# Mems cantilevers vs. Euler-Bernoulli

1. Sep 28, 2009

### bifurcator

Greetings
I am trying to understand the greatly differing results between calculating the theoretical deflection of a cantilver and simulating it in ANSYS.

The cantilever is 1 um thick, 5 um wide and 300um long, one end is attached, the other end receives a force of 100uN in the y-direction (just at the end). E = 1.69e9, v = 0.066

Using E-B deflection = (4 * F * L^3)/(w * E * t^3)
The solution is ...well a lot (>1m)

In ANSYS, depending upon the meshing, etc deflection seems to be around 26-29um.

So does Euler-Bernoulli become invalid when applied to MEMS structures or when the length is much greater than the width?
is there are better way to calculate a theoretical value for the deflection?

cheers
Andrew

2. Sep 28, 2009

### Mapes

Hi bifurcator, welcome to PF. I get 13 mm deflection by using your equation; is there a chance you've made a numerical error somewhere?

Even still, this is larger than the simulated value; do you have the large displacements flag in ANSYS turned on or off?

3. Sep 28, 2009

### nvn

bifurcator: Perhaps tell us the units on your E value, so we would know what you are asking about. Also, check the units on all of your values, and ensure you are listing them correctly for your question in post 1.

Your Poisson ratio looks strange. Are you sure it is correct? What is the material?

If you intended E = 1.69 GPa, then your cantilever cannot support more than, roughly, F = 20 nN. Therefore, why do you think it should support 5000 times this load?

Last edited: Sep 29, 2009
4. Sep 28, 2009

### bifurcator

Hi Mapes
I get 1.278m, it has been checked, other students in my group had the same result. Either way, 13mm is too large as well. Is there an alternative to E-B using the supplied values? Or should I just report the cantilever is in snapdown/maximum deflection?

It is the first I have heard of the large displacements flag, will try again this morning and report back.

Thanks for helping
Andrew

5. Sep 28, 2009

### bifurcator

Hi nvn

E = 1.69 GPa.
I agree the values are strange. The material is supposedly silicon, in which case E should be more like 169GPa and v ~ 0.25

It would also be nicer if F = 100nN. I have questioned the given values to our lecturer but received no response, which tends to mean my question was wrong.

Thanks for your response, I am thinking that the given (given to me) values are incorrect. Although the results in ANSYS are feasible, which does't help me understand this.

6. Sep 28, 2009

### Mapes

(I used 169 GPa to get the 13 mm value, assuming you were working with single crystal silicon and had made a typo when listing E = 1.69e9. I didn't even realize I'd made the switch; I've worked a lot of MEMS calculations!)

Last edited: Sep 28, 2009
7. Sep 29, 2009

### nvn

bifurcator: I agree that the value listed by Mapes, E = 169 GPa, sounds correct. What is the flexural ultimate strength, Sfu, of your material? If E = 169 GPa and Sfu = ~4 GPa, then a practical upper limit for your applied load is F = 2.3 uN, not 100 uN. And a practical limit for the deflection is 0.3 mm.

After you get the above discrepancies resolved, ensure you are inputting all of your Ansys input values in correct, consistent units.

8. Sep 30, 2009

### bifurcator

hi
Thankyou to everybody that helped.
As suspected the supplied values were incorrect.
It should of been E = 169GPa and a force of 100uN was, of course, too large for the poor beam. So the instructor told us to choose an appropriate value for the force (wish he had told us this 2 weeks ago when I started working on this and asked at that time).
I picked 1uN
Get a deflection of 127.81um and similar results in ANSYS using Brick 20 Node 186 for meshing.
Again - Thankyou!!

9. Sep 30, 2009

### nvn

bifurcator: If you want to make life simpler and better for yourself, you could choose an applied load of F = 100 nN. If you choose an applied load greater than F = 115 nN, then you are going outside the range of small deflection theory, where hand calculations versus actual stresses no longer match well, although it depends on what you input for the stress-strain curve. But if you do want to exceed F = 115 nN, turn on the large displacement static flag, mentioned by Mapes. I don't know the tensile yield strength and tensile ultimate strength of your material.