This study develops closed-form equations to calculate yield and ultimate moments and curvatures under combined axial load and uniaxial bending. These equations can be used to determine the force-displacement capacity of bridge piers, accounting for shear capacity and P-delta effects. To verify the accuracy of the proposed equations, a comparison is conducted with results from fiber-section analyses in OpenSeesPy, showing satisfactory agreement. The equations enable efficient multi-parametric and stochastic analyses, allowing for the rapid calculation of fragility curves, which is particularly beneficial for large-scale territorial risk studies. A typical application includes risk analysis of railway and highway networks, where the piers of statically determined bridges are critical components.