FORME ∆H

R=150

Demi Cercle R=150 2PR3/EI

R=75

L=150

DIMENSION (mm)

Quart de cercle

PR3/EI

Boussoirs courbe

PR(L+R)2/2EI

Boussoirs anguleux

L1=150 [PL1L2(0,707L1+L2)/2+PL22/6]/EIL2=105

∆V

∏PR3/4EI

∏PR3/2EI

PR2(4L+R)/4EI

PL22(3L1+L2)/6EI

Boussoir courbé (mm) Boussoir anguleux (mm)

Masse (g) Charge (N) ∆V(mm) ∆H(mm) ∆V(mm)Experim. Theorique Experim. Theorique Experim. Theorique

0 0 0 0 0 0 0 0100 1 0.24 0.066 0.39 0.132 0.12 0.071200 2 0.56 0.132 0.91 0.264 0.4 0.142300 3 0.9 0.198 1.5 0.396 0.92 0.212400 4 1.35 0.264 2.25 0.528 1.15 0.284500 5 1.64 0.33 2.79 0.66 1.47 0.355

Demi cercle (mm) Quart de cercle (mm)

Masse (g) Charge (N) ∆V(mm) ∆H(mm) ∆V(mm)Experim. Theorique Experim. Theorique Experim. Theorique

0 0 0 0 0 0 0 0100 1 0.58 0.37 0.22 0.47 0.52 0.18200 2 2.68 0.74 3.29 0.94 1.36 0.36300 3 3.32 1.11 3.43 1.41 2.17 0.54400 4 4.43 1.48 4.59 1.88 3.19 0.72500 5 6.26 1.85 7.19 0.895 3.67 0.9

0 0.12 0.4 0.92 1.15 1.470

2

4

6

8

10

12

Boussoir anguleux (∆V)

∆V (Exp.)∆V (Th.)

∆V

Charge

0 0.12 0.4 0.92 1.15 1.470

2

4

6

8

10

12

Boussoir anguleux (∆V)

∆V (Exp.)∆V (Th.)

∆V

Charge

0 0.37 0.74 1.11 1.48 1.850

2

4

6

8

10

12

Demi cercle (∆V)

∆V (Th.)∆V (Exp.)

∆V

Charge

0 0.18 0.36 0.54 0.72 0.90

2

4

6

8

10

12Quart de cercle (∆V)

∆V (Th.)∆V (Exp)

∆V

Charge

0 0.24 0.56 0.9 1.35 1.640

2

4

6

8

10

12

Boussoir courbé (∆V)

∆V (Exp.)∆V (Th.)

∆V

Charge

0 0.39 0.91 1.5 2.25 2.790

2

4

6

8

10

12

Boussoir courbé (∆H)

∆H (Exp.)∆H (Th.)

∆V

Charge

Boussoir anguleux (mm)

∆H(mm)Experim. Theorique

0 00.16 0.1150.66 0.231.62 0.3451.98 0.462.53 0.6

Quart de cercle (mm)

∆H(mm)Experim. Theorique

0 00.31 0.120.84 0.241.36 0.361.99 0.482.3 0.6

0 0.12 0.4 0.92 1.15 1.470

2

4

6

8

10

12

Boussoir anguleux (∆V)

∆V (Exp.)∆V (Th.)

∆V

Charge

0 0.115 0.23 0.345 0.46 0.60

2

4

6

8

10

12

Boussoir anguleux (∆H)

∆H (Th.)∆H (Exp.)

∆H

Charge

0 0.47 0.94 1.41 1.88 0.8950

2

4

6

8

10

12Demi cercle (∆H)

∆H (Th.)∆H (Exp.)

∆H

Charge

0 0.12 0.4 0.92 1.15 1.470

2

4

6

8

10

12

Boussoir anguleux (∆V)

∆V (Exp.)∆V (Th.)

∆V

Charge

0 0.37 0.74 1.11 1.48 1.850

2

4

6

8

10

12

Demi cercle (∆V)

∆V (Th.)∆V (Exp.)

∆V

Charge

0 0.47 0.94 1.41 1.88 0.8950

2

4

6

8

10

12Demi cercle (∆H)

∆H (Th.)∆H (Exp.)

