Why is a crowned back favorable?

[joachimM]

Hi folks,

I've been thinking on the exact reasons why a crowned back is often/sometimes favorable to bow design. Maybe this explanation has been given before on this forum (and suppose it's right, it was probably a better one than mine), but since I always have the feeling that to understand something really well, you should be able to explain it someone else. So here's my attempt at doing so. If you disagree, please tell me, and tell me why so I might learn something from my mistakes.

Most if not all woods are much stronger in tension than in compression. As far as I interpret the data from wood databases correctly, most woods are about twice as strong in tension than in compression. That is, a beam of wood that is bent takes set (non-elastic deformation of the belly due to compression) way before the back breaks. This is also what comes out of the standardized bending tests of Tim Baker (and later bowyers). So typically, the thickness of a bow is dictated by the ability of the wood to withstand compression rather than tension, unless you allow set to occur. At least, that's the principle behind no set tillering. In flight bows one evidently wants to avoid set, so the belly dictates the thickness. 
In David Dewey’s (aka Woodbear) spreadsheet the amount of strain the back can take before the belly starts to take set is termed the proportional limit, which is on average half of the rupture modulus (the tension at which the back starts to fail). If I'm not mistaken, this difference in tension and compression strength can also be deduced by the difference in crushing strength (compression failure) and modulus of rupture (tension failure).

When wood is stronger in tension than in compression and the surface of belly and back is equally large (rectangular cross-section), the neutral plane will not be exactly in the middle, but shifted towards the back of the bow: more belly wood is required to withstand the compression dictated by the back, because the wood is stronger in tension. The thicker the part that has to deal with compression, the easier the wood takes set at the same bending radius.
This can be avoided by having a broader belly than back, which is the case in bows with a crowned back. It doesn’t stop there: in a crowned back design, the neutral plane of the bow shifts more towards the belly (because back and belly are more in balance), as a result of which there is less belly thickness in which set can occur, as a result of which the bow can bend further before taking set. Or put otherwise, a bow with a crowned back and flat belly has a higher proportional limit than one with a rectangular cross-section.
Toasting the belly has a similar effect. For reasons not entirely clear to me, toasting increases the the crushing strength of wood.

I'm sure there's plenty of leeway for improvement of my reasoning, so shoot!

[Badger]

  I am not convinced that the thinner parts of the limb are doing their share of the work. Just a small change in thickness equals a large change in beam strength. I sometimes trap when I want lateral stability but want an area slightly bending more. I am not conviced trapping or crowning has any benefit.

[bradsmith2010]

well I have no proof other than what experts have written,,, that the flat back and belly can bend further without taking a set or breaking,,, but I am open minded as to what you say if someone can show me an example of such design and performance obtained 

[gianluca100]

i use a lot of ash, which is known to be excellent in tension and rater bad in compression. since i started trapping the back i noticed that bows on te narrow side tend to get quite a bit less follow. with wide limbed bows this effect seems to be smaller.

this is of course by no means a scientific research but only some observations from a guy who is a bit mathematically challenged but has a few hundred bows under his belt 

i'd like to see the engeneers jump in and tell te true story...

[joachimM]

Badger, I'm not sure I understand your comment on thinner parts of the limb not doing their share of work.
Suppose you have no thickness taper at all (pyramid design), and you trap the back. If that indeed means that the neutral plane shifts a bit to the belly, there's  also less risk of set at same bend radius. This allows you to make a slightly thicker (and therefore stronger) bow or to draw the bow further without taking set (because trapping increases the proportional limit due to the shift of the neutral plane). As you say, a slight increase in thickness will strongly increase draw weight (and that's because draw weight follows a cubic relation with thickness; this makes tillering such a critical procedure: a small error has large consequences).

