Memo
Date: September 19, 2018
To: Dr. Kadri Parris and GTA Sheng Zhu
From: Alexander Leslie, Group I
Subject: Beam Bending Lab
Introduction
The purpose of the lab is to expose students to dial calipers and the concepts of Moment of Inertia,
Young’s Modulus, as well as technical documentation through a lab memo. This lab memo contains
Group I’s results from the Beam Bending Lab. The lab asked the group to use dial calipers to record
deflection data on four separate beams. We first recorded the data on Excel, then used our data to create
visual representation through graphs. The group then used the measured data to calculate Young’s
Modulus of an unknown beam. The group was then asked to suggest possible solutions to a client who
wants to know if the beam material could be reused in future properties. The client is located at a
brownfield site, with a metal factory in which they can pool resources from.
Results and Discussion
The group first began by measuring deflection on an Aluminum Rectangular Beam. As all of us began
recording our measurements, we first discovered an error of 7 percent (From Excel Tables). However, the
rest of our data was under 3 percent error. The error could be accounted for by potential swinging of the
structure, and the fact that the group was not asked to weigh each weight to verify that it was 50 grams
and variability could be introduced as a result.
Deflection vs. Force for Aluminum Cantilever Beam
Absolute Deflection (Inch)
0,300
0,250
0,200
0,150
Measured
0,100
Theory
0,050
0,000
0,00
0,20
0,40
0,60
0,80
1,00
1,20
Force (Weight) applied to Beam (lbf)
FIGURE A
The second measured beam was a Copper Rectangular Beam. Our percent error was under 5 percent
(Excel Tables) until the final measurement, where the screw holding the weights was loosened. This
resulted in a large amount of swinging. Other variability could be accounted for by inconsistent tapping of
the structure, and what was mentioned previously. The graph appears to be a line
Absolute Deflection
(in)
Direction vs. Force for Copper Rectangular Beam
0,15
0,1
Measured
0,05
0
-0,050,000
Theory
0,200
0,400
0,600
0,800
1,000
1,200
Force (Weight) Applied to Beam (lbf)
FIGURE B
The third measurement was a Copper Square Beam. The percent error for this beam ranged from 16
percent at its highest, to 2 percent at its lowest. (Excel Tables) The percent error was also negative, which
could be accounted for in our initial measurement for the dimensions of the beam. Our measured value
was lower than the theoretical value for the thickness of the beam. The percent error could be accounted
for by all previously stated reasons.
Absolute Deflection(in)
Deflection vs. Force for Copper Square Beam
0,04
0,03
0,02
Measured
0,01
0
-0,010,000
Theory
0,200
0,400
0,600
0,800
1,000
1,200
Force(Weight) Applied to Beam (lbf)
FIGURE C
Our final measurement was of an unknown beam, and we had to solve for the Young’s Modulus of the
beam to determine what material it was. We had to create a graph of deflection vs. force of our
measurements. We then inserted a best fit line. The slope of the graph was then calculated, and we then
solved for Young’s Modulus.
Measured Deflection vs. Force of Unknown Beam
0,1
Absolute Deflection (in)
0,09
y = 0,08x - 0,0013
R² = 0,9949
0,08
0,07
0,06
0,05
0,04
0,03
0,02
0,01
0
-0,010,000
0,200
0,400
0,600
0,800
Force (Weight) Applied to Beam (lbf)
FIGURE D
Slope of graph (m)=0.08
We used the equation for theoretical deflection, and solved it for Young’s Modulus:
𝜎 = 𝐹𝑆 2
3𝐿−𝑆
6𝐸𝐼
(Equation 1)
1,000
1,200
𝐸=
𝑆 2 (3𝐿−𝑆)
6𝐼𝑚
(Equation 1 - Rearranged Theoretical Deflection Equation)
Using this equation, a value for Young’s Modulus is 2710832.922lbf/in^2. See appendix for calculation.
After researching online, the metal appears to have a Young’s Modulus that closely matches steel.
Discussion
We expected to find a linear pattern in each of our data sets, as the theoretical deflection is given by:
𝜎 = 𝐹𝑆 2
3𝐿 − 𝑆
6𝐸𝐼
I = Moment of Inerta
S= Distance to Dial Indicator
F= Force (lbf)
E= Young’s Modulus
L=Distance to Force
The equation simplifies into a linear form, and as such resulted in a linear theoretical graph. Our
measured values closely matched our theoretical values, and our percent error was relatively low. This
indicated that our results closely matched those of the theoretical values given our measurements.
At first It appeared that the unknown rectangular beam had the least amount of deflection as it’s line
has the highest slope. It was next followed by the rectangular copper beam. The Aluminum beam
however had a higher deflection than the copper beams. However, the copper square beam had the
most amount of deflection. We based our results on the slopes of the graphs.
Conclusions & Recommendations
Possible recommendations to the client include, using the unknown identified material, steel, in their new
buildings. The steel has the lowest deflection of the four beams. Materials with water damage, and those
that appear strained must not be used in future projects. The copper and aluminum can be used for interior
projects as the copper will not have to withstand elements such as wind, rain, and snow. The steel should
mainly be used for exterior projects. All materials should be thoroughly examined at a site away from the
metal plant so that accurate data can be recorded. When re-using the material, the client should opt for
rectangular beams as opposed to square beams, as they have a lower deflection. Any remaining metals
that cannot be reused should be melted and sold to further fund the construction project.
We can conclude that overall, our measurements were accurate. Our highest percent error was 16 percent.
Despite the R^2 value being very close to 1 and showing a strong linear correlation (Figure D), additional
methods to decrease the percent error could include weighing each weight to ensure that it is 50 grams, as
well as having a set number of taps to reset the caliper each measurement. Random error can be
accounted for by human error at different angles of sight on the dial caliper.
References
-
Course materials
Ohio State Fundamentals of Engineering Program. (n.d.) Beam Bending Lab Procedure [Course
documentation]. Retrieved September 19, 2018 from carmen.osu.edu
-
Course materials
Ohio State Fundamentals of Engineering Program. (n.d.) Beam Bending Lab PowerPoint [Course
documentation]. Retrieved September 19, 2018 from carmen.osu.edu
-
Data
Lab 04 Beam Bending Lab, Nate Johnson and Morgan Beebe’s Measurements for Task 1 and
Task 2. Retrieved September 12, 2018 from class.
-
Online
a
Engineering Tool Box – Modulus of Elasticity for Metals and Alloys
Accessed September 19, 2018.
Appendix
Calculations for Young’s Modulus:
𝛿=𝐹
𝑚=
𝑆 2 (3𝐿 − 𝑆)
6𝐸𝐼
𝑆 2 (3𝐿 − 𝑆)
6𝐸𝐼
𝑆 2 (3𝐿 − 𝑆)
𝐸=
6𝐼𝑚
L=8.750in
S=7.500in
I=(w*t^3)/12=((0.498in)(0.125in)^3)/12=8.1055E-5 in^4
F=Total Weight (lbf)
m=0.08
𝐸=
(7.5𝑖𝑛)2 (3(8.75𝑖𝑛) − (7.5𝑖𝑛)
6(0.08)(8.1055𝐸 − 4𝑖𝑛4 )
𝐸 = 2710832.922lbf/in^2