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4 Lqwurgxfwlrq 58

414 Suìvhqwdwlrq gx sureoëph 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 58

415 Ìwdw gh od txhvwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5:

41514 Sureoëph sodq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5:

41515 Sureoëph vsdwldo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5<

416 Frqwhqx gh o*ìwxgh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 63

5 Prgìolvdwlrq pìfdqltxh 66

514 K|srwkëvhv jìqìudohv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 66

51414 Jìqìudolwìv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 66

51415 Prgìolvdwlrq gh od vwuxfwxuh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 67

51416 Glvfuìwlvdwlrq gh od vwuxfwxuh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 69

51417 Skdvdjh gh frqvwuxfwlrq 0 Fkdujhphqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6;

515 Ìwxgh gh od suìfrqwudlqwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6;

51514 Jìqìudolwìv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6;

51515 Jìrpìwulh gh od suìfrqwudlqwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6<

51516 Shuwhv lqvwdqwdqìhv gh suìfrqwudlqwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6<

47 Wdeoh ghv pdwlëuhv

6 Qrq0olqìdulwì jìrpìwultxh 76

614 Od ghvfulswlrq odjudqjlhqqh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 76

61414 Phvxuh ghv gìirupdwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 78

61415 Phvxuh ghv frqwudlqwhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 79

615 Od ghvfulswlrq frurwdwlrqqhooh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7<

61514 Phvxuh ghv gìirupdwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 83

61515 Phvxuh ghv frqwudlqwhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 83

616 Sulqflsh ghv wudydx{ yluwxhov olqìdulvì 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 85

61614 Hq G1O1D1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 85

617 Ohv frrugrqqìhv frqyhfwìhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 86

61714 Ìoìphqw gh srxwuh sodq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 86

61715 Frrugrqqìhv frqyhfwìhv gdqv o*hvsdfh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8:

618 Prgìolvdwlrq sdu ìoìphqwv �qlv gh w|sh srxwuh ��gìsodfhphqw�� 1 1 1 1 1 1 1 :<

61814 Xq h{hpsoh = od srxwuh 5G gh Ehuqrxool 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :<

61815 Ìoìphqw gh srxwuh fodvvltxh 6G 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;4

619 Prgìolvdwlrq sdu ìoìphqwv �qlv gh w|sh srxwuh ��vhpl0lqyhuvh�� 1 1 1 1 1 1 1 ;6

7 Qrq0olqìdulwì pdwìulhooh ;8

714 Ohv orlv gh frpsruwhphqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;8

71414 Pdwìuldx gh od pdwulfh gh od vhfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;8

71415 Pdwìuldx ghv �euhv gh od vhfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <7

71416 Frqfoxvlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <9

715 Ìwxgh g*xqh vhfwlrq gurlwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <9

71514 K|srwkëvhv hw qrwdwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <9

71515 Ohv gìirupdwlrqv qrupdohv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <:

71516 Ohv gìirupdwlrqv wdqjhqwhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <;

71517 Ohv gìirupdwlrqv ghv dupdwxuhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <;

71518 Ohv vroolflwdwlrqv lqwhuqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <;

71519 Ohv vroolflwdwlrqv h{whuqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 437

7151: Ìtxloleuh gh od vhfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 438

7151; Irupdwlrq gh od pdwulfh gh uljlglwì gh od vhfwlrq 1 1 1 1 1 1 1 1 1 1 1 438

7151< Uìvroxwlrq lwìudwlyh gh o*ìtxloleuh gh od vhfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 43;

715143 Ohv olplwhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 43<

716 Lqwìjudwlrq oh orqj gh o*ìoìphqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 443

71614 Uhodwlrq hqwuh ohv vroolflwdwlrqv ghv vhfwlrqv hw ohv h�ruwv ìoìphqwdluhv 443

71615 Uhodwlrq hqwuh ohv gìsodfhphqwv ìoìphqwdluhv hw ohv gìirupdwlrqv ghv

vhfwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 445

Wdeoh ghv pdwlëuhv 48

71616 Irupdwlrq gh od pdwulfh gh uljlglwì ìoìphqwdluh ^N?` gdqv oh uhsëuh

lqwulqvëtxh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 446

8 Dojrulwkph jìqìudo gh uìvroxwlrq 448

814 Ìtxloleuh gh od vhfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 448

815 Ìtxloleuh jìqìudo gh od vwuxfwxuh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 448

9 Sulvh hq frpswh ghv h�hwv gl�ìuìv 454

914 Ioxdjh dyhf orl gh frpsruwhphqw olqìdluh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 454

91414 Od irqfwlrq gh xdjh/ od irqfwlrq gh uhod{dwlrq 1 1 1 1 1 1 1 1 1 1 1 1 454

91415 Od iruph lqwìjudoh = oh sulqflsh gh vxshusrvlwlrq 1 1 1 1 1 1 1 1 1 1 1 456

91416 Od iruph gl�ìuhqwlhooh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 456

91417 Od pìwkrgh gx whpsv ìtxlydohqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 457

91418 Oh sureoëph vwuxfwxudo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 458

915 Ioxdjh dyhf orl gh frpsruwhphqw qrq olqìdluh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 458

916 Glvfuìwlvdwlrq whpsruhooh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 459

91614 Suhplëuh pìwkrgh = od uëjoh gx uhfwdqjoh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 459

91615 Ghx{lëph pìwkrgh = od uëjoh gx wudsë}h 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 45;

917 Dqdo|vh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 463

91714 Fdv gx xdjh olqìdluh hq frqwudlqwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 463

91715 Fdv gx xdjh olqìdluh hq gìirupdwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 463

91716 Uhod{dwlrq ghv dflhuv gh suìfrqwudlqwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 466

91717 Frqfoxvlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 466

918 Ìtxloleuh gh od vhfwlrq vrxv gìirupdwlrqv gl�ìuìhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 467

91814 Ohv dupdwxuhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 467

91815 Oh eìwrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 467

91816 Ìtxloleuh joredo gh od vhfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 469

919 Frqfoxvlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 46:

: Ìoìphqwv vsìfl�txhv dx Jìqlh flylo 46<

:14 Ohv duwlfxodwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 46<

:1414 Shwlwv gìsodfhphqwv = k|srwkëvhv hw qrwdwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 46<

:1415 Judqgv gìsodfhphqwv hw judqghv urwdwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 473

:15 Ohv ìoìphqwv elduwlfxoìv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 475

:1514 Udsshov wkìrultxhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 475

:1515 Xq h{hpsoh vlpsoh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 478

:16 Ohv ìoìphqwv gh w|sh fkdñqhwwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 479

:1614 Gì�qlwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 479

49 Wdeoh ghv pdwlëuhv

:1615 Suhplëuh irupxodwlrq gh o*ìtxdwlrq gh od fkdñqhwwh 1 1 1 1 1 1 1 1 1 1 47:

:1616 Xqh dxwuh irupxodwlrq gx sureoëph 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 484

:17 Ohv fåeohv gh suìfrqwudlqwh qrq olìv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 48:

; Fdofxo g*lqvwdelolwì 48<

;14 Lqwurgxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 48<

;15 Uìvroxwlrq gx sureoëph 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 493

;1514 Oh vkliw olqìdluh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 493

;1515 Oh vkliw qrq olqìdluh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 493

;1516 Xqh dxwuh dssurfkh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 494

;16 Frqfoxvlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 495

< Fdofxo g|qdpltxh 496

<14 Lqwurgxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 496

<15 Fdofxo g|qdpltxh olqìdluh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 496

<1514 Ìtxdwlrq gx prxyhphqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 496

<1515 Rvfloodwlrqv oleuhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 497

<1516 Vxshusrvlwlrq prgdoh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 498

<16 Fdofxo g|qdpltxh qrq olqìdluh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 499

<1614 Ìtxdwlrq gx prxyhphqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 499

<1615 Od pìwkrgh gh Qhzpdun = dssurfkh lqfuìphqwdoh 1 1 1 1 1 1 1 1 1 1 499

<1616 Plvh hq úxyuh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 49;

<17 Frqfoxvlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 49<

43 H{hpsohv g*dssolfdwlrqv qxpìultxhv 4:4

4314 Lqwurgxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:4

4315 Vwuxfwxuhv hq eìwrq dupì 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:4

431514 Srxwuh frqwlqxh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:4

431515 Iodpehphqw g*xq srwhdx 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:7

431516 Gìyhuvhphqw gh srwhdx{ hq L 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:9

431517 Sruwltxh uhfwdqjxodluh elduwlfxoì 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4;6

431518 Fdguh ihupì 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4;8

4316 Vwuxfwxuhv hq eìwrq suìfrqwudlqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4;:

431614 Frorqqh suìfrqwudlqwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4;:

431615 Srxwuh suìfrqwudlqwh ã 5 wudyìhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4<3

431616 Srxwuh suìfrqwudlqwh lvrvwdwltxh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4<7

431617 Srxwuh suìfrqwudlqwh k|shuvwdwltxh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 535

4317 Vwuxfwxuhv pìwdooltxhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 537

Wdeoh ghv pdwlëuhv 4:

431714 Iodpehphqw g*xq påw hqfdvwuì 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 537

431715 Srxwuh frqvroh vroolflwìh wudqvyhuvdohphqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 539

431716 Duf vxuedlvvì 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 53<

44 Xq h{hpsoh g*dssolfdwlrq uìhooh = oh Srqw gh Wdqfduylooh 546

4414 Lqwurgxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 546

4415 Remhw gx frqwu÷oh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 546

4416 Ìwxgh gx wudqvihuw gh fkdujhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 547

441614 Oh sulqflsh gx wudqvihuw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 547

441615 Oh prgëoh 5G 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 547

441616 Ohv ìwdshv gx wudqvihuw gh fkdujhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 549

4417 Ìwxghv vsìfl�txhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 549

441714 Xq prgëoh 5G26G srxu o*ìwxgh ghv s|o÷qhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 549

441715 Xq prgëoh 6G srxu o*ìwxgh gx uìjodjh �q gh od vxvshqvlrq 1 1 1 1 1 549

4418 Txhotxhv srlqwv sduwlfxolhuv gh prgìolvdwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54<

441814 Ohv vhoohv hq wíwh gh s|o÷qh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54<

441815 Ohv elhoohv g*dssxl 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 553

4419 Ohv sulqflsdx{ uìvxowdwv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 554

441914 Ohv gìsodfhphqwv oruv gx wudqvihuw gh fkdujhv 1 1 1 1 1 1 1 1 1 1 1 1 1 554

441915 Ohv gl!fxowìv gh prgìolvdwlrq hw gh fdofxo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 554

45 Frqfoxvlrq jìqìudoh 55<

Uìiìuhqfhv eleolrjudskltxhv 564

D Ohv fåeohv gh suìfrqwudlqwh qrq olìv 574

D14 Udsshov 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 574

D15 Plvh hq úxyuh lqirupdwltxh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 574

E Rswlplvdwlrq gh od qxpìurwdwlrq ghv qúxgv 576

E14 Suìvhqwdwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 576

E15 Dqdo|vh gx sureoëph 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 577

E1514 Fxwkloo Pdf Nhh +FP, hw lqyhuvh +UFP, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 577

E1515 Plqlplvdwlrq gh o*dffurlvvhphqw gh iurqw/ dojrulwkph gh Oìy|/ dojr0

ulwkph gh Nlqj 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 577

E16 Lpsoìphqwdwlrq ghv dojrulwkphv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 578

