# FORMULAS FOR EXTENSION SPRING FATIGUE DESIGN

Extension 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 Spring index -1 [C1] $${ C }_{ 1 }=\frac { 2{ R }_{ 1 } }{ d }$$ Spring index -2 [C2+] $${ C }_{ 2 }=\frac { 2{ R }_{ 2 } }{ d }$$ Curvature correction factor for point A [KA ] $${ K }_{ A }=\frac { 4{ C }_{ 1 }^{ 2 }-{ C }_{ 1 }-1 }{ 4{ C }_{ 1 }({ C }_{ 1 }-1) }$$ Stress correction factor for point B [KB ] $${ K }_{ B }=\frac { 4{ C }_{ 2 }-1 }{ 4{ C }_{ 2 }-4 }$$ Wahl factor [Kw] $${ K }_{ W }=\frac { 4C-1 }{ 4C-4 } +\frac { 0.615 }{ C }$$ Torsional rupture strength [Ssu] Ssu=0.67Sut Tensile stress amplitude at point A [σA_a] $${ \sigma }_{ A\_ a }={ F }_{ a }\left[ { K }_{ A }\frac { 16D }{ \pi { d }^{ 3 } } +\frac { 4 }{ \pi { d }^{ 2 } } \right]$$ Midrange tensile stress at point A [σA_m] $${ \sigma }_{ A\_ m }={ F }_{ m }\left[ { K }_{ A }\frac { 16D }{ \pi { d }^{ 3 } } +\frac { 4 }{ \pi { d }^{ 2 } } \right]$$ 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 [Sse_Gerber] $${ S }_{ se\_ gerber }=\frac { { S }_{ sa\_ zim } }{ 1-{ ({ S }_{ sm\_ zim }/{ S }_{ su }) }^{ 2 } }$$ Shear endurance limit according to Goodman [Sse_Goodman] $${ S }_{ se\_ goodman }=\frac { { S }_{ sa\_ zim } }{ 1-({ S }_{ sm\_ zim }/{ S }_{ su }) }$$ Endurance limit according to Gerber [Se_Gerber] $${ S }_{ e\_ gerber }=\frac { { S }_{ r }/2 }{ 1-{ ({ S }_{ r }/{ 2S }_{ ut }) }^{ 2 } }$$ Endurance limit according to Goodman [Se_Goodman] $${ S }_{ e\_ goodman }=\frac { { S }_{ r }/2 }{ 1-{ ({ S }_{ r }/{ 2S }_{ ut }) } }$$ Factor of safety at point-A according to Gerber [fosA_gerber] $${ fos }_{ A\_ gerber }={ (\frac { { S }_{ ut } }{ { \sigma }_{ A\_ m } } ) }^{ 2 }\frac { { \sigma }_{ A\_ a } }{ 2{ S }_{ e\_ gerber } } \left[ -1+\sqrt { { 1+\left( \frac { 2{ \sigma }_{ A\_ m }{ S }_{ e\_ gerber } }{ { \sigma }_{ A\_ a }{ S }_{ ut } } \right) }^{ 2 } } \right]$$ Factor of safety at point-A according to Goodman [fosA_goodman] $${ fos }_{ A\_ goodman }=\frac { { S }_{ e\_ goodman }{ S }_{ ut } }{ { \sigma }_{ A\_ a }{ S }_{ ut }+{ \sigma }_{ A\_ m }{ S }_{ e\_ goodman } }$$ Factor of safety at point-B according to Goodman [fosB_goodman] $${ fos }_{ B\_ gerber }={ (\frac { { S }_{ su } }{ { \tau }_{ B\_ m } } ) }^{ 2 }\frac { { \tau }_{ B\_ a } }{ 2{ S }_{ se\_ gerber } } \left[ -1+\sqrt { { 1+\left( \frac { 2{ \tau }_{ B\_ m }{ S }_{ se\_ gerber } }{ { \tau }_{ B\_ a }{ S }_{ su } } \right) }^{ 2 } } \right]$$ Factor of safety at point-B according to Goodman [fosB_goodman] $${ fos }_{ B\_ goodman }=\frac { { S }_{ se\_ goodman }{ S }_{ su } }{ { \tau }_{ B\_ a }{ S }_{ su }+{ \tau }_{ B\_ m }{ S }_{ se\_ goodman } }$$ Factor of safety at spring body according to Gerber [fosBody_gerber] $${ fos }_{ body\_ gerber }={ (\frac { { S }_{ su } }{ { \tau }_{ Body\_ m } } ) }^{ 2 }\frac { { \tau }_{ Body\_ a } }{ 2{ S }_{ se\_ gerber } } \left[ -1+\sqrt { { 1+\left( \frac { 2{ \tau }_{ Body\_ m }{ S }_{ se\_ gerber } }{ { \tau }_{ Body\_ a }{ S }_{ su } } \right) }^{ 2 } } \right]$$ Factor of safety at spring body according to Goodman [fosBody_goodman] $${ fos }_{ Body\_ goodman }=\frac { { S }_{ se\_ goodman }{ S }_{ su } }{ { \tau }_{ Body\_ a }{ S }_{ su }+{ \tau }_{ Body\_ m }{ S }_{ se\_ goodman } }$$

Note 1 : + Recommended practice is to make C2 greater than 4.

 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 Sr Allowable bending strength for the spring end for cycling loading