# FORMULAS FOR COMPRESSION SPRING DESIGN

Compression spring formulas including spring index formula, Wahl factor formula, spring rate formula, shear stress at spring body formula, spring outer diameter at solid height formula, spring stability condition and Hooke's law.

 Parameter Formula Spring outer diameter [OD] OD = D + d Spring inner diameter [ID] ID = D - d Spring index [C] C = D/d Wahl factor [Kw] $${ K }_{ W }=\frac { 4C-1 }{ 4C-4 } +\frac { 0.615 }{ C }$$ Shear stress at spring body (corrected with Wahl factor)- used for unprestressed springs [τs] $$\tau ={ K }_{ W }\frac { 8FD }{ \pi { d }^{ 3 } }$$ Shear stress at spring body  (uncorrected )- used for prestressed springs [τs] $$\tau =\frac { 2C+1 }{ 2C } \times \frac { 8FD }{ \pi { d }^{ 3 } }$$ Spring rate [k] $$k=\frac { { d }^{ 4 }G }{ 8{ D }^{ 3 }{ N }_{ a } }$$ OD at solid height [ODat solid] $${ OD }_{ at\quad solid }=\sqrt { { D }^{ 2 }+\frac { { p }^{ 2 }-{ d }^{ 2 } }{ { \pi }^{ 2 } } } +d$$ Spring stability condition $${ L }_{ f }<\frac { \pi D }{ \alpha } { \left[ \frac { 2(E-G) }{ 2G+E } \right] }^{ 1/2 }$$ Hooke's Law $$\Delta F=k\Delta x$$

 Type of Spring Ends Parameter Open or plain (Not ground) Open or plain (Ground) Squared or closed (Not ground) Squared or closed (Ground) Total coils [Nt] Na Na+1 Na+2 Na+2 Free height [Lf] pNa+d p(Na+1) pNa+3d pNa+2d Solid height [Ls] d(Nt+1) dNt d(Nt+1) dNt Pitch [p] (Lf - d) / Na Lf / (Na+1) (Lf -3d) / Na (Lf -2d) / Na Guidelines for Dimensional Characteristics of Compression Springs Source : From Design Handbook [Ref 1] page 32

 List of Parameters Symbol Definition OD Spring outer diameter ID Spring inner diameter D Spring mean diameter d Wire diameter p Pitch Lf Spring free length Ls Spring solid height F Axial force Fs Force at solid length ΔF Force difference (Ex: F2-F1) Δx Deflection (Ex: L2-L1)