CalcHandbook

Rectangular Plate, 3 Edges Fixed + 1 Free — Varying Load

Plates · Rectangular · Three Edges Built In, One Edge Free · Linearly Varying (Hydrostatic / Trapezoidal) Load — Kirchhoff Thin-Plate Theory, Ritz Solution
1

Geometry, material and load

a = short side, b = long side (b ≥ a); which edge is free; thickness, E, ν and the pressure at the two ends of the load ramp
m
m
mm
kN/m²
kN/m²
MPa
—
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 (Mn = 0, Vn = 0), pressure varying linearly from q₁ at the built-in edge opposite the free one to q₂ at the free edge and constant along it (q₁ = 0 or q₂ = 0 gives the hydrostatic triangle, q₁ = q₂ the uniform case); coefficients α, β are referred to qmax = max(q₁, q₂). 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·Tk(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 limits (triangular load: w = 11qb⁴/120D or qb⁴/30D, root moment qb²/3 or qb²/6). 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 when the load grows toward it; with the heavy end at the root it moves inside the plate
w(x, y) = ΣmΣn cmn·φm(x)·φn(y) φm = clamped–clamped modes (cos λ·cosh λ = 1), ψn = ξ²·Tn(2ξ−1); cmn from δΠ = 0, Π = ½D∫[w,xx² + w,yy² + 2νw,xxw,yy + 2(1−ν)w,xy²] − ∫q(x,y)·w, the load ramp integrated exactly
wmaxat … — = α·qmax·a⁴/D = —mm a / wmax = —
Compare (square, ν = 0.3, three built in + one free): uniform load α = 0.0030 at the free edge; hydrostatic triangle growing toward the free edge 0.0024 (free edge), reversed 0.0008 (inside the plate, the free edge itself only 0.0006). As a/b grows the plate tends to a cantilever strip across b: w → 11qmaxb⁴/120D (root moment qmaxb²/3) for the triangle growing toward the tip, qmaxb⁴/30D (qmaxb²/6) reversed. 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; the built-in edges govern, the free edge carries a sagging moment along it
Mx,edgebuilt-in edges x = 0 and x = a, peak at y = … — hogging, = βf·qmax·a² = —kN·m/m
My,edgebuilt-in edge y = 0, peak at x = … — hogging, = βf2·qmax·a² = —kN·m/m
Mfreemiddle of the free edge, parallel to it — sagging, = βe·qmax·a² = —kN·m/m
Mx,centre(a/2, b/2), spanning the short side — = β·qmax·a², β = — = —kN·m/m
My,centre(a/2, b/2), spanning the long side — = β₁·qmax·a², β₁ = — = —kN·m/m
σedge= 6·|Medge|/t² — top face in tension at the governing built-in edge = —MPa
σx,centre= 6·Mx/t² = —MPa σy,centre= 6·My/t² = —MPa
Square plate, ν = 0.3, hydrostatic triangle growing toward the free edge: βf ≈ −0.076 on the two opposite built-in edges (peak close to the free end), −0.021 at the root, +0.034 along the free edge; reversed: −0.030, −0.035 and +0.010. Built-in edge moments are Richardson-extrapolated (error ∝ 1/N²); the free-edge quantities converge directly with the Chebyshev family.
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) — 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
RcornerMxy = 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, 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