Understanding Mohr’s Circle 2D: How to Find Principal Stresses and Directions

Mohr’s Circle 2D: A Step-by-Step Guide for Stress Transformation

Overview

Mohr’s Circle is a graphical method to transform 2D stress components (σx, σy, τxy) into stresses on an arbitrary plane, and to find principal stresses and maximum shear stress.

Given

• σx = normal stress on the x-face
• σy = normal stress on the y-face
• τxy = shear stress (positive when tending to rotate the element counterclockwise on the positive x-face)

Step-by-step procedure

1. Compute center and radius

   • Center: C = (σx + σy) / 2
   • Radius: R = sqrt(((σx – σy)/2)^2 + τxy^2)

2. Principal stresses

   • σ1 = C + R
   • σ2 = C – R

3. Maximum shear stress

   • τmax = R

4. Angles

   • Angle to principal plane (2θp):
     tan(2θp) = 2τxy / (σx – σy)
     (Use atan2 to get correct quadrant; θp = 0.5atan2(2τxy, σx – σy).)
   • Angle to plane of maximum shear: θs = θp ± 45°

5. Constructing Mohr’s Circle

   • Plot normal stress σ on the horizontal axis and shear stress τ on the vertical axis.
   • Mark points A(σx, τxy) and B(σy, -τxy).
   • Circle through A and B with center C and radius R.
   • Any point on the circle corresponds to stresses on a plane at some angle θ from the x-axis; the horizontal coordinate is σn and vertical is τn.

6. Transforming to an arbitrary plane θ

   • Normal stress: σn = C + R * cos(2θ)
   • Shear stress: τn = -R * sin(2θ)
     (Sign convention depends on chosen positive shear direction; adjust sign if needed.)

Quick worked example

Given σx = 60 MPa, σy = 20 MPa, τxy = 30 MPa:

• C = (60+20)/2 = 40 MPa
• R = sqrt(((60-20)/2)^2 + 30^2) = sqrt(20^2 + 30^2) = sqrt(400+900)= sqrt(1300)= 36.06 MPa
• σ1 = 40 + 36.06 = 76.06 MPa; σ2 = 40 – 36.06 = 3.94 MPa
• τmax = 36.06 MPa
• 2θp = atan2(2*30, 60-20) = atan2(60,40)=56.31° → θp = 28.15°

Tips and common pitfalls

• Always use consistent units.
• Use atan2(y,x) to get correct angle quadrant.
• Keep track of shear sign conventions when plotting and interpreting τ.
• For computer implementation, compute C and R algebraically first, then produce stresses with cos/sin of double angles to avoid numerical issues.

References for further reading

• Standard mechanics of materials textbooks (e.g., Beer & Johnston, Hibbeler)
• Engineering mechanics lecture notes and solved problem sets