Plate bending
A thin rectangular plate with each edge simply supported, fixed or free, under uniform and patch pressures: moments, deflection and first yield. Use it for floor plate, hatches, covers, tank panels and stair treads. No code sets this method: it is plate theory, solved by finite elements.
Inputs of the plate #
- Spans a and b, thickness t.
- Edges: simply supported (no deflection, free to rotate), fixed (no deflection and no slope), or free. A real bolted or tack-welded plate is usually between the first two, and which applies is your call. Two simply supported edges or one fixed edge are needed to hold the plate.
- E, ν, Fy: steel prefilled. For another metal, enter its values; the φ and Ω used are AISC's, so choose an Fy that fits your code.
- Own weight: on by default, as load type D.
- Pressures: up to 8, each with a load type, uniform or on a patch. A wheel or post is its force over its contact area. A patch needs to be at least 1 % of the span in each direction.
- Deflection limit: shorter span over the value you enter. No code sets it for a plate; 120 is prefilled.
Auto-design varies the thickness. Spans, edges, loads, material and the deflection limit are kept.
Checks performed #
| Check | Basis |
|---|---|
| D, Mx, My, Mxy | Thin-plate theory; finite elements with cubic Hermite functions, at least 24 per direction, closer around patches |
| First yield, von Mises at the worst point | 6 M / t² per stress; AISC 360-22 Chapter F factors φ 0.90, Ω 1.67. Load combinations ASCE 7-22 2.3.1 or 2.4.1 |
| Deflection | Largest of the ASD combinations, against your limit |
| Advisory: span under 10 × thickness | Shear deformation, ignored here, starts to count |
| Advisory: deflection over half the thickness | Membrane action stiffens the plate; linear theory overstates this |
| Advisory: patch smaller than the thickness | Thin-plate theory overstates the moment |
First yield is a conservative limit on purpose: a plate has reserve past it by redistribution and membrane action, which you may count in your own calculation.
Limitations #
- Linear, thin, flat, of constant thickness, on continuous edges. No stiffeners, openings, interior supports or discrete fasteners. A simply supported plate's corners lift in reality; here the edges hold against uplift.
- No point load: a true point load has an infinite moment in this theory. Enter its contact patch.
- A long span over 20 × the short span stops: use a strip of unit width.
- No fatigue, no tread pattern, no plate-to-support design. Tread plate: use the base thickness.
Verification examples #
Verification status: 37 values compared in 9 examples (9 published, 0 hand calculations from the cited equations): 37 within tolerance, 0 recorded deviations. Code basis: the provisions it uses were read in the target edition.
| Example | Source | Values |
|---|---|---|
| All edges simply supported, b/a 1, 2, 5 | Batista, arXiv 1001.3016 (2010), Table 2a | Deflection, Mx, My at the center |
| x edges simply supported, y edges fixed, b/a 1, 2 | Tables 3a, 3b | With My at the fixed edge |
| x edges simply supported, y edges free, b/a 1, 2 | Tables 4a, 4b | With deflection and Mx at the free edge |
| All edges fixed, b/a 1, 2 | Tables 5a, 5b | With moments at the fixed edges |
These are series solutions printed to eight places, and the paper compares them with Timoshenko and Woinowsky-Krieger. Roark and Timoshenko themselves are not on hand. Patch loads have no published example here: they are checked against the Navier double series, a hand method for simply supported edges. Down to the smallest patch tried, 1/16 of the span, the moment here is within 2 % of that series. Patch loads on fixed or free edges are not checked against anything else. First yield is a plain limit with no published plate example.