CalcHandbook

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

Plates · Rectangular · Three Edges Simply Supported, One Edge Built In · Uniform Load q — Kirchhoff Thin-Plate Theory, Lévy Solution
1

Geometry, material and load

a = short side, b = long side (b ≥ a); which edge is built in; thickness, E, ν and the uniform pressure q
m
m
mm
kN/m²
MPa
—
Dflexural rigidity per unit width = E·t³12(1 − ν²) = —kN·m
b/aaspect ratio = —
Model: Kirchhoff thin plate, three edges simply supported (w = 0, Mn = 0) and one edge built in (w = 0, ∂w/∂n = 0), uniform pressure q. Solved exactly with the Lévy method: a sine series between the two parallel simply supported edges, hyperbolic functions across (odd terms to 99, each fixed by the four boundary conditions) — no coefficient tables, any aspect ratio and any ν. Sign convention: w positive downward, M positive when the bottom face is in tension; the built-in edge carries a negative (hogging) moment.
2

Deflection

Lévy series — the maximum sits between the centre and the simply supported edge opposite the built-in one
w(x, y) = Σm Ym(y)·sin(mπx/a) m = 1, 3, 5, … — Ym = 4qa⁴/(π⁵Dm⁵) + hyperbolic terms fixed by Y(0) = Y′(0) = 0 (built in) and Y(b) = Y″(b) = 0 (simply supported)
wmaxat … — = α·q·a⁴/D = —mm a / wmax = —
wcentre(a/2, b/2) — = αc·q·a⁴/D, αc = — = —mm
Compare the all-simply-supported plate: α = 0.00406 for a square at ν = 0.3, against 0.0028 at the centre here (Timoshenko Table 32) — one built-in edge cuts the deflection by about a third and moves the peak away from that edge. 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 edge governs
Mfixedmidpoint of the built-in edge — hogging, = βf·q·a², βf = … = —kN·m/m
Mx,centre(a/2, b/2), spanning the short side — = β·q·a², β = — = —kN·m/m
My,centre(a/2, b/2), spanning the long side — = β₁·q·a², β₁ = — = —kN·m/m
σfixed= 6·|Mfixed|/t² — top face in tension along the built-in edge = —MPa
σx,centre= 6·Mx/t² = —MPa σy,centre= 6·My/t² = —MPa
The built-in edge takes the largest moment — for a square plate at ν = 0.3, βf = −0.084 against 0.034 / 0.039 at the centre (Timoshenko Table 32). The centre moments are read at (a/2, b/2); the span moment peak lies slightly toward the free simply supported edge.
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

Edge reactions and corner forces

Kirchhoff edge reactions V = Q + ∂Mxy/∂s; corner forces only at the two corners between simply supported edges
Vfixedmidpoint of the built-in edge — = δf·q·a, δf = — = —kN/m
Vssmidpoint of the opposite simply supported edge — = δ₁·q·a, δ₁ = — = —kN/m
Vsidelargest along the two simply supported side edges — = δ·q·a, δ = — = —kN/m
Rcorner= 2·Mxy at the two SS–SS corners, downward — = n·q·a², n = — (zero at the built-in edge, where Mxy = 0) = —kN
ΣVequilibrium check: edge reactions − 2·Rcorner = q·a·b = —kN vs —kN
The built-in edge attracts the largest reaction; the two corners next to it need no hold-down (Mxy vanishes on a clamped edge), the two far corners do. The equilibrium sum closes to within ~0.5 % — the edge-reaction series converges slowly (∝ 1/m).
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