# Bicycle

There is something delightful about riding a bicycle. Once mastered, the simple action of pedaling to move forward and turning the handlebars to steer makes bike riding an effortless activity. In the demonstration below, you can guide the rider with the slider, and you can also drag the view around to change the camera angle:

Compared to internal combustion engines or mechanical watches, bicycles are fairly simple machines – most of their parts operate in plain sight. What’s not directly visible are all the forces that make it possible to ride and control a bicycle without compromising the structural integrity of its components.

In this article, I’ll focus on the delicate interplay between many of the forces that act on a bicycle and its parts when riding. We’ll witness how forces applied through tires make a bicycle accelerate, brake, and turn, and we’ll also investigate how the wheels and the frame handle those different forces without breaking.

Before we understand these more complicated interactions, we’ll play with much simpler objects. The basic scenarios we’re about to explore will help us develop an intuition about the behavior of a real bicycle.

# Forces

Let’s introduce the unassuming protagonist of this section – a plain wooden box. In the demonstration below, you can use the slider to apply a force to this box. The little speedometer in the bottom part tracks the velocity of the crate. You can restart the simulation using the button:

The red arrow represents the pushing force, which could be applied by a hand, or a strong gust of air. The size of that arrow reflects the magnitude of that force – the bigger the arrow, the stronger the pushing action.

Notice that once the block is moving, it will continue to move with the same velocity even when you stop applying the force. In the magical world of these simulations I removed all friction or air resistance – the only forces present are the ones we see.

As you apply that force on the block, it increases its velocity – the bigger the force, the faster the acceleration. Moreover, for the same force, the bigger the mass of the box, the slower it will accelerate. You can experience this in the demonstration below, where the bigger box is twice as heavy as the smaller box, but the applied forces are the same:

While in this article I’ll mostly avoid the underlying mathematical details, the relation tying the applied force F, the mass of the object m, and the output acceleration a of that object, deserves to be presented in one of the most fundamental equations in physics:

F = m · a

Let’s complicate things a little by adding a second force that is trying to push the box to the left. You can still control the magnitude of the right-pushing force with the slider, but to make our experiments a little easier, I’ll restore that force to its original magnitude after you let go:

Notice, that it’s not the mere presence of forces that causes an object to accelerate – the velocity changes only when these forces are not balanced. Recall that we’re representing the magnitude of forces using arrows of different sizes, so as long as the length of the right-pointing arrow is equal to the length of the left-pointing arrow, the box won’t change its velocity.

It’s worth pointing out once more that when the box starts to move due to force imbalance, just letting the slider go to balance the two forces again will not stop the box – it will keep moving with some velocity. To slow the crate, we need to change the balance of forces to work against the current motion, and do it carefully enough to avoid overshooting.

Let’s explore one more concept related to these simple forces. In the demonstration below, the box is resting against a wall and you can once more control the force pushing that box to the right:

Similarly to the previous examples, as we apply a force on the box we’d expect it to move, but it doesn’t budge at all, which implies that there must be some other force acting on it to balance that pushing force. Let’s visualize all the other forces present in this scenario:

Notice that as we push the box, that box pushes on the wall, but the wall pushes the box back, and it’s that force that balances the pushing force, which keeps the box static. Let’s draw the box and the wall separately to show the forces acting on each one:

The wall is firmly attached to the ground, so the force pushing on the wall is effectively trying to push the entire Earth. Since our planet is very heavy, the acceleration of the Earth and the wall attached to it is effectively non-existent.

You may wonder how the wall knows how much back-force to apply, so let’s look at the interaction between these two objects up close and in slow motion. As we apply the force, the box actually starts accelerating into to the wall, pushing its surface to the right:

As the box moves to the right, it compresses the molecules in the wall, which create a spring-like force that pushes the box back. If that force is too small to balance the pushing force, the box will continue to move to the right, which compresses the wall even more, creating an even larger push-back force.

If that force grows too large and overpowers the pushing force, it will start to accelerate the box to the left, which will slow it down and even move it back to the left. This will then reduce the compression of the wall, which in turn decreases that resisting force. The entire system very quickly finds its balance, and since the wall and box are very stiff, the final displacement isn’t perceptible.

In this next demonstration we’re complicating things a little by applying an inclined force that’s pushing the box into the corner:

As the box is not moving, the forces acting on it must be balanced, which we can see in the right side, where I’ve tallied up head-to-tail all the arrows representing the forces acting on the box to show that they form a closed loop. We can also decompose the angled pushing force into its horizontal and vertical components – these are balanced by the horizontal and vertical reaction forces from the wall.

Before we finish up with these simple forces, let’s explore what happens when we line up two boxes next to each other and try to push the left one towards the wall:

As we push the first box, it pushes on the second box, which then pushes on the wall. Notice that the wall doesn’t know that there are two boxes there, it just feels the effects of that single pushing force. The wall pushes back on that second box, which in turn pushes on the first box. To make things clearer, we could even separate the boxes and draw forces acting on each one, showing that both crates are in balance:

Although I’ve shown them explicitly here, there is nothing special about the two forces between the boxes. Equivalently, we could draw forces between every plank of wood in these crates – each plank pushes on its neighbor and is also pushed back by it. If we were to look closely enough, we could draw the arrows between the individual molecules constituting these boxes – they’re all steadily kept in balance by the surrounding forces exerted by the neighboring particles.

In this article, I’ll sometimes visualize these internal forces, as they will help us understand the behavior of individual bicycle components, but note that it’s up to us to decide where we want to draw these artificial separating boundaries to expose the internal reactions.

The ideas we’ve explored in this section are collectively known as Newton’s laws of motion, and they are fundamental for understanding how objects move and interact with each other when forces are applied. However, there is a little more we need to explore to understand the behavior of a bicycle.

# Moments

You may have realized that the forces we’ve been applying to the boxes have been conspicuously pointing at the centers of those objects. Let’s see what happens to this box when we exert a force that isn’t so neatly aligned:

As you can see, the box still moves to the right, but it also starts to rotate and continues to rotate after the force disappears. Notice that I marked the box with a small black and white symbol that symbolizes the center of mass of this object.

I’m also drawing a dashed line through which the force operates. In this example, the line doesn’t go through the box’s center of mass, but it’s placed at a distance from that center. This creates a lever arm through which the force rotates the object.

The further away that line is from the center of mass, the easier it is for the force to rotate the object. In the following demonstration, you can apply two forces of the same magnitude to two identical boxes. The only difference is the distance to the center of mass at which these forces act:

When the distance between the force-line and the center of mass is large, the box spins faster as well. That distance doesn’t change the acceleration of the box to the right and both boxes move with the same linear speed. However, that distance affects the angular acceleration of a box – the longer that arm, the faster the box spins.

The product of the force F and the length r of the arm at which it acts is known as moment of force or torque, which we can represent with a letter M:

M = r × F

While in three dimensions the moment of force can rotate an object around an arbitrary axis, in these simple two dimensional scenarios we just care if torque rotates the object clockwise or counterclockwise.

Naturally, every off-center force applied to an object creates torque. In the demonstration below you can shift the position of two forces acting on the block:

The forces remain balanced and the object’s center of mass stays in place, but the moments of forces are not balanced and the object changes its angular velocity.

While it’s harder to visualize the balance of moments, we can still do it by drawing colored rectangles spanning the length of the acting force arrow and the force arm:

The red force wants to rotate the box in the counterclockwise direction. The blue force wants to rotate the box clockwise and that force is twice as large as the red force, but it also operates at half the distance. The resulting red and blue areas are the same, which indicates that the moments are balanced. The green force goes through the center of mass so it doesn’t rotate the crate, it just balances the other two forces to prevent the box from moving to the right.

Before we put the boxes back in a warehouse, let’s take a final look at one more trick we’re going to use to make it easier to visualize forces. In the simulation below, I added Earth-like gravity that attracts the cube to the bottom of your screen. Both left and right view show the same cube and the same gravity and ground reaction forces acting on that cube – they’re just visualized differently:

When put in this gravitational field, every individual molecule of that cube is being pulled down to the bottom, and once the cube lands, every molecule of the ground under that cube pushes back on it – you can see that roughly presented on the left side.

While more realistic, that soup of arrows is a little messy and hides the simpler, but equally valid notion of gravity and a ground reaction force acting on that cube – you can see them on the right side.

