Effective Bending Stiffness of Flexible Sword Trainers

Right before WMAW 2013, I got to thinking about certain aspects of equipment safety as discussed a bit in an earlier post. Part of this was about starting an on-going investigation into the behavior of flexible sword trainers from an engineering mechanics standpoint. The hope is that it may inform safety requirements for groups and competitions as well as equipment design, and if nothing else be a bit of science-y fun.

This post will focus on the first part of that investigation: the determination of an effective bending stiffness for flexible sword trainers. I'll present data based on weapons I was able to measure myself, and provide the methodology so others may do the same. I'll take any and all data reported to me, cross-verify it where possible and make it freely available here.

Back in October, I made available a pdf of the procedure I'll discuss here, but this post will get into more detail and present more data so should be thought of as the 'real deal'.

Edit, 10 Dec 13: I fixed a sign error in the flexure formula. Signs be crazy, bro.

Motivation

Many fencing events are governed by organizations such as the Society for Creative Anachronism (SCA), USA Fencing, the Federation Internationale d'Escrime (FIE), or the International Kendo Federation (IKF). For these events, there are rules laid out by the governing body regarding the requirements of both protective equipment and weapons. The nature and source of these rules vary, but for the SCA, FIE and USA Fencing, one aspect of these rules is a test to determine if a participant's weapons fit into an acceptable range of bending stiffness.

For the FIE and USA Fencing, a single test is used for all weapons though the specific requirements differ. The test involves fixing the blade 70 cm from the point and applying a 200 g-f weight 3 cm behind the point of the blade. The deflection of the point under these conditions must fall into a weapon-specific range (e.g. 5.5 cm to 9.5 cm for foils)[1]. The SCA makes use of a similar test but fixes the handle of the weapon, uses a 3 oz (85 g-f) weight applied at the point, and has minimum deflections based on the length of the blade (blade >= 18 in: 0.5 in deflection. blade < 18 in :  0.25 in)[2]. The loads are applied in the direction of the narrowest dimension of the blade's cross section (i.e. on the 'flat' for typical swords).

HES, however, is a relatively young activity with a variety of groups and events, each with their own internal metrics and rules. As there is no single governing body responsible for the homologation of equipment (as is the case with Kendo and Olympic Fencing), manufacturers do not have specific guidelines for stiffness and participants must rely on experience (either their own, or the community's) in selecting a trainer since there is usually no advertised metric for the stiffness of a given weapon.

In addition to my own efforts, there have been at least two articles by other authors on related topics. One brought engineering mechanics to bear on the reaction of fencing trainers by way of a comparison between bending stiffness and buckling loads[3]. The other was a discussion of the standard the FIE uses as a basis for the approval of masks and protective equipment, in the context of its applicability to HES[4].

Bending of a Cantilever Beam

The tests used by the FIE, USA Fencing and the SCA are all based on a straightforward mechanical system: the cantilever beam with a transverse load. Originally I was going to discuss this problem in gory detail, along with a couple of other problems, in Mechanics 101 part 5. But then Blogger ate it and I changed my workflow. So I'll do it here, in a slightly less gory manner (probably still R-rated though). If you're willing to take my word for it, go ahead and skip to the end of this section for the important stuff.

The tests all describe a member that is fixed at one end and free at the other. At the free end, a transverse load is applied. This load results in a deformation in the direction of the applied load. This system is referred to as a cantilever beam, and is represented schematically below:

Diagram of a simple cantilever beam, where \(l\) is the length of the beam, \(P\) is the applied load (static point force), and \(\Delta\) is the displacement caused by the applied load.

 This problem is one of the classic problems in strengths of materials courses and is an incredibly well traveled road with a number of uses. So let's dig into it. For Mechanics!

Just like in the other mechanics examples, our first step is to draw a FBD to determine the unknown reaction forces in the beam:

FBD for the static equilibrium of the cantilever beam.

