Maximum Stress and Deflection of Rectangular Plates Under Hydrostatic Load - Tall Plates

Question

I am hoping someone here can help point me in the right direction. I am trying to understand the fundamental behavior of rectangular plates under load and have run up against a wall. Can anyone help with this?

The practical problem I am trying to understand better is design of tanks to hold liquids. They are rectangular tanks, but they present a number of different problems. The bottom of the tank has uniform load consistent with the depth and density of the liquid in the tank and the walls have hydrostatic load from the liquid, but depending on the design of the tank all walls may be "wider" than the are tall, all may be taller than they are wide, and in some cases you may have both. Imagine, for instance, an aquarium where the front and back are typically wider than they are tall but the sides are often taller than they are wide.

I understand the top level formulas intended for use with tabulated values and I have access to all of the tables, but both the relatively low precision of the numbers in the tables (usually only two digits) and the relatively sparse entries were such that I wanted much better precision and higher sampling. See, for instance, in referring to Roark's Formulas for Stress and Strain (I'm looking at the Seventh Edition), the tables I am referring to are 11.4.1a (for a plate simply supported on all sides under uniform load), 11.4.1e (for a plate simply supported on all sides under hydrostatic load with the load varying on the shorter side - "wide" tank walls), and 11.4.1d (for a plate simply supported on all sides under hydrostatic load with the load varying on the longer side - "tall" tank walls).

Here is the curve that Roark's uses to derive the values in Table 11.4.1e. I have colored the curve in red to highlight it. If you check Roark's you'll see that the values in the table agree with this curve. I can derive this curve from first principles.

The curves (here and below) are Roark's reference (Ref. 8 in the Seventh Edition) for the values in it's tables: https://asmedigitalcollection.asme.org/appliedmechanics/article-abstract/3/2/A71/1101081/Stress-and-Deflection-of-Rectangular-Plates?redirectedFrom=fulltext

I've used Navier's solution to calculate the stresses and deflections to any precision needed for any ratio of length/width, and it is also possible with Navier's approach to also prove that when the length/width is infinite, that the transverse shear deformation factor (β) is exactly 3/4 (for all Poisson ratios) and the axial stress deformation factor (α) is exactly 5(1-ν²)/32, where nu is the Poisson ratio for the material. So that is under control.

I've used Levy's solution for 11.4.1e with similar success. I can calculate α and β to any precision desired at any a/b ratio (for width greater than height) and can prove that when the width/height is infinite that β for this case is also the same for all ν, and the closed form is $2 \sqrt3/9$. At infinite a/b ratio α is dependent upon nu here as before, constant*(1-ν²) again, but the constant is $(4/255)\sqrt([15 + 4\sqrt(6/5)]$. So that is also under control.

These results can be replicated to any desired precision. With a little work after studying the literature, the actual mathematical description of the curve (α) above is as follows (in Mathematica):

Poisson's ratio is ν, which is set at 0.3 (similar to glass) for the curves shown here. In the formula above, "b" is the a/b ratio. At 201 summation terms this converges to better than a part per million versus the sum at infinite limit. This is the maximum deflection factor. The formula gives the deflection factor at a given position x, which is the direction that the hydrostatic load varies (x = 0 corresponds to no load, x = 1 corresponds to maximum load), so the position where α is a maximum must be searched, which is what "FindMaximum" does. By symmetry, the position of the maximum deflection is on the y axis if the problem is set up that way (as I did in this case), so y=0 and drops out of the equations I started from. The result for a/b = 1 is shown here to be 0.224317 (0.22 in Roark's table) and the position of maximum deflection is {x, y} = {0.548744, 0}.

I first solved this last case following Timoshenko's prescription in Chapter 5 of Theory of Plates and Shells (2nd Edition) where he lays this last case out in detail, then on page 196 of that edition he suggests that the Roark's case 11.4.1d, with the plate "taller" than wide, can be solved using the same approach as before with some suggested modifications and leaves it up to the reader to work out the rest of the details. I made several attempts at that and have been able to generate values similar to, but not identical to those in Roark's tables. That's where I am having difficulty completing this exercise. In trying to understand this better I went to Roark's ultimate source for these values, referenced in the table: Wojtaszak, I. A.: Stress and Deflection of Rectangular Plates, ASME Paper A-71, J. Appl. Mech., vol. 3, no. 2, 1936.