As Bradsmith2010 says: the proof of the pudding is in the eating:

I mostly make bows with a crowned back, and nearly all of them weigh way (20 to 30%) below what the mass principle tells me to expect, probably because I take off excess back wood, shift the neutral plane to the belly and make the bow thicker for the same draw weight and barely taking set.
If I take Woodbear's spreadsheet for a no-set bow (I have a few bows without any set), the intended mass of the bow is a lot more than what I actually have with my trapped or crowned backs. this makes my recent hawthorn branch bow (heavily crowned) shoot very well (173 fps at 9 gpp for a 33# draw at 27"), weighing 216 g instead of the expected (mass principle) 297 g. With a proper string (I chronied it with a nylon string that stretches enormously) I could probably gain a few fps. It doesn't seem to me the thinner parts of the limbs aren't bending (see for a pic http://joachimm.paleoplanet69529.yuku.com/photos/view/pid/3856371)

Another bow I tillered today is another crowned branch bow (plum). It draws 40# at 27.5", is 151 cm long (just under 60") and weighs 333 g (with the bark still on; removing that will reduce mass further, it's starting to flake anyway). It has very little set (it keeps 1.5 " of resting reflex after heat treating the entire belly; immediately after shooting it still has 0.75" of reflex), outside the fades it's 1.85 cm thick (.73") (by 2.8 cm or 1.1" wide) and even midlimb that's still 1.4 cm or .55" thick. Intended weight according to the mass principle is approx 439 g. Woodbear's spreadsheet intends such a no set bow with flat back and belly to weigh even 510 g (at a pyramid design). My crowned design is thus 24 to 35% lighter in mass than expected with no crowned back. Yet it doesn't break, because the wood is stronger in tension than compression. The reason for this mass difference is that the allowed thickness (at no set tillering; Woodbear's spreadsheet) for the limbs is only 0.4 cm with a max width of 5.4 cm. My bow limbs are at most 2.8 cm wide but much thicker.

I think they take little set because the crowned back shifts the neutral plane to the belly (because the crown forces more wood to take the tension load). So for the same draw weight I have a bow that is much lighter than a flat back bow because I made it thicker because I could do so.  And I could do so because the proportional limit was increased due to this shift of the neutral plane.
 
Need more convincing? 70" bow, red oak. trapped back (back is 67% width of belly). 29" draw, 77#. Pyramid design 4.5 cm wide at belly (3 cm at back) to 0.8 cm wide at nocks, 1.85 cm thick throughout the toasted limbs (tension of 0.95% compared to expected neutral plane, but likely even higher due to shifting of the neutral plane! That's what you would expect for an osage or yew bow). Bow mass=588 g (still needs sanding to remove tool marks) instead of expected 737 g. That's 20% lighter than suggested by the mass principle. I admit, it shows 1" of set right after shooting and has 0.4" of permanent set. I still need to find a 800 grain arrow to chrony it at 10 gpp. This thickness is four times the allowed thickness by the no-set method.

You said it yourself: increasing thickness a bit makes a bow far heavier. So the eternal trade off is how to make the bow as thick as allowed by set. Shifting the neutral plane to the belly shifts the proportional limit, allows for thicker bows, with lower bow mass for the same draw weight. Crowning or trapping the back (according to my train of thought) does exactly that.

I may not  be able to convince you (right now), but as long as I'm thinking it works and it allows me to make better bows I'm satisfied :-)

[bradsmith2010]

shoot the 70# bow with any arrow,, and it will give a good idea of performance ,,,sounds like a great bow,,, I am old school so if you shot it with a 500 grain arrow, I could compare with many of my own results,,, thanks for posting the info B

[George Tsoukalas]

Too tired to digest the science and watching a Hallmark movie with my wife.

But heavily crowned saplings have only worked for me when I've left them a few inches longer. In a crowned back I believe the tension stress to be concentrated down the middle.

I like my bows to be 64-66" long unless heavily crowned then I go 2 inches longer.

Jawge

[missilemaster]

When you build a bow from a sapling with a high crown you have basically flat sawn wood on the back and quartersawn wood on the belly. The best of both worlds. Not to mention the closer to the center of a log you get the better the rings generally are.

[Hrothgar]

So conversely, when should a person de-crown a stave? Last year I de-crowned a hackberry stave and flipped the tips. No particular reason, just wanted to experiment.  It turned out to be a nice, light weight shooter, but don't know if anything was gained from flattening the back.

[George Tsoukalas]

I would not decrown. Just leave it longer. Jawge

[Jim Davis]

Jawge, you are certainly correct about the tension being concentrated along the center of a crowned back. As long as that center has somewhere near 1/3 as much width as the belly, all will be well. (Most wood is closer to 3 times stronger in tension, not 2 times, as offered up top.)

joachimM, Wow. Way too many words (not too much science, just loquacious).

