young's modulus uniform bending experiment calculations

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Young’s Modulus Formula and Example

Young’s modulus  ( E ) is the modulus of elasticity under tension or compression. In other words, it describes how stiff a material is or how readily it bends or stretches. Young’s modulus relates stress (force per unit area) to strain (proportional deformation) along an axis or line.

The basic principle is that a material undergoes elastic deformation when it is compressed or extended, returning to its original shape when the load is removed. More deformation occurs in a flexible material compared to that of a stiff material.

• A low Young’s modulus value means a solid is elastic.
• A high Young’s modulus value means a solid is inelastic or stiff.

The behavior of a rubber band illustrates Young’s modulus. A rubber band stretches, but when you release the force it returns to its original shape and is not deformed. However, pulling too hard on the rubber band causes deformation and eventually breaks it.

Young’s Modulus Formula

Young’s modulus compares tensile or compressive stress to axial strain. The formula for Young’s modulus is:

E = σ / ε = (F/A) / (ΔL/L 0 ) = FL 0 / AΔL = mgL 0 / π r 2 ΔL

• E is Young’s modulus
• σ is the uniaxial stress (tensile or compressive), which is force per cross sectional area
• ε is the strain, which is the change in length per original length
• F is the force of compression or extension
• A is the cross-sectional surface area or the cross-section perpendicular to the applied force
• ΔL is the change in length (negative under compression; positive when stretched)
• L 0 is the original length
• g is the acceleration due to gravity
• r is the radius of a cylindrical wire

Young’s Modulus Units

While the SI unit for Young’s modulus is the pascal (Pa). However, the pascal is a small unit of pressure, so megapascals (MPa) and gigapascals (GPa) are more common. Other units include newtons per square meter (N/m 2 ), newtons per square millimeter (N/mm 2 ), kilonewtons per square millimeter (kN/mm 2 ), pounds per square inch (PSI), mega pounds per square inch (Mpsi).

Example Problem

For example, find the Young’s modulus for a wire that is 2 m long and 2 mm in diameter if its length increases 0.24 mm when stretched by an 8 kg mass. Assume g is 9.8 m/s 2 .

First, write down what you know:

• Δ L = 0.24 mm = 0.00024 m
• r = diameter/2 = 2 mm/2 = 1 mm = 0.001 m
• g = 9.8 m/s 2

Based on the information, you know the best formula for solving the problem.

E = mgL 0 / π r 2 ΔL = 8 x 9.8 x 2 / 3.142 x (0.001) 2 x 0.00024 = 2.08 x 10 11 N/m 2

Despite its name, Thomas Young is not the person who first described Young’s modulus. Swiss scientist and engineer Leonhard Euler outlined the principle of the modulus of elasticity in 1727. In 1782, Italian scientist Giordano Riccati’s experiments led to modulus calculations. British scientist Thomas Young described the modulus of elasticity and its calculation in his  Course of Lectures on Natural Philosophy and the Mechanical Arts  in 1807.

Isotropic and Anisotropic Materials

The Young’s modulus often depends on the orientation of a material. Young’s modulus is independent of direction in isotropic materials . Examples include pure metals (under some conditions) and ceramics. Working a material or adding impurities forms grain structures that make mechanical properties directional. These anisotopic materials have different Young’s modulus values, depending on whether force is loaded along the grain or perpendicular to it. Good examples of anisotropic materials include wood, reinforced concrete, and carbon fiber.

Table of Young’s Modulus Values

This table contains representative Young’s modulus values for various materials. Keep in mind, the value depends on the test method. In general, most synthetic fibers have low Young’s modulus values. Natural fibers are stiffer than synthetic fibers. Metals and alloys usually have high Young’s modulus values. The highest Young’s modulus is for carbyne, an allotrope of carbon.

Modulii of Elasticity

Another name for Young’s modulus is the elastic modulus , but it is not the only measure or modulus of elasticity:

• Young’s modulus describes tensile elasticity along a line when opposing forces are applied. It is the ratio of tensile stress to tensile strain.
• The bulk modulus (K) is the three-dimensional counterpart of Young’s modulus. It is a measure of volumetric elasticity, calculated as volumetric stress divided by volumetric strain.
• The shear modulus or modulus of rigidity (G) describes shear when opposing forces act upon an object. It is shear stress divided by shear strain.

