Preload Accuracy by Tightening Method Calculator

What each tightening method actually guarantees from the same bolt. VDI 2230’s tightening factors turned into a guaranteed minimum clamp load, the capacity the scatter wastes, the preload embedding costs computed from the joint’s own stiffness, and how much more clamp load a better method buys — with four published scatter tables side by side and none of them averaged.

Last updated 24 Sep 2026

Preload accuracy by method

Bolt and method → clamp load you can count on

Thread size

Property class or gradeThe bolt’s own limit is what the scatter has to fit under, so a stronger bolt makes the wasted capacity larger in absolute terms and no smaller in proportion.

Tightening methodVDI 2230 Table A8. Thirteen techniques, each with a tightening factor α = F_max/F_min and the setting technique it depends on.

Utilisation of the yield strength at the MAXIMUM preload (%)VDI 2230 sizes the assembly preload so that the highest bolt in the population reaches 90% of the minimum yield strength. Lower it if the bolt also has to carry an external load on top, and note this is a percentage of YIELD, not of proof.

Embedding amount f_Z (micrometres)VDI 2230’s worked example uses 5 µm for a clamping length ratio of 3.5. SAE data quoted for embedment give up to 12.7 µm at each surface mate. Most of it happens during tightening and costs nothing; only what settles afterwards is lost preload.

Series stiffness of the joint (kN/mm)k_b k_m/(k_b + k_m). The joint stiffness page reports it directly. Embedding is a displacement, so the preload it costs is this stiffness times that displacement — a stiff short joint loses much more preload to the same settling than a long one.

Clamp load the joint actually needs (N)The minimum clamp load your design depends on: enough to carry the shear by friction, or to keep a seal, or to stop the joint separating. The whole point of this page is whether the method can guarantee it.

Not a circuit: thirteen bars on one preload axis, in per cent of the bolt's own ceiling. Each bar runs from the LOWEST preload that tightening method may deliver up to the ceiling at 100, so the bar's LENGTH is the scatter and the gap between its left-hand end and zero is the clamp load you can count on. Every bar ends at the same place, because the bolt does not know what tool you used. The method you chose is outlined twice. The double line at 100 is the ceiling and the single line across the stack is the clamp load you said the joint needs — any bar whose left-hand end is to the right of it will do the job, and any bar whose end is to the left of it will not. Read the stack from the bottom: hand tightening and impact wrenches guarantee a quarter of the bolt, an ordinary torque wrench on an estimated friction coefficient half of it, the same wrench with its setpoint found by test on the real joint about 63 per cent, and ultrasonic length measurement 87 per cent. Nothing about the fastener changes between the top bar and the bottom one.

14.84kNExample

M10 class 8.8, the highest bolt allowed to reach 90% of the minimum yield strength, tightened with a torque wrench on an estimated friction coefficient, 5 µm of embedding on a joint with a series stiffness of 372 kN/mm, and a joint that needs 15 kN of clamp load

One bolt, one ceiling, and what the method lets you keep

F_max = ν · A_s · R_p0.2 · F_min = F_max / α · scatter = ±(α − 1)/(α + 1) · F_Z = f_Z · k_b k_m/(k_b + k_m) · F_guaranteed = F_max/α − F_Z · wasted = 1 − 1/α

α: VDI 2230’s tightening factor, F_max/F_min for the method. Everything on this page is a consequence of it
ν: how much of the minimum yield strength the HIGHEST bolt in the population is allowed to reach. VDI 2230 uses 0.9. Note it is yield, not proof load
F_max: the bolt’s ceiling. Set by the bolt and the design, not by the tool, and the same for every method
F_min: the lowest preload the method may deliver. This is what the design has to work with, and it is the number almost nobody prints
f_Z: embedding — how far the joint settles after tightening. A displacement of a few micrometres
F_Z: the preload that displacement costs. The joint’s SERIES stiffness times f_Z, so a short stiff joint pays much more for the same settling
1 − 1/α: the fraction of the bolt you bought and cannot count on. 37.5% at α = 1.6, 13% at α = 1.15, 75% at α = 4

Worked example

M10 class 8.8, the highest bolt allowed to reach 90% of the minimum yield strength, tightened with a torque wrench on an estimated friction coefficient, 5 µm of embedding on a joint with a series stiffness of 372 kN/mm, and a joint that needs 15 kN of clamp load
The ceiling: 0.90 × 57.99 × 640 = 33.4 kN. That is what the STRONGEST bolt in the population is allowed to reach, and it is the same whatever tool is used
The method: VDI 2230 Table A8 gives a torque wrench on an estimated friction coefficient α = 1.6 to 2.0. Take the worse end. F_min = 33.4 kN/2.0 = 16.7 kN
Which is a scatter of ±(2.0−1)/(2.0+1) = 33.3% about the mean of 22.27 kN. VDI print ±23 to 33% for this row, and 33% is exactly what that expression gives at α = 2.0 — the printed column is the identity, at all thirteen rows
Embedding: 5 µm against a series stiffness of 372 kN/mm costs 0.005 × 372,000 = 1.86 kN, which is 11% of the minimum preload. Note that this is computed from the displacement and the joint's stiffness, not quoted as a percentage — the same settling on a joint twice as long would cost about half as much
So the clamp load you can actually count on is 16.7 kN − 1.86 kN = 14.84 kN, against a ceiling of 33.4 kN. You bought a bolt with 33.4 kN of capacity and you can rely on 44% of it. The joint needs 15 kN, so it is short by 159 N — this combination does not work
NOW CHANGE ONLY THE TOOL. Ultrasonic length measurement is α = 1.1 to 1.2, so F_min becomes 27.84 kN and after the same embedding you can count on 25.98 kN — 1.75 times as much clamp load from the same bolt, and the requirement is now met with a margin of 1.73. Nothing about the fastener changed
Or keep the wrench and calibrate it. VDI's next row up is a torque wrench whose setpoint was determined by TEST on the real joint: α 1.4 to 1.6, so F_min becomes 20.88 kN and the guaranteed clamp load 19.02 kN. Measuring the nut factor on your own joint once is the cheapest 30% of clamp load available

VDI 2230 Table A8, and what each method guarantees from the same M10 class 8.8 bolt

Compare the third and fourth columns. The ± scatter VDI prints beside each tightening factor is not independent data: it is exactly (α − 1)/(α + 1) rounded to a whole per cent, at every one of the thirteen rows. That identity is what confirmed the column mapping of the extract this batch worked from before any of it was used, and it is worth knowing for its own sake — the band is symmetric about the arithmetic mean of F_max and F_min, which is a modelling choice rather than a measurement. Now read the fifth column, which is the point of the page. The bolt’s ceiling is 33.4 kN whatever tool you use. Estimate the friction and set a torque and you can count on 14.8 kN. Measure the bolt’s length ultrasonically and you can count on 26.0 kN — from the SAME BOLT. A better method is not a refinement; it is 75% more clamp load. These dimensions come from a published standard’s table, not from a formula. The standard itself is cited below and the printed values are attributed to the catalogue they were taken from; a different publisher may round differently in the last digit.
Tightening technique	α	Scatter VDI prints (%)	(α−1)/(α+1), computed (%)	Guaranteed clamp load (kN)	Fraction of the bolt you can count on (%)	Against estimated-friction torque control	Set by
Ultrasonic length measurement	1.1 – 1.2	±5 – 9	4.8 – 9.1	25.98	83	1.75	sound travel time
Mechanical elongation, pressure screws	1.1 – 1.3	±5 – 13	4.8 – 13.0	23.83	77	1.61	prespecified bolt elongation, set by a forcing torque
Mechanical elongation measurement	1.1 – 1.5	±5 – 20	4.8 – 20.0	20.41	67	1.38	direct length measurement, or indirectly via axial play
Hydraulic tensioning	1.1 – 1.4	±5 – 17	4.8 – 16.7	22.00	71	1.48	pressure or length measurement, or a further rotation angle
Multipartite nut with a threaded bushing	1.2 – 1.5	±9 – 20	9.1 – 20.0	20.41	67	1.38	torque of the tightening tool
Yield-point controlled	1.2 – 1.4	±9 – 17	9.1 – 16.7	22.00	71	1.48	preset relative torque / rotation-angle coefficient
Rotation-angle controlled (torque plus angle)	1.2 – 1.4	±9 – 17	9.1 – 16.7	22.00	71	1.48	experimentally determined snug torque and angle
Hydraulic impulse driver	1.2 – 2.0	±9 – 33	9.1 – 33.3	14.84	50	1.00	rotation angle or a further torque
Torque-controlled hydraulic, gradual	1.4 – 1.6	±17 – 23	16.7 – 23.1	19.02	63	1.28	pressure measurement
Torque wrench, dynamic measurement, torque found by test	1.4 – 1.6	±17 – 23	16.7 – 23.1	19.02	63	1.28	setpoint torque determined experimentally on the real joint
Torque-controlled, friction coefficient ESTIMATED	1.6 – 2.0	±23 – 33	23.1 – 33.3	14.84	50	1.00	estimated coefficient of friction, good surfaces
Torque-controlled, friction estimated, rough surfaces	1.7 – 2.5	±26 – 43	25.9 – 42.9	11.50	40	0.77	estimated coefficient of friction, hard or rough surfaces
Impact wrench or hand tightening	2.5 – 4.0	±43 – 60	42.9 – 60.0	6.49	25	0.44	retightening moment or subjective assessment

