<p>The progressive degradation of joints in rock masses may occur through time-dependent failure of intact rock bridges along discontinuity planes, leading to a reduction in joint cohesive strength and, ultimately, to the onset of instability in underground excavations. This paper investigates the role of rock bridges in controlling the stability of a symmetrical roof block in a circular opening, under the assumption of temporally evolving bridge failure and the associated reduction in joint cohesion, in order to estimate the stand-up time of the block. The analytical framework is developed under plane-strain conditions. Five parametric scenarios are defined based on distinct fracture toughness values, which are interpreted as representative of different rock-quality classes. The results indicate that cohesion decays over time to a critical threshold at which the block attains limit equilibrium, thereby defining the stand-up time for each scenario. A global sensitivity analysis, combining the Morris elementary-effects method with Monte Carlo–based uncertainty propagation, demonstrates that the in situ stress ratio, the bridge-to-joint length ratio, and the joint friction angle are the primary controlling parameters. These factors exhibit comparable influence and are markedly more important than the parameters governing subcritical crack-growth kinetics. Complementary finite-element simulations conducted with RS2 illustrate the stress distribution mechanism that drives bridge-localized failure. Incorporating both time-dependent degradation mechanisms and reliability-based uncertainty propagation facilitates a probabilistic estimation of stand-up time, thereby providing a more robust basis for the design and safety assessment of underground excavations.</p>