## 3 Torsional Shear Flow Calculations
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Successive Normalised Length Ratios** & & & & \\ \hline \(z/L\) & 0 & \(1/4\) & \(1/2\) & \(3/4\) & 1 \\ \(1-z/L\) & 1 & \(3/4\) & \(1/2\) & \(1.4\) & 0 \\ (A) \(\mu L(1-z/L)\) & 1.9685 & 1.4764 & 0.9843 & 0.4921 & 0 \\ \(\cosh\mu L(1-z/L)\) & 3.65 & 2.303 & 1.5248 & 1.1236 & 1 \\ \(\cosh\mu L(1-z/L)\) & 1 & 0.631 & 0.4178 & 0.3078 & 0.274 \\ \(\cosh\mu L\) & & & & & \\ (\(G/T\)) \(\mathrm{d}^{3}\theta/\mathrm{d}z^{3}\times 10^{6}\) & \(-0.6357\) & \(-0.4011\) & \(-0.2656\) & \(-0.1957\) & \(-0.1742\) \\ (B) \(\mu L(1-z/L)\) & 4.55 & 3.4125 & 2.275 & 1.1375 & 0 \\ \(\cosh\mu L(1-z/L)\) & 47.32 & 15.187 & 4.9154 & 1.7198 & 1 \\ \(\cosh\mu L(1-z/L)\) & 1 & 0.3209 & 0.1039 & 0.03634 & 0.02113 \\ \(\overline{\cosh\mu L}\) & & & & & \\ (\(G/T\)) \(\mathrm{d}^{3}\theta/\mathrm{d}z^{3}\times 10^{9}\) & \(-20.4\) & \(-6.533\) & \(-2.1155\) & 0.7399 & \(-0.4302\) \\ (C) \(\mu L(1-z/L)\) & 0.936 & 0.702 & 0.468 & 0.234 & 0 \\ \(\cosh\mu L(1-z/L)\) & 1.471 & 1.2567 & 1.1115 & 1.0275 & 1 \\ \(\overline{\cosh\mu L(1-z/L)}\) & 1 & 0.8543 & 0.7556 & 0.6985 & 0.6798 \\ \(\mathrm{cosh\mu L}\) & & & & & \\ (\(G/T\)) \(\mathrm{d}^{3}\theta/\mathrm{d}z^{3}\times 10^{6}\) & \(-0.167\) & 0.1426 & \(-0.1261\) & \(-0.1166\) & \(-0.1135\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Torsional shear flow dependence upon length within three channel sections A, B and C with torsion applied at E.  
Figure 3: Torsional shear flow dependence upon length and section. where, as calculated previously,  
\[Q_{1}=0\text{, }Q_{3}=-(11/32)a^{3}\text{, }Q_{4}=0\text{, }Q_{5}=(11/32)a^{3}\text{, }Q_{7}=0\]  
giving  
\[q_{1}=q_{4}=q_{7}=0\] \[\therefore q_{3}=-q_{5}=-(1.265\times 10^{-3})\times(-11/32)(1/2 \times 25.4)^{3}=0.891\text{ N/mm}\]  
The torsional \(q_{T}\) distribution has convex curvature within the channel reflecting at the web centre (e.g., see Figure 6c). To this, there is to be added the flexural shear flow from \(F_{x}\) and \(F_{y}\) re-applied together at E (in the negative \(x\) and \(y\) directions). Recall that this summation of torsion and shear amounts to adopting a force system equivalent to that of applying these forces to the centroid directly. That is, when, as here, \(F_{x}\) and \(F_{y}\) are applied to G in Figure 4a, they are equivalent to re-applying said forces at E with an accompanying torque, \(T=F_{y}(e_{x}+X^{\prime})\), as shown in Figure 4b. Torsion is absent only when the transverse forces are applied directly to E. Thus, either equivalent force system in (b) and (c) provides an alternative calculation for \(q_{net}\) in (a). Here, with section symmetry, bending moments arising from \(F_{y}\), which act about axis \(x\), pass through both G are E. This paper later questions in Section 9.8 whether bending moments arising from \(F_{x}\) and acting about axis y through G in Figure 4a are altered by the parallel transfer of the y-axis to E. With section asymmetry, such a 'trans-moment' would appear possible with the transfer of each non-coincident axis between G and E. Normally, the static equivalence that Figure 4a,b show does not account for any change to the bending moments that a shift in their forces might produce, i.e., Figure 4a,b do not include bending. The trans-moment is introduced here as a novel theoretical feature for which stress calculations for each type of section are to follow (see Section 9.8).