÞ; ½r s uÞ deterioration process to proceed in one direction, we can ¼ f ðxÞ dx ¼ 0 , ð3Þ XðtÞ GðvðtÞÞ best consider it as a gamma process (see, e.g. [8]). x¼r0 s R The gamma process is a stochastic process with 1 a 1 t where Gða; xÞ¼ t¼x t e dt is the incomplete gamma independent non-negative increments (e.g. the increments function for xX0anda40. Using the chain rule for of crest-level decline of a dike) having a gamma distribu- differentiation, the probability density function of the tion with identical scale parameter. In the case of a gamma lifetime is deterioration, dikes can only sink. To the best of the q GðvðtÞ; ½r0 s uÞ q Gðv~; ½r0 s uÞ 0 authors’ knowledge, Abdel-Hameed [1] was the first to f ðtÞ¼ ¼ v ðtÞ propose the gamma process as a proper model for qt GðvðtÞÞ qv~ Gðv~Þ v~¼vðtÞ deterioration occurring random in time. The gamma (4) process is suitable to model gradual damage monotonically under the assumption that the shape function vðtÞ is accumulating over time, such as wear, fatigue, corrosion, differentiable. The partial derivative in Eq. (4) can be crack growth, erosion, consumption, creep, swell, a calculated by the algorithm of Moore [14]. Using a series degrading health index, etc. For the mathematical aspects expansion and a continued fraction expansion, this of gamma processes, see Dufresne et al. [9], Ferguson and algorithm computes the first and second partial derivatives Klass [10], Singpurwalla [11], and van der Weide [12].An with respect to x and a of the incomplete gamma integral advantage of modelling deterioration processes through Z 1 x GðaÞ Gða; xÞ gamma processes is that the required mathematical Pða; xÞ¼ ta 1e t dt ¼ . calculations are relatively straightforward. The stochastic GðaÞ t¼0 GðaÞ differential equation from which the gamma process In modelling the temporal variability in the deteriora- follows is presented by Wenocur [13]. tion, a key input is the trend of the expected deterioration In mathematical terms, the gamma process is defined as increasing over time. Empirical studies show that the follows. Recall that a random quantity X has a gamma distribution with shape parameter v40 and scale para- meter u40 if its probability density function is given by Resistance R(t) v Initial u r0 resistance Gaðxjv; uÞ¼ xv 1 expf uxg 1 ðxÞ, r0 GðvÞ ð0;1Þ Deterioration X(t) where 1AðxÞ¼1 for x 2 A and 1AðxÞ¼0 for xeA. Furthermore, let vðtÞ be a non-decreasing, right contin- L2 X L5 uous, real-valued function for t 0withvð0Þ 0. The L1 gamma process with shape function vðtÞ40 and scale L3 parameter u40 is a continuous-time stochastic process Stress/ Failure Load L4 fXðtÞ; tX0g with the following properties: Load l0 threshold l0 ð Þ¼ 0 (1) X 0 0 with probability one, 0 t time (2) XðtÞ XðtÞ GaðvðtÞ vðtÞ; uÞ for all t4tX0, (3) XðtÞ has independent increments. Fig. 1. Stochastic processes of deterioration and stress/load. ARTICLE IN PRESS J.M. van Noortwijk et al. / Reliability Engineering and System Safety 92 (2007) 1651–1658 1653 expected deterioration at time t often follows a power law: and the probability of failure in unit time i reduces to a shifted Poisson distribution with mean 1 þ uðr sÞ (see vðtÞ atb 0 EðXðtÞÞ ¼ ¼ (5) e.g. [16]): u u i 1 i for some physical constants u40, a40, and b40. Some pi ¼ Pr oTp examples are the expected degradation of concrete due to a a corrosion of reinforcement (linear: b ¼ 1), sulfate attack 1 i 1 ¼ ½uðr0 sÞ expf uðr0 sÞg, ð7Þ (parabolic: b ¼ 2), and diffusion-controlled aging (square ði 1Þ! root: b ¼ 0:5) studied by Ellingwood and Mori [15]. i ¼ 1; 2; 3; ... : Note that this probability function is a Because there is often engineering knowledge available shifted Poisson distribution because it is defined for i ¼ about the shape of the expected deterioration (for example, 1; 2; ...rather than for i ¼ 0; 1; ... : A physical explanation in terms of the parameter b in Eq. (5)), this parameter may for the appearance of the Poisson distribution (7) is that it be assumed constant. In the event of expected deterioration represents the probability that exactly i exponentially in terms of a power law, the parameters a and u yet must be distributed jumps with mean u 1 cause the component to assessed by using expert judgment and/or statistics. It fail; that is, cause the cumulative amount of deterioration should be noted that the analysis of this paper is not to exceed r0 s. restricted to using a power law for modelling the expected The unit time for which the increments of deterioration deterioration over time. As a matter of fact, any shape are exponentially distributed, facilitates the algebraic function vðtÞ suffices, as long as it is a non-decreasing, right manipulations considerably and, moreover, often results continuous, and real-valued function. in a very good approximation of the optimal design or An important special case is the gamma process with maintenance decision, especially when the expected cost stationary increments, which is defined by a linear shape must be calculated. Examples are determining optimal dike X function at40, t 0, and scale parameter u40. The heightenings and optimal sand nourishments whose corresponding probability density function of XðtÞ is then expected discounted costs are minimal ([17,18], respec- given by tively) and determining cost-optimal rates of inspection for berm breakwaters and the rock dumping of the Eastern- f XðtÞðxÞ¼Gaðxjat; uÞ. (6) Scheldt barrier (see [8,16,19]). Due to the stationarity, both the mean value and the variance of the deterioration are linear in time; that is, 3. Stochastic process of load at at ð ð ÞÞ ¼ ð ð ÞÞ ¼ E X t ; Var X t 2 . u u In this section, the design stress is replaced by loads A stochastic process has stationary increments if the occurring at random times according to a Poisson process. probability distribution of the increments Xðt þ hÞ XðtÞ We consider load threshold exceedances rather than the depends only on h for all t; hX0. A gamma process with loads themselves, because the Poisson assumption is in this a ¼ u ¼ 1 is called a standardised gamma process. context better justified. An important property of the gamma process is that it is For probabilistic design, the distribution of extreme a jump process (for an illustration, see Fig. 1). For a loads (such as water levels, discharges, waves, winds, gamma process, the number of jumps in any time interval is temperatures and rainfall) is of considerable interest. For infinite with probability one. Nevertheless, EðXðtÞÞ is finite, example, the probability distribution of high sea water as the majority of jumps are extremely small. Dufresne et levels is important for designing sea dikes. In order to al. [9] showed that the gamma process can be regarded as a determine the probability distribution of extreme load limit of a compound Poisson process. Note that for a quantities, we can use extreme-value theory. Extreme-value compound Poisson process, the number of jumps in any analysis can be based on either maxima in a given time time interval is finite with probability one. period (block maxima), or on peaks over a given threshold. Because deterioration should preferably be monotone, The advantage of using peaks over threshold rather than we can choose the best deterioration process to be a block maxima is the availability of more observations. compound Poisson process or a gamma process. In the Using only block maxima generally leads to loss of presence of inspection data, the advantage of the gamma information. process over the compound Poisson process is evident: In the peaks-over-threshold method, all load values measurements generally consist of deterioration increments larger than a certain threshold are considered. The rather than of jump intensities and jump sizes. differences between these values and a given threshold A very useful property of the gamma process with are called exceedances over the threshold. These excee- stationary increments is that the gamma density in Eq. (6) dances are typically assumed to have a generalised Pareto transforms into an exponential density if t ¼ a 1. When the distribution (Section 3.1). The stochastic process of load unit-time length is chosen to be a 1, the increments of threshold exceedances can be considered as a Poisson deterioration are exponentially distributed with mean u 1 process (Section 3.2). ARTICLE IN PRESS 1654 J.M. van Noortwijk et al. / Reliability Engineering and System Safety 92 (2007) 1651–1658
3.1. Generalised Pareto distribution Define the sequence of partial sums by Si ¼ T1 þ þT i, i ¼ 1; 2; ... : The homogeneous Poisson process fNðtÞ; A random variable Y has a generalised Pareto distribu- tX0g with intensity l is defined as tion with scale parameter s40 and shape parameter c if the NðtÞ¼maxfnjSnptg. (11) probability density function of Y is given by 8 hi Hence, the following two events are equivalent: ð1=cÞ 1 > 1 cy a < 1 I ð0;1ÞðyÞ; c 0; s40; NðtÞ¼n 3 S t S s s þ np o nþ1. (12) Paðyjs; cÞ¼ no > 1 y :> exp I ðyÞ; c ¼ 0; s40; According to Karlin and Taylor [7, pp. 126–127], the joint s s ð0;1Þ probability density function of the occurrence times (8) S1; ...; Sn is given by n f ðs1; ...; snÞ¼l expf lsng1f0 s s s g. (13) where ½y þ ¼ maxf0; yg. The survival function is defined by S1;...;Sn o 1o 2o o n 8 hi > 1=c The Poisson process implies that the probability that > cy a < 1 I ð0;1ÞðyÞ; c 0; s40; exactly n load threshold exceedances occur in time interval ¯ nos þ PrfY4yg¼F Y ðyÞ¼ ð0; t can be written as > y : exp I ð0;1ÞðyÞ; c ¼ 0; s40: s ðltÞn PrfNðtÞ¼ng¼ expf ltg; n ¼ 0; 1; 2; ..., (14) (9) n! X The range of y is 0oyo1 for cp0and0oyos=c for for all t 0; that is, the random quantity NðtÞ follows a c40. The case c ¼ 0, which is the exponential distribution, Poisson distribution with parameter lt. This parameter is the limiting distribution as c ! 0. The generalised Pareto equals the expected number of threshold exceedances in distribution mathematically arises as a class of limit time period ð0; t . The Poisson process fNðtÞ; tX0g with distributions for exceeding a certain threshold, as the Nð0Þ¼0 has independent increments. threshold increases toward the distribution’s right tail. The conditions that must be satisfied in order to assure 4. Combination of deterioration and load the existence of such limit distributions are rather mild. For the mathematical details, we refer to Pickands III [4] Given the definition of the stochastic process of and de Haan [5]. deterioration and the stochastic process of load, the question arises how we can combine them. The mathema- tical derivation and numerical computation of time- 3.2. Homogeneous Poisson process dependent failure probabilities are studied in Sections 4.1 and 4.2, respectively. Apart from the probability distribution of load threshold exceedances, the stochastic process of the occurrence times of these load exceedances needs to be specified. We define 4.1. Derivation of failure probability the load threshold exceedance to be Y ¼ L l0X0 with To combine the stochastic processes of deterioration and threshold l0. Threshold exceedances Y 1; ...; Y n are as- sumed to be mutually independent and to have a general- load, we follow the approach of Ellingwood and Mori [15]. On the basis of a useful property of the Poisson process, ised Pareto distribution, where Y i ¼ Li l0, i ¼ 1; ...; n. In most hydrological applications, the occurrence process they proposed an elegant way to combine stochastic load of exceedances of large thresholds can be regarded as a with a resistance decreasing over time. The first assumption Poisson process (see, e.g. [20]). To overcome the problem of is that the stochastic processes of deterioration and load dependence between successive threshold exceedances in are independent. Mori and Ellingwood [21] claim that the very small time periods, threshold exceedances are defined deterioration can be treated as deterministic, because ‘‘its to be the peaks in clusters of threshold exceedances. variability was found to have a second-order effect on the Approximating the occurrence process of threshold ex- structural reliability’’. This