Row of Round Holes in Infinite Plate under General In-plane Load

Stresses around an infinite row of equal round holes 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 holes lie along x at a pitch s. The field is the periodic Muskhelishvili solution (Howland 1935, Schulz 1941, Meijers 1967) carried to convergence: the stress components, principal stresses and maximum shear at a user input point about the central hole, the hoop-stress extremes on the hole edge with Kt referred to the maximum far-field principal stress S1, the net-section factor for the load across the row, and the stress field around the holes.

  • Infinite row of round holes along x at pitch s, infinite plate, 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, Ktg = σedge,max / S1 and the net-section factor Ktn for the load across the row

  • Fringe plots and line plots of the stress field between the holes

Loads and Hole Geometry

Unit System: /

Convention

Infinite row of round holes along x at pitch s in an infinite plate under N_xx, N_yy, N_xy; point P at (r, theta) about the central hole, theta from the y axis

Far-field Loads and Thickness





Row of Holes



Stress State Location



Row of Holes General Load
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Row of Round Holes in Infinite Plate under General In-plane Load

Row of Round Holes in an Infinite Plate under General In-plane Load

Problem Statement

An infinite thin plate of thickness t contains an infinite row of equal round holes of diameter D (radius a = D/2) whose centers lie on the x axis at the pitch s. Far from the row the plate carries a uniform in-plane stress state given by the resultants Nxx, Nyy, Nxy, so that σxx∞ = Nxx/t acts along the row, σyy∞ = Nyy/t across it and τxy∞ = Nxy/t in shear. The holes interact: a hole shields its neighbors from a load along the row and crowds the ligaments between them under a load across the row. The questions are the stress at a point P(r, θ) about the central hole, θ measured from the y axis towards x, the peak hoop stress on the hole edge, the gross factor Ktg = σmax/S1 referred to the maximum far-field principal stress, and for a load across the row the net factor Ktn = σmax/σn with σn = σyy∞ s/(s − D).

Infinite row of round holes along x at pitch s under N_xx, N_yy, N_xy; P at (r, theta) about the central hole

Figure A: Convention. The holes lie along x at the pitch s; θ is measured from the y axis towards x, the same convention as the other hole tools, so θ = ±90° are the points of the central hole facing its neighbors.

Key Assumptions

Infinite plate, infinite row of equal holes, linear elastic, plane stress, uniform far field. Every hole is traction-free and carries no resultant force, so the solution is periodic in x with the period s and does not depend on the elastic constants (it is the same in plane strain). The material stays elastic at the peak. The series is carried to convergence for 0 < D/s ≤ 0.9. A finite number of holes, a finite plate width or a second row (staggered or square lattice) are outside the tool.

Periodic Solution

With Muskhelishvili's potentials Φ(z) and Ψ(z) the stresses are σxx + σyy = 4 Re Φ and σyy − σxx + 2iτxy = 2[z̅ Φ′ + Ψ]. A single traction-free hole without a resultant force has the Laurent expansions Φ = Γ + ∑ αn z−n, Ψ = Γ′ + ∑ βn z−n (n ≥ 2; Kirsch's solution is the case n = 2, 4). By periodicity every hole of the row carries the same coefficients, and summing the single-hole terms over the hole centers ks (k = 0, ±1, ±2, …) gives, with the lattice sums Pn(z) = ∑k (z − ks)−n,

Φ(z) = Γ + ∑n αn Pn(z), z̅ Φ′ + Ψ = Γ′ + ∑n [ βn Pn(z) + 2i n y αn Pn+1(z) − n αn Pn(z) ].

