To examine the seismic resistance performance of a steel arch bridge during a strong earthquake, an approach is proposed in this paper to determine the ultimate strain of an arch rib using the static elasto-plastic large deformation theory. A steel fixed-end deck arch bridge with a span length of 200 m was used as an example in this study. The ultimate strain of the arch rib was calculated using the elasto-plastic finite element model. The seismic responses of the arch bridge exposed to strong earthquakes in two and three directions were also simulated. Moreover, the effect of earthquake motion in the lateral direction on the elasto-plastic earthquake response was discussed, and the seismic resistance of the structure was analyzed. The results show that steel arch bridges have preferable seismic resistance under strong earthquake conditions. While lateral earthquake motion has some influence on the damage to an arch bridge, it has little effect on the displacement, axial force, and bending moment responses of the arch rib. It is feasible to examine the seismic performance of the arch bridge by ascertaining the out-of-plane and in-plane earthquake responses of the bridge.