Potential good news for Mt. Washington access. - Page 22 - Cycling Mob: Road Cycling, Mountain Biking and More

01-12-2005, 10:50 AM

nobody <[Only registered and activated users can see links. ]> writes:

> On Tue, 11 Jan 2005 06:37:07 GMT, Joe Riel <[Only registered and activated users can see links. ]> wrote:
>
>
> [snip maths re: spoke tension vs. rim temp]
>
> Following on from your (Joel) and Carl's brief dialogue,
> ISTR that at least two people who frequent this group
> have constructed FE models of the bicycle wheel
> hub-spokes-rim, primarily to investigate the validity
> of the "stands on its spokes" statement.
>
> Maybe, if asked nicely, they could be persuaded to
> superpose a thermal or equivalent initial strain loadcase
> on the rim elements and report their findings thereafter?

Do you recall who they were? I don't. I also don't recall
seeing any followup to that, that is, whether they actually
built the models.

Joe

01-12-2005, 07:53 PM

In article <[Only registered and activated users can see links. ]>,
Joe Riel <[Only registered and activated users can see links. ]> wrote:
> 1/Feff = 1/Fs + (n/2/pi)/Fr

This is, I would say, the most important single equation of your
analysis, and represents the meeting of the spokes' spring constant with
the equivalent constant presented by the rim - everything else is just
baby steps of definition. But, you skipped over a very interesting
number - this constant for the rim - which should give some idea of the
relative magnitude of what happens in the rim vs what goes on with the
spokes.
I'll use your cross sectional areas (2mm^2 for the spokes,
80mm^2 for the rim) but for my own sanity, I'll keep forces in N and
use metric values for material constants.

Fspoke = 2mm^2 * 200GPa = 400kN
Frim(eff) = 80mm^2 * 70GPa * 2*pi/36 ~ 977kN

This suprised me somewhat: for a uniform radial load across the
entire rim, the rim's share of that load is more than twice that of the
spokes! This is quite different from the intuition Jobst gives us of the
spokes in the LAZ taking all the load with the rim serving as a
pretensioning element for the spokes.
I still need to think a bit about what's going on, but I don't
see this as necessarily contradicting anything - the 'spokes do all the
work' intuition is for when the load is on a couple of spokes. Increased
rim compression is a secondary effect, but accumulates as you add
loads around the rest of the circumference.

It'd be interesting to take another look at Jobst's FEM results,
but my copy of the book is three time zones away.

Another interesting excercise would be to look at the relative
effective yield strengths are for each. ISTR the tensile yield strength
of spoke steel to be around 1GPa, while 7178-T6's compressive yield
stength is around 530MPa according to:

[Only registered and activated users can see links. ]

Yspoke = 2mm^2 * 1GPa = 2kN
Yrim = 80mm^2 * 530MPa * 2*pi/36 ~ 7.4kN

This would suggest that in tensioning a wheel, the spokes would
yield way before the rim does! Something doesn't seem right to me,
but, I have no idea how representative the yield strength I dug up
would be for a typical rim, nor do I have a good feel for how good the
80mm^2 estimate of the sectional area is (though it doesn't strike me
as being too far off).

Hmm... for comparison, a 6061-O alloy at:

[Only registered and activated users can see links. ]

lists a yield strength (though tensile) of 55.2MPa, an order of
magnitude less than the 7178. This would have the rim yielding ~
..77kN, well before the spokes. This seems more reasonable.

>So a 100degC rise in the rim temperature increases the spoke
>tension by 75kgf (165lbf). This is not a trival amount, it
>represents about 25% of the ultimate strength of a spoke.

I get to pretty much the same tension too (though for the
yield strength I worked out, it's an even scarier 38%, so if there's an
error in your reasoning, I've made the same folly.) I should point out
though, this increase in tension doesn't happen alone - when you have
an inflated tire on the rim, the tire pressure on the rim bed can be
pretty significant:

622mm*pi/36 * 20mm ~ 1.7 in^2 for a 20mm wide rim

With a tire inflated to 100 psi, this would be 170lbs of
pressure per spoke. But, while this number is remarkably close to the
tension increase we got earlier, this force is supported by both the
rim and spoke, of which the spoke sees:

170 * 40/(40+97.7) ~ 50lbs

with the other 120 being supported by the hoop strength of the rim.

