The erosion of soil around bridge shallow foundation piers can substantially increase the fragility of bridges during flood events. The available research for assessing the fragility of shallow foundation piers under floods is predominantly qualitative, while ignoring the influence of coupling effects arising from complex hydrodynamic-soil-structure interactions. To fill these gaps, a numerical solver for coupled simulations accounting for hydrodynamic-soil-structure interaction is developed, facilitating boundary conversions for different physical fields and updating them over time. The numerical solver is validated in terms of subgrade reactions, pier displacements and scour extent based on the results of open-channel tests of a reference shallow foundation pier. The validation results show that the developed coupled solver has satisfactory accuracy in predicting the morphology of scour-hole and failure process of shallow foundation piers. Then all potential failure modes of shallow foundation piers under floods are analyzed within the framework of deterministic analysis. Finally, a probabilistic fragility analysis using the coupled solver accounting for uncertainties in hydraulic, structural, and geological parameters is performed through the Latin Hypercube Sampling method. The study concludes that the deterministic analysis without considering the uncertainties in model parameters leads to underestimating the risk due to overturning failure and vertical failure.