Mohr's Circle is a graphical method for visualizing how normal stress and shear stress change when you look at different planes inside a material. It matters in engineering because parts often fail on planes that are not aligned with the original x and y axes. Instead of testing every possible angle, one circle shows all possible stress states at a point.
The diagram also makes principal stresses and maximum shear stress easy to identify.
Key Facts
- Center of Mohr's Circle: C = (σx + σy)/2
- Radius of Mohr's Circle: R = sqrt[((σx - σy)/2)^2 + τxy^2]
- Principal stresses: σ1,2 = C ± R
- Maximum in-plane shear stress: τmax = R
- Stress transformation: σθ = C + ((σx - σy)/2)cos(2θ) + τxy sin(2θ)
- Angle relation: rotating a physical plane by θ corresponds to moving 2θ on Mohr's Circle
Vocabulary
- Normal stress
- Normal stress is stress that acts perpendicular to a plane and tends to stretch or compress the material.
- Shear stress
- Shear stress is stress that acts parallel to a plane and tends to slide one layer of material past another.
- Principal stress
- Principal stress is a normal stress acting on a plane where the shear stress is zero.
- Maximum shear stress
- Maximum shear stress is the largest shear stress value found among all possible plane orientations at a point.
- Stress transformation
- Stress transformation is the calculation of normal and shear stress on a plane rotated from the original coordinate axes.
Common Mistakes to Avoid
- Using θ instead of 2θ on Mohr's Circle, which gives the wrong plane orientation because the circle angle is twice the physical angle.
- Forgetting the sign convention for shear stress, which can place the starting point on the wrong side of the circle and reverse the rotation direction.
- Calling the circle radius a normal stress, which is wrong because the radius represents the maximum in-plane shear stress and the offset from the center to each principal stress.
- Assuming principal stress always occurs on the original x or y plane, which is wrong because principal planes are often rotated from the original axes.
Practice Questions
- 1 A plane stress state has σx = 80 MPa, σy = 20 MPa, and τxy = 30 MPa. Find the center, radius, principal stresses, and maximum in-plane shear stress using Mohr's Circle.
- 2 For σx = 120 MPa, σy = 40 MPa, and τxy = -25 MPa, calculate the principal stresses and the angle to the principal plane using tan(2θp) = 2τxy/(σx - σy).
- 3 Explain why a material can fail in shear even if the largest applied stresses seem to be normal stresses along the x and y directions.