# Statics

## π¦Β  4.1 Connections are used to prevent motion

Are you designing a building, a bridge, a machine, a rocket, or a robot? You're going to have many components that must be properly connected.

We define engineering connections by their effects on the system:

For instance, think about using a pushpin to hang a piece of paper from a bulletin board made of cork. Someone lightly pushes on the paper as shown. Does the pushpin constrain translation (rightwards motion)? Yes, it does. For that reason, it has the ability to develop a force reaction (a mechanism to transfer force). Does the pushpin constrain rotation? No, it does not. For that reason, it does not have the ability to develop a moment reaction (a mechanism to transfer moment).

How about the taped paper? The tape prevents the paper from rotating. It also prevents translation. Therefore the taped connection constrains (prevents) both translation and rotation. It develops both a force reaction and a moment reaction.

## π¦Β  4.2 Overview of engineering connections

The main types of engineering connections we use are the (external) pin, the roller (or pin-roller), and the fixed connection. We use these types of connections to connect our design to the context.

The interactive provides an introductory overview of these three common types of engineering connections. This is how we connect our design to the context.

Inside our design, we often have to connect one member to another. Those connections can be complex, but generally fall into one of two categories. If force is transferred but moment is released, we model an internal pin (sometimes called a hinge). If both force and moment are transferred over the joint, we call it a continuous connection.

Each of the three main connection types is discussed in more detail below. These are all considered two-dimensional (planar) engineering connections. Three-dimensional connections are introduced later in this course.

## π¦Β  4.3 A single pin

The most common type of engineering connection is called the pin.

One element (in this case, a yellow bar) has a hole. We insert a cylindrical element (in this case, a bolt) through the hole, and secure it (in this case, with a square nut).

Are you wondering what this pin is doing? Why is it there? What is its purpose? Well ... it's not doing much of anything; it's just here to show you what a pin looks like in real life.

Of course, engineers need to understand how nuts and bolts work to make good connections. For now, though, it may be useful to think of the pin as a solid cylinder.

A pin can transfer a force. The force can be at any angle. For that reason, many texts will state that a pin can transfer two forces (one in the x-direction and one in the y-direction). The force points in the direction required for static equilibrium. It could be upwards, leftwards, or at a 30 degree angle.

### β Flipbook: A single pin

In real-world applications, we securely tighten the bolt and nut assembly. In fact, we often make the connection so tight that we engage significant friction between the components.

In this Statics course, we will neglect all friction in the connection. For this reason, you can imagine all pin connections (in this class) as loosely-fitted bolts. Depending on your major, some of you will go on to learn to design bolted connections in future courses.

## π¦Β  4.4 The pin vs. the pinned connection

Now, let's build a structure out of two bars and a pin. I'll use the simple representation of a pin: a solid cylinder. (You can use your imagination if you want to think of it as a bolt.)

This concept is incredibly important for learning Statics and the classes that follow. You need to understand the difference between a pin and a pinned connection.

The pin transfers a force, but is not constrained from movement.Β

A pinned connection transfers a force and is constrained from movement.Β

Whenever a connection constraints translation while allowing rotation, we model it as a pinned connection.

In the flipbook, the orange cylinder is the pin. When the pins act in combination with the blue plates and the hands, we simulate a pinned connection.

If you connect two bodies with one pin, both tend to rotate about the pin.

Conversely, if you connect two elements with multiple pins, they tend to create a rigid body that acts like a single unit.

### β Pinned connection

Photo by S. Reynolds, Amsterdam Centraal Station, Summer 2023

### β Multi-pin connection

The two bars act like a single unit due to being connected by two pins. They don't pivot about either pin, because the partner pin provides a force that prevents that.

## π¦Β  4.5 Rollers (also called pin-rollers)

You can think of the roller connection (or pin-roller) by thinking about a skateboard.

The wheels of the skateboard can provide upward force reactions, but they do not provide resistance to the horizontal force the skateboarder supplies by pushing against the ground. Horizontal translation is unconstrained; vertical motion is impeded.

Since the roller prevents translation perpendicular to the surface, it has the ability to develop a force reaction perpendicular to the surface. It's a normal, compressive force.

### β Flipbook: roller connections

Here is the tricky thing about roller connections. Different engineers use the roller symbol slightly differently.

Let's compare the skateboard type of roller to a wheel that rolls in a track. You will find the wheel-in-track connection in an overhead garage door, a dishwasher, and a "chest of drawers" ("dresser").

On this site, we will assume that unless noted otherwise, all of our pin-rollers act like wheels in a track.

### β There is no consensus on roller connectionsΒ

Mechanical engineers tend to think of the roller as the sphere on a surface. Structural engineers tend to think of the roller as the wheel-in-a-track. Software tends to use the wheel-in-a-track default (although you can further constrain the computer model if you need a ball bearing type of connection.)

The roller has a close cousin, called the rocker. The one depicted here develops a vertical force reaction, but does not provide a horizontal force reaction. Instead, if there is a horizontal force in the system, the rocker pivots to accommodate it.

This type of connection is used in bridges. It combats thermal contraction and expansion. This topic is studied in Mechanics of Materials.

Rockers aren't a focus of this Statics course, but it's good for you to know that they exist.

## π¦Β  4.6 Fixed connections

Let's begin with a thought experiment.Β

Two bodies are supported by pushpins as shown. A single force is applied to each.

Can the pushpins create static equilibrium?

Pushpin A, acting in isolation, cannot create static equilibrium all by itself. The body will rotate clockwise about A due to the applied force.

Pushpins B and C both deliver normal forces to the body. Together, their force reactions impede both rotation and translation of the body. They prevent the body from rotating clockwise by applying an equal and opposite moment counterclockwise. They can create static equilibrium.

### β Interactive: effects of multiple pushpins

Together, pins B and C help us conceptualize the idea of a fixed connection.

They combine to supply a net force in the y-direction that counteracts the applied force. They also create a reaction moment. The fixed connection is the only one that can develop a reaction moment. That's what makes it so special.

The step-by-step way to think about the reaction moment is cartooned below.

### β FBD with moment reaction

Another way to think about a fixed connection is to imagine a built-in support. In the image at right, imagine that the piece of sawn lumber has been wedged (tightly) into a cast-in-place concrete support.

If you applied a force horizontally, the member wouldn't move. Since horizontal translation is constrained, the fixed connection can develop a horizontal force reaction.

What about if you apply a force vertically? Again, the member wouldn't translate vertically due to the constraint. The fixed connection can develop a vertical force reaction too.

Finally, if you tried to rotate the beam by applying a moment, you wouldn't be able to do so. This is because the concrete support pushes back in the same way that the pushpins do above. Since rotation is constrained, the fixed support can generate a moment reaction.

### β Illustration of a "built-in" fixed connection

You can think about the fixed connection as an βupgradedβ pin connection. It does everything the pin connection does, except that it can also develop a reaction moment. For this reason, the fixed connection is the most powerful type of connection we have in our arsenal.

## π¦Β  4.7Β  Connections in software

As engineers, we talk about pinned connections, roller connections, and fixed connections as we talk to colleagues.

When we use software to direct a computer to perform an analysis for us, we simply toggle the constraints we want. Each constraint creates the potential to develop a reaction force (or moment).

The screenshots here are from SAP2000, a structural analysis program commonly used in professional practice. While this dialog box has the ability to program 3D connections, it should still be understandable in a 2D planar context.

Note that the dialog box uses 1/2/3 instead of x/y/z to reference the coordinate system.Β

### β When there is no connection, we say it's "free"

If a node (point, or particle) is not supported at all, we can say it's "free" to translate and rotate. It's not constrained in any way. It won't develop any reactions either.

In other words, the lack of connection can be called "free."

## π¦Β  4.8 Graphic summary of engineering connections

In this last section, you'll see the three main types of connections in a few different ways. On the left is a 3D sketch of how the connection might appear in the real world. Then, you'll see the proper engineering symbol for each idea. (Sometimes engineers use "two-line" drawings to show thickness or width. Other times, we use a simplified "one-line" drawing instead.) Lastly, on the right, you'll see the force (and moment) reactions cartooned in 2D and 3D.

### β The symbol for a pinned connection is a triangle

The pinned connection can generate component forces in the x- and y-directions.

### β The symbol for a roller connection is either a circle or a triangle on wheels

The roller (or pin-roller) connection can only generate a force reaction that is perpendicular or normal to the rolling surface.

### β The symbol for a fixed connection is a heavy line and short, parallel hatch marks

The fixed connection is like an upgraded pinned connection. It does everything the pinned connection does, but also can develop a reacting moment. This is our most robust connection (and our most expensive). Make sure you account for the reacting moment in your calculations!

We have already learned that loading diagrams and free-body diagrams are related, but exist for different purposes:

These are the "fancy" symbols for pinned and roller connections. You'll see this type of representation on this website.

You are welcome to use these simpler symbols in your problem-solving practice. They are extremely common and used fairly universally.

### β The free-body diagram (FBD)

Remember to free the body from the supports (the context) when drawing the FBD. The supports have been replaced by their effects on the body.

### β Key concepts and takeaways from the images:

1) Did you notice how the reaction forces (and moments) are portrayed differently on the FBD above? I am making an effort to update my teaching materials so that applied forces and moments are pink, reacting forces and moments are purple. Once we get to internal forces and moments, you'll see me color them orange. You don't have to color-code your work, of course, unless the color-coding helps you learn.

2) Did you notice the little hatch marks applied to the reaction forces (and moments) in the FBD? This technique is common in structural engineering, but less common in other engineering disciplines. It's a way to keep track of which forces are applied vs. reacting when you're not color-coding the vectors. On this site, sometimes I use the hatchmarks (when I really want to emphasize the difference between applied and reacting forces) and sometimes I omit them. That's OK.

3) Did you note that I used the symbol P for applied forces in this problem instead of the more generic F? The P stands for point load and is the most common symbol used for applied forces (or loads) in solid mechanics. You'll learn more about point loads in the next lesson - stay tuned for more on that front! π

4) Did you notice that Ax, Ay, Dx, and Dy are shown as x- and y- components? We would say that those vectors are uncoupled (we don't know the angle of inclination of their resultant force). The normal force at B also has components, but they are coupled: we know the angle of inclination due to the problem geometry. Solve one and you can figure out the other through geometry. At C, since the component is known to align with the global y-direction, Cy is used. It would also be OK to use NC (N sub C) for that vector.

## β Practice Problems

Fall 2024 students: No practice problems are assigned on this lesson. If you're behind on prior practice problems, please use this time to get all caught up! -S

note to self: in the future, this would be a good opportunity to give students more targeted practice here with multiple coordinate systems. also, maybe better to put a section on pins vs continuous joints here?Β