CalcHandbook

Rectangular Plate, All Edges Fixed — Central Point Load

Plates · Rectangular · All Four Edges Built In · Concentrated Load P at the Centre — Kirchhoff Thin-Plate Theory, Navier + Lévy Superposition
1

Geometry, material and load

a = short side, b = long side (b ≥ a); thickness, E, ν, the load P and the side u of the small square it bears on
m
m
mm
kN
mm
MPa
—
Dflexural rigidity per unit width = E·t³12(1 − ν²) = —kN·m
b/aaspect ratio = —
Model: Kirchhoff thin plate, all four edges built in (w = 0, ∂w/∂n = 0), load P at the centre spread uniformly over a small square u × u (a true point load gives a finite deflection but a logarithmically infinite moment under it — the bearing area, typically the contact patch or about the plate thickness, sets the peak moment). Solved exactly by superposition (Timoshenko §38 generalised): the simply supported plate under the patch load (Navier series, odd terms to 299) plus edge-moment distributions En·sin(nπy/b) and Fm·sin(mπx/a) on the two pairs of edges (Lévy functions), the 40 coefficients chosen so that the edge slopes vanish. Checked against Timoshenko's coefficients for the clamped plate with a concentrated load (square: α = 0.00560, edge moment −0.1257·P). Sign convention: w positive downward, M positive when the bottom face is in tension; the built-in edges carry negative (hogging) moments.
2

Deflection

Doubly symmetric — the maximum is at the centre, wmax = α·P·a²/D
w(x, y) = wSS(x, y) + Σn En·Xn(x)·sin(nπy/b) + Σm Fm·Ym(y)·sin(mπx/a) wSS = Navier patch-load series; Xn, Ym = Lévy functions for unit edge moments; En, Fm from ∂w/∂n = 0 on the edges
wmaxat the centre — = α·P·a²/D = —mm a / wmax = —
α is Timoshenko's coefficient for the clamped plate with a concentrated load (0.00560 for a square plate, 0.00722 for b/a = 2; the simply supported plate gives 0.0116 — clamping halves the deflection). The small bearing patch changes the deflection only in the 4th decimal. 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 moment under the load and the midpoints of the long built-in edges govern
Mx,edgelong edges x = 0 and x = a, at y = b/2 — hogging, = βf·P = —kN·m/m
My,edgeshort edges y = 0 and y = b, at x = a/2 — hogging, = βf2·P = —kN·m/m
Mx,centreunder the load, spanning the short side — = β·P, β = — = —kN·m/m
My,centreunder the load, spanning the long side — = β₁·P, β₁ = — = —kN·m/m
σedge= 6·|Medge|/t² — top face in tension at the governing built-in edge = —MPa
σx,load= 6·Mx/t² = —MPa σy,load= 6·My/t² = —MPa
Timoshenko (ν = 0.3, concentrated load): edge moment at the midpoint of the long edges βf = −0.1257 / −0.1490 / −0.1674 for b/a = 1 / 1.4 / 2 — practically independent of the bearing size. The moment under the load grows like (1+ν)·P/4π·ln(a/u) as the bearing square shrinks (halving u adds about 0.07·P); square plate, ν = 0.3, u = 0.1a: β = β₁ = 0.231; u = 0.05a: 0.302.
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

All edges built in — 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 load P is shared by the four built-in edges, the long edges taking the larger part.
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