## 🟦 9.1 Introduction to trusses

Trusses are structures that are composed of members connected at their ends with pin connections.

For stability, each truss panel is comprised of three members at the perimeter. As a result, trusses tend to take the shape of a sequence of connected triangles. This phenomenon is called triangulation.

True trusses are only loaded at the nodes (at the pins).

As an introduction to trusses, let's consider the structure in the images below. While few people would describe this particular structure as a truss, it behaves like one, and is a good introduction to the topic.

Let's assume that the three rubber bands are identical in every way, and stretched equal amounts. That means that each carries the same magnitude of tension force.

Let's also assume that the system is in static equilibrium.

A number of FBDs follow. All are in static equilibrium. Inspect them carefully.

### ❏ Global equilibrium: the structure as a whole is in static equilibrium

### ❏ Exploded FBDs: each node and each member is in equilibrium

Did you notice the Newton's Third Law pairs in the exploded FBD? They are color-coded for you. Remember that Newton's Third Law pairs are equal in magnitude but opposite in direction.

This first example is a simple one, but the takeaways are important:

(1) Global equilibrium is still important (the process of solving the equations of equilibrium on the entire structure, intact).

(2) We can also investigate static equilibrium at each and every node. Since the forces at each nodal FBD are concurrent, in a planar (2D) truss, we can solve for unknowns using two equations of equilibrium:

the summation of forces in the x-direction equals zero

the summation of forces in the y-direction equals zero

## 🟦 9.2 Trusses in the wild

Trusses can be of any size or scale, but we often associate them with bridge structures, such as the railroad crossing depicted at left.

Trusses act a little like beams -- except that most of the material is cut away. A truss that spans a certain distance is significantly more efficient (cheaper, uses less material) than a beam under equivalent conditions.

Source: https://images-na.ssl-images-amazon.com/images/I/81AGStBikFL._SX679_.jpg

## 🟦 9.3 "Two-force" or axial members

True trusses are comprised solely of two-force members or axial members. The former terminology is common in lower-level courses (like Statics) and the latter is common in higher-level courses (like Structural Analysis).

### ❏ What are 2FMs?

A two-force member (2FM) is defined as one that:

has pin connections at both ends

does not have any internal or intermediate connections to other members

does not have any applied forces or moments along the member's length.

### ❏ Why are 2FMs important?

It's simple. When you have the ability to identify two-force members early in your problem-solving process, you will save yourself lots of time. This is because:

each pin has only one unknown (either axial tension or axial compression)

that is, the x-direction and y-direction components of that force are coupled -- they are similar to the x-direction and y-direction components of the member's geometry

### ❏ Two-force members are either in tension or in compression:

### ❏ Which members are 2FM?

### ❏ How to use the ratio of lengths to solve for force components for a 2FM

## 🟦 9.4 Method of Joints

In a truss, we are usually interested in solving for unknown reactions, as well as the forces transferred between various members at the pin connections. The Method of Joints is a systematic way to solve for all of the unknowns in a truss.

The flipbook below shows the procedure. Start by using global equilibrium to solve unknown reactions. Then, move systematically from joint to joint (or node to node) throughout the structure. At each node, use two equations:

the sum of forces in the x-direction equals zero

the sum of forces in the y-direction equals zero

You have to work strategically: start at a node that is solvable (no more than two unknowns). After each node is solved, apply Newton's Third Law to transfer equal and opposite forces to the next node.

## 🟦 9.5 Zero-force members

Sometimes, under a given loading, a truss member carries no load. When the internal force in a moment (under a given loading) is zero, we call that member a zero-force member (ZFM).

Why would we design a truss with members that don't carry any force? One application is long-span trusses that support moving traffic. Cars can be modeled as point loads (forces), and as they drive across the bridge, certain truss members transfer the weight to the supports. Others simply do not.

How do you find the ZFMs?

First, look for unloaded nodes (or joints or pins). More specifically, look for nodes that have neither an external reaction force nor an applied load.

Then, find patterns similar to the ones sketched in the diagram:

Members 1 and 2 are pin-connected at A. Member 1 can only exert a horizontal force on A, while Member 2 can only exert a vertical force on A. Since static equilibrium is impossible if either member carries force, both must be zero-force members.

Members 3 and 4 are pin-connected at B. Member 3 can exert horizontal and vertical force components on the node, while member 4 can only exert a vertical force. Therefore, both must be zero-force members.

Members 5, 6, and 7 are pin-connected at C. Members 5 and 6 are aligned and can easily transfer tension or compression through the pin connection. Member 7, however, is a zero-force member.

## 🟦 9.6 Problem-solving approach and mapping

In this flipbook, you'll see a mapping approach to fully solving a truss problem using the Method of Joints. Remember that each joint is a concurrent force problem, and that we can use ratios of the lengths to determine force components of inclined forces.

## 🟦 9.7 Method of Sections

All (statically determinate) trusses can be solved with the Method of Joints. That is a slow, systematic, dogmatic way to solve for all unknowns.

Sometimes, though, we aren't asked to solve for the entire truss. Maybe we just need to solve one particular unknown, such as the internal force in just one member.

The Method of Sections is a useful tool for solving a specific unknown, with surgical precision. Use the flipbook to see how this works.

Important: it is only because all truss members are two-force members that we can cut members and represent their internal workings with a single axial force.

## 🟦 9.8 Triangulation and stability

A truss panel is stable when it consists of 3 members connected by three pins. This is due to triangulation.

If you were to try to design a truss with a panel with four members and four pins, you will discover that it is not in static equilibrium at all. If this is an accident, we might call it a collapse mechanism. If it is intentional, we are designing a machine that moves.

Some trusses have more members than are necessary. The extra members can be called "redundant." This type of truss is called statically indeterminate, meaning that it cannot be solved with statics alone (meaning that we won't learn how to solve it in this class).

## ➜ Practice Problems

Solve each truss. Use the Method of Joints to put each node (or joint, or pin) in static equilibrium. Solve for all numeric unknowns. When you have fully solved the truss, draw an exploded FBD that shows each node in static equilibrium. Bring your work to class - we will figure out solutions together.

Hint: use symmetry to your advantage!!! Sometimes you only have to solve for half of the truss...

Problem 1.

Problem 2