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# Cive 302 solid mechanics lab report

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Data:

 X (in) 100lb at P1 (30 in) 100lb at P2 (60 in) 100lb at P1/P2 d1 5 -0.011 -0.0085 -0.0195 d2 25 -0.0455 -0.04 -0.085 d3 45 -0.054 -0.055 -0.1085 d4 65 -0.038 -0.046 -0.084 d5 85 -0.0075 -0.0095 -0.0175

Table 1: Experimental displacement readings

 LC1 LC2 LC3 P1, lbs 100 0 100 P2, lbs 0 100 100 C1 -50000 -40000 -90000 E, ksi 10000 I, in^4 2.5 L, in 90 c, in 1.5

Table 2: Given values

Calculations:

 x (in) Moment LC1, lb-in Moment LC2, lb-in Moment LC3, lb-in LC1 disp LC2 disp LC3 disp 0 0 0 0 0 0 0 0.5 33.3333 16.6667 50 -0.0010 -0.0008 -0.0018 1 66.6667 33.3333 100 -0.0020 -0.0016 -0.0036 1.5 100.0000 50.0000 150 -0.0030 -0.0024 -0.0054 2 133.3333 66.6667 200 -0.0040 -0.0032 -0.0072 2.5 166.6667 83.3333 250 -0.0050 -0.0040 -0.0090 3 200.0000 100.0000 300 -0.0060 -0.0048 -0.0108 3.5 233.3333 116.6667 350 -0.0070 -0.0056 -0.0126 4 266.6667 133.3333 400 -0.0080 -0.0064 -0.0144 4.5 300.0000 150.0000 450 -0.0090 -0.0072 -0.0161 5 333.3333 166.6667 500 -0.0099 -0.0080 -0.0179 5.5 366.6667 183.3333 550 -0.0109 -0.0088 -0.0197 6 400.0000 200.0000 600 -0.0119 -0.0096 -0.0215 6.5 433.3333 216.6667 650 -0.0129 -0.0103 -0.0232 7 466.6667 233.3333 700 -0.0138 -0.0111 -0.0250 7.5 500.0000 250.0000 750 -0.0148 -0.0119 -0.0267 8 533.3333 266.6667 800 -0.0158 -0.0127 -0.0285 8.5 566.6667 283.3333 850 -0.0167 -0.0135 -0.0302 9 600.0000 300.0000 900 -0.0177 -0.0142 -0.0319 9.5 633.3333 316.6667 950 -0.0186 -0.0150 -0.0336 10 666.6667 333.3333 1000 -0.0196 -0.0158 -0.0353 10.5 700.0000 350.0000 1050 -0.0205 -0.0165 -0.0370 11 733.3333 366.6667 1100 -0.0214 -0.0173 -0.0387 11.5 766.6667 383.3333 1150 -0.0223 -0.0181 -0.0404 12 800.0000 400.0000 1200 -0.0232 -0.0188 -0.0420 12.5 833.3333 416.6667 1250 -0.0241 -0.0196 -0.0437 13 866.6667 433.3333 1300 -0.0250 -0.0203 -0.0453 13.5 900.0000 450.0000 1350 -0.0259 -0.0211 -0.0470 14 933.3333 466.6667 1400 -0.0268 -0.0218 -0.0486 14.5 966.6667 483.3333 1450 -0.0276 -0.0225 -0.0502 15 1000.0000 500.0000 1500 -0.0285 -0.0233 -0.0518 15.5 1033.3333 516.6667 1550 -0.0293 -0.0240 -0.0533 16 1066.6667 533.3333 1600 -0.0302 -0.0247 -0.0549 16.5 1100.0000 550.0000 1650 -0.0310 -0.0254 -0.0564 17 1133.3333 566.6667 1700 -0.0318 -0.0261 -0.0579 17.5 1166.6667 583.3333 1750 -0.0326 -0.0268 -0.0594 18 1200.0000 600.0000 1800 -0.0334 -0.0275 -0.0609 18.5 1233.3333 616.6667 1850 -0.0342 -0.0282 -0.0624 19 1266.6667 633.3333 1900 -0.0350 -0.0289 -0.0638 19.5 1300.0000 650.0000 1950 -0.0357 -0.0296 -0.0653 20 1333.3333 666.6667 2000 -0.0364 -0.0302 -0.0667 20.5 1366.6667 683.3333 2050 -0.0372 -0.0309 -0.0681 21 1400.0000 700.0000 2100 -0.0379 -0.0315 -0.0694 21.5 1433.3333 716.6667 2150 -0.0386 -0.0322 -0.0708 22 1466.6667 733.3333 2200 -0.0393 -0.0328 -0.0721 22.5 1500.0000 750.0000 2250 -0.0399 -0.0335 -0.0734 23 1533.3333 766.6667 2300 -0.0406 -0.0341 -0.0747 23.5 1566.6667 783.3333 2350 -0.0412 -0.0347 -0.0759 24 1600.0000 800.0000 2400 -0.0419 -0.0353 -0.0772 24.5 1633.3333 816.6667 2450 -0.0425 -0.0359 -0.0784 25 1666.6667 833.3333 2500 -0.0431 -0.0365 -0.0796 25.5 1700.0000 850.0000 2550 -0.0436 -0.0371 -0.0807 26 1733.3333 866.6667 2600 -0.0442 -0.0377 -0.0819 26.5 1766.6667 883.3333 2650 -0.0447 -0.0383 -0.0830 27 1800.0000 900.0000 2700 -0.0453 -0.0388 -0.0841 27.5 1833.3333 916.6667 2750 -0.0458 -0.0394 -0.0851 28 1866.6667 933.3333 2800 -0.0462 -0.0399 -0.0862 28.5 1900.0000 950.0000 2850 -0.0467 -0.0405 -0.0872 29 1933.3333 966.6667 2900 -0.0472 -0.0410 -0.0881 29.5 1966.6667 983.3333 2950 -0.0476 -0.0415 -0.0891 30 2000.0000 1000.0000 3000 -0.0480 -0.0420 -0.0900 30.5 1983.3333 1016.6667 3000 -0.0484 -0.0425 -0.0909 31 1966.6667 1033.3333 3000 -0.0488 -0.0430 -0.0917 31.5 1950.0000 1050.0000 3000 -0.0491 -0.0435 -0.0926 32 1933.3333 1066.6667 3000 -0.0494 -0.0439 -0.0934 32.5 1916.6667 1083.3333 3000 -0.0498 -0.0444 -0.0941 33 1900.0000 1100.0000 3000 -0.0500 -0.0448 -0.0949 33.5 1883.3333 1116.6667 3000 -0.0503 -0.0452 -0.0956 34 1866.6667 1133.3333 3000 -0.0506 -0.0457 -0.0962 34.5 1850.0000 1150.0000 3000 -0.0508 -0.0461 -0.0969 35 1833.3333 1166.6667 3000 -0.0510 -0.0465 -0.0975 35.5 1816.6667 1183.3333 3000 -0.0512 -0.0469 -0.0981 36 1800.0000 1200.0000 3000 -0.0514 -0.0472 -0.0986 36.5 1783.3333 1216.6667 3000 -0.0516 -0.0476 -0.0992 37 1766.6667 1233.3333 3000 -0.0517 -0.0479 -0.0997 37.5 1750.0000 1250.0000 3000 -0.0518 -0.0483 -0.1001 38 1733.3333 1266.6667 3000 -0.0520 -0.0486 -0.1006 38.5 1716.6667 1283.3333 3000 -0.0520 -0.0489 -0.1010 39 1700.0000 1300.0000 3000 -0.0521 -0.0492 -0.1013 39.5 1683.3333 1316.6667 3000 -0.0522 -0.0495 -0.1017 40 1666.6667 1333.3333 3000 -0.0522 -0.0498 -0.1020 40.5 1650.0000 1350.0000 3000 -0.0522 -0.0500 -0.1023 41 1633.3333 1366.6667 3000 -0.0523 -0.0503 -0.1025 41.5 1616.6667 1383.3333 3000 -0.0522 -0.0505 -0.1028 42 1600.0000 1400.0000 3000 -0.0522 -0.0507 -0.1030 42.5 1583.3333 1416.6667 3000 -0.0522 -0.0509 -0.1031 43 1566.6667 1433.3333 3000 -0.0521 -0.0511 -0.1033 43.5 1550.0000 1450.0000 3000 -0.0521 -0.0513 -0.1034 44 1533.3333 1466.6667 3000 -0.0520 -0.0515 -0.1034 44.5 1516.6667 1483.3333 3000 -0.0519 -0.0516 -0.1035 45 1500.0000 1500.0000 3000 -0.0518 -0.0518 -0.1035 45.5 1483.3333 1516.6667 3000 -0.0516 -0.0519 -0.1035 46 1466.6667 1533.3333 3000 -0.0515 -0.0520 -0.1034 46.5 1450.0000 1550.0000 3000 -0.0513 -0.0521 -0.1034 47 1433.3333 1566.6667 3000 -0.0511 -0.0521 -0.1033 47.5 1416.6667 1583.3333 3000 -0.0509 -0.0522 -0.1031 48 1400.0000 1600.0000 3000 -0.0507 -0.0522 -0.1030 48.5 1383.3333 1616.6667 3000 -0.0505 -0.0522 -0.1028 49 1366.6667 1633.3333 3000 -0.0503 -0.0523 -0.1025 49.5 1350.0000 1650.0000 3000 -0.0500 -0.0522 -0.1023 50 1333.3333 1666.6667 3000 -0.0498 -0.0522 -0.1020 50.5 1316.6667 1683.3333 3000 -0.0495 -0.0522 -0.1017 51 1300.0000 1700.0000 3000 -0.0492 -0.0521 -0.1013 51.5 1283.3333 1716.6667 3000 -0.0489 -0.0520 -0.1010 52 1266.6667 1733.3333 3000 -0.0486 -0.0520 -0.1006 52.5 1250.0000 1750.0000 3000 -0.0483 -0.0518 -0.1001 53 1233.3333 1766.6667 3000 -0.0479 -0.0517 -0.0997 53.5 1216.6667 1783.3333 3000 -0.0476 -0.0516 -0.0992 54 1200.0000 1800.0000 3000 -0.0472 -0.0514 -0.0986 54.5 1183.3333 1816.6667 3000 -0.0469 -0.0512 -0.0981 55 1166.6667 1833.3333 3000 -0.0465 -0.0510 -0.0975 55.5 1150.0000 1850.0000 3000 -0.0461 -0.0508 -0.0969 56 1133.3333 1866.6667 3000 -0.0457 -0.0506 -0.0962 56.5 1116.6667 1883.3333 3000 -0.0452 -0.0503 -0.0956 57 1100.0000 1900.0000 3000 -0.0448 -0.0500 -0.0949 57.5 1083.3333 1916.6667 3000 -0.0444 -0.0498 -0.0941 58 1066.6667 1933.3333 3000 -0.0439 -0.0494 -0.0934 58.5 1050.0000 1950.0000 3000 -0.0435 -0.0491 -0.0926 59 1033.3333 1966.6667 3000 -0.0430 -0.0488 -0.0917 59.5 1016.6667 1983.3333 3000 -0.0425 -0.0484 -0.0909 60 1000.0000 2000.0000 3000 -0.0420 -0.0480 -0.0900 60.5 983.3333 1966.6667 2950 -0.0415 -0.0476 -0.0891 61 966.6667 1933.3333 2900 -0.0410 -0.0472 -0.0881 61.5 950.0000 1900.0000 2850 -0.0405 -0.0467 -0.0872 62 933.3333 1866.6667 2800 -0.0399 -0.0462 -0.0862 62.5 916.6667 1833.3333 2750 -0.0394 -0.0458 -0.0851 63 900.0000 1800.0000 2700 -0.0388 -0.0453 -0.0841 63.5 883.3333 1766.6667 2650 -0.0383 -0.0447 -0.0830 64 866.6667 1733.3333 2600 -0.0377 -0.0442 -0.0819 64.5 850.0000 1700.0000 2550 -0.0371 -0.0436 -0.0807 65 833.3333 1666.6667 2500 -0.0365 -0.0431 -0.0796 65.5 816.6667 1633.3333 2450 -0.0359 -0.0425 -0.0784 66 800.0000 1600.0000 2400 -0.0353 -0.0419 -0.0772 66.5 783.3333 1566.6667 2350 -0.0347 -0.0412 -0.0759 67 766.6667 1533.3333 2300 -0.0341 -0.0406 -0.0747 67.5 750.0000 1500.0000 2250 -0.0335 -0.0399 -0.0734 68 733.3333 1466.6667 2200 -0.0328 -0.0393 -0.0721 68.5 716.6667 1433.3333 2150 -0.0322 -0.0386 -0.0708 69 700.0000 1400.0000 2100 -0.0315 -0.0379 -0.0694 69.5 683.3333 1366.6667 2050 -0.0309 -0.0372 -0.0681 70 666.6667 1333.3333 2000 -0.0302 -0.0364 -0.0667 70.5 650.0000 1300.0000 1950 -0.0296 -0.0357 -0.0653 71 633.3333 1266.6667 1900 -0.0289 -0.0350 -0.0638 71.5 616.6667 1233.3333 1850 -0.0282 -0.0342 -0.0624 72 600.0000 1200.0000 1800 -0.0275 -0.0334 -0.0609 72.5 583.3333 1166.6667 1750 -0.0268 -0.0326 -0.0594 73 566.6667 1133.3333 1700 -0.0261 -0.0318 -0.0579 73.5 550.0000 1100.0000 1650 -0.0254 -0.0310 -0.0564 74 533.3333 1066.6667 1600 -0.0247 -0.0302 -0.0549 74.5 516.6667 1033.3333 1550 -0.0240 -0.0293 -0.0533 75 500.0000 1000.0000 1500 -0.0233 -0.0285 -0.0518 75.5 483.3333 966.6667 1450 -0.0225 -0.0276 -0.0502 76 466.6667 933.3333 1400 -0.0218 -0.0268 -0.0486 76.5 450.0000 900.0000 1350 -0.0211 -0.0259 -0.0470 77 433.3333 866.6667 1300 -0.0203 -0.0250 -0.0453 77.5 416.6667 833.3333 1250 -0.0196 -0.0241 -0.0437 78 400.0000 800.0000 1200 -0.0188 -0.0232 -0.0420 78.5 383.3333 766.6667 1150 -0.0181 -0.0223 -0.0404 79 366.6667 733.3333 1100 -0.0173 -0.0214 -0.0387 79.5 350.0000 700.0000 1050 -0.0165 -0.0205 -0.0370 80 333.3333 666.6667 1000 -0.0158 -0.0196 -0.0353 80.5 316.6667 633.3333 950 -0.0150 -0.0186 -0.0336 81 300.0000 600.0000 900 -0.0142 -0.0177 -0.0319 81.5 283.3333 566.6667 850 -0.0135 -0.0167 -0.0302 82 266.6667 533.3333 800 -0.0127 -0.0158 -0.0285 82.5 250.0000 500.0000 750 -0.0119 -0.0148 -0.0267 83 233.3333 466.6667 700 -0.0111 -0.0138 -0.0250 83.5 216.6667 433.3333 650 -0.0103 -0.0129 -0.0232 84 200.0000 400.0000 600 -0.0096 -0.0119 -0.0215 84.5 183.3333 366.6667 550 -0.0088 -0.0109 -0.0197 85 166.6667 333.3333 500 -0.0080 -0.0099 -0.0179 85.5 150.0000 300.0000 450 -0.0072 -0.0090 -0.0161 86 133.3333 266.6667 400 -0.0064 -0.0080 -0.0144 86.5 116.6667 233.3333 350 -0.0056 -0.0070 -0.0126 87 100.0000 200.0000 300 -0.0048 -0.0060 -0.0108 87.5 83.3333 166.6667 250 -0.0040 -0.0050 -0.0090 88 66.6667 133.3333 200 -0.0032 -0.0040 -0.0072 88.5 50.0000 100.0000 150 -0.0024 -0.0030 -0.0054 89 33.3333 66.6667 100 -0.0016 -0.0020 -0.0036 89.5 16.6667 33.3333 50 -0.0008 -0.0010 -0.0018 90 0.0000 0.0000 0 0.0000 0.0000 0.0000

