<p>You are correct in your intuition regarding the signs of the forces.  The reason for two terms is due to the direction of the force.  The y-direction is not aligned with the face AB (it is aligned with y') so the force on this face is a combination of the forces normal to AB ($-\sigma_{x'x'}AB$) and the force along the face ($-\sigma_{x'y'}AB$).  Because the question wants only the force in the y-direction, we need to find the projection of these forces along that direction. </p>

<p>If the angle between the coordinate axes is $\theta$, then the contribution of $-\sigma_{x'x'}AB$ in the y-direction is $-\sigma_{x'x'}AB\sin\theta$ and the contribution of $-\sigma_{x'y'}AB$ in the y-direction is $-\sigma_{x'y'}AB\cos\theta$, thus the total force in the y-direction is the sum of these contributions:</p>

<p>$$-\sigma_{x'x'}AB\sin\theta-\sigma_{x'y'}AB\cos\theta$$</p>

<p>Here is a quick figure to help visualize the geometry:</p>

<p><a href="https://i.sstatic.net/jOuIR.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/jOuIR.png" alt="enter image description here"></a></p>

CC BY-SA 3.0
2015-08-11T15:21:39.960Z
2015-08-11T15:21:39.960Z
casey
<p>I'm working on the rotation of the state of stress using the book <a href="https://books.google.com/books?id=heJSAwAAQBAJ&amp;pg=PA100&amp;source=gbs_toc_r&amp;cad=4#v=onepage&amp;q&amp;f=false" rel="nofollow">Geodynamics of Turcotte</a> page 101. The 2D coordinate system $x'$ and $y'$ is inclined at an angle $\phi$ with respect to the $x$, $y$ coordinate system (see Figure 2.14).</p>

<p>The force in the $y$-direction on face $AO$ is</p>

<p>$$ \sigma_{yy}AO $$</p>

<p>and the force in the $y$-direction on face $OB$ is</p>

<p>$$ \sigma_{xy}OB $$</p>

<p>The force in the $y$-direction on face $AB$ is </p>

<p>$$ -\sigma_{x'x'}AB\cdot\sin\theta -\sigma_{x'y'}AB\cdot\cos\theta$$</p>

<p>I don't understand why the force in the $y$-direction on face $AB$ is equal to the above equation? </p>

<p>I think the negative signs are there because the stresses are not oriented in the same direction than the $y$-direction. I don't know why it is necessary to subtract the two lengths.</p>

<geophysics><geodynamics><structural-geology>
State of Stress in rotated 2D system
user1166251
# State of Stress in rotated 2D system

(Asked by: username_0 on 2015-08-11)

I'm working on the rotation of the state of stress using the book [Geodynamics of Turcotte](https://books.google.com/books?id=heJSAwAAQBAJ&pg=PA100&source=gbs_toc_r&cad=4#v=onepage&q&f=false) page 101. The 2D coordinate system $x'$ and $y'$ is inclined at an angle $\phi$ with respect to the $x$, $y$ coordinate system (see Figure 2.14).

The force in the $y$-direction on face $AO$ is

$$ \sigma\_{yy}AO $$

and the force in the $y$-direction on face $OB$ is

$$ \sigma\_{xy}OB $$

The force in the $y$-direction on face $AB$ is

$$ -\sigma\_{x'x'}AB\cdot\sin\theta -\sigma\_{x'y'}AB\cdot\cos\theta$$

I don't understand why the force in the $y$-direction on face $AB$ is equal to the above equation?

I think the negative signs are there because the stresses are not oriented in the same direction than the $y$-direction. I don't know why it is necessary to subtract the two lengths.

**Tags:** geophysics, geodynamics, structural-geology


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## 1 answers

### Answer by username_1 on 2015-08-11. Score: 1 | Accepted answer

You are correct in your intuition regarding the signs of the forces. The reason for two terms is due to the direction of the force. The y-direction is not aligned with the face AB (it is aligned with y') so the force on this face is a combination of the forces normal to AB ($-\sigma\_{x'x'}AB$) and the force along the face ($-\sigma\_{x'y'}AB$). Because the question wants only the force in the y-direction, we need to find the projection of these forces along that direction.

If the angle between the coordinate axes is $\theta$, then the contribution of $-\sigma\_{x'x'}AB$ in the y-direction is $-\sigma\_{x'x'}AB\sin\theta$ and the contribution of $-\sigma\_{x'y'}AB$ in the y-direction is $-\sigma\_{x'y'}AB\cos\theta$, thus the total force in the y-direction is the sum of these contributions:

$$-\sigma\_{x'x'}AB\sin\theta-\sigma\_{x'y'}AB\cos\theta$$

Here is a quick figure to help visualize the geometry:

[![enter image description here](https://i.sstatic.net/jOuIR.png)](https://i.sstatic.net/jOuIR.png)


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