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Experiment III
The Beam
OBJECTIVES
The objectives of this experiment are (a) to determine the stress, deflection and strain of a simply
supported beam under load, and (b) to experimentally verify the beam stress and flexure
formulas.
THEORY
Structural members are usually designed to carry tensile, compressive, or transverse loads. A
member which carries load transversely to its length is called a beam. In this experiment, a beam
will be symmetrically loaded as shown in Fig. III-1(a), where P is the applied load. Note that at
any cross section of the beam there will be a shear force V (Fig. III-1(b)) and moment M (Fig.
III-1c). Also, in the central part of the beam (between the loads P/2) V is zero and M has its
maximum constant value. Notice the sign convention of a positive moment, M, causing a
negative (downward) deflection, y.
If in this part a small slice EFGH of the beam is imagined to be cut out, as shown, then it is clear
that the external applied moment, M, must be balanced by internal forces (stresses) at the
sections (faces) EF and GH. For M applied as shown in Fig. III-2(a), these forces would be
compressive near the top, EG, and tensile near the bottom, FH. Since the beam material is
considered elastic, these forces would deform the beam such that the length EG would tend to
become shorter, and FH would tend to become longer. The first fundamental assumption of the
beam theory can be stated as follows:
“Sections, or cuts, which are plane (flat) before deformation remain plane after
deformation.”
Thus, under this assumption, the parallel and plane sections EF and GH will deform into plane
sections E ′F ′ and G ′H ′ which will intersect at point O, as shown in Fig. III-2(b). Since E ′F ′
and G′H ′ are no longer parallel, they can be thought of as being sections of a circle at some
radial distance from O. Convince yourself of this by drawing a square on an eraser and observe
its shape when you bend the eraser. Since the forces near E ′G ′ are compressive, and those near
F ′H ′ are tensile, there must be some radial distance r where the forces are neither compressive
nor tensile, but zero. This axis, N-N, is called the neutral axis. Notice that N-N is not assumed to
lie in the center of the beam.
Consider an arc of distance +η, from the neutral axis, or distance r + η from O (Fig. III-2(b)). At
this radius, the length of arc is l ′ =(r + η) Δθ. As shown in Fig. III-2(a), the length of the arc was
l before the deformation. This length is also equal to rΔθ (because at N-N there are no forces to
change the length). Thus, the strain at distance +η from the neutral axis can be found by:
ε =
η
l′ - l
(r + η )Δθ - rΔθ
=
=
l
rΔθ
r
III-1
(III-1)
L
bb
a
c
P
2
P
2
ym
(a)
V
P
2
+
M
-
x
(b)
Pa
2
x
(c)
Figure III-1. Symmetrically Loaded Beam (a), with Shear Force Diagram (b)
and Bending Moment Diagram (c)
III-2
Figure III-2. Stresses and Strains of a Beam
III-3
In other words, the axial strain is proportional to the distance from the neutral axis. It is remarked
that this strain is positive, because positive η was taken on the tensile side of N-N in Fig. III2(b). Had η been taken in the opposite direction, then the strain would have been negative, as
appropriate for the compressive side.
The second fundamental assumption is that Hooke’s Law applies both in tension and
compression with the same modulus of Elasticity. Thus, from Eqs. (I-3) and (III-1),
σ = E
η
(III-2)
r
If c is the maximum distance from the neutral axis (largest positive or negative value of η), then
the maximum stress (compressive or tensile) is given by σm = Ec/r, and Eq. (III-2) can also be
written as
σ = σm
η
(III-3)
c
That is, the stress at a section EF or GH, due to applied moment M, varies linearly from zero at
the neutral axis to some maximum value σm (positive or negative) when η = c. To obtain the
beam stress formula, it remains to define where the neutral axis is located, and to relate σm to M.
To locate the neutral axis, it is observed that the tensile and compressive forces on a section are
equal to the stress times a differential element of area, as shown in Fig. III-2(c). For static
equilibrium, the sum (or integral) of all these internal forces must be zero. That is,
dF = ∫ σ dA =
A
σ m η dA = 0
∫
c
A
where, the integrals are over the whole cross-sectional area. Thus, it is seen that the neutral axis
is located such that the first moment of area about it is zero; that is, the neutral axis passes
through the centroid of the cross-sectional area. In Fig. III-2(c), a rectangular area was used for
illustration; however, any shape of vertically symmetric cross-sectional area is valid for the area
integral.
