What is the difference between Kt and Kf in stress concentration analysis?

Kt (theoretical stress concentration factor) and Kf (fatigue notch factor) serve different purposes in structural steel design. Kt = sigma_peak / sigma_nominal is purely geometric — it depends only on the shape and dimensions of the discontinuity, not the material. For a circular hole in an infinitely wide plate under tension, Kt = 3.0 exactly. For finite plates, Kt varies with the d/w ratio (hole diameter to plate width). As d/w increases, the net section shrinks and Kt actually decreases from 3.0 (at d/w = 0) to about 2.0 (at d/w = 0.6). For shoulder fillets, Kt ranges from 1.2 (r/d = 0.5, generous radius) to 3.4 (r/d = 0.02, sharp corner). Kf (fatigue notch factor) accounts for material notch sensitivity and is always less than or equal to Kt. Kf = 1 + q*(Kt - 1), where q is the notch sensitivity (0 <= q <= 1). For ductile structural steels (A36, A992), q ranges from 0.6 to 0.9 depending on the notch radius and material ultimate tensile strength. A sharp notch (small radius) has q approaching 0.9, meaning Kf is close to Kt. A large-radius hole has q around 0.6-0.7, meaning Kf is significantly less than Kt. For static loading of ductile steel, stress concentrations have limited effect because local yielding redistributes the peak stress — the net section strength governs (with the shear lag factor U where applicable). However, for fatigue loading, stress concentrations dominate: a bolt hole with Kt = 3.0 and q = 0.8 gives Kf = 1 + 0.8*(3.0 - 1) = 2.6, meaning the fatigue life is reduced by a factor of roughly (1/2.6)^3 = 1/17.6 (since fatigue life is proportional to the inverse cube of stress range for typical S-N curves). Weld toes are the worst stress concentrators in steel structures: Kt at a fillet weld toe can range from 2.5 to 5.0 depending on the weld profile, undercut, and toe radius. This is why the AISC fatigue categories place weld details in low categories (Category C, D, E) with correspondingly low allowable stress ranges.

What is the stress concentration factor for a bolt hole in a steel plate?

The stress concentration factor for a circular hole in a finite-width steel plate under tension depends on the hole diameter to plate width ratio (d/w). From Peterson's Stress Concentration Factors, 4th edition: at d/w = 0 (infinitesimal hole in an infinite plate), Kt = 3.00 exactly — this is the classic elasticity solution. At d/w = 0.1, Kt = 3.03 (slightly higher due to width effect). At d/w = 0.2, Kt = 2.51. At d/w = 0.3, Kt = 2.24. At d/w = 0.4, Kt = 2.10. At d/w = 0.5, Kt = 2.03. At d/w = 0.6, Kt = 2.01. The counter-intuitive decrease in Kt with increasing d/w occurs because: as the hole gets larger relative to plate width, the stress redistribution involves a larger portion of the remaining net section, and the stress gradient from the hole edge to the far-field stress becomes less severe. However, the net section stress (P/A_net) increases as d/w increases, so the peak stress at the hole edge still increases despite the lower Kt. For bolt holes in structural steel connections: a 3/4 inch bolt hole (13/16 inch actual) in a 3-inch wide plate gives d/w = 0.81/3.0 = 0.27, Kt ≈ 2.3. Under bending, Kt values are similar to tension for the same d/w ratio (Kt = 3.00 at d/w = 0, 2.51 at d/w = 0.2, 2.10 at d/w = 0.4). Under torsion of a circular shaft with a transverse hole, Kt = 2.0 at d/w = 0, increasing to about 2.5 at d/w = 0.4. For static design of bolted connections per AISC 360, the stress concentration at bolt holes is not explicitly calculated — instead, the code uses the net section area with a shear lag factor U (Table D3.1) that accounts for the non-uniform stress distribution. For fatigue design, the AISC Appendix 3 Table A-3.1 assigns Fatigue Category B to base metal with bolted connections (allowable stress range = 16 ksi at 2,000,000 cycles for redundant members), which inherently accounts for the Kt ≈ 2.5-3.0 at bolt holes. For comparison, base metal with mill scale (no holes) is Category A (24 ksi at 2e6 cycles), demonstrating that bolt holes approximately halve the fatigue strength.

Why are weld toes the most critical stress concentration in steel structures?

Weld toes are the most severe stress concentrators in structural steel because they combine multiple stress-raising features at a single location. The Kt at a fillet weld toe typically ranges from 2.5 to 5.0, substantially higher than a drilled hole (Kt = 2.0-3.0). Four mechanisms compound at the weld toe: (1) Geometric discontinuity — the weld creates an abrupt change in section from the attached plate to the base metal, similar to a shoulder fillet with a very small effective radius. The weld reinforcement angle (typically 30-60 degrees) determines the severity — flatter weld profiles (30 degrees) have lower Kt than steeper profiles (60 degrees). (2) Weld toe radius — the actual toe radius is typically only 0.005 to 0.020 inches (0.1-0.5 mm), giving r/d ratios of 0.001-0.01, which produce Kt values of 4-6. This is equivalent to a very sharp notch. (3) Intrusions and defects — microscopic slag inclusions, lack of fusion at the toe, and undercut (a groove melted into the base metal at the toe) create additional stress raisers. Undercut of 0.010 inch depth acts like a sharp crack and can increase local Kt to 8-10. (4) Tensile residual stresses — welding creates high tensile residual stress at the toe (often approaching Fy), which adds to the applied stress and accelerates fatigue crack initiation. The combination means that the fatigue strength of a welded detail is controlled almost entirely by the weld toe. AISC Appendix 3 fatigue categories reflect this: Category C (12 ksi at 2e6) for transverse stiffener welds, Category D (10 ksi) for CJP groove welds with backing bars, Category E (8 ksi) for fillet-welded attachments in shear, Category E' (5.4 ksi) for intermittent fillet welds. Stress concentration at the weld toe is the reason weld quality matters for fatigue-sensitive applications: grinding the weld toe flush (removing the geometric stress concentration) can improve fatigue life by a factor of 2-3; TIG dressing the toe (remelting to create a smoother radius) improves life by 3-5x; and ultrasonic peening (introducing compressive residual stress at the toe) can improve life by 5-10x. These improvements are captured in AISC 360 by allowing higher fatigue categories for improved weld profiles.

How does the AISC fatigue category system relate to stress concentration factors?

The AISC 360-22 Appendix 3 fatigue categories (A through F) represent an empirical integration of stress concentration effects, crack growth behavior, and statistical variation for common structural details. Each category defines an S-N curve of the form log(N) = b - m*log(S_r), where S_r is the applied stress range. Category A (base metal, plain rolled surfaces, no attachments): allowable stress range = 24 ksi at 2,000,000 cycles, m = 3, constant amplitude fatigue threshold = 24 ksi. Category B (base metal with bolted connections, CJP welds ground flush): 16 ksi, m = 3. Category B' (base metal with thermal cut edges, roughness < 1,000 micro-inches): 12 ksi, m = 3. Category C (CJP groove welds with backing not removed, transverse stiffeners welded to flange): 12 ksi, m = 3. Category D (fillet welds loaded in transverse direction, intermittent CJP welds): 10 ksi, m = 3. Category E (fillet welds loaded in longitudinal shear, base metal at welded shear studs): 8 ksi, m = 3. Category E' (intermittent fillet welds, plug/slot welds): 5.4 ksi, m = 3. Category F (fillet welds in shear on throat area): 8 ksi, m = 3. The relationship between stress concentration and fatigue category is approximately: Kf = (24 / F_TH)^(1/m) where 24 ksi is the Category A threshold. For Category C (12 ksi): apparent Kf = (24/12)^(1/1) = 2.0 relative to Category A. For Category E (8 ksi): Kf = 3.0. For Category E' (5.4 ksi): Kf = 4.4. These apparent Kf values incorporate not just the geometric stress concentration but also residual stress effects, statistical scatter, and crack growth behavior. For structural engineers, the key practical implication is: when detailing fatigue-sensitive connections (crane runways, bridge girders, vibrating machinery supports), select details from Category B or C wherever possible. Avoid Category D, E, and E' details in the tension flange of fatigue-loaded members. If a Category E detail is unavoidable, increase the section to reduce stress range below the corresponding threshold.

