Subject

Preload and tightening torque

RevisionE 2026-09-26

StandardsISO 898-1:2013ISO 68-1:2023ISO 261:1998ISO 262:2023

T Tightening torque

62.6N·m

F_i Preload

31.3kN

F_p Proof load

48.1kN

Within proof load

source: As and Sp, ISO 898-1:2013
assumed: K = 0.20, assumed
input: 65 % of proof load
excluded: Not hot-dip galvanized or structural
formula: T = K·Fi·d

Show the formula

$F_{p} = A_{s} \cdot S_{p}$ (1)
$F_{i} = \frac{target}{100} \cdot F_{p}$ (2)
$T = K \cdot F_{i} \cdot d$ (3)

$A_{s}$: Tensile stress area, as ISO 898-1:2013 defines it, computed from the ISO 68-1:2023 basic profile; mm²
$S_{p}$: Nominal stress under proof load for the property class, ISO 898-1:2013, at the thread's diameter; MPa
$F_{p}$: Proof load; N
$target$: Target preload as a percentage of proof load; %
$F_{i}$: Preload; N
$K$: Nut factor. Assumed, not derived; common shop practice, not standardized
$d$: Nominal diameter; mm
$T$: Tightening torque. With d in mm, K·Fi·d is in N·mm and is divided by 1000; N·m

Preload sensitivity to the nut factor

Rev B, 2026-10-02

The joint, as entered

Equations

F_{p} = A_{s} \cdot S_{p} = 58.0 mm² \cdot 830 MPa = 48.1 kN

F_{i} = \frac{target}{100} \cdot F_{p} = \frac{65}{100} \cdot 48.1 kN = 31.3 kN

T = K \cdot F_{i} \cdot d = 0.20 \cdot 31.3 kN \cdot 10 mm = 62.6 N·m

Notes

$F_{p} = A_{s} \cdot S_{p}$ ISO 898-1:2013, Sp from Table 3
$T = K \cdot F_{i} \cdot d$ short-form torque equation; common shop practice, not standardized

Preload reached at T = 62.6 N·m

Equations

F_{i} = \frac{T}{K' \cdot d} = \frac{62.6 N·m}{0.20 \cdot 10 mm} = 31.3 kN

K_{proof}^{'} = K \cdot \frac{F_{i}}{F_{p}} = 0.20 \cdot \frac{31.3 kN}{48.1 kN} = 0.130

K_{yield}^{'} = K \cdot \frac{F_{i}}{F_{y}} = 0.20 \cdot \frac{31.3 kN}{54.5 kN} = 0.115

Notes

$T = K \cdot F_{i} \cdot d$ short-form torque equation; common shop practice, not standardized
$F_{p} = A_{s} \cdot S_{p}$ ISO 898-1:2013, Sp from Table 3
$F_{y} = A_{s} \cdot (yield min)$ yield min from ISO 898-1:2013, Table 3

K ±25 %Proof loadMin. yield

Torque against preload

Equation

T = K \cdot F_{i} \cdot d = 0.20 \cdot 31.3 kN \cdot 10 mm = 62.6 N·m

Notes

$T = K \cdot F_{i} \cdot d$ short-form torque equation; common shop practice, not standardized

Assumed KK ±25 %Proof load

input: Torque held at T = 62.6 N·m
assumed: Assumed K = 0.20, actual K shaded ±25 %
source: Min. yield load As × Rp0.2 min, ISO 898-1:2013 Table 3
excluded: Tension only, torsion during tightening not included
formula: Fi = T / (K·d)
drawing scale: Joint diagram schematic, not to scale

Show the formula

$F_{i} (K') = \frac{T}{K' \cdot d}$ (1)
$K' at proof = \frac{K \cdot F_{i}}{F_{p}}$ (2)
$K' at yield = \frac{K \cdot F_{i}}{F_{y}}$ (3)
$F_{y} = A_{s} \cdot (yield min)$ (4)

$T$: Tightening torque from the preload calculator, held fixed. With d in mm, T is taken in N·mm; N·m
$K$: The nut factor the torque was calculated with. Assumed, not derived; common shop practice, not standardized
$K'$: The joint's actual nut factor
$d$: Nominal diameter; mm
$F_{i}$: Preload; N
$F_{p}$: Proof load, As·Sp, as the preload calculator computes it; N
$F_{y}$: Tensile load at the class's minimum yield; N
$yield min$: Minimum ReL, Rpf or Rp0.2 for the property class, whichever ISO 898-1:2013 Table 3 specifies; MPa

What this shows

A calculated torque gives the target preload only at the nut factor assumed. The preload calculator gives one torque for one assumed nut factor. Tightened to that torque, a joint whose real nut factor is lower reaches more preload, and one whose nut factor is higher reaches less, in inverse proportion. The joint diagram shows the case entered in the calculator; the two charts show the relationship from both sides, and the first marks the actual nut factor at which the preload reaches each limit.

Both thresholds are the assumed nut factor scaled by a ratio of loads. The two thresholds follow from it directly. Preload reaches proof load when the actual nut factor falls to the assumed one times the target fraction, and reaches the minimum yield load, in tension, when it falls to the assumed one times Fi / Fy.

The yield threshold counts tension only, not the torsion of tightening. While a bolt is being tightened it also carries torsion from thread friction, which raises the stress at that moment; this visualizer does not include it.

Preload stress against the class's strengths

Rev A, 2026-10-20

Preload stress, class 10.9

Equations

σ_{i} = \frac{F_{i}}{A_{s}} = \frac{31.3 kN}{58.0 mm²} = 540 MPa

\frac{σ_{i}}{S_{p}} = \frac{target}{100} = \frac{540 MPa}{830 MPa} = 65.0 %

Notes

$σ_{i} = \frac{F_{i}}{A_{s}}$ stress on the tensile stress area, ISO 898-1:2013 §9.1.6.1
$S_{p}, yield min, R_{m} min$ ISO 898-1:2013, Table 3; the curve between them is schematic

input: Preload Fi = 31.3 kN
source: Sp, Rp0.2 min and Rm min, ISO 898-1:2013 Table 3
drawing scale: Curve shape schematic, not to scale
excluded: Tension only, torsion during tightening not included
formula: σi = Fi / As

Show the formula

$σ_{i} = \frac{F_{i}}{A_{s}}$ (1)
$\frac{σ_{i}}{S_{p}} = \frac{target}{100}$ (2)

$σ_{i}$: Preload stress, the preload over the tensile stress area; MPa
$F_{i}$: Preload, from the preload calculator; N
$A_{s}$: Tensile stress area, as the preload calculator computes it; mm²
$S_{p}$: Nominal stress under proof load for the class, ISO 898-1:2013 Table 3; MPa
$target$: Target preload as a percentage of proof load, from the preload calculator; %

What this shows

The preload stress is the target fraction of the proof stress. Preload is the target fraction of proof load, and proof load is the stress area times the proof stress, so the stress the preload puts in the stress area is that same fraction of the proof stress. At 65 % of proof load it is 65 % of Sp, whatever the thread.

The stresses are to scale; the curve between them is a sketch. ISO 898-1:2013 Table 3 gives each property class a stress under proof load, a minimum yield and a minimum tensile strength. It gives no curve. The curve here is drawn to pass through the minimum yield and the minimum tensile strength, and its strain axis carries no values: read the stresses off it, not the shape.

The yield comparison is in tension only. While a bolt is being tightened it also carries torsion from thread friction, which raises its stress at that moment; this chart does not include it.

Thread geometry and the stress area

Rev A, 2026-09-26

Axial profile, M10 × 1.5

Equations

H = \frac{\sqrt{3}}{2} \cdot P = \frac{\sqrt{3}}{2} \cdot 1.500 mm = 1.299 mm

d_{2} = d - \frac{3}{4} \cdot H = 10.00 - \frac{3}{4} \cdot 1.299 = 9.026 mm

d_{1} = d - \frac{5}{4} \cdot H = 10.00 - \frac{5}{4} \cdot 1.299 = 8.376 mm

d_{3} = d_{1} - \frac{H}{6} = 8.376 - \frac{1.299}{6} = 8.160 mm

Notes

$H = \frac{\sqrt{3}}{2} \cdot P$ ISO 68-1:2023, the basic profile
$d_{2} = d - 0.649519 \cdot P, d_{1} = d - 1.082532 \cdot P$ ISO 724:2023, the same as d − ¾H and d − 5⁄4H
$d_{3} = d_{1} - \frac{H}{6}$ ISO 898-1:2013, §9.1.6.1, for the stress area

Section across the axis

Equation

A_{s} = \frac{π}{4} \cdot {(\frac{d_{2} + d_{3}}{2})}^{2} = \frac{π}{4} \cdot {(\frac{9.026 + 8.160}{2})}^{2} = 58.0 mm²

Notes

$A_{s} = \frac{π}{4} \cdot {(\frac{d_{2} + d_{3}}{2})}^{2}$ ISO 898-1:2013, §9.1.6.1

input: M10 × 1.5
source: Basic profile, ISO 68-1:2023
source: d2 and d1 per ISO 724:2023
source: d3 = d1 − H/6 and As per ISO 898-1:2013
excluded: No tolerance class or coating
drawing scale: Drawn to scale

Show the formula

$H = \frac{\sqrt{3}}{2} \cdot P$ (1)
$d_{2} = d - \frac{3}{4} \cdot H$ (2)
$d_{1} = d - \frac{5}{4} \cdot H$ (3)
$d_{3} = d_{1} - \frac{H}{6}$ (4)
$A_{s} = \frac{π}{4} \cdot {(\frac{d_{2} + d_{3}}{2})}^{2}$ (5)

$d$: Nominal (major) diameter; mm
$P$: Pitch; mm
$H$: Height of the fundamental triangle; mm
$d_{2}$: Pitch diameter, where the thread ridge and the groove are equal in width; mm
$d_{1}$: Basic minor diameter, at the root flat of the basic profile; mm
$d_{3}$: The minor diameter ISO 898-1:2013 uses to define As. Drawn as a line; it is not a root on the basic profile; mm
$A_{s}$: Tensile stress area, the area of a circle whose diameter is the mean of d2 and d3; mm²

What this shows

The basic profile is the major diameter plus a shape fixed by the pitch. The drawing is the ISO 68-1 basic profile for the thread selected in the calculator: a crest flat of P/8 at the major diameter, flanks at 30° to the radial, and a root flat of P/4 at the basic minor diameter. The far side is drawn half a pitch along, as the helix puts it.

