**2. Three-Point Bending**

First of all, the terminology for three-point bending straightening should be introduced. The simplified situation of the three-point bending case is shown in Figure 1, where the initial shape of the billet is depicted by a dotted line. The maximal deflection *w* caused by the applied force *F* can be visualised by the deformed shape of the billet drawn with a dashed line. In the ideal case, the billet shape is straight after spring back, as displayed by the solid line.

**Figure 1.** A scheme of a three-point bending case.

In accordance with the additive rule, the total deflection *w* is composed of plastic deflection *wpl* and elastic deflection *wel*, thus

$$w = w\_{cl} + w\_{pl} \tag{1}$$

The irreversible deflection *wpl* is also important input for the algorithm, and it is supposed that it can be accurately measured by a sensory system of an automatic straightening machine for the given distance of supports *L*.

The plastic deflection *wpl* can be calculated considering an elastic stiffness *kel* and applied bending force *F* according to the analogy to Hooke's law.

$$w\_{pl} = w - w\_{el} = w - \frac{F}{k\_{cl}}\tag{2}$$

The elastic stiffness *kel* is a function of the Young modulus *E*, moment of inertia *Iz*, and support distance *L*. We will consider just a square cross-section of the billet in this work, i.e., *Iz* = *D*4/12, where *D* is the dimension of the square cross-section.

A prediction of required total deflection (output quantity of the algorithm) is proposed to be determined from the linear relationship.

$$w = k\_{w}w\_{pl} + w\_{y\_{\prime}} \tag{3}$$

where *wy* and *kw* are material parameters. Substituting (3) into (2), one can obtain the linear relation between the bending force and total deflection.

$$F = k\_{\rm el} \frac{w\_{\rm y}}{k\_{\rm w}} + k\_{\rm el} \left(1 - \frac{1}{k\_{\rm w}}\right) w = A + B \times w. \tag{4}$$

It can be noted that the parameter *wy* expresses the total deflection of the billet corresponding to the maximal bending stress in the cross-section for the elastic region of loading, i.e., yield stress *σy*.

#### **3. Laboratory Experiments and Their Numerical Simulations**

In order to show the idea of the approximation of material response during straightening by three-point bending, an experimental study on three-point bending performed on 51CrV4 material at room temperature will be presented. First, the basic mechanical properties were determined by tensile test; see Table 1. The bending tests were realised on specimens with the square cross-section of variety of dimensions *D* and distances of supports *L*. The proper ratio of *D*/*L* for each bending test had to be determined analytically or numerically.

**Table 1.** Mechanical properties of 51CrV4 material obtained from the tensile tests.


In this study, finite element method (FEM) was used. The material model introduces the nonlinear kinematic hardening rule of Chaboche [30]. According to Chaboche's superposition, two back-stress parts are considered to express the back-stress.

$$\alpha = \sum\_{i=1}^{2} \alpha\_i = \alpha\_1 + \alpha\_2 \tag{5}$$

and the evolution equation of Armstrong and Frederick [31] for uniaxial loading is

$$d\alpha\_i = \mathbb{C}\_i d\varepsilon\_p - \gamma\_i \mathbb{a}\_i dp \tag{6}$$

where *Ci* and *γ<sup>i</sup>* are material parameters, *dε <sup>p</sup>* is the increment of longitudinal plastic strain, and *dp* is the increment of accumulated plastic strain.

The constitutive equation of the Chaboche model for uniaxial tension is

$$
\sigma = \sigma\_y + \mathfrak{a}\_1 + \mathfrak{a}\_2 = \sigma\_y + \frac{\mathbb{C}\_1}{\gamma\_1} (1 - e^{-\gamma\_1 \varepsilon\_p}) + \frac{\mathbb{C}\_2}{\gamma\_2} (1 - e^{-\gamma\_2 \varepsilon\_p}) \tag{7}
$$

The tensile curve of the investigated material is used to calibrate the Chaboche model [30] for preliminary simulations by FEM; see Figure 2. All material parameters resulting from a non-linear least-square method application are stated in Table 2. Poisson's ratio ν = 0.3 was considered in the simulations too.