Table 3a: Calculated theoretical moment, displacement, and bending stress along the beam length

 x (in) LC1 bending stress (psi) LC2 bending stress (psi) LC3 bending stress (psi) 0 0 0 0 0.5 20 10 30 1 40 20 60 1.5 60 30 90 2 80 40 120 2.5 100 50 150 3 120 60 180 3.5 140 70 210 4 160 80 240 4.5 180 90 270 5 200 100 300 5.5 220 110 330 6 240 120 360 6.5 260 130 390 7 280 140 420 7.5 300 150 450 8 320 160 480 8.5 340 170 510 9 360 180 540 9.5 380 190 570 10 400 200 600 10.5 420 210 630 11 440 220 660 11.5 460 230 690 12 480 240 720 12.5 500 250 750 13 520 260 780 13.5 540 270 810 14 560 280 840 14.5 580 290 870 15 600 300 900 15.5 620 310 930 16 640 320 960 16.5 660 330 990 17 680 340 1020 17.5 700 350 1050 18 720 360 1080 18.5 740 370 1110 19 760 380 1140 19.5 780 390 1170 20 800 400 1200 20.5 820 410 1230 21 840 420 1260 21.5 860 430 1290 22 880 440 1320 22.5 900 450 1350 23 920 460 1380 23.5 940 470 1410 24 960 480 1440 24.5 980 490 1470 25 1000 500 1500 25.5 1020 510 1530 26 1040 520 1560 26.5 1060 530 1590 27 1080 540 1620 27.5 1100 550 1650 28 1120 560 1680 28.5 1140 570 1710 29 1160 580 1740 29.5 1180 590 1770 30 1200 600 1800 30.5 1190 610 1800 31 1180 620 1800 31.5 1170 630 1800 32 1160 640 1800 32.5 1150 650 1800 33 1140 660 1800 33.5 1130 670 1800 34 1120 680 1800 34.5 1110 690 1800 35 1100 700 1800 35.5 1090 710 1800 36 1080 720 1800 36.5 1070 730 1800 37 1060 740 1800 37.5 1050 750 1800 38 1040 760 1800 38.5 1030 770 1800 39 1020 780 1800 39.5 1010 790 1800 40 1000 800 1800 40.5 990 810 1800 41 980 820 1800 41.5 970 830 1800 42 960 840 1800 42.5 950 850 1800 43 940 860 1800 43.5 930 870 1800 44 920 880 1800 44.5 910 890 1800 45 900 900 1800 45.5 890 910 1800 46 880 920 1800 46.5 870 930 1800 47 860 940 1800 47.5 850 950 1800 48 840 960 1800 48.5 830 970 1800 49 820 980 1800 49.5 810 990 1800 50 800 1000 1800 50.5 790 1010 1800 51 780 1020 1800 51.5 770 1030 1800 52 760 1040 1800 52.5 750 1050 1800 53 740 1060 1800 53.5 730 1070 1800 54 720 1080 1800 54.5 710 1090 1800 55 700 1100 1800 55.5 690 1110 1800 56 680 1120 1800 56.5 670 1130 1800 57 660 1140 1800 57.5 650 1150 1800 58 640 1160 1800 58.5 630 1170 1800 59 620 1180 1800 59.5 610 1190 1800 60 600 1200 1800 60.5 590 1180 1770 61 580 1160 1740 61.5 570 1140 1710 62 560 1120 1680 62.5 550 1100 1650 63 540 1080 1620 63.5 530 1060 1590 64 520 1040 1560 64.5 510 1020 1530 65 500 1000 1500 65.5 490 980 1470 66 480 960 1440 66.5 470 940 1410 67 460 920 1380 67.5 450 900 1350 68 440 880 1320 68.5 430 860 1290 69 420 840 1260 69.5 410 820 1230 70 400 800 1200 70.5 390 780 1170 71 380 760 1140 71.5 370 740 1110 72 360 720 1080 72.5 350 700 1050 73 340 680 1020 73.5 330 660 990 74 320 640 960 74.5 310 620 930 75 300 600 900 75.5 290 580 870 76 280 560 840 76.5 270 540 810 77 260 520 780 77.5 250 500 750 78 240 480 720 78.5 230 460 690 79 220 440 660 79.5 210 420 630 80 200 400 600 80.5 190 380 570 81 180 360 540 81.5 170 340 510 82 160 320 480 82.5 150 300 450 83 140 280 420 83.5 130 260 390 84 120 240 360 84.5 110 220 330 85 100 200 300 85.5 90 180 270 86 80 160 240 86.5 70 140 210 87 60 120 180 87.5 50 100 150 88 40 80 120 88.5 30 60 90 89 20 40 60 89.5 10 20 30 90 0 0 0

