The stress concentration is one of the major causes of failure around the tunnel in a transversely isotropic rock mass. Knowledge about the features of damage helps tunnel engineers develop suitable countermeasures. In this paper, failure modes around the tunnel are studied through numerical analyses using the three-dimensional (3D) particle flow code (PFC3D). The transversely isotropic rock masses are generated with seven joint dip angles of 0°, 15°, 30°, 45°, 60°, 75°, and 90°. The tunnel excavation is simulated in 3D to study the effect of tunneling direction relative to the joint orientation on the tunnel stability. Various scenarios involving the tunneling direction, the joint strike, and joint dip are considered. Concerning the displacement around the tunnel, the displacement field (DF) may be grouped into one of the following four types: the tensile displacement field (DF-I and DF-II), the shear and tensile displacement field (DF-III), and the shear displacement along the joint (DF-IV). The failure mode may be described with one of the following five terms: (1) Detaching and Buckling, (2) Sliding, (3) Bending and Spalling, (4) Slabbing and Spalling, and (5) Falling. The failure modes at the crown, the sidewalls, and the invert of the tunnel in Scenarios 2 and 3 are similar. However, the failure modes in the tunnel face are different; the “Falling” mode is observed in Scenario 2, while the “Sliding” model is a risk in Scenario 3.