c If in the condition shown, the edge of the sheet just comes to the point F , estimate
approximately the initial semi blank width. d
If, in the position shown, the side wall is about to split, estimate the punch force P and the strain at the centre ε
1
.
Solutions
1. Tension at the centreline T
O
= σ
1
t = Kε
n 1O
t exp
−ε
1O
= 750 × 10
6
× 0.8 × 0.03
0.23
exp −0.03 = 260kNm
2. Arc length sin θ
= a
− R
P
R
f
− R
P
= 0.33
− 0.01 2.8
− 0.01 ,
θ = 0.115rad = 6.6
◦
arc OA = R
f
− R
p
θ = 2.8 − 0.01 × 0.115 = 0.321m
arc AB = R
P
π 2
− θ = 0.01 × π
2 − θ = 0.015m
CB = 0.028
Arc CD = π2 × 0.01 = 0.015m
DF = 0.080m
∴ arc length OF = 0.460m
3. Tensions Using Equation 4.13b
T
A
= T
O
expμθ = 260 exp0.1 × 0.115 = 263kNm
T
B
= T
O
expμπ2 = 263 exp0.1 × 3.142 = 304kNm
T
C
= T
B
T
C
= T
D
expμπ2 = T
D
exp 0.1 × 3.142 = 304kNm
T
D
= T
E
= 308 expμπ2 = 260kNm 4. Blankholder force
From Equation 4.14, we have T
D
= 2μB, ∴B
= T
D
2μ =
260 2
× 0.1 = 1300kN
5. Strains T
A
= Kε
n 1A
t exp
−ε
1A
= 600ε
1A
exp −ε
1A
= 263 ∴ by trial and error, ε
1A
= 0.032 Simplified stamping analysis 55
similarly, ε
1B
= ε
1C
= 0.078 ε
1D
= ε
1E
= 0.032 6. Average strains
OA: 0.031; AB: 0.055; BC: 0.078; CD: 0.055; EF: 0.016 7. Original arc length
OA: L = 0.321 exp−0.031 = 0.311m
AB: L = 0.0157 exp−0.055 = 0.015m
BC: L = 0.028 exp−0.078 = 0.026m
CD: L = 0.0157 exp−0.055 = 0.015m
DE: L = 0.080 exp−0.016 = 0.079m
8. Original blank semi-width b
= L
= 0.446m Overall average strain
= ln 0.4600.446 = 0.031 9. Ultimate side-wall tension
T
ult
= Kt n
n
exp −n = 340kNm
10. Maximum centre-line strain T
0 max
= T
ult
expμπ2 = 290kNm
ε
0.23 10
exp −ε
10
= 290Kt = 0.48
ε
10 max
= 0.05 11. Pressure
p = T R
p =
260 × 10
3
2.8 = 0.093MPa
p
A
= P
D
= 263
× 10
3
0.01 = 26.3MPa = P
D
p
B
= P
C
= 308
× 10
3
0.01 = 31MPa
p
EF
= Barea = 1300
× 10
3
0.08 × 1
= 16.3MPa 56 Mechanics of Sheet Metal Forming
12. Thickness t
= t exp
−ε
1
t
A
= 0.8 exp−0.03 = 0.78mm t
B
= t
C
= 0.74mm t
o
= t
D
= t
E
= 0.78mm
4.3 Stretch and draw ratios in a stamping
In stampings of simple shape, it is often useful to determine the stretch and draw ratios at a section. Figure 4.10 shows the blank and die at the start of the operation. The points D
are the tangent points of the sheet at the die corner radius. As the blank is drawn in, the die will mark the sheet slightly so that the position of the material point that was originally
at D can be seen on the sheet. The line on the stamping indicated by this marking or scratching is called the die impact line and in Figure 4.10b it is indicated by the point
D
′
. The length measured around the sheet, 2d, as shown in Figure 4.10b, is the current length of 2d
and the stretching ratio is defined as SR
= d
− d d
× 100 4.20
a
b 2
b
2 b
2 d
2 d
D′ D′
D D
Figure 4.10 Section of a stamping illustrating the drawing and stretching ratios.
The drawing ratio is DR
= b
− b b
o
× 100 4.21
It is often found that problems will occur in stamping if these ratios change too rapidly with successive sections along the tool.
4.4 Three-dimensional stamping model
A number of stampings resemble the rectangular pan shown in Figure 4.11. In the corners, shown shaded, the material is drawn inwards in converging flow. This mode of deformation
Simplified stamping analysis 57
is considered in a subsequent chapter. The straight sides can be modelled approximately as two-dimensional sections, except that the deformation over the face of the punch is
no longer plane strain, but biaxial stretching with a strain ratio in the range 0 β 1. Useful information can be determined from a simple model and this is demonstrated in
the following worked example.
4.4.1 Worked example Stamping a rectangular panel
A rectangular pan as shown in Figure 4.11 is drawn in a die. The base of the panel is flat and it is specified that the base should be stretched uniformly so that a lip of 4 mm height
should be drawn up around the edge as shown in Figure 4.12. Determine the side-wall tensions T
1
and T
2
and the punch force required for the following conditions: initial sheet thickness
0.90 mm stress strain law
σ = 700 0.009 + ε
0.22
MPa
Figure 4.11
One half of a typical rectangular stamping.
800 400
400 4
1 2
4 200
4 mm
T
1
T
2
Figure 4.12 Diagram of a rectangular pan drawn to a specified strain.
Solution.
For the semi-axes in Figure 4.12b, the sheet is stretched so that ε
1
≈ ln 204
200 = 0.0198
and ε
2
≈ ln 404
400 = 0.010
4.22 the strain ratio is
β =
ε
2
ε
1
= 0.010
0.0198 = 0.505
4.23
58 Mechanics of Sheet Metal Forming
