Out of plane bending by an end couple applied to a 90° elbow or pipe bend of circular cross-section is considered. There is conversion of bending to twist so that at 90° the elbow is held in equilibrium by a twist couple. Using the methods of toroidal elasticity, the six stresses of the first order stress field are determined from the zero-order field. The zero-order field constitutes a permissible initial deformation as verified by strain compatibility equations. The equations in the text are derived from the general theory of toroidal elasticity by applying the method of successive approximations. The equations of the general theory are expanded in a series in 1 R (where R is the toroidal radius). This leads to the working equations given in the paper.