Mohr Circle: A Thorough Guide to Plane Stress, Principal Stresses and Shear

Webadmin Misc 21. August 2025 | 0

The Mohr Circle is a timeless tool in engineering and materials science, offering a visual and intuitive method to analyse planar stress states. By representing normal and shear stresses on a two‑dimensional circle, the Mohr Circle lets you read off principal stresses, maximum shear stresses, and the rotation that aligns your material plane with its principal directions. This article walks you through the theory, construction, practical steps, and real‑world applications of the Mohr Circle, with clear examples and discussion of its limitations.

What is the Mohr Circle?

The Mohr Circle is named after Christian Otto Mohr, a German engineer who introduced a graphical method for transforming stresses in a thin, flat plane—what is commonly called plane stress. In the standard two‑dimensional setting, the circle exists in the plane defined by normal stress (σ) on the horizontal axis and shear stress (τ) on the vertical axis. Each physical state on a material surface corresponds to a point on the circle, and rotating the material element by an angle θ corresponds to moving to a different point on the circle through an angle of 2θ. This doubling of the rotation angle is a key feature of the Mohr Circle and a helpful reminder of the deep link between geometric motion and stress transformation.

The central idea

Centre of the circle: C = ((σx + σy)/2, 0).
Radius of the circle: R = sqrt(((σx − σy)/2)^2 + τxy^2).
Principal stresses: σ1 = Cx + R, σ2 = Cx − R.
Maximum shear stress: τmax = R.

Key relationships

From the Mohr Circle, the transformation of stresses from one plane to another can be written succinctly. If you rotate the physical plane by an angle θ, the corresponding point on the Mohr Circle is rotated by 2θ. The standard transformation equations are:

σx′ = (σx + σy)/2 + (σx − σy)/2 cos 2θ + τxy sin 2θ

τx′y′ = −(σx − σy)/2 sin 2θ + τxy cos 2θ

Alternative but equivalent expressions for the same transformation are:

σx′ = σx cos^2θ + σy sin^2θ + 2τxy sinθ cosθ

τx′y′ = (σy − σx) sinθ cosθ + τxy (cos^2θ − sin^2θ)

Constructing the Mohr Circle: a practical guide

Constructing the Mohr Circle from a given stress state is straightforward once you know σx, σy and τxy. Here is a step‑by‑step approach you can follow on paper or with digital tools.

Step 1: Gather the plane stress components

Collect the normal stresses on the x and y faces (σx and σy) and the shear stress on the xy plane (τxy). All quantities should be in the same units, typically MPa or MPa units in structural analysis.

Step 2: Calculate the circle centre and radius

Compute the centre and radius of the Mohr Circle using:

Centre, Cx = (σx + σy)/2

Radius, R = sqrt(((σx − σy)/2)^2 + τxy^2)

Step 3: Read off principal stresses and maximum shear

The principal stresses are the horizontal extrema on the Mohr Circle:

σ1 = Cx + R

σ2 = Cx − R

The maximum shear stress is the radius:

τmax = R

Step 4: Determine the orientation of principal planes

The angle θp that aligns the material with principal stresses satisfies:

tan 2θp = 2τxy / (σx − σy)

From this, you can compute θp and rotate the material plane accordingly. Note that two orthogonal planes will exhibit the same principal stresses but in swapped order, which is why the circle representation is so helpful.

Worked example: reading a Mohr Circle with numbers

Suppose a flat plate experiences plane stress with:

σx = 120 MPa
σy = 60 MPa
τxy = 40 MPa

Calculate centre and radius

Centre: Cx = (120 + 60)/2 = 90 MPa

Radius: R = sqrt(((120 − 60)/2)^2 + 40^2) = sqrt(30^2 + 40^2) = sqrt(900 + 1600) = sqrt(2500) = 50 MPa

Principal stresses and maximum shear

σ1 = 90 + 50 = 140 MPa

σ2 = 90 − 50 = 40 MPa

τmax = 50 MPa

Orientation of principal planes

tan 2θp = 2×40 / (120 − 60) = 80 / 60 = 4/3 ≈ 1.333

2θp ≈ arctan(1.333) ≈ 53.13°, so θp ≈ 26.57°

In this example, rotating the material by approximately 26.6 degrees places the material in its principal directions, with stresses σ1 = 140 MPa and σ2 = 40 MPa and a maximum shear of 50 MPa.

The mathematics behind the Mohr Circle

The Mohr Circle is a geometric representation of the stress transformation equations. It encodes two scalar invariants of the 2D stress state: the average normal stress and the shear magnitude. The circle’s centre is the mean normal stress, while its radius reflects the disparity between σx and σy and the magnitude of τxy. This makes the Mohr Circle a compact way to capture all the necessary information about a planar stress state in one visual object.

Invariants and their interpretation

The key invariants in plane stress are:

The normal stress average: σ̄ = (σx + σy)/2
The radius: R = sqrt(((σx − σy)/2)^2 + τxy^2)
The principal stresses: σ1 = σ̄ + R, σ2 = σ̄ − R

From circle to transformation formulas

As θ varies, the Mohr Circle traces how σ and τ on a plane change. The transformation equations shown earlier are simply the algebraic form of this geometric movement: moving a point around the circle as the plane rotates, while the circle’s centre and radius remain fixed for a given stress state.

Mohr Circle and the 3D stress state

In real engineering problems, stresses are often three‑dimensional. While a single Mohr Circle suffices for pure plane stress, a three‑dimensional state is commonly analysed with Mohr’s circle method extended into Mohr’s sphere or multiple Mohr Circles for the three principal plane pairs (xy, yz, xz).

