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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 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

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 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.

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.

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.