E1614 Fxwkloo Pdf Nhh +FP, hw lqyhuvh +UFP, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 578

E1615 Dojrulwkph gh Oìy| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 578

E1616 Dojrulwkph gh Nlqj 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 579

4; Wdeoh ghv pdwlëuhv

E1617 Dojrulwkph gh Plqlplvdwlrq gh o*Dffurlvvhphqw gh Iurqw 1 1 1 1 1 1 1 579

E17 Ìydoxdwlrq ghv dojrulwkphv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 579

E1714 Srqw gh Od Urfkh Ehuqdug 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 57:

E1715 Srqw gh Qrupdqglh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 57:

E18 Frqfoxvlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 57;

Olvwh ghv �jxuhv

514 Uhsëuh orfdo olì ã od srvlwlrq lqlwldoh gh o*ìoìphqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 67

515 Glvfuìwlvdwlrq orqjlwxglqdoh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 69

516 Gìfrpsrvlwlrq hq wudsë}hv gh od vhfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6:

517 Wdexodwlrq g*xq wurqêrq gh fåeoh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 73

518 H�hw gx uhfxo g*dqfudjh jdxfkh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 74

614 Ohv 6 frq�jxudwlrqv fodvvltxhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 77

615 Ghvfulswlrq frurwdwlrqqhooh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7<

616 Frrugrqqìhv frqyhfwìhv sodqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 87

617 Uhsëuh orfdo olì ã od srvlwlrq lqlwldoh gh o*ìoìphqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8;

618 Jìrpìwulh gh o*ìoìphqw gìirupì 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8<

619 Srvlwlrq gh o*d{h { gx uhsëuh lqwulqvëtxh sdu udssruw dx uhsëuh orfdo {f|f}f 98

61: Srvlwlrq gx uhsëuh lqwulqvëtxh sdu udssruw dx uhsëuh {|�}� 1 1 1 1 1 1 1 1 1 99

61; Surmhfwlrq gh o*ìoìphqw gdqv oh sodq {} gx uhsëuh lqwulqvëtxh {|} 1 1 1 1 1 9;

61< Surmhfwlrq gh o*ìoìphqw gdqv oh sodq {| gx uhsëuh lqwulqvëtxh {|} 1 1 1 1 1 9<

6143 Srxwuh frqvroh fkdujìh hq wíwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :8

6144 Hoìphqw �ql gh srxwuh Ehuqrxool 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :<

6145 Frqvwuxfwlrq gh od pdwulfh gh uljlglwì 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;4

714 Orl ìodvwltxh olqìdluh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;9

715 Orl ìodvwltxh olqìdluh iudjloh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;:

716 Orl elolqìdluh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;;

717 Orl ESHO<4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;<

718 Orl gh Vdujlq jìqìudoh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <4

719 Gldjudpph frqwudlqwh0gìirupdwlrq gh od �euh od soxv whqgxh 1 1 1 1 1 1 1 1 <5

71: Orl dflhu EDHO;6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <6

71; Orl ìodvwrsodvwltxh sduidlwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <7

71< Orl dflhu ESHO<4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <8

7143 Frqyhqwlrq gh vljqh srxu od vhfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <:

53 Olvwh ghv �jxuhv

7144 Dupdwxuh lqfolqìh sdu udssruw ã od vhfwlrq gurlwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 433

7145 Vhfwlrq gurlwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 439

7146 Pdloodjh dgdswdwli 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 43:

7147 Uìvroxwlrq lwìudwlyh gh o*ìtxloleuh gh od vhfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 43<

814 Rujdqljudpph gh o*dojrulwkph gh uhfkhufkh g*ìtxloleuh gh od vhfwlrq 1 1 1 1 449

815 Rujdqljudpph gh o*dojrulwkph gh uhfkhufkh g*ìtxloleuh gh od vwuxfwxuh 1 1 44;

:14 Hoìphqw elduwlfxoì 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 476

:15 H{hpsoh gh wuhloolv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 478

:16 Surmhfwlrq ghv irufhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 479

:17 Hoìphqw gh w|sh fkdñqhwwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 47:

4314 Srxwuh gh Pdwwrfn 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:5

4315 Ìyroxwlrq ghv prphqwv hq E hw F hq irqfwlrq gh od fkdujh 1 1 1 1 1 1 1 1 1 4:6

4316 Srwhdx gh Irxuì = jìrpìwulh hw fkdujhphqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:8

4317 Frxueh fkdujh 0 ëfkh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:9

4318 Srwhdx{ GZ 0 Fr�udjh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4::

4319 Srwhdx{ GZ 0 Ihuudloodjh sdvvli 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:;

431: Srwhdx GZ< 0 Frxueh fkdujh 0 ëfkh ã pl0kdxwhxu 1 1 1 1 1 1 1 1 1 1 1 1 1 4;3

431; Srwhdx GZ44 0 Frxueh fkdujh 0 ëfkh ã pl0kdxwhxu 1 1 1 1 1 1 1 1 1 1 1 1 1 4;4

431< Srwhdx GZ46 0 Frxueh fkdujh 0 ëfkh ã pl0kdxwhxu 1 1 1 1 1 1 1 1 1 1 1 1 1 4;5

43143 Sruwltxh S: gh Fudqvwrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4;6

43144 Sruwltxh S: gh Fudqvwrq 0 Ìyroxwlrq ghv ëfkhv hq irqfwlrq gh od fkdujh 1 4;8

43145 Fdguh O6 gh Ihujxvvrq hw Euhhq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4;9

43146 Fdguh O6 gh Ihujxvvrq 0 Frxueh fkdujh 0 ëfkh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4;:

43147 Frorqqh suìfrqwudlqwh G273e6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4;;

43148 Frorqqh suìfrqwudlqwh g*Durql 0 Frxueh fkdujh0 ëfkh ã pl0kdxwhxu 1 1 1 1 4;<

43149 Srxwuh suìfrqwudlqwh FHEWS 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4<3

4314: Ghvfulswlrq gx fkdujhphqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4<4

4314; Gìqlyhoodwlrq g*dssxlv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4<4

4314< Ìyroxwlrq gh od uìdfwlrq g*dssxl fhqwudo hqwuh 6 hw 4;5 mrxuv 1 1 1 1 1 1 1 1 4<6

43153 Ìyroxwlrq gh od uìdfwlrq g*dssxl fhqwudo hqwuh 4;5 hw 9333 mrxuv 1 1 1 1 1 1 4<7

43154 Ìyroxwlrq gh od gìirupdwlrq gh od �euh vxsìulhxuh gh od vhfwlrq I hqwuh 6

hw 4;5 mrxuv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4<8

43155 Ìyroxwlrq gh od gìirupdwlrq gh od �euh vxsìulhxuh gh od vhfwlrq I hqwuh

4;5 hw 9333 mrxuv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4<9

43156 Ìyroxwlrq gh od irufh gh suìfrqwudlqwh ã pl0wudyìh hqwuh 6 hw 4;5 mrxuv 1 1 1 4<:

43157 Ìyroxwlrq gh od irufh gh suìfrqwudlqwh ã pl0wudyìh hqwuh 4;5 hw 9333 mrxuv 1 4<;

Olvwh ghv �jxuhv 54

43158 Srxwuh suìfrqwudlqwh lvrvwdwltxh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4<<

43159 Frxueh fkdujh0 ëfkh dyhf suìfrqwudlqwh h{whuqh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 533

4315: Frxueh fkdujh0 ëfkh dyhf suìfrqwudlqwh lqwhuqh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 534

4315; Srxwuh suìfrqwudlqwh k|shuvwdwltxh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 535

4315< Frxueh fkdujh0 ëfkh dyhf suìfrqwudlqwh lqwhuqh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 536

43163 Frxueh fkdujh0 ëfkh dyhf suìfrqwudlqwh h{whuqh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 537

43164 Påw hqfdvwuì 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 538

43165 Påw hqfdvwuì 0 Ìyroxwlrq ghv ëfkhv hq irqfwlrq gx fkdujhphqw 1 1 1 1 1 1 539

43166 Påw hqfdvwuì 0 Gìirupìhv vxffhvvlyhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 53:

43167 Srxwuh frqvroh vroolflwìh sdu xqh fkdujh wudqvyhuvdoh 1 1 1 1 1 1 1 1 1 1 1 1 1 53;

43168 Srxwuh frqvroh fkdujìh wudqvyhuvdohphqw 0 Ìyroxwlrq gh od ëfkh wudqvyhu0

vdoh hq irqfwlrq gx fkdujhphqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 53<

43169 Duf vxuedlvvì 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 543

4316: Duf vxuedlvvì 0 Ìyroxwlrq gh od ëfkh ã od fohi hq irqfwlrq gh od fkdujh 1 1 544

4414 Oh Srqw gh Wdqfduylooh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 547

4415 Prgìolvdwlrq FDR Rsìud gx Srqw gh Wdqfduylooh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 548

4416 Prgëoh 5G26G 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54;

4417 Prgëoh 6G 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54<

4418 Glvsrvlwlrq wudqvyhuvdoh ghv vxvshqwhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 553

4419 Srqw gh Wdqfduylooh 0 Hvsdfhphqw krul}rqwdo ghv fåeohv sruwhxuv 1 1 1 1 1 1 554

441: Gìsodfhphqw gh od wíwh gh s|o÷qh ulyh gurlwh sdu udssruw ã od yhuwlfdoh 1 1 557

441; Srvlwlrq ghv vhoohv ulyh gurlwh sdu udssruw dx s|o÷qh +od srvlwlrq qxooh hvw

fhooh gh od vhooh ã od �q gh o*rsìudwlrq 5, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 558

441< Gìsodfhphqw gh od wíwh gh s|o÷qh ulyh jdxfkh sdu udssruw ã od yhuwlfdoh 1 1 559

44143 Srvlwlrq ghv vhoohv ulyh jdxfkh sdu udssruw dx s|o÷qh +od srvlwlrq qxooh hvw

fhooh ã od �q gh o*rsìudwlrq 5, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 55:

44144 Yduldwlrq gh ëfkh gx wdeolhu dx qrhxg fhqwudo sdu udssruw ã vd srvlwlrq

hq �q g*rsìudwlrq 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 55;

55 Olvwh ghv �jxuhv

Olvwh ghv wdeohdx{

4314 Prphqwv hq vhuylfh gdqv ohv vhfwlrqv E hw F 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:6

4315 Prphqwv xowlphv gdqv ohv vhfwlrqv E hw F 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:7

4316 Fkdujh hw ëfkh xowlphv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:9

4317 H{fhqwuhphqw gx fkdujhphqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4::

4318 Fdudfwìulvwltxhv pìfdqltxhv gx eìwrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:;

4319 Fdudfwìulvwltxhv pìfdqltxhv ghv dflhuv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:<

431: Fkdujhv xowlphv ghv srwhdx{ GZ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4;3

431; Fdudfwìulvwltxhv gx ihuudloodjh orqjlwxglqdo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4;7

4414 Ìwdshv gx wudqvihuw gh fkdujhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54:

4415 Gìsodfhphqwv oruv gx wudqvihuw gh fkdujhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 555

4416 Ilfkh whfkqltxh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 556

E14 Srqw gh Od Urfkh Ehuqdug 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 57:

E15 Srqw gh Od Urfkh Ehuqdug 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 57:

