Principal Stress: A Beginner’s Guide

Principal stress is a fundamental concept in solid mechanics. It is often over complicated and taught in a confusing manner.

In this post, I aim to present the important elements of principal stress in a simple and easy to understand way.

What is principal stress?

Principal stress refers to the maximum and minimum normal stresses that act on a plane at a particular point within a material when subjected to an external force.

It is a fundamental concept in the field of mechanics and engineering, particularly in understanding a material’s behavior under various loading conditions.

In general, there are three principal stresses – the maximum (σ₁), intermediate (σ₂), and minimum (σ₃) principal stress. Although, the number of principal stresses depends on the dimensionality of the problem (2D, 3D).

Understanding these stress values allows engineers and scientists to predict the behavior of materials, evaluate their strength, and design structures accordingly.

How to calculate principal stress

There are various methods to calculate principal stresses. The exact method to use is based on the complexity of the given scenario.

Calculating 2-D principal stresses by hand will be covered in a later section.

Analytical methods

Analytical methods for calculating principal stress are based on mathematical equations and analytical procedures.

One common approach is using Mohr’s Circle method, which involves plotting stress states on a graphical representation and determining the transformed stresses at specific orientations.

Another widely employed analytical method involves using the principal stress equations. These equations are derived from the equilibrium and boundary conditions of the studied material.

These might include:

• Normal Stress Equations
• Shear Stress Equations
• Strain Equations

It’s important to note that these methods work effectively for simpler, well-defined problems where the geometry, material properties, and loading conditions are known and not too complex.

Computational methods

For more complex problems, computational methods are used to calculate principal stresses.

These methods employ numerical techniques and computer simulations to solve stress-related problems.

Some of the most common computational methods include:

1. Finite Element Method (FEM): This method involves dividing the material’s domain into small, interconnected discrete elements. By analyzing each element’s behavior, the overall stress field in the material is determined. FEM is useful for problems with complex geometries, material compositions, and loading conditions.
2. Boundary Element Method (BEM): BEM is a numerical technique used to solve linear partial differential equations formed by the material’s boundary conditions. It reduces the dimensionality of the problem, making it computationally less demanding compared to FEM.
3. Finite Difference Method (FDM): FDM is a numerical method that discretizes continuous functions by approximating the differential equations describing the material’s stress-strain relationship. It is usually applied to simple, regular geometries and is less flexible than FEM or BEM.

Each computational method has its merits and limitations, which depend on the complexity of the problem, the material properties, and the desired accuracy.

Therefore, choosing the appropriate method is essential for obtaining accurate principal stress calculations.

Key concepts and definitions

Before we learn to calculate principal stress, it’s important to cover some of the key terms involved.

Principal planes

Principal planes are the planes where the stress state is entirely constituted by normal stresses and no shear stresses are present.

These planes play a significant role in understanding the behavior of materials subjected to external forces.

In a two-dimensional state, there are two mutually perpendicular principal planes – defined by the principal angles θ_p1 and θ_p2.

They correspond to the two principal stresses: maximum (σ₁) and minimum (σ₂).

Principal directions

The principal directions relate to specific orientations at which shear stress is eliminated, isolating only the normal stress components. These directions are defined by the eigenvectors of the stress tensor.

They are pivotal as they signify the directions of the maximum and minimum normal stresses within a material.

Geometrically, the principal directions are represented by normal vectors, which are orthogonal to the principal planes where the principal stresses are exhibited.

This provides a clear analysis and visualization of the stress in materials.

Shear stress and normal stress

The two main types of stress acting on a material are normal stress (σ) and shear stress (τ).

Normal stress occurs when a force is applied perpendicular to a surface, leading to elongation or compression in the material. It can be either tensile (pulling) or compressive (pushing) in nature.

Shear stress, on the other hand, acts parallel to the surface of a material and causes deformation in the form of a change in shape without any change in volume.

The concept of shear and normal stresses is essential in determining the reaction of a material when subjected to different loading conditions.

Maximum in-plane shear stress

Maximum in-plane shear stress is the maximum value of shear stress acting on a plane within a material, considering an in-plane stress state (only two dimensions).

It can be determined by maintaining specific relationships between normal stresses and shear stresses on the plane and can be used for predicting failure or estimating the stability of a structure.

To calculate the maximum in-plane shear stress, the difference and average of the normal stresses acting on the material need to be determined.

These values are then used in conjunction with the associated shear stress values, allowing for the determination of the maximum in-plane shear stress and its orientation.

This information is crucial for characterizing the stress state in a material and analyzing its potential to fail or deform under specific loading conditions.

Mathematical foundations of principal stress

Stress tensor

The stress tensor is a key concept in understanding principal stress. It is a mathematical representation of the forces acting on a material at a certain point.

The tensor consists of nine components arranged in a 3 x 3 matrix and describes the distribution of forces along various planes in a three-dimensional space.

\(\Large\sigma_{ij} = \begin{pmatrix}
\sigma_{xx} & \tau_{xy} & \tau_{xz} \\
\tau_{yx} & \sigma_{yy} & \tau_{yz} \\
\tau_{zx} & \tau_{zy} & \sigma_{zz}
\end{pmatrix}\)

Each entry in the stress tensor corresponds to a force in a specific direction acting on a particular plane.

