American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) 
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  

http://asrjetsjournal.org/ 
 

Determination of Springback Values in Bending I-sections 

with Tresca Criteria 

Reham Saleha*, Gamal Ali b, Abla El-Megharbelc 

aSpeacliast third engineer at El-Max Salines, El-Max Salines, Port Said 42523, Egypt 
b,cAssocaite Prof. Prod. Eng. Dept.,Port said University, PortSaid 42523, Egypt 

aEmail: rsaleh@yahoo.com 
bEmail: gamal_osmun@yahoo.com 

cEmail: aelmegharbel@eng.psu.edu.eg 

 

Abstract 

The springback ratio for bending I-section beams is introduced analytically in the present study. A complete 

analysis using the strength of materials approach is carried out with Tresca yielding criteria. Analytical methods 

are given in the form of equations for two cases according to the positions of the yield point along the height of 

the beam. The results represent the effect of different parameters affecting the springback ratio. 

Keywords: Bending sections; Springback; I sections; Tresca. 

1. Introduction  

Bending of different sections shape is one of the most widely applied metal forming operations. In the 

fabrication of metals, the elastic recoveries after unload causing the springback phenomenon in which the radius 

of curvature in bending increases after the bending moment is removed. Therefore, the precise prediction of 

springback is a key to assessing the accuracy of part geometry. Many analyses have been carried done to 

calculate and investigate the springback phenomenon in bending different sections. Moon and his colleagues [1] 

have investigated the effect of tool temperature on the reduction of springback amount of aluminum 1050 sheet. 

Emphases are placed on the reduction of springback amount in terms of combination of die and punch 

temperature. It was found that the combination of hot die and cold punch can reduce the amount of springback 

up to 20% when compared to conventional room temperature bending test. While T. Liu and his colleagues [2] 

have studied the springback behaviors of the age-hardened 2196-T8511 and 2099-T83 Al–Li alloys extrusions 

with Z-section and T-section under displacement controlled cold stretch bending (DCSB).  

------------------------------------------------------------------ 

* Corresponding author.  

187 
 

http://asrjetsjournal.org/


American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  3, pp 187-201 

 

Analytical model as well as FEM were used. It was found that increasing pre-stretching or post-stretching strain 

can significantly reduce the radius springback and time-dependent springback. Moreover [3] have introduced a 

study to springback in sheet metal forming and discussed the parameters affecting the springback phenomenon 

such as punch angle, grain direction of sheet metal material, die opening, ratio of die radius to sheet thickness, 

sheet thickness, punch radius, punch height, coining force, pre bend condition of strip etc. reference [4] have 

evaluated the elastic springback and the residual stress after stretch bending of a rectangular section of 7075 

aluminum sheet. Also [5] have introduced a theoretical analysis of the elastic–plastic bending of tubes and 

sections with different shapes. Analytical methods are given in the form of equations to provide a quantitative 

method for predicting the moment for forming the section of the tube to a specific radius of curvature. Reference 

[6] has studied the mechanism of the springback-free phenomenon observed at warm forming temperatures 

higher than 750K. It was found that the abrupt decrease in springback at approximately 750K is mainly caused 

by a momentary increase of high-temperature creep strain just after the forming process. Z. Damián-Noriega and 

his colleagues [7] have presented a new equation to predict the springback in the bending and rolling processes 

of metallic sheet. The resulted equation has been experimentally applied in the design of a wood truncated cone 

to form an aluminum sheet truncated cone. Wenjuan Liu and his colleagues [8] have introduced a process 

optimal control scheme of sheet metal forming springback based on evolutionary strategy. The experiment 

results indicated that the best process scheme can be achieved with the process optimal springback control 

method based on evolutionary strategy; the springback can be controlled and reduced effectively. Feng Ruan 

and his colleagues [9] have tried to predict and control the springback for sheet metal forming. In this paper, 

back propagation (BP) neural network and genetic algorithm (GA) was introduced to predict springback of 

complex sheet metal forming parts. The model can be used to predicate springback and provides a theoretical 

guide for complex sheet metal parts forming, tools designing and die modification. ZHAO Jun and his 

colleagues [10] have addressed an analytical method for springback of small curvature plane bending with 

unloading rule of classical elastic- plastic theory and principle of strain superposition. The springback equation 

of plane bending is considered to analyze the expanding and setting round process, and the results agree with the 

experimental data. Sanjay Kumar Patel and his colleagues [11] were interested in the springback analysis in 

sheet metal forming using modified Ludwik stress-strain relation. Using the deformation theory of plasticity, 

formulation of the problem and springback ratio is derived using modified Ludwik stress strain relationship with 