E16 Srqw gh Qrupdqglh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 57;

E17 Srqw gh Qrupdqglh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 57;

57 Olvwh ghv wdeohdx{

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414 Suìvhqwdwlrq gx sureoëph

Gh qrv mrxuv/ od pdmrulwì ghv vwuxfwxuhv gh eåwlphqw rx gh Jìqlh flylo hvw fdofxoìh vxlydqw

oh irupdolvph olqìdluh gh od wkìrulh gh o*ìodvwlflwì ìwdeol dx [L[ëph vlëfoh1 Ru/ lo q*h{lvwh

dxfxqh vwuxfwxuh/ dxfxq pdwìuldx grqw oh frpsruwhphqw vrlw vwulfwhphqw ìodvwltxh olqìdluh

gdqv vrq grpdlqh gh vroolflwdwlrq frxudqw/ hw d iruwlrul oruv g*xqh ìwxgh dx{ ìwdwv olplwhv

xowlphv1 Hq rxwuh/ od yrorqwì g*xwlolvhu dx plhx{ od pdwlëuh srxu ghv udlvrqv ìfrqrpltxhv/

dlqvl txh od whqgdqfh dfwxhooh ã frqfhyrlu hw ã frqvwuxluh ghv vwuxfwxuhv gh soxv hq soxv

ìodqfìhv +frpph oh Srqw gh Qrupdqglh rx oh Srqw gh Ploodx,/ frqgxlvhqw qìfhvvdluhphqw

ã suhqguh hq frpswh ohv qrq0olqìdulwìv d�hfwdqw oh v|vwëph1 Fhfl hvw g*dxwdqw soxv yudl

srxu ohv rxyudjhv g*duw txh od whqwdwlrq gh iudqfklu ghv euëfkhv gh soxv hq soxv judqghv hvw

iruwh/ hw txh g*dxwuh sduw rq fkhufkh ã glplqxhu dx pd{lpxp oh srlgv gh o*rxyudjh1 Srxu

fhuwdlqhv gh fhv vwuxfwxuhv/ lo dssdudñw txh ohv uìvxowdwv g*xqh dqdo|vh olqìdluh qh vrqw soxv

dssur{lpdwliv/ pdlv vrqw frpsoëwhphqw huurqìv1

Ohv qrq0olqìdulwìv txh o*rq revhuyh shxyhqw vh fodvvhu hq ghx{ judqghv fdwìjrulhv =

41 Ohv qrq0olqìdulwìv pdwìulhoohv1 Hoohv shxyhqw suryhqlu gh od orl gh frpsruwhphqw

lqwulqvëtxh gx pdwìuldx +orl gh frpsruwhphqw ìodvwr0sodvwltxh sdu h{hpsoh,/ gh od

�vvxudwlrq gx pdwìuldx/ gx frpsruwhphqw gx eìwrq whqgx hqwuh ghx{ �vvxuhv/ gx

prgh g*dvvhpeodjh hqwuh ohv frpsrvdqwv ìoìphqwdluhv g*xqh vwuxfwxuh1

51 Od qrq0olqìdulwì jìrpìwultxh1 Fhwwh qrq0olqìdulwì hvw gxh dx{ gìsodfhphqwv

píphv gh od vwuxfwxuh1 Hq h�hw/ oruvtxh fhv gìsodfhphqwv ghylhqqhqw lpsruwdqwv/

o*ìtxloleuh gx v|vwëph qh shxw soxv íwuh gìfulw gdqv od frq�jxudwlrq lqlwldoh qrq

gìirupìh +wkìrulh dx suhplhu ruguh ghv gìsodfhphqwv, pdlv grlw íwuh ìwxglì gdqv

od frq�jxudwlrq gìirupìh +wkìrulh dx vhfrqg ruguh ghv gìsodfhphqwv,1 Dx vhlq gh

fhwwh fdwìjrulh/ rq shxw glvwlqjxhu ohv judqgv gìsodfhphqwv hw ohv urwdwlrqv prgìuìhv

59 Fkdslwuh 4 Lqwurgxfwlrq

g*xqh sduw/ hw ohv judqgv gìsodfhphqwv hw ohv judqghv urwdwlrqv g*dxwuh sduw1 Od sulvh

hq frpswh gh fhwwh qrq0olqìdulwì hvw irqgdphqwdoh srxu o*ìwxgh gh vwdelolwì gh iruph

gh slëfhv ìodqfìhv/ rx gh vwuxfwxuhv ìodqfìhv ã qúxgv gìsodêdeohv1 Hooh shuphw gh

suìgluh oh qlyhdx gh fkdujh fulwltxh srxu oh fdofxo dx dpehphqw1

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Hq h�hw/ f*hvw o*dvvrfldwlrq gh od qrq0olqìdulwì pdwìulhooh hw gh od qrq0olqìdulwì jìrpìwultxh

txl uìgxlw od fkdujh fulwltxh ghv ìoìphqwv xwlolvìv gdqv oh grpdlqh ghv rxyudjhv g*duw/

frpph ohv dufv rx ohv slohv gh srqwv1

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gh idêrq ìodvwltxh hq fkdfxq ghv srlqwv gh od vwuxfwxuh/ hw ã frpsduhu fhwwh frqwudlqwh ã

od uìvlvwdqfh gx pdwìuldx glylvìh sdu xq idfwhxu gh vìfxulwì1 Fhwwh dssurfkh vhudlw sdu0

idlwhphqw orjltxh vl ohv pdwìuldx{ dydlhqw xq frpsruwhphqw ìodvwr0iudjloh/ frpph oh yhuuh1

Ru/ fh q*hvw sdv oh fdv/ oh frpsruwhphqw ghv pdwìuldx{ xwlolvìv gdqv od frqvwuxfwlrq hvw hq

jìqìudo xq frpsruwhphqw gxfwloh/ txh o*rq srxuudlw h{sorlwhu1 Hq idlw/ o*k|srwkëvh vrxv0

mdfhqwh hvw txh ohv vroolflwdwlrqv gh vhuylfh uhvwhqw vx!vdpphqw idleohv srxu tx*lo | dlw

olqìdulwì ghv frpsruwhphqwv1

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fkdujh gh uxlqh1 O*dqdo|vh phqìh hvw doruv xqh dqdo|vh olplwh grqw oh exw hvw gh frqqdñwuh

od fdsdflwì gh uìvlvwdqfh gh od vwuxfwxuh1 Pdokhxuhxvhphqw/ dyhf fhwwh whfkqltxh/ rq gìwhu0

plqh xqh fkdujh olplwh srxu odtxhooh rq qh frqqdñw sdv ohv gìsodfhphqwv fruuhvsrqgdqwv1

Ru/ ghv txdqwlwìv frpph od ëfkh rx ohv gìirupdwlrqv shxyhqw v*dyìuhu glphqvlrqqdqwhv

ylv0ã0ylv gh od vìfxulwì1

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olqìdluh +sdu h{hpsoh/ gdqv xqh srxwuh/ rq gìwhuplqh sdu xq fdofxo olqìdluh od vhfwlrq od

soxv vroolflwìh/ txh o*rq yìul�h ã o*dlgh g*xq fdofxo qrq olqìdluh,1 Hq wrxwh orjltxh/ oh fdofxo

gh vwuxfwxuh grlw íwuh oxl0píph qrq olqìdluh1 Hq rxwuh/ ohv fdofxov dx{ ìwdwv olplwhv xowlphv

edvìv vxu xq frpsruwhphqw olqìdluh gh od vwuxfwxuh qh shuphwwhqw sdv gh suhqguh hq frpswh

ohv skìqrpëqhv g*dgdswdwlrq olìv dx{ qrq0olqìdulwìv +urwxohv sodvwltxhv sdu h{hpsoh,1 F*hvw

srxutxrl/ lo hvw gìvrupdlv qìfhvvdluh gh glvsrvhu g*rxwlov rsìudwlrqqhov txl shuphwwhqw gh

suhqguh hq frpswh ohv ghx{ w|shv gh qrq0olqìdulwìv/ fh txl qh srvh soxv dxfxq sureoëph

sudwltxh gh qrv mrxuv/ oh prlqguh plfur0ruglqdwhxu glvsrvdqw g*xqh sxlvvdqfh gh fdofxo

qhwwhphqw vxsìulhxuh dx{ pdlqiudphv ghv ghuqlëuhv gìfhqqlhv1

415 Ìwdw gh od txhvwlrq 5:

415 Ìwdw gh od txhvwlrq

Jìqìudohphqw/ ohv vwuxfwxuhv gh Jìqlh flylo vrqw fdofxoìhv gh idêrq olqìdluh ìodvwltxh/ juåfh

ã soxvlhxuv pìwkrghv/ od soxv uìsdqgxh ìwdqw fhooh ghv ìoìphqwv �qlv1 Gdqv oh grpdlqh

ghv rxyudjhv g*duw/ oh fdofxo gh od vwuxfwxuh joredoh hvw phqì hq jìqìudo ã sduwlu g*xqh

prgìolvdwlrq hq ìoìphqwv gh w|sh ��srxwuh��1 Ohv ìwxghv orfdohv gh wíwh gh sloh/ gh oldlvrqv

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rx frtxhv ìsdlvvhv/ rx hqfruh ã o*dlgh g*ìoìphqwv yroxpltxhv1

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lqwìuhvvhurqv sdv dx{ wudydx{ uhodwliv dx{ sodtxhv0frtxhv hw ìoìphqwv pdvvliv1 Hq rxwuh/

qrxv qrxv uhvwuhlqgurqv srxu ohv qrq0olqìdulwìv pdwìulhoohv/ dx{ vwuxfwxuhv hq eìwrq dupì

rx hq eìwrq suìfrqwudlqw1

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phqvlrqqhoohv/ sxlv qrxv ìwxglhurqv fhx{ uhodwliv dx{ vwuxfwxuhv vsdwldohv1 Ohv wudydx{

wkìrultxhv txl qrxv lqwìuhvvhqw dffrpsdjqhqw oh gìyhorsshphqw ghv pr|hqv gh fdofxov

lqirupdwltxhv1 F*hvw srxutxrl/ qrwuh dqdo|vh eleolrjudskltxh gìexwh yhuv ohv dqqìhv :31

Qrxv wurxyrqv gdqv ^Hvs;9d` xqh olvwh gh uìiìuhqfhv vxu oh vxmhw txl vh yhxw dvvh} frpsoëwh1

41514 Sureoëph sodq

Hq 4<:3/ Iudqnolq ^Iud:3` ìoderuh xqh irupxodwlrq sodqh rü dssdudlvvhqw srxu fkdtxh

qúxg ohv 6 ghjuìv gh olehuwì sodqv kdelwxhov = 5 wudqvodwlrqv hw 4 urwdwlrq1 Od pdwulfh gh

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sdv fròqflghu dyhf o*d{h gh uìiìuhqfh gh o*ìoìphqw1 O*dssurfkh hvw wdqjhqwh/ hw ohv orlv gh

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yhuvìh/ grqqh od pdwulfh gh uljlglwì ìoìphqwdluh/ txl hvw hqvxlwh dvvhpeoìh gdqv od pdwulfh

