# Mohr's Circle

Homework Statement
Sketch the element for the stress state indicated and then draw Mohr's circle.

Given: Uniaxial compression, i.e. $$\sigma_{x} = -p$$ MPa

The attempt at a solution

Below I have the sketch and a partially complete Mohr's circle:

[PLAIN]http://img710.imageshack.us/img710/6001/civek.jpg [Broken]

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Any ideas?

vela
Staff Emeritus
Homework Helper
First, tell us what $\sigma_y$ and $\tau_{xy}$ are equal to.

The thing is, they don't provide $$\sigma_{y}$$ or $$\tau_{xy}$$, which is why I was confused.

vela
Staff Emeritus
Homework Helper
Well, the problem says the compression is uniaxial. What does uniaxial mean?

Well, the problem says the compression is uniaxial. What does uniaxial mean?

That would mean having a single axis, so $$\sigma_{y}$$ is not involved here. But how about $$\tau_{xy}$$? I simply assumed it existed, as you can see in my drawing of the Mohr's circle.

vela
Staff Emeritus
Homework Helper
I'd take it to be 0 as well.

I'd take it to be 0 as well.

If $$\tau_{xy} = 0$$ then there wouldn't even be a circle. Would it be a straight line?

vela
Staff Emeritus
Homework Helper
No, you always get a circle. The two points you know are on the circle will be $(\sigma_y,\tau_{xy}) = (0,0)$ and $(\sigma_x,-\tau_{xy})=(-p,0)$. Now you go about the same procedure as before and find the location of the center of the circle, its radius, etc.

So in my sketch I should remove the shear stress arrows?

vela
Staff Emeritus
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Sure, or label them as being equal to 0.

I found the centre to be $$(\frac{-p}{2},0)$$ and the radius to be $$\frac{p}{2}$$. Is this correct.

vela
Staff Emeritus
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Yes, that's correct.

How do I find the line X'Y' since I don't know the angle $$\theta$$?

vela
Staff Emeritus
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What are X, Y, X', and Y' supposed to denote?

What are X, Y, X', and Y' supposed to denote?

These 4 variables are points on the Mohr's circle denoted by:
X:($$\sigma_{x},-\tau_{xy}$$)

Y:($$\sigma_{y},+\tau_{xy}$$)

X':($$\sigma_{x}',-\tau_{xy}'$$)

Y':($$\sigma_{y}',+\tau_{xy}'$$)

There are equations used to solve for X' and Y', but one of the variables is $$\theta$$, which isn't given.

vela
Staff Emeritus
Homework Helper
OK. Did the problem ask you to find the axial and shear stresses for some plane?

OK. Did the problem ask you to find the axial and shear stresses for some plane?

That's part b of the question, which askes me to determine the maximum shear stresses that exist and to identify the planes on which they act by drawing the orientation of the element for these normal stresses.

But actually, $$\theta=0$$ because the angle between the line XY and the x-axis is 0.

vela
Staff Emeritus
Homework Helper
What points on the circle correspond go the orientation when the shear stress is maximized?

What points on the circle correspond go the orientation when the shear stress is maximized?

Would that be the points where the circle is at the highest and lowest in the y direction?

vela
Staff Emeritus
Homework Helper
OK, I'm still not clear on exactly what you're trying to do with (X', Y') and θ.

OK, I'm still not clear on exactly what you're trying to do with (X', Y') and θ.

2θ is what separates the lines XY an X'Y'. I think since θ = 0, there isn't an X'Y' line.

vela
Staff Emeritus
Homework Helper
When you draw Mohr's circle, typically you start with the axial and shear stresses for a given orientation of the element, so you know where the points X and Y lie on the diagram. Where X' and Y' lie depend on what you're trying to find. For instance, if you're interested in the principal axes, you'd choose to have X'Y' lie on the horizontal axis. If you wanted to find where the shear stress is maximized, you'd choose X'Y' so that it was vertical.

For part c of the question, it asks me to sketch the element for the stress state and draw the Mohr's circle for the case of a biaxial compressive stress, i.e., $$\sigma_{x}=\sigma_{y}=-p$$ MPa.

I found the radius to be 0. Does that mean it's simply a point, instead of a cricle?

vela
Staff Emeritus