1. ME 612 Metal Forming and Theory of Plasticity
17. Sheet Metal Bending
Assoc.Prof.Dr. Ahmet Zafer Şenalp
Mechanical Engineering Department
Gebze Technical University

2. 17.1. Elastic Plane Strain Bending
17. Sheet Metal Bending
Figure Coordinate System for Analysis of Bending
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

3. Mechanical Engineering Department, GTU
17.1. Elastic Plane Strain Bending
17. Sheet Metal Bending
For elastic bending from plane strain asumption elastic strain in y direction is; And when thickness is small the stress in z direction is; Using the above is obtained. Let r be the radius of curvature measured to the mid-plane and z the distance of an element from the mid-plane then elastic strain in x direction is; (17.4) true strain in x direction is;
(17.1)
(17.2)
(17.3)
(17.5)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

4. Mechanical Engineering Department, GTU
17.1. Elastic Plane Strain Bending
17. Sheet Metal Bending
For small strains: But: Placing Eq (17.2) into the above eq.; is obtained. In this case; and E’ is defined as the plane strain elastic modulus.
(17.6)
(17.7)
(17.8)
(17.9)
(17.10)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

5. Mechanical Engineering Department, GTU
17.2. Springback in Sheet Bending
17. Sheet Metal Bending
Here ideal elastic plastic material assumpttion is made. (Yield stregth; Y=constant). For plane strain y= 0. Yield criteria yeids With the sheet metal assumption is obtained. If the above equality is placed in the von Mises equivalent stress equation; is obtained. Thus from yield condition; is obtained.
(17.11)
(17.12)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

6. Mechanical Engineering Department, GTU
17.2. Springback in Sheet Bending
17. Sheet Metal Bending
Figure Strain and stress distribution across sheet thickness. Bending strain (a) varies linearly across the section. For the non-work hardening stress-strain relation (b), the bending moment causes the stress distribution in (c). Elastic unloading after removal of the loads results in the residual stresses shown in (d).
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

7. Mechanical Engineering Department, GTU
17.2. Springback in Sheet Bending
17. Sheet Metal Bending
In the Figure 17.2 (c) in order to form stress distribution we need to calculate bending moment; M. Thus: For ideal plastic material with the assumption of neglectable central elastic deformation; Here is defined to simplfy the formulation. According to this; (17.15) term is obtained.
(17.13)
(17.14)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

8. Mechanical Engineering Department, GTU
17.2. Springback in Sheet Bending
17. Sheet Metal Bending
Example 1:
A steel sheet, inches thick, is bent to a radius of curvature of 5.0 inches. The flow stress; Y=33*103 psi (i.e. o=33*103 psi ).E’= 33*106 psi.
What fraction of the cross section remains elastic?
What percent error does neglecting the elastic core cause in the calculation of the bending moment?
Solution :
1. The elastic strain at yielding is ex=0 /E,
where E’ is the plane-strain modulus, E/(1-n2).
The limit of the elastic core will be at z = r ex = r 0 / E’ .
E’= 33*106 psi,
z= 5*33*103 /33*106= 0,005 inch. The elastic fraction is 2*0.005/0.036 =0.28 or 28 %.
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

9. Mechanical Engineering Department, GTU
17.2. Springback in Sheet Bending
17. Sheet Metal Bending
2. To calculate the bending moment, for the elastic portion (0 ≤ z ≤ 0.005) x = εxE’=zE’/r and for the plastic portion( ≤ z ≤ ) , sx = s0 and bending moment; Using the equation which neglects the elastic core, Here, The error is ( – 10,.42 ) / = or 2.6 %.
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

10. Mechanical Engineering Department, GTU
17.2. Springback in Sheet Bending
17. Sheet Metal Bending
The external moment applied by the tools and the internal moment resisting bending must be equal, so Eq. (17.15) applies to both. When the external moment is released, the internal moment must also vanish. As the material unbends (springs back) elastically, the internal stress distribution results in a zero bending moment. Since the unloading is elastic, M+∆M=0 (17.16) Since the unloading is elastic, ∆x =E’ ∆ex (17.17) The change in strain is given by where r’ is the radius of curvature after springback. This causes a change in bending moment, DM of
(17.18)
(17.19)
(17.20)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

