Mohr's Circle Explained: A Visual Guide for Engineering Students
Mohr's circle is one of the most elegant and practically useful tools in engineering mechanics, yet it confuses more students than almost any other topic in a first course on stress analysis. This guide explains what Mohr's circle represents, how to construct it step by step, and — critically — the sign convention pitfalls that cause most of the confusion.
What Does Mohr's Circle Represent?
Consider a small element of material under a general state of plane stress. The element experiences normal stresses and shear stresses on its faces. Now imagine rotating that element to a different orientation. The normal and shear stresses on the rotated faces change — even though the physical state of stress at that point has not changed at all. The material is still under the same loading; you are simply looking at it from a different angle.
Mohr's circle is a graphical representation of how the normal stress and shear stress on a plane through the point vary as the orientation of that plane changes. Every point on the circle corresponds to the stress state (normal stress, shear stress) on a plane at a particular angle. The entire circle represents all possible orientations of the plane through that point.
This is not just a mathematical curiosity. It answers questions that are directly relevant to engineering design and failure analysis:
- What is the maximum normal stress at this point, and on which plane does it act? (These are the principal stresses and principal planes.)
- What is the maximum shear stress, and on which plane does it act?
- What are the stresses on a plane at a specific angle — for instance, a potential failure plane, a weld line, or a joint?
In geotechnical engineering, Mohr's circle is particularly important because soil failure criteria — most notably the Mohr-Coulomb failure criterion — are defined in terms of the stress state on the failure plane. Understanding Mohr's circle is therefore essential for understanding how and why soils fail.
The Sign Convention: Where Most Confusion Begins
Before constructing the circle, you must establish a sign convention. This is where most errors originate, because different textbooks use different conventions, and mixing them produces wrong answers.
The convention used in this guide (and in most geotechnical engineering and mechanics of materials textbooks) is:
- Normal stress: Tensile is positive, compressive is negative. In geotechnical engineering, where almost all stresses are compressive, some authors reverse this convention (compressive positive). Be aware of which convention your course or textbook uses.
- Shear stress: A shear stress that would cause clockwise rotation of the element is plotted as positive (upward) on the Mohr's circle diagram. A shear stress causing anticlockwise rotation is plotted as negative (downward). This is the convention that ensures the angle of rotation on Mohr's circle corresponds correctly to the physical angle of the plane.
The most important thing is consistency. Pick one convention and use it throughout the entire problem. Do not switch mid-calculation.
Step-by-Step Construction of Mohr's Circle
Suppose you have a plane stress element with the following known stresses: normal stress σx on the vertical faces, normal stress σy on the horizontal faces, and shear stress τxy on both pairs of faces (with appropriate signs according to the convention above).
Step 1: Set Up the Axes
Draw a set of axes with normal stress (σ) on the horizontal axis and shear stress (τ) on the vertical axis. Tensile (positive) normal stress is to the right; compressive (negative) is to the left. Positive shear stress (clockwise rotation on the element) is plotted upward.
Step 2: Plot the Two Known Stress Points
Plot two points representing the stress states on the two perpendicular faces of the element:
- Point A: Represents the stress on the face with outward normal in the x-direction. Its coordinates are (σx, τxy), where τxy is plotted with the sign convention for the x-face.
- Point B: Represents the stress on the face with outward normal in the y-direction. Its coordinates are (σy, −τxy). Note the sign reversal on the shear stress — if the shear on the x-face causes clockwise rotation, the complementary shear on the y-face causes anticlockwise rotation, and vice versa.
Step 3: Find the Centre
The centre of Mohr's circle lies on the σ-axis at the average of the two normal stresses:
σavg = (σx + σy) / 2
This is also the midpoint of the line segment connecting points A and B. The centre has coordinates (σavg, 0).
Step 4: Determine the Radius
The radius of Mohr's circle is the distance from the centre to either point A or point B. Using the Pythagorean theorem:
R = √[ ((σx − σy) / 2)2 + τxy2 ]
This radius has a direct physical meaning: it equals the maximum shear stress at the point.
Step 5: Draw the Circle
Draw a circle with centre (σavg, 0) and radius R. This circle passes through both points A and B. Every point on this circle represents the stress state on a plane at some orientation through the point of interest.
Step 6: Read Off the Results
The circle immediately reveals several important quantities:
- Principal stresses: The two points where the circle intersects the σ-axis (where τ = 0). The rightmost intersection is the major principal stress σ1 = σavg + R. The leftmost is the minor principal stress σ2 = σavg − R.
- Maximum shear stress: The topmost point of the circle, with τmax = R. The associated normal stress on this plane equals σavg.
