Stress Concentration Factors for Simple Tubular X Joints

Axial load in one brace only - fixed chord ends

X image
$$ \beta = \frac{d}{D} $$
$$ \alpha = \frac{2L}{D} $$
$$ \gamma = \frac{D}{2T} $$
$$ \tau = \frac{t}{T} $$
Location Equation Short chord correction
Chord saddle $\left(1-0.26\beta^3\right)\left(\left(\gamma\tau^{1.1}\left(1.11-3\left(\beta-0.52\right)^2\right)\left(\sin\theta\right)^{1.6}\right)+C_1\left(0.8\alpha-6\right)\tau\beta^2\left(1-\beta^2\right)^{0.5}\left(\sin\left(2\theta\right)\right)^2\right)$ F2
Chord crown $\gamma^{0.2}\tau\left(2.65+5\left(\beta-0.65\right)^2\right)+\tau\beta\left(C_2\alpha - 3\right)\sin \theta$ None
Brace saddle $\left(1-0.26\beta^3\right)\left(1.3 + \gamma \tau^{0.52}\alpha^{0.1}\left(0.187 - 1.25 \beta^{1.1}\left(\beta - 0.96\right)\right)\left(\sin \theta\right)^{2.7 - 0.01 \alpha}\right)$ F2
Brace crown $3 + \gamma^{1.2} \left( 0.12 \exp\left(-4 \beta \right) + 0.011 \beta^2 - 0.045\right) + \beta \tau \left( C_3 \alpha - 1.2 \right)$ None

Short chord correction factor $\left(\alpha \lt 12 \right)$

$F1 = 1 - \left( 0.83 \beta - 0.56 \beta^2 - 0.02 \right) \gamma^{0.23} \exp \left( -0.21 \gamma^{-1.16} \alpha^{2.5}\right)$

Chord-end fixity parameter

$C_1 = 2(C-0.5)$

$C_2 = C/2$

$C_3 = C/5$

$0.5 \le C \le 1.0$, Typically $C=0.7$