1.1. Planes of Principal Stresses in Mohr’s Circle

We now want to find the angle of the principal planes with the x plane. We can see in Figure 2 that we need to go by an angle of (2φ) clockwise from the x plane in the Mohr’s circle to reach the first principal plane, i.e., where λ is acting. That means that in the original coordinate system (Figure 1), we need to go by an angle of φ in the anticlockwise direction from the x plane to get to the first principal plane.

To find the direction of the principal planes using Mohr's Circle, we first establish the x plane as a reference. The angle (2φ) represents the movement in the circular representation (Mohr’s circle) to locate the first principal stress. This clockwise rotation corresponds to an anticlockwise shift of φ degrees in the original coordinate system, which helps in visualizing the relationship between the angles in both systems.

Imagine you're standing on a roundabout. If you want to meet a friend located at an off-road café directly connected to the roundabout (your x plane), you might first need to go around the roundabout clockwise to get closer before stepping off in the opposite (anticlockwise) direction to reach the café.

Similarly, for the second principal plane, we have to go (2φ + 180°) clockwise in the Mohr’s circle from the x plane. Thus, in the actual coordinate system, we need to go by an angle of (φ + 90°) in the anticlockwise direction from the e plane.

The second principal plane requires another adjustment. Here, the (2φ + 180°) indicates a complete half-circle rotation plus the additional angle needed to signify the correct principal plane direction. The offset transformation (φ + 90°) showcases how it converts the rotational movements in Mohr's circle back into the original coordinate setting to find the overall direction of principal stresses.

Using the roundabout analogy again, think of your friend waiting at the opposite café on the roundabout. To reach them, after locating the first café, you would need to turn around completely (180°) and then account for a small anticlockwise correction (90°) to navigate effectively.

Earlier, when we were finding principal planes and principal stress components by algebraic techniques, we had to solve for the eigenvalues and eigenvector of the stress matrix. The stress matrix being a 3 × 3 matrix, to get its eigenvalues, we have to find the roots of its characteristic equation which would be cubic. This mathematical process turns out to be complex and time-consuming when compared to the graphical method, i.e., by using the Mohr’s circle. But, Mohr’s circle procedure has a limitation that one of the principal axes must be aligned with any of the coordinate axis.

Finding the principal stresses algebraically involves solving a cubic equation, which can be cumbersome and intricate. Mohr's Circle, on the other hand, offers a visual method to grasp principal stresses more easily. However, this method does impose a constraint: the principal axes must align with the existing coordinate axes for the graphical approach to work. If the conditions aren’t met, using the characteristic equation remains necessary for accurate results.

Think of using a map for navigation. If you're in a familiar area where the streets align perfectly with your map's coordinates, navigating is much simpler and faster. However, if you venture into a complex neighborhood where the streets twist and turn unpredictably (analogous to not aligning with axes), you need a detailed street guide (characteristic equation) to find your way accurately.

If we have a general stress matrix, Mohr’s circle does not work, but the method to obtain eigenvalues from characteristic equation works for every case.

The distinction emphasized here is between the utility of graphical and algebraic methods in stress analysis. While Mohr's Circle provides a simplified approach that works under specific circumstances, the algebraic solution using eigenvalue analysis is universally applicable, regardless of the matrix configurations.

Consider riding a bike on a flat road (Mohr's Circle), where it's easy to pedal and navigate. However, if you're on a rugged trail with rocks and obstacles (general stress matrix), it's better to rely on a sturdy mountain bike geared up for tough terrain (algebraic method), which can handle any situation without limitations.

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