CalcHandbook

Rectangular Plate, Simply Supported — Uniform Load

Plates · Rectangular · All Four Edges Simply Supported · Uniform Load q — Kirchhoff Thin-Plate Theory, Navier 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 (t ≪ a, deflection ≪ t), all four edges simply supported (w = 0, Mn = 0), uniform pressure q over the whole plate. Solved exactly with the Navier double sine series (odd terms up to m, n = 199) — no coefficient tables, any aspect ratio and any ν. Sign convention: w positive downward, M positive when the bottom face is in tension.
2

Deflection

Navier series — maximum at the centre
w(x, y) = 16qπ⁶D ΣmΣn sin(mπx/a)·sin(nπy/b)m·n·(m²/a² + n²/b²)² m, n = 1, 3, 5, …
wmaxcentre (a/2, b/2) — = α·q·a⁴/D, α = — = —mm a / wmax = —
α is the Timoshenko deflection coefficient (0.00406 for a square plate at ν = 0.3, 0.01302 for b/a → ∞ where the plate acts as a one-way strip: 5qa⁴/384D ÷ (1−ν²)). 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
Mx,maxcentre, spanning the short side — = β·q·a², β = — = —kN·m/m
My,maxcentre, spanning the long side — = β₁·q·a², β₁ = — = —kN·m/m
σx,max= 6·Mx/t² — centre, top/bottom face = —MPa
σy,max= 6·My/t² = —MPa
The larger moment spans the short side (Mx); as b/a grows it tends to qa²/8 (one-way strip) while My tends to ν·qa²/8. For a square plate at ν = 0.3, β = β₁ = 0.0479.
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)
4

Edge reactions and corner forces

Kirchhoff edge reaction V = Q + ∂Mxy/∂s; corners must be held down
Vx,maxmiddle of the long edges (x = 0, a) — = δ·q·a, δ = — = —kN/m
Vy,maxmiddle of the short edges (y = 0, b) — = δ₁·q·a, δ₁ = — = —kN/m
Rcorner= 2·Mxy at each corner, downward (anchorage) — = n·q·a², n = — = —kN
ΣVequilibrium check: edge reactions − 4·Rcorner = q·a·b = —kN vs —kN
Twisting moments Mxy along the edges become extra distributed reactions and concentrated corner forces pulling the plate up; without hold-downs the corners lift and the moments rise. For a square plate at ν = 0.3: δ = 0.420, n = 0.065. 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