How to determine gradient of load/extension graph?

Question

Here's the excercise: A tensile test is carried out on a mild steel specimen of gauge length 40mm and cross-sectional area 100mm². The results obtained for the specimen up to it's yield point are given below:

Load (kN)     | 0 |   8   |  19   |  29   |  36
Extension(mm) | 0 | 0.015 | 0.038 | 0.060 | 0.072

I need to determine the gradient of the load / extension graph. When the data is plotted (in the book) on a graph it gives a straight line therefore the gradient should be any load (of the given values) divided by the corresponding extension gradient = load/(extension/1000) except e.g. the first one (8kN) gives 533*10⁶ while the third one (19kN) gives 500*10⁶.

Why and which one do I use for the calculations?

Answer 1

Theoretically, the elastic range of a material is linear. But as Yogi Berra said, "In theory there is no difference between theory and practice. In practice there is." This data is obtained from a physical experiment, so errors are natural.

Perhaps the piece or the strain gauge was incorrectly placed. Or, even more likely (given the data), you were dealing with a specimen which simply wasn't perfect (none ever is). Slight microimperfections meant that even in the elastic regime you might have a result which isn't perfectly linear.

So you need to find a best estimate of what the "theoretical" Young's modulus is.

Here are the results for each of the points, take your pick:

disp (mm) | F (kN)  | F/d (kN/mm)
----------+---------+------------
0         | 0       | -
0.015     | 8       | 533.3
0.038     | 19      | 500
0.060     | 29      | 483.3
0.072     | 36      | 500

It is visually clear that the result is hovering around 500.

I'm unsure of how serious the statistical analysis needs to be.

• You can simply say 500 looks reasonable.
• You can take an average, which gives you 504
• You can do a least-squares analysis, which gives you 490.6, but with a line crossing the y-intercept at 0.25, as opposed to zero, as would be expected.
• You can force the least-squares to cross through zero, which gives you 495.

They all give you the following results: