Time-Dependent Reliability Assessment

Abstract

This chapter discusses the approaches for structural time-dependent reliability assessment. The significant difference between the time-dependent reliability and the classical reliability (c.f. Chap. 4) is the involvement of the time-variant characteristics in the analysis, where the variation of both the structural resistance and the external loads on the temporal scale should be reasonably modelled. This chapter starts from the motivation of time-dependent reliability assessment, followed by the modelling techniques of the resistance deterioration and the external load processes. Both the discrete and continuous load processes are discussed. Subsequently, the time-dependent reliability assessment approaches in the presence of both the discrete and the continuous loads are addressed. The comparison between the reliabilities associated with the two types of load processes is also presented.

Problems

5.1

In Problem 4.20, if each bar has an identical circular cross-section, and its diameter degrades with time according to $D(t)=D(0)\cdot \frac{40}{40+t}$, where t is in years and D(0) is the initial diameter,

(1) What is the resistance deterioration model of each bar?

(2) What is the system failure probability at the end of 10 years?

5.2

For a specific type of load within a reference period of T years, its occurrence is modeled by a Poisson process with a time-variant occurrence rate $\lambda (t)=\lambda _0+\kappa _\lambda t$, and the load intensity, conditional on occurrence, is a Weibull variable with a constant shape parameter $\alpha$ and a time-variant scale parameter $u(t)=u_0+\kappa _u t$. Derive the CDF of the maximum load within time interval [0, T].

5.3

In Problem 5.2, if the load is applied to a structure whose resistance is lognormally distributed with mean $\mu _R$ and standard deviation $\sigma _R$, what is the structural failure probability for a reference period of T years? Use the following values to evaluate your result: $T=10$, $\lambda _0=1$, $u_0=0.5$, $\kappa _\lambda =\kappa _u=0.1$, $\alpha =1.5,\mu _R=4,\sigma _R=1$.

5.4

Suppose that the initial resistance of a structure is deterministically 3. The resistance degrades linearly with time, which is modeled by a Gamma process. The deterioration function at the end of 50 years has a mean of 0.7 and a standard deviation of 0.3. The load occurrence is a Poisson process with a constant occurrence rate of 0.2/year. The load effect, conditional on occurrence, is an Extreme Type I variable with a mean of 0.5 and a COV of 0.6.

(1) Use a simulation-based approach to find the time-dependent failure probabilities for reference periods up to 50 years.

(2) Show that simply considering the maximum of the load effect and the minimum of resistance over a reference period of interest may introduce a significant error in the estimate of structural failure probability.

5.5

In Problem 4.22, if the resistances of each bar independently degrades by 10% linearly over 10 year, what is the time-dependent failure probability of the truss for a reference period of 10 years?

5.6

In Problem 5.5, we further know that the resistance deterioration of each bar is a linear process and is fully correlated on the temporal scale, and that the deterioration function at the end of 10 years is a Beta variable with mean 0.9 and COV 0.2.

(1) If the deterioration process of each bar is mutually independent, what is the time-dependent failure probability of the truss for a reference period of 10 years?

(2) If the deterioration processes of all the bars are correlated, and the linear correlation coefficient of the deterioration functions evaluated at the end of 10 years is 0.7 for any two bars, what is the failure probability of the truss for a reference period of 10 years?

5.7

Recall Example 5.11. If in Eq. (5.43), $\hat{d}_i(t_i)=k\cdot t_i^m \cdot |t_i-t_{i-1}|$, and $\epsilon (t_i)$ is a Gamma variable with mean 1 and standard deviation $\frac{p\cdot t_i^q}{\sqrt{t_i-t_{i-1}}}$, where k, m, p, q are four positive constants, derive the mean and standard deviation of the deterioration function G(t) [50].

5.8

In Problem 5.4, what are the mean value and standard deviation of the structural service life?

5.9

In Problem 5.4, if the threshold for the failure probability is 0.01, what is the predicted service life of the structure?

5.10

The load occurrence is modeled by a Poisson process with a constant occurrence rate. The load effect, conditional on occurrence, follows an identical Extreme Type I distribution. For a reference period of [0, T], show that the CDF of the maximum load effect within [0, T] can be reasonably modeled by an Extreme Type I distribution at the upper tail.

