Abstract Today’s experiment objectives are to determine the stress, deflection, and the strain of a simply supported beam under load. Moreover, experimentally verify the beam stress and flexure formulas. In this week’s experiment we had to use the MTS machine in order to apply a load to a simply supported beam and measure the deflection and strain that comes out from it. As a result from the graphs we plotted, we saw that whenever the load increases, the deflection and strain also increases. We used the strain to find the theoretical stress in our calculations, and we also used the moment, moment of inertia, and the neutral axis to find the experimental stress. We calculated the moment of inertia, which came out to be 0.05122 . Also, we found the neutral axis to be 0515 in , and the maximum deflection also came out to be 0.000013 in. The maximum load applied on the beam came out to be 40049.5 psi, which we calculated from the maximum stress. Table of Contents Abstract……………………………………………………………..………..2 Table of Contents……………………………………….……………………3 Introduction and Theory…………………………………………………….4-6 Procedure………………………………………………….……………….7-9 Summary of Important Results…………………...………………………..10-12 Sample Calculations and Error Analysis……………….………………….13 Discussion and Conclusion………………………………………………..14-15 References……………………………………..…………………………….16 Appendix……………………………………………………….……………17 Introduction and Theory Engineers use beams to support loads over a span length. These beams are structural members that are only loaded non-axially causing them to be subjected to bending. “A piece is said to be in bending if the forces act on a piece of material in such a way that they tend to induce compressive stresses over one part of a cross section of the piece and tensile stresses over the remaining part” (Ref. 1). This definition of bending is illustrated below in Figure 1. It can be seen from Figure 1 that the compressive force, C, and the tensile force, T, acting on the member are equal in magnitude because of equilibrium. Therefore, the compressive force and the tensile force form a force couple whose moment is equal to either the tensile force multiplied by the moment arm or the compressive force multiplied by the moment arm. The moment arm is denoted, e, in Figure 1. This is why structural members usually carry the center of the load into the tensile, compressive, or transverse loads. A beam usually carries the load transversely. During today’s experiment the load will be forced onto the beam in a symmetric order. We also must know that any cross section of the beam there will be a shear force V and a moment M. When we see in the middle of the beam we realize that the shear force diagram is zero and the moment reaches its maximum constant value. When a beam is cur in to slices we see that if we want the moment the internal forces must be equal to the moment on the outside. So, M must be equal to the internal forces applied.

1. Abstract
Today’s experiment objectives are to determine the stress,
deflection, and the strain of a simply supported beam under
load. Moreover, experimentally verify the beam stress and
flexure formulas. In this week’s experiment we had to use the
MTS machine in order to apply a load to a simply supported
beam and measure the deflection and strain that comes out from
it. As a result from the graphs we plotted, we saw that whenever
the load increases, the deflection and strain also increases. We
used the strain to find the theoretical stress in our calculations,
and we also used the moment, moment of inertia, and the
neutral axis to find the experimental stress. We calculated the
moment of inertia, which came out to be 0.05122 . Also, we
found the neutral axis to be 0515 in , and the maximum
deflection also came out to be 0.000013 in. The maximum load
applied on the beam came out to be 40049.5 psi, which we
calculated from the maximum stress.

2. Table of Contents
Abstract……………………………………………………………..
………..2
Table of
Contents……………………………………….……………………3
Introduction and
Theory…………………………………………………….4-6
Procedure………………………………………………….…………
…….7-9
Summary of Important
Results…………………...………………………..10-12
Sample Calculations and Error
Analysis……………….………………….13
Discussion and
Conclusion………………………………………………..14-15
References……………………………………..……………………
……….16
Appendix……………………………………………………….……
………17

3. Introduction and Theory
Engineers use beams to support loads over a span length. These
beams are structural
members that are only loaded non-axially causing them to be
subjected to bending. “A piece is said to be in bending if the
forces act on a piece of material in such a way that they tend to
induce compressive stresses over one part of a cross section of
the piece and tensile stresses over
the remaining part” (Ref. 1). This definition of bending is
illustrated below in Figure 1.
It can be seen from Figure 1 that the compressive force, C, and
the tensile force, T, acting on the member are equal in
magnitude because of equilibrium. Therefore, the compressive
force and the tensile force form a force couple whose moment is
equal to either the tensile force multiplied by the moment arm
or the compressive force multiplied by the moment arm. The
moment arm is denoted, e, in Figure 1.
This is why structural members usually carry the center of
the load into the tensile, compressive, or transverse loads. A
beam usually carries the load transversely. During today’s
experiment the load will be forced onto the beam in a symmetric
order. We also must know that any cross section of the beam
there will be a shear force V and a moment M. When we see in
the middle of the beam we realize that the shear force diagram
is zero and the moment reaches its maximum constant value.
When a beam is cur in to slices we see that if we want the
moment the internal forces must be equal to the moment on the
outside. So, M must be equal to the internal forces applied at

4. the ends of each cross section. “Sections, or cuts, which are
plane (flat) before deformation, remain plane after
deformation.”
At this radius, the length of arc is l’=(r + η) Δθ. This
length is also equal to rΔθ. Therfore, the strain at distance +η
from the neutral axis can be found by:
(III-1)
We must also assume that Hook’s law applies in tension and
compression with the same value of modulus of elasticity.
Therefore;
(III-2)
If the stress is given by σm = Ec/r. Then Equation (III-2) can
also be written as:
(III-3)
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,
(III-4)
If I is defined as the second moment of area about the neutral
axis, commonly called the moment of inertia,
(III-5)
Then Eq. (III-4) can be written as:
(III-6)
From calculus, it can be shown that the curvature of a function
y(x) is given by
the differential equation of the elastic curve:
(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

5. And that the maximum deflection at x = a + b/2 is
(III-8)
In particular, for a = b = L/3,
(III-9)
Procedure
Before we began with our experiment we studied the
information needed to complete the lab. The professor made
sure we knew all the details of the experiment before we entered
the experimental room. We knew that the test was going to be
performed on a 1018 steel beam (E=30*10^6 psi) using the MTS
testing machine that we used for the previous five weeks in this
course. So, it was very clear to how we were supposed to use
the MTS machine since we have used it several times before.
The position of the beam on the MTS testing machine must be
same as the picture shown in figure (1-1). We had to take the
cross sectional area of the beam in order to use it in further
calculations. For us to get the cross sectional are we had to get
the dimensions of the beam. We had to make sure that we took
all the dimensions of all ends, since some flanges of the beam
are thicker than others, so we took the average of them at the
end. If the beam was not installed in the MTS machine (which it
was in our case) go ahead and place the beam aligned to the 12-
inch black marks on the beam with the roller supports of the
lower bending fixture. You will have to make sure that the beam
is placed in the middle of the lower support with the strain
gauge facing down.
If the software on the computer was not opened up already
(everything was set up in our case) go ahead and click on the
icon on the desktop to open up TestWorks 4. After doing that
make sure that the name field under the user login says
‘’306A_lab’’, click OK to login, Under the Open Method
dialog, select “exp-3 4 Point Flex Mod X”; after doing all that
click on the motor reset button in the bottom right corner. Zero

6. the load reader by clicking gin the ‘Load cell’ icon and
selecting ‘zero channel’. Now you must position the upper
bending fixture over the beam. In order to do that you must first
enable the handset by pressing the unlock button. After that
slowly lower the crosshead using the down arrow until the
fixture is close to touching the beam. 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. Now, slowly raise the
fixture with the thumb wheel until only a very slight pre-load of
approximately 0.2 lb is applied. When finished, return control
to the computer software by locking the handset using the same
button as before.
Now if you do not have the LabVIEW software open (it
was open for us) double click the LabVIEW icon on the
desktop. Start the strain gauge acquisition. Zero the strain
indicator by pressing the ‘Zero Strain” button.
Now it is time to start the experiment, we had to press the
green arrow on the TestWorks 4. Then we entered the sample ID
number that we got. We knew that we had to load the beam to
1000 lb in increments of 100 lb. In order to do this we had to
press OK on the program to begin. After every 100 lb we
stopped the reader so we can get the actual load and record it on
our paper. We also had to record the strain reading from
LabVIEW and the deflection from the dial indicator. We did
that at every increment of 100 lb until it reached 1000 lb.
In order to make sure we did the experiment the right way,
we had to repeat the experiment over again. If we got similar
data as the first experiment that would mean that our
experiments were done the right way. Before we started our
second experiment we had to zero all the channels on the load
readout, strain indicator, and the dial indicator.

7. We can see from all the graphs that we obtained that
relationship was always proportional. The theoretical deflection
vs. load graph gave us a straight line with no curves on the
slope. Moreover, we saw that the experimental deflection vs.
load also had a proportional relationship. Thus, we obtained the
stress vs. strain graph to be also proportional in their data. We
had calculated the length of the beam before and therefore we
got the theoretical deflection, which was 1.044in, and the are
moment of inertia, which is 0.05122 in4. . These calculations
gave us the second graph that is shown above. By dividing the
moment over the section modulus we obtained stress, and the
force applied multiplied by the distance 4 in gave us the
moment. This information led to our third graph shown above.
The moment of inertia over the
neutral axis gave us the section modulus, which came up to be
0.0995.

8. Discussions and conclusions
Now that we have finished our experiment, we calculated our
data points and plotted our graphs we have come to many
conclusions that we obtained from this experiment. Since the
graphs that we plotted were all proportional we came to an
understanding that whenever the load increases, the deflection
and the stress increases also. The theoretical stress can be
obtained from the strain; also the moment of inertia is obtained
from the moment. Moreover, the experimental stress can be
obtained from the neutral axis. All the graphs that we obtained
had proportional relationships. We used the dial indicator to get
the deflections that we used in our calculations. Some human
error could have occurred during the laboratory, however our
results match up to the reference values, which lead me to
believe that error could not have played a big factor in this lab.
Overall a greater understanding of deflection and how different
materials react to the same applied load was achieved.