To meet the requirements of your contract, your team will:
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:
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).
The two geometric factors (y and I) can be combined into the seciton modulus (z) where:
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).
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:
Or, for Crazy Eights
This figure shows a more detailed analysis with the same result.
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.
The formula for the second moment of area (I) of a rectangle about its own centroid is:
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.
**Needs examples here***
Kelvin Bridge and Tower Testers
Source of Balsa