Lid Deflection Calculator
Pump a chamber down and the atmosphere presses on the lid with about 10 N per square centimeter — over a tonne on a 40 cm lid. This calculator applies the standard flat circular-plate formulas to give the center deflection and peak stress under 1 atm of uniform load, for a lid resting on an O-ring (simply supported) or bolted rigidly all around (clamped).
Results
| Flexural rigidity D | — |
|---|---|
| Center deflection wmax | — |
| Deflection / thickness | — |
| Maximum bending stress | — |
| Stress location | — |
| Safety factor (strength / stress) | — |
| Total load on lid | — |
Validity envelope: these numbers assume an ideal flat, unperforated plate with uniform load, textbook edge conditions, and nominal material properties. Real lids have ports, O-ring grooves, scratches, and welds that concentrate stress and reduce strength — a comfortable-looking safety factor here is a preliminary sizing result, not a structural qualification. See the assumptions below, and treat any lid near people or thin margins conservatively.
Note: the computed deflection exceeds half the thickness. Small-deflection plate theory is unreliable beyond w ≈ t/2 (membrane stiffening makes the real plate stiffer, but stresses redistribute); thicken the lid rather than trusting this number.
Note: the safety factor is below the recommendation for this material — this site suggests at least 3–4 for metals and at least 8 for acrylic, which creeps and crazes under sustained load and fails without warning.
The math
For a flat circular plate of radius a, thickness t, under uniform pressure q (here 1 atm = 101 325 Pa), define the flexural rigidity
Roark's cases for a uniformly loaded circular plate (Roark & Young, Table 11.4, cases 10a/10b; identical in Timoshenko §16) give:
Simply supported edge — deflection and stress both peak at the center:
Clamped edge — deflection peaks at the center, stress at the edge (radial, top surface):
A lid resting on an O-ring is close to simply supported — the seal cannot apply a bending moment at the rim — and this is the conservative case: about four times the deflection and higher center stress than clamped. Use it unless the lid is genuinely bolted stiff against a heavy flange. Real lids with viewport holes, weld seams, or off-center ports concentrate stress beyond these formulas.
Material values used
| Material | E (GPa) | ν | Strength (MPa) |
|---|---|---|---|
| Mild steel (A36) | 200 | 0.29 | 250 (yield) |
| Aluminum 6061-T6 | 68.9 | 0.33 | 276 (yield) |
| Acrylic (PMMA) | 3.1 | 0.37 | 70 (tensile) |
Assumptions and limits
- Small-deflection thin-plate theory: valid for w ≲ t/2 and t ≲ a/4. The calculator flags the first condition.
- Solid, flat, uniform plate — no holes, ribs, or welds; room temperature.
- Acrylic values are short-term; under sustained vacuum load acrylic creeps, so its effective long-term modulus and strength are substantially lower. The conservative safety-factor floor of 8 reflects common vacuum-practice guidance for brittle plastics.
Worked check
Steel lid, 300 mm span, 10 mm thick: D = 18 197 N·m. Clamped: w = 101 325 × 0.15⁴ / (64 × 18 197) = 0.044 mm, σ = 17.1 MPa — matching the textbook qa⁴/64D case by hand. Simply supported: w = 0.181 mm (4.1× more), σ = 28.1 MPa, safety factor ≈ 8.9 on A36 yield.
Sources
- W. C. Young & R. G. Budynas, Roark's Formulas for Stress and Strain, 7th/8th ed., McGraw-Hill — Table 11.4, circular plate, uniform load, cases 10a (simply supported) and 10b (fixed).
- S. Timoshenko & S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed., McGraw-Hill, 1959 — §16.
- Material properties: ASM Handbook Vol. 2 (aluminum), ASTM A36 (steel), typical published PMMA data.
Educational reference, not an operating procedure: results reflect the stated model and assumptions. Verify anything safety-critical against primary sources, and read the safety fundamentals before applying numbers to real hardware. Last reviewed: · Report a correction.