Tools › Structures › Plates and Shells › Orthotropic Plate Bending › Example
Rib-stiffened aluminium plate as an orthotropic plate: Huber's equation by the Lévy series, the smeared rigidities of Timoshenko Art. 86 and the affine reduction to Table 8
| Tool | Orthotropic Plate Bending |
| Source | Timoshenko, S. P., and Woinowsky-Krieger, S. (1959). Theory of Plates and Shells, 2nd ed. McGraw-Hill, New York. Chapter 11: Art. 85 (eq. 212, Huber's equation), Art. 86 (rigidities of a plate reinforced by equidistant stiffeners and of a gridwork), Art. 91 (the Lévy solution and its three root cases); Table 8 (Art. 30); Libove, C., and Batdorf, S. B. (1948). A General Small-Deflection Theory for Flat Sandwich Plates. NACA Report 899, eq. (8) and Appendix B. |
| Level | Graduate |
Problem
A 40 in by 20 in aluminium plate 0.1 in thick (E = 10.5 Msi, ν = 0.33) carries rectangular ribs 1.5 in deep and 0.125 in wide along its long side at a pitch of 5 in, on the face away from the load. All four edges are simply supported and the plate carries a uniform pressure of 2 psi. Idealize the stiffened plate as an orthotropic plate with Timoshenko's smeared rigidities (rib inertia about the plate middle plane, rib torsion counted), solve Huber's equation by the Lévy series, and find the deflection, the moments, the plating and rib-tip stresses, the reactions, and the deflection Huber's affine reduction and Timoshenko's Table 8 would give.
Inputs to Enter
- Build: plate stiffened by ribs; E = 10.5 Msi, ν = 0.33, t = 0.1 in; ribs along x at s_x = 5 in, h_r = 1.5 in, b_r = 0.125 in, E_r = 10.5 Msi, rib torsion counted, ribs on the bottom face
- a = 40 in along x, b = 20 in along y, edges 3 and 4 simply supported (edges 1 and 2 always simply supported); uniform pressure q = 2 psi
- Plate allowable 40 ksi, rib allowable 40 ksi, factor of safety 1.0, limit load; deflection limit 0.25 in
Answers
- Rib: I_own = 0.03516 in⁴, e = 0.8 in, I_r = I_own + A e² = 0.1552 in⁴, J_r = 0.000925 in⁴; plating D_p = 981.9 lb-in
- Rigidities: D_x = D_p + E_r I_r/s_x = 326,810 lb-in, D_y = D_p = 981.9 lb-in, D_1 = ν D_p = 324.0 lb-in, D_xy = (1 − ν) D_p/2 + G_r J_r/(4 s_x) = 511.6 lb-in, H = 1,347 lb-in; H/√(D_x D_y) = 0.0752 (Timoshenko's case 3, complex roots)
- w_max = 0.2250 in at the centre (Lévy series; independent numpy Navier 2001 × 2001 terms 0.225042948 in, finite differences with Richardson 0.22504295 in); α = w D_x/(q a⁴) = 0.01436; strip 5 q a⁴/(384 D_x) = 0.2040 in
- M_x = 443.4 lb-in/in and M_y = 3.364 lb-in/in at the centre; largest |M_y| = 6.925 lb-in/in at (20, 3.44) in; corner forces 8.05 lb; reactions 660.6 lb on edges 1 and 2, 155.5 lb on edges 3 and 4, sum 1,632 lb = 1,600 lb load + 32.2 lb of corner forces
- Stresses: plating faces σ_x = 1.377 ksi, σ_y = 4.155 ksi; rib tip E_r κ_x (t/2 + h_r) = 22.03 ksi (κ_x = 1.354 × 10⁻³ 1/in); local plating bending between ribs q s²/12 → 2.50 ksi; M.S. plate +8.63, rib +0.82, deflection +0.11 (limit)
- Huber's affine reduction: b′ = b (D_x/D_y)^¼ = 85.42 in, Table 8 at b′/a = 2.136 gives α = 0.01041 and w = 0.1632 in, 27 % below the exact orthotropic value because H/√(D_x D_y) = 0.075, not 1
Notes
The tool's default inputs are this plate; press Submit. The case is the tool's own default checked in test_orthotropic_plates.php (275 checks) against an independent numpy Navier sum, a finite-difference solution and a scipy boundary-value integration of the Lévy ordinary differential equation; the isotropic limit of the same library reproduces the Plate Bending tool and Timoshenko's Table 8 and Art. 41 values, and Seydel's corrugated-sheet rigidities of NACA TM 705 are reproduced within 1 % for D_x and H. The brief's 1 in deep ribs would put the rib tips at 43 ksi under 2 psi, above the yield strength of 2024-T3, so the default uses 1.5 in ribs (w/depth = 0.14).