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Analytical solution for a partially loaded rectangular elastic thin plate
Tue, 2011-05-17 04:22 - Hongyuan Zhou
Hi, everyone:
A rectangular plate, simply supported at four edges, subjected to a partial load- a uniformly distributed pressure over a round area at the plate center. What is the analytical solution for the plate deflection?
Since we only concern the thin plate, elastic response and small deflection, I believe the analytical solution exists (may be in the form of single or double series of sinusoidal or sinh functions). However, I search many relevant books including the classic book by Timoshenko and cannot find it. If the exact solution is not available, an approximate solution is also OK.
Could you please tell me where I can find it?
Many thanks!
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I know the solution may be
I know the solution may be found in an indirect way- summation or integration of patch load response in the round area. However, it is not concise and not easy to simplify.So here I am wondering whether there is some elegant and concise solution obtained in a direct way?
Roark rules
The solution is given in:
Roark's Formulas for Stress and Strain
editors: WARREN C. YOUNG, RICHARD G. BUDYNAS
TABLE 11.4 Formulas for flat plates with straight boundaries and constant thickness
original reference:
Timoshenko, S., and S. Woinowsky-Krieger; ‘‘Theory of Plates and Shells,'' 2d ed., McGraw-Hill, 1959.
Regards
Frank
------------------------------------------
Ruhr-University
Bochum
Germany
Thank you very much,
Thank you very much, Frank!Just now I browse the reference book you mentioned and find that the prediction in Table 11. 4 (1b) in fact is an approximation of a point load in the rectangular plate center. The radius of the loading should be not too much greater than half the plate thickness.This may be not suitable for my problem, in which the radius of the uniform pressure is much larger than the plate thickness, although the loading radius is smaller than the plate dimensions.Therefore I will use integration of small patch solution to determine the center defection. Thanks again!Regards, Hongyuan
Re: Analytical solution for a partially loaded rectangular plate
For a possible solution see Section 37, p. 158 "Theory of plates and shells", Timoshenko and Woinowsky-Krieger. The solution you seek is given on p. 162.
-- Biswajit
Many thanks, Biswajit!The
Many thanks, Biswajit!The solution in P162 is for the moment and force. And what I seek is the deflection. Now, I use the basic formulations in P109 of the book you mentioned then integrate it to get the a(mn), although the integration cannot be analytically obtained. Hongyuan
Check if Square plate with
Check if
Square plate with clamped edges under normal pressure producing large deflections
Levy, Samuel
naca-report-740
1942
http://naca.central.cranfield.ac.uk/report.php?NID=1848
or
Square plate with clamped edges under normal pressure producing large deflections
Levy, Samuel (National Bureau of Standards)
naca-tn-847
1942
http://naca.central.cranfield.ac.uk/report.php?NID=1881
meet your requirement.
Frank
------------------------------------------
Ruhr-University
Bochum
Germany
thank you!
Thanks again, Frank!
I will check it.
Regards,
Hongyuan
More on plate solutions
It might actually be easier to do the calculation yourself than to search for it in the literature :)
Steps:
1) Find w(x,y) for a point load at location (x0,y0).
2) Convert the solution into cylindrical coordinates with origin at (x0,y0)
3) Integrate w(x,y) over the circle of applied load.
4) Convert the solution back to the (x,y) coordinate system.
-- Biswajit
Hi I think the
Hi
I think the following reference will help you to solve your problem.
Vibration of Plates - A. W. Leissa...
Link: http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19700009156_1970009156.pdf
Sreenivas