Now that I think of it, yes, too much science in one post. To answer all the points would take an equally extensive reply.

What's more, there is at least one source of all those issues that was written by engineer Paul Klopsteg in June of 1935 and reprinted in the book "Archery, the Technical Side." "I don't have that book!" comes the collective lament. Okay, neither did I for years,  but I managed to get a copy and so can you.

Trying to rediscover those established principles without engineering background seems to me certain to produce a less dependable understanding.

If you can get the use of Archery the Technical Side, one of the best of several pertinent sections begins on page 146 with the title, "Getting the Most out of the Bowstave." Worth every word.

Jim Davis

[George Tsoukalas]

Jim, that makes sense. Jawge

[joachimM]

Thanks everyone for your suggestions. Will try to find that book, will chrony with 500 gr arrow.
And for the record: no I'm not an engineer, I'm a scientist (with a PhD). Yes that makes me write lengthy and maybe tedious messages. Sorry for that :-)

[Badger]

 Jawge, I agree with you, I tend to use more working limb on high crowned bows than I do flatter staves. I feel the same way that the tension is concentrated down to middle. Wood is much more elastic in compression, there is no reason to try and put the nuetral plane somewhere near the middle. If you do manage to get the tension side to stretch a little bit it is right at failure. I like to use as little working limb as I can get away with. I find flatter allows me to get away with less working limb.

[joachimM]

Steve,

You seem convinced that wood is more elastic in compression than in tension. Permanent set in a bow is caused by the collapse of wood cells in the belly, hence by failure to resist compression. The observation that bows take set before they break, isn't that evidence that wood is stronger in tension? That is why the proportional limit (the degree of tension above which there is set) tends to be lower than the breaking strength.

Take for example the wood database (http://www.wood-database.com/) and pick any tree species. You will see that the max tension strength (given by the modulus of rupture) is roughly twice as high as the crushing strength (aka Compression Strength parallel to the grain).
This clearly shows beyond any doubt that most woods are twice as strong in tension than compression. Osage is right in that ball park. In many conifers the ratio is a bit smaller, and is smallest in juniper (which explains why Juniper is a good bow wood for sinew backing: it's light in weight, but withstands more compression relative to its SG than most other species). In Hickory this ratio is higher, which makes hickory a better than average wood for backing.
Chief RID: Sassafras has the lowest ratio I've seen so far (tension/compression strength = 1.36 instead of 2), so whenever that wood starts to take set in a crowned design, you know you are close to breaking the back of the bow.

So far I haven't read any sound argument against my reasoning. I've read arguments like "I do things differently", but not the actual theory behind why that is supposed to work better. So I'll give it a try myself :-)

I do realize that a thin broad limb (as you advocate) allows you to have less working limb, and thereby reduce hysteresis (internal friction among wood fibers in the bow), as a smaller portion of the fibers are moving, which in turns increases efficiency. But in another way than a crowned limb, which increases efficiency by reducing (inner and outer) mass (at the cost of higher hysteresis). In a thin limb, trapping doesn't reduce as much mass as in a thicker narrower limb, so I agree that in thin limbs trapping will have little effect. I also agree that outer mass is what matters most, so broad, flat and thin backs in the inner part of the limb are allowed to be heavy, as long as the outer limbs are light. The mass principle isn't only about mass, but also about how that mass is distributed.

So in no way does what I am saying question the performance of your bows or the quality of their design (the numbers speak for themselves...). It's just another pathway to the same goal: having a faster bow. And it all depends on what bow wood you start with and what design you aim for. If you start with a thin stave with narrow limbs and crowned back you can still get a fast bow by taking advantage of the crown. As Marc St Louis mentioned in your flight bow post: it's all about balance (between back tension and belly compression). How you achieve that balance will depend on which design you are constrained by (due to the stave properties).

Does that reconcile the apparently contradicting bow designs? Either flat, thin and broad inner limbs are one proven optimum. You can also get to another peak (it may not be as high) with thicker narrow but heavily crowned backs and a flat (toasted) belly. 
 
Joachim Mergeay