# Spring Constant

Written by Jerry Ratzlaff on . Posted in Constants

Spring force constant, abbreviated as $$k_s$$, also called spring constant, is the ratio of opposing force to the displacement from the origional position or how much force is needed to change a springs distance.

## Spring Constant formulas

$$\large{ k_s = - F \; d_s }$$

$$\large{ k_s = \frac {F} {d_s} }$$

$$\large{ k_s = \frac {2 \; E}{ d^2 } }$$

$$\large{ k_s = \frac { F }{ x \;-\; x_0 } }$$

$$\large{ k_s = \frac { 2 \; PE_s }{ x^2 } }$$

$$\large{ k_s = \frac {G \; d^4} {8 \; n_a \; D^3} }$$

$$\large{ k_s = \frac { 2 \; C \;+\; 1 }{ 2 \; C } }$$     $$\large{ C = \frac{ D }{ d_w } }$$

Symbol English Metric
$$\large{ k_s }$$ = spring force constant $$\large{lbf}$$ $$\large{N}$$
$$\large{ x }$$ = distance from equilibrium $$\large{in}$$ $$\large{mm}$$
$$\large{ D }$$ = mean coil diameter $$\large{in}$$ $$\large{mm}$$
$$\large{ n_a }$$ = number of active coils $$\large{dimensionless}$$
$$\large{ G }$$ = shear modulus of material $$\large{\frac{lbf}{in^2}}$$ $$\large{Pa}$$
$$\large{ d_s }$$ = spring displacement $$\large{in}$$ $$\large{mm}$$
$$\large{ E }$$ = spring energy $$\large{lbf-ft}$$ $$\large{J}$$
$$\large{ F }$$ = spring force $$\large{lbf}$$  $$\large{N}$$
$$\large{ x_0 }$$ = spring equilibrium position $$\large{in}$$ $$\large{mm}$$
$$\large{ PE_s }$$ = spring potential energy $$\large{lbf-ft}$$  $$\large{J}$$
$$\large{ d_w }$$ = wire diameter $$\large{in}$$ $$\large{mm}$$
$$\large{ d }$$ = wire size $$\large{in}$$ $$\large{mm}$$

$$\large{ D/N }$$ = index correction

•  $$\large{ G }$$ value for common spring materials
• Copper = 6.5 x 10^6
• Beryllium Copper = 6.9 x 10^6
• Inconel = 11.5 x 10^6
• Monel = 9.6 x 10^6
• Music Wire = 11.5 x 10^6
• Phospher Bronze = 5.9 x 10^6
• Stainless Steel = 11.2 x 10^6
$$\large{dimensionless}$$