Mohr's Circle
Mohr's circle formulas: sigma_avg = (sx+sy)/2, R = sqrt[((sx-sy)/2)^2 + tau^2].
Graphical stress transformation method for engineers.
The Formula
R = √[((σx − σy) / 2)² + τxy²]
σ1,2 = σavg ± R
Mohr's circle is a graphical method — and an algebraic set of formulas — used to determine the state of stress at a point when the coordinate system is rotated to a different orientation. Developed by German engineer Christian Otto Mohr in 1882, it remains one of the most powerful and intuitive tools in structural and mechanical engineering. Rather than solving rotation matrix equations directly, engineers can draw a circle on a stress diagram and read off transformed stress components geometrically.
The center of the circle is located at the average normal stress σavg, which is the arithmetic mean of the two normal stress components σx and σy. The radius R of the circle represents the maximum shear stress at the point and is calculated from the difference between normal stresses and the shear stress component τxy.
The principal stresses σ1 and σ2 are the maximum and minimum normal stresses, found at the left and right ends of the circle where shear stress equals zero. The maximum shear stress τmax equals the radius R and occurs at the top and bottom of the circle. The angle between the original stress state and the principal stress orientation is half the angle measured on the circle — a key insight that Mohr's graphical approach makes visually clear.
In three-dimensional stress analysis, three Mohr's circles are used simultaneously — one for each pair of principal planes. The largest circle defines the absolute maximum shear stress in 3D space, which is critical for failure analysis using criteria like Tresca's yield criterion.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| σx, σy | Normal stress components on x and y faces | Pa or MPa |
| τxy | Shear stress component on the x-y plane | Pa or MPa |
| σavg | Average normal stress — center of Mohr's circle | Pa or MPa |
| R | Radius of Mohr's circle — equals maximum shear stress | Pa or MPa |
| σ1,2 | Principal stresses — maximum and minimum normal stress | Pa or MPa |
Example 1
A stress element has σx = 80 MPa, σy = 20 MPa, and τxy = 30 MPa. Find the principal stresses and maximum shear stress.
σavg = (80 + 20) / 2 = 50 MPa
R = √[((80 − 20)/2)² + 30²] = √[30² + 30²] = √[900 + 900] = √1800 = 42.43 MPa
σ1 = 50 + 42.43 = 92.43 MPa
σ2 = 50 − 42.43 = 7.57 MPa
Principal stresses: 92.43 MPa and 7.57 MPa. Maximum shear stress = 42.43 MPa
Example 2
A shaft cross-section has σx = 60 MPa, σy = −40 MPa, and τxy = 50 MPa. Find the radius and principal stresses.
σavg = (60 + (−40)) / 2 = 10 MPa
R = √[((60 − (−40))/2)² + 50²] = √[50² + 50²] = √5000 = 70.71 MPa
σ1 = 10 + 70.71 = 80.71 MPa, σ2 = 10 − 70.71 = −60.71 MPa
Principal stresses: 80.71 MPa (tension) and −60.71 MPa (compression)
When to Use It
Mohr's circle is applied in engineering whenever stress transformation is needed:
- Finding principal stresses and their orientations in structural members
- Determining maximum shear stress for yield analysis using Tresca criteria
- Analyzing stress states in pressure vessels, shafts, and beams under combined loading
- Designing mechanical components against fatigue by identifying critical stress planes
- Civil engineering applications: retaining walls, soil mechanics, and foundation design
- Transforming strain gauge data from rosette gauges into principal strains