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Research Papers

Analytical Prediction of Large Radius Bending by Circular Approximation

[+] Author and Article Information
Vitalii Vorkov

Department of Mechanical Engineering,
KU Leuven,
Celestijnenlaan 300B,
Leuven 3001, Belgium
e-mail: vitalii.vorkov@kuleuven.be

Richard Aerens, Dirk Vandepitte, Joost R. Duflou

Department of Mechanical Engineering,
KU Leuven,
Celestijnenlaan 300B,
Leuven 3001, Belgium

1Corresponding author.

Manuscript received March 26, 2018; final manuscript received September 12, 2018; published online October 5, 2018. Assoc. Editor: Yannis Korkolis.

J. Manuf. Sci. Eng 140(12), 121010 (Oct 05, 2018) (12 pages) Paper No: MANU-18-1183; doi: 10.1115/1.4041496 History: Received March 26, 2018; Revised September 12, 2018

An accurate analytical method is normally the preferred choice in engineering practice since this approach usually does not require additional software and can be applied for different situations. A number of analytical methods have been proposed for the air bending process, however, none of them has the capacity to deal with large radius bending. Large radius bending is characterized by a high ratio of the punch radius to the die opening and it is often applied for high-strength steels because of their limited bendability. This bending mode is used to fulfill the imposed level of maximum strain during the forming process. This contribution develops an analytical solution based on the assumption that the bent plate profile can be represented by two straight lines and a circular segment. Three different hardening laws—linear, Swift, and Aerens—are used for the bending moment calculation. Unit moment measurements are used in order to avoid extrapolation of hardening curves obtained by tensile testing. The model is compared with a wide range of experiments using the coefficient of determination, relative and absolute average errors, in addition to the mean standard error. The analytical prediction based on the circular approximation is found to be an accurate and robust tool for the calculation of the major bending characteristics for large radius air bending of high-strength steels.

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References

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Figures

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Fig. 1

Bending characteristics: (a) contact point positions; (b) forming angles and springback; and (c) bend allowance, where l0 is the initial plate length

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Fig. 2

Contact point positions and their influence on the bending moment

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Fig. 3

The circular model closely resembles the bent profile for the construction steel St-37 (a) and for the high-strength steel Strenx 1300 (b)

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Fig. 4

Circular approximation model for large radius bending: (a) geometrical scheme; (b) forces and their levers; and (c) die opening calculation

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Fig. 5

Representation of the stress distribution through the thickness for different models of work hardening

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Fig. 6

Stress–strain curves for AISI 304

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Fig. 7

Stress–strain curves for Strenx 700 MC

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Fig. 8

Stress–strain curves for St-37

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Fig. 9

Stress–strain curves for Strenx 1300

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Fig. 10

Unit moment for Strenx 700 MC

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Fig. 11

Unit moment for Strenx 1300

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Fig. 12

(a) Division of the bent plate according to the strain levels and (b) approximation of the bending moment for the large radius air bending, with the depiction of the assigned values of the elasticity modulus

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Fig. 13

Definition of secant elasticity modulus Esec and initial elasticity modulus Einit

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Fig. 14

Evolution of Esec versus the prestrain

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Fig. 15

Scheme for the determination of the bend allowance. The plate shape (a) before and (b) after springback.

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Fig. 16

Overview of used plates and tooling

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Fig. 17

Comparison of results for contact points position between analytical model and experimental data. Plate: AISI 316 L; thickness: 8 mm; die opening: 50 mm; and punch radius: 10 mm.

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Fig. 18

Comparison of results for contact points position between analytical model and experimental data. Plate: Strenx 700 MC; thickness: 6 mm; die opening: 80 mm; and punch radius: 40 mm.

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Fig. 19

Comparison of results for the bending force between analytical model and experimental data. Plate: St-37; thickness: 2 mm; die opening: 40 mm; and punch radius: 20 mm.

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Fig. 20

Comparison of results for the bending force between analytical model and experimental data. Plate: Strenx 1300; thickness: 4 mm; die opening: 60 mm; and punch radius: 30 mm.

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Fig. 21

Comparison of results for the springback angle between analytical model and experimental data. Plate: AISI 316 L; thickness: 4 mm; die opening: 40 mm; and punch radius: 10 mm.

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Fig. 22

Comparison of results for springback angle between analytical model and experimental data. Plate: Strenx 700 MC; thickness: 4 mm; die opening: 60 mm; and punch radius: 30 mm.

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Fig. 23

Comparison of results for bend allowance position between analytical model and experimental data. Plate: St-37; thickness: 8 mm; die opening: 80 mm; and punch radius: 20 mm.

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Fig. 24

Comparison of results for bend allowance position between analytical model and experimental data. Plate: Strenx 1300 MC; thickness: 6 mm; die opening: 60 mm; and punch radius: 20 mm.

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Fig. 25

Bend allowance depends on the hardening exponent and yield strength

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