1903 bicycle wheel FEA

Carl Fogel said:

It turns out that the FEA for a bicycle wheel was done before 1903:

Quoted message said:

[After an impressive page of equations for an 8-spoke bicycle wheel]

Quoted message said:

"As a particular instance let W [weight] = 1000 pounds... Then the
stress for all the spokes except [lowermost spoke] is 76 pounds
tension, and that in [lowermost spoke] is 924 pounds compression;
also the stress in each segment of the rim is about 98 pounds
compression. It thus appears that the vertical spoke under the hub
carries nearly all the load as a compressive stress, and that as the
wheel slightly turns this is turned into a small tensile stress."

Quoted message said:

--"A Text Book on Roofs and Bridges," 5th ed., 1903, Merriman & Jacoby
p. 149-150

http://tinyurl.com/yrqx6d

Unfortunately this escaped notice and the wire spoked wheel escaped
mention in this work. I didn't understand how the author entered the
structural parameters of the rim and spokes, but that is less
important than the conclusions... that did not carry over into the
world of bicycling.

Jobst Brandt

Carl Fogel said:

It turns out that the FEA for a bicycle wheel was done before 1903:

Quoted message said:

[After an impressive page of equations for an 8-spoke bicycle wheel]

Quoted message said:

"As a particular instance let W [weight] = 1000 pounds... Then the
stress for all the spokes except [lowermost spoke] is 76 pounds
tension, and that in [lowermost spoke] is 924 pounds compression;
also the stress in each segment of the rim is about 98 pounds
compression. It thus appears that the vertical spoke under the hub
carries nearly all the load as a compressive stress, and that as the
wheel slightly turns this is turned into a small tensile stress."

Quoted message said:

--"A Text Book on Roofs and Bridges," 5th ed., 1903, Merriman & Jacoby
p. 149-150

http://tinyurl.com/yrqx6d

Unfortunately this escaped notice and the wire spoked wheel escaped
mention in this work. I didn't understand how the author entered the
structural parameters of the rim and spokes, but that is less
important than the conclusions... that did not carry over into the
world of bicycling.

Jobst Brandt

Quoted message said:

Carl Fogel said:

It turns out that the FEA for a bicycle wheel was done before 1903:

Quoted message said:

[After an impressive page of equations for an 8-spoke bicycle wheel]

Quoted message said:

"As a particular instance let W [weight] = 1000 pounds... Then the
stress for all the spokes except [lowermost spoke] is 76 pounds
tension, and that in [lowermost spoke] is 924 pounds compression;
also the stress in each segment of the rim is about 98 pounds
compression. It thus appears that the vertical spoke under the hub
carries nearly all the load as a compressive stress, and that as the
wheel slightly turns this is turned into a small tensile stress."

Quoted message said:

--"A Text Book on Roofs and Bridges," 5th ed., 1903, Merriman & Jacoby
p. 149-150

http://tinyurl.com/yrqx6d

Unfortunately this escaped notice and the wire spoked wheel escaped
mention in this work. I didn't understand how the author entered the
structural parameters of the rim and spokes, but that is less
important than the conclusions... that did not carry over into the
world of bicycling.

Jobst Brandt

Dear Jobst,

I'd be astonished if anyone could have dug it up back in the
pre-Google era. You'd have to read every engineering book back to the
1890's to find that 2-page example--and it's in a book about roofs and
bridges, not about bicycles.

I stumbled over it while searching for references to tying and
soldering and thought damn, but that looks like somebody doing an FEA
for 8 spokes by hand instead of with a computer!

I didn't follow the equations at all--it may not even be an FEA in the
modern sense. But the conclusion and example tensions sure look like
they agree with "The Bicycle Wheel."

And that final sentence is awfully good: "It thus appears that the
vertical spoke under the hub carries nearly all the load as a
compressive stress, and that as the wheel slightly turns this is
turned into a small tensile stress."

Cheers,

Carl Fogel

Quoted message said:

Carl Fogel said:

It turns out that the FEA for a bicycle wheel was done before 1903:

Quoted message said:

[After an impressive page of equations for an 8-spoke bicycle wheel]

Quoted message said:

"As a particular instance let W [weight] = 1000 pounds... Then the
stress for all the spokes except [lowermost spoke] is 76 pounds
tension, and that in [lowermost spoke] is 924 pounds compression;
also the stress in each segment of the rim is about 98 pounds
compression. It thus appears that the vertical spoke under the hub
carries nearly all the load as a compressive stress, and that as the
wheel slightly turns this is turned into a small tensile stress."

Quoted message said:

--"A Text Book on Roofs and Bridges," 5th ed., 1903, Merriman & Jacoby
p. 149-150

http://tinyurl.com/yrqx6d

Unfortunately this escaped notice and the wire spoked wheel escaped
mention in this work. I didn't understand how the author entered the
structural parameters of the rim and spokes, but that is less
important than the conclusions... that did not carry over into the
world of bicycling.

Jobst Brandt

Dear Jobst,

I'd be astonished if anyone could have dug it up back in the
pre-Google era. You'd have to read every engineering book back to the
1890's to find that 2-page example--and it's in a book about roofs and
bridges, not about bicycles.

I stumbled over it while searching for references to tying and
soldering and thought damn, but that looks like somebody doing an FEA
for 8 spokes by hand instead of with a computer!

I didn't follow the equations at all--it may not even be an FEA in the
modern sense. But the conclusion and example tensions sure look like
they agree with "The Bicycle Wheel."

And that final sentence is awfully good: "It thus appears that the
vertical spoke under the hub carries nearly all the load as a
compressive stress, and that as the wheel slightly turns this is
turned into a small tensile stress."

Cheers,

Carl Fogel

Quoted message said:

It turns out that the FEA for a bicycle wheel was done before 1903:

[After an impressive page of equations for an 8-spoke bicycle wheel]

"As a particular instance let W [weight] = 1000 pounds . . . Then the
stress for all the spokes except [lowermost spoke] is 76 pounds
tension, and that in [lowermost spoke] is 924 pounds compression; also
the stress in each segment of the rim is about 98 pounds compression.
It thus appears that the vertical spoke under the hub carries nearly
all the load as a compressive stress, and that as the wheel slightly
turns this is turned into a small tensile stress."

--"A Text Book on Roofs and Bridges," 5th ed., 1903, Merriman & Jacoby
p. 149-150

http://books.google.com/books?id=kBM5AAAAMAAJ&pg=PA149&dq=%22bicycle+wheel+with+stiff+spokes%22&as_brr=1&ei=yMguR4_UIZ3epQKkr63vCQ#PPA149,M1
or http://tinyurl.com/yrqx6d

Cheers,

Carl Fogel

Here's a vastly more intimidating (and slightly earlier) set of
calculations about how loading a bicycle wheel changes spoke-tension:

http://books.google.com/books?id=4rc4AAAAMAAJ&pg=PA197&dq=%22we+assume+that+the+number+of+spokes%22&ei=bb8vR6uaMpSQsgP02rCpCQ
or http://tinyurl.com/2hgdl4

B.A. Smith's calculations run from page 197 to 203--unless you love
equations, skip to page 203.

Smith concludes that spokes on a normally tensioned and loaded wheel
will see only a slight tension increase in the spokes outside the
bottom 60 degree arc under the hub, but the five spokes under the hub
will lose tension:

"The practical result of the above investigation is that, with steel
rim and spokes of the usual proportions, if the initial tension is
about half the weight of the rider the spokes will always be in
tension, and the tension of spokes at a greater angle than about 30
degrees from the bottom is only very little increased above its
intitial value by the weight of the rider."

Here are the bottom 7 spokes, showing the tension changes indicated by
the table just above the conclusion. The initial spoke tension is 100
lbs and rises or falls to 0.[censored] * 200 lbs. (The other spokes all rise
slightly to 0.522 in the theoretical wheel.)

hub
spokes
/ | \
/ / / | \ \ \
0.500 0.500 0.500 0.500 0.500 0.500 0.500 initial tension
100.0 100.0 100.0 100.0 100.0 100.0 100.0 * 200 lbs

add 100-lb load to wheel

0.522 0.427 0.240 0.093 0.240 0.427 0.522 loaded tension
104.4 85.4 48.0 18.6 48.0 85.4 104.4 * 200 lbs

+4.4 -14.6 -52.0 -71.4 -52.0 -14.6 +4.4 lbs-change

In other words, Smith's theory matches "The Bicycle Wheel."

