Abstract: A straightforward and concise approach is proposed to analyze the Saint-Venant’s torsion of a circular shaft containing multiple holes or cracks based on a complex variable method. The semi-analytic solutions for shear stresses and torsional rigidity are first given by introducing the Faber series expansion, Fourier series expansion and conformal mappings. Then，when the elliptical holes degenerate into cracks, the expression for the stress intensity factors of cracks is also derived. Finally, the effects of the parameters on the holes stress concentration, torsional rigidity and the stress intensity factors are studied through several numerical examples. It is shown from the obtained results for special examples that the method presented in this paper has many advantages such as high accuracy and good convergence.