Shear check for timber

According to EN 1995-1-1 §6.1.7 (1)P the shear check implies:

$\tau_d \leq f_v_._d$

with

$\tau_d the design shear stress</li> <li>$ f_v_._d
$the design shear strength (= a material property)</li> </ul> From any course on mechanics, you can find that the maximal shear stress in a rectangular section equals: <img src="https://support.buildsoft.eu/wp-content/ql-cache/quicklatex.com-67920c7763b862871049acb8de018d00_l3.png" height="37" width="80" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[ \tau_d=\frac{3 V_E_d}{2 A} \]" title="Rendered by QuickLaTeX.com"/> Combining both equations gives: <img src="https://support.buildsoft.eu/wp-content/ql-cache/quicklatex.com-2ca2f387ee4a9fb4597011deffa5215f_l3.png" height="37" width="94" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[ \frac{3 V_E_d}{2 A} \leq f_v_._d \]" title="Rendered by QuickLaTeX.com"/> or <img src="https://support.buildsoft.eu/wp-content/ql-cache/quicklatex.com-a5449bf6a801bac2a254feebe26f7220_l3.png" height="37" width="108" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[ V_E_d \leq \frac{2 f_v_._d A}{3} \]" title="Rendered by QuickLaTeX.com"/> The area$

A $of a rectangular section equals$ b h

$: <img src="https://support.buildsoft.eu/wp-content/ql-cache/quicklatex.com-4a63a90d39adf85e9db882c9d032b49b_l3.png" height="36" width="114" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[ V_E_d \leq \frac{2}{3} b h f_v_._d \]" title="Rendered by QuickLaTeX.com"/> Yet EN 1995-1-1: 2004/A1:2008 §6.1.7 (2) states that the effective width$

b_e_f_f $=$ k_c_r $should be used instead of the width$ b $to take the influence of cracks into account.$ k_c_r = 0,67 $for solid and glued laminated timber,$ k_c_r= 1,0$ for all other wood based products.

$V_E_d \leq \frac{2}{3} b_e_f_f h f_v_._d = \frac{2}{3} k_c_r b h f_v_._d = \frac{2}{3} \frac{2}{3} b h f_v_._d = \frac{2}{3} A_e_f_f f_v_._d$