Offhand, I'm not sure what the forces that a tire casing applies
to a clincher rim would look like and what it contributes to this
picture.

Other numbers aside, the 2mm^2 spoke area would be for a 1.6mm
diameter spoke. 15ga, which I'm more comfortable with would give a spoke
section of 2.5mm^2, and 14ga gives us 3.14mm^2. Going to these higher
gauges would change how the numbers balance.

-Luns

01-12-2005, 07:53 PM

In article <[Only registered and activated users can see links. ]>,
Joe Riel <[Only registered and activated users can see links. ]> wrote:
> 1/Feff = 1/Fs + (n/2/pi)/Fr

This is, I would say, the most important single equation of your
analysis, and represents the meeting of the spokes' spring constant with
the equivalent constant presented by the rim - everything else is just
baby steps of definition. But, you skipped over a very interesting
number - this constant for the rim - which should give some idea of the
relative magnitude of what happens in the rim vs what goes on with the
spokes.
I'll use your cross sectional areas (2mm^2 for the spokes,
80mm^2 for the rim) but for my own sanity, I'll keep forces in N and
use metric values for material constants.

Fspoke = 2mm^2 * 200GPa = 400kN
Frim(eff) = 80mm^2 * 70GPa * 2*pi/36 ~ 977kN

This suprised me somewhat: for a uniform radial load across the
entire rim, the rim's share of that load is more than twice that of the
spokes! This is quite different from the intuition Jobst gives us of the
spokes in the LAZ taking all the load with the rim serving as a
pretensioning element for the spokes.
I still need to think a bit about what's going on, but I don't
see this as necessarily contradicting anything - the 'spokes do all the
work' intuition is for when the load is on a couple of spokes. Increased
rim compression is a secondary effect, but accumulates as you add
loads around the rest of the circumference.

It'd be interesting to take another look at Jobst's FEM results,
but my copy of the book is three time zones away.

Another interesting excercise would be to look at the relative
effective yield strengths are for each. ISTR the tensile yield strength
of spoke steel to be around 1GPa, while 7178-T6's compressive yield
stength is around 530MPa according to:

[Only registered and activated users can see links. ]

Yspoke = 2mm^2 * 1GPa = 2kN
Yrim = 80mm^2 * 530MPa * 2*pi/36 ~ 7.4kN

This would suggest that in tensioning a wheel, the spokes would
yield way before the rim does! Something doesn't seem right to me,
but, I have no idea how representative the yield strength I dug up
would be for a typical rim, nor do I have a good feel for how good the
80mm^2 estimate of the sectional area is (though it doesn't strike me
as being too far off).

Hmm... for comparison, a 6061-O alloy at:

[Only registered and activated users can see links. ]

lists a yield strength (though tensile) of 55.2MPa, an order of
magnitude less than the 7178. This would have the rim yielding ~
..77kN, well before the spokes. This seems more reasonable.

>So a 100degC rise in the rim temperature increases the spoke
>tension by 75kgf (165lbf). This is not a trival amount, it
>represents about 25% of the ultimate strength of a spoke.

I get to pretty much the same tension too (though for the
yield strength I worked out, it's an even scarier 38%, so if there's an
error in your reasoning, I've made the same folly.) I should point out
though, this increase in tension doesn't happen alone - when you have
an inflated tire on the rim, the tire pressure on the rim bed can be
pretty significant:

622mm*pi/36 * 20mm ~ 1.7 in^2 for a 20mm wide rim

With a tire inflated to 100 psi, this would be 170lbs of
pressure per spoke. But, while this number is remarkably close to the
tension increase we got earlier, this force is supported by both the
rim and spoke, of which the spoke sees:

170 * 40/(40+97.7) ~ 50lbs

with the other 120 being supported by the hoop strength of the rim.

Offhand, I'm not sure what the forces that a tire casing applies
to a clincher rim would look like and what it contributes to this
picture.

