Reliability of a single component
Consider the reliability of a component over time interval [0,T]. Significant load events (e.g., earthquake events) occur randomly in time with random intensities. Using a Poisson process to model the occurrence of the load events, let λ(t) be the time-variant occurrence rate (t denotes time), and FS(s,t) the cumulative distribution function (CDF) of the load effect at time t conditional on occurrence. Mathematically, if the load effect at time t is S(t), then Pr(S(t)≤s)=FS(s,t), where Pr() denotes the probability of the event in the brackets.
The resistance at time t, R(t), is modeled as follows,
$$ R(t) = R_{0}\cdot g(t) $$
(1)
where g(t) is the deterioration function, and R0 is the initial resistance. The deterioration function may take different shapes (e.g., linear, square-root and parabolic), depending on the dominant deterioration mechanism [2,39].
With this, the time-dependent reliability within [0,T],L(T), is estimated by [13]
$$ \begin{aligned} L(T)=\int_{0}^{\infty} & \exp\left\{-\int_{0}^{T}\lambda(t)\left[1-F_{S}(r\cdot g(t),t)\right]\mathrm{d}t\right\} \\ & \cdot f_{R}(r)\mathrm{d}r \end{aligned} $$
(2)
where fR(r) is the probability density function (PDF) of the initial resistance.
Reliability of a series or parallel system
For a series system with n components (structures), let Rj be the initial resistance of component j (j=1,2,…n), and gj(t) the deterioration function of component j=1,2,…n, which is assumed to be independent of the load process. Suppose that the load process is modeled by a Poisson process with a time-variant occurrence rate of λ(t) and a time-variant CDF of load effect FS(s,t). An occurring load event with magnitude s induces a structural action cj·s in component j (e.g., moment, shear, etc) for j=1,2,…n. With this, the time-dependent series system reliability, Lss(T), is given by [20]
$$ \begin{aligned} L_{ss}(T) = & \int\ldots\int \exp\left[-\int_{0}^{T}\lambda(t)\cdot\left\{1- F_{S}\left(\min_{j=1}^{n}\frac{r_{j} g_{j}(t)}{c_{j}},t\right)\right\}\mathrm{d}t\right] f_{\mathbf{R}}(\mathbf{r})\mathrm{d}\mathbf{r} \end{aligned} $$
(3)
where fR(r) is the joint PDF of the initial resistances R={R1,R2,…Rn}, and r={r1,r2,…rn}.
For a parallel system consisting of n components with the same configuration as that of the series system as discussed earlier, if each cj is independent of the number of working components, and the deterioration function of each component is identical, denoted by g(t), the time-dependent parallel system reliability, Lps(T), is given by [40]
$$ \begin{aligned} L_{ps}(T) = & \int\ldots\int \exp\left[-\int_{0}^{T}\lambda(t)\cdot\left\{1- F_{S}\left(\max_{j=1}^{n}\frac{r_{j} g(t)}{c_{j}},t\right)\right\}\mathrm{d}t\right] f_{\mathbf{R}}(\mathbf{r})\mathrm{d}\mathbf{r} \end{aligned} $$
(4)
In Eqs. (3) and (4), if the component resistances are statistically independent, then \(f_{\mathbf {R}}(\mathbf {r}) = \prod _{i=1}^{n} f_{R_{i}}(r_{i})\), where \(f_{R_{i}}(r_{i})\) is the PDF of Ri. However, taking into account the correlation between different component resistances, one would need to reasonably model the probabilistic behaviour of fR(r). With this regard, the copula function is a promising tool to describe correlated random variables [40,41].
Reliability of a k-out-of-n system
For a k-out-of-n system with n components (structures), the system survives if at least k components work, as mentioned before. Its time-dependent reliability will be discussed in this section. Assume that the resistance deterioration function for each component is identical, denoted by g(t), and is independent of the load process. It is also assumed that the value of cj(j=1,2,…n) is independent of the number of working components.
We introduce \(\mathcal {M}(\mathbf {a},k)\), which is a function of vector a (with n elements) and integer k≤n. The function \(\mathcal {M}\) returns the kth largest element of a, which will be used in the following derivation. It is straightforward to observe that \(\mathcal {M}(\mathbf {a},1)=\max (\mathbf {a})\) and \(\mathcal {M}(\mathbf {a},n)=\min (\mathbf {a})\).
The hazard function, denoted by h(t) at time t, represents the probability of structural failure at time t provided structural survival up to time t. It can be related to the time-dependent reliability L(t) according to [3,40]
$$ h(t) = -\frac{\mathrm{d}\ln L(t)}{\mathrm{d}t} $$
(5)
or equivalently,
$$ L(t) = \exp\left(-\int_{0}^{t} h(\tau)\mathrm{d}\tau\right) $$
(6)
Consider the hazard function hkn(t) for the k-out-of-n system at time t, conditional on R={R1,R2,…Rn}=r={r1,r2,…rn}. Let \(\mathbf {a}(t) = \left [\frac {r_{1} g(t)}{c_{1}},\frac {r_{2} g(t)}{c_{2}},\ldots \frac {r_{n} g(t)}{c_{n}}\right ]\). By the definition of hazard function, for Δt→0,hkn(t)Δt equals the probability that the system fails within time interval (t,t+Δt] provided the system’s survival within [0,t]. The system failure is caused by the occurrence of a significant load event with probability λ(t)Δt, with which
$$ \begin{aligned} h_{kn}(t) & =\lambda(t)\left\{1-\text{Pr}[S(t)\leq \mathcal{M}(\mathbf{a}(t),k)]\right\} \\ & = \lambda(t)\left\{1-F_{S}[\mathcal{M}(\mathbf{a}(t),k),t]\right\} \end{aligned} $$
(7)
where S(t) is the load effect at time t conditional on load occurrence as before. It is noticed that in Eq. (7), hkn(t) represents the instantaneous failure probability of the system. The condition of system survival before time t refers to the case that at least k components work at any time τ∈[0,t] (not necessarily all the n components working at time τ).
Remark 1
Letting \(\mathcal {N}_{f}(\tau)\)denote the number of working components at time τ, the probability of structural survival up to time t, L(t), is determined by
$$ L(t) = \text{Pr}\left(\mathcal{N}_{f}(\tau)\geq k, \forall \tau\in[0,t]\right) $$
(8)
If there are nt load events occurring within [0,t] at times \(\phantom {\dot {i}\!}t_{1},t_{2},\ldots t_{n_{t}}\) respectively, it follows that
$$ L(t) = \text{Pr}\left(\bigcap_{i=1}^{n_{t}} S(t_{i})\leq \mathcal{M}(\mathbf{a}(t_{i}),k)\right) $$
(9)
Eq. 9 is explained by the fact that, with an identical deterioration function for each component, the condition of \(\bigcap _{i=1}^{n_{t}} S(t_{i})\leq \mathcal {M}(\mathbf {a}(t_{i}),k)\) guarantees that the k components with the greatest resistances at the initial time survive at any time τ1∈[0,t]. Furthermore, at time τ2∈(t,t+dt], where dt→0, if one load event occurs (with probability λ(t)dt) and the magnitude satisfies \(S(\tau _{2})> \mathcal {M}(\mathbf {a}(\tau _{2}),k)\), then the system fails at time τ2.
According to Eq. (7), the time-dependent reliability for a reference period of [0,T], conditional on R=r, is estimated by
$$ L_{kn}(T)=\exp\left[-\int_{0}^{T}\lambda(t)\cdot \left\{1-F_{S}\left[\mathcal{M}(\mathbf{a}(t),k),t\right]\right\}\mathrm{d}t\right] $$
(10)
which is further rewritten as follows taking into account the uncertainties associated with the component initial resistances,
$$ \begin{aligned} L_{kn}(T)=\int\ldots\int & \exp\left[-\int_{0}^{T}\lambda(t)\cdot \left\{1- F_{S}\left[\mathcal{M}(\mathbf{a}(t),k),t\right]\right\}\mathrm{d}t\right] f_{{\mathbf{R}}}({\mathbf{r}})\mathrm{d}\mathbf{r} \end{aligned} $$
(11)
where fR(r) is the joint PDF of R={R1,R2,…Rn}, and r={r1,r2,…rn} as before. Furthermore, Eq. (11) can be simplified as follows,
$$ \begin{aligned} L_{kn}(T)=\int & \exp\left[-\int_{0}^{T}\lambda(t)\cdot \left\{1- F_{S}\left[r\cdot g(t),t\right]\right\}\mathrm{d}t\right] f_{\mathcal{M}(\mathbf{a}(0),k)}(r)\mathrm{d}r \end{aligned} $$
(12)
where \(f_{\mathcal {M}(\mathbf {a}(0),k)}(r)\) is the PDF of \(\mathcal {M}(\mathbf {a}(0),k)\). Eq. 12 implies that the (n+1)-fold integral in Eq. (11) can be converted into a two-fold integral if \(f_{\mathcal {M}(\mathbf {a}(0),k)}(r)\) is known. In Eq. (12), if treating \(\mathcal {M}(\mathbf {a}(0),k)\) as a generalized initial resistance, then Eq. (12) is consistent with Eq. (2) by considering an equivalent component having a resistance of \(\mathcal {M}(\mathbf {a}(0),k)\) for the system.
