Plate cantilever stress and force theory

1. Jun 15, 2011

chan40

Hi all,

I have a cantilever that is a plate (not a beam) with dimensions 2mm wide x 4mm long x 0.3mm thick. It is fixed only at one side, with the other 3 sides free. I want to be able to find the intrinsic stress on it after it is fabricated.

To complicate matters, I also want to put biological cells (which can be assumed as a thin film) on the cantilever plate and calculate the stress that the cells exert on it based on the measured deflection. Further, these cells also beat and exert a force on the cantilever.

Is it possible to be able to:
(1) Find the intrinsic stress based on measured deflection
(2) Find the stress caused by cells (as a thin film) based on measured deflection
(3) Find the force caused by beating cells (as a thin film) based on measured deflection
all for a 'plate cantilever'?

I've tried to study Timoshenko's book and look for similar equations in Rourke's book, but could not find one similar in this case. I know this sounds like 3 complicated cases, but would really appreciate some help if anyone has insight.

Thanks!

2. Jun 15, 2011

Studiot

You need to describe the distribution of the cells on the plate for loading calculation purposes.

I do not understand what you mean by the cells loading the plate by beating?

What deflections do you propose to measure?

3. Jun 15, 2011

chan40

Hi Studiot,

You need to describe the distribution of the cells on the plate for loading calculation purposes.

It can be assumed as a distributed load (uniformly seeded on top of the cantilever plate).

I do not understand what you mean by the cells loading the plate by beating?

Sorry, I did not describe it fully. These are cardiac cells which have the ability to contract synchronously (a thin sheet of cells ~10um thick which beats in unison on the plate and causes it to bend further). So I would like to measure the deflection that the beating causes on the cantilever plate and calculate the force that the cells exert.

What deflections do you propose to measure?

I'm not sure if I understand this question correctly - but I plan to measure the z-deflection at the free end of the cantilever from it's original position (ex. before and after cells are loaded) to its end position (after the cells exert the stress and bend it down).

Thanks.

4. Jun 15, 2011

DickL

It sounds as though you want to use the plate as a scale to measure the weight of the cells and the force their beating applies to the beam/plate. I'm not clear why you need the stress in the plate to do this. A distributed load from something that only 1/30th as thick (and probably is a lot less dense) means this is a pretty delicate scale.

For case 2, the deflection can be used to measure the additional loading. Finding the zero load deflection maybe a bit harder, as the beam/plate's weight would need to be re leaved to get the no load / no deflection case. You might be able to turn the beam/plate vertically for that, but the deflection measurement would probably be tricky.

Its not clear that case 3 would impose a change in loading. Sounds as though the beating is similar to flexing a muscle, which would not change the loading on the surface the muscle is resting upon. The beating doesn't add mass does it; what would the beating force react against in order to also push against the plate.

5. Jun 15, 2011

chan40

Hi DickL,

This kind of gets away from the focus of my question, but, despite its thickness, the contractile forces of cardiac cells are quite strong, especially when grouped together. When the cells contract, they do so inward toward their cell body and not only in the XY, but the Z as well. I've already recorded how each beat (contraction, followed by relaxation) can and does push against the plate to cause a temporary deflection.

I've measured the deflections for each of these cases already. Are there any closed-form plate cantilever equations that I can use to calculate the stresses/forces in these cases? Can I assume it a beam cantilever and use those equations instead?

Thanks.

6. Jun 15, 2011

Mapes

Look at Ben Freund's Thin Film Materials. He handles the deflection of materials under tangential traction (which is what cells exert on their substratum). Have you looked at Scott Manalis' work on characterizing individual cells by cantilever resonant frequency? It may give you some ideas.

How do you plan on measuring cantilever deflection in solution?

7. Jun 16, 2011

Studiot

I am still not sure about the 'beating' question. I am guessing that you are thinking of resonance with all the organisms beating in synchronicity a la soldiers marching over a bridge and their step inducing vibrations in the structure?

If this is the case there are published tabulated solutions.

The attachments ( an extract from Pilkey) shows solutions for your particular case and the notations for the tables.

You mention Timoshenko. But which book? He has written quite a few.

On page 210 of his "Theory of Plates and Shells" he treats a cantilever plate loaded with a point load at the centre of the free edge. Sorry he does not explicitly treat a distributed load.

I will investigate further.

Have you access to a University library if I post further references?
Can you set up an FE model?

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Last edited: Jun 16, 2011
8. Jun 16, 2011

Mapes

chan40 is talking about the tangential forces that attached cells exert on an adjacent surface. Cells pull at their surroundings; each cell is like a little compression dipole. If one had a sufficiently compliant cantilever, one could estimate the magnitude of the compression by seeding cells on it and looking at how much the cantilever curls up (slightly). The same measurement is performed in the microfabrication context to check the intrinsic stress of thin films.

9. Jun 16, 2011

SteamKing

Staff Emeritus
Here's the \$64 question: What material is this plate fabricated out of?