Fatigue Endurance Limit and S-N Curve Calculator

S_e′ = 0.5·S_ut capped at 700 for steel, all five Marin factors, the S-N line read both ways, and Goodman, Gerber, Soderberg and ASME-elliptic on one cycle — with aluminium’s lack of any endurance limit treated as the point rather than a footnote.

Last updated 24 Sep 2026

Fatigue endurance limit and S-N curve

Strength, finish, size, duty → endurance limit and life

Material family — ASK THIS FIRSTSteel has a true endurance limit: below a stress level of about half its tensile strength it will run indefinitely, and its S-N line goes flat at around a million cycles. Aluminium and the other non-ferrous metals DO NOT. Their S-N line keeps falling past 10⁸ cycles, so there is no stress below which the part is safe forever, and the only honest number is a fatigue STRENGTH at a stated cycle count. That distinction is the most important sentence on this page.

Ultimate tensile strength S_ut (N/mm²)Everything on this page scales from this. The endurance limit is half of it for steel, capped at 700; the surface factor is a power of it; the 10³-cycle anchor is a fraction of it; and Goodman’s mean-stress limit is it. Get it from the specification for the actual heat-treat condition, not from a generic table.

Yield strength S_y (N/mm²)Used by Soderberg and by the ASME-elliptic criterion, and for the first-cycle yield check — because a part can pass every fatigue criterion and still yield on the very first load application if the mean plus the amplitude exceeds the yield strength.

Surface finishTHE BIGGEST of the Marin factors and the one most worth spending money on. At 700 N/mm² a ground surface keeps about 92 per cent of the endurance limit, machined 73 per cent, hot-rolled 46 and as-forged 27. That is a factor of three and a half between the best and worst finish on identical metal. Surface ROUGHNESS conversion between Ra, Rz and the rest belongs to the roughness converter in the converters plugin, not here.

Diameter or section size d (mm)The size factor. A larger part is weaker in fatigue than a small one of identical material, because more surface is exposed to high stress and the chance of a critical defect rises. Ignored for axial loading, where there is no size effect at all. For a non-round section, Shigley’s rule is an effective diameter of 0.37d for a non-rotating round member and a comparable-volume rule otherwise; this page takes the diameter you give it.

LoadingBending is the reference case, which is why k_c is 1. Axial loading is worse (0.85) because the whole section sees the peak stress rather than just the outer fibre. Torsion is 0.59, which is close to the 0.577 shear-yield ratio and is not a coincidence — for a torsional load it is often cleaner to set k_c = 1 and compare the von Mises equivalent stress instead.

Operating temperature (°C)Shigley’s temperature factor. Note that it rises slightly above 1 between about 50 and 250 °C — a steel is marginally stronger warm than cold — and then falls away steeply: 0.90 at 400 °C and 0.77 at 500. Above roughly 400 °C creep starts to matter more than fatigue and this page stops being the right tool.

ReliabilityA published endurance limit is a MEDIAN, so designing to it means half the population fails before the design life. k_e = 1 − 0.08·z_a knocks it down to the survival probability you want, on an assumed 8 per cent standard deviation. 99 per cent costs you 19 per cent of the endurance limit; 99.9 per cent costs 25.

Fatigue stress concentration factor K_fLeave at 1 if you have already applied the stress concentration to the stresses you enter below. Otherwise put K_f here and the page divides the endurance limit by it, which is the equivalent treatment. K_f is NOT K_t: notch sensitivity removes part of the penalty, and the shaft fillet and stress concentration page computes both for a shoulder, a groove and a keyway. This page consumes its answer and does not duplicate it.

ALTERNATING stress amplitude σ_a (N/mm²)Half the peak-to-peak stress range, not the range itself. This is the number that drives fatigue; the mean matters much less, and that asymmetry is the whole of the Goodman diagram.