Eq. 12 (or (11)) is the proposed method for time-dependent reliability of an aging k-out-of-n system, where the non-stationarity of loads can be reflected by the time-variation of λ(t) and FS(·,t).
Remark 2
Note that Eq. (11) is consistent with Eqs. (3) and (4), since a series system is equivalent to an n-out-of-n system and a parallel system is simply a 1-out-of-n system. In fact, in Eq. (11), letting k=n gives \(\mathcal {M}(\mathbf {a}(t),k)= \min _{j=1}^{n}\frac {r_{j} g(t)}{c_{j}}\), and similarly, \(\mathcal {M}(\mathbf {a}(t),1)= \max _{j=1}^{n}\frac {r_{j} g(t)}{c_{j}}\).
Remark 3
For an n-dimensional vector a and an integer k≤n, if \(\overline {\mathcal {M}}(\mathbf {a},k)\) returns the kth smallest element of a, then \(\mathcal {M}(\mathbf {a},k) = \overline {\mathcal {M}}(\mathbf {a},n+1-k)\) and \(\overline {\mathcal {M}}(\mathbf {a},k) = \mathcal {M}(\mathbf {a},n+1-k)\). The implementation of \(\mathcal {M}\) and \(\overline {\mathcal {M}}\) can be realized through some commercial software such as Matlab (https://www.mathworks.com). For example, the built-in function mink(a,k) in Matlab returns a vector containing the k smallest elements of a. Illustratively, if a=[1 2 3 4 5], then \(\mathcal {M}(\mathbf {a},2)=\overline {\mathcal {M}}(\mathbf {a},4)=\texttt {max(mink(a,4))}=4\).
Remark 4
Note that Eq. (12) has been based on a k-out-of- n:G system. If considering a k-out-of- n:F system (that is, the system fails if at least k components fail), then the hazard function for the system is
$$ \begin{aligned} h_{kn}(t) & =\lambda(t)\left\{\text{Pr}[S(t)> \overline{\mathcal{M}}(\mathbf{a}(t),k)]\right\} \\ & = \lambda(t)\left\{1-F_{S}[\overline{\mathcal{M}}(\mathbf{a}(t),k),t]\right\} \\ & = \lambda(t)\left\{1-F_{S}[\mathcal{M}(\mathbf{a}(t),n+1-k),t]\right\} \end{aligned} $$
(13)
which is consistent with that in Eq. (7).
Remark 5
It is noticed that in Eq. (12), the deterioration process of each component has been assumed to be deterministic. This is applicable for many engineering cases where the variation of the deterioration process is small [42]. However, for a deterioration process with large variation, Eq. (12) needs to be modified slightly to incorporate the effect of uncertainty associated with component deterioration. With this regard, it was shown in [13] that a fully-correlated process is reasonable to describe the component resistance deterioration. That being the case, letting fG(T)(g) be the PDF of g(T), the two-fold integral in Eq. (12) would become three-fold by replacing the item \(f_{\mathcal {M}(\mathbf {a}(0),k)}(r)\mathrm {d}r\) with \(f_{\mathcal {M}(\mathbf {a}(0),k)}(r)f_{G(T)}(g) \mathrm {d}g\mathrm {d}r\), if taking into account the uncertainty associated with the component deterioration process.