If we tally up all the small ground forces and moments they create we can replace them with a single arrow centered at the bottom of the cube. For gravity forces we can replace them all with a force located at the center of gravity, which for most purposes is equivalent to the location of the center of mass .

The rules we’ve explored here are universal – unbalanced forces change linear velocity of an object, and unbalanced moments change angular velocity of an object. We’re finally ready to understand how these ideas apply to a bicycle and a rider. Let’s draw some of the forces that act on them during a steady turn:

As you can see, it’s easy to get overwhelmed with all the arrows pointing in different directions. We’ll eventually understand all of these forces, but we’d preferably start with a more welcoming approach.

Instead of analyzing all the complex 3D interactions in a single shot, we’ll first look at simpler situations that can be easily presented in flat diagrams. We’ll also decompose the motion of the bike and rider into three primary directions, and only then will we see how they interact with each other.

In fact, we’ll begin by exploring the direction in which there’s commonly no movement at all – the vertical one.

# Up and Down

Let’s look at a diagram of vertical forces acting on a bicycle and its rider that are coasting down a road. A chunk of the rider’s weight rests on the seat, which pushes it back, but the pedals and the handlebars support some of that weight too. The weight of the rider and the bike is then carried by the ground through the two contact points under the wheels:

As you’ve probably realized, the yellow slider allows you to change the posture of the rider. Notice, that the reaction forces from the ground shift back and forth when this happens.

It may appear that the ground is aware of the rider and the weight of his torso and limbs, but note that the ground only feels the two forces exerted by the wheels. Although that load could be slightly different during the dynamic part of the rider’s leaning, once settled, these two forces perfectly sum up to the combined weight of the bike and the rider, keeping them in balance. After all, the bike and the rider neither fly away nor sink into the ground.

To understand why these ground forces change their distribution as the rider leans, we first have to visualize the center of mass of the bike , the center of mass of the rider , and the center of mass of both bike and rider . The rider is many times heavier than the bike, so it dominates that shared center of mass :

As the rider leans forwards or backwards, the position of his center of mass changes, which in turn shifts the center of mass of the combined system . This causes the moments of forces to no longer be balanced, and the bicycle and the rider start to rotate around their center of mass. Let’s see this in a vastly exaggerated form in the demonstration below where the slider controls the time progress of this event:

When the rider’s position shifts, the arm lengths for the reaction of the ground also change, which creates a net torque that rotates the bicycle counter-clockwise. In practice, the actual rotation is very tiny, but as it happens, one of the wheels lifts a little, which decreases the ground reaction force in that area. Simultaneously, the other wheel bites into the ground a little more, which increases the reaction force under that wheel.

It may once again feel a bit magical that the ground somehow knows how much force to distribute to each wheel, but this behavior follows from a careful balance of the ground’s springiness, very similar to the one we’ve seen with a wall resisting a box pushing on it. In the demonstration below, you can once more change the rider’s position, but this time we’re watching the contact areas between the tires and the road from up close and in slow motion:

The entire system very quickly finds the right balance that equalizes the two torques and the ground forces also keep ensuring the equilibrium in the vertical direction – they add up to counteract the weight of the bike and rider.

So far we’ve completely ignored any forces in the forwards and backwards direction, which allowed the rider to coast forever without any pedaling. Naturally, this was a simplifying assumption, and during a typical ride on a flat ground the rider has to pedal to keep moving forward:

On their own, these dynamic forces exerted by the rider’s legs barely affect anything about the distribution of the ground reaction forces – one can’t make oneself heavier by just pressing harder on the ground.

However, through the cranks, chain, and the rear wheel, that pedaling creates a force that pushes the bicycle forward. As we’ll see, these horizontal forces can also affect vertical balance, but let’s first understand how the pedaling action propels the bicycle.

# Forwards and Backwards

Let’s start by looking at what happens to various bike components as the rider presses a pedal:

The pedal-pressing force generates a torque that acts on the front sprocket, which is studded with teeth. That sprocket then pulls on the chain, which then pulls on the rear sprocket generating a torque on the rear wheel. That rear wheel torque can then apply a force that will try to push the ground back.

Notice that the forces and moments change their magnitude as they pass through each component. That transformation is the consequence of different radii and lever arms involved in the pedaling process. For example, the cranking action forms a simple lever:

The crank serves as the input lever, and since the crank’s length is larger than the radius of the sprocket, the output force is also larger than the input force. Since the sprocket is rigid, every point on its circumference can exert this force, so the chain is also pulled equally strongly.

The chain then pulls on the rear sprocket, which has a different radius than the front one. As a result the chain force also acts on a different arm length:

The rear sprocket is smaller on our bike, so the torque on the rear wheel created by the pulling chain is smaller than the torque on the front wheel, but more complicated bicycles allow the rider to change the gearing ratio – this can magnify or minify the rider’s pedaling torque.

Through the friction between the tire and the road, the bottom part of the wheel stays in contact with the ground, so that as the wheel rotates, it pushes the ground to the left. As a reaction to that force, the ground pushes back on the wheel to the right, which pushes the bike and the rider forwards:

The little speedometer in the top left corner shows the current speed. I’m using imperialmetric units here, but you can to the metricthe imperial system if you prefer.

Although these forces apply in a different direction than the weight-related forces we’ve looked at before, they’re yet another manifestation of the principle of action and reaction. The force that the wheel exerts tries to push the Earth back, but its mass is enormous, so the planet’s acceleration is microscopic, but the force of the same magnitude applied to the bike easily speeds it up.

Notice that the pressing forces generated by the rider vary across a single rotation. Moreover, the arm length at which these forces act also changes during each cycle, so the final torque and the outcome pushing force generated by the rider also has a periodic nature.

Looking at the little speedometer you may have noticed that the bike stays at roughly the same speed, even though it’s being pushed forward by the pedaling action. There are other forces slowing it down, which prevent more velocity from building up, but before we see those forces we should talk about a more direct way to slow down a bicycle.

A force that pushes forwards on a wheel accelerates the bicycle and its rider, and, similarly, a force that pushes backwards on a wheel will decelerate the bicycle. This idea underlies how braking works:

As the rider pulls on the brake lever, that lever pulls with mechanical advantage on the cable. The cable then pulls on one of the caliper arms of the brake, causing them to close, which presses the brake pads against the wheel.

When a wheel is spinning, that pressing force resists the rotational motion of the wheel by applying a friction force acting in the direction opposite to the wheel’s rotation.

Let’s experience this in the demonstration below. When the rider pulls on the brake, the brake pads create friction, which creates a torque that slows the wheel and the tire down. Through friction with the road, the tire introduces a force that tries to push the ground forward, and the ground responds by pushing the tire and the entire bicycle back:

Clearly, both acceleration and braking requires some interaction between the tires and the ground. Let’s look at the details of how a tire actually generates these forces.

A typical bicycle tire has a donut-like toroidal shape and is pressurized, but it still flattens a little when loaded with weight of the bike and rider, so the area where it touches the road has a roughly elliptical shape. You can see that shape on the right side of the following demonstration, in the zoomed-in view of the small section of the road on which the tire is resting:

This region is known as the contact patch. It may seem obvious once pointed out, but the two elliptical regions under the two tires are the only places where the bicycle interacts with the road and almost all of the rider-controlled forces have to act through them.

To understand how forces in the contact patch get developed during acceleration and braking, we have to first look at a rolling motion. In the demonstration below, you can play with a lone wheel rolling down the road. The wheel is covered in white paint, so it leaves marks on the ground. The sliders let you control two different types of velocity. The first one controls the forward motion or the linear velocity, and the second one controls the rotational motion or the angular velocity:

You may have noticed that in the wheel looks like it’s mostly sliding, in it looks like it’s mostly spinning. These two situations may look a little unusual, but you’ve probably seen a vehicle’s wheels sliding when braking, or spinning when accelerating too quickly. When we keep the two velocities at the , the wheel looks like it’s rolling smoothly, and the paint marks it leaves are more or less as long as the ones on the tire.

When the wheel rotates “just right”, its angular velocity denoted with a Greek letter omega ω, is tied to its linear velocity v with a simple equation involving the effective radius r of that wheel, which is slightly smaller than the outer radius of the tire as it compresses a little under load:

ω = v / r

Since a wheel is attached to a bicycle, the forward linear velocity v of the wheel is always equal to the velocity of that bike. When the external forces are balanced and the bicycle rolls freely, the angular velocity ω of the wheels will be equal to the natural, free-rolling angular velocity for that speed v.