Tresca and von Mises yielding criteria. The mechanical behavior for beams has an important value for studying 

because of their widely used in engineering components. Moreover the use of I- beam sections in forming 

structural components for cars, trains, aircraft, boats and ships has grown exponentially. Thus, the aim of this 

paper is to present a theoretical analysis for springbackin I-section beams under bending. In addition, studying 

the affecting parameters is also one of the targets of this paper. 

2. Analysis 

In the present work an analysis was made for I-beams under bending to evaluate the springback values. The 

stress- strain relationship is given by: 

𝜎 = �
𝐸𝜀               𝜀 ≤ 𝑌

𝐸

𝐾𝜀𝑛              𝜀 ≥ 𝑌
𝐸

�          (1) 

188 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  3, pp 187-201 

 

Where σ is the bending stress,ε is the bending strain, Y is the yield strength, E is the modulus of elasticity, K is 

the strength coefficient and n is the strain hardening exponent. 

Figure 1 shows the schematic plot of applied bending moment versus curvature during the formation of a wide 

beam of metal around a portion of a cylindrical die. At point A, the material yields and plastic deformation 

continuous until the inside surface of the material conforms to die at point B when the applied moment is 

released elastic springback occurs from B to C. The change in curvature due to this elastic springback is (1/𝑅𝑅o)-

(1/Rf), which is from figure 1, 

1
𝑅𝑅𝑜

−
1
𝑅𝑅𝑓

=
𝑀𝑚𝑎𝑥

𝜕𝑀𝐸 𝜕(1 𝑅𝑅⁄ )⁄  

          (2) 

 

Figure 1: Schematic representation of curvature versus applied moment during bending 

The figure 2 shows the I-section considered in this study. 

The following assumptions are considered to simplify the analysis of the bending process: 

(1) The stress-strain characteristics of material are the same in tension and compression. 

(2) The friction effect at the interface between the beam and the die is neglected. 

(3) The cross section dimensions of the beam are such that the width to height ratio is large. 

(4)The natural surface is always in the centre of the beam, and plane sections remain plane during bending. 

(5) The cross section dimensions of the beam do not change significantly in bending. 

(6) The radius of bending is large compared to the height of the beam so radial stresses are assumed to be 

negligible. 

 

189 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  3, pp 187-201 

 

(7) The circumferential strains are sufficiently small so that the conventional strain and the true strain are 

approximately equivalent. 

(8)The transverse strain is zero at any point in the beam. 

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

The relationship between the principal stresses and strain for the elastic according to Hooke's law , 

 

 

Figure 2: The I-section 

𝜀𝑥 = 1
𝐸
�𝜎𝑥 − 𝒱𝒱�𝜎𝑦 + 𝜎𝑧�� , 𝜀𝑦 = 1

𝐸
�𝜎𝑦 − 𝒱𝒱(𝜎𝑥 + 𝜎𝑧)� , 𝜀𝑧 = 1

𝐸
�𝜎𝑧 − 𝒱𝒱�𝜎𝑦 + 𝜎𝑥��,   (3) 

According to the assumptions that the bending radius is large compared to the height of the beam so radial 

stresses is zero at any point along the beam, 

𝜎𝑦 = 𝜀𝑧 = 𝛿𝑧 = 0          (4) 

As well as the circumferential strain for any fiber does not vary along the bent section; then 

𝜀𝑥 = 𝛿𝑥            (5) 

Substituting the values of Eq. (4) in Eq.(3); yields to  

H h 

B 

b/2 

x 

z 

y
 

 

 

 

 

 

 

  

 

  

b/2 

190 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  3, pp 187-201 

 

𝜎𝑧 = 𝒱𝒱𝜎𝑥           (6) 