MEAN stress σ_m (N/mm²)The midpoint of the cycle. Tensile mean stress reduces fatigue life because it holds cracks open; compressive mean stress helps, which is why shot peening works. Standard practice takes a compressive mean as zero rather than crediting it, and this page does the same. For a preloaded bolt the mean is dominated by the preload and only a fraction of the external load range reaches the bolt at all — the bolt fatigue page owns that case.

Target life N (cycles)The page reads the S-N line both ways: the fatigue strength at this life, and separately the life at the alternating stress you entered. 10³ to 10⁶ is the finite-life region the line is drawn for; below 10³ is low-cycle fatigue, which is strain-controlled and is a different calculation.

Non-ferrous only: fatigue strength at 5×10⁸ cycles (N/mm²)Ignored for steel. For a non-ferrous metal there is no endurance limit to compute, so this page will not invent one — you have to supply a measured fatigue strength at a stated life from the material’s own data. 6061-T6 is published at 96.5 N/mm² at 5×10⁸ cycles fully reversed on an R.R. Moore machine, which is the default here. Note that this is a rotating-beam figure and the Marin factors still apply to it.

Not a circuit: two graphs. The top one is the S-N line on a five-decade logarithmic cycle axis, drawn in normalised coordinates so that it always starts at f·S_ut in the top-left corner whatever your material is. What the normalisation leaves free is the SLOPE, and the slope is everything: a steep line means fatigue strength falls away quickly with life. For a steel the line goes flat at 10⁶ cycles — that horizontal part is the endurance limit and the reason a steel part can be designed for infinite life. Switch the material family to non-ferrous and the flat part disappears and the line keeps falling to 10⁸ and beyond, because that is what aluminium actually does. The horizontal line is your own alternating stress; where it crosses is the life, and if it lies below the flat part of a steel line there is no crossing. The lower panel is the Haigh diagram, also normalised — mean stress in units of S_ut and amplitude in units of S_e — so the straight Goodman line always runs corner to corner, the Gerber parabola always bulges above it, and the Soderberg line always cuts inside because it runs to yield rather than to ultimate. Your own cycle is the crosshairs, and how far inside the lines it sits is the factor of safety.

183.5N/mm²Example

A machined steel shaft, 25 mm diameter, S_ut = 700 and S_y = 550 N/mm², in reversed bending at 20 °C, 99 per cent reliability

The endurance limit, the line, and the mean

S_e′ = 0.5·S_ut (≤1400), else 700 · S_e = k_a k_b k_c k_d k_e S_e′ / K_f · S_f = a·N^b, a = (f·S_ut)²/S_e, b = −⅓·log(f·S_ut/S_e) · σ_a/S_e + σ_m/S_ut = 1/n

S_e′: the endurance limit of a polished rotating-beam SPECIMEN, not of your part. Half the tensile strength for a steel, capped at 700 N/mm². For a non-ferrous metal there is no such thing and this page will not invent one
k_a: surface finish. The biggest factor and a power law in S_ut — note that a STRONGER steel is punished MORE by a rough surface, which is why a high-strength part with a poor finish can be no better in fatigue than a mild steel one
k_b: size. A big part is weaker in fatigue than a small one of the same metal. Exactly 1 for axial loading
k_c: load type: 1 bending, 0.85 axial, 0.59 torsion
k_d, k_e: temperature and reliability. k_e is 1 − 0.08 z_a, which is the identity the published table satisfies at all eight of its rows
f: the fraction of S_ut the line starts from at 10³ cycles. NOT 0.9 except below 490 N/mm²; it falls to about 0.77 at 1,400. See the table
K_f: the fatigue stress concentration factor, from the shaft fillet page. Applied here as a DIVISOR on the endurance limit, which is equivalent to multiplying the stress

