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lets say you have two tubes of steel, exactly the same material, one is 40mm in diameter and the other 50mm. wich tube is going to be stiffer? less likely to deflect under load?

Thank you,

assuming the wall thickness is the same the 50 mm will be stronger/stiffer

I guess you are talking about Thin (not thick) wall tubes
(classification depends on behavior against "pandeo" -sory dont know english word for it )

Generally:
-THICK: high relationship D/t, transverse section ovalizes, local "pandeo" forms while resistent momentum decreases, here resistence can exceed theoric plastic momentum due too deformation hardening
-THIN: low relationship D/t, many "pandeo" waves forms quickly with almost no ovalization, resistence falls quickly

To calculate flexion in thin wall tubes quickly (simplifications made):

d^2 w(x) / dx^2 = M (x) / EI

So, you apply a momentum M, E depends on material (for steel use 2.100.000 kg/cm2 or 200.000 MPa) and I = (π/4)*(Re^4 - Ri^4).

You do the math, have fun