From Tresca yield criteria (the maximum shear stress theory) 

Y = 𝜎𝑥 − 𝜎𝑧           (7) 

or by using Eq. (6) 

Y = 𝜎𝑥(1−𝒱𝒱)           (8) 

At yield point the stress in elastic and plastic region is equal, so 

E𝜀𝑜 = 𝐾εon           (9) 

Which yields to 

𝜀𝑜 = �𝐾
𝐸
�

1
1−𝑛           (10) 

Therefore the stress at the yield point is 

𝜎𝑜 = K𝜀𝑜𝑛 = 𝐾 �𝐾
𝐸
�

𝑛
1−𝑛          (11) 

Substation with Eq. (8) in Eq. (11) 

𝜎𝑜 = 𝜎𝑥(1− 𝒱𝒱) = 𝐾 �𝐾
𝐸
�

𝑛
1−𝑛         (12) 

𝜎𝑥 =
𝐾�𝐾𝐸�

𝑛
(1−𝑛)

(1−𝒱𝒱)
           (13) 

Thus, 

𝜀𝑥 = �1
𝐸
�𝜎𝑥(1−𝒱𝒱2)          (14) 

In addition the circumferential strain at the yield point is 

𝜀𝑜𝑥 = �1
𝐸
�𝜎𝑜(1−𝒱𝒱2)          (15) 

Substitute with Eq.(11), get 

𝜀𝑜𝑥 = �𝐾
𝐸
�
1 (1−𝑛)⁄

(1 + 𝒱𝒱)          (16) 

191 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  3, pp 187-201 

 

The previous value is the circumferential strain at elastic-plastic interface. Therefore the identical stress is 

𝜎𝑥 = 𝐸
(1−𝒱𝒱2) 𝜀𝑥           (17) 

Where𝜀𝑥 = 𝑦
𝑅𝑜

, then 

𝜎𝑥 = 𝐸
(1−𝒱𝒱2)

� 𝑦
𝑅𝑜
�           (18) 

From the deformation theory of plasticity 

𝛿𝑥 = 1

𝐾
1
𝑛
�𝜎𝑥2 + 𝜎𝑦2 + 𝜎𝑧2 − 𝜎𝑥𝜎𝑦 − 𝜎𝑦𝜎𝑧 − 𝜎𝑧𝜎𝑥�

(1−𝑛)/2𝑛
∗ �𝜎𝑥 −

𝜎𝑦
2
− 𝜎𝑧

2
�    (19) 

𝛿𝑦 = 1

𝐾
1
𝑛
�𝜎𝑥2 + 𝜎𝑦2 + 𝜎𝑧2 − 𝜎𝑥𝜎𝑦 − 𝜎𝑦𝜎𝑧 − 𝜎𝑧𝜎𝑥�

(1−𝑛)/2𝑛
∗ �𝜎𝑦 −

𝜎𝑥
2
− 𝜎𝑧

2
�    (20) 

𝛿𝑧 = 1

𝐾
1
𝑛
�𝜎𝑥2 + 𝜎𝑦2 + 𝜎𝑧2 − 𝜎𝑥𝜎𝑦 − 𝜎𝑦𝜎𝑧 − 𝜎𝑧𝜎𝑥�

(1−𝑛)/2𝑛
∗ �𝜎𝑧 −

𝜎𝑥
2
− 𝜎𝑦

2
�    (21) 

Substituting with Eqs. (4) and (5) in Eqs. (21)yields to;  

𝜎𝑧 = 𝜎𝑥
2

            (22) 

Replacing the previous value of σzin Eq. (19), yields to the following  

𝜎𝑥 = 𝐾

�34�
(1+𝑛)/2 𝛿𝑥

𝑛          (23) 

Since the initial assumption state that the circumferential strain is sufficiently small so that the axial strain and 

true strain are approximately the same, 

𝜀𝑥 = 𝑦
𝑅𝑜

= 𝛿𝑥           (24) 

𝜎𝑥 = 𝐾

�34�
1+𝑛
2
� 𝑦
𝑅𝑜
�
𝑛
           (25) 

Then, the maximum bending moment for I-section could be introduced as 

𝑀𝑚𝑎𝑥 = 2∫ 𝜎𝑥
𝐻
2
0 𝑑𝐴          (26) 

