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Theoretical Considerations

Here is an example of studying sheet metal roll forming.

Metal sheets and plates are produced by rolling a long workpiece through a set of rolls.

The curved surface of contact makes the computation of forces and stress distribution involved in the rolling process very complicated.

Using the slab method of analysis for plane strain, the stresses in rolling can be analyzed as follows.

Let ho be the thickness of the strip entering the roll gap
and hf be the reduced thickness of the strip.

As in the case of a fluid flowing through a converging channel, the volume rate of metal flows is kept constant by increasing the velocity of the strip as it moves through the roll gap.

Let Vr be the surface speed of the roll, Vo and Vf be the velocities of the strip at the entrance and at the exit, respectively, of the roll gap. As Vr is constant while the velocity of the strip changes, sliding occurs between the roll and the strip.

The point of the arc of contact at which the velocity of the strip is equal to the velocity of the roll, Vr, is called the neutral point or the no-slip point. The neutral point sets the boundary between the region of the workpiece where the roll moves faster than the workpiece and the region where the workpiece moves faster than the roll.

The frictional forces acting on the strip surface are greater in the region where the roll moves faster than the workpiece. The difference between the frictional forces in the two regions produces a net frictional force that pulls the strip into the roll gap.

Note that the net frictional force must be in the same direction as the roll velocity so that work is supplied to the workpiece. Hence, the neutral point should be toward the exit.

A measure of the relative velocities involved is defined as the forward slip

Roll Pressure Distribution

The stresses acting on an element in the entry and exit zones, respectively, are shown in the figure below:

Stresses on an element in Rolling

The balance of horizontal forces in the entry zone gives:

( + d)(h + dh) - 2pR d sin - h + 2 µ p R d cos = 0

Where R is the radius of the roller. The balance of horizontal forces in the exit zone gives:

( + d)(h + dh) - 2pR d sin - h + 2 µ p R d cos = 0

Neglecting second order terms and rearranging, the above two equations become, respectively:

d( h) / d = 2pR ( - )	(1)
d( h) / d = 2pR ( + )	(2)

Since the angles involved are small, p is assumed to be a principal stress. Hence, the relationship between the two principal stresses and the flow stress Yf of the material follows:

p - = [2 / (3) .5 ] Yf = Yf'

Note that the flow stress Yf in the above equations corresponds to the strain that the material has undergone at that particular location in the roll gap. Also, note that as the thickness of the material decreases, Yf increases due to cold working.

Substituting for as a function of p and Yf' in equations (1) and (2), and after some manipulations, we get the following differential equations for p / Yf':

d( p / Yf' ) / d = ( p / Yf' ) (2 R / h )( - )	(3)
d( p / Yf' ) / d = ( p / Yf') ( 2 R / h ) ( + )	(4)

If hf is the final thickness of the workpiece, h can be written as:

h = hf +2R(1- cos )

or

h = hf + R 2

Substituting for h in equations (3) and (4), and solving for p:

p = C Yf' (h /R) e- H

and

p = C Yf' (h /R) e H

Where H is given by:

H = 2 ( R / h ).5 Tan -1[ ( R / h ).5 ]

The Boundary conditions are:

At entry,	= a	H = Ho
At exit,	= 0	H = Hf =0

Hence,

In the entry zone,	C = (R / hf) e Ho	p = Yf' ( h/ho )e-(Ho-H)
In the exit zone,	C = R/hf	p = Yf' ( h/hf )eH

From the above expressions for p, note that it increases with increasing strength of the material, increasing coefficient of friction, and increasing R / hf ratio.