# FORMULAS FOR COMPRESSION SPRING FATIGUE DESIGN

Compression spring formulas for fatigue design.

 Parameter Formula Force amplitude [Fa] Fa = (Fmax - Fmin)/2 Midrange force [Fm] Fm = (Fmax + Fmin)/2 Spring outer diameter [OD] OD = D + d Spring inner diameter [ID] ID = D - d Spring index [C] C = D/d Whal factor [Kw] $${ K }_{ W }=\frac { 4C-1 }{ 4C-4 } +\frac { 0.615 }{ C }$$ Shear stress amplitude [τa] $${ \tau }_{ a }={ K }_{ W }\frac { 8{ F }_{ a }D }{ \pi { d }^{ 3 } }$$ Midrange shear stress [τm] $${ \tau }_{ m }={ K }_{ W }\frac { 8{ F }_{ m }D }{ \pi { d }^{ 3 } }$$ Torsional rupture strength [Ssu] Ssu=0.67Sut Slope line [r] $$r=\frac { { \tau }_{ a } }{ { \tau }_{ m } }$$ Amplitude strength component for infinite life (Shot peened)-(Zimmerli Data) [Ssa_zim] Ssa = 398 MPa (57.5 kpsi) Midrange strength component for infinite life (Shot peened)-(Zimmerli Data) [Ssm_zim] Ssa = 534 MPa (77.5 kpsi) Amplitude strength component for infinite life (Unpeened)-(Zimmerli Data) [Ssa_zim] Ssa = 241 MPa (35 kpsi) Midrange strength component for infinite life (Unpeened)-(Zimmerli Data) [Ssm_zim] Ssa = 379 MPa (55 kpsi) Shear endurance limit according to Gerber (with Zimmerli Data) [Sse] $${ S }_{ se }=\frac { { S }_{ sa\_ zim } }{ 1-{ ({ S }_{ sm\_ zim }/{ S }_{ su }) }^{ 2 } }$$ Shear stress amplitude limit (According to Gerber failure criteria) [Ssa] $${ S }_{ sa\_ lim }=\frac { { r }^{ 2 }{ S }_{ su }^{ 2 } }{ 2{ S }_{ se } } [-1+\sqrt { 1+{ (\frac { 2{ S }_{ se } }{ r{ S }_{ su } } ) }^{ 2 } } ]$$ Shear endurance limit according to Goodman (with Zimmerli Data) [Sse] $${ S }_{ se }=\frac { { S }_{ sa\_ zim } }{ 1-({ S }_{ sm\_ zim }/{ S }_{ su }) }$$ Shear stress amplitude limit (According to Goodman failure criteria) [Ssa] $${ S }_{ sa\_ lim }=\frac { r{ S }_{ se }{ S }_{ su } }{ r{ S }_{ su }{ S }_{ se } }$$ Shear stress amplitude limit (According to Sines failure criteria) [Ssa] Ssa_lim = Ssa_zim Factor of safety against fatigue [fosf] $${ fos }_{ f }=\frac { { S }_{ sa\_ lim } }{ { \tau }_{ a } }$$

 List of Parameters Symbol Definition Fmax Maximum cyclic force Fmin Minimum cyclic force (preload) D Spring mean diameter d Wire diameter Sut Ultimate tensile strength of material