1. For shear area to find shear stress, if you punch a hole into a material, you multiply the circumference by the thickness of the material. What if you don't fully punch a hole into the material, or is that not related to shear stress?

2. I'm looking at an example question: "A rectangular plate for a truss bridge is to be punched from 10mm sheet steel with a force of 60kN. During the punching operation two 25mm diameter holes are also punched. Find the shear stress if the plate has dimensions 150mm by 120mm."

The answers took the total perimeter (including the sheet and the holes punched into: 2(0.15) + 2(0.12) + 2(pi)(0.025) = 0.697m), which was then used to find the shear area (= total perimeter x thickness = 0.697 x 0.01)

My question is; why? Why did they take the total perimeter and not just the circumference of the circles together x the thickness of the material? Shear area is usually found by the circumference of the circle x the thickness of the material. I don't get it.

3. For stress/strain graphs - what's the difference between definite yield points and progressive yield. They look different but why? And what's a yield point and progressive yield mean?

4. What's the difference between Hooke's Law and Young's Modulus? I feel like they say the same thing but Young's Modulus is more sophisticated and detailed.

Sorry for the amount of questions! I have just been confused with some things.

Thank you very much!

Sorry for the very late reply, but I'll give those questions my best shot.

So,

1. Shear stress is calculated using:

**σ = P/A**, where σ = stress (Pa), P = load (N) and A = cross sectional area (m^2)

I don't know if you can punch half a hole and still use this equation because it depends what happens to the cut out - does it crumple, is it pushed all the way through, does it require any extra force etc. Possibly you just multiply it by the thickness that it goes through with the punch (if that makes sense) since it uses the area sheared. Not too sure, but I doubt you need it for HSC.

2. The formula mentioned above, σ = P/A, requires the shear area as you have said. Shear area is just the area that the hole punch has to actually cut through. So, like your example (which I also found in the Copeland textbook

) you must find the total area that was punched through. This can be found by multiplying the thickness by the total surface perimeter that was cut through. Then you can use the calculations you have, and the given values to find that Shear Area = 0.697 x 0.01 = 6.97 x 10^-3

and then the Shear Stress = P/A = 60000/(6.97 x 10^-3) = 8.61 MPa

I think you might just be confused, the hole puncher actually cuts out the rectangular plate and the two circles AT THE SAME TIME.

3.

*Yield point*/yield stress is where there is a marked increase in strain without a corresponding increase in stress. In mild steel, for example, there are definite yield points, but most materials show progressive yield. This point will always be more than the elastic limit, but less than UTS (although often not by much). On a stress-strain diagram, the point will be at the top of a jagged downward movement at the end of the constant rate bit (where is undergoing elastic deformation. Also, after this point it will begin to work harden.

*Progressive Yield* is the same thing, but just describes when it has no distinct point where is passes the yield point. On the graph, this looks like a smooth continuation from the straight line, but it begins to curve slightly, slowly curving increasingly.

*Proof Stress* measures the progressive yield of a material. This is the amount of stress required to cause a permanent strain in the material (ie. past the yield point). Normally it'll give you something like find the proof stress of 0.2% strain in the material. Then you've got to find 0.2% strain past the yield point, and match it with the corresponding stress.

4. Hooke's Law says: "Stress is proportional to strain up to the elastic limit." Therefore, any increase in stress causes a proportional increase in strain. It can be represented by this equation:

**E = σ/ε**, where E = constant, σ = stress (Pa) and ε = strain (a ratio).

It just so happens that this constant, E, is called the

*modulus of stiffness*, or

*Young's Modulus* which relates to elasticity.

Also, E = PL/eA, which you can look up... but it's basically just replacing stress and strain with their respective equations.

Hope that helps