What is Mohr's Circle?
Mohr's Circle is a two-dimensional graphical representation used in mechanics to visualize the state of stress at a point in a material. It allows engineers to determine principal stresses, maximum shear stresses, and the orientation of principal planes from a known stress state. This powerful tool was developed by German civil engineer Otto Mohr and is extensively used in mechanical engineering, civil engineering, and materials science.
When to Use Mohr's Circle Calculator
A Mohr's Circle Calculator is valuable in the following scenarios:
- Analyzing stress distribution in structural components under complex loading conditions
- Determining critical stress points that might lead to material failure in mechanical parts
- Designing components that need to withstand specific stress states in multiple directions
Examples
Example 1: Analyzing Stress in a Beam
Calculate the principal stresses and maximum shear stress for a point in a beam with normal stresses σx = 80 MPa, σy = 20 MPa, and shear stress τxy = 30 MPa.
| Parameter | Value |
|---|---|
| Normal stress σx | 80 MPa |
| Normal stress σy | 20 MPa |
| Shear stress τxy | 30 MPa |
| Center of Mohr's Circle (σavg) | 50 MPa |
| Radius of Mohr's Circle (R) | 36.06 MPa |
| Principal stress σ1 | 86.06 MPa |
| Principal stress σ2 | 13.94 MPa |
| Maximum shear stress τmax | 36.06 MPa |
| Angle to principal plane θp | 22.5 degrees |
Example 2: Stress Analysis in a Pressure Vessel
For a point on a pressure vessel with normal stresses σx = 120 MPa, σy = 60 MPa, and shear stress τxy = 40 MPa, determine the principal stresses and maximum shear stress.
| Parameter | Value |
|---|---|
| Normal stress σx | 120 MPa |
| Normal stress σy | 60 MPa |
| Shear stress τxy | 40 MPa |
| Center of Mohr's Circle (σavg) | 90 MPa |
| Radius of Mohr's Circle (R) | 50 MPa |
| Principal stress σ1 | 140 MPa |
| Principal stress σ2 | 40 MPa |
| Maximum shear stress τmax | 50 MPa |
| Angle to principal plane θp | 26.57 degrees |
Example 3: Pure Shear Analysis
Analyze a state of pure shear where σx = 0 MPa, σy = 0 MPa, and τxy = 50 MPa.
| Parameter | Value |
|---|---|
| Normal stress σx | 0 MPa |
| Normal stress σy | 0 MPa |
| Shear stress τxy | 50 MPa |
| Center of Mohr's Circle (σavg) | 0 MPa |
| Radius of Mohr's Circle (R) | 50 MPa |
| Principal stress σ1 | 50 MPa |
| Principal stress σ2 | -50 MPa |
| Maximum shear stress τmax | 50 MPa |
| Angle to principal plane θp | 45 degrees |
Common Stress States and Mohr's Circle Characteristics
| Stress State | Characteristics | Mohr's Circle Properties |
|---|---|---|
| Uniaxial Tension | σx > 0, σy = 0, τxy = 0 | Center at σx/2, Radius = σx/2 |
| Pure Shear | σx = 0, σy = 0, τxy ≠ 0 | Center at origin, Radius = τxy |
| Biaxial Tension | σx > 0, σy > 0, τxy = 0 | Center at (σx+σy)/2, Radius = |σx-σy|/2 |
| Hydrostatic Stress | σx = σy, τxy = 0 | Reduces to a point (no shear) |
| Complex Stress | σx ≠ σy, τxy ≠ 0 | Center at (σx+σy)/2, Radius as per formula |
Related Calculators
For more stress and structural analysis tools, you might find these calculators useful:
- Pythagorean Theorem Calculator - For basic geometric calculations
- Friction Calculator - For analyzing forces in mechanical systems
- Potential Energy Calculator - For energy analysis in structures