Tightening torque and screw loads

Calculate the tightening torque and the maximum permissible preload of a steel or stainless steel metric screw, following VDI 2230. You can compare several cases side by side. Everything updates instantly with each input.

To compare cases side by side, open the page on a computer or a tablet.

Data

Screw head

Size

Pitch

Screw property class

Screw finish

Lubrication

Tool

This covers tightening only (steps R1, R7, R10 and R13 of VDI 2230), not the load in service, settling or fatigue. The check of the part under the head is in the Advanced mode.

Results

Tightening torque

36.4N·m

Clamping force (preload)

31.1kN

How the torque is shared

Creates the clamping force

20%

Friction in the thread

36%

Friction under the head

44%

Tightening

Smallest preload to expect

15.5kN

Lowest torque the tool may give

34.9N·m

Highest torque the tool may give

37.8N·m

Only 20 % of the torque goes into the clamping force. The rest goes in friction. That is why lubrication affects the result so much.

A design guide, based on the standards listed below. For critical joints, check against them or with a specialist.

Partly verified data

We have checked part of this data against reliable sources. The values marked with a have fewer sources than we require: use them as a guide and check them (they may not appear in this view).MechLab Geek accepts no liability for any use of this data or its consequences. Check it before you use it. Legal notice.Spotted a mistake? Let us know.

How this calculator works

This calculator lets you work out the torque and the preload of a screw according to VDI 2230, the reference guideline for calculating bolted joints.

You choose the screw, its finish, the lubrication and the tool. You get the wrench torque with its tolerance, the minimum preload to expect with that tool, the pressure under the head against what the part withstands, and the proof and breaking loads of the screw.

Assumptions and limitations

Scope

Tightening during assembly only: one screw tightened by torque, at room temperature.
Full-shank metric screw. The maximum preload brings it to 90 % of its yield strength.
The pressure under the head, under static load, is compared with what the material of the part withstands.
Proof and breaking loads are of the screw alone, not of the joint.

Simplifications

Friction is constant during tightening; a friction class takes its lowest value.
The washer changes the hole and limits the bearing, but does not spread the load: a wider one does not lower the pressure shown.
The tool tolerance is that of the instrument alone, without the scatter of the friction, and assumes a click-type torque wrench (type II).
In the reverse mode, the band is that of the friction class, without the scatter of the wrench.

Out of scope

Operating load, embedding, joint stiffness and fatigue.
Thread stripping, locking nuts or patches, coatings and retightening.

Sources: VDI 2230 values come from reproductions by manufacturers (Bossard, Würth, Reyher, Schaeffler), not from the guideline itself; the results match their tables to within 1 %.

What VDI 2230 is

VDI 2230 is a guideline developed by the Verein Deutscher Ingenieure (the Association of German Engineers) that explains how to calculate bolted joints systematically.

Before it appeared, general rules and wide safety margins were used. VDI 2230, by contrast, starts from a mathematical model that treats the bolted joint as an elastic system of springs, accounting for every force, deformation and friction involved.

It was first published in 1974 and has been revised several times since. Today it is widely used as the reference technical guide for calculating bolted joints.

How the guideline works

The guideline proposes a systematic step-by-step calculation (R1 to R13) that starts from the loads acting on the joint and checks that the screw and the parts withstand them, from assembly to service.

How this calculator applies it

This calculator does not apply the method step by step, since it does not start from a joint design and known loads. What it does is provide the torque and the preload for a given screw, using the mathematical models of steps R1, R7, R10 and R13 of VDI 2230.

Formulas and fundamentals

This section is for those who like to understand the sense of things. Here we develop, step by step, the tightening theory of the VDI 2230 method.

What we are after

A screw can be seen as a very stiff spring. When we tighten it, we force it to stretch a little. And as it tries to regain its shape, it pulls on the parts it holds and clamps them against each other.

A screw behaves like a very stiff spring.

