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Helical spring with round cross section

Determine mechanical properties of a helical spring with wire diameter d, radius r, shear modulus G and nw as number of active windings. Calculate also deflection v and tension σ as a function of force F.

The shear modulus can be calculated using Young's Modulus and Poisson's ratio of a material.

$$G=\dfrac {E}{2(1+ν)}$$

$$k=\dfrac {\ Gd^{3}}{64n_{w}r^{3}}$$

$$v_{(F)}=\dfrac {\ 64Fn_{w}r^{3}} {Gd^{4}}$$

$$v_{max}=\dfrac {\ 4n_{w}r^{2}πσ} {Gd}$$

$$σ_{(F)}=\dfrac {\ 16Fr} {πd^{3}}$$