Mohr’s circle is a graphical and analytical method for finding stresses on rotated planes in a stressed material element. This cheat sheet helps engineering students connect the stress transformation equations to the circle’s center, radius, and key points. It is especially useful in mechanics of materials, machine design, civil engineering, and failure analysis when principal stresses and maximum shear stresses must be found quickly.
The core idea is that any 2D stress state can be represented by a circle on a stress plot with normal stress on the horizontal axis and shear stress on the vertical axis. The circle center is the average normal stress, and the radius is the maximum in-plane shear stress. Principal stresses occur where shear stress is zero, while maximum shear stress occurs at the top and bottom of the circle.
Key Facts
- For plane stress, the circle center is C = (sigma_x + sigma_y) / 2.
- The radius of Mohr’s circle is R = sqrt(((sigma_x - sigma_y) / 2)^2 + tau_xy^2).
- The principal stresses are sigma_1 = C + R and sigma_2 = C - R.
- The maximum in-plane shear stress is tau_max = R.
- The average normal stress on the maximum shear planes is sigma_avg = C.
- The principal plane angle satisfies tan(2 theta_p) = 2 tau_xy / (sigma_x - sigma_y), using the sign convention of the course or text.
- The maximum shear plane angle is theta_s = theta_p + 45 degrees in the physical material element.
- A rotation of theta in the physical element corresponds to a rotation of 2 theta on Mohr’s circle.
Vocabulary
- Plane stress
- A 2D stress condition where sigma_x, sigma_y, and tau_xy are considered while out-of-plane stresses are assumed negligible.
- Principal stress
- A normal stress acting on a plane where the shear stress is zero.
- Maximum shear stress
- The largest shear stress value for the stress state, equal to the radius of Mohr’s circle for in-plane analysis.
- Mohr’s circle center
- The point on the normal stress axis located at the average normal stress C = (sigma_x + sigma_y) / 2.
- Stress transformation
- The process of calculating normal and shear stress on a plane rotated from the original x-y coordinate system.
- Principal plane
- A plane orientation in the material element where only normal stress acts and shear stress is zero.
Common Mistakes to Avoid
- Using theta instead of 2 theta on Mohr’s circle is wrong because angular movement on the circle is double the physical rotation of the material element.
- Forgetting the sign convention for shear stress is wrong because different textbooks plot positive tau_xy upward or downward, which changes the direction of rotation on the circle.
- Calling the radius a normal stress is wrong because R represents the maximum in-plane shear stress magnitude, not the average or principal normal stress.
- Assuming sigma_1 is always sigma_x is wrong because principal stress depends on the combined effect of sigma_x, sigma_y, and tau_xy.
- Dropping the square on tau_xy in the radius formula is wrong because the radius is based on a distance calculation: R = sqrt(((sigma_x - sigma_y) / 2)^2 + tau_xy^2).
Practice Questions
- 1 Given sigma_x = 80 MPa, sigma_y = 20 MPa, and tau_xy = 30 MPa, find the circle center, radius, sigma_1, and sigma_2.
- 2 Given sigma_x = -40 MPa, sigma_y = 10 MPa, and tau_xy = 25 MPa, calculate the maximum in-plane shear stress.
- 3 For a stress state with sigma_x = 100 MPa, sigma_y = 60 MPa, and tau_xy = 0 MPa, identify the principal stresses and explain why the original planes are principal planes.
- 4 Explain why principal stress planes and maximum shear stress planes are separated by 45 degrees in the physical material element.
Understanding Mohr's Circle Stress Analysis Reference
Stress at a point is not a single arrow. It depends on the imaginary cut made through the material. A cut perpendicular to the x direction has one normal stress and one shear stress.
When the cut is turned, the internal force on that cut must still balance the surrounding material. The normal part and shear part therefore change together. Mohr’s circle packages this balance requirement into a picture.
Every point on the circle represents one possible orientation of a pair of perpendicular planes. Opposite ends of a diameter belong to those perpendicular planes.
The most common source of errors is the shear stress sign convention. Some courses draw positive shear upward on the physical element. Others define it by the direction that tends to rotate the element.
On the circle, many engineering texts plot positive shear downward, which can feel backward compared with an ordinary graph. This choice is not wrong if it is used consistently from start to finish. Mark the stresses on the x face and y face directly from the element before drawing anything.
Then check that the two plotted points are opposite ends of a diameter. If they are not, a sign or coordinate has been mixed up.
The double angle rule comes from the geometry of stress transformation. A small change in the material plane changes the resolved normal and shear components through squared direction terms and products of direction terms. Those terms vary over twice the physical angle.
This is why a forty five degree turn of the material takes a ninety degree trip around the circle. It is useful to sketch the material element beside the circle and label the direction of rotation on both. Do not assume the direction around the circle without checking the sign convention used in the class notes.
Principal stresses matter because shear stress vanishes on principal planes. These planes often identify the directions in which a material is pulled or compressed most strongly. Maximum in plane shear matters in ductile metals, where yielding is closely linked to shear distortion.
Engineers use these results when checking a shaft near a keyway, a bracket around a bolt hole, a beam at a point carrying bending and torsion, or a thin pressure vessel with combined loading. The circle gives stresses at one location only. The calculation must be repeated at other locations where the loading or geometry changes.
Plane stress is an approximation with limits. It works well near a free surface of a thin plate because the stress normal to that surface is usually small. A thick part or a point deep inside a body can have three important normal stresses.
In that case, three dimensional stress analysis is needed. When solving problems, separate the tasks. First identify the stress components and their signs.
Next construct or calculate the circle. Then translate the chosen point back to a real plane in the material. A neat circle alone is not enough unless the physical plane and stress directions are stated clearly.