Beam Deflection Lab

Leah Perez of 209/21/10Lab Report: Lab #3 Beam Bending Experiment

Lab #3 Beam Bending Experiment

Introduction:

The objective of this lab is to develop the skills needed to conduct an experiment, gather the data, and organize it into a format that is easy to communicate using Microsoft Excel. This lab experiment will measure the deflections of wooden beams that are loaded at the midpoint of their lengths, or span. This experiment will also then compare the observed deflections with the ones that are actually calculated.

There will be two beams, each of which will be loaded under four different beam scenarios. Two of the scenarios will also be repeated, for a total of ten scenarios to test for deflection. Once we have calculated these, then we can make tables and plots that fit the data.

Beam Orientations:

I) II)h

b h

b

Formulas Used:

Formula 1: for the deflection of a beam which is supported at each end and that is loaded at its midpoint.

P*L3

=δ _________48 *E* I

Where: = Deflection of the beam at its midpoint (in.)δ

P= Load applied at the midpoint (lbs)L= Length of the beam (in.)E= Young’s Elastic Modulus of the beam material (lbs/in.3)I= Moment of Inertia of the beam (in4)

Formula 2: used to determine E using the formula derived from the plot of each scenario.

Eplot= (P/ )*(Lδ 3 /48 I) Where:(P/ ) =m δ(L3 /48 I)=x

Leah Perez Page 2 of 20

9/21/10Lab Report: Lab #3 Beam Bending Experiment

b= 0

Formula 3: used to determine E using the values that will be found for b, h, L, , and Iδ to back calculate E.

Ecalc= PL3 / 48IδWhere:I= bh3 /12

Procedures:

In order to perform this lab, the procedures below should be followed in this order.

1) In the laboratory, calculate the dimensions of the wooden beams using a caliper (round to 3 decimal places) and insert them into the following table to find their averages:

2) Now using the values for h and b calculated above and the given values of L, create a new table that shows all of the different beam loading scenarios for each beam:

Table 1: Beam Loading Scenarios

Scenario # Beam Orientation Height [h] (inches) Width/Base [b] (inches) Span [L] (feet)#1 l 1.481 3.465 5#2 l 1.481 3.465 7#3 ll 3.465 1.481 5#4 ll 3.465 1.481 7#5 l 1.539 5.505 5#6 l 1.539 5.505 7#7 ll 5.505 1.539 5#8 ll 5.505 1.539 7

3) Now that you know the orientations of the beams and their corresponding values of b, h and L, you can start the actual experiment of calculating the deflections for each of the different beam orientations. You will need to load your wooden beams at the midpoint of the span and make sure that they are supported simply on each end to represent 5 or 7 feet as needed. Now you


Table 2: Calculated Dimensions of the Wooden Beams Beam A "2 in. x 4 in." Beam B- "2 in. x 6 in."

PositionHeight [h] (inches) Width/Base [b] (inches) Position

Height [h] (inches) Width/Base [b] (inches)

1st End 1.491 3.451 1st End 1.552 5.506Middle 1.475 3.466 Middle 1.546 5.5172nd End 1.476 3.479 2nd End 1.518 5.493AVERAGE 1.481 3.465 AVERAGE 1.539 5.505


will load the beam with weights in 25 lb increments, starting at 25 lbs and ending at 400 lbs or until you reach 0.25 in deflection. That is to mean whichever comes first. An example of this table will be included in the next section and is entitled Table 3: Summary of Measured Dimensions of wooden Beams & Observed (Measured) Deflections of Wooden Beams.

4) You will now insert the data into a table as it is calculated with the use of a dial gauge. Make

sure to write down the initial reading of the dial gauge when there is no load. It will be important when calculating the deflection. An example of this table is Table 3: Summary of Measured Dimensions of wooden Beams & Observed (Measured) Deflections of Wooden Beams, which will be included in the next section.

5) Once you have created these tables, which show the deflections that were observed in the experiment, you can now create ten corresponding scatter plots for each of the eight scenarios plus the two replicas. Use the values for the x axis and the load values, P, for the y axis. Once δyou have made these plots, make sure to right click on the plot and add a “linear” trend line, set “intercept to 0”, and “display equation on chart”. You should now have ten scatter plots along with the three other tables you have created. Examples of these plots are included in the next section and are entitled Plot 1 through Plot 10.

6) Now create a fourth table to calculate the Moment of Inertia for each of the ten scenarios. Use the values of b, h and L already given and use the equation for I found in the introduction. An example of this table is included in the next section, entitled Table 5: Moment of Inertia Calculations for All Scenarios.

