Force, Stress, and the Hoof, Part 2

So far, we’ve been talking about the hoof as if it’s a single structure, even though we know it’s really a composite of different parts.  The terminal phalanx that I wrote about in my October 18 post, known as the horse’s coffin bone, is sheathed in a highly complex nest of tissues that are designed to safely handle stress. You can see in the stylized cross-section to the left that the “hoof” is made of hoof material itself, ligaments, tendons, bone, connective tissue, and skin.  Many of these tissues are made of combinations of collagen and elastin, two types of fibers that both stretch and absorb energy in different ways.  Cartilege is also a large component of these tissues, increasing with the age of the horse.  (See this paper by Robert Bowker, VMD and PhD at Michigan State for more details.)

It is exceedingly difficult to measure the strength of such compound structures. In our last post we therefore generalized about the hoof as a whole to get a rough idea of what’s going on. And we’ll do that again in future posts.  But in this post, we’re going to consider the issue of how various parts of the hoof transmit stress.  Then we’ll return to considering the hoof as a whole.

In that first diagram above, the terminal phalanx or coffin bone is labeled p3 (for phalanx 3).  You can see the hoof wall material all down the front of the hoof in front of that bone, and the frog is beneath it.  A thick pad of special tissue called the digital cushion is also between the p3 and the ground.  It’s important to remember that the hoof wall extends all the way around the foot, though, which you can’t see in the section shown above.  It’s also important to remember that the horse has a sole across the bottom of its foot that isn’t very visible in the section above.  Both are shown in this bottom-of-the-hoof view from the University of Missouri Extension website.

Roman “hipposandal” boot-like hoof covering made of metal, about the first century CE.  From Wikipedia.

The history of putting protective coverings over the hooves of horses is nearly as contentious as contemporary discussions of whether horses should be shod, go barefoot, or wear boots.  In 1934, Professor A. D. Fraser wrote in Classical Journal about the academic wrangling going on even then — 80 years ago! — about whether the Romans shod their horses or the practice began in much later Medieval times.  But one thing that is clear is that two different types of hoof coverings have been used by Eurasian cultures when they did anything at all to a horse’s hoof.  There were boot-type coverings that went across the entire bottom of the hoof, and shoes that were nailed only to the hoof itself, leaving the sole bare and elevated above the ground. Of course, we have both types of hoof coverings available to us today, for our horses, although it’s only in the last couple of decades that boot-type coverings have reappeared after being absent for quite a long time.

One of the most striking things about the underside of a horse’s bare hoof is the flatness or concavity of the sole.  Some barefoot horses have flat soles that touch the ground all the way across, whereas others have slightly concave soles that are domed so that only the frog touches the ground.  Of course, the imprint of the frog is visible in a barefoot horse’s print, regardless, because it is made of different material that leaves a distinctive impression in the dirt.  You can see the impression of the frog and of the outer hoof wall on the modern hoofprint image (from a barefoot horse) in the top picture of the set to the right.

What’s interesting to me is that you see much the same shape in the image directly beneath that one.  That image was carved into rock about 24,000 years ago in Le Cellier Cave in France, and is almost always identified by archeologists as the picture of a vulva.  (You can see several other similar images carved into the same rock in other areas if you look closely.)  But in that same cave, there is another carving of the same shape made over the top of a horse’s head and neck.  A drawing of the shape is third in the column of images, and a photo of the original stone is beneath it at the bottom.  An archeological description of the piece says “. . . it is a stone much less voluminous, carrying a bizarre design. The image appears to be the head of a horse, and on the right a vulva not well represented. These designs are united by a line which seems to indicate a real relation existed between the two.”  One is tempted to speculate that the “real relation” might be that both images represent parts of the same animal.  (You can read more about these shapes and other items found in the cave at a page of the Don’s Maps archeology website.  The hole in the hoof/vulva is thought to have held oil, the stone being a small portable lamp.)

There’s no particular reason here to argue that these hoof-like shapes are actually hooves, that their having been carved into stone tells us something about how these ancient people felt about horses, or that the level of assurance with which contemporary archeologists identify the shapes as vulvas is also meaningful.  But the carvings at least suggest the fact that people have been paying attention to the shape a horse’s hoof makes in dirt for quite a long time.  And of course, horse hooves are important symbols to people even today — as you can tell by looking up “horse jewelry” online and counting the number of horse-hooves and horse-shoes that come up.  (I should note in passing that there’s a good deal of professional and lay argument about shapes like the ones from Le Cellier even in ancient Celtic art that’s only about 2000 years old, with an additional level of discussion about a possible metaphor that may have linked women, fertility, and horses.  But that’s fodder for a different post.)

The point is that the sole of a horse’s foot doesn’t always touch the ground, even when the horse is barefoot.  And we all know that the sole can be bruised or “sored up” in some way by sharp or stony surfaces.  In fact, one of the reasons shoes or boots are used is to protect this sole.  However, it’s interesting to notice that there is a likely biomechanical reason for any concavity of a horse’s sole.

At left is a diagram (modified from one on a Federal Transportation Authority webpage) of two different types of reinforced concrete beams.  The one on top is a “regular” beam that is cast in a flat shape.  When force is applied to the beam (because it is helping to hold up a bridge, for example), the stress causes the beam to bend or bow in such a way that it cracks on the underside.  The black arrows in that diagram show the force being applied, and you can see the way the cracks are represented.  This really does happen in beams that experience loads greater than they can bear.

However, if the beam is cast in a bow or curve that faces against the direction in which it will be loaded, there’s a different outcome.  This type of beam is called “prestressed” to indicate the way it can “resist” the stresses caused by bearing a load.  Now when the load is applied (again, see the arrows), the stress flattens out the beam instead of bending it.  So cracks do not form.  Many domed or bent structures are actually designed to resist force.  Skulls, for instance, are domed outward to resist the stress of fairly large force against the head from the outside (as from falling and hitting the ground).  So you can see that the degree to which a horse’s foot is slightly concave on the underside allows it to carry stress better without failing.  This doesn’t mean that a flat sole that touches the ground all the time is “bad”, though.  In fact, a hoof that’s slightly concave when you pick it up off the ground to look at it might flatten out and touch the ground when it’s bearing the horse’s weight.  We are not establishing a value system here, but merely considering the possible adaptiveness of certain aspects of hoof structure.

One of the more interesting things to consider is how a typical metal horseshoe affects the way that stress is transmitted through the horse’s leg.  Gail Snyder, a hoof care professional who also has a Mechanical Engineering degree, wrote an excellent overview of this difference in a 2012 issue of Natural Horse (that you can download here).  She assumed that a typical barefoot horse has a flat sole and therefore distributes force across the entire surface area of the hoof — just as we assumed in our calculations of Part 1.  She assumed a smaller sized foot for her calculations than I did, so her figures came out slightly different, but the important issue is how adding a horse shoe changes the stress in a horse’s foot.  Once a horse shoe is in place, as shown in the figure to the left (from her paper), the surface of the foot that’s in actual contact with the ground is much smaller, restricted to the bottom of the shoe itself.  The stress in the weight-bearing part of the hoof therefore goes up — by over 230% — simply because the shoe is so much smaller than the whole foot.  (Remember, Stress = Force divided by area.  The shoe is smaller, so the area is smaller, which makes the Stress larger for the same Force.)

There’s an intriguing corrollary to this, too.  To understand it, you have to realize that force also comes up into a horse’s foot and leg from the ground.  If you hit the wall next to you with your knuckles, reading this, you will feel what I’m talking about:  you hit the wall, but the wall also “hits” you back and smacks your knuckles.  (If the wall did not “hit” you back with the same force that you used against it, your knuckles would go through the wall.  The wall would not resist your force with one of its own.)  If you hit the wall harder, the wall “hits” back harder.  A person who’s really angry may hit a wall so hard as to suffer bruises and possible broken bones as a result of the fact that the wall will hit them back with the same huge force.  You might remember this returned force from studies in high school: Newton’s Third Law of Motion states that for every action there is an equal and opposite reaction.  When it comes to forces that react to an animal standing on the ground, such a reactive force is called the ground reaction force.

So not only does a horse experience force, and therefore stress, from gravity pulling down on its body mass so that its feet strike the ground, a horse also experiences force and therefore stress from the impact of hoof on ground in which the ground exerts force against the horse.  You can see this, and very clearly, in an astonishing series of slow-motion video images made by Sky Sports UK of Hickstead horse jumping, at YouTube, here.  I tried, however, to capture some small piece of what’s visible in a series of three images that are reproduced to the right.  When the horse’s hoof comes down and hits the ground, you can see a wave of force returning up the horse’s pastern, from the ground impact.  I have pointed to one of those waves with a red arrow in the second image.  The apparent distortion of the top of the pastern area in the last image is due to similar waves continuing to propogate through the soft tissues.  If you watch the video, you will see that these waves move upward and are caused by force coming from the ground, due to impact.

(Yes, the fetlock drops very low in the last image.  We will not discuss that here, as it’s an adaptation related to elastic storage of energy.  All you need to know right now is that this image does not show a dangerous degree of flexion.)

The interesting thing about what you see in these images is that the ground reaction force is being transmitted upward through the skin layers of the horse’s foot and leg, as well as through the stronger bones in the “core” area.  It’s possible that there’s proportionately more “shallow tissue transmission” when the ground reaction force is directed ONLY through the shoe, since the shoe is going to transmit  force through the hoof wall it’s nailed to, and this force is then very likely to be transmitted on upward through the skin and deep dermal tissues that are in line with the hoof wall.  If so, then the wraps or boots around the cannon of such a horse may well keep those forces from adversely impacting tendons that aren’t designed specifically to transmit those forces but are quite coincidentally sitting in their path.  (If anyone who designs such boots and knows the biophysics of their design elements wants to educate me about this, I’d love to know what data exist.)

On the other hand (and there is always an “other” hand), remember that the terminal phalanx sits just inside the front portion of the hoof wall.  So forces being transmitted through the hoof wall will also, at least to some extent, enter the chain of bones nearby.  The question is how ground reaction forces are transmitted through the legs and feet of shod and unshod — and also booted — horses.  And as far as I know, we don’t have good data on that.  But it’s certainly something to bear in mind when you’re weighing the decision each of us has to make about shoeing, booting, barefooting, or whatever.  While you are taking into account the kinds of surfaces you ride on, your horse’s personal history and past injuries, and the sorts of activities in which your horse engages, remember that the different kinds of things we put on our horse’s feet change the way that force is transmitted — to the ground and also back up into the horse.  There aren’t any easy answers.  But it is probably safe to say that it’s more important to take extra steps to protect the hoof wall and tendons if your horse wears standard shoes than if it goes barefoot.

Next up:  What happens when the horse starts moving

Force, Stress, and the Hoof, Part I

How does a horse who is outstanding in his field carry his weight?  Well, if he’s out standing in his field, as opposed to trotting or running, he carries it on all four of his legs (bum-dum-ching).  Here’s an example, to the left.  Whatever this horse weighs, gravity is acting on his mass to pull it towards the earth with a calculable force.  That force travels through each of his four feet to the ground.  So a very simple way to begin thinking about force and stress at a horse’s hooves is to think about the force caused by gravity pulling on the whole horse’s body mass, and then simply dividing this force by four to estimate how much is supported by one of the hooves.

You can try this out for yourself on a bathroom scale.  Stand on it with both feet and see what it says you weigh.  Then balance yourself so you’re standing on it with just one foot, the other off the ground somehow.  (Don’t balance yourself by touching a hand to the sink or a wall right now, as it will mess up your experience.  You’ll do that in a moment for another reason.)  Although you will see your weight wobble a bit on the scale as you move around to get to the one-footed position, in the end you will weigh the same amount.  Only now you can feel that your weight is all being held up by that one leg and foot rather than two.  Your body weight is your body weight, and it is going to push down the same amount on the scale whether it’s held up by one leg and foot or two of them.  The difference is how much the leg/foot you’re standing on, that’s holding you, has to hold.

