The out-of-plane secondary bifurcation buckling load-displacement equilibrium paths of the elastic circle pipe arch with and without out-of-plane brace at the top of the arch are traced using a new numerical tracing strategy. The out-of-plane secondary bifurcation buckling loads of the arch with the same sections and different rise-span ratios are obtained under the concentrated load at the top of the arch and the full span uniformly distributed load, which are compared with out-of-plane linear buckling load and in-plane primary buckling load. The calculation results show: for the same section circle pipe arches without the out-of-plane brace and under the concentrated load at the top the arch, the out-of-plane secondary buckling load is always less than the in-plane primary buckling load and the out-of-plane buckling will occur before the in-plane primary buckling. The out-of-plane secondary bifurcation buckling load of the arch with 0.2 rise-span ratio is the biggest. The bigger the rise-span ratio is, the bigger the difference between out-of-plane and in-plane buckling load. When the arch is subjected to full span uniformly distributed load, the out-of-plane buckling will also occur before the in-plane primary buckling and the out-of-plane secondary bifurcation buckling load of the arch with 0.4 rise-span ratio is the biggest. The difference between out-of-plane and in-plane buckling load of the arch with 0.2 rise-span ratio is the biggest. For the circle pipe arch with the out-of-plane brace at the top of the arch, the out-of-plane buckling load of the arch with 0.4 rise-span ratio is the biggest under the two load conditions. The brace can raise the out-of-plane buckling load significantly especially for the arch with big rise-span ratio and under full span load. The out-of-plane buckling will occur before the in-plane primary buckling when the arch is under full span uniformly distributed load. The out-of-plane buckling will occur before the in-plane primary buckling only when the arch is under concentrated load and the rise-span ratio of the arch is less than 0.3. No matter there is or not brace for the arch, the ultimate load carry capacity of the arches increase a little bit after the out-of-plane secondary buckling occurs.