Mohr's Circle for the FE Exam
Mohr's circle is a graphical method for performing stress transformations. It allows you to find the normal and shear stresses on any plane through a stressed point, identify the principal stresses, and determine the maximum shear stress. On the FE exam, Mohr's circle problems are common and very scoreable once you understand the construction method.
Why Stress Transformation Matters
When you calculate stress at a point using sigma = P/A or sigma = M*y/I, you find the stress on a specific plane (usually a cross-sectional plane). But the material may fail on a different plane. Stress transformation tells you the stress state on any oriented plane through that point, which is essential for failure analysis.
The Stress State at a Point
The 2D stress state at a point is defined by three components:
- sigma_x: Normal stress on the x-face
- sigma_y: Normal stress on the y-face
- tau_xy: Shear stress on both faces (equal in magnitude due to equilibrium)
Sign convention: Normal stress is positive in tension. Shear stress is positive when it tends to rotate the element clockwise on the face with an outward normal in the positive direction.
Constructing Mohr's Circle
Step 1: Set Up the Axes
Draw a horizontal axis (sigma, normal stress) and a vertical axis (tau, shear stress). Normal stress is positive to the right (tension) and negative to the left (compression). Shear stress sign convention varies by textbook; the most common convention for Mohr's circle is: shear that rotates the element clockwise is plotted upward.
Step 2: Plot the Two Known Stress Points
Plot point X representing the stress on the x-face: (sigma_x, tau_xy)
Plot point Y representing the stress on the y-face: (sigma_y, -tau_xy)
Note: The shear stress on the y-face has the opposite sign from the x-face in the Mohr's circle sign convention.
Step 3: Find the Center
The center of Mohr's circle lies on the sigma-axis at:
C = (sigma_x + sigma_y) / 2
This is the average normal stress.
Step 4: Calculate the Radius
The radius of the circle is:
R = sqrt[((sigma_x - sigma_y)/2)^2 + tau_xy^2]
Step 5: Draw the Circle
Draw a circle centered at (C, 0) with radius R passing through points X and Y.
Reading Results from Mohr's Circle
Principal Stresses
The principal stresses are the maximum and minimum normal stresses at the point:
- sigma_1 (maximum principal stress) = C + R
- sigma_2 (minimum principal stress) = C - R
Principal stresses occur on planes where shear stress is zero (the points where the circle intersects the sigma-axis).
Maximum Shear Stress
The maximum in-plane shear stress equals the radius of the circle:
tau_max = R = sqrt[((sigma_x - sigma_y)/2)^2 + tau_xy^2]
Maximum shear stress occurs on planes oriented 45 degrees from the principal planes. The normal stress on these planes equals the average normal stress C.
Stress on an Arbitrary Plane
To find the stress on a plane oriented at angle theta from the x-axis:
- Rotate 2*theta from point X on Mohr's circle (the angle on the circle is twice the physical angle).
- Read the normal stress (horizontal coordinate) and shear stress (vertical coordinate) at the new point.
Principal Plane Orientation
The angle from the x-axis to the major principal plane is:
tan(2*theta_p) = 2*tau_xy / (sigma_x - sigma_y)
This gives two angles (90 degrees apart) corresponding to the two principal planes.
Using Formulas Instead of the Graph
On the FE exam, you may not need to draw the full circle. The key formulas derived from Mohr's circle are:
- sigma_1, sigma_2 = (sigma_x + sigma_y)/2 +/- sqrt[((sigma_x - sigma_y)/2)^2 + tau_xy^2]
- tau_max = sqrt[((sigma_x - sigma_y)/2)^2 + tau_xy^2]
- tan(2*theta_p) = 2*tau_xy / (sigma_x - sigma_y)
These formulas are in the reference handbook and can be applied directly without drawing the circle.
Common Mohr's Circle Problem Types
- Given sigma_x, sigma_y, tau_xy, find principal stresses: Apply the principal stress formula or construct the circle.
- Given sigma_x, sigma_y, tau_xy, find maximum shear stress: Calculate the radius.
- Given sigma_x, sigma_y, tau_xy, find stresses on a specific plane: Use the rotation formula or rotate on the circle.
- Given principal stresses, find maximum shear stress: tau_max = (sigma_1 - sigma_2) / 2.
Special Cases
- Pure tension/compression: If tau_xy = 0, the given stresses are already the principal stresses.
- Pure shear: If sigma_x = sigma_y = 0, the principal stresses equal +/- tau_xy.
- Uniaxial stress: If sigma_y = 0 and tau_xy = 0, sigma_1 = sigma_x and sigma_2 = 0. The maximum shear stress is sigma_x/2.
Practice Recommendations
- Practice constructing Mohr's circle for various stress states until the process is automatic
- Verify your graphical results with the analytical formulas
- Focus on finding principal stresses and maximum shear stress, as these are the most commonly asked questions
- Remember: angles on Mohr's circle are doubled compared to physical angles
Mohr's circle is an elegant tool that becomes intuitive with practice. Once you understand the construction, you can solve stress transformation problems quickly and confidently on the FE exam.