We call that force the preload, $F$ , and it is what holds the joint: it stops the parts separating or slipping and the screw coming loose.

In practice we usually cannot measure the force inside a screw. But what we can measure is the tightening torque on the wrench, $\MA$ . Almost everything that follows is translating from that torque to that force.

First we will see how much torque a preload costs. Then, how much preload a screw can take. And last, what happens when the tightening torque is not exact.

The torque in the thread

A thread is a ramp in the shape of a helix.

If we unroll one turn of thread, we get an inclined plane. This plane rises a height equal to the pitch $P$ , and runs a perimeter of $\pi\,d_2$ , where $d_2$ is the mean diameter of the thread flanks.

If we represent the screw as an object on that ramp, we can say that turning the screw is equivalent to pushing the object up the ramp with a force $H$ .

One turn of thread unrolled, with the forces on the screw. The scale is exaggerated: the real pitch is far smaller than the perimeter.

The drawing shows all the forces acting on the screw: the preload $F$ , the push $H$ , the normal force $N$ , with which the ramp supports the screw, and the friction, which opposes the motion.

Writing the equilibrium of forces, we get:

H\cos\alpha = F\sin\alpha + \muG\,N \qquad N = F\cos\alpha + H\sin\alpha

where

$H$: Tangential push that lifts the screw up the ramp
$\alpha$: Helix angle: the slope of the ramp, with tan α = P / (π d₂)
$N$: Normal force with which the flank supports the screw
$\muG$: Friction coefficient between the thread flanks

Solving for the push, we arrive at:

H = F\,\frac{\tan\alpha + \muG}{1 - \muG\,\tan\alpha}

However, the ramp is very gentle ( $\alpha$ is about 3°) and $\muG$ is around 0.15, so the product $\muG\,\tan\alpha$ is tiny and the denominator is almost 1. So we can simplify it like this:

H \approx F\left(\frac{P}{\pi\,d_2} + \muG\right)

One more effect is missing: the metric thread has its flank tilted 30° from the axis (60° between flanks). It is not a straight ramp, it is a wedge. So the real normal force $N$ grows by a factor $1 / \cos 30^\circ = 1.155$ , and the friction with it. It is enough to replace $\muG$ with $1.155\,\muG$ :

H = F\left(\frac{P}{\pi\,d_2} + 1.155\,\muG\right)

A torque is a tangential force times the radius it acts at, and $H$ acts at the mean radius $d_2 / 2$ . So the torque in the thread is:

\MG = H\,\frac{d_2}{2} = F\,\frac{d_2}{2}\left(\frac{P}{\pi\,d_2} + 1.155\,\muG\right) \approx F\left(0.16\,P + 0.58\,d_2\,\muG\right)

where

$\MG$: Torque in the thread: what has to be applied to the screw to turn it up the thread

Let us look at how it comes out: the preload $F$ times two coefficients, which are the two effects we have seen.

$0.16\,P$ : the slope. It is the torque it costs to climb the ramp without friction and, so, the one that really stretches the screw. It is tiny, a fraction of the pitch.

$0.58\,d_2\,\muG$ : the friction of the flanks. It is the torque lost overcoming it: it stretches nothing, and grows with the coefficient $\muG$ and with the diameter $d_2$ .

The torque at the head

Besides the torque in the thread, there is another one to account for: the one produced by the friction of the head turning against the bearing surface.

No equilibrium is needed here: the normal force is simply the preload $F$ , so the friction force is $\muK\,F$ . This force acts at the mean diameter of the bearing ring, $\DKm$ (the average of the bearing diameter and that of the hole), that is, at a radius $\DKm / 2$ . Its torque is, as before, force times radius:

\DKm = \frac{\dw + \dh}{2} \qquad \MK = F\,\frac{\DKm}{2}\,\muK

where

$\muK$: Friction coefficient between the head and the part
$\dw$: Bearing diameter of the head
$\dh$: Diameter of the clearance hole
$\DKm$: Mean diameter of the bearing ring
$\MK$: Torque under the head

The torque needed

Now it only remains to add the torque of the thread, $\MG$ , and that of the head, $\MK$ .

Key formula

\MA = \MG + \MK = F \left(0.16\,P + 0.58\,d_2\,\muG + \frac{\DKm}{2}\,\muK\right)

This is the formula that gives the torque, and now we know where each term comes from: $0.16\,P$ is the one that stretches the screw, $0.58\,d_2\,\muG$ is the friction in the thread and $\DKm / 2 \cdot \muK$ is that of the head.

With usual frictions, only about a tenth of the torque stretches the screw; the rest is lost to friction. That is why lubrication changes so much: with the same torque, less friction leaves more torque for stretching and the preload goes up.

Stretches the screw≈ 10 %
In the thread≈ 40 %
Under the head≈ 50 %

Where the tightening torque goes. The percentages are indicative; the calculator gives those of each case.

The K factor shortcut

In some tables and calculators, a much more compact version of the torque equation we have seen is used:

\MA = K\,F\,d

where

$K$: Torque factor
$d$: Nominal diameter of the screw

That factor $K$ comes from dividing the tightening torque by $F\,d$ :

K = \frac{\MA}{F\,d} = 0.16\,\frac{P}{d} + 0.58\,\frac{d_2}{d}\,\muG + \frac{\DKm}{2\,d}\,\muK

The ratios $P/d$ , $d_2/d$ and $\DKm/d$ change little from one size to another, so $K$ depends mostly on the frictions. That is why a single value, near 0.2, works for dry or lightly oiled screws and is far off in other conditions.

In this calculator we do not use $K$ to calculate the torque. We think that, while $K$ can be useful for quick manual calculations, it is not worth using when the full model is at hand.

However, we give it as a result, so you can compare it with other tables and calculators.

How far it can be tightened

A screw takes what its section takes. The part that really works is called the stress area, and it is tabulated for each thread.

The tensile stress is the preload spread over that area:

\sigma = \frac{F}{\As}

where

$\sigma$: Tensile stress in the screw
$\As$: Stress area of the screw

The material is given by the property class. In steels, the first number of “8.8” times 100 is the tensile strength $\Rm$ in MPa (800), and the product of the two numbers times 10 is the yield strength $\Rp$ (640 MPa): the stress beyond which the screw no longer returns to its shape.

The name comes from how it is measured: R is strength, p permanent deformation, and 0.2 the 0.2 % of elongation left after unloading the screw.

Tightening right up to the yield strength would be a gamble: no margin would be left for the working load or the scatter of the tightening. VDI 2230 proposes using only a part $\nu$ of it, 90 % in its tables:

\sigma \le \nu\,\Rp

where

$\nu$: Fraction of the yield strength that is used
$\Rp$: Yield strength of the screw

But something is still missing: when tightened, the screw is not only stretched.

Stretching and twisting at once

The torque in the thread, $\MG$ , does not only stretch the screw: it also twists it.

This torque produces a shear stress $\tau$ in the section. For a stress area of diameter $d_0$ and an elastic material, it would be $\tau = 16\,\MG / (\pi\,d_0^3)$ . VDI 2230 lowers it to three quarters, because with the screw at its limit the material starts to yield and the twist is shared better. Divided by the tensile stress $\sigma = F / \As$ , the factor for the elastic case is 2; with that reduction, 1.5:

\frac{\tau}{\sigma} = 1.5\,\frac{d_2}{d_0}\left(\frac{P}{\pi\,d_2} + 1.155\,\muG\right)

The bracket is $H / F$ , the push of the thread per unit of preload: the friction of the flanks not only costs torque, it also twists the screw.

where

$\tau$: Shear stress produced by the twist
$d_0$: Diameter of the circular section with the area of the stress area

Tension and shear act together, and combine into an equivalent stress, von Mises, which is the one compared with the yield strength:

