﻿ Springback of I-Section Beam after Pure Bending with von Mises Criteria

World Journal of Engineering and Technology
Vol.06 No.01(2018), Article ID:82251,15 pages
10.4236/wjet.2018.61006

Springback of I-Section Beam after Pure Bending with von Mises Criteria

Reham Saleh1, Gamal Ali2, Abla El-Megharbel2

1Port Said, Egypt

2Department of Production Engineering, Port Said University, Port Said, Egypt    Received: December 21, 2017; Accepted: January 30, 2018; Published: February 2, 2018

ABSTRACT

The objective of the present paper is to introduce a theoretical analysis of bending I-sections after pure bending. The springback values are determined to provide a quantitative method for predicting the springback using von Mises criteria. The analytical methods for the I-section are given for two cases according to the positions of the yield point along the height of the beam. The controlling parameters on the springback of I-sections are studied. The results obtained are quite successful for the prediction of springback for bending I-sections.

Keywords:

Bending Sections, Beam, Springback, von Mises 1. Introduction

Springback values for I-section beam have been determined with the assumption of yielding according to Tresca criteria as shown by  . This assumption is not completely valued for large plastic, therefore von Mises criteria are used in the present work. The controlling parameters such as, material properties, strain hardening exponent and height of the beam have been studied, as well as their influence on the springback behavior. Moreover, a comparison between Tresca and von Mises yielding criteria has been carried out.

2. Numerical Analysis

In this paper, analysis of springback for I-section under bending is done to predict the springback values. The stress-strain relationship is $\sigma =E\epsilon$ is in the elastic range and $\sigma =K{\epsilon }^{n}$ is in the plastic range. Where $\sigma$ is the bending stress, $\epsilon$ is the bending strain, Y is the yield strength, E is the modulus of elastically, K is the strength coefficient and n is the strain hardening exponent.

Figure 1 shows the relation between the applied bending moments on I-section to the change in curvature. At point A the material yields, at maximum bending moment the material undergoes to plastic deformation. At point B, when the applied moment is released elastic springback occurs from point B to

Figure 1. The relation between applied bending moments and curvature.

point C. The change in curvature due to elastic springback is given by the following equation:

$\frac{1}{{R}_{o}}-\frac{1}{{R}_{f}}=\frac{{M}_{\mathrm{max}}}{\partial {M}_{E}/\partial \left(1/R\right)}$ . (1)

where Ro is the required radius and Rf is the final radius

Assumptions are considered to derivation the springback equation for I-section beam (Figure 2) according to von Mises yielding criteria:

1) The friction effect in the interface between the beam and the die is neglected.

2) The cross section dimension of the beam are such as the width to height ratio is height.

3) The stress-strain characteristic of material is the same in the tension and compression.

4) The cross section dimension of the beam do not change significantly in bending.

5) The radius of bending is large compared to the height of the beam so radial stresses are assumed is negligible.

6) The natural surface is always in the center of the beam, and plane section remains plane during bending.

7) The transverse strain is zero at any point in the plane.

8) The circumferential strains are sufficiently small so that the conventional strain and the strain are approximately equivalent.

9) The circumferential strain for any fiber does not vary along the bent section.

The general relationship between principle stresses and strain for elastic deformation is given by:

${\epsilon }_{x}=\frac{1}{E}\left({\sigma }_{x}-\nu \left({\sigma }_{y}+{\sigma }_{z}\right)\right)$ . (2-a)

Figure 2. I-section of the beam as a case study  .

${\epsilon }_{y}=\frac{1}{E}\left({\sigma }_{y}-\nu \left({\sigma }_{x}+{\sigma }_{z}\right)\right)$ . (2-b)

${\epsilon }_{z}=\frac{1}{E}\left({\sigma }_{z}-\nu \left({\sigma }_{y}+{\sigma }_{x}\right)\right)$ . (2-c)

From the assumption that:

${\sigma }_{y}={\epsilon }_{z}={\delta }_{z}=0$ . (3)

Then,

${\sigma }_{z}=\nu {\sigma }_{x}$ . (4)

From maximum shear stress theory of failure (von Mises yield criteria),

${\sigma }_{o}^{2}=\frac{1}{2}\left({\left({\sigma }_{x}-{\sigma }_{y}\right)}^{2}+{\left({\sigma }_{y}-{\sigma }_{z}\right)}^{2}+{\left({\sigma }_{z}-{\sigma }_{x}\right)}^{2}\right)$ . (5)

Substitute Equations ((2) and (4)) in Equation (5), yields

${\sigma }_{o}={\sigma }_{x}{\left(1+{\nu }^{2}-\nu \right)}^{\frac{1}{2}}$ . (6)