In the demonstration below, you can experience what happens in the contact patch area during those free-rolling conditions. I put little markers on the inner and outer parts of the tire to show their relative movement. The slider lets you scrub back and forth in time:

In this free-rolling scenario, the inner and outer parts of the tire stay in sync as the wheel smoothly rolls over the road.

Things change a little when the rider starts to pedal. Recall that the torque that the chain exerts on the rear sprocket makes that rear wheel rotate faster – the angular velocity of that wheel increases a little, but the forward velocity stays the same, as the bike itself hasn’t yet accelerated.

In the demonstration below, you can see what happens between the road and the tire, when the wheel has a slightly increased angular velocity:

As this wheel rotates, the high friction between the road surface and the rubber causes the outer part of the tire to stick to the ground, while the inner part of the tire continues to turn with the wheel. Since the wheel rotates a little faster than its free-rolling speed, it pulls the inner parts away from the stuck outer sections.

This causes the tire to deform near the contact patch area. It’s that deformation that makes the tire want to push the ground back – you can almost see it trying to sweep the road to the left. The ground reacts by pushing the tire and the entire bike forward.

This force is created because of the difference between the free-rolling angular velocity of the wheel and the actual angular velocity of the wheel – the latter has been increased by the pedaling action. The generated pushing force accelerates the bike, which increases the natural free-rolling speed of the wheel until it matches the actual speed of the wheel, at which point the deformation will no longer be present and no additional forces will be created.

Similarly, as the rider presses the brakes, causing the wheel to rotate slower than its free-rolling speed, the inner side of the tire also starts to shift relative to the outer side that sticks to the ground. This deforms the tire in the opposite direction:

This deformation creates a force pushing the bike backwards, which slows it down, which then reduces the free-rolling speed of its wheel.

Unfortunately, there are limits to the deformation a tire can handle without locally loosing contact with the ground and starting to skid – you may have noticed these limits in the rear part of the tire, where the outer sections eventually break away from the road and start sliding.

Let’s explore the magnitude of forces that a tire applies a little closer. Intuitively, the more deformed the tire, the bigger the forces it exerts. That deformation, and thus the developed forces, depends on the difference between the actual angular velocity of the wheel ω and its free-rolling angular velocity ω0 joined in the parameter s known as the slip ratio:

s = ω − ω0 / ω0 = ω − v/r / v/r

Let’s explore these dependencies in practice. In the demonstration below, the free-rolling angular velocity ω0 depends on the forward speed v of the wheel. Just like before, you can tweak the actual angular velocity of the wheel ω with the second slider:

When the wheel rotates at the of the current velocity, the slip ratio is 0. When the wheel than the free-rolling speed, which happens when we pedal, the slip ratio is positive. Finally, during braking, when the wheel than the free-rolling speed the slip ratio is negative. When , the theoretical slip ratio may become infinite, but in practice, as the rear wheel starts turning, the force developed by the tire immediately starts pushing the bicycle forward.

Other than the slip ratio, the magnitude of forces a tire can exert depends on road conditions, speed, tire pressure and wear, and, most importantly, the vertical load on that tire. The plot below roughly demonstrates that dependence of the output force on the slip ratio, and the vertical load, which you can control with the slider:

When the wheel rotates faster than the free-rolling speed and the slip ratio is positive, the generated force is positive and it pushes the bicycle forward. While initially that force grows proportionally to the slip ratio, it quickly reaches a maximum and tapers out.

Similarly, when the rider brakes, the wheel rotates slower than the free-rolling speed and the slip ratio is negative, the output force is negative, as it pushes the bicycle backward. It’s worth pointing out that a tire generates the largest braking force for smaller values of slip – braking is actually less efficient when a wheel completely locks up and slides over the road.

Let me also point out that the forces generated by the tire against the ground affect the wheel itself. For example, when the wheel rotates slower than the free-rolling velocity, the forces that the tire generates are not only slowing the bike down, but they also want to accelerate the wheel. Ultimately, this is how a front wheel starts to rotate as the bike starts moving, even though it’s only the rear wheel that’s powered.

The larger the vertical load on the wheel, the bigger the braking or accelerating forces. We’ve already seen how the position of the rider can affect the vertical loads on the wheels, but those vertical forces are also affected by the braking and accelerating forces themselves. The latter are typically much smaller, so let’s focus on braking.

In the demonstration below, the slider controls the force applied by the rider on the rear brake lever. As the brake pads start rubbing against the rotating wheel, they create friction, which makes that rear wheel rotate slower. As a result, the rear tire introduces braking forces. The speedometer in the top left corner shows the current velocity of the bike:

Notice that the backward force applied at the rear wheel not only slows the bike down, but it also wants to rotate the bike and rider clockwise around their center of mass . Similar to our previous examples, the ground reaction forces adjust by increasing the load on the front wheel and decreasing the load on the rear wheel to compensate for this additional torque.

At some point, that braking force reaches its maximum value. Recall that the forces generated by a tire depend on the vertical load, so any further increase in that braking force would unload the rear wheel more, which then would decrease the tire-generated braking force. Additionally, the rear wheel can lock up quite easily, as the clamping friction can get much stronger than the relatively weak braking force that tries to keep that wheel rotating.

As we’ve just seen, the rear wheel can only apply a limited amount of braking force, so let’s explore what happens when we brake with just the front brake instead:

The braking force also wants to rotate the bicycle clockwise, which increases the load on the front wheel, which then increases the braking force, because a loaded tire generates more force. Notice that by braking with the front brake we can halt to a stop much faster, but there is a certain risk involved with applying too much of the front-braking force.

As that braking force grows larger, it unloads the rear wheel more and more. At some point it can even cause the rear wheel to get completely unloaded, and the braking force will eagerly try to rotate the bike and rider further:

Since the center of mass is relatively high, the torque induced by the braking force overpowers the torque from the front wheel reaction and the bicycle starts to rotate, but simultaneously that ground reaction force lifts the bicycle and the rider.

Naturally, braking forces are not the only ones that slow a bicycle down. At typical speeds the primary resistance to perpetual motion comes from the air drag, which is proportional to the square of the velocity, but it also depends on the shape and size of the rider and the bike facing the incoming air – professional cyclists will often change their position when riding downhill to minimize drag.

In the demonstration below, you can control the cadence and strength of the rider’s pedaling using the slider to see how the air drag rises as the rider gains speed:

Although the aerodynamic resistance pushes on every front-facing part of the bike and the rider, we can aggregate these effects into a single force acting on the center of pressure, which I’m visualizing it with a blue dot.

While pedaling accelerates the bicycle, that effort is quickly countered by the increased air drag caused by increased speed. A bicycle also gets slowed down by the rolling resistance of the tires and other frictional losses in the rotating parts. To keep a steady forward speed, a cyclist has to keep pedaling to provide a pushing force that counteracts these regressive forces.

Recall that with the same force, a heavier object accelerates slower, so the reduction of the weight of the bicycle is important for improving its responsiveness to more intense pedaling or braking. Low weight is even more vital when riding uphill:

Recall from our wooden box playground that we can always decompose an angled force into more convenient directions. The force of gravity still acts straight down, and the vast majority of it pushes the bike and rider perpendicularly into the tilted ground, but a small component of that gravity is parallel to the road and it pulls them downhill. The lighter the bike and rider, the smaller that force and the easier it is to keep riding uphill.

Throughout all these examples we’ve witnessed how the two ground forces underneath the wheels redistribute themselves as the rider changes position, or when aerodynamic and braking forces slow the bicycle down. It was only in the quite extreme front-braking conditions when the bicycle lost its front-to-back balance and flipped over.

Unfortunately, if we look at a bicycle from behind, the situation gets drastically different. Let’s explore how different forces affect the side-to-side behavior of a bicycle.

# Left and Right

In the demonstration below, you can witness what happens to the bicycle and its inattentive rider who is holding the handlebars in the straight-ahead direction, as they together tilt away from the vertical even by a small amount. By dragging the slider, you can directly control that leaning angle, but once you let go, the forces will take over:

The intuitive way to understand why this happens is to notice that the friction between the tires and the ground makes the contact patches act like hinges on which the bicycle can pivot. As the gravity develops an arm against that pivot, it rotates the bicycle more and more until it eventually falls down.