The present analysis went through two cases according to the position of the yield point (h*) which is the 

distance from the neutral surface up to the layer at which yielding occurs. For that reason, The stress and strain 

distributions across the I-section beam loaded by bending moment M will be derived for the following two cases 

192 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  3, pp 187-201 

 

according to the position of yielding along the beam height:  

case (1): when yielding occur in the flange height 

case (2): when yielding occur in the web height 

2.1. case: 1 ℎ
2
≤ ℎ∗ ≤ 𝐻

2 

The first case is when yielding occurs in the flange i.e. h
2
≤ h∗ ≤ H

2
, thus, for this case, Eq.(26) could rewritten as 

follows: 

𝑀𝑚𝑎𝑥 = 2 �∫ 𝜎𝑥𝑒𝑙𝑎𝑠𝑡𝑖𝑐(𝐵 − 𝑏)𝑦𝑑𝑦
ℎ
2
0 + ∫ 𝜎𝑥𝑒𝑙𝑎𝑠𝑡𝑖𝑐 

ℎ∗
ℎ
2

𝐵𝑦𝑑𝑦 + ∫ 𝜎𝑥𝑝𝑙𝑎𝑠𝑡𝑖𝑐
𝐻
2
ℎ∗ 𝐵𝑦𝑑𝑦�    (27) 

Replacing the value of σx in the previous equation with the equivalent values given in Eqs. (18) and (25) 

𝑀𝑚𝑎𝑥 = 2 �∫ 𝐸
(1−𝒱𝒱2) ∗

𝑦2

𝑅𝑜
(𝐵 − 𝑏)𝑑𝑦

ℎ
2
0 + ∫ 𝐸

(1−𝒱𝒱2) ∗
𝑦2

𝑅𝑜

ℎ∗
ℎ
2

𝐵𝑑𝑦 + ∫ 𝐾

�34�
1+𝑛
2

𝑦
Ron

𝑛+1𝐻
2
ℎ∗ 𝐵𝑑𝑦�   (28) 

Where, 

ℎ∗ = 𝑅𝑅𝑜 �
𝐾
𝐸
�

1
1−𝑛 (1 + 𝒱𝒱)          (29) 

Thus,  

𝑀𝑚𝑎𝑥 = 2 � 𝐸𝐵𝑅𝑜2

3(1−𝒱𝒱2)
�𝐾
𝐸
�

3
1−𝑛 (1 + 𝒱𝒱)3 − 𝐸𝑏

3(1−𝒱𝒱2)𝑅𝑜
�ℎ
2
�
3

+ 𝐾

(3 4)⁄ (1+𝑛) 2⁄ (𝑛+2)
� 1
𝑅𝑜
��𝐻

2
�
𝑛+2

− 𝑅𝑅𝑜2 �
𝐾
𝐸
�
𝑛+2
1−𝑛 (1 + 𝒱𝒱)𝑛+2�

            (30) 

While the elastic bending moment could be expressed as; 

𝑀𝐸 = 𝜎𝑥.𝐼𝑁.𝐴
𝑦

           (31) 

Where IN.A is the moment of inertia for the I-section about the neutral axis, which is equal to 

𝐼𝑁.𝐴 = 𝐵𝐻3

12
− 𝑏ℎ3

12
           (32) 

Therefore Eq. (3) could be rewritten as  

𝑀𝐸 = 𝐸
(1−𝒱𝒱2).𝑅𝑜

� 1
12
� (𝐵𝐻3 − 𝑏ℎ3)         (33) 

193 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  3, pp 187-201 

 

Then,  

𝜕𝑀𝐸
𝜕(1 𝑅)⁄ = 𝐸

(1−𝒱𝒱2)
� 1
12
� (𝐵𝐻3 − 𝑏ℎ3)         (34) 

Referring to Eq. 2 which could be rewritten as 

𝑅𝑜
𝑅𝑓

= 1− 𝑀𝑚𝑎𝑥
𝜕𝑀𝐸
𝜕(1 𝑅)⁄

∗ 𝑅𝑅𝑜          (35) 

Replacing in the previous equation the value of bending moment in both Eqs. (30)and Eq. (34), gives 