Stress in yield point is

${\sigma }_{o}=K{\left(\frac{K}{E}\right)}^{\frac{n}{1-n}}={\sigma }_{x}{\left(1+{\nu }^{2}-\nu \right)}^{\frac{1}{2}}$ . (7)

${\sigma }_{ox}=\frac{K{\left(\frac{K}{E}\right)}^{\frac{n}{1-n}}}{{\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{1}{2}}}$ . (8)

${\epsilon }_{x}=\left(\frac{1}{E}\right){\sigma }_{x}\left(1-{\nu }^{2}\right)$ . (9)

So, in the yield point the axial strain is,

${\epsilon }_{ox}=\left(\frac{1}{E}\right){\sigma }_{ox}\left(1-{\nu }^{2}\right)$ . (10)

Substitute with the yield point stress value in the equation

${\epsilon }_{ox}={\left(\frac{K}{E}\right)}^{\frac{1}{1-n}}\frac{\left(1-{\mathcal{V}}^{2}\right)}{{\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{1}{2}}}$ . (11)

This is approximate value of axial strain elastic-plastic interface, then, in the elastic region the axial stress is,

${\sigma }_{x}=\frac{E}{\left(1-{\mathcal{V}}^{2}\right)}{\epsilon }_{x}$ . (12-a)

where

${\epsilon }_{x}=\frac{y}{{R}_{o}}$ . (12-b)

Substitute with the circumferential strain, gives

${\sigma }_{x}=\frac{E}{\left(1-{\mathcal{V}}^{2}\right)}\left(\frac{y}{{R}_{o}}\right)$ for $0\le {\epsilon }_{x}\le {\left(\frac{K}{E}\right)}^{\frac{1}{1-n}}\frac{\left(1-{\mathcal{V}}^{2}\right)}{{\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{1}{2}}}$ .

which (σx) is the axial stress in the elastic region.

From the assumptions, substitute with Equations ((4), (5)) in the previous equation, yields

${\sigma }_{z}=\frac{{\sigma }_{x}}{2}$ . (13)

${\sigma }_{x}=\frac{K}{{\left(\frac{3}{4}\right)}^{\frac{1+n}{2}}}{\delta }_{x}^{n}$ . (14)

where σx is valid for plastic region; that is

${\left(\frac{K}{E}\right)}^{\frac{1}{1-n}}\frac{\left(1-{\mathcal{V}}^{2}\right)}{{\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{1}{2}}}\le {\epsilon }_{x}\le \frac{H}{2}$ . (15)

During the bending forming of the I-section beam the springback behavior occur according to position of the yield point ( ${h}^{*}$ ) the distance from the neutral surface up to the layer at which the yielding occurs so, this point is controlled in the springback behavior. Thus that, the applied bending moment is analyzes to two cases according the position of the yielding point along the beam height.

Case (1), the plastic region is in flange of I-section beam.

Substituting the values of elastic and plastic stress in the general equation of bending moment with considerable the integration limits as the following.

$M=2{\int }_{0}^{\frac{H}{2}}{\sigma }_{x}\text{d}A$ . (16)

The limits of integration are $0\to h/2$ the distance from the neutral axis to the lower layer of the flange, $h/2\to {h}^{*}$ the distance from the lower layer of the flange to the yielded point where the plastic deformation occurs and finally, ${h}^{*}\to H/2$ the distance from the yielded point to the upper layer of the flange.

${M}_{\mathrm{max}}=2\left({\int }_{0}^{\frac{h}{2}}{\sigma }_{{x}_{\text{elastic}}}\left(B-b\right)y\text{d}y+{\int }_{\frac{h}{2}}^{{h}^{*}}{\sigma }_{{x}_{\text{elastic}}}By\text{d}y+{\int }_{{h}^{*}}^{\frac{H}{2}}{\sigma }_{{x}_{\text{plastic}}}By\text{d}y\right)$ . (17)

$\begin{array}{c}{M}_{\mathrm{max}}=2\left({\int }_{0}^{\frac{h}{2}}\frac{E}{\left(1-{\mathcal{V}}^{2}\right)}\ast \frac{{y}^{2}}{{R}_{o}}\left(B-b\right)\text{d}y+{\int }_{\frac{h}{2}}^{{h}^{*}}\frac{E}{\left(1-{\mathcal{V}}^{2}\right)}\ast \frac{{y}^{2}}{{R}_{o}}B\text{d}y\begin{array}{c}\begin{array}{l}\\ \\ \end{array}\\ \end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{{h}^{*}}^{\frac{H}{2}}\frac{K}{{\left(\frac{3}{4}\right)}^{\frac{1+n}{2}}}\frac{{y}^{n+1}}{{R}_{o}^{n}}B\text{d}y\right)\end{array}$ (18)

where:

${h}^{*}={R}_{o}{\left(\frac{K}{E}\right)}^{\frac{1}{1-n}}\frac{\left(1-{\mathcal{V}}^{2}\right)}{{\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{1}{2}}}$ . (19)

Thus

$\begin{array}{c}{M}_{\mathrm{max}}=2\left(\frac{EB{R}_{o}^{2}}{3\left(1-{\mathcal{V}}^{2}\right)}{\left(\frac{K}{E}\right)}^{\frac{3}{1-n}}\frac{{\left(1-{\mathcal{V}}^{2}\right)}^{3}}{{\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{3}{2}}}-\frac{Eb}{3\left(1-{\mathcal{V}}^{2}\right){R}_{o}}{\left(\frac{h}{2}\right)}^{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{KB}{{\left(3/4\right)}^{\left(1+n\right)/2}\left(n+2\right)}\left(\left(\frac{1}{{R}_{o}^{n}}\right){\left(\frac{H}{2}\right)}^{n+2}-{R}_{o}^{2}{\left(\frac{K}{E}\right)}^{\frac{n+2}{1-n}}\frac{{\left(1-{\mathcal{V}}^{2}\right)}^{n+2}}{{\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{n+2}{2}}}\right)\right)\end{array}$ (20)

The bending equation in the elastic range is

${M}_{E}=\frac{{\sigma }_{x}\cdot {I}_{z}}{y}$ . (21)

where ${I}_{z}$ , is the moment of inertia for the I-section about the neutral axis.

${I}_{z}=\frac{B{H}^{3}}{12}-\frac{b{h}^{3}}{12}$ . (22)

Referring to the springback ratio

$\frac{{R}_{o}}{{R}_{f}}=1-\frac{{M}_{\mathrm{max}}}{\partial {M}_{E}/\partial \left(1/R\right)}\ast {R}_{o}$ . (23)

Yielding:

$\begin{array}{c}\frac{{R}_{o}}{{R}_{f}}=1-\left(\left(\frac{K}{E}\right)\frac{24B\left(1-{\mathcal{V}}^{2}\right)}{{\left(3/4\right)}^{\left(1+n\right)/2}\left(n+2\right)\left(B{H}^{3}-b{h}^{3}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left(\left(\frac{1}{{R}_{o}^{n}}\right){\left(\frac{H}{2}\right)}^{n+2}-{R}_{o}^{2}{\left(\frac{K}{E}\right)}^{\frac{n+2}{1-n}}\frac{{\left(1-{\mathcal{V}}^{2}\right)}^{n+2}}{{\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{n+2}{2}}}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{b{h}^{3}}{\left(B{H}^{3}-b{h}^{3}\right)}-\frac{8B{R}_{o}^{3}}{\left(B{H}^{3}-b{h}^{3}\right)}{\left(\frac{K}{E}\right)}^{\frac{3}{1-n}}\frac{{\left(1-{\mathcal{V}}^{2}\right)}^{3}}{{\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{3}{2}}}\right)\end{array}$ (24)

Assume

$\frac{b}{B}=\beta ,\text{\hspace{0.17em}}\frac{h}{H}=\alpha$ . (25)

$\begin{array}{c}\frac{{R}_{o}}{{R}_{f}}=1-\frac{1}{\left(1-B{\alpha }^{3}\right)}\left(\frac{3\left(1-{\mathcal{V}}^{2}\right)}{{\left(3/4\right)}^{\left(1+n\right)/2}\left(n+2\right)}{\left(\frac{2{R}_{o}}{H}\right)}^{1-n}{\left(\frac{{\sigma }_{o}}{E}\right)}^{1-n}-B{\alpha }^{3}\underset{\begin{array}{l}\\ \\ \end{array}}{\overset{}{}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\left(\frac{2{R}_{o}}{H}\right)}^{3}{\left(\frac{{\sigma }_{o}}{E}\right)}^{3}\left(\frac{{\left(1-{\mathcal{V}}^{2}\right)}^{3}}{{\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{3}{2}}}-\frac{3{\left(1-{\mathcal{V}}^{2}\right)}^{n+3}}{{\left(3/4\right)}^{\left(1+n\right)/2}\left(n+2\right){\left(1-\mathcal{V}+{\mathcal{V}}^{2}\right)}^{\frac{n+2}{2}}}\right)\right)\end{array}$ (26)

This is the model equation of the springback ratio for I-section beam which the plastic deformation is in the flange according to von Mises yield criteria.

