Traditionally, shallow spread footing-type foundations are used for medium span bridges supported on hard soil and rock strata. Such bridges are modelled with fixed supports and no soil–foundation–structure interaction (SSI) is normally considered. The investigation presented herein utilized a sub-structuring technique and finite element method (FEM) model for the analysis of a four-span bridge designed for five different rock classes and subjected to an ensemble of actual ground motions. SSI was incorporated through Winkler springs while non-linear behaviour of reinforced concrete pier column was modelled by an equivalent linear model. The results of the study were evaluated to delineate the effect of SSI and pier column non-linearity on seismic response parameters of bridges founded on rock. It was found that the soil–foundation–structure interaction couldn’t be neglected in all cases of rock classes and input ground motions. Furthermore, pier column non-linearity influenced bridge displacement and base shear more significantly than SSI. Impact of foundation rocking was also examined and was found to be rather insignificant on bridge response parameters due to high rocking impedance of properly designed bridge foundations.