Table 3b: Continued theoretical calculations

 X (in) 100lb at P1 (%) 100lb at P2 (%) 100lb at P1/P2 (%) 5 10.61 6.62 8.84 25 5.68 9.51 6.81 45 4.35 6.28 4.83 65 4.03 6.84 5.55 85 5.92 4.47 2.33

Table 4: Error analysis

Figure 1: Load case 1 moment diagram

Figure 2: Load case 2 moment diagram

Figure 3: Load case 3 moment diagram

Figure 4: Load case 1 displacement

Figure 5: Load case 2 displacement

Figure 6: Load case 3 displacement

Figure 7: Super position proof

Discussion:

Our theoretical values very closely matched the data gathered during the experiment and were within the margin of error that comes from the theoretical calculations not factoring in the deformation caused by shear which was included in the measurements by the dial indicators. The differences between the calculated theoretical displacement was only off by a few thousandths of an inch at any given point where a dial indicator was located, which is to be expected. The displacement of the beam in the vertical direction is mostly caused by flexural deformation, but shear deformation factors into that as well, albeit a very small amount. This is why the displacement measured by the dial indicator is slightly larger than the values that were calculated. This accounts for most of the error found between the two values. Aside from this error, the experimental and theoretical data line up very closely and follow nearly identical trends when graphed against each other, which can be seen in figures 4-6. Another source of error is the high likelihood that the dial indicators were not perfectly perpendicular to the beam when placed. Once zeroed on the beam, they measure the distance changed on only one axis which would be the vertical axis in this case. If they were not perfectly perpendicular, they would be measuring the hypotenuse of a triangle rather than the vertical distance traveled. All of the length measurements and positioning of the loads as well as dial indicators were placed using a tape measure, which adds yet another source of error. During the experiment, point load 2 shifted slightly between runs two and three as the weights were loaded and unloaded. The weights used not all being exactly 20 pounds may also add more error to the calculations.

Comparing load cases one and two, they display essentially equal and opposite deformation reactions in the beam. To prove the theory of superposition, the deformation reactions from case one and two were added together and should be equal to case three since case three was the loads from cases one and two applied at the same time. This is proven to be true as evidenced by the graph in Figure 7 where the sum of cases one and two overlap the line of case three. The moment diagrams for all three cases are also related and the visual representations make sense since the internal moment should peak where the single loads are applied and when both loads are applied in case three, the moment between the loads should stay constant like it does.

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