Torsion is the twisting of an object caused by a moment acting about the longitudinal axis of the object. A moment that tends to cause twisting is called torque.
The video below takes an in-depth look at torsion, including how to calculate the angle of twist, and the shear stresses and shear strains associated with torsional deformation. It also covers how ductile and brittle materials fail under torsional loads.
Stress and strain are fundamental concepts in engineering, and in strength of materials in particular, that describe how an object responds to applied loads. This page covers the basics, which are also summarised in the following video:
When a body like the bar shown below is loaded by external forces, internal forces develop within it to resist the applied forces. We can visualise these forces by making an imaginary cut through the object. The internal forces develop in such a way that they balance the external forces, to maintain equilibrium.
Instead of discussing the magnitude of these imaginary internal forces, it’s more useful to talk about how the internal forces are distributed over a surface, using a parameter called stress, that quantifies the internal force per unit area. Stress describes the distribution of internal forces within a body.
The concept of stress gives us a way of describing the internal state that develops within a body as it responds to externally applied loads. This is important because it allows us to predict when an object will fail. By comparing the stress in a body with the yield or ultimate strengths of the material, obtained from tests, we can estimate how close an object is to deforming permanently or to fracturing. Using stress to predict failure is covered in more detail in the page on failure theories.
Stress is split into two different types, normal stress and shear stress, that depends on whether the internal forces are perpendicular or parallel to the surface of interest.
The type of stress where the internal forces act perpendicular to a surface, like for the bar shown above, is called normal stress. This type of stress is denoted using the symbol $\sigma$ (the greek letter sigma), and is calculated as the force divided by the area over which it acts.
Stress is a measure of the internal force per unit area, so it has units of Newtons per square meter ($\mathrm{N/m^2}$) in SI units and pounds per square inch in US units. Units of $N/m^2$ are also called Pascals ($\mathrm{Pa}$).
Normal stresses can be either tensile, when the object is getting stretched, or compressive, when it is getting compressed.
The type of stress where the internal forces act parallel to a surface is called shear stress.
It can be calculated in a similar way to normal stress, as the applied force $F$ divided by the cross-sectional area $A$. Shear stress is denoted using the symbol $\tau$ (the greek letter tau).
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You might be wondering why the stress at a single point should change from a shear stress to a normal stress or the other way around depending on how we choose to orient our imaginary cut through the bar.
The truth is that it doesn’t. The stress state at a single point within a body will actually have components in the normal and shear directions. The magnitude of the normal and shear components depends on the angle of the plane being used to observe the stresses, as illustrated below.
This is because stress is actually a tensor quantity – at each point within a body the stress state can be represented by three normal stresses and six shear stresses. These stress components are shown on the stress element, a small cube that is used to represent the stress state at a single point within a larger body. You can learn more about the stress tensor and stress element by having a read of our page on stress transformation and Mohr’s circle.
Strain is a quantity that describes the deformations that occur within a body. Like with stress, we can differentiate between normal strain and shear strain.
Normal strain is defined as a change in length $\Delta L$ divided by the original length $L_0$. This makes it a dimensionless quantity, that is often expressed as a percentage.
Shear strain is a measure of the deformation caused by a shear stress. It corresponds to the change in angle between two lines that are initially perperpendicular to one another, as shown in the image below. Like normal strain, shear strain is a dimensionless quantity that is usually expressed as a percentage.
Stress and strain are two closely linked parameters – it makes sense that the internal forces that develop within a body (stress) depend on how much the body is being deformed (strain).
The relationship between these two parameters can be described using a stress-strain diagram. Stress-strain diagrams are different for different materials. The image below shows typical diagrams for glass, steel and rubber.
The diagram for a specific material can be obtained by performing a tensile test, which involves applying a known force to a test piece and measuring the normal stress and normal strain in the test piece as the applied force is increased.
For many materials the relationship between stress and strain is linear for relatively low values of stress and strain. In this region the linear relationship between normal stress and normal strain is defined by Hooke’s law, where $E$ is Young’s modulus, a material property that defines the stiffness of a material.
Other important material properties that can be determined from the stress-strain curve include strength, ductility and toughness.
A steel bar is being designed to support a suspended 2000 kg mass, as illustrated below. The bar is 1 m in length and has a circular cross-section. The maximum allowable stress in the bar is 80% of the material yield strength and the selected steel has a yield strength of 300 MPa. What is the minimum allowable radius of the bar, to the nearest 0.1 mm?
0.4 mm
1.1 mm
5.1 mm
6.4 mm
The tensile stress in the bar is:
$$\sigma = \frac{F}{A} = \frac{9.81 \cdot 2000}{\pi r^2}$$
The maximum allowable stress is:
$$\sigma_{max} = 0.8 \sigma_y$$
The limiting radius can be calculated as follows by equating the tensile stress in the bar with the allowable stress:
$$\frac{9.81 \cdot 2000}{\pi r^2} = 0.8\sigma_y$$
$$r = \sqrt{\frac{9.81 \cdot 2000}{\pi 0.8 \sigma_y}} = 5.1 \mathrm{mm}$$
Young's Modulus
Young's modulus is a measure of the stiffness of a material – it describes the relationship between stress and strain.
Strength, Ductility & Toughness
Strength, ductility and toughness are three important material properties that can be determined from a stress-strain curve.
Internal stresses develop within any body in response to externally applied loads. At any given point within the body, these internal stresses have components acting in both the normal and the shear directions.
