The influence of the vertical component of ground motion on the probabilistic treatment of the rocking response of free-standing blocks

The influence of the vertical component of ground motion is investigated for assessing the distribution of the seismic response of unanchored rigid blocks. Multiple stripes of site-hazard-consistent ground motions are employed for calculating the seismic response of rigid rocking blocks with and without the inclusion of the vertical component. The comparison of the resulting response is being made both for single records and full suites, employing a paired record versus an ensemble-statistics comparison, respectively. It is shown that on a single record basis, the vertical component may have a non-negligible but highly variable influence on the rocking response, sometimes detrimental, sometimes beneficial. Still, when considering any large ensemble of records, the effect becomes statistically insignificant, except for the very specific case of rocking uplift for stocky blocks. To this end, for cases where the appearance of uplift is associated with damage, closed-form expressions are proposed to modify the lognormal fragility function of rocking initiation given the block slenderness and the ratio of the peak vertical over the peak horizontal ground acceleration.