- Principal plane orientations: The angle from point A to the σ1 point, measured around the circle, is 2θp, where θp is the physical angle you would rotate the element to align with the principal planes. The factor of two is fundamental: angles on Mohr's circle are always twice the physical rotation angle.
The 2θ Relationship: Why Angles Are Doubled
The fact that angles on Mohr's circle are twice the physical angles is the single most important concept to grasp, and also the one that causes the most mistakes. It arises directly from the stress transformation equations, which contain terms in 2θ (cos 2θ and sin 2θ). As the physical plane rotates by an angle θ, the corresponding point moves around Mohr's circle by 2θ.
This has practical consequences:
- The two perpendicular faces of the element (physically 90° apart) are represented by points that are 180° apart on the circle — they are diametrically opposite. This is why points A and B always form a diameter.
- The principal planes (physically at an angle θp from the original orientation) are at 2θp from point A on the circle.
- The maximum shear stress planes are at 45° to the principal planes physically, which is 90° on the circle — exactly at the top and bottom of the circle, which is 90° from the σ-axis intersections. This is consistent.
Worked Example: Conceptual Walkthrough
To solidify the method, consider a general example. Suppose a soil element at some depth is under a vertical normal stress σy (compressive, hence negative in our tension-positive convention), a horizontal normal stress σx that is smaller in magnitude than σy (also compressive), and a shear stress τxy on the faces.
Constructing the Circle
Plot point A at (σx, τxy) and point B at (σy, −τxy). Since both normal stresses are compressive (negative), both points lie to the left of the origin on the σ-axis. The centre of the circle is at the average of the two normal stresses, also to the left of the origin.
The radius extends from the centre to either point. The circle lies entirely in the compressive (left) half of the diagram if both normal stresses are compressive and the shear stress is not so large as to push the circle past the origin. This is the typical situation in geotechnical problems, where the stress state is predominantly compressive.
Reading the Principal Stresses
The major principal stress σ1 (the least compressive, or closest to zero in our sign convention) is at the right intersection of the circle with the σ-axis. The minor principal stress σ2 (the most compressive) is at the left intersection. In geotechnical convention with compressive positive, these labels would be reversed — the larger compressive stress would be σ1. Again, be consistent with your convention.
Finding the Failure Plane
If we apply the Mohr-Coulomb failure criterion, we draw a line on the same diagram with slope tan(φ') and intercept c' (the cohesion). Failure occurs when Mohr's circle touches this line. The point of tangency gives the stress state on the failure plane, and the angle from the principal stress point to the tangent point (measured around the circle) gives twice the angle of the failure plane from the principal direction. For a Mohr-Coulomb material, this angle works out to 45° + φ'/2 from the major principal plane — a result that every geotechnical engineer should know, and Mohr's circle makes it visually obvious.
Mohr's Circle in Three Dimensions
In three-dimensional stress states, there are three principal stresses (σ1 ≥ σ2 ≥ σ3, using the geotechnical convention of compressive positive, or equivalently σ1 ≤ σ2 ≤ σ3 in tension-positive convention). Three Mohr's circles can be drawn, one for each pair of principal stresses. The stress state on any plane through the point lies within the region bounded by the outermost circle and outside the two inner circles.
The absolute maximum shear stress is the radius of the largest circle:
τmax = (σ1 − σ3) / 2
(in tension-positive convention, where σ1 is the most tensile and σ3 is the most compressive). This is important in three-dimensional failure analysis and in FEA post-processing, where you need to assess whether the stress state at each integration point is approaching failure.
Mohr's Circle and the Mohr-Coulomb Failure Criterion
In geotechnical engineering, the connection between Mohr's circle and the Mohr-Coulomb failure criterion is fundamental. The failure criterion states that shear failure occurs on a plane when the shear stress on that plane reaches a critical combination of the cohesion and the normal stress:
τf = c' + σn' tan(φ')
where τf is the shear stress at failure, c' is the effective cohesion, σn' is the effective normal stress on the failure plane, and φ' is the effective friction angle.
On the Mohr's circle diagram, this failure criterion plots as a straight line (the failure envelope). As the stress state changes (for example, as the soil is loaded), Mohr's circle grows. Failure occurs at the instant when the circle becomes tangent to the failure envelope. The point of tangency identifies the stress state on the failure plane, and the geometry of the tangent determines the failure plane orientation.
This graphical interpretation makes the Mohr-Coulomb criterion intuitive in a way that the algebraic formulation alone does not. You can immediately see how increasing confining pressure (moving the circle to the right, further into compression) increases the shear stress required for failure — this is the physical origin of the friction angle concept. You can see how cohesion raises the failure envelope, allowing the soil to sustain shear stress even at zero normal stress. And you can see how changing the stress path (the trajectory of the stress point as loading progresses) determines when and how failure is reached.