5.11

Consider a structure subjected to the combined effect of live load L and dead load D. The dead load is deterministically $1.05D_{\mathrm {n}}$, where $D_{\mathrm {n}}$ is the nominal dead load. The live load occurrence is a stationary Poisson process with an occurrence rate of 1/year. Conditional on occurrence, the live load effect is an Extreme Type I variable with a mean value of $0.4L_{\mathrm {n}}$ and a COV of 0.3, where $L_{\mathrm {n}}$ is the nominal live load. The structure was designed according to the criterion $1.0R_{\mathrm {n}} = 1.25D_{\mathrm {n}}+1.75L_{\mathrm {n}}$, where $R_{\mathrm {n}}$ is the nominal initial resistance. The initial resistance is a lognormal variable with mean $1.1R_{\mathrm {n}}$ and COV 0.1. The resistance deterioration is modeled by a Gamma process with a linearly-varying mean, and the deterioration function evaluated at the end of 40 years has a mean 0.8 and a standard deviation 0.2. Compute the time-dependent failure probabilities of the structure for reference periods up to 40 years.

5.12

Suppose that the initial resistance of a structure is deterministically 3. The resistance degrades linearly with time from time $T_i$ according to

$$\begin{aligned} G(t)=\left\{ \begin{aligned}&1,&t\le T_i \\&1-K(t-T_i),&t>T_i\end{aligned}\right. \end{aligned}$$

(5.297)

where K is a normal variable with mean 0.006 and COV 0.2. The time of deterioration initiation, $T_i$, is a normal variable with mean 5 years and COV 0.15. The load occurrence is a Poisson process with a constant occurrence rate of 0.3/year. The load effect, conditional on occurrence, is an Extreme Type I variable with a mean of 0.5 and a COV of 0.6. Compute the time-dependent failure probabilities for reference periods up to 50 years.

5.13

Consider the time-dependent reliability of a 5-out-of-8 system. Suppose that the performance of each component is identical and independent of each other. The annual failure probability of each component increases with time according to $p(t)=0.01(1+0.05t)$, where $t=1,2,\ldots$ is in years. What is the system failure probability for a reference period of 10 years?

5.14

Consider a structure subjected to wind hazard. The post-hazard damage state can be classified into four categories: none ($D_0$, when $V<20$ m/s), moderate ($D_1$, when 20 m/s $\le V<35$ m/s), severe ($D_2$, when 35 m/s $\le V<50$ m/s) and total ($D_3$, when 50 m/s $\le V$), where V denotes the wind speed. Suppose that the occurrence of wind events is a Poisson process with an occurrence rate of $\lambda =0.5$/year. Conditional on occurrence, the wind speed is a Weibull variable with mean 40 m/s and COV 0.3. The economic costs associated with states $D_0,D_1,D_2,D_3$ are 0, 1, 5 and 25 respectively. Suppose that the structure is restored to the pre-damage state before the occurrence of the next wind event. What are the mean and standard deviation of the accumulative wind-induced economic costs for a reference period of 20 years?

5.15

In Problem 5.14, if the occurrence of wind events is modeled by a renewal process, where the time interval between two adjacent wind events is a Gamma variable with mean 2 (years) and COV 0.3, reevaluate the mean and standard deviation of the accumulative wind-induced economic costs for a reference period of 20 years.

5.16

In Problem 5.14, let $\Delta _i$ be the time interval between the $(i-1)$th and the ith wind events for $i=1,2,\ldots$ (suppose that the 0th event occurs at the initial time with zero wind speed). If $\Delta _i$ and $\Delta _j$ have a correlation efficient of $0.8^{|i-j|}$ for $i,j\in \{1,2,\ldots \}$, reevaluate the mean and standard deviation of the accumulative wind-induced economic costs for a reference period of 20 years.

5.17

In Problem 5.14, let $t_i$ (in years) be the occurring time of the ith successful event for $i=1,2,\ldots$. If the wind speeds associated with the ith and the jth wind events are correlated with a correlation coefficient of $\exp \left( -\frac{|t_i-t_j|}{10}\right)$ for $i,j\in \{1,2,\ldots \}$, reevaluate the mean and standard deviation of the accumulative wind-induced economic costs for a reference period of 20 years.

5.18

Consider a structure subjected to wind hazard. The post-hazard damage state can be classified into four categories: none ($D_0$, when $V<20$ m/s), moderate ($D_1$, when 20 m/s $\le V<35$ m/s), severe ($D_2$, when 35 m/s $\le V<50$ m/s) and total ($D_3$, when 50 m/s $\le V$), where V denotes the wind speed. In terms of the annual extreme winds actioned on the structure, we classify the wind load into two levels (1 and 2). Suppose that the wind level in the $(k+1)$th year depends on that in the kth year only for $k=1,2,\ldots$. Given a level 1 wind in the kth year, the probability of a level 1 wind in the $(k+1)$th year is 0.7; if the wind load is level 2 in the kth year, then the probability of a level 1 wind in the $(k+1)$th year is 0.2. The level 1 wind speed is a Weibll variable with mean 30 m/s and COV 0.2, while the level 2 wind speed is a Weibull variable with mean 50 m/s and COV 0.35. In the first year, the wind is classified as level 1. The economic costs associated with states $D_0,D_1,D_2,D_3$ are 0, 1, 5 and 25 respectively. Suppose that the structure is restored to the pre-damage state before the occurrence of the next wind event. What are the mean and standard deviation of the accumulative wind-induced economic costs for a reference period of 20 years?

5.19

Consider a continuous load process $X(t)=\sum _{j=1}^n A_j \sin (\omega t)+\sum _{k=1}^n B_k \cos (\omega t)$, where $\omega$ is a constant, n is a positive integer, and $\{A_1,A_2,\ldots A_n, B_1, B_2,\ldots B_n\}$ are 2n independent standard normal variables. Show that X(t) is a covariance stationary process.

5.20

For a weakly stationary process X(t) with an autocorrelation function of $\mathbb {R}(\tau )=\frac{a}{k\tau ^2+a}$, where $a>0,k>0$, find the PSDF of X(t).

5.21

Reconsider Example 5.23. For a zero-mean stationary Gaussian load process S(t) whose PSDF is as in Eq. (5.140) with $k = 1$, use a simulation-based approach to discuss the accuracy of Eq. (5.152). Use $T = 10$ and $T = 50$ respectively.

5.22

For the case in Problem 5.21, what are the mean value and variance of the maximum load effect within [0, T], $S_{\max }$, conditional on $S_{\max }>2$? Use $T = 10$ and $T = 50$ respectively.

5.23

If the hazard function is

$$\begin{aligned} h(t)=\left\{ \begin{aligned}&\frac{a_1-a_2}{t_1}t+a_2,&0\le t<t_1 \\&a_1,&t_1\le t<t_2 \\&\frac{a_2-a_1}{t_3-t_2}(t-t_2)+a_1,&t_2\le t\le t_3\end{aligned}\right. \end{aligned}$$

(5.298)

where $0<a_1<a_2$ and $0<t_1<t_2<t_3$, derive the time-dependent reliability $\mathbb {L}(t)$ for $t\in [0,t_3]$.

5.24

For a k-out-of-n system with independent component performance, if the hazard function for each component is identically h(t), what is the hazard function for the system?

5.25

If the service life follows a Gamma distribution with a shape parameter of $a > 0$ and a scale parameter of $b > 0$, and the corresponding hazard function represents a DFR, what are the ranges for a and b?

5.26

Consider a series or parallel system with four components having the same physical configuration and load conditions as summarized in Table 5.3. Two live load models are considered, with an occurrence rate of 1.0/year. The first represents a stationary load process with a constant mean value and a constant COV with time, while the second is a nonstationary load process with an increasing trend of load magnitude, whose mean value, $\mu _2(t)$, increases with time as

$$\begin{aligned} \mu _2(t)=\mu _2(0)\cdot (1+\epsilon \cdot t) \end{aligned}$$

(5.299)

in which $\epsilon$ is the scale factor, $\mu _2(0)$ is the initial load intensity, which equals 500kN$\cdot$m so that the mean initial load intensity of model 2 is the same as that of model 1. It is assumed that each load event will induce identical load effect to each component, with which $c_i=1$ for $\forall i$ in Eq. (5.195) or (5.199).

Table 5.3 Statistics of resistance and load effects of a single component

The deterioration of resistance for each component is assumed to be deterministic and is given by $g(t)=1-0.004 t$, where t is in years. The resistances of the components are mutually correlated, and it is assumed that the resistances are identically distributed and are equally correlated pairwise with a correlation coefficient of $\rho =0.6$. In the presence of the two live load models in Table 5.3, compute the time-dependent failure probabilities for reference periods up to 50 years, considering a series or parallel system.

