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.
Understanding Engineering: Mohr's Circle
Stress at a point is not a single force. It describes the internal force spread over an imagined cut through a material. A cut perpendicular to its own surface carries normal stress.
This stress pulls the faces apart or pushes them together. A cut can carry shear stress, which tries to make one face slide past the other. When the cut is turned, the same internal loading is resolved in new directions.
The amounts of normal and shear stress therefore change, even though the material point and its external loading have not changed. For a small element in balance, shear stresses on perpendicular faces must match in size. This equilibrium condition is what allows a two dimensional stress state to be represented by one geometric picture.
The horizontal direction on the circle represents normal stress, while the vertical direction represents shear stress. A stress state on one face of the element is plotted as one point. The stress state on the perpendicular face lies at the opposite end of a diameter.
This is a useful check during construction. If the two plotted points do not lie opposite each other, a sign or coordinate error has probably occurred. Different textbooks choose different positive directions for shear stress.
Some draw positive shear upward and some draw it downward. Neither choice changes the physical answer, but mixing conventions produces incorrect angles. Students should state their convention first and use it consistently in the element sketch and the circle.
The doubled angle has a physical reason. A real plane is unchanged when it is turned through one hundred eighty degrees, because its two faces simply exchange places. A point on the circle returns to the same stress state after a full three hundred sixty degree trip.
The diagram therefore needs twice the physical rotation to match these repeating patterns. This detail is one of the most common sources of mistakes.
The direction of travel around the circle can differ from the direction used in a physical sketch, depending on the chosen shear convention. It is safer to verify a result by checking the transformed stresses on the rotated element than by relying only on a memorized clockwise rule.
Principal planes are important because shear stress vanishes on them. They show directions where the material experiences only direct pulling or compression. Brittle materials such as concrete, rock, and some cast metals often crack in directions linked to large tensile principal stress.
Ductile metals often begin yielding when shear stress becomes large, so the maximum shear condition matters for shafts, brackets, pressure vessels, and machine parts. Mohr's Circle is mainly a two dimensional tool. A real component may have stress in three dimensions, bending, contact loads, temperature effects, or stress concentrations near holes and sharp corners.
The circle gives the local state at one point, not a complete safety judgment. Engineers combine it with material data, failure criteria, and a careful choice of the critical location.
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.