In a similar fashion, the moment due to all the forces is the sum (or integral) of the forces times
their moment arms about the neutral axis, and this must be equal to the external applied moment.
Thus,
M = ∫ ηdF = ∫ ησdA =
σm
c
∫ η dA
2
(III-4)
If I is defined as the second moment of area about the neutral axis, commonly called the moment
of inertia,
I = ∫ η dA
2
(III-5)
then Eq. (III-4) can be written as:
σm =
M
Mc
=
I
Z
III-4
(III-6)
where Z = I/c is the section modulus, which depends only on the cross-sectional geometry of the
beam. Equation (III-6) is the beam stress equation which relates the maximum (compressive or
tensile) stress to the applied moment. Notice its similarity to Equation (I-1), the stress equation
for uniaxial tension. It is understood, of course, that σm is the maximum bending stress at a
particular location, x, along the beam. In general, both σm and M are functions of x, and are
related by Eq. (III-6).
The remaining question about the beam concerns its degree of deformation, or flexure. That is,
how is the radius of curvature, r, related to the moment M (or load P)? From calculus, it can be
shown that the curvature of a function y(x) is given by
d2 y
1
dx 2
=
d2 y 3
r
)2
(1 +
dx 2
Thus, if x is the distance along the beam, y will be the deflection as indicated in Fig. III-1(a). For
most beams of practical interest, this deflection will be small, so that the slope dy/dx will be very
small compared to 1. Hence, a very good approximation is
1
d2 y
=
r
dx 2
But, since σm = Ec/r = Mc/I, there results the differential equation of the elastic curve:
d2 y
EI 2 = M(x)
dx
(III-7)
To obtain the elastic curve of the beam, y(x), and the maximum deflection, ym, it is necessary to
integrate Eq. (III-7) using the moment function M(x) in Fig. III-1(c). Thus, using M(x) = Px/2 for
0 ≤ x ≤ a and M(x) = Pa/2 for a ≤ x ≤ a + b, it is found that
y(x) =
y ( x) =
P ⎛ x 3 ax(a + b) ⎞
⎟,
2EI ⎜⎝ 6
2
⎠
for 0 ≤ x ≤ a
P ⎛ a 3 ax(2a + b) ax 2 ⎞
+
,
2EI ⎜⎝ 6
2
2 ⎟⎠
for a ≤ x ≤ a + b
and that the maximum deflection at x = a + b/2 is
- ym =
Pa a 2 ab b2
( + + )
2EI 3
2
8
(III-8)
In particular, for a = b = L/3,
- ym =
23 Pa 3
23 PL3
=
48EI
1296EI
III-5
(III-9)
Although the above stress and flexure formulas are quite simple, it took some of the best minds
of the 17th and 18th centuries to derive them correctly. Part of the difficulty in obtaining the
correct results at that time was that there were no methods of verifying the results
experimentally. Today, with the advent of sensitive displacement dial gauges, the verification is
more convenient.
Beam
Deflection
Dial indicator
Beam to
Be Tested
Figure III-3. MTS Insight Tensile Testing Machine
PROCEDURE
1. The test will be conducted on a 1018 steel beam (E = 30x106 psi) using the MTS testing
machine. The position of the beam in the testing machine is shown in Fig. III-3.
2. Observe that the cross-section of the I-beam is not a symmetric; one flange of the beam is
thicker than the other. Measure carefully the cross-sectional dimensions and the location of
the loading and support points.
3. Carefully place the beam into the MTS machine if it is not already installed. Align the 12inch black marks on the beam with the roller supports of the lower bending fixture. Make
sure the beam is squarely resting and centered on the lower support with the strain gauge
facing down.
III-6
4. Enter the TestWorks 4 software by double clicking on the icon on the desktop.
a. When prompted, make sure the Name field under the User Login says “306A_lab”.
b. Click OK to login.
c. Under the Open Method dialog, select “exp-3 4 Point Flex Mod X”.
d. Now, select the Motor Reset button in the bottom right corner by clicking on it.
5. Zero the “load” readout by right clicking on the “Load cell” icon and selecting “zero
channel”.
6. Next we will use the handset to position the upper bending fixture over the beam.
a. Enable the handset by pressing the “unlock” button at the top right of the handset
b. Slowly lower the crosshead using the down arrow until the fixture is NEARLY touching
the beam.
c. Make sure not to pinch the strain gauge lead wires between the fixture and the beam.
d. While observing the digital load readout on the screen, use the thumb wheel of the
handset to lower the fixture onto the beam. Watch for the load reading to increase when
the upper loading component of the fixture makes contact with the beam.
e. Now, slowly raise the fixture with the thumb wheel until only a very slight pre-load of
approximately 0.2 lb is applied.
f. When finished, return control to the computer software by locking the handset using the
same button as before.
Note that the beam has a strain gauge bonded to its surface. See Figure III-4. This strain
gauge has a strain gauge factor of 2.14 ± 0.5% and 350 ohm resistance. Connect the strain
gauge lead wires to the “#1 Strain” channel of the grey DAQ box. Data acquisition is
conducted by the LabVIEW software.
7. Take the magnetic base holding the dial indicator, and position it with the dial indicator in
the center of the beam on the bottom side. Make sure the dial indicator is not touching the
strain gauge. Also, be VERY CAREFUL not to damage the dial indicator while doing this!
When the dial indicator is in position, lock it to the MTS frame by activating the magnetic
base. Finally, zero the dial indicator by turning the dial of the indicator. Note the divisions
on the dial indicator and the accuracy with which it can be read.
8. Start the LabVIEW software.
a. Double click the LabVIEW icon on the desktop, or follow the procedure suggested for
Experiment II.
b. Select “Open” and double click “exp2&3-Strain Mod 9-15 LV7.1”.
c. Press the white arrow near the upper left corner of the screen to start the strain gauge
acquisition.
d. Press the “Zero Strain” button to zero-out the strain indicator.
9. You are now ready to run the experiment.
a. Press the Green Arrow on the TestWorks 4 GUI.
b. Give the sample ID for your group.
III-7
c. Under the Required Inputs dialog, enter 0.125 for the thickness and 0.500 for the width.
DO NOT press OK yet!
10. You will load the beam up to 1000 lb in increments of 100 lb. To do this:
a. “OK” the Required Inputs dialog from above and the test will begin.
b. When the digital load readout reaches approximately 100 lb, press the “Pause” button to
temporarily stop the test.
c. Record the actual load on your data sheet from the digital readout.
d. Toggle over to the LabVIEW screen and record the strain reading from the software.
e. Record the deflection of the beam from the dial indicator.
f. Return to the TestWorks 4 screen and press the “Pause” button again to resume the test.
g. Continue pausing and taking readings approximately every 100 lb from 0 to 1000 lb.
11. At 1000 lb the test will stop. Collect your last data point before pressing “OK” at the last
dialog.
12. Before repeating the experiment, make sure your load readout, strain indicator, and dial
indicator are all re-zeroed.
13. Repeat steps 7 through 9.
Figure III-4. Beam with Bonded Strain Gauge
REPORT REQUIREMENTS
1.
Determine the neutral axis, moment of inertia, and section modulus of the beam crosssection.
2.
Draw shear force and bending moment diagrams for the beam for the maximum load.
3.
Compute the maximum bending stress and deflection of the beam for the maximum load.
4.
Plot deflection versus load, and also stress versus load. Each graph should contain a curve
based completely on theoretical calculations, and another using the experimental data points.
III-8
5.
By error calculation, determine if the beam theory is vindicated within the precision of the
instruments.
REFERENCES
[1] Beckwith, T. G., Buck, N. L. and Marangoni, R. D., Mechanical Measurements, AddisonWesley.
[2] Popov, E.P. Mechanics of Materials, Prentice-Hall Inc.
[3] Thomas, G.B., and Finney R.L., Calculus and Analytical Geometry, Addison-Wesley,
III-9
•
An ability to communicate effectively with a range of audiences.
•
An ability to function effectively on a team whose members together provide
leadership, create a collaborative and inclusive environment, establish goals, plan tasks,
and meet objectives.
•
An ability to use the techniques, skills and modern engineering tools necessary for
engineering practice (k).
INSTRUCTIONAL MATERIALS, HANDOUTS AND TEXTBOOKS
• Required: Experiment instructions provided by the instructor in PDF format*
• Recommended: J. P. Holmam, “Experimental Methods for Engineers”, 8th Edition,
McGraw-Hill, 2012
• Recommended: J. Chaudhuri and A. Marathe, “Materials and Mechanics Laboratory
Experiments”, Cognella Academic Publishing, 2014
* Students are required to bring a printed copy of the experiment instructions to the class, or
any device capable of reading and opening the PDF files (cell phones are not allowed)
GRADING STANDARDS, AND CRITERIA
Grading
Individual Laboratory Reports
60 %
Class Participation and Attendance
10%
Final Exams
30 %
Final Grade Scheme
Over/or 90.00%
A
Between 89.99 % and 80.00%
B
Between 79.99 % and 70.00%
C
Between 69.99 % and 60.00%
D
Less Than 59.99 %
F
Note: The above ranges for the course letter grades may be adjusted to lower limits based on
the overall performance of the class that will be determined at the end of the semester. +/grading will not be used in this course.
ATTENDANCE POLICY
• Attendance: Attendance is mandatory. Attending the class on time and participating in
class lectures and discussions are essential to successful completion of this class. Since much
of the learning in this course is experiential, you must be in class to experience it. If you are
not in class, you cannot participate; thus, any more than 2 unexcused absences will result in a
reduction in the class participation and attendance component of the grade and may endanger
your ability to complete the course.
• Students who must miss class to represent the university or to participate in a universitysponsored activity must notify the class instructor in writing a minimum of two weeks in
advance of the absence.
2|Page Fall 2018
• Whether an absence is excused or unexcused is determined solely by the instructor with the
exception of absences due to religious observance and officially approved activities described
above.
Late Assignments
Late submissions of reports will not be collected unless prior arrangements are made with the
instructor in a timely manner.
INSTRUCTION FOR LAB REPORT WRITING
• Submit your report in .PDF formation through Titanium.
• Use professional language.
• DO NOT use content/figures, from any source (Books or Internet) without proper
referencing.
• DO NOT copy word for word from the textbook or experiment instructions. Use your own
wording.
Lab Report Writing Content (Total Point: 100)
Section
Points
Contents
▪ Course Name and Number
▪ Title of Experiment
▪ Lab Sec.
▪ Name
▪ Lab partner’s name
▪ Date
▪ Submitted to
An overview of the entire experiment including:
▪ Should be no more than 200 words
▪ Objectives, approach and primary results
▪ Briefly answer:
o What was done?
o How was it done?
o What were your basic results?
o How does your result compare to that of
theory and/or other sources?
Title / Cover page
2
Abstract
10
List of Symbols
and Units
2
▪
▪
▪
Variables
Name of Variables
Units
Introduction
10
▪
▪
A brief review of the problem
Objectives
Theory
15
▪
▪
Theoretical background
Proper figures & complete set of Equations
3|Page Fall 2018
Test Setup
and
Procedure
Results
▪
▪
▪
▪
Complete description of the setup
Schematic of setup / image of the setup with proper
labelling
Specimen description using test standards used
Complete step by step procedure
▪
▪
▪
Explanations of how reduced results were obtained
Results (tabulated form)
Plots with titles
▪
▪
▪
Interpretation of data
Error analysis
Comparison of results to establish standard values
and discussion
▪
▪
Conclusions reached based on results & discussion
Recommendations if appropriate
▪
Numerically number list of all used references and
cite within the report body
▪
Datasheets, Sample Calculations, additional figures,
charts etc.
▪
All reports must follow the ‘’ Lab Report Writing
Format’’.
15
18
Discussion
15
Conclusions
4
References
2
Appendix
2
Format
5
Lab Report Writing Format
•
•
•
•
•
•
•
•
•
Use 1 1⁄2 spacing for texts and equations.
For all texts and equations, you must use the font Times New Roman, font size 12.
For section titles, use Times New Roman, size 14 with boldface.
Use 1-inch left margin, and 3⁄4 margin on all other sides (right, top, and bottom).
Use Justification on both left and right margins.
All equations must be centred with equations number: (1), (2), (3) with right
justification.
Try to use present tense in writing your report.
For all pages except the cover page, you should have headers/footers such that:
o Upper left corner: EGME 306A
o Upper right corner: Experiment name
o Lower left corner: Your name
o Lower right corner: Page #/Total pages
For all graphics and tables:
o Numbered consecutively and captioned
o Caption = 9 pt. boldface sans serif typeface, uppercase, centered below the
graphic
4|Page Fall 2018
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10/10/2018
Experiment 3
3 trials
30 screenshots total
don't put the screenshots in the lab report
only use the measurements in the screenshots in the
report
Measure the lengths of the beam at the end of
the experiment
x 1,80 0.252
2, 0.494
32297
40,197
9, 1.011
6.0.405
7.1.008