Since we have no axial forces, we only need to sum forces in y-direction, and moments about the z-axis:
\[\sum{F_y} = F_R - P = 0 \rightarrow F_R = P\]
\[\sum{M_z}\bracevert_{x=0} = M_R - Pl = 0 \rightarrow M_R = Pl\]

Our next goal is to determine the distribution of shear forces and moments in the beam, and from that the stresses. For such a simple system, it is very straightforward to use a graphical approach called shear and moment diagrams. But before we get into the development of the diagrams, there's a slightly different sign convention used when looking at the internal forces in a beam than in our static equilibrium equations. The convention for positive internal shear and bending moments at different ends of a section is as follows:

Sign convention for positive internal shear force (V) and bending moments (M).

We start with the shear diagram. Starting from \(x=0\), we have the shear load at the fixed support (\(F_R = P \)) and don't have a change in shear force until we reach the applied load that acts in the opposite direction, \(-P\) at \(x=l\). Therefore the shear diagram looks like the following:

Shear diagram for the simple cantilever beam.

 For the internal moment, we know we have a reaction moment at \(x=0\) of magnitude \(Pl\) in the counter-clockwise direction. From the beam sign convention, we know this means we have a moment at x=0 of \(-Pl\). Also, we know that there are no other applied moments and that the internal shear distribution will correspond to a moment at each point along the axis of the beam, with magnitude \(Px\), acting in the positive direction. In other words, we know the shear is the derivative of the moment distribution. This results in a moment diagram that looks like the following:

Moment diagram for the simple cantilever beam.

 Based on our earlier discussion about the definition of shear stress, we can readily define the nominal transverse shear stress in this beam based on the shear diagram:
\[\tau_{xy} = P/A\]
Where \(A\) is the cross-sectional area of the beam.

To get the normal stress, we can recall that moments act as linearly distributed normal forces. Therefore at a given section, we'd expect that at a distance \(y\) from the bending axis, the stress would also be linearly distributed:
\[\sigma(y) = \frac{y}{c} \sigma_{max}\]
Where \(c\) is the maximum possible distance above the bending axis and \(\sigma_{max}\) is the maximum normal stress above the bending axis. For a positive bending moment, the top surface of the beam is under compression (negative normal stress) while the bottom is in tension (positive normal stress). The bending axis, also called the neutral axis or centroid, is defined by the axial plane in the beam that does not change length during bending (called the neutral plane). For cross-sections that are symmetric in the transverse plane (like an I-beam, a round rod, etc.), this neutral axis passes through the center of the cross-section.

But we don't yet know a relationship between the bending moment and the maximum stress. To get one, we can start by looking at the sum of moments about the neutral axis of an arbitrary section of our beam (static equilibrium). At each distance \(y\) from the neutral axis, there acts an increment of force \(dF\) on an incremental area \(dA\), such that the sum of moments acting on the surface of the section can be written in integral form as follows:
\[\sum{M_z} = M = \int{y\:dF} = \int{y\:(\sigma} \:dA) = \int{y\:(\frac{y}{c} \sigma_{max})} \:dA \]
Which can be re-arranged into:
\[M = \frac{\sigma_{max}}{c} \int{y^2}\:dA\]
We can define \(I \equiv \int{y^2}\:dA\), which is the formal definition for the area moment of inertia. I discussed some interesting aspects of the area moment of inertia in a previous post. Finally, we arrive the relationship between the bending moment and the maximum normal stress:
\[M = -\frac{\sigma_{max}I}{c} \rightarrow \sigma_{max} = -\frac{Mc}{I}\]
This is called the flexure formula. For an arbitrary position in the cross-section, the formula is as follows:
\[\sigma = -\frac{My}{I}\]
In the last two equations, the sign comes about to ensure that a positive bending moment results in the top surface (y > 0) of the beam being under compression (negative normal stress), while the bottom surface is in tension.

What we really want though, is a relationship between the applied load and the displacement at the point. The above work will actually be required to determine it. The next thing we need to do is look at the curvature of the bending beam. Let's consider an initially straight bar being bent upwards. As described earlier, there exists some axis along which the length of the beam does not change (the neutral axis). If we were to watch an arbitrary cross-section during bending, we would expect to see a rotation of the section, and possibly also a deformation (the cross-section could change). For modest deformations o fslender sections, whose thickness is much less than their length, each section tends to rotate and deform very little: the cross-sections can be said to remain planar and the cross section can be said to remain constant during bending. These assumptions are the key assumptions to the simplest (yet incredibly useful) form of beam theory: Euler-Bernoulli Beam Theory. Using these assumptions, we can develop a relationship for the radius of curvature, \(\rho\), of an arbitrary beam section as follows:

Schematic of undeformed section (left) and section deformed by a bending moment (right)

We can write the strain of the arbitrary line length \(ds\) in terms of its final length, \(ds'\), as well as in terms of the radius of curvature \(\rho\) and the distance from the neutral axis \(y\):
\[ds = dx = \rho d\theta ; \; ds'=(\rho-y) d\theta\]
\[\epsilon=\frac{(ds'-ds)}{ds} = \frac{((\rho-y)d\theta - \rho d\theta}{\rho d\theta} = \frac{-yd\theta}{\rho d\theta} \rightarrow \frac{1}{\rho} = -\frac{\epsilon}{y}\]
Where the last relationship, \(\frac{1}{\rho} = -\frac{\epsilon}{y}\), is the definition of the curvature of the deformed section. If we assume our material obeys Hooke's Law, then we know that \(\sigma = E\epsilon\) and therefore:
\[\frac{1}{\rho} = -\frac{\epsilon}{y} = -\frac{\sigma}{Ey}\]
Using the flexure formula, we arrive at the following:
\[\frac{1}{\rho} = -\frac{\sigma}{Ey} = \frac{M}{EI}\]
which expresses the curvature of a beam in terms of the bending moment \(M\), the Young's modulus \(E\) and the area moment of inertia \(I\). The product \(EI\) is called the bending stiffness, and is a measure that accounts for both material and geometric contributions to the beam's resistance to elastic deformation.

Our next step is to obtain a relationship between the curvature and the transverse deflection, \(u(x)\) of the beam. From calculus, this relationship is as follows:
\[\frac{1}{\rho} = \frac{d^2v/dx^2}{[1 + (dv/dx)^2]^{3/2}}\]
Which is a nonlinear second-order differential equation that defines the exact relationship between the curvature and the deflection for this beam. This type of equation is very difficult to solve analytically in the general case… Let's see what we can do about that.

First, let's do some substitutions to get a relation in terms of the bending moment and the bending stiffness:
\[\frac{1}{\rho} = \frac{M}{EI} = \frac{d^2v/dx^2}{[1 + (dv/dx)^2]^{3/2}}\]
Next, if we assume that the slope of the deflected beam (\(dv/dx\)) is small compared to 1, we can greatly simplify this equation by assuming the denominator to be 1. This assumption is very reasonable for small deflections of long slender beams, and can still be reasonable for moderate deflections. On the up side, we can also verify it after we perform our calculation (or if we perform an experiment). With this assumption, our equation now looks like the following:
\[\frac{M}{EI} = \frac{d^2v}{dx^2}\]
Now we have a linear second-order differential equation that can be easily solved if the term on the left hand side is not too complex. With some re-arrangement, we arrive at the following equation:
\[EI\frac{d^2v}{dx^2} = M\]
Where, in general, the moment and bending stiffness are functions of position \(x\).

We can obtain the analytical solution for the deflection of our cantilever beam by considering the free-body diagram for an arbitrary section of the beam:

FBD for arbitrary section of the cantilever beam, I've changed coordinates to make some of the math easier because we only really care about the deflection at the free end.

Doing a sum of moments, we arrive at the following:
\[\sum{M_z\bracevert_x} = -M - Px = 0 \rightarrow M = -Px\]
So our differential equation becomes:
 \[EI\frac{d^2v}{dx^2} = -Px\]
a linear second-order differential equation. By integrating both sides twice, we arrive at the general solution:
\[v(x) = -\frac{P}{6EI}x^2 + C_1x + C_2\]
Because we know the beam is fixed at its base (\(x=l\) in this coordinate system), \(v(l) = 0\) and \(dv/dx(l) = 0\). Therefore:
\[v(x) = \frac{P}{6EI}(-x^3 + 3l^2x - 2l^3)\]
The maximum deflection occurs at \(x=0\) and therefore:
\[\Delta \equiv v(0) = -\frac{Pl^3}{3EI}\]
Where \(P\) is the magnitude of the applied downward load, and a downward displacement is taken to be negative. This equation right here qualifies as the important stuff.