The axial modulus, P-wave modulus, and Lamé’s first parameter are other modulii of elasticity. Poisson’s ratio may be used to compare the transverse contraction strain to the longitudinal extension strain. Together with Hooke’s law, these values describe the elastic properties of a material.

• ASTM International (2017). “ Standard Test Method for Young’s Modulus, Tangent Modulus, and Chord Modulus “. ASTM E111-17. Book of Standards Volume: 03.01.
• Jastrzebski, D. (1959).  Nature and Properties of Engineering Materials  (Wiley International ed.). John Wiley & Sons, Inc.
• Liu, Mingjie; Artyukhov, Vasilii I.; Lee, Hoonkyung; Xu, Fangbo; Yakobson, Boris I. (2013). “Carbyne From First Principles: Chain of C Atoms, a Nanorod or a Nanorope?”.  ACS Nano . 7 (11): 10075–10082. doi: 10.1021/nn404177r
• Riccati, G. (1782). “Delle vibrazioni sonore dei cilindri”. Mem. mat. fis. soc. Italiana . 1: 444-525.
• Truesdell, Clifford A. (1960).  The Rational Mechanics of Flexible or Elastic Bodies, 1638–1788 : Introduction to Leonhardi Euleri Opera Omnia, vol. X and XI, Seriei Secundae. Orell Fussli.

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Here’s How to Calculate Young’s Modulus

And why it’s important to know this equation

Whether you’re fresh out of college or it’s been some time since you’ve taken math classes, it’s always helpful to review equations that frequently show face in the work that you do. One example of this that we’d like to cover in our engineering fundamental series is Young’s modulus, which we’ll define below and then break down how to calculate it.

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Young’s modulus points to the slope of a stress strain curve and can help show when a material is simply stretching and holding up okay under stress versus when it’s bound to deform or break. It’s an essential insight into the behavior of a material, and can help determine which materials are best to use based on this number and other factors. 

As you can see in this image, this style of graph can help show Young’s modulus, the ultimate strength point of a material, and at what point you can expect deformation or breakage to occur.

Steps to Calculate Young’s Modulus

To calculate Young’s modulus of elasticity, follow these steps:

1. Measure the Original Length of Your Material with a Micrometer

Use a micrometer to measure the original length (L0) of the material you’re working with, but do this before any force is applied or stretching occurs so that you can get the most accurate answer.

2. Use the Micrometer to Measure the Cross-Sectional Area and Other Diameters

Measure the width and height of your material, then measure out any other diameters it has and note those down. Doing so will help you clock any irregularities and get the most accurate number for the cross-sectional area, which you’ll need to calculate Young’s modulus.

3. Get Slotted Masses Into Place and Apply Force to the Material

Set up several slotted masses and attach them to your material. These will help you better control the tension and deformation process.

4. Use a Vernier Scale to Measure the Different Lengths

Once you start applying force and tension with the slotted masses, you can measure the various lengths of your material as it extends with a vernier scale.

5. Graph Your Measurements

Using a graphing calculator or a computer, create a graph using your measurements so you have a visual representation of your experimental data and can accurately show the important points on the graph.

6. Calculate Young’s Modulus 

Now that you have all the necessary information, you can complete the calculation to find your material’s stiffness and elasticity. You’ll get it by dividing the tensile stress by the tensile strain and then adding in other important information, which you’ll see examples of below. 

Use this equation to do find Young’s modulus: E = Tensile Stress / Tensile Strain = (FL) / (A * Change in L)

7. Analyze Your Graph and Note the Most Important Points

Once you have the graph and equation calculated, double-check your graph and ensure that you can see the clear linear region where elastic behavior for your specific material is shown. If a material has already been stretched and deformed, it’s not likely that you’ll be able to correctly and accurately complete this equation. 