Notice that the fall is self-perpetuating – as soon as the gravity develops an arm force, the bike will lean even further, which increases that arm and the rotating moment. In the front-to-back direction, the two wheels provided two points of support that could trade the load to keep the bike balanced. In the side-to-side direction, the contact patch under a tire is very narrow and it can’t provide much support, which makes the entire system very easy to tip over.

While this situation often happens when learning to ride a bike, most riders have very little trouble keeping a bicycle upright. One could assume that riders simply balance their bodies left and right to keep the center of mass over the tires to minimize the force arm of gravity. After all, a skilled cyclist can easily ride a bike hands-free using only the body motions to keep the bicycle straight.

Unfortunately, merely tilting the body to change the position of the center of mass is often not enough, which you may have experienced trying to balance yourself on a bicycle that’s not moving and has its brakes locked to prevent any subtle forwards or backwards motions. In the simplified simulation below, you can try to balance such a stationary bicycle by tilting the rider’s body with the slider. Be ready for action as soon as you start over with button:

Despite the labored contortions of the rider, the center of mass doesn’t move that much, making it very difficult to balance a bicycle this way. Before we solve the puzzle of hands-free stability in motion, let’s explore a more typical way of keeping the bike upright when riding by simply gently turning the handlebars.

In the demonstration below, you can witness what happens to the bike and rider as the handlebars are turned. I keep the camera directly behind the bike so that you can see how it tilts away from the vertical. The slider controls the progress of time:

Unfortunately, we’ve reached the limits of what we can easily visualize using these flat illustrations. Even though we leave the green pastures behind and enter the kingdom of infinite checkerboard pavement, we at least make it easier to show these forces from different angles. The demonstration below shows the exact same situation as the one we’ve just seen:

When the handlebars are turned and the front wheel doesn’t point straight at the direction of travel, the front tire generates a side force. The rear tire follows very quickly, and both forces start to counter the tilt of the bicycle, which straightens it up. The indicator in the bottom right corner shows the current steering angle of the front wheel.

Notice that instead of trying to shift the center of mass on top of the contact patches of the tires, this gentle steering action pushes these contact patches under the center of mass – this also reduces the arm length of gravity to zero.

In general, as the bicycle starts to tilt to a side, the rider can gently steer into that leaning side, which develops a force that restores the bike and rider to the vertical. As the bike straightens, the rider also repositions the handlebars.

Naturally, even if the bike gets back to a perfectly balanced position, it will quickly get disturbed away from it, so during a normal ride the correction process happens continuously as the rider subconsciously reacts to the changes in the tilt of the bike and his body.

The development of this side force may seem a little mysterious, so let’s take a closer look at what happens between the tire and the ground after the rider turns the handlebars. Firstly, notice that it’s possible for a wheel to be traveling at a slightly different direction than it’s currently aiming at:

The angle spanned between these two directions is known as the slip angle. As you may suspect, a non-zero slip angle is a little unnatural for a tire, so it will undergo some deformation when forced to roll like this.

In the demonstration below, I put a bunch of small points on the circumference of the tire, I also made that tire semitransparent to make it easier to see what’s going on in the contact patch area:

Recall that the contact patch has a shape of a long ellipse. When a tire rolls with a non-zero slip angle, the first point of contact with the ground happens to the side of the central axis, that I’m showing with a thin blue line. Due to friction, the tire sticks to the ground in that area, so that part of the tire doesn’t follow the motion of the wheel and is gets deformed away as the wheel rolls forward.

This deformation generates a force that wants to push the wheel to a side to minimize that deformation. Eventually, these deforming forces become large enough to overcome the friction, and the rear section of the tire slides across the ground.

I need to note that when you change the slip angle in the demonstration, I’m showing you the deformation of the tire in its fully developed state for that angle. In the physical world it can take a small amount of travel for the tire to deform and exert full force.

As we may expect, the magnitude of the side force depends on the slip angle, but similarly to the forward and backward forces developed by the accelerating and braking tires, the side forces developed by a tire are also strongly related to the vertical load on the wheel:

Naturally, the bigger the slip angle, the bigger the deformation and the resulting force in one direction or the other, but at some point the tire gets too deformed and it will start mostly sliding, which limits the side forces it can generate.

There are two other aspects of tire behavior that are worth mentioning. Firstly, a tire can only sustain a certain amount of deformation in any direction without breaking contact with the ground, so it’s much easier to start skidding when braking and turning at the same time. Secondly, a leaning tire pointed in the direction of travel can still generate some side forces through an effect known as camber thrust, but it’s usually a few times smaller than the regular slip angle forces, at least at typical angles of side tilt.

While vital for stability, the side force generated by tires also allow a bicycle to turn. In the demonstration below, you can witness a bicycle and a rider in a steady turn:

Looking at this situation , we can see that the side tire forces try to restore the bicycle to the vertical position by rotating clockwise around the center of mass . The moment these forces generate helps to balance the reaction forces trying to tilt the bicycle towards the ground. Although this bicycle is not vertical, it doesn’t lean any further because the moments are balanced – this explains why the bike and rider tilt during a turn.

If we look at this situation , we can also notice that the forces generated by the tires push the bicycle to the left. Since these forces are tied to the bicycle, they continuously keep changing their direction, which makes the bicycle move in a circle.

Throughout these examples you may have found it a little surprising that both tires generate a side force. That behavior is a consequence of a circular motion. In the demonstration below, you can witness that during a steady cornering, the current velocity of the front wheel and of the rear wheel is tangent to the circle that the rider is taking:

To straighten a bicycle that’s turning left, the rider has to turn left even more, which will increase the slip angles and the forces that the tires generate. This is the exact same behavior that happens when the rider tries to remain stable:

Finally, let’s explore how a rider enters a turn. One could assume that to start turning left from a vertical position, the rider also has to turn the handlebars to the left. In the demonstration below, you can see what happens in that scenario:

Notice that tires generate forces that lean the bicycle to the right. However, to turn left we need to lean to the left as well. To engage a left lean from a straight position, the rider must turn the handlebars to the right, and then, once the bicycle starts tilting to the left, the handlebars can be turned left:

This may sound quite surprising, but this is exactly how a bicycle turns! This initial countersteer in the opposite direction is usually something that a rider does subconsciously. However, this explains why it’s so uncomfortable to ride close to a curb, because getting away from the curb first requires getting a little closer to it. It’s all very unintuitive, but some special bicycles show that it’s impossible to turn when we prevent that countersteer.

This whole discussion brings us a step closer to understanding how a skilled rider can control a bicycle without touching the handlebars. In fact, we’ll go one step further by looking into what causes many bicycles to remain stable without a rider, at least in a certain range of speeds.

# Stability

The self-stabilizing behavior of bicycles has fascinated researches for over hundred years, and over time, two explanations became popular, but as we’ll see, the stability story is a bit more complicated.

It is sometimes believed that what gives a moving bicycle its stability is the rotation of its wheels that could be causing some sort of gyroscope-related stabilization that prevents a bicycle from tilting to the sides, similarly to how a spinning top can maintain its balance without falling.

While it’s true that rotation of the wheels helps, this stabilizing effect is a little more subtle. In the demonstration below, a red string is tied to the handlebars to prevent the front wheel from steering. By dragging the slider you can see what happens to this bicycle after it is brought up to speed and let go:

The bicycle still very easily falls to the side despite the rotational motion of its wheels. For a bicycle it’s not the mere presence of spinning wheels that helps with balance, but it’s the way a spinning object behaves when forced to rotate around some other axis. These effects are quite unintuitive so let’s briefly run two experiments.

In the demonstration below, you can see a stationary bicycle wheel visualized with its three major axes. By using the slider you can apply a torque that rotates the wheel around the green axis:

This follows what we’ve learned about torques creating angular acceleration, which then makes the wheel rotate faster and faster.

In this next demonstration, the wheel is spinning around the red axis, and you can also apply a torque that rotates the wheel around the green axis:

Unexpectedly, the wheel ends up rotating around the blue axis! These three dimensional rotational motions of already spinning bodies are a bit more complicated than what we’ve seen so far, but for our use it suffices to say that this gyroscopic effect couples the three axes of rotation together.

While that scenario may have seem a little abstract, the exact same situation happens in a moving bicycle as it leans. The front wheel spins around the red axis, the tilting induces rotation around the green axis, and the gyroscopic effect wants to rotate that front wheel and the entire front assembly around the blue axis into the turn:

Naturally, the construction of the bicycle prevents the front from rotating exactly around the blue axis, but it can freely rotate around the steering axis. As the front wheel steers into the turn, the bike’s tires develop side forces, which straighten the bicycle up, similar to how it works when a rider does the steering.

Given what we’ve experimented with, one could assume that it’s the gyroscopic effect that’s responsible for stability, but it’s possible to construct special bicycles that remove the gyroscopic effect by adding an additional wheel that rotates in the opposite direction. A bicycle like this is still somewhat rideable without hands, so there must be some other factors contributing to stability.

Let’s take a closer look at the geometry of the bicycle from a side. If we draw a dashed line extending from the steering axis all the way to where it intersects the ground we can notice that the place where the front wheel touches the ground is behind that point :

This distance is known as trail and it depends on the geometry of the front part of the bike – the steering axis angle and the fork offset. Notice that some combination of these parameters would require a more invasive redesign of the bicycle frame to make the front wheel fit.

When a bicycle is falling to a side, an upwards-pointing ground reaction and sideways-pointing friction form combined reaction forces that are not pointing at the bike’s center, but they’re tilted at a small angle to the bicycle:

Let’s take a closer look at the reaction force acting on the front wheel. In the demonstration below, you can track its direction as the bicycle leans to a side:

Notice that we can decompose this force into the component acting in the plane of the wheel and into the component acting perpendicular to the plane of the wheel. It’s that perpendicular component that acts at a trail-related arm relative to the steering axis, causing the front wheel to steer.

Similarly to the gyroscopic effect, this trail effect makes the front wheel steer into the fall, which creates tire side forces that straighten the bicycle up. This may all sound convincing, but notice that once the tire develops a side force, the trail also acts as an arm that helps to straighten a turned wheel:

For typical bicycles gyroscopic and trail effects are important, but one can’t explain the influence of these actions with individual examples – both effects form a dynamic system that evolves as the bicycle leans to a side. Moreover, it’s possible to create very unique bicycles that are self-stable without gyroscopic effects and without any trail.

Bicycle stability can’t be explained using just one or two mechanisms. It’s a combination of many different intertwined factors, like the mass distribution of individual components, size of the tires, geometry of the frame, and others.

For hands-free riding, the rider can tilt his body to gently change the position of the center of mass, which then induces a lean to a side. As the bicycle starts leaning, the effects we’ve just explored kick in – the front wheel steers into the turn, and the bicycle straightens up or enters a gentle turn.

Stability also heavily depends on the bicycle’s speed – a bicycle that moves too slow or too fast can become unstable. What keeps bicycles balanced with or without a rider is still an active area of research, and even the seemingly basic idea that, for a bicycle to be self-stable, it needs to turn the handlebars into the fall, has not yet been proven.

At this point we’re done describing how forces and moments affect the motion and stability of a bicycle. In the next two sections of this article, we’ll look into the forces transferred inside the components of a bicycle, starting with the parts that give bicycle its name – the two wheels.

# Wheels

In our discussions so far we’ve only focused on external forces acting on the entire bicycle and the rider. For example, when we looked at gravity, all we cared about was that their weight gets distributed between the rear and front wheel and then it’s countered by the reaction forces from the ground:

While this level of detail was useful for analyzing what happens to the bike and rider as a single unit, it unfortunately hides the interactions between the individual components of the bike. It’s time we start looking at these internal forces.

If we pay a close attention to how this bicycle is constructed, we can notice that the weight of the rider and the bicycle frame actually rests on the axles of the wheels, and it’s those axles that push back on the frame and the rider to prevent them from falling down. Let’s separate these components to visualize these inner forces:

Both wheels experience similar loads, just of a different magnitude, so let’s focus on the rear wheel. It’s worth pointing out that while the red and blue forces have the same magnitude, and the green and white forces are also equal, but the red and green forces are not equal, because the wheel carries part of the weight of the frame and the rider, but the ground also has to support the weight of that wheel. Thankfully, that difference is quite small as wheels are quite light.

With these forces exposed we can now see that a bicycle wheel is compressed between the force acting on that wheel’s axle and the reaction force from the ground. In three dimensions we can now show that each end of the axle carries half of the total load we put on that wheel:

Notice that most of the wheel is completely empty, and it’s only the thin spokes that connect the outer and inner part of the wheel together. The position of the spokes and their attachment points may seem arbitrary, but they’re carefully designed to make the wheel sturdy, reliable, and lightweight.

To understand the design and construction of this modern bicycle wheel we first have to go back in history and look at the wheels used in some of the early precursors of bicycles. In the demonstration below, you can assemble a simple wheel one could find in vehicles built in a bygone era:

This wheel consists of the central hub, eight rounded spokes, and four arched felloes. These wooden parts are then wrapped with an iron rim. In its normal state, that rim is a little undersized and it has to be heated before assembly – after cooling down it shrinks to keep all the pieces tightly together. The hub rests and rotates on the central axle.

When a wheel like this is loaded with weight, the hub presses down on the spokes in the bottom part of the wheel, and each spoke gets compressed. As the wheel rotates, different spokes become loaded and unloaded:

Notice that in this wheel, the top spokes don’t carry any load because they are only loosely inserted into the holes in the hub – the spokes can slide out with any pulling force.

These spokes work like little columns that resist the compressive forces. Intuitively, the thicker the spoke, the bigger force it can handle before breaking. To compare loads on differently sized objects we use the concept of stress denoted with lowercase Greek letter sigma σ. For pure compressive loads that stress is simply equal to the pushing force F divided by the cross section area A:

σ = F / A

Let’s experience this directly. In the demonstration below, you can control the cross section of a solid wooden cylinder and the magnitude of a compressive force applied to its ends.

You’ve probably noticed that, when compressed, the cylinder gets a little shorter. That shrinkage is proportional to the properties of the material and the stress σ the rod experiences. That’s why a shrinks less than a with the same compressive force – its cross section is bigger, so the stress is lower, which results in a smaller size change. As a side note, the stress is expressed in units of pressure, so we’ll use pound per square inch or psimegapascals or MPa for short.

With a fixed amount of force, the smaller the cross section the bigger the stress. Every material has a certain limit of stress after which it will either yield and deform irreversibly or downright break. However, this also means that all stresses that are kept at a safe margin below these critical limits are permissible to use. As a result, as long as we stay away from the danger zone, we can actually make the spokes thinner and increase the stresses in them for the purpose of saving weight.

Unfortunately, if we make spokes in our wooden wheel too thin we’ll open ourselves up to a nasty surprise. In the demonstration below, you can witness what would happen to a wheel with thin wooden spokes as the hub is loaded:

The thin spokes buckle under compressive load and the axis drops down a lot – a further increase in the pressing force could cause the spokes to break completely.

This buckling behavior is not restricted to wood. Any slender column, even one made from steel, would buckle when compressed with a large enough force:

While thin and long objects aren’t effective at resisting forces pushing them on both ends, they’re typically much better at handling forces that want to stretch them. You can easily experience this yourself – a single strand of uncooked spaghetti provides little resistance when gently pressed at its ends, but it’s fairly difficult to break it by just pulling its ends apart.

Thin metal rods are no different. In the demonstration, below you can apply a pulling force on a rod and see how it reacts to that force, depending on the rod’s cross section:

When this thin rod is pulled like a string, its tension increases and we say that it is experiencing a tensile load. These forces also create stress σ that, just like compression, depends on the magnitude of this force F and the cross section A of the stretched rod:

σ = F / A

In the previous demonstration we’ve used relatively small loads, and the rod effectively acted like a very stiff spring that returns to its original shape when unloaded, but when the stresses grow too high, these rods would also eventually yield and break. More importantly, however, that critical tensile load is significantly larger than a compressive load at which this rod buckles.

Let’s try to exploit the pulling resistance of metal rods by building a lighter, modern bicycle wheel. Firstly, instead of thick wooden spokes, we’ll use thin steel spokes. Each spoke has a little bent knee with a flat cap at one end, and a screw-like thread on the other end:

We also need a new hub that will be the central part of the wheel. The spokes can be guided through the holes in the new hub, with the little knees acting like little hooks and the caps preventing spokes from going completely through:

The aluminum rim forms the outer part of the wheel. It has a set of openings through which the threaded ends of the spokes can go. Notice that the spokes are so thin that they can be easily bent by hand to get into a hole:

The threaded end of a spoke is capped with a so-called nipple, which also has a thread inside. In the demonstration below, I made the rim semitransparent, so that you can how these parts interact. At first, let’s just loosely screw the nipples onto the spokes right until the point where the nipples meet the rim:

Let’s see what happens if we now put a vertical load on this wheel with nipples just loosely tightened:

When we increase the load, the hub is mostly pulling on the spokes in the top part of the wheel. As the rim deforms and flattens in its bottom part, the spokes and nipples in that area just fall through the holes in the rim. Those bottom spokes are simply dangling from the hub without supporting any of the load, but even if the nipples were somehow locked to the rim, these spokes would just buckle.

While seemingly functional, a wheel like this wouldn’t be very useful – the axis of the wheel drops a lot, and the deformation of the rim is quite severe. With large enough load, the rim itself could buckle. We could make the rim sturdier and thicker to limit these deformations, but that would also increase its weight.

Notice, however, that the large number of spokes doesn’t participate in the load bearing at all. If we could make sure that these spokes are working too, we’d potentially distribute the pulling forces more evenly, which would help the rim maintain its circular shape.

To understand how we can achieve that, let’s take look at the cross section of the parts to see the interaction between a spoke, a nipple, and the rim, as the nipple is screwed with the slider:

As we turn the nipple, it initially moves further down onto the spoke, but it eventually gets stopped by the rim. At that point the nipple braces against the rim and it starts pulling on the spoke, trying to screw it into itself. The spoke gets pulled into the nipple.

If we just turn a single nipple, its spoke will simply move the hub with it, but if we turn all the nipples we can cause all the spokes to be under tension and pull with more or less the same force:

The rim stays rounded because the tension of the spokes is well balanced across the entire circumference, which keeps the whole structure tightly together. Even without any external forces every spoke is already under tension, because it’s pulled between the hub and the rim. By principle of reaction, it’s also pulling back on these parts – this is all similar to how in a game of tug of war the rope’s ends are pulled away from each other, but the rope also pulls the players towards each other.

For the sake of clarity, let’s visualize these internal forces, because sometimes we’ll care about the forces on the spokes, and sometimes the forces on the rim and the hub will be important to us:

Because a spoke can freely rotate in its hole in the hub, and a nipple can freely rotate like a ball in the rim, these tensile loads are the only forces that the spokes can carry.

Let’s see what happens when we then load this wheel. In the demonstration below, you can witness how a weight applied on the hub ends up distributing over the spokes as the wheel rotates. Note that I’m only visualizing the additional forces induced by the weight without showing you any of the preexisting tension:

A vertical load put on a wheel primarily tries to compress the few spokes under the hub. It’s almost impossible to see, but the other spokes are also gently pulled on as the hub is pushed downwards.

To see the total forces that the spokes experience, we need to add these weight-induced forces to the tension forces already present in the spokes:

Notice, that although the bottom spokes are compressed by the load, that compression just reduces the initial tension that we applied by turning the nipples. All the spokes are still getting pulled, but as the wheel turns, some of the spokes experience less tension.

It’s mostly the few spokes directly under the hub that carry most of the load through a decrease in their tension – all the other spokes just very gently increase their tension. Naturally, without the spokes in the upper part of the wheel the lower ones couldn’t do their work, so ultimately all the spokes serve an important role in keeping a bicycle wheel functional.

You may have noticed that we’ve been rolling the rim directly on the ground without the tire present. A pressurized tire will deform much more easily than a rim, but ultimately all the forces acting on a tire are carried onto the rim too. By removing the tire, we can just focus on the rim and spokes themselves.

In principle, the wheel we’ve build could handle pure vertical loads just fine, but when biking there are also other forces at play. As we’ve seen, when a bicycle is turning, it’s the side forces generated by the tires that make the bicycle change its direction:

In a wheel, that force ends up pushing sideways on the tire, which then transfers that load onto the rim. Let’s see what happens if we apply a side force to the wheel that we’ve assembled so far:

While the hub doesn’t move relative to the rest of the bike, the rim quite easily deflects from its original plane, which I’m showing with the semi-transparent circle. While some amount of deflection is always expected, if the rim tilts too much, it can start rubbing on the other parts of the bicycle, which could prevent the wheel from rotating.

To remedy this we first have to see what happens to the spokes as the rim gets deflected. In the demonstration below, you can see a side view of two neighboring spokes attached to the rim and the hub. In the bottom part of the demonstration I’m drawing the same two spokes side be side letting you can compare their lengths as the rim gets deflected with the first slider:

The second slider controls the “thickness” of the hub, which affects the distance between the spokes' attachment points. To allow the rim to deflect by a certain amount, the two spokes have to change their lengths much more when the than when the .

By elongating a spoke we make its pull on the rim stronger, and by shrinking it we reduce the pretension, which makes its pull on the rim weaker. When we tilt the rim, the force imbalance in the spokes makes them want to prevent that deflection by pulling the rim back in place. Overall, the bigger the difference in lengths of the spokes, the more strongly they resist the deflection of the rim.

For a the spokes' lengths don’t change that much as the rim tilts, so even a small side force can easily deflect the rim pretty far, before the forces in spokes prevent any further displacement. For a , the spokes change their lengths much more quickly as the rim deflects, so they will be able to resist the force pushing on the rim at much smaller rim displacement.

We can experience these effects in the demonstration below, where you can adjust the thickness of the hub and see how it affects the deflection of the rim for a given side force. Notice that in practice we’re adding thickness by offsetting a second flange onto which the spokes on the other side of the wheel are attached – we now have left and right set of spokes:

As you can see, by keeping the spokes at the hub we make the wheel significantly more resistant to side loads. By making the hub wide enough we end up with a modern wheel, with spokes arranged in a so-called radial configuration.

One could hope that this was the final step of building a bicycle wheel. If we were to use our creation as a front wheel of a bike with typical caliper brakes, we’d indeed be done. However, to get a fully robust rear wheel we still have one more improvement to make.

To understand the problem we need to solve let’s look at what it takes to make this wheel rotate. In the demonstration below, you can grab one of the gray points on the rim of the wheel and then drag it in any direction. Once you let go, I’ll apply a short-lasting force that pulls on the wheel in that direction. The longer the dragging distance, the bigger the force:

Notice that when you apply the pulling force directly towards the axis of rotation in the middle of the hub, the wheel starts turning very slowly, even when the applied force is very big. However, when you pull in the tangent, or perpendicular to the “towards-center” direction, it’s much easier to make this wheel rotate.

The difference in that behavior can be easily explained when we look at the effective radius at which the pulling force acts. In the demonstration below, you can observe how the angle of the applied force changes that radius:

The smaller that radius, the smaller the arm force and thus the resulting torque on the rim, which makes the wheel rotate more slowly. Let’s look at how these concepts translate to the rear wheel of a bike.

When a rider pedals, the chain pulls the sprocket attached to the hub, which exerts a torque on that hub. The hub will rotate a little relative to the rim, which changes the location of the inner ends of the spokes. This makes the spokes a little longer, which increases their tension, but it also creates a tiny arm on which the spokes can act to turn the rim. Let’s visualize that force exerted by one of the spokes on the rim:

Recall, that the spokes can rotate in their holes, so they can only apply forces along the length of the spoke itself. Here, however, lies the problem. As we’ve already seen in the wheel turning demonstration, because of that tiny arm, these new forces will have a very hard time rotating the rim.

If we only cared about rotating the rim itself then our existing arrangement of spokes would be good enough as the rim is quite light. However, when the hub is forcibly turned by the pedaling action, the spokes attached to that hub have to pull and accelerate not only the relatively light rim, but that wheel also has to accelerate the entire bike and rider, which is a significantly harder task. That difficulty manifests in an increased spoke tension and rim stresses, so radial spokes are very rarely used on a rear wheel.

To make it easier for the hub and the spokes to rotate the rim, we have to make sure that the effective arm length at which the spokes pull the rim is larger. We can easily do that by using longer spokes that are positioned away from the radial direction by applying some twist to the hub. Firstly, let’s just set things up using bare long spokes:

Unfortunately, if we then screw on the nipples and try to build up tension in the spokes, that pulling force would just rotate the hub back, preventing any tension from building up until the spokes are back at the radial configuration:

The brilliant solution that bicycle wheel designers came up with is to flip the orientation of every other spoke. Let’s remove the nipples and rearrange half of the spokes to point in the other direction:

If we now put the nipples back in and we tighten them, one half of the spokes wants to rotate the hub clockwise, and the other half wants to rotate the hub counterclockwise, keeping everything in balance and allowing the tension in the spokes to build up:

This configuration uses a so-called “3 cross” because each spoke crosses three other spokes on its way from the hub to the rim. This and other patterns form a family of tangential lacing, because the spokes are, more or less, tangent to the hub circle instead of coming out of it radially.

With all the pieces in place, let’s see how external loads end up affecting the tension in the spokes. With just the weight load, these spokes behave very similarly to radial spokes, with the spokes close to ground contact carrying the vast majority of the load:

With every turn, each spoke passing under the axle gets loaded and unloaded. This contributes to mechanical fatigue of spokes, which over a long period of time can eventually cause the spokes to snap.

When the rider is pedaling and a torque gets applied on the hub, the “trailing” spokes, which are pulled by the hub increase their tension, and the leading spokes decrease their tension:

Our bicycle is using the traditional caliper brakes that squeeze the rim, which results in friction that applies torque to that rim. Some bicycles use disc brakes, which work by applying torque directly on the hub. These braking systems apply forces similar to pedaling, just in the other direction, which also prevents use of radial spokes in those bicycles.

Finally, let’s explore how a side force that a wheel experiences during turns affects the tension in the spokes:

During a ride the spokes experience a mix of all of these loads, many of them happening at the same time. Spokes are typically pretensioned strongly enough so that none of them lose tension when riding, but during an accident the damaging loads can exceed the safe limits and the wheel can collapse and bend under radially unbalanced tension. Interestingly enough, tightening the spokes too much is also unsafe, as they can snap under too much load or cause the rim to buckle.

Some high-end bicycle wheels abandon the traditional tensioned spokes and instead use a shell-like structure – these rims work similar to the traditional wagon wheels, but the carbon composites used for their construction allows them to be very lightweight.

As we’ve seen, the spokes in a modern bicycle wheel are always being pulled, so the stresses they experience are of a tensile nature. In the last part of this article we’ll look into bicycle frames to see how they handle a whole range of other kinds of stresses.

# Simple Frame

Let’s take a look at the shape of a typical bicycle frame:

A frame like this is commonly made from steel or aluminum tubes of different lengths and diameters that are welded together to form a single, rigid structure.

Fundamentally, a bicycle frame has to provide support and mounting place for four major points of interest: the seat, the rear axle, the crankset, and the rotation axis of the fork:

It may be then a little puzzling why a typical frame connects these four locations with two triangular patterns. After all, there are many other ways one could join these points to form a frame:

Moreover, it may be unclear why the tubes forming the frame are hollow, and why their diameter varies across different sections of the frame.

Let’s try to understand what factors influence the design of a bicycle frame by trying to build one from scratch. We’ll keep the wheels, pedals, seat, and the entire front assembly as is – the shape and placement of these components have been perfected over the years to ensure that the bike is stable and that they fit the human anatomy well.

We’ll start with a very simple idea of a cross-like frame shape constructed from solid steel rods. The slider controls the diameter of these rods – you can see their cross section in the bottom right corner:

In the bottom part of the demonstration you can see the total weight of the frame as constructed. The thinner the rods in the frame, the lighter the bike and the easier it is to accelerate and slow down, so we definitely want to keep the rods as thin as possible.

This simple frame joins all the important parts we care about, so it would appear that it could work just fine, but this scenario doesn’t consider any loads one can put on a bike. Let’s see what happens to this frame when we load it up with the weight of the rider on the seat:

Even though the frame is made from steel, it’s not perfectly rigid and it bends when loaded. Although some amount of deflection is unavoidable in any frame, there are also hard limits that we can’t cross. For example, if the frame sinks too much, it may prevent the pedals from rotating without hitting the ground.

Unfortunately, the amount of deflection under typical load is not the only concern we have when designing a bicycle frame, but to understand these other considerations we need talk about bending.

# Bending

Let’s start with a simple example. In the following demonstration you can use the slider to apply a moment of force at the two ends of a beam made of metal – you can choose which metal using the control below:

The applied moment bends the beam – it becomes curved. While the material from which the beam is made affects how much it bends, the nature of the bending behavior is identical, so we’ll just focus on simple steel elements.

Let’s take a closer look at what happens to different sections of a beam when they’re bent. In the demonstration below, you can once more apply a bending moment to the beam to see it deform, but I’m also drawing colored lines on its side. The bottom part shows the same lines that have been straightened out, but kept at the same length as the lines in the top part. Notice that when we bend this beam its top sections become shorter and its bottom sections become longer.

The upper parts are shortened so they’re under compression – bending squeezes these areas. The lower sections of the beam are getting longer so they’re under tension – bending stretches these sections of the beam.

Notice, that there is also a dashed neutral line that doesn’t change its length as we bend the beam. For this simple rectangular cross section the neutral axis goes right through the middle of the beam, but in more complicated forms the different distribution of pushing and pulling sections may shift this neutral line away from the midpoint.

In our discussion of wooden and steel spokes we’ve witnessed how compression and tension create stresses in a material and these beams are no different. In the demonstration below, you can bend the same beam using an adjustable bending moment. Different sections of this beam are colored accordingly to the stresses they experience – either increasing tension, or increasing compression:

As we’ve seen, the further away from the neutral axis a slice of the beam is, the more it changes its length, so the forces it exerts are larger, and so are the stresses. If these stresses grow too big, the beam can deform permanently, or even break completely.

At first glance, it may appear that we should change the cross section of our beam to move the material closer to the neutral axis to make sure it’s not overstressed. In fact, we want to do the exact opposite.

In the demonstration below, you can see a closeup of a small part of a beam and the forces exerted by a slice of it – you can control the location of the investigated slice with the slider. Notice that the forces that a slice exerts act at a small arm, thus creating a counter-moment that opposes the bending moment:

The sum of the moments exerted by every slice of the beam is what balances the external bending moment. If we were to move one of these slices further away from the neutral axis, we would not only increase the arm length at which this slice exerts its force, but that force would also grow larger since the slice would get stretched more, as it would have to fit a longer arc.

Both the arc length and the force arm are proportional to the distance from the neutral line, so the moment that a slice exerts grows with a square of that distance. A beam slice placed further away from the neutral axis is more effective at countering the bending moment, so by moving it away, the overall curvature and the elongation of individual slices would decrease too.

Let’s see this behavior in practice. In the demonstration below, you can change the proportions of the beam to make its cross section shorter or taller. Throughout these changes I’m keeping the area of that cross section the same, which means that the beam’s mass doesn’t change, we’re just redistributing that mass around:

Notice that a slab bends very easily and experiences high stresses, but a one barely changes shape and is also lightly stressed. You’ve probably experienced this yourself trying to bend a school ruler – it’s much easier to do it in one direction than the other.

While these tall shapes are great at resisting bending in one direction, they’re naturally weak in the perpendicular direction. A turning bicycle undergoes side loads too, so the elements we use for building a frame have to be capable of handling forces in multiple directions, which requires a more proportional shape. We also don’t want our bicycle frame to have sharp edges, which brings us back to the solid round rods we’ve initially used for our frame. Let’s see the stresses and deformation in rods of different diameters as we apply a bending moment:

These rods are no different than the square beams we’ve seen before – the sections furthest away from the neutral axis carry the most load. Conversely, the sections closest to center carry the least load. In fact, we can completely remove these central sections of the rod and move that material more to the outside – instead of solid rods, we’ll use hollow tubes.

In the demonstration below you can control the hollowness of a simple rounded rod. I’m keeping the area of the rod’s cross section constant so that as the radius of the tube grows, the walls get thinner. This also means that all the tubes used in this demonstration have the same mass:

By increasing the tube’s diameter and thinning its walls, we’re making it more resistant to bending. It may seem that by making the walls very thin and the tube diameter very large we could make this cylinder arbitrarily strong, but if the walls are too thin they’ll simply buckle on the compressed side – you can easily test this by trying to bend an empty aluminum can.

We’ve already seen that the way a beam bends depends on the geometrical properties of its cross section, the mechanical properties of the material used, and on the applied loads. That last aspect deserves a little more attention.

In our examples we’ve been bending the rods with a simple moment of force, but the loads on a bicycle frame are a little more complicated, with various forces applied at many different locations. These forces and the moments they create are carried through all the particles in the frame, and each section of the frame ends up experiencing a slightly different load.

Let’s experiment with a simple rod that is loaded with three balanced forces. You can change their magnitude using the slider:

Notice that the tensile and compressive stresses are no longer uniform across the rod’s length. To understand why this happens, we need to draw a diagram of forces acting on that beam, which you can see in the next demonstration. The bottom part shows the same rod, but this time we’re virtually slicing it at the location determined by the second slider, letting us see the forces that the cut part feels:

Notice that neither the whole rod nor any of its parts are moving, which implies that the forces and moments on every section have to be perfectly balanced. This lets us calculate the internal force and the internal moment that the particles at the sliced location have to exert to keep the remaining section of the beam static.

As you can see, the bending moment varies across the length. Since the bending-related stresses are proportional to that moment, different sections of the beam experience different amounts of stress.

In general, the forces that a tube in a frame experiences will be quite varied, and every load may cause that tube to be bent and stretched or compressed at the same time. These stresses will combine to form a more complicated distribution. From this point on I’ll stop separating compressive and tensile stresses, and instead I’ll combine them into a single equivalent stress:

The brighter the element the higher the stress. Once more, when these stresses get too high, they can permanently deform or break the material. Let’s see how the concepts we’ve explored here apply to our simple bicycle frame.

# Stresses in the Frame

In the demonstration below, you can revisit the previous frame constructed from solid rods with controllable diameter. This time, however, I’m also visualizing the stresses in the rods:

I need to note that this analysis is simplified, as we’re only analyzing the stresses within the tubular sections of the frame – we’re ignoring more complicated stress distributions in the joints between these elements. However, it’s still useful for doing the first pass analysis of a frame.

Notice that for a we need to increase the diameter of the rods quite a lot before we – they indicate areas of higher stress. Unfortunately, this makes the frame quite heavy.

To reduce this weight we can make the tubes hollow – we’ve seen how this mass redistribution can actually reduce the stresses. In the demonstration below, we’re once again starting with solid rods, but this time we can increase their diameter, while also making the walls thinner. I’m keeping the area of the cross section of the pipes constant, which fixes the weight of the frame:

If we make the tube cross sections we can make this frame handle vertical seat loads with little deflection and within the limits of acceptable stress.

However, as we get ready to ride this bike, new problems emerge. In the demonstration below, you can witness what happens to the frame when the rider starts to vigorously pedal by shifting all his weight from the seat to standing on the right pedal:

The pedals are positioned at a certain distance from the frame, so when a pedal is pressed, that distance acts as a lever that tries to twist the frame out of its position, making many of the tubes become more stressed again.

This time, however, the stresses are a little different as the tubes also undergo a torsional deformation. Let’s explore this concept a little closer. In the demonstration below, we’re once again encountering a solid steel rod. Instead of bending it with a force or a moment, we’re applying a twisting moment to it:

With a solid rod like this, it’s a little hard to see whats going on inside of it, so let’s focus on a short section of that rod and employ a magic see-through vision to let us observe how the ring sections of this rod behave. The second slider lets you choose which ring of this rod we’re looking at:

As we twist this beam, its different sections become deformed, which you can see by observing the horizontal lines I drew on each ring. Notice that lines in the center are , but the ones close to the surface are .

The further away from the central axis a part of the rod is, the more deformed it becomes, as it has to sweep a longer arc as it gets twisted. The bigger the deformation, the larger the stress that section experiences:

This stress is a little different than the ones we’ve explored before, as these elements gets sheared. These shearing stresses add another contribution to the total stress that frame elements experience.

Notice that the central parts of this solid rod barely get deformed and stressed. This is yet another reason why hollow pipes are practical for reducing mass of a bicycle frame without compromising its load carrying capabilities. You can experience these loads for pipes with the same mass, but different hollowness in the demonstration below:

Let’s go back to our bicycle frame. As we’ve seen, despite our clever attempts to use hollow tubes, the frame we’ve built still was experiencing high stresses. If we kept trying to redistribute the mass inside individual tubes, we could eventually end up with a cross-like frame that’s quite robust, but unfortunately heavy.

However, instead of shifting the mass inside the rods, we can shift the mass inside the frame itself. Let’s start by adding three new tubes at the bottom of the frame. To reduce weight we’ll use hollow tubes across the board, so now we can just control their diameter:

Conceptually, these new tubes redistribute the mass further away from the neutral bending axis. For , the central part of this frame now acts similarly to a very tall element as the upper and lower tubes are quite far apart. For , this new distribution also helps by bracing the sprocket area and preventing bending stresses from accumulating in the central part.

This new design lets us use thinner and lighter tubes, but there are still a few things we can improve. Notice that the tube that supports the seat still can get pretty stressed because the bending moment inside it can grow quite large. Let’s see if we can improve it by changing the location of the central joints using the third slider:

Notice that in the we’ve almost completely minimized the stresses from the . While the area close to the sprocket is still somewhat stressed , many of the tubes are now too strong for their purpose. Let’s tweak the size of the individual tubes by thinning the ones that are underutilized, and thickening the more stressed ones:

We’ve more or less arrived at the modern bicycle frame. One could take things even further by making the tubes oval to make them more rigid for bending with vertical forces, or by butting the tubes by making them thinner in the central part, while maintaining thicker walls close to joints, where the stresses are larger.

I need to emphasize that we’ve only conducted a very simplified investigation of a bicycle frame. A more rigorous mechanical analysis requires calculation of all the stresses caused by the bending, torsional, and normal loads to make sure that the materials used will not fail with these and other acting forces a bicycle is required to handle. We also haven’t looked at loads in welds connecting the frame elements – these are often one of the more stressed parts of the frame.

While the classic two-triangle “diamond” frame is mostly commonly used, many bicycles tweak that timeless construction by changing position or even removing some of the tubes. All of these changes have to consider the stresses in the final design to ensure that the bike is safe.

Thankfully, the use of computer aided design tools and modern composite materials lets bicycle manufacturers create much fancier frames with different shapes and thicknesses of individual tubes with varied location of their connection points.

Bicycle stability is an actively researched topic, with new discoveries being made in the last two decades. Andy Ruina’s lecture serves as a gentle introduction into the topic before diving into the canonical paper on the subject written by a group of bicycle researchers. These collaborators also took a closer look into some mistakes made by previous inquirers in this area. For a very long but enjoyable and down-to-earth read, I highly recommend Jason Moore’s PhD thesis on human control of bicycles.

Matthew Ford did extensive research of bicycle wheels, which culminated in his great PhD thesis that thoroughly covers theoretical background and experimental results of what happens to spoked wheels under load. Additionally, Matthew wrote a few more entry level articles on his blog and also made a wheel simulator that lets you see forces in spokes under different types of loads.

While exhaustive resources on the influence of forces on the motion of bicycles are relatively scarce, in the sibling field of motorcycle physics, Vittore Cossalter’s Motorcycle Dynamics serves as a good stand-in. While some of the topics in the book aren’t directly applicable to bicycles, the publication still covers a lot of phenomena relevant to all single-track vehicles.

Finally, Bicycling Science by David Gordon Wilson covers many of other bicycle-related topics like human power generation, rolling resistances, and a deeper dive into aerodynamics, all written in a very approachable style.

# Final Words

In the real world, the forces that we have conveniently been visualizing with arrows are not directly visible, but their effects are very tangible. Sometimes that action is, quite literally, straightforward, like when we pedal to propel the bicycle forwards. On the other hand, the interplay of forces that keeps a bicycle stable is much more complicated.

All of these forces are ultimately exerted and carried out by individual atoms that move around to affect their neighbors. The interactions of a single particle in the ground, a tire, or a frame, are completely insignificant, but once we accumulate the effects of all the molecules in these bodies, they end up forming a visible manifestation that makes bicycles rideable.

Perhaps the next time you’re on a bicycle, you’ll be able to conceptualize some of those invisible forces that make the ride so enjoyable.