Mohr's Circle and Stress Transformations

Combined stresses in materials engineering refers to situations where multiple types of stresses act simultaneously on a single point. These stresses include normal stresses (σ) acting perpendicular to surfaces and shear stresses (τ) acting parallel to surfaces. In many real-world applications, especially in thin plates and structural surfaces, these stresses exist in a state called plane stress, where stresses along one direction are zero.

Mohr's circle serves as a powerful graphical tool for visualizing and analyzing these stress states. This method helps engineers determine principal stresses (maximum and minimum normal stresses), extreme shear stresses, and their corresponding angles. The circle provides a complete visualization of how stresses transform when an element is rotated, with the circle's radius representing the magnitude of extreme shear stresses. Understanding these concepts is crucial for structural analysis, mechanical design, and materials science applications, where knowing the maximum stresses helps prevent material failure and optimize designs.

Principal stresses (σ₁ and σ₂) represent the maximum and minimum normal stresses acting on a material at any given point. To calculate these values, you first need to know the normal stresses in the x and y directions (σx and σy) and the shear stress (τ) on the original plane. The principal stresses can then be determined using the following equation: σ₁,₂ = (σx + σy)/2 ± √[(σx - σy)²/4 + τ²]. The term (σx + σy)/2 represents the center of Mohr's circle, while the square root term represents its radius. The '+' gives you the maximum principal stress (σ₁), and the '-' gives you the minimum principal stress (σ₂). These principal stresses always occur on planes where the shear stress is zero, and their directions are perpendicular to each other. The angle θp to the maximum principal stress can be calculated using: θp = ½ arctan[2τ/(σx - σy)].

The maximum shear stress (τₘₐₓ) can be calculated once you know either the principal stresses or the original stress components. Using the original stress components (σx, σy, and τ), the maximum shear stress is given by the equation: τₘₐₓ = ±√[(σx - σy)²/4 + τ²]. Alternatively, if you already know the principal stresses (σ₁ and σ₂), you can use the simpler equation: τₘₐₓ = (σ₁ - σ₂)/2. This maximum shear stress always occurs at angles that are 45° from the principal stress directions, or θs = θp ± 45°, where θp is the angle to the maximum principal stress. The planes of maximum shear stress always experience an average normal stress of (σx + σy)/2, or equivalently, (σ₁ + σ₂)/2. This relationship is clearly visible on Mohr's circle, where the maximum shear stress represents the radius of the circle.

To find the stresses on a plane rotated by an angle θ from the original x-axis (counterclockwise being positive), two transformation equations are used. The normal stress (σθ) on the rotated plane is calculated using: σθ = (σx + σy)/2 + [(σx - σy)/2]cos(2θ) + τxysin(2θ). The shear stress (τθ) on the rotated plane is calculated using: τθ = -[(σx - σy)/2]sin(2θ) + τxycos(2θ). Note that these equations use 2θ rather than θ because a 180° rotation of the stress element corresponds to a full 360° rotation in Mohr's circle. These transformation equations effectively describe how stresses redistribute as the plane of reference rotates, and their results can be visualized as points along Mohr's circle, where each point represents the normal and shear stress values at a particular rotation angle. The term (σx + σy)/2 represents the center of Mohr's circle, while the remaining terms determine the position along the circle's circumference for any given angle θ.