Plate bending deflection in 3D 1

[corus]

Does anyone know of a formula (or a reference) for the bending of a plate under a uniformly applied load simply supported at the two extreme edges? I'm not only interested in simple beam type bending of the plate but also the lateral deflection caused by poisson's effect. The only solution I can find is to solve the fourth order PDE by Levy's method. I've also tried

http://www.xcalcs.com

but this gives inconsistent answers, although it does try to offer a numerical solution of sorts.

corus

 

[prex]

Which sheet specifically were you using at

http://www.xcalcs.com

? There is (was) none corresponding to your description.
Now it is there (freshly added): you find it under Plates -> Simple bending -> Rectangular -> 2 opp.supp,2 free
Please come back if you find any inconsistency.

prex

http://www.xcalcs.com

Online tools for structural design

 

[vonlueke]

Very impressive web site, prex. Thanks.

 

[Bree]

Prex, how would I go about solving this problem on paper? Where did you find the fomula for deflection of a plate with these boundary conditions? I saw the formula on the xcalcs.com website but I'm confused as to how I can calculate a deflection using it.


Cheers

 

[prex]

You can't do it by hand, or at least there is no simple formula (not even a series expansion) that I know of.
The solution at xcalcs is an approximate one obtained by energy minimization: it requires the solution of a large linear system of equations.
But why would you want to do it by hand, as you can have it done for you by a computer?

prex

http://www.xcalcs.com

Online tools for structural design

 

[rb1957]

you could consider your plate as three sections, the middle one supporting the load. the reactions (for the middle strip) being pinned (or cantilevered) at the supports and a distributed load into the adjacent strips; the adjacent strips react this distributed load at the supports.

the middle strip is then a beam with a point load and an opposite sense distributed load; easy enough to calculate the deflected shape for different distributed loads (different reactions into the adjacent strips). The adjacent strips are beams with a distributed load, simple enough.

good assumptions (the load distribution into the adjacent strips) would have very similar displaced shapes. if you wanted a challenge you could include the torsion effect (the adjacent strips have the appled load offset from the edge reactions. and/or include more strips.

or you could assume a deflected shape, say every point at a distance from the load deflects the same (circles); with zero slope at the load point and some power series (at least a cubic, to fit with the distributed loads, possible quartic) defining the displaced shape. this will tell you something of the twist on the adjacent strips.

 

[Cockroach]

Actually, you can do it by hand. The solution set is based on boundary values imposed by the fixed edge and point placement of the load, partial differential equation with constant coefficients.

For sure the Fourier Series will work, but there are more classical methods which involve the steady state and transient solutions in the form X(x)Y, that is, single valued functions that are separable in the differential form.

I would try advanced textbooks in mathematics such as Partial Differential Equations.

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada

 

[corus]

prex, the web site is good, and eventually I managed to get the answer and the result is the same as that obtained by FE. Unfortunately I need to pass on the formula for a general case to someone who will not have access to the internet while at work. Text book solutions I've found only apply for other cases, such as simply supported on all four sides. The only resource I've had is to work through the series solution of the PDE myself. The problem now is what boundary condition to impose at the free edge. Is it d^2w/dx^2 = d^3w/dx^3 = 0?

corus

 

[corus]

Forget that last line. Are the boundary conditions at the free edge that d^2w/dy^2=0 and that Syy=0 ie. u.d^2w/dx^2+d^2w/dy^2=0 ? where u is poisson's ratio.

confused

corus

 

[prex]

In using energy minimization it is not necessary to impose any boundary condition on free sides, as the free side condition corresponds to an energy minimum with respect to any other condition, and is hence automatically generated.
For a free edge parallel to y axis the boundary conditions should be:
[tt]
[∂]²w [∂]²w
--- + [ν] --- = 0
[∂]x² [∂]y²
[∂]³w [∂]³w
--- + (2-[ν]) ----- = 0
[∂]x³ [∂]x[∂]y²
[/tt]
Good luck!

prex

http://www.xcalcs.com

Online tools for structural design

 

[Hookem]

It is ALWAYS a good idea to check any computer calculation by hand. For the hand calculation, consider looking up Formulas for Stress and Strain by O'Rourke. They have formulas for bending if square, rectangular, round, oval, etc. plate for many, many, many boundry conditions.

Happy Hunting.

 

[corus]

Thanks prex, I've confirmed your expressions in a book by Timoshenko and Gere today and will now be able to solve the PDE, hopefully.

Hookem, Roarke doesn't have such formulae for plates with only 2 sides simply supported as normal practise would be to consider the plate as a beam and ignore the poisson's effect.

Many thanks to all.

corus

 

[GregLocock]

Corus - can you give me an idea of the sort of application where this effect is important?



Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[corus]

Greg,
Bending of sheet metal, levelling, uncoiling thin plate etc.

corus

 

[corus]

In producing an analytical solution to the problem I find that the numeric solution using both MathCad and Excel give slightly different results (about 1 or 2%) to those obtained from Xcalcs.com and an Abaqus FE model (for which both give the asme answer). Do both MathCad and Excel use single precision numbers such that the difference I find can be explained by rounding errors?

corus

 

[prex]

Excel uses for sure double precision numbers, and so does, I suppose, MathCad.
As I understand you are using a series expansion, what about a very slow convergence, as is often found in similar situations?

prex

http://www.xcalcs.com

Online tools for structural design

 

[corus]

nope prex, I looked at that. The solution converges rapidly and only the first two terms are needed, if that. The problem lies with the multiplication of a number raised to the power 4 multiplied by a very small number giving an answer that has a small but not insignificant error. I'll look at the maths again to see if an alternative exists.



corus

 

[vonlueke]

corus: Have you tried it in Octave yet? Octave would be a serious number cruncher.

 

[IRstuff]

There are several options to get more precision. Mathcad, for example, has a much higher numerical precision in its symbolic engine.

Excel has a variety of add-ins that support extended precision calculations.

There are some free extended precision calculators:

http://www.engineering.com/content/...isciplineID=mechanical&tabID=600&subTabID=610

TTFN

 

[jdspear]

corus, et. al.,

This is a relevant topic for the work in which I am involved, whereby a mirror in the form of a flat solid bar is polished, and then bent into an elliptical shape.

One of the problems is the so-called "anticlastic" bending, in the perpendicular axis. A typical mirror might be around 5 inches long, by 0.6 inches wide, by 0.15 inches thick.

I am new to this sort of problem, but I am interested in seeing how the effect might be altered by making the cross sectional shape of the mirror trapezoidal rather than rectangular.

Jon

http://ssg.als.lbl.gov/ssgdirectory/spear/spear.htm

Jon Spear
Berkeley Lab

http://www.lbl.gov

http://bcsb-web.als.lbl.gov/Staff/Newstaff2.htm