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Simply supported 20 × 10 in aluminium plate under 2 psi: Timoshenko's Table 8 coefficients from the Navier and Lévy series
| Tool | Plate Bending |
| Source | Timoshenko, S. P., and Woinowsky-Krieger, S. (1959). Theory of Plates and Shells, 2nd ed. McGraw-Hill, New York. Article 30 and Table 8 (simply supported rectangular plates under uniform load, ν = 0.3, b/a = 2: α = 0.01013, β = 0.1017, β1 = 0.0464, n = 0.092); Article 28 (Navier's solution). |
| Level | Undergraduate |
Problem
A rectangular aluminium plate 20 in by 10 in and 0.16 in thick (E = 10.5 Msi, ν = 0.3) is simply supported on all four edges and carries a uniform pressure of 2 psi. Find the flexural rigidity, the maximum deflection, the bending moments per unit width at the centre, the largest bending stress, the edge reactions and the corner forces, and compare the deflection and moment coefficients with Timoshenko's Table 8 for b/a = 2.
Inputs to Enter
- Shape rectangular, a = 20 in along x, b = 10 in along y, t = 0.16 in; E = 10.5 Msi, ν = 0.30; all four edges simply supported
- Load type uniform pressure, q = 2 psi; allowable stress 40 ksi, factor of safety 1.0, load type limit; deflection limit 0.1 in
Answers
- D = E t³ / [12 (1 − ν²)] = 3,938 lb-in
- w_max = 0.05143 in at the centre (Lévy series, 120 harmonics; Navier 135 × 135 terms 0.051435 in); α = w D / (q s⁴) = 0.010129 with s = 10 in the shorter side (Table 8: 0.01013 → 0.05144 in)
- M_y (across the short span) = 20.34 lb-in/in, β = 0.1017 (Table 8: 0.1017); M_x = 9.270 lb-in/in, β1 = 0.04635 (Table 8: 0.0464)
- σ_max = 6 M_y / t² = 4.766 ksi on the faces at the centre; M.S. = 40 / 4.766 − 1 = +7.39 (limit); deflection M.S. = 0.1 / 0.05143 − 1 = +0.94
- Edge reactions 72.59 lb (edges 1 and 2) and 164.4 lb (edges 3 and 4), sum 474.0 lb = load 400 lb + four corner forces of 18.51 lb (n = R / (q s²) = 0.0925; Table 8 prints 0.092 for b/a = 2)
Notes
The tool's default inputs are this plate; press Submit. The series values are checked in test_plate_bending.php against an independent numpy Navier sum to 4001 × 4001 terms (w_c = 0.0514346171 in, M_y = 20.33661705 lb-in/in, R = 18.50682186 lb) and every row of Timoshenko's Tables 8 and 35 is reproduced within 0.6 % (the tables were computed with a few series terms; the exact clamped-square coefficients 0.0012653, −0.05133 and 0.02291 of Taylor and Govindjee 2004 are reproduced to five figures). The brief's 0.1 in thickness under 10 psi would deflect ten times its thickness, outside small-deflection theory, so the default uses 0.16 in and 2 psi (w/t = 0.32).