Principal stresses are the normal stresses acting on specific planes within a material where shear stress equals zero. At any point inside a loaded object, you can always find an orientation where the internal forces act purely as push or pull, with no sliding component. The stress values on those planes are the principal stresses, and the planes themselves are called principal planes.
This concept is foundational in structural and mechanical engineering because principal stresses represent the maximum and minimum normal stresses a material experiences at a given point. Knowing them tells you how close a component is to failure.
Why Principal Stresses Matter
When forces act on a structure, every internal point experiences a combination of normal stresses (pushing or pulling perpendicular to a surface) and shear stresses (sliding along the surface). The exact values of these stresses change depending on which angle you “slice” through the material. At most orientations, you get a mix of both normal and shear stress.
But there is always a set of orientations, perpendicular to one another, where shear stress drops to zero and only normal stress remains. These are the principal planes. The normal stresses on those planes are the largest and smallest the material will see at that point, making them critical for predicting whether a material will yield, crack, or buckle. Most failure criteria in engineering, like von Mises or Tresca, are expressed in terms of principal stresses.
Principal Stresses in Two Dimensions
In a two-dimensional stress state, there are two principal stresses, typically written as σ₁ (the larger value) and σ₂ (the smaller value). You find them by starting with the known normal stresses in the x and y directions (σₓ and σᵧ) plus the shear stress (τ) on those planes, then solving for the orientation where shear vanishes.
The formulas work out to:
- σ₁ = (σₓ + σᵧ)/2 + √[(σₓ − σᵧ)²/4 + τ²]
- σ₂ = (σₓ + σᵧ)/2 − √[(σₓ − σᵧ)²/4 + τ²]
The first term, (σₓ + σᵧ)/2, is the average normal stress. The square root term represents how far the principal stresses deviate from that average. When shear stress is zero and the two normal stresses are equal, both principal stresses collapse to the same value.
Finding the Principal Angle
The principal planes sit at a specific angle θₚ relative to your original x-y axes. That angle satisfies the relationship: the tangent of twice the angle equals twice the shear stress divided by the difference between the two normal stresses, or tan(2θₚ) = 2τ / (σₓ − σᵧ). Because this equation gives two solutions 90° apart, the two principal planes are always perpendicular to each other.
Visualizing With Mohr’s Circle
Mohr’s circle is a graphical tool that plots every possible combination of normal stress and shear stress at a point as you rotate through all orientations. The horizontal axis represents normal stress (σ), and the vertical axis represents shear stress (τ). Every orientation of a plane through the material corresponds to a point on the circle.
The center of the circle sits on the horizontal axis at the average normal stress: (σₓ + σᵧ)/2. The radius equals √[(σₓ − σᵧ)²/4 + τ²]. Principal stresses appear where the circle crosses the horizontal axis, because those are the points where shear stress is zero. The rightmost crossing is σ₁ (center plus radius), and the leftmost is σ₂ (center minus radius). The maximum shear stress is simply the radius of the circle, and it acts on planes oriented 45° from the principal planes.
Principal Stresses in Three Dimensions
Real structures are three-dimensional, so every point actually has three principal stresses: σ₁, σ₂, and σ₃. By convention, they are ordered so that σ₁ ≥ σ₂ ≥ σ₃. The three corresponding principal planes are all mutually perpendicular.
Mathematically, the stress state at a point is described by a 3×3 symmetric matrix called the stress tensor. Because this matrix is symmetric, it always has three real eigenvalues. Those eigenvalues are the principal stresses, and the eigenvectors point in the principal directions. Finding them requires solving a cubic characteristic equation:
σ³ − I₁σ² + I₂σ − I₃ = 0
The coefficients I₁, I₂, and I₃ are called stress invariants because their values don’t change no matter how you orient your coordinate system. I₁ is the sum of the three normal stress components (a quantity related to the average pressure on the point). I₂ and I₃ are more complex combinations of all six stress components, including shear terms. The three roots of this cubic equation give you σ₁, σ₂, and σ₃.
Maximum Shear Stress
Once you know the three principal stresses, the maximum shear stress at that point is straightforward to calculate. It equals the largest of these three quantities: |σ₁ − σ₂|/2, |σ₂ − σ₃|/2, or |σ₁ − σ₃|/2. Because σ₁ is the largest and σ₃ is the smallest by convention, the maximum shear stress simplifies to:
τ_max = (σ₁ − σ₃) / 2
This maximum shear stress acts on a plane oriented 45° from the principal planes associated with σ₁ and σ₃, and perpendicular to the σ₂ principal plane. This relationship is the basis of the Tresca failure criterion, which predicts that yielding begins when the maximum shear stress reaches a critical value.
Tensile, Compressive, and Mixed States
Principal stresses can be positive (tensile, meaning the material is being pulled apart) or negative (compressive, meaning it’s being squeezed). How the three values combine tells you a lot about the stress state:
- All three positive: The material is in triaxial tension, being pulled in every direction.
- All three negative: The material is in triaxial compression, like rock deep underground under hydrostatic pressure.
- Mixed signs: The material is being pulled in some directions and compressed in others. This is common in bending, where one side of a beam is in tension and the other in compression.
The sign and magnitude of the principal stresses determine which failure mode is most likely. Brittle materials like concrete and glass tend to fail when the maximum tensile principal stress exceeds their tensile strength. Ductile materials like steel tend to fail based on a combination of all three principal stresses, which is why engineers use criteria like von Mises that account for the differences between them rather than any single value alone.
Practical Applications
In finite element analysis (FEA), software packages compute the full stress tensor at thousands or millions of points within a structure, then extract principal stresses at each point. Color-coded plots of σ₁ or σ₃ across a part help engineers quickly identify where a design is most vulnerable. A pressure vessel, for instance, will show its highest principal stress along the hoop direction of the cylinder wall, which is why tanks tend to split lengthwise rather than around their circumference.
Principal stress directions are also used in reinforced concrete design. Engineers align rebar along the directions of maximum tensile principal stress, since concrete handles compression well but cracks easily under tension. In geological applications, the three principal stresses in the earth’s crust determine what type of faulting occurs: normal faults, reverse faults, or strike-slip faults each correspond to a different ordering of vertical and horizontal principal stresses.