You need to read the bit on the previous page about Linear Superposition:

The first issue to address is that of *linear
superposition*. What this means is that I'm not analysing all
the stresses in the wheel. I know that all the spokes start out with
a uniform tension, and I don't really care about it. For the purposes
of the analysis I simply ignore it, and it goes away!

Therefore, when the analysis shows a force in a spoke,
the *real* force in the spoke is whatever the preload (the
initial tension) was, plus the force calculated. If the force was
tension, we end up with a more highly stressed spoke. If the
calculated load was compression we end up with a less tensile spoke.
That is, a reference to a 'compressive' spoke could be read as a 'less
tensile' spoke. To get the true state in the wheel you need to
superimpose (ie, add) the results of this analysis on the initial state.

None of this affects the analysis. If we ignore buckling (or restrain against it), a spoke under tension which is subject to (say) 100N less tension contracts just as much as an unstressed spoke subject to 100N compression (or for that matter a compressive spoke subject to 100N more compression). In the analysis I simply don't tell the computer to let anything buckle, so I can analyse a spoke with compression without worrying that it's really a spoke that started out with a tension and now has less tension.

That is to say, in all these analyses, the initial preload
state of the wheel is neglected,
ignored. The initial tensile load in every spoke and compression load in
the rim is not included. The true stress state in a spoke is not the load
these calculations show divided by the cross-section - it's the load these
calculations show *added to the initial load* divided by the cross
section. All references in the text to a 'compressive spoke' mean
a 'spoke which is now stressed to a lower tension than it was when teh wheel
is not externally loaded'. It's just a shorthand, becasue that's a bit of
a mouthful.

Try this analogy:

What is fourteen million and twenty-seven minus six (14,000,027 - 6)?

If you have any sense for numbers, you didn't just do any arithmetic operations on 8-digit numbers, you identified that the 14 million could be set aside (neglected, ignored) and did twenty-seven minus six in your head. At the end, you put the 14 million back on, but it didn't really affect the processing you did.

So, to develop the analogy, suppose you have five boxes of widgets. All boxes start identical, and each box contains 14,000,027 widgets. There is a uniform and constant distribution of widgets.

Now, you do something, the effect of which is as follows:

- Remove 5 widgets from box 'A'
- Remove 12 widgets from box 'B'
- Remove 6 widgets from box 'C'
- Add 5 widgets to box 'D'
- Add 8 widgets to box 'E'

Now, you can actually examine widget numbers quite usefully without doing 8-digit arithmetic, without even putting it back on at the end:

- Which box has most widgets?
- Which box has least widgets?
- Which box had the greatest change in number of widgets?
- Was the greatest change an increase or decrease?
- Did more boxes increase or decrease in contents?
- How many widgets are 'spare' (or left over) after the operations?
- What is the average change in widget numbers in a box?
- What is the change in total number of widgets in the five boxes?

All those questions can be answered without 8-digit arithmetic, by using what is effectively linear superposition.

In particular, that remove / add process can be carried out with real, physical widgets without violating any laws of physics, even though when you do the maths you might make use of the concept of a negative widget (the change in box A is five negative widgets). You don't need any actual ant-matter widgets, even though the maths has some 'negative widgets'.

None of these compressive loads exceed the preload in the spoke. Every spoke in the wheel, in all these analyses, stays tensile at all times.

Spokes in tension don't buckle.

If you're now worrying about standing on something in tension, you need to
read the semantics bit on the main page.

back to the main wheel analysis page

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To comment on anything (please do) email ian@astounding.org.uk