4. Sign convention while using Mohr's circle

Let us draw the Mohr’s circle again as shown in Figure 6. We know the point corresponding to the e plane. To get to the e plane, α should be 90° in the counter-clockwised direction. So, in the Mohr’s circle, we need to go 180° in the clockwise direction from e plane. Thus, we get the e plane at the diametrically opposite point with respect to e plane point.

In Mohr's circle, each plane corresponds to a specific angle α that represents the orientation of the stress in the plane. When considering the transformation from one plane to another, particularly from the e plane to the e plane in the context of Mohr's circle, it is essential to recognize how the angles affect the positions on the circle. Specifically, to visualize the e plane, you need to rotate the angle α by 180° clockwise from the e plane position, reflecting the relationship between the two orientations on the circle.

Imagine you're using a compass to point out different directions. If you point north (the e plane), to find south (the opposite point, e plane), you'd need to turn around 180°. Similarly, in Mohr's circle, to find the opposing shear component, you rotate through a specific angle, representing how forces translate through different orientations.

The two right-angled triangles in Figure 6 are similar and thus we get σ at this point as σ as required but τ as −τ. However, in Figure 2, we see that one plane, shear component is τ. So, why are we getting −τ from the Mohr’s circle? This is because of our convention for the shear component of traction. In Figure 2, we had defined positive τ when we go 90° in the counterclockwise direction from n.

In the analysis of Mohr's circle, shear components are defined with a specific sign convention. The convention states that positive shear (τ) occurs when it is oriented counterclockwise from the normal. However, when calculating values from Mohr's circle for a specific orientation, the shear values obtained can sometimes yield a negative result. This negative sign indicates that the shear direction is opposite to our defined positive shear, revealing the importance of understanding the orientation of stress components in relation to our defined coordinate system.

Think about driving a car on a road. If you consider turning left as positive movement, then a right turn (the opposite direction) would be considered negative. Just like how the direction of your turn impacts your position on the road, the orientation of stress on the Mohr's circle determines whether the shear component is positive or negative.

So, to get shear component on one plane, we go 90° in the anticlockwise direction from e direction and thus, get to the −e direction. So, Mohr’s circle is giving us τ on the e plane in the −e direction whereas τ, by definition, is the shear component in the +e direction.

This chunk discusses how physical angles in Mohr's circle correspond with our defined stresses. When you analyze the shear components, there are prescribed directions to take into account. Flipping the perspective based on where you are defining 'positive' can alter the understanding of the stresses acting on a plane. In this case, when the calculation indicates a negative shear, it relates to our chosen reference axis, underscoring the importance of conventions in stress analysis.

Imagine you're at a party where the only positive reaction to an event is clapping. If someone booed instead of clapping, it would be a negative reaction relative to the positive ones. Similarly, when Mohr's circle presents negative shear, it reflects the orientation against what we've defined as a positive direction in our coordinate system.

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