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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.

Production of different sections has a great significance in the industry, because of the importance of these sections in various industries such as aerospace, trains, ships and various industrial purposes. In every industry, quality and productivity are major issues for being competitive. For example, a car frame needs to be designed to achieve strength requirements and aesthetic aspects; on the other hand, cost of production and repeatability is crucial to the business. A bending process has been one solution used in practice to achieve these goals in the sheet metal fabrication business. However, springback, a shape discrepancy between the fully loaded and unloaded configurations, undermines the stamping benefits, since a major effort on the tooling design is needed to compensate springback, so the bending process of forming is the most important operations at all. During bending process, the elastic recovery after unloading causes springback phenomenon. Therefore determination of springback in bending process is of great significance as it is used in the die design, geometry of the parts. This leads us to try to find the appropriate prediction of springback to reduce and control it. Many researchers have studied springback in different sections and factors controlling it. Researcher [

Springback values for I-section beam have been determined with the assumption of yielding according to Tresca criteria as shown by [

In this paper, analysis of springback for I-section under bending is done to predict the springback values. The stress-strain relationship is σ = E ε is in the elastic range and σ = K ε n is in the plastic range. Where σ is the bending stress, ε 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.

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

1 R o − 1 R f = M max ∂ M E / ∂ ( 1 / R ) . (1)

where R_{o} is the required radius and R_{f} is the final radius

Assumptions are considered to derivation the springback equation for I-section beam (

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:

ε x = 1 E ( σ x − ν ( σ y + σ z ) ) . (2-a)

ε y = 1 E ( σ y − ν ( σ x + σ z ) ) . (2-b)

ε z = 1 E ( σ z − ν ( σ y + σ x ) ) . (2-c)

From the assumption that:

σ y = ε z = δ z = 0 . (3)

Then,

σ z = ν σ x . (4)

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

σ o 2 = 1 2 ( ( σ x − σ y ) 2 + ( σ y − σ z ) 2 + ( σ z − σ x ) 2 ) . (5)

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

σ o = σ x ( 1 + ν 2 − ν ) 1 2 . (6)

Stress in yield point is

σ o = K ( K E ) n 1 − n = σ x ( 1 + ν 2 − ν ) 1 2 . (7)

σ o x = K ( K E ) n 1 − n ( 1 + V 2 − V ) 1 2 . (8)

ε x = ( 1 E ) σ x ( 1 − ν 2 ) . (9)

So, in the yield point the axial strain is,

ε o x = ( 1 E ) σ o x ( 1 − ν 2 ) . (10)

Substitute with the yield point stress value in the equation

ε o x = ( K E ) 1 1 − n ( 1 − V 2 ) ( 1 + V 2 − V ) 1 2 . (11)

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

σ x = E ( 1 − V 2 ) ε x . (12-a)

where

ε x = y R o . (12-b)

Substitute with the circumferential strain, gives

σ x = E ( 1 − V 2 ) ( y R o ) for 0 ≤ ε x ≤ ( K E ) 1 1 − n ( 1 − V 2 ) ( 1 + V 2 − V ) 1 2 .

which (σ_{x}) is the axial stress in the elastic region.

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

σ z = σ x 2 . (13)

σ x = K ( 3 4 ) 1 + n 2 δ x n . (14)

where σ_{x} is valid for plastic region; that is

( K E ) 1 1 − n ( 1 − V 2 ) ( 1 + V 2 − V ) 1 2 ≤ ε x ≤ 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 ∫ 0 H 2 σ x d A . (16)

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

M max = 2 ( ∫ 0 h 2 σ x elastic ( B − b ) y d y + ∫ h 2 h * σ x elastic B y d y + ∫ h * H 2 σ x plastic B y d y ) . (17)

M max = 2 ( ∫ 0 h 2 E ( 1 − V 2 ) ∗ y 2 R o ( B − b ) d y + ∫ h 2 h * E ( 1 − V 2 ) ∗ y 2 R o B d y + ∫ h * H 2 K ( 3 4 ) 1 + n 2 y n + 1 R o n B d y ) (18)

where:

h * = R o ( K E ) 1 1 − n ( 1 − V 2 ) ( 1 + V 2 − V ) 1 2 . (19)

Thus

M max = 2 ( E B R o 2 3 ( 1 − V 2 ) ( K E ) 3 1 − n ( 1 − V 2 ) 3 ( 1 + V 2 − V ) 3 2 − E b 3 ( 1 − V 2 ) R o ( h 2 ) 3 + K B ( 3 / 4 ) ( 1 + n ) / 2 ( n + 2 ) ( ( 1 R o n ) ( H 2 ) n + 2 − R o 2 ( K E ) n + 2 1 − n ( 1 − V 2 ) n + 2 ( 1 + V 2 − V ) n + 2 2 ) ) (20)

The bending equation in the elastic range is

M E = σ x ⋅ I z y . (21)

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

I z = B H 3 12 − b h 3 12 . (22)

Referring to the springback ratio

R o R f = 1 − M max ∂ M E / ∂ ( 1 / R ) ∗ R o . (23)

Yielding:

R o R f = 1 − ( ( K E ) 24 B ( 1 − V 2 ) ( 3 / 4 ) ( 1 + n ) / 2 ( n + 2 ) ( B H 3 − b h 3 ) × ( ( 1 R o n ) ( H 2 ) n + 2 − R o 2 ( K E ) n + 2 1 − n ( 1 − V 2 ) n + 2 ( 1 + V 2 − V ) n + 2 2 ) − b h 3 ( B H 3 − b h 3 ) − 8 B R o 3 ( B H 3 − b h 3 ) ( K E ) 3 1 − n ( 1 − V 2 ) 3 ( 1 + V 2 − V ) 3 2 ) (24)

Assume

b B = β , h H = α . (25)

R o R f = 1 − 1 ( 1 − B α 3 ) ( 3 ( 1 − V 2 ) ( 3 / 4 ) ( 1 + n ) / 2 ( n + 2 ) ( 2 R o H ) 1 − n ( σ o E ) 1 − n − B α 3 + ( 2 R o H ) 3 ( σ o E ) 3 ( ( 1 − V 2 ) 3 ( 1 + V 2 − V ) 3 2 − 3 ( 1 − V 2 ) n + 3 ( 3 / 4 ) ( 1 + n ) / 2 ( n + 2 ) ( 1 − V + V 2 ) n + 2 2 ) ) (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. ( 0 → h * ) is the distance from the neutral axis to the yield point in the web of the I-section beam where the plastic deformation occurs, ( h * → h 2 ) is the distance from the yielding point to the lower surface of the flange, ( h 2 → H 2 ) is the distance from the lower surface of the flange to the upper surface of the flange of the beam.

M max = 2 ( ∫ 0 h o e x σ x elastic ( B − b ) y d y + ∫ h o e x h 2 σ x plastic ( B − b ) y d y + ∫ h 2 H 2 σ x plastic B y d y ) . (27)

where

h * = R o ( K E ) 1 1 − n ( 1 − V 2 ) ( 1 + V 2 − V ) 1 2 . (28)

M max = 2 ( E ( B − b ) R o 2 3 ( 1 − V 2 ) ( K E ) 3 1 − n ( 1 − V 2 ) 3 ( 1 + V 2 − V ) 3 2 − K b ( 3 4 ) 1 + n 2 ( n + 2 ) R 0 n ( h 2 ) n + 2 + K B ( 3 4 ) 1 + n 2 ( n + 2 ) R 0 n ( H 2 ) n + 2 − K ( B − b ) R o 2 ( 3 4 ) 1 + n 2 ( n + 2 ) ( K E ) n + 2 1 − n ( 1 − V 2 ) n + 2 ( 1 + V 2 − V ) n + 2 2 ) (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

R o R f = 1 − 1 ( 1 − β α 3 ) ( 3 ( 1 − V 2 ) ( 3 4 ) 1 + n 2 ( n + 2 ) ( E σ o ) 1 − n ( 2 R 0 H ) 1 − n ( 1 − β α n + 2 ) + ( ( 2 R o H ) 3 ( σ o E ) 3 ( 1 − β ) ) ( ( 1 − V 2 ) 3 ( 1 + V 2 − V ) 3 2 − 3 ( 1 − V 2 ) n + 3 ( 3 4 ) 1 + n 2 ( n + 2 ) ( 1 + V 2 − V ) n + 2 2 ) ) (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.

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 R_{o}/H, Y/E the strain hardening coefficient (n), the geometrical coefficients α, β and the Poisson’s ratio (ν).

_{o}/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) [^{−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).

_{o}/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. ^{−3}, 2.4 × 10^{−3}). From _{o} = 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 R_{o} and

radius of curvature after bending R_{f}, it is noticed that the radius of curvature after bending R_{f} is increase with the increase of R_{o}/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 [^{−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 R_{o}/H is more than 60 for small values of R_{o}/H (less than

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

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.

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