Cool… now why should I care?

The last equation is what the FIE and SCA bending tests are based on and now you know where it comes from. In a bending test, \(P\) is known (the applied weight), and \(l\) and \(\Delta\) can be measured. Therefore, you can solve for the effective bending stiffness \(EI\) for the weapon. I say effective because it will provide a single value that smears out any changes in Young's modulus or cross-section along the length of the weapon. The way the FIE and SCA rules work, the acceptable displacement \(\Delta\) is specified because it is an indirect way of specifying an acceptable bending stiffness range. The main advantage is that the indirect specification is far more clear to a layperson than the explicit specification of a bending stiffness range.

So why not just measure the bending stiffness directly by determining \(E\) and \(I\) then multiplying them together? Well, to be blunt: because it is terribly impractical compared to a bending test. The Young's modulus can be determined through materials testing using very expensive tensile test equipment and a great deal of time, money and knowledge. The area moment of inertia is difficult to determine for arbitrary sections, though it can be found analytically for simple ones as I discussed before. By comparison, slapping a weight on the end of a sword and measuring how far the point moves is dead simple. It's also cheap, fast, repeatable and provides you the same (or close enough) answer at a significantly lower level of effort. In engineering, we call this a win.

There is one final reason you should care about the bending of a cantilever beam: the bending stiffness \(EI\) is the same term that will govern the onset of flex during a thrust. This is because blade flex occurs through a process called buckling, where an axial load leads to a transverse deformation. I'll discuss this process in detail in a later post.

Measuring the Effective Bending Stiffness

Now that we've been through the derivation of the bending stiffness, I'm going to lay out a method by which we can determine it for a flexible sword trainer. Simply put, we're going to create a cantilever beam and apply a transverse load. We'll then measure the deflection and do a bit of math. Then we'll do a little happy dance and talk about it.

Equipment

The device used to measure the deflection of the sword should have the following components:

• A clamping area that is sufficiently large that it can accommodate the blade in a position parallel to the ground. The clamp should be such that there is no motion of the section of blade clamped nor in the hilt when the weight is applied to the point.
• A known weight that can be fixed to the point of the blade (or very close). This weight should be on the order of 200-500 g-f, to ensure that the loading is not too large to cause excessive deformation, but yet sufficiently large that a measurable deflection of the point will occur. One option is a plumb-bob type arrangement, or a C-clamp of the appropriate weight. Some experimentation with the weight may be necessary in general.
• A ruler, meterstick, tape measure etc. that can be used to measure the initial and final (after the weight is added) positions of the point, as well as other miscellaneous measurements. 1mm resolution preferred.
• A level to ensure that your deflection measurement is taken consistently with respect to the vertical and that the un-loaded clamped blade is approximately level.

Here's a picture of a basic setup that I made from stuff I had lying around. It was my first attempt and is pretty crude but produced repeatable results. The level can be seen on the lower left.

Basic homebrew setup.

It is of course possible to build a purpose-built device, but I haven't really had a need. For complex hilted weapons, a riser block is likely to be required to accommodate the furniture of the weapon.

Measurement Procedure

The following is the procedure for using the above apparatus to measure the point deflection due to the applied load. The next section will deal with the reporting of the data and associated calculations.