5.27

We reconsider the rigid-plastic portal frame as shown in Fig. 2.1a, which is subjected to horizontal load H and vertical load V. The structure may fail due to one of the following three limit states,

$$\begin{aligned} \begin{aligned}&G_1 = M_1+2M_3+2M_4-H-V \\&G_2 = M_2+2M_3+M_4-V \\&G_3 = M_1+M_2+M_4-H \end{aligned} \end{aligned}$$

in which $M_1$ through $M_4$ are the plastic moment capacities at the joints. It is assumed that at the initial time, $M_1$ through $M_4$ are independent normal variables with mean 1 and COV 0.3. Furthermore, $M_1$ through $M_4$ degrade with time independently with a constant rate of 0.003/year. The vertical load V is constantly applied to the frame, which is a normal variable with mean 1.2 and COV 0.3. The occurrence of the horizontal load H is a Poisson process with occurrence rate $\lambda =0.5$/year. The magnitude of H, conditional on occurrence, follows an Extreme Type I distribution with mean 1 and COV 0.4. Compute the time-dependent failure probabilities of the frame for reference periods up to 50 years.

5.28

Consider a bridge girder that has an initial resistance (moment) of 2500 kN$\cdot$m and a successful service history of 20 years. The in-situ inspection suggests that the current resistance, having served for 20 years, is a lognormal variable with mean 2000 kN$\cdot$m and COV 0.2. The annual live load causes a moment that follows an Extreme Type I distribution with mean 900 kN$\cdot$m and COV 0.3. The dead load causes a deterministic moment of 600 kN$\cdot$m. Find the mean and standard deviation of the updated current resistance taking into account the impact of successful service history.

5.29

We use a compound Poisson process to model the accumulation of structural damage Y(T) with a reference period of [0, T], that is, $Y=\sum _{i=1}^{N(T)} X_i$, where $X_i$ is the magnitude of the ith shock deterioration, and N(T) is the number of shock deteriorations within [0, T], which is a Poisson variable with a mean occurrence rate of $\lambda$. Assume that each $X_i$ is independent of N(T). Suppose that each $X_i$ is statistically independent and identically Gamma-distributed, with a shape parameter of $a > 0$ and a scale parameter of $b > 0$. Define a function $W(x,z)=\sum _{n=1}^{\infty }\frac{x^n}{n!\Gamma (zn)},x\ge 0,z>0$. Derive the PDF of Y(T) for a reference period of [0, T] [48].

5.30

In Problem 5.29, if the occurrence of shock deterioration is a nonstationary Poisson process with mean occurrence $\lambda (t)$, what is the PDF of Y(T)?

5.31

Consider a structure that has successfully served for T years. The PDF of the initial resistance, $R_0$, is $f_{R_0}(r)$. The resistance deterioration function, G(t), takes a form of $G(t)=1-A\cdot t^\alpha$, where $\alpha >0$ is a constant while $A>0$ is a random variable that reflects the uncertainty associated with G(t). Let $f_A(x)$ be the PDF of A. For the past service history of T years (and the future years), let $F_{S,i}$ be the CDF of the maximum load effect within the ith year ($i=1,2,\ldots T,\ldots$). Suppose that the structural resistance is constant within each year, and that the load process is independent of the resistance deterioration. At the current time (with a service history of T years), if the structure survives from a proof load with a deterministic magnitude of w, derive the time-dependent reliability of the structure for a subsequent reference period of $T'$ years [56].

5.32

In Problem 5.31, we further assume that the annual maximum load effect, $S_i$, is identically distributed for each year. Each $S_i$ ($i=1,2,\ldots T$) is correlated with G(T), and the correlation is modeled by a copula function C(u, v). What is the updated PDF of the current resistance (with a service history of T years) taking into account the successful service history (without considering the impact of the proof load) [49]?

5.33

Reconsider Example 5.35. Taking into account the impact of discount rate, denoted by r, the cumulative damage index is rewritten as $D_c^*=\sum _{j=1}^{N}\frac{Q(t_j)}{(1+r)^{t_j}}$. With this, derive the mean and variance of $D_c^*$.

5.34

Reconsider Example 5.11. If the resistance deterioration is described by the model in Problem 5.7 with $q=0.5$, recalculate the time-dependent failure probabilities for reference periods up to 40 years.

5.35

In Problem 5.4, if we additionally consider the impact of Type II failure mode (c.f. Sect. 5.4.5.2) with a permissible level of 1.2 for the degraded resistance, recompute the time-dependent failure probabilities for reference periods up to 50 years.

5.36

Reconsider Example 5.38. If the PSDF of S(t) is as in Eq. (5.140) with $k = 1$, recompute the time-dependent failure probability for T being up to 50.