How to calculate Young's Modulus on trapezoidal shape ? Can't find the answer .

[klausner]

How to calculate Young's Modulus on trapezoidal shape ? Can't find the answer...

Hi guys.

I'm really stumbling over this question.

If I'm trying to calculate the Young's modulus of a specimen that is being compressed from the top, but the contact area on top is smaller than the bottom area, which area do I use for the cross section ? The smaller area from the top, or the larger area from the bottom ??
Viewed from the side, it would like a trapezoid being compressed from the top, while sitting on a hard surface.

Thanks for your help.
If anyone can recommend a good textbook or article where that is explained, I would greatly appreciate it.


Rick

 

[Studiot]

What do you mean you are trying to calculate young's modulus?
YM is a characteristic property of the material, not the shape.

What are you trying to calculate it from ie what do you know?

I understand the Xsection you have described.

 

[klausner]

I know the force, the initial vertical height of the sample and the vertical compression distance, and am trying to calculate the Young's modulus, assuming the shape is made up of a homogeneous material.
So I know h and h(0), so I can calculate strain.
But I don't know how to calculate the stress properly. To find the stress, I know force F, but am not sure which area to use (the smaller one on top, or the larger one on the bottom??).

Thanks for your help.

 

[Studiot]

You need to divide your trapezoid into horizontal strips, the width of each will be a linear function of height above the base.

The total load in each strip will be the same so the stress in each strip will be proportional to the width, which in turn will be proportional to the height.

Young's modulus will be the same for each strip.

The compression in each strip will be proportional to the vertical size of the strip, say $$\delta$$h
The total compression is obviously the sum of the individual compressions in each strip.

Armed with the above you can allow the vertical size to tend to zero and the sum becomes an integral in terms of height h.

Does this help?

 

[klausner]

Wow, great.

So is the average elastic modulus equal to the force divided by the AVERAGE area ((A-top + A-bottom) /2) , all divided by the strain?
Since the width of the trapezoid is a linear function of its height, does that mean I can use the average width to get the average area to use in the Elastic modulus calculation??

Thanks so much!