Circular Plate, Simply Supported Edge

1

Geometry, material and load

Radius R, thickness t, E, ν and the uniform pressure q — edge simply supported all round

Radius R

m

Thickness t

mm

Uniform load q

kN/m²

Modulus E

MPa

Poisson's ratio ν

—

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 simply supported all round (w = 0, M_r = 0 at r = R), uniform pressure q. The problem is axisymmetric and has the exact closed-form solution w = q(R² − r²)·[(5 + ν)R²/(1 + ν) − r²]/64D (Timoshenko & Woinowsky-Krieger §16): 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; here every moment is sagging.

2

Deflection

Axisymmetric closed form — maximum at the centre, the edge rotates freely

w(r) = q·(R² − r²)64·D [ 5 + ν1 + ν R² − r² ] satisfies ∇⁴w = q/D with w(R) = 0 and M_r(R) = 0

w_maxcentre, r = 0 — = (5 + ν)·q·R⁴ / 64(1 + ν)D = —mm 2R / w_max = —

The deflection is (5 + ν)/(1 + ν) times that of the clamped plate — about 4 times for ν = 0.3 — because the edge is free to rotate: w′(R) = −qR³/8(1 + ν)D. Thin-plate theory holds while w_max stays below about t/2.

Deflection along the diameter — section A–A (−R ≤ r ≤ R)

3

Bending moments and stresses

M_r = −D(w″ + ν·w′/r), M_t = −D(w′/r + ν·w″) — per unit width; the centre governs, M_r vanishes at the edge

M_r(r) = q16 (3 + ν)·(R² − r²)

M_t(r) = q16 [ (3 + ν)·R² − (1 + 3ν)·r² ]

M_centrer = 0, M_r = M_t — the maximum, sagging, = (3 + ν)·q·R²/16 = —kN·m/m

M_t,edger = R, tangential — sagging, = (1 − ν)·q·R²/8 = —kN·m/m

M_r,edger = R, radial — the simply supported edge carries no moment = —kN·m/m

σ_centre= 6·M_centre/t² — bottom face in tension, the maximum stress = —MPa

σ_t,edge= 6·M_t,edge/t² — tangential stress at the edge = —MPa

Both moments are sagging everywhere: M_r falls parabolically from (3 + ν)qR²/16 at the centre to zero at the edge, M_t only to (1 − ν)qR²/8. For ν = 0.3 the centre moment is 0.206·qR², 1.65 times the clamped-edge value of the built-in plate (0.125·qR²).

M_r (radial) along the diameter — section A–A

M_t (tangential) along the diameter — section A–A

4

Shear and edge reaction

Q_r(r) = q·r/2 from radial equilibrium; the whole load goes to the supported edge — no corner forces, no twisting moments

V_edgereaction per unit length of the edge — = q·R/2 = —kN/m

ΣVequilibrium check: V_edge·2πR = q·πR² = —kN vs —kN

Axisymmetry means M_rt = 0 everywhere, so the Kirchhoff edge reaction equals the shear Q_r(R) exactly and there are no concentrated corner forces (unlike the simply supported rectangular plate). The edge carries no moment.

5

At a given radius

Read w, M_r, M_t, Q_r at any r — marked in amber on the diagrams

r (from the centre)

m

0R/2R

w= —mm M_r= —kN·m/m

M_t= —kN·m/m Q_r= —kN/m