An Introduction to Stress and Strain

Summary

This video explores the fundamental concepts of stress and strain, using a solid metal bar under uniaxial loading as an example. Stress, measured in Pascals, is the internal force per unit area and can be normal or shear, affecting the body's response to external loads. Strain, a dimensionless quantity, describes deformation, such as the change in length of the bar. The video explains how stress and strain are calculated, their relationship through Hooke's Law and Young's Modulus, and the importance of understanding these concepts for predicting material failure and analyzing more complex scenarios like torsion and beam bending.

Takeaways

• 🔍 Stress and strain are fundamental in understanding how materials respond to external loads.
• 📚 Uniaxial loading is a scenario where all applied loads act along the same axis, causing a bar to stretch.
• ⚖️ Internal forces within a material develop to resist external forces, and these can be visualized by imagining a cut through the material.
• 📏 Stress is defined as the internal force per unit area, measured in Pascals (or Newtons per square meter in SI units).
• 🔑 The ability to calculate stress is crucial for predicting material failure, as it relates to the material's strength.
• 📉 Normal stress, which acts perpendicular to the material's surface, can be tensile (stretching) or compressive (shortening).
• 🔄 Stress and strain are closely related, and their relationship can be depicted through a stress-strain diagram, which varies with material type.
• 📈 Hooke's law describes the linear relationship between stress and strain for small deformations, defined by Young's modulus.
• 🔄 Shear stress, denoted by the Greek letter tau, is the internal force that acts parallel to the material's cross-section and is common in applications like bolts.
• 🔗 Both normal and shear stresses can be present at a point within a material, and their magnitudes depend on the observation plane's angle.
• 📚 Understanding stress and strain is foundational for more advanced topics in mechanics, such as torsion and beam bending.

Q & A

What are stress and strain, and how do they relate to the response of a body to external loads?

Stress and strain are fundamental concepts used to describe how a body responds to external loads. Stress is a measure of the internal force per unit area that develops within a body to resist applied forces, while strain is a measure of the deformation that occurs within the body as a result of these forces.

What is uniaxial loading, and how does it affect the stress distribution in a bar?

Uniaxial loading refers to a condition where all applied loads act along the same axis. In the case of a solid metal bar, uniaxial loading causes the bar to stretch, leading to the development of internal forces that resist the applied forces. The stress distribution in such a bar is uniform and perpendicular to the direction of the applied force.

How is stress defined and what are its units in SI and US units?

Stress is defined as the internal force per unit area. In SI units, it is measured in Newtons per meter squared, which is also known as Pascals. In US customary units, stress is measured in pounds per square inch.

What is normal stress, and how is it calculated for an axially loaded bar?

Normal stress is the type of stress that acts perpendicular to the cross-section of a body. For an axially loaded bar, normal stress is calculated as the applied force (F) divided by the cross-sectional area (A) of the bar. It is denoted by the Greek letter sigma (σ).

Why is it important to calculate stress, and how does it relate to material failure?

Calculating stress is important because it allows us to predict when an object will fail under applied loads. When the stress within a material exceeds its strength, the material will fail. For example, a bar made from mild steel with a strength of 250 MPa will fail if the applied force exceeds a certain threshold calculated based on its cross-sectional area.

What is the difference between tensile and compressive stress?

Tensile stress occurs when forces act to stretch or elongate a material, while compressive stress occurs when forces act to shorten or compress a material. Tensile stresses are considered positive, and compressive stresses are considered negative according to the conventional sign convention.

How is strain calculated, and what does it represent?

Strain is calculated as the change in length (ΔL) of a body divided by its original length (L). It represents the deformation that occurs within a body due to the application of force. Strain is a non-dimensional quantity and is often expressed as a percentage.

What is Hooke's law, and how does it relate to stress and strain?

Hooke's law states that within the elastic limit, the relationship between stress and strain is linear. This linear relationship is defined by Young's modulus, which is the ratio of stress to strain. Hooke's law typically applies for small strains, and deformations in this region are reversible when the load is removed.

What is shear stress, and how does it differ from normal stress?

Shear stress is the type of stress that acts parallel to the cross-section of a body, as opposed to normal stress which acts perpendicular. It is caused by forces that tend to slide one part of the material over another. Shear stress is denoted by the Greek letter tau (τ) and is calculated as the applied force (F) divided by the cross-sectional area (A).