7) Your fifth table will now use the values of the moments of inertia you just calculated to find the Eplot values for all of the scenarios. The Eplot can be calculated using the formula for Eplot from the introduction. An example of this table is included in the next section and is entitled Table 4: Calculation of Eplot for All Scenarios.

8) Now you will create a series of ten tables that calculate the Ecalc average using the equation from the introduction for Ecalc. You will have to use the load and deflection values from each scenario to make a table for each of the scenarios’ corresponding Ecalc average values. An example of these tables is included in the next section and is entitled Table 6: Calculations of E Average for Each of the Eight Scenarios.

9) Once all of the values of Eplot and Ecalc have been determined, you will create a table that compares their values for each of the beam scenarios. An example of this table is included in the next section and is entitled Table 7: Comparison of Ecalc & Eplot for Each Scenario.

10)Lastly, create two new plots that will compare the deflection of the two original scenarios to the deflection of the replicate scenarios. So, one of the plots will include the values of the deflection of the original scenario #3 compared to the deflection values of the replica of scenario #3. The second plot will represent the same data except it will be for scenario #5 and its replica. Once you have made these plots, make sure to right click on the plot and add a “linear” trend line, set “intercept to 0”, and “display equation on chart”. Examples of these two plots are included in the next section and are entitled Comparison of Replicate to Original Scenario #3 and Comparison of Replicate to Original Scenario #5.



Data Analysis and Discussion:

The tables and graphs generated from this lab, with the aide Excel, are shown in this section as well as a short discussion of the data represented by them.

Table 1 and Table 2 show the calculated orientations of each beam and organize the data into tabular form.

Table 1: Beam Loading Scenarios

Scenario # Beam Orientation Height [h] (inches) Width/Base [b] (inches) Span [L] (feet)#1 l 1.481 3.465 5#2 l 1.481 3.465 7#3 ll 3.465 1.481 5#4 ll 3.465 1.481 7#5 l 1.539 5.505 5#6 l 1.539 5.505 7#7 ll 5.505 1.539 5#8 ll 5.505 1.539 7

Table 2: Calculated Dimensions of the Wooden Beams Beam A "2 in. x 4 in." Beam B- "2 in. x 6 in."

PositionHeight [h] (inches)

Width/Base [b] (inches) Position

Height [h] (inches)

Width/Base [b] (inches)

1st End 1.491 3.451 1st End 1.552 5.506Middle 1.475 3.466 Middle 1.546 5.5172nd End 1.476 3.479 2nd End 1.518 5.493AVERAGE 1.481 3.465 AVERAGE 1.539 5.505

Table 3 is a series of ten tables that show the measurements of the ten beam scenarios and their observed deflections that were found during the experiment.

Table 3: Summary of Measured Dimensions of

wooden Beams & Observed (Measured) Deflections of Wooden BeamsLeah Perez of 209/21/10

Lab Report: Lab #3 Beam Bending Experiment

Scenario #3 Replicate of Scenario #3L (feet) L (feet)

5 5h (inches) b (inches)

h (inches) b (inches)

3.465 1.481 3.465 1.481

P (lb)Readings (inches)

δ (inches) P (lb)

Readings (inches)

δ (inches)

0 0.31 0 0 0.321 025 0.325 0.015 25 0.333 0.012

Scenario #1 Scenario #2L (feet) L (feet)



1.481 3.465 1.481 3.465


δ (inches) P (lb)

Readings (inches)

δ (inches)

0 0.252 0 0 0.292 025 0.309 0.057 25 0.45 0.15850 0.367 0.115 50 0.621 0.32975 0.422 0.17 75

100 0.481 0.229 100 125 0.538 0.286 125 150 150 175 175 200 200 225 225 250 250 275 275 300 300 325 325 350 350 375 375 400 400

50 0.339 0.029 50 0.345 0.02475 0.352 0.042 75 0.358 0.037

100 0.363 0.053 100 0.37 0.049125 0.376 0.066 125 0.381 0.06150 0.388 0.078 150 0.391 0.07175 0.399 0.089 175 0.403 0.082200 0.409 0.099 200 0.415 0.094225 0.42 0.11 225 0.426 0.105250 0.432 0.122 250 0.437 0.116275 0.445 0.135 275 0.449 0.128300 0.458 0.148 300 0.461 0.14325 0.469 0.159 325 0.473 0.152350 0.48 0.17 350 0.483 0.162375 0.492 0.182 375 0.494 0.173400 0.503 0.193 400 0.506 0.185





3.465 1.481 1.593 5.505


δ (inches) P (lb)

