Post-reconnaissance findings after recent earthquakes have shown that the effect of the vertical acceleration component of the ground motion can be more damaging than typically considered thus far, particularly in near-field regions where the ratio of vertical-to-horizontal ground accelerations may exceed considerably the 2/3 scaling factor used in design. Besides, the usual period range for the constant-acceleration region of vertical acceleration spectra often overlaps with the range of vertical vibration periods in reinforced concrete (RC) structures. These reasons explain the appearance of large dynamic axial forces, which can reduce the column shear strength due to tensile demands or alternatively promote direct compressive failures. Estimating the change in the member axial forces during nonlinear seismic response due to the vertical ground motion component is therefore of paramount significance and can only be simulated through dynamic analyses, for which a specific damping model needs to be assigned. Using a cantilever bridge pier with a top mass as an illustrative example, the present paper assesses the effects of the most commonly used damping models and damping values on the simulation of axial forces. Distributed plasticity beam elements with distinct formulations are employed, and a range of top masses is considered. The results show that, even for very low damping values, distinct damping models can have a very significant influence on the simulation of the seismically-induced axial forces, which increases considerably for larger values of the top mass.