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] |
|
$${ fos }_{ Body\_ goodman }=\frac { { S }_{ se\_ goodman }{ S }_{ su } }{ {
\tau }_{ Body\_ a }{ S }_{ su }+{ \tau }_{ Body\_ m }{ S }_{ se\_ goodman } } $$ |