## Introduction: Analyzing a Simple Truss by the Method of Joints

**What is a truss?**

A truss is one of the major types of engineering structures and is especially used in the design of bridges and buildings. Trusses are designed to support loads, such as the weight of people, and are stationary. A truss is exclusively made of long, straight members connected by joints at the end of each member.

**What to expect from this Instructable**

This Instructable will explain how to calculate the effects of a force on a truss. It will teach you how engineers determine the strength of bridges and determine their maximum weight capacity on a small scale. This Instructable will use concepts from classical physics and math.

__Expect to use__

- Trigonometry

- Algebra

- Scientific calculator

- The Pythagorean Theorem

- Application of Newton's Third Law

- Addition of forces in the horizontal and vertical directions

__How long will it take?__

The Instructable should take 30 minutes to an hour to work through, depending on your prior math knowledge

**Supplies**

To complete your truss analysis you will need:

- A piece of paper (or two)

- A pencil

- Scientific calculator ( can calculate sine, cosine, and tangential angles)

## Step 1: Examples of Trusses

Trusses are used in the construction of nearly every road bridge you will encounter in your city's highway system. The 3 main types of trusses used in bridge design are Pratt, Warren and Howe. Truss type differs only by the manner and angle in which the members are connected at joints.

Bridge trusses can also be unique, and made of multiple types of truss designs.

The Golden Gate Bridge has a unique truss incorporated into its design.

## Step 2: Trusses, Joints and Forces

**Example of simple truss**

This diagram is an example of a simple truss. A simple truss is one that can be constructed from a basic triangle by adding to it two new members at a time and connecting them at a new joint.

**Example of joint **

In this diagram, points A,B,C,D,E,F and G are all joints. A joint is any point at which a member is connected to another, typically by welding, pins or rivets.

**Forces**

The weight that each joint bears can be represented by a force. A force is defined by physics as an objects' mass multiplied by it's acceleration. In the case of a stationary truss, the acceleration taken into account is that of gravity. Therefore, the forces that a truss absorbs are the weight (equal to mass multiplied by gravity) of its members and additional outside forces, such as a car or person passing over a bridge. In the diagram of the simple truss, the forces are represented by black arrows in units of **Newtons**.

A Newton is the International System of Units (SI) derived unit of force.

**Method of joints**

The method of joints analyzes the force in each member of a truss by breaking the truss down and calculating the forces at each individual joint. Newton's Third Law indicates that the forces of action and reaction between a member and a pin are equal and opposite. Therefore, the forces exerted by a member on the two pins it connects must be directed along that member.This will be more clearly seen in the next few steps.

The analysis of the truss reduces to computing the forces in the various members, which are either in tension or compression.

## Step 3: Using Trigonometry

To calculate forces on a truss you will need to use trigonometry of a right triangle. A** right triangle** is a triangle in which one angle is equal to 90 degrees. If the angle is 90 degrees, the two sides of the triangle enclosing the angle will form an "L" shape. A 90 degree angle is typically denoted in diagrams as a square in the corner of the triangle. A right triangle is the basis for trigonometry.

The side of the triangle opposite the 90 degree angle is known as the** hypotenuse**. The hypotenuse is always the longest.

Using either of the remaining angles, you can name the other sides of the triangle. We will declare the other angle as the Greek letter **theta **until we calculate its value.

the "**opposite**" side is opposite angle theta. (diagram)

the "**adjacent**" side is always next to angle theta. (diagram)

**Sine, Cosine** and **Tangent **are the three main functions in trigonometry and are shortened to **sin, cos** and** tan** (as they are displayed on your calculator).

As shown in the diagrams, each function can be represented by an equation using the side lengths of the triangle.

With these equations, you can calculate the side length of a triangle if the angle theta is known.

You can also calculate the angle theta if the side lengths of the triangle are known. to do this you will use the **inverse** of the sin, cos or tan function. The inverse trig functions are denoted by "sin−1 (x), cos−1 (x), tan−1 (x)," and can be found on most scientific calculators. An example of calculating the inverse is shown in a photo above.

You can also calculate the side length of a triangle if two of the side are known by using **Pythagorea's Theorem** which says the square of the hypotenuse is equal to the sum of the square of the adjacent side plus the square of the opposite (a^2+b^2=c^2). You can plug in the known side lengths and solve for the unknown.

This trigonometry will be applied in the Instructable when solving for forces. Inverse functions will be used frequently to determine angles based off the dimensions of the truss.

## Step 4: Draw a Free-body Diagram of the Entire Truss

A free-body diagram is a diagram that clearly indicates all forces acting on a body, in this case the body being the truss.

As an example, consider this crate suspended from two cords. The forces exerted at point A are the force of tension from the cord on the left, the force of tension from the cord on the right and the force of the weight of the crate due to gravity pulling down. These forces are represented in the free body diagram as Tab, Tac, and 736 Newtons, respectively.

As an example of a free body diagram of an entire simple truss, consider this truss with joints A,B,C,D. **This truss will be used as an example for the next few steps**. Force P, represented as the downward arrow, is representing the weight of the truss and it is located at the truss' center of gravity. Point A is connected to the ground and cannot move up, down, or left-right.

Therefore, point A experiences what is called a ** reactionary force**. This is a force is that is exerted on point A that prevents A from moving.

Point B also experiences a reactionary force, but the support at point B only prevents the structure from moving up or down. Therefore, the reactionary force at B is only directed upward.

-supports that have only an upward or downward reactionary force are represented in the diagrams with a rounded bottom or round wheels.

-you will use trigonometry to break the reactionary force at A into horizontal and vertical components

## Step 5: Solve for Reactionary Forces of Truss

Using the free-body diagram you have just drawn of the entire truss you will solve for the reactionary forces.

To do this you will write three equations. These equations come from the fact that the truss is stationary, or unmoving. In order for the truss to remain stationary, the forces it experiences in the horizontal direction must cancel each other out, and the forces in the vertical direction must also cancel out.

The first equation is written for the forces in the vertical direction. We will denote downward forces to be negative and upward forces to be positive. The vertical forces are all added together and set equal to zero.

The second equation will be written for the forces on the truss in the horizontal direction. We will denote forces to the right to be positive and to the left to be negative. Similarly, the horizontal forces will be added and set equal to zero.

The third equation is the sum of the moments of the forces acting on the truss. A **moment** is a measurement of the tendency of a force to make the object rotate around a fixed point. A moment is equal to the force multiplied by its perpendicular distance from the fixed point.

For our fixed point, we have chosen A. The point at which the moments are summed is arbitrary, but the best choice is a point that has multiple forces acting directly on it. Forces that act directly on the point not considered in it's moment equation. We chose point A because the vertical and horizontal components of Ra are therefore not considered in the equation. The sum of the moments about the fixed point are added together and set equal to zero.

**If the force acting on the body will cause the body to rotate counterclockwise, such as Rb in this case, it is considered positive. if the force causes the body to rotate clockwise, it is considered negative.

Using these three equations and substitution we can solve for reactionary forces of the truss.

## Step 6: Locate a Joint With Only Two Members

After solving for the reactionary force, the next step is to locate a joint in the truss that connects only two members, or that has only 2 unknown forces.** Based on the simple truss used in the last step**, this joint would be either A or B. The choice of this joint is up to you, as long as it only connects two members.

We will choose joint B.

After choosing your joint, you will draw another free-body diagram. This free body diagram will correspond to the joint alone and not the entire truss.

Each member is represented as a force arrow.

## Step 7: Determine the Unknown Forces of the Joint

This is the step that will also involve the use of your calculator and trigonometry. In order for the truss to remain stationary, the forces on each joint from every direction must cancel each other out. If a force is directed at an angle, like in the case of some members of a truss, the force can be broken into a vertical and a horizontal component.

To calculate the forces on the joint, you will sum the horizontal forces and set them equal to zero. Seperately, you will sum the vertical forces and set them equal to zero.

A force directed to the right will be positive and a force directed to the left will be negative. A force directed upward will be positive and downward will be negative.

Joint B is only acted on by one purely horizontal force, represented by Fbd. Force Fbc is acting on the joint at and angle, which means it has both horizontal and vertical components (blue and orange dashed lines in photo denoted as FbcX and FbcY) . To determine the components separately we will use trigonometry of a right triangle.

Using your calculator and the sine and cosine functions, you will be able to solve for FbcY and FbcX. The unknown angle,Z, can also be calculated by using Sines and Cosines and the length of the members .

Now that the forces on the joint have been broken into horizontal and vertical components the two summation equations can be written as shown. You can now solve for the forces at joint B.

## Step 8: Repeat This Procedure Until the Forces in All the Members of the Truss Have Been Found

Using the free body diagrams of the other joints, as shown in the diagram, you will repeat the process on the next joint with only two unknown force components.

You should continue with this procedure until you have calculated the force in each member.

** note: when drawing free-body diagrams of the joints with unknown force members, the direction( the force pointing away from the joint or towards the joint) in which you draw the force is arbitrary. Your calculations will give you a negative or a positive number designating the real direction of the force.

## Step 9: Full Example

This is an example of a full analysis of a simple truss by the method of joints

## Step 10: Your Turn

Here is a simple truss to solve on your own.

A step-by-step solution will be provided in the next step if you get stuck.

Goodluck!

answers:

__Force on members:__

**Member AC= 64.0 kN Tension**

**Member BC= 80 kN Compression**

**Member AB= 52 kN Tension**

## Step 11: Entire Solution

* see attached pictures for step-by step solution*

## Step 12: Conclusion

You have now learned how to analyze a simple truss by the method of joints. Engineers, designers and architects use these calculations to determine which materials will hold the anticipated load for a particular truss. They also use these calculations to develop a safety ratio, known as the** factor of safety.** A factor of safety for bridges tells tell the public how many people, cars, etc. the bridge can safely hold with out collapsing.

Using this process and trigonometry, you may also be able to construct your own small scale truss.

**For more information on building simple trusses, you may be interested in the website's below:**

http://pages.jh.edu/~virtlab/bridge/truss.htm