Deflection of an aluminium plate

[lavecchiasignora]

Dear Engineers,

I have a rectangular plate (3 x 2 m) made of aluminium (12 mm thickness). I also have a evenly distributed load of 2500 kg spread over the entire plate. How can I calculate the maxiumum deflection at a certain point on the plate? The plate is fixed in all directions on the edges.

Thank you.

 

[SteamKing]

Dear Engineers,

I have a rectangular plate (3 x 2 m) made of aluminium (12 mm thickness). I also have a evenly distributed load of 2500 kg spread over the entire plate. How can I calculate the maxiumum deflection at a certain point on the plate? The plate is fixed in all directions on the edges.

Thank you.

How are the edges of this plate supported? Simple supports, fixed supports, what?

 

[lavecchiasignora]

Hi, The plate is fixed in all directions on the edges, i.e. fixed. :)

 

[NumericalFEA]

One way is to use an analytical solution; you would probably find some useful formulae in the works of Stephen Timoshenko, who published several books on the topic. Another approach would be to use a finite element program, for example like the one described below (the link also contains an example very similar to yours, including stress distribution contours, etc.):
http://members.ozemail.com.au/~comecau/quad_plate.htm

 

[lavecchiasignora]

Thanks for your reply. I forgot to write that I shall verify my numerical FE model with an analytical calculation. I have done an numerical but need to verify it now.

 

[Nidum]

Theory of Plates and Shells - Timoshenko .

Page 197 et al is directly relevant .

 

[lavecchiasignora]

Those equations are very very complicated. I am just trying to find an simple analytical formula for calculating deflection and stress.. should it be so hard?

 

[NumericalFEA]

There is no simple formula for that problem.

 

[SteamKing]

Those equations are very very complicated. I am just trying to find an simple analytical formula for calculating deflection and stress.. should it be so hard?

Have you tried looking in a structural handbook like Roark's Formulas for Stress and Strain?

You can find a copy online if you search for it. This book is chock full of simple formulas which can be evaluated by a calculator or spreadsheet.

 

[lavecchiasignora]

Have you tried looking in a structural handbook like Roark's Formulas for Stress and Strain?

You can find a copy online if you search for it. This book is chock full of simple formulas which can be evaluated by a calculator or spreadsheet.

Thank you! Do you know if these formulas can be applied on aluminium?

 

[SteamKing]

Thank you! Do you know if these formulas can be applied on aluminium?

I don't see why not. Most structural metals are isotropic. The formulas in Roark use Young's modulus and Poisson's ratio to calculate deflection.

 

[lavecchiasignora]

Hi again, I have used Roarks equation for deflection of a plate with uniform distributed load and all clamped edges. The answer I get is larger than the palte thickness so these Roark equations do only apply for small deflection theory and not large defcletion theory... Now I need equations for large deflection theory.. any idea how to do it?

 

[NumericalFEA]

You have a geometrically non-linear problem, i.e. the membrane forces cannot be ignored, since they contribute (significantly) to the deflections and to the overall stress fields. The way to go is to use a commercial-quality FEA system.

 

[lavecchiasignora]

Thank you for your reply... I have already done a FE analysis of the plate, now I need to verify it by hand calculations.. I was thinking that I could use "Large deflection of plates" in Timochenko and calculate (p*b^4) / (D*h) but I get a value of 1600 and the graph only shows values up to 250...

 

[NumericalFEA]

Analytical solutions for large displacements mean that the relations between strains and deflections are non-linear. From the mechanical point of view, when the edges of the plate are jammed, non-linearity naturally leads to the fact that membrane (tensile) forces appear, which significantly reduces the deflection. I think you should check if the Timoshenko's solution takes into account the membrane forces in this case. If it does, another reason for the discrepancy may be that Timoshenko and the FEA developers have used different plate theories, for example Kirchhoff-Love and Mindlin theories.

 

[Nidum]

You haven't told us what actual deflections your FE analysis and other calculations give but let's say they are of same order as plate thickness .

Deflection of order of 12mm for a plate 3000 mm by 2000 mm by 12 mm thick with more than two tons load on it does not seem unreasonable .