11. Mechanical Engineering Department, GTU
17.2. Springback in Sheet Bending
17. Sheet Metal Bending
Since after springback, if Eq (17.15) and Eq (17.20) are equated: or: The resulting residual stress, This is plotted in Figure (17.2.d). Note that on the outside surface where z=t/2 the residual stress is compressive, ’x = -o/2 dir. and on the inside surface z=-t/2 it is tensile ’x = + o/2.
(17.21)
(17.22)
(17.23)
(17.24)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

12. Mechanical Engineering Department, GTU
17.2. Springback in Sheet Bending
17. Sheet Metal Bending
A similar development can be made for a work-hardening material. If then, where dir. Finally, Since ∆M is still described as before and after spring back,
(17.25)
(17.26)
(17.27)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

13. Mechanical Engineering Department, GTU
17.2. Springback in Sheet Bending
17. Sheet Metal Bending
The variations of x, ∆x, and ’x through the section are shown in Figure The magnitude of the spring back predicted can be very large.
Figure Stress distribution under bending moment and after unloading for a work-hardening material.
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

14. Mechanical Engineering Department, GTU
17.2. Springback in Sheet Bending
17. Sheet Metal Bending
Example 2: Find the tool radius necessary to produce a final bend radius of r’=10 in. in a part made from a steel of thickness 0.03 inches. Assume a yield stress of psi (0= psi) . Solution : , t=0.03, psi, Using above equation: r=4.2 in. Note: The springback problem is actually greater, since at a bend radius of 4.2 inches, the elastic core is z=r0/E’=4.2x45x103/33x106= in. i.e., 38% of the cross section. This introduces 5 error in the moment value.
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

15. Mechanical Engineering Department, GTU
17.3. Bending with Superimposed Tension
17. Sheet Metal Bending
Such allowances for spring back would cause severe problems in tool design, but fortunately there is a relatively simple solution. Often, as in stretch forming, the tooling does not apply a pure bending moment as assumed above. Rather, tension is applied simultaneously with bending. With increasing tensile forces, Fx , the neutral plane shifts towards the inside of the bend and in most operations, this tension is sufficient to move the neutral plane completely out of the sheet so that the entire cross section yields in tension. For such a case, the strain and stress distributions are sketched in Figure Figure 17.4.
Figure Bending with superimposed tension. With sufficient tension, the neutral axis moves out of the sheet so the strain is tensile across the entire section, (a). With the stress-strain curve shown in (b), the stress distribution in (c) results. After removal of the moment, elastic unloading leaves very minor residual stresses, as shown in (d).
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

16. Mechanical Engineering Department, GTU
17.4. Sheet Bendability
17. Sheet Metal Bending
If bend radius is too sharp, excessive tensile strain on outside surface can cause cracking, while buckling can occur on the inside surface. The limiting values of r/t have been shown to be function of the tensile ductility (% of elongation at fracture or % of area reduction at fracture). where R = inside radius of curvature (i.e. R/t=r/t-1/2), And ( ).
(17.28)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

17. Mechanical Engineering Department, GTU
17.4. Sheet Bendability
17. Sheet Metal Bending
The above correlation is not accurate for sharp bends (low r/t), because the neutral axis shifts from the mid-plane and the amount of shift depends upon the applied tension and the frictional conditions. With tight bends (small r/h) the neutral axis shifts toward the inside; there are several reasons for this.
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

18. Mechanical Engineering Department, GTU
17.4. Sheet Bendability
17. Sheet Metal Bending
Figure Correlation of limiting bend severity, R/t, with tensile ductility
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

19. Mechanical Engineering Department, GTU
17.4. Sheet Bendability
17. Sheet Metal Bending
The cross section at the inside will increase while the outside decreases and the magnitude of the true strain (and hence the flow stress in a work-hardening material) increases faster with z in compression than tension. As a consequence, the neutral axis moves inward to compensate for the higher stresses and greater cross section. In non-symmetric sections, transition from elastic to plastic flow will not occur simultaneously on both sides of the bend and, consequently, as yielding starts, there will be a shift of the neutral axis toward the heavier sections.
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU