Statics

Lesson 17

Internal Forces (and Moments)

🟦  17.1  What are internal forces (and moments)?

We can subcategorize forces (and moments) into three groups:

🟦  17.2 Prior lessons that incorporated internal forces

This isn't our first foray into internal forces or moments.

In the truss lesson, we learned to use the Method of Sections to slice through a truss and cut it into two pieces. 

Then, we were able to use the equations of equilibrium to solve for unknown tension or compression forces.

The internal forces are shown in orange.

In the cable lesson, we learned that we could cut a cable at any point and expose the internal tension in that particular fiber.

For straight-line segments, tension is constant.

For a cable supporting a uniform live load (uniformly distributed load), tension varies through the cable. This is depicted below.

The internal forces are shown in orange.


🟦  17.3 What exactly are internal forces (and moments) anyway?

In this flipbook, we will slice through this cantilever beam with two cuts, so that we can extract a dx slice.

By using the equations of equilibrium, we can solve for unknown forces (and moments) on the dx slice.

We can imagine that the dx slice of solid material is replaced with springs. 

This gives us a way to visualize the deformations from each force and moment.

Lastly, we take what we learned about a single dx slice, and extrapolate that to the entire length of the beam, so that we can visualize:

🟦  17.4 Terminology and sign conventions for 2D (planar) problems

There are three types of internal forces and moments that occur in a 2D (planar) problem.

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.

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 learn:

Moment or Bending Moment (M)

For a 2D (planar) problem in the x-y 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:

❏ Flipbook: Tricky Sign Conventions to Memorize


Normal? Positive TENSION

Shear? Positive ⬆️⬇️

Moment? Positive SMILE 😊


Note: these tricky sign conventions can vary significantly by engineering discipline, by subfield of engineering, by software package, and by country / geographic region. Watch out! Always make sure that you understand whatever sign convention is being used when you take a new class, work with a new person, or open a new piece of software! That said, the sign conventions explained here are fairly consistent within the United States in courses in Statics and Mechanics of Materials. They are also 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.

🟦  17.5 Terminology and sign conventions for 3D problems

In Statics, we won't focus on internal forces in 3D. However, it is useful to know the basics.

There are six internal forces and moments in 3D space:


N = normal force

V1 = shear in one direction

V2 = shear in the other direction

M1 = moment about one axis

M2 = moment about the other axis

T = torsion (twisting) about the longitudinal axis

🟦  17.6 Avoid common mistakes (signs, signs, signs!)

Sign errors on these types of problems are too common. 

We need a deep understanding of how to use these three different sets of sign conventions.

🟦  17.7 Solving for discrete values of N, V, and M . . . at a plane

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

    • assume directions of reactions
    • use signs for the E.o.E.
    • interpret your results using signs for unknowns
    • cut it into two pieces
    • pick whichever piece looks easiest to solve
    • use the signs for internals
    • if the cut plane is on the RIGHT of your FBD, N is rightward, V is downward, and M is counterclockwise
    • if the cut plane is on the LEFT of your FBD, N is leftward, V is upward, and M is clockwise
    • use the signs for the E.o.E.

🟦  17.8 Writing equations for V(x) and M(x) . . . over a domain

We will do this in class together!



➜ Practice Problems

Problem 1. Solve for the internal forces and moments at planes c-c and d-d.

Note: plane c-c is just to the left of the reaction at A (x=2.9999999'). Plane d-d is just to the right of the reaction at A.


Problem 2. Solve for the internal forces and moments at plane c-c.

answer:

Problem 3. Solve for the internal shear force and bending moment for the planes shown in the image. (Note: all of the normal forces are zero.)


I am not planning on providing a full solution for this one, but here are scrambled answers, without units, so that you can check your work:-6, -4, -2, 0, 0, 0, 2, 4, 6, 10, 10, 16, 16, 18

Problem 4. Solve for the internal shear force and bending moment for the planes shown in the image. (Note: all of the normal forces are zero.)


I am not planning on providing a full solution for this one, but here are scrambled answers, without units, so that you can check your work:0, 0, -4, -8, -8, -12, -32, -72