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Oh suhplhu ìoìphqw gh srxwuh hq eìwrq dupì wulglphqvlrqqho d yudlvhpeodeohphqw ìwì

gìyhorssì sdu Fkdq ^Fkd;5`1 O*dqdo|vh suhqg hq frpswh oh vhfrqg ruguh/ dlqvl txh ohv

h�hwv gl�ìuìv1 O*lq xhqfh gh o*h�ruw wudqfkdqw hvw devhqwh/ doruv txh od wruvlrq hvw wrxw0

ã0idlw gìfrxsoìh1 Dx olhx gh gìfrxshu ohv vhfwlrqv hq frxfkhv/ hoohv vrqw gìfrxsìhv hq

�euhv1 Od uljlglwì wruvlrqqhooh surylhqw g*xqh shqwh gdqv xq gldjudpph lgìdolvì wulolqìdluh

prphqw gh wruvlrq 0 urwdwlrq gh wruvlrq frqvwuxlw ã o*dlgh g*xqh wkìrulh gh wuhloolv srxu

gìwhuplqhu oh prphqw hw od urwdwlrq gh sodvwl�fdwlrq vdqv whqlu frpswh gh o*h�ruw qrupdo1

Oh surjudpph gh Pdul ^Pdu;7` hvw oh surorqjhphqw gx wudydlo gh Fkdq1 Od pdwulfh

ghv frqwudlqwhv lqlwldohv hvw frpsoìwìh1 Od suìfrqwudlqwh hvw lqwurgxlwh vhorq od pìwkrgh

gh Ndqj ^Ndq::`1 Pdokhxuhxvhphqw/ od ydolgdwlrq gx surjudpph qh vh idlw txh vxu ghv

h{hpsohv wkìrultxhv1 Ohv gìyhorsshphqwv hq ìoìphqwv �qlv fodvvltxhv yrqw íwuh qrpeuhx{1

Elhq txh qh uhohydqw sdv gx grpdlqh gx eìwrq dupì/ pdlv soxw÷w ghv vwuxfwxuhv pì0

wdooltxhv/ rq shxw flwhu oh wudydlo h�hfwxì hq 4<;</ sdu gh Ylooh gh Jr|hw ^GhY;<`1 Idlvdqw

vxlwh dx{ wudydx{ gh Iuh| ^Iuh::`/ gh Ylooh gh Jr|hw ìwdeolw xq elodq ghv gl�ìuhqwhv irupx0

odwlrqv hq judqgv gìsodfhphqwv hw vh olyuh ã xqh frpsdudlvrq wuëv frpsoëwh gh gl�ìuhqwv

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txl prgìolvh oh jdxfklvvhphqw ghv vhfwlrqv1 Fhw ìoìphqw hvw lqwìjuì ã xqh ghvfulswlrq

frurwdwlrqqhooh/ txl hvw gìyhorssìh hq gìwdlov/ hw txl suìvhqwh gh qrpeuhx{ dydqwdjhv1

O*ìoìphqw dlqvl sursrvì v*dyëuh wuëv shuirupdqw/ oh vhxo ìoìphqw vxssruwdqw od frpsdudl0


vrq ìwdqw fhoxl gh Mhqqlqjv/ pdokhxuhxvhphqw olplwì dx{ sureoëphv sodqv1 Od ydolgdwlrq

v*h�hfwxh vxu xq judqg qrpeuh g*h{hpsohv/ dxvvl elhq wkìrultxhv tx*h{sìulphqwdx{1

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hvw edvì vxu ohv frrugrqqìhv frqyhfwìhv lqwurgxlwhv sdu Mhqqlqjv ^Mhq9;`1 Oh surjudpph hvw

ìwhqgx dx{ sureoëphv wulglphqvlrqqhov1 Od sulvh hq frpswh gh od qrq0olqìdulwì pdwìulhooh

hvw frpsoëwh1 Ohv h�hwv gxv ã o*h�ruw wudqfkdqw hw dx prphqw gh wruvlrq vrqw lqwurgxlwv/

pdlv hq idlvdqw o*k|srwkëvh g*xq frpsruwhphqw olqìdluh gx pdwìuldx vrxv vroolflwdwlrqv wdq0

jhqwhv1 Sdu frqwuh/ od qrq0olqìdulwì jìrpìwultxh q*hvw txh sduwlhoohphqw dssuìkhqgìh1 Hq

h�hw/ ohv gìsodfhphqwv grlyhqw uhvwhu prgìuìv/ hw ohv urwdwlrqv shwlwhv1 Oh surjudpph hvw

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qrxyhoohv orlv gh frpsruwhphqw v*dssx|dqw vxu od wkìrulh gh o*hqgrppdjhphqw1 Fhwwh

dssurfkh hvw glfwìh sdu oh fdofxo dx vìlvph/ srxu ohtxho oh fkdujhphqw hvw f|foltxh1 Oh

prgëoh g*hqgrppdjhphqw txl idlw uìiìuhqfh hq Iudqfh hvw fhoxl gìyhorssì sdu Oderu0

ghulh ^Ode<4`/ idlvdqw vxlwh dx{ wudydx{ gh Pd}duv ^Pd};7`1 Fh prgëoh hvw lvvx g*xqh

irupxodwlrq wkhuprg|qdpltxh ghv plolhx{ frqwlqxv1 Od orl gh frpsruwhphqw/ gdqv oh fdv

xqld{ldo/ v*h{sulph vhorq % @�"

��rü " hvw o*ìqhujlh oleuh gh Jleev1 O*xwlolvdwlrq gh fh

prgëoh qìfhvvlwh o*lghqwl�fdwlrq/ sdv wrxmrxuv dlvìh/ gh < sdudpëwuhv1 O*hqgrppdjhphqw

hq frpsuhvvlrq hvw glvwlqjxì gh fhoxl hq wudfwlrq1

Hq 4<<7/ Xop ^Xop<7` xwlolvh ìjdohphqw xq prgëoh g*hqgrppdjhphqw1 Lo dgrswh

xqh dssurfkh vhpl0joredoh = o*ìoìphqw srxwuh tx*lo xwlolvh hvw hq idlw glvfuìwlvì hq �euhv

sdudooëohv ã od oljqh pr|hqqh gh o*ìoìphqw1 Fkdtxh �euh glvsrvh g*xqh orl gh frpsruwhphqw/

g*xq hqgrppdjhphqw hw g*xq klvwrultxh1 Fhw ìoìphqw hvw lqvìuì gdqv xqh ghvfulswlrq

frurwdwlrqqhooh d�q gh suhqguh hq frpswh ohv h�hwv gh od qrq0olqìdulwì jìrpìwultxh1

416 Frqwhqx gh o*ìwxgh

Od suìvhqwh ìwxgh v*hvw gìurxoìh dx Fhqwuh ghv Whfkqltxhv Rxyudjhv g*Duw +FWRD, gx

Vhuylfh g*Ìwxghv Whfkqltxhv ghv Urxwhv hw Dxwrurxwhv +VHWUD,1 O*remhfwli �{ì ìwdlw g*lp0

sodqwhu gdqv xq surgxlw h{lvwdqw/ oh v|vwëph SFS +Srqwv Frqvwuxlwv sdu Skdvhv, ohv irupx0

odwlrqv qrq olqìdluhv ghv ìoìphqwv gh w|sh ��srxwuh��1 Dx gìsduw/ SFS hvw xq surjudpph

gh fdofxo olqìdluh g*rxyudjhv g*duw/ glvsrvdqw gh wrxwhv ohv irqfwlrqqdolwìv qìfhvvdluhv ã od

vlpxodwlrq gh od frqvwuxfwlrq ghv rxyudjhv = fkdujhphqwv txhofrqtxhv/ skdvdjh gh frq0

vwuxfwlrq/ plvh hq suìfrqwudlqwh skdvìh/ sulvh hq frpswh ghv h�hwv gl�ìuìv/ plvh hq whqvlrq

ghv kdxedqv111 O*hqvhpeoh gh fhv irqfwlrqqdolwìv ghydlw elhq hqwhqgx íwuh pdlqwhqx gdqv

oh fdguh g*xq fdofxo qrq olqìdluh1

Qrxv qrxv sursrvrqv g*| lqwurgxluh od qrq0olqìdulwì jìrpìwultxh hq judqgv gìsodfh0

416 Frqwhqx gh o*ìwxgh 64

phqwv hw hq judqghv urwdwlrqv ã sduwlu gh o*dqdo|vh qrq olqìdluh pdwìulhooh gh Qdlw0Udedk

^Qdl<3`1 Dx sdvvdjh/ fhwwh ghuqlëuh d ìwì dpìolruìh hq whuphv gh shuirupdqfhv1

Ã fhod/ qrxv vxshusrvhurqv ohv h�hwv gl�ìuìv/ ã vdyrlu oh xdjh hw oh uhwudlw gx eìwrq

dlqvl txh od uhod{dwlrq ghv dflhuv gh suìfrqwudlqwh/ gdqv oh fdguh ghv orlv gh frpsruwhphqw

uìjohphqwdluhv1

Hq rxwuh/ qrxv gìyhorsshurqv xqh sduwlh frqvdfuìh dx{ ìoìphqwv vsìfl�txhv dx Jìqlh

flylo = g*xqh sduw/ qrxv qrxv lqwìuhvvhurqv dx{ duwlfxodwlrqv +dx vhqv gh SFS/ lo v*djlw

g*xqh frqqh{lrq hqwuh ghx{ qúxgv lqlwldohphqw frqirqgxv jìrpìwultxhphqw, hq judqgv

gìsodfhphqwv hw hq judqghv urwdwlrqv> g*dxwuh sduw/ qrxv wudlwhurqv oh fdv ghv ìoìphqwv

elduwlfxoìv/ hw elduwlfxoìv dyhf h�hw gh fkdñqhwwh juåfh dx{txhov qrxv srxuurqv wudlwhu oh

sureoëph ghv srqwv vxvshqgxv1

Ã sduwlu gh od irupxodwlrq hq judqgv gìsodfhphqwv hw judqghv urwdwlrqv/ qrxv h{0

klehurqv od pdwulfh ghv frqwudlqwhv lqlwldohv/ txl shuphw g*h�hfwxhu ghv fdofxov g*lqvwdelolwì

olqìdluhv1

Qrxv yhuurqv txh o*dojrulwkph gh uìvroxwlrq plv hq sodfh shxw íwuh dydqwdjhxvhphqw

uìxwlolvì srxu h�hfwxhu xq fdofxo hq g|qdpltxh wudqvlwrluh/ ã o*dlgh g*xq vfkìpd gh Qhz0

pdun1

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gdqv ohv uëjohphqwv iudqêdlv hw lqwhuqdwlrqdx{1 F*hvw srxutxrl qrxv qh qrxv lqwìuhvvrqv sdv

dx{ prgëohv g*hqgrppdjhphqw/ gh gìyhorsshphqw wurs uìfhqw1 Hq rxwuh/ ìwdqw grqqì oh

qrpeuh vrxyhqw lpsruwdqw gh sdudpëwuhv ã lghqwl�hu/ hw oh qrpeuh surklelwli gh �euhv tx*lo

idxgudlw/ sdu h{hpsoh/ phwwuh hq sodfh srxu prgìolvhu oh Srqw gh Ploodx +f*hvw xqh �qdolwì

d!upìh gh SFS,/ lo qh qrxv d sdv sdux rssruwxq/ ã o*khxuh dfwxhooh/ gh ohv lpsodqwhu gdqv

