[pianotech] Breaking bass string

[John Delacour]

At 09:10 -0700 10/04/2011, David Love wrote:


>The maximum safe tension for a given string that I have used is T = 
>.557d^1.667 representing 60% of the breaking strength,  d given in 
>mils so for your #13 gauge I would come up with .557*31^1.667 which 
>equals about 170 lbs.  That's quite a bit different from your 137 
>lbs even if you figure in the differences for metric  versus English 
>diameters.

David, your formula is ambiguous.  Your first statement of it 
disagrees with the second.

The first would mean to me (0.557 * d)^1.6666666

The second would mean 0.557 * d^1.666666

For mwg.13 (0.775 mm) this would give

in the first case 112 lbs.

in the second 166 lbs.

So which is it?  And how is the formula arrived at?

At all events it is not possible to use one equation for all sizes 
because the tensile strength per unit cross-sectional area varies - 
the thinner the wire, the greater the tensile strength per unit.  So 
for example if the 'mechanical resistance' of 13 mwg is 548 lbs per 
mm2, 23 mwg. is only 505 lbs.

But to continue.  So far as I can tell, without spending half an hour 
puzzling out of what formula yours is a simplification, (0.557 * 
d)^1.6666666 makes sense and the other does not, because it gives the 
wrong curve.  And if, as you say, this gives 60% of the "breaking 
strain" (whatever that is!), then, as Del says, this is very 
conservative.  If we raise that to 75% we get figures far more 
similar to mine and Paulello's...

But the breaking strain to start with is not the 'mechanical 
resistance' or 'nominal breaking load', to use Paulello's terms, that 
is the tension under which the wire breaks on the testing machine has 
to be reduced by 25% to arrive at the 'practical breaking load'.  The 
elastic limit of the wire is another 20% below that, in other words 
it is 80% of the 'practical breaking load', and we need to make sure 
the tension of the string never exceeds the elastic limit, so we need 
another margin.  To quote from Paulello:

"Since Mersenne's work about gut strings until the recent researches 
by Claude Valette and Christian Cuesta in their book dedicated to 
vibrating strings, it is generally accepted that a string provides 
the fullness of its vibrating power when it is stressed to around 60 
to 75% of its practical breaking load, or close to its elastic limit 
(Re). Stressing a string in this way actually limits internal damping 
and offers the best balance between the fundamental frequency and 
higher partials."

At 09:43 -0700 12/04/2011, Delwin D Fandrich wrote:
>For example, the breaking strength of Mapes IG No. 13 gage wire 
>(U.S., or 0.031"/0.787 mm) ...averages 296 lbs.

I have asked Mapes to send me a full list.  Perhaps someone here can 
send me it.

Now this is the 'mechanical resistance' or 'nominal breaking load' of 
the wire.  To arrive at the 'practical breaking load' we multiply by 
0.75 and get 222 lbs.  Multiplying by 0.80 we get 178 lbs. as the 
elastic limit. We then need a safety margin to be sure the string 
will not exceed 178 lbs. at any point in its life.

By comparison, Paulello's No. 13 would have a 'mechanical' 
resistance' of about 265 lbs.  I stupidly got some of my numbers 
wrong in my last message.  The nominal strength is less than Mapes 
but not all that much.

JD