## 8 Yield Criteria  
The yield stresses for the aluminium alloy are \(Y=250\) MPa in tension and \(k=150\) MPa in shear. The former is employed here (\(Y=250\) MPa), given that the net axial stress \(\sum\sigma_{z}\) appears more dominant than the net shear stress \(\sum\tau\) in each channel cross section of Table 9A-C above. The Tresca and von Mises yield criteria appear with their respective safety factors \(S_{T}\) and \(S_{V}\) as follows:  
Figure 10: Net axial stress distributions at the fixing from bending and warping constraint. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Channel A** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline
1 st term, MPa & 37.95 & 4.12 & \(-\)7.04 & \(-\)23.89 & \(-\)7.04 & 4.12 & 37.95 \\ \hline
2 nd term, MPa & 253.46 & 43.9 & \(-\)25.93 & \(-\)69.85 & \(-\)113.75 & \(-\)43.9 & 165.65 \\ \hline \(\sum\sigma_{z}\), MPa & 291.4 & 48.02 & \(-\)32.97 & \(-\)93.74 & \(-\)120.79 & \(-\)39.78 & 203.60 \\ \hline \(\sum\tau\), MPa & 0 & 5.805 & 5.876 & 2.784 & \(-\)2.163 & \(-\)3.021 & 0 \\ \hline \(S_{T}\) & 0.859 & 5.06 & 7.144 & 2.662 & 2.068 & 6.213 & 1.23 \\ \hline
**S\({}_{V}\)** & 0.859 & 5.096 & 7.24 & 2.663 & 2.069 & 6.231 & 1.23 \\ \hline
**Channel B** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline
1 st term, MPa & 10.36 & 1.14 & \(-\)2.22 & \(-\)7.09 & \(-\)2.22 & 1.14 & 10.36 \\ \hline
2 nd term, MPa & 110.4 & 21.57 & \(-\)10.79 & \(-\)32.36 & \(-\)53.93 & \(-\)21.57 & 67.43 \\ \hline \(\sum\sigma_{z}\), MPa & 120.76 & 22.71 & \(-\)13.0 & \(-\)39.45 & \(-\)56.15 & \(-\)20.44 & 77.8 \\ \hline \(\sum\tau\), MPa & 0 & 1.758 & 1.736 & 0.7877 & \(-\)0.64 & \(-\)0.8758 & 0 \\ \hline \(S_{T}\) & 2.07 & 10.88 & 18.58 & 6.332 & 4.451 & 12.165 & 3.213 \\ \hline \(S_{V}\) & 2.07 & 10.91 & 18.736 & 6.329 & 4.452 & 12.197 & 3.213 \\ \hline
**Channel C** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline
1 st term, MPa & 39.04 & 4.34 & \(-\)4.34 & \(-\)18.8 & \(-\)4.34 & 4.34 & 39.04 \\ \hline
2 nd term, MPa & 218.8 & 25.18 & \(-\)23.24 & \(-\)48.42 & \(-\)73.59 & \(-\)25.18 & 168.5 \\ \hline \(\sum\sigma_{z}\), MPa & 257.8 & 29.52 & \(-\)27.58 & \(-\)67.22 & \(-\)77.93 & \(-\)20.84 & 207.54 \\ \hline \(\sum\tau\), MPa & 0 & 5.587 & 5.60 & 2.058 & \(-\)3.247 & \(-\)3.767 & 0 \\ \hline \(S_{T}\) & 0.97 & 7.92 & 8.40 & 3.712 & 3.196 & 11.30 & 1.205 \\ \hline \(S_{V}\) & 0.97 & 8.05 & 8.55 & 3.714 & 3.20 & 11.46 & 1.205 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Net axial stress summations in sections (**A**), (**B**) and (**C**) from end constraint and bending. \[\text{Tresca: }(\sum\sigma_{z})^{2}+4(\sum\tau)^{2}=(\gamma/S_{\gamma})^{2}\] \[S_{T}=Y/\sqrt{[(\sum\sigma_{z})^{2}+4(\sum\tau)^{2}]} \tag{15a}\] \[\text{von Mises: }(\sum\sigma_{z})^{2}+3(\sum\tau)^{2}=(\gamma/S_{ \
u})^{2}\] \[S_{V}=Y/\sqrt{[(\sum\sigma_{z})^{2}+3(\sum\tau)^{2}]} \tag{15b}\]  
The yield criteria in Equation (15a,b) when applied show that Tresca is more conservative. The safety factors attached to Table 9A-C indicate yielding when \(S_{T}\) and \(S_{V}\) are less than unity by either criterion, i.e., at point 1 for channels A and C. Despite this, it appears that all three structures are safe for remaining points 2, 3 \(\ldots\) 7. at the fixed end. Given that large elastic safety factors apply to each of these points, the yielding that is seen to occur at point 1 is likely to be localised, thus forming a partially plastic hinge that does not result in tensile failure. Compressive and/or shear buckling failure may be local or global. In the absence of an applied axial compression, Euler global buckling is unlikely. Local buckling of the section's thin rectangular web plate is possible where edge shear and axial compression exist. For each mode to appear in the web, the critical elastic buckling stress appears in terms of the plate geometry aspect ratio, \(d/L\), the respective elastic constants \(E\) and \(G\) and the buckling coefficient, \(K_{e}\)[15; 16; 17; 18; 19].