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FAQ 004281 EN

12 December 2019

# Does the program RF-LAMINATE consider the shear correction factor for cross-laminated timber slabs?

The shear correction factor is taken into account in the RF-LAMINATE program using the following equation.

$k_{z}=\frac{{\displaystyle\sum_i}G_{xz,i}A_i}{\left(\int_{-h/2}^{h/2}E_x(z)z^2\operatorname dz\right)^2}\int_{-h/2}^{h/2}\frac{\left(\int_z^{h/2}E_x(z)zd\overline z\right)^2}{G_{xz}(z)}\operatorname dz$

with $\ int _ {- h/2} ^ {h/2} E_x (z) z ^ 2 \ operatorname dz = EI _ {, net}$

The calculation of the shear stiffness itself can be found on page 15 of the English version to the manual of RF-LAMINATE as follows:

For the 10 cm thick plate in Figure 1, the calculation of the shear correction factor is shown. The equations used here are only valid for the simplified symmetrical plate structures!

 Layer z_min z_max E_x (z) (N/mm²) G_xz (z) (N/mm²) 1 -50 -30 11000 690 2 -30 -10 300 50 3 -10 10 11000 690 4 10 30 300 50 5 30 50 11000 690

$\sum_iG_{xz,i}A_i=3\times0,02\times690+2\times0,02\times50=43,4N$

$EI_{,net}=\sum_{i=1}^nE_{i;x}\frac{\mbox{$z$}_{i,max}^3-\mbox{$z$}_{i,min}^3}3$

$=11000\left(\frac{-30^3}3+\frac{50^3}3\right)+300\left(\frac{-10^3}3+\frac{30^3}3\right)$

$+11000\left(\frac{10^3}3+\frac{10^3}3\right)+300\left(\frac{30^3}3-\frac{10^3}3\right)+11000\left(\frac{50^3}3-\frac{30^3}3\right)$

$=731,2\times10^6Nmm$

$\int_{-h/2}^{h/2}\frac{\left(\int_z^{h/2}E_x(z)zd\overline z\right)^2}{G_{xz}(z)}\operatorname dz=\sum_{i=1}^n\frac1{G_{i;xz}}\left(χ_i^2(z_{i;max}-z_{i,min})\;χ_iE_{i,x}\frac{z_{i,max}^3-z_{i,min}^3}3+E_{i,x}^2\frac{z_{i,max}^5-z_{i,min}^5}{20}\right)$

$χ_i=E_{i;x}\frac{z_{i;max}^2}2+\sum_{k=i+1}^nE_{k;x}\frac{z_{k,max}^2-z_{k,min}^2}2$

 χ1 13.75 106 χ2 8.935 106 χ3 9.47 106 χ4 8.935 106 χ5 13.75 106

$\sum_{i=1}^n\frac1{G_{i;yz}}\left(χ_i^2(z_{i,max}-z_{i,min})-χ_iE_{i,y}\frac{z_{i,max}^3-z_{i,min}^3}3+{E^2}_{i,y}\frac{z_{i,max}^5-z_{i,min}^5}{20}\right)=$

 8.4642 1011 3.147 1013 2.5 1012 3.147 1013 8.4642 1011

Total 6.7133 x 1013

$k_z=\frac{43,4}{{(731,2e^6)}^2}6,713284\;e^{13}=5,449\;e^{-3}$

$D_{44}=\frac{{\displaystyle\sum_i}G_{xz,i}A_i}{k_z}=\frac{43,4}{5,449\;e^{-3}}=7964,7N/mm$

This corresponds to the value output in RF-LAMINATE (Figure 2).