Reliability of a weighted k-out-of-n system
In this section, the time-dependent reliability of a weighted k-out-of-n system will be discussed, which is by nature a generalized form of Eq. (12). Recall that the system works if the sum of weights associated with the working components is no less than the threshold k.
As before, let \(\mathbf {a}(t) = \left [\frac {r_{1} g(t)}{c_{1}},\frac {r_{2} g(t)}{c_{2}},\ldots \frac {r_{n} g(t)}{c_{n}}\right ]\), and as(t) the sorted a(t) in a descending order. Let b be an n-dimensional vector representing the weights of the elements in as(0). We can determine such a positive integer ks that
$$ k_{s} = \min\left\{j:\sum_{i=1}^{j}b_{i}\geq k\right\} $$
(14)
Clearly, ks=k if bi=1 for ∀i=1,2,…n.
Note that \(\mathcal {M}(\mathbf {a}(t),k_{s})=\mathcal {M}(\mathbf {a}_{s}(t),k_{s})\). With this, the hazard function hwkn(t) at time t, conditional on R=r, is determined by
$$ \begin{aligned} h_{wkn}(t) & =\lambda(t)\left\{1-\text{Pr}[S(t)\leq \mathcal{M}(\mathbf{a}(t),k_{s})]\right\} \\ & = \lambda(t)\left\{1-F_{S}[\mathcal{M}(\mathbf{a}(t),k_{s}),t]\right\} \end{aligned} $$
(15)
Similar to Eq. (10), the time-dependent reliability for a service period of [0,T], conditioned on R=r, is estimated by
$$ {}L_{wkn}(T)=\exp\left[-\int_{0}^{T}\lambda(t) \left\{1-F_{S}\left[\mathcal{M}(\mathbf{a}(t),k_{s}),t\right]\right\}\mathrm{d}t\right] $$
(16)
which, using the law of total probability, is rewritten as follows considering the uncertainty of R,
$$ \begin{aligned} L_{wkn}(T)=\int\ldots\int & \exp\left[-\int_{0}^{T}\lambda(t)\cdot \left\{1- F_{S}\left[\mathcal{M}(\mathbf{a}(t),k_{s}),t\right]\right\}\mathrm{d}t\right] f_{{\mathbf{R}}}({\mathbf{r}})\mathrm{d}\mathbf{r} \end{aligned} $$
(17)
By referring to Eq. (12), Eq. (17) can be further reduced to
$$ \begin{aligned} L_{wkn}(T)=\int & \exp\left[-\int_{0}^{T}\lambda(t)\cdot \left\{1- F_{S}\left[r\cdot g(t),t\right]\right\}\mathrm{d}t\right] f_{\mathcal{M}(\mathbf{a}(0),k_{s})}(r)\mathrm{d}r \end{aligned} $$
(18)
where \(f_{\mathcal {M}(\mathbf {a}(0),k_{s})}(r)\) is the PDF of \(\mathcal {M}(\mathbf {a}(0),k_{s})\).
Eq. 18 (or (17)) presents the proposed time-dependent reliability method for a weighted k-out-of-n system in the presence of resistance deterioration and correlation. It has a similar form to Eq. (12) except the item ks involved, due to the impact of the weights for each component.
Implementation of time-dependent reliability assessment
For the (weighted) k-out-of-n system reliability in Eqs. (11) and (17), if the expressions of \(f_{\mathcal {M}(\mathbf {a}(0),k)}(r)\) and \(f_{\mathcal {M}(\mathbf {a}(0),k_{s})}(r)\) are unaccessible, one would need to solve the multi-fold integrals numerically. This is usually time-consuming to conduct, and one could alternatively assess the reliability via Monte Carlo simulation, which is especially powerful in dealing with multi-dimensional problems with mathematical simplicity and robustness [20,40,43,44]. Technically, one can first generate a sample for R, denoted by r, and then compute the cores of Eqs. (11) and (17) numerically with R=r, with which the multi-fold integration can be evaluated via simulation according to
$$ L_{*}(T)=\mathrm{E}\left\{ L_{*}(T|{\mathbf{R}})\right\} $$
(19)
where ∗=kn or wkn, and E() denotes the mean value of the random variable in the brackets.