**Basic Formulas**

$f_b = \dfrac{Mc}{I}$

Horizontal Shear Stress

$f_v = \dfrac{VQ}{Ib}$

Formula for Spacing of Bolts and Nails

$s = \dfrac{RI}{VQ} = \dfrac{R}{q}$

**Notching of Beams Formulas**

$F_v = \dfrac{3V}{2bd'}\left( \dfrac{d}{d'} \right)$

Beams notched at points of support on the tension side

$F_v = \dfrac{3V}{2A_n}\left( \dfrac{d}{d_n'} \right)$

Beams notched at point of support on the compression side

$V = \dfrac{2}{3}F_v b \, \left[ d - \dfrac{e(d - d')}{d'} \right]$

**Allowable Bending Stress**

$C_s = \sqrt{\dfrac{L_e d}{b^2}}$

Short Unbraced Beam (*C _{s}* ≤ 10)

$F_b ' = F_b$

Intermediate Unbraced Column (10 < *C _{s}* ≤

*C*)

_{k}$F_b ' = F_b \left[ 1 - \dfrac{1}{3}\left( \dfrac{C_s}{C_k} \right)^4 \right]$

$C_k = 0.811 \sqrt{\dfrac{E}{F_b}}$

Long Unbraced Beam For (*C _{k}* <

*C*< 50)

_{s}$F_b ' = \dfrac{0.438 E}{{C_s}^2}$

**Allowable Compression at an Angle to Grain**

$P_n = P\sin^2 \phi + Q\cos^2 \phi$

Hankinson's Formula

$P_n = \dfrac{PQ}{P\sin^2 \theta+ Q\cos^2 \theta}$

**Allowable Compressive Stress for Columns**

*L*/

_{e}*d*≤ 11)

$F_c ' = F_c$

Intermediate column (11 < *L _{e}*/

*d*<

*K*)

$F_c ' = F_c \left[ 1 - \dfrac{1}{3}\left( \dfrac{L_e / d}{K} \right)^4 \right]$

$K = 0.671\sqrt{\dfrac{E}{F_c}}$

Long column (*L _{e}*/

*d*≥

*K*)

$F_c' = \dfrac{0.30E}{(L_e / d)^2}$

**Combined Flexure and Axial Stress**

$\dfrac{f_t}{F_t} + \dfrac{f_b}{F_b} \le 1.0$

$\dfrac{f_b - f_t}{F_b'} \le 1.0$

Flexure and Axial Compression

$\dfrac{f_c}{F_c'} + \dfrac{f_b}{F_b' - Jf_c} \le 1.0$

$J = \dfrac{L_e/d - 11}{K - 11}$

$K = 0.671\sqrt{\dfrac{E}{F_c}}$

**Wind Load for the Design of Purlins**

$P = 0.0000473v^2$

Wind pressure normal to the roof (Duchemins Formula)

$P_n = \dfrac{2P \sin \theta}{1 + \sin^2 \theta}$