B A
A D
B C
C q
D y
t r
Figure 6.2 Deformation of longitudinal fibres in bending and tension.
6.2.1 Geometry and strain in bending
In bending a thin sheet to a bend radius more than three or four times the sheet thickness, it may be assumed that a plane normal section in the sheet will remain plane and normal
and converge on the centre of curvature as shown in Figure 6.2. In general, a line CD
at the middle surface may change its length to CD if, for example, the sheet is stretched during bending; i.e. the original length l
becomes l
s
= ρθ 6.1
A line AB at a distance y from the middle surface will deform to a length
l = θ ρ + y = ρθ
1 +
y ρ
= l
s
1 +
y ρ
6.2 The axial strain of the fibre AB is
ε
1
= ln l
l = ln
l
s
l + ln
1 +
y ρ
= ε
a
+ ε
b
6.3 where ε
a
is the strain at the middle surface or the membrane strain and ε
b
is the bending strain. Where the radius of curvature is large compared with the thickness, the bending
strain can be approximated as, ε
b
= ln 1
+ y
ρ ≈
y ρ
6.4 The strain distribution is approximately linear as illustrated in Figure 6.3.
6.2.2 Plane strain bending
If the flat sheet on either side of the bend in Figure 6.1 is not deforming it will constrain the material in the bend to deform in plane strain; i.e. the strain parallel to the bend will
be zero. In this work, plane strain conditions will be assumed, unless stated otherwise. The deformation process in bending an isotropic sheet is therefore
ε
1
; ε
2
= 0; ε
3
= −ε
1
σ
1
; σ
2
= σ
1
2 ; σ
3
= 0 6.5
Bending of sheet 83
y e
b
e
a
2 t
2 t
Figure 6.3 Assumed strain distribution in bending.
Following Equations 2.18b and 2.19c, for, β = 0, α = 12, we obtain
σ
1
= 2
√ 3
σ
f
= S and
ε
1
= √
3 2
ε 6.6
where S is the plane strain flow stress. Equation 6.6 assumes the von Mises yield condi- tion. If a Tresca yield criterion is assumed, σ
1
= σ
f
= S. The stresses on a section along the bend axis are illustrated in Figure 6.4. Clearly, at the edge of the sheet, the stress
along the bend axis will be zero at the free surface and plane strain will not exist. It is usually observed that the edge of the sheet will curl as illustrated. This happens because
the stress state is approximately uniaxial tension near the edges of the sheet; the minor strain will be negative near the outer surface and positive near the inner surface giving rise
to the anticlastic curvature as shown. Within the bulk of the sheet, however, plane strain deformation is assumed with the minor strain along the axis of the bend equal to zero.
s
1
s
2
, e
2
= 0 Free edge
Figure 6.4 Stress state on a section through the sheet in plane strain bending.
6.3 Equilibrium conditions
We consider a general stress distribution on a normal section through a unit width of sheet in bending, as shown in Figure 6.5. The force acting on a strip of thickness dy across
the unit section is σ
1
× dy × 1. The tension T on the section is in equilibrium with the integral of this force element, i.e.
T =
t 2
−t2
σ
1
dy 6.7
84 Mechanics of Sheet Metal Forming
R
r
y t
2
t
2 s
1 ×
d
y
×
1
M T
dy
a b
Figure 6.5 Equilibrium diagram a for a section through a unit width of sheet and b a typical
stress distribution.
Integrating the moment of the force element, we obtain M
=
t 2
−t2
σ
1
dy1y =
t 2
−t2
σ
1
y dy
6.8 We note too that there is a third equilibrium equation for forces in the radial direction
arising from the tension T . This is given in Section 4.2.5 by Equation 4.11.
6.4 Choice of material model
For the strain distribution given by Equation 6.3, the stress distribution on a section can be determined if a stress strain law is available. In general, the material will have an elastic,
plastic strain-hardening behaviour as shown in Figure 6.6a. In many cases, it is useful to approximate this by a simple law and several examples will be given. The choice of
material model will depend on the magnitude of the strain in the process. The strain will depend mainly on the bend ratio, which is defined as the ratio of the radius of curvature
to sheet thickness, ρt.
6.4.1 Elastic, perfectly plastic model
If the bend ratio is not less than about 50, strain-hardening may not be so important and the material model can be that shown in Figure 6.6b. This has two parts, i.e. if the stress
s
1
e
1
E ′ S
S s
1
= K′e
n 1
a s
1
e
1
b s
1
e
1
c s
1
e
1
d
Figure 6.6 Material models for bending. a An actual stress–strain curve. b An elastic, perfectly
plastic model. c A rigid, perfectly plastic model. d A strain-hardening plastic model.
Bending of sheet 85
is less than the plane strain yield stress, S σ
1
= E
′
ε
1
6.9 where the modulus of elasticity in plane strain is slightly different from the uniaxial
Young’s modulus, E; i.e. E
′
= E
1 − ν
2
6.10 where ν is Poisson’s ratio.
For strains greater than the yield strain, σ
1
= S 6.11
where S is constant. In isotropic materials, S is related to the uniaxial flow stress by Equation 6.6 for the von Mises yield condition.
6.4.2 Rigid, perfectly plastic model
For smaller radius bends, and where we are not concerned with elastic springback, it may be sufficient to neglect both elastic strains and strain-hardening. A rigid, perfectly plastic
model is shown in Figure 6.6c, where
σ
1
= S 6.12
and S is a value averaged over the strain range as indicated in Section 3.5.4.
6.4.3 Strain-hardening model
Where the strains are large, the elastic strains may be neglected and the power law strain- hardening model used, where
σ
1
= K
′
ε
n 1
6.13 For a material having a known effective stress–strain curve of the form
σ = σ
f
= Kε
n
6.14 the strength coefficient K
′
can be calculated using Equations 6.6. This model is illustrated in Figure 6.6d.
6.5 Bending without tension
Where sheet is bent by a pure moment without any tension being applied, the neutral axis will be at the mid-thickness. This kind of bending is examined here for several types of
material behaviour. In these cases, a linear strain distribution as illustrated in Figure 6.3 is assumed and the equilibrium equations, Equations 6.7 and 6.8, will apply.
86 Mechanics of Sheet Metal Forming