𝑅𝑜
𝑅𝑓

= 1− 1
1−𝛽𝛼3

� 3�1−𝜈2�

�34�
1+𝑛
2 (𝑛+2)

�𝜎𝑜
𝐸
�
1−𝑛

�2𝑅𝑜
𝐻
�
1−𝑛

− 𝛽𝛼3 + �2𝑅𝑜
𝐻
�
3
�𝜎𝑜
𝐸
�
3

(1 + 𝜈)3 �1− 3(1−𝜈)(1+𝜈)𝑛

�34�
1+𝑛
2 (𝑛+2)

��  (36) 

Where,  

𝛼 = ℎ
𝐻

,   𝛽 = 𝑏
𝐵
          (37) 

Which is nothing but the equation for springback ratio derived for the first case 

2.2. case: 2 0 ≤ ℎ∗ ≤ ℎ
2 

The second case is when yielding occurs in the web i.e. when 0 ≤ h∗ ≤ h
2, thus Eq.(26) could be rewritten as 

follows for this case 

𝑀𝑚𝑎𝑥 = 2 �∫ 𝜎𝑥𝑒𝑙𝑎𝑠𝑡𝑖𝑐(𝐵 − 𝑏)𝑦𝑑𝑦ℎ∗

0 + ∫ 𝜎𝑥𝑝𝑙𝑎𝑠𝑡𝑖𝑐 

ℎ
2
ℎ∗ (𝐵 − 𝑏)𝑦𝑑𝑦+ ∫ 𝜎𝑥𝑝𝑙𝑎𝑠𝑡𝑖𝑐

𝐻
2
ℎ
2

𝐵𝑦𝑑𝑦�   (38) 

Replacing the value of σx in the previous equation with the equivalent values given in Eqs. (18) and (25) 

𝑀𝑚𝑎𝑥 = 2 �∫ 𝐸
(1−𝒱𝒱2) ∗

𝑦2

𝑅𝑜
(𝐵 − 𝑏)𝑑𝑦ℎ∗

0 + ∫ 𝐾

�34�
1+𝑛
2

𝑦𝑛+1

Ron

ℎ
2
ℎ∗ (𝐵 − 𝑏)𝑑𝑦 + ∫ 𝐾

�34�
1+𝑛
2

𝑦
Ron

𝑛+1𝐻
2
ℎ
2

𝐵𝑑𝑦�  (39) 

Therefore the springback ratio could be introduced in the following form 

𝑅𝑜
𝑅𝑓

= 1− 1
1−𝛽𝛼3

� 3�1−𝜈2�

�34�
1+𝑛
2 (𝑛+2)

�𝜎𝑜
𝐸
�
1−𝑛

�2𝑅𝑜
𝐻
�
1−𝑛

(1 − 𝛽𝛼𝑛+2) + �2𝑅𝑜
𝐻
�
3
�𝜎𝑜
𝐸
�
3

(1 + 𝜈)3(1− 𝛽) �1− 3(1−𝜈)(1+𝜈)𝑛

�34�
1+𝑛
2 (𝑛+2)

��

            (40) 

This is the equation for springback ratio derived for the second case 

 

194 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  3, pp 187-201 

 

3. Results and discussion 

For bending I-sections the springback ratio is calculated for the previous two cases in Eqs. (36) and (40). It is 

noticed that both equations have a final term that is very negligible because the Y/E=5.5 x 10-4. Therefore, the 

last term of both equations could be neglected. Thus the springback ratio will be introduced, for the first and 

second case respectively; as the following equations: 

𝑅𝑜
𝑅𝑓

= 1− 1
1−𝛽𝛼3

� 3�1−𝜈2�

�34�
1+𝑛
2 (𝑛+2)

�𝜎𝑜
𝐸
�
1−𝑛

�2𝑅𝑜
𝐻
�
1−𝑛

− 𝛽𝛼3�       (41) 

𝑅𝑜
𝑅𝑓

= 1− 1
1−𝛽𝛼3

� 3�1−𝜈2�

�34�
1+𝑛
2 (𝑛+2)

�𝜎𝑜
𝐸
�
1−𝑛

�2𝑅𝑜
𝐻
�
1−𝑛

(1 − 𝛽𝛼𝑛+2)�      (42) 

This equations (Eqs. (41) and (42)) are depending on the ratio Ro/H, Y/E, the strain exponent (n), the 

geometrical coefficients α, β and the Poisson's ratio (ν). Figures 3-5 shows the variation of springback ratio with 

(Ro/H) and different values of (n) at constant value of α= 0.9 and β= 0.8, with different values of (Y/E): 5.5x10-

4, 1.522x10-3 and 2.4x10-3 (for different material c10100 copper, 1100al, and 1065steel) [11]. 

