Saturday, 20 July 2013

Post 109 The science of wire

When we wire a tree branch to bend and hold it in a new position, we want the bending resistance of the branch to be overcome by the willingness of the wire to take a permanent deformation; but what does the science tell us so that we can improve our chances of success. We all know from our own experiences and from watching others that wiring is far from a simple proposition.

When you bend a piece of wire a little bit it will elastically come back to where you started. Bend it some more and you can eventually feel it yield, come back a little and then stay in a  new position. Now try to bend it back and it requires more effort to do so than the original bend. All metalic materials behave this way but each in different quantities some requiring more effort than others for example to get to yield.

Mechanical material science tells us that materials have a measurable property called the Modulus of Elasticity which defines how that material behaves when a bending force is applied. This Modulus is the slope of the stress/strain curve. For stress think force and for strain think displacement or distance.

This is what a typical curve looks like. The bending starts at the intersection of the axes. As the stress is increased the strain does also etc.

The elastic region is where you bend and it comes right back. In this region a stiffer material like copper will have a steeper angle than for a softer material like aluminium, which says that for the same force the stiffer material will bend a smaller distance before reaching yield.
Yield is where it begins to take on a permanent new set.

We know that from experience but it's all in the curve. The slope of this region is the Modulus of Elasticity and measurements tell us that Copper's modulus is about 1.7 times that of Aluminium.
So for a Copper wire and an Aluminium wire of the same diameter the Copper one needs 1.7 times the effort to bend the wire the same distance as the Aluminium one. Or more importantly for bonsai the Aluminium wire will bend 1.7 times as much as the Copper BEFORE reaching yield, which is where we need to get to bend the branch. That's a pretty important factor if you are looking for fine changes in ramification.

What else does the curve tell us.  Once we pass the Yield Point we get to the Plastic Region where the long term shape is plastically changed; but it's not as clearcut as that.

One of the other dominant wiring observations and challenges is that when you bend the wired branch, even after reaching the end of the purely elastic region, when you release the pressure there is certain amount of recovery in position. That is to get to a position you have to bend it further than you want to compensate for this recovery. The two reasons for this are that the wire characteristics retain a level of elasticity applying from the new point and under the counter pressure from the branch which is still in its elastic range the wire is bent back  towards its starting position.

On the following graph we track along the red line to Yield, a little further and then release the pressure. As the pressure comes back the displacement (Strain) tracks down the short line to X which will be the starting point for any subsequent bend. This shows how there can be some recovery in the old position which is also exacerbated by the elasticity of the branch.

Having to bend further than you want to achieve wire yield and overcome the branch elastic recovery is when most branch fractures occur. The answer of course is heavier wire, but there comes a point where applying heavy wire, bending it and the branch and then getting it off eventually may be a few too many risks to the branch.

The size of the wire is interesting. It is self evident that heavier wire requires more effort to bend than lighter, but how much?  The cross-sectional shape of a piece of wire is circular and we know that the area of a circle is Pi * Diameter squared / 4 so if we double the diameter of a piece of wire its cross-sectional area is 4 times greater. The interesting thing about bending however is that Bending Effort is proportional to both the Elastic Modulus by the Area Moment of the material shape. For a circle that Area Moment is proportional to the diameter to the 4th power. So double the diameter and the Bending Effort is actually 2 to the power of 4 greater, ie 2*2*2*2 which is 16 times greater.

So double the wire diameter and this needs, and you thus apply to the branch, 16 times the effort or force. That's the power of scale and why just small increases in wire size are so effective.

Now to compare Aluminium and Copper if we use a copper wire twice the diameter of an aluminium one it requires 16 *1.7 or about 27 times the effort to move it the same distance and by the same deduction applies that level of resistance to the branch elastic recovery!!!! And what size of Aluminium wire would you need to apply the same force as a piece of Copper wire? Well the answer will be the fourth root of 1.7, which is about 1.15 which says that an Aluminium wire just 15% larger diameter than a Copper wire will offer the same bending resistance. One thing that doesn't change however is the Modulus, the slope of the curve, which means that you still have to bend the Aluminium wire, even the bigger piece, further to get the same effect.

We need to think about the branch in somewhat the same way. A branch of more than a couple of seasons age will have its own elastic bending characteristic, recovering from a bend if you only go so far and then if you go too far as well all know you will reach yield. Unfortunately the yield we are talking about is failure, because timber generally does not have a plastic region. When we wire a branch and hope to achieve a new shape we need to exert sufficient force to achieve both the yielding of the wire and to overcome the elastic recovery of the branch. Unfortunatley I've seen too many demonstrations where branches are put at risk from over bending where the wire will never hold the position sought because of the characteristic of elasticity in the plastic recovery and insufficient wire weight.

 Now let's take a moment and talk about springs, actually to be specific Helical Springs, things like these.

The reason we need to think about these is that when we wire a branch we actually end up with a wound spring on the branch. So how do springs stay springy? Well because of the spiral nature of their design and the true length of the wire used to make them they enable the material to stay in its elastic region with a large displacement. This is true axially, because that is where they are normally deisgned to bear load from, but also laterally too, ie from the side.
Now the best way to be clear about that is to take it to the extreme of a 'slinky'.

This helical spring structure has no resistance at all to lateral forces with so many tight windings that the material will never approach yield. Let's define spring Pitch as the distance between any two consecutive windings. So with a slinky the pitch is the width of the wire and for the other springs above it is closer to half the diameter of the spring.

The issue of resistance to lateral forces is very important for our bonsai application because that is exactly what we rely on to hold the position of a wired bonsai branch. So what pitch should we apply when wiring a branch. Clearly a closely wound wire with small pitch will offer little resistance to the branch and the wire will be less likely to reach yield, ie useless. A very long pitch with only a couple of windings will give the best bending result but with little branch contact the wire may be ineffictive at containing the branch. The middle ground is best with as long a pitch as possible while still wrapping the branch.

A really interesting question after all this then is to ask if you don't have heavy enough wire to do the job what is the best way to proceed.

One approach is to double up with the wire you have. Ok so what do the numbers say about that. If one piece of wire applies a load of F then two will apply 2F.
Now two pieces of wire of diameter d would be equivalent to one piece of wire of diameter 1.4d, cross sectional area wise. We know from the bending stress calculation that the bending stress of the larger piece of wire would be the increase in diameter to the 4th power, or about 3.8 times ie 3.8F. So if you are going to use strands of the same wire you need to use almost 4 strands, not two, to get the same effect from one piece of wire of the combined weight of two strands of the smaller wire.
Heavier wire is always the best answer, more of the same will rearely do it and heavier wire, kilo for kilo gives you much better bang for your buck.

There is another approach and that is the one often used in extreme bending practise where straight sections of wire are raffiaed to the branch and then followed by a helical winding. This can work just as well without the raffia and as the straight sections are bent they will offer much greater impact than the same number of additional helical windings.

The other option is to use a branch bending clamp. I like these because you only have to bend a branch to the position you want and no further to provide for any recovery. You can also do it in a slow and progressive manner and get the bend exactly where you want it. Alternatively guy wires offer the same benefit when an appropriate anchor point can be provided. For a simple big bend I would much prefer the control and limited bending requirement of a clamp or guy wire in preference to the uncertainties of heavy wired freehand branch bending.

 The summary of all that is as follows; the Happy rules of wiring!
  • Heavier wire is many times more effective than its equivalent weight in smaller strands and with wire sold by weight the heavier wire will give the best bang for your buck.
  • If you have to bend a branch past the point you want it to stay your wire is not the right weight for the job.
  • If you need but don't have heavy enough wire better to apply straight strands along the branch under one helical application than additional helical bands.
  • When wiring apply with the longest pitch possible while containing the branch.
  • Stiffer wire is always better but the increase in stiffness can be countered by only a very small increase in diameter of the softer alternative.
  • If simple bends can be accomplished with bending clamps or guy wires, these will most likely offer controllable alternatives to getting a branch in the position you want.

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