Rectangular Plate, Two Adjacent Edges Fixed

1

Geometry, material and load

a = short side, b = long side (b ≥ a); edges x = 0 and y = 0 built in, x = a and y = b simply supported

Short side a

m

Long side b

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

b/aaspect ratio = —

Model: Kirchhoff thin plate, the two adjacent edges x = 0 (length b) and y = 0 (length a) built in (w = 0, ∂w/∂n = 0), the other two simply supported (w = 0, M_n = 0), uniform pressure q. No separable exact series exists for this case, so the plate is solved by the Rayleigh–Ritz method with products of clamped–simply-supported beam eigenfunctions (16 × 16 terms; edge moments Richardson-extrapolated from the 8- and 16-term solutions) — checked against a fine finite-difference solution to better than 1 %. Sign convention: w positive downward, M positive when the bottom face is in tension; the built-in edges carry negative (hogging) moments.

2

Deflection

Ritz solution — the maximum sits toward the corner between the two simply supported edges

w(x, y) = Σ_mΣ_n c_mn·φ_m(x)·φ_n(y) φ = clamped–SS beam modes (tan λ = tanh λ); c_mn from δ[½D∫(∇²w)² − ∫q·w] = 0

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

Compare (square, ν = 0.3): all edges simply supported α = 0.00406; one edge built in 0.0028; two opposite edges built in 0.00192; two adjacent edges built in 0.0022 at the peak, 0.0021 at the centre. 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, each with its own peak

M_x,edgebuilt-in edge x = 0 (length b), peak at y = … — hogging, = β_f·q·a² = —kN·m/m

M_y,edgebuilt-in edge y = 0 (length a), peak at x = … — hogging, = β_f2·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 the two built-in edges are alike: β_f = −0.069 at 0.57 of the edge length from the corner between the built-in edges, against 0.030 at the centre. For b/a > 1 the long built-in edge (x = 0, spanning the short direction) governs. Edge moments are extrapolated from the 8- and 16-term Ritz solutions (error ∝ 1/N²) and agree with a 200-division finite-difference solution to 0.1 %.

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 force

Only the corner between the two simply supported edges (a, b) needs holding down

R_corner= 2·M_xy at (a, b), downward — = n·q·a², n = — = —kN

M_xy vanishes along a built-in edge, so the three corners touching one carry no concentrated force; the SS–SS corner does. Edge reactions are not reported for this case — third derivatives of a Ritz expansion are not reliable enough.

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