How is shear strain defined, and how does it relate to the deformation of a body?

Shear strain is defined as the change in angle between two originally perpendicular lines within a body due to shear stress. It represents the deformation that occurs in the shape of the body without changing its volume. Shear strain is denoted by the Greek letter gamma (γ).

What is a stress element, and why is it used in stress analysis?

A stress element is a graphical representation used to depict the normal and shear stresses acting at a single point within a body. It is used in stress analysis to visualize and understand the complex stress state at a point, which can have components in both normal and shear directions depending on the orientation of the plane used to observe the stresses.

Outlines

00:00

🔧 Introduction to Stress and Strain

This paragraph introduces the fundamental concepts of stress and strain, which are essential for understanding how materials respond to external forces. It uses the example of a solid metal bar under uniaxial loading to explain how internal forces develop to resist the applied forces. The concept of stress, defined as the internal force per unit area, is explored, with a focus on normal stress, which acts perpendicular to the cross-section of the bar. The video also discusses how stress can be calculated and its importance in predicting material failure. The normal stress in the bar is calculated as the applied force divided by the cross-sectional area, represented by the Greek letter sigma. The paragraph concludes with a brief mention of the difference between tensile and compressive stresses and the stress distribution in more complex scenarios like bending beams.

05:04

📏 Further Exploration of Stress and Strain

The second paragraph delves deeper into the relationship between stress and strain, particularly for ductile materials. It explains the process of creating a stress-strain diagram through tensile tests, which involves applying a known force to a test piece and measuring the resulting stress and strain. The initial linear region of the stress-strain curve, where deformations are elastic and reversible, is described, along with Hooke's law and Young's modulus. The paragraph then transitions to discuss shear stress, which occurs when forces are applied perpendicular to the axis of a bar, causing internal forces parallel to the cross-section. Shear stress is represented by the Greek letter tau and is calculated similarly to normal stress. The concept of shear strain is introduced as the change in angle due to deformation. The video concludes with a discussion of the combined normal and shear stress components at a single point within a body, using the stress element to illustrate these stresses in both two and three-dimensional cases. The importance of understanding these concepts for advanced topics in material science is emphasized.

Keywords

💡Stress

Stress, in the context of the video, refers to the internal resistance forces that develop within a material when it is subjected to external forces. It is a measure of the force per unit area and is crucial for understanding how materials respond to loads. The video uses the example of a solid metal bar under uniaxial loading to explain how stress is calculated as the applied force divided by the cross-sectional area, denoted by the Greek letter sigma. Stress is a fundamental concept in material science and engineering as it helps predict material failure.

💡Strain

Strain is a measure of the deformation or change in shape of a material under the influence of an applied force. It is a non-dimensional quantity, often expressed as a percentage change in length. In the video, strain is illustrated by the change in length of the bar (Delta-L) divided by its original length (L). Strain is essential for understanding material behavior, as it relates to how much a material can stretch or compress before failure.

💡Uniaxial Loading

Uniaxial loading is a type of stress condition where all the applied loads act along the same axis. The video uses a solid metal bar being pulled by two equal but opposite forces as an example of uniaxial loading. This type of loading is simple and allows for the development of normal stress, which is perpendicular to the direction of the applied force.

💡Normal Stress

Normal stress is the internal force acting perpendicular to the cross-section of a material. It is named 'normal' because it acts at a right angle to the surface. In the video, normal stress is calculated by dividing the applied force by the cross-sectional area of the bar, and it is denoted by the Greek letter sigma. Normal stress can be either tensile (when the material is being stretched) or compressive (when the material is being compressed).

💡Tensile Stress

Tensile stress is a type of normal stress that occurs when a material is being pulled or stretched. The video explains tensile stress by describing the forces that cause the metal bar to stretch. Tensile stress is positive according to the sign convention used in the video, and it is a critical factor in determining the strength and ductility of materials.

💡Compressive Stress

Compressive stress is the opposite of tensile stress, where the material is subjected to forces that try to push it together or shorten it. Although not explicitly detailed in the provided script, compressive stress would be relevant if the forces were trying to shorten the bar instead of stretching it. It is mentioned in the context of normal stress being either tensile or compressive.