Other numbers aside, the 2mm^2 spoke area would be for a 1.6mm
diameter spoke. 15ga, which I'm more comfortable with would give a spoke
section of 2.5mm^2, and 14ga gives us 3.14mm^2. Going to these higher
gauges would change how the numbers balance.

-Luns

01-12-2005, 07:53 PM

In article <[Only registered and activated users can see links. ]>,
Joe Riel <[Only registered and activated users can see links. ]> wrote:
> 1/Feff = 1/Fs + (n/2/pi)/Fr

This is, I would say, the most important single equation of your
analysis, and represents the meeting of the spokes' spring constant with
the equivalent constant presented by the rim - everything else is just
baby steps of definition. But, you skipped over a very interesting
number - this constant for the rim - which should give some idea of the
relative magnitude of what happens in the rim vs what goes on with the
spokes.
I'll use your cross sectional areas (2mm^2 for the spokes,
80mm^2 for the rim) but for my own sanity, I'll keep forces in N and
use metric values for material constants.

Fspoke = 2mm^2 * 200GPa = 400kN
Frim(eff) = 80mm^2 * 70GPa * 2*pi/36 ~ 977kN

This suprised me somewhat: for a uniform radial load across the
entire rim, the rim's share of that load is more than twice that of the
spokes! This is quite different from the intuition Jobst gives us of the
spokes in the LAZ taking all the load with the rim serving as a
pretensioning element for the spokes.
I still need to think a bit about what's going on, but I don't
see this as necessarily contradicting anything - the 'spokes do all the
work' intuition is for when the load is on a couple of spokes. Increased
rim compression is a secondary effect, but accumulates as you add
loads around the rest of the circumference.

It'd be interesting to take another look at Jobst's FEM results,
but my copy of the book is three time zones away.

Another interesting excercise would be to look at the relative
effective yield strengths are for each. ISTR the tensile yield strength
of spoke steel to be around 1GPa, while 7178-T6's compressive yield
stength is around 530MPa according to:

[Only registered and activated users can see links. ]

Yspoke = 2mm^2 * 1GPa = 2kN
Yrim = 80mm^2 * 530MPa * 2*pi/36 ~ 7.4kN

This would suggest that in tensioning a wheel, the spokes would
yield way before the rim does! Something doesn't seem right to me,
but, I have no idea how representative the yield strength I dug up
would be for a typical rim, nor do I have a good feel for how good the
80mm^2 estimate of the sectional area is (though it doesn't strike me
as being too far off).

Hmm... for comparison, a 6061-O alloy at:

[Only registered and activated users can see links. ]

lists a yield strength (though tensile) of 55.2MPa, an order of
magnitude less than the 7178. This would have the rim yielding ~
..77kN, well before the spokes. This seems more reasonable.

>So a 100degC rise in the rim temperature increases the spoke
>tension by 75kgf (165lbf). This is not a trival amount, it
>represents about 25% of the ultimate strength of a spoke.

I get to pretty much the same tension too (though for the
yield strength I worked out, it's an even scarier 38%, so if there's an
error in your reasoning, I've made the same folly.) I should point out
though, this increase in tension doesn't happen alone - when you have
an inflated tire on the rim, the tire pressure on the rim bed can be
pretty significant:

622mm*pi/36 * 20mm ~ 1.7 in^2 for a 20mm wide rim

With a tire inflated to 100 psi, this would be 170lbs of
pressure per spoke. But, while this number is remarkably close to the
tension increase we got earlier, this force is supported by both the
rim and spoke, of which the spoke sees:

170 * 40/(40+97.7) ~ 50lbs

with the other 120 being supported by the hoop strength of the rim.

Offhand, I'm not sure what the forces that a tire casing applies
to a clincher rim would look like and what it contributes to this
picture.

Other numbers aside, the 2mm^2 spoke area would be for a 1.6mm
diameter spoke. 15ga, which I'm more comfortable with would give a spoke
section of 2.5mm^2, and 14ga gives us 3.14mm^2. Going to these higher
gauges would change how the numbers balance.

