A fully smoothed finite element method to solve the torsion problem of axisymmetric elastic body is proposed. Due to axial symmetry of geometry and boundary conditions, computing the torsion problem of axisymmetric elastic body requires extracting one cross section for meshing and analysis. In order to address the problem of high sensitivity to mesh distortion, the edge-based smoothing domains are further constructed based on the triangular mesh and smoothing operations are performed on the strain in each smoothing domain. The smoothing strain technique is combined with the quasi-weak form of smoothed integral for treatment of the partial derivative and non-partial derivative of shape functions. Accordingly, all the smoothed domain integrals can be simplified as boundary integrals of the smoothing domains and there is no need for the coordinate mapping and calculation of Jacobian matrix. Numerical examples demonstrate that the proposed method for the torsion problems of axisymmetric elastic body can produce accurate solutions even for irregular elements.