The length of the element can be expressed in terms of the tool radius and the angle subtended, i.e.
ds = R dθ
and the surface area for a unit width of sheet is R
dθ 1 The force acting on the element radially outward is
pR dθ
The force tangential to the sheet due to friction is μpR
dθ The tension forces are T
1
and T
1
+ dT
1
. As the direction of the tension forces differs by an angle, dθ , there is a radially inward component of force, T
1
dθ , as shown in Figure 4.5d. The equilibrium equation for forces in the radial direction is
T
1
dθ = pR dθ
or p
= T
1
R 4.11
It is useful at this stage to re-arrange Equation 4.11. Recalling that T
1
= σ
1
t , the contact
pressure is p
= σ
1
Rt 4.12
The contact pressure as shown in Equation 4.12 is inversely proportional to the bend ratio Rt
. The radius of curvature of the punch face is likely to be several orders of magnitude greater than the thickness and even at most corner radii, it will be 5 to 10 times the
thickness. The principal stress σ
1
is at most only 15 greater that the flow stress σ
f
, and so the contact pressure will be a small fraction of the flow stress, justifying the assumption
of plane stress, except at very small radii in the tooling. The equilibrium condition for forces along the sheet is, from Figure 4.5,
T
1
+ dT
1
− T
1
= μp1R dθ or, combining the above equations,
dT
1
T
1
= μ dθ 4.13a
From Equation 4.12, the contact pressure depends on the radius ratio, but the change in tension as given in Equation 4.13a is independent of curvature and is a function of the
coefficient of friction and the angle turned through, sometimes called the angle of wrap. If the tension at one point, j , in the section is known, then the tension at some other
point, k, can be found by integrating Equation 4.13a, i.e.
T
1k
T
1j
dT
1
T
1
=
θ
j k
μ dθ
50 Mechanics of Sheet Metal Forming
or T
1k
= T
1j
exp μθ
j k
4.13b where, θ
j k
, is the angle turned through between the two points. Care must be taken in using this relation to ensure that the material is sliding in the same direction everywhere
between the two points and that there is no point of inflection in the surface profile.
4.2.6 Force equilibrium at the blank-holder and punch
At the region EF in Figure 4.3, the sheet is clamped between two flat surfaces by the blank- holder force. The force is expressed in terms of a force per unit length, B, as shown. A
friction force, μB, acts on each side of the sheet and hence the equilibrium condition, as shown in Figure 4.6a is,
T
1E
= 2μB 4.14
B
B m
B m
B T
1E
E F
S T
1E
T
1
a b
Figure 4.6 Equilibrium of the sheet under the blank-holder.
It is sufficiently accurate to assume that the tension will fall off linearly over the distance, EF, as shown in Figure 4.6b.
As already mentioned, there is frequently a draw-bead instead of flat faces at a region such as EF. The effect will be similar in that the tension force will increase sharply over
this region. The mechanics of draw-beads is given in a subsequent chapter, but in die design, a step change in tension determined from experience is often used to model the
draw-bead action in an overall analysis of the process.
4.2.7 The punch force
The force acting on the punch, as shown in Figure 4.7, is in equilibrium with the tension in the side-wall. The angle of the side-wall θ
B
can be obtained from the geometry. The vertical component of this tension force is T
1B
sin θ
B
and therefore the punch force per unit width, considering both sides of the sheet, is
F = 2T
1B
sin θ
B
4.15 Simplified stamping analysis 51
q
B
q
B
O A
B F
C T
1B
T
1B
sin q
B
Figure 4.7 Diagram showing the relation between punch force and side-wall tension.
4.2.8 Tension distribution over the section
It is now possible to determine the tension at each point along a strip as illustrated in Figure 4.8. If the strain at the mid-point ε
1O
is known or specified, the centre-line tension T
1O
can be calculated from Equation 4.8. As the sheet between O and B is sliding outwards against an opposing friction force from B to O, the tension in the sheet will increase. The
angle of wrap θ
B
can be determined from the punch depth h and the tool geometry. The tension at B can be found from Equation 4.13b, i.e.
T
1B
= T
1O
expμ.θ
B
O q
B
A B D
F G B
E B
C a
C O
A B D E F G
T
1
s b
Figure 4.8 Distribution of tension forces across the sheet in a draw die.
In the side-wall, between B and C, the sheet is not in contact with the tooling and the tension is constant, i.e. T
1C
= T
1B
. If the surface of the sheet under the blank-holder is horizontal as shown, the angle turned through between C and D will be the same as θ
B
and hence the tension at D, and also at E, will be equal to that at the centre-line, i.e. T
1D
= T
1E
= T
1O
. From E to F, the tension falls to zero as indicated in Section 4.2.6. 52 Mechanics of Sheet Metal Forming
