a Formula for Limb Width (@the fades)

[KS51]

I see the question about how wide to make a limb pretty regularly and have considered the question myself when working with a new species of wood.  So, I sat down and came up with this quasi-emperical equation to give myself a starting point.  I present it here for comment and discussion.

G=specific gravity
D=draw length (in)
L=working limb length (in)
P=draw weight (lbs)

Width = (P*D^2) / (L*G^1.13*1400)

* is multiply
/ is divide
^ is to the power

Now, for those that want more info, this equation is based upon:

Cantilever beam stress equation
67% of the MOR as calculated from the Specific Gravity
A basic limb thickness of 0.60" (this is on the thin side, but that makes the answer more conservative)
A string to tip angle of 90deg
A string to arrow angle of 45deg
Basic "do-able" bow geometry,  IE. you can't have a 10" limb with a 28" draw

I did write a more involved equation that includes the angles and thickness but only wanted to present the simplified version for review.

I tend to make laminated bows so I start with boards.  With a caliper, tape measure, scale and RH; I can calculate the specific gravity pretty well.  I can post that process for anyone interested.

Ken

Note: I add 1/8" to the calculated width for safety.

[NonBacked]

I ran the numbers for some bows I've already built:  Osage (0.82), hickory (0.72), and elm (0.53). All are pretty close in poundage and length: appx. 50 lb. at 26 in. and close to 28 in. working limbs (about 65 in. long). NOT BAD! Pretty good actually. I probably don't need the formula for wood I'm familiar with, but its an easy way to verify the width while I'm drawing the dimensions; and for unfamiliar wood...I'll use it for sure. However, I did have to find batteries for my Scientific calculator to  determine the ^1.13 power. It fooled me - the S.G. decreased. Another thing, the 1/8 in. safety factor really is a good idea for roughing out the stave - not all wood exactly matches the book values.

Nice job! Thanks for sharing.
H     

[Badger]

  Thats interesting, could you put that to an excel spread sheet formula.

[KS51]

I can write up an Excelmspreadsheet for it.

Ken

[mikekeswick]

I like your thinking. 
I'll be writing that on my wall for reference.

[redhawk55]

Great thinking!
It will help a lot if you're not familiar with a wood.
Don't underrate our bowyers here, some of them are such crazy to make bows with a 10" working limb at 28"!?

Michael

[KS51]

Here is a simple little spreadsheet for folks to toy around with.

It was saved in Excel 2003 format.

Ken

[Knoll]

Thanks for the spreadsheet.
Regarding limb length .... most of my bows are stiff for 3-5" at tips .... that length should be subtracted to arrive at "working" limb length?

[KS51]

Limb length is from fade to nock. Working or non-working is not a consideration for this purpose.  IE the wood at the fade does not care what is happening 2" or 12" away, only the overall lever length matters to the moment generated.

Ken

[Badger]

  Ken, I just did a 140# english longbow and the width came out at over 3". Have you made any allowances for this.

[JW_Halverson]

It had to happen sooner or later.  Someone HAD to come along, take something fun, and make work out of it!   

But seriously, thanks for marrying your passion for numbers with your passion for bows.  While some of us knuckledraggers will assiduously avoid your calculations in our work, this will ultimately make other's journey easier or more fulfilling. 

Passion does not make one a master.  But I have yet to see a Master that lacked passion.

[KS51]

  Ken, I just did a 140# english longbow and the width came out at over 3". Have you made any allowances for this.

I wrote up a more advanced version of the equation that would incorporate the estimated limb thickness, the arrow-to-string angle and the arrow-to-tip angle.  This would likely give a better estimate for the english longbow design, since they tend to be much thicker.  A quick/dirty adjustment could be done by multiplying the answer by (.6/t)^2.  IE if your actual expected thickness is .75 inches then you would multiply the width by .64 and your 3" becomes 1.92

Though this still doesn't account for the string angle differences.

The simple version of the equation is best suited to american flatbow or pyramid bow design.  More knowledge of the full draw geometry (expected) becomes necessary and more variables are used with the advanced form of the equation.

It is still just an approximated starting point.  True evaluation requires a whole other branch of mathematics tools.  This was intended to be used with basic variables we could reasonably know going into the build.

Ken