If you know the radial and axial expansion amounts ahead of time, then it's not too difficult if you have enough detailed data about the bearing, although that data can be very hard to get, depending on the type of bearing. However, if you want to predict the expansion, it can get very complicated because you can run into the situation where the growth changes the bearing loads, which produces more heat generation, which further changes the bearing loads etc - sometimes leading to bearing destruction. So it depends on how you frame the problem. But most people simply rely on the bearing manufacturers to help them with the calculations.
If you are using tapered roller bearings, don't neglect to consider "lateral loss" (axial) resulting from assembly, i.e. shrink of cup from press fit and expansion of cone from press fit. This can add significantly for large bearings.
The bearings I am using are Deep groove single row ball bearings SKF 16011. 55mm Bore Diameter, 90mm Outside Diameter, 11mm wide. There are two bearings used in a Overrunning clutch assembly for a hydraulic turbine starter drive. The input shaft enters the clutch and the output shaft is integrated into the clutch housing. The bearings are on either side of the clutch and and the clutch, with both bearings inserted into the clutch housing. Make sense? Yeah it's hard to put into words.
When there is a large thermal growth issue in the axial direction, it is general practice to constrain a shaft at one end (with something like a deep groove ball bearing) and let the other end float (using a roller bearing).
As for radial fit with a large CTE mismatch between bearing race and housing, there are several things you can do. Say you have a magnesium housing and a steel bearing. If you design for a proper fit at room temp, you will have a loose fit (and fretting) at an operating temp of say 200 degF. Coversely, if you design for a proper fit at an operating temp of 200 degF, you may have a condition where the intereference between the housing and bearing at room temp or lower, removes any running clearance in the bearing and causes it to seize. To avoid this problem you can press and pin a steel sleeve into the nonferrous housing, and then machine the steel sleeve for a proper fit with the bearing. If there is room, you can also get bearings that have a bolt flange on the outer race, eliminating the need for a press fit.
If you want to know how bad the axial load on your bearing can get try this:
Assume steel shaft with a coefficient of thermal expansion equal to 6.5x10^-6 in/in°F. Assume temperature differential equal 200°F. E equal 2.9 x 10^7 lb/in^2.
For convenience sake assume shaft length equal 10 inches and diameter equal 1 in. (area = 0.7854 in^2)
If the shaft is constrained between two fixed bearings then there will be no expansion and thus the strain will be:
e=10in x 6.5 x 10^-6 in/in°F x 200°F
e=.013 in/in
and since stress = e x E
stress = .013in/in x 2.9 x 10^7 lb/in^2
stress = 377,000 lb/in^2
stress = Force / area
then
Force = stress x area
Force = 377,000lb/in^2 x 0.7854 in^2
Force = 296,096 lb
Obviously this is oversimplified. The setscrews on a mounted bearing will slip and/or the shaft will buckle in addition to overloading the bearing, but the forces generated by thermal expansion are very high. You should use an expansion bearing and a fixed bearing for shafts subjected to thermal loads.