Readings (inches)

δ (inches)

0 0.34 0 0 0.43 0.4325 0.371 0.031 25 0.454 0.45450 0.404 0.064 50 0.485 0.48575 0.436 0.096 75 0.513 0.513

100 0.467 0.127 100 0.545 0.545125 0.496 0.156 125 0.575 0.575150 0.526 0.186 150 0.603 0.603175 0.556 0.216 175 0.633 0.633200 0.586 0.246 200 0.663 0.663225 0.617 0.277 225 0.692 0.692250 250 275 275 300 300 325 325 350 350 375 375 400 400

Replicate of Scenario #5 Scenario #6L (feet) L (feet)



1.593 5.505 1.593 5.505


δ (inches) P (lb)

Readings (inches)

δ (inches)

0 0.438 0.032 0 0.68 0.28125 0.482 0.076 25 0.763 0.36450 0.513 0.107 50 0.846 0.44775 0.543 0.137 75 0.93 0.531

100 0.574 0.168 100 125 0.604 0.198 125 150 0.634 0.228 150 175 0.665 0.259 175 200 0.695 0.289 200 225 225 250 250 275 275 300 300 325 325 350 350 375 375 400 400



5 7h (inches) b (inches) h (inches) b (inches)

5.505 1.593 5.505 1.593


δ (inches) P (lb)

Readings (inches)

δ (inches)

0 0.385 0 0 0.432 025 0.387 0.002 25 0.434 0.00250 0.39 0.005 50 0.439 0.00775 0.393 0.008 75 0.445 0.013

100 0.396 0.011 100 0.453 0.021125 0.399 0.014 125 0.461 0.029150 0.403 0.018 150 0.469 0.037175 0.406 0.021 175 0.477 0.045200 0.409 0.024 200 0.484 0.052225 0.413 0.028 225 0.493 0.061250 0.416 0.031 250 0.5 0.068275 0.419 0.034 275 0.509 0.077300 0.422 0.037 300 0.516 0.084325 0.426 0.041 325 0.524 0.092350 0.43 0.045 350 0.532 0.1375 0.433 0.048 375 0.54 0.108400 0.437 0.052 400 0.548 0.116

The following are the plots that correspond with the deflections of each of the beam scenarios seen in the last set of tables. This information can now be seen graphically.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

20

40

60

80

100

120

140

f(x) = 437.473205984043 x

Deflection of Scenario #1

Deflection of Scenario #1Linear (Deflection of Scenario #1)

Deflection δ (inches)

Load

P (

lb)

Plot 1


0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

10

20

30

40

50

60

f(x) = 153.147404376713 x



Deflection δ (inches)

Load

P (

lb)

Plot 2

0 0.05 0.1 0.15 0.2 0.250

100

200

300

400

500

f(x) = 2035.76560753632 x



Deflection (inches)δ

Load

P (

lb)

Plot 3


0 0.05 0.1 0.15 0.2 0.25 0.30

50

100

150

200

250

f(x) = 807.363624718649 x




Load

P (

lb)

Plot 4

0 0.05 0.1 0.15 0.2 0.25 0.30

50

100

150

200

250

f(x) = 863.548998999209 x




Load

P (

lb)

Plot 5


0 0.05 0.1 0.15 0.2 0.25 0.30

1020304050607080

f(x) = 300.428077776059 x




Load

P (

lb)

Plot 6

0 0.01 0.02 0.03 0.04 0.05 0.060

100

200

300

400

500

f(x) = 7960.63793688497 x




Load

P (

lb)

Plot 7


0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

100

200

300

400

500

f(x) = 3583.00237189034 x




Load

P (

lb)

Plot 8

00.02

0.040.06

0.08 0.10.12

0.140.16

0.18 0.20

100

200

300

400

500

f(x) = 2148.58505705694 x

Deflection of Replicate of Scenario #3

Deflection of Replicate of Scenario #3Linear (Deflection of Replicate of Scenario #3)


Load

P (

lb)

Plot 9


0 0.05 0.1 0.15 0.2 0.25 0.30

50

100

150

200

250

f(x) = 759.635954893216 x

Deflection of Replicate of Scenario #5

Deflection of Replicate of Scenario #5Linear (Deflection of Replicate of Scenario #5)


Load

P (

lb)

Plot 10

Next is the table that calculates the moment of inertia for each scenario. Then in the next table we use these values to find the Eplot.