💡Shear Stress

Shear stress is the internal force that acts parallel to the surface of a material, causing it to slide or deform in a direction parallel to the applied force. In the video, shear stress is introduced by imagining a load perpendicular to the bar's axis, which would cause shear forces within the material. Shear stress is denoted by the Greek letter tau and is calculated similarly to normal stress but accounts for the force's orientation.

💡Stress-Strain Diagram

A stress-strain diagram is a graphical representation that illustrates the relationship between the stress applied to a material and the resulting strain. The video mentions that these diagrams are different for different materials and can be obtained through tensile tests. The initial linear region of the diagram, where stress and strain are directly proportional, is described by Hooke's law.

💡Hooke's Law

Hooke's law states that within the elastic limit, the stress and strain in a material are directly proportional. This linear relationship is described by Young's modulus, which is a measure of a material's stiffness. The video explains that Hooke's law typically applies to small strains and that the relationship between stress and strain is linear in the initial part of the stress-strain diagram.

💡Young's Modulus

Young's modulus is a measure of a material's stiffness, defined as the ratio of stress to strain in the elastic region of the material. It is an important material property that helps predict how much a material will deform under a given stress. The video mentions Young's modulus in the context of the linear relationship between stress and strain as described by Hooke's law.

Highlights

Stress and strain are fundamental in describing a body's response to external loads.

Uniaxial loading is when all applied loads act along the same axis, causing the bar to stretch.

Internal forces develop within a bar to resist applied forces, maintaining equilibrium.

Stress is the internal force per unit area, measured in Pascals (Newtons per meter squared).

Normal stress is calculated as the applied force divided by the cross-sectional area of the bar.

Material strength dictates when an object will fail under stress, such as 250 MPa for mild steel.

Normal stress can be tensile (stretching) or compressive (shortening), with positive and negative values respectively.

Strain measures deformation within a body, calculated as change in length over original length.

Stress-strain diagrams, obtained from tensile tests, show the relationship between stress and strain for different materials.

Hooke's law defines the linear relationship between stress and strain for small deformations.

Young's modulus is the ratio of stress to strain, an important material property for elasticity.

Shear stress is caused by forces oriented parallel to the cross-section, common in bolts.

Shear stress is calculated as the applied force divided by the cross-sectional area, resulting in an average value.

Shear strain is the change in angle caused by shear stress, denoted by the Greek letter gamma.

The stress state at a point includes both normal and shear stress components, varying with the observation plane's angle.

The stress element represents the normal and shear stresses acting at a single point within a body.

Understanding stress and strain is crucial for advanced topics like torsion and beam bending.

For more on normal and shear stresses, the video on stress transformation is recommended.

Transcripts

Stress and strain are fundamental concepts that are used to describe how a body responds

to external loads.

In this video we'll explore these concepts using the simple example of a loaded bar.

Here we have a solid metal bar that is loaded by two equal but opposite forces.

We refer to this as uniaxial loading, because all of the applied loads are acting along

the same axis.

The two forces are pulling the bar, causing it to stretch.

Internal forces will develop within the bar to resist these applied forces.

We can expose these internal forces by making an imaginary cut through the bar.

I chose to remove the right side of the bar, but I could have removed the left side instead.

For any imaginary cut like this one, the internal forces develop in such a way that equilibrium

will be maintained.

In this case the effect of the internal forces acting on the cross-section created by our

cut will be equal to the effect of the applied external force.

I have represented the internal forces as 4 separate forces here, but I could have represented

them as one or even 20 forces.

In reality the internal forces are distributed over the entire surface of the cross-section.

For this reason it doesn't make much sense to talk about specific internal forces.

Instead it is better to talk about stress.

Stress is a quantity that describes the distribution of internal forces within a body.

It makes it easier to discuss the internal state that develops within a body as it responds

to externally applied loads.

Stress is a measure of the internal force per unit area, and so has units of

Newtons per meter squared in SI units and pounds per square inch in US units.

Newtons per meter squared are also called Pascals.

In the case of our axially loaded bar, the internal forces are acting perpendicular to

the direction of the cut we made.

This type of stress is called normal stress.

We can calculate the normal stress in our bar as the applied force F divided by the

cross-sectional area A of the bar.

It is denoted by the Greek letter sigma.

One reason being able to calculate stresses is important is because it allows us to predict

when an object will fail.

Let's say our bar is made from mild steel, which has a strength of 250 MPa.

