This paper investigates the in-plane nonlinear elastic and inelastic buckling behaviour and the strength of fixed circular steel arches by using a rational finite element model. It is found that the elastic and inelastic buckling behaviour of a fixed arch is quite different from that of a pin-ended arch. The design equation for pin-ended steel arches in uniform compression cannot be used directly for the design of fixed steel arches, nor can the design interaction equation for pin-ended steel arches be used for the design of fixed steel arches that are subjected to combined axial compressive and bending actions produced by general in-plane loading. A design equation for the strength of fixed steel arches that are subjected to uniform compression is proposed. The finite element investigations show that this proposed design equation provides good predictions for the strengths of fixed steel arches in uniform compression. An interaction equation for strength design of fixed steel arches that are subjected to combined bending and axial compressive actions against in-plane failure is also proposed. The finite element investigations show that the proposed design equation provides good lower bound predictions for the strengths of fixed steel arches in combined compressive and bending actions.