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It is an ancient Mariner,
And he stoppeth one of three.
'By thy long gray beard and glittering eye,
Now wherefore stoppst thou me?
'The Bridegroom's doors are opened wide,
And I am next of kin;
The guests are met, the feast is set:
May'st hear the merry din.'
He holds him with his skinny hand,
'There was a ship,' quoth he.
'Hold off! unhand me, graybeard loon!'
Eftsoons his hand dropt he.
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Continuing the Ancient Mariner's tale of the concrete prism
stressed by the steel prism......
[Think of this as a serial in parts, like the Sherlock Holmes
stories in the nineteenth century Strand Magazine.
Appropriately enough, my director with whom I had a grand
falling out was named Dr Watson <g> ]
The next job is to get rid of the testing machine and to
have a closed system which consists merely of the steel
block load cell (stress = alias for strain) and the
concrete prism (strain). In this case the load cell will
be in tension and the concrete prism in balancing compression.
So we can look at this either as the steel stressing the
concrete in compression or the concrete stressing the
steel in tension.
Over the "elastic" range we can write,
Ps = k1.dLc/Lc or Pc = k2.dLs/Ls
Better still, and far more revealing we can write
k1.dLc/Lc = k2.dLs/Ls
In short we can forget about stress altogether and see
this as a relationship between strains.
Now the beauty of strain as a variable is that it is
DOMOMETRIC variable. I'm not fond of neologisms myself
but in this case I don't know an existing word which conveys
the meaning I want, so I had better define domometric variable.
Domo is Latin for home or house, so a domometric variable
is a variable where the metric, the measuring scale in
other words, is an in-house scale, i.e. one which relates
to a characteristic measure of the object under consideration.
In this case the metric is the length of the specimen.
This is an analogous situation to that pertaining to
temperature and the three water vapour power phases.
In that case the domometric variables were temperature.
The huge advantage of the domometric variable of strain
is that it can be measured with any other metric one cares
to think of - feet, centimeters, sachine, verst, vara or
light years. It doesn't matter what foreign scale one uses,
the value for strain will always be the same. Put another
way, strain is dimensionless.
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A small digression. When I was a young Scientific Officer
I was having a conversation with Dr Randall Wood on the
subject of dimensional analysis. He said, when he was
working at NPL. there was a chap who has the knack of
solving the most amazingly intractable problems in heat
transfer, etc. by using dimensional analysis techniques.
I am confident that the reason he was so successful is
that dimensional analysis implicitly bypasses the jerry
built conceptual variables, like FORCE, etc., of
traditional physics.
As Clayton remarks in a recent e-mail where he comments
on Buchanan's UBIQUITY
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>He observes that "the world is simpler
>than it seems" (p72). As you always
>said, it is just that we look at it in
>the wrong terms.
>Nigel
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If I did say that I think I must have pinched the
phrase from Stafford Beer.
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BALANCING TENSILE AND COMPRESSIVE STRAINS.
Now let us balance the tensile and compressive strains
by arranging the load cell as the tensile flange of a
beam and the concrete as the compressive flange of a beam.
Let compressive strain be positive,
i.e. strain energy input to the material.
Thus the concrete has positive strain energy relative
to ambient Beta-atmosphere pressure.
Let tensile strain be negative,
i.e. strain energy output from the material.
Thus the steel has negative strain energy relative
to ambient Beta-atmosphere pressure
Now when we load such a beam two strain energy changes
take place.
The steel stretches.
Its negative strain energy is increased arithmetically
but decreased algebraically.
The concrete shrinks.
Its positive strain energy is increased arithmetically,
and also increased algebraically.
If the stiffnesses of the two flanges is virtually the
same (in other words when subjected to an external
standard strain, a one percent change in length of the
standard metre, say, they deform the same amount) then
the strain energy input needed by the concrete will be
provided almost wholly by the strain energy output of
the steel.
I don't know if any Vortexians have ever walked across
a small prestressed concrete bridge, the single beam
type that is often used to span rural streams. But if
they have they might have noticed how flexible the beams
were. Indeed my Screwtape devil often tempts me to
resonate the beam to destruction by jumping up and down.
Fortunately, my guardian angel manages to talk me out
of it <g>.
Though he didn't realise it, by inventing prestressed
concrete Eugène Fressinet provided us with a synthetic
model for understanding the behaviour of all materials
whatsoever. For all materials can be partitioned into
two phases. A phase in a state of negative strain
relative to its ambient strain outside the material,
and a phase in a state of positive strain relative to
its strain outside the ambient material.
But what about all those bonds and thingees? Forget
'em. As Buchanan so correctly observes on page 72 of
UBIQUITY,
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All in all, science took nearly four
centuries to consume the iron magnet.
Yet in unravelling the mystery physicists
learned a profound lesson: the world is
simpler than it seems. And that when it
comes to understanding some things the
details most certainly do not matter.
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For a structural engineer these two phases are best
thought of as struts (the positive strain phase) and
ties (the negative strain phase).
As a materials scientist I find the best mnemonic for
the two phases is to see one as a quasi solid and the
other as a quasi fluid. Each phase in turn may be
partitioned into quasi-solid and quasi-fluid phases,
and this process may be carried on indefinitely (yes,
its elephants all the way down) giving us a binary
hierarchy of strains. These iterative hierarchical
strains provide us with a binary conceptual structure
to replace what one might describe as the imperial
conceptual structure of second millennium physics,
the analogue of the imperial weights and measures
structure of our forebears.
If a century ago I had prophesied the in 100 years
the most important machine on earth would simply be
taking in and pumping out long strings of ones and
zeros I'm sure I would have got some very funny
looks, so I don't really expect any but a tiny
minority to take my Iterative Hierarchical Mechanics
manifesto any more seriously than early 20th century
man would have taken the proposal for a binary digital
computer. Fortunately Parkie doesn't have the power to
close down Vortex and to ship me off to Siberia in a
closed train. 8-)
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As I was going to St. Ives,
I met a man with seven wives.
Each wife had seven sacks,
Each sack had seven cats,
Each cat had seven kits.
Kits, cats, sacks, and wives,
..........................
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Since my grandchildren have no difficulty in grasping
the nature of a septenary system, Vortexians should
have no problem with binary. 8-)
And if I don't stop there, I wont get any tea.
Cheers
Grimer
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Who is she that cometh forth as the morning rising
Fair as the moon
Bright as the sun
Terrible as an army set in battle array
- King Solomon -
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