For instance, at $\theta_1$ the normal stress could be either minimum or maximum, if it is max then $\theta_3$ value will give the same max value, and then $\theta_2$ and $\theta_4$ will give the minimum normal stress.

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Principalplane in Mohr's circle

In the problem above you calculated that the angle $\theta_1$ is 23.7 degrees (there is a small error in your solution -- you mention 27.3).

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Consider that a state of plane stress exists at a point and the stresses acting on an element taken at that point are given as,

There is a more generic form of those equations which allows for converting from coordinate system xy to x'y', given that the later is rotated by $\theta$. See below for said equations.

If you plug $\theta_1$ to the stress transformation equation you get -41.6 MPa, but if you plug $\theta_2 $ you will end up with 116.4 MPa.

I found this query is somewhat difficult to explain, I did my best to explain the problem, if the question makes sense to anyone please help. Thank You.

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To determine the principal stresses we first need to determine the location of them, i.e. the $\theta$ for them, this can be done by equating the derivative of above equation w.r.t $\theta$ to zero to obtain

Principal planesexplained

"Since for the value θ1 the normal stress could be either a max or a min, i expect when I put θ1 in σθ , it should yield two values of normal stress i.e., one for σ1 and other for σ2 , but I only get one value i.e. of σ1.

Principal planesexamples

The transformation equation you are using is only giving you the normal stress at a direction $\theta$ from the current coordinate system. So you need to apply them at different angles to get different values.

Principal planespdf

Principal planesandprincipalstresses

Per the given stresses and orientation of the element, the sketch below shows the results and the rotation of the principal axes with respect to the original position.

My Trouble starts from here: If I put $\theta_1$ in the stress transformation equation i.e. in the equation for $\sigma_{\theta}$ then I get

What isprincipalplane in Optics

$$\sigma_{\theta_1} = {\sigma_{x} + \sigma_{y} \over 2} + \sqrt{ \left( {\sigma_{x} - \sigma_{y} \over 2} \right)^2 + \tau_{xy}^2 }$$

The normal stress acting at any plane of this element inclined at an angle $\theta$ (+ve when measured antickw from horizontal) is given by the stress transformation equation

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Principal planeslist

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which is the expression for maximum normal stress (principal stress $\sigma_1)$. This equation tell us that we are getting maximum normal stress at $\theta_1$. However, I solved some numerical problems on the same topic and when I found $\theta_1$, the maximum normal stress wasn't coming in its plane but rather in $\theta_2$ plane. (the numerical problem is given below)

Principalplane formula

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This angle is with one of the principal axis and its complementary angle $\theta_2$ is between angle $x$ and the other principal axis $x_2$. i.e. $\theta_2 = 90- \theta_1= 66.3 deg$

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