Time-Dependent Reliability

What changes in a time-dependent problem

Static reliability evaluates a limit state for a defined set of random variables. Time-dependent reliability asks whether a structure remains acceptable over an interval while load and resistance evolve. Examples include fatigue accumulation under random loading, corrosion reducing wall thickness, fluctuating pressure exceeding a degrading capacity, or wind and wave peaks acting repeatedly over a design life. The key distinction is that failure can occur at any time. Evaluating Pf at the final time alone can miss a temporary excursion across the failure boundary earlier in life. The mathematical object of interest is therefore often a first-passage or survival probability.

Survival function and hazard rate

Let T_f denote the random time to failure. The survival function R(t)=P(T_f>t) gives the probability of surviving beyond time t, while F_T(t)=1-R(t) is the cumulative probability of failure by time t. The hazard rate describes the instantaneous failure tendency conditional on survival to that time. These quantities are related but answer different questions. A rising hazard may indicate ageing or degradation; a constant hazard corresponds to a memoryless exponential life model, which is rarely appropriate for a structure whose resistance deteriorates systematically.

R(t) = P(T_f > t)
F_T(t) = 1 - R(t)
h(t) = f_T(t) / R(t)

First-passage limit states

For an evolving limit state g(X,t), failure by time T occurs if g crosses zero at any time in [0,T]. The relevant probability is P[min g(t)≤0], not simply P[g(T)≤0]. This matters when demand is fluctuating, because the largest excursion may occur well before the final time. For monotonic degradation with approximately constant demand the two can be close, but that should be demonstrated rather than assumed. Numerical simulation records the first crossing time for every realisation and builds the distribution of T_f directly.

Stochastic loads and stochastic resistance

Loads can be modelled as stochastic processes, sequences of random events or annual maxima depending on the physics. Resistance can also be stochastic in time because of degradation, uncertain maintenance or temperature-dependent properties. Temporal correlation matters: hourly wind loads are not independent, and a corrosion rate measured this year is informative about next year. Simplifying a process to independent samples at every time step can greatly distort the number of effective extremes. Choose a temporal model whose correlation structure is consistent with the actual environment and with the resolution at which the structural response is evaluated.

Outcrossing and extreme-event approaches

For smooth stochastic processes, outcrossing methods estimate how often the response crosses a limit-state boundary. Extreme-value models can be efficient when reliability is governed by maxima over blocks such as annual wind, wave or pressure peaks. These approaches are not interchangeable: an extreme-load model combined with a degrading resistance still needs a coherent lifecycle formulation. Where rare events and degradation interact, simulation or specialised time-dependent FORM methods may be preferable. The analyst should understand whether the governing uncertainty lies in event occurrence, event magnitude, structural capacity, or their combination.

Simulation strategy and discretisation

Monte Carlo lifecycle simulation is conceptually simple: generate a coherent history of loads and degradation, calculate response at each required time, detect the first limit-state crossing, and repeat. The time step must be fine enough to capture peaks and damage evolution; otherwise failures can be missed. Event-driven simulation may be more efficient when loads occur as discrete shocks or inspections. For expensive FEA, reduced-order models or response surfaces can map time-varying inputs to response. Validate the surrogate at extreme combinations because lifecycle failure is a tail event.

Inspection, repair and conditional survival

When inspection or maintenance occurs, reliability becomes conditional on the information and intervention history. An inspection at time t_i updates the current damage state; a repair may partially or fully restore capacity. Subsequent survival should be calculated from that updated state rather than from the original unconditional population. This creates a natural framework for planning inspection intervals, life extension and condition-based maintenance. The same model can compare strategies by plotting risk trajectories and identifying when a target reliability threshold would be crossed without intervention.

Reporting and engineering interpretation

Report reliability or cumulative Pf as a function of time, not just at one date. Show the assumed load-process model, degradation model, temporal correlation, inspection events and failure definition. Include numerical time-step convergence and uncertainty in the probability estimate. Sensitivity should distinguish variables that affect early-life risk from those that dominate late-life degradation. If the decision is a life limit or inspection date, state the target reliability and how the chosen date follows from the calculated trajectory. Avoid extrapolating far beyond the data range without highlighting the resulting epistemic uncertainty.

For non-monotonic demand, Pf at the end of life is not generally the probability of failure during life. A first-passage formulation is needed to capture earlier excursions across the limit state.

Engineering judgement — governing sensitivities

For Time-Dependent Reliability, the most useful review question is not simply whether the solver has produced a plausible contour or scalar result, but whether the model preserves how stochastic load, resistance degradation, inspection information and time correlation combine. Treating yearly states as independent can overstate the rate at which risk accumulates. This is where apparently small modelling choices can change the engineering conclusion. The analyst should identify the variables that can move the governing response, separate physical uncertainty from deliberate conservatism, and show that the selected modelling fidelity is proportionate to the decision being supported. Where the response is close to an acceptance boundary, sensitivity cases should bracket credible changes rather than apply arbitrary percentage perturbations.

Key takeaways

• Time-dependent reliability is usually a first-passage problem rather than a final-time snapshot.
• Temporal correlation in loads and degradation must be modelled at a physically meaningful scale.
• Inspection and repair naturally enter as conditional updates to the survival trajectory.