12 mm deflection is only 0.4 % of 3000 mm . Small deflection theory probably gives results of adequate accuracy .

 

[Nidum]

Quick calculations using two different methods give deflections of 12 and 13 mm .

If this was a real job I would check the stress levels as well to ensure that they were within safe limits .

 

[Satonam]

Theory of Plates and Shells - Timoshenko .

Page 197 et al is directly relevant .

It seems your link is no longer available. I'm curious to understand how to design the thickness of a plate with a perpendicular load. Here's what I have so far:

In reality, the plate is welded at x = 0 and x = 25". A 570-lb pump is placed in the shaded area. I want to make sure the plate can hold this load. Can I simplify this problem by treating it like a simply supported beam? In the image, I have shown my calculations.

P.S.
I know I didn't include shear stress in the calculation. That's because:
1. I wanted a quick answer to post here
2. In my experience (albeit limited), shear stress is very small compared to bending stress -therefore the von mises stress is assumed to be identical to the bending stress in this draft calculation
.

 

[jrmichler]

Yes, you can treat this as a simply supported beam. The answer will be somewhat conservative, but that is a good thing. The true case is somewhere between simply supported and fixed because the weld and supporting structure are not infinitely rigid.

Yes, you can ignore shear stress in this case. If the beam length was two or three times the beam depth, then it would be necessary to include shear stress in your calculations.

Treating the pump as a point load also increases your safety factor.

In this case, your simplifications significantly increase the safety factor. Normally, a safety factor of 1.9 would not be enough for a vibrating machine because you would eventually get fatigue cracks.

You should also check the deflection. Machine bases need to hold equipment in position, and a flexible base is not good. Many most times, deflection is more important than strength in machine design.

Quick calculations are good. They will tell you if your safety factor is less than or near one, in which case you need to change the design, or really large, in which case you are finished. If in between, you just need to do the full calculation.

 

[Satonam]

In that case, using the maximum deflection formula for a simply supported beam and a young's modulus of 29000 ksi, I'm coming up with a max deflection of 0.22"

How do you know if the deflection is too large or small enough?

If my simplifications increase the FS, isn't that a bad thing? Doesn't that mean that the actual FS upon refinement is smaller?

I know how to calculate the FS based on fatigue with the Goodman Failure Theory given a range of stresses -however, how would I determine that range of stress? I imagine that I would have to determine the weight of the pump at its "highest" or "lowest" points in the cycle (minimum contact and maximum contact respectively). I'd imagine that, in theory, this might be feasible with a vibrational analysis that most likely requires FEA -but it might also be overkill for small deflections.

Another option that comes to mind is using strain rosettes to record the pump's influence over time and deduce the range of stress from that data. With that said, I don't think I have the resources to invest in all the necessary equipment.

Blindly making the plate thicker to achieve a higher FS is probably the most cost effective solution, however, I have no idea what a reasonable FS increase would be since I'm not familiar with the magnitude of influence that fatigue would play in that calculation. For example, maybe I think that a FS of 5 would be reasonable, but if I had actually computed the FS with real fatigue values, I might come up with a 10. I guess my question here is, does fatigue tend to drastically affect the FS?

 

[jrmichler]

Fatigue calculations are important in the aerospace industry, where everything is designed for minimum weight. The rest of us can usually use a safety factor of five, and call that good enough.

Allowable deflection depends on exactly what the base is required to do. If the motor and pump are both mounted to the base, and connected with a coupling, then the base needs to maintain alignment between the motor and pump. Alignment tolerances vary, and a good place to start is by search terms shaft alignment tolerances. The base needs to be rigid enough to maintain alignment within tolerances under all conditions of motor torque, pressure variation, thermal expansion, and other external loads. On the other hand, if the motor is directly mounted to the pump (as with a face mount motor), then the base just needs to support the assembly.

The base also needs to support the pump without loading the piping. You want the pump to be supported by the base, not hanging from the piping. The base needs to be more rigid than the piping so that the piping can be aligned to the pump at installation. Pump bases properly designed for deflection are typically much stronger than needed. There is a reason that industrial pumps typically have a heavy steel or cast iron base mounted and grouted to a concrete pad. I suggest the following search terms, followed by some reading: industrial pump installation.