Shear correction factor

The shear correction factor is used in technical mechanics to take account of the change due to warping due to the shear force of the shear surface compared to the actually flat cross-sectional area of ​​the beam . ${\displaystyle \kappa }$ ${\displaystyle A_{S}}$ ${\displaystyle A}$

Derivation for thick-walled cross-sections

When deriving the shear correction factor , the deformation energy of the shear force (internal size) is equated with the deformation energy of the real shear stress . ${\displaystyle \kappa }$ ${\displaystyle \Pi _{Q}}$ ${\displaystyle Q}$ ${\displaystyle \Pi _{\tau }}$ ${\displaystyle \tau _{(z)}}$

The deformation energy of the shear force can be determined with the mean slip : ${\displaystyle \Pi _{Q}}$ ${\displaystyle Q}$ ${\displaystyle \gamma _{m}}$

${\displaystyle \Pi _{Q}={\frac {1}{2}}\cdot Q\cdot \gamma _{m}}$

For the mean glide we use the elasticity law of the shear force: ${\displaystyle \gamma _{m}}$

${\displaystyle Q=\kappa \cdot G\cdot A\cdot \gamma }$     →         →     ${\displaystyle \gamma ={\frac {Q}{\kappa \cdot G\cdot A}}}$ ${\displaystyle \Pi _{Q}={\frac {1}{2}}\cdot {\frac {Q^{2}}{\kappa \cdot G\cdot A}}}$

The deformation energy of the real shear stress is obtained by integrating the real shear stress over the cross-sectional area of ​​the beam: ${\displaystyle \Pi _{\tau }}$ ${\displaystyle \tau _{(z)}}$ ${\displaystyle \tau _{(z)}}$

${\displaystyle \Pi _{\tau }=\int _{A}{\frac {1}{2}}\cdot \tau _{(z)}\cdot \gamma \cdot dA}$

For that is Hooke's law with inserted: ${\displaystyle \gamma }$ ${\displaystyle \tau =G\cdot \gamma }$

${\displaystyle \Pi _{\tau }=\int _{A}{\frac {1}{2}}\cdot {\frac {\tau _{(z)}^{2}}{G}}\cdot dA}$

Furthermore, the equation is used for the real shear stress distribution : ${\displaystyle \tau _{(z)}}$ ${\displaystyle \tau _{(z)}={\frac {Q\cdot S_{y(z)}}{I_{y}\cdot b_{(z)}}}}$

${\displaystyle \Pi _{\tau }=\int _{A}{\frac {1}{2}}\cdot {\frac {Q^{2}\cdot S_{y(z)}^{2}}{G\cdot I_{y}^{2}\cdot b_{(z)}^{2}}}\cdot dA={\frac {1}{2}}\cdot {\frac {Q^{2}}{G\cdot I_{y}^{2}}}\cdot \int _{A}{\frac {S_{y(z)}^{2}}{b_{(z)}^{2}}}\cdot dA}$

With:

${\displaystyle A}$ Beam cross-sectional area

${\displaystyle S_{y(z)}}$ Static moment

${\displaystyle G}$ Shear modulus

${\displaystyle I_{y}}$ axial geometrical moment of inertia

${\displaystyle b_{(z)}}$ Section width at the point ${\displaystyle z}$

If both deformation energies are equated:

${\displaystyle \Pi _{Q}=\Pi _{\tau }={\frac {1}{2}}\cdot {\frac {Q^{2}}{\kappa \cdot G\cdot A}}={\frac {1}{2}}\cdot \int _{A}{\frac {Q^{2}\cdot S_{y(z)}^{2}}{G\cdot I_{y}^{2}\cdot b_{(z)}^{2}}}\cdot dA}$

can be resolved directly after the shear correction factor for thick-walled cross-sections: ${\displaystyle \kappa }$

${\displaystyle {\frac {1}{\kappa }}={\frac {A}{I_{y}^{2}}}\cdot \int _{A}{\frac {S_{y(z)}^{2}}{b_{(z)}^{2}}}\cdot dA}$

Derivation for thin-walled cross-sections

The shear correction factor for thin-walled cross-sections can also be derived in the same way. Here only the real shear stress has to be used. It follows for the shear correction factor: ${\displaystyle \tau _{(z)}={\frac {Q\cdot S_{y(\zeta )}}{I_{y}\cdot b_{(\zeta )}}}}$

${\displaystyle {\frac {1}{\kappa }}={\frac {A}{I_{y}^{2}}}\cdot \int _{\zeta }{\frac {S_{y(\zeta )}^{2}}{t_{(\zeta )}^{2}}}\cdot d\zeta }$

This contains the running coordinate along the profile center line of the thin-walled cross-section and the cross-section width at the respective running coordinate. ${\displaystyle \zeta }$ ${\displaystyle t_{(\zeta )}}$

Examples

cross-section	Shear correction factor
rectangle	${\displaystyle {\frac {5}{6}}=0{,}83{\overline {3}}}$
Full circle	${\displaystyle {\frac {3}{4}}=0{,}75}$
thin-walled circular ring	${\displaystyle 0{,}5}$
I-profile (DIN 1025-1)	${\displaystyle \approx 0{,}35...0{,}45}$
I-profile, medium-wide (DIN 1025-2)	${\displaystyle \approx 0{,}1...0{,}25}$
I-profile, wide flange (DIN 1025-3)	${\displaystyle \approx 0{,}18...0{,}45}$
T-profile (DIN 59051)	${\displaystyle \approx 0{,}45...0{,}5}$

The approximation introduced by Robert Land can also be used for thin-walled profiles : ${\displaystyle \kappa \approx {\frac {A_{Steg}}{A}}}$

annotation

In some literature, the reciprocal is used. This would z. B. be the deformation energy of the transverse force . ${\displaystyle \kappa }$ ${\displaystyle {\frac {1}{\kappa }}}$ ${\displaystyle \Pi _{Q}={\frac {1}{2}}\cdot {\frac {\kappa \cdot Q^{2}}{G\cdot A}}}$

literature

Christian Spura: Technical Mechanics 2. Elastostatics . 1st edition. Springer Vieweg, Wiesbaden 2019, ISBN 978-3-658-19978-4 .