Solid Mechanics · Mohr’s Circle
Mohr’s Circle – principal stresses, maximum shear & plane stress transformation explained
Mohr’s Circle gives you a fast, visual way to transform plane stresses, find principal stresses, compare in-plane and absolute maximum shear stress, and connect stress states directly to yield and failure checks.
Quick answer: what Mohr’s Circle tells you
Core Mohr’s Circle relationships (plane stress)
Mohr’s Circle takes any 2D stress state \( (\sigma_x, \sigma_y, \tau_{xy}) \) and maps it to a circle in \( \sigma\!-\!\tau \) space, letting you read off principal stresses, transformed stresses, and maximum shear stress with geometry instead of repeated algebra.
In mechanics of materials, Mohr’s Circle is one of the fastest ways to answer “what do the stresses look like on another plane?” Instead of plugging into several transformation equations each time, you sketch a circle from the original stress state and then move around that circle to read off transformed normal and shear stress directly. Principal stresses \( \sigma_1 \) and \( \sigma_2 \), as well as the maximum shear stress \( \tau_{\max} \), come straight from the circle’s geometry.
For a given in-plane stress state, you first compute the average normal stress \( \sigma_{\text{avg}} \) and radius \(R\). The rightmost point of the circle is \( \sigma_1 \), the leftmost point is \( \sigma_2 \), and the vertical distance from the center is the maximum in-plane shear stress. The most important conceptual step is remembering that if the material plane rotates by \( \theta \), the corresponding point on Mohr’s Circle rotates by \( 2\theta \). Once that mapping clicks, the circle becomes a visual stress-transformation calculator.
Most introductory engineering uses focus on plane stress: plates, beam surfaces, thin-walled members, gusset plates, and many machine elements. But Mohr’s Circle also connects naturally to failure envelopes, stress invariants, and 3D stress states, which is why it remains valuable long after students move from hand calculations to software and FEA.
Why engineers keep using Mohr’s Circle
- It gives a fast visual check on signs, magnitudes, and critical orientations.
- It links directly to yield criteria such as Tresca and to brittle/soil failure envelopes such as Mohr–Coulomb.
- It helps explain why a shear crack or slip plane forms on a different physical plane than the original loading direction.
Symbols, sign convention & units in Mohr’s Circle
Mohr’s Circle is a geometric picture of the plane stress transformation equations. To use it correctly, you need consistent notation, a clear shear sign convention, and a habit of checking units before interpreting the circle.
Common Mohr’s Circle notation (plane stress)
| Symbol | Quantity | Typical units | Description |
|---|---|---|---|
| \( \sigma_x \) | Normal stress on x-face | MPa, ksi, psi | Normal stress acting on the face whose outward normal points in the \(+x\) direction. |
| \( \sigma_y \) | Normal stress on y-face | MPa, ksi, psi | Normal stress acting on the face whose outward normal points in the \(+y\) direction. |
| \( \tau_{xy} \) | Shear stress on x-face in y-direction | MPa, ksi, psi | In-plane shear stress whose sign depends on the convention used for positive element rotation. |
| \( \sigma_{\text{avg}} \) | Average normal stress | MPa, ksi, psi | The circle center on the \(\sigma\)-axis: \( \sigma_{\text{avg}} = (\sigma_x + \sigma_y)/2 \). |
| \( R \) | Circle radius | MPa, ksi, psi | \( R = \sqrt{((\sigma_x – \sigma_y)/2)^2 + \tau_{xy}^2} \), equal to the maximum in-plane shear stress. |
| \( \sigma_1, \sigma_2 \) | Principal stresses | MPa, ksi, psi | Normal stresses on planes where shear stress is zero: \( \sigma_{1,2} = \sigma_{\text{avg}} \pm R \). |
| \( \tau_{\max} \) | Maximum in-plane shear stress | MPa, ksi, psi | For plane stress, \( \tau_{\max,\text{in-plane}} = R \). |
| \( \theta \) | Physical plane angle | deg, rad | Rotation of the material plane from the original \(x\)-face. |
| \( 2\theta \) | Angle on Mohr’s Circle | deg, rad | The corresponding rotation of the point on the circle. |
| \( I_1, I_2 \) | Stress invariants | stress, stress² | The center and radius are tied to invariant combinations of the stress tensor, which is why the circle is physically meaningful and coordinate-independent. |
Sign conventions & unit-system notes
- Use a consistent normal stress sign convention: tension positive and compression negative is standard.