∆H

Charge

0 0.18 0.36 0.54 0.72 0.90

2

4

6

8

10

12Quart de cercle (∆V)

∆V (Th.)∆V (Exp)

∆V

Charge

0 0.12 0.24 0.36 0.48 0.60

2

4

6

8

10

12Quart de cercle (∆H)

∆H (Th.)∆H (Exp.)

∆H

Charge

0 0.24 0.56 0.9 1.35 1.640

2

4

6

8

10

12

Boussoir courbé (∆V)

∆V (Exp.)∆V (Th.)

∆V

Charge

0 0.39 0.91 1.5 2.25 2.790

2

4

6

8

10

12

Boussoir courbé (∆H)

∆H (Exp.)∆H (Th.)

∆V

Charge

0 0.115 0.23 0.345 0.46 0.60

2

4

6

8

10

12

Boussoir anguleux (∆H)

∆H (Th.)∆H (Exp.)

∆H

Charge

0 0.47 0.94 1.41 1.88 0.8950

2

4

6

8

10

12Demi cercle (∆H)

∆H (Th.)∆H (Exp.)

∆H

Charge

0 0.47 0.94 1.41 1.88 0.8950

2

4

6

8

10

12Demi cercle (∆H)

∆H (Th.)∆H (Exp.)

∆H

Charge

0 0.12 0.24 0.36 0.48 0.60

2

4

6

8

10

12Quart de cercle (∆H)

∆H (Th.)∆H (Exp.)

∆H

Charge

F (KN)

1.2 10.6100795756 0.002

2.1 18.5676392573 0.004

2 17.6834659593 0.003

4.3 38.0194518126 0.0154.4 38.9036251105 0.0145.2 45.9770114943 0.0175.6 49.5137046861 0.0186.6 58.3554376658 0.0257.4 65.4288240495 0.0278.5 75.1547303271 0.0368.7 76.9230769231 0.0399.9 87.5331564987 0.045

11.4 100.795755968 0.05612.5 110.521662246 0.06

A 13.9 122.900088417 0.071

14 123.784261715 0.17314.5 128.205128205 0.26114 123.784261715 0.276

14.2 125.552608311 0.3

14.4 127.320954907 0.42614.5 128.205128205 0.53814.6 129.089301503 0.65714.6 129.089301503 0.72514.6 129.089301503 0.83115.3 135.278514589 0.98314.7 129.973474801 1.14915.1 133.510167993 1.26515.1 133.510167993 1.44815.1 133.510167993 1.57615.6 137.931034483 1.83115.4 136.162687887 2.00215.4 136.162687887 2.098

B 16.3 144.120247569 2.61

σ = F/So (MPA) Allongement ΔL (mm)

15.1 133.510167993 3.16516.1 142.351900973 3.32513.2 116.710875332 4.393

0.0050.01

0.0075

0.03750.035

0.04250.045

0.0625

0.06750.09

0.0975

0.11250.14

0.15

0.1775

0.4325

0.65250.69

0.751.065

1.345

1.6425

1.8125

2.0775

2.4575

2.8725

3.16253.62

3.94

4.57755.005

5.2456.525

7.9125

8.3125

10.98250

20

40

60

80

100

120

140

160

Courbe σ-ε

ε%

σ (MPA)

A

B

C

0.00177342654914178

0.00431567406760243

0.00650380433866607

0.00687630393943206

0.00747201483870106

0.0105936882108699

0.0133603517018364

0.0162915687768858

0.0179627003741725

0.0205621427128776

0.0242778924595235

0.0283201713772869

0.0311352289307113

0.0355601753985509

0.0386436235922045

0.0447582373417735

0.0488377820833001

0.0511207795077142

0.0632095133978844110

115

120

125

130

135

140

145

150

Courbe rationnelle

ε*

σ (MPA)

A

B

0.00177342654914178

0.00431567406760243

0.00650380433866607

0.00687630393943206

0.00747201483870106

0.0105936882108699

0.0133603517018364

0.0162915687768858

0.0179627003741725

0.0205621427128776

0.0242778924595235

0.0283201713772869

0.0311352289307113

0.0355601753985509

0.0386436235922045

0.0447582373417735

0.0488377820833001

0.0511207795077142

0.0632095133978844110

115

120

125

130

135

140

145

150

Courbe rationnelle

ε*

σ (MPA)

A

B

F (KN)

0.005 A 13.9 0.071

0.01 14 0.173

0.0075 14.5 0.261

0.0375 14 0.2760.035 14.2 0.3

0.0425 14.4 0.4260.045 14.5 0.538

0.0625 14.6 0.6570.0675 14.6 0.725

0.09 14.6 0.8310.0975 15.3 0.9830.1125 14.7 1.149

0.14 15.1 1.2650.15 15.1 1.448

0.1775 15.1 1.576

0.4325 15.6 1.8310.6525 15.4 2.002

0.69 15.4 2.098

0.75 B 16.3 2.61

1.0651.345

1.64251.81252.07752.45752.87253.1625

3.623.94

4.57755.0055.245

6.525

Allongement relatif ε %

Allongement ΔL (mm)

7.91258.3125

10.9825

0.0050.01

0.0075

0.03750.035

0.04250.045

0.0625

0.06750.09

0.0975

0.11250.14

0.15

0.1775

0.4325

0.65250.69

0.751.065

1.345

1.6425

1.8125

2.0775

2.4575

2.8725

3.16253.62

3.94

4.57755.005

5.2456.525

7.9125

8.3125

10.98250

20

40

60

80

100

120

140

160

Courbe σ-ε

ε%

σ (MPA)

A

B

C

0.00177342654914178

0.00431567406760243

0.00650380433866607

0.00687630393943206

0.00747201483870106

0.0105936882108699

0.0133603517018364

0.0162915687768858

0.0179627003741725

0.0205621427128776

0.0242778924595235

0.0283201713772869

0.0311352289307113

0.0355601753985509

0.0386436235922045

0.0447582373417735

0.0488377820833001

0.0511207795077142

0.0632095133978844110

115

120

125

130

135

140

145

150

Courbe rationnelle

ε*

σ (MPA)

A

B

0.00177342654914178

0.00431567406760243

0.00650380433866607

0.00687630393943206

0.00747201483870106

0.0105936882108699

0.0133603517018364

0.0162915687768858

0.0179627003741725

0.0205621427128776

0.0242778924595235

0.0283201713772869

0.0311352289307113

0.0355601753985509

0.0386436235922045

0.0447582373417735

0.0488377820833001

0.0511207795077142

0.0632095133978844110

115

120

125

130

135

140

145

150

Courbe rationnelle

ε*

σ (MPA)

A

B

0.001773426549 122.90008841733

0.004315674068 123.7842617153

0.006503804339 128.20512820513

0.006876303939 123.78426171530.007472014839 125.552608311230.010593688211 127.320954907160.013360351702 128.205128205130.016291568777 129.089301503090.017962700374 129.089301503090.020562142713 129.089301503090.02427789246 135.27851458886

0.028320171377 129.973474801060.031135228931 133.510167992930.035560175399 133.51016799293

0.038643623592 133.51016799293

0.044758237342 137.931034482760.048837782083 136.162687886830.051120779508 136.16268788683

0.063209513398 144.12024756852

Déformation rationnelle ε* σ = F/So (MPA)

0.0050.01

0.0075

0.03750.035

0.04250.045

0.0625

0.06750.09

0.0975

0.11250.14

0.15

0.1775

0.4325

0.65250.69

0.751.065

1.345

1.6425

1.8125

2.0775

2.4575

2.8725

3.16253.62

3.94

4.57755.005

5.2456.525

7.9125

8.3125

10.98250

20

40

60

80

100

120

140

160

Courbe σ-ε

ε%

σ (MPA)

A

B

C

0.00177342654914178

0.00431567406760243

0.00650380433866607

0.00687630393943206

0.00747201483870106

0.0105936882108699

0.0133603517018364

0.0162915687768858

0.0179627003741725

0.0205621427128776

0.0242778924595235

0.0283201713772869

0.0311352289307113

0.0355601753985509

0.0386436235922045

0.0447582373417735

0.0488377820833001

0.0511207795077142

0.0632095133978844110

115

120

125

130

135

140

145

150

Courbe rationnelle

ε*

σ (MPA)

A

B

0.00177342654914178

0.00431567406760243

0.00650380433866607

0.00687630393943206

0.00747201483870106

0.0105936882108699

0.0133603517018364

0.0162915687768858

0.0179627003741725

0.0205621427128776

0.0242778924595235

0.0283201713772869

0.0311352289307113

0.0355601753985509

0.0386436235922045

0.0447582373417735

0.0488377820833001

0.0511207795077142

0.0632095133978844110

115

120

125

130

135

140

145

150

Courbe rationnelle

ε*

σ (MPA)

A

B

σ = F/So (MPA) ln(σ) ln(ε*)

0.001773426549 122.90008841733 4.8113717 -6.33484170.004315674068 123.7842617153 4.8185402 -5.445501750.006503804339 128.20512820513 4.8536315 -5.035367990.006876303939 123.7842617153 4.8185402 -4.979673990.007472014839 125.55260831123 4.8327249 -4.896590590.010593688211 127.32095490716 4.8467111 -4.547496910.013360351702 128.20512820513 4.8536315 -4.315463790.016291568777 129.08930150309 4.8605044 -4.117107560.017962700374 129.08930150309 4.8605044 -4.019457870.020562142713 129.08930150309 4.8605044 -3.884303630.02427789246 135.27851458886 4.9073357 -3.71818912

0.028320171377 129.97347480106 4.8673304 -3.564180960.031135228931 133.51016799293 4.8941776 -3.469415340.035560175399 133.51016799293 4.8941776 -3.336528940.038643623592 133.51016799293 4.8941776 -3.25337350.044758237342 137.93103448276 4.9267538 -3.106479780.048837782083 136.16268788683 4.9138504 -3.019251040.051120779508 136.16268788683 4.9138504 -2.973564220.063209513398 144.12024756852 4.970648 -2.76130046

Déformation rationnelle ε*

-6.33-5.45

-5.04-4.98

-4.90-4.55

-4.32-4.12

-4.02-3.88

-3.72-3.56

-3.47-3.34

-3.25-3.11

-3.02-2.97

-2.764.7

4.75

4.8

4.85

4.9

4.95

5

f(x) = 0.00692286479534031 x + 4.80440115054378

ln(σ)=f(ln(ε*))

ln(ε*)

ln(σ)

-6.33-5.45

-5.04-4.98

-4.90-4.55

-4.32-4.12

-4.02-3.88

-3.72-3.56

-3.47-3.34

-3.25-3.11

-3.02-2.97

-2.764.7

4.75

4.8

4.85

4.9

4.95

5

f(x) = 0.00692286479534031 x + 4.80440115054378

ln(σ)=f(ln(ε*))

ln(ε*)

ln(σ)

Matière ACIER

Diamètre initial a (mm) 4.1

5LDC : Lecture du comparateur

Diff. : Différence par rapport à la mesure précédente

0° 15° 30° 45°

LDC Diff. LDC Diff. LDC Diff. LDC

0 51 56 48 45

4 91 40 103 47 98 50 96

8 138 47 155 52 155 57 151

12 187 49 210 55 214 59 209

16 229 42 264 54 271 57 270

20 273 44 350 86 344 73 350

24 324 51 386 36 430 86 398

28 394 70 429 43 502 72 493

32 481 87 562 133 703 201 570

36 520 39 639 77 802 99 694

40 536 16 924 122 778

44 544 8 1374 450

48 554 10 1352 -22

50 560 6

0° 15° 30°LDC M T LDC M T LDC

0 51 0 5.6 56 1.58 5.89 48

Tmax

Mmax

Poids du bras de chargement (N)

Charge P (N)

Charge P (N)

0 51 0 5.6 56 1.58 5.89 484 91 0 9.6 103 2.79 10.43 988 138 0 14.3 155 4.14 15.46 155

12 187 0 19.2 210 5.56 20.77 21416 229 0 23.4 264 6.96 25.98 27120 273 0 27.8 350 9.18 34.29 34424 324 0 32.9 386 10.11 37.77 43028 394 0 39.9 429 11.23 41.92 50232 481 0 48.6 562 14.67 54.77 70336 520 0 52.5 639 16.66 62.21 80240 536 0 54.1 92444 544 0 54.9 137448 554 0 55.9 135250 560 0 56.5

Tmax

Mmax

LDC : Lecture du comparateur

Diff. : Différence par rapport à la mesure précédente

45° 60° 75° 90°

Diff. LDC Diff. LDC Diff. LDC Diff.

56 46 49

51 93 37 82 36 83 34

55 136 43 122 40 123 40

58 183 47 157 35 165 42

61 227 44 200 43 203 38

80 271 44 247 47 243 40

48 325 54 290 43 286 43

95 379 54 340 50 330 44

77 440 61 382 42 369 39

124 466 26 394 12 378 9

84 486 20 409 15 388 10

515 29 419 10 396 8

556 41 433 14 405 9

571 15 439 6 409 4

30° 45° 60°M T LDC M T LDC M

2.65 4.59 45 3.53 3.53 56 5.28

Tmax

Mmax

2.65 4.59 45 3.53 3.53 56 5.285.15 8.92 96 7.14 7.14 93 8.488.00 13.86 151 11.03 11.03 136 12.21

10.94 18.97 209 15.13 15.13 183 16.2813.79 23.91 270 19.44 19.44 227 20.0917.44 30.23 350 25.09 25.09 271 23.8921.74 37.68 398 28.49 28.49 325 28.5725.34 43.91 493 35.20 35.20 379 33.2535.38 61.32 570 40.64 40.64 440 38.5340.33 69.90 694 49.41 49.41 466 40.7846.43 80.47 778 55.34 55.34 486 42.5168.92 119.44 515 45.0267.82 117.54 556 48.57

571 49.87

119.44

68.92

max

max

60° 75° 90°T LDC M T LDC M T

3.05 46 4.93 1.32 49 5.4 0

3.05 46 4.93 1.32 49 5.4 04.90 82 8.40 2.26 83 8.8 07.06 122 12.27 3.30 123 12.8 09.41 157 15.65 4.20 165 17 0

11.61 200 19.80 5.32 203 20.8 013.81 247 24.34 6.54 243 24.8 016.52 290 28.49 7.65 286 29.1 019.22 340 33.32 8.95 330 33.5 022.27 382 37.37 10.04 369 37.4 023.57 394 38.53 10.35 378 38.3 024.57 409 39.98 10.74 388 39.3 026.02 419 40.95 11.00 396 40.1 028.08 433 42.30 11.36 405 41 028.83 439 42.88 11.52 409 41.4 0

Matière ACIER

Diamètre initial a (mm) 4.1

5LDC : Lecture du comparateur

Diff. : Différence par rapport à la mesure précédente

0° 15° 30° 45°

LDC Diff. LDC Diff. LDC Diff. LDC

0 54 68 18 204 110 56 69 1 66 48 488 184 74 86 17 116 50 59

12 250 66 109 23 126 10 8016 325 75 159 50 170 44 10920 425 100 210 51 200 30 12524 500 75 307 97 242 42 15328 525 25 334 27 286 44 17532 550 25 387 53 330 44 23336 602 52 440 53 394 64 27840 670 68 474 34 434 40 32144 729 59 526 52 515 81 34148 744 15 609 83 626 111 37250 744 0 611 2 665 39 389

0° 15° 30°LDC M T LDC M T LDC

0 54 0 5.9 68 1.89 7.05 18

4 110 0 11.5 69 1.91 7.15 66

Tmax

Mmax

Poids du bras de chargement (N)

Charge P (N)

Charge P (N)

4 110 0 11.5 69 1.91 7.15 66

8 184 0 18.9 86 2.35 8.79 116

12 250 0 25.5 109 2.95 11.01 126

16 325 0 33 159 4.24 15.84 170

20 425 0 43 210 5.56 20.77 200

24 500 0 50.5 307 8.07 30.14 242

28 525 0 53 334 8.77 32.75 286

32 550 0 55.5 387 10.14 37.87 330

36 602 0 60.7 440 11.51 42.99 394

40 670 0 67.5 474 12.39 46.27 434

44 729 0 73.4 526 13.74 51.29 515

48 744 0 74.9 609 15.88 59.31 626

50 744 0 74.9 611 15.94 59.50 665

Tmax (N.m)

Mmax (N.m)

LDC : Lecture du comparateurDiff. : Différence par rapport à la mesure précédente

45° 60° 75° 90°

Diff. LDC Diff. LDC Diff. LDC Diff.

