Beam Deflection Lab

Leah Perez Page 1 of 20 9/21/10 Lab 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: δ= Deflection of the beam at its midpoint (in.) 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/δ)*(L3 /48 I) Where: (P/δ) =m (L3 /48 I)=x Leah Perez Page 2 of 20 9/21/10 Lab 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, δ, and I to back calculate E. Ecalc= PL3 / 48Iδ Where: I= bh3 /12

Procedures:

Positio n 1st End Middle 2nd End AVERA GE

Table 2: Calculated Dimensions of the Wooden Beams Beam A "2 in. x 4 in." Beam B- "2 in. x 6 in." Height [h] Width/Base [b] Positio Height [h] Width/Base [b] (inches) (inches) n (inches) (inches) 1.491 3.451 1s 1 st End 1.552 5.506 1.475 3.466 Mi Middle 1.546 5.517 2nd 1.476 3.479 End 1.518 5.493 AVERA 1.481 3.465 GE 1.539 5.505

In order to perform this lab, the procedures below should be followed in this order. 1) In the laboratory, laboratory, calculate the dimension dimensions s of the wooden beams using using a caliper (round to 3 decimal places) and insert them into the following table to find their averages:

2) Now using using the values values for h and b calculated above and the given values 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 # #1 #2 #3 #4

Beam Orientation l l ll ll

Height [h] (inches) 1.481 1.481 3.465 3.465

Width/Base [b] (inches)

Span [L] (feet) 3.465 3.465 1.481 1.481

5 7 5 7

#5 #6 #7 #8

l l ll ll

1.539 1.539 5.505 5.505

5.505 5.505 1.539 1.539

3) Now Now that that you you know know the orienta orientatio tions ns of the the beam beams s and their their corre correspo spond nding ing 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 Leah Perez Page 3 of 20 9/21/10 Lab Report: Lab #3 Beam Bending Experiment 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 now insert the the data into a table as it it is calculated calculated with the use 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 have created these tables, tables, which show show the deflections deflections that were 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 δ values for the x axis and the load values, P, for the y axis. 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”. 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 table to calculate calculate the Moment Moment of Inertia for 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 table will now use use the values values of the moments moments of inertia inertia you just 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 will create a series series of ten tables tables that calculate calculate the Ecalc Ecalc average 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’

5 7 5 7

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 the values of of Eplot and Ecalc Ecalc have been determined, determined, you will create 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. Leah Perez Page 4 of 20 9/21/10 Lab Report: Lab #3 Beam Bending Experiment 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 # #1 #2 #3 #4 #5 #6 #7 #8

Beam Orientation l l ll ll l l ll ll

Height [h] (inches) 1.481 1.481 3.465 3.465 1.539 1.539 5.505 5.505

Width/Base [b] (inches) 3.465 3.465 1.481 1.481 5.505 5.505 1.539 1.539

Span [L] (feet) 5 7 5 7 5 7 5 7

Table 2: Calculated Dimensions of the Wooden Beams Beam A "2 in. x 4 in." Beam B- "2 in. x 6 in." Height [h] Width/Base Height [h] Position (inches) [b] (inches) Position (inches) 1st End 1.491 3.451 1st End 1.552 Middle 1.475 3.466 Middle 1.546 Scenario #1 Scenario #2 2nd End 1.476 3.479 2nd End 1.518 L (feet) L (feet) AVERAGE 1.481 3.465 AVERAGE 1.539 5 7 h h (inches) b (inches) (inches) b (inches) 1.481 3.465 1.481 3.465 Readings (inches)

P (lb) 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400

0.252 0.309 0.367 0.422 0.481 0.538

Width/Base [b] (inches) 5.506 5.517 5.493 5.505

δ δ (inches (inches Readings ) ) P (lb) (inches) 0 0 0 0.292 0.057 0.158 25 0.45 0.115 0.329 50 0.621 0.17 75 0.229 100 0.286 125 150 175 200 225 250 275 300 325 350 375 400

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 Beams Leah Perez Page 5 of 20 9/21/10 Lab Report: Lab #3 Beam Bending Experiment

Scenario #3 L (feet) 5 h (inches) 3.465

P (lb) 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400

Replicate of Scenario #3 L (feet) 5

b (inches) 1.481 Readings (inches) 0.31 0.325 0.339 0.352 0.363 0.376 0.388 0.399 0.409 0.42 0.432 0.445 0.458 0.469 0.48 0.492 0.503

h (inches) 3.465

b (inches) 1.481

δ δ (inche (inche Readings s) s) P (lb) (inches) 0 0 0 0.321 0.015 25 0.333 0.012 0.029 50 0.345 0.024 0.042 75 0.358 0.037 0.053 100 0.37 0.049 0.066 0.06 125 0.381 0.078 0.07 150 0.391 0.089 175 0.403 0.082 0.099 200 0.415 0.094 0.11 225 0.426 0.105 0.122 250 0.437 0.116 0.135 275 0.449 0.128 0.148 0.14 300 0.461 0.159 325 0.473 0.152 0.17 350 0.483 0.162 0.182 375 0.494 0.173 0.193 400 0.506 0.185

Leah Perez Page 6 of 20 9/21/10 Lab Report: Lab #3 Beam Bending Experiment

Scenario #4 L (feet) 7 h (inches) 3.465 P (lb)

b (inches) 1.481 Readings δ (inches) (inche

Scenario #5 L (feet) 5 h (inches) 1.593 P (lb)

b (inches) 5.505 Readings δ (inches) (inche

s) 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400

0 0.031 0.064 0.096 0.127 0.156 0.186 0.216 0.246 0.277

0.34 0.371 0.404 0.436 0.467 0.496 0.526 0.556 0.586 0.617

s) 0 25 50 75 10 0 12 5 15 0 17 5 20 0 22 5 2 50 2 75 3 00 3 25 3 50 3 75 4 00

Replicate of Scenario #5 L (feet) 5 h (inches) 1.593

P (lb) 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400

Readings (inches) 0.438 0.482 0.513 0.543 0.574 0.604 0.634 0.665 0.695

δ (inche s) 0.032 0.076 0.107 0.137 0.168 0.198 0.228 0.259 0.289

0.43 0.454 0.485 0.513 0.545 0.575 0.603 0.633 0.663 0.692

Scenario #6 L (feet) 7 h (inches) 1.593

b (inches) 5.505

0.43 0.454 0.485 0.513 0.545 0.575 0.603 0.633 0.663 0.692

P (lb) 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400

b (inches) 5.505

δ (inche s)

Readings (inches) 0.68 0.763 0.846 0.93


Scenario #7 L (feet) 5 h (inches)

b (inches)

Scenario #8 L (feet) 7 h (inches)

b (inches)

0.281 0.364 0.447 0.531

5.505

P (lb) 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400

1.593 Readings (inches) 0.385 0.387 0.39 0.393 0.396 0.399 0.403 0.406 0.409 0.413 0.416 0.419 0.422 0.426 0.43 0.433 0.437

5.505

1.593

δ δ (inches (inches Readings ) ) P (lb) (inches) 0 0 0 0.432 0.002 0.002 25 0.434 0.005 0.007 50 0.439 0.008 0.013 75 0.445 0.011 0.021 100 0.453 0.014 0.029 125 0.461 0.018 0.037 150 0.469 0.021 0.045 175 0.477 0.024 0.052 200 0.484 0.028 0.061 225 0.493 0.031 0.068 250 0.5 0.034 0.077 275 0.509 0.037 0.084 300 0.516 0.041 0.092 325 0.524 0.045 0.1 350 0.532 0.048 0.108 375 0.54 0.052 0.116 400 0.548

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. Plot 1 Leah Perez Page 8 of 20 9/21/10 Lab Report: Lab #3 Beam Bending Experiment

Plot 2

Plot 3


Plot 4

Plot 5


Plot 6

Plot 7

Leah Perez Page 11 of 20

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

Plot 8

Plot 9


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 # #1 #2 #3 #4 #5 #6 #7 #8

Base (inches) Height (inches) Length (inches) Moment of Inertia (in4) 3.465 1.481 60 0.937966156 3.465 1.481 84 0.937966156 1.481 3.465 60 5.134327051 1.481 3.465 84 5.134327051 5.505 1.539 60 1.672214314 5.505 1.539 84 1.672214314 1.539 5.505 60 21.39584011 1.539 5.505 84 21.39584011

Replicate of #3 Replicate of #5

1.481 5.505

3.465 1.539

60 60

5.134327051 1.672214314


Table 4: Calculation of Eplot for All Scenarios Scenario # #1 #2 #3 #4 #5 #6 #7 #8 Replicate of #3 Replicate of #5

Slope Eplot 437.47 2098812.4 153.15 2016166.774 2035.8 1784284.466 807.36 1941691.906 863.55 2323849.86 300.43 2218441.505 7960.6 1674283.404 3583 2067826.446 2148.6 1883148.444 759.64 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 Scenarios Load P (lbs) 25 50 75 100 125 150 175 200 225 250 275 300 325 350

Scenario #1 Deflection δ (inches) 0.057 0.115 0.17 0.229 0.286

Ecalc (psi) 2104216.871 2085919.333 2116594.617 2095028.151 2096859.469

375 400 AVERAGE E

2099723.688


Load P (lbs) 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400

Load P (lbs) 25 50 75 100 125 150 175 200 225 250 275 300 325

Scenario #2 Deflection δ (inches) 0.158 0.329

Ecalc (psi) 2083014.888 2000707.309

AVERAGE E

2041861.099

Scenario #3 Deflection δ (inches) 0.015 0.029 0.042 0.053 0.066 0.078 0.089 0.099 0.11 0.122 0.135 0.148 0.159

Ecalc (psi) 1460756.186 1511127.089 1565095.913 1653686.248 1659950.211 1685487.907 1723364.039 1770613.559 1483652.051 1796011.704 1785368.672 1776595.361 1791493.435

350 375 400

0.17 0.182 0.193 AVERAGE E

1804463.524 1805879.9 1816484.376 1693126.886


Load P (lbs) 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400

Load P (lbs) 25 50 75 100 125 150 175 200 225 250

Scenario #4 Deflection δ (inches) 0.031 0.064 0.096 0.127 0.156 0.186 0.216 0.246 0.277

Ecalc (psi) 1939507.245 1878897.644 1878897.644 1893692.114 1927074.507 1939507.245 1948486.446 1955275.597 1953510.908

AVERAGE E

1923872.15


Ecalc (psi) 2803169.402 2446402.387 2431665.023 2340037.066 2319864.333 2333273.953 2319864.333 2309907.833 2311009.889

275 300 325 350 375 400 AVERAGE E

2401688.247


Load P (lbs) 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400

Load P (lbs) 25 50 75 100 125 150 175 200 225

Scenario #6 Deflection δ (inches) 0.083 0.166 0.25

Ecalc (psi) 2224162.941 2224162.941 2215266.29

AVERAGE E

2221197.391


Ecalc (psi) 2629015.721 2103212.577 1971761.791 1912011.433 1877868.372 1752677.147 1752677.147 1752677.147 1690081.535

250 275 300 325 350 375 400

0.031 0.034 0.037 0.041 0.045 0.048 0.052 AVERAGE E

1696139.175 1701127.819 1705307.495 1667180.701 1635832.004 1643134.825 1617855.828 1819285.045


Load P (lbs) 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400

Load P (lbs) 25 50 75 100 125 150 175

Scenario #8 Deflection δ (inches) 0.002 0.007 0.013 0.021 0.029 0.037 0.045 0.052 0.061 0.068 0.077 0.084 0.092 0.1 0.108 0.116 AVERAGE E

Replicate of Scenario #3 Deflection δ (inches) 0.012 0.024 0.037 0.049 0.06 0.07 0.082

Ecalc (psi) 7214019.138 4122296.65 3329547.294 2748197.767 2487592.806 2339681.883 2244361.51 2219698.196 2128726.959 2121770.335 2061148.325 2061148.325 2038744.539 2019925.359 2003894.205 1990074.245 2695676.721

Ecalc (psi) 1825945.232 1825945.232 1776595.361 1788681.044 1825945.232 1878115.096 1870480.482

200 225 250 275 300 325 350 375 400

0.094 0.105 0.116 0.128 0.143 0.152 0.162 0.173 0.185 AVERAGE E

1864795.131 1878115.096 1888908.861 1883006.021 1838714.08 1873996.423 1893572.833 1899827.409 1895035.052 1856729.912


Load P (lbs) 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400

Replicate of Scenario #5 Deflection δ (inches) 0.044 0.075 0.105 0.136 0.166 0.196 0.227 0.257

AVERAGE E

Ecalc (psi) 1529001.492 1794028.417 1922173.304 1978707.813 2026387.52 2059471.397 2074592.333 2094196.596

1934819.859

Table 7: Comparison of Ecalc & Eplot for Each Scenario Scenario Ecalc Eplot 2099723.688 2098812.4 #1 2041861.099 2016166.774 #2 1712445.272 1784284.466 #3 1923872.15 1941691.906 #4

#5 #6 #7 #8 Replicate #1 Replicate #2

2401688.247 2221197.391 1819285.045 2695676.721 1856729.912 1934819.859

2323849.86 2218441.505 1674283.404 2067826.446 1879116.757 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.


Plot 11 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. Leah Perez Page 20 of 20 9/21/10 Lab Report: Lab #3 Beam Bending Experiment

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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