TO BEND A BEAM AT VARIABLE RADIUS OF CURVATURE TO ATTAIN THE REQUIRED PROFILE

BYVIGNESH UDAY PAI

VEERMATA JIJABAI TECHNOLOGICAL INSTITUTE

ATBHABHA ATOMIC RESEARCH CENTER

CONTENTS Problem Statement Introduction Need for Springback Prediction Development of Springback Prediction Model Finite Element Modeling Radius of curvature of the die Determination of the die shape Results Co-ordinates and radius of curvature of the die Conclusion and future works

PROBLEM STATEMENT To determine the shape of the die required to

bend the beam to attain the required profile.

INTRODUCTION Springback prediction in bending of beams is an important

factor that needs to be considered. Springback allows us to determine how many times the beam

must be bent in order to obtain the required profile. For bending of the beam in the desired radius of curvature it

is important to understand the springback that occurs due to unloading.

A springback perdition model has been developed and the results were verified using finite element method.

The model was then used to develop a model to determine the die shape to bend a beam.

NEED FOR SPRINGBACK PREDICTION MODEL

The springback prediction model will allow us to determine how much a particular type of beam will springback on loading/unloading.

This will be useful further to determine the radius of curvature at which the beam should be bent so that the required profile of the beam is attained on springback.

The springback perdition model is an important tool in determining the die shape in which the beam has to be bent.

DEVELOPMENT OF SPRINGBACK PERDITION MODELINITIAL CONDITION

Franc Kosel, Tomaz Videnic, Tadej Kosel and Mihael Brojan in their paper have explained the springback phenomenon occurring due to bending/unbending in rectangular cross section of beams.

Bending of a straight beam: Moment for plastic loading (Mpl):

Moment for elastic unloading (Mel):

Radius of curvature after first cycle:

SECONDARY CONDITION Bending of curved beam: Moment in plastic loading (Mol):

Moment in elastic unloading (Mel):

Radius of curvature after second cycle:

FINITE ELEMENT MODELLING Finite element modeling is one of the most efficient methods

for finding approximate solutions to boundary value problems.

The entire solid model is modeled in a static environment with structural preference.

The beam and the die are modeled and the element chosen is 8 node 183.

The die is made of a rigid material with the degrees of freedom constraint on all sides.

The beam is made of a softer material and the front end of the beam is constraint along the x-axis and the y-axis.

FINITE ELEMENT MODELLING

The bending and unbending of a beam resulting in springback.

FINITE ELEMENT MODELLING

FINITE ELEMENT MODELLING

The bending and unbending of a beam resulting in springback.

FINITE ELEMENT MODELLING

RADIUS OF CURVATURE OF THE DIE

The knowledge of secant method was used to develop a model to find the radius at which the beam must be bent in order to obtain the desired radius of curvature.

Consider two initial guesses of the radius of curvature Ri1 and Ri2 at which the beam will be bent. Rfi1 and Rfi2 is the radius of curvatures obtained by bending of beam once and considering springback.

Δ1 and Δ2 are the two differences between the required radius of curvature R and the radii of curvature Rfi1 and Rfi2.

RADIUS OF CURVATURE OF THE DIE

The value of Ri3 can be verified by modeling the equation such that the value of R and Rfi3 converges upto a value of 0.0001. Rfi3 is the radius of curvature obtained by bending the beam at Ri3.

Rfi3 is the final radius of curvature that is required.

Ri3 is the radius of curvature of the die along which the beam should be bent.

DETERMINATION OF THE DIE SHAPE

Once the radius of curvature of the die has been determined the shape of the die is determined.

The shape of the die can be obtained by equating the curve to the equation of radius of curvature. The radius of curvature equation is gives as:

The radius of curvature formula for a curve is used to calculate the co-ordinates of the die. Using the knowledge of trapezoidal rule and basic numerical integration the z-coordinate of the die is calculated.

RESULTS

The results obtained from the mathematical model and the results of the finite element modelling are compared in the table below.

The results obtained from the finite element model are within an error of 1.30% of the mathematical model.

Radius of curvature (mm) Mathematical model (mm) Finite Element Modeling (mm)

Error (%)

110 163.4 163.7 -0.18%

60 71.9 72.4 -0.70%

37.5 52.7 53.4 -1.30%

DIE SHAPE

CO-ORDINATES AND RADIUS OF CURVATURE OF THE DIE

X(m) Z(m) R(curve)

0 0 21.29

1.6 0.06 21.92

3.2 0.23 23.86

4.8 0.52 27.23

6.4 0.90 32.02

8.0 1.38 37.91

9.6 1.93 44.41

11.2 2.55 52.32

12.8 3.24 61.97

14.4 3.98 73.54

16 4.75 87.23

DIE SHAPE

CO-ORDINATES AND RADIUS OF CURVATURE OF THE DIE

X(m) Z(m) R(curve)

0 0 24.9239

1.6 0.051 25.4135

3.2 0.21 26.9011

4.8 0.45 29.4424

6.4 0.79 33.1263

8.0 1.22 38.0705

9.6 1.72 44.4169

11.2 2.29 52.3261

12.8 2.92 61.9734

14.4 3.60 73.5454

16 4.32 87.2373

CONCLUSION AND FUTURE WORKS

Conclusion: Springback prediction is an important issue that needs to

be considered after bending process. The springback prediction model is a useful tool that can

be used to find the radius of curvature of the die. The radius of curvature of the die can be found for

different cross-sections by modifying the numerical analysis.

The die shape for two cross-sections have been shown graphically.

Future Work: The springback prediction model can be further modified

for materials with hardening. Theoretical results of the die will be compared by

modelling and FEM analysis.