Basic Formulas

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

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

Rectangular beams notched at points of support on the tension side:
$F_v = \dfrac{3V}{2bd'}\left( \dfrac{d}{d'} \right)$

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

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

Short Unbraced Beam (Cs ≤ 10)
$F_b ' = F_b$

Intermediate Unbraced Column (10 < CsCk)
$F_b ' = F_b \left[ 1 - \dfrac{1}{3}\left( \dfrac{C_s}{C_k} \right)^4 \right]$

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

Long Unbraced Beam For (Ck < Cs < 50)
$F_b ' = \dfrac{0.438 E}{{C_s}^2}$

Allowable Compression at an Angle to Grain

Jacoby's Formula
$P_n = P\sin^2 \phi + Q\cos^2 \phi$

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

Allowable Compressive Stress for Columns

Short column (Le/d ≤ 11)
$F_c ' = F_c$

Intermediate column (11 < Le/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 (Le/dK)
$F_c' = \dfrac{0.30E}{(L_e / d)^2}$

Combined Flexure and Axial Stress

Flexure and Axial Tension
$\dfrac{f_t}{F_t} + \dfrac{f_b}{F_b} \le 1.0$

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

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

Wind pressure perpendicular to vertical surface
$P = 0.0000473v^2$

Wind pressure normal to the roof (Duchemins Formula)
$P_n = \dfrac{2P \sin \theta}{1 + \sin^2 \theta}$

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