# Torsion and tension in a steel bar

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HI there.

I've been doing some repairs on my car recently and I found myself wondering about my torque wrench.

It isn't a precision tool, but good enough for bolts in the chassis, or the wheel nuts.

Anyway, I was thinking about what happens to a steel bar if you fix it horizontally at one end and hang a weight at the other.

It bends down at the weighted end and this end is now at an angle to the horizontal. So what does this angle represent?

I know a torque is perpendicular to the applied force, but where do I go from there?

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On 2/13/2022 at 10:10 AM, SuperSlim said:

Anyway, I was thinking about what happens to a steel bar if you fix it horizontally at one end and hang a weight at the other.

It bends down at the weighted end and this end is now at an angle to the horizontal. So what does this angle represent?

It's called the 'angle of deflection' and for an ideal weightless cantilever is proportional to the end load times the square of the cantilever length.

Since this is a fixed displacement (the beam doesn't continue to move) mechanics in particular will tend to refer to the loading force as a moment rather than a torque. They are sort of the same thing, but torque is used more in the context of rotating bodies like crankshafts, and this can lead to confusion if the terms are applied haphazardly.

Useful Wikipedia lpages to browse through may be Deflection (engineering) and Euler-Bernoulli beam theory.

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On 2/15/2022 at 9:22 AM, sethoflagos said:

It's called the 'angle of deflection' and for an ideal weightless cantilever is proportional to the end load times the square of the cantilever length.

Ok thanks. What I'm actually trying to figure out is why a torque wrench has a graduated scale that measures this angle, in mass x distance units. Do I start with forces, or is it a derivative of the momentum? I realise that the time interval in the bending isn't in the result because it's time-independent and depends only on the mass of the weight and the 'tensility' of the bar.

I suppose I want to understand how a bending moment gives this kind of measurement of the torque.

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I might have worked it out.

If you attach a weight and let go it accelerates; now there are Newtons and a second derivative to deal with. Exclude this by not lettling the weight accelerate (duh!).

Lower the weight at more or less constant velocity, declare you in fact acheived this, and eliminate the time factor in the usual way with s = vt. What's left has units of kg.m.

In other words an integral over time of the momentum. With no acceleration I just "transfer" some to the bar and bend it.

Edited by SuperSlim
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13 hours ago, SuperSlim said:

Ok thanks. What I'm actually trying to figure out is why a torque wrench has a graduated scale that measures this angle, in mass x distance units. Do I start with forces, or is it a derivative of the momentum? I realise that the time interval in the bending isn't in the result because it's time-independent and depends only on the mass of the weight and the 'tensility' of the bar.

I suppose I want to understand how a bending moment gives this kind of measurement of the torque.

I'm more familiar with the sprung ball detent type wrench, so notwithstanding ... :

As stated in my previous post, deflection angle is proportional to the applied moment for a given wrench length, so providing you dont try to apply load from somewhere other than the end grip, the two should be linearly correlated.

I see no need to go any further than the static force balance here. Serious bolt tensioning is best done without significant dynamic loadings.

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14 hours ago, SuperSlim said:

I suppose I want to understand how a bending moment gives this kind of measurement of the torque.

It's called the area-moment method for the bending of cantilevers (which is what your type of torque wrench is)

3 minutes ago, sethoflagos said:

+1

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