Although arches and curved beams have very attractive structure behavior in the plane of curvature, the behavior in the out of plane is very critical. This is due to the coupling effect of the torsional, bending and axial straining actions due to the geometry of the arch / curved beam. In this paper, coupling between the torsional moment and out-of-plane moment in arches was studied. A parametric study was conducted to identify the effect of different parameters on the coupling behavior. It was found that the rise of the arch, the support configuration and the rigidity of the supports had a considerable effect on the value of the out-of-plane moment produced due to torsional load. Similar to arches, horizontally curved beams experienced the same coupling effect in the plane perpendicular to the plane of curvature, i.e. in the vertical plane. In order to emphasize the importance of this phenomenon, a practical example comparing a curved beam and a straight beam was presented. Both beams were subjected to uniform torsional moments. It was shown that the design demand on the curved beam increased by more than 250% compared to the straight beam due to the secondary moment perpendicular to the plane of curvature induced due to the torsional load. This result showed that if the curved beam is subjected to torsional load, simplifying the design by ignoring the curvature could lead to serious failures even in cases where the curvature is not big.