Example Calculations with Different Materials

To really help this concept stick, take a look at the following calculations that find the elasticity modulus of different materials:

In this example, we’ll look at a steel rod and find Young’s modulus for it. Consider the equation if the following factors are true:

• The rod has an original length (L0) of 2 meters
• It has a final length (Ln) of 2.04 meters
• The cross-sectional area (A) is 1 square centimeter (0.0001 square meters)
• To get the elongation you’ll subtract L0 from Ln (which will be 4 cm in this case)
• The force (F) applied is 1,000 Newtons (N)

Here’s how you would calculate this: 

Equation: Young’s modulus = (F x L0) / (A x (Ln - L0))

With numbers inputted: Young’s modulus = (1,000 N x 2 m) / (0.0001 m2 x 0.04 m)

Answer: Young’s modulus ≈ 5 x 108 Pascal (Pa)

2. Aluminum

If you’d like to calculate the same equation but with an aluminum rod, here’s how to do so using sample numbers and with less force:

• The cross sectional area (A) is 1 square centimeter (0.0001 square meters)
• To get the elongation you’ll subtract L0 from Ln (which will be 4 cm)
• The force (F) applied is 800 Newtons (N)

With numbers inputted: Young’s modulus = (800 N x 2 m) / (0.0001 m2 x 0.04 m)

Answer: Young’s modulus ≈ 4 x 108 Pascal (Pa)

Here is how this calculation could be determined if we use a rubber band and possible measurements for this type of material: 

• The rubber band has an L0 of 10 centimeters
• It has an Ln of 15 centimeters
• The elongation is 5 centimeters (Ln-L0)
• The force applied is 20 Newtons

With numbers inputted: Young’s modulus = (20 N x 0.1 m) / (0.0001 m2 x 0.05 m)

Answer: Young’s modulus ≈ 4 x 105 Pascal (Pa)

When you compare all of these different materials, you’ll notice that the Young’s moduli for rubber is a much lower magnitude than steel and aluminum, meaning that the former material is more flexible and elastic, whereas the latter materials are stiffer.

Why It’s Useful to Calculate Young’s Modulus

Engineers use this equation in real-world scenarios for several reasons: 

• Understanding materials: Knowing the Young’s modulus for a certain material will help you get to know it better and help predict what environments and situations it can be put through with success.
• Design considerations : When a structure is relying on a specific material or a human needs a particular implant, calculating Young’s modulus will help determine which materials to use, as you’ll have a better understanding of which ones can withstand the stresses. 
• Stress calculation: You’ll need to know Young’s modulus in order to calculate the stress in a material — another vital factor in engineering and design. 
• Predicting material failure: With the modulus of elasticity, you can have a much better and more accurate prediction of the point at which a material will break or deform for good under pressure. 

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Young's Modulus Calculator

What is the modulus of elasticity.

• Young's modulus equation
• How do I calculate Young's modulus?

Example using the modulus of elasticity formula

• How to calculate Young's modulus from a stress-strain curve

With this Young's modulus calculator, you can obtain the modulus of elasticity of a material, given the strain produced by a known tensile/compressive stress .

We will also explain how to automatically calculate Young's modulus from a stress-strain curve with this tool or with a dedicated plotting software.

Keep reading to learn more about:

• What the modulus of elasticity is;
• How to calculate Young's modulus with the modulus of elasticity formula;
• What Young's modulus unit is;
• What material has the highest Young's modulus; and more.

Young's modulus , or modulus of elasticity , is a property of a material that tells us how difficult it is to stretch or compress the material in a given axis.

This tells us that the relation between the longitudinal strain and the stress that causes it is linear. Therefore, we can write it as the quotient of both terms.

💡 Read more about strain and stress in our true strain calculator and stress calculator !

However, this linear relation stops when we apply enough stress to the material. The region where the stress-strain proportionality remains constant is called the elastic region .

If we remove the stress after stretch/compression within this region, the material will return to its original length .

Because of that, we can only calculate Young's modulus within this elastic region, where we know the relationship between the tensile stress and longitudinal strain.

🙋 If you want to learn how the stretch and compression of the material in a given axis affect its other dimensions, check our Poisson's ratio calculator !

Young's modulus equation

Before jumping to the modulus of elasticity formula, let's define the longitudinal strain ϵ \epsilon ϵ :

• L 0 L_{0} L 0 ​ is the material's initial length; and
• L L L is the length while being under tensile stress.

And the tensile stress σ \sigma σ as:

• F F F is the force producing the stretching/compression; and
• A A A is the area on which the force is being applied.

Thus, Young's modulus equation results in:

Since the strain is unitless, the modulus of elasticity will have the same units as the tensile stress (pascals or Pa in SI units).

