Plate Deflection: Calculating Central & Eccentric Loads

[minger]

Deflection in "Plate"

Hi, question is kind of based on another problem I'm trying to solve. I'm trying to find a max force from a falling object without knowing deflection or time, but that thread is in College Homework Help, because the problem to me resembles it.

Anyways, my approach to that problem involves calculating deflection in a plate. I basically have a long narrow plate, about 40' x 1' with a central load. Come to think of it, I'm going to need equations for a central load, and eccentric. Anyways, if anyone knows of those, or a good reference, I would greatly appreciate it.

 

[FredGarvin]

In stead of using a plate, I would recommend using a long, thin beam that you would configure as a cantilever. Drop your object on the end of the beam. Beam equations are much less involved, easier to calculate and much more readily available.

If you insist on going with a plate, I would suggest a circular plate. There is much more documentation done on loading of circular plates.

 

[FredGarvin]

Here is the calculation for max deflection of a plate with dimensions of a (long side) and b (short side) with a uniform load over a small circle of radius at the center of the plate, and simply supported sides:

$$Max \Delta = \frac{-\alpha Wb^2}{Et^3}$$

Where:
Max $$\Delta = \mbox{maximum deflection}$$
W = $$\mbox{load}$$
$$\alpha$$ = $$\mbox{constant based on geometry of plate}$$
$$b = \mbox{short plate dimension}$$
$$E = \mbox{Young's Modulus}$$
$$t = \mbox{plate thickness}$$

For $$\alpha$$ use the following values:

a/b = 1.0, $$\alpha$$= .1267
a/b = 1.2, $$\alpha$$= .1478
a/b = 1.4, $$\alpha$$= .1621
a/b = 1.6, $$\alpha$$= .1715
a/b = 1.8, $$\alpha$$= .1770
a/b = 2.0, $$\alpha$$= .1805
a/b = $$\infty$$, $$\alpha$$= .1851




Ref: Roark's Formulas For Stress and Strain. Table 26 "Formulas For Flat Plates With Straight Boundaries and Constant Thickness."
Load case 1b.

 

[Q_Goest]

Just wanted to add to what Fred mentioned. Roark's is the bible of stress analysis, and it's unfortunate that all college stress analysis courses don't use it. I wonder why? Certainly once you graduate, you'll quickly become introduced to it if you do any stress analysis. Perhaps because it's basically a short cut and there's not a lot of theory in it... or at least the practical side is emphasized more than the theoretical.

 

[brewnog]

In addition to what Fred said, I might just say that with an aspect ratio of 40:1, you would be quite justified in modelling your problem as a beam rather than a plate. If your load is applied centrally, the bending along the length will be far more significant than the bending across the width, even if the load isn't applied across the thickness of the beam.

 

[minger]

Yes, thank you guys very much. A structural engineer let me borrow that text, it seems amazing. I just now need to go out into the field to make sure I know how the "plate" is supported, thanks again.

 

[kashoo]

I think u should better see Timoshenko Modified Method ,Roark,s Method for Plates and Shells...and in case of sandwich plates Military Handbook 23A approach...These are very useful techniques to find out deflection,shear stress,stiffness of plates...if u need some help regarding SAndwich plates of Honeycomb...i can give u a lot of details including FEA ,Roarks,Timoshenko,Military Handbook 23A ,& ASTM approaches.

 

[FredGarvin]

Kashoo,
In case you missed it, my reference was from Roark's. The "Methods of Plates and Shells" as you put it does not exist in that reference. The reference I gave was from flat plate bending theory (chapt. 10). The closest is "Shells of Revolution; Presure Vessels and Pipes" which doesn't really apply, does it?

 

[minger]

I seen the equation you used, and instead of simply supported, I decided to go with a plate that is fixed on all 4 sides, (as I was told that the floor was pretty welded all around) and the load is applied uniformly over a small cocentric circle. I figured this better approximated my load as opposed to uniformly on the entire area.

I ended up getting reasonable numbers after all was said and done. Factor of safety is a little over 2, so hopefully I'm good.

 

[FredGarvin]

Good call on the constraints. I am interested to find out the overall results of your method. It's one of those ideas that sounds good but have never had the opportunity to try out.

 

[brewnog]

Yeah, it seems your loading conditions *aren't* how we imagined them to be!

 

[minger]

Well, I am "done" with my analysis. It seems to be very skewed, so I'm not sure if it's correct or not. The problem is that the load causes a quite significant displacement of the plate. This significant displacement ends up LOWERING the force because of the energy to work relationship (assuming all of the kinetic energy is transferred to work done on the plate). Because of this, making the thickness of the plate smaller, ends up lowering the pressure applied.

So...I'm not sure what I'm going to do right now. The structural expert won't be available for a couple more days, I hope they don't need an answer soon.

 

[FredGarvin]

Are you also sure that you are not going into plastic deformation range of the material? That could also be a source of error.

Is it possible to do a test drop? If so, perhaps you could look into dropping a mass on a similar type of surface and use an accelerometer to measure the impulse waveform. That would give you your highest acceleration experienced and thus max force. Just a thought.

 

[minger]

I don't think that's feasible. Our mass is too much for our testing capabilities. Just this morning however, I was given a couple of pages out of a 1940's textbook that had approximations for impact force, nothing that I've ever seen before. They use impact factors which mulitiply the static force, and the impact factor is based on static deflection, which I can find. Here are the equations (sorry, haven't mastered the laxtype thing yet)
Impact Factors for Loads on Members
Static Load: 1.0
Suddenly Applied: 2.0
Suddenly Applied and Reversed: 3.0
Dropped from a height h: k
where k = 1 + ((d² + dv²)^.5)/d
and v = velocity
From here I should be able to find my stresses. Has anyone else ever seen these formulas?

 

[FredGarvin]

I have not seen that before. I'm taking d to be deflection?

 

[minger]

Yes sir, d is the static deflection. I showed them to the other guys here in the office and none of them have seen these either. It's weird because the entire book is handwritten, it's pretty cool. It has a lot of useful stuff in it too. I'd kinda like to see where this is derived from. I'm almost sure it's just an approximation, but I'd still like to find out where it comes from.

 

[brewnog]

I've seen some very similar formulae, but only for a bar in tension.

 

[multipeditalist]

Deflection Help!

Dear friendes, I have just measured the strain of a composite plate at it's surface. It was an impact test. I have only epslon x. How can I find the plate defection? Plate size is equal to 10*10*0.2 cm.