Table 5: Moment of Inertia Calculations for All Scenarios

Scenario # Base (inches) Height (inches) Length (inches) Moment of Inertia (in4)#1 3.465 1.481 60 0.937966156#2 3.465 1.481 84 0.937966156#3 1.481 3.465 60 5.134327051#4 1.481 3.465 84 5.134327051#5 5.505 1.539 60 1.672214314#6 5.505 1.539 84 1.672214314#7 1.539 5.505 60 21.39584011#8 1.539 5.505 84 21.39584011Replicate of #3 1.481 3.465 60 5.134327051Replicate of #5 5.505 1.539 60 1.672214314


Table 4: Calculation of Eplot for All Scenarios

Scenario # Slope Eplot#1 437.47 2098812.4#2 153.15 2016166.774#3 2035.8 1784284.466#4 807.36 1941691.906#5 863.55 2323849.86#6 300.43 2218441.505#7 7960.6 1674283.404#8 3583 2067826.446Replicate of #3 2148.6 1883148.444Replicate of #5 759.64 2044223.62

Below are the ten calculations of Ecalc average for each of the beam scenarios. These are accompanied by a table that compares the Ecalc values to those of the Eplot for all ten scenarios.

Table 6: Calculations of Ecalc Average for Each of the Eight ScenariosScenario #1

Load P (lbs) Deflection δ (inches) Ecalc (psi)25 0.057 2104216.87150 0.115 2085919.33375 0.17 2116594.617

100 0.229 2095028.151125 0.286 2096859.469150 175 200 225 250 275 300 325 350 375 400

AVERAGE E 2099723.688


Scenario #2Load P (lbs) Deflection δ (inches) Ecalc (psi)

25 0.158 2083014.88850 0.329 2000707.30975

100 125 150 175 200 225 250 275 300 325 350 375 400

AVERAGE E 2041861.099


25 0.015 1460756.18650 0.029 1511127.08975 0.042 1565095.913

100 0.053 1653686.248125 0.066 1659950.211150 0.078 1685487.907175 0.089 1723364.039200 0.099 1770613.559225 0.11 1483652.051250 0.122 1796011.704275 0.135 1785368.672300 0.148 1776595.361325 0.159 1791493.435350 0.17 1804463.524375 0.182 1805879.9400 0.193 1816484.376

AVERAGE E 1693126.886



25 0.031 1939507.24550 0.064 1878897.64475 0.096 1878897.644

100 0.127 1893692.114125 0.156 1927074.507150 0.186 1939507.245175 0.216 1948486.446200 0.246 1955275.597225 0.277 1953510.908250 275 300 325 350

375 400

AVERAGE E 1923872.15


25 0.024 2803169.40250 0.055 2446402.38775 0.083 2431665.023

100 0.115 2340037.066125 0.145 2319864.333150 0.173 2333273.953175 0.203 2319864.333200 0.233 2309907.833225 0.262 2311009.889250 275 300 325 350 375 400

AVERAGE E 2401688.247



25 0.083 2224162.94150 0.166 2224162.94175 0.25 2215266.29

100 125 150 175 200 225 250 275 300 325 350

375 400

AVERAGE E 2221197.391


25 0.002 2629015.72150 0.005 2103212.57775 0.008 1971761.791

100 0.011 1912011.433125 0.014 1877868.372150 0.018 1752677.147175 0.021 1752677.147200 0.024 1752677.147225 0.028 1690081.535250 0.031 1696139.175275 0.034 1701127.819300 0.037 1705307.495325 0.041 1667180.701350 0.045 1635832.004375 0.048 1643134.825400 0.052 1617855.828

AVERAGE E 1819285.045



25 0.002 7214019.13850 0.007 4122296.6575 0.013 3329547.294

100 0.021 2748197.767125 0.029 2487592.806150 0.037 2339681.883175 0.045 2244361.51200 0.052 2219698.196225 0.061 2128726.959250 0.068 2121770.335275 0.077 2061148.325300 0.084 2061148.325325 0.092 2038744.539350 0.1 2019925.359

375 0.108 2003894.205400 0.116 1990074.245

AVERAGE E 2695676.721

Replicate of Scenario #3Load P (lbs) Deflection δ (inches) Ecalc (psi)

25 0.012 1825945.23250 0.024 1825945.23275 0.037 1776595.361

100 0.049 1788681.044125 0.06 1825945.232150 0.07 1878115.096175 0.082 1870480.482200 0.094 1864795.131225 0.105 1878115.096250 0.116 1888908.861275 0.128 1883006.021300 0.143 1838714.08325 0.152 1873996.423350 0.162 1893572.833375 0.173 1899827.409400 0.185 1895035.052

AVERAGE E 1856729.912


Replicate of Scenario #5Load P (lbs) Deflection δ (inches) Ecalc (psi)

