Calculating Wrench Rotation Angle for Specific Bolt Tightening Force: Theoretical Considerations

Is it possible to calculate the angle by which the key should be turned so that the bolt is tightened with the appropriate force? It seems so, because the same bolt, in the same place, will always have a constant angle corresponding to some force. Unless, of course, the thread is not damaged, because then, of course, the screw will no longer have the factory properties.

How is it calculated? What data do you need for this? Maybe there are just some calculators for this?

I know that's what torque wrenches are for, it's just a purely theoretical question.

Such calculations are used, for example, when tightening the head at the last stage. Of course, without a torque wrench, it will not be possible, and with the wrench itself, it will not be possible, because the greater the tightening force, the greater the friction on the thread and between the head and the material. This friction can be different for e.g. 10 head bolts. Therefore, the last stage of tightening is performed at a given angle and in practice it looks like that some bolts go relatively light and e.g. the bolt next to it goes hard. If you used a torque wrench for this, the first bolt would work properly, but the second bolt might not even twitch. Just the angular tightening might make sense under ideal conditions, but how would you know where to start counting the angle?

When the bolt begins to resist tightening by hand
Of course, we are talking about a fairly clean thread.
Either way, ideally there must be an angle suitable for the given force. How to calculate it? I am very curious about it, besides, it can be useful for emergency tightening when the torque wrench is not available.

I really don't know what you're up to and why you're making your life difficult
It is difficult to create ideal conditions for all sockets in the head and the screws for them
That is why someone came up with a torque wrench and manufacturers of seals additionally recommend angular tightening so that the assembly process is performed as precisely and evenly as possible, e.g. head
If you want so much, maybe do it by observation, tighten the screws with different strength and keep notes on the observation how many degrees the key will turn at 10, 20, 30 Nm and so on and you will have your pattern voluntarily speaking
But in my opinion there will be discrepancies with different thread diameters, but maybe I am wrong

Well, yes and no.
The screw tightening torque is the sum of the two friction torques: friction on the threads and friction of the screw head against the bearing surface. Thread friction:

Mt = 0.5 * ds * F * tg (? + ? ')

where ds - mean thread diameter, F - tensile force of the bolt, ? - helix angle of the bolt and ? 'is the apparent friction angle in the thread. how to calculate it, I can give you a formula later if you are interested, but you probably won't.

Now the friction on the shoulder surface - and here is the problem, because it depends on what surface. It is different for the surface of the ring washer, different for the circular washer, and different for the conical washer. In addition, various sources provide different information, but e.g. for a ring washer or a collar of a nut, the formula looks like this:

Mt = 0.25 * (Dp + dp) * F * u

where Dp - outer diameter of the washer, dp - internal, F - tensile force of the bolt, and u is, of course, the friction coefficient.

Now, knowing these two formulas, you can determine the tensile force F from them. And this force results directly from stretching the bolt while tightening - because, for example, one turn of a bolt by 360 degrees means stretching it by the length of one thread pitch. And now, if you know this force F, then from Hook's law you determine the extension of the bolt ?l:

?l = (l * F) / (S * E)

Where l is the original length of the bolt, F is the force you determined it a moment ago, S is the cross-sectional area of the bolt - but on the inside diameter, i.e. measured at the bottom of the thread - and E is the Young's modulus for the bolt material. And once you know ?l, you know that you will calculate it from the thread pitch P and the rotation angle you are asking for.

? = ?l * 360 ° / P

What's the catch in all of this? Well, all these diameters, angles and, above all, the coefficients of friction are average and approximate values. We never really know how much force was put into the friction. Lubricate one bolt with oil more, another less, sometimes some dirt will come up, etc. And the more you tighten, the greater the discrepancies.

marszałekkom wrote:

What's the catch in all of this? Well, all these diameters, angles and, above all, the coefficients of friction are average and approximate values. We never really know how much force was put into the friction. Lubricate one bolt with oil more, another less, sometimes some dirt will come up, etc. And the more you tighten, the greater the discrepancies.

I've been waiting for that ending .
All formulas and calculations, not only average, but also prove themselves in laboratory conditions. Here, even the temperature matters. Such an exemplary pad may be of different materials or covered with different materials. The screws may not have a perfect thread as well. Some may be "tighter" others not, and then we have different friction surfaces. It could be like that for a long time

No, it's not just such fairy tales. It is from these dependencies that various diameters, thread lengths, tightening torques, etc. are calculated in the construction of machines. Only that in factories with factory-new elements, the conditions are more similar to theoretical ones. And of course, the fact that the washer is made of a different material does not matter, you just need to have tables with friction coefficients for different material configurations.

@ marshals and could you show a simple example based on these formulas? And what are the constants 0.5 and 0.25?

If I had to decode the patterns while tightening the head and feel the customer's breath on my back ... I would rather buy a dynamometer (it's not a big cost), use the vehicle manufacturer's instructions and that's it.

szekel wrote:

@ marshals and could you show a simple example based on these formulas? And what are the constants 0.5 and 0.25?

For example, tighten an M10 bolt with a length of l = 100 mm using a torque of 100 Nm. Let us assume that the bolt presses directly with its head against the friction surface, the dimensions of the friction surface for a standard M10 bolt with a key 17: Dp = 17 mm, dp = 11.2 mm. We take the coefficient of friction 0.1 and the Young's modulus 210 GPa.

Such a screw has a pitch P = 1.5 mm and an average thread diameter ds = 9.026 mm. The tangent of the helix angle is:

tg? = P / (? * ds) = 0.0529

hence

? = 3.025 °

Apparent friction angle tangent:

tg ? '= u / cos30 °

30 °, it follows that it is half the angle of the metric thread, such a surface is affected by the frictional force in the thread.

tg ? '= 0.1155
? '= 6.589 °

Having these two slightly abstract parameters, we transform the formula into the moments of friction so as to determine the tension force for the bolt. It's a bit hard to write formulas here so I'll give you the result right away:

F = 68,041 N

A standard M10 bolt has a cross-sectional area of S = 52.266 mm?. E = 210 GPa, while the length l = 100 mm, let's assume that this is only the length of the tensile part. From Hook's law:

?l = (l * F) / (S * E)
?l = 0.62 mm

So if:

? = ?l * 360 ° / P

then for ?l = 0.62 mm:

? = 148.8 °

So you only need to turn the screw almost 150 degrees. Except that it is 150 ° of pure stretch, i.e. when the slack is removed, the dirt is out, etc. Also note that we stretch 100 mm of the bolt by only 0.6 mm, and such a small stretch causes a force of over 68 kN, which is almost 7 tons . In fact, this force is so great that even a 10.9 bolt would not hold out, I should have taken the M12 as an example, but I did not want to count again.
The constant 0.5 means half, because the torque is the force multiplied by the radius on which the force acts, and the radius is half the diameter. In this case, the diameter is ds, and half of ds is 0.5 ds. However, 0.25 is half of a half, which results from the calculation of the mean radius of the washer, which is half the distance between the outer diameter Dp and the inner dp, and the radius is half the diameter .

Oh, maybe I made a mistake, I learned it a long time ago, and if there is any constructor here, let him check me.

I admire you felt like it, but this is valuable information for the inquisitive .
Coming back to the head for a moment. "Today's" screws have a special design. I do not know, I did not go into detail, but they cannot be used a second time because they are stretched "screw flows" during tightening, and reusing them may break.

It's not just today, they all have it. Even in Sam, I repair a Mercedes 190, a car from the early 90s, it is written to measure the bolts before assembly and if they exceed a certain length, replace it. The point is, you stretch an element within acceptable limits, within an elastic range, and then expose it to a high temperature, but still much lower than the softening point of the material. And although there is neither high stress nor high temperature, after a few years in such conditions, the elastic stress disappears - the bolt ceases to tighten the screwed elements. Therefore, among others the gaskets under the head let go.

All calculations are cool and true, but there is an assumption that twisted materials do not deform. Unfortunately, this is not the case.

"Mechanics" from pegs turn large engines without a torque wrench and somehow the Ursus and Zetors worked.
It was worse when a Perkins engine hit their paws. In English instructions, they give two different head tightening torques for one motor, depending on the material from which the bolts are made. Overall bolts are identical, only the material is different and the problem is how to recognize what material these bolts are made of.
Tightening the head with a torque wrench makes the bolts that are tightened last on the circumference compress the head more tightly than those that were tightened first. As someone understands it, then the bolt tension is evened out, especially on the soft head gaskets.

I always have freaks when tightening the head screws. Especially the long threads, although I dip them in oil, I give the YATO key for legalization every year. I tighten the bolts in sequence, about four times, and you will still get one of two that it goes slightly easier, as if the thread is loosening.

I bought my first torque wrench when I had to repair a Polonaise with a 1.4 Rover engine. Plastic construction, long head bolts. I twisted one, it puffed like a carrot, and then I understood what tightening torques were. Previously, it was probably a good chunk (felt) and it was spinning. You can feel when the screw springs up and the thread no longer pulls. The old engines were much more tolerant.