Tools › Structures › Stability › Plate Buckling › Example
Timoshenko and Gere: buckling coefficients of clamped plates, plates in shear and in pure bending
| Tool | Plate Buckling |
| Source | Timoshenko, S. P., and Gere, J. M. (1961). Theory of Elastic Stability, 2nd ed. McGraw-Hill, New York. Arts. 9.5 to 9.7 (Table 9-2 for clamped edges, Art. 9.6 pure bending, Art. 9.7 shear). |
| Level | Graduate |
Problem
Find the buckling coefficient k = N_cr b² / (π² D) of a rectangular isotropic plate (ν = 0.3) for: all four edges clamped under uniaxial compression at a/b = 1, 2 and 3; simply supported and clamped plates in pure shear at a/b = 1; and a simply supported plate under in-plane bending at a/b = 2/3 and 1.
Inputs to Enter
- Energy method (Rayleigh-Ritz) with 12 to 14 series terms; unit plates b = 1 at the stated a/b
- Fixity C C C C with N_x; S S S S and C C C C with N_xy; S S S S with N_xb
Answers
- Clamped, compression: k = 10.07 (a/b 1), 7.88 (a/b 2), 7.37 (a/b 3) (Table 9-2; reproduced within 0.5–1 %)
- Shear: k_s = 9.34 simply supported square, 14.71 clamped square; 6.55 at a/b 2 simply supported (Art. 9.7; within 0.5–1.5 %)
- Pure bending, simply supported: k_b = 23.9 at a/b = 2/3 and at a/b = 2 (three half-waves), 25.6 at a/b = 1 (Art. 9.6; within 1–1.5 %)
- Bending with the compressed edge clamped: least k_b = 39.6 (Bleich; the tool's closed form keeps Timoshenko's 41.8)
Notes
These are library checks (test_plate_ritz.php) of the energy method against the published coefficients; on the tool the same cases are run by entering the plate and one load. The closed-form method uses the handbook long-plate values 4.0 / 6.98 / 5.35 + 4 (s/l)² / 23.9 / 41.8, so it differs from the energy method on short clamped plates.