Round Hole in a Thin Plate under Bending

Moments and surface stresses around a round hole in a thin plate under uniform remote bending and twisting moments Mxx, Myy, Mxy per unit width (Kirchhoff plate theory, Goodier 1936). Closed-form moment field, the hole-edge extremes with the bending stress concentration factor K = (5 + 3ν)/(3 + ν) for uni-axial bending, surface stresses 6M/t², and the field plotted around the hole.

  • Thin plate with a round hole under remote bending Mxx, Myy and twisting Mxy (Kirchhoff theory, Goodier 1936)

  • Closed-form moment field; surface stresses 6M/t² at a user input point

  • Hole-edge moment extremes; K = (5 + 3ν)/(3 + ν) ≈ 1.8 for uni-axial bending, far below the membrane value 3

  • Fringe plots and line plots of the surface stress field

Moments, Hole and Plate

Unit System: /

Convention

Round hole in a thin plate under remote bending moments M_xx, M_yy, M_xy

Far-field Moments per Unit Width and Thickness





Hole and Material



Stress State Location



Hole In Plate Bending
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Round Hole in a Thin Plate under Bending

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 Mxx, Myy and a twisting moment Mxy. Wanted are the moments and surface stresses at a point, and the concentration of the bending stress at the hole edge.

Round hole in a plate under bending moments

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: Mr = 0 and the effective shear Vr = Qr + (1/r)∂Mrθ/∂θ = 0 (the twisting moment itself need not vanish). The result, with q = a²/r² and θ from the moment's x axis:

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

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

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

On the edge Mr = 0 and Mθ = 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 Mθ = 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 Mr = 0 and Vr = 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.


References

Goodier, J. N. (1936). "The Influence of Circular and Elliptical Holes on the Transverse Flexure of Elastic Plates". Philosophical Magazine, Series 7, Vol. 22, No. 145, pp. 69–80. The Kirchhoff-plate solution for a circular hole under uniform bending: the moment field and the edge factor (5 + 3ν)/(3 + ν).

Timoshenko, S. P., and Woinowsky-Krieger, S. (1959). Theory of Plates and Shells, 2nd ed. McGraw-Hill, New York. Chapter 9, "Plates of various shapes", article on the bending of a plate with a circular hole: Goodier's result and the Kirchhoff free-edge conditions used in the derivation.

Reissner, E. (1945). "The Effect of Transverse Shear Deformation on the Bending of Elastic Plates". Journal of Applied Mechanics, Vol. 12, pp. A69–A77. Thick-plate theory; the bending stress concentration at a hole rises from the Kirchhoff value towards 3 as the hole diameter approaches the thickness.

Pilkey, W. D., and Pilkey, D. F. (2008). Peterson's Stress Concentration Factors, 3rd ed. John Wiley & Sons, Hoboken. Chapter 4, Section 4.6, "Bending": circular hole in a plate under simple and cylindrical bending, with the Kirchhoff and Reissner curves against D/t.

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