The stress area is a convention, and proof load is computed on it. The tensile stress area is a defined quantity, not a measured one: the area of a circle whose diameter is the mean of the pitch diameter and d3. It is smaller than the nominal diameter suggests, and it is the area the proof load is computed on.

The drawing is the basic profile, not a manufactured thread. The drawing shows no tolerance class, no coating allowance and no root radius. A real external thread’s root is rounded and sits below the basic root.

How it computes

The proof load comes first: the tensile stress area of the thread times the nominal stress under proof load for the class, both from ISO 898-1:2013.

F_{p} = A_{s} \cdot S_{p}

The target preload is a fraction of it, and the torque follows from the short-form torque equation:

F_{i} = \frac{target}{100} \cdot F_{p}

T = K \cdot F_{i} \cdot d

The verdict compares Fi with Fp. Proof load is the load a fastener must withstand without permanent set, so a preload equal to it is within, and only a preload above it exceeds.

Below the result, the page also shows the proof load ISO 898-1:2013 prints for the same thread and class, in Table 5 (coarse) or Table 7 (fine), with the difference. The two are not the same number: the calculator uses the exact As, while the table rounds As to three significant figures and rounds the load again. The difference is small but not zero, and the verdict uses the calculated value.

What it assumes

K is assumed

The nut factor lumps thread friction, underhead friction and geometry into one number. It is not a property of the bolt, and it is the largest uncertainty in the result. The presets beside the field are common shop practice, not standardized: 0.20 for dry, unplated steel (the default), 0.15 for oiled, unplated steel, and 0.11 for steel lubricated with molybdenum disulphide paste. Where the torque matters, measure K on the actual parts, finish and lubricant.

T = K·F·d is common shop practice, not standardized

It does not separate thread and underhead friction, and it takes no account of the joint.

Nominal values

As is the defined tensile stress area, and Sp is the nominal proof stress. Neither is a measured property of a particular bolt.

Not hot-dip galvanized, not structural bolting

ISO 898-1:2013 Table 5 qualifies two groups of its proof loads, and the calculator uses the general value for both. Hot-dip galvanized M8 and M10 bolts are cut to thread tolerance 6az to make room for the zinc, and reduced values from ISO 10684:2004, Annex A apply instead (footnote c): here the calculator overstates the proof load. Structural class 8.8 bolts from M12 take the higher-strength band, so their proof loads are higher, 50 700 N for M12 rather than 48 900 N (footnote d): here it understates it.

Tension only

The verdict compares the preload that remains after tightening with proof load. It does not include the torsional stress the bolt carries while it is being tightened, which adds to the tensile stress at that moment.

ISO metric threads, steel property classes

The thread and class tables cover M3 to M24, and the classes ISO 898-1:2013 specifies for carbon and alloy steel. Stainless steel (ISO 3506) is out of scope.

Worked example

M10 × 1.5, class 10.9, $K = 0.20$ , target 65 % of proof load.

Given

Quantity	Source
$d = 10 mm$	Nominal diameter, M10 × 1.5
$A_{s} = 57.9896 {mm}^{2}$	From the ISO 68-1:2023 basic profile, as ISO 898-1:2013 defines it
$S_{p} = 830 MPa$	Class 10.9, ISO 898-1:2013
$K = 0.20$	Assumed; common shop practice, not standardized
$target = 65$	Percent of proof load

Find

$F_{p}, F_{i}, T$

Solution

$F_{p} = A_{s} \cdot S_{p} = 57.9896 {mm}^{2} \cdot 830 MPa = 48 131.4 N$ (1)
$F_{i} = \frac{target}{100} \cdot F_{p} = \frac{65}{100} \cdot 48 131.4 N = 31 285.4 N$ (2)
$T = K \cdot F_{i} \cdot d = 0.20 \cdot 31 285.4 N \cdot 10 mm = 62 570.8 N \cdot m m = 62.6 N \cdot m$ (3)

The calculator shows 31.3 kN preload and 62.6 N·m tightening torque.

The figures above are the calculator’s own, computed when the site is built, and the numbers in brackets are the relations in the derivation. The calculator is pinned to this case, worked by hand, in a test: if the two disagree, the site is not published.

Sources

Rev	Date	By	Description
Rev E	2026-09-26	SD	Added the thread-geometry visualizer below the result, verified separately

4 earlier revisions

Rev D	2026-09-26	SD	Added the preload-sensitivity visualizer below the result, verified separately
Rev C	2026-09-26	SD	Added K presets for dry, oiled and MoS₂-lubricated steel, common shop practice, not standardized
Rev B	2026-09-26	SD	Added M7, the one ISO 262:2023 size from M3 to M24 that was missing
Rev A	2026-09-25	SD	Initial issue