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
$f_v_._d$ the design shear strength (= a material property)

From any course on mechanics, you can find that the maximal shear stress in a rectangular section equals:

$\tau_d=\frac{3 V_E_d}{2 A}$

Combining both equations gives:

$\frac{3 V_E_d}{2 A} \leq f_v_._d$

$V_E_d \leq \frac{2 f_v_._d A}{3}$

The area $A$ of a rectangular section equals $b h$ :

$V_E_d \leq \frac{2}{3} b h f_v_._d$

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.