The bar will fail when the stress within it exceeds the strength of the material.

If our bar has a diameter of 20 mm, for example, we can calculate that it will

fail if the applied force is larger than 79 kN.

Normal stress can be either tensile or compressive.

In this case the stress is tensile because the forces are stretching the bar.

If the forces were trying to shorten the bar, we would have a compressive stress.

The sign convention that is normally used is that tensile stresses are positive values

and compressive stresses are negative values.

In the case of our bar it is reasonable to assume that the stresses are distributed uniformly

across the cross-section and along the length of the bar, but this is a very simple scenario.

The stress distribution in a beam that is bending, for example, will be more complex.

Stresses will be tensile on one side of the cross-section, but compressive on the other.

Strain is a quantity that describes the deformations that occur within a body.

If we fix our bar at one end and apply a force to the other end, the force will cause the

bar to deform.

The normal strain within our bar associated with this deformation can be calculated as

the change in length of the bar Delta-L divided by the original length L.

Strain is a non-dimensional quantity, and is often expressed as a percentage.

Normal strains can be tensile or compressive.

I mentioned earlier that the concepts of stress and strain are closely linked.

The relationship between the two can be described using a stress-strain diagram.

Stress-strain diagrams are different for different materials.

We can obtain the diagram for a specific material by performing a tensile test.

This involves applying a known force to a test piece, and measuring the stress and strain

in the test piece as the applied force is increased.

Stress-strain diagrams for ductile materials like this one show that there is an initial

region for low strain values where the relationship between stress and strain is linear.

Deformations occurring in this region are fully reversed when the load is removed, and

so are said to be elastic.

This linear relationship between stress and strain is defined by Hooke's law.

The ratio between stress and strain is called Young's modulus,

which is an important material property.

Hooke's law usually only applies for small strains.

For larger strains, the relationship between stress and strain is no longer linear.

Deformations are not reversed when the load is removed,

and we have permanent plastic deformation.

You can learn more about stress strain curves in my videos about Young's modulus, and about

material strength, ductility and toughness.

So far we have only talked about normal stress, which is stress acting

perpendicular to a surface.

The other type of stress is shear stress.

If our bar isn't loaded along its axis, but instead perpendicular to its axis, like this,

the internal forces that develop within it are oriented parallel to the bar's cross section.

These internal forces are called shear forces.

Shear loading is common in bolts, for example.

Once again it is helpful to use the concept of stress to talk about the internal shear

forces within the bar.

Shear stress is denoted by the Greek letter tau, and can be calculated in a similar way

to normal stress, as the applied force F divided by the cross-sectional area A.

This is actually an average shear stress, since the internal forces will not be distributed

evenly across the cross-section.

We can better understand shear stresses by looking at the stresses acting on a small

element within our bar.

We have a shear stress on one face of the element.

But the element needs to be in equilibrium, so we must also have shear stress on the opposite

face, in the opposite direction.

And to maintain rotational equilibrium we must also have two additional shear stresses,

as shown here.

These four stresses all have a magnitude equal to tau, and define the shear stresses acting

at a single location.

Shear stresses cause a rectangular object to deform like this.

We have deformation, so of course we also have strain.

Shear strain is defined as the change in angle shown here,

and is denoted by the Greek letter gamma.

Hooke's law also applies for shear stresses and shear strains, but the ratio between them

is the shear modulus G instead of Young's modulus.

Although I have discussed normal and shear stresses separately so far, the stress state

at a single point within a body will actually have components in both the normal and the

shear directions.

The magnitudes of the normal and shear components will depend on the angle of the plane we are

using to observe the stresses.

In our bar with uniaxial loading, the plane we used to make the imaginary cut was perpendicular

to the axis of the bar, and so we had normal stresses but no shear stresses.

If we instead use an inclined plane to cut the bar, we will have both normal and

shear stress components.

The stress element is commonly used to represent the stresses acting at a single point within

a body.

This is the stress element showing the normal and shear stresses acting at a single point

for a two dimensional case.

For a three dimensional case the stress element looks like this.

That's it for this introduction to stress and strain.

Having a solid understanding of these concepts will be important for grasping more advanced

topics like torsion and beam bending, that I will cover in separate videos.

If you are interested in learning more about normal and shear stresses I recommend that

you watch my video about stress transformation next!

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