Bending and Hoop Stresses-Cylinder

As long as you are in the linear region, you can add components. Once nonlinearity sets in (e.g. buckling) all bets are off.

"The Principle of Superposition is a method used to solve complex problems with multiple loads and/or reactions acting on the member. Superposition helps us solve these problems by breaking the member down as many times as necessary for each force acting on it. Once all the stresses or deflections for the point of interest are found, they can then be added all together to get a final answer."

http://www.engineeringwiki.org/wiki/Principle_of_Superposition

Typically vonMises stress formula is used to combine longitudinal and hoop stresses in pressurized cylinders. Longitudinal stresses are axial and bending. Pipeline codes address this for example ASME B31.8. Only limitation is that basis is in thin shell theory (i.e. Barlows equation for hoop stress) so when applied to heavy wall problems yields unreasonably conservative results in some instances .

I suggest that you review some of the information available on the development of piping and pressure vessel codes. Specifically you should look for historical development of the Code equations for evaluating stresses.

Remember, the equations incorporated into the Codes are the subjective result of a committee of engineers....

When these Codes were developed compromises were made.

Here are some places to start:

https://www.osti.gov/servlets/purl/4031361

https://www.scribd.com/document/173934962/Companion-Guide-to-the-ASME-Boiler-Pressure-Vessel-Code-Volume-1