This ASME reference has plots of curves for α and β for plates under various conditions, but while they are single curves for the other cases, it is not a single curve in this case as for the others and as is expressed by the results of the above approaches, but in fact composed of multiple curves. There is absolutely nothing in Timoshenko that would suggest multiple curves are required to fully describe the physics, or why. There is nothing in Reddy's text that would suggest multiple curves are required, or why. I have actually reached out to both Reddy (Theory and Analysis of Elastic Plates and Shells) and the current editor of Roark's but have received no responses. I have been through all the references I can obtain from the ASME paper that claims this behavior is described in the references, but except for a copy of "Elastische Platten" published in Berlin by A. Nadai in 1925 and "Theory of Structures of Ships" published in St. Petersburg in 1914 by J Boobnoff, neither of which I have been able to obtain, none of the references in that paper discuss this issue at all.

Here is the curve used by Roark's for the values in its Table 11.4.1d. I have shaded the two separate curves that make up the composite curve in red and green for each portion. As before, the values in Roark's table correspond to this composite curve.

So here is what I am asking. Does anyone have any idea why there would have to be multiple curves to describe this behavior? Can anyone point me to a reference that discusses it that might help me figure out how to handle it?

Also, is anyone aware of any resources that calculate these values numerically (but from first principles, FEM, not using the tables!!) which I might use to calculate values to several digit precision so I could see: a) are the results described by multiple curves, b) if so, where do those curve break/meet and if I have more precise descriptions of the curves can that inform my own rederivation of the unique mathematical description of each curve.

I am also interested in the same cases under hydrostatic load, but with one edge free (the top edge) - but again the values in Roark's tables for these cases are derived from the same ASME paper and in both cases there are multiple curves describing the α and β values. It's a vexing problem.

Thanks for any insights or suggestions...

[30 March 2022 Update]

OK - here is a preliminary update that confirms the proper behavior for the simply supported plate, under hydrostatic load, with the plate being taller than it is wide is definitely described by a single curve and not a composite of multiple curves. For the deflection of the plate I was able to model this simply with partial differential equations. The problem is pretty straight forward. The displacement function is described by the biharmonic equation. The Laplacian of the Laplacian of the deflection function is equal to the load. For a plate with uniform load the load is just a constant, q. For a plate with hydrostatic load the load is just q*x/a (where a is the width of the plate in the x direction). So at x=0 the load is zero, at x=a the load is q. This is the way the hydrostatic force works. In both cases q = (ρ g h), where ρ is the density of the fluid.

I used Mathematica's numerical differential equation solver. I thought there was a bug in Mathematica that delayed the results, but it turns out it was a change in syntax that wasn't properly documented, so they explained to me what the new syntax was and are correcting the documentation. Using the differential equation I described and the necessary boundary conditions I derived the displacement function then searched for the maximum displacement on that function (which in turn is proportional to α). In this way I repeated calculations for the plate under uniform load and the plate under hydrostatic load that I had previously solved analytically and was able to demonstrate that the numerical solutions agreed within the reasonable precision of the method (on the order of a part in about 10^-5 to 10^-6). It was then easy to extend the problem to plates that are taller than wide and I was able to show two things:

1. The data from Roark's tables (for Case 1d) and the reference above are wrong, but reasonably close to right. It seems clear these were mathematical approximations made at a time when computing numerical sums of slowly converging series was challenging.

2. The new numerical result also agrees well with expectations. Let me show the results first and then talk about that a little more.

Here is a plot of the alpha values for each of the three cases I described.

The green lines are the new numerical results I have and the red dots on the lines are the values from Roark's for those cases - as labeled. The blue line is the new result for plates taller than wide and the orange dots are the values from Roark's that clearly show the deviations between the table values and the line. There is a lot of confidence the blue line is correct because the same code for the other cases is clearly correct and the agreement with the tables for the case in question is close as you'd expect it would have to be since people have been using those tabulated values without disaster. But two other pieces of evidence confirm the blue line is correct. The first is that for a square plate case 1d and 1e are identical. And those two curves meet at a/b = 1 == square. The second is that for infinitely tall plates the problem should converge on the same result as Case 1a at infinitely large ratios - and for large values (ratios greater than 100) the two results converge - and converge to the 5(1-ν²)/32 values described before. So this is all very strong evidence this result is correct - and it's not a surprise. Numerical solutions of differential equations of this sort should be reliable.

