Stress Concentration Factors: the Hole in the Flange and the Number the Textbook Gives You

A student drills a bolt hole in the tension flange of a steel member, opens the strength of materials textbook, and finds the rule for a discontinuity: σ_max = K · σ_nom. The chart hands back a factor near 2.5, the arithmetic gives a peak stress around 300 MPa, and the conclusion looks unavoidable: the flange is overstressed.

The calculist at the next desk runs the same joint through a steel specification and it passes at roughly half capacity. Same geometry, same load, two verdicts that disagree by a factor of two. The gap is not an arithmetic slip. It is a difference in what each number is for.

This article walks the hole all the way through: what the stress concentration factor measures, where the famous value of 3 comes from, why a ductile steel member under a static load does not design around the peak, and the situations where that same peak is precisely the thing that breaks the part.

A stress concentration factor is the ratio of the true local peak stress to a chosen nominal stress:

K_t = σ_max / σ_nom

Three properties decide how you should read it. First, it is elastic. The theory behind K_t assumes linear elastic material all the way up, with no yielding anywhere. Second, it is geometric. K_t is fixed by the shape of the discontinuity and by ratios such as the hole diameter over the width, not by the steel grade or by how hard you pull. Third, it is tied to a reference area. σ_max is a single physical value, but σ_nom is a convention, and a K_t only means something when you know whether its nominal stress was taken on the gross section or the net section.

The subscript t stands for theoretical. It is a property of the drawing. Later we meet K_f, the fatigue notch factor, which is smaller and does depend on the material.

The value everyone remembers, K_t = 3, is the Kirsch solution of 1898. For an infinite plate with a small circular hole pulled in one direction, the tangential stress at the hole edge, on the axis perpendicular to the load, is exactly three times the far field stress. That is the origin of the 3, and it is referenced to the gross (far field) stress.

Real flanges are not infinite. For a finite width strip the chart in Peterson or Roark gives K_t as a function of the ratio d/w. Referenced to the net section, it falls from 3.0 as the hole grows:

K_t,net = 3.00 − 3.14(d/w) + 3.667(d/w)² − 1.527(d/w)³

The two conventions are linked by K_t,gross = K_t,net / (1 − d/w). Mixing them is the single most common error in reading the chart, and we return to it in the mistakes list. For the worked flange below, d/w = 0.22 and K_t,net = 2.47.

Take a flange plate 100 mm wide and 10 mm thick, with one standard 22 mm hole for an M20 bolt, carrying an axial tension of P = 100 kN.

Now the tension. That 317 MPa peak is above the yield stress of ordinary structural steel: F_y = 250 MPa for ASTM A36, 275 MPa for EN S275. Read at face value, the hole edge is overstressed by a quarter. Yet the average stress across the net ligament is only 128 MPa, about half of yield. The member is nowhere near failing, and one edge fibre is nominally past yield. Both statements are true. The next two sections explain how.

The first temptation is to wave the peak away as a theoretical artifact. Do not. Photoelastic tests and finite element models both show the concentration plainly, and a fatigue crack really does start at the hole edge. The elastic peak is a genuine feature of the stress field.

Two facts put it in proportion. It is local: the elevated stress decays back to the nominal within roughly one hole diameter, so only a thin rim of material ever sees anything close to σ_max. And it is elastic theory: K_t assumes the material stays linear all the way up to the peak, which stops being true the moment that rim yields. What happens after the rim yields is the whole story, and it depends on whether the material is ductile.

Raise the load on the worked flange. The hole edge reaches F_y first and yields. A ductile steel does not fracture there. It flows at roughly constant stress, the yield plateau, and sheds the extra force to the still elastic material beside it. The stress profile across the ligament flattens: the peak clips at F_y while the rest of the net section climbs to meet it.

By the time the member is near its limit, the net section is close to fully plastic and the average stress, not the vanished spike, sets the capacity. The governing quantity becomes the net section against F_u (rupture) or the gross section against F_y (yielding). This redistribution is the physical reason a steel specification folds the hole into an area rather than a K_t. Ductility is doing the work the factor seemed to demand.

AISC 360, and equally NBR 8800 and Eurocode 3, size a tension member with two limit states, neither of which contains a stress concentration factor:

• Gross section yielding: φR_n = φ F_y A_g, with φ = 0.90.
• Net section rupture: φR_n = φ F_u A_e, with φ = 0.75 and A_e = U·A_n, where U captures shear lag.

For the worked A36 flange, with U = 1: gross yielding gives 0.90 × 250 × 1000 = 225 kN, and net rupture gives 0.75 × 400 × 780 = 234 kN. The capacity is 225 kN, governed by gross yielding, against a demand of 100 kN, so the flange runs at a utilisation of 0.44. The 317 MPa peak appears in neither equation. The φ factors and the use of F_u on the reduced area already carry the reserve the peak seemed to demand.

The redistribution argument rests on two conditions: the material must be able to yield, and the load must be static. Remove either one and the factor comes straight back.

• Fatigue. A cyclic load never lets the peak redistribute for good, and the crack nucleates exactly at the hole edge. Fatigue uses the notch factor K_f = 1 + q(K_t − 1), where q is the notch sensitivity, about 0.8 to 0.9 for structural steel and a hole this size, giving K_f ≈ 2.25 here. Steel codes deliver this through detail categories rather than K_f directly, but the physics is the peak.
• Brittle or cast material. No yield plateau, so nothing redistributes and the peak is simply the fracture stress.
• Low temperature or low toughness steel. Cold service or a high strength low ductility grade weakens the safe assumption of redistribution.
• Threads, keyways, sharp reentrant corners. High local K_t on details that usually also see cyclic load.

In each case the concentration is not a curiosity. It is the design driver.

The number the whole story multiplies is σ_nom, and that comes straight from the analysis: the axial force or the bending moment divided by the section that carries it. Model the member, read the stress the flange actually sees, and let the design report run the net section checks the way the code does, with no K_t multiplier bolted on top.

1. Multiplying a static ductile steel design stress by K_t. Double counting. The code already checks the net section, and the peak has redistributed.
2. Mixing the reference areas. Using the gross area for σ_nom while reading a net referenced K_t, or the reverse. The two differ by (1 − d/w), and mixing them corrupts the peak.
3. Ignoring K_t in fatigue because steel is ductile. Fatigue is precisely the case where it does not redistribute.
4. Confusing K_t with K_f. K_t is theoretical, elastic, geometric. K_f is the fatigue value, smaller, and material dependent through notch sensitivity. Always K_f ≤ K_t.
5. Reading the chart wrong. Using the bolt diameter instead of the hole diameter, or a hole chart for a fillet, lands on a factor that belongs to a different part.

For a static, ductile steel member: use the gross section and net section checks with F_y and F_u, and no stress concentration factor. For fatigue, brittle, or cold service: the peak governs, and you move to K_f and the detail categories.

The textbook number is not wrong. It answers a different question. K_t tells you how high the elastic stress spikes, not how much load the member carries. Knowing which of those two questions you are actually asking is the whole skill.