 

Figure 3:Thespringback ratio with 𝑅𝑅𝑜/𝐻With different values of n at Y/E=5.5*10-4,𝒱𝒱 = 0.33 

The preceding figures showed that: decreasing of the springback ratio (Ro/Rf) occurs with increasing values of 

(n) (strain hardening exponent) as well as increasing of the values of (Y/E). Figures 6-9 show the variation of 

springback ratio with (Ro/H) and different values of (Y/E) at constant value of α= 0.9 and β= 0.8 with different 

values of n (strain hardening exponent).It addition, It was noticed that the springback ratio (Ro/Rf) decreases 

with the increasing of the value of (Y/E) as well as the value of (n) (strain hardening exponent).Whereas figures 

10-12 show the variation of the springback ratio (Ro/Rf) at constant value of α= 0.9 and β= 0.8with different 

 

195 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  3, pp 187-201 

 

values of Poisson's ratio (ν) and at fixed value of (Y/E) (5.5x10-4, 1.522x10-3 and 2.4x10-3). It was confirmed 

that the springback ratio (Ro/Rf) increases with the increasing of the value of (ν). Figures 12-15 show the 

variation of the springback ratio (Ro/Rf) with different values of (H) (the height of the beam) as well as different 

values of n=0.1, 0.2, 0.3 and 0.4 and Ro= 40, β= 0.8, ν=0.33.It could be noticed that the rate of the springback 

ratio is increasing rapidly from range 1 to 2, then it becomes gradually less after wards. 

 

 

Figure 4: The springback ratio with 𝑅𝑅𝑜/𝐻With different values of n at Y/E=1.5*10-3,𝒱𝒱 = 0.33 

 

Figure 5: The springback ratio with 𝑅𝑅𝑜/𝐻With different values of n at Y/E=2.4*10-3,𝒱𝒱 = 0.33 

 

 

196 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  3, pp 187-201 

 

 

Figure 6: The springback ratio with 𝑅𝑅𝑜/𝐻With different values of Y/E at n=0.1,𝒱𝒱 = 0.33 

 

Figure 7: The springback ratio with 𝑅𝑅𝑜/𝐻With different values of Y/E at n=0.2,𝒱𝒱 = 0.33 

 

Figure 8: The springback ratio with 𝑅𝑅𝑜/𝐻With different values of Y/E at n=0.3,𝒱𝒱 = 0.33 

 

 

 

197 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  3, pp 187-201 

 

 

Figure 9: The springback ratio with 𝑅𝑅𝑜/𝐻With different values of Y/E at n=0.4,𝒱𝒱 = 0.33 

 

Figure 10: The springback ratio with 𝑅𝑅𝑜/𝐻With different values of ν at Y/E=5.5*10-4 

 

Figure 11: The springback ratio with 𝑅𝑅𝑜/𝐻With different values of ν at Y/E=1.5*10-3 

 

 

 

198 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  3, pp 187-201 

 

 

Figure 12: The springback ratio with 𝑅𝑅𝑜/𝐻With different values of ν at Y/E=2.4*10-3 

 

Figure 13: The springback ratio with 𝑅𝑅𝑜/𝐻With different values of n at Y/E=5.5*10-4 

 

Figure 14: The springback ratio with 𝑅𝑅𝑜/𝐻With different values of n at Y/E=1.5*10-3 

 

 

 

199 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  3, pp 187-201 

 

 

Figure 15: The springback ratio with 𝑅𝑅𝑜/𝐻With different values of n at Y/E=2.4*10-3 

4. Conclusions 

The springback equations were derived for I-section beams taking into consideration the Tresca criteria. The 

Springback ratio for 𝑅𝑅𝑜𝑜/H less than 20, is assumed to be slightly insignificant but for high values of 𝑅𝑅𝑜𝑜/𝑡𝑡or more 

than 20, the differences in springback ratio is significant. Moreover, the following conclusions about the 

springback analysis in I-section beams using nonlinear constitutive equation are discovered: 

(i) The theoretical analysis for the I-sections under pure bending has been done, and it was found that the 

prediction of springback was quite successful. 