How do I calculate Young's modulus?

To calculate the modulus of elasticity E of material, follow these steps:

Measure its initial length, L₀ without any stress applied to the material.

Measure the cross-section area A .

Apply a known force F on the cross-section area and measure the material's length while this force is being applied. This will be L .

Calculate the strain ϵ felt by the material using the longitudinal strain formula: ϵ = (L - L₀) / L₀ .

Calculate the tensile stress you applied using the stress formula: σ = F / A .

Divide the tensile stress by the longitudinal strain to obtain Young's modulus: E = σ / ϵ .

Let's say we have a thin wire of an unknown material, and we want to obtain its modulus of elasticity.

Assuming we measure the cross-section sides, obtaining an area of A = 0.5 × 0.4 mm . Then we measure its length and get L₀ = 0.500 m .

Now, we apply a known force, F = 100 N for example, and measure, again, its length, resulting in L = 0.502 m .

Before computing the stress, we need to convert the area to meters:

A = 0.5×0.4 mm = 0.0005 × 0.0004 m

With those values, we are now ready to calculate the stress σ = 100/(0.0005 × 0.0004) = 5×10⁸ Pa and strain ϵ = (0.502 - 0.500) / 0.500 = 0.004 .

Finally, if we divide the stress by the strain according to the Young's modulus equation, we get: E = 5×10⁸ Pa / 0.004 = 1.25×10¹¹ Pa or E = 125 GPa , which is really close to the modulus of elasticity of copper ( 130 GPa ). Hence, our wire is most likely made out of copper!

How to calculate Young's modulus from a stress-strain curve

Our Young's modulus calculator also allows you to calculate Young's modulus from a stress-strain graph !

To plot a stress-strain curve, we first need to know the material's original length , L 0 L_{0} L 0 ​ . Then, we apply a set of known tensile stresses and write down its new length , L L L , for each stress value.

Lastly, we calculate the strain (independently for each stress value) using the strain formula and plot every stress-strain value pair using the Y Y Y -axis and X X X -axis, respectively.

Analysing the stress-strain chart

As you can see from the chart above, the stress is proportional (linear) to the strain up to a specific value . This is the elastic region, and after we cross this section, the material will not return to its original state in the absence of stress.

Since the modulus of elasticity is the proportion between the tensile stress and the strain, the gradient of this linear region will be numerically equal to the material's Young's modulus.

We can then use a linear regression on the points inside this linear region to quickly obtain Young's modulus from the stress-strain graph.

Our Young's modulus calculator automatically identifies this linear region and outputs the modulus of elasticity for you . Give it a try!

Is stiffness the same as Young's modulus?

No, but they are similar . Stiffness is defined as the capacity of a given object to oppose deformation by an external force and is dependent on the physical components and structure of the object. Young's modulus is an intensive property related to the material that the object is made of instead.

Is tensile modulus the same as Young's modulus?

Yes . Tensile modulus is another name for Young's modulus, modulus of elasticity, or elastic modulus of a material. It relates the deformation produced in a material with the stress required to produce it.

What material has the highest Young's modulus?

Diamonds have the highest Young's modulus or modulus of elasticity at about ~1,200 GPa . Diamonds are the hardest known natural substances, and they are formed under extreme pressures and temperatures inside Earth's mantle.

Is the modulus of elasticity constant?

Yes . Since the modulus of elasticity is an intensive property of a material that relates the tensile stress applied to a material, and the longitudinal deformation it produces, its numerical value is constant. The resulting ratio between these two parameters is the material's modulus of elasticity.

12 volt wire size

Bmr - harris-benedict equation, flat vs. round earth, led resistor.

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Determination of young’s modulus of a bar by Bending method

Abstract 

Introduction   [1]

Experimental diagram

Experimental Data

Table: Data for load vs depression

Calculation

From   the graph (y 0   vs   m) when, load, m = 1400 gm De p r e s si o n,   y 0   =   0.3 Length of the beam, l = 90.3 cm 

Breadth of the beam, b = 2.52 cm 

Depth of the beam, d = 0.617 cm

Acceleration due to gravity, g = 980 cms -2

Young modulus,

Percentage of Error

Conclusion [5]

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