Common Mistakes to Avoid
Having taught stress analysis and seen hundreds of student attempts at Mohr's circle problems, the following mistakes appear repeatedly:
1. Inconsistent Sign Convention
This is the most common error. Students mix the geotechnical convention (compressive positive) with the mechanics of materials convention (tensile positive), or they forget to reverse the sign of the shear stress when plotting the complementary face. The result is a circle that is in the wrong location, with incorrect principal stresses and incorrect plane orientations.
Prevention: Write down your sign convention explicitly at the start of every problem. Check that the two points you plot are diametrically opposite on the circle (they must be, because perpendicular faces are 180° apart on Mohr's circle).
2. Confusing Physical Angles with Mohr's Circle Angles
Forgetting the factor of two is extremely common. If you measure a 60° angle on Mohr's circle, the physical rotation is 30°. If a problem asks for the physical angle to the principal plane, and you read 2θp from the circle, you must divide by two to get the actual rotation.
3. Wrong Direction of Rotation
When finding the stress on a plane at a specific angle, the direction of rotation on Mohr's circle must correspond to the direction of rotation in physical space (with the factor of two). With the convention used in this guide, a counterclockwise rotation of the physical element corresponds to a counterclockwise movement around Mohr's circle (by twice the angle). Reversing this gives stresses on the wrong plane.
4. Forgetting That the Maximum Shear Plane Is Not the Failure Plane
In a Mohr-Coulomb material, the failure plane is not the plane of maximum shear stress. The failure plane is the plane where the combination of shear and normal stress first satisfies the failure criterion, which occurs at the tangent point between Mohr's circle and the failure envelope. This plane is at 45° + φ'/2 from the major principal plane, not at 45° (which is where the maximum shear stress acts). Students who assume that failure occurs on the maximum shear stress plane get the wrong failure angle.
5. Applying Plane Stress Results to Plane Strain Problems
In plane stress (thin plates), σ3 = 0. In plane strain (long structures like retaining walls, tunnels, and embankments — the typical geotechnical case), σ3 is not zero; it is determined by the Poisson effect. Using plane stress Mohr's circle analysis for a plane strain problem gives incorrect results for the out-of-plane stress and potentially for the maximum shear stress if the intermediate principal stress is actually the out-of-plane stress.
Why This Matters Beyond the Exam
Mohr's circle is not merely an examination topic. It appears throughout professional engineering practice:
- Interpreting triaxial test results: The standard way to plot results from triaxial compression tests in geotechnical engineering is as Mohr's circles at failure, with the Mohr-Coulomb envelope drawn tangent to the circles. The intercept and slope of the envelope give c' and φ' — the fundamental strength parameters for design.
- FEA post-processing: When reviewing stress results from a finite element analysis, you need to understand what principal stresses, von Mises stress, and maximum shear stress represent. These are all derived from the stress tensor using exactly the same transformation that Mohr's circle visualises.
- Failure assessment: Whether you are checking a steel connection against yield (von Mises criterion), a concrete element against cracking (principal tensile stress), or a soil element against shear failure (Mohr-Coulomb), you are applying a failure criterion in principal stress space — the space that Mohr's circle maps out.
- Strain analysis: Mohr's circle applies to strains as well as stresses. The construction is identical, with normal strain replacing normal stress and half the shear strain (γ/2) replacing shear stress. This is useful for interpreting rosette strain gauge measurements and for understanding strain transformations in monitoring data.
Summary
Mohr's circle is a graphical method for visualising how normal and shear stresses vary with plane orientation at a point. It reveals the principal stresses, maximum shear stress, and the orientations of these critical planes. The key concepts to remember are: angles on the circle are twice the physical angles; the centre of the circle is at the average normal stress; the radius equals the maximum shear stress; and the sign convention must be applied consistently throughout the problem.
In geotechnical engineering, Mohr's circle connects directly to the Mohr-Coulomb failure criterion, making it essential for understanding soil strength, interpreting triaxial test data, and assessing stress states in numerical models. Master it once, and it will serve you throughout your engineering career.
Want to Learn More?
If you are a student or early-career engineer looking for tutoring in geotechnical engineering, soil mechanics, or stress analysis, I offer one-to-one sessions covering these topics and more. For practising engineers interested in how these concepts apply in numerical modelling, see my articles on finite element analysis in geotechnical engineering and ABAQUS soil-structure interaction modelling.