Determining Young's Modulus by Bending a Piece of Wood

In summary, the formula used to calculate Young's Modulus in the experiment involves bending a simply supported beam with a central load applied. The formula is d = PL3/(48EI), where d is the central deflection, P is the applied load, L is the distance between supports, E is the modulus of elasticity, and I is the second moment of area of the cross section. This formula can be derived from the mechanics of bending, where the axial strain on the outside of the bend is tensile and the axial strain on the inside of the bend is compressive, resulting in a bending moment at each cross section. It is recommended to study Strength of Materials to fully understand this derivation.
  • #1
Kenny Wong
10
0
There is an experiment to determine the Young's Modulus of the wood which involves bending it.
The equation used to calculate it is like this
E=WX3/4ZDY3
Where D=amount of bending
W=force (N) which causes the bending
Z=width of the wood
Y=thickness of the wood
X=horizontal distance between the support
I am curious about the derivation of this equation.

Attempts:
Young's modulus = Stress/Strain
= (F/A)/(E/L)
F=W, A=YZ, L=X?, E=??
 

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  • #2
Kenny Wong said:
There is an experiment to determine the Young's Modulus of the wood which involves bending it.
The equation used to calculate it is like this
E=WX3/4ZDY3
Where D=amount of bending
W=force (N) which causes the bending
Z=width of the wood
Y=thickness of the wood
X=horizontal distance between the support
I am curious about the derivation of this equation.

Attempts:
Young's modulus = Stress/Strain
= (F/A)/(E/L)
F=W, A=YZ, L=X?, E=??

This is a bending problem, rather than a tensile test.

The equation used to determine E is derived from the bending of a simply supported beam with a central load applied.

d = PL3/(48EI)

where

d = central deflection
P = applied load
L = distance between supports
E = modulus of elasticity
I = second moment of area of the cross section

In this case I = bt3/12

where

b = breadth of the plank
t = thickness of the plank

Substitute into the equation for deflection, do the algebra, and viola.
 
  • #3
The formula that SteamKing presented was derived on the basis of a Strength of Materials approach. Mechanistically, it involves noting that, when a beam is bent, the axial strain on the outside of the bend is tensile, and the axial strain on the inside of the bend is compressive. So unlike pure tension, where the tensile strain is uniform throughout the beam, in bending the tensile strain varies linearly through the thickness of the beam. This gives rise to a bending moment at each cross section. The bending moment and the bending strain increases in proportion to the load that is applied. I hope this simple mechanistic picture makes sense to you.

Chet
 
  • #4
So the formula is an original formula? Is it possible to explain and derive it in simple terms?

I have checked up the wikipedia on 'Bending' and the equations and explanations are all non-human level. LOL
 
  • #5
Kenny Wong said:
So the formula is an original formula? Is it possible to explain and derive it in simple terms?

I have checked up the wikipedia on 'Bending' and the equations and explanations are all non-human level. LOL

I thought I just did. If you haven't taken a basic strength of materials course, then you probably won't understand the regular procedure to determine Young's modulus either.
 
  • #6
Hey Kenny,

Get yourself a book on Strength of Materials. Every book on Strength of Materials has lots of material on how to solve beam problems.

Chet
 

1. How do you determine Young's Modulus by bending a piece of wood?

To determine Young's Modulus, also known as the modulus of elasticity, you will need a piece of wood, a ruler, and weights. First, measure the dimensions of the wood, including its length, width, and thickness. Then, place the wood horizontally on two supports, with a gap between them. Add weights to the center of the wood until it bends a certain amount. Measure the amount of deflection and use this value, along with the dimensions of the wood and the weight, to calculate Young's Modulus using the formula: E = (3FL) / (4bd3), where E is the modulus of elasticity, F is the applied force, L is the length of the wood, b is the width, and d is the thickness.

2. What is Young's Modulus and why is it important?

Young's Modulus is a measure of the stiffness of a material, specifically its ability to resist deformation under an applied force. It is important in engineering and materials science as it allows us to predict how materials will behave under different conditions and helps us design structures that can withstand various forces and loads.

3. How does bending a piece of wood help determine Young's Modulus?

Bending a piece of wood allows us to apply a known force and measure the resulting deflection, which can then be used to calculate Young's Modulus. This method is known as the flexural test and it is commonly used to determine the stiffness of materials.

4. Are there any factors that can affect the accuracy of the results when determining Young's Modulus by bending a piece of wood?

Yes, there are several factors that can affect the accuracy of the results. These include the type of wood being used, the dimensions and shape of the wood, the accuracy of the measurements, and external factors such as temperature and humidity. It is important to control these variables as much as possible to obtain accurate results.

5. Are there any alternative methods for determining Young's Modulus?

Yes, there are other methods for determining Young's Modulus, such as tension and compression tests, torsion tests, and ultrasonic testing. Each method has its own advantages and limitations, and the choice of method will depend on the material being tested and the desired accuracy of the results.

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