oh surjudpph1

Od suìvhqwh ìwxgh frpsruwh dx wrwdo 45 fkdslwuhv1

Oh fkdslwuh 4 lqwurgxlw oh sureoëph jìqìudo/ hw sursrvh xq ìwdw gh od txhvwlrq1

Oh fkdslwuh 5 srvh xq fhuwdlq qrpeuh g*k|srwkëvhv frqfhuqdqw oh sureoëph txh o*rq vh

sursrvh gh uìvrxguh1 Qrxv suìvhqwrqv hq gìwdlov oh v|vwëph SFS/ dlqvl txh od idêrq grqw

| hvw lqwurgxlwh od suìfrqwudlqwh1

Oh fkdslwuh 6 suìvhqwh ohv gl�ìuhqwv irupdolvphv frxudpphqw xwlolvìv gdqv oh fdguh

ghv judqgv gìsodfhphqwv1 Sxlv/ gdqv xq suhplhu whpsv/ qrxv gìyhorssrqv o*dssurfkh hq

frrugrqqìhv frqyhfwìhv hq 5G/ sxlv/ gdqv xq ghx{lëph whpsv/ vrq h{whqvlrq hq 6G1

Oh fkdslwuh 7 gìyhorssh od sulvh hq frpswh gh od qrq0olqìdulwì pdwìulhooh1 Ã sduwlu

gh o*ìtxloleuh qrq olqìdluh g*xqh vhfwlrq gurlwh gh od srxwuh/ sdu lqwìjudwlrq oh orqj gh

o*ìoìphqw/ qrxv iruprqv od pdwulfh gh uljlglwì wulglphqvlrqqhooh gh o*ìoìphqw1

Oh fkdslwuh 8 suìvhqwh g*xqh sduw o*dojrulwkph gh uìvroxwlrq gh o*ìtxloleuh ghv vhfwlrqv/

hw g*dxwuh sduw o*dojrulwkph shuphwwdqw gh wurxyhu o*ìtxloleuh joredo gh od vwuxfwxuh/ dyhf


wrxv ohv uìjodjhv txl hq dvvxuhqw od urexvwhvvh1

Oh fkdslwuh 9 prqwuh frpphqw ohv h�hwv gl�ìuìv/ ã vdyrlu oh xdjh hw oh uhwudlw gx eìwrq/

dlqvl txh od uhod{dwlrq ghv dflhuv gh suìfrqwudlqwh/ vrqw lqwìjuìv gdqv o*dojrulwkph g*ìtxlol0

euh lqvwdqwdqì1 Soxvlhxuv pìwkrghv fodvvltxhv g*lqwìjudwlrq gx xdjh | vrqw suìvhqwìhv/

pdlv xqh vhxoh d ìwì frgìh1

Oh fkdslwuh : hvw frqvdfuì dx{ ìoìphqwv txh o*rq wurxyh kdelwxhoohphqw gdqv ohv sur0

judpphv rulhqwìv rxyudjhv g*duw1 Gdqv xq suhplhu whpsv/ qrxv dqdo|vrqv oh frpsruwhphqw

ghv frqqh{lrqv fhqvìhv ghphxuhu srqfwxhoohv hq shwlwv gìsodfhphqwv/ fh txl q*hvw soxv oh

fdv hq judqgv gìsodfhphqwv1 Qrxv qrxv lqwìuhvvrqv hqvxlwh dx{ ìoìphqwv elduwlfxoìv/ dlqvl

tx*dx{ ìoìphqwv elduwlfxoìv shvdqwv vxmhwv ã o*h�hw gh fkdñqhwwh1 Qrxv sursrvrqv xqh pì0

wkrgh srxu vlpxohu ohv skdvhv gh whqvlrq hw gh gìwhqvlrq gh fhv ìoìphqwv/ txl vxuylhqqhqw

sdu h{hpsoh gdqv oh fdv ghv vxvshqwhv ghv srqwv vxvshqgxv1

Oh fkdslwuh ; gìyhorssh oh fdofxo g*lqvwdelolwì ã sduwlu gh od pdwulfh ghv frqwudlqwhv

lqlwldohv lvvxh gh qrwuh irupxodwlrq hq judqgv gìsodfhphqwv1

Oh fkdslwuh < prqwuh o*h{whqvlrq srvvleoh gh qrwuh dojrulwkph gh uìvroxwlrq gh fkdujh0

phqw vwdwltxh dx fdv gh fkdujhphqw g|qdpltxh/ ã o*dlgh g*xq vfkìpd gh Qhzpdun1

Oh fkdslwuh 43 hvw frqvdfuì ã od ydolgdwlrq gx prgëoh1 Ohv uìvxowdwv rewhqxv sdu oh

surjudpph vrqw frqiurqwìv ã ghv uìvxowdwv h{sìulphqwdx{/ hw2rx srxu ohvtxhov rq glvsrvh

gh vroxwlrqv wkìrultxhv rx gh vlpxodwlrqv g*dxwuhv dxwhxuv1 Ohv h{hpsohv fkrlvlv vrqw ghv

srxwuhv frqwlqxhv/ ghv srwhdx{ lvrvwdwltxhv hw ghv sruwltxhv1 Ohv pdwìuldx{ xwlolvìv vrqw

oh eìwrq dupì/ oh eìwrq suìfrqwudlqw/ dlqvl txh oh pìwdo1

Od fkdslwuh 44 prqwuh od plvh hq úxyuh lqirupdwltxh gx prgëoh sursrvì/ hw o*xwlolvd0

wlrq gx surjudpph srxu xq surmhw uìho = od vlpxodwlrq gx uhpsodfhphqw gx v|vwëph gh

vxvshqvlrq gx Srqw gh Wdqfduylooh/ skdvh sdu skdvh1

Hq�q/ oh fkdslwuh 45 wluh ohv frqfoxvlrqv jìqìudohv ghv uìvxowdwv rewhqxv1

Fkdslwuh 5

Prgìolvdwlrq pìfdqltxh

514 K|srwkëvhv jìqìudohv

51414 Jìqìudolwìv

O*remhfwli gh fh wudydlo hvw g*lqwìjuhu dx vhlq g*xq surjudpph h{lvwdqw +oh v|vwëph SFS, xq

fhuwdlq qrpeuh gh irqfwlrqqdolwìv shuphwwdqw gh phqhu ghv fdofxov glwv ��qrq olqìdluhv��1

Ohv qrq0olqìdulwìv sulvhv hq frpswh frqfhuqhqw g*xqh sduw od qrq0olqìdulwì jìrpìwultxh hw

g*dxwuh sduw od qrq0olqìdulwì pdwìulhooh1

Oh v|vwëph SFS +Srqwv Frqvwuxlwv sdu Skdvhv, hvw xq orjlflho jìqìudo gh fdofxo gh

vwuxfwxuhv sduwlfxolëuhphqw ghvwlqì ã o*ìwxgh hw ã od yìul�fdwlrq uìjohphqwdluh ghv rxyudjhv

g*duw ^Sfs<9`1

Irqgì vxu od wkìrulh ghv srxwuhv/ lo wudlwh ohv vwuxfwxuhv wulglphqvlrqqhoohv txhofrqtxhv/

ìyhqwxhoohphqw suìfrqwudlqwhv1 Oh pdwìuldx frqvwlwxdqw fhv vwuxfwxuhv hvw vrlw oh eìwrq

vrlw o*dflhu1

Oh v|vwëph SFS vh fdudfwìulvh sdu =

� O*dftxlvlwlrq gluhfwh ghv grqqìhv jìrpìwultxhv gh fr�udjh/ gh suìfrqwudlqwh hw gh

ihuudloodjh sdvvli1

� Od srvvlelolwì gh vlpxohu od soxsduw ghv prghv gh frqvwuxfwlrq1

� Od gì�qlwlrq gh fdv gh fkdujh txhofrqtxhv/ qrwdpphqw ohv fkdujhv urxwlëuhv gì�qlhv

sdu oh uëjohphqw iudqêdlv hw sdu ohv Hxurfrghv1

Mxvtx*ã pdlqwhqdqw/ oh v|vwëph SFS xwlolvdlw xqh irupxodwlrq hq shwlwv gìsodfhphqwv

hq ìodvwlflwì olqìdluh1

67 Fkdslwuh 5 Prgìolvdwlrq pìfdqltxh

51415 Prgìolvdwlrq gh od vwuxfwxuh

Od vwuxfwxuh ìwxglìh hvw gìfrpsrvìh fodvvltxhphqw hq ìoìphqwv �qlv gh srxwuh 6G/ fkdtxh

qúxg srvvìgdqw 9 ghjuìv gh olehuwì = ohv 6 wudqvodwlrqv hw ohv 6 urwdwlrqv gdqv o*hvsdfh1

Ohv vwuxfwxuhv wudlwìhv sdu SFS shxyhqw íwuh frqvwlwxìhv g*xqh rx soxvlhxuv srxwuhv ã �euh

pr|hqqh gurlwh rx frxueh/ dsshoìhv srxwuhv vsdwldohv/ hw g*ìoìphqwv sulvpdwltxhv gurlwv

uholdqw fhv srxwuhv rx frqvwlwxdqw xq uìvhdx dxwrqrph1

Fkdtxh srxwuh frxueh hvw hq idlw prgìolvìh frpph xq dvvhpeodjh gh wurqêrqv gh srxwuhv

gurlwhv/ od frxuexuh ìwdqw frqfhqwuìh dx{ srlqwv gh glvfuìwlvdwlrq1 Dlqvl/ hooh q*dssdudñw

sdv gdqv ohv ìtxdwlrqv frqvwlwxwlyhv1 Fhwwh k|srwkëvh irxuqlw xqh erqqh dssur{lpdwlrq

oruvtxh o*rxyhuwxuh dqjxodluh gh fkdtxh wurqêrq qh gìsdvvh sdv 8 ghjuìv hw txh ohv glphq0

vlrqv wudqvyhuvdohv gh od srxwuh vsdwldoh vrqw lqiìulhxuhv rx ìjdohv dx gl{lëph gx ud|rq gh

frxuexuh gh od �euh pr|hqqh1

Ohv uhsëuhv gh wudydlo

Oh uhsëuh joredo hvw sdu sulqflsh xq uhsëuh �{h gdqv ohtxho rq gì�qlw jìrpìwultxh0

phqw od vwuxfwxuh1 F*hvw gdqv fh uhsëuh txh vhurqw srvlwlrqqìv ohv srxwuhv vsdwldohv hw ohv

ìoìphqwv sulvpdwltxhv gurlwv1

x0J 0

y0

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0z

Iljxuh 514 = Uhsëuh orfdo olì ã od srvlwlrq lqlwldoh gh o*ìoìphqw

Oh uhsëuh orfdo gh o*ìoìphqw hvw olì ã od srvlwlrq lqlwldoh gh o*ìoìphqw gdqv oh uhsëuh

joredo +�jxuh 514,1 O*ìoìphqw gh od vwuxfwxuh d srxu qúxg ruljlqh Lf/ oh fhqwuh gh judylwì gh

od vhfwlrq ruljlqh/ hw srxu qúxg h{wuìplwì Mf/ oh fhqwuh gh judylwì gh od vhfwlrq h{wuìplwì1

514 K|srwkëvhv jìqìudohv 68

Oh uhsëuh orfdo gh uìiìuhqfh olì ã od srvlwlrq lqlwldoh gh o*ìoìphqw vhud qrwì {f|f}f1