16 15 6

28 36 20 18 3 17 11

11 54 18 30 12 33 16

21 70 16 44 14 47 14

29 92 22 60 16 62 15

16 115 23 72 12 97 35

28 121 6 88 16 97 0

22 136 15 105 17 102 5

58 154 18 125 20 126 24

45 172 18 133 8 132 6

43 190 18 185 52 145 13

20 208 18 210 25 159 14

31 224 16 215 5 174 15

17 234 10 220 5 182 8

30° 45° 60°M T LDC M T LDC M T

1.15 1.99 20 1.77 1.77 16 1.82 1.05

3.55 6.15 48 3.75 3.75 36 3.55 2.05

Tmax

Mmax

3.55 6.15 48 3.75 3.75 36 3.55 2.05

6.05 10.48 59 4.52 4.52 54 5.11 2.95

6.55 11.35 80 6.01 6.01 70 6.49 3.75

8.75 15.16 109 8.06 8.06 92 8.40 4.85

10.25 17.76 125 9.19 9.19 115 10.39 6.01

12.34 21.39 153 11.17 11.17 121 10.91 6.31

14.54 25.21 175 12.72 12.72 136 12.21 7.06

16.74 29.02 233 16.82 16.82 154 13.77 7.96

19.94 34.56 278 20.00 20.00 172 15.32 8.86

21.94 38.02 321 23.04 23.04 190 16.88 9.76

25.99 45.04 341 24.46 24.46 208 18.44 10.66

31.54 54.65 372 26.65 26.65 224 19.83 11.46

33.48 58.03 389 27.85 27.85 234 20.69 11.96

74.90

33.48

(N.m)

(N.m)

75° 90°LDC M T LDC M T

15 1.93 0.52 6 1.1 0

18 2.22 0.60 17 2.2 0

18 2.22 0.60 17 2.2 0

30 3.38 0.91 33 3.8 0

44 4.73 1.27 47 5.2 0

60 6.28 1.69 62 6.7 0

72 7.44 2.00 97 10.2 0

88 8.98 2.41 97 10.2 0

105 10.62 2.85 102 10.7 0

125 12.55 3.37 126 13.1 0

133 13.33 3.58 132 13.7 0

185 18.35 4.93 145 15 0

210 20.76 5.58 159 16.4 0

215 21.25 5.71 174 17.9 0

220 21.73 5.84 182 18.7 0

θ 0° 15° 30° 45° 60°

566 639 1374 778 571

744 611 665 389 234

R 1 1 1 1 1

1 1 1 1 1

744 611 665 389 234

M(A) (N.m) 0 16.66 68.92 55.34 49.87

T(A) (N.m) 56 62.21 119.44 55.34 28.83

M(B) (N.m) 0 15.94 33.48 27.85 20.69

T(B) (N.m) 74.9 59.5 58.03 27.85 11.96

M(A) (N.m) T(A) (N.m) M(B) (N.m)

0 56 0

16.66 62.21 15.94

41.4 0 33.48

42.88 11.52 27.85

49.87 28.83 20.69

55.34 55.34 21.73

68.92 119.44 18.7

Pmax(A) (1ér éprouvette)

Pmax(B) (2éme éprouvette)

R3

B' = R3 x Pmax(B)

0.0 16.7 41.4 42.9 49.9 55.3 68.90

20

40

60

80

100

120

140

Eprouvette A

M (A)

T (A)

0.0 16.7 41.4 42.9 49.9 55.3 68.90

20

40

60

80

100

120

140

Eprouvette A

M (A)

T (A)

0 15.94 18.7 20.69 21.73 27.85 33.480

10

20

30

40

50

60

70

80

Eprouvette B

M (B)

T (B)

75° 90°

439 409

220 182

1 1

1 1

220 182

42.88 41.4

11.52 0

21.73 18.7

5.84 0

T(B) (N.m)

74.9

59.5

58.03

27.85

11.96

5.84

0

0.0 16.7 41.4 42.9 49.9 55.3 68.90

20

40

60

80

100

120

140

Eprouvette A

M (A)

T (A)

0.0 16.7 41.4 42.9 49.9 55.3 68.90

20

40

60

80

100

120

140

Eprouvette A

M (A)

T (A)

0 15.94 18.7 20.69 21.73 27.85 33.480

10

20

30

40

50

60

70

80

Eprouvette B

M (B)

T (B)