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Shear force Analysis:

The first process for obtaining the forces withheld by the die is to figure out how much pressure is actually being used by the press in order to press the copper sample. The press is capable of five hundred tons. In order to obtain this, material properties of the sample (copper) had to be studied. Shown in Table 4.2 is the approximate resistance to shear of copper.


Figure 4.1: Material resistance to shear property chart recorded from Marks Standard Handbook. [3]

From this value of 28000lb/in2 resistance to shear, a press force can be obtained by using the equation shown below:
      [1]   (2)
where:
t = sample thickness (in.)
D = sample diameter (in.)
sCU = resistance to shear of Cu (psi.)

Using Equation 1, assuming D = XXX and t = YYY, a force of 49480 lbf was calculated, which is approximately twenty-five tons.

Force applied at 45 degrees turn.

The concern now is to find the force applied at the forty-five degree turn, which will experience the bulk loading of the press force. In order to calculate this, a free body force diagram was drawn of the channel, which is shown below in Figure 4.2.



Figure 4.2: Force Diagram of the ECAP press channel

Equation for the sum of the forces in the x & y directions were then derived from the figure 2 shown above. The two force equations are shown below:

                (2)
                          (3)

Where “Vs” is the shear force being caused by the press, “P” is the force caused by the press, and “N’ is the normal force caused by the die reacting opposite from the forces applied. Assuming that the sum of the forces in the x & y direction are equal to zero, the shear force (Vs) can be solved from the force equations 2 and 3 above. A shear force of 37870 lbf was found on die at the forty-five degree angle.

After determining the shear stress a check can be made by using the equation for shear strain. The strain of the sample as it is distorted through the angle is directly proportional to its shear stress. The relationship is shown below in equation 4.

                                     [4]
                          (2)                [5]

Where the change in gamma is the shear strain, “P” is the press force applied, and “Vs” is the shear force caused by the die. A shear force of 37870lbf was calculated. This verifies the sum of the forces method.