**Appendix E**

The integral for the torsion moment *M*<sup>T</sup> for the profile geometry reads

$$\mathcal{M}\_{\mathrm{T}} = \int\_{-\left(\frac{b}{2} - z\_0 - t\right)}^{\frac{b}{2} + z\_0} \int\_{\frac{b}{2} - y\_0 - t}^{\frac{b}{2} - y\_0} \pi\_{\mathrm{yz}, \mathrm{RT}} t \, dydz + \int\_{-\left(\frac{b}{2} - z\_0\right)}^{\left(\frac{b}{2} - z\_0 - t\right)} \int\_{-\left(\frac{b}{2} + y\_0\right)}^{\frac{b}{2} - y\_0} \pi\_{\mathrm{yz}, \mathrm{ht}} t \, dydz \tag{A5}$$

where *z*<sup>0</sup> and *y*<sup>0</sup> are the correspondent coordinate distances to the profile center to the center of gravity, *τ*yz,RT is the shear stress in the room temperature area, and *τ*yz,h is the shear stress at the heated area. Evaluation of Equation (9) yields

$$\tau\_{yz, \text{RT}} = \frac{Q \cdot \left( b - \frac{b^2 t + (b-t)t^2}{2(bt + (b-t)t)} - z \right) \left( \frac{b^2 t + (b-t)t^2}{2(bt + (b-t)t)} - \frac{t}{2} \right)}{\frac{tb^3}{3} + \frac{(b-t)t^3}{3} - \frac{(b^2 t + (b-t)t^2)^2}{4(bt + (b-t)t)}} \tag{A6}$$

$$\tau\_{\rm yz,h} = -\frac{\mathcal{Q} \cdot \left( b - \frac{b^2 t + (b-t)t^2}{2(bt + (b-t)t)} + \mathcal{y} \right) \left( \mathcal{y} - \frac{t}{2} \right)}{\frac{tb^3}{3} + \frac{(b-t)t^3}{3} - \frac{\left( b^2 t + (b-t)t^2 \right)^2}{4(bt + (b-t)t)}}. \tag{A7}$$

for the corresponding profile geometry. Solving the torsion moment (Equation (A5)) using both shear stress from force equilibrium *M*T,<sup>e</sup> (Equation (A6)) and shear stress from strains *M*T,<sup>s</sup> (Equation (10)), the warping angle α can be calculated numerically using the relation

$$M\_{\rm T,e} = M\_{\rm T,s}.\tag{A8}$$
