The in-plane nonlinear elastic buckling and post-buckling of circular arches have been investigated extensively and the corresponding problems of arches in the Cartesian coordinate system have not attracted enough attention although these arches are applied extensively in the arch bridge engineering. This paper explored the in-plane nonlinear membrane strain, bending strain and shear strain for curve arches in the Cartesian coordinate system based on the Timoshenko beam hypothesis, and the in-plane nonlinear elastic buckling and post-buckling of catenary arches subjected to infill gravity are investigated based on these nonlinear strains and analytical solutions for the stability are derived. Comparisons with finite element method results show that the analytical predictions are in good agreement with numerical solutions, and the proposed in-plane nonlinear strains can be applied to derive analytical solutions for the nonlinear stability of arches in the Cartesian coordinate system.