Tools › Structures › Plates and Shells › Circular and Annular Plates › Example
Annular steel plate, simply supported outside with a free central hole, under uniform pressure: exact solution against the Wahl–Lobo coefficients
| Tool | Circular and Annular Plates |
| Source | Timoshenko, S. P., and Woinowsky-Krieger, S. (1959). Theory of Plates and Shells, 2nd ed. McGraw-Hill, New York. Chapter 3, Art. 15–17 (general solution of the axisymmetric plate equation, Table 3 after Wahl and Lobo 1930); Young, W. C., and Budynas, R. G. (2002). Roark's Formulas for Stress and Strain, 7th ed. Table 11.2 case 2a. |
| Level | Undergraduate |
Problem
A steel annular plate (E = 29 Msi, ν = 0.3) of 10 in outer radius, 3 in inner radius and 0.25 in thickness is simply supported round its outer edge, free at the hole, and carries a uniform pressure of 5 lbf/in². Find the deflection, the moments and the bending stresses, the largest stress and where it occurs, the edge reaction, and the Timoshenko Table 3 coefficient form of the results.
Inputs to Enter
- a = 10 in, b = 3 in, t = 0.25 in; E = 29 Msi, ν = 0.3; outer edge simply supported, inner edge free
- Uniform pressure q = 5 lbf/in²; radius of interest 6 in; allowable 36 ksi at limit, factor 1.0
Answers
- D = E t³/(12(1 − ν²)) = 41,490 lb-in; general solution w = C_1 + C_2 (r/a)² + C_3 ln(r/a) + C_4 (r/a)² ln(r/a) + q r⁴/(64 D) with C_1 = 0.06654, C_2 = −0.08537, C_3 = −0.02596, C_4 = −0.01356 in
- Largest deflection 0.09173 in at the hole edge (0.367 t); w(6 in) = 0.05400 in; edge slope −0.01349
- Hole edge: M_r = 0, M_t = 163.6 lb-in/in → σ_t = 15.71 ksi, the largest stress; r = 6 in: M_r = 45.13, M_t = 94.89 lb-in/in, Q_r = −11.25 lb/in; outer edge M_t = 50.95 lb-in/in
- Reaction 1,429 lb = q π (a² − b²); Timoshenko form k = w_max E t³/(q a⁴) = 0.8313, k_1 = σ_max t²/(q a²) = 1.963 against Table 3's k_1 = 1.88 (a/b = 3) and 2.08 (a/b = 4), interpolated 1.947 → 15.57 ksi
- M.S. = 36/15.71 − 1 = +1.29 at limit
Notes
The tool's default inputs are this plate; press Submit. The case is the tool's own default checked against an independent scipy boundary-value solution (References/Circular-Plates/checks/circplate_check.py) and the Plate Bending tool's annular closed form (test_circular_plates.php); the Table 3 figures kept on the page are those the exact solution reproduces within their printed rounding.