Interesting ponding/deflection problem

[bugbus]

There was a previous thread about water ponding on slabs that got me thinking about an interesting problem. I have not had any luck solving this theoretically, so for now it remains just a thought experiment.

Imagine a massless plate with elastic modulus E and thickness t that is simply supported on two edges with span of L and width B (shown in (1)). In its unloaded state, the plate is perfectly flat.

The plate is then subjected to some initial bending causing it to deflect by Δ (shown in (2)), and a liquid with unit weight γ is gradually poured onto it. (Ignore the fact the sides of the plate are open).

Finally, the external bending is released so that only the weight of the liquid acts on the plate (shown in (3)).

Questions:

• What happens to the plate once the initial bending is released?
• (A) - Will it spring back to its original flat position, displacing the liquid in the process? Or...
• (B) - Will it settle into a particular deflected shape and achieve equilibrium?
• Is there some critical combination of initial deflection, plate stiffness, etc. that this outcome hinges on?

 

[Denial]

I believe the theoretical answer is B, regardless of the magnitude of the initial (pre-liquid) deflection. Of course you will never get this answer through conventional linear elastic analysis. You would need to allow for the non-linear behaviour associated with large displacement theory. The linearity or non-linearity of the material's behaviour is irrelevant.

 

[Denial]

As an aside, can you give a reference to the "previous thread about water ponding on slabs"? I missed it.

 

[canwesteng]

Well the linearity of the material would matter, since something like reinforced concrete would be become less stiff as it is loaded, though the question seems more theoretical. I suspect for any practical case you don't really need to account for large displacements causing membrane stresses in the panel, only for geometric non linearity (maybe large delta P-delta is what is meant in the comment).

I think either the stiffness of the roof is such that deflection under some unit load (k) is either (1) greater or (1) lesser than the density of water times that deflection, in which case 1. it should flatten out 2. it's a ponding instability and keeps going. Maybe a third case where they happen to be exactly balanced and the roof stays put, though this doesn't seem possible theoretically, since some load is applied to deflect the roof at first. This leads to other problem with scenario 2 - surely if ponding instability occurs after the load is removed, it would have occured while the load was applied as well. So my position is that provided the structure initially and subject to some continuous rainfall, option A is the result of removing the load.

 

[bugbus]

I think I have found a solution (I only say this tentatively) - the derivation is a work in progress (and may be wrong). Essentially this boils down to a 4th order ODE where (d4Δ/dx4)=(Bγ/EI)Δ(x), where Δ(x) is the deflection. The solution is in the form of a sine curve.

Solving, with some algebra, etc., leads to a critical case of: (12γ/Et^3)=(π/L)^4.

Expressed in terms of a critical span length: Lcrit = π(Et^3/12γ)^0.25

I assume that this is the only stable configuration where the weight of the liquid balances the deflection of the beam.

E.g. for a 10 mm thick steel (E = 200,000 MPa) plate filled with water (γ = 10 kN/m3), the span would need to be 3.57 m to reach some equilibrium state.

Anything less and it would spring back to its flat position; anything more and it will become unstable and continue deflecting (assuming there is a continuous supply of liquid to fill the gap that forms).

I will keep working on it. Maybe test it with some FEA to check an example case.

Also interestingly, if the above is all correct, it means that for a plate with the critical span, the actual value of the deflection is arbitrary. The plate will stabilise on whatever initial deflection it is given because the loading is always balanced with the deflection of the beam.

 

[prex]

Not sure to understand your setup: in (1) you show the plate/membrane held (supported and held, I suppose) in only one direction. If that is true, then the problem becomes really complex, as the level of the retained water will depend on the deflection of side L, and won't be as depicted in (3). Look here to see how a plate in bending, with uniform load, deflects when only two sides are supported.
The membrane version, with 4 held sides and liquid filled, is solved here. This is not a fully analytic solution, it is obtained from the minimization of strain energy with a polynomial expansion of deflections.
I don't see this issue of only one stable configuration: with 4 supported sides (the only configuration of practical interest), even a small amount of liquid will cause a deflection and a liquid filled stable condition will be attained. The deflection will of course vary with plate bending/membrane properties (and with liquid density).

 

[bugbus]

I played around with a Strand7 model for the above example, i.e., 10 mm thick steel (E = 200,000 MPa) plate filled with water (γ = 10 kN/m3).

I checked a plate with less than the critical span length (L = 3.4 m), more than the critical span length (L = 3.8 m), and then the theoretical critical span length of 3.5695... m.

The loading is updated automatically each iteration to be proportional to the deflected shape. I haven't shown all iterations, only the first two for each case, but you can see the trend occurring as expected. For the short span, the deflection immediately starts reducing and approaches zero. For the long span, the deflection immediately starts increasing and presumably goes on infinitely.

Interestingly, the critical span length does not seem to be quite correct. Its deflection is pretty stable but it gradually starts to increase.

Fine tuning this a bit, the critical span (at least according to the model) turns out to be something around 3.54 m. However, it's very close to the theoretical number and is probably due to the fact that for each iteration, the software saves the deformed shape and re-analyses (effectively nonlinear geometry). So the plate is not subject to pure bending but also has some small axial component along its length.

 

[bugbus]

Not sure to understand your setup: in (1) you show the plate/membrane held (supported and held, I suppose) in only one direction. If that is true, then the problem becomes really complex, as the level of the retained water will depend on the deflection of side L, and won't be as depicted in (3). Look here to see how a plate in bending, with uniform load, deflects when only two sides are supported.
The membrane version, with 4 held sides and liquid filled, is solved here. This is not a fully analytic solution, it is obtained from the minimization of strain energy with a polynomial expansion of deflections.
I don't see this issue of only one stable configuration: with 4 supported sides (the only configuration of practical interest), even a small amount of liquid will cause a deflection and a liquid filled stable condition will be attained. The deflection will of course vary with plate bending/membrane properties (and with liquid density).

I agree, the initial condition is a little poorly worded.

Maybe a neater way of asking the question is just whether there is a stable configuration for an elastic plate where the distributed load is proportional to its deflected shape.

However, disagree that any amount of liquid will cause the plate to assume a stable condition. I have not checked the four-sided support case, but I assume it is similar to the two-sided case above. For a stiff enough plate, it should rebound back to a flat arrangement and displace all the water.

Thanks for the link, interesting stuff

 

[prex]

If you consider a beam (so not a plate) in bending, then your equation (d4Δ/dx4)=(Bγ/EI)Δ(x) is correct, but the solution is not in the form of a sine curve, it is instead a polynomial of the fifth order, so no critical case can exist.