## Introduction: How to Solve a Shear and Moment Problem

This instructable will teach you how to solve for forces and moments on a beam and how to draw a shear and moment diagram for that beam. A shear and moment problem is a common problem found in an engineering course that uses the various fundamentals of engineering to solve. You will learn how to take those fundamentals and use them together to solve a shear and moment problem.

This instructable will walk you through a simple beam problem with only variables. Then it will ask you to solve a beam problem on your own to test your knowledge.

To begin you will need:

- A pencil
- Blank paper
- A calculator
- Some skill in basic calculus
- Some knowledge of forces and how they work

## Step 1: Draw a Free Body Diagram

First we take a look at the beam and identify where the reaction forces and moments are located. The reason why we draw a free body diagram (FBD) is to identify the forces and moments from the picture so we know exactly what we are working with. To draw an "FBD" for a beam, all you really need is to draw a line and place the forces and moments on the line as they appear in the given picture. Forces are drawn with straight arrows and moments are drawn with circular arrows. A fixed pin has forces in the x and y direction. A roller pin only has a force in the y direction.

***Note: The sketch on the right is what is given to us. The sketch on the left is the FBD we draw.***

## Step 2: Solve the Sums of Forces and Moments

Next we solve for the sums of forces and moments. With these types of problems we assume that the forces and moments are in equilibrium, meaning that the sums are equal to zero. We will first solve for the sum of the forces in the x direction. Since there is only one force in the x direction, it is equal to zero. Now we will solve for the forces in the y direction. We assume both the reaction forces at A and C are positive so we add them together and set them equal to zero. When we solve for one in terms of the other, it turns out that the forces are equal to each other but are pointed in different directions. When solving for moments, we choose an arbitrary point on the beam to take the moment about. In this case we will take the moment about point A. When solving for moments, we look at the forces causing the beam to spin about that point. So when summing the moments, we multiply force and the distance to the point we chose. In this case we only take into account the moment given in the picture and the force at C. The force at A is not shown because the force at A is zero distance away from A, thus canceling it out.

***Note: When summing the forces, an arrow pointing down on the FBD is a negative force. An arrow pointing up is a positive force. When summing the moments, an arrow pointing counter clockwise is a positive moment. An arrow pointing clockwise is a negative moment. The convention in determining signs for forces and moments can be flipped as long as the signs are consistent throughout the problem.***

## Step 3: Draw a Section of the Beam

Now we will look at only one part of the beam. In this case we will cut the beam in half from length zero to length L/2. When the beam is cut, we have to take into account the shear force (V) and moment (M) inside of the beam. Once we have our section drawn, we will solve for the unknown shear and moment in the beam. To solve for the shear, we once again use sum of the forces in the y direction to solve for "V". To solve for the moment, we choose an arbitrary point x on our beam and use sum of the moments around point x to solve for "M".

## Step 4: Draw Another Section of the Beam.

The reason for looking at another section of the beam is to solve for as many points as we can so we can complete our shear and moment diagrams. Now we will look at an arbitrary length x that extends past length L/2. We solve for the shear and the moment here the same way we did in the last step.

## Step 5: Draw a Shear and Moment Diagram

We now have enough information to draw a shear and moment diagram! First we will draw the shear diagram. As you may notice in the picture, we draw the diagrams underneath the given sketch so we can line up the shear and moments with the beam. Looking at the information we found when analyzing two different sections of the beam, we found that the shear force is the same throughout the beam. For the moment diagram we found that M=-Mo(x/L) and M=Mo(1=(x/L)). When we evaluate the first equation at x=0 and x=L/2 we get M=0 and M=-Mo/2 respectively. Next we will evaluate the second equation at x=L/2 and x=L. We get M=Mo/2 and M=0 respectively. Your diagram should look like the picture above if plotted correctly.

Now you will attempt a shear and moment diagram on your own.

## Step 6: Prompt

Solve for the shear and moment diagrams given the picture above.

## Step 7: Answers

All of the answers to complete the problem are given in the picture above.

Congratulations! You now know how to solve a shear and moment diagram problem.