- For shear, your course or textbook may treat positive \( \tau_{xy} \) as the shear that tends to rotate the element clockwise or counterclockwise. The important rule is consistency between the element sketch and the circle.
- Keep all stresses in the same units. Mohr’s Circle is unit-agnostic, but mixed units make the result meaningless.
The most common student mistake is not the math — it is plotting the correct values with the wrong sign convention. The interactive sketch and sign-convention toggle below is designed to make that issue much easier to see.
How Mohr’s Circle works: center, radius & angles
At its core, Mohr’s Circle is a re-plot of the plane stress transformation equations. If you start from the algebra for transformed normal and shear stress and group terms correctly, every possible transformed stress state falls on one circle in \( \sigma\!-\!\tau \) space. That one geometric object contains the complete in-plane transformation behavior.
From plane stress equations to a circle in \( \sigma\!-\!\tau \) space
For a plane rotated by angle \( \theta \), the transformed stresses are:
If you square and add those equations, the cross terms cancel and you obtain:
- The center is the average normal stress.
- The radius is the maximum in-plane shear stress.
- The principal stresses are the two \(\sigma\)-axis intercepts where \( \tau = 0 \).
Enhanced visual-physical mapping: why the element rotates by \( \theta \) but the circle rotates by \( 2\theta \)
The hardest part of Mohr’s Circle is usually not finding the center or radius. It is understanding why rotating the physical stress element by \( \theta \) corresponds to moving around the circle by \( 2\theta \). This doubling comes directly from the trigonometric structure of the transformation equations, which use \( \sin 2\theta \) and \( \cos 2\theta \), not \( \sin \theta \) and \( \cos \theta \).
| Physical interpretation | Material view | Mohr’s Circle view |
|---|---|---|
| Original x-face stress state | Plane at \( \theta = 0^\circ \) | Point \( (\sigma_x, \tau_{xy}) \) |
| Rotate the element by \( \theta \) | New plane orientation | Move around circle by \( 2\theta \) |
| Reach a principal plane | Shear on that plane is zero | Point lands on \(\sigma\)-axis |
| Reach a max-shear plane | Plane is 45° from principal plane | Point is 90° from principal point |
A good way to remember the mapping is this: the physical element tells you which plane in the material you are on, and the circle tells you what normal and shear stress exist on that plane. The circle is not a picture of the material itself. It is a transformed coordinate space that encodes how stresses vary with angle.
In-plane vs. absolute maximum shear stress
This is one of the most searched points of confusion around Mohr’s Circle. In a plane stress problem, the circle radius gives the maximum in-plane shear stress:
But the absolute maximum shear stress in 3D depends on all three principal stresses:
For plane stress, \( \sigma_3 = 0 \). That means the absolute maximum shear could come from \( (\sigma_1, 0) \), \( (0, \sigma_2) \), or \( (\sigma_1, \sigma_2) \), depending on the signs and magnitudes. This matters whenever you use Mohr’s Circle as an input to a yield criterion like Tresca.
Pole method (method of planes)
In geotechnical engineering and some older mechanics texts, you may see the Pole Method or Method of Planes. Instead of relying only on angle calculations, the pole method uses a special point on the circle — the pole — to connect a point on the circle to the actual orientation of a plane in the element. It is especially useful when rapidly identifying stress components on multiple planes or when interpreting graphical constructions in soil mechanics.
Even if you mainly solve Mohr’s Circle analytically, knowing the pole method broadens the page’s practical value because many field-oriented and geotechnical references still use it extensively.
Stress invariants: why the circle still matters in advanced analysis
Mohr’s Circle is not just a classroom sketch. Its center and radius are tied to combinations of the stress tensor that do not depend on how you rotate your coordinate axes. In that sense, the circle is a geometric expression of stress invariants. The average stress reflects part of the first invariant, while the radius is tied to the deviatoric part of the stress state. That deeper connection is why the circle remains meaningful even when engineers move on to tensor notation, 3D stress states, and computational mechanics.
