CalcHandbook

Rectangular Plate, 2 Edges SS + 2 Fixed — Uniform Load

Plates · Rectangular · Two Opposite Edges Simply Supported, the Other Two Built In · Uniform Load q — Kirchhoff Thin-Plate Theory, Lévy Solution
1

Geometry, material and load

a = short side, b = long side (b ≥ a); which pair of edges is built in; 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, two opposite edges simply supported (w = 0, Mn = 0) and the other two built in (w = 0, ∂w/∂n = 0), uniform pressure q. Solved exactly with the Lévy method: a sine series between the simply supported pair, hyperbolic functions across (odd terms to 99, each fixed by the four boundary conditions) — no coefficient tables, any aspect ratio and any ν. Sign convention: w positive downward, M positive when the bottom face is in tension; the built-in edges carry negative (hogging) moments.
2

Deflection

Lévy series — symmetric about both centrelines, maximum at the centre
w(x, y) = Σm Ym(y)·sin(mπx/a) m = 1, 3, 5, … — Ym = 4qa⁴/(π⁵Dm⁵) + hyperbolic terms fixed by Y = Y′ = 0 at both built-in edges
wmaxat … — = α·q·a⁴/D = —mm a / wmax = —
Compare: all edges simply supported α = 0.00406, one edge built in 0.0028, two opposite edges built in 0.00192 (square, ν = 0.3, Timoshenko Table 35). 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 built-in edges govern
Mfixedmidpoint of each built-in edge — hogging, = βf·q·a², βf = … = —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
σfixed= 6·|Mfixed|/t² — top face in tension along the built-in edges = —MPa
σx,centre= 6·Mx/t² = —MPa σy,centre= 6·My/t² = —MPa
The built-in edges take the largest moment — for a square plate at ν = 0.3, βf = −0.070 against 0.024 / 0.033 at the centre (Timoshenko Table 35). With the built-in pair on the short sides and b/a large, the middle of the plate tends to a one-way strip between the simply supported edges (β → 0.125).
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

Edge reactions and corner forces

Kirchhoff edge reactions V = Q + ∂Mxy/∂s; every corner touches a built-in edge, so there are no corner forces
Vfixedmidpoint of each built-in edge — = δf·q·a, δf = — = —kN/m
Vssmidpoint of each simply supported edge — = δ₁·q·a, δ₁ = — = —kN/m
ΣVequilibrium check: edge reactions = q·a·b (Mxy = 0 on every corner, no corner forces) = —kN vs —kN
The built-in edges attract the larger reactions. Because Mxy vanishes along a clamped edge and every corner lies on one, no corner hold-down forces arise. The equilibrium sum closes to within ~0.5 % — the edge-reaction series converges slowly (∝ 1/m).
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