Case (2): the plastic region in the web of I-section beam,

Substituting the values of elastic stress and plastic stress; moreover change the integration limits according to the position of yields point. $\left(0\to {h}^{*}\right)$ is the distance from the neutral axis to the yield point in the web of the I-section beam where the plastic deformation occurs, $\left({h}^{*}\to \frac{h}{2}\right)$ is the distance from the yielding point to the lower surface of the flange, $\left(\frac{h}{2}\to \frac{H}{2}\right)$ is the distance from the lower surface of the flange to the upper surface of the flange of the beam.

${M}_{\mathrm{max}}=2\left({\int }_{0}^{{h}^{oex}}{\sigma }_{{x}_{\text{elastic}}}\left(B-b\right)y\text{d}y+{\int }_{{h}^{oex}}^{\frac{h}{2}}{\sigma }_{{x}_{\text{plastic}}}\left(B-b\right)y\text{d}y+{\int }_{\frac{h}{2}}^{\frac{H}{2}}{\sigma }_{{x}_{\text{plastic}}}By\text{d}y\right)$ . (27)

where

${h}^{*}={R}_{o}{\left(\frac{K}{E}\right)}^{\frac{1}{1-n}}\frac{\left(1-{\mathcal{V}}^{2}\right)}{{\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{1}{2}}}$ . (28)

$\begin{array}{c}{M}_{\mathrm{max}}=2\left(\frac{E\left(B-b\right){R}_{o}^{2}}{3\left(1-{\mathcal{V}}^{2}\right)}{\left(\frac{K}{E}\right)}^{\frac{3}{1-n}}\frac{{\left(1-{\mathcal{V}}^{2}\right)}^{3}}{{\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{3}{2}}}-\frac{Kb}{{\left(\frac{3}{4}\right)}^{\frac{1+n}{2}}\left(n+2\right){R}_{0}{}^{n}}{\left(\frac{h}{2}\right)}^{n+2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{KB}{{\left(\frac{3}{4}\right)}^{\frac{1+n}{2}}\left(n+2\right){R}_{0}{}^{n}}{\left(\frac{H}{2}\right)}^{n+2}-\frac{K\left(B-b\right){R}_{o}^{2}}{{\left(\frac{3}{4}\right)}^{\frac{1+n}{2}}\left(n+2\right)}{\left(\frac{K}{E}\right)}^{\frac{n+2}{1-n}}\frac{{\left(1-{\mathcal{V}}^{2}\right)}^{n+2}}{{\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{n+2}{2}}}\right)\end{array}$ (29)

Substituting with the Equation (25) in Equation (29), substituting the result equation of the maximum bending moment in the Equation (23), we get the following equation

$\begin{array}{c}\frac{{R}_{o}}{{R}_{f}}=1-\frac{1}{\left(1-\beta {\alpha }^{3}\right)}\left(\frac{3\left(1-{\mathcal{V}}^{2}\right)}{{\left(\frac{3}{4}\right)}^{\frac{1+n}{2}}\left(n+2\right)}{\left(\frac{E}{{\sigma }_{o}}\right)}^{1-n}{\left(\frac{2{R}_{0}}{H}\right)}^{1-n}\left(1-\beta {\alpha }^{n+2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({\left(\frac{2{R}_{o}}{H}\right)}^{3}{\left(\frac{{\sigma }_{o}}{E}\right)}^{3}\left(1-\beta \right)\right)\left(\frac{{\left(1-{\mathcal{V}}^{2}\right)}^{3}}{{\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{3}{2}}}-\frac{3{\left(1-{\mathcal{V}}^{2}\right)}^{n+3}}{{\left(\frac{3}{4}\right)}^{\frac{1+n}{2}}\left(n+2\right){\left(1+{\mathcal{V}}^{2}-\mathcal{V}\right)}^{\frac{n+2}{2}}}\right)\right)\end{array}$ (30)

This is the equation of springback ratio using maximum shear stress (von Mises) theory for I-section where the plastic deformation is in the web.

3. Results and Discussion

For bending I-sections the springback calculation for the previous two cases in Equations ((26), (30)). These Equations ((26) and (30)) are depending on the ratio Ro/H, Y/E the strain hardening coefficient (n), the geometrical coefficients α, β and the Poisson’s ratio (ν).

Figure 3 and Figure 4 show the variation of springback ratio with the ratio (Ro/H) for different values of n, strain hardening coefficient, at constant value of α = 0.8, β = 0.9 with different values of Y/E which are 1.522 × 10−3 and 2.4 × 10−3 (for different material 1100 al, 1065 steel)  . The range of the yield strain values (Y/E) from 1.522 × 10−3 to 2.4 × 10−3, is recommended to investigate its effect on springback. It noticed that the springback ratio decreasing with increasing of Y/E and increasing the values of strain hardening coefficient (n).