-Luns

01-12-2005, 10:41 PM

[Only registered and activated users can see links. ] (Luns Tee) writes:

> In article <[Only registered and activated users can see links. ]>,
> Joe Riel <[Only registered and activated users can see links. ]> wrote:
>> 1/Feff = 1/Fs + (n/2/pi)/Fr
>
> This is, I would say, the most important single equation of your
> analysis, and represents the meeting of the spokes' spring constant with
> the equivalent constant presented by the rim - everything else is just
> baby steps of definition.

Agreed.

> But, you skipped over a very interesting
> number - this constant for the rim - which should give some idea of the
> relative magnitude of what happens in the rim vs what goes on with the
> spokes.
> I'll use your cross sectional areas (2mm^2 for the spokes,
> 80mm^2 for the rim) but for my own sanity, I'll keep forces in N and
> use metric values for material constants.

80mm^2 was for a tubular rim (I had it handy); the material
cross-section for a clincher rim is larger.

> Fspoke = 2mm^2 * 200GPa = 400kN
> Frim(eff) = 80mm^2 * 70GPa * 2*pi/36 ~ 977kN
>
> This suprised me somewhat: for a uniform radial load across the
> entire rim, the rim's share of that load is more than twice that of the
> spokes! This is quite different from the intuition Jobst gives us of the
> spokes in the LAZ taking all the load with the rim serving as a
> pretensioning element for the spokes.

I'm confused as to your interpretation. The ratio of rim compression
over spoke tension is n/2/pi = 36/2/pi ~ 5.7. You are referring to the
ratio of spring constants...

> Another interesting excercise would be to look at the relative
> effective yield strengths are for each. ISTR the tensile yield strength
> of spoke steel to be around 1GPa, while 7178-T6's compressive yield
> stength is around 530MPa according to:
>
> [Only registered and activated users can see links. ]
>
> Yspoke = 2mm^2 * 1GPa = 2kN
> Yrim = 80mm^2 * 530MPa * 2*pi/36 ~ 7.4kN
>
> This would suggest that in tensioning a wheel, the spokes would
> yield way before the rim does! Something doesn't seem right to me,

You [and I] are ignoring the bending stress in the rim due to the
discrete spoke locations. Isn't that where the yield problem for
the rim lies?

Joe

01-12-2005, 10:41 PM

[Only registered and activated users can see links. ] (Luns Tee) writes:

> In article <[Only registered and activated users can see links. ]>,
> Joe Riel <[Only registered and activated users can see links. ]> wrote:
>> 1/Feff = 1/Fs + (n/2/pi)/Fr
>
> This is, I would say, the most important single equation of your
> analysis, and represents the meeting of the spokes' spring constant with
> the equivalent constant presented by the rim - everything else is just
> baby steps of definition.

Agreed.

> But, you skipped over a very interesting
> number - this constant for the rim - which should give some idea of the
> relative magnitude of what happens in the rim vs what goes on with the
> spokes.
> I'll use your cross sectional areas (2mm^2 for the spokes,
> 80mm^2 for the rim) but for my own sanity, I'll keep forces in N and
> use metric values for material constants.

80mm^2 was for a tubular rim (I had it handy); the material
cross-section for a clincher rim is larger.

> Fspoke = 2mm^2 * 200GPa = 400kN
> Frim(eff) = 80mm^2 * 70GPa * 2*pi/36 ~ 977kN
>
> This suprised me somewhat: for a uniform radial load across the
> entire rim, the rim's share of that load is more than twice that of the
> spokes! This is quite different from the intuition Jobst gives us of the
> spokes in the LAZ taking all the load with the rim serving as a
> pretensioning element for the spokes.

I'm confused as to your interpretation. The ratio of rim compression
over spoke tension is n/2/pi = 36/2/pi ~ 5.7. You are referring to the
ratio of spring constants...

> Another interesting excercise would be to look at the relative
> effective yield strengths are for each. ISTR the tensile yield strength
> of spoke steel to be around 1GPa, while 7178-T6's compressive yield
> stength is around 530MPa according to:
>
> [Only registered and activated users can see links. ]
>
> Yspoke = 2mm^2 * 1GPa = 2kN
> Yrim = 80mm^2 * 530MPa * 2*pi/36 ~ 7.4kN
>
> This would suggest that in tensioning a wheel, the spokes would
> yield way before the rim does! Something doesn't seem right to me,

You [and I] are ignoring the bending stress in the rim due to the
discrete spoke locations. Isn't that where the yield problem for
the rim lies?

