## 7 Net Axial Stress
### Constrained Axial Stress Distribution  
The axial stress distribution induced within the length due to constraining the torsional warping displacements fully at the fixed end has been described in [1]. There, it was shown that the fixed end, where \(z=0\), provides the maximum contribution to the axial stress for the full constraint  
\[\sigma_{z}=-[(\omega-\varpi)\ T/(\mu\Gamma_{1})]\ (1-\mathrm{e}^{-2\mu L})/(1+ \mathrm{e}^{-2\mu L})\] (13a) where each channel section's properties have been given previously (see Section 5 and Table 3 above). The specific properties required when applying Equation (13a) to channel cross-section A (with torque applied to E) appear as follows:  
Channel A:  
\(d/a=2\), \(e_{x}/a=3/8\), \(L=300\) mm, \(a=12.7\) mm, \(t=1.5875\) mm, \(\Gamma_{1}/a^{5}t=0.66275\),  
\(\varpi/a^{2}=0.8438\), \(X^{\prime}/a=1/4\), \(\mu=0.8189t/a^{2}\), \(\mu L=0.8189(t/a)(L/a)=2.418\),  
\((1-\mathrm{e}^{-2\mu L})/(1+\mathrm{e}^{-2\mu L})=0.9842\)  
Substituting into Equation (13a) leads to a parametric form for \(\sigma_{z}\):  
\[\sigma_{z}=-[(\omega-\varpi)T/(0.8189t/a^{2}\times 0.66275a^{5}t)]\times 0.9842\]  
\[(at^{2}/T)\sigma_{z}=1.8134\ (0.8438-\ \omega/a^{2}) \tag{13b}\]  
Taking the ordinates \(\omega/a^{2}\) from in the swept area plot in Figure 2a,b, Equation (13b) is applied to points 1, 3 and 4 (NA) as follows:  
at 1 \(\omega_{1}=0\), \(\therefore(at^{2}/T)\sigma_{z1}=1.8135(0.8438-0)=1.5303\)  
at 3, \(\omega_{3}=ad/2\), \(\therefore(at^{2}/T)\sigma_{z3}=1.8135(0.8438-(1/2)(d/a)]=-0.2834\)  
at 4, \(\omega_{4}=(a+e)d/2\), \(\therefore(at^{2}/T)\sigma_{z4}=1.8135(0.8438-(1/2)(d/a)(1+e/a)]=-0.9633\)  
With symmetry evident from Figure 2a, it follows that \(\sigma_{z1}=\sigma_{z7}\) and \(\sigma_{z3}=\sigma_{z5}\). Torque \(T\) upon axis E, corresponding to transverse shear forces \(F_{x}=F_{y}=100\) N at G, follows from  
\[T/a=F_{y}(e_{x}/a+X^{\prime}/a)=100(3/8+1/4)=62.5\ \mathrm{N}\]  
Hence, the stresses at these positions within the channel fixing are as follows:  
\(\sigma_{z1}=1.5303\ (T/at^{2})=1.5303\ \times\ 62.5/1.5875^{2}=37.95\ \mathrm{MPa}\)  
\(\sigma_{z3}=0.2834\ (T/at^{2})=-0.2834\ \times\ 62.5/1.5875^{2}=-7.04\ \mathrm{MPa}\)  
\(\sigma_{z4}=-0.9633\ (T/at^{2})=-0.9633\ \times\ 62.5/1.5875^{2}=-23.89\ \mathrm{MPa}\)  
The swept area diagram shows that the stresses at points 1, 3 and 4 are connected with linear segments and extended symmetrically to the section's bottom half. Figure 9 shows the full 3D constrained axial stress distribution at the fixing. _Note_: Figures 8 and 9 (revised) contribute to the net axial stress within different distributions. Figure 9: Constrained axial stress distribution at the channel fixing with torque applied to an axis through the shear centre. A similar stress distribution to Figure 9 will apply to other positions within the length \(0<z<L\) but with reduced magnitude [1]. Maximum tension appears at the flange ends and maximum compression at the web centre. The stress levels are all relatively low compared to those from bending (see Figure 8) but are equally important to the safe design of this structure (see Section 8). Further calculations follow below for the constrained axial stress within the fixed ends of channels B and C:  
Channel B:  
\(d/a=7/4\), \(e_{x}/a=12/31\), \(L=1000\) mm, \(a=25.4\) mm, \(t=3.175\) mm,  
\(\Gamma_{1}/a^{5}t=0.448\), \(\varpi/a^{2}=0.7206\), \(X^{\prime}/a=4/15\), \(\mu=0.924t/a^{2}\),  
\(\mu L=0.924(t/a)(L/a)=4.5474\), \((1-\mathrm{e}^{-2\mu L})/(1+\mathrm{e}^{-2\mu L})=0.99978\).