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

Answer

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!

Layerz_minz_maxE_x(z)(N/mm²)G_xz(z)(N/mm²)
1-50-3011,000690
2-30-1030050
3-101011,000690
4103030050
5305011,000690

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

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

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

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

$=731.2\times10^6 Nmm$

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

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


χ113.75 106
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).

Keywords

Shear correction Factor Shear

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Deflection analysis and stress design of laminate and sandwich surfaces

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