Pressurized Round Hole in Infinite Plate under General In-plane Load

Stresses around a round hole in an infinite plate that carries a uniform internal pressure (fluid pressure, or the contact pressure of an interference-fit pin or bushing) together with any uniform in-plane far-field state given as Nxx, Nyy, Nxy and the thickness t. The Lamé pressure field (σrr = −p a²/r², σθθ = +p a²/r²) superposes on the Kirsch field: stresses at a user input point, hoop-stress extremes on the hole edge, and the stress field around the hole. The interference-fit pressure of a rigid pin can be computed from the radial interference.

  • Round hole with internal pressure p (fluid or interference fit) plus far-field Nxx, Nyy, Nxy

  • Lamé pressure field superposed on the Kirsch field; rigid-pin interference pressure from the radial interference

  • Stress components and principal stresses at a user input point; hoop-stress extremes on the hole edge

  • Fringe plots and line plots of the combined stress field

Loads, Pressure and Hole Geometry

Unit System: /

Convention

Pressurized round hole in an infinite plate under N_xx, N_yy, N_xy; point P at (r, theta), theta from the y axis

Far-field Loads and Thickness





Hole and Internal Pressure






Stress State Location



Pressurized Hole
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Pressurized Round Hole in Infinite Plate under General In-plane Load

Pressurized Round Hole under General In-plane Load

Problem Statement

A round hole of radius a = D/2 in an infinite plate carries a uniform radial pressure p on its surface, from a fluid or from a press-fit pin or bushing, while the plate carries the uniform far-field state Nxx, Nyy, Nxy over the thickness t. Wanted are the combined stresses at a point, the hoop-stress extremes on the hole edge and the stress field.

Pressurized hole under far-field load

Key Assumptions

Uniform pressure, infinite plate, linear elastic, plane stress. The pressure is the same all round the hole, as for a fluid or a well-fitted bushing; a pin that bears on one side of the hole (a loaded lug) produces a cosine-like bearing distribution and a resultant force, a different problem. The interference pressure formula takes the pin as rigid; an elastic pin reduces p.

Solution

The pressure alone gives the Lamé field, the thick-cylinder solution with the outer radius at infinity: σrr = −pa²/r², σθθ = +pa²/r², σrθ = 0. It is a pure shear state (σrr + σθθ = 0), carries no net load, decays as 1/r², and moves the hole edge outward by u = (1 + ν)pa/E in plane stress. A rigid pin of radial interference δ therefore produces p = Eδ/[(1 + ν)a]. The far-field state gives the general Kirsch field of the round-hole tool. Both are linear, so they add: on the edge the radial stress is −p, the shear vanishes and the hoop stress is the Kirsch edge value plus p, so its extremes are 3S1 − S2 + p on the S2 axis and 3S2 − S1 + p on the S1 axis. The pressure raises the hoop stress everywhere on the edge by the same amount, which is why a tight press fit lowers the fatigue life of a plate in tension less than one might expect at the sides (the mean stress rises but the range does not) while it can turn the compressive points at the load axis tensile.

Reference for Kt

With a far-field load present the factors are referred to the maximum far-field principal stress S1, as in the companion tools (so Kt = 3 − S2/S1 + p/S1); with pressure only they are referred to p (hoop stress = 1.0p all round).

Limits

  • Pin bearing load (one-sided contact with a resultant) is not a uniform pressure; see Peterson's pin-loaded hole charts or a contact analysis.
  • Elastic pin, finite width, yielding: not included.


References

Timoshenko, S. P., and Goodier, J. N. (1970). Theory of Elasticity, 3rd ed. McGraw-Hill, New York. Chapter 4: the Lamé thick-cylinder solution (the pressurized hole in an infinite plate is its outer-radius limit, σr = −pa²/r², σθ = +pa²/r²) and the Kirsch solution that it superposes with.

Kirsch, G. (1898). "Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre". Zeitschrift des Vereines deutscher Ingenieure, Vol. 42, No. 28, pp. 797–807. The far-field part of the solution; scan filed in the site's References folder.

Boresi, A. P., and Schmidt, R. J. (2003). Advanced Mechanics of Materials, 6th ed. John Wiley & Sons. Thick-wall cylinders and shrink fits: the interference pressure between a pin or bushing and a plate, and its superposition with the plate stresses.

Pilkey, W. D., and Pilkey, D. F. (2008). Peterson's Stress Concentration Factors, 3rd ed. John Wiley & Sons, Hoboken. Chapter 4: pressurized holes and pin-loaded (bearing-loaded) holes; the latter, with its non-uniform bearing distribution, is not covered by this tool.

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