To reveal the failure mechanisms and strength anisotropy of layered rock, uniaxial compression tests were conducted on dolomite samples with seven different bedding dip angles, and the compressive strength and the Young’s modulus with dip angle β were analyzed. Meanwhile, five failure modes (rock tensile cracking, rock shearing failure, bedding cracking, bedding sliding failure, and rock bending) were observed from laboratory phenomena, and the corresponding anisotropy constitutive model was further developed based on the transversely isotropic model and the plastic mechanics flow rules. Finally, the self-defined anisotropic constitutive model was coded in FLAC3D to carry out a numerical simulation of uniaxial compression. Results showed that the numerical simulation agreed well with experimental tests results, which proved that the assumption of a transversely isotropic model was reasonable. Furthermore, the proposed model could be applied to evaluate the stability of rock slopes with different bedding dip angles, and the bedding dip angle played an important role on failure modes and the stability of slope.