CalcHandbook

Circular Plate, Clamped Edge — Uniform Load

Plates · Circular · Edge Built In All Round · Uniform Load q — Kirchhoff Thin-Plate Theory, Exact Axisymmetric Solution
1

Geometry, material and load

Radius R, thickness t, E, ν and the uniform pressure q — edge built in all round
m
mm
kN/m²
MPa
—
Dflexural rigidity per unit width = E·t³12(1 − ν²) = —kN·m
q·πR²total load on the plate = —kN
Model: Kirchhoff thin plate (t ≪ R, deflection ≪ t), edge built in all round (w = 0, dw/dr = 0 at r = R), uniform pressure q. The problem is axisymmetric and has the exact closed-form solution w = q(R² − r²)²/64D (Timoshenko & Woinowsky-Krieger §15): every quantity below is a polynomial in r — no series, no coefficient tables. Sign convention: w positive downward, M positive when the bottom face is in tension; the built-in edge carries a negative (hogging) radial moment.
2

Deflection

Axisymmetric closed form — maximum at the centre, zero slope at the edge
w(r) = q·(R² − r²)²64·D satisfies ∇⁴w = q/D with w(R) = 0 and w′(R) = 0
wmaxcentre, r = 0 — = q·R⁴/64D = —mm 2R / wmax = —
The simply supported circular plate deflects (5 + ν)/(1 + ν) times more — about 4 times for ν = 0.3 — so clamping is far more effective here than for rectangular plates. Thin-plate theory holds while wmax stays below about t/2.
Deflection along the diameter — section A–A (−R ≤ r ≤ R)
3

Bending moments and stresses

Mr = −D(w″ + ν·w′/r), Mt = −D(w′/r + ν·w″) — per unit width; the built-in edge governs
Mr(r) = q16 [ (1 + ν)·R² − (3 + ν)·r² ] Mt(r) = q16 [ (1 + ν)·R² − (1 + 3ν)·r² ]
Mr,edgebuilt-in edge, r = R — hogging, = −q·R²/8 = —kN·m/m
Mt,edgebuilt-in edge, tangential — = ν·Mr,edge = −ν·q·R²/8 = —kN·m/m
Mcentrer = 0, Mr = Mt — sagging, = (1 + ν)·q·R²/16 = —kN·m/m
σedge= 6·|Mr,edge|/t² — top face in tension at the built-in edge = —MPa
σcentre= 6·Mcentre/t² — bottom face in tension = —MPa
The edge moment −qR²/8 is independent of ν and is 2/(1 + ν) times the centre moment (1.54× for ν = 0.3). Mr changes sign at r = R·√[(1 + ν)/(3 + ν)] ≈ 0.63R, Mt at r = R·√[(1 + ν)/(1 + 3ν)] ≈ 0.83R; the tangential moment stays hogging over the outer ring only.
Mr (radial) along the diameter — section A–A
Mt (tangential) along the diameter — section A–A
4

Shear and edge reaction

Qr(r) = q·r/2 from radial equilibrium; the whole load goes to the built-in edge — no corner forces, no twisting moments
Vedgereaction per unit length of the edge — = q·R/2 = —kN/m
ΣVequilibrium check: Vedge·2πR = q·πR² = —kN vs —kN
Axisymmetry means Mrt = 0 everywhere, so the Kirchhoff edge reaction equals the shear Qr(R) exactly and there are no concentrated corner forces. The edge also carries the hogging moment Mr,edge per unit length.
5

At a given radius

Read w, Mr, Mt, Qr at any r — marked in amber on the diagrams
m
0R/2R
w= —mm Mr= —kN·m/m
Mt= —kN·m/m Qr= —kN/m