Statics

When do we include applied forces (and moments) on a FBD?

It's when they are applied to the portion of the member that is cut or disconnected from the context.

When do we include reacting forces (and moments) on a FBD? 

It's when we disconnect a support from the context. For example, we remove a pin and replace it with x-direction and y-direction forces. These are the actions of the support on the body.

When do we include internal forces (and moments) on a FBD?

It's whenever we pass a cutting plane through the solid material. For instance, beam AB has been cut at plane c-c.

If you don't cut through the structure, you do not show any internal forces (and moments) on the FBD.

Through this unit, we will use the generic term force to refer to both forces and moments. This may seem a little strange at first. Forces and moments aren't the same thing; they don't even share the same units. 

We can portray a couple moment as a force couple (two forces, opposite in direction, equal in magnitude, and parallel). 

Therefore, in the subsequent sections, you'll no longer see the parenthetical note "and moments" when internal forces are mentioned.

In this flipbook, we will slice through a cantilever beam with two cuts. This allows us to extract and investigate a dx slice of the solid material.

We use the equations of equilibrium to solve for unknown forces (and moments) on the dx slice:

• N = normal (perpendicular) force

• V = in-plane (parallel) force

• M = moment

We can imagine that the dx slice of solid material is replaced with springs. This helps us visualize the deformations that occur due to each internal force.:

• normal force results in elongation (under tension) or shortening (under compression)

• shear force results in angular warping

• bending moment results in curvature

After you finish Statics, you'll probably go on to study stress in a subsequent course. A simple definition for stress is force normalized over (divided by) area.

This flipbook is a preview of what you'll learn about stress after Statics.

The takeaway for this course is that internal forces are applied at a point while stresses lie in a plane. Force is a vector, while stresses are a vector field.

In Statics, we will cut through a plane to expose the force. That force results from integrating the stress distribution over the cross-sectional surface area.

For notation, lowercase letters indicate planes while uppercase letters continue to indicate points (or nodes, or particles).

Each point should be interpreted as the centroid of the cross-sectional area, if no other indication is provided.

❏ Normal force (N)

When the internal force is perpendicular (normal) to the cut plane, we call it a normal force and use the symbol N.

• arrows that point away from the body create tension, which we define as positive

• arrows that point toward the body create compression, which we define as negative

❏ Shear force (V)

When the internal force is parallel (in-plane) to the cut, we call it a shear force and use the symbol V.

This sign convention is notoriously tricky to apply:

• Method 1: when the shear force on the left (negative x face) goes up (in the positive y direction), V is positive; when the shear force on the right (positive x face) goes down (in the negative y direction), V is positive

• Method 2: when a shear force tends to rotate the body clockwise (yes, clockwise, that is not a typo), V is positive

Shear vectors are often drawn with half arrowheads in 2D drawings. This signifies that the force is located in the adjacent plane. We offset it slightly so that it's easy to see.

❏ Moment or Bending Moment (M)

For a 2D (planar) problem in the xy plane, the internal moment will always be about the z-axis. We use the symbol M. 

You can think about the sign convention in two ways:

• Method 1: when the internal moment makes the beam "smile" at you (concave up), it's positive

• Method 2: a clockwise moment on the negative x face is positive; a counterclockwise moment on the positive x face is positive

1. These sign conventions can vary significantly by engineering discipline, by subfield of engineering, by software package, and by country / geographic region.
2. Always make sure that you understand whatever sign convention is being used when you take a new class, work with a new person, use a new website, or open a new piece of software!
3. The sign conventions explained here are fairly consistent within the United States in courses in Statics and Mechanics of Materials. 
4. The sign conventions used here are consistent with the Fundamentals of Engineering (FE) standardized exam that is required for future licensed (professional) engineers that wish to practice in the United States.

An example problem for solving for discrete values of N, V, and M is shown in the flipbook. The basic steps are:

1. Construct a global FBD.

2. Solve unknown reactions with the E.o.E.

   • assume directions of reactions
   • use signs for the E.o.E.
   • interpret your results using signs for unknowns

3. Cut through the member.

   • cut it into two pieces
   • pick whichever piece looks easiest to solve

4. Draw the unknown internal forces.

   • always draw these in the positive direction, using the signs for internal forces

5. Solve the unknown internal forces.

   • use the signs for the E.o.E.

6. Interpret your results.

   • use the signs for internal forces

Here is an example of how to write functions for V(x) and M(x) for a simply-supported beam that supports a uniformly distributed line load. There is only one domain (0<x<L).

Here is another example of the same concept. This time, there are two domains in play.