Rectangular Plate, Three Edges Fixed, One Free

1

Geometry, material and load

a = short side, b = long side (b ≥ a); which edge is free; thickness, E, ν and the uniform pressure q

Short side a

m

Long side b

m

Thickness t

mm

Uniform load q

kN/m²

Modulus E

MPa

Poisson's ratio ν

—

Free edge

Dflexural rigidity per unit width = E·t³12(1 − ν²) = —kN·m

b/aaspect ratio = —

Model: Kirchhoff thin plate, three edges built in (w = 0, ∂w/∂n = 0) and one edge free (M_n = 0, V_n = 0), uniform pressure q. Solved by the Rayleigh–Ritz method with the full strain energy (the free edge deflects, so the Gaussian-curvature term is kept): clamped–clamped beam modes between the two opposite built-in edges, and across them a clamped-root Chebyshev family ξ²·T_k(2ξ−1) that leaves the free edge unconstrained (16 × 16 terms; built-in edge moments Richardson-extrapolated from the 8- and 16-term solutions). Checked against an independent trial space and the cantilever-strip limit. Sign convention: w positive downward, M positive when the bottom face is in tension.

2

Deflection

Ritz solution — the maximum is at the middle of the free edge

w(x, y) = Σ_mΣ_n c_mn·φ_m(x)·φ_n(y) φ_m = clamped–clamped modes (cos λ·cosh λ = 1), ψ_n = ξ²·T_n(2ξ−1); c_mn from δΠ = 0, Π = ½D∫[w_,xx² + w_,yy² + 2νw_,xxw_,yy + 2(1−ν)w_,xy²] − ∫q·w

w_maxat … — = α·q·a⁴/D = —mm a / w_max = —

Compare (square, ν = 0.3): three edges built in and the fourth simply supported gives 0.0016; leaving it free raises the peak to 0.0030 at the free edge (the centre stays at 0.0019). As a/b grows the plate tends to a cantilever strip across b: w → qb⁴/8D, root moment → qb²/2. Thin-plate theory holds while w_max 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

M_x = −D(w_,xx + ν·w_,yy), M_y = −D(w_,yy + ν·w_,xx) — per unit width; the built-in edges govern, the free edge carries a sagging moment along it

M_x,edgebuilt-in edges x = 0 and x = a, peak at y = … — hogging, = β_f·q·a² = —kN·m/m

M_y,edgebuilt-in edge y = 0, peak at x = … — hogging, = β_f2·q·a² = —kN·m/m

M_freemiddle of the free edge, parallel to it — sagging, = β_e·q·a² = —kN·m/m

M_x,centre(a/2, b/2), spanning the short side — = β·q·a², β = — = —kN·m/m

M_y,centre(a/2, b/2), spanning the long side — = β₁·q·a², β₁ = — = —kN·m/m

σ_edge= 6·|M_edge|/t² — top face in tension at the governing built-in edge = —MPa

σ_x,centre= 6·M_x/t² = —MPa σ_y,centre= 6·M_y/t² = —MPa

For a square plate at ν = 0.3: β_f ≈ −0.092 on the two opposite built-in edges (peak close to the free end of the edge), −0.056 at the midpoint of the third, and +0.044 along the free edge. Built-in edge moments are Richardson-extrapolated (error ∝ 1/N²); the free-edge quantities converge directly with the Chebyshev family.

M_x along the short span — section A–A (y = b/2, 0 ≤ x ≤ a)

M_y along the long span — section B–B (x = a/2, 0 ≤ y ≤ b) — built-in end at y = 0

4

Corner forces

Every corner touches a built-in edge — no concentrated corner forces; the free edge carries no reaction

R_cornerM_xy = 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 built-in edges carry the bulk of the load.

5

At a given point

Read w, M_x, M_y, M_xy anywhere — marked in amber on the diagrams

x (along a)

m

0a/2a

y (along b)

m

0b/2b

w= —mm M_x= —kN·m/m

M_y= —kN·m/m M_xy= —kN·m/m