Seismic fragility of free-standing rocking-sliding rigid blocks under earthquake excitation

The seismic fragility of free-standing rigid blocks under earthquake excitation is investigated, focusing on the combined effect of rocking and sliding in their dynamic response. A variational-based formulation of the non-smooth contact dynamics method is employed for the problem solution, requiring the solution of a quadratic programming problem at each time step in the three unknown block scalar velocities. Such an approach efficiently captures all possible block response modes, including rocking, sliding, rocking-sliding, and free flight. The seismic fragility of both slender and stocky blocks is evaluated through a multiple-stripe analysis, introducing overturning-sliding fragility curves that account for overturning and excessive-sliding limit states. Numerical validation against benchmark results confirms the accuracy of the proposed formulation, showing that while a no-sliding assumption is reliable for slender blocks, it may prove unsafe for stocky blocks, even assuming high friction at the foundation. The resulting overturning-sliding fragility curves highlight the dependence of the block response on the normalized friction coefficient, defined as the ratio of the friction coefficient to the block slenderness. Blocks with a normalized friction coefficient below unity exhibit an isolation effect, preventing overturning at the cost of significant sliding. Conversely, blocks with normalized friction coefficient above unity undergo rocking-sliding or pure-rocking motion, which can lead to overturning. Accordingly, the overturning-sliding fragility curves are determined by the interplay between overturning and excessive-sliding limit states, as governed by a prescribed sliding displacement capacity. For typical values of the friction coefficient at the block-foundation interface, slender blocks are largely unaffected by sliding, whereas overturning-sliding or sliding fragility curves are crucial for accurately assessing the seismic performance of stocky blocks.