conclusion is based on Mori ceedances by a Poisson process is supported by asymptotic and Ellingwood [6] in which they modelled the uncertainty extreme-value theory [2,3]. It follows that the higher the in the deterioration process by posing a probability threshold, the better the approximation [4]. distribution on the damage growth rate a=u of the power Let the inter-occurrence times of threshold exceedances law in Eq. (5). However, in Section 5, we will show this conclusion is not always justified. T1; T 2; ... be an independent, identically distributed, sequence of random quantities having an exponential In order to combine the load events with the deteriorat- distribution with mean l 1. Hence, the cumulative ing resistance, Mori and Ellingwood [21] assumed that the distribution function of the inter-occurrence time of duration of extreme load events is generally very short and threshold exceedances, T, is given by is negligible in comparison with the service life of a structure. Therefore, they regard the load threshold PrfTptg¼F T ðtjlÞ¼1 expf ltg. (10) exceedances as a series of pulses having an ‘‘infinitely ARTICLE IN PRESS J.M. van Noortwijk et al. / Reliability Engineering and System Safety 92 (2007) 1651–1658 1655 small’’ duration (see Fig. 1). As a consequence, we may function of the lifetime is: Z then suppose that the resistance does not degrade during t the occurrence of an extreme load event. Furthermore, they FðtÞ¼1 E exp kðRðuÞÞ du , (17) assume that failure can only occur during exceedances of u¼0 the load threshold. where RðtÞ¼r0 XðtÞ and the physics-based expression Assume that there are n changes of the Poisson process ¯ in time interval ½0; t ; i.e., consider 0pS1pS2p pSnpt. kðrÞ¼lF Y ðr l0Þ¼l PrfL4rg (18) Karlin and Taylor [7, pp. 126–127] show that S ; ...; S 1 n can be interpreted as a killing rate for resistance r in the can be regarded as a set of order statistics of size n sense of Wenocur [13]. In our time-dependent stress- associated with a sample of n independent uniformly strength model, the killing rate is exactly the frequency distributed random quantities on the interval ½0; t . The of the stress L ¼ l þ Y exceeding the strength r. Hence, conditional probability distribution of the occurrence time 0 Eq. (18) gives a nice justification of Wenocur’s definition of of a load threshold exceedance, given one exceedance of the the killing rate. Note that, by differentiating Eq. (17), the load threshold occurred in ð0; t , directly follows from the lifetime probability density function becomes memoryless property of the exponential distribution: it is a Z uniform distribution on the interval ½0; t . t f T ðtÞ¼EkðRðtÞÞ exp kðRðtÞÞ dt . (19) The conditional probability of no failure in time interval t¼0 ð0; t , when n independent threshold exceedances of the load Y are given with cumulative distribution function Wenocur [13] considers a very interesting extension of a stationary gamma process for the deterioration state under PrfYpyg¼F Y ðyÞ, can now be formulated as [7, p. 180] two different failure modes. A system is said to fail either Prfno failure in ð0; t jNðtÞ¼ng when a traumatic event (such as an extreme load) destroys Z t F ð½r s XðuÞÞ n it or when its condition reaches a failure level. Suppose that ¼ E Y 0 0 du , ð15Þ t the traumatic events occur as a Poisson process with a rate u¼0 (called the ‘killing rate’) which depends on the system’s where s0pr0 and the expectation is defined with respect to condition. The latter is a meaningful assumption, since the the stochastic process fXðtÞ; t40g. This result was obtained worse the condition, the more vulnerable it is to failure due by Ellingwood and Mori [15] for deterministic deteriora- to trauma. A fixed failure level is often used in tion and extended by Mori and Ellingwood [6] for mathematical models to optimise inspection intervals for deterioration with a random damage growth rate. Math- systems subject to gamma-process deterioration such as ematically, the last step follows by conditioning