*3.4. Calculation of Profile Warping*

Using the mean slope *E*<sup>p</sup> (T) (Appendix D) of the flow curve defined in Equation (1) and using temperature dependent Young's modulus definition (Equation (2)), a relation

between shear stress and shear strain rate in the plastic zone can be described through constant temperature as [19]

$$\pi\_{\rm Yz} = \frac{1}{3} \frac{E(T) \cdot E\_{\rm P}(T)}{E(T) + E\_{\rm P}(T)} \tan(\alpha). \tag{8}$$

To describe the correlation between shear strains and forming force, the equation

$$
\pi\_{\rm yz} = \frac{\mathbf{Q} \cdot \mathbf{S}}{I\_{\mathbf{z}} \cdot \mathbf{t}} \tag{9}
$$

is used with the cross-sectional force *Q*, the first moment of area *S*, the second moment of area *I*<sup>z</sup> and the profile thickness *t*. Using the relation between torsion moment and shear stress

$$M\_{\rm T} = \int \tau\_{\rm Yz} \cdot t \, dA\_{\prime} \tag{10}$$

(see Appendix E for the formulated integral) and using Equations (8) and (9) the warping angle α can be calculated numerically.

#### *3.5. Calculation of the Bending Moment*

The equation for the bending moment of the process can be expressed as

$$M\_b = \int \sigma\_\mathbf{B} \cdot y \, dA\_\prime \tag{11}$$

where the bending stress in the plastic area *σ*B,pl can be resolved through the Mises equation and the flow rule

$$
\sigma\_{\rm B} = \frac{2}{\sqrt{3}} \sqrt{k\_{\rm f} \left( \overline{\varepsilon}\_{\rm pl} \dot{\overline{\varepsilon}}\_{\rm pl} \, T \right)^2 - 4 \left| \tau\_{\rm yz} \right|^2} \tag{12}
$$

Considering the profile cross-section (Figure 7a), stresses and strains can develop partially plastic over the course of the *y* coordinate (Figure 7b). Additionally, in the partial heated case, assumption 6 must be generalized. Wolter [20] firstly determined that it is possible for the stress- and strain-free fiber to deviate in position and that the position of stress and strain symmetry changes during the bending process. For example, an added normal compressive stress shifts the position of the stress-free fiber in the direction of the compressive area [21]. A similar effect is expected for a partially heated profile. As temperature increases in the heated area, the flow stress is reduced. In the room temperature area, flow stress remains constant. As the force equilibrium needs to be fulfilled, the symmetry axis between the tensile and the compressive zone (the stress-free fiber) needs to shift in direction of the compressive zone. Due to the coupling of stress and strain, the strain free fiber will move in direction of the compressive area too. To solve the integral correlation in Equation (11), it is necessary to calculate the position of the stress-free fiber (*y*m) and the position at which the material starts plastic forming (*y*pl,l in the lower and *y*pl,u in the upper area).

The fibers for plasticity onset *y*pl,l and *y*pl,u at *x* ≥ *l*<sup>p</sup> can be calculated using assumption 9 and Equation (3) through the relation

$$\pm \left( \frac{2\left(1 - \nu^2\right)}{\sqrt{3}E(T)} k\_\mathrm{f} \left( 0, \dot{\overline{z}}\_{\mathrm{pl}}, T \right) - a\_\mathrm{T} \Delta \mathrm{T} \right) r\_\mathrm{mC} = y\_\mathrm{Pl} \tag{13}$$

with the Poisson's ratio *υ*. The shift of the stress-free fiber *y*<sup>m</sup> from the center of gravity results from a shift due to normal stresses *y*m,N and a shift due to temperature *y*m,T yielding in

$$y\_m = y\_{\rm m,N} + y\_{\rm m,T} \,. \tag{14}$$

The shift due to normal stresses of the stress-free fiber occurs by the superposition of normal stresses [5] as they are added in the Mises stress. The position of the stress-free fiber is temperature dependent because of the reduced flow stress in the heated area. As the flow stress in the room temperature area remains constant, the stress-free fiber has to shift to the compressive zone so that force equilibrium between the tensile and the compressive zone of the cross section is fulfilled. As the normal stresses are negligible (assumption 5), the stress-free fiber position is *y*<sup>m</sup> = *y*m,T. <sup>୫</sup> ୮୪,୪ ୮୪,୳

,୮୪

ଶ

୷ =

் = න୷ ∙ ,

= න ∙ ,

ට(୮୪̅ , <sup>୮୪</sup> ሶ̅ , )<sup>ଶ</sup> − 4 ୷

α

=

2 √3 ∙ ∙

**Figure 7.** (**a**) profile cross-section of the partially heated profile, (**b**) longitudinal stress and strain progression for ideal plastic and linear elastic material behavior and partially plastic behavior over the cross section.

The position of the stress-free fiber due to temperature *y*m,T needs to be evaluated to calculate the bending moment. To evaluate *y*m,T, some considerations are necessary. Mechanical equilibrium between tensile and compressive zones for the room temperature case is fulfilled if the stress-free fiber is in the origin of the profile cross section (Figure 8a). Through the lower flow stress, the stress in the heated section is reduced (Figure 8b). As the force equilibrium between both areas still needs to be fulfilled, a shift in stress-free fiber position *y*m,T is caused. To calculate the position of the stress-free fiber, the forces in the compressive area (*F*RT and *F*h) are related to their corresponding distance at which they act upon. In detail, *<sup>b</sup>* <sup>2</sup> <sup>−</sup> *<sup>y</sup>*<sup>0</sup> is related to the room temperature force *<sup>F</sup>*RT and *<sup>b</sup>* <sup>2</sup> − *y*<sup>0</sup> − *y*m,T is related to the force in the partially heated case *F*h. *y*<sup>0</sup> is the distance between the middle of the profile cross section and the center of gravity. Through geometrical considerations, it is then assumed that the share of *y*m,T on the distance of the compressive zone in the room temperature case ( *<sup>b</sup>* <sup>2</sup> − *y*0) is the same as the share of the force *F*RT − *F*<sup>h</sup> on the force at room temperature (*F*RT) (Figure 8c). This means that the force change between the room temperature case and the partial heated case is related to the stress-free fiber shift. ଶ − ଶ − − ୫, ଶ −

**Figure 8.** Relation between cross-sectional forces and stress-free fiber. (**a**) Room-temperature case, (**b**) partially heated case, (**c**) Relation between force and stress-free fiber.

=

୫, 2 − .

ୖ − <sup>୦</sup> ୖ

The resulting equation for the stress-free fiber can then be expressed as

$$\frac{F\_{\rm RT} - F\_{\rm h}}{F\_{\rm RT}} = \frac{y\_{\rm m,T}}{\frac{b}{2} - y\_0}.\tag{15}$$

Using the integral relation

$$F = \int \sigma\_{\mathsf{B}} dA \tag{16}$$

the forces *F*RT and *F*<sup>h</sup> can be calculated. With *y*m, *y*pl, and α, the bending moment (Equation (11)) for the process can be calculated (see Appendix F).
