The existing of joints has great effect on the mechanical properties and strength characteristics of rock mass. In this study, four different failure modes of rock mass with one group of joints and the corresponding ranges of joint dip angles are obtained according to the strength criterion of statistical mechanics of rock mass and the Mohr-Coulomb failure criterion. An expression of the critical confining pressure is deduced for representing the transformation of rock mass strength from structure control to stress control. Moreover, rock masses with a group of joints are divided into four classes by the relationship between rock and joint parameters. Then the potential failure modes and the corresponding failure conditions are discussed. Finally, the strength characteristics of diorite with a group of joints are analyzed. The results show that the diorite belongs to class I rock mass. The diorite fails firstly along with the direction of joints and then fractures in the rock blocks at a critical confining pressure of 9.12 MPa under the axial pressure condition. It is found that the diorite has significant anisotropy from the spatial analysis, and the increase of confining pressure would result in the transfer of rock mass strength from structure control to stress control under the particular loading directions.