(ii) Springback ratio increases with decreasing the ratio of the yield point stress to Young’s modulus of 

elasticity 

(iii) Springback ratio increases with increasing Possion’s  ratio. 

(iv) Springback ratio increases with increasing beam height 

(v) The elastic recovery is found to be more effective with decreasing values of work hardening 

coefficient. While with lower values of n, the material will approach an elasto-ideal plastic behavior. 

It is also recommended to investigate the springback behavior in bending I-sections with another criteria such as 

von Mises criteria.  

References  

[1] Y.H.Moon, S.S. Kang, J.R. Cho, T.G. Kim."Effect of tool temperature on the reduction of the springback of 

aluminum sheets" Journal of Materials Processing Technology 132 (2003) 365–368. 

[2] TianjiaoLiua, YongjunWanga, JianjunWua, XiaojiaoXiaa, JunbiaoWanga, Wei Wangb and Shunhong 

 

200 
 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No  3, pp 187-201 

 

Wang."Springback analysis of Z & T-section 2196-T8511 and 2099-T83 Al–Lialloys extrusions in displacement 

controlled cold stretch bending" Journal of Materials Processing Technology 225 (2015) 295–309. 

[3] S B Chikalthankar, G D Belurkar and V M Nandedkar."Factors Affecting on Springback in Sheet Metal 

Bending: A Review" International Journal of Engineering and Advanced Technology (IJEAT), Volume-3, 

Issue-4, April 2014. 

[4] A. El-Megharbel, A. El-Domiaty, M. Shaker: "Springback and residual stresses after stretch bending of 

workhardening sheet metal". Journal of Materials Processing Technology 12/1990; 24(1):191-200. 

[5] A. El Megharbel, G.A. El Nasser, Aly El Domiaty. "Bending of tube and section made of strain-hardening 

materials". Journal of Materials Processing Technology 07/2008; 203(1-3):372-380. 

[6] J. Yanagimoto and K. Oyamada, "Mechanism of Springback-Free Bending of High-Strength Steel Sheets 

under Warm Forming Conditions", Annals of the CIRP Vol. 56/1/2007. 

[7] Z.Damián-Noriega, R. Pérez-Moreno, S.A. Villanueva-Pruneda, V.M. Domínguez-Hernández, J.P.A. 

Puerta-Huerta and C. Huerta-Muñoz."A New Equation to Determine the Springback in the Bending Process of 

Metallic Sheet", 2008 ,ICCES, vol.8, no.1, pp.25-30. 

[8] Wenjuan Liu, Zhiyong Liang, Tao Huang, Yuanbin Chen and  JinpingLian, "Process Optimal Control of 

Sheet Metal Forming Springback Based on Evolutionary Strategy", Proceedings of the 7th World Congress on 

Intelligent Control and Automation June 25 - 27, 2008, Chongqing, China. 

[9] FengRuan, Yang Feng and Wenjuan Liu."Springback Prediction for Complex Sheet Metal Forming Parts 

Based on Genetic Neural Network", Second International Symposium on Intelligent Information Technology 

Application, 2008 IEEE. 

[10] ZHAO Jun, YIN Jing, MA Rui and MA LiXia, "Springback equation of small curvature plane bending", 

Science China Press and Springer-Verlag Berlin Heidelberg, September 2011 Vol.54. 

[11] Sanjay Kumar Patel, Radha Krishna Lal, J. P. Dwivedi, and V. P. Singh."Springback Analysis in Sheet 

Metal Forming Using Modified Ludwik Stress-Strain Relation", Hindawi Publishing Corporation ISRN 

Mechanical Engineering Volume 2013, Article ID 640958, 11 pages. 

201