I am unsure of your terms here.
Stress intensity is one of many ways to look at stress. Here, stress intensity is the diameter of Mohr's Circle, which is twice maximum shear stress. This value can can be compared with the yield allowable limit in satisfying the Tresca theory of failure.
Stress range is not a stress equation. Stress range is a change in stress as opposed to a state of stress. This range is used for the simplified fatigue evaluation found in most piping codes. Stress state is normally used to evaluate collapse loads (sustained & occasional).
I don't know what CAESAR II text you are referencing regarding ASME Section VIII Division 2 fatigue evaluation. The fatigue curves have been updated in VIII-2 and these have not been changed yet in CAESAR II. But you can enter your own.
CAESAR II, as a piping program, calculates stress in accordance with the piping codes. The stress we calculate is a nominal stress - using a calculated bending moment along the pipe centerline and a stress intensification factor(SIF). This code-defined SIF accounts for the shell distortion (ovalization) caused by elbow bending. The Code does not accommodate the change in hoop stress associated with this distortion. The Code is a simplified method of assuring piping safety. You can employ more rigorous evaluation if you wish (i.e. FEA), but not with CAESAR II alone.