Mohr's Circle Calculator

Mohr's circle is a graphical method to find stresses on any plane through a point.

How the Math Works

Mohr's Circle is a geometric representation used to determine principal stresses and maximum shear stress in a material. The center of the circle, calculated as (σx + σy)/2, represents the average normal stress between two perpendicular planes. The radius, derived from √(((σx−σy)/2)² + τ²), combines the difference in normal stresses and the shear stress (τ) to show the maximum deviation from the average stress. This radius directly corresponds to the maximum shear stress experienced by the material, while the horizontal axis (σ) and vertical axis (τ) plot stress states for different orientations of the material's cross-section.

Practical Applications

Engineers use Mohr's Circle to analyze stress in structures, components, and materials under complex loading conditions. By calculating the center and radius, they can identify principal stresses (the maximum and minimum normal stresses) and determine the orientation of planes where these occur. This is critical in designing bridges, aircraft components, or pressure vessels to ensure they can withstand applied forces without failure. The maximum shear stress value helps assess material ductility and predict potential yielding or fracture points during deformation.

Day-to-Day Use

Understanding stress analysis through Mohr's Circle contributes to safer, more efficient designs in everyday items. For example, it ensures that car tires, building foundations, or bicycle frames can handle real-world stresses without breaking. By optimizing material usage, it reduces waste and costs in manufacturing while improving product durability. Additionally, it aids in creating safer infrastructure, like earthquake-resistant buildings, by predicting how materials respond to forces like wind, vibrations, or impacts.

Worked example

σx 80, σy 20, τ 30 → center 50, radius ≈ 42.4 MPa.

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