# Cantilever Beam

#### Physics_wiz

I'm an undergrad. student trying to do research. Today was my first day working with grad students and one grad student gave me something to find. We have a cantilever beam made of aluminum nitride (AlN). The only things known about the beam are the dimensions and young's modulus. He wants to find the max load that could be applied to the free end of the beam without the beam breaking. I looked in my mechanics of materials book and found a formula for the displacement of the free end but I don't know how to find the max load. I don't even know if this problem is doable given that we only know the material's young's modulus. I'll try to find some online information and read some more in the mechanics of materials book. In the mean time, any help would be greatly appreciated.

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#### Clausius2

Science Advisor
Gold Member
Physics_wiz said:
I'm an undergrad. student trying to do research. Today was my first day working with grad students and one grad student gave me something to find. We have a cantilever beam made of aluminum nitride (AlN). The only things known about the beam are the dimensions and young's modulus. He wants to find the max load that could be applied to the free end of the beam without the beam breaking. I looked in my mechanics of materials book and found a formula for the displacement of the free end but I don't know how to find the max load. I don't even know if this problem is doable given that we only know the material's young's modulus. I'll try to find some online information and read some more in the mechanics of materials book. In the mean time, any help would be greatly appreciated.
Ok, I congrat you for starting your researching career!. When sometime passes, you will laugh at these first steps!.

1) You need a failure criterion. Usually, it is considered for engineering purposes that if in any section of the beam is reached the plasticity stress, then the beam gets into technical failure. There are another criterions such as Von-Mises and Tresca.

2) Once you have the maximum stress your beam can deal, you have to look for the section in which it will be probably reached. This question is left for you, try to think of it and solve on your own.

3) Knowing the beam dimensions and the Young modulus, you can obtain the displacement which causes that maximum stress in the section chosen, and then the load which should be applied for failure.

Good luck.

EDIT: changed "fracture" by "failure".

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#### Physics_wiz

Thanks for the reply.

1) Is that a property of the material? If I don't have it, do you know what tests or resources it could be obtained from?

2) I'm guessing the max stress will happen in the middle top part.

3) The equation to use is: Max stress = (Young modulus) (change in length/original length). Right? I have the max stress from (1) and young's modulus and the original length of the beam (top part). The change in length is the horizontal change in length of the whole beam (top part) right? After that is found I have to find the max downwards displacement of the free end and plug it in the equation in my book to get the max force. Did I get all that right? Also, if I know the change in length of the top part of the beam, how do I find the max downward displacement of the free end?

Thanks again.

#### Physics_wiz

Hmmmm I take number 2 back. The max stress occurs at the fixed end where the moment is maximum. I don't think I know how to find #3 yet though.

#### Q_Goest

Science Advisor
Homework Helper
Gold Member
Hey physics_wiz. Calculating the stress in a cantalivered beam is one thing. Calculating the load required to break the beam is a different issue because to break the beam, things must happen that are not easily calculable. In a typical beam, deformation occurs along the fibers farthest from the neutral axis because the tensile stress is linearly proportional to the distance from the neutral axis. This can be seen from the equation for stress:

S = Mc/I

Where S = stress
M = Moment
c = distance from neutral axis
I = moment of inertia

As the beam deforms, the stress no longer remains linearly proportional as the equation suggests. As the beam yeilds, the stress in the outermost fibers remains almost constant and stretches, while the fibers at a lesser distance to the neutral axis begin to yield. It would be easier to draw a picture here, but those are the limitations of a message board.

A second issue also needs attention. If the beam is long in relation to the depth, the moment does not increase linearly with increased force. That's because the distance between the wall where the beam is mounted and the force decreases as the beam bends.

Taking these two factors into account, one has a very difficult time determining the breaking point.

But if your beam is relatively short and your material relatively brittle (aluminum nitride is very brittle), it could be a bit easier to determine. Unfortunately, you don't have enough information to determine the breaking point of the beam. You need the ultimate tensile strength. Without that, you have no way of doing this. You could look up text book data for the material, but I wouldn't trust that to within 20%, even if you know everything about the material, so you can assume the calculation you will perform will be off by a fairly large margin.

If I were you, I'd ask the grad student why he wanted to know what load can be applied before breaking. Because that load should never be even close to being applied under normal use. If the beam is being used for something, there are allowable stresses that should never be exceeded in the material. Those allowable stresses are below yield stress by some factor of safety. So you need to find out what the yield and ultimate tensile strength is to work that out.

Just a final note, aluminum nitride is a type of ceramic. See this reference:
http://www.accuratus.com/alumni.html

If the beam is a conventional structural support, I'd be very surpised if it were made from aluminum nitride. Beams aren't generally made from that material. There are aluminum beams, but they are typically aluminum alloy of some type, not aluminum nitride. I'd suggest you verify the material before going any farther.

#### Physics_wiz

Yes it's aluminum nitride. I forgot to mention that this was for a micro switch kindda thing. They're working on nano stuff now (I really don't know what it's for...yet) but this is a part of what they're doing. The Beam will actually deflect by applying a voltage to the fixed end (don't know how that works either...yet). That's what I understood from the guy.

#### Q_Goest

Science Advisor
Homework Helper
Gold Member
Ah - ha.. <lightbulb goes on> That makes a lot more sense now.

If someone wants an estimate of the load needed to break the beam, use the flexure strength to estimate the stress it will break at with the equation I gave above.

If someone wants to know what load can safely be applied, chose a factor of safety below that estimated load. I'd suggest a factor of safety of 3 to 5. (ie: don't exceed a load of 1/3 to 1/5 that which will break the beam) You could use a smaller value, but that should really only be done after you and the others understand what benefits you can get from increasing stress and what possible reprecussions there will be if the beam breaks.

If this beam is under a fatigue load, that's a different story altogether.

So before you go any farther, you'll need to look into material properties. I wouldn't use textbook data if you don't have to. Find out where the beam came from and see if the supplier can give you more accurate strength data.

Better yet, if you have a whole bunch of these things laying around, try breaking them and use that data!

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