NC State

Learning Module: Moments of Inertia (including 3D printed model)

By Anna Howard

Area Moments of Inertia by Integration

Moments of Inertia (both area and mass) can be difficult to understand — or remember. This module and those to come will help you work through the concepts and calculations for area moments of inertia.

Materials:

  • 12-inch wooden ruler
  • 3D printer with flexible materials
  • 3 straight pins or nails
  1. Reading: Area Moment of Inertia Background and Motivation (3 pages)
  2. Video: https://youtu.be/omrmA-QGkcw (7 minutes)
  3. Hands-on: Bend your ruler. There are three axes: one along the length, one along the inch-marks, and one through the thickness. Remember that when we’re talking about bending something, the axis you’re bending about is where your thumb would be. So a moment along the length of the beam twists the beam. A moment in either of the other two axes (flat-wise or width-wise) bends the ruler.
    1. Which way bends more easily?
  4. Reading: Note about Axes (1 page)
    1. Draw axes along your ruler so that z is along the beam. (Make your axes a right-handed coordinate axis system. If you look at the cross-section, you should see a regular x-y coordinate axes system.)
    2. Which moment bends more, the one about the x-axis or the one about the y-axis?
    3. Which way would you want to install the ruler in a structure to hold up more weight?
    4. Since the material and the length don’t change, what does?
  5. Reading: Ruler Bending About its Own Centroid (just over 1 page)
  6. Example (read or watch): Rectangular Beam Bending about Centroidal Axes (2 pages) or https://youtu.be/BvEu0fpLfdc (just over 1 minute)
  7. Write a brief paragraph for your own notes explaining the axes and the bending loads to ruler in your own hands.
  8. Print the 3D flexible i-beam.stl. Print this at 100% density with flexible material.
    1. If funds permit, print a copy with inflexible material so you can feel the difference.
  9. Hands-on: Bend the I-beam in your own hands in two directions.
    1. Draw the axes directly onto your printed beam so that the z axis is along your beam. Alternately, put pins into your beam so that you can see where the axes are. (Make sure it’s a right-handed coordinate system so that when you look at the cross-section you see a regular x-y axis system.)
    2. Draw an FBD of your beam to specify which direction your loads are acting in when you bend the beam in each of the directions. Relate your loads to a simply supported-beam.
    3. Which way does the beam bend more? About which axis is that? (Remember the direction the moment is applied is where your thumb points.)
    4. Which area moment of inertia are we talking about for each bending direction? (Ix and Iy)
  10. Hands-on: Draw a one-inch line on the top and bottom of the I-beam. Bend the I-beam about the x-axis. Note that one of your inch lines is longer than the other side.
  11. Reading: Neutral Axis (1 page)
  12. Video: watch the video animation of the stress in the bending I-beam from both directions. Bending as an I-beam and bending the same beam as an H-beam.
    1. Lay your printed I-beam on top of your pencil so you can push down on either side and bend the beam over the pencil as is shown in the video.
    2. Which direction deflects more easily?
  13. Example: Rectangular Beam Bending About Baseline (2 pages) or https://youtu.be/nD1ZMlrgJ_k (2 minutes)
  14. Hands-on: Lay your flexible I-beam flat on the table. If you push on the top two ends, the beam experiences a bending moment; the top surface of the beam will bend a bit but the bottom surface can’t deform. The neutral axis is where the deformation is zero.
  15. Write a brief paragraph for your own notes explaining why the moment of inertia about a rectangle’s baseline is bigger than about its centroid. Relate this to the ruler and your I-beam.
  16. Reading: Parallel Axis Theorem (1 page)
  17. Practice: Find the moment of inertia of the triangle about its baseline. Solution (watch only after you’ve worked it yourself): https://youtu.be/UjC4xjr0jCc (2 minutes)
    1. Relate this to a triangular beam in your head where you are trying to bend a beam with a triangular cross-section which is also sitting on a table.
  18. Practice: think about what would be required to find the area moment of inertia about the centroidal axes of a triangle. Solution: https://youtu.be/MApovKX7CIc (just over 4 minutes)
    1. Explain to yourself why the moment of inertia about the centroid of the shape is lower.
  19. Practice: use the parallel axis theorem to relate the moment of inertia about the baseline in #17 to the moment of inertia about the centroid in #18. Solution (1 page)
  20. Practice: Find the moment of inertia for the green shape about the x and y axes. The units shown are in cm; the drawing is not to scale. Solution: Area Between Two Curves (2 pages).
  21. Practice: Find the moment of inertia for the turquoise shape about the y axis. The units shown are in inches; the drawing is not to scale.
    1. This would be an odd-shaped beam, admittedly, but if you can find the moment of inertia by integration for this shape, you should be in reasonable condition for any shape. Answer = 5.53×10^7 in^4.
    2. Note: to find the moment of inertia about the x axis for this shape is complicated. (Students can sometimes get creative here. Be careful.)

Once you can find the area moment of inertia by integration the next part is to learn the method of composite bodies. That will come in a future post.