The mechanical connection was the main connection means of composite structures. In this paper, a three-dimensional finite element model of progressive failure was established by business software ABAQUS. The model considered the contact between bolt and hole, progressive damage, large deformation theory, and nonlinear shear stress/strain relations. HASHIN failure criterion and TAN material property degradation criterion were adopted to study the progressive failure process of mechanical connection structure, and the effects of different interferences (0%, 0.5%, 3%) on the strength and stiffness of mechanical connection structures were investigated. The results showed that the proper interference fit could improve the bearing strength of the mechanical connection, while the excessive interference fit could reduce the bearing strength of the connection, which was in consistent with the test results.