You can use the equation for hoop tension in a hose versus pressure inside the hose (or hoop stress versus pressure in a thin walled pressure vessel). Another example would be rope tension vs. contact pressure for a rope over a pulley. You can Google these.

Look at a cross-section showing the spring around the shaft (circle in a circle). Now cut the shaft in half so it is two half-circles. The two half circles are held together by two times the spring tension (T). The contact "pressure" (actually,in this case, force per inch of diameter (D)) is 2T/D. If the shaft were a fluid, the pressure must be the same in all direction, so the radial force of the spring is also 2T/D.

You could also use calculus to show that a radial force on a cylinder resolves to the same force across the diameter.

Design Handbook Engineering Guide To Spring Design, 1987 Edition, Associated Spring, Barnes, Page 56-59 Garter Springs.