To: Rachel Warren
From: John Maloney
Subject: Beam Bending Lab
The Beam Bending Lab was a group project that gave students’ knowledge of the pragmatic properties of different materials. In the lab, students tested and measured the change in deflection of different materials by applying more force. The purpose of this lab was to conduct an experiment that involves measuring the difference in deflection of different materials through the change of weight. The lab had an aluminum rectangular beam, a copper rectangular beam, a copper square beam, an unknown rectangular beam, a pulley, a dial indictor, and weights to add force to the vertical beams.
Results and Discussion
The lab data was collected on four different types of beams. The beams were tested vertically, with one end in a fixed holder and the other end was free to move with the force against it. Figure 1, below, shows the absolute deflection of the rectangular aluminum beam versus the applied force against the beam. The aluminum beam had an absolute deflection of 0.259 inches when the maximum force, 1.102 lbf was applied. The beam then deflected an average of 0.026 inches each increment of force applied.
The aluminum rectangular beam served as a good first example test. The data was easy to retrieve and was accurate enough to compare to the other materials tested later. The beam deflected 0.028 inches more than it theoretically was supposed to. Since the data collected was consistent and relatively close to the expected value, the error between the theoretical and the measured can be considered negligible. If the experiment was repeated the measurement error would likely vary from 0.028 inches.
Figure 2 displays the absolute deflection of the rectangular copper beam versus the applied force against the beam. The copper beam had an absolute deflection of 0.154 inches when the maximum force, 1.102 lbf was applied. The beam then deflected an average of 0.015 inches each increment of force applied.
Based off Figure 2, we can see that there is a difference in absolute deflection from Figure 1. Therefore, changing the material type changes the strength of a beam. In this instant, the strength of the material increased. Young’s Modulus can help explain this. Young’s Modulus is a specific constant for a given material. It is used to determine the deflection of a beam under the stress from an applied force. Since Young’s Modulus is inversely proportional to deflection, if it increases the absolute deflection would then decrease. Comparing aluminum and copper’s Young’s Modulus gives me evidence to verify this trend, copper (greater Young’s Modulus) has less deflection than aluminum. Changing the width of the beam would vary the percent error for the deflection. This error can also vary by human error. The error is negligible; however, because the error does not disturb the data significantly.
Shown below, Figure 3 illustrates the absolute deflection of the square copper beam versus the force applied to the beam. Notice the shape of the beam has changed, this means the width and thickness will differ. The implied comparison is with the rectangular copper beam. The square beam had an absolute deflection of 0.039 inches when the maximum force of 1.102 lbf was applied. The beam then deflected an average of 0.004 inches each increment of force applied.
As stated earlier, material type plays a role in the strength of a beam, but now it is seen that shape too can affect the strength of a beam. This is seen by the decrease in absolute deflection of the square copper beam. The decrease of deflection can be interpreted by the moment of inertia. The moment of is described as an aspect of an object that is relied on by the width and thickness of the object. The moment of inertia is dependent of where it is evaluated and the point at which the object is rotated. In this experiment all the rods are ‘rotated’ at the same point, so this part of inertia does not factor in. The moment of inertia is inversely proportional to deflection. That is when the moment of inertia of an object increases, the deflection will then decrease. To prove this, comparing the absolute deflection of both the copper beams follows this trend. Comparing both the copper beams cause the material to stay constant, which implies the Young’s Modulus is constant. The square beam (greater moment of inertia) had experienced less deflection than the rectangular copper beam.
Figure 4, shown below, represents the deflection of an unknown beam versus the force applied to the beam. The shape for the unknown beam was rectangular, so the unknown beam is compared to the rectangular copper beam and the rectangular aluminum beam. Based off the graph the Young’s Modulus for the unknown material looks to be greater than both the rectangular copper and aluminum beams, as the absolute deflection for the unknown beam is 0.076 inches when the maximum force of 1.102 lbf was applied. The unknown beam deflected an average of 0.008 inches each increment of force applied.
After going through the data, not including the unknown beam, an increase of the moment of inertia correlated to a decrease in the deflection. The moment of inertia is inversely proportional to deflection, so as the applied force was identical in each case, the numerator became a constant. The constant divided by the increasing values equals lower values due to the Theoretical Deflection Equation. The square copper beam had the highest moment of inertia, 3.255E-4, and also had the smallest absolute deflection, 0.039.