# Deflection of constrained rectangular plate

Does a standard formula exist for calculating the (maximal) deflection of a constrained rectangular plate that is loaded with a evenly distributed pressure?

I found this Useful tables of mechanics, however I'm not sure if it's correct:

deflection of a constrained rectangular plate

By this I assume you mean that all edges are fixed.

My general go-to for these types of formulations is Roark's Formulas for Stress and Strain, 7th Edition. These formulations assume a flat plate with straight boundary conditions and constant thickness. Also, Poisson's ratio for the material is assumed to be $\nu = 0.3$.

From Table 11.4 of that text, Case No. 8a, Uniform load over entire plate:

$$y_{max} = \frac{\alpha q b^4}{Et^3}$$

The $\alpha$ value is found by considering the dimensions of the plate:

Note that this formula is really only strictly valid for materials with $\nu = 0.3$. If the material has a different Poisson's ratio, they may need to be adjusted. Roark's lists the following references next to this formulation that may be of some help if you have a differing $\nu$ value:

• Timoshenko, S., and J. M. Lessells: "Applied Elasticity," Westinghouse Technical Night School Press, 1925.

• Evans, T. H.: Tables of Moments and Deflections for a Rectangular Plate Fixed at All Edges and Carrying a Uniformly Distributed Load, ASME J. Appl. Mech., vol. 6, no. 1, March 1939.

• Timoshenko, S., and S. Woinowsky-Krieger; "Theory of Plates and Shells," 2d ed., McGraw-Hill, 1959.

I do not have copies of these documents, so I cannot verify their utility for this problem.

• That's indeed a different formula..I'm using rubber as material (Poisson's ratio is about 0.5) should that differ a lot? Aug 19 '15 at 15:19
• @Alexm, please see my edit. The formulation will probably be a close enough approximation for your rubber case, but you might need to do further reading. Aug 19 '15 at 15:27
• @Alexm, I just realized if you plug in your different $b/a$ ratios in the equation you provided, you will find the same $\alpha$ constants as tabulated by Roark Aug 19 '15 at 16:04
• @Alexm, Yes, a nearly incompressible material ($\nu \approx 0.5$) will differ "a lot" from results tabulated for $\nu= 0.3$. Timoshenko & Krieger referenced in the answer is one of "the bibles" and is available here archive.org/details/TheoryOfPlatesAndShells. The formulas in T&K are valid for isotropic materials with any value of $\nu$, but numerical constants are usually only tabulated for $\nu = 0.3$. Aug 19 '15 at 17:02
• Link to the book: materiales.azc.uam.mx/gjl/Clases/MA10_I/… Aug 15 '19 at 23:54