Memorandum
To: Meagan Ita
From: Hank Rumpke (Seat 18), Joshua Flick (Seat 8), Maggie Wilson (Seat 14), Zachary Dick
(Seat 5)
Date: 11/7/2012
Re: Beam Bending & Material Joining
1. Introduction
The office supply company Office Spaces is looking to renew their chair product line.
Materials to be used for the new arm supports have already been narrowed and a group of
engineers have been paid to make deflection vs. force curves for the pre-selected materials
under cantilever conditions. Along with the curves, the engineers are to provide a
recommendation for the material type and cross-section of the material to minimize
deflection.
Within this memo contains the results and description, discussion, and summary and
conclusion of the experiment. In the Results and Description, data acquisition for the
cantilever beams is described. For the discussion, the curves, empirical and theoretical
Young’s Modulus values, and the behavior of the two brass beams is compared. Welding
proper practices and the effect they could have on the deflection of the materials as well as
sources of error are also stated in the discussion. Lastly, the Summary and Conclusion
restates the purpose of the experiment and briefly describes the results of the experiment.
Besides the purpose and results, the Summary and Conclusion includes a material and crosssection recommendation for the arm of the new office chair line.
1 of 5
2. Results and Description
After the experimental set up was complete, the deflections of each beam were measured by
reading the scale on the Dial Indicator of the Beam Bender Kit pictured below in Figure 1.
The maximum and minimum deflections for the masses from 47.5g to 475g of each material,
Brass, Aluminum, and Steel, were recorded in inches. In order to get clean data in metric
Figure 1: Kit used to read deflection of cantilever beams [1].
units, the maximum and minimum deflections for each material at each mass needed to be
𝑚
average and inches converted to meters by a factor of 0.0254𝑖𝑛. Also, the mass must be
𝑚
multiplied by the gravitational constant of 9.81𝑠2 in order to know the force acting on the
beam. A chart of these deflection values with a certain force applied for each material can be
found in the Figures and Tables Attachment. From this data table, a deflection versus force
plot was made for each material. The plot can also be seen in the Figures and Tables
Attachment. Calculating the Empirical value for Young’s Modulus for each material
involved using a variation of the deflection equation seen in the Sample Calculations
Attachment [1]. The equation depended on the inverse of the slopes in the deflection versus
force graph, the moment of inertia of the beam, and some length variables. A sample solution
to solving the Empirical Young’s Modulus can be seen in the Sample Calculations and a
2 of 5
table comparing the Empirical and Theoretical Moduli of each of the materials can be seen in
Table 2 in the Figures and Tables Attachment [2].
3. Discussion
In the data in the Figures and Tables Attachments, the Brass beam with a square cross section
and the Brass beam with a rectangular cross section were made of the same material yet had
quite different deflection results. The gap in deflection results was because of the two beams
different moment of inertia. A higher moment of inertia for the square Brass beam led to
lower deflection and a lower moment of inertia for the rectangle Brass beam led to higher
deflection. However, the difference in the deflection values should not result in different
Empirical Young’s Modulus for the two Brass beams, but it does according to Table 2 in the
Figures and Tables Attachment.
Under similar conditions, PVC would have behaved fairly well with deflections that were
greater than Steel and the square cross section Brass beams, but less than the Aluminum and
rectangular cross section Brass beams. Welding two pieces of PVC together makes a solid
bond, if done the right way, and reduces the cantilever deflection of the extended piece that
has force being applied to it. Welded PVC, if used in the experiment, would have had an
even lower deflection than the normal cantilever PVC.
To obtain a good weld, the welder must evenly heat each section of the joint without burning
or deforming the object. A way to evenly weld the joint is to slowly move over the joint in
one direction. Welding parts of the joints multiple times hurts instead of helps the bond
between two objects. When welding two objects together, make sure the welding device is
3 of 5
only pointed at the joint and does not heat or deform any other parts of those objects. If an
engineer does not practice good welding tips, such as being careful only to weld at the joint,
then materials are wasted therefore money is wasted.
One possible source of error could have been that the Dial Indicator in the Beam Bender Kit
was not zeroed after every new beam. Even though there are zeros as the default deflections
for each beam as seen by the deflection versus force applied table in the Figures and Tables
Attachment, they were just kind of put into the data of our group and was not always
measured that there was zero deflection with zero force applied.
4. Summary and Conclusions
A group of engineers was to determine whether Brass, Aluminum, or Steel would work best
under cantilever conditions and be a suitable material for the arm supports for the office chair
product line for the office supply company Office Space. With the recorded data from the
different beams, a deflection versus force plot was made and analyzed. After studying the
results, the engineers suggested that Office Space use the square cross section brass beams
that are welded into their arm support position. This material had the lowest reflection and
welding one side of the support to the main part of the chair can only reduce the deflection.
4 of 5
Attachment: Sample Calculations, 1.
Attachment: Figures and Tables, 2.
Attachment: Symbols, 1.
Attachment: References, 1.
5 of 5
Sample Calculations
𝐸=
𝐹 𝑠2
(3𝐿 − 𝑠)
𝛿 6𝐼
(1)
𝑏ℎ3
12
(2)
𝐼=
Sample Calculation for the Empirical Young’s Modulus
𝐹 𝑠2
𝐹 𝑠2
0.19052
(3𝐿
(3𝐿
𝐸=
− 𝑠) = 𝐸 =
− 𝑠) = 2000 ×
× (0.47625)
𝛿 6𝐼
𝛿 𝑏ℎ3
2.03 × 10−10
2
= 1.02252 × 1012
1 of 3
Figures and Tables
Table 1: Deflection of each beam for the given force applied
Deflection vs. Force
Deflection (m)
Force (N)
Brass Square
Brass Rectangle
Aluminum
Steel
0
0
0
0
0.0001524
0.000635
0.0005461 0.0002286
0.465975
0.0002794
0.0012954
0.0012319 0.0004572
0.93195
0.0004064
0.0020193
0.001778
1.397925
0.0004953
0.0026924
0.0024765 0.0008636
1.8639
0.0006223
0.0033401
0.0030861 0.0011049
2.329875
0.0007493
0.0039878
0.0037338 0.0013081
2.79585
0.0008763
0.0047625
0.0043434 0.0015494
3.261825
0.0010033
0.0053975
0.0050165 0.0018034
3.7278
0.0011176
0.005969
0.0056134 0.0020066
4.193775
0.0012446
0.0066675
0.006223
4.65975
1 of 3
0.0006731
0.0022098
0
Figure 2: A graph that plots the deflection of each material as it relates to the forces applied.
Table 2: Empirical Young’s modulus value for each material compared with the Theoretical values
Material
Brass Square
Brass Rectangle
Aluminum
Steel
Young's Modulus
Empirical (N/m^2)
4.25197E+11
3.62281E+11
3.94859E+11
1.02252E+12
Theoretical (N/m^2)
1.10E+11
1.10E+11
7.00E+10
2.07E+11
Symbols
𝐸
Young’s Modulus
𝐹
Force/Load
𝐼
Moment of Inertia
𝑏
Length of Cross Section
ℎ
Width of Cross Section
𝐿
Distance to Force/Load
𝑠
Distance to Dial Indicator
𝛿
Deflection
1 of 3
References
[1]
Beam Bending Slides 11-1. 2012, November 6. http://carmen.osu.edu.
[2]
Engineered Materials: Young’s Modulus. 2012, November 6.
http://invsee.asu.edu/nmodules/engmod/propym.html
1 of 3