Ocean platforms are subjected to a variety of environment loads, such as those from winds, waves, currents, etc. In this study, the torsion problems of a gravity platform column with cracks under wind load were investigated. The column was assumed to be a composite cylinder. Therefore the torsion fracture problem of a composite cylinder was considered, and new boundary integral equations for the Saint-Venant torsion problem of a composite cylinder with curvilinear cracks were derived. The problem was reduced to solving the boundary integral equations on every boundary. By using the new boundary element method, the torsion problem of the gravity platform column with a straight crack under various wind loads was calculated. The obtained results were compared with those obtained for a torsion problem of the same column without cracks to prove the applicability of the present method. The comparison showed that the detrimental effect of cracks in a column should be considered in marine engineering.