\sred = \sqrt{\sigma^2 + 3\,\tau^2} = \sigma\,\sqrt{1 + 3\left(\frac{\tau}{\sigma}\right)^2}

where

$\sred$: Equivalent von Mises stress

If we ask it not to exceed $\nu\,\Rp$ and solve for the tension, we get the permitted one. Times the stress area, it is the maximum preload:

Key formula

\sMzul = \frac{\nu\,\Rp}{\sqrt{1 + 3\left(\frac{\tau}{\sigma}\right)^2}} \qquad \FMzul = \sMzul\,\As

where

$\sMzul$: Maximum permitted tension when tightening
$\FMzul$: Maximum permitted preload

The more friction in the thread, the more twist and the less tension fits: the permitted preload goes down.

The wrench is not perfect

Everything above assumes we know the frictions and the wrench hits its mark. Neither is quite true: with the same torque, two identical screws end up with different preloads, mostly because the friction varies from one to the other.

The tightening factor says how far apart the largest and the smallest possible preload are. The more precise the method, the closer to 1 it is:

\alphaA = \frac{\FMmax}{\FMmin}

where

$\alphaA$: Tightening factor
$\FMmax$: Largest preload that can result
$\FMmin$: Smallest preload that can result

We aim for the largest to be the permitted one, so as never to overshoot. The smallest is then:

Key formula

\FMmin = \frac{\FMzul}{\alphaA}

With $\alphaA = 2$ , the preload can end up at half of the calculated one. That is why the joint must work with the smallest, not the largest.

So the parts are not crushed

The whole preload rests on the ring under the head, between the bearing diameter and the hole. Its area is that of the large circle minus that of the hole:

\Ap = \frac{\pi}{4}\left(\dw^2 - \dh^2\right)

where

$\Ap$: Area of the bearing ring

The pressure on the part is the preload over that area, and must not exceed what the material takes:

p = \frac{F}{\Ap} \le \pG

where

$p$: Pressure on the part under the head
$\pG$: Limit pressure the material takes

If it does, the head sinks into the part and the joint loses preload. A washer enlarges the bearing, but this calculator does not count that and is therefore conservative (see “Assumptions and limitations”).

The loads of the property class

These are loads of the screw alone, not of the joint: the stress area times the stress of each level. In steel, the proof load uses the proof stress of the property class; in stainless steel, the yield strength.

\Fp = \As\,\Sp \qquad \Fpz = \As\,\Rp \qquad \Fm = \As\,\Rm

where

$\Fp$: Proof load
$\Fpz$: Load at the yield strength
$\Fm$: Breaking load
$\Sp$: Proof stress of the property class

So far, the tightening. What happens when the joint works, with external loads, the stiffness of the parts or fatigue, is left out. What this calculator takes for granted is in “Assumptions and limitations”.

References

The parts

ISO 4017:2022 Fasteners — Hexagon head screws — Product grades A and B
ISO 4762:2004 Hexagon socket head cap screws
ISO 4032:2012 Hexagon regular nuts (style 1) — Product grades A and B
ISO 7089:2000 Plain washers — Normal series — Product grade A
ISO 7092:2000 Plain washers — Small series — Product grade A
ISO 7093-1:2000 Plain washers — Large series — Part 1: Product grade A

The method and the holes

VDI 2230 Part 1:2015 Systematic calculation of highly stressed bolted joints — Joints with one cylindrical bolt
ISO 898-1:2013 Mechanical properties of fasteners made of carbon steel and alloy steel — Part 1: Bolts, screws and studs
ISO 3506-1:2020 Fasteners — Mechanical properties of corrosion-resistant stainless steel fasteners — Part 1: Bolts, screws and studs
ISO 273:1979 Fasteners — Clearance holes for bolts and screws
ISO 68-1:1998 ISO general purpose screw threads — Basic profile — Part 1: Metric screw threads
ISO 262:1998 ISO general purpose metric screw threads — Selected sizes for screws, bolts and nuts

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