How do you reduce stress concentration at a coped beam or web opening?

Coped beams and web openings create severe stress concentrations at re-entrant corners where the cut meets the beam flange or web. The default fabrication method — a flame-cut cope with a sharp re-entrant corner — produces Kt values of 3-5, which can initiate brittle fracture or fatigue cracks at load levels well below the beam's design capacity. The primary mitigation is a generous radius at the re-entrant corner. AISC 360-22 Section J1.6 requires a minimum radius of 1/2 inch (12.7 mm) at coped beam re-entrant corners, but for fatigue-sensitive or dynamically loaded applications, larger radii are justified. The effect of radius on Kt follows Peterson's data: a zero-radius (sharp) corner: Kt = 5-10 (theoretically infinite for a perfectly sharp crack). r = 1/4 inch: Kt ≈ 3.5. r = 1/2 inch: Kt ≈ 2.5. r = 3/4 inch: Kt ≈ 2.0. r = 1.0 inch: Kt ≈ 1.8. Each additional 1/4 inch of radius beyond 1/2 inch buys about 20% reduction in Kt. Additional mitigation strategies: (2) Drill a stress-relief hole at the cope termination — a 3/4 inch diameter hole drilled at the tip of the cope cut removes the flame-cut surface and replaces it with a smooth machined surface with lower Kt. (3) Grind the flame-cut surface smooth — flame-cut surfaces have roughness of 500-2,000 micro-inches that acts as a series of micro-notches. Grinding to 125 micro-inches or better removes these stress raisers. (4) For web openings, use circular or elongated shapes with generous corner radii (minimum r/Do = 0.20 where Do is the opening diameter). Rectangular openings with sharp corners should never be used in fatigue-sensitive applications. (5) For welded reinforcement around web openings, extend the reinforcement plate beyond the opening with a smooth taper (30-degree included angle) to avoid creating a new stress concentration at the end of the reinforcement. (6) Post-fabrication inspection: magnetic particle testing (MT) of cope radii and web opening corners in fracture-critical members can detect cracks before they propagate.

Stress Concentration — Kt, Kf & Fatigue Design Implications

Stress concentration factors (Kt) for holes, fillets, notches, and welds in structural steel. Fatigue notch sensitivity (Kf), Peterson chart values, and AISC fatigue category mapping.

What is stress concentration?

Stress concentration is the localized increase in stress that occurs at geometric discontinuities — holes, notches, sharp corners, cross-section changes, and weld toes. The theoretical stress concentration factor Kt is the ratio of peak local stress to nominal (average) stress in the net section:

Kt = sigma_peak / sigma_nominal

For static loading of ductile steel, stress concentrations have limited effect because local yielding redistributes stress. However, for fatigue loading and brittle fracture assessment, stress concentrations dominate the design. A bolt hole with Kt = 3.0 means the local stress at the hole edge is three times the average stress — this is where fatigue cracks initiate.

Kt for circular holes by d/w ratio

d/w Ratio	Kt (Tension)	Kt (Bending)	Kt (Torsion)	Effective Width Ratio
0.0	3.00	3.00	2.00	1.00
0.1	3.03	3.03	2.05	0.90
0.2	2.51	2.51	1.72	0.80
0.3	2.24	2.24	1.55	0.70
0.4	2.10	2.10	1.44	0.60
0.5	2.03	2.03	1.38	0.50
0.6	2.01	2.01	1.36	0.40

Source: Peterson's Stress Concentration Factors. d/w = hole diameter / plate width. For infinite plate (d/w = 0), Kt = 3.0 exactly.

