# Mohr's Circle

Precursor
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]

What am I missing on the Mohr's circle? Did I even go about this correctly?

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## Answers and Replies

Precursor
Any ideas?

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

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

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

Precursor
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.

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

Precursor
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?

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.

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

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

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

Staff Emeritus
Homework Helper
Yes, that's correct.

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

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

Precursor
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.

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

Precursor
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.

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

Precursor
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?

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

Precursor
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.

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.

Precursor
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?

Staff Emeritus
Homework Helper
Yes.

Precursor
For the case of uniaxial compression($\sigma_{x}=-p$) I am asked to determine the maximum shear stresses, and to draw the orientation of the element for these normal stresses. Below I have what I think it should be. Is it correct?

[PLAIN]http://img404.imageshack.us/img404/116/cive2.jpg [Broken]

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Staff Emeritus
Homework Helper
No, that's not correct. First, what are the axial and shear stresses equal to when the shear stress is maximized? What angle do you have to rotate by on Mohr's circle to reach those points? How does that translate to the orientation of the element?

Precursor
Here is my Mohr's circle for this case.

[PLAIN]http://img710.imageshack.us/img710/4182/cive3.jpg [Broken]

The centre point corresponds to $(\frac{-p}{2},0)$

Therefore, the maximum shear stress occurs when the normal stress is $\frac{-p}{2}$. So does that mean the angle of rotation is 90 degrees clockwise?

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Staff Emeritus
Homework Helper
Yes, you rotate by 90 degrees on Mohr's circle (clockwise or counterclockwise doesn't really matter).

Precursor
So will it look something like this:

[PLAIN]http://img190.imageshack.us/img190/8683/cive4.jpg [Broken]

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Staff Emeritus
Homework Helper
No, that's not right. First, what are $\sigma'_x$, $\sigma'_y$, and $\tau'_{xy}$ equal to? Second, since you have to rotate by 90 degrees on Mohr's circle, that means $2\theta=90^\circ$, so $\theta=45^\circ$. What do you suppose this 45 degrees corresponds to?

Precursor
I believe there are formulas to solve for $\sigma_x$', $\sigma_y$', and $\tau_{xy}$'.

Would the 45 degress correspond to the rotation of the element?

Staff Emeritus