Worked example

A machined steel shaft, 25 mm diameter, S_ut = 700 and S_y = 550 N/mm², in reversed bending at 20 °C, 99 per cent reliability
The specimen's endurance limit first: S_e′ = 0.5 × 700 = 350 N/mm². The 0.5 holds up to S_ut = 1,400 N/mm², above which it is capped at a flat 700 — a stronger steel does not keep buying endurance limit indefinitely
Surface: machined, so k_a = 3.04 × 700^−0.217 = 0.7336 on the 11th edition's constants. The 10th edition gives 4.51 × 700^−0.265 = 0.7947, which is 8.3 per cent higher. Both are printed on this page; neither is a misprint
Size: 25 mm is in the 2.79-to-51 mm band, so k_b = 1.24 × 25^−0.107 = 0.8787. A 100 mm shaft of the same steel would get 0.7328 — six per cent less endurance limit for being bigger, with no other change
Load: reversed bending is the reference case, k_c = 1.00. Axial would be 0.85 and torsion 0.59
Temperature: 20 °C gives k_d = 0.9994, essentially 1. Worth knowing that this factor RISES to about 1.025 near 150 °C — a steel is slightly stronger warm — and then falls to 0.90 at 400 °C and 0.77 at 500
Reliability: 99 per cent needs z_a = 2.326, so k_e = 1 − 0.08 × 2.326 = 0.814. That is the single largest deliberate knock-down here, and it is there because a published endurance limit is a median: design to it and half the population fails early
Multiply: S_e = 350 × 0.7336 × 0.8787 × 1 × 0.9994 × 0.814 = 183.5 N/mm². That is 26 per cent of the tensile strength, not 50 — the Marin factors have taken away nearly half of the specimen value, and that is normal
NOW THE LINE, and here is where the usual shortcut goes wrong. The 10³-cycle anchor is f·S_ut, and f is NOT 0.9 at this strength: the relation behind Shigley's figure gives f = 0.8417, so the anchor is 589 rather than 630 N/mm². Then a = (f·S_ut)²/S_e = 1,891.4 and b = −⅓·log(f·S_ut/S_e) = -0.168843
Read it either way. At 100,000 cycles the fatigue strength is a·N^b = 1,891.4 × 100,000^(-0.1688) = 270.7 N/mm². Or inversely, an alternating stress of 100 N/mm² gives a life of (100/a)^(1/b) = 36,466,581 cycles — which is above 10⁶, so for a steel it is infinite life
THE MEAN STRESS, which the alternating figure alone ignores. At σ_a = 100 and σ_m = 120 N/mm² the four criteria give: Soderberg 1.311, Goodman 1.396, ASME-elliptic 1.704, Gerber 1.683. The order never changes — Soderberg runs to yield rather than to ultimate so it is always lowest, and Gerber is a parabola that fits wrought steel data best so it is always highest. The spread here is 28 per cent, which is the honest uncertainty in a mean-stress correction
And the check nobody does: first-cycle yield. σ_m + σ_a = 220 N/mm² against S_y = 550 gives 2.50, so this part will not take a set on its first application. A part with a high mean stress can pass every fatigue criterion above and still yield once, on cycle one

The surface factor — and the two editions that disagree about it

Surface finish is the largest of the Marin factors and the one worth spending money on: at 700 N/mm² a ground surface keeps 92 per cent of the endurance limit and an as-forged one keeps 27, a factor of three and a half on identical metal, from nothing but how the surface was made. And here is something a single-source page would not tell you. The 10th and 11th editions of the same textbook print DIFFERENT constants for the same four finishes, and the last column is the size of it — up to nine per cent at 700 N/mm². Neither is a misprint: the 11th edition’s pairs were checked through the a_MPa = a_kpsi × 6.8948^−b unit conversion against the kpsi values used in the same edition’s own solutions, and all four are self-consistent. It is a revised fit to revised data. This page computes the headline from the 11th edition and prints the 10th beside it, because a fatigue calculation that changes by nine per cent depending on which book is on the shelf should say so. The same designation can mean different dimensions in different standards families — ANSI against ISO, inch against metric, one national standard against another. The family used here is named beside every figure; check which one your part was made to.
Finish	a / b, 11th ed (MPa)	a / b, 10th ed (MPa)	k_a at 400 (11e)	(10e)	k_a at 700 (11e)	(10e)	k_a at 1200 (11e)	(10e)	Gap at 700
Ground	1.38 / -0.067	1.58 / -0.085	0.9237	0.9495	0.8897	0.9054	0.8582	0.8648	1.8 %
Machined or cold-drawn	3.04 / -0.217	4.51 / -0.265	0.8284	0.9218	0.7336	0.7947	0.6527	0.6890	8.3 %
Hot-rolled	38.60 / -0.650	57.70 / -0.718	0.7857	0.7814	0.5461	0.5229	0.3847	0.3551	-4.3 %
As-forged	54.90 / -0.758	272.00 / -0.995	0.5851	0.7007	0.3828	0.4015	0.2544	0.2348	4.9 %

f at 10³ cycles is not 0.9, and here is what it actually is

The 10³-cycle anchor of the S-N line is f·S_ut, and f is very commonly quoted as 0.9. That is correct only for steels weaker than about 490 N/mm², which is where Shigley states it in words. Above that it is a curve on a FIGURE, and every published worked solution reads it off by eye — which is not a citable number, so this page reconstructed the relation that generates the figure instead: f = (σ′_F/S_ut)(2×10³)^b with b = −log(σ′_F/S_e′)/log(2×10⁶) and σ′_F = S_ut + 345 N/mm², which is Shigley’s own equation for the true fracture strength of a steel. It returns 0.896 at 490, 0.830 at 770 and 0.772 at 1,400 — the three values the published solutions to problems 6-4 and 6-13 take off that figure, to within 0.004. Assuming 0.9 at 900 N/mm² would put the 10³-cycle anchor eleven per cent high, and because the line is steep that is a large error in finite life.
S_ut (N/mm²)	σ′_F = S_ut+345	S_e′	f computed	f·S_ut	0.9·S_ut	Error from assuming 0.9	How f is known
400	745	200	0.9000	360	360	0.0 %	0.90 stated in words
490	835	245	0.9000	441	441	0.0 %	0.90 stated in words
550	895	275	0.8769	482	495	2.6 %	read off the figure
620	965	310	0.8586	532	558	4.8 %	read off the figure
690	1,035	345	0.8436	582	621	6.7 %	read off the figure
770	1,115	385	0.8296	639	693	8.5 %	read off the figure
900	1,245	450	0.8117	731	810	10.9 %	read off the figure
1,100	1,445	550	0.7920	871	990	13.6 %	read off the figure
1,400	1,745	700	0.7724	1,081	1,260	16.5 %	figure’s right-hand edge
1,600	1,745	700	0.7724	1,236	1,440	16.5 %	figure’s right-hand edge

Four mean-stress criteria on the same cycle, at σ_a = 100 N/mm²

At zero mean stress all four agree exactly, because they all pass through the endurance limit on the amplitude axis — the disagreement is entirely about how much a tensile mean stress costs you. The ORDER never changes: Soderberg is the most conservative because it runs to YIELD on the mean-stress axis rather than to ultimate, and yield is well below ultimate for any ductile metal. Goodman is a straight line to the ultimate strength and is the usual choice; it is conservative against most data. Gerber is a parabola to the ultimate strength and fits experimental data for wrought steels best, which makes it the least conservative of the four. ASME-elliptic runs to yield like Soderberg but as a quarter-ellipse rather than a straight line, so it sits between. The last column is the check people forget: none of the four sees first-cycle yielding, so a part with a high mean can pass every fatigue criterion and still take a permanent set on its very first load application. This page sizes a part; it does not certify one. Where the answer carries a consequence — a load path, a lifting duty, a pressure boundary, a fastener holding something that can fall — confirm it against the design code that governs the application, and against the manufacturer’s own rating, before relying on it.
Cycle	σ_m	Soderberg	Goodman	ASME-elliptic	Gerber	Gerber above Soderberg	First-cycle yield
Zero mean (fully reversed)	0	1.835	1.835	1.835	1.835	0 %	5.500
σ_m = 60	60	1.529	1.586	1.800	1.792	17 %	3.438
σ_m = 120	120	1.311	1.396	1.704	1.683	28 %	2.500
σ_m = 240	240	1.019	1.126	1.433	1.408	38 %	1.618
σ_m = 400	400	0.786	0.896	1.100	1.104	40 %	1.100

What this page owns and what it hands over

A general fatigue page is useful precisely because it is general, and it earns that by not pretending to be the specialised pages. The first two rows are the ones that matter most: stress concentration and preloaded bolts each have a page in this plugin that does the job properly, and this page’s job is to take their outputs and put them on an S-N line. Everything here is a first-pass estimate on a textbook method. A real fatigue assessment in Europe is made against the FKM guideline or, for welds, BS 7608, and both are structured assessments with their own material data and their own statistical basis. This page sizes a part; it does not certify one. Where the answer carries a consequence — a load path, a lifting duty, a pressure boundary, a fastener holding something that can fall — confirm it against the design code that governs the application, and against the manufacturer’s own rating, before relying on it.
The question	Where it belongs	Why
What is K_t for this shoulder, groove or keyway? And what is K_f after notch sensitivity?	Shaft fillet and stress concentration	that page owns K_t, the notch sensitivity q and K_f = 1 + q(K_t − 1) for the three shaft features. This page CONSUMES its K_f — put the number in the field above — and does not recompute it
Will this preloaded BOLT survive a cyclic external load?	Bolt fatigue under alternating load	a preloaded bolt is a different problem, not a special case of this one: only a fraction Φ of the external load range reaches the bolt at all, the mean stress is dominated by the preload, and VDI 2230 gives an endurance amplitude by thread size rather than by Marin factors. That page owns it and this page does not duplicate it
Will this GEAR TOOTH survive?	Gear tooth bending stress	gear rating is its own discipline. Lewis plus the Barth factor is the classical estimate and AGMA 2001 or ISO 6336 is the real rating, with its own load, dynamic, size and reliability factors that are not these ones
Will this ROLLING BEARING survive?	Bearing life L10	rolling contact fatigue is a subsurface phenomenon under enormous triaxial compression, rated by the L10 cubic law rather than by an S-N curve. A von Mises or S-N treatment of a contact patch is the wrong model
What roughness does Ra 1.6 correspond to in Rz or in N grades?	the surface roughness converter in the converters plugin	this page’s surface factor takes a finish CATEGORY, not a roughness number. Converting between roughness scales is a separate job and lives in a separate plugin
Is this part safe against a STATIC overload?	von Mises and failure criteria	fatigue and static strength are different checks and a part needs both. The first-cycle yield figure on this page is a reminder, not a substitute

No endurance limit for aluminium, f is not 0.9, and what the Marin factors actually cost

Aluminium has no endurance limit, and that is the most important sentence here. A steel’s S-N curve goes flat at around a million cycles: there is a stress below which the part will run indefinitely, and “infinite life” is a design target you can actually hit. Aluminium and the other non-ferrous metals do not behave that way. Their S-N line keeps falling past 10⁸ cycles with no sign of a knee, so there is no safe stress — only a fatigue STRENGTH at a stated life. Everything follows from that. An aluminium part has to be designed to a finite life with an inspection or replacement interval attached. The 0.5·S_ut rule does not apply (published ratios are nearer 0.25 to 0.4 and vary by grade). And this page will not compute an endurance limit for a non-ferrous metal: it asks you for a measured fatigue strength at 5×10⁸ cycles, because no general relation exists to derive one from.

The Marin factors, and which one to spend money on. The endurance limit of a polished rotating-beam specimen is not the endurance limit of your part, and the five Marin factors are the published knock-downs between them. Surface finish is by far the biggest: at 700 N/mm² a ground surface keeps 92 per cent and an as-forged one 27, a factor of three and a half on identical metal. It is also the one with a shape worth noticing — the exponent is strongly negative, so a STRONGER steel is punished MORE by a rough surface, and a high-strength alloy left unfinished can end up with a lower endurance limit than a mild steel that was machined. Size costs a few per cent to about fifteen. Load type is 1 for bending, 0.85 for axial and 0.59 for torsion. Temperature is nearly 1 up to 250 °C and then falls away. Reliability is the largest deliberate one, because a published endurance limit is a median and designing to a median means half the population fails early.

f is not 0.9, and that matters more than it looks. The S-N line is anchored at f·S_ut at 10³ cycles and at S_e at 10⁶, and f is very widely quoted as 0.9. It is 0.9 only for steels below about 490 N/mm². Above that it falls: about 0.83 at 770 and 0.77 at 1,400. Shigley gives it as a FIGURE, and every published worked solution reads it off by eye — which is not a number this page can cite, so the relation that generates the figure was reconstructed instead, from Shigley’s own true-fracture-strength equation σ′_F = S_ut + 345 N/mm². It reproduces the three values the published solutions use, to within 0.004. Assuming 0.9 at 900 N/mm² puts the top of the line eleven per cent high, and because the line is steep on log-log axes an eleven per cent stress error is a factor of nearly two in life.

Mean stress: four criteria, one order, and a check none of them makes. A cycle with a tensile mean stress is worse than the same amplitude about zero, because a tensile mean holds cracks open. The four published corrections all pass through S_e on the amplitude axis and differ only in where they land on the mean-stress axis. Soderberg runs to YIELD, which makes it the most conservative of the four — and note that this is a genuinely different idea rather than a more cautious version of the same one. Goodman runs as a straight line to the ultimate strength and is the usual engineering choice. Gerber runs to the ultimate as a parabola and fits experimental data for wrought steels best, so it is the least conservative. ASME-elliptic runs to yield as a quarter-ellipse and sits between. What none of the four sees is first-cycle yielding: a part with a high mean stress can satisfy every one of them and take a permanent set the first time it is loaded. This page computes that check separately and reports whichever governs.

What this page owns, and what it hands over. It owns the general endurance limit, the Marin factors, the S-N line and the mean-stress diagram. It does NOT own stress concentration: the shaft fillet and stress concentration page computes K_t, the notch sensitivity q and K_f = 1 + q(K_t − 1) for a shoulder, a groove and a keyway, and this page consumes its K_f rather than recomputing it. It does NOT own the preloaded bolt: the bolt fatigue page handles that, and it is a genuinely different problem rather than a special case, because only a fraction of an external load range reaches a preloaded bolt at all and VDI 2230 gives an endurance amplitude by thread size rather than by Marin factors. Gear teeth belong to the gear tooth page and AGMA or ISO rating; rolling contact belongs to the bearing life page and the L10 law. Surface roughness conversion between Ra, Rz and N grades lives in the converters plugin and not here.

And what fatigue analysis of this kind cannot see at all. A crack that is already there — that is fracture mechanics and it depends on the crack length rather than the nominal stress. Corrosion, which removes a steel’s endurance limit altogether and makes its S-N curve behave like aluminium’s. Fretting at a clamped or press-fitted interface, which initiates cracks at stresses far below any plain fatigue limit and is a common cause of shaft failures at a hub. Variable-amplitude loading, which needs a cumulative damage rule (Miner’s, usually) and a load spectrum. Welds, which have their own classification-based S-N curves in BS 7608 and whose fatigue strength depends far more on joint geometry than on the parent metal’s strength. And residual stress, which is what shot peening manipulates and which can change a fatigue life by more than any dimension on the drawing.

