🟦 17.1 What are internal forces (and moments)?
We can subcategorize forces (and moments) into three groups:
applied forces and moments (loads)
reacting forces and moments (at supports and connections)
internal forces and moments (the effects of the fibers of the solid material, which stay connected under load and therefore must transfer force from each cross-sectional plane to its neighboring plane)
🟦 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:
effects of force perpendicular to the cross-section
effects of force parallel to the cross-section
effects of bending moment
🟦 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.
arrows that point away from the body create tension, which we will define as positive
arrows that point toward the body create compression, which we will 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 learn:
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
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:
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
❏ 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:
Draw a global FBD and 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
Cut through the member at the desired plane.
- cut it into two pieces
- pick whichever piece looks easiest to solve
Draw the unknown internals (N, V, and M) in the positive direction
- 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
Solve the unknown internals with the E.o.E.
- use the signs for the E.o.E.
Use the signs for unknowns to interpret your results.
🟦 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.
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