CalcHandbook

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

Plates · Rectangular · Three Edges Built In, One Simply Supported · 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 simply supported; 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 simply supported (w = 0, Mn = 0), pressure varying linearly from q₁ at the built-in edge opposite the simply supported one to q₂ at the simply supported 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). No separable exact series exists, so the plate is solved by the Rayleigh–Ritz method with products of beam eigenfunctions — clamped–clamped modes between the two opposite built-in edges, clamped–simply-supported modes across (16 × 16 terms; edge moments Richardson-extrapolated from the 8- and 16-term solutions) — checked against a fine finite-difference solution to better than 0.5 %. 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 on the centreline, shifted toward the simply supported edge and toward the heavier end of the load
w(x, y) = ΣmΣn cmn·φm(x)·φn(y) φm = clamped–clamped modes (cos λ·cosh λ = 1), φn = clamped–SS modes (tan λ = tanh λ); cmn from δ[½D∫(∇²w)² − ∫q(x,y)·w] = 0, the load integral taken with the linear ramp
wmaxat … — = α·qmax·a⁴/D = —mm a / wmax = —
Compare (square, ν = 0.3, three built in + one SS): uniform load α = 0.0016; hydrostatic triangle with the zero at the built-in edge and qmax at the SS edge 0.00091, the other way round 0.00073 — roughly half the uniform value, as the triangle carries half the load. 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
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
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: with the zero at the built-in edge and qmax at the SS edge βf = −0.036 on the two opposite built-in edges (peak at 0.66 of the edge from the built-in end) and −0.020 at the third; with the triangle reversed −0.029 (peak at 0.45) and −0.035. Edge moments are extrapolated from the 8- and 16-term Ritz solutions (error ∝ 1/N²) and agree with a 120-division finite-difference solution to 0.5 %.
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
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