A versatile geometrically non-linear curved shell finite element formulation includes exact Green's strain displacement for the inplane strains and linear strain displacement for the transverse shear strains. Additional coupling results for membrane-bending and membrane-transverse shear due to non-linear curvature strain-displacement relations. Additional non-linear terms that are a function of the shell's radius of curvature are also present. The approach is applied to the bifurcation buckling of axially compressed flat plates and transversely pressure-loaded cylindrical shell panels. The method gives a more flexible result for the plates compared to von Karman non-linearity. Additionally, larger buckling pressures are found for shell panels. The effects of transverse shear flexibility are evident.