O*ruljlqh gx v|vwëph g*d{hv hvw hq Lf hw o*d{h {f srlqwh yhuv Mf1 O*rulhqwdwlrq gx uhsëuh

dxwrxu gh {f gìfrxoh gh fhooh g*xq uhsëuh pr|hq/ txl hvw od pr|hqqh hqwuh oh uhsëuh

sulqflsdo g*lqhuwlh gh od vhfwlrq ruljlqh gh o*ìoìphqw +hq Lf, hw oh uhsëuh sulqflsdo g*lqhuwlh

gh od vhfwlrq h{wuìplwì gh o*ìoìphqw +hq Mf,1

Ohv srxwuhv vsdwldohv

Ohxuv vhfwlrqv vrqw ã sdurlv plqfhv hw gh sur�o ihupì +fdlvvrq prqr rx pxowlfhooxodluh,

rx rxyhuw +vhfwlrqv qhuyxuìhv,1 Ohv fdudfwìulvwltxhv jìrpìwultxhv rx pìfdqltxhv gh fhv

vhfwlrqv shxyhqw ydulhu orqjlwxglqdohphqw1

Oh v|vwëph SFS glvfuìwlvh ohv srxwuhv vsdwldohv vhorq xq gìfrxsdjh/ suìdodeohphqw

gì�ql sdu o*xwlolvdwhxu/ hq ìoìphqwv sulvpdwltxhv gurlwv grqw ohv h{wuìplwìv frqvwlwxhqw

dxwdqw gh qúxgv gh mrqfwlrq ìyhqwxhov dyhf g*dxwuhv ìoìphqwv1

Vhxohv ohv srxwuhv vsdwldohv shxyhqw íwuh suìfrqwudlqwhv/ od jìrpìwulh gh od vhfwlrq ìwdqw

qìfhvvdluh ã od ghvfulswlrq gx ihuudloodjh dfwli hw sdvvli1

Oh fdofxo lqwhuyhqdqw dx qlyhdx ghv vhfwlrqv idlw dssho ã xq fhuwdlq qrpeuh g*k|0

srwkëvhv =

� Ohv vhfwlrqv sodqhv uhvwhqw sodqhv/ píph dsuëv gìirupdwlrq1

� Ohv vhfwlrqv vrqw vxssrvìhv wudqvyhuvdohphqw lqgìirupdeohv1

� Od wkìrulh ghv sur�oìv ã sdurlv plqfhv hvw xwlolvìh srxu oh fdofxo ghv frqwudlqwhv gh

flvdloohphqw/ pdlv ohv h�hwv gxv dx{ elprphqwv vrqw qìjoljìv1 Fhwwh k|srwkëvh hvw

oìjlwlph gdqv oh fdv ghv sur�ov ihupìv1

� Oh fhqwuh gh judylwì hw oh fhqwuh gh wruvlrq ghv vhfwlrqv +rx fhqwuh gh flvdloohphqw,

shxyhqw íwuh gl�ìuhqwv1 Oh eudv gh ohylhu dlqvl lqgxlw hvw fruuhfwhphqw sulv hq frpswh

srxu oh fdofxo hq wruvlrq1

Ìoìphqwv sulvpdwltxhv gurlwv

Fhv ìoìphqwv/ grqw rq qh frqqdñw sdv od jìrpìwulh gh od vhfwlrq pdlv xqltxhphqw ohv

fdudfwìulvwltxhv vhfwrulhoohv/ vrqw glwv vwdqgdugv vl ohv wurlv urwdwlrqv dx{ ghx{ h{wuìplwìv

vrqw eortxìhv hw elduwlfxoìv vl hoohv vrqw oleuhv1 Oh fhqwuh gh wruvlrq gh fhv ìoìphqwv shxw

íwuh glvwlqfw gx fhqwuh gh judylwì1 Od jìrpìwulh gh od vhfwlrq q*ìwdqw sdv frqqxh/ oh

pdwìuldx frqvwlwxwli gh fhv ìoìphqwv qh srxuud sdv glvsrvhu g*xqh orl gh frpsruwhphqw

qrq olqìdluh1 Fhshqgdqw/ frpph ohv ìoìphqwv elduwlfxoìv qh wudydloohqw tx*hq h�ruw qrupdo/

ohxu pdwìuldx srxuud dyrlu xq frpsruwhphqw qrq olqìdluh1 O*h�hw gþ ã od vwulfwlrq gh od

vhfwlrq vhud fhshqgdqw qìjoljì1


Od prgìolvdwlrq gh od vwuxfwxuh srxuud íwuh frpsoìwìh sdu ghv duwlfxodwlrqv hw ghv

dssxlv grqw oh frpsruwhphqw hvw vrlw uljlgh/ vrlw ìodvwltxh olqìdluh1 Fhuwdlqv ìoìphqwv

elduwlfxoìv srxuurqw íwuh gìfulwv frpph ghv fkdñqhwwhv d�q gh prgìolvhu hqfruh soxv �qh0

phqw ohv kdxedqv1 O*lqwurgxfwlrq g*duwlfxodwlrqv hqwuh ghx{ qúxgv shuphw ìyhqwxhoohphqw

gh oleìuhu od oldlvrq hqwuh ghx{ ìoìphqwv gh od vwuxfwxuh/ dxvvl elhq hq wudqvodwlrq tx*hq

urwdwlrq1

51416 Glvfuìwlvdwlrq gh od vwuxfwxuh

Glvfuìwlvdwlrq orqjlwxglqdoh

Iljxuh 515 = Glvfuìwlvdwlrq orqjlwxglqdoh

D�q gh suhqguh hq frpswh fruuhfwhphqw od qrq0olqìdulwì pdwìulhooh/ o*xwlolvdwhxu gì�qlw

od glvfuìwlvdwlrq orqjlwxglqdoh gx wurqêrq gh srxwuh hq lqgltxdqw oh qrpeuh gh vhfwlrqv

lqwhupìgldluhv ã jìqìuhu dxwrpdwltxhphqw +�jxuh 515,1 Od jìrpìwulh hw oh ihuudloodjh ghv

514 K|srwkëvhv jìqìudohv 6:

vhfwlrqv lqwhupìgldluhv vrqw gìgxlwv sdu lqwhusrodwlrq gh od jìrpìwulh hw gx ihuudloodjh

ghv vhfwlrqv g*h{wuìplwì gh o*ìoìphqw1 Fhwwh ghuqlëuh q*hvw srvvleoh txh vl ohv vhfwlrqv

g*h{wuìplwì rqw od píph pruskrorjlh1 Gdqv oh fdv frqwudluh/ dxfxqh vhfwlrq lqwhupìgldluh

q*hvw jìqìuìh hw o*lqwìjudwlrq gh od qrq0olqìdulwì pdwìulhooh oh orqj gh o*ìoìphqw vh idlw ã

sduwlu ghv ghx{ vhfwlrqv h{wuíphv vhxohphqw1 O*lqwhusrodwlrq vh idlw dydqw wrxw fkdujhphqw/

gdqv oh uhsëuh orfdo gh o*ìoìphqw1

Glvfuìwlvdwlrq ghv vhfwlrqv

y0

0z

I 0

Poin ts de déf in i t ion du contour

Trapèze é lémenta i re

Iljxuh 516 = Gìfrpsrvlwlrq hq wudsë}hv gh od vhfwlrq

Oh wudlwhphqw gh od qrq0olqìdulwì pdwìulhooh lqwhuylhqw h{foxvlyhphqw dx qlyhdx gh od vhf0

wlrq1

Od jìrpìwulh gh od vhfwlrq hvw lqwurgxlwh sdu o*xwlolvdwhxu gdqv oh uhsëuh jìqìultxh gh

fhwwh ghuqlëuh1 Od vhfwlrq uìhooh hvw dssur{lpìh sdu xqh vhfwlrq ã frqwrxuv sro|jrqdx{/ txl

suìvhqwh xq rx soxvlhxuv ìylghphqwv1 Oh surjudpph fdofxoh dxwrpdwltxhphqw od srvlwlrq

gx fhqwuh gh judylwì hw ohv d{hv sulqflsdx{ g*lqhuwlh gh od vhfwlrq qrq gìirupìh/ srxu gì�qlu

oh uhsëuh sulqflsdo g*lqhuwlh gh od vhfwlrq1 Fkdtxh vhfwlrq frqvwlwxwlyh g*xq ìoìphqw hvw

srvlwlrqqìh gdqv oh uhsëuh orfdo gh o*ìoìphqw sdu wudqvodwlrq hw urwdwlrq1 Gdqv fh uhsëuh/

fkdtxh vhfwlrq hvw gìfrpsrvìh hq wudsë}hv grqw ohv f÷wìv sdudooëohv vrqw krul}rqwdx{ +�jxuh

6; Fkdslwuh 5 Prgìolvdwlrq pìfdqltxh

516,1 Ohv whuphv gh od pdwulfh gh uljlglwì gh od vhfwlrq vrqw rewhqxv sdu lqwìjudwlrq vxu

wrxv ohv wudsë}hv frqvwlwxwliv gh od vhfwlrq ã o*dlgh g*xq pdloodjh g|qdpltxh txl vhud gìfulw

soxv wdug dx fkdslwuh 71

51417 Skdvdjh gh frqvwuxfwlrq 0 Fkdujhphqw

Oh v|vwëph SFS shuphw gh vlpxohu xq judqg qrpeuh g*rsìudwlrqv txl h{lvwhqw oruv gh od

uìdolvdwlrq g*xq rxyudjh/ ã vdyrlu =

� Od plvh hq sodfh hw oh uhpsodfhphqw dyhf rx vdqv yìulqdjh/ rx od vxssuhvvlrq ghv

dssxlv/ hq q*lpsruwh txho qúxg gx prgëoh1

� Oh gìsodfhphqw/ o*dfwlydwlrq hw od vxssuhvvlrq ghv ìoìphqwv gx prgëoh1

� Od plvh hq whqvlrq/ o*lqmhfwlrq hw od gìwhqwh ghv fåeohv gh suìfrqwudlqwh1

� O*dssolfdwlrq ghv fkdujhphqwv h{wìulhxuv/ l=h= =

� oh srlgv sursuh>

� ohv h�ruwv qrgdx{>

� ohv h�ruwv uìsduwlv vxu ohv ìoìphqwv>

� ohv glvwruvlrqv lpsrvìhv vxu ohv ìoìphqwv>

� ohv fkdujhphqwv wkhupltxhv/ dyhf qrwdpphqw oh judglhqw wkhupltxh>

� ohv gìirupdwlrqv g*dssxlv1

Oh skdvdjh gh frqvwuxfwlrq idlw grqf sduwlh lqwìjudqwh gx irqfwlrqqhphqw gx v|v0

wëph SFS hw frqvwlwxh xq gh vhv srlqwv iruwv/ qrwdpphqw sdu udssruw dx{ surjudpphv

jìqìudx{ g*ìoìphqwv �qlv1 O*lpsodqwdwlrq ghv irqfwlrqqdolwìv gh fdofxo qrq olqìdluh qh

grlw sdv shuwxuehu od sklorvrsklh gx surjudpph lqlwldo/ hw hooh grlw v*dgdswhu ã o*dvshfw

wuëv rsìudwlrqqho gx surgxlw1 Od urexvwhvvh gh o*dojrulwkph gh uìvroxwlrq grlw dlqvl íwuh

sduidlwhphqw dvvxuìh1

515 Ìwxgh gh od suìfrqwudlqwh

51514 Jìqìudolwìv

Od suìfrqwudlqwh hvw gì�qlh jìrpìwultxhphqw fåeoh sdu fåeoh/ srxwuh vsdwldoh sdu srxwuh

vsdwldoh1 Hooh hvw vsìfl�txh ã xqh srxwuh vsdwldoh/ hw q*djlw txh vxu ohv vhfwlrqv gh fhwwh

ghuqlëuh1

515 Ìwxgh gh od suìfrqwudlqwh 6<

Lo hvw srvvleoh gh gì�qlu xqh suìfrqwudlqwh h{wìulhxuh dx eìwrq = lo vx!w g*d�hfwhu dx

fåeoh hq txhvwlrq xq gldpëwuh gh jdlqh hw xq frh!flhqw gh shuwhv sdu iurwwhphqwv wuëv

idleohv1

51515 Jìrpìwulh gh od suìfrqwudlqwh

O*xwlolvdwhxu gì�qlw ghv srlqwv gh sdvvdjh ghv fåeohv gdqv ohv vhfwlrqv ghv srxwuhv vsd0

wldohv/ dyhf oh fdv ìfkìdqw/ ghv wdqjhqwhv lpsrvìhv1 Gx idlw gx fdudfwëuh wulglphqvlrqqho

gx sureoëph/ fkdtxh wurqêrq gh fåeoh fruuhvsrqgdqw ã xq ìoìphqw hvw lqwhusroì sdu ohv

ìtxdwlrqv sdudpìwultxhv vxlydqwhv =

[ +w, @ Y%� . Y%2 =w. Y%�=w2. Y%e =w

� +514,

\ +w, @ Y+� . Y+2 =w. Y+� =w2. Y+e =w

� +515,

] +w, @ Y5� . Y52 =w. Y5� =w2. Y5e =w

� +516,

srxu w 5 ^d�> e�`1 Oh sdudpëwuh duelwudluh w hvw who txh o*rq sdufrxuw gh idêrq elmhfwlyh oh

fåeoh oruvtxh w sdufrxuw o*lqwhuydooh ^d�> e�`1 Ohv 45 sdudpëwuhv shuphwwhqw gh yìul�hu ohv

frqglwlrqv lpsrvìhv dx{ srlqwv gh sdvvdjhv1

51516 Shuwhv lqvwdqwdqìhv gh suìfrqwudlqwh

Oruv gh od plvh hq whqvlrq ghv fåeohv gh suìfrqwudlqwh/ od whqvlrq lpsrvìh sdu ohv yìulqv

gdqv oh fåeoh yd vxelu ghv shuwhv lqvwdqwdqìhv txh o*rq shxw fodvvhu hq ghx{ fdwìjrulhv1

G*xqh sduw/ ohv shuwhv sdu iurwwhphqwv/ txl vrqw olìhv ã od jìrpìwulh gx fåeoh hw/ g*dxwuh

sduw/ ohv shuwhv gxhv dx uhfxo g*dqfudjh1

Shuwhv sdu iurwwhphqwv

Ã sduwlu ghv ìtxdwlrqv sdudpìwultxhv ghv fåeohv/ rq surfëgh ã xqh ìydoxdwlrq �qh ghv

devflvvhv fxuyloljqhv hw ghv gìyldwlrqv dqjxodluhv vsdwldohv ã o*dlgh gh srlqwv gh wdexodwlrq

+�jxuh 517,1 Ohv yhfwhxuv�$Y w& vrqw ghv yhfwhxuv wdqjhqwv qrupìv1 Od gìyldwlrq dqjxodluh

hqwuh�$Y w& hw

��$Y w&n� hvw ìydoxìh sdu =

��& @ dufvlq

��$Y w& a��$Y w&n�

�� +517,

Od gìyldwlrq dqjxodluh wrwdoh vxu oh wurqêrqP�P�n� hvw rewhqxh hq vrppdqw ohv ��&1 Rq

xwlolvh doruv od orl gh Frroh| srxu oh fdofxo gh fhv shuwhv1 Soxvlhxuv fdv vrqw ã hqylvdjhu =

41 Fåeoh whqgx sdu o*ruljlqh =

�f +v, @ �Rfh3Es�kEr�n)�r� +518,


Vti

V tk

V tk+1 V t

k+2

V ti+1

i

i+1

M

M

Iljxuh 517 = Wdexodwlrq g*xq wurqêrq gh fåeoh

51 Fåeoh whqgx sdu o*h{wuìplwì =

�u +v, @ �Rfh3Es�kEu�n)�u�nEs�kEr�n)�r� +519,

61 Fåeoh whqgx sdu ohv 5 h{wuìplwìv =

� +v, @ plq +�f +v, > �u +v,, +51:,

Dyhf =

�Rf = whqvlrq ã o*ruljlqh>

i = frh!flhqw gh iurwwhphqw hq frxueh gh o*dupdwxuh vxu od jdlqh�udg3�

�>

* = frh!flhqw gh shuwh gh whqvlrq sdu xqlwì gh orqjxhxu�p3��>

v = devflvvh fxuyloljqh gx srlqw rü o*rq fdofxoh od whqvlrq gdqv oh fåeoh>

� +v, = vrpph ghv ydohxuv devroxhv ghv gìyldwlrqv dqjxodluhv hqwuh o*ruljlqh hw oh

srlqw g*devflvvh fxuyloljqh v1

Shuwhv sdu uhfxo g*dqfudjh

Od shuwh gh whqvlrq/ txl shxw uìvxowhu gx jolvvhphqw gh o*dupdwxuh sdu udssruw ã vrq

dqfudjh/ gx wdvvhphqw rx gh od gìirupdwlrq gh o*dqfudjh/ hvw dsshoìh ��shuwh sdu uhfxo

g*dqfudjh�� ^Esho<4`1

Rq glvwlqjxh ohv píphv fdv tx*dxsdudydqw =

41 Oh fåeoh hvw whqgx sdu o*ruljlqh rx sdu o*h{wuìplwì = rq d doruv xq dqfudjh dfwli txl

hvw fhqvì íwuh vlwxì ã jdxfkh/ hw xq dqfudjh sdvvli txl hvw vlwxì ã gurlwh>

515 Ìwxgh gh od suìfrqwudlqwh 74

51 Oh fåeoh hvw whqgx sdu ohv ghx{ h{wuìplwìv1 O*ruguh gh uhoåfkhphqw ghv fåeohv hvw

doruv sulprugldo hw o*rq vxssrvhud txh o*dqfudjh gh jdxfkh hvw uhoåfkì oh suhplhu1

O*ruguh gh uhoåfkhphqw ghv fåeohv hvw gì�ql sdu o*xwlolvdwhxu1 Rq vh udpëqh dx

wudlwhphqw gx fdv suìfìghqw/ pdlv hq ghx{ irlv1

Oh gldjudpph gh whqvlrq lvvx gh od orl gh Frroh| hvw olqìdulvì sdu prufhdx{/ ã o*dlgh

gh srlqwv gh wdexodwlrq1 Rq glvsrvh dlqvl g*xqh oljqh eulvìh txl uhsuìvhqwh oh gldjudpph

gh od frqwudlqwh hq irqfwlrq gh o*devflvvh fxuyloljqh1 Srxu fkdtxh srlqw gh wdexodwlrq/ rq

wudqviruph oh gldjudpph gh od idêrq vxlydqwh +�jxuh 518, =

s

σDiagramme in i t ia l

D iagramme t rans fo rmé

(s)

Iljxuh 518 = H�hw gx uhfxo g*dqfudjh jdxfkh

41 Wrxv ohv vhjphqwv gh shqwh qìjdwlyh vlwxìv ã jdxfkh gx srlqw gh wdexodwlrq vrqw

wudqvirupìv hq vhjphqwv ã shqwh rssrvìh1

51 Wrxv ohv vhjphqwv ã shqwh srvlwlyh vrqw wudqvodwìv yhuv oh edv1

O*dluh vlwxìh hqwuh oh gldjudpph lqlwldo hw oh gldjudpph dlqvl wudqvirupì hvw qrwìhD,1

Rq fkhufkh oh suhplhu srlqw gx gldjudpph who txh =

D, A j=HR +51;,

rü j hvw oh uhfxo g*dqfudjh hw HR oh prgxoh g*\rxqj gh o*dflhu frqvwlwxdqw oh fåeoh gh

suìfrqwudlqwh1 Vl o*rq dwwhlqw o*dqfudjh sdvvli vdqv vdwlvidluh od frqglwlrq +51;,/ rq wudqvodwh

yhuv oh edv oh ghuqlhu gldjudpph rewhqx gh idêrq ã vdwlvidluh fhwwh frqglwlrq1


Oh fdv gh o*dqfudjh gurlw hvw wudlwì dyhf oh píph dojrulwkph/ sdu uhqyhuvhphqw gx vhqv

ghv devflvvhv fxuyloljqhv1

Vl dx frxuv gh fhw dojrulwkph/ rq rewlhqw xqh whqvlrq qxooh gdqv oh fåeoh/ oh wudlwhphqw

v*duuíwh1

Oh gldjudpph gh whqvlrq dlqvl rewhqx hvw oh gldjudpph gh whqvlrq lqvwdqwdqìh gx fåeoh

gh suìfrqwudlqwh1 Od suìfrqwudlqwh dlqvl gì�qlh hvw dssoltxìh vxu fkdfxqh ghv vhfwlrqv gh

od srxwuh vsdwldoh sdu od pìwkrgh lqwhuqh gìfulwh sdu Idxfkduw ^Idx:5`1 Od suìfrqwudlqwh

h{hufh vxu od vhfwlrq gx eìwrq xq h�ruw gh frpsuhvvlrq ìjdo hw rssrvì ã o*h�ruw gh wudfwlrq

gdqv oh fåeoh1 Fhw h�ruw hvw dssoltxì vxlydqw od wdqjhqwh ã od oljqh pr|hqqh gx fåeoh dx

srlqw rü lo wudyhuvh od vhfwlrq1

Xq frxolv gh flphqw shxw íwuh lqmhfwì gdqv od jdlqh gh suìfrqwudlqwh d�q gh uhqguh oh

fåeoh vrolgdluh gh od vhfwlrq hq eìwrq1 Vl who hvw oh fdv/ oh fåeoh hvw glw ��olì�� ã od vhfwlrq

gh eìwrq hw wrxwhv ohv gìirupdwlrqv xowìulhxuhv ã o*lqmhfwlrq vhurqw dssoltxìhv dx fåeoh txl

uìdjlud frpph xq dflhu sdvvli1 Vlqrq/ oh fåeoh srxuud jolvvhu gdqv vd jdlqh hw od yduldwlrq

gh frqwudlqwh gdqv oh fåeoh gxh dx jolvvhphqw vhud dssoltxìh ã od vhfwlrq gh eìwrq1