Frequently asked questions

Does aluminium have an endurance limit?

No, and the distinction is the most important one on this page. A steel’s S-N curve flattens at around 10⁶ cycles, so below that stress it runs indefinitely. Aluminium’s keeps falling past 10⁸ cycles with no knee, so no stress is safe forever — only a fatigue strength at a stated cycle count is meaningful. Published aluminium data is usually quoted at 5×10⁸ cycles (6061-T6 is 96.5 N/mm², fully reversed, R.R. Moore machine). In practice that means an aluminium part gets a finite design life and an inspection or replacement interval, where a steel one can get an infinite-life design.

Is the endurance limit really half the tensile strength?

For a polished rotating-beam steel SPECIMEN, roughly yes — 0.5 S_ut is the standard estimate, capped at 700 N/mm² for steels above 1,400. For your PART, no: the Marin factors typically take it down to 25 to 40 per cent of S_ut. On this page’s default — a machined 25 mm shaft at 99 per cent reliability — the fully corrected endurance limit is 26 per cent of the tensile strength, not 50. The gap between the specimen and the part is the entire reason the Marin method exists.

Why is f not just 0.9?

Because 0.9 is only right for soft steels. f is the fraction of S_ut that the S-N line starts from at 10³ cycles, and it falls with strength: about 0.90 at 490 N/mm², 0.83 at 770, 0.77 at 1,400. Shigley states 0.9 in words for steels below 490 and gives the rest as a curve on a figure. This page reconstructs the relation behind that figure rather than reading it by eye, and reproduces the values the published solutions use. Assuming 0.9 throughout puts the top of the line too high, and the error in finite life is roughly twice the error in stress.

Should I use Goodman, Gerber or Soderberg?

Goodman for general design, unless a code says otherwise. It is a straight line, it is conservative against most data, and it is what most machine design practice uses. Gerber fits experimental wrought-steel data best and is therefore the least conservative — use it when you have good data and need the margin. Soderberg is the most conservative because it runs to yield rather than to ultimate; it is really answering a different question (will the part yield anywhere in the cycle) and the separate first-cycle yield check on this page does that job more directly. All four agree exactly at zero mean stress.

Do I apply K_f to the stress or to the endurance limit?

Either, for the endurance limit itself — they are arithmetically equivalent. This page divides S_e by K_f, so enter your nominal stresses and put K_f in its own field, or apply the factor to the stresses and leave K_f at 1. Do not do both. The two are NOT equivalent for the whole S-N line: the 10³-cycle anchor is much less affected by a notch than the endurance limit is, because at high stress the notch yields locally and the concentration is partly relieved. Applying K_f only to S_e, as here, is the conservative reading of that.

What is the difference between K_t and K_f?

K_t is the elastic stress concentration factor — pure geometry, from the shape of the notch. K_f is the fatigue-effective factor, and it is smaller: K_f = 1 + q(K_t − 1), where q is the notch sensitivity. A small notch in a soft, ductile steel has a low q, so much of the K_t penalty does not materialise; a high-strength steel has q approaching 1 and pays nearly all of it. The shaft fillet and stress concentration page in this plugin computes K_t, q and K_f for a shoulder, a groove and a keyway, and this page consumes its answer.

Can I use this for a bolt?

No — use the bolt fatigue page, which exists precisely because a preloaded bolt is a different problem and not a special case. Two reasons. Only a fraction Φ of an external load range actually reaches a preloaded bolt, because the joint carries the rest; so the bolt’s alternating stress is much smaller than the external range until the joint separates, at which point it jumps. And VDI 2230 gives a bolt’s endurance amplitude by thread size and manufacturing route rather than through Marin factors, because the thread root is the notch and it is a known quantity. Applying this page to a bolt would get both halves wrong.

Why does reliability cost so much endurance limit?

