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

The shear correction factor is considered in the RF‑LAMINATE add-on module by 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 shear stiffness can be found in the English version of the RF-LAMINATE manual, page 15 ff.

For a plate with the thickness of 10 cm in Figure 01, the calculation of the shear correction factor is shown. The equations used here are only valid for simplified symmetrical plate structures!

 Layer z_min z_max E_x(z)(N/mm²) G_xz(z)(N/mm²) 1 -50 -30 11,000 690 2 -30 -10 300 50 3 -10 10 11,000 690 4 10 30 300 50 5 30 50 11,000 690

$\sum_iG_{xz,i}A_i=3\times0.02\times690+2\times0.02\times50=43.4N$

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

$=11,000\left(\frac{-30^3}3+\frac{50^3}3\right)+300\left(\frac{-10^3}3+\frac{30^3}3\right)$

$+11,000\left(\frac{10^3}3+\frac{10^3}3\right)+300\left(\frac{30^3}3-\frac{10^3}3\right)+11,000\left(\frac{50^3}3-\frac{30^3}3\right)$

$=731.2\times10^6 Nmm$

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

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

 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

$k_z=\frac{43.4}{{(731.2e^6)}^2}6.713284\;e^{13}=5.449\;e^{-3}$

$D_{44}=\frac{{\displaystyle\sum_i}G_{xz,i}A_i}{k_z}=\frac{43.4}{5.449\;e^{-3}}=7,964.7 N/mm$

This corresponds to the resulting value in RF‑LAMINATE (Figure 02).