Mohr’s Sphere: a brief overview

Mohr’s Sphere is a three‑dimensional generalisation that represents normal and shear stresses on all possible planes passing through a point in a 3D material. The approach uses a sphere in a three‑axis space (σ, τ1, τ2) to capture the correlations between normal and shear components for all orientations. While more abstract, Mohr’s Sphere provides a comprehensive visual framework for 3D stress transformation.

Applications of the Mohr Circle

The Mohr Circle finds broad use across engineering disciplines, from civil and structural engineering to materials science and mechanical design. Here are some of the principal applications you’ll encounter.

Principal stresses and failure analysis

One of the most important uses is identifying principal stresses, which are the normal stresses on planes where shear is zero. In design, these stresses are critical for predicting failure modes, whether by yielding or fracture, as many failure criteria (such as the maximum normal stress criterion or the via the maximum shear) rely on σ1, σ2, and τmax.

Design against fatigue and fracture

In fatigue analysis, the amplitude of principal stresses can drive crack initiation and growth. The Mohr Circle provides a convenient visual tool to assess how multi‑axial loading translates into principal stress ranges and to compare them against endurance limits.

Material characterisation and anisotropy

For anisotropic materials, the Mohr Circle can simplify the interpretation of stresses in different directions. By plotting the relevant stress components, engineers can identify orientations that optimise strength or minimise risk of failure under a given load path.

Educational value and intuitive understanding

Beyond practical design, the Mohr Circle offers a powerful pedagogical aid. Students and professionals new to stress analysis often grasp the concepts of transformation, rotation of planes, and principal directions more quickly when they can visualise the circle as a map of stresses rather than an abstract algebraic manipulation.

Common pitfalls and tips for using the Mohr Circle effectively

Like any modelling tool, the Mohr Circle has its caveats. Being aware of these helps you avoid common mistakes and misinterpretations.

Plane stress versus plane strain

The Mohr Circle described here assumes plane stress, where out-of‑plane stresses are negligible (σz ≈ 0, τxz ≈ τyz ≈ 0). In thin structures or surfaces where this assumption does not hold, you may need to use a three‑dimensional approach or Mohr’s Circle variants for plane strain, or a 3D Mohr sphere.

Units and consistency

Ensure that all stress components are expressed in consistent units. Mixing units (e.g., MPa with psi) can lead to misinterpretation or calculation errors.

Interpreting the angular results

The angle obtained from tan 2θp = 2τxy / (σx − σy) may have multiple valid angles within a 0–180° range depending on which plane you label as x or y. The Mohr Circle clarifies these orientations, but when reporting principal directions, maintain a consistent convention across calculations and diagrams.

Limitations for transient or non‑linear loading

The classical Mohr Circle assumes linear elastic behaviour and static loading. For nonlinear materials, significant plasticity, or dynamic loading, the circle may only provide approximate guidance, or you may need to advance to incremental analysis or finite element methods for accuracy.

Practical tips for students and professionals

Practice with a few numerical examples to build intuition about how σx, σy, and τxy shape the Mohr Circle.
Draw both the circle and the corresponding σ–τ plot to reinforce the relationship between geometry and transformation.
Use the circle to verify transformation results obtained from algebraic equations; the visual check is often a quick assessor of correctness.
When teaching or learning, label principal planes carefully and consistently to avoid confusion about σ1/σ2 ordering.

Tips for using the Mohr Circle in the lab

In an experimental setting, the Mohr Circle can help interpret strain gauge data or optical measurements of surface stresses. Steps include:

Measure or estimate in‑plane normal and shear stresses on a given surface.
Construct the Mohr Circle from the measured values.
Extract principal stresses and the corresponding rotation angle to guide design changes or to identify critical loading directions.

Historical context and the evolution of the Mohr Circle

The Mohr Circle emerged in the late 19th century as a visually intuitive approach to stress analysis. It has endured because it connects geometry, linear algebra and physical insight in a compact form. Over time, the method has been refined and extended, but its core idea remains the same: a simple circle that encodes complex information about stress states, rotations, and failure criteria in a single, readable diagram.

Alternatives and complements to the Mohr Circle

While the Mohr Circle is a powerful tool, engineers periodically rely on complementary approaches for three‑dimensional or highly non‑linear problems.

Numerical methods and finite element analysis

Finite element analysis (FEA) can accommodate complex geometries, nonlinear material behaviour, and dynamic loads that exceed the scope of the Mohr Circle. FEA yields stress tensors at numerous points, from which principal stresses can be computed numerically.

Mohr’s Sphere for 3D stress

As discussed, Mohr’s Sphere extends the circle concept to three dimensions, offering a visual method for three principal stresses and a full set of shear components associated with all possible planes.

Critical plane analysis

In some engineering problems, failure criteria are more nuanced and depend on the orientation of cracks or defects relative to the loading. Critical plane methods go beyond the Mohr Circle by evaluating the most damaging plane according to the chosen criterion.

Summary: why the Mohr Circle matters

The Mohr Circle remains a foundational tool in engineering practice and education. It provides a compact, intuitive mechanism to transform stresses, identify principal stresses, and visualise shear interactions on a plane. Its simplicity belies its power: with a few known stresses, you can reveal the entire transformation behaviour of a surface and make informed decisions about design, safety and performance.

Closing thoughts: applying the Mohr Circle in practice

Whether you are an undergraduate student, a practising engineer, or a researcher, the Mohr Circle offers an accessible entry point into the complexities of stress analysis. By translating a 2D stress state into a geometric object, it becomes easier to reason about rotations, principal directions and failure. With practice, the Mohr Circle becomes a reliable companion in design reviews, material investigations and classroom explanations alike.

Mohr Circle: A Thorough Guide to Plane Stress, Principal Stresses and Shear