Foundationally Fundamental Series 5: The concept of Principal Stress in Mohr Circle.

In this series, we further investigate the Mohr Circle's role in understanding the influence of normal and shear stresses across distinct planes within geotechnical frameworks. Our initial focus is on principal stress—a pivotal concept for those engaged in geotechnical studies or practice. Defined precisely, principal stresses are the magnitude of normal stress on planes in which shear stress is zero. Grasping principal stresses is crucial, as they significantly affect the mechanical responses of soils and rocks under varied loading conditions, thereby guiding the evaluation of material strength and deformation properties. This concept is essential for understanding the execution and implications of laboratory tests such as uniaxial compression, direct shear, and triaxial tests, which are instrumental in determining the material strength parameters needed for assessments like slope stability.

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Now we will do another exercise with Mohr circle based on a study case. In the accompanying figure below, we observe a soil element subjected to rotation, exhibiting a state of stress at 60 degrees from the horizontal. This rotation reveals that normal stresses on the planes are 90 kPa and 30 kPa, while shear stresses are equal to zero. Remember that normal stress is stress acting perpendicular to the plane, while shear stress is acting parallel with the plane orientation. Illustration below is exhibiting principal stress conditions. The maximum principal stress (σ1) is 90 kPa, and the minimum principal stress (σ3) is 30 kPa. Our objective is to find the magnitude of normal and shear stresses on a horizontal plane within this soil element. This analysis is crucial for geotechnical evaluations as it impacts the design and safety of soil-structure interactions.

Revisiting the Mohr Circle concept, we will get back to the drawing board & employing the graphical method to solve the stated problem by charting a circle that represents the stress state of a soil element. On the Cartesian coordinate system, the X-axis corresponds to normal stress (σ), while the Y-axis denotes shear stress (τ). We begin by plotting the maximum principal stress (σ1) at 90 kPa along the normal stress axis, with zero shear stress. Similarly, we identify the minimum principal stress (σ3) at 30 kPa, also at zero shear stress. With these pivotal points established, we determine their midpoint, constructing a circle through them to represent the stress state. This circle, known as the Mohr Circle, provides a visual and analytical means to ascertain the stress condition at any plane within the soil element, enhancing our understanding of soil mechanics and behaviour under various stress conditions.

Moving forward with the Mohr Circle analysis, we proceed now to locate the pole of origin. Begin by constructing a line starting from the σ3 point at a 60-degree angle from the horizontal axis; this line simulates the orientation of the plane within the soil element, as indicated by the red line in the figure. Extend this line until it intersects the Mohr Circle at a point we will identify with a green dot. From this intersection, draw a line to the σ1 point, which we will highlight in blue. Comparing this construction within the Mohr Circle to the soil element's rotation illustrates the similarity between the graphical method and the actual stress conditions.

Now finally we can find the answer to our problem: what is the magnitude of normal stress & shear stress acting on a horizontal plane inside the soil element? To get into the answer, extend a horizontal line from the pole of origin to intersect with the Mohr Circle. The intersection point, identified by its coordinates, reveals the required stress values. For the normal stress (σ), we read a value of 77 kPa, and for shear stress (τ), a value of 23 kPa. These values represent the stresses acting on the horizontal plane in the soil element, thus solving our initial problem and showcasing the practicality of the Mohr Circle in geotechnical applications.

In laboratory tests like the triaxial test, the concept of principal stresses is critical for assessing the soil's strength under various conditions. The triaxial test subjects a cylindrical soil sample to axial stress (σ1) while confining pressure (σ3) is applied. This simulates the conditions soils experience in situ. By incrementally increasing the axial stress and measuring the sample's response, we obtain the failure envelope of the soil. The Mohr Circle is then used to interpret these results, helping to identify the principal stresses, in particular the maximum principal stress (because the σ3 is predefined) at the onset of failure. This information provides the cohesion and the angle of internal friction of the soil, fundamental parameters for designing safe and effective geotechnical solutions.

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To summarize the discussion thus far, the article meticulously explores the Mohr Circle’s application in geotechnical engineering, with a focus on principal stresses and their practical significance. Through graphical representation, it explains how to interpret normal and shear stresses on any given plane within a soil element. The detailed walkthroughs demonstrate the methodology for determining these stresses, particularly under rotated stress conditions, and culminate in the practical implications for laboratory tests such as the triaxial test. This series not only enhances theoretical understanding but also bridges the gap to practical application, underscoring the importance of principal stresses in soil mechanics and stability analyses.