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Cive 302 solid mechanics lab report

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

Load case 1 moment diagram

Figure 1: Load case 1 moment diagram

Load case 2 moment diagram

Figure 2: Load case 2 moment diagram

Load case 3 moment diagram

Figure 3: Load case 3 moment diagram

Load case 1 displacement

Figure 4: Load case 1 displacement

Load case 2 displacement

Figure 5: Load case 2 displacement

Load case 3 displacement

Figure 6: Load case 3 displacement

Super position proof

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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