The Khotan bow and the P-effect

[Kviljo]

I just red through the old discussion about the design of the Khotan bow:
http://www.atarn.org/letters/khotan_bow ... iscuss.htm
The bending limbs of it has a pronounced concave back. Adam explained how the limbs could have gone from flat to concave by being affectet by the poisson effect for a long time.

I do see how that would work, but at the same time I am puzzeled by why it is said that one can prevent some of the P-effect by adding sinew to the sides of the bow. Yes, the extra sinew on the sides would work against the poisson effect once the bow is strung and drawn. It would secure the edges of the limbs against the extra stress.

What I'm not able to wrap my mind around, is how one can shape the limb in a concave manner (adding sinew to sides) to prevent the bow from aquiring the same shape... In other words, I am not convinced the Khotan bow was made with flat bending limbs and not made with concave ones from the start.

If the bow was made with a concave back and convex belly, wouldn't that prevent the horn slats on the belly from separating longitudionally? If the bow was made with so much concavity on the back that the poisson effect would be reversed? If it is possible in a real bow, and not just in my mind, would it not also give the bow extra early draw weight, by having the limbs to flatten out while being bent? - like the ACS technology of O.L. Adcock?

Have the reconstruction been strung/shot for long enough to aquire permanent concavity of the back?


As the bow is so well documented, and because I really fancy it's interesting flight bow design ? la extreme Holmeg?rd, I hope to make a lookalike.

[Adam Karpowicz]

In order to have the Poisson limbs as in the Khotan bow, the bow has to have a substantial built-in reflex. The bow you mention made to be merely 50lb and to the original dimensions could not have such reflex. The effect would show immediately, not after a long time.

I am not sure if you read in my book about the other situation - and quite common for bows with limbs of moderate width, with the sinew built out at the edges to form a tapered, sharper edge and the belly very round (common in Turkish bows). There most of the resistance to compression bending is within a narrow line along the belly at the apex of the horn convexity. This, plus the relatively low stiffness of sinew at the edges creates the situation where the edges are no longer as supported by the belly and bend down towards the neutral plane of limbs. You will see it easily if you follow the cross-sections of old bows. It can be imitated by bending a piece of a plastic T-bar. It is not that the sinew at the edges somehow "forces" the edges down. It is that the sinew has nothing under it to hold the edges up...

I have not seen bows with sinew deliberately built up into the concavity on the back, although many bows show some of it due to the transverse shrinkage of sinew and glue. There are the Tatar bows with ridges formed ove the edges probably on purpose, but I have not been able to solve this puzzle. Perhaps then the bow would look thicker in profile and therefore stronger to scare the opponents?

So, in bows with very wide limbs and flat horn (typically flat for the most part anyway in this construction) there is the Poisson effect. In narrow limbs with round belly the (unsupported) edges plus the concentation of stiffness at the center of the belly cause the opposite curvature on the back.

I do not see how the Poisson-like limb could flatten when bent. A limb of the opposite construction - convex on the back and concave on the belly will flatten.

Adam

[Stephen Selby]

I'd just like to point out one thing. I have a complete, two-piece bow core from the time of the Khotan bow. It is painted and has evidently never been sinewed or finished. There is no horn on it.

But it still has the 'poisson effect' curvature of the broad part of the limbs. It seems to have been an artefact of the manufacture.

[Bede]

Stephen,
Could that have been built in to make it look like composite bows?

Adam,
Sind bows often have two or three ridges of sinew on the back, but not on the edges. Could they take all of the tension in the back?

[Kviljo]

As discussed in the old thread too, couldn't the all-wood bow(s) also have been strung, and by that aquired the poisson curvature? Depending on how thick it is, it could perhaps also have been symbolicly shot with some 10-30 pounds? If it was left strung in the grave, it would aquire the shape permanently.

The effect I am talking about is basically the same as you (Adam) mention with the round-belly heavily sinewed sides. The transverse bending of the limbs might also be storing energy and by that giving better efficiency? In the same way as the compressional load is centered in such a bow, would the five horn laminates in the Khotan bow get some transverse compression and by that securing the longitudinal glue lines between the horn slivers - if the bow was built with some back concavity. On the other hand, if the limb was glued up flat, the poisson effect would want those glue lines to open up. Such cracking might not have any major short term effect on the bow, but durability must be affected.

I am not suggesting that a limb with built in P-effect would flatten completely when bent, but would it not flatten a little if it was sufficiently transversely bent?

[Adam Karpowicz]

Ivar, why not make an experiment with the wide limbs? As I said, I do not believe the Poisson limbs will flatten under stress, they would rather go into more Poisson, not less. So, the glue lines in the belly will open again... I see no way the transversly bent limbs storing energy unless the transverse bending is somehow restricted or counteracted.

