Plate bending deflection in 3D
Plate bending deflection in 3D
(OP)
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 www.xcalcs.com but this gives inconsistent answers, although it does try to offer a numerical solution of sorts.
corus





RE: Plate bending deflection in 3D
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
RE: Plate bending deflection in 3D
RE: Plate bending deflection in 3D
Cheers
RE: Plate bending deflection in 3D
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
RE: Plate bending deflection in 3D
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.
RE: Plate bending deflection in 3D
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(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
RE: Plate bending deflection in 3D
corus
RE: Plate bending deflection in 3D
confused
corus
RE: Plate bending deflection in 3D
For a free edge parallel to y axis the boundary conditions should be:
∂2w ∂2w
--- + ν --- = 0
∂x2 ∂y2
∂3w ∂3w
--- + (2-ν) ----- = 0
∂x3 ∂x∂y2
Good luck!
prex
http://www.xcalcs.com
Online tools for structural design
RE: Plate bending deflection in 3D
Happy Hunting.
RE: Plate bending deflection in 3D
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
RE: Plate bending deflection in 3D
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
RE: Plate bending deflection in 3D
Bending of sheet metal, levelling, uncoiling thin plate etc.
corus
RE: Plate bending deflection in 3D
corus
RE: Plate bending deflection in 3D
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
RE: Plate bending deflection in 3D
corus
RE: Plate bending deflection in 3D
RE: Plate bending deflection in 3D
Excel has a variety of add-ins that support extended precision calculations.
There are some free extended precision calculators:
http://
TTFN
RE: Plate bending deflection in 3D
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
h
Jon Spear
Berkeley Lab
www.lbl.gov
http://bcsb-web.als.lbl.gov/Staff/Newstaff2.htm
RE: Plate bending deflection in 3D
For analytical solutions other than rectangular then you'd be better using finite elements. I'm not sure how a trapezoidal section would help (presuming that is the section about which you'd be bending the plate) but perhaps starting with a curved plate that would then bend back into a flat shape might be better. If you were starting with a flat plate then you'd have to bend it about two axis in order to eventually get a curved plate that was flat in the other direction, if you know what I mean.
corus
RE: Plate bending deflection in 3D