Because a published endurance limit is a MEDIAN, and the population scatter in fatigue is large. k_e = 1 − 0.08·z_a assumes an 8 per cent standard deviation on the endurance limit, so 99 per cent survival needs 2.326 standard deviations and costs 19 per cent; 99.9 per cent costs 25 per cent. That is not pessimism, it is what the test data looks like. Fatigue life scatter is larger still — a factor of two or three in life at a given stress is normal — which is why a fatigue calculation is a screening tool and a test programme is the evidence.

Related calculators

References

R. G. Budynas and J. K. Nisbett, Shigley’s Mechanical Engineering Design, 10th and 11th editions (McGraw-Hill). The source for the endurance limit S_e′ = 0.5 S_ut capped at 700 MPa, for the Marin factors k_a to k_e, for the S–N line S_f = aN^b, and for the Goodman, Gerber, Soderberg and ASME-elliptic mean-stress criteria. BOTH editions are cited because they disagree: the 10th edition’s Table 6-2 gives the surface factor constants as 1.58/−0.085 ground, 4.51/−0.265 machined, 57.7/−0.718 hot-rolled and 272/−0.995 as-forged in MPa, while the 11th edition’s own published solutions use 1.38/−0.067, 3.04/−0.217, 38.6/−0.650 and 54.9/−0.758. At S_ut = 700 MPa on a machined surface that is k_a = 0.795 against 0.734, an eight per cent difference in endurance limit from nothing but the edition on the shelf. Both are printed here. Each 11th-edition pair was checked through the a_MPa = a_kpsi × 6.8948^−b unit conversion against the kpsi pair used in the same solutions, which all four satisfy.
The fatigue strength fraction f at 10³ cycles is not a table and not an equation in Shigley: it is a FIGURE (Fig. 6-18 in the 10th edition, 6-23 in the 11th) and every published solution reads it off by eye. Reading a figure by eye is not a citable value, so this batch reconstructed the relation that generates it instead: f = (σ′_F/S_ut)(2×10³)^b with b = −log(σ′_F/S_e′)/log(2×10⁶) and σ′_F = S_ut + 345 MPa, which is Shigley’s own Eq. (6-44). It returns 0.896 at 490 MPa, 0.830 at 770 MPa and 0.772 at 1,400 MPa — the three values the published solutions to problems 6-4 and 6-13 read off that figure, to within 0.004. Below 490 MPa Shigley says in words to use f = 0.9 and that is what this page does.
ISO 12107, Metallic materials — Fatigue testing — Statistical planning and analysis of data, and ASTM E466 (constant-amplitude axial fatigue tests) and E739 (statistical analysis of linear S–N data). Cited by number. They are the reason this page’s reliability factor exists at all: a published endurance limit is a median, and designing to a median means half the population fails early.
FKM-Richtlinie, Analytical Strength Assessment of Components in Mechanical Engineering (Forschungskuratorium Maschinenbau), and BS 7608 for welded steel. Cited by number as the two documents a real fatigue assessment in Europe is actually made against. Both are structured assessments with their own material data, their own statistical basis and their own treatment of welds and residual stress; the Marin-factor method on this page is a first-pass estimate and is not a substitute for either.
VDI 2230 Part 1, Systematic calculation of highly stressed bolted joints. Cited by number and NOT used here: it is the document behind Bolt fatigue under alternating load, which owns the preloaded-bolt case in this plugin. This page hands that case over rather than re-deriving it.
Aluminium 6061-T6 is taken at a minimum yield of 240 MPa with typical values near 270, a minimum tensile of 290 MPa with typical values near 310, and E = 68–69 GPa. Stainless 304 and 316 in the annealed condition are taken at 215 and 205 MPa yield. Grey cast iron is carried with NO yield strength at all, deliberately: it is a brittle material with no yield point, its tensile strength is a fraction of its compressive strength, and a von Mises check on it is the wrong check. That is the whole reason the failure-criteria page asks for the material class before it asks for anything else.