The matrix elements are represented as σ_ij, where i and j range from 1 to 3, corresponding to the x, y, and z axes of a coordinate system.

Stress transformation

Stress transformation refers to the process of expressing the stress tensor in a new coordinate system, which is rotated relative to the original one.

This is useful for determining the principal stresses and their orientations. The purpose of stress transformation is to find a coordinate system where the shear stresses are zero, and only the normal stresses exist.

Mohr’s circle method is a well-known technique for finding the principal stresses and their directions.

The transformed stress tensor can be obtained by combining the original stress tensor with the transformation matrix, which is formed by the rotation angles.

Coordinate systems

Different coordinate systems can be used in the analysis of principal stress.

The principal coordinate system is particularly relevant, as it is aligned with the directions of the principal stresses, and the shear stresses vanish.

In this system, the stress tensor becomes a diagonal matrix, with the principal stresses along the diagonal elements.

Coordinate systems can be rotated using transformation matrices, such as the one introduced in the stress transformation section.

Rotating the original coordinate system to a principal coordinate system is essential for understanding the material’s behavior and optimizing designs for various engineering applications.

2-D principal stress

Finding the principal stresses in a two-dimensional space is the most common scenario in undergraduate classwork.

In two-dimensional (2-D) stress analysis, the focus is on the plane stress state.

This is where stress components acting perpendicular to the plane (in and out of the page) are negligible compared to the in-plane stress components.

This simplification allows engineers to determine critical information like failure points in mechanical components and materials.

The principal stresses, denoted as σ₁ and σ₂, are the maximum and minimum normal stresses acting on a stressed element in its plane.

The two-dimensional stress state can be described by a stress tensor containing normal stresses (σ_x and σ_y) along the x and y directions, and the shear stress (τ_xy) acting on the x-y plane.

We are going to cover this process step-by-step in the next section.

Determining in-plane principal stresses (step by step)

In the following steps, I will determine the principal stresses for a state of plane stress. I will use the rotated stress element and notations seen in the figure below.

This plot is a result of transforming a right-hand coordinate system by an angle θ.

Stress transformation is a topic covered in a separate post.

Step one

The first step is defining the equations for the transformed stresses: σ_x’, σ_y’, and τ_x’y’.

Equation 1:

\(\Large{\sigma}_{x’} = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x – \sigma_y}{2} \cos(2\theta) + \tau_{xy} \sin(2\theta)\)

Equation 2:

\(\Large{\sigma}_{y’} = \frac{\sigma_x + \sigma_y}{2} – \frac{\sigma_x – \sigma_y}{2} \cos(2\theta) – \tau_{xy} \sin(2\theta)\)

Equation 3:

\(\Large{\tau}_{x’y’} = -\frac{\sigma_x – \sigma_y}{2} \sin(2\theta) + \tau_{xy} \cos(2\theta)\)

Step two

We can find the values for the maximum normal stress (and minimum) by taking the derivative of Equation 1 with respect to theta.

Using the differentiation of trigonometric functions, we get the following equation:

\(\Large\frac{d \sigma_{x’}}{d \theta} = -\frac{\sigma_x – \sigma_y}{2} (2\sin(2\theta)) + 2\tau_{xy} \cos(2\theta)\)

We then set it equal to zero and solve for theta:

\(\Large -\frac{\sigma_x – \sigma_y}{2} (2\sin(2\theta)) + 2\tau_{xy} \cos(2\theta) = 0\)

Equation 4:

\(\Large\tan(2\theta_p) = \frac{\tau_{xy}}{(\sigma_x – \sigma_y)/2}\)

Equation 5:

\(\Large 2\theta_p = \tan^{-1}\left(\frac{\tau_{xy}}{0.5(\sigma_x – \sigma_y)}\right)\)

This gives us the angle in which the principal stresses act along. They are called the principal planes, θ_p1 and θ_p2.

Step three

Using the trig relation, tan = opposite/adjacent, we can construct a right triangle with an angle 2θ_p:

From the triangle, we get the relationships for sin(2θ_p1) and cos(2θ_p2):

\(\Large \sin(2\theta_{p1}) = \frac{\tau_{xy}}{\sqrt{\left(\frac{\sigma_x – \sigma_y}{2}\right)^2 + (\tau_{xy})^2}}\)

\(\Large \cos(2\theta_{p1}) = \frac{\frac{(\sigma_x – \sigma_y)}{2}}{\sqrt{\left(\frac{\sigma_x – \sigma_y}{2}\right)^2 + (\tau_{xy})^2}}\)

Step four

Now, we can substitute the trigonometric relations into Equation 1. This will give us the equation for the maximum and minimum in-plane normal stresses aka principal stresses.

Equation 6:

\(\Large\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left( \frac{\sigma_x – \sigma_y}{2} \right)^2 + \tau_{xy}^2}\)

Once the principal stresses are determined, it is possible to assess the potential failure risks of materials and components subjected to 2-D stresses.

