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The German engineer Otto Mohr (1835-1918) developed a useful pictorial interpretation of the equations for finding principal stresses and maximum shearing stress at a point in a stressed member.
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This method, commonly called Mohr's Circle, involves constructing a circle where the coordinates of each point represent the normal and shearing stresses on a specific plane. The angular position of the radius gives the orientation of that plane.
Understanding the Plot
Figure 1: The geometric relationship between Normal Stress (σ) and Shear Stress (τ).
Coordinate Rules:
- Normal Stresses (σ): Plotted on the horizontal axis. Tensile (+) is right; Compressive (-) is left.
- Shearing Stresses (Ï„): Plotted on the vertical axis. Clockwise rotation is above the axis; Counter-clockwise is below.
The results obtained from Mohr's circle are identical to the equations derived from the free-body diagram. Consequently, it provides an extremely useful visual aid for determining stresses on mutually perpendicular planes.
Although it can be drawn to scale for direct measurement, today it is primarily used as a conceptual aid for analysts performing analytical determinations of stress direction and magnitude.
Visualizing Mohr's Circle with Excel
While analytical calculation is standard, visualizing how the circle changes dynamically with load inputs is powerful. The video below demonstrates how a Mohr's Circle calculator can be built and visualized using Microsoft Excel VBA.
Figure 2: Excel VBA implementation for dynamic stress visualization.
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Beyond the Basics: From Mohr's Circle to FEA
While Mohr's Circle is excellent for a single point, modern engineering design deals with complex 3D components where stress varies continuously.
1. Failure Theories: Tresca vs. Von Mises
Mohr's Circle allows us to easily calculate the Maximum Shear Stress. This corresponds directly to the Tresca Failure Criterion, which is conservative and often used for ductile materials.
However, most modern Finite Element Analysis (FEA) software (like Ansys, Abaqus, or SolidWorks Simulation) defaults to the Von Mises yield criterion (Distortion Energy Theory).
| Criterion | Basis | Application |
|---|---|---|
| Mohr's / Tresca | Max Shear Limit | Conservative; simple manual calc. |
| Von Mises | Distortion Energy | More accurate for ductile metals (Steel, Alum). |
2. Essential Tools for Stress Analysts
If you are performing stress analysis professionally, you cannot rely solely on Excel. Here are the industry-standard references you should have on your desk.
Roark's Formulas for Stress & Strain
The "Bible" of stress analysis. Contains thousands of pre-solved formulas for beams, plates, and shells.
Check PriceAdvanced CAS Calculators
For the PE exam, the TI-Nspire CX II CAS can solve stress matrices symbolically, saving hours of algebra.
Shop Calculators3. Next Steps in Design
Once you have determined the stresses are safe, the next step is ensuring the part fits in the assembly. Check out our guide on Geometric Dimensioning and Tolerancing (GD&T) to master tolerance stacks.
Comments
You have to "Enable macros" when opening the excel file.
Please note that if you don't follow these steps, you may not be able to run the macro.
1) Tools > Macro > Security
2) Security Level tab > Medium. > OK
3) Close Microsoft Excel
4) Open the file again and you will see the pop-up
5) Select Enable Macros
Note: step 1 -3, you have to do only once, later you can skip them
I hope this helps.
Ake
is it possible for you to include a square diagram with normal stress, shear stress and angels?
i get really confused when relating angle and rotation on square TO morh's circle's angles and rotation.
could you contact me if you can be bothered? slxia1@student.monash.edu.au
I'm a french student and I would like to have this program on excel for a project in mechanical engineering.
Please can someone send me this program.
Thanks