SME
Manufacturing Processes · Problem type
Flow stress and forming force
Use this for “Steel with K = 275 MPa, n = 0.15 is rolled to ε = 0.223 over 300 × 35.4 mm at 50 rpm. Find the roll force and power.”
Diagram
σ
=
Kεⁿ:
at
ε
=
0.2231
the
metal
flows
at
219.6
MPa.
Ȳf
=
Kεⁿ
1 + n
=
191
MPa:
the
rectangle
with
the
same
area
as
the
shaded
work
under
the
curve.
F = Ȳf × w × L = 2.025 MN.
This is an example. Type your own values to start a new problem.
Ȳf (MPa)
=
K (MPa)
×
ε
1
+
F (MN)
=
Ȳf (MPa)
×
w (mm)
×
L (mm)
N
Roll speed
example · rpm
P
Roll power
calculated
Formulas
P
=
2
×
π
×
N
×
F
×
L
60,000,000
374.9319
=
2
×
π
×
50
×
2,025,343.851
×
35.3553
60,000,000
Ȳf
=
K
×
ε
n
1 +
n
190.9512
=
275
×
0.2231
0.15
1 + 0.15
F
=
Ȳf
×
w
×
L
2,025,343.851 = 190.9512 × 300 × 35.3553
Formulas are worked in N. The step-by-step shows the conversions.
Assumptions
•
The metal is formed cold and follows the flow curve σ = Kεⁿ, with no strain-rate effect.
•
The average flow stress over the pass is Kεⁿ ÷ (1 + n), the curve averaged from 0 to ε.
•
The force is that stress over the contact area wL, with friction and roll flattening left out.
•
Power for both rolls: P = 2πNFL, the force acting at half the contact length on each roll.
Step-by-step
WE KNOW
K = 275 MPa, n = 0.15, ε = 0.2231, w = 300 mm, L = 35.3553 mm, N = 50 rpm
FIND
average flow stress (Ȳf), forming force (F), roll power (P)
Step 1 · Find average flow stress (Ȳf)
Use Ȳf = K × εⁿ ÷ (1 + n)
The flow curve averaged over the strain the pass gives.
Ȳf
=
275
×
0.22314355
0.15
1 + 0.15
Ȳf
=
275
×
0.2231
0.15
1.15
Ȳf
=
275
×
0.79852
1.15
Ȳf
=
219.593
1.15
Ȳf = 190.9512 MPa
Step 2 · Find forming force (F)
Use F = Ȳf × w × L
The
average
flow
stress
times
the
area
of
roll
in
contact
(MPa
×
mm²
=
N
).
F
=
190.9512
×
300
×
35.3553
F
=
57,285.36
×
35.3553
F
=
2,025,343.851
N
Step 3 · Find roll power (P)
Use P = 2 × π × N × F × L ÷ 60,000,000
Each
roll’s
torque
is
F
at
half
the
contact
length;
two
rolls
at
2π
N
60
rad/s
(÷
1000
for
m,
÷
1000
for
kW).
P
=
2
×
π
×
50
×
2,025,343.9
×
35.355339
60,000,000
P
=
314.2
×
2,025,343.9
×
35.3553
60,000,000
P
=
636,400,000
×
35.3553
60,000,000
P
=
22,500,000,000
60,000,000
P
=
374.9319
kW
Step 4 · Convert the answers
Change each answer back to the unit you chose.
F = 2,025,343.851 N = 2.0253439 MN (1 MN = 10⁶ N)
Check
Put the numbers back in. Both sides should match.
374.9319
=
2
×
π
×
50
×
2,025,343.9
×
35.355339
60,000,000
✓
190.9512
=
275
×
0.22314355
0.15
1 + 0.15
✓
2,025,343.851 = 190.9512 × 300 × 35.3553 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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