The adjustable localization operator (ALO) method allows for the local analytical calculation of complex structures under multiaxial confined plasticity cyclic loadings. This study proposes a theoretical framework for the extension of the method to cover anisotropic yield surface materials. The application is carried out on steel tubes that have been axially compressed beforehand to create a bulge that acts as a stress concentration factor during the fatigue loading process. The ALO method is compared to the reference finite element anisotropic analysis at different steps of the fatigue design chain. Results have shown that although a slight gap on the absolute strain values exists, the strain amplitude and thus the fatigue life are correctly predicted with a reduction of the calculation time by 100.