**Figure 2.** Deformation curve of 51CrV4 material and its approximation by Equation (7).

**Table 2.** Material parameters of the Chaboche model for 51CrV4 material.


All FE simulations within this paper were done in ANSYS 2020R1. The goal of the numerical study was to find a proof of the relationship between the total deflection and the plastic deflection described by Equation (3). The square cross-sections of 6 × 6, 8 × 8, 10 × 10, 12 × 12, and 14 × 14 were considered.

For the discretisation of geometry, the BEAM188 element was used. Boundary conditions applied to the FE model are shown in Figure 3. All nodes of the model are fixed in rotations around the *x*-axis. Ramped displacement with time is applied in the middle of the model in the *y*-direction, leading to maximal displacement of *Uy* = 4 mm at the end of the computation.

**Figure 3.** Finite element (FE) model with boundary conditions.

An optimization task was done (parametric study) to get proper distance of the supports for each cross-section dimension *D*. Initially, the distance of supports of 80 mm was chosen for the 6 × 6 specimen. After performing the FE analysis for this case, the dependency of the total deflection on the plastic deflection was evaluated using Equation (2)

and approximated by the linear function (3). Then, the largest cross-section of 14 × 14 was considered for simulations by trial and error to gain acceptable correlation with the approximated curve of the first case (total deflection vs. plastic deflection). Other cases, 8 × 8, 10 × 10, and 12 × 12, were solved by repeated FE simulation with an initial guess of the support length supposing the linear relationship between the support distance and cross-section dimension from previous two limit cases.

The resulting curves, which describe the relation between the total deflection and the plastic deflection, are shown in Figure 4. Good overall correlation is achieved for particular cases of cross-sectional dimensions. The dependency is pretty linear in the interval between 0.5 and 2.5 mm of plastic deflection, which confirms the validity of Equation (3). The optimal distances of supports are as follows: 80, 90, 100, 110, and 120 mm (for cross-sectional dimensions of 6 × 6, 8 × 8, 10 × 10, 12 × 12, and 14 × 14).

**Figure 4.** The dependency of total deflection on plastic deflection from the numerical study concerning optimal distances of supports (*L* in the legend) for each cross-section size *D* × *D.*

Based on the numerical study, the curve describing the dependency of the optimal support distance *L* on the cross-sectional dimension *D* of specimens is constructed; see Figure 5. It is clear that the idea of linear dependency of the optimal support distance on the cross-sectional dimension is true. Concerning the available material of billet, the following appropriate dimensions of specimens were selected: 2.9, 4.75, 7.45, 9.55, and 14 mm. The corresponding support distances are as follows: 65, 73, 89, 98, and 120 mm. The specimens for experiments were made by electric discharge machining (EDM) using a portion of the material chosen from the same position of the billet cross-section as for tensile tests.

**Figure 5.** The dependency of the optimal distance of supports *L* on cross-section size *D.* FEM, finite element method.

All experiments were realized using a TESTOMETRIC M500-50CT universal testing machine. The position rate was 5 mm per minute. Deflection was measured as the position of the crossbar. A photo from a three-point bending test realization is shown in Figure 6, where the deformed shapes of specimens are also presented.

**Figure 6.** Photos from the three-point bending test: the whole setup (**a**) and selected deformed specimens (**b**).

Obtained bending force versus total deflection diagrams are shown in Figure 7. The target total deflection (position of crossbar) was 3 mm for *D* = 2.9, 5 mm for *D* = 14, and 4 mm for all others.

**Figure 7.** Force response to total deflection for all considered cases.

For eventual straightening of billets with different cross-sectional dimensions, it is important to investigate how the dependences of the total deflection on the plastic deflection differ for individual cross-sections, as presented in Figure 8.

An important finding from the performed experimental study is the fact that the slope *kw* remains approximately the same even though the cross-sections are significantly different in their dimensions. The curves on the graph shown in Figure 8 differ only in the vertical offset. It should be noted that a slight nonlinearity is present in the initial part of the curve of total deflection versus plastic deflection. However, the straightening of billets will be done only in positions where it makes sense. The interventions will be proposed only for significant deviation from a straight line created between supports based on billet shape captured by the camera system.

**Figure 8.** Dependences of total deflection on plastic deflection evaluated from three-point bending tests.
