Yes I understand how to do this. I was merely suggesting that you were asking a question that it was in your own best interest to research for yourself. And your post immediately above this one reinforces my answer that yes, you do know how to do this.
There are phenomena that occur in such systems that are functions of the properties of the two metals, the relative thicknesses of the two layers of metal, the pipe OD, the operating temperatures, the assumed radial thermal gradient, and other factors. This is not an area for new stress analysts to enter. Incorrect application of cladding, and/or poor welding procedures or welding QA/QC in the shop and field, have probably been responsible for more fatalities than we would care to count.
If I just answer one question about a very basic case, that answer may be used as the basis for an analysis for which the phenomena you refer to become very significant, and damage may occur to facilities, plant staff, and the general public. I choose not to give specific answers to simple problems if I am afraid that the answer that I give may have such a result. Sorry to let my personal code of conduct intrude in your life.
I have had a couple of occasions in my career to work with the design of systems >600bar, and the assumptions on which the piping code is based are suspect in that range. As your second post suggests, it becomes necessary to account for radial thermal gradients and local stresses even in systems made of one materail For clad systems, the relative differential expansion of the two metals also becomes more significant as the ratio of the pipe OD to the wall thickness decreases.
I think you will find that cladding a CS line with a nickel alloy that has a higher coefficient of expansion is, in general, helpful to the mechanical strength of the system. But in extreme cases you MUST do as you suggest - consider a 2D slice of the piping to determine temperature gradient effects and the radial distribution of pressure stresses.
And your results would be very different for, say, a 200 mm pipe with a 12 mm carbon steel wall and a 3 mm nickel clad lining vs. a 50 mm pipe with a 12 mm carbon steel wall and a 3 mm nickel clad lining. So your initial post did not define a problem that a responsible engineer would want to solve in a forum of this sort, as the subsequent posts (including your own) show.
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CraigB