Above And Below the Neutral Plain.

[greybeard]

Please note that the following refers to bows made from milled boards.

This is the second longbow bow that I have made which goes against the accepted principle.

I have used the standard ‘D’ limb cross section where the ‘flat’ of the ‘D’ becomes the belly of the bow.

In the conventional design the belly is longer than the back which means the back has to stretch more and the belly has to compress more.

By utilising my method the belly is shorter than the back which means less stretch and compression is experienced when the bow is being drawn.

Although we are talking very small numbers I believe it makes a difference to the integerity of the timber.

If the neutral plain lies in the geometrical centre of the bow compare the amount of timber above and below this plain.

With my configuration there is less timber to stretch and more timber to handle compression. To date there is no evidence of chrysaling on the belly of either of the bows.

Presently I do not have hard data to back up my theory but field testing would suggest that I am on the right track.

The bow bends through the handle and was made from spliced hickory lengths reinforced with timber slats on the back and belly and wrapped with jute string glued with PVA.

Your comments would be welcomed.

I will get extra photos when the weather fines up.

Flat Belly.JPG (27.37 KiB) Viewed 4587 times

D Section Back.JPG (27.6 KiB) Viewed 4587 times

Daryl.

[hunterguy1991]

Nice bow Daryl!

I would have to do the calcs on the cross section but from memory the neutral axis of bending is located where there are equal area's in compression and tension.

Interesting theory for a bow in any case. Looking forward to seeing some drawn photos of it.

Colin

[Nezwin]

Is this a similar concept to 'trapping the back' that is sometimes referred to on US boards? Reducing the surface area of the back while maintaining a "standard" surface area on the belly.

Either way, bravo on the experiment and, as always, meticulous attention to detail in the build.

[Dennis La Varenne]

Daryl,

You are on the right track there.

As Nezwin suggests, the principle has to do with the reason why the concept of trapezoiding the limb cross section to make the back surface narrower. That follows from principles worked out by Hickman et al, back in the 1930s where it was found that generally, most (bow) wood is stronger in tension by something approacing 20% than it is in compression leading to the conclusion that trapezoiding the limbs with a sidewall slant to the back of around 17 degrees evens out this disparity for the most part.

I have a 63 inch Yew static recurve in my collection from the 1940s built to those principles. It isn't a pretty bow, being very angular surfaced but obviously has been designed using all the best scientific knowledge of the times and I am very eager to shoot it. However, it still has 1 1/4 inches of set just below the recurves which are 4 inches long despite the trapezoiding, suggesting that its 24 inches of working limb was still of insufficient length or width to take the bending load placed on them.

Colin will probably work out the relevant maths, but in 'Archery - The Technical Side' pp34 - 38, Hickman has a chapter entitled 'The Neutral Plane of Bending of a Bow' where he provides the maths to calculate the neutral layer in a bow which is that line through a bow where there is an equal volume of material on either side of that line. It is not of equal distance from either the belly or back except in bows which are symmetrical in cross section.

At the end of the chapter, Hickman suggests that for the mathematically disenclined, by drawing an accurate scaled up cross-sectional drawing on some stiff card, cutting it out and simply balancing that on a point until it balances will give the position of the neutral axis for any cross section.

For example, it has long been my contention that the D-section bow has the most inefficient cross-section of any design and it is clear that mechanically, if it were reversed to put the bowl of the D as the back as Daryl has just done, the belly would be much better able to withstand the compression load so long as the bowl of the D is of sufficient width to withstand the tension.

As Colin and some others already know, I have started to make and alter my present ELBs to have a symmetrical cross-section with a back cambered to the same profile as the belly to place the neutral axis close to or at the bow's centre and distributing the bending load equally between back and belly. This is how the Mary Rose bows were made rather than the long held belief that they were D-section. Most of them are actually oval in section.

However, I don't understand where Daryl suggests that the belly is shorter than the back unless the reference is to a drawn bow. A resting bow is of equal length unless it has taken a significant set.

[Dennis La Varenne]

It is not of equal distance from either the belly or back except in bows which are symmetrical in cross section.

I should also add that the degree of load taken by the back and belly of a bow DOES relate directly to the distance from the neutral axis to the back and belly surfaces.

On an ELB of conventional Victorian design with a Roman arch belly, an easy way (I hope) that I have found to calculate the neutral plane of an ELB to a reasonable degree of accuracy is to calculate the area of the rectangular section added to the area of the semi-circular section. The square root of that sum will yield an area which can be divided by the width of the rectangular section of the bow to give a depth from the back surface. That depth number is where the neutral layer is situated.

ELB cross section.jpg (9.45 KiB) Viewed 4484 times

Presuming that the drawing above is of an ELB whose width is 40mm x 32mm deep (80% stack) and the red rectangular section is 40mm wide x 10mm deep.

To calculate the area of the black semi-circular section, we use the equation (Πr^2) ÷ 2, so the area of this semi-circle is -
(22/7 x 20 x 20) ÷ 2 = 628.57mm^2,

The area of the red box section is 40 x 10 = 400mm^2.

Thus the total area of the cross-section is 400 + 628.57 = 1028.57mm^2.

So, √1028.57 = 32.07mm.

Therefore, if the total area of the rectangular cross-section (400) is divided by 32.07, we get 12.47 which is the position of the neutral layer from the back surface as in the illustration in the drawing below.

ELB cross section_2.jpg (10.68 KiB) Viewed 4484 times

So, the distance from the back surface to the neutral plane is close to 12.5mm and the distance from the belly surface is 16.5 mm, which is 32% greater distance than from the back to the neutral plane. However, because the belly surface of the ELB is also so narrow, the compression load is even greater because that stress is spread over such a narrow width.

If this ELB cross-section had the more conventional Gothic arch shaped belly of the first quarter of the 20th century, the problem of excessive compression and consequent string-follow would be even worse.

[greybeard]

Dennis,

Thank you for the calculations and diagrams.

Please note that the following refers to bows made from milled boards.
In the conventional design the belly is longer than the back which means the back has to stretch more and the belly has to compress more.
By utilising my method the belly is shorter than the back which means less stretch and compression is experienced when the bow is being drawn.

........However, I don't understand where Daryl suggests that the belly is shorter than the back unless the reference is to a drawn bow. A resting bow is of equal length unless it has taken a significant set.

....Although we are talking very small numbers I believe it makes a difference to the integerity of the timber.

Hopefully the diagram [not to scale] illustrates my point.

D Section Comp.jpg (32.64 KiB) Viewed 4471 times

Daryl.

[Nezwin]

It is not of equal distance from either the belly or back except in bows which are symmetrical in cross section.

I should also add that the degree of load taken by the back and belly of a bow DOES relate directly to the distance from the neutral axis to the back and belly surfaces.

On an ELB of conventional Victorian design with a Roman arch belly, an easy way (I hope) that I have found to calculate the neutral plane of an ELB to a reasonable degree of accuracy is to calculate the area of the rectangular section added to the area of the semi-circular section. The square root of that sum will yield an area which can be divided by the width of the rectangular section of the bow to give a depth from the back surface. That depth number is where the neutral layer is situated.

ELB cross section.jpg

Presuming that the drawing above is of an ELB whose width is 40mm x 32mm deep (80% stack) and the red rectangular section is 40mm wide x 10mm deep.

To calculate the area of the black semi-circular section, we use the equation (Πr^2) ÷ 2, so the area of this semi-circle is -
(22/7 x 20 x 20) ÷ 2 = 628.57mm^2,

The area of the red box section is 40 x 10 = 400mm^2.

Thus the total area of the cross-section is 400 + 628.57 = 1028.57mm^2.

So, √1028.57 = 32.07mm.

Therefore, if the total area of the rectangular cross-section (400) is divided by 32.07, we get 12.47 which is the position of the neutral layer from the back surface as in the illustration in the drawing below.

ELB cross section_2.jpg

So, the distance from the back surface to the neutral plane is close to 12.5mm and the distance from the belly surface is 16.5 mm, which is 32% greater distance than from the back to the neutral plane. However, because the belly surface of the ELB is also so narrow, the compression load is even greater because that stress is spread over such a narrow width.

If this ELB cross-section had the more conventional Gothic arch shaped belly of the first quarter of the 20th century, the problem of excessive compression and consequent string-follow would be even worse.

Dennis,

It has been some years since I studied any Structural Engineering and this is all straight from memory of broad principles that I was never particularly good with, so I am happy to be corrected in this matter. But, truthfully, you are very close to, or on, the jackpot in your theorising. I believe this is taught differently here in Australia, but in the UK we were taught to use 'Second Moment of Area', which is essentially what you have described.

While balancing proportions of mass is important, the further that mass is from the neutral plane the more internal moments act upon it (and, accordingly, internal forces). Consider an I-Beam - the depth of that beam effects how stiff it is while the thickness of each flange effects how much ultimate load it can hold (among other factors, all other things being equal). For bow making, the width of your limb equates here to the thickness of the flange while the limb cross-section equates to your depth of beam.

If the I-Beam depth is increased, the distance of the centre of mass of each flange from neutral plane increases, thus increasing stiffness but potentially reducing the maximum allowable load, as each flange would now be experiencing greater internal moments upon it. This would be much like building a 1" width bow - no problem if you're using Osage (a strong material) and the bow is only 25lb@28". If you're using Red Oak (a weaker material) and the bow is 50lb@28", it is likely to fail.

If the flanges are thickened, the centre of mass for each section above & below the neutral plane moves away from the centre of mass for the beam as a whole, thus stiffening the beam and increasing the maximum allowable load to the beam. Similarly, if one flange changes material to something stiffer (like bamboo when compared to a standardish timber) the neutral plane will move toward it. However, in this case, the maximum allowable load remains that of the weaker flange.

I've not explained very well there, but here it is in summary -

Centre of Mass multiplied by Distance to Neutral Plane is equal for each section of the bending beam assuming normal, uniform bending (under which bows bend).

For a bow with z-value internal forces, a point 5mm (say, on the very outer edge of a flat bellied bow) from the neutral plane will experience 5mm x z-value internal moments. A point 7.5mm from the neutral plane (say, on the very outer edge of an oval shaped back) would experience 7.5mm x z-value internal moments, an additional 50% loading.

Better explanation found here - http://www.learneasy.info/MDME/MEMmods/ ... oment.html

Colin has completed his degree more recently and can equate this to Moments of Intertia, which I believe is how the same principle is taught in Australia.

Having just reread your post, I realise I actually skimmed some important details and see that you've pretty much covered exactly what I've written and I am essentially repeating what you've already explained, albeit I am using the I-Beam example. Please accept it as a compliment when I say that that is impressive Engineering for a retired Registered Nurse!

Neil

[Dennis La Varenne]

Neil,

I am embarrassed by your kindness. My formal skill level at mathematics is first year secondary school. Anything else is gleaned by dint of pure persistence and trying to understand. Thank you. I am glad I am on the right track.

Daryl,
Now I get what you are saying. I have never thought of it like that, but I think that with bending and shortening of the belly, there would be very little in it as you suggest.

[Nezwin]

Not at all, Dennis. I'm a strong believer in credit given where credit due.

I don't mean to labour this point too much - as I think Daryl has demonstrated this very well & Dennis has provided an excellent Engineering breakdown of the principles - but it has been on my mind a bit. I'm posting my thoughts on it in an effort to get my brother to sign up to this board... He's encouraged me to consider the limb movement in terms of energy storage, which is not dissimilar to structural work. What I've scribbled below is a quick run through on the principles above but, with my dyslexia, done in such a way that it perhaps makes it more complicated for other people!

Essentially, what Daryl has done is take a tension-strong timber (Hickory) and reduced cross sectional area on the tension side/back to equalise the internal forces acting within the limb, thereby optimising the energy storage. The way this has been done works particularly well with Hickory in a selfbow, for other timbers other approaches would be required. It reaffirms Daryl's comments on appropriate design for specific timbers.

If my brother isn't interested, someone may get some benefit from these ideas somewhere down the line. As always, I'm welcome to correction - it's been a long time since I even considered these concepts...

Note: I've used CSA as an abbreviation for Cross Sectional Area. Optimal energy release/storage for a given material occurs just prior to Proportional Limit.

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[Dennis La Varenne]

Neil,

That all makes perfect sense to me and pretty much iterates everything I have been able to understand from Hickman, Klopsteg and Nagler. Your explanation is a lot shorter as well. There is some pretty complicated maths in the publications of the 3 archery greats (they are to me anyway, even more so than the great marksmen).

Thank you.

[hunterguy1991]

I very much enjoyed reading all this

Been over 6 months since looking at any engineering and it was a nice little change!!

Neil I didn't know you had a Structural Eng background!! Seems like there's a fair few engineers hiding on this forum.

Im seriously considering taking some of this back to uni for a good long chat with my structural lecturers as there is some really interesting things going on in bow limbs.

I agree completely with Dennis's analysis of the Second Moment of Inertia for calculations of the neutral axis, and with Neil's statements about neutral axis not being the same distance from back and belly except in symmetrical cross section limbs. I also like the Spring model Neil has provided, I had never considered breaking it down that far before. You have given me LOTS to think about mate!! haha

I have often thought about the link between load at the tips from the string and the Stress in the extreme back and belly fibres in the equation Sigma (stress MPa) = M (bending load Kn.m) x y (distance from neutral axis to extreme fibre mm) all divided by Ixx (second moment of Inertia mm^4)

Equation : Sigma = (My)/ Ixx or simplified Sigma = M/Z where Z = Ixx/y

This stress is can be used to assess how close to failure a bow may be depending on its material properties.

So many interesting things happening!!

Colin

[perry]

I'm certainly no Engineer so some is not so clear When I started making Selfbows I was taught the advantages of a Trapazoid Cross Section for a Selfbow, also could never understand why so many Fibreglass Laminated Flatbow's where Trapped opposite with the Back broader than the Belly, still don't even though I understand the process is safe and works due to Fibreglass's inherent strength, just looks odd to my Eyes

I worked out early on that it was so much rubbish that a small diameter Tree with a high Crown of the Back is a poor choice for a Selfbow. If one is less fussy a small Tree yield's a Stave with a very similar Back to Daryl's Bow's, its simply a matter of working the Stave to have a Flat broad Belly. Pink Ash is particular often grows with an Oval Cross Section that allows this

A bit out of left field but back in the 90's I played around with Cable Backed Bows, they raise the Neutral Plane above the Back of the Bow. Messing with Cable Tension is key to making a Cable Back Bow Shoot, too much though and the Cable applies so much compression the Belly's Chrysal's badly. My poor version of a Penobscot is still Shooting, they are an interesting thing. This one shoots very well, I have the Cables twisted to apply moderate Tension



regards Jacko

[Nezwin]

I had no idea a few scribbled thoughts would wager such interest! Thanks to Daryl for prompting the discussion with his build & to Dennis for starting the theoretical discussion.

Neil I didn't know you had a Structural Eng background!! Seems like there's a fair few engineers hiding on this forum.

I have a Civil Engineering degree (including Structural Engineering - in the UK this can be done as a separate Masters subject but is generally included within Civil also), but Structures were never my forte. My career has been primarily within a Geotechnical arena, but I'm a manager now. It happens to the best of us

I have often thought about the link between load at the tips from the string and the Stress in the extreme back and belly fibres in the equation Sigma (stress MPa) = M (bending load Kn.m) x y (distance from neutral axis to extreme fibre mm) all divided by Ixx (second moment of Inertia mm^4)

Equation : Sigma = (My)/ Ixx or simplified Sigma = M/Z where Z = Ixx/y

This stress is can be used to assess how close to failure a bow may be depending on its material properties.

That would work too, just a more 'engineering' approach! I tend to work in general concepts than in detailed design (I've never worked in design, it's not really my thing). Bigger picture stuff as opposed to detailed problem solving.

I wouldn't advocate any use my concept to design a bow, but it is a model that can be used to consider how a material might be used. It's that medium between the art of bowmaking and the science of bending materials. I might be an Engineer, but my heart lies in the functioning art that is produced by some of the great bowmakers on this forum. I'll get there one day.

A bit out of left field but back in the 90's I played around with Cable Backed Bows, they raise the Neutral Plane above the Back of the Bow. Messing with Cable Tension is key to making a Cable Back Bow Shoot, too much though and the Cable applies so much compression the Belly's Chrysal's badly. My poor version of a Penobscot is still Shooting, they are an interesting thing. This one shoots very well, I have the Cables twisted to apply moderate Tension

That's a pretty cool bow! I saw some similar designs with reflex risers somewhere, but they were pretty sterile to look at - all actionwood & fibreglass. Not that there isn't a place for that, just the natural materials are flavour of the year for me right now. Cables being good in tension would work well with the strong compression strength of most timbers. By shifting the neutral plane out of the limb, you're reducing some areas for internal shear.

Anyway, some better Engineers than me came up with these thoughts -

Tbh I can't see anything outrageously wrong but it isn't the approach I would have taken. I think in practical terms the theoretical location of the neutral axis is fairly irrelevant as wood is really variable so actual bows will end up quite different to theory

I'd argue it's important where the neutral plane is being as how that affects your internal shear distribution, but the point about wood being variable is spot on. That's why it's an art & not a science

For a bow of a fixed length, fixed cross section (unlikely) and fixed force applied at the centre you have a fixed moment applied through the centre of that bow, evenly.
Now this moment is REACTED to by the "springs" in your bow thus keeping it in equilibrium.
Call this moment M. For equilibrium to take place:
Fb*Db + Fc*Dc = M
Like a car in roll where Fb and Fc are the vertical forces at the tyres and Db and Dc are the distances from the centre of mass.
However you don't have two point forces, they are distributed over an area.
To save me some writing I think this diagram explains a lot of what I'm trying to say.

http://www.codecogs.com/users/23287/Cur ... ms-004.png

The sum of the forces on one side has to equal the sum of the forces on the other, or the bow flys away, hence the force distribution looks something like this. However the stress distribution would differ as the area is larger on the side with the flat face and higher peak force.
SO REALLY, its a function of the shape more than anything.
I don't think your wrong nez I just think you made a fair few assumptions. An excel sheet should sort out all the issue of Shape vs. Peak Stress though.

He's right, I made a lot of assumptions!

The three smart brothers are discussing modelling limbs on computers now, which is way beyond my area of interest, but there are those on the board who might be interested. I think I've hijacked this thread enough though, so if anyone wants to carry on the theorising it might be appropriate to start a new thread?

Neil

[hunterguy1991]

Funny what a little discussion can do to a group of engineers isn't it!!