5. Other conclusions that can be drawn using Mohr's circle

We can also get the maximum and minimum values of σ and τ using Mohr’s circle. The maximum and minimum values of σ are attained on the σ axis itself. These are plotted in Figure 6. The maximum and minimum values of τ are on the top and bottom of the circle respectively as highlighted in Figure 6. We can verify that these values match with the ones derived in the last lecture.

Using Mohr's circle, we can visually determine the maximum and minimum normal stress (C3) values through the σ axis. The maximum shear stress (C4) is found at the top of the circle, while the minimum shear stress is at the bottom. This means that by drawing Mohr's circle, we can get these extreme values directly without solving complex equations. The values obtained from the circle correspond to those derived through theoretical calculations in previous lessons.

Imagine you are at an amusement park and want to find the highest and lowest points on a roller coaster ride. By looking at a map of the ride (analogous to Mohr's circle), you can quickly identify those peak points without needing to ride it multiple times. Similarly, Mohr's circle gives you a fast way to find maximum and minimum stress values without complicated calculations.

Suppose that the principal stress components λ₁, λ₂, and λ₃ are defined such that λ₁ > λ₂ > λ₃. Thus, λ₁ and λ₂ will correspond to the points of maximum σ and minimum σ respectively. From the circle, λ₁ and λ₂ will be obtained by adding and subtracting R to the σ for center respectively, i.e., (18) (19).

In Mohr's circle, the principal stresses (λ₁, λ₂) can be easily determined from the center and radius of the circle. When the stress state is represented in the circle, the maximum principal stress λ₁ is found at the outer edge farthest from the center along the σ axis, while the minimum principal stress λ₂ is at the nearest edge. Thus, λ₁ is obtained by adding the radius R to the center value of σ, and λ₂ by subtracting R from it.

Think of a hot air balloon that's floating in the air. The highest point the balloon reaches is like λ₁, whereas the lowest point it can go before touching the ground is like λ₂. By observing how high the balloon can rise and how low it can dip, you can easily find its peaks (maximum stress) and dips (minimum stress) just like we find principal stresses from Mohr's circle.

We can also get the radius of the circle in terms of principal stress components by subtracting equation (19) from equation (18), i.e., (21). Notice that in the previous lecture, we had derived the maximum value of the shear component of traction to be R. And from Mohr’s circle too, we get.

The radius of Mohr's circle represents the maximum shear stress in the system, which can be derived directly from the principal stresses. By calculating the difference between the maximum and minimum normal stress, we arrive at the radius. This allows students to understand that knowing the principal stresses gives them straight access to the radius of the circle, reinforcing the relationship between these parameters.

Imagine you're baking a cake. The height of the cake (like the radius) represents how fluffy and rising your cake is compared to the initial batter (the center). The more ingredients you add to make the cake fluffier (akin to principal stresses), the higher the cake rises, and you can easily tell how high it has gone just by looking at it.

From the Mohr’s circle, we can also see that σ corresponding to the point where we have maximum shear will be the same as σ for the center of the circle, i.e. . This is the same value that we had derived in the previous lecture. We have thus verified our results of the Mohr’s circle.

Mohr's Circle acts as a verification tool for stress calculations. It confirms that the normal stress at points of maximum shear directly corresponds to the average value of the normal stresses represented by the center of the circle. This cross-verification ensures that our analytical calculations align with graphical interpretations, reinforcing our understanding of stress states.

Consider finding both the length of your shoe using a ruler (theoretical calculation) and then measuring it with a tape measure (visual confirmation). If both methods give you the same result, you have verified your findings, much like how Mohr’s circle allows us to verify stress calculations through graphical representation.

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