1. Remove any rubber or plastic blunts, tape, etc. from the blade.
2. Measure the weight and any attachment hardware to be used to the nearest gram. This is your applied load, P. This value should be recorded and reported.
3. Measure overall blade length, \(l_0\), to the nearest millimeter, from the point down to the intersection of the cross and base of the blade. For weapons with guards or crosses that result in some areas of the blade being covered, or have a non-uniform shape (such as a pointed portion of the cross at the center of the blade), measure to the least protruding section of the guard that directly contacts the blade. If there is an angle to the guard that contacts the blade the entire way, use the average of the longest and shortest lengths. (I'll get some pictures to demonstrate this more clearly, at some point)
4. Determine 1/3 of the blade length (to the nearest millimeter), measured from the point on the blade near the cross used in the previous step. Mark it on the weapon with chalk, tape or some other easily removable marking. This will be the position that will be aligned with edge of the clamp area that faces the point. If your blade is known to have a specific area that flexes while the rest remains rigid, use that portion instead and adjust your length measurements as needed.
5. Clamp the weapon with the flat of the blade parallel to the ground (use a level if needed), ensuring the clamp is tight enough that you cannot move the sword in the fixture without significant effort. A very snug hand-tightness worked in my basic setup. The marking placed on the blade in the previous step should be aligned with the edge of the clamp towards the point. Re-measure the length from the point down to the edge of the clamp. This is your effective blade length, \(l_{eff}\). Write this number down, as it will be used in the calculations and should be reported along with the overall length.
6. Measure the distance from some reference surface directly below the point (such as the floor, or a stack of books) to the top of the point (i.e. the surface away from the ground, at the point), to the nearest millimeter. This is your initial position \(d_0\), which should be recorded and reported.
7. Attach the weight as close to the point as possible. Gently release the weight (the blade will likely wobble a bit). Steady the wobble gently with your finger without pushing or pulling on the blade until the wobble stops. Remove your finger from the blade.
8. Measure the distance from the same reference surface used in step 6 to the top of the point, to the nearest millimeter. This is your final position \(d_f\), which should be recorded and reported.
9. Remove the weight, steady the blade, and re-measure the initial position. If it has changed, repeat the above process to ensure a good deflection value has been measured.

Calculation of the Effective Bending Stiffness

Now that we have the data, we can plug the values into the following formula:
\[EI (g*mm^2) = \frac{P(l_{eff}) {}^3}{3(d_f - d_0)}\]
The value can then be converted into the more customary unit, \(\mathrm{N*m}^2\) as follows (the prefactor is just the condensed unit conversions):
\[EI (N*m^2) = \frac{9.806}{1000^3} EI (g*mm^2)\]
The converted value should be reported along with the raw measurement data, as it will make it easier to cross-verify the results.

You can also short-cut the last step by converting your measurements into Newtons and meters beforehand. I did this in my spreadsheet because it is really easy to do so.

Data Reporting (and where to send it)

Here is a handy list of values that should be recorded:

• Weapon make and model
• Weight used (\(P\), in g-f)
• Overall blade length (\(l_0\), in mm)
• Effective blade length (\(l_{eft}\), in mm)
• Initial position (\(d_0\), in mm)
• Final (loaded) position (\(d_f\), in mm)
• Calculated Effective Bending Stiffness (\(EI\), in N*m^2)

If you know the precision of your measurement instruments, please include that as well. For example, if your scale only measures to the nearest gram or if the smallest division on your ruler is 2 mm. Even better is if you know the accuracy of them as well: some scales will give you the expected measurement error on a label, such as +/- 0.5 gram. If you can provide a picture of your weapon, that will be a fantastic.

If you are wondering why I am asking for all of this: it's so that if I get someone's data, I can verify that they did the math right and that there is nothing wonky about the setup and results. That's right, your chem prof wasn't just being a douche: there's actually reason to record the values used in a calculation instead of just the result.

I invite anyone who reads this to go ahead and measure whatever trainers they have access to. If you provide me the above data, I will publish all of the valid data I receive (assuming it passes the sniff test) here on this blog for the community at large. I'd like to get as many measurements on as many trainers as I can: even if people do the same ones, it'll provide useful data. You can get in touch with me using the contact form in the sidebar, or simply send me plain text data. Please don't post it in the comments.

Results of My Own Measurements

Thanks to a good friend and some trainers from my own collection, I've been able to take measurements of 9 different sword trainers and 1 dagger trainer. My weight measurement was within 1 gram, and my distance measurements were within about 1 mm. I also took the allowed displacement values for FIE homologated weapons, their regulation length and calculated the allowed range of bending stiffnesses in the same manner as the weapons I actually measured.

Here are the weapons I measured (I also measured an old Del Tin rapier blade, but it isn't pictured):

Photograph of the longsword trainers used in this study. A) Albion Swords Lichtenauer, B) Tinker-Hanwei blunt longsword, C) Hanwei Practical bastard sword, D) Arms & Armor Fechterspiel, E) Swordcrafts longsword.

Photograph of the arming sword (one-handed sword) trainers used in this study. A) Tinker-Hanwei blunt arming sword, B) Hanwei Practical one-handed sword (4th gen), C) Darkwood Armory-Hanwei rondel dagger.

And here is a plot of the effective bending stiffnesses I determined, along with the FIE stiffness ranges:

Dat data.

Discussion

My data provides a nice illustration of the wide range of stiffnesses available in flexible metal sword trainers. At the top end, we find the longsword trainer produced by Swordcrafts, which is approximately 1.8 times stiffer than the second stiffest trainer measured, the Albion Swords Lichtenauer (part of their `Maestro' line). The Swordcrafts longsword is nearly 3 times stiffer than the least stiff longsword measured, the Arms & Armor Fechterspiel. Only one single-handed sword was in the stiffness range of the longswords: the Hanwei Practical one-handed sword (4th generation). This one handed sword is about 20% stiffer than the Arms & Armor Fechterspiel, while also being significantly shorter (by approximately 20 cm). The least stiff of the weapons tested was the Darkwood Armory rondel built with a Hanwei flexible dagger blade, which we can see falls significantly below the other weapons tested.

Based on this data, we can see a trend in stiffnesses: longswords tend to be more stiff than single-handed swords, and both are significantly above the FIE homologation range. From a user's and mechanics stand point, this makes sense: the longer the weapon, the greater the bending stiffness that can be allowed for a given amount of blade deflection under the same load (see the equation for the deflection we derived earlier).  In other words, if the bending stiffness was kept the same, but the length increased, you would get a larger deflection for the same applied load. This translates in user terms to a 'whippy' blade that may bend while performing certain basic actions (winding, deflections, etc.) if the flexibility is taken to an extreme. Similarly, if a short one-handed sword was made with the same bending stiffness of a longer weapon, you'd expect a decrease in the deflection for a given load. This would translate to an increase in energy transfer to the target compared to a lower stiffness blade. So for a given length, there is a balance to be struck: stiff enough to allow for actions using the blade, but flexible enough to remove a reasonable amount of energy from a stout thrust by flexing.

An interesting finding to consider is the case of the Del Tin rapier blade and Hanwei Practical bastard sword (the trainer with the next closest effective length): the Del Tin rapier blade is a long single handed sword blade, approximately 2 cm longer than the Hanwei Practical bastard sword (a two-handed weapon). The Del Tin is about half the stiffness of the Hanwei and appears to be a counter-point to the idea just discussed that a longer training sword should be stiffer than a shorter one. However, the intended use of a rapier and the intended use of the longsword differ greatly: Firstly, the rapier is primarily a thrusting weapon and so long as it is stiff enough for basic blade-on-blade actions and to maintain point-presence, you are free to make it more flexible as a way to make a safer trainer. This is because any losses in cutting ability or the ability to use the blade as a lever arm are generally less important than in the typical usage of a longsword. Whereas with a longsword, if the blade bends too much or too easily, you may end up with strange artifacts in certain common actions, such as binds and cuts.

The effective bending stiffness for a trainer can also serve to differentiate between trainers based on relative flexibility. For example, if I know that a particular trainer isn't stiff enough for my tastes, I can look for ones with a higher effective bending stiffness, just as I would look for a lighter trainer if I felt mine was too heavy. Therefore, the effective bending stiffness can be a useful metric for flexible sword trainers much like the more typical attributes of weight, length, center-of-balance, etc. Finally, The effective bending stiffness also comes into play when we consider what happens to a blade during a thrust, something I'll discuss in detail in a future post.

References

1. USA Fencing Rules. United States Fencing Association, January 2013 edition, 2013. 
2. Rapier Marshal’s Handbook. The Society for Creative Anachronism, Inc., 2006.
3. L.B. Eldred. Engineering analysis of the deformation of fencing swords. 2011. Available at: http://www.baronllwyd.org/articles/Sword bending.pdf, accessed Oct 26 2013.
4. K. Farrell. Construction of a fencing mask. 2012. V. 01. Avail- able at http://www.historical-academy.co.uk/files/research/keith-farrell/Construction of a Fencing Mask.pdf, accessed Oct 25, 2013.

As a note, I'm going to see about putting my spreadsheet up on google docs, so that anyone can view it, and if I can figure out some controlled access scheme, I'll let folks put data in there. I'll update when I get that sorted.