Row of Holes: Method

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 N_xx, N_yy, N_xy, so that σ_xx^∞ = N_xx/t acts along the row, σ_yy^∞ = N_yy/t across it and τ_xy^∞ = N_xy/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 K_tg = σ_max/S₁ referred to the maximum far-field principal stress, and for a load across the row the net factor K_tn = σ_max/σ_n with σ_n = σ_yy^∞ s/(s − D).

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 σ_xx + σ_yy = 4 Re Φ and σ_yy − σ_xx + 2iτ_xy = 2[z̅ Φ′ + Ψ]. A single traction-free hole without a resultant force has the Laurent expansions Φ = Γ + ∑ α_n z⁻ⁿ, Ψ = Γ′ + ∑ β_n 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 P_n(z) = ∑_k (z − ks)⁻ⁿ,

Φ(z) = Γ + ∑_n α_n P_n(z), z̅ Φ′ + Ψ = Γ′ + ∑_n [ β_n P_n(z) + 2i n y α_n P_n+1(z) − n α_n P_n(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 = (σ_xx^∞ + σ_yy^∞)/4 and Γ′ = (σ_yy^∞ − σ_xx^∞)/2 + iτ_xy^∞ 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 σ_rr = τ_rθ = 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) z^j s^−n−j ζ_K(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 K_tg = σ_max/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 N_yy s per pitch, so the net-section stress is σ_n = σ_yy^∞/(1 − D/s) and K_tn = σ_max/σ_n; the tool also prints σ_yy 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): K_tn = 3.000 − 3.095λ + 0.309λ² + 0.786λ³,
tension along the row (Meijers 1967): K_tg = 3.0000 − 0.9916λ − 2.5899λ² + 2.2613λ³,
equal biaxial tension (Hütter 1942): K_tn = 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 (K_tn on σ/(1 − λ), K_tg on the gross stress)

D/s	across: Ktg	across: Ktn	Schulz fit	along: Ktg	Meijers fit	biaxial: Ktn	Hütter fit
0.1	3.000	2.700	2.694	2.936	2.877	1.830	1.850
0.2	3.006	2.405	2.400	2.768	2.716	1.704	1.715
0.3	3.031	2.122	2.121	2.547	2.531	1.605	1.596
0.4	3.096	1.858	1.862	2.326	2.334	1.516	1.489
0.5	3.241	1.621	1.628	2.139	2.139	1.427	1.393
0.6	3.546	1.419	1.424	1.995	1.961	1.335	1.305
0.7	4.204	1.261	1.255	1.887	1.813	1.241	1.224
0.8	5.755	1.151	1.124	1.802	1.707	1.151	1.148
0.9	10.706	1.071	1.038	1.733	1.658	1.071	1.073

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 σ_yy^∞ s across each ligament (Gauss integration of σ_yy 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.