The two extra terms come from writing the distance to each hole, z̅ − ks, as z̅ − z + (z − ks); they make the stress field periodic even though z̅ is not. All sums with n ≥ 2 converge absolutely and decay as exp(−2π|y|/s) away from the row, so Γ = (σxx∞ + σyy∞)/4 and Γ′ = (σyy∞ − σxx∞)/2 + iτxy∞ give the far field exactly. A uniform far field on a centrally symmetric row excites even n only; the normal loads give real coefficients and the shear load imaginary ones, so the two are solved separately and superposed. The coefficients are found by imposing σrr = τrθ = 0 at 3N + 60 points of the hole edge in a least-squares sense, with the order N chosen from D/s (8 at D/s = 0.1, 20 at 0.5, 104 at 0.9); the lattice sums are evaluated exactly, the nearest holes directly and the rest by the expansion ∑|k|>K (z − ks)−n = 2(−1)n ∑j C(n+j−1, j) zj s−n−j ζK(n+j) in truncated zeta functions. This is the construction of Horii and Nemat-Nasser (1985) and Isida and Igawa (1991) applied to the single row, and the problem first solved by Howland (1935) with periodic biharmonic functions.

Stress Concentration Factors

The hoop stress on the hole edge is scanned for its extremes. The gross factor Ktg = σmax/S1 is referred to the maximum far-field principal stress, as in the other general-load tools (S2 when S1 is not tensile). For a load across the row the ligaments of width s − D carry the load Nyy s per pitch, so the net-section stress is σn = σyy∞/(1 − D/s) and Ktn = σmax/σn; the tool also prints σyy midway between two holes, the lowest ligament stress. For a load along the row no load crosses the ligaments and only the gross factor is meaningful. The published charts (Peterson Charts 4.57–4.59) are reproduced by the fitted polynomials of Pilkey's tables, printed for comparison:

tension across the row (Schulz 1941): Ktn = 3.000 − 3.095λ + 0.309λ² + 0.786λ³,
tension along the row (Meijers 1967): Ktg = 3.0000 − 0.9916λ − 2.5899λ² + 2.2613λ³,
equal biaxial tension (Hütter 1942): Ktn = 2.000 − 1.597λ + 0.934λ² − 0.337λ³, λ = D/s.

Table 1 compares the converged series with these fits. The series agrees with the Schulz fit within 0.5 % up to λ = 0.7 and lies 2 to 3 % above it for the two closest spacings, where the original chart was drawn from a truncated solution. The Meijers fit matches at λ = 0.4 to 0.5 but drops too steeply from 3 near λ = 0: a first-order estimate with the neighbors' Kirsch fields summed over the row (ζ(2) = π²/6) gives 2.934 at λ = 0.1, confirming the series value 2.936 against the fit's 2.877. The Hütter fit is within 1 % up to λ = 0.4. The tool's values are the converged series.

Table 1: Series factors against the published fits (Ktn on σ/(1 − λ), Ktg on the gross stress)
D/sacross: Ktgacross: KtnSchulz fitalong: KtgMeijers fitbiaxial: KtnHütter fit
0.13.0002.7002.6942.9362.8771.8301.850
0.23.0062.4052.4002.7682.7161.7041.715
0.33.0312.1222.1212.5472.5311.6051.596
0.43.0961.8581.8622.3262.3341.5161.489
0.53.2411.6211.6282.1392.1391.4271.393
0.63.5461.4191.4241.9951.9611.3351.305
0.74.2041.2611.2551.8871.8131.2411.224
0.85.7551.1511.1241.8021.7071.1511.148
0.910.7061.0711.0381.7331.6581.0711.073

Checks

The lattice sums were verified against a brute-force sum over 400,000 holes and against the closed form P2 = (π/s)² csc²(πz/s). The solved field is periodic in x to 10−16, returns to the far field as exp(−2πy/s), leaves a root-mean-square traction on the hole edge below 10−8 of the load (10−13 for D/s ≥ 0.5), and carries exactly σyy∞ s across each ligament (Gauss integration of σyy from the hole edge to mid-ligament). At D/s = 0.001 the factor is 3.000 (Kirsch) and the pure-shear factor is 4.0; the results do not change when the series order is doubled.

Limitations

The tool stops at D/s = 0.9 (ligament one ninth of the diameter), where the series needs about 100 terms and a page takes a few seconds. It does not cover a finite number of holes (two or three interacting holes, Peterson Charts 4.21–4.25, differ from the infinite row by several percent at close spacing), a finite plate width (Schulz's finite-width chart), staggered or square lattices (Meijers), or holes of unequal size. The hoop stress in Figure 1 is referred to the nearest hole center, so under a shear load the color field has a seam midway between the holes where the reference center changes.



References

Howland, R. C. J. (1935). "Stresses in a Plate Containing an Infinite Row of Holes". Proceedings of the Royal Society of London, Series A, Vol. 148, pp. 471–491. The first solution of the infinite row of equal circular holes in a plate under tension, by periodic biharmonic functions; the problem the tool solves.

Schulz, K. J. (1941). Over den spanningstoestand in doorboorde platen (On the State of Stress in Perforated Plates). Doctoral thesis, Technische Hoogeschool Delft. Stress concentration factors for a row of holes with the tension across the row, the source of Peterson's Chart 4.57 and of the net-section polynomial the tool prints.

Meijers, P. (1967). Doubly-Periodic Stress Distributions in Perforated Plates. Dissertation, Technische Hogeschool Delft. Rows and lattices of holes under tension along and across the row by the complex-potential method; the source of Peterson's Chart 4.58 for the tension along the row.

Hütter, A. (1942). "Die Spannungsspitzen in gelochten Blechscheiben und Streifen". Zeitschrift für angewandte Mathematik und Mechanik, Vol. 22, p. 322. Row of holes under equal biaxial tension, the source of Peterson's Chart 4.59.

Pilkey, W. D., and Pilkey, D. F. (2008). Peterson's Stress Concentration Factors, 3rd ed. John Wiley & Sons, Hoboken. Section 4.3.12 and Charts 4.57–4.59: infinite row of circular holes in an infinite thin element under uniaxial tension across the row, along the row, and biaxial tension; the gross and net definitions.

Pilkey, W. D. (2005). Formulas for Stress, Strain, and Structural Matrices, 2nd ed. John Wiley & Sons, Hoboken. Chapter 6, Table 6-1, case 5 "Single row of circular holes in infinite plate": the fitted polynomials Kt = 3.000 − 3.095(d/L) + 0.309(d/L)² + 0.786(d/L)³ with σnom = σ/(1 − d/L), Kt = 3.0000 − 0.9916(d/L) − 2.5899(d/L)² + 2.2613(d/L)³ on the gross stress, and Kt = 2.000 − 1.597(d/L) + 0.934(d/L)² − 0.337(d/L)³ for equal biaxial tension (0 ≤ d/L ≤ 1), printed by the tool for comparison. The series identifies the net-section polynomial with the tension across the row and the gross-stress polynomial with the tension along the row. Copy of the chapter filed in the site's References folder.

Muskhelishvili, N. I. (1953). Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen. The complex potentials Φ(z), Ψ(z) and the stress combinations σxx + σyy = 4 Re Φ, σyy − σxx + 2iτxy = 2[z̅Φ′ + Ψ] on which the periodic solution is built.

Horii, H., and Nemat-Nasser, S. (1985). "Elastic Fields of Interacting Inhomogeneities". International Journal of Solids and Structures, Vol. 21, No. 7, pp. 731–745. Interacting holes by superposed single-hole Laurent expansions with the coefficients fixed on each boundary; the tool applies the same construction to the periodic row with lattice sums of the Laurent terms.

Isida, M., and Igawa, H. (1991). "Analysis of a Zig-Zag Array of Circular Holes in an Infinite Solid under Uniaxial Tension". International Journal of Solids and Structures, Vol. 27, No. 7, pp. 849–864. Periodic arrays of holes by the same complex-potential and lattice-sum approach; the single row is its limiting case.

Timoshenko, S. P., and Goodier, J. N. (1970). Theory of Elasticity, 3rd ed. McGraw-Hill, New York. Chapter 4, "Effect of circular holes on stress distributions in plates": the Kirsch limit Kt = 3 for the single hole and the discussion of rows of holes.

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