By comparing the principal stresses to the material’s strength criteria, engineers can identify if a material or component is expected to fail under the given stress conditions.

Example: Calculating principal stresses in an element

The state of plane stress at a point on a body is represented by the element below. Determine the principal stresses σ₁ and σ₂. Determine the principal planes defined by the rotations θ_p1 and θ_p2.

Sign convention

For this problem, we assume the typical right-hand coordinate system where +y is pointing upwards and +x is pointing to the right.

This allows us to write the normal and shear stresses on the element such as:

σ_x = 21 MPa
σ_y = 74 MPa
τ_xy = 48 MPa

Determine the orientation of the element

The next step is to determine the orientation of the element using the principal angles.

Plugging in the stress values into Equation 4 we get the following:

\(\Large \tan(2\theta_p) = \frac{48}{(21 – 74)/2}\)

Simplifying into the format of Equation 5:

\(\Large 2\theta_p = \tan^{-1}(-1.811)\)

Solving for θ would give you the following.

\(\Large \theta_{p2} = -30.54^\circ\)

Using the difference between θ_p1 and θ_p2, you can solve for θ_p1:

\(\Large 2\theta_{p1} = 180^\circ + 2\theta_{p2}\)

\(\Large \theta_{p1} = 59.46^\circ\)

Now, we can rotate our element to correspond with its principal angles.

Solving for the principal stresses

We can solve for the principal stresses by plugging in the provided stress values into Equation 6:

\(\Large \sigma_{1,2} = \frac{(21 + 74)}{2} \pm \sqrt{\left(\frac{(21 – 74)}{2}\right)^2 + (48)^2}\)

Now, use the principal stress equations to find σ₁ and σ₂:

σ₁ = 47.5 + 54.825 ≈ 102.325 MPa

σ₂ = 47.5 – 54.825 ≈ -7.325 MPa

So, the principal stresses are σ₁ ≈ 102.325 MPa and σ₂ ≈ -7.325 MPa.

From here, we can use the principal angles along with the provided stresses to determine which plane each normal stress acts.

We do this by plugging in the values into Equation 1:

\(\Large{\sigma}_{x’} = \frac{(21 + 74)}{2} + \frac{(21 – 74)}{2} \cos(2(-30.54^\circ)) + 48 \sin(2(-30.54^\circ))\)

Solving the equation, we get σ_x’ = -7.325 MPa, which is equal to σ₂. This tells us that σ₂ is acting along the -30.54 degree plane.

3-D principal stress

In real-world applications, stress often occurs in three dimensions.

Three-dimensional stress analysis helps engineers and scientists better understand and predict the behavior of materials subjected to complex loading scenarios.

The stress state in three-dimensional cases are represented by the 3×3 stress tensor we covered in an earlier section.

Mathematical dive

The 3D principal stresses can be solved using the stress invariants and eigenvector approach.

In general, to find the principal stresses in a three-dimensional stress state, we solve the following equation related to the stress tensor:

I is the identity matrix (a special square matrix). Solving this equation gives us the three principal stresses, which we usually arrange in order, like

A similar approach is used to determine principal strain.

Why it matters

Knowing the 3-D principal stresses is key for figuring out if a material will hold up under different loads without deforming or breaking.

These stresses are the backbone of some important failure theories, like von Mises and Tresca, which help engineers predict when a material might fail.

They’re also super handy for analyzing and designing structures to make sure they’re safe and will last a long time.

Calculating 3D principal stresses can become math intensive. There’s a lot more to the calculation and theory of 3-D principal stresses than what was covered here.

I will save that discussion for a different article.

Practical applications

Below are two examples of how principal stresses are being used in modern research.

3-D crack growth

An area of interest is the influence of intermediate principal stress on 3-D crack growth.

One study focuses on the qualitative influence of the intermediate principal stress on the shape of 3-D cracks.

Analyzing this effect can help engineers predict and prevent failure in various structures and materials.

Principal stress in soil

Monitoring principal stress in soil is essential for geotechnical engineering and assessing soil failure.

Scholars have developed a fiber Bragg grating (FBG) and micro-electromechanical systems (MEMS) based device that monitors 3-D principal stress in soil.

Advancements in this field will help engineers analyze and design better foundations for buildings and infrastructure projects.

Advanced topics

Complex loading scenarios

Complex loading scenarios arise in various engineering applications, such as tunnel modeling in geotechnical engineering and residual stresses in composites.

These scenarios may involve combinations of tensile, compressive, and shear stresses acting on a material, potentially leading to failure or deformation.

A thorough understanding of the principal stresses under complex loading conditions helps engineers develop adequate designs and prevent failures.

Stress invariants

In the study of stress, it is useful to analyze stress invariants – quantities that remain unchanged under different coordinate transformations.

Stress invariants are functions of the principal stresses and are key to simplifying stress analyses.

Incorporating stress invariants in stress analysis can involve the use of principal invariants of the stress tensor, which are instrumental in understanding material behavior under various loading conditions.

Utilizing stress invariants helps engineers determine critical stress levels, evaluate the stability of structures, and perform failure analysis more effectively.

Further reading

>> Stress In Engineering

>> Failure Theories