In order to change load by changing wire size, the basic compression and extension spring formulas for spring manufacturers and spring designers is:
Rate= Modulus times wire diameter to the fourth power divided by eight 8 times the number of active coils times the mean diameter to the third power.
To double the spring rate the spring designer first changes the spring wire size, doubling the D4 number for a given spring wire size.

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Katy Spring & Mfg. Inc.