Statics

Recall that all vectors have four attributes:

• the line of action

• the point of application

• the magnitude (and units)

• the direction 

The Principle of Transmissibility states that you can move a vector along its line of action to create a statically equivalent system.

Now, consider a box. in scenario A, the weight of the box is counteracted with a tensile force above. In scenario B, the weight is held in equilibrium with a compressive force below.

Forces F1 and F2 are statically equivalent. They have the same magnitude, the same direction, and the same line of action.

F1 results from a cable above (a pull) and F2 results from a cable below (a push). The represent different actions on the box, yet they are statically equivalent.

Example: 

Let's say that you are asked to compute the moment of the 5 kip force about Point A.

You create a statically equivalent system by sliding the vector along its line of action from C to B. The vertical 3 kip component causes a moment about A (-30 kip-feet). The horizontal 4 kip component is coincident with line AB and does not create a moment.

Imagine placing your cell phone on your desk and applying the loads from system I, II, III, and IV. Which systems are equivalent?

Systems I and II have the same amount of force, but since the force is on a different line of action, they are not equivalent.

Systems II and III are combined into System IV.

Systems I and IV are statically equivalent. They both have:

• a net force of 5N rightwards

• no net moment

You can move a force off of its line of action and create an equivalent system, provided that you introduce a couple moment to counteracts the rotation you caused.

You can compute the magnitude and sign of the moment by setting the tendency to rotate in System I equal to the tendency to rotate in System II. That is, the net moment about an axis in System I must equal the net moment about the same axis in System II.

You can do the moment calculation about any axis, but it's most intuitive when you sum moments about the destination of the force you're moving.

You can also combine a force and a moment into a single force, by moving the force off of its line of action to a new location.

Again, you want to ensure that both systems have the same tendency to rotate about an axis. 

You can do the moment summation about any axis, but it's most intuitive when you sum moments about the original location of the force you're moving.

We can use the same principle to combine multiple parallel forces into a single force.

The concept is the same as before: in order for two systems to be statically equivalent, they must have the same net force (tendency to translate) and the same net moment (tendency to rotate).

We use the symbol x bar to solve for an unknown distance in the +x direction from the origin (shown as a solid circle).

Sometimes, it is also useful to take a force, and divide by a length. The result is called a line load. 

We will use the symbol w for line load.

You can also think of this as "force per distance" or "force intensity."

We will often be converting line loads to equivalent forces by taking the area under the curve.

NOTE! In a 2D projection, a line load and an area load look identical. The only way to tell them apart is by inspecting the units. Line loads are force per distance and area loads are force per area.

One of the most common applications of static equivalency in Statics is to convert a line load to an equivalent point load.

You must solve for the magnitude of the equivalent force as well as its location. 

Remember that a definite integral is simply the area under the curve. When your load function is a simple geometric shape (a rectangle, a triangle, etc.), don't integrate: just compute the area under the curve.

For a uniformly distributed load, the resultant force is equal to area under the curve: wL. 

The force is located at midspan, at L/2.

Here is the derivation for a load function that varies linearly. We often call it a triangle load.

The equivalent force is the area under the curve of the loading diagram: wL/2.

The force is located at the thirdpoint of the triangle, on the "heavy" side of the triangle.

In this interactive model, you can explore a number of different equivalent systems in 3D space.

You can truly create an infinite number of equivalent systems!