Crazy Eights Teacher Resource Page

Achievement

To meet the requirements of your contract, your team will:

  1. Design and construct a beam to span an eight inch gap while supporting a force downward at the center
  2. Demonstrate the performance of each beam by loating it to failure. For each test
    • Sketch the cross-section of the beam
    • Determine the cross-sectional area of the beam
    • Record the amount of force required to break the beam
    • Calculate the second moment of area and section modulus
    • Use the elastic flexure formula to estimate the stress at failure
  3. Design and construct different beams with the goal of making a stronger beam.
  4. Test each new beam as before.
  5. Do the assigned problems to learn more about material properties.
  6. Plot the many beam properties you have learned against force at failure to see what factors make one beam stronger than another.
  7. Provide your results in an informational report in the required format.


Strength of a Beam — From the End to the Beginning

Elastic Flexure Formula

The stress (σ) at any point on a beam depends on the distance from the neutral axis (y), the applied moment (M), and the resistance to bending (I, the second moment of area of the cross-section).

In the model that we will use, the stress is zero at the neutral axis and increases linearly with the distance away; tension below, compression above.

The elastic flexure formula, which predicts the stress at any point, is then:

σ = M y / I

In the specific case where the beam breaks, the stress will be the fracture or yield stress (σyield), the moment will be the moment measured when the beam breaks which is the maximum moment (Mmax), and the y will be the maximum possible y (ymax).

σyield = Mmax ymax / I

The two geometric factors (y and I) can be combined into the seciton modulus (z) where:

Z = I / ymax

And so:

σyield = Mmax / Z

Units

The units of stress are pounds per square inch (lbs/in2). The units of Moment are inch pounds (in·lbs). The units of y are inches. The units of I are inches to the fourth power (in4). The units of Z are inches to the third power (in³).

How to find Maximum Moment

In our specific case, the beam is supported at the ends and the force (F) presses down in the middle. The reaction forces at the end of the beam are then half the force (F/2). The span (S) in our particular case is 8 inches. The moment arm will be half the span (S/2) so that, if we use the force at which the beam broke:

Mmax = (S/2)(F/2)

Or

Mmax = (8 inches/2)(F/2)

Or, for Crazy Eights

Mmax = 2 F , with the resultant units of in·lbs

This figure shows a more detailed analysis with the same result.

How to Find ymax

Find the centroid of the cross-section of the beam. The neutral axis goes through the centroid. Find the distance to the top and to the bottom of the beam from the neutral axis. The greater of these two distances is ymax.

How to Find I

The formula for the second moment of area (I) of a rectangle about its own centroid is:

I = b h³ / 12

Example

Suppose a beam is three sticks stacked up so that the base is 1/8 inch and the height it 3/8 inches. The centroid is half-way up the stack, so ymax is 3/16 in.

b = 1/8 in
h = 3/8 in

I = b h³ / 12

I = (1/8 in) (3/8 in)³ / 12

I = 5.49 x 10-4 in4

Z = I / ymax

Z = 5.49 x 10-4 in4 / (3/16 in)

Z = 2.9 x 10-3 in³

If the cross-section of the beam can be created by adding two or more rectangles that are centered on the neutral axis, then the I's can simply be added. "Holes" can also be subtracted to find the combined I

If the various rectangles are not on the same axis, then the parallel axis theorm must be used to find the total I for the beam.

I = ¯I + Ad²

**Needs examples here***


Links

Kelvin Bridge and Tower Testers

Source of Balsa

 

Return to the Beginning