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Levy's simply supported square plate under uniform pressure: large-deflection centre deflection and stresses at p a⁴/(E t⁴) = 132

Tool	Large-Deflection Plates
Source	Levy, S. (1942). Bending of Rectangular Plates with Large Deflections. NACA Report 737, National Advisory Committee for Aeronautics, Washington, DC. Tables 9, 10 and 11 (square plate, edge displacement zero, μ = 0.316).
Level	Graduate

Problem

A 10 × 10 in aluminium plate 0.05 in thick (E = 10.5 Msi, ν = 0.316) is simply supported on edges that stay straight, carry no shear and cannot move in plane, and carries a uniform pressure of 0.866 lbf/in², so that p a⁴/(E t⁴) = 132, a row of Levy's Table 9. Find the centre deflection, how far it falls short of small-deflection theory, the membrane and bending stresses at the centre and at an edge midpoint, and the edge tensions the frame must react.

Inputs to Enter

a = b = 10 in, t = 0.05 in; E = 10.5 Msi, ν = 0.316; edges simply supported, immovable in plane; 4 odd harmonics per direction (16 modes)
p = 0.866 lbf/in² (load parameter 132.0); allowable 40 ksi at limit, factor 1.0

Open the Large-Deflection Plates tool (loaded with this example)

Answers

D = E t³/(12(1 − ν²)) = 121.5 lb-in; λ = p a⁴/(E t⁴) = 132.0
Centre deflection w_c = 0.06997 in = 1.399 t (Levy Table 9: w_c/h = 1.402, series −0.16 %); small-deflection value 0.2895 in, ratio 0.242
One-mode closed form λ = (π⁶/16)[W/(3(1 − ν²)) + 0.4905 W³] gives W = 1.496 (Table 11, one equation: 133.0 at W = 1.5)
Centre: membrane 1.626 ksi (σ a²/(E t²) = 6.19), bending 2.033 ksi (7.75), total 3.659 ksi; edge midpoint membrane 1.807 ksi (6.88, Levy's coefficients give 6.22 / 6.88 / 8.25)
Average edge tension 1.084 ksi, N_x = 54.2 lb/in on every edge; margin against 40 ksi +9.93; Wang's riveted-edge plate (TN 1425) deflects 1.243 t at this load

Notes

The tool's default inputs are this plate; press Submit. With Levy's own six coefficients the library reproduces every entry of Table 9 to the printed digits (test_large_deflection_plates.php, 314 checks: Report 737 Tables 9–11, Report 740 Tables 3–5, Wang's TN 1425 / 1462 tables, Timoshenko's small-deflection coefficients, scipy solve_bvp for the circular plate and an independent numpy quadrature Galerkin in References/Large-Deflection/checks/).