Interactive sketch & sign convention toggle
Students often understand Mohr’s Circle only after they see the circle update as the stress state changes. The lightweight sketch below lets you change \( \sigma_x \), \( \sigma_y \), and \( \tau_{xy} \) and switch between shear-positive-clockwise and shear-positive-counterclockwise conventions. That makes it easier to see how sign conventions affect the plotted point without changing the underlying physical state.
Interactive Mohr’s Circle sketch
How to use this sketch
- Change \( \sigma_x \), \( \sigma_y \), and \( \tau_{xy} \) to see the center, circle radius, and principal stresses update in real time.
- Flip the sign convention toggle to see how the plotted point moves above or below the axis.
- Use the widget as a quick intuition-builder before working the full algebra by hand.
Worked Mohr’s Circle examples
The examples below show the standard workflow: compute the center, compute the radius, read the principal stresses, and then connect the result to an engineering interpretation instead of stopping at the raw numbers.
Example 1 – Principal stresses & maximum shear from a general state
A point in a steel plate is subjected to plane stresses: \( \sigma_x = 60~\text{MPa} \), \( \sigma_y = -20~\text{MPa} \), and \( \tau_{xy} = 30~\text{MPa} \). Use Mohr’s Circle to find \( \sigma_1 \), \( \sigma_2 \), and \( \tau_{\max,\text{in-plane}} \).
Result: The principal stresses are \( \sigma_1 = 70~\text{MPa} \) and \( \sigma_2 = -30~\text{MPa} \). The maximum in-plane shear stress is \( 50~\text{MPa} \).
Example 2 – Checking a principal stress against an allowable limit
Suppose the same state occurs in an aluminum component with an allowable tensile principal stress of \( 85~\text{MPa} \). Check the result using principal stress.
Result: The tensile principal stress uses about 82% of the allowable limit. That is acceptable for this simplified check, but in real design you would usually continue into a ductile or brittle failure criterion depending on the material and loading history.
Example 3 – Mohr’s Circle for pure shear
Consider a thin plate in pure shear: \( \sigma_x = 0 \), \( \sigma_y = 0 \), \( \tau_{xy} = 40~\text{MPa} \).
Result: Pure shear corresponds to equal-magnitude principal tension and compression. This is why cracks associated with shear often align on planes near 45° to the original loading orientation.
Example 4 – Finding the principal plane angle
Using the Example 1 stresses, find the principal plane angle.
Interpretation: The physical principal plane is rotated about \( 18.4^\circ \) from the original \(x\)-face, while the corresponding movement on Mohr’s Circle is \( 36.9^\circ \). This is the exact visual-physical doubling that trips up many students.
Tresca, von Mises & Mohr–Coulomb: using Mohr’s Circle with failure envelopes
Engineers rarely use Mohr’s Circle in isolation. In practice, the circle is a way to get principal stresses, shear stress levels, and critical plane orientations that are then fed into a failure criterion. That connection is what turns Mohr’s Circle from a classroom graphic into a design tool.
Tresca yield criterion: why the radius matters
The Tresca or Maximum Shear Stress Theory says yielding begins when the maximum shear stress in the material reaches a critical value tied to uniaxial yield. In a plane stress setting, the circle radius immediately gives the maximum in-plane shear stress:
That is why Mohr’s Circle pairs so naturally with Tresca. Once you know the radius, you already know the shear driving force that the theory cares about. For a full 3D check, you then compare all three principal stresses and use the largest pairwise difference:
von Mises stress from Mohr’s Circle inputs
Mohr’s Circle does not directly plot von Mises equivalent stress, but it gives principal stresses that make the von Mises calculation straightforward. In plane stress:
For ductile metals, that is often the next step after using Mohr’s Circle to identify the principal stress state.
Mohr–Coulomb failure envelope in geotechnical engineering
In soils and rocks, Mohr’s Circle is commonly used with the Mohr–Coulomb failure envelope. Instead of asking whether ductile yielding begins, the question becomes whether the stress circle touches or crosses a strength line defined by cohesion and friction angle.
Here, \( c \) is cohesion, \( \phi \) is the friction angle, and \( \sigma’ \) is the effective normal stress. In geotechnical practice, effective stress matters because pore-water pressure changes the stress actually transmitted through the soil skeleton. That makes Mohr’s Circle useful not only in structural mechanics but also in slope stability, shear strength testing, and foundation engineering.
In a triaxial or direct shear interpretation, the Mohr circle is drawn in normal-stress/shear-stress space and compared to the Mohr–Coulomb envelope. If the circle is tangent to the envelope, failure is imminent. This is one of the most important places where Mohr’s Circle becomes more than a stress-transformation exercise.
Advanced note: 3D Mohr’s Circle and where to go next
The page so far focuses on plane stress because that is what most students and many practicing engineers need most often. But for a full 3D stress state, there is not just one circle. There are three overlapping Mohr circles, built from the three principal stresses \( \sigma_1 \ge \sigma_2 \ge \sigma_3 \).
The largest of those three radii corresponds to the absolute maximum shear stress. This is the complete picture that students often miss when they learn only the 2D version. If your component is thick, highly constrained, or loaded in all directions, the single plane-stress circle is no longer enough by itself.
Where to go next after 2D Mohr’s Circle
- Principal stress extraction from a full 3D stress tensor
- Stress invariants and deviatoric stress
- Tresca and von Mises comparisons in 3D
- Mohr–Coulomb and effective stress in soil mechanics
Design tips, limits & checks when using Mohr’s Circle
Mohr’s Circle is powerful, but it works best when you stay clear on assumptions and on what the circle does — and does not — tell you. It transforms and organizes the stress state. It does not replace engineering judgment, loading realism, or the correct material failure theory.
The form used on this page assumes plane stress, meaning one normal stress component is negligible. That is often appropriate for thin plates, surface stresses in beams, and many connection details. It is not sufficient for thick pressure vessels, strong through-thickness restraint, or fully 3D stress states.
Many Mohr’s Circle mistakes happen before any algebra starts. Draw the small stress element, label the signs on each face, and make sure the point you plot on the circle matches that element.
Once you have principal stresses and shear stresses, continue into the criterion that matches the material:
- Tresca or von Mises for many ductile metals
- Maximum normal stress for some brittle checks
- Mohr–Coulomb for soils, rock, and frictional materials
Modern FEA packages can calculate principal stresses instantly, but Mohr’s Circle remains useful because it helps you verify whether the reported stress directions, principal values, and shear levels make physical sense.
Mohr’s Circle – FAQ
What is Mohr’s Circle in simple terms?
Mohr’s Circle is a graphical tool that shows how normal and shear stresses at a point change as you rotate the plane through that point. One circle in \(\sigma\!-\!\tau\) space lets you read principal stresses, maximum shear, and transformed stresses on any plane.
Why does the circle angle use \(2\theta\) instead of \( \theta \)?
Because the stress transformation equations contain \( \sin 2\theta \) and \( \cos 2\theta \). A physical rotation of the material plane by \( \theta \) therefore maps to a rotation of the point on the circle by \( 2\theta \).
What is the difference between in-plane and absolute maximum shear stress?
The in-plane maximum shear stress is the radius of the 2D circle used for plane stress. The absolute maximum shear stress is a 3D quantity based on the largest difference between principal stresses, which can include the third principal stress even when it is zero.
How does Mohr’s Circle connect to Tresca and von Mises?
Mohr’s Circle gives principal stresses and maximum shear stress. Tresca uses the maximum shear stress directly, while von Mises uses combinations of the principal stresses to calculate an equivalent stress for ductile yielding.
What is the pole method in Mohr’s Circle?
The pole method, also called the method of planes, is a graphical construction that helps identify the stresses on differently oriented planes without repeatedly solving angle equations. It is especially common in geotechnical applications and traditional graphical stress analysis.
How is Mohr’s Circle used in soil mechanics?
In soil mechanics, Mohr’s Circle is often compared to the Mohr–Coulomb failure envelope. The circle shows the current stress state, while the envelope represents shear strength defined by cohesion and friction angle. Effective stress is usually the key normal stress used in that comparison.
References & further reading
- Mechanics of materials and solid mechanics textbooks covering plane stress transformation, principal stress, and Mohr’s Circle.
- University lecture notes and open courseware on stress transformation, failure theories, and graphical methods in mechanics.
- Geotechnical engineering texts covering effective stress, friction angle, and Mohr–Coulomb failure.
- Design references and engineering software documentation that report principal stresses and equivalent stress measures for validation and interpretation.