on the [8,23–26]. sample paths of the gamma process, using the fact that the Two straightforward possible extensions of the time- load threshold exceedances are independent, and applying dependent stress-strength model are the following. First, in the law of total probabilities with respect to the sample situations where the initial resistance R0 is unknown, the paths of the gamma process. Because failure can only occur lifetime distribution in Eq. (17) can be rewritten as Z Z at load threshold exceedances, and these exceedances are 1 t independent and uniformly distributed on ½0; t ,we F T ðtÞ¼1 E exp kðr0 XðtÞÞ dt primarily can focus solely on these n exceedances. r0¼l0 t¼0 Invoking the law of total probabilities with respect to the f ðr Þ r ð Þ R0 0 d 0, 20 number of load threshold exceedances, the probability of where f ðr Þ is the probability density function of R , no failure in time interval ð0; t or survival probability can R0 0 0 be written as R0Xl0. Second, if we take a non-homogeneous Poisson process with rate function lðtÞ as the model for the times of F¯ ðtÞ¼1 FðtÞ¼Prfno failure in ð0; t g load threshold exceedances, we can calculate in the same X1 way as before the probability of the event that there is no ¼ Prfno failure in ð0; t jNðtÞ¼ng PrfNðtÞ¼ng failure during the time-interval ð0; t . The survival prob- n¼ 0 Z ability in Eq. (16) can then be revised to t ¼ E exp l F¯ Y ð½r0 l0 XðuÞÞ du . ð16Þ Prfno failure in ð0; t g u¼0 Z t The expectation in Eq. (16) is defined with respect to the ¼ E exp lðuÞF¯ Y ð½r0 l0 XðuÞÞ du . ð21Þ non-decreasing stochastic process fXðtÞ; t40g. This result u¼0 was obtained by Ellingwood and Mori [15] for determinis- In the remaining part of the paper, we will confine tic deterioration and extended by the same authors in [6] ourselves to homogeneous Poisson processes for modelling for deterioration with a random damage growth rate. the process of load threshold exceedances. This is mainly Eq. (16) generalises the lifetime model of van Noortwijk for notational convenience, because the proposed algo- and Klatter [22] in which the load was assumed to be equal rithm can be easily applied to non-homogeneous Poisson to the constant design stress s. The cumulative distribution processes as well. ARTICLE IN PRESS 1656 J.M. van Noortwijk et al. / Reliability Engineering and System Safety 92 (2007) 1651–1658
4.2. Computation of failure probability Xðt=2Þ and Xð3t=4Þ given the value of XðtÞ. Similarly, we can simulate the values of Xðt=8Þ, Xð3t=8Þ, Xð5t=8Þ, In order to compute the time-dependent probability of Xð7t=8Þ, and so on. An example of gamma-bridge sampling failure, the following two-dimensional integral remains to is given in Fig. 1. be solved numerically: The advantage of gamma-bridge sampling over gamma- Z t increment sampling is that old iteration results can still be E exp kðRðtÞÞ dt used if we would decide to increase the accuracy. Gamma- t¼0 Z Z ()bridge sampling allows the accuracy to be pre-defined and 1 1 Xn the numerical integration to be terminated when that ¼ lim exp kðr0 xiÞðti ti 1Þ n!1 accuracy is achieved. 0 0 i¼1 f ðx x Þ x x ð Þ Xðt1Þ;...;XðtnÞ 1; ...; n d 1 ...d n, 22 5. Example: Den Helder sea defense where ti ¼ði=nÞt, i ¼ 0; ...; n. This integral is a special case of the so-called Kac functional equation (see [13,27]) and can be solved numerically in two combined steps. The first For the purpose of illustration, we study the probability step is to approximate the integral over time by applying of failure of a dike section subject to crest-level decline. The numerical integration with respect to the time grid dike section is part of the sea defense at Den Helder in the North-West of the Netherlands and was constructed as 0; t1; t2; ...; tn 1; tn. The second step is to approximate the integral over the sample paths of the gamma process by early as in 1775. applying Monte Carlo simulation with respect to the The failure mechanism that we regard is overtopping of the sea dike by waves. Both sea level and crest-level decline