 In the previous video, which you can find right here, we looked at the concept of a free body diagram and how you can go about creating one. One of the steps in that process was to examine the structural supports that are severed in the process of freeing a body from its environment and replace them with appropriate reaction forces. In this video, we will take a closer look at the process as well as look at some of the more common support types you may encounter in engineering structures. I keep talking about supports, but what is a support, really? Take a look at this model aircraft. We can clearly see that the aircraft is supported by the metal arm connected to the base that sits on a table. If I wanted to create a free body diagram of the model, I would cut away the surroundings like this. But if I did that in reality without the support, the model would simply fall, so support can be viewed as something that limits the motion of a structure. In engineering mechanics, when we talk about the possible motion of a body, we typically use the term degree of freedom. If you consider the degrees of freedom of a rigid body in two dimensions, you can see that the body can translate in two different coordinate directions, as well as rotate in a third direction. Any motion of this body in 2D space can be described by these two translations and one rotation. It thus has three degrees of freedom. In 3D space, we have six possible degrees of freedom, three translations and three rotations, and the motion of any body in 3D space can be described by a combination of these three translations and three rotations. We can thus easily evaluate the appropriate reaction forces and moments that can act at a support that we remove in our free body diagrams by looking at the motion that support would have restricted. Let's look at the procedure for this using a welded bicycle frame. Imagine we wanted to create a free body diagram of only the top tube of the frame. We would have to cut through the welded joint as shown here. To determine the appropriate reaction forces, there are some simple steps we can follow. First, we need to determine if the problem will be analyzed as two-dimensional or three-dimensional. As we saw earlier, this will influence the number of degrees of freedom to consider in the problem, and thus the potential number of reaction forces and moments. For this bike frame, we will consider it as a 2D problem, only considering forces acting in the plane of the frame itself, which we will label the XY plane. Next, we will look at what translational degrees of freedom are restricted by the support. For the bike frame, the weld prevents translation in the X and Y directions. Thus, we need a reaction force in the X direction to prevent translation in this direction, and similarly, we will need a reaction force in the Y direction to prevent translation in that direction. Translation is not the only possible degree of freedom, so we also have to look at what rotations are constrained by the support and add appropriate moments to replace that constraint. As a welded connection of the bike restricts rotation within the XY plane, we need to add a moment to prevent that motion, a moment in the Z direction. If the problem was analyzed as three-dimensional, then we would have to consider the other degrees of freedom as well. This welded connection would thus impart a reaction force in the Z direction and moments in the X and Y direction as well, but the overall procedure remains the same. Now that we know the procedure for determining appropriate reaction forces and moments, let's examine some of the more common engineering supports you will encounter in engineering problems. The first type of support we will look at are contact surfaces. A contact surface refers to two objects that are simply in contact with each other, but not otherwise joined or connected. We tend to further subcategorize contact surfaces as smooth contact surfaces with negligible friction, such as an ice gate gliding on a sheet of ice, or as rough contact surfaces where non-negligible friction forces can develop, such as in the landing struts of the SpaceX Falcon 9 rocket. First, let's consider a smooth contact surface where friction is negligible. If we cut away the contact surface when creating the free body diagram, what reaction forces and moments need to be applied in order to emulate the constraints it places on the motion, or degrees of freedom, of the body? As the contact surface is frictionless, the body would be free to translate along the contact plane. Thus, there should be no reaction force parallel to the contact surface. However, a reaction force normal to the surface is necessary as the body would be constrained from moving through the contact surface. We even know the direction of this normal force as the body is only constrained from passing through the contact surface, but is free to be lifted off of it. For this particular contact surface, we can also see that the rotation would not be constrained. For a rough contact surface, friction is not negligible, or at least the contact surface has a potential to generate non-negligible friction if there is a large enough normal force acting on it. How would this affect the expected reaction forces acting on the contacting body relative to what we saw for smooth contact? The situation would be similar to smooth contact in that a normal force would prevent motion of the body through the contact surface. However, now we would also have a friction force parallel to the contact surface that would prevent the body from sliding along it. The total reaction force R would be the vector sum of these two components. So you can alternatively view the reaction at a rough contact surface as a single reaction force R acting at an angle theta relative to the contact surface. The next group of supports we will look at are wheel, roller, rocker, and ball supports. The reason we look at these together is that they all provide the same constraint to the motion of the body they support. They are simply different engineering solutions to provide that constraint. What constraint do they all share in common? Each of these supports is in fact an engineering solution to the reality of friction. They utilize rolling motion as a means to introduce a translational degree of freedom along a contact surface where there is friction. This is best exemplified by an inline skate which utilizes wheel supports to glide along a rough road surface, mimicking the smooth contact between the blade of an ice skate and a sheet of ice. As these supports mimic smooth contact surfaces, we indeed expect that they will produce the same reaction forces, a reaction force acting on the body which is normal to the supporting surface. We could be satisfied with this as a proof, but I find it useful to look a little more closely at exactly how one of these supports achieves this in order to prove it. Let's take the wheel support for instance. This support consists of a contacting surface, a wheel, a pin or bearing, and the body the wheel is supporting. We can draw free body diagrams for each of these components to see just how it is able to mimic a smooth contact surface. Starting with the wheel and contact surface, we can see this looks exactly like our previous scenario of a rough contact surface. However, as the wheel can roll, there will not be any reaction force parallel to the contact surface. Looking at the wheel, this normal force has to be balanced by an action force from the pin if it is in equilibrium. As the pin sits in a hole within the wheel, it can form a contact surface with the wheel in any direction and thus can support an action force in any direction. But since the force is only required to balance the reaction force present on the wheel, we will get an equal but opposite action force on the wheel from the pin. Using similar logic, we see that the reaction force on the pin from the wheel will be balanced by an action force from the body on the pin. This action force then leads to a reaction force on the body that is normal to the original contact surface. You can follow a similar line of reasoning to demonstrate how the other three support types achieve the same result, albeit with a slightly different series of contact surfaces. The next category is pin supports. You have likely seen such supports in many modern buildings constructed with tubular structural elements. You have almost certainly interacted with one of these on a daily basis as well whenever you open a door, although you might refer to it by an alternative name in such a case, a hinge. Looking at our pin support, we see that it is similar to our previous category of support, except that it is not free to translate. Thus, in addition to the normal force we had before, we also need to include a reaction force parallel to its base. Alternatively, you can also view this as a single resultant reaction force acting at an angle theta. No moments are generated at the pin connection as the pin permits the body to rotate, as it is in fact a hinge. The next category is known as a built-in or fixed support. The body is rigidly fixed to another body, preventing it from translating or rotating. How many degrees of freedom would such a constraint have in a 2D problem then? A fixed support constrains all degrees of freedom. In a 2D problem, this means we need to constrain the two translations with reaction forces and the one rotation with a reaction moment. This is precisely the case we analyzed earlier with the top tube of the bike frame. This covers some of the more basic supports we see in engineering structures, but it is by no means an exhaustive list. However, the process for determining what reaction forces a support can impart on a body remains the same. Look at the constraints the support places on the motion of the body and add the appropriate reaction forces and moments to maintain the constraint when the support is removed.