h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

If you pay out the tape horizontally from a self-retracting reel-type tape measure, it's very stiff when it's one way up (with the dished profile in 'smile' orientation) and extends a long way before finally collapsing. The other way up, in the 'frown' orientation, it collapses almost immediately.

Anyone got a simple explanation for this?

Have you just been watching 'The Wrong Trousers', Pinn?

Err...no.
I watched it a long time ago, but I don't remember anything relevant. I seem to remember a penguin with a rubber glove on its head, but that's about it.

So what havw Wallace + Gromit got to say about steel tape measures?

Good question, Pinn. As an engineer, I should know the answer to this, but I don't. I'll have to do some experiments.

How does this sound:

The cross section of the tape might be imagined as contained within rectangle b(readth) x d(epth).

As you load it, b at the bottom (in compression) will increase, while b at the top (in tension) will decrease, i.e., the cross-section tends towards a trapezium.

If the cross-section is a smile, this has the effect of increasing the convexity of the cross-section, thereby increasing its second moment of inertia (and hence its resistance to bending).

If the cross-section is a frown, increasing b(bottom) and decreasing b(top) has the effect of decreasing the convexity of the cross-section (tending towards flat) and hence decreasing the second moment of inertia.

AB will describe it better I'm sure.

Who's AB?

U197657

TM : Post 4

"How does this sound:
The cross section of the tape might be imagined as contained within rectangle b(readth) x d(epth).
As you load it, b at the bottom (in compression) will increase, while b at the top (in tension) will decrease, i.e., the cross-section tends towards a trapezium.
If the cross-section is a smile, this has the effect of increasing the convexity of the cross-section, thereby increasing its second moment of inertia (and hence its resistance to bending).
If the cross-section is a frown, increasing b(bottom) and decreasing b(top) has the effect of decreasing the convexity of the cross-section (tending towards flat) and hence decreasing the second moment of inertia."

That's actually the wrong way round.

When the cross section forms a smile, you increase the breadth, which increases the tension across the tape, which, I assume, stiffens the arch, and acts to prevent deformation at 90 degrees (along the length of the tape).

As a frown, the curve becomes more pronounced, (increasing the convex shape), and reducing the tension across the apex of the curve, which reduces the resistance to lateral deformation, which is what causes the collapse of the arch.

Thanks so far, guys.

If 2nd Moments of Area are needed to understand it, so be it, but:
i) that's not quite as simple an explanation as I'd hoped for, and
ii) I don't think it's the whole story either.

Done any experiments, Gnomon? If so, you might have found that the longest reach with a horizontal pay-out comes from neither smile nor frown, but a dish-uppermost skew feed at about 30 degrees from vertical. If you pay out fully vertical (ie with one edge directly above the other), it just twists into a frown and then collapses. You get the longest run out if you turn it over far enough to force it to twist the other way, and finish up a smile.

I thought for a while that the explanation lay in this torsional instability of the tape, but it now looks like it's a secondary factor. What's more certain is that this is an elastic buckling problem, and that the critical plane is bending across the chord of the smile/frown due to the cantilevered load of the payed-out tape. I'm also sure that there's significance in the fact that it goes with a click (ie it undergoes an abrupt and elastic profile change, like a spring-steel catch does) when it finally collapses in the smile, but that the frown collapse has no click, just a smooth and progressive curl into oblivion. This click-difference must be something beyond mere 2nd MoAs, mustn't it?

Do you think we can take this far enough to write an EG Entry? OMG, this is even more exciting than dropping Slinkies or snapping spaghetti...

A. Can anyone here derive "I" for the four cross-sections:

(i) u
(ii) u tilted to 30 degrees from vertical
(iii) c
(iv) n

Just for some base data.

B. It would be interesting to see where the centroid of the cross-section sits. As the tape twists back to a smile when it is paid out with the profile 30 degrees from vertical, it's clearly most stable at rest as a U.

C. The tape is cold pressed into the U profile yes? For the U to fail, it needs to overcome the springiness that that cold pressing introduces. That happens catastrophically and comes with a clack.

D. Is the method of support is contributing to failure, i.e., a point load at the underside of the invert of the U at the point of cantilever is assisting the arch to flatten as per U1309606's proposal?

E. It would be good to get some empirical data from different tapes. Mass/lin. m & cross-sectional profile are variable, but to what extent do they influence the results?

It's all down to ' Section Modulus ' - Try it in Google.
This is about as simple as it gets.
http://toolboxes.flexiblelearning.net.au/demosites/series10/10_01/content/bcgbc4010a/03_properties/02_section_properties/page_007.htm

That link has one of the most confusing pictures I've ever come across.

It goes on about the section modulus Z and then gives a diagram containing lots of x and y dimensions

I was thinking about this on the way home from work last night, and it occured to me that the tape, when inside the reel is curled up, into the smile cross section, rather than down, into the frown shape, which would probably pre-tension the tape, giving it additional structural strength in the smile orientation, and weaken it in he frown.

Or is that all cobblers?

Sorry Orcus the 'Section Modulus' link was taken out of context.
The text being relevant. Try reading links on the left of the page.

Pinniped
There was a 'clicking' frog toy and dog training devices among other things based on the sound that sometimes comes from the 'collapsing' tape. I think that you may find that the sound that emanates from Rolf Harris's wobble board has some connection.

It's cobblers, ThatPrat.

The tape when rolled up is actually flattened so that its cross section is a straight line. So it is a flat tape rolled into a spiral.