The safety of high-potential risk facilities concerned with fault displacement hazards is a complex technical issue, especially if the fault is revealed beneath the facility during the operation. Applying simple conservative engineering hazard evaluation methods is rational if an urgent decision should be made to continue operation or implement protective measures. Engineering methods are being published for strike-slip on-fault sites and structures. Their crucial point is to estimate the probability of the rupture at the site intersection and consider the displacement distribution over the rupture length relative to the site’s on-fault location. It is shown in the paper that strict geometrical relations between the site location, length and initial point of the rupture determine whether the principal fault displacement intersects the site. The paper considers these geometrical parameters as independent random variables and applies a screening of ruptures contributing to the hazard. Magnitude- and on-fault coordinate-dependent empirical relations have been analysed and selected to evaluate the site displacements. The procedure resulted in realistic but conservative hazard curves for different on-fault site locations using data from the Paks site in Hungary. The results were compared to those obtained by the conservative engineering method for the same site and some published analyses.