The normal and shear stress components are shown in the image below acting on a 3D stress element that represents a single point within the body. They can also be written in a matrix form, which is called the stress tensor. A tensor is a mathematical object, in this case a 3×3 matrix, that has special properties and is used to represent certain physical quantities, like the stress state at a given point in a body.
For plane stress conditions the stress tensor is simplified to a 2×2 matrix because the stresses in one of the three directions are close to zero. This is usually a valid assumption for thin objects that are only loaded in the plane of the material.
The magnitude of the normal and shear stress components will change depending on how the stress element is oriented. This is illustrated below for a bar under plane stress conditions subjected to uniaxial tension.
If the stress element is oriented as shown in the top part of the image, where it is aligned with the direction of the applied load, then there is only one stress component, a normal stress $\sigma_x$ acting in the $x$ direction – the $\sigma_y$ and $\tau_{xy}$ components are equal to zero. If the stress element is rotated by an angle $\theta$, as shown in the bottom part of the image, the components denoted as $\sigma_{x’}$, $\sigma_{y’}$ and $\tau_{x’y’}$ are now non-zero.
It’s important to understand that the overall stress state at the point of interest isn’t changing as the stress element is rotating. The only thing that is changing is the coordinate system used to observe the stresses.
The equations shown below can be used to compute the stress components for any orientation of the stress element, where $\theta$ is the angle through which the stress element is rotated. The process of determining the normal and shear stress components for different orientations of the stress element is called stress transformation, and as such these are called the stress transformation equations.
The purpose of stress transformation is to obtain the normal and shear stress components acting on a particular plane. There are quite a few different scenarios where you might need to do this. Here are a few examples:
Mohr’s circle is a powerful graphical method used to visualise and analyze the stress state at a single point within a body. It allows you to determine the normal and shear stress components for different orientations of the stress element graphically, instead of using the stress transformation equations.
Mohr’s circle is a circle drawn on a graph that has normal stress on the horizontal axis and shear stress on the vertical axis. An example for a plane stress case (i.e. two-dimensional stress) is shown below. Each point on the circle defines the normal and shear stress components for a certain orientation of the stress element. The image shows the stress elements for three different points on Mohr’s circle, corresponding to three different orientations of the stress element.
Mohr’s circle can be used to, for example:
An important thing to note is that angles on Mohr’s circle are doubled compared to the angle the stress element is rotated by. For example there is a 90 degree angle between the stresses on the X and Y faces of the stress element. However on Mohr’s circle there is a 180 degree angle between these stresses.
When angles are shown on Mohr’s circle they are often denoted as $2 \theta$. $\theta$ is the angle the stress element is rotated by, and $2 \theta$ is the corresponding angle on Mohr’s circle. In the image above rotating the stress element by an angle of $\theta = 80^\text{o}$ corresponds to an angle of $2 \theta = 160^\text{o}$ on Mohr’s circle.
Stresses are usually considered to be positive or negative based on the following sign convention:
On Mohr’s circle the normal stress component is shown on the horizontal axis and the shear stress component is shown on the vertical axis. The most common convention for the vertical axis is to plot positive shear stresses (i.e. shear stresses that tend to rotate the stress element counter-clockwise) in the downwards direction.
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The Efficient Engineer summary sheets are designed to present all of the key information you need to know about a particular topic on a single page. It doesn’t get more efficient than that!
The video below covers stress transformation and how to construct Mohr’s circle in detail.
To construct Mohr’s circle all you need to know is the normal and shear stresses for one orientation of the stress element. Here are the steps:
A lot of useful information can be determined from Mohr’s circle, like the maximum shear stress $\tau_{max}$, which is equal to the radius of the circle, or the principal stresses $\sigma_1$ and $\sigma_2$.
For certain orientations of the stress element the shear stresses will be zero, and the normal stresses will be at their maximum and minimum values. These maximum and minimum normal stresses are called the principal stresses, and they are denoted as $\sigma_1$ and $\sigma_2$ respectively. If the normal stress is at it’s maximum value on the X face of the element, it will be at it’s minimum value on the Y face of the element, and vice-versa.
It is easy to identify the principal stresses on Mohr’s circle – they occur where the shear stress component is zero, i.e. where the circle crosses the horizontal axis.
The planes (i.e. the orientations of the stress element) where the principal stresses occur are called the principal planes.
Being able to determine the principal stresses is often an important first step in predicting failure of a material.
Determine the maximum shear stress for the stress state defined by the Mohr's circle shown below.
90.1 MPa
155.1 MPa
0 MPa
220.3 MPa
The maximum shear stress is the lowest point on the vertical axis (postive shear stresses have been plotted in the downwards direction). As can be seen from the image below, this corresponds to 90.1 MPa.
So far we’ve only discussed Mohr’s circle for plane stress conditions, where the stress state is two dimensional. But Mohr’s circle can also be drawn for a more generic three-dimensional stress state, where it is made up of three different circles, as shown below. All possible combinations of normal and shear stresses for the 3D stress element lie on the boundary of, or within, the shaded area.
For a three-dimensional stress state there are three principal stresses, which by convention are numbered as follows: $\sigma_3 < \sigma_2 < \sigma_1$.
Mohr’s circle can also be applied to strains instead of stresses. It works in exactly the same way, except the normal stresses $\sigma$ and shear stresses $\tau$ are replaced by normal strains $\varepsilon$ and shear strains $\gamma$.
Stress and Strain
Stress and strain are fundamental concepts that relate to the internal forces and deformations within a body in response to applied loads.
Plane Stress
A component is said to be in a condition of plane stress when all of the stresses acting on it are in the same plane.
Understanding Mohr’s Circle and Stress Transformation Read More »