The Poisson effect can be seen in all-wood bows too. Place a straight edge against a strung, wide limb, rectangular section bow with flat back and see the gap in the middle.

In response to Stephen: the core could permanently acquire the Poisson limbs after long time strung and in storage. I am tempted to make a bow like that to prove the point - the limbs were made flat and the Poisson came during use of bow, not before.

Bede: the ridges at the back formed in sinew will be under more stress and will add energy to the bow too, although an added full layer would be more effective and safer. I doubt if the ridges can truly take all tension on the back, because sinew is too elastic and the tension would drop as the sinew deforms into the extended position. Then the neutral plane of limb will move closer to belly, unless the belly is also weakened by ridges.

Adam

[Orda Khan]

I've seen Grozer's replica but I'm sure we have better bowyers here. As a big fan of the Khotan bow, I'd love to get my hands on a bow like this.

[Daniel T.]

I admit I am a bit confused about this discussion. What is a "Poisson limb"?

To clarify the Poisson effect, a small comment. Please excuse, it's "trade disease".
There is a material parameter called Poisson's ratio. It is the negative ratio of the perpendicular strain and the axial strain, when a material is stretched or compressed along a certain axis. Or conversely, the strain in the directions perpendicular to the axis (there are two such directions) is proportional to minus the axial strain. The minus sign is important.

When the material is stretched (axial strain is positive) the strain in the perpendicular directions (transverse strain) is negative. This means that the material becames smaller in the transverse direction as if compressed.

When the material is compressed (axial strain is negative) the strain in the perpendicular directions (transverse strain) is positive. This means that the material becames larger in the transverse direction as if pulled.

For the back of the bow, the negative transverse strains (from back to belly and from one side to the other) causes the back to dip in the center (apparent compression in back to belly direction) and curl inwards (apparent compression towards the center line from each side).
The opposite things occur at the belly.

All of this combined leads to this arced cross-section of a bow that is rectangular when it is not bent. When the unbraced cross-section is changed, a different bent cross-section occurs, but it is again defined by the strains created by the Poisson effect. There is no negating this effect by choosing a certain cross-section.

The final cross-section due to the Poisson effect is such that there is no stress in the lateral direction involved because there is no constraint. The bow can deform freely in the lateral directions. No stress means no energy stored.

[Adam Karpowicz]

This is all true Daniel. A good explanation of the Poisson effect on the limbs of bows.

Adam

[Kviljo]

Oh, I'm afraid my keyboard is not able to convey my thoughts as well as I'm hoping for. I'll try the late night version and hope for new thoughts in the morning:

Ever tried bending a transversely curved piece of anything? Does it not take a larger effort to bend it than a flat piece of similar dimentions? - and that with the same "limb" mass. It should mean a more effective limb.

The poisson effect will always be there, but the way the transverse curve of the limb acts when bent can be reversed if the limb is sufficiently curved.

[Bede]

O. L. Adcock makes modern flight bows that way and swears by them. He also wins flight tournaments too. He makes limbs from advanced materials and builds in a transverse curve.

[Adam Karpowicz]

I see now what you mean. You are saying a half tube is harder to bend than a flat piece of the same wall thickness and width, which is true. It has nothing to do with the Poisson effect, but is caused by the cross-sectional geometry. And I understand you believe that bending limbs curved into the concavity through the Poisson effect are then progressively harder to bend, because they become progressively more concave. Why not making an experiment to see what happens? You could use a piece of rubber (rubber has a high Poisson ratio to follow the post above).

Adam

[Kviljo]

I do think the effect would be more noticable in longbows, as the flattening of the limb is mostly going on in the first part of the bending. In a hornbow it would perhaps add slightly to the pre-stressing of the limbs, plus it would have the advantage of securing the glue lines between the horn pieces.

Such a configuration in a hornbow might require a bunch of extra sinew at the sides of the limbs, though. Perhaps it would also need some transverse/diagonally laid sinew to prevent the back from splitting. I'm not sure if I want to make a full-scale test before I have tried the flat limb, but I do think a rubber experiment would be worth the effort.

[Adam Karpowicz]

"as the flattening of the limb... "

?? There is no flattening.

Adam

[Daniel T.]

The Poisson effect creates a tensile transverse strain, which acts to pull the horn strips apart rather than push them together.

I believe that Poisson's ratio of horn is about 0.4 and of glue 0.3 (although that may change at higher strains). The lateral strains are then between 30% and 40% of the axial strain. Not sure if this enough to cause a permanent cross-sectional shape change (transverse set if you will), but that depends on the amount of set. When the axial stress is enough to cause set, it's possible that there is also transverse set.
You had that in bows already, Adam?