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 planesandprincipalstresses

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).

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)

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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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.

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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.

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

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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.

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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.

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

"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.

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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

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Principalplane in Mohr's circle

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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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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.

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,

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