Two failure modes are addressed for cylinder-cone junctions under internal or external pressure: axisymmetric yielding and low-cycle fatigue. If the junction fails to meet the failure criterion of any one of the two modes, then it must be strengthened by reinforcement. It is shown in the paper that the degree to which a junction is strengthened depends on the distribution of the reinforcement. A placement of reinforcement on the cylinder alone, leaving the actual connection between the cylinder and cone unreinforced, adds strength with regard to axisymmetric yielding, but may not strengthen the junction sufficiently with regard to low-cycle fatigue. This means that the junction may appear reinforced, but is not strengthened. It is pointed out that the design rules of Section VIII, Div. 1 of the ASME B & PV Code (1992) set the need for reinforcement according to the failure criterion of low-cycle fatigue, while the distribution of the reinforcement is guided by the criterion of axisymmetric yielding. There is no assurance that the reinforced junction will meet the failure criterion of low-cycle fatigue. This means that the safety margin on the number of allowed cycles is less than that which is expected and that the junction may be unfit for cyclic service. It is also shown in the paper that a reinforcement distribution that requires minimum thicknesses for sections of both the cylinder and cone near the junction can satisfy criteria for both failure modes. This approach is already used in Code Case 2150 of Section VIII, Div. 1, for half-apex cone angles from 30 to 60 deg, and required in Div. 2 for cone angles from 0 to 30 deg. Its extension to angles from 0 to 60 deg for both internal and external pressure is recommended.