The reactions in paragraph 119.10 are loads caused by thermal strain. If I wish to mimic these reactions in CAESAR II, I would run a T1 or T1+D1 analysis using the (default) cold modulus. But the system must be linear. This will give R. Then Rh would be as you show but Rc would be 0 (Rc=-CR, with C=0, Rc=0).
CAESAR II provides a comprehensive method of analysis. Using CAESAR II, I would run the operating case using Eh and any cold spring and any nonlinear support. and also run the installed case with Ec. These reactions would also include weight.

The last paragraph in 119.10 implies that (18) & (19) are for simple systems through the sentence "In this case, the piping system shall be analyzed by a comprehensive method."