Figure 3. Springback ratio with Ro/H with different values of Y/E at n = 0.2, ν = 0.35.

Figure 4. Springback ratio with Ro/H with different values of Y/E at n = 0.4, ν = 0.35.

Figure 5 and Figure 6 show the variation of springback ratio with the Ro/H ratio at different value of Y/E at constant value of α = 0.8, β = 0.9 with different values of n (strain hardening coefficient. Figure 7 and Figure 8 show the variation of springback ratio at constant value of α = 0.8 and β = 0.9 with different values of Poisson’s ratio (n = 0.25, 0.35, 0.45) and at fixed value of Y/E (1.522 × 10−3, 2.4 × 10−3). From Figure 5 and Figure 6, it is observed that the springback ratio decrease with the decreasing of Poisson’s ratio and with the increasing of Y/E ratio. Figure 9 and Figure 10 show the variation of radius of curvature after springback with different values of H (the height of the beam) and different values of strain hardening coefficient (n = 0.2, 0.4) at constant values of Ro = 40, α = 0.8 and β = 0.9. It is shown that the radius of curvature after springback is decreasing rapidly from rang 1 to 2 mm height, and then the decrease of the finial radius of curvature is become stable. In the next figures showed that: Figures 11-14 show the relation between radius of curvature before bending Ro and

Figure 5. Springback ratio with Ro/H with different values of n at Y/E = 1.5 × 10−3, ν = 0.35.

Figure 6. Springback ratio with Ro/H with different values of n at Y/E = 2.4 × 10−3, ν = 0.35.

Figure 7. Springback ratio with Ro/H with different values of n at Y/E = 1.5 × 10−3, n = 0.2.

Figure 8. Springback ratio with Ro/H with different values of n at Y/E = 2.4 × 10−3, n = 0.2.

Figure 9. Springback ratio with H (mm) with different values of n at ν = 0.35, Y/E = 1.5 × 10−3.

Figure 10. Springback ratio with H (mm) with different values of n at ν = 0.35, Y/E = 2.4 × 10−3.

Figure 11. Relation between Ro, Rf (mm) at values of Ro/H at n = 0.35, Y/E = 1.5 × 10−3, n = 0.1.

Figure 12. Relation between Ro, Rf (mm) at values of Ro/H at n = 0.35, Y/E = 1.5 × 10−3, n = 0.4.

Figure 13. Relation between Ro, Rf (mm) at values of Ro/H at n = 0.35, Y/E = 2.4 × 10−3, n = 0.1.

Figure 14. Relation between Ro, Rf (mm) at values of Ro/H at n = 0.35, Y/E = 2.4 × 10−3, n = 0.4.

radius of curvature after bending Rf, it is noticed that the radius of curvature after bending Rf is increase with the increase of Ro/H ratio, the strain hardening coefficient n and decrease of Y/E ratio at constant values of α = 0.8, β = 0.9 and n = 0.35, also the radius of curvature after bending increase with the increase of radius of curvature before bending.

Mathematical model for springback equation with I-section beam is derived using Tresca in reference  . In this paper, a comparison between the springback equation according to Tresca and von Mises yield criteria has done. Figure 15 shows the comparison between springback equation using Tresca and von Mises criteria at Y/E = 2.4 × 10−3, n = 0.1, α = 0.8, β = 0.9 and n = 0.25. From this figure it is shown that the springback according to von Mises is little high than according to Tresca when Ro/H is more than 60 for small values of Ro/H (less than

Figure 15. Comparison between Tresca and von Mises yielding criteria at n = 0.25, Y/E = 2.4 × 10−3, n = 0.1.

60) this difference is negligible. But using von Mises for very ductile materials is more useful.

4. Conclusion

The mathematical model of springback equations were derived in this paper according to von Mises yield criteria using nonlinear constitutive equation as the following conclusions: the theoretical analysis for I-section beam under bending has been carried out, and it was found that the prediction of springback is quite successful. Springback ratio increases with increasing Poisson’s ratio. Springback ratio increases with increasing the beam height. Springback ratio increases with decreasing the ratio of yield point stress to young’s modulus of elasticity. The springback is found to be more with decreasing values of strain hardening coefficient.

Cite this paper

Saleh, R., Ali, G. and El-Megharbel, A. (2018) Springback of I-Section Beam after Pure Bending with von Mises Criteria. World Journal of Engineering and Technology, 6, 104-118. https://doi.org/10.4236/wjet.2018.61006

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