Von Mises Truss, deltacrit derivation

The following reformulates Bazant’s critical rotation formula for a Von Mises Truss in terms of a critical deflection.

Figure 1
Figure 1

\[\begin{align} \cos q_{0} & = (\cos \alpha)^{\frac{1}{3}} \\ \frac{1}{\sqrt{\tan^{2} q_{0} + 1}} & = \Bigg( \frac{1}{\sqrt{\tan^{2} \alpha + 1}} \Bigg)^\frac{1}{3} \\ (\tan^{2} q_{0} + 1)^{-\frac{1}{2}} & = (\tan^{2} \alpha + 1)^{-\frac{1}{2}\frac{1}{3}} \\ \tan^{2} q_{0} + 1 & = (\tan^{2} \alpha + 1)^{\frac{1}{3}} \\ \Bigg(\frac{H - \delta_{crit}}{L} \Bigg)^{2} + 1 & = \Bigg[ \bigg(\frac{H}{L}\bigg)^{2} + 1 \Bigg]^{\frac{1}{3}} \\ (H - \delta_{crit})^{2} & = L^{2} \Bigg[ \bigg( \frac{H^{2}}{L^{2}} + 1 \bigg)^{\frac{1}{3}} - 1 \Bigg] \\ H - \delta_{crit} & = \sqrt{L^{2} \Bigg[ \bigg( \frac{H^{2}}{L^{2}} + 1 \bigg)^{\frac{1}{3}} - 1 \Bigg]} \\ \delta_{crit} & = H - \sqrt{L^{2} \Bigg[ \bigg( \frac{H^{2}}{L^{2}} + 1 \bigg)^{\frac{1}{3}} - 1 \Bigg]} \\ \delta_{crit} & = H - L \sqrt{\Bigg[ \bigg( \frac{H^{2}}{L^{2}} + 1 \bigg)^{\frac{1}{3}} - 1 \Bigg]} \end{align}\]