Determining Young's Modulus by Bending a Piece of Wood

[Kenny Wong]

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=??

 

[SteamKing]

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.

 

[Chestermiller]

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

 

[Kenny Wong]

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

 

[SteamKing]

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.

 

[Chestermiller]

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