Kt for shoulder fillets by r/d ratio

r/d Ratio	D/d = 1.5	D/d = 2.0	D/d = 3.0
0.02	2.6	3.0	3.4
0.05	2.2	2.4	2.7
0.10	1.7	1.9	2.1
0.15	1.5	1.6	1.7
0.20	1.4	1.5	1.6
0.30	1.3	1.3	1.4
0.50	1.2	1.2	1.2

Larger fillet radii (higher r/d) dramatically reduce Kt. A radius increase from r/d = 0.02 to 0.10 cuts Kt by 35-40%.

Common Kt values for structural details

Detail	Kt (approximate)	Key Parameter	Source
Circular hole in wide plate (tension)	3.0	d/w ratio	Peterson, exact for infinite plate
Circular hole, d/w = 0.2	2.5	d/w = 0.2	Peterson chart
Circular hole, d/w = 0.5	2.1	d/w = 0.5	Peterson chart
Shoulder fillet, r/d = 0.1, D/d = 2.0	1.9	r/d = 0.1	Peterson chart
Shoulder fillet, r/d = 0.25, D/d = 1.5	1.5	r/d = 0.25	Peterson chart
V-notch (60 deg, depth 0.2d)	3.0-4.0	Notch angle	Peterson chart
U-notch (semicircular)	2.0-2.5	Notch radius	Peterson chart
Square notch	4.0-5.0	Sharp corner	Peterson chart
Keyway in shaft	2.0-3.0	Fillet radius	Peterson chart
Thread root (V-thread)	3.0-5.0	Thread profile	Machine design texts
Butt weld (flush ground)	1.0-1.2	Grinding quality	Experimental data
Butt weld (as-welded, cap intact)	1.5-2.5	Weld profile	Experimental data
Fillet weld toe	2.5-4.0	Weld angle and profile	Experimental data
Coped beam flange	2.0-4.0	Cope radius	Experimental data

Static vs fatigue — why it matters

Under static (monotonic) loading, structural steel grades (A36, A992, Grade 300) are ductile enough that local yielding at stress concentrations simply redistributes load to adjacent material. The member reaches its full plastic capacity regardless of the hole or notch. This is why AISC 360 and AS 4100 allow net section tension rupture checks based on uniform stress across the net area, without applying Kt.

Under cyclic (fatigue) loading, the story is different. Fatigue cracks initiate at the point of highest stress range, which is always the stress concentration. Even if the nominal stress range is well below yield, the local stress range at Kt = 3 exceeds yield, and repeated plastic cycling drives crack growth. This is why fatigue design is governed by detail category, not member strength.

Fatigue notch factor Kf

Not all of the theoretical Kt is effective for fatigue because the stress gradient at a sharp notch is steep — the high stress only exists in a tiny volume that may not contain a critical flaw. The fatigue notch factor Kf is:

Kf = 1 + q x (Kt - 1)

where q is the notch sensitivity factor (0 to 1). For structural steel:

Mild steel (Fy = 250 MPa): q is approximately 0.75-0.85 for r >= 3 mm, dropping to 0.5-0.6 for r = 1 mm.
High-strength steel (Fy = 690 MPa): q approaches 0.95 for r >= 3 mm. High-strength steels are more notch-sensitive.

Notch sensitivity by material strength

Fy (MPa)	q at r=0.5mm	q at r=1mm	q at r=2mm	q at r=5mm	q at r=10mm	q at r=25mm
250	0.35	0.50	0.65	0.80	0.88	0.95
350	0.45	0.60	0.75	0.88	0.93	0.97
450	0.55	0.70	0.82	0.92	0.96	0.99
550	0.65	0.78	0.88	0.95	0.98	1.00
690	0.78	0.88	0.94	0.98	0.99	1.00

Higher-strength steels are more notch-sensitive because they have less capacity for plastic redistribution at the notch root.

For a 20 mm diameter bolt hole (r = 10 mm) in Grade 350 steel: Kt = 2.5, q = 0.93, Kf = 1 + 0.93 x (2.5 - 1) = 2.40.

AISC fatigue detail categories

Category	CAFT (ksi)	Details	Implied Kt Range
A	24.0	Base metal, rolled surfaces	1.0
B	16.0	Base metal at welded transverse stiffeners	1.5-2.0
B'	12.0	Base metal at groove welds, ground flush	1.5-2.0
C	10.0	Base metal at transverse groove welds	2.0-2.5
D	7.0	Base metal at groove welds, as-welded	2.5-3.0
E	4.5	Base metal at copes, short attachments	3.0-4.0
E'	2.6	Base metal at long attachments, > 12x thickness	4.0-5.0

The category is essentially a codified Kt. Moving from Category E' to Category D (by grinding a weld toe smooth) doubles the allowable stress range.

Worked example — fatigue life at a cope

A W16x40 beam with a bottom flange cope (radius r = 25 mm) at the connection. The cope creates Kt = 2.5. The beam supports a crane trolley inducing a nominal stress range at the cope of delta_sigma_nom = 50 MPa.

Local stress range = Kt x delta_sigma_nom = 2.5 x 50 = 125 MPa.

Per AISC 360 Appendix 3, Table A-3.1, a coped beam is Fatigue Category E (if the cope is flame-cut and not ground smooth). The allowable stress range for Category E at 500,000 cycles is 44.8 MPa (nominal). Since 50 MPa > 44.8 MPa, the detail fails the fatigue check at 500,000 cycles.

Solution options: (1) grind the cope radius smooth to upgrade to Category D (allowable 55.2 MPa at 500,000 cycles — marginal), (2) increase the cope radius to r = 50 mm to reduce Kt, or (3) reinforce the cope with a reinforcement plate.

Worked example — plate with central hole in tension

Given: 12 in. wide x 1/2 in. thick A572 Gr 50 plate with a 3 in. diameter central hole. Pu = 200 kip (static, factored). Check capacity with and without stress concentration.

Step 1 — Nominal stress: Gross area Ag = 12 x 0.5 = 6.0 in^2. Net area An = (12 - 3) x 0.5 = 4.5 in^2. sigma_gross = 200 / 6.0 = 33.3 ksi. sigma_net = 200 / 4.5 = 44.4 ksi.

Step 2 — Static capacity: Yielding: phi x Fy x Ag = 0.90 x 50 x 6.0 = 270 kip > 200 kip. OK. Rupture: phi x Fu x An = 0.75 x 65 x 4.5 = 219 kip > 200 kip. OK. No Kt needed for static design — local yielding redistributes stress.

Step 3 — Fatigue check (if cyclic): d/w = 3/12 = 0.25. Kt = 2.37 (interpolated). Peak stress = 2.37 x 44.4 = 105 ksi. Even though the nominal stress is well below Fy, the local stress exceeds Fy. Cyclic loading at this detail will initiate fatigue cracks at the hole edge.

Step 4 — Mitigation: If the hole is reamed smooth (vs. punched), the surface finish improves fatigue life by reducing surface defects. For critical fatigue applications, specify reamed holes.

Mitigation strategies for stress concentration

Strategy	Kt Reduction	Fatigue Improvement	Cost Impact	When to Use
Increase fillet radius	30-50%	+1 to +2 categories	Low	Design stage
Grind weld toes smooth	30-40%	+1 category	Moderate	Fatigue-critical details
Peening (UIT/HFMI)	20-30%	+1 to +2 categories	Moderate	Existing structures, retrofits
Drill stop-holes at cracks	N/A (arrest)	Extends life	Low	Crack repair
Add reinforcement plates	Reduces load	Improves capacity	High	Coped beams, notches
Use lower-strength steel	Reduces Kf	More forgiving	Neutral	Brittle fracture concern
Redesign to avoid sharp corners	Eliminates Kt	Major improvement	Low (design stage)	New designs

Fracture toughness connection

Stress concentration also affects brittle fracture resistance. When the peak stress at a Kt location exceeds the fracture toughness K_IC, rapid brittle fracture can occur without warning. This is temperature-dependent:

Steel Grade	K_IC at -40F (ksi-in^0.5)	K_IC at 70F (ksi-in^0.5)	CVN at -40F (ft-lb)
A36	80-120	150-200	15-25
A992	100-150	180-250	20-30
A572 Gr 50	100-140	170-230	20-30
A514	60-100	120-180	15-25

AISC 360 Appendix 3 requires minimum CVN toughness for certain applications. Below the transition temperature, steel becomes brittle and Kt-driven fracture becomes the governing failure mode.