25 0.044 1529001.49250 0.075 1794028.41775 0.105 1922173.304

100 0.136 1978707.813125 0.166 2026387.52150 0.196 2059471.397175 0.227 2074592.333200 0.257 2094196.596225 250 275 300 325

350 375 400

AVERAGE E 1934819.859

Table 7: Comparison of Ecalc & Eplot for Each ScenarioScenario Ecalc Eplot

#1 2099723.688 2098812.4#2 2041861.099 2016166.774#3 1712445.272 1784284.466#4 1923872.15 1941691.906#5 2401688.247 2323849.86#6 2221197.391 2218441.505#7 1819285.045 1674283.404#8 2695676.721 2067826.446

Replicate #1 1856729.912 1879116.757Replicate #2 1934819.859 2044223.62

Lastly, two more plots will be included to show the comparison of deflections in the first testing of beams #3 and #5 and the second their second testing. The graph shows that even under the same conditions the results varied some.


0 0.05 0.1 0.15 0.2 0.250

50

100

150

200

250

300

350

400

450

f(x) = 2144.0407934463 xf(x) = 2144.0407934463 xf(x) = 2035.76560753632 x

Comparison of Replicate to Original Scenario #3

Original Scenario #3

Linear (Original Scenario #3)

Replicate Scenario #3

Linear (Replicate Scenario #3)


Load

P (

lbs)

Plot 11

0 0.05 0.1 0.15 0.2 0.25 0.30

50

100

150

200

250

f(x) = 759.635954893216 x

f(x) = 863.548998999209 x

Comparison of Replicate to Original Scenario #5

Original Senario #5

Linear (Original Senario #5)

Replicate Scenario #5

Linear (Replicate Scenario #5)


Loa

d P

(lb

s)

Plot 12

Discussion of Data:It is interesting to note that the two methods used for calculating E, Eplot and Ecalc, yield very similar results, but they are not identical. This is due to the fact that Eplot used the slope for finding the Modulus of Elasticity. By using the slope from the plots to substitute for the (P/ )δ , we may have reached less accurate results than with Ecalc which used the individual values in the deflection equation.

For this reason it is better to use the Ecalc method for calculating E. By this method there is less room for error. There is very little error with Ecalc because we found the exact value of each deflection of beam scenario and then averaged them to get E average. By using such a tedious method we gain more accuracy.


When comparing the values of E between the four beam orientation scenarios for each of the two beams, we see that the values are very close between them. This is good to know because it confirms that our results are reasonable.

Also, when we compare the replicate scenario calculations to the original scenario calculations we can see that they are not very different. There is enough error present to where it is noticeable, but the results of the replicate are not contrary to the original results. This shows that the experiments are not highly reproducible, but they are reproducible enough to use for confirmation.

It is also important to notice that the values that were calculated with Eplot and Ecalc are quite different from the accepted value of Young’s Modulus of Elasticity for wood, which is about 1,600,000. This value varies however for different types of wood, which could cause some of the difference in values that we calculated in the experiment. Our value for the Young’s Modulus of Elasticity was off from the standard by ~300,000 at the least. However, since we do not know the type of wood used, we cannot discredit our results. They are therefore reasonable values for Young’s Modulus of Elasticity for wood.

The main source of error, which may have caused this difference, may have been that in the experiment the dial gauge could not be placed directly at the midpoint of the span. This was due to physical barriers during experimentation. If the dial gauge had measured the deflection at the midpoint of the beam our calculations may have been more accurate. Also, human error must be accounted for. Mistakes may have been made when creating the Excel document. But overall, the E that was calculated in our experiment is a very reasonable value for wood, so the experiment was not a failure.

Conclusion:In summary, our experiment calculated the deflections of eight different beam

scenarios and confirmed that the results were reasonable using the equation for deflection and the slope of the plots to find Young’s Modulus of Elasticity (E). Our results confirmed that the experiment was successful.

Our objectives were also met. We successfully measured all of the dimensions of the beams and their deflection in the laboratory. Then we organized all of the data into tables in Excel. And lastly we made this information highly communicable by creating the scatter plots that represent our data graphically. The tables that compared values of the experiment also helped to make the Excel document simpler to communicate to others.

Laboratory work and experimentation are extremely important to Civil Engineering. The only way to confirm the ideas that engineers come up with is to test them and analyze the data. This lab experiment was just one example of many situations that may face a Civil Engineer in the work place everyday. Being able to test, organize, and communicate data is fundamental for any Civil Engineer.

Beam Deflection Lab

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