Four published scatter tables, side by side and not averaged

These disagree, and the disagreement is the most useful thing on the page. For plain torque control: the Industrial Fasteners Institute say ±25%, Machine Design say ±15 to 30%, Sandia’s guideline says ±35% unlubricated and ±25% lubricated, and VDI 2230 says ±23 to 33% when the friction coefficient is estimated and ±17 to 23% when the setpoint torque has been determined by test on the real joint. That last distinction is the one the other three tables do not make and it is the actionable one. For measuring the bolt’s stretch the gap is wider still: Machine Design say ±1 to 8% and VDI say ±5 to 20%, a factor of 2.5 at the top end. Nothing here is averaged. If your design depends on which figure is right, the answer is to measure your own joint — which is exactly what VDI’s better rows describe doing. 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.
Published by	Method	Preload scatter (%)
Industrial Fasteners Institute	Feel — the operator’s judgement	±35
Industrial Fasteners Institute	Torque wrench	±25
Industrial Fasteners Institute	Turn of the nut	±15
Industrial Fasteners Institute	Load-indicating washers	±10
Industrial Fasteners Institute	Fastener elongation	±3 to 5
Industrial Fasteners Institute	Strain gauges	±1
Machine Design (control accuracies)	Torque	±15 to 30
Machine Design (control accuracies)	Turn	±15 to 30
Machine Design (control accuracies)	Torque and turn	±10 to 25
Machine Design (control accuracies)	Torque past yield	±3 to 10
Machine Design (control accuracies)	Bolt stretch	±1 to 8
Sandia SAND2008-0371 (torque wrench)	Unlubricated	±35
Sandia SAND2008-0371 (torque wrench)	Cadmium plated	±30
Sandia SAND2008-0371 (torque wrench)	Lubricated	±25
VDI 2230 Table A8	Torque-controlled, friction estimated	±23 to 33
VDI 2230 Table A8	Rotation-angle controlled (torque plus angle)	±9 to 17
VDI 2230 Table A8	Mechanical elongation measurement	±5 to 20
VDI 2230 Table A8	Ultrasonic length measurement	±5 to 9

Why the better methods are better — what each one actually measures

One idea runs through the whole table: the scatter falls as the measured quantity gets closer to the bolt’s stretch. Torque is two friction coefficients away from it; angle is one; length is the thing itself. The Chinese Journal of Mechanical Engineering’s review puts the gap between the first two at a factor of four: “the scatter in the bolt preload induced by a given tightening torque may be over four times larger than that for angle-rotation control”, because angle control’s preload is “proportional to the angle of turn regardless of the frictional variables”. Note what every angle method needs and what therefore limits it: a snug point. The angle has to be counted from somewhere, that somewhere is found by torque, and an inconsistent snug torque puts the friction back in through the front door.
Method	Measured quantity	Why the scatter is what it is
Torque, friction estimated	Torque at the tool	About 90% of the torque is friction, and the friction coefficients are properties of the surfaces on the day. The measurement is only weakly connected to the thing you want. α 1.6 to 2.0, or 1.7 to 2.5 on hard or rough surfaces.
Torque, setpoint found by test on the real joint	Torque at the tool, calibrated against measured preload	The same measurement, but the nut factor is no longer a guess. This is the cheapest real improvement available and it halves the scatter: α 1.4 to 1.6 against 1.6 to 2.0.
Angle past a snug torque	Rotation of the nut past a datum	Turning the nut a measured angle past snug is a DISPLACEMENT measurement, and preload is a displacement — the bolt’s stretch. Friction still sets the snug point, which is why the snug torque has to be consistent and why the method is not friction-free. α 1.2 to 1.4.
Yield-point control	The slope of the torque-angle curve	The tool watches the gradient and stops when it collapses. Past yield the torque-angle slope is nearly flat, so the stopping point is set by the bolt’s own material and, as the Chinese Journal of Mechanical Engineering’s review puts it, “the scatter of the final preload is relatively small because the material properties vary negligibly”. α 1.2 to 1.4, and the bolt is deliberately taken past yield.
Hydraulic tensioning	Hydraulic pressure, or length	The bolt is stretched axially and the nut run down behind it, so thread friction never enters the preload at all. α 1.1 to 1.4. Used on large flanges and foundation bolts where the tool can be got around the bolt.
Bolt stretch, measured	The bolt’s length	Directly the quantity that IS the preload, divided by the bolt’s own compliance. α 1.1 to 1.5 mechanically, 1.1 to 1.2 ultrasonically. What is left is the uncertainty in the effective length and, for ultrasound, in the acoustoelastic coefficient.

Embedding, computed from the displacement rather than quoted as a percentage

Embedding is the joint settling: local yielding at the bearing faces and in the thread flanks flattens the asperities, the bolt gets shorter in effect, and the preload falls by the settling distance times the joint’s series stiffness. That is why it is computed here from f_Z and k_b k_m/(k_b + k_m) rather than quoted as a percentage — the same 5 µm costs four times as much preload in a stiff short joint as in a long one, and a percentage cannot know which you have. VDI 2230’s own worked example uses f_Z = 5 µm at a clamping length ratio of 3.5, which on the joint assumed in this table costs 1.86 kN, or 11% of what torque control guarantees. The commonly quoted 2 to 10% loss follows for f_Z of one to five micrometres. SAE data quoted for embedment give up to 12.7 µm at each surface mate; the Chinese Journal of Mechanical Engineering’s review of preload control reports cases losing 13% immediately after tightening. And note the timing, which is the one piece of good news: most embedding happens DURING tightening and costs nothing, because the tool goes on turning. Only what settles afterwards is lost preload — which is the whole argument for a retightening pass. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see.
f_Z (µm)	Preload lost (N)	% of the guaranteed minimum	% of the ceiling	Guaranteed clamp load left (kN)
1.0	372	2.2	1.1	16.33
2.5	930	5.6	2.8	15.77
5.0	1,860	11.1	5.6	14.84
7.5	2,790	16.7	8.4	13.91
10.0	3,720	22.3	11.1	12.98
12.7	4,724	28.3	14.1	11.98
20.0	7,440	44.5	22.3	9.26

What the scatter costs, and why a better method is 75% more clamp load

This is the page that makes the other six honest. A bolt has one ceiling: the preload at which the highest bolt in the population reaches its limit. That ceiling does not depend on the tool. What the tool decides is how far BELOW the ceiling the design has to work, because the design must function at the minimum preload the method may deliver while the bolt survives the maximum. The gap between those two is capacity you paid for and cannot count on.

VDI 2230 Table A8 puts numbers on it, thirteen of them. The tightening factor α is F_max/F_min for the method: 1.1 to 1.2 for ultrasonic length measurement, 1.2 to 1.4 for angle or yield control, 1.4 to 1.6 for a torque wrench whose setpoint was found by test on the real joint, 1.6 to 2.0 for one set on an estimated friction coefficient, and 2.5 to 4.0 for hand tightening or an impact wrench. The ± scatter VDI prints beside each one turns out not to be independent data at all: it is exactly (α − 1)/(α + 1), rounded to a whole per cent, at every single row. That identity is how this batch confirmed it had read the table correctly before using it, and it says something about the model too — the band is symmetric about the arithmetic mean of the two extremes, which is a choice rather than a measurement.

What the scatter costs, in kilonewtons. Take an M10 class 8.8 and let the highest bolt reach 90% of its minimum yield, which is VDI’s rule: 33.4 kN. Tighten it with a torque wrench on an estimated friction coefficient and you can count on 16.7 kN before losses, or 14.8 kN after 5 µm of embedding. Measure the bolt’s length ultrasonically and you can count on 26.0 kN. Same bolt, same joint, 75% more guaranteed clamp load. That is the argument for a better method and it is not a small one — and the cheapest version of it is not a better wrench but a calibrated one: determining the setpoint torque by test on the real joint moves α from 1.6–2.0 to 1.4–1.6 and buys about 30%.

Why the better methods are better: they measure something closer to the bolt’s stretch. Preload IS a stretch. Torque is two friction coefficients away from it, and about 90% of the torque is friction. Turning the nut a measured angle past a snug point is a displacement measurement, which is why the review in the Chinese Journal of Mechanical Engineering reports that torque control’s scatter “may be over four times larger than that for angle-rotation control”. Yield control goes one step further and watches the torque-angle gradient collapse; past yield the slope is nearly flat, so the stopping point is set by the bolt’s own material, which varies very little. Hydraulic tensioning removes thread friction from the preload path entirely. And measuring the length measures the thing itself. Every angle-based method needs a snug torque to count from, which is where the friction gets back in, and that is why the snug torque has to be specified and consistent.

And then the joint settles. Embedding is local yielding at the bearing faces and the thread flanks: the asperities flatten, the joint gets shorter, and the preload falls by that displacement times the joint’s SERIES stiffness. This page computes it from f_Z rather than quoting a percentage, because the same 5 µm costs four times as much preload in a short stiff joint as in a long one and a percentage cannot know which you have. VDI 2230’s worked example uses 5 µm; SAE data quoted for embedment give up to 12.7 µm per surface mate; the commonly quoted 2 to 10% loss follows for a few micrometres on a typical joint. The one piece of good news is the timing — most embedding happens while the tool is still turning and costs nothing. Only what settles afterwards is lost, which is the whole case for a retightening pass. The stiffness this acts against comes from the bolted joint stiffness and separation calculator, the bolt’s ceiling from the proof load and tensile stress area calculator, and the torque itself from the bolt torque calculator.

Frequently asked questions

How accurate is a torque wrench at setting bolt preload?

Not accurate, and four published tables agree about that while disagreeing on the number. VDI 2230 gives ±23 to 33% when the friction coefficient is estimated and ±17 to 23% when the setpoint torque has been determined by test on the real joint. The Industrial Fasteners Institute say ±25%. Machine Design say ±15 to 30% for torque control. Sandia’s guideline gives ±35% unlubricated, ±30% cadmium plated and ±25% lubricated. This page prints all four side by side and averages none of them. Note that the wrench’s own accuracy — ±3 to 15% for a bar wrench, better for a digital one — is not the limiting factor; the friction is.

What does the tightening factor α mean?

F_max/F_min: the ratio of the highest preload the method may deliver to the lowest, for the same nominal setting. VDI 2230 tabulates it for thirteen techniques. Its practical meaning is that you can only count on 1/α of the bolt’s capacity, so α = 1.6 wastes 37.5% of the bolt and α = 4 wastes 75%. The ± scatter figures quoted in the literature are the same information written differently: (α − 1)/(α + 1) reproduces VDI’s own printed scatter column at all thirteen rows.

Why does angle control work better than torque control?

Because it measures a displacement, and preload is a displacement — the bolt’s stretch. Turning the nut a counted angle past a snug point stretches the bolt by pitch × angle/360, whatever the friction happens to be doing. The review in the Chinese Journal of Mechanical Engineering puts the improvement at over four times less scatter, and notes that angle control’s preload is “proportional to the angle of turn regardless of the frictional variables”. The catch is the snug point: the angle has to be counted from somewhere, that somewhere is found with a torque, and an inconsistent snug torque puts the friction straight back in. Specify it.

How much preload is lost to relaxation and embedment?

Commonly quoted at 2 to 10%, and this page computes it instead of quoting it, because the answer depends on your joint’s stiffness. Embedding is a displacement f_Z of a few micrometres — VDI 2230’s worked example uses 5 µm, SAE data quoted for embedment give up to 12.7 µm at each surface mate — and the preload it costs is that displacement times the joint’s series stiffness. On a short stiff joint 5 µm can cost 11% of the guaranteed minimum preload; on a long one, half that. Most of the embedding happens during tightening and costs nothing, so a retightening pass after the first settling recovers much of it.

Is it better to use a better tightening method or a bigger bolt?

It depends on which is cheaper in your situation, and this page computes both sides. Going from estimated-friction torque control (α 2.0) to ultrasonic (α 1.2) multiplies the guaranteed clamp load by about 1.75 from the same bolt. Getting the same increase from a bigger bolt needs 1.75 times the stress area, which is roughly one and a half size steps and brings a bigger head, a bigger hole, more edge distance and a deeper tapped hole with it. The results report the stress area you would need. On a one-off, the bigger bolt usually wins; in production, or where the joint is inaccessible, the better method does.

Should the preload target be a percentage of proof load or of yield?

Both conventions are in use and they differ by about a third, so say which you mean. Machinery’s Handbook, as quoted in Sandia’s guideline, says 75% of the PROOF strength for removable fasteners and 90% for permanent ones. VDI 2230 sizes the assembly preload so the highest bolt reaches 90% of the minimum YIELD strength, which for a class 8.8 is 576 N/mm² against a proof stress of 580 — that is 99% of proof load, not 75%. The difference is that VDI’s figure is the maximum of the scattered population and Machinery’s Handbook’s is the target; with α = 1.6 a 90%-of-yield maximum corresponds to a target of about 76% of proof. This page uses VDI’s convention and reports the proof-load percentage beside it.

