ASME PVP Code stress linearization is needed for assessment of primary and primary-plus-secondary stresses. The linearization process is not precisely defined by the Code; as a result, it may be interpreted differently by analysts. The most comprehensive research on stress linearization is documented in the work of Hechmer and Hollinger [1998, “3D Stress Criteria Guidelines for Application,” WRC Bulletin 429.] Recently, nonmandatory recommendations on stress linearization have been provided in the Annex [Annex 5.A of Section VIII, Division 2, ASME PVP Code, 2010 ed., “Linearization of Stress Results for Stress Classification.”] In the work of Kalnins [2008, “Stress Classification Lines Straight Through Singularities” Proceedings of PVP2008-PVT, Paper No. PVP2008-61746] some linearization questions are discussed in two examples; the first is a plane-strain problem and the second is an axisymmetric analysis of primary-plus secondary stress at a cylindrical-shell/flat-head juncture. The paper concludes that for the second example, the linearized stresses produced by Abaqus [Abaqus Finite Element Program, Version 6.10-1, 2011, Simulia Inc.] diverge, therefore, these linearized stresses should not be used for stress evaluation. This paper revisits the axisymmetric analysis discussed by Kalnins and attempts to show that the linearization difficulties can be avoided. The paper explains the reason for the divergence; specifically, for axisymmetric models Abaqus inconsistently treats stress components, two stress components are calculated from assumed formulas and all other components are linearized. It is shown that when the axisymmetric structure from Kalnins [2008, “Stress Classification Lines Straight Through Singularities” Proceedings of PVP2008-PVT, Paper No. PVP2008-61746] is modeled with 3D elements, the linearization results are convergent. Furthermore, it is demonstrated that both axisymmetric and 3D modeling, produce the same and correct stress Tresca stress, if the stress is evaluated from all stress components being linearized. The stress evaluation, as discussed by Kalnins, is a primary-plus-secondary-stresses evaluation, for which the limit analysis described by Kalnins [2001, “Guidelines for Sizing of Vessels by Limit Analysis,” WRC Bulletin 464.] cannot be used. This paper shows how the original primary-plus-secondary-stresses problem can be converted into an equivalent primary-stress problem, for which limit analysis can be used; it is further shown how the limit analysis had been used for verification of the linearization results.