Varmint Al (google him) has a succinct explanation. I lifted this from his website (giving credit where credit is due):
ONE OR TWO PIECE SCOPE BASE.... The question of one piece or two piece scope base is a tricky question. There are a number of other factors that need to be considered to optimize the situation. Let's assume that that both bases fit the action reasonably well. First, the action is usually 4140 steel or equivalent with a Coefficient of Thermal Expansion (CTE) of 6.2E-6 in/(in-°F). One exception I know of is the Stolle Panda action which is aluminum. Aluminum 6061-T6 has a CTE of 13.1E-6 or more than twice that of steel. The difference in thermal expansion is the major problem with a one piece aluminum base on a steel action. When the base is screwed onto the action at say 72 °F and assuming it fits well, it is stress free in the axial direction. If the assembly is then heated to 120 °F (which is possible in full sun), the differential in expansion between the base and action for a 5 inch span is:
dL=L * da * dT,
where dL is the length difference (in), L is the span length (5 in), da is the difference in CTE's between steel and aluminum (6.9E-6 in/(in-°F)), and dT is (48 °F) the temperature change.
Running the numbers, the 5 inch section of a free base would lengthen about 0.00166 inch more than the action would. The cross-section area of the B-Square base on my Savage 12BVSS is approximately 0.12 sq-in and force required to compress the 5 inch section of the base 0.00166 inch is,
Force = K * dL = 398 lb
Where the spring constant, K = Area * Elastic-Modulus / Length
Area = .12 sq-inch
Elastic-Modulus = 10E6 lb/sq-inch
L = 5 inch
K = 240,000 lb/inch and the force required to shorten the base 0.00166 inch is 398 lb. This is a lot of force! The stress in the aluminum base is 3,300 psi which is well below the yield stress.
ASSUMING THE ACTION IS RIGID.... This assumes that the much larger steel action is rigid and is a reasonable assumption because the action is much larger than the base. Taking into account the flexibility of the action involves a statically indeterminate problem. This would require very sophisticated Finite Element Analysis including the geometry of each action and one piece base to determine the deformations of both. Now assume that your rifle scope is made of aluminum. If you use a two piece base, then a 0.00166 inch displacement goes directly into the scope tube. If the section area of the scope tube is half that of the base, then the force would be half or 199 lb. But in either event you are going to shorten the center of the scope. Using an aluminum one piece base, resists some of the load and lowers the load transmitted to the scope. I think you would change the point of impact more by stressing the scope than stressing the action, all else being equal.
But, if you have a scope with a steel tube, and use a two piece base, then the mounted scope goes through temperature swings, matching the action's length change, without being stressed. This is the best of all worlds. It is also similar with an aluminum Stolle Panda action and an aluminum scope.
SUMMARY.... So, in summary, I would suggest:
Steel action, aluminum scope: use a one piece aluminum scope base.
Steel action, steel scope: use a two piece steel or aluminum scope bases.
FWIW!