Od vlpxowdqìlwì gh plvh hq whqvlrq gh soxvlhxuv fåeohv gh suìfrqwudlqwh hvw dlqvl dx0

wrpdwltxhphqw sulvh hq frpswh sdu oh surjudpph/ píph vl oh skdvdjh gh plvh hq suìfrq0

wudlqwh hvw frpsoh{h1

Ohv h�hwv gxv ã od uhod{dwlrq ghv fåeohv vrqw sulv hq frpswh hw vhurqw gìwdlooìv gdqv oh

fkdslwuh 9 wudlwdqw ghv h�hwv gl�ìuìv1

Fkdslwuh 6

Qrq0olqìdulwì jìrpìwultxh

D�q g*ìwxglhu oh sureoëph gh od qrq0olqìdulwì jìrpìwultxh/ txhotxhv udsshov gh flqìpd0

wltxhv yrqw íwuh suìvhqwìv1 Dsuëv dyrlu udsshoì ohv gl�ìuhqwhv ghvfulswlrqv odjudqjlhqqhv

gx prxyhphqw/ qrxv ìwdeolurqv od irupxodwlrq gx sulqflsh ghv wudydx{ yluwxhov gdqv ohv

gl�ìuhqwhv ghvfulswlrqv ^Iuh::`/ ^Hvs;9d`/ ^GhY;<`1

Gdqv xq hvsdfh hxfolglhq/ rq ìwxglh oh prxyhphqw g*xq frusv frqvwlwxì g*xqh lq�qlwì

gh srlqwv pdwìulhov1 Vrxv o*h�hw gh vroolflwdwlrqv pìfdqltxhv/ fh frusv yd vh gìiruphu hw

vh gìsodfhu1 Qrxv doorqv vxlyuh oh wudmhw gh o*hqvhpeoh ghv srlqwv pdwìulhov frqvwlwxwliv

gx frusv1 O*hqvhpeoh ghv srvlwlrqv tx*rffxshqw fhv srlqwv ã fkdtxh lqvwdqw v*dsshooh xqh

frq�jxudwlrq1

614 Od ghvfulswlrq odjudqjlhqqh

Rq glvwlqjxh 6 frq�jxudwlrqv +�jxuh 614, =

� od frq�jxudwlrq lqlwldoh gh o*ìoìphqw1 Hooh fruuhvsrqg ã o*ìwdw dx uhsrv gx frusv/

txl qh v*hvw ql gìirupì/ ql gìsodfì1 Hooh vhud qrwìhf

�1 Ohv frrugrqqìhv g*xq srlqw

pdwìulho gdqv fhwwh frq�jxudwlrq vhurqw qrwìhvf{�>

� od frq�jxudwlrq frxudqwh1 Hooh vhud qrwìh��1 Ohv frrugrqqìhv gx píph srlqw pdwìulho

gdqv fhwwh frq�jxudwlrq vhurqw qrwìhv�{�>

� od frq�jxudwlrq srvwìulhxuh ã od frq�jxudwlrq frxudqwh/ lqfrqqxh d sulrul1 Hooh vhud

qrwìh�n{�� 1 Ohv frrugrqqìhv gx srlqw pdwìulho gdqv fhwwh frq�jxudwlrq vhurqw qrwìhv

�n{�{� 1

Rq gì�qlw fodvvltxhphqw ohv gìsodfhphqwv gh od idêrq vxlydqwh =

�x� @

�{� � f

{� +614,�n{�x� @

�n{�{� � f

{� +615,

77 Fkdslwuh 6 Qrq0olqìdulwì jìrpìwultxh

M o u v e m e n t1 1 1

2 2 2i0

Γ

τ+∆τ

τ

τ+∆ττ

0

x ,x ,x

0 τ+∆ττx ,x ,x

333

0 τ+∆ττx ,x ,x

γ

γ

x0

ix

ix

τ

τ+∆τ

Iljxuh 614 = Ohv 6 frq�jxudwlrqv fodvvltxhv

hw o*lqfuìphqw gh gìsodfhphqw hqwuh�� hw

�n{�� v*ìfulw =

x� @�n{�x� � �

x�

Vl o*rq suhqg srxu frq�jxudwlrq gh uìiìuhqfh od frq�jxudwlrq lqlwldohf

�/ od ghvfulswlrq

hvw glwh odjudqjlhqqh wrwdoh +G1O1W1,1 Vl o*rq suhqg srxu frq�jxudwlrq gh uìiìuhqfh od

frq�jxudwlrq frxudqwh��/ od ghvfulswlrq hvw glwh odjudqjlhqqh dfwxdolvìh +G1O1D1,1

Hq G1O1W1/ ohv gìulyìhv sduwlhoohv gh�n{�{� iruphqw ohv frpsrvdqwhv gx whqvhxu mdfrelhq =

�n{�

M��f

@C�n{�{�

Cf{�

+616,

Hq G1O1D1/ rq d =�n{�

M��

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^S�f ` @ ^L` .vlq ��f��f

k��f�

l.

4� frv ��f

�2�f

k��f�

l2+61445,

Oh gìyhorsshphqw olplwì gh ^S�f ` ã o*ruguh 4 hq ��f grqqh =

^S�f` @ ^L` .k��f�

l@

5997

4 ��5f ��+f

��5f 4 ��%f��+f ��%f 4

6::8 +61446,

Rq uhwurxyh doruv od pdwulfh elhq frqqxh ghv shwlwhv urwdwlrqv1 Qxpìultxhphqw/ o*ìyd0

oxdwlrq gh +61445, srvh xq sureoëph oruvtxh ��f hvw idleoh/ doruv txh pdwkìpdwltxhphqw/

lo q*| d dxfxqh dpeljxòwì1 Rq d frqvwuxlw dlqvl 5 h{suhvvlrqv gh ^S�f ` hq irqfwlrq gh od

ydohxu gh ��f1 Vl ��f hvw idleoh/ doruv ^S�f ` @ ^L` .k��f�

l.

�2

k��f�

l21 Vlqrq/ ^S�f ` hvw ìydoxì

dyhf +61445,1

Rq frqvwuxlw ìjdohphqw =

^S�f ` @ ^L` .vlq ��f��f

k��f�

l.

4� frv ��f

�2�f

k��f�

l2+61447,

dyhf

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6::8 +61448,


hw

��f @

t�2�%f

. �2�+f. �2�5f

+61449,

G*rü hq Lf =

^S�f `�c� @

�2�%f

�2�f+4� frv ��f, . frv ��f +6144:,

^S�f `�c2 @��%f��+f

�2�f+4� frv ��f,�

��5f��f

vlq ��f +6144;,

^S�f `�c� @��%f��5f

�2�f+4� frv ��f, .

��+f��f

vlq ��f +6144<,

^S�f `2c� @��%f��+f

�2�f+4� frv ��f, .

��5f��f

vlq ��f +61453,

^S�f `2c2 @

�2�+f

�2�f+4� frv ��f, . frv ��f +61454,

^S�f `2c� @��+f��5f

�2�f+4� frv ��f,�

��%f��f

vlq ��f +61455,

^S�f `�c� @��%f��5f

�2�f+4� frv ��f,�

��+f��f

vlq ��f +61456,

^S�f `�c2 @��+f��5f

�2�f+4� frv ��f, .

��%f��f

vlq ��f +61457,

^S�f `�c� @

�2�5f

�2�f+4� frv ��f, . frv ��f +61458,

Hw hq Mf =

^S�f `�c� @

�2�%f

�2�f+4� frv ��f, . frv ��f +61459,

^S�f `�c2 @��%f��+f

�2�f+4� frv ��f,�

��5f��f

vlq ��f +6145:,

^S�f `�c� @��%f��5f

�2�f+4� frv ��f, .

��+f��f

vlq ��f +6145;,

^S�f `2c� @��%f��+f

�2�f+4� frv ��f, .

��5f��f

vlq ��f +6145<,

^S�f `2c2 @

�2�+f

�2�f+4� frv ��f, . frv ��f +61463,

^S�f `2c� @��+f��5f

�2�f+4� frv ��f,�

��%f��f

vlq ��f +61464,


^S�f `�c� @��%f��5f

�2�f+4� frv ��f,�

��+f��f

vlq ��f +61465,

^S�f `�c2 @��+f��5f

�2�f+4� frv ��f, .

��%f��f

vlq ��f +61466,

^S�f `�c� @

�2�5f

�2�f+4� frv ��f, . frv ��f +61467,

Sdvvdjh gx uhsëuh orfdo dx uhsëuh lqwulqvëtxh

Srxu suhqguh hq frpswh/ gdqv od pdwulfh gh uljlglwì gh o*ìoìphqw/ ohv h�hwv ghv qrq0

olqìdulwìv gxhv dx frpsruwhphqw sk|vltxh ghv pdwìuldx{ frqvwlwxwliv hw gh ohxu dvvrfldwlrq

hw ohv h�hwv gh vhfrqg ruguh gxv dx{ gìsodfhphqwv ghv qúxgv/ o*ìwdw gh gìirupdwlrq gh

o*ìoìphqw hvw h�hfwxì gdqv oh uhsëuh {|} glw ��lqwulqvëtxh��/ olì ã od srvlwlrq gh o*ìoìphqw/

hq qìjoljhdqw od gìirupdwlrq sursuh gh o*ìoìphqw vxssrvì gh idleoh orqjxhxu1 Ohv h�hwv gh

od qrq0olqìdulwì jìrpìwultxh v*lqwurgxlvhqw gdqv oh sdvvdjh gx uhsëuh orfdo {f|f}f gì�ql

suìfìghpphqw dx uhsëuh lqwulqvëtxh {|}1

Oh uhsëuh lqwulqvëtxh {|} hvw gì�ql frpph vxlw = o*ruljlqh ghv d{hv hvw dx qúxg L1

O*d{h { srlqwdqw yhuv oh qúxg M / hvw sduidlwhphqw srvlwlrqqì sdu udssruw dx uhsëuh orfdo

{f|f}f/ ã o*dlgh ghv ghx{ dqjohv g*Hxohu � hw � txl gìshqghqw ghv gìsodfhphqwv uhodwliv gh

wudqvodwlrq ghv qúxgv L hw M +�jxuh 619,1

O*rulhqwdwlrq gx uhsëuh lqwulqvëtxh hvw rewhqxh hq frqvlgìudqw xqh urwdwlrq � dxwrxu

gh o*d{h { +�jxuh 61:,1

Oh sdvvdjh gx uhsëuh orfdo {f|f}f dx uhsëuh lqwulqvëtxh {|} hvw h�hfwxì dx pr|hq gh

od pdwulfh ^Skq�` gì�qlh dx vhqv g*Hxohu/ f*hvw0ã0gluh txh o*rq wrxuqh dxwrxu ghv qrxyhdx{

d{hv =

+[,E%f+f5f� @ ^Skq� ` =+[,E%+5� +61468,

Rq glvsrvh grqf gh wurlv dqjohv gh urwdwlrqv hw gh wurlv pdwulfhv dvvrflìhv =

^Sk` @

5997

frv+�, � vlq+�, 3

vlq+�, frv+�, 3

3 3 4

6::8 +61469,

O*dqjoh � uhsuìvhqwh od urwdwlrq dxwrxu gh o*d{h }f/ gdqv oh sodq shushqglfxodluh ã o*d{h

}f =

� @ dufwdq

�y

Of . x

�+6146:,

rü Of uhsuìvhqwh od orqjxhxu lqlwldoh gh o*ìoìphqw dydqw gìirupdwlrq1 ^Sk` hvw gì�qlh sdu =

+[,E%f+f5f�@ ^Sk` = +[,E%�+�5f�

+6146;,


α y

z

y

α

x

x

xβ

β

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Iljxuh 619 = Srvlwlrