Two Holes: Method

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

Problem Statement

An infinite thin plate of thickness t contains two round holes of diameters D₁ and D₂ (radii a₁, a₂) whose centers lie on the x axis a distance s apart, hole 1 at the origin and hole 2 at x = s. Far from the holes 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 line of centers, σ_yy^∞ = N_yy/t across it and τ_xy^∞ = N_xy/t in shear. The holes interact: under a load across the line of centers the ligament between them crowds the stress and the peaks on the x axis of each hole rise above Kirsch's 3σ (at the outer points for moderate spacing, where the other hole's σ_xx lowers the facing point, and at the facing points once the holes are close); under a load along the line each hole shields the other and the peaks fall below 3σ and move away from the ligament; a small hole beside a large one sees the large hole's own concentration as its far field and can reach twice Kirsch's factor. The questions are the stress at a point P(r, θ) about hole 1, θ measured from the y axis towards x, the peak hoop stress on each hole, the gross factors K_tg = σ_max/S₁ referred to the maximum far-field principal stress, and the stresses in the ligament.

Figure A: Convention. Hole 1 is at the origin and hole 2 at x = s; θ is measured from the y axis towards x about each hole's own center, the same convention as the other hole tools, so θ = 90° on hole 1 and θ = −90° on hole 2 are the two points facing each other.

Key Assumptions

Infinite plate, two round holes, linear elastic, plane stress, uniform far field. Both holes are traction-free and carry no resultant force; the solution 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 as long as the gap between the edges is at least a fifth of the larger diameter (equal holes: D/s ≤ 0.833). A finite plate, a third hole or a row of holes are outside the tool (the row has its own tool).

Solution by Superposed Single-hole Expansions

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]. Each traction-free hole without a resultant force contributes the Laurent terms α_n (z − c)⁻ⁿ to Φ and β_n (z − c)⁻ⁿ to Ψ, n ≥ 2 (Kirsch's solution is n = 2, 4), about its own center c. For two holes the two expansions are added to the far field,

Φ(z) = Γ + ∑_j=1,2 ∑_n α_n^(j) (z − c_j)⁻ⁿ, z̅ Φ′ + Ψ = Γ′ + ∑_j ∑_n [ β_n^(j) − n α_n^(j) e^−2iφ_j ] (z − c_j)⁻ⁿ,

with φ_j the polar angle about hole j, Γ = (σ_xx^∞ + σ_yy^∞)/4 and Γ′ = (σ_yy^∞ − σ_xx^∞)/2 + iτ_xy^∞. The geometry is symmetric about the x axis, so the normal loads give real coefficients and the shear load imaginary ones; the two are solved separately and superposed. Each hole's terms leave tractions on the other hole, so the coefficients of both holes are found together by imposing σ_rr = τ_rθ = 0 at N_j + 12 points of the upper half of each hole in a least-squares sense (Horii and Nemat-Nasser 1985), the order N_j following a_j/(s − a_k) (8 at D/s = 0.1, 19 at 0.5, 52 at 0.8 for equal holes; a large hole beside a small one needs the higher order on the large hole). Ling (1948) solved the equal-hole problem exactly in bipolar coordinates and Haddon (1967) the unequal-hole problem by complex potentials; the series reproduces both.

Stress Concentration Factors

The hoop stress on each edge is scanned for its extremes. The gross factors K_tg = σ_max/S₁ are referred to the maximum far-field principal stress, as in the other general-load tools (S₂ when S₁ is not tensile). In an infinite plate the ligament has no net-section stress of its own; Peterson's charts for the load across the line of centers use the convention σ_nom = σ √(1 − λ²)/(1 − λ), λ = D/s, which the tool prints for equal holes together with the fitted polynomials of Pilkey's tables:

tension along the line of centers (Chart 4.21): K_tg = 3.000 − 0.712λ + 0.271λ²,
tension across the line of centers (Chart 4.22): K_tn = 3.0000 − 3.0018λ + 1.0099λ²,
equal biaxial tension (Chart 4.23): K_tn = 2.000 − 2.119λ + 2.493λ² − 1.372λ³.

Table 1 compares the converged series with these fits and Table 2 with the values of Ling, Haddon and Chiang collected by Badiger and Ramakrishna (2024), for equal holes and for a hole beside a larger one. The series reproduces the exact values to every printed digit; the fits are within 1 % of it for λ ≤ 0.6 and the across-the-line fit falls below it by 2 % at λ = 0.7 and 8 % at 0.8. The tool's values are the converged series.

Table 1: Two equal holes, series against the fits (K_tn on σ√(1 − λ²)/(1 − λ), K_tg on the gross stress)

D/s	along: Ktg	fit	across: Ktg	across: Ktn	fit	biaxial: Ktn	fit
0.1	2.980	2.932	3.001	2.714	2.710	1.819	1.812
0.2	2.927	2.868	3.004	2.453	2.440	1.673	1.665
0.3	2.853	2.811	3.015	2.212	2.190	1.557	1.552
0.4	2.777	2.759	3.035	1.987	1.961	1.465	1.464
0.5	2.715	2.712	3.066	1.770	1.752	1.392	1.392
0.6	2.670	2.670	3.116	1.558	1.563	1.330	1.330
0.7	2.642	2.634	3.381	1.420	1.394	1.272	1.268
0.8	2.623	2.604	4.034	1.345	1.245	1.219	1.198

Table 2: Gross factor of the pair against the published values (gap between the edges g in units of the radius a of hole 1; hole 2 has the radius b)

b/a	g/a	load	series	Ling 1948	Haddon 1967	Chiang	finite elements
1	1	along	2.650	2.623	2.650	2.625	2.682
1	2	along	2.715	2.703	2.715	2.703	2.752
1	4	along	2.827	2.825	2.827	2.825	2.863
1	1	across	3.264	3.264	3.264	3.264	3.316
1	2	across	3.066	3.066	3.066	3.066	3.108
1	4	across	3.020	3.020	3.020	3.020	3.062
2	1	along	2.925			2.924	2.961
3	1	along	2.982			2.982	3.021
4	1	along	2.996			2.993	3.035
2	1	across	4.051			4.051	4.112
3	1	across	4.854			4.854	4.899
4	1	across	5.545			5.537	5.583

For the unequal pairs the peak of the load across the line sits on the small hole (4.051 at b/a = 2 against 3.263 on the large one), and the peak of the load along the line sits on the large hole, which is nearly undisturbed (2.925 to 2.996) while the small hole in its shadow falls to 2.0, 1.5 and 1.1.

Checks

A center distance of 2000 radii gives Kirsch's 3 and −1 on each hole; the field of two equal holes is symmetric about x = s/2 to 10⁻¹⁴; the root-mean-square traction left on the two edges is below 3 × 10⁻⁸ of the load (10⁻¹³ for close holes, where the order is higher); pure shear at s = 4a gives 4.55 against 4 for a single hole; the results do not change when the order is doubled.

Limitations

The tool stops when the gap between the edges is less than a fifth of the larger diameter (equal holes: D/s > 0.833), where the series needs more than 60 terms per hole and the page takes several seconds; Ling's and Haddon's solutions reach closer spacings. It does not cover a finite plate, a hole near a straight edge, three or more holes, or holes of different shape. The hoop stress in Figure 1 is referred to the nearer hole center, so the color field has a seam where the reference center changes.