The variations in composition will come from carbon and alloy
segregation inherited from the solidification structure,and modified
by the soaking and hot working practice used to make the shaft.
Variations in hardness will come from how heat treatment and resultant
microstructure are affected by the composition variations. You are at
a large enough diameter to make this difficult to predict, since you
are beyond Jominy bar cooling rates. Actually, you are asking for a
whole course in physical metallurgy of steel!
Some insight into composition variations can be gotten by looking at
standards such as SAE Standard J409 "Product Analysis - Permissible
Variations from Specified Chemical Analysis of a Heat or Cast of
Steel". The variations depend both on the element and its specified
range though, even at a given cross sectional area. You can find J409
in a copy of the SAE Handbook. Look at your friendly local engineering
Would ASTM A914 be of use?
This gives the expected range of hardness (presumably mostly variation
between casts), at the surface,the centre and an intermediate position for
some alloys, I can not remember if it goes to large enough diameters.
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