Joe

01-12-2005, 10:41 PM

[Only registered and activated users can see links. ] (Luns Tee) writes:

> In article <[Only registered and activated users can see links. ]>,
> Joe Riel <[Only registered and activated users can see links. ]> wrote:
>> 1/Feff = 1/Fs + (n/2/pi)/Fr
>
> This is, I would say, the most important single equation of your
> analysis, and represents the meeting of the spokes' spring constant with
> the equivalent constant presented by the rim - everything else is just
> baby steps of definition.

Agreed.

> But, you skipped over a very interesting
> number - this constant for the rim - which should give some idea of the
> relative magnitude of what happens in the rim vs what goes on with the
> spokes.
> I'll use your cross sectional areas (2mm^2 for the spokes,
> 80mm^2 for the rim) but for my own sanity, I'll keep forces in N and
> use metric values for material constants.

80mm^2 was for a tubular rim (I had it handy); the material
cross-section for a clincher rim is larger.

> Fspoke = 2mm^2 * 200GPa = 400kN
> Frim(eff) = 80mm^2 * 70GPa * 2*pi/36 ~ 977kN
>
> This suprised me somewhat: for a uniform radial load across the
> entire rim, the rim's share of that load is more than twice that of the
> spokes! This is quite different from the intuition Jobst gives us of the
> spokes in the LAZ taking all the load with the rim serving as a
> pretensioning element for the spokes.

I'm confused as to your interpretation. The ratio of rim compression
over spoke tension is n/2/pi = 36/2/pi ~ 5.7. You are referring to the
ratio of spring constants...

> Another interesting excercise would be to look at the relative
> effective yield strengths are for each. ISTR the tensile yield strength
> of spoke steel to be around 1GPa, while 7178-T6's compressive yield
> stength is around 530MPa according to:
>
> [Only registered and activated users can see links. ]
>
> Yspoke = 2mm^2 * 1GPa = 2kN
> Yrim = 80mm^2 * 530MPa * 2*pi/36 ~ 7.4kN
>
> This would suggest that in tensioning a wheel, the spokes would
> yield way before the rim does! Something doesn't seem right to me,

You [and I] are ignoring the bending stress in the rim due to the
discrete spoke locations. Isn't that where the yield problem for
the rim lies?

Joe

01-13-2005, 12:00 AM

Joe Riel writes:

> You [and I] are ignoring the bending stress in the rim due to the
> discrete spoke locations. Isn't that where the yield problem for
> the rim lies?

The rim fails in column buckling at the spoke holes from the
compressive load. the valve hole in some rims being the weakest
location. That's why the rim goes into a buckling "S" bend, being
constrained by the spokes from any single large lateral excursion as a
free standing column (pole vault pole) would.

Jobst Brandt
[Only registered and activated users can see links. ]

01-13-2005, 12:00 AM

Joe Riel writes:

> You [and I] are ignoring the bending stress in the rim due to the
> discrete spoke locations. Isn't that where the yield problem for
> the rim lies?

The rim fails in column buckling at the spoke holes from the
compressive load. the valve hole in some rims being the weakest
location. That's why the rim goes into a buckling "S" bend, being
constrained by the spokes from any single large lateral excursion as a
free standing column (pole vault pole) would.

Jobst Brandt
[Only registered and activated users can see links. ]

01-13-2005, 12:00 AM

Joe Riel writes:

> You [and I] are ignoring the bending stress in the rim due to the
> discrete spoke locations. Isn't that where the yield problem for
> the rim lies?

The rim fails in column buckling at the spoke holes from the
compressive load. the valve hole in some rims being the weakest
location. That's why the rim goes into a buckling "S" bend, being
constrained by the spokes from any single large lateral excursion as a
free standing column (pole vault pole) would.

Jobst Brandt
[Only registered and activated users can see links. ]