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.