OK - so that's great confirmation that the α values should behave as expected and even provides a means to calculate α values for other materials with other Poisson ratios. The Roark's tables all provide values calculated for materials with a Poisson ratio of 0.3. But, for instance, in the case of tanks to hold water made of acrylic or glass, acrylic Poisson ratios are on the order of 0.4 and glass values are typically 2-2.3. So the table values are not as ideal as you would like them to be. In practical application the "engineering factor" chosen here is often quoted as 3.8, which is a little excessive, and probably because the tabulated values are not that great.

But, as great as this progress is it is a little anticlimactic because I still seek the more analytical result. But far more importantly, this is only the α values, which describe the deflection behavior of the plate. To properly engineer the plate thickness to ensure the tank won't shatter the β values are required, which describe the plate stresses. With an analytical solution to the deflection problem determining the stresses is straight forward, since the bending moment (and therefore the stress) is proportional to the second derivative of the deflection function. If the deflection function is a numerical approximation instead of an analytical function the problem is challenging at best, and fraught with numerical imprecision as numerical methods are applied on top of numerical methods.

So I am still looking for anyone who can offer any hints on how to address this problem analytically.

[02 April 2022 Update]

So, I knew this would not be super pretty - but I had to see the results. Falling back to plate theory, if you have a function describing the curvature of the plate under stress (the deflection function) then you can calculate the moments - the x moments are the second derivative of the deflection function with respect to x and the y moments are the second derivative of the deflection function with respect to y. The maximum stress, which is proportional to the beta constant, is the maximum of the x moment function plus the Poisson's ratio times the maximum of the y moment function:

σ_max = M_{x, max} +ν M_{y, max}

So, recapping, we are deriving the curvature function, which is the numerical solution of a 4th order differential equation, then we're taking two separate numerical second derivatives of that original numerical function, searching for the maximum of each of those functions, then the desired result is proportional to the result immediately above.

I did this first for Case 1a - a simply supported plate with uniform load - because I have the exact answers, and compared those results to exact results and as you can see from the graph immediately below, those results are extremely good. Two things to note for these numbers: the maximum stress is guaranteed by symmetry to be at {x = a/2, y = b/2} so there is no need to search for the maxima (simply calculate them); and the immediate result, σ_max, is only proportional to the β value you want. I found the value needed to scale these numbers is 0.5522. From Roark's:

σ_max = (β q b²) / t²

That scalar should be t² / (q b²), but it's not. So I am scratching my head about that a little bit. In the plot, the green line is the numerical fit I just described, the black dots are the exact results derived analytically. So the numerical method is doing what it is supposed to do.

The final thing to point out about the Case 1a results is that you will notice some "jagged" behavior of the numerical results shown in the green line. This is because the number of dependently coupled equations is so high that the results are literally non-deterministic. If I rerun the calculation the line is extremely similar, but not identical. And I could not use the canned Mathematica routines to search for the maxima of the moment curves because this non-deterministic behavior caused the routines to fail. I had to write my own routine to search out the maxima.

Case 1e is also shown, again, the green line is the numerical fit just described but the Case 1e conditions and the black dots are the exact results determined analytically. Other than the more pronounced "excursions" from the multi-step/multi-numerical approach, I think it's clear the physics is also being correctly modelled here. Again the scalar, which in this case is about 30, isn't directly relatable to the analytical relation above.

Finally, Case 1d, which is the "white whale" I have been after (1 of 2 white whales), using the same scalar as that derived for Case 1d, is shown in blue with the black dots there representing the values in Roark's Table 11.4.1d (because I have no analytical solution for that yet). This is more important than the α values - the β factor is used to decide how thick of a plate you need to guarantee the plate doesn't fail under load. I think it's very clear here that the present numerical fit is modelling the physics correctly, and other than the "jagged" behavior is getting the correct values, but even with that behavior, and now that the Roark's β values are posted, I think the graph makes quite clear the tabulated values are actually quite poor and clearly do not lie on a single curve, which the physics clearly indicates they do.

A direct finite element analysis would get more deterministic and presumably higher precision results. I guess I have to start looking at that.

So, again, my hope is someone can help me with any references or suggestions on how to solve the problem analytically.

Answer 1

""Elastische Platten" published in Berlin by A. Nadai in 1925" is available at the library of the "Technische Universität München" (Technical University of Munich, Germany). However, the book contains some serious errors, probably committed by the typesetter and not corrected by Nadai. The book has been scanned, so the university library (www.ub.tum.de) may be able to send you a pdf version.