Now, put a hand on the bathroom sink while you’re standing on the scale.  You will see your weight go down.  That’s because you are supporting part of your weight through your arm and the sink, so less of your body weight is going through your legs to the floor.  If a horse “leans” towards its front end, more weight goes through its front legs and less through its hind legs, just like you can lean more of your weight on the sink and watch the scales show you less and less weight going through your legs.

What you probably did not realize is that if you live in the United States and measure your weight in pounds, your bathroom scale does not measure your body’s mass, but the force with which your body’s mass is pressing down on the scale.  A bathroom scale is a very simple type of machine that measures how much force pushes against it. A spring inside the scale deflects (gets squished) when you step on it and thereby apply force to it.  A rack-and-pinion mechanism inside the scale turns the spring’s deflection into the motion of a needle on a dial, so that it points to a certain number of “pounds” depending on how far the spring has been deflected.  So weight, in the US, in pounds is a measure of FORCE. (Also, see the section “spring scales” on this page about weighing scales.)

To explain how this happens, consider the equation for force, F = ma.  F stands for the “force” an object exerts; “m” is the mass of the object; and “a” is that object’s acceleration.  So the equation can be written in English, with specific reference to you on a scale, this way:  the force your body exerts against the ground (or the scale) is equal to how much mass your body has, times how quickly your body is accelerating.

At this point, you are liable to say, “Wait a minute.  My body is not accelerating if I’m simply standing on the bathroom scale.”  But it is.  The Earth’s gravitational attraction is pulling down on your body all the time.  It pulls so hard that if your foot slips on some water as you get off the scale, you’ll crash to the tile floor hard enough to regret it. And it pulls so hard that when you hit middle age, various parts of your body start sagging down towards the earth’s surface because of gravity’s pull on them.

Gravity pulls on things enough to cause them to accelerate if they fall, and anything in freefall accelerates at exactly the same amount:  32 feet per second per second.  That means that with each second that passes, the falling object goes faster than it did before.  The diagram at the left (adapted from one on a UC Berkeley website) shows the position of a falling ball as gravity makes it go faster and faster, or accelerate, in free-fall.  You can see where the ball is at the end of 1 second, 2 seconds, 3 seconds, and so on.  Notice how much farther it goes in the 4th second.  The speed of the ball, or its velocity, is recorded at the left as 32 feet per second at the end of the first second of its fall, and then 160 feet per second at the end of 5 seconds.  You can see, here, what it means to say that gravity accelerates objects by pulling them towards the earth. The ball goes faster and then even faster, second by second, as it falls.

No matter what you drop, the speed of descent is going to be same.  That’s what Galileo demonstrated in his famous experiment at the Leaning Tower of Pisa (which may have been just a thought experiment), where he dropped two objects of different masses from the balcony to see when they would hit the ground.  “Common sense” held that the heavier object would fall faster.  But in fact the two objects hit the ground at the same time.  This is because the gravitational constant is simply and only 32 feet per second per second — mass isn’t in there at all.  So the mass of the object gravity is pulling on has nothing to do with the speed at which it falls.

However, we all know intuitively that although a bowling ball and a tennis ball dropped from the Leaning Tower of Pisa might hit the ground at the same time, they will not hit the ground with the same force.  We might be willing, if suitably encouraged, to stand where the falling tennis ball could hit us on the head.  We would run like mad to get out of range of the shards of ball and pavement that would fly through the air from the much bigger force of impact with which the bowling ball would hit the ground.  The force with which the tennis ball and the bowling ball hit the ground is described by the equation F = ma.  Mass, m, is a part of Force.  So the bigger the object that the earth’s gravity is pulling on, the more force that object applies to the ground.  And this brings us back not only to the force at a horse’s foot, but our own bathroom scale.

American bathroom scales already factor in the acceleration due to gravity when we get on the scale.  They do not measure our body’s mass.  They measure the FORCE that results from gravity acting upon our body mass.  That is what a pound measures on a bathroom scale in the U.S.

But, as you learned if you stood on the scale on just one foot long enough to do the exercise I described above, force isn’t the only thing that matters.  Standing on the bathroom scale on two feet isn’t difficult.  Standing for a long time on the bathroom scale on just one foot is.  If you tried this part of the exercise, you might have noticed that you had trouble keeping your balance, that your joints hurt, and that your foot felt more “smashed” by having all your weight (force, remember!) on that one leg.  The difference you felt was in the amount of stress that was in your foot.

In physics, stress is not a measure of how desperate you feel when you’ve missed a project deadline, have an overdue bill, and your car breaks down all on the same day.  Stress is a measure of how much force is acting on a given area of an object.  Stress, S, is also defined in an equation:  Stress = Force divided by unit area.  In other words, stress is how much force is being transmitted through any given part of an object.  It is stress, not force, that causes structures to fail (if they fail).  The area you measure — and this is important — is at a right angle to the force.  So in your foot, the picture looks something like this:

What matters, in measuring the stress in your foot, is how much force is pushing down on it and how large the cross-sectional area is. In the picture on the left, we are looking at how much stress there is in the lower part of the shin area, just above the ankle.  That’s where I’ve drawn a green ellipse that represents the cross-sectional area of the leg right at that point, as if you sliced through it horizontally.  That plane is the one that the body’s weight, or force (the red arrow labeled “F”), is acting through.  If the diameter of the person’s leg here is 3-1/2 inches (not too uncommon in a woman), then the area of the leg’s cross-section there is the area of a circle, pi times the radius squared.  (Do I hear groans as thoughts of geometry flood back?  Hang in there!  We’re about to get back to horses, and this will all be worthwhile!)  The radius of this part of the leg is half the diameter, or 1.75 inches, and pi times this radius squared is 9.6 square inches.  So if our woman weighs 130 pounds, the stress this part of her leg is experiencing is 130 pounds divided by 9.6 square inches (because Stress = Force/area).  That’s roughly 130/10, which is 13 pounds per square inch.  That is the amount of stress going through this part of the woman’s leg in our example.

IF she is standing on one foot.  If she is, all the force of her body (created by gravity acting on her mass) is being supported by one foot.  But if she’s standing on TWO feet, then they are sharing the force.  In that case, the stress in this part of the leg — the other one also being on the ground — is half her weight, or 65 pounds, divided by about 10 square inches, for just 6.5 pounds per square inch of stress.  As you can see, stress is twice as high when she stands on one foot as when she stands on both — 13 pounds per square inch compared to only 6.5 pounds per square inch.

Let me pause here and say that if you’d like a reasonably good idea about how much stress this is, measure off a one-inch by one-inch square on a piece of paper.  Look at how big it is. Now think about something that weighs 13 pounds — a medium-sized watermelon, for example.  And imagine balancing the entire weight of that watermelon on that one-inch sized piece of paper.  That’s how much stress is on the leg of that 130 pound woman if she stands on one foot.  If she puts the other foot on the ground, the stress is half of that (about large cantaloupe-sized).

Soles of humans, male on the left and female on the right. From Wikipedia.

But remember we are heading for a horse’s HOOF here, not its leg.  So let’s go on down to the part of a human leg and foot that is against the ground:  the sole of the foot.  How much stress is in that part of the body?  The force or body weight is the same:  it’s still 130 pounds if the woman is standing on just one foot, and 65 pounds (half of 130) if she’s standing on two.  But the surface area of the sole of a foot is a lot bigger than the cross-sectional area of a shin.  Look at the bottom of your own foot, and compare it to an imaginary slice through your shin bone, and you’ll see what I mean.

A common estimate for the surface area of the bottom of a woman’s foot is about 25 square inches. So our woman who weighs 130 pounds (who is, remember, exerting a FORCE of 130 pounds on her foot) experiences only 130/25 = 5.2 pounds per square inch of stress on the sole of her foot if she’s standing on one leg.  And it’s half that — 2.6 pounds per square inch of stress — if she puts the other foot on the ground to hold up half the load.

Notice that the stress in the woman’s foot is much lower than it is in her shin, even though her weight (or force) is the same, regardless:  she weighs 130 pounds.  But since her shin has a smaller cross-sectional area than does the sole of her foot, the stress is higher there than in her foot:  6.5 pounds per square inch in her shin compared to 2.6 pounds per square inch in her foot (if she is standing on both of her feet).  So now you understand one reason why it’s more common to hear of someone breaking their shin bone than breaking a bone in their foot.  The basic stresses in a shin are higher than they are in the foot because the shin is smaller in diameter.

And now we can talk about force and stress in a horse‘s foot . . . er, hoof.

Let’s consider a horse that weighs 1100 pounds.  If this horse is standing at a tie rail and has 4 feet on the ground, each of those four feet carries about 1/4 of his total body weight, or 1100/4 = 275 pounds.

I have to pause here to point out that this means if you want to lift up his hoof to pick it out, and if he doesn’t do anything to change the situation, you are going to have to pull up against those 275 pounds to get that hoof off the ground.  That’s what he’s putting through that leg and foot if he’s just standing there eyeing you.  I point this out to remind you that horses are actually fairly cooperative.  Most of them “lean away” when we go to pick up their hoof, and redirect a lot of their weight to the other three legs so we can lift the one we’re after.  But a horse that’s determined not to pick up a foot can not only brace itself but lean more heavily on the foot the person wants to lift up.  I’m sure you’ve seen a horse do this.  And when you see that, you’re not looking at a force-and-stress problem, but a relationship one.

So, back to force and stress.  We have a horse hoof with 275 pounds of force going through it.  The amount of stress on the hoof is going to be this amount of force (275 pounds) divided by the area of the hoof. This means we need to estimate the area of the bottom of a horse’s hoof, that’s against the ground.

The Davis Boot company makes standard sizes of boots for horse hooves to soak in, and they provide the boot dimensions on their website.  They say that “size 2” is for average horses but that the boots are intended to go over a bare (unshod) hoof and to fit loosely enough for soaking solutions to be added.  So we’ll select the boot that’s one size smaller, the size 1, to represent the dimensions of our 1100 pound horse’s foot.  That boot’s dimensions are 5-1/4 inches wide by 5-5/8 inches long.  Notice that the boot is not perfectly round, but longer than it is wide. This reflects the shape of horse hooves, as you can see on the right.

Since the shape of a hoof is not a circle, we are going to average the two measurements (5-1/4 inches by 5-5/8 inches averages to 5.35 inches) to approximate a circle of roughly the same size. The radius (half the diameter) is therefore about 2.7 inches.  We can now calculate the surface area using the standard “area of a circle equals pi times the radius squared” equation.  Multiplying this out gives us a figure of about 23 square inches for the bottom surface of this hoof that is against the ground.

And now we can calculate the stress there.  We have a hoof with 275 pounds of force going through it, and its area is 23 square inches.  So the hoof is experiencing 275/23 = 11.9 pounds per square inch of stress. . . if all four feet are equally on the ground and the horse is not moving.  That’s quite a bit more than the 2.6 pounds per square inch of stress that we calculated in a 130 pound woman’s foot if she’s standing with both feet on the ground.  Notice why:  the surface area of a hoof (23 square inches) is about the same as the surface area of the sole of a woman’s foot (25 square inches).  Horses are carrying much more weight, and experiencing much greater forces, through feet that have about the same amount of surface area in contact with the ground that our feet do. Granted, horses have four feet instead of two to carry the load, but they also weigh a great deal more — even proportionally.

However, there is obviously far more to the story of stress in horse hooves than this. Horses don’t simply stand still at tie rails, and they often have only one or two feet on the ground at a given moment.  Now that we have gone through this set of calculations, though, we can take things to the next step (pardon the pun!).

More coming up in Part 2!