Elliptic Hole in Infinite Plate under General In-plane Load

Stresses around an elliptic hole, aligned with the plate axes or inclined, in an infinite plate under any uniform in-plane far-field state given as stress resultants Nxx, Nyy, Nxy and the plate thickness t (far-field stresses N/t). The Inglis / Kolosov-Muskhelishvili solution is linear, so the general field is the superposition of the uni-axial solutions along the far-field principal directions: stress components, principal stresses and maximum shear at a user input point, the hoop-stress extremes on the hole edge with Kt referred to the maximum far-field principal stress S1, and the stress field around the hole.

  • Infinite plate with an elliptic hole under Nxx, Nyy, Nxy (biaxial tension or compression and shear)

  • Stress components, principal stresses and principal angle at a user input point

  • Hoop-stress extremes on the hole edge and Kt = σedge,max / S1

  • Fringe plots and line plots of the stress field

Loads and Hole Geometry

Unit System: /

Convention

Elliptic hole in an infinite plate under N_xx, N_yy, N_xy; semi-axes a (along x for beta = 0) and b; point P at (r, theta), theta from the y axis

Far-field Loads and Thickness





Hole




Stress State Location



Elliptic Hole General Load
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Elliptic Hole in Infinite Plate under General In-plane Load

Elliptic Hole in an Infinite Plate under General In-plane Load

Problem Statement

A large, thin, flat plate with an elliptic hole (semi-axes a and b, a inclined by β from the x axis) carries a uniform in-plane membrane state far from the hole, given as the stress resultants Nxx, Nyy, Nxy and the thickness t: σxx∞ = Nxx/t, σyy∞ = Nyy/t, τxy∞ = Nxy/t. The plate is linear elastic, isotropic, in plane stress and infinite. The question is the stress at a point P(r, θ), θ from the y axis towards x, and the hoop-stress extremes on the hole edge.

Elliptic hole under N_xx, N_yy, N_xy; P at (r, theta), theta from the y axis

Figure A: Convention (β = 0 shown). Nxx and Nyy positive in tension; Nxy positive when it acts in +x on the top edge; β rotates a counterclockwise from x; θ is measured from the y axis towards x.

Key Assumptions

Infinite plate, linear elastic, plane stress, uniform far field, open traction-free hole. Only membrane resultants are handled; bending moments are a different problem. The material stays elastic at the peak. The solution does not depend on the elastic constants.

Solution by Superposition

The far-field state is resolved into its principal values S1 ≥ S2 and the direction φ of S1. The Inglis solution is linear, so the field is the sum of the uni-axial fields for S1 at φ and S2 at φ + 90°. Each is evaluated with the Kolosov–Muskhelishvili potentials of the uni-axial elliptic tool on the map z = R(ζ + m/ζ), R = (a + b)/2, m = (a − b)/(a + b), with the load angle in the hole's frame α = φ − β:

φ(ζ) = (SR/4) [ζ + A/ζ],   A = 2e2iα − m;    ψ(ζ) = −(SR/4) [1/ζ + Āζ] − (SR/4) (1 + mζ²)(ζ² − A) / [ζ(ζ² − m)]

σxx + σyy = 4 Re Φ,    σyy − σxx + 2iτxy = 2 [z̄ Φ′ + ψ′] / ω′,    Φ = φ′/ω′.

The two fields are summed in the hole's frame and the tensor is rotated to the plate frame. The result is exact: the only arithmetic is complex algebra. With a = b it reproduces the general Kirsch field of the round-hole tool.

Hole Edge and Stress Concentration Factor

On the free edge only the hoop stress remains, equal to σxx + σyy = 4 Re Φ there. For an aligned ellipse under a single tension S along y it is (1 + 2a/b)S at the ends of a and −S at the ends of b; under equal biaxial tension 2(a/b)S at the ends of a and 2(b/a)S at the ends of b; for a round hole 3S1 − S2 and 3S2 − S1. In general (any a/b, β, S2/S1) the extremes leave the axis ends, so the tool evaluates the hoop stress around the whole edge (3600 points, refined by a golden-section search) and reports the largest and most compressive values with the polar angle of each.

Reference for Kt. There is no single applied stress, so Kt = σedge,max/S1, referred to the maximum far-field principal stress (uni-axial aligned: 1 + 2a/b; round hole in pure shear: 4). When the far field has no tensile principal stress the ratios are formed with S2.

Principal Stresses at the Probed Point

From σxx, σyy, τxy at P the tool gives the polar components about the hole center, σmax, σmin, τmax, and the direction of σmax from the y axis (φx) and from the radial direction (φp), each within ±90°.

Reading the Plots

  • σηη fringe. The tangential stress along the confocal ellipses, the hoop stress on the edge; the color scale spans the two edge extremes in units of the reference stress.
  • σmax fringe. The maximum principal stress; uniform S1 far from the hole.
  • Line plots. The hoop stress around the edge versus the polar angle of the edge point, and the plate-frame stresses along the hole's own a and b axes.

Limits

  • Finite width, yielding and very slender holes (a/b above about 20): as for the uni-axial tool.
  • Bending resultants are not handled.


References

Inglis, C. E. (1913). "Stresses in a Plate Due to the Presence of Cracks and Sharp Corners". Transactions of the Institution of Naval Architects, Vol. 55, pp. 219–241. The elliptic-hole solution in elliptic coordinates, including the biaxial case (Part II treats the plate under tension in two directions); scan filed in the site's References folder.

Muskhelishvili, N. I. (1953). Some Basic Problems of the Mathematical Theory of Elasticity, 3rd ed., translated by J. R. M. Radok. P. Noordhoff, Groningen. The complex potentials for an elliptic hole under tension at an arbitrary angle on the map z = R(ζ + m/ζ) (§§ 82–83 in the 1963 printing); the general far-field state is the sum of two such solutions along the far-field principal directions, which is how this tool evaluates it.

Savin, G. N. (1961). Stress Concentration Around Holes, translated by E. Gros. Pergamon Press, Oxford. Chapter II: elliptic hole under uni-axial, biaxial and pure shear loading, aligned and inclined, with the hoop-stress distributions around the edge and tables of the extremes.

Timoshenko, S. P., and Goodier, J. N. (1970). Theory of Elasticity, 3rd ed. McGraw-Hill, New York. Chapter 6, article "Elliptic hole in a plate under simple tension": the Inglis solution and the superposition for a second tension at right angles (hoop stress 2S(a/b) at the ends of a for equal biaxial tension).

Sadd, M. H. (2009). Elasticity: Theory, Applications, and Numerics, 2nd ed. Academic Press, Burlington. Chapter 10, "Complex variable methods": the Kolosov–Muskhelishvili stress formulas and the elliptic hole under uniform tension at an angle.

Pilkey, W. D., and Pilkey, D. F. (2008). Peterson's Stress Concentration Factors, 3rd ed. John Wiley & Sons, Hoboken. Chapter 4, "Holes": Kt for an elliptic hole in an infinite thin element under uni-axial and biaxial stress and under shear, and the inclined-ellipse charts.

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