Proposed model

In order to develop a mathematical model of a bent shaft deformation based on geometry and mechanical behavior of materials, some assumptions have to be made. The final goal is to compute the force necessary to straighten a bent shaft.

First, the bending moment required to straighten the shaft will be computed with Equation (1) (^{Ugural, and Kenster, 1995}).

Where, y is the height of the cross section, b is the width, d is the diameter and o is the normal stress. Figure 1

In this case the cross section is circular, it can be shown that the width of the section in terms of height is given by Equation (2).

The normal stress, as a function of height, is a piecewise continuous and bounded function. The fact that it is a continuous piecewise function is due to the transition from elastic flow, determined by Hooke's law, to plastic flow which can be approximated by the Ludwik equation (^{Lubliner, 2008}); and it is bounded by 0 (no stress applied) and sut (ultimate strength). Then, the normal stress as a function of height is given by:

Where E is the Young modulus of the material, R is the average radius of curvature of the shaft in the cross-section analyzed (Where R ≠ ∞ due to shaft deflection), σ_ys is the yield strength of the material, K is the strain hardening coefficient of the material, n is the strain hardening exponent of the material and y_f is the height which divides the elastic and plastic region in the cross section.

On the other hand, once the shaft is deformed plastically, the stress-strain curve changes (Strain hardening - Cold work) and the new function σ'(y) has to be computed. The material tries to recover its original shape (This phenomenon is called "springback"). This recovery is approximately linear with a slope equal to the Young modulus of the material. Also the material suffers a strain hardening due to plastic deformation and the new yield stress σ'_ys is equal to the strain that caused the deformation observed in the component. Figure 2 shows this process.

Then, considering the new stress-strain curve, the new normal stress function σ' is given by:

Where R' is the radius of curvature after deformation with stress level σ '_ys and the load still applied, e is the strain corresponding to σ'_ys and R(y) is the radius of curvature of the shaft at height y. This latter term is also dependent on the variation of the stress σ' but this is a function of y, then R ends a function of y.

Now, the values of R' and ε have to be computed. The strain ε is determined using equation (5).

Where ε' is the strain corresponding to the springback, and in this case it is given by:

Now R' can be calculated using Equation (7).

The stress corresponding to R(y) = ∞ is denoted by σ₁ It is important to take into account the material springback again, so an extra stress (overbending) has to be added, this extra stress is denoted by σ₂Figure 3 illustrates both stresses.

Given that elastic strain recovery is approximately linear with a slope equal to the Young's modulus and ε₁ is the strain corresponding to R = ∞, then:

Where ε₂ can be obtained from Ludwik equation:

Taking advantage of the symmetry for stress distribution and width of the cross section the equation for the bending moment Mf is given by:

The goal is to know the bending moment Mf when R(y) approaches infinity. The integral is a Riemann series evaluated over a range in which the function to be integrated is uniformly continuous, then:

For the design of straightening machines is imperative to know the maximum force required to straighten a shaft. In order to compute the maximum force is necessary to define the maximum diameter dmax, the maximum length of the shaft Lmax and the maximum radius of curvature Rmax. The latter can be calculated as follows using Equation (12) and Equation (13):

Finally, the maximum force required for a straightening process (this is the force the machine has to applied to the shaft) is given by: