Arrow Selection & Slender Column Buckling

[Joe Tapley]

In The appendix to Blythe's paper "Ballistic Properties in Ancient Egyptian Arrows" the Euler buckling criteria appears to be used to define the maximum loading that can be applied to the arrow.  Exceed the buckling load and the arrow structurally collapses and it's a gonner.

My own take on this is the complete opposite. In order to get around the "archer's paradox" problem the arrow has to bend and the way to get the arrow to bend is to exceed the Euler buckling critical load. The arrow isn't inevitably a gonner (as explained in the Archers Paradox section on the main site) as it has a get out clause. The front of the arrow is free to accelerate and relieve the bending stress.

If I apply the Blythe methodology to my own arrows (26.5" ACC 3L-04 with 100 grain points) it gives me a critical string load of 109 Newtons (25 pounds). This is equivalent to a launch speed of around 45 m/s (146 fps). I estimate that my actual launch speed is more like 185 fps. This equates to a maximum string force on the arrow of around 138 Newtons and assuming 80% bow efficiency to a 39 pound draw weight (which is in perfect agreement with the Easton arrow selection tables).

So in my case it seems that the string force on the arrow significantly exceeds the buckling critical loading (which is what you would expect, otherwise you are stuck with having to come up with some explanation of why the arrow bends in the first place).

The above graph includes a calculated string force on the arrow curve (E) from Kooi. The initial force, while below the static draw force value is still high but rapidly drops initially (as the arrow buckles). The string force variation oscillates (based on a point mass arrow) during the power stroke.

The following table compares the shaft critical loading value with the draw force based on the Easton 2014 arrow selection chart (assumes a point weight of 100 grain). As can be seen the shaft critical buckling load is around 70% of the draw force. This is in agreement with the experiment reported by Liston who found that when statically loading an arrow using the bow string the arrow would fail before reaching the full bow draw weight.

and for comparison

Nibb point	Shaft length	Arrow	Shaft gpi grains	Spine	Draw Weight	Critical Load	CL/DW %
grains	inches	Type	grains	Easton	Pounds	Pounds
71	28	X7 2014	9.56	579	43	31.88	74.14
64	28	X7 1914	9.28	658	38	28.34	74.58
64	28	X7 1914	9.28	658	33	28.34	85.88
60	28	X7 1814	8.57	799	28	23.29	83.18
102	30	X7 2212	8.84	505	43	30.15	70.12
100	30	X7 2114	9.9	510	38	30.47	80.18
71	30	X7 2014	9.56	579	33	28.03	84.94
64	30	X7 1914	9.28	658	28	24.91	88.96

The attached graph is from Kooi's Archers Paradox review indicating the calculated bow weight versus stiffness (spine) for a variety of arrow behaviour models. The graph relates to a wooden (longbow) arrow found in Westminster Abbey in London. The Blythe curve is based on the critical buckling criteria. The predicted (bow draw weight) values are notably lower then the other models. This is because the buckling criteria only indicates the threshold draw weight for an arrow of given stiffness to bend not the draw weight for the arrow to "match" the bow.

As with all mechanical models of arrow flexing  behaviour there are major technical simplifications in this Euler Buckling approach i.e.

• With Euler's formula the rod is weightless and the load acts at one end so the compressive stress in the rod is a constant. With an arrow the weight is distributed between point and shaft and as the force is generated by acceleration the axial stress (and hence stiffness) varies along the rod.
• With Euler's formula the ends of the rod are pinned and only move along an axis aligned with the applied force. With an arrow while the point can be regarded as being pinned the nock end moves a significant distance laterally wrt the force direction. The shape of the buckling arrow is very complex and not the simple Sine configuration of the classical Euler approach.

Having said that,  the Euler critical load as calculated by the Blythe method could be quite a good measurement of what archers refer to as the "Dynamic Spine". The higher the Euler critical load the "stiffer" the arrow is likely to be in a tuning sense. An example application could be calculating the change in critical buckling load (Dynamic Spine) by changing the arrow point weight. An increased point mass will act to weaken the arrow. Any  increase in insert length associated with the heavier point will stiffen the arrow as the shaft length will be reduced. You can use the buckling calculator, referenced below, to predict whether overall the heavier point will make the arrow stiffer, weaker or make little difference.

Here are a some nice related links:

An Excel spreadsheet to estimate the arrow Euler critical load is available from the downloads page.

Euler Buckling Video - An arrow is generally taken to be the pin-pin scenario.
Arrow failure - What is interesting here is that it is the excessive Archers paradox (arrow to bow handle force) that is the ultimate cause of failure not the simple buckling process.
Euler Buckling Theory - introduction to basic theory (See the Blythe reference above for how the basic Euler equation is modified for an accelerating arrow on a bow).