Hole in Plate Bending: Method

Round Hole in a Thin Plate under Bending

Problem Statement

A thin plate of thickness t with a round hole of radius a = D/2 carries uniform remote bending moments per unit width M_xx, M_yy and a twisting moment M_xy. Wanted are the moments and surface stresses at a point, and the concentration of the bending stress at the hole edge.

Key Assumptions

Kirchhoff thin-plate theory, infinite plate, free hole edge, linear elastic. Transverse shear deformation is neglected, which holds for D/t above about 5; for smaller holes Reissner's theory gives a larger factor, approaching 3. Surface stresses are 6M/t²; the mid-plane is unstressed, so membrane loads (the companion tools) add linearly on each face.

Solution

For a single remote moment M on the faces normal to x, the far-field deflection is w₀ = −M(x² − νy²)/[2D(1 − ν²)], D the flexural rigidity. The hole adds the biharmonic terms ln r, r⁻²cos 2θ and cos 2θ, whose coefficients follow from the two Kirchhoff free-edge conditions at r = a: M_r = 0 and the effective shear V_r = Q_r + (1/r)∂M_rθ/∂θ = 0 (the twisting moment itself need not vanish). The result, with q = a²/r² and θ from the moment's x axis:

M_r = (M/2)(1 − q) + (M/2)[1 − 4νq/(3 + ν) − 3(1 − ν)q²/(3 + ν)] cos 2θ

M_θ = (M/2)(1 + q) + (M/2)[−1 − 4q/(3 + ν) + 3(1 − ν)q²/(3 + ν)] cos 2θ

M_rθ = −(M/2)[1 − 2(1 − ν)q/(3 + ν) + 3(1 − ν)q²/(3 + ν)] sin 2θ

On the edge M_r = 0 and M_θ = M[1 − 2(1 + ν)/(3 + ν) cos 2θ], largest at θ = 90°: K = (5 + 3ν)/(3 + ν), Goodier's factor (1.79 for ν = 0.3), and (1 − ν)/(3 + ν) at θ = 0. The membrane counterpart is 3 and −1: in bending the hole disturbs the plate much less, because the free edge can shed its moment through the Kirchhoff shear and both faces share the curvature. Equal biaxial bending gives M_θ = 2M all round, the same as the membrane case. A general state is the superposition over the two principal moments.

Checks built into the development: the derived field satisfies M_r = 0 and V_r = 0 on the edge to 10⁻⁹ (the shear condition evaluated by finite differences of the moments), recovers the far-field moments, reproduces Goodier's factor, and the superposition equals the single-moment field.

Limits

• Thick plates (D/t < 5): Reissner theory, Peterson's charts.
• Elliptic holes in bending: Goodier (1936) also; not implemented.
• Combined membrane and bending: add the surface stresses of this tool to those of the membrane tools on each face.