Problem

A circular bar of magnesium alloy is 750 mm long. The stress-strain diagram for the material is shown in the figure. The bar is loaded in tension to an elongation of 6.0 mm, and then the load is removed.

(a) What is the permanent set of the bar?

(b) If the bar is reloaded, what is the proportional limit?

(Hint: Use the concepts illustrated in Figs. 1-36b and 1-37.)

Step-by-step solution

1. Step 1 of 4

   Calculate the total strain at point B by using the following relation.

   

   Here,  is the strain at point B,  is the elongation of the bar, and l is the original length of the bar.

   Substitute 6 mm for  and 750 mm for l.

   

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2. Step 2 of 4

   Draw the line from point B parallel to the tangent at point O, which meets at point C.

   

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3. Step 3 of 4

   (a)

   Obtain the residual strain in the bar from the stress strain diagram after removal of load.

   

   Here,  is the residual strain in the bar.

   Calculate the permanent set of the bar.

   

   Here,  is the residual elongation of the bar.

   Substitute 0.0056 for  and 750 mm for l.

   

   Therefore, the residual elongation or permanent set of the bar is, .

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4. Step 4 of 4

   (b)

   Calculate the proportional limit, if the bar is reloaded.

   From the stress-strain diagram, the proportional limit of the bar after reload is the yield stress at B.

   

   Therefore, the proportional limit of the bar due to reload is, .

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