Again, Jobst could never have tracked this article down in the
pre-Google era. It's from the 8th annual report of the Australasian
Association for the Advancement of Science, which met at Melbourne,
Australia, in 1900.

Cheers,

Carl Fogel

Quoted message said:

It turns out that the FEA for a bicycle wheel was done before 1903:

[After an impressive page of equations for an 8-spoke bicycle wheel]

"As a particular instance let W [weight] = 1000 pounds . . . Then the
stress for all the spokes except [lowermost spoke] is 76 pounds
tension, and that in [lowermost spoke] is 924 pounds compression; also
the stress in each segment of the rim is about 98 pounds compression.
It thus appears that the vertical spoke under the hub carries nearly
all the load as a compressive stress, and that as the wheel slightly
turns this is turned into a small tensile stress."

--"A Text Book on Roofs and Bridges," 5th ed., 1903, Merriman & Jacoby
p. 149-150

http://books.google.com/books?id=kBM5AAAAMAAJ&pg=PA149&dq=%22bicycle+wheel+with+stiff+spokes%22&as_brr=1&ei=yMguR4_UIZ3epQKkr63vCQ#PPA149,M1
or http://tinyurl.com/yrqx6d

Cheers,

Carl Fogel

Here's a vastly more intimidating (and slightly earlier) set of
calculations about how loading a bicycle wheel changes spoke-tension:

http://books.google.com/books?id=4rc4AAAAMAAJ&pg=PA197&dq=%22we+assume+that+the+number+of+spokes%22&ei=bb8vR6uaMpSQsgP02rCpCQ
or http://tinyurl.com/2hgdl4

B.A. Smith's calculations run from page 197 to 203--unless you love
equations, skip to page 203.

Smith concludes that spokes on a normally tensioned and loaded wheel
will see only a slight tension increase in the spokes outside the
bottom 60 degree arc under the hub, but the five spokes under the hub
will lose tension:

"The practical result of the above investigation is that, with steel
rim and spokes of the usual proportions, if the initial tension is
about half the weight of the rider the spokes will always be in
tension, and the tension of spokes at a greater angle than about 30
degrees from the bottom is only very little increased above its
intitial value by the weight of the rider."

Here are the bottom 7 spokes, showing the tension changes indicated by
the table just above the conclusion. The initial spoke tension is 100
lbs and rises or falls to 0.[censored] * 200 lbs. (The other spokes all rise
slightly to 0.522 in the theoretical wheel.)

hub
spokes
/ | \
/ / / | \ \ \
0.500 0.500 0.500 0.500 0.500 0.500 0.500 initial tension
100.0 100.0 100.0 100.0 100.0 100.0 100.0 * 200 lbs

add 100-lb load to wheel

0.522 0.427 0.240 0.093 0.240 0.427 0.522 loaded tension
104.4 85.4 48.0 18.6 48.0 85.4 104.4 * 200 lbs

+4.4 -14.6 -52.0 -71.4 -52.0 -14.6 +4.4 lbs-change

In other words, Smith's theory matches "The Bicycle Wheel."

Again, Jobst could never have tracked this article down in the
pre-Google era. It's from the 8th annual report of the Australasian
Association for the Advancement of Science, which met at Melbourne,
Australia, in 1900.

Cheers,

Carl Fogel

Quoted message said:

Quoted message said:

Carl Fogel writes:

Quoted message said:

Quoted message said:

It turns out that the FEA for a bicycle wheel was done before 1903:

Quoted message said:

Quoted message said:

[After an impressive page of equations for an 8-spoke bicycle wheel]

Quoted message said:

Quoted message said:

"As a particular instance let W [weight] = 1000 pounds... Then the
stress for all the spokes except [lowermost spoke] is 76 pounds
tension, and that in [lowermost spoke] is 924 pounds compression;
also the stress in each segment of the rim is about 98 pounds
compression. It thus appears that the vertical spoke under the hub
carries nearly all the load as a compressive stress, and that as the
wheel slightly turns this is turned into a small tensile stress."

Quoted message said:

Quoted message said:

--"A Text Book on Roofs and Bridges," 5th ed., 1903, Merriman & Jacoby
p. 149-150

Quoted message said:

http://tinyurl.com/yrqx6d

Quoted message said:

Unfortunately this escaped notice and the wire spoked wheel escaped
mention in this work. I didn't understand how the author entered the
structural parameters of the rim and spokes, but that is less
important than the conclusions... that did not carry over into the
world of bicycling.

Quoted message said:

Jobst Brandt

Dear Jobst,

I'd be astonished if anyone could have dug it up back in the
pre-Google era. You'd have to read every engineering book back to the
1890's to find that 2-page example--and it's in a book about roofs and
bridges, not about bicycles.

I stumbled over it while searching for references to tying and
soldering and thought damn, but that looks like somebody doing an FEA
for 8 spokes by hand instead of with a computer!

I didn't follow the equations at all--it may not even be an FEA in the
modern sense. But the conclusion and example tensions sure look like
they agree with "The Bicycle Wheel."

And that final sentence is awfully good: "It thus appears that the
vertical spoke under the hub carries nearly all the load as a
compressive stress, and that as the wheel slightly turns this is
turned into a small tensile stress."

Cheers,

Carl Fogel

Seeing how it was a good ten years before Boris Galerkin proposed his
weighted residual method it wasn't an FEA in any sense, modern or
other.

Quoted message said:

Quoted message said:

Carl Fogel writes:

Quoted message said:

Quoted message said:

It turns out that the FEA for a bicycle wheel was done before 1903:

Quoted message said:

Quoted message said:

[After an impressive page of equations for an 8-spoke bicycle wheel]

Quoted message said:

Quoted message said:

"As a particular instance let W [weight] = 1000 pounds... Then the
stress for all the spokes except [lowermost spoke] is 76 pounds
tension, and that in [lowermost spoke] is 924 pounds compression;
also the stress in each segment of the rim is about 98 pounds
compression. It thus appears that the vertical spoke under the hub
carries nearly all the load as a compressive stress, and that as the
wheel slightly turns this is turned into a small tensile stress."

Quoted message said:

Quoted message said:

--"A Text Book on Roofs and Bridges," 5th ed., 1903, Merriman & Jacoby
p. 149-150

Quoted message said:

http://tinyurl.com/yrqx6d

Quoted message said:

Unfortunately this escaped notice and the wire spoked wheel escaped
mention in this work. I didn't understand how the author entered the
structural parameters of the rim and spokes, but that is less
important than the conclusions... that did not carry over into the
world of bicycling.

Quoted message said:

Jobst Brandt

Dear Jobst,

I'd be astonished if anyone could have dug it up back in the
pre-Google era. You'd have to read every engineering book back to the
1890's to find that 2-page example--and it's in a book about roofs and
bridges, not about bicycles.

I stumbled over it while searching for references to tying and
soldering and thought damn, but that looks like somebody doing an FEA
for 8 spokes by hand instead of with a computer!

I didn't follow the equations at all--it may not even be an FEA in the
modern sense. But the conclusion and example tensions sure look like
they agree with "The Bicycle Wheel."

And that final sentence is awfully good: "It thus appears that the
vertical spoke under the hub carries nearly all the load as a
compressive stress, and that as the wheel slightly turns this is
turned into a small tensile stress."

Cheers,

Carl Fogel

Seeing how it was a good ten years before Boris Galerkin proposed his
weighted residual method it wasn't an FEA in any sense, modern or
other.

Quoted message said:

Quoted message said:

On 05 Nov 2007 16:21:05 GMT, [email hidden] wrote:

Quoted message said:

Quoted message said:

Carl Fogel writes:

Quoted message said:

Quoted message said:

> It turns out that the FEA for a bicycle wheel was done before 1903:

Quoted message said:

Quoted message said:

> [After an impressive page of equations for an 8-spoke bicycle wheel]

Quoted message said:

Quoted message said:

> "As a particular instance let W [weight] = 1000 pounds... Then the
> stress for all the spokes except [lowermost spoke] is 76 pounds
> tension, and that in [lowermost spoke] is 924 pounds compression;
> also the stress in each segment of the rim is about 98 pounds
> compression. It thus appears that the vertical spoke under the hub
> carries nearly all the load as a compressive stress, and that as the
> wheel slightly turns this is turned into a small tensile stress."

Quoted message said:

Quoted message said:

> --"A Text Book on Roofs and Bridges," 5th ed., 1903, Merriman & Jacoby
> p. 149-150

Quoted message said:

Quoted message said:

http://tinyurl.com/yrqx6d

Quoted message said:

Quoted message said:

Unfortunately this escaped notice and the wire spoked wheel escaped
mention in this work. I didn't understand how the author entered the
structural parameters of the rim and spokes, but that is less
important than the conclusions... that did not carry over into the
world of bicycling.

Quoted message said:

Quoted message said:

Jobst Brandt

Quoted message said:

Dear Jobst,

Quoted message said:

I'd be astonished if anyone could have dug it up back in the
pre-Google era. You'd have to read every engineering book back to the
1890's to find that 2-page example--and it's in a book about roofs and
bridges, not about bicycles.

Quoted message said:

I stumbled over it while searching for references to tying and
soldering and thought damn, but that looks like somebody doing an FEA
for 8 spokes by hand instead of with a computer!

Quoted message said:

I didn't follow the equations at all--it may not even be an FEA in the
modern sense. But the conclusion and example tensions sure look like
they agree with "The Bicycle Wheel."

Quoted message said:

And that final sentence is awfully good: "It thus appears that the
vertical spoke under the hub carries nearly all the load as a
compressive stress, and that as the wheel slightly turns this is
turned into a small tensile stress."

Quoted message said:

Cheers,

Quoted message said:

Carl Fogel

Seeing how it was a good ten years before Boris Galerkin proposed his
weighted residual method it wasn't an FEA in any sense, modern or
other.

It's an discretization of a continuous problem, and does appear
to be a finite element method. I think your objection is relevant
in that the discretization is ad hoc, rather than derived in a way
that assures you get an efficiently linearized problem that converges
to the solution of the original continuous problem.

Ben

Quoted message said:

Quoted message said:

On 05 Nov 2007 16:21:05 GMT, [email hidden] wrote:

Quoted message said:

Quoted message said:

Carl Fogel writes:

Quoted message said:

Quoted message said:

> It turns out that the FEA for a bicycle wheel was done before 1903:

Quoted message said:

Quoted message said:

> [After an impressive page of equations for an 8-spoke bicycle wheel]

Quoted message said:

Quoted message said:

> "As a particular instance let W [weight] = 1000 pounds... Then the
> stress for all the spokes except [lowermost spoke] is 76 pounds
> tension, and that in [lowermost spoke] is 924 pounds compression;
> also the stress in each segment of the rim is about 98 pounds
> compression. It thus appears that the vertical spoke under the hub
> carries nearly all the load as a compressive stress, and that as the
> wheel slightly turns this is turned into a small tensile stress."

Quoted message said:

Quoted message said:

> --"A Text Book on Roofs and Bridges," 5th ed., 1903, Merriman & Jacoby
> p. 149-150

Quoted message said:

Quoted message said:

http://tinyurl.com/yrqx6d

Quoted message said:

Quoted message said:

Unfortunately this escaped notice and the wire spoked wheel escaped
mention in this work. I didn't understand how the author entered the
structural parameters of the rim and spokes, but that is less
important than the conclusions... that did not carry over into the
world of bicycling.

Quoted message said:

Quoted message said:

Jobst Brandt

Quoted message said:

Dear Jobst,

Quoted message said:

I'd be astonished if anyone could have dug it up back in the
pre-Google era. You'd have to read every engineering book back to the
1890's to find that 2-page example--and it's in a book about roofs and
bridges, not about bicycles.

Quoted message said:

I stumbled over it while searching for references to tying and
soldering and thought damn, but that looks like somebody doing an FEA
for 8 spokes by hand instead of with a computer!

Quoted message said:

I didn't follow the equations at all--it may not even be an FEA in the
modern sense. But the conclusion and example tensions sure look like
they agree with "The Bicycle Wheel."

Quoted message said:

And that final sentence is awfully good: "It thus appears that the
vertical spoke under the hub carries nearly all the load as a
compressive stress, and that as the wheel slightly turns this is
turned into a small tensile stress."

Quoted message said:

Cheers,

Quoted message said:

Carl Fogel

Seeing how it was a good ten years before Boris Galerkin proposed his
weighted residual method it wasn't an FEA in any sense, modern or
other.

It's an discretization of a continuous problem, and does appear
to be a finite element method. I think your objection is relevant
in that the discretization is ad hoc, rather than derived in a way
that assures you get an efficiently linearized problem that converges
to the solution of the original continuous problem.

Ben