CalcHandbook

Rectangular Plate, All Edges Fixed — Uniform Load

Plates · Rectangular · All Four Edges Built In · Uniform Load q — Kirchhoff Thin-Plate Theory, Ritz Solution
1

Geometry, material and load

a = short side, b = long side (b ≥ a); thickness, E, ν and the uniform pressure q
m
m
mm
kN/m²
MPa
—
Dflexural rigidity per unit width = E·t³12(1 − ν²) = —kN·m
b/aaspect ratio = —
Model: Kirchhoff thin plate, all four edges built in (w = 0, ∂w/∂n = 0), uniform pressure q. No separable exact series exists, so the plate is solved by the Rayleigh–Ritz method with products of clamped–clamped beam eigenfunctions in both directions (16 × 16 terms; edge moments Richardson-extrapolated from the 8- and 16-term solutions) — checked against a fine finite-difference solution to better than 0.5 % and against Timoshenko's Table 35 (square: α = 0.00126, βf = −0.0513, β = 0.0231). Sign convention: w positive downward, M positive when the bottom face is in tension; the built-in edges carry negative (hogging) moments.
2

Deflection

Ritz solution — doubly symmetric, the maximum is at the centre
w(x, y) = ΣmΣn cmn·φm(x)·φn(y) φm, φn = clamped–clamped beam modes (cos λ·cosh λ = 1: λ = 4.7300, 7.8532, …); cmn from δ[½D∫(∇²w)² − ∫q·w] = 0
wmaxat the centre — = α·q·a⁴/D = —mm a / wmax = —
Compare (square, ν = 0.3): all edges simply supported α = 0.00406; one edge built in 0.0028; two opposite 0.00192; two adjacent 0.0022; three built in 0.0016; all four built in 0.00126 — the stiffest case; for b/a ≥ 2 the plate behaves as a fixed-ended strip across a (α → 1/384 = 0.00260). Thin-plate theory holds while wmax stays below about t/2.
Deflection along the short span — section A–A (y = b/2, 0 ≤ x ≤ a)
Deflection along the long span — section B–B (x = a/2, 0 ≤ y ≤ b)
3

Bending moments and stresses

Mx = −D(w,xx + ν·w,yy), My = −D(w,yy + ν·w,xx) — per unit width; the midpoints of the long built-in edges govern
Mx,edgelong edges x = 0 and x = a, at y = b/2 — hogging, = βf·q·a² = —kN·m/m
My,edgeshort edges y = 0 and y = b, at x = a/2 — hogging, = βf2·q·a² = —kN·m/m
Mx,centre(a/2, b/2), spanning the short side — = β·q·a², β = — = —kN·m/m
My,centre(a/2, b/2), spanning the long side — = β₁·q·a², β₁ = — = —kN·m/m
σedge= 6·|Medge|/t² — top face in tension at the governing built-in edge = —MPa
σx,centre= 6·Mx/t² = —MPa σy,centre= 6·My/t² = —MPa
Timoshenko Table 35 (ν = 0.3): βf = −0.0513 / −0.0757 / −0.0829 at the midpoint of the long edges for b/a = 1 / 1.5 / 2 (limit −1/12 = −0.0833), β = 0.0231 / 0.0368 / 0.0412 at the centre. Edge moments are extrapolated from the 8- and 16-term Ritz solutions (error ∝ 1/N²) and agree with a 120-division finite-difference solution to 0.5 %.
Mx along the short span — section A–A (y = b/2, 0 ≤ x ≤ a)
My along the long span — section B–B (x = a/2, 0 ≤ y ≤ b) — built-in end at y = 0
4

Corner forces

All edges built in — no concentrated corner forces
RcornerMxy = 0 along every built-in edge, and each corner lies on one = 0
Edge reactions are not reported for this case — third derivatives of a Ritz expansion are not reliable enough for design; the total load q·a·b is shared by the four built-in edges, the long edges taking the larger part.
5

At a given point

Read w, Mx, My, Mxy anywhere — marked in amber on the diagrams
m
0a/2a
m
0b/2b
w= —mm Mx= —kN·m/m
My= —kN·m/m Mxy= —kN·m/m