Durability and service life 25
Reaching the service life within the time interval (t, t + dt) implies that
R < S within (t, t + dt) and that R > S in (0, t). Indeed, if this latter condition
is not fulfilled, the service life would already have been reached before the
interval (t, t + dt). Consequently, it can be written that:
f L (t) dt = P [R < S in (t, t + dt) and R > S in (0, t)]
(1.6)
The PDF of the service life L can be considered as the failure probability
per time unit or the failure rate. In this respect, the terminology ‘unconditional failure rate’ is also applied, in contrast to the ‘conditional failure
rate’ or ‘hazard function’ r(t). This hazard function
r(t) dt = P [R < S in (t, t + dt) | R > S in (0, t)]
(1.7)
is the probability that a system or component, which was functioning in
the time interval (0, t), will fail in the following elementary time interval
dt. The hazard function gives an indication whether it is becoming more
probable for an element to fail with aging.
It can be shown that the following relations exist between r(t) and the
distribution function of the service life L:
r(t) = f L (t)/(1 – F L (t))
(1.8)
F L (t) = 1 – exp [−
0
t
∫
r (τ) dτ]
(1.9)
Figure 1.15 shows some typical cases for F L (t), f L (t) and r(t). In the first
case (a), the hazard function r(t) is constant. The conditional failure rate is
constant in time. Applying equation (7), this leads to an exponential distribution of the service life:
F L (t) = 1 – exp (−λt)
(t ≥ 0, λ = constant)
(1.10)
In case (b) of Figure 1.15, the hazard function r(t) decreases with time
due to ongoing hydration and increased strength of the concrete. In case
(c), an increasing function of r(t) is shown, which is typical in case of degradation, ageing or damage. In reality, a combination of the different situations can occur, including a first period with decreasing hazard function,
a second period with constant values, and a third period with increasing
hazard function. The resulting overall evolution of the hazard function is
then typically as shown in Figure 1.16.
Instead of considering the service life as a stochastic variable, a more
classical reliability analysis typically starts from a well-defined reference
Reaching the service life within the time interval (t, t + dt) implies that
R < S within (t, t + dt) and that R > S in (0, t). Indeed, if this latter condition
is not fulfilled, the service life would already have been reached before the
interval (t, t + dt). Consequently, it can be written that:
f L (t) dt = P [R < S in (t, t + dt) and R > S in (0, t)]
(1.6)
The PDF of the service life L can be considered as the failure probability
per time unit or the failure rate. In this respect, the terminology ‘unconditional failure rate’ is also applied, in contrast to the ‘conditional failure
rate’ or ‘hazard function’ r(t). This hazard function
r(t) dt = P [R < S in (t, t + dt) | R > S in (0, t)]
(1.7)
is the probability that a system or component, which was functioning in
the time interval (0, t), will fail in the following elementary time interval
dt. The hazard function gives an indication whether it is becoming more
probable for an element to fail with aging.
It can be shown that the following relations exist between r(t) and the
distribution function of the service life L:
r(t) = f L (t)/(1 – F L (t))
(1.8)
F L (t) = 1 – exp [−
0
t
∫
r (τ) dτ]
(1.9)
Figure 1.15 shows some typical cases for F L (t), f L (t) and r(t). In the first
case (a), the hazard function r(t) is constant. The conditional failure rate is
constant in time. Applying equation (7), this leads to an exponential distribution of the service life:
F L (t) = 1 – exp (−λt)
(t ≥ 0, λ = constant)
(1.10)
In case (b) of Figure 1.15, the hazard function r(t) decreases with time
due to ongoing hydration and increased strength of the concrete. In case
(c), an increasing function of r(t) is shown, which is typical in case of degradation, ageing or damage. In reality, a combination of the different situations can occur, including a first period with decreasing hazard function,
a second period with constant values, and a third period with increasing
hazard function. The resulting overall evolution of the hazard function is
then typically as shown in Figure 1.16.
Instead of considering the service life as a stochastic variable, a more
classical reliability analysis typically starts from a well-defined reference