Statics: Equilibrium in 2 Dimensions Part 1

In equilibrium the sum of the Forces is equal to 0. We don't want things to have an angular acceleration either. This means that we don't want any angular velocity either, but those aren't guaranteed. 

That's basically what the class is, so let's apply this to equilibium in 2 dimensions. Looks like there's 2 equations, where I can handle 2 unknowns.   

However, most of our problems can be 2D. With no Z forces, we'll have no moments about the X axis or moments about the Y axis, due to cross product. We need a Z component of force to have a moment of the X axis or Y axis. 

For example there's going to be a beam only subject to one direction of force. Design everything within the elastic potential. 

Assuming points A and B won't fail, we can assume that the moments summing up to A and moments summing up to B are both equivalent to 0. 

But XA can't equal to XB. So, as a result, we can handle that type of thing. 

We could sum the moment about any 3 points if those 3 points are not colinear and can't be such that any 2 X components or (possible Y components) are of the same magnitude. However, this isn't super common. 

Freely pinned means something can't go up, can't go sideways, but it can turn. We need to take the beam and embed it in a wall, meaning it can't go up, can't go sideways, nor can it turn. The second thing is called cantilever. Cantilever is static, but the pinned support can rotate on a moment. We can add a roller support in the bend, to make the beam not go up or down, but left or right instead. Here are the respective diagrams: 

So, as the beam starts to sag, the end can move in a little bit because the total length of the beam is the same, because it just bends a little bit. Trouble is, when we replace the support with the supporting forces, we are going to take the supports and put in the forces they offer. The left keeps it from being up or down or left and right, which there is a single force that is pinned. Because it's pinned, there are 2 unknowns, in our eyes, This type only keeps things from going up or down not left or right. So there's Ax, Ay, Bx, By, etc, and I will draw a diagram representing this.  This is our simplest problem with 3 unknowns.  In dynamics, we will have an equation knowing how these pieces will actually deform, since I don't want this pin to move at all, and we'll be able to do 4 unknowns as a result. Right now we only have 3 equations in 2D, which explains a roller. 

In real life, there can be thermal expansion where it expands during the summer and contracts during the winter. Here's an expansion joint for a bridge: 

As the loads try to turn the cantilever one way the support turns the other way such that the total sum of the moments is equal to zero (moments is in equilibrium). 

The supporting forces are called the reaction because of the support reaction and the fact that there's a load on it. These are known as the reactions.

To find the 3 unknowns, we can sum the forces in the X direction, where the 1 force in the X direction must be 0, otherwise the forces in the X direction will not sum to 0. 

In the Y direction, all of the down forces will equal all of the up forces. 

So there's one equation, 2 unknowns, and we have to sum all the moments equal to 0 about some point. It doesn't matter which point we are trying to use.