Mohr's circle provides a graphical way to visualize the state of stress at a point in a loaded body. For plane stress, only two normal stresses and one shear stress are considered. The circle's center represents the average normal stress, and its radius represents the maximum shear stress. Rotating the coordinate axes corresponds to moving around the circle, revealing principal stresses at specific angles. Engineers use this tool to determine failure criteria, evaluate material safety, and design components subject to complex loading.
The principal stresses \(\sigma_1\) and \(\sigma_2\) are derived from the plane-stress components by solving a quadratic equation. The average normal stress \(\sigma_{avg}\) is given by:
The maximum shear stress \(\tau_{max}\) equals the radius of Mohr's circle and follows from:
The principal stresses then become:
The orientation of the first principal plane relative to the x-axis, \(\theta_p\), is found via:
These relationships form the foundation for this calculator's numeric output.
\(\sigma_1\) represents the largest normal stress acting on the point, while \(\sigma_2\) is the smallest. When both have the same sign, the material is either entirely in tension or compression. If their signs differ, shear effects become more pronounced and the risk of yielding or fracture may increase depending on material strength. The angle \(\theta_p\) indicates how much to rotate the element to align with these principal directions, often relevant for strain gauges or experimental setups.
Suppose a thin plate experiences \(\sigma_x=50\) MPa, \(\sigma_y=20\) MPa, and \(\tau_{xy}=10\) MPa. The average normal stress equals 35 MPa, and the maximum shear stress computes to about 22.9 MPa. The principal stresses are therefore roughly 57.9 MPa and 12.1 MPa. The principal angle comes out to approximately 13°. Comparing these results to the allowable stresses of the material reveals whether the part can safely withstand the applied loads.
| Material | Yield Strength (MPa) | Young's Modulus (GPa) |
|---|---|---|
| Aluminum 6061-T6 | 275 | 69 |
| A36 Structural Steel | 250 | 200 |
| Polycarbonate | 65 | 2.3 |
| Wood (Douglas Fir) | 40 | 12 |
When you fill out the stresses in the form and click Calculate, the JavaScript reads each value, ensuring they are valid numbers. The script calculates \(\sigma_{avg}\), \(\tau_{max}\), the two principal stresses, and the orientation angle. Because the arithmetic only involves square roots and trigonometric functions, the result is displayed instantly in your browser without contacting a server. This allows repeated experimentation with different load combinations, providing quick insight into how changes affect the state of stress.
Structural engineers depend on stress transformation to analyze components like beams, pressure vessels, and machine parts. Many failure theories—such as maximum shear stress or von Mises—rely on the principal stresses to predict when a part will yield or fracture. Mohr's circle also serves as a teaching aid in mechanics courses, giving students a visual understanding of how shear and normal stresses interact. By putting the calculations in a convenient form, this tool aims to make that abstract concept more tangible.
Real-world components may not fit the simple plane-stress assumption. Thick plates or three-dimensional bodies require a full stress tensor. Yet the 2D case still offers valuable intuition about how loads redirect through a material. Finite element software often outputs stresses in each element that can be transformed using the same formulas. Whether you are cross-checking results or learning the fundamentals, quick access to principal stress calculations empowers better design choices.
Mohr's circle elegantly relates normal and shear stresses. By encapsulating the geometry of stress transformation in a few equations, this calculator reveals where and how the extremes occur within a loaded plane. Experiment with different numbers to see how the principal stresses shift, and apply that knowledge to ensure your own designs remain secure.
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