Code comparison — fatigue provisions

Aspect	AISC 360 App. 3	AS 4100 Cl. 11	EN 1993-1-9	CSA S16 Cl. 26
Categories	A through F, F2	Detail Categories 36-160	Detail Categories 36-160	A through E2
S-N curve slope	m = 3 (all categories)	m = 3	m = 3 (N <= 5x10^6), m = 5 (N > 5x10^6)	m = 3
Threshold	CAFT per category	Cut-off limit at 10^8 cycles	Fatigue limit at 5x10^6, cut-off at 10^8	CAFL per Table
Partial factor	phi = 1.0 (implied)	Capacity factor phi = 1.0	gamma_Mf = 1.0 to 1.35	phi = 1.0

AS 4100 and EN 1993-1-9 use the same detail category numbering system (the number represents the characteristic stress range at 2 million cycles in MPa).

Common pitfalls

Ignoring stress concentrations at fatigue-loaded details. Static design does not require Kt, but fatigue design is entirely governed by detail category, which is a proxy for Kt. Treating a fatigue check like a static check leads to grossly unconservative designs.
Assuming grinding removes stress concentration entirely. Grinding a weld toe reduces Kt from perhaps 3.5 to 1.5-2.0 and upgrades the fatigue category by one or two steps. It does not eliminate the stress concentration.
Using theoretical Kt for fatigue instead of the detail category approach. AISC Appendix 3 and EN 1993-1-9 already incorporate the stress concentration effect into the detail category S-N curves. Applying Kt on top of the detail category double-counts the effect.
Specifying tight cope radii without considering fatigue. A 12 mm cope radius at the bottom flange of a crane beam creates Kt approximately 4.0 and drops the fatigue category to E or E'. Specify minimum 25-50 mm radii and drill the end of the cope.

Frequently asked questions

Can I ignore stress concentration for static loads? Yes, for ductile steel under static (monotonic) loading. Local yielding at the Kt location redistributes stress to adjacent material. AISC 360 and AS 4100 allow net section checks without applying Kt for tension rupture.

What is Peterson's book? "Peterson's Stress Concentration Factors" by Walter Pilkey is the standard reference for Kt values. It provides charts for hundreds of geometric configurations — holes, notches, fillets, shafts, and more. Most structural engineers use Peterson values as their primary Kt source.

How does weld toe grinding improve fatigue? Grinding removes the sharp transition at the weld-to-base-metal interface, reducing Kt from approximately 3.5 to 1.5-2.0. This upgrades the AISC fatigue category by one or two steps. The grinding must be done in the direction perpendicular to the weld toe.

What Kt does a bolt hole have? For a circular hole in a wide plate, Kt = 3.0 exactly. For finite width plates, Kt decreases as d/w increases (hole diameter / plate width). At d/w = 0.5, Kt = 2.03.

When do I need to consider stress concentration? For fatigue design, fracture assessment, and brittle materials. You do NOT need Kt for static design of ductile steel — the codes already account for this through the net section approach.

What is the difference between Kt and Kf? Kt is the theoretical (elastic) stress concentration factor. Kf is the fatigue notch factor, which accounts for the notch sensitivity of the material. Kf = 1 + q(Kt - 1), where q ranges from 0 (no sensitivity) to 1 (fully sensitive). For structural steel with moderate notch radii, Kf is typically 85-95% of Kt.

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This page is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against the applicable standard and project specification before use. The site operator disclaims liability for any loss arising from the use of this information.