Module 8: Principal Stresses and Mohr's Circle

Finding the absolute maximum stresses regardless of orientation.
In previous modules, we calculated normal and shear stresses acting on specific horizontal or vertical cross-sectional planes of a member. However, materials don't necessarily fail along those convenient geometric axes. The absolute maximum stresses within a material might actually occur on an angled plane.
This module explores how to mathematically and graphically transform known stresses to find these absolute maximum normal and shear stresses, known as Principal Stresses, using a powerful visual tool called Mohr's Circle.

Key Concepts

Checklist

Plane Stress

A state of stress where all stresses act within a single 2D plane (e.g., the xyxy-plane), and the stresses in the perpendicular direction (the zz-axis) are assumed to be exactly zero. This is a very common and highly accurate assumption for analyzing the surface of structural members or thin plates.

Stress Transformation

Rotating the axis of analysis
Imagine a small, infinitesimal square element extracted from the surface of a stressed beam. It has known normal stresses (σx\sigma_x, σy\sigma_y) and a known shear stress (τxy\tau_{xy}) acting on its xx and yy faces based on our standard calculations (P/AP/A, My/IMy/I, VQ/IbVQ/Ib).
If we conceptually rotate this square element by an angle θ\theta, the values of the normal and shear stresses on its new angled faces will change, even though the external loads on the overall beam remain exactly the same. The internal force must be resolved into new perpendicular and parallel components relative to the new angled plane.
The stress transformation equations allow us to calculate the specific stresses (σx′,τx′y′\sigma_{x'}, \tau_{x'y'}) on any plane rotated by an angle θ\theta:
σx′=σx+σy2+σx−σy2cos⁡(2θ)+τxysin⁡(2θ) \sigma_{x'} = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2} \cos(2\theta) + \tau_{xy} \sin(2\theta)
τx′y′=−σx−σy2sin⁡(2θ)+τxycos⁡(2θ) \tau_{x'y'} = -\frac{\sigma_x - \sigma_y}{2} \sin(2\theta) + \tau_{xy} \cos(2\theta)
Key Takeaways
  • Stresses on an element change depending on the angle of the plane being analyzed.
  • Stress transformation equations mathematically resolve stresses onto any rotated plane.

Principal Stresses

The absolute maximum and minimum
As we rotate the element through all possible angles (θ\theta from 0∘0^\circ to 180∘180^\circ), there is a specific, critical angle (θp\theta_p) where the normal stress reaches its absolute maximum value, and another angle 90∘90^\circ away where it reaches its absolute minimum value. These extreme normal stresses are called Principal Stresses (σ1\sigma_1 and σ2\sigma_2).
Crucially, on the specific planes where these principal normal stresses act, the shear stress is exactly zero. This is a fundamental property of principal planes.
The formula for calculating the magnitude of the principal stresses directly from the initial x−yx-y stresses is:
σ1,2=σx+σy2±(σx−σy2)2+τxy2 \sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}

Principal Stresses

The absolute maximum (σ1\sigma_1) and minimum (σ2\sigma_2) normal stresses acting at a specific point within a loaded material, regardless of the orientation. They occur on "principal planes", which are unique planes where the shear stress is mathematically proven to be exactly zero.
Key Takeaways
  • Principal stresses (σ1,σ2\sigma_1, \sigma_2) are the absolute maximum and minimum normal stresses at a point.
  • On principal planes, the shear stress is always exactly zero.

Maximum In-Plane Shear Stress

Similarly, there is an angle (θs\theta_s) where the shear stress reaches its absolute maximum value (τmax\tau_{max}). The planes of maximum shear stress are always oriented exactly 45∘45^\circ away from the principal planes.
τmax=(σx−σy2)2+τxy2 \tau_{max} = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}

Note

Notice that τmax\tau_{max} is exactly equal to the radical (square root) term in the principal stress equation. Therefore, we can also write a very useful relationship: τmax=σ1−σ22\tau_{max} = \frac{\sigma_1 - \sigma_2}{2}.
Key Takeaways
  • Maximum shear stress occurs on a plane oriented 45 degrees away from the principal planes.
  • It is equal to the radius of Mohr's Circle: τmax=(σ1−σ2)/2\tau_{max} = (\sigma_1 - \sigma_2) / 2.

Mohr's Circle

A brilliant graphical solution
Memorizing and properly using the complex trigonometric transformation equations can be tedious and prone to sign errors. In 1882, the German civil engineer Christian Otto Mohr developed a brilliant, elegant graphical method to visually represent the entire state of stress at a point.
Mohr's Circle plots normal stress (σ\sigma) on the horizontal axis and shear stress (τ\tau) on the vertical axis. Every single point on the continuous circumference of the circle represents the exact state of stress (σ,τ\sigma, \tau) on a differently angled plane passing through the element.

Properties of Mohr's Circle

  1. Center (CC): Always located exactly on the horizontal normal stress axis at the average normal stress value: C=σx+σy2C = \frac{\sigma_x + \sigma_y}{2}.
  2. Radius (RR): Exactly equal to the maximum in-plane shear stress: R=τmax=(σx−σy2)2+τxy2R = \tau_{max} = \sqrt{(\frac{\sigma_x - \sigma_y}{2})^2 + \tau_{xy}^2}.
  3. Principal Stresses (σ1,σ2\sigma_1, \sigma_2): The two points where the circle intersects the horizontal normal stress axis (where shear τ=0\tau = 0). They are simply calculated graphically as C+RC + R and C−RC - R.
  4. Angles: A physical rotation angle of θ\theta on the actual stress element corresponds exactly to a rotation angle of 2θ2\theta on Mohr's Circle.

Mohr's Circle for Plane Stress

Input Stresses (MPa)

Principal Results

Center (C)
50.0
Radius (R)
50.0
Max Principal (σ1\sigma_1)
100.0
Min Principal (σ2\sigma_2)
0.0
Max In-Plane Shear (τmax\tau_{max})
50.0
Loading chart...
Red dots indicate principal stresses, max shear stresses, the center, and the original state of stress faces (X and Y).
Key Takeaways
  • Mohr's Circle is a graphical representation of the stress state at a point.
  • The horizontal axis represents normal stress; the vertical axis represents shear stress.
  • A rotation of θ\theta on the physical element corresponds to a 2θ2\theta rotation on the circle.

3D State of Stress and Absolute Maximum Shear Stress

General state of stress in three dimensions
The 2D plane stress (x−yx-y) analysis covered so far assumes that all stresses in the third (zz) direction are zero. However, a general point in a body actually experiences a 3D state of stress with three orthogonal principal stresses: σ1\sigma_1, σ2\sigma_2, and σ3\sigma_3.
By convention, these are ordered algebraically as σ1≥σ2≥σ3\sigma_1 \ge \sigma_2 \ge \sigma_3.
Even if a member is loaded only in a 2D plane (e.g., σz=0\sigma_z = 0), the true absolute maximum shear stress might not occur in that x−yx-y plane. The Absolute Maximum Shear Stress (τabs,max\tau_{abs, max}) for any 3D state of stress is determined by the largest difference between any two of the three principal stresses, divided by two:
τabs,max=σmax−σmin2=σ1−σ32 \tau_{abs, max} = \frac{\sigma_{max} - \sigma_{min}}{2} = \frac{\sigma_1 - \sigma_3}{2}

Important

If the in-plane principal stresses σ1\sigma_1 and σ2\sigma_2 have the same sign (e.g., both tensile), the absolute maximum shear stress will actually occur out-of-plane, involving the zero stress in the zz-direction (σ3=0\sigma_3 = 0).
Key Takeaways
  • The absolute maximum shear stress is always half the difference between the algebraically largest and smallest of the three 3D principal stresses (τabs,max=(σ1−σ3)/2\tau_{abs, max} = (\sigma_1 - \sigma_3)/2).
  • For 2D plane stress, if the principal stresses share the same sign, the absolute maximum shear stress occurs on a 3D plane inclined at 45 degrees to the unloaded surface.

Failure Theories

Applying Principal Stresses to design

Predicting Material Failure

Finding the principal stresses (σ1,σ2,σ3\sigma_1, \sigma_2, \sigma_3) and maximum shear stress (τmax\tau_{max}) is crucial because materials fail in different ways under multi-axial loading. Failure Theories use these principal values to predict if a component will fail compared to a simple uniaxial tension test (SyS_y).
Maximum Shear Stress Theory (Tresca): Used for ductile materials. It predicts yielding when the absolute maximum shear stress reaches the shear yield strength (τmax≥Sy/2\tau_{max} \ge S_y / 2). Maximum Distortion Energy Theory (von Mises): Also for ductile materials, often more accurate than Tresca. It predicts yielding based on the distortional strain energy involving all principal stresses. Maximum Normal Stress Theory (Rankine): Used for brittle materials. It predicts failure when the maximum principal stress reaches the ultimate strength (σ1≥Sut\sigma_1 \ge S_{ut}).
Key Takeaways
  • Principal stresses are inputs for Failure Theories used to design safe structures under complex loading.
  • Tresca and von Mises theories are used for ductile materials; Rankine theory is used for brittle materials.
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