Foundationally Fundamental Series 4: Introduction to Mohr Circle

In our journey to understand stress in soil and rock, we have now arrived at a pivotal concept: the Mohr Circle. This tool is instrumental in visualizing and analyzing the state of stress at a point within a material. Imagine a soil element subjected to different stresses: on one plane, it experiences a combination of normal stress (σx) and shear stress (τxy), and on another plane, it experiences a different set of normal stress (σy) and shear stress (τyx). The Mohr Circle allows us to graphically represent these stresses, providing a clear picture of the stress state at any orientation within the soil.

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To begin, let's consider how these stresses interact and how they can be effectively plotted on the Mohr Circle. This method simplifies the complex relationships between different stress components and lays the groundwork for further understanding material behavior under various loading conditions.

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Imagine you have a block of cheese, and you push on it from the top and the sides. The way the cheese squishes and slides is a lot like how soil and rock respond under the ground when forces act on them. In our picture, think of Plane A (marked by the red dashed line) as the top of the cheese where you're pushing down, and Plane B (marked by the green dashed line) as the side where you're pushing inwards. The block wants to stay still, so it pushes back against your hands; that's what we call stress.

Now, σ (sigma) is like the push straight down on the top of the cheese, and τ (tau) is like the push sideways at the edges. If we know how hard we're pushing on the top and the sides (that's Plane A and Plane B), we can use the Mohr Circle to figure out what's happening at any other slice inside the cheese, like our Plane C.

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Let's envision our block of cheese, which we've been using as a stand-in for soil or rock, to better understand how to plot the Mohr Circle. Imagine we're applying pressures on this cheese—squeezing it tightly (which is our normal stress, σ) and also trying to twist it (which is our shear stress, τ).

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On our graph, see the figure below, the x-axis is where we measure how much we're squeezing (σ), and the y-axis is where we measure how much we're twisting (τ). For Plane A, we see how much we're squeezing and plot that as a positive value on the x-axis because in our world of soils and rocks, more squeeze means a higher number. Then, we look at the twist: if it's clockwise, we go down on our graph because that's the agreed-upon direction for a negative twist in the world of stress.

That's why on the graph, τyx is plotted below the x-axis—it's our soil element twisting clockwise. For normal stress, which is our squeeze, we're pressing together, not pulling apart, so that’s a positive value, and we plot that to the right.

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At the point where our amount of squeezing and twisting meet, that's point A on the graph. We do the same for Plane B—measure the squeeze, measure the twist (this time it’s counterclockwise, so we go up!), and where they meet, that's point B.

Once we have points A and B plotted from our known stresses, we draw our Mohr Circle through these points, as can be seen from the figure above. This circle isn't just a shape; it's a powerful tool that shows us all possible states of stress on our cheese—or more practically, within a sample of soil or rock.

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To find the stresses on Plane C, we can now graphically determine them on the Mohr Circle. Here's how it's done: First, we translate the vertical Plane B from our soil element diagram to the Mohr Circle, marked by a green dashed line on the graph. Similarly, we transfer Plane A, our horizontal plane, to the circle.

The point where these two planes intersect on the graph is marked by an orange dot. This special spot is known as the pole or origin of planes in Mohr Circle terminology. It's a key reference point because from here, you can draw a line at any angle representing a new plane in the soil. Extend this line until it meets the circle, and voilà, you find the new values of normal stress (σ) and shear stress (τ) acting on Plane C—just like finding an exact location on a map using lines of latitude and longitude. This intersection on the Mohr Circle gives us a direct reading of the stresses that would act on any plane within our soil element, just as if we could slice through the soil at any angle and measure how much force is squeezing and twisting it.

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By plotting the Mohr Circle, we've graphically determined the stresses on Plane C. You find σn and τn by locating the coordinates where line C intersects the circle. This graphical approach, supported by trigonometry, simplifies complex calculations.

As our soil element is on a horizontal plane, this process has been straightforward. In the next series, we'll tackle more challenging scenarios, including rotated soil elements. We'll also discuss why knowing the principal stresses is so vital in geotechnical engineering. Stay tuned for more practical applications in our upcoming series.