Related calculators

References

VDI 2230 Blatt 1 Table A8 (“Guide values for the tightening factor α_A”), from the extract published by pvp-software.de. Thirteen tightening techniques with their α_A ranges, their setting technique and the ± scatter each implies. That scatter column is not independent data: it is exactly (α_A − 1)/(α_A + 1) rounded to a whole percent, at every one of the thirteen rows, which is what confirmed the column mapping of the extract before any of it was used.
VDI 2230 Blatt 1, Systematic calculation of highly stressed bolted joints — Joints with one cylindrical bolt. Cited by clause and table; the guideline is copyrighted and was not reproduced. The tightening-torque terms used here are its own form, M_A = F_M[0.16 P + 0.58 d₂ μ_G + 0.5 D_Km μ_K], where the 0.58 is 0.5/cos 30° written out. Table A8 gives the tightening factor α_A; clause R12 gives the endurance limit; clause R4 gives the embedding amount f_Z.
NASA Fastener Design Manual (RP-1228), as reproduced by the Engineering Library. Source for the torque coefficient table against equal thread and head friction — K = 0.074, 0.133, 0.189 and 0.250 at μ = 0.05, 0.10, 0.15 and 0.20 — and for the two accuracy tables: the Industrial Fasteners Institute’s (feel ±35%, torque wrench ±25%, turn of the nut ±15%, load-indicating washers ±10%, fastener elongation ±3 to 5%, strain gauges ±1%) and Machine Design’s control accuracies (torque ±15 to 30%, turn ±15 to 30%, torque and turn ±10 to 25%, torque past yield ±3 to 10%, bolt stretch ±1 to 8%).
Sandia National Laboratories, SAND2008-0371, Guideline for Bolted Joint Design and Analysis: Version 1.0, as reproduced by the Engineering Library. Source for the target preload rule, attributed there to Machinery’s Handbook: “use 75% of the proof strength … for removable fasteners and 90% of the proof strength for permanent fasteners”; for the torque-wrench accuracies by lubrication state (unlubricated ±35%, cadmium plated ±30%, lubricated ±25%); and for Shigley’s frustum form of the member stiffness with the statement that the cone half-angle “should be between 25 and 33 degrees and in general recommends 30 degrees”.
Y. Liu et al., Preload Control Method of Threaded Fasteners: A Review, Chinese Journal of Mechanical Engineering 37 (2024) 82. Source for the comparison between methods: “the scatter in the bolt preload induced by a given tightening torque may be over four times larger than that for angle-rotation control”; that angle control’s preload is “proportional to the angle of turn regardless of the frictional variables”; that in yield control “the scatter of the final preload is relatively small because the material properties vary negligibly”; and that preload can fall by as much as 13% immediately after tightening.
Embedment in a bolted joint: the Wikipedia article Embedment, which cites SAE data for “values of up to 0.0005 inches … at each surface mate” and notes that “most of the embedment occurs during torquing” so only what happens afterwards costs preload, together with VDI 2230’s own worked example, where “for a clamping length ratio of 3.5 there is a total amount of embedding of f_Z = 5 × 10⁻³ mm”, as reported in PCB Load & Torque’s Review of the Application of Design Guideline VDI 2230. This page computes the preload loss from f_Z and the joint’s own stiffness rather than quoting a percentage.
Portland Bolt. Bolt Torque Chart and Tension vs. Torque. Source for the other three nut factors: “waxed (e.g. pressure wax as supplied on high strength nuts) = .10, hot dip galvanized = .25, and plain non-plated bolts (as received) = .20”, tension taken at 75% of proof load. It also states the range plainly: K “can range from 0.10 for a well lubricated/waxed assembly, to over 0.30 for one that is dirty or rusty”, and that torque values “that claim to relate torque to pretension without verification shall not be used”.
Fastenal. Torque-Tension Reference Guide. Source for four of the published nut factors: 0.12 for a PTFE-type coating on nut, bolt and washer, 0.15 for lubricated conditions including oil and tapping fluid, 0.17 for some anti-seize, thread lockers and some plain conditions, and 0.20 for “zinc and dry conditions”. It states the target clamp load as “75% of the proof loads specified by the standard” and warns that “torque is only an indirect indication of tension”.
ISO 898-1:2013, Mechanical properties of fasteners made of carbon steel and alloy steel — Part 1: Bolts, screws and studs with specified property classes. Cited by clause; the standard is copyrighted and was not fetched. Clause 9.1.6.1 defines the nominal stress area as the circle on the mean of the pitch diameter d₂ and the minor diameter d₃ of the basic profile, which is where the 0.938194 on this page comes from — it is derived here from d₂ = d − 0.649519 P and d₃ = d − 1.226870 P, not copied. Table 3 carries the property classes.
Kova Fasteners. ISO 898 Part 1 – 2013 (Extract). The source for the minimum tensile strength R_m, the minimum yield or 0.2% proof strength, and the stress under proof load S_p for every property class used here. It also carries the note that for classes 4.8, 5.8 and 6.8 “the values for R_pf min are under investigation” — which is why the second digit of those three designations is a label and not a property.
Bolt Science. Torque Control and Torque-Angle Control Tightening Methods (technical note). Listed as consulted for the snug-torque threshold and the torque-angle slope; the PDF could not be fetched through the proxy (robots.txt could not be read), so nothing is attributed to it. What this page says about the flatness of the torque-angle curve past yield is taken from the Springer review above and from VDI 2230’s own description of yield-point control as setting a “relative torque / rotation-angle coefficient”.