independent increments Xðt1Þ; Xðt2Þ Xðt1Þ; ...; XðtnÞ are considered random, whereas wave run-up is considered Xðtn 1Þ. Although we can approximate a gamma process with a fixed. The wave run-up is represented by the height z2% limit of a compound Poisson process, it is not efficient to (this is the wave run-up level in metres, which is exceeded simulate a gamma process in such a way. This is because by 2% of the number of incoming waves). Furthermore, there are infinitely many jumps in each finite time interval. the effect of seiches, gust surges and gust oscillations is A better approach for simulating a gamma process is taken into account by the factor b0 (m). For the Den simulating independent increments with respect to very Helder sea defense, the wave run-up height is computed as small units of time. For sampling independent gamma- z2% ¼ 7:52 m and the effect of seiches and gusts as process increments there are basically two methods: b0 ¼ 0:2 m. The Den Helder sea defense has a crest level nap gamma-increment sampling and gamma-bridge sampling. of h ¼ 12:33 m þ (normal Amsterdam level). Including Gamma-increment sampling is defined as drawing the effects of waves, seiches and gusts, the initial resistance independent samples d ¼ x x from the gamma of the dike can be defined as r0 ¼ h z2% b0 ¼ i i i 1 nap density 4:61 m þ . To account for the temporal variability of the sea level, GaðdjvðtiÞ vðti 1Þ; uÞ we assume that the occurrence of extreme sea levels can be
vðtiÞ vðti 1Þ u ½vðt Þ vðt Þ 1 ¼ d i i 1 expf udgð23Þ Cumulative distribution and survival function of time to failure GðvðtiÞ vðti 1ÞÞ 1 ¼ ¼ ¼ for every i 1; 2; ...; n, where t0 x0 0. 0.9 Gamma-bridge sampling is described in Dufresne et al. 0.8 [9]. It consists of the following steps. First, draw a sample F(t) S(t)=1-F(t) of the cumulative amount of deterioration XðtÞ at the end 0.7 of the time interval ð0; t that we are considering. Second, draw a sample of the cumulative deterioration Xðt=2Þ from 0.6 the conditional distribution of Xðt=2Þ given XðtÞ¼x: 0.5
1 y t t probability f ðyjxÞ¼ Be v ; vðtÞ v 0.4 Xðt=2ÞjXðtÞ x x 2 2 hi 0.3 GðvðtÞÞ y vðt=2Þ 1 ¼ Gðvðt=2ÞÞGðvðtÞ vðt=2ÞÞ x 0.2 hi y ½vðtÞ vðt=2Þ 1 1 0.1 1 ð24Þ x x 0 0 50 100 150 200 250 300 350 400 for 0pypx. This is a transformed beta density on the interval ½0; x with parameters vðt=2Þ and vðtÞ vðt=2Þ. time [year] Third, subdivide this time interval ð0; t into two intervals Fig. 2. Cumulative distribution and survival function of the lifetime with ð0; t=2 and ðt=2; t , and sample Xðt=4Þ given the value of CVðXð100ÞÞ ¼ 0:3. ARTICLE IN PRESS J.M. van Noortwijk et al. / Reliability Engineering and System Safety 92 (2007) 1651–1658 1657 modelled with a homogeneous Poisson process for the relative sea-level rise. In van Dantzig [29], the expected occurrence times of load threshold exceedances and a crest-level decline was assumed to be 0.7 m per century. We peaks-over-threshold distribution for the load threshold regard the stochastic process of crest-level decline as a exceedances themselves. In particular, we use the general- stationary gamma process with mean EðXð100ÞÞ ¼ 0:7 and ised Pareto distribution estimated by Philippart et al. [28] a corresponding coefficient of variation of 0.3; that is, for observed sea levels at Den Helder. They estimated the a 100 following parameter values: l ¼ 0:5, l0 ¼ 2:19, s ¼ 0:3245, EðXð100ÞÞ ¼ ¼ 0:7, and c ¼ 0:05465. Using this generalised Pareto distribu- u tion, the design water level with an exceedance probability pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 nap VarðXð100Þ 1 of 10 is 4:40 m þ . Crest-level decline consists of a CVðXð100ÞÞ ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:3. combination of settlement, subsoil consolidation, and EðXð100ÞÞ a 100
Probability density function of time to failure 0.01 f (t) 0.009
0.008
0.007
0.006
0.005
0.004 probability density 0.003
0.002
0.001
0 0 50 100 150 200 250 300 350 400 time [year]
Fig. 3. Probability density function of the lifetime with CVðXð100ÞÞ ¼ 0:3.
Probability density function of time to failure 0.01 f (t) 0.009
0.008
0.007
0.006
0.005
0.004 probability density 0.003
0.002
0.001
0 0 50 100 150 200 250 300 350 400 time [year]
Fig. 4. Probability density function of the lifetime with deterministic deterioration. ARTICLE IN PRESS 1658 J.M. van Noortwijk et al. / Reliability Engineering and System Safety 92 (2007) 1651–1658
Solving these equations for a and u leads to the parameter [8] van Noortwijk JM, Kok M, Cooke RM. Optimal maintenance values a ¼ 0:1111 and u ¼ 15:8730. The 5th and 95th decisions for the sea-bed protection of the Eastern-Scheldt barrier. In: percentile of Xð100Þ are 0.39 and 1.08 m, respectively. Cooke R, Mendel M, Vrijling H, editors. Engineering probabilistic design and maintenance for flood protection. Dordrecht: Kluwer In Fig. 2, the cumulative distribution function and the Academic Publishers; 1997. p. 25–56. survival function are shown based on Eq. (17). In this [9] Dufresne F, Gerber HU, Shiu ESW. Risk theory with the gamma computation, the number of gamma-process sample paths process. ASTIN Bull 1991;21(2):177–92. is 10,000 with numerical integration step size of ti ti 1 ¼ [10] Ferguson TS, Klass MJ. A representation of independent increment 2 years for all i. The probability density function of the processes without Gaussian components. Ann Math Stat 1972;43(5): 1634–43. lifetime in Eq. (19) is graphed in Fig. 3. In order to [11] Singpurwalla N. Gamma processes and their generalizations: an investigate the sensitivity of the probability density overview. In: Cooke R, Mendel M, Vrijling H, editors. Engineering function of the time to failure with respect to the probabilistic design and maintenance for flood protection. Dor- uncertainty in the deterioration, we also computed Eq. drecht: Kluwer Academic Publishers; 1997. p. 67–75. (19) with a deterministic deterioration. Obviously, Fig. 4 [12] van der Weide H. Gamma processes. In: Cooke R, Mendel M, Vrijling H, editors. Engineering probabilistic design and maintenance shows that the uncertainty in the deterioration really for flood protection. Dordrecht: Kluwer Academic Publishers; 1997. matters in quantifying the uncertainty in the lifetime. p. 77–83. [13] Wenocur ML. A reliability model based on the gamma process and 6. Conclusions its analytic theory. Adv Appl Probab 1989;21(4):899–918. [14] Moore RJ. Algorithm AS 187: derivatives of the incomplete gamma integral. Appl Stat 1982;31(3):330–5. In this paper, a method is presented to compute the time- [15] Ellingwood BR, Mori Y. Probabilistic methods for condition dependent reliability of structures where both the deterior- assessment and life prediction of concrete structures in nuclear power ating resistance and the fluctuating load are modelled as plants. Nucl Eng Des 1993;142(2–3):155–66. stochastic processes. The stochastic process of deteriora- [16] van Noortwijk JM, Cooke RM, Kok M. A Bayesian failure model tion is modelled as a gamma process having independent based on isotropic deterioration. Eur J Oper Res 1995;82(2):270–82. [17] Speijker LJP, van Noortwijk JM, Kok M, Cooke RM. Optimal gamma-distributed increments with identical scale para- maintenance decisions for dikes. Probab Eng Inf Sci 2000;14(1): meter. The stochastic process of load is modelled as a 101–21. combination of a peaks-over-threshold distribution (such [18] van Noortwijk JM, Peerbolte EB. Optimal sand nourishment as the generalised Pareto distribution) and a Poisson decisions. J Water Port Coastal Ocean Eng 2000;126(1):30–8. process for the threshold exceedances. It is shown that the [19] van Noortwijk JM, van Gelder PHAJM. Optimal maintenance decisions for berm breakwaters. Struct Safety 1996;18(4):293–309. cumulative distribution function of the time to failure, the [20] Buishand TA. Statistics of extremes in climatology. Stat Neerl time at which the resistance drops below the load, can be 1989;43(1):1–30. formulated as a Kac functional equation. This equation [21] Mori Y, Ellingwood BR. Time-dependent system reliability analysis can be solved by combining numerical integration and by adaptive importance sampling. Struct Safety 1993;12(1):59–73. simulating sample paths of the gamma process. The [22] van Noortwijk JM, Klatter HE. Optimal inspection decisions for the block mats of the Eastern-Scheldt barrier. Reliab Eng Syst Safety advantages of the proposed method are that a gamma 1999;65(3):203–11. process properly models the monotonic behaviour of aging [23] Dieulle L, Be´renguer C, Grall A, Roussignol M. Sequential and that peaks-over-threshold distributions is consistent condition-based maintenance scheduling for a deteriorating system. with extreme-value statistics. Eur J Oper Res 2003;150(2):451–61. [24] Grall A, Be´renguer C, Dieulle L. A condition-based maintenance policy for stochastically deteriorating systems. Reliab Eng Sys Safety References 2002;76(2):167–80. [25] Grall A, Dieulle L, Be´renguer C, Roussignol M. Continuous-time [1] Abdel-Hameed M. A gamma wear process. IEEE Trans Reliab predictive-maintenance scheduling for a deteriorating system. IEEE 1975;24(2):152–3. Trans Reliab 2002;51(2):141–50. [2] Pickands III J. The two-dimensional Poisson process and extremal [26] Park KS. Optimal continuous-wear limit replacement under periodic processes. J Appl Probab 1971;8(4):745–56. inspections. IEEE Trans Reliab 1988;37(1):97–102. [3] Leadbetter MR. Extremes and local dependence in stationary [27] Beghin L, Hochberg KJ, Orsingher E. Conditional maximal sequences. Z Wahrscheinlichkeitstheorie Verw Geb 1983;65:291–306. distributions of processes related to higher-order heat-type equations. [4] Pickands III J. Statistical inference using extreme order statistics. Stochastic Processes Appl 2000;85(2):209–23. Ann Stat 1975;3(1):119–31. [28] Philippart ME, Dillingh D, Pwa ST. De basispeilen langs de [5] de Haan L. Fighting the arch-enemy with mathematics. Stat Neerl Nederlandse kust; de ruimtelijke verdeling en overschrijdingslijnen 1990;44(2):45–68. [The base levels along the Dutch coast; the spatial distribution and [6] Mori Y, Ellingwood BR. Maintaining: reliability of concrete exceedance curves]. Technical Report RIKZ-95.008, Ministry of structures. I: role of inspection/repair. J Struct Eng 1994;120(3): Transport, Public Works and Water Management, The Hague, The 824–45. Netherlands; 1995. [7] Karlin S, Taylor HM. A first course in stochastic processes, second [29] van Dantzig D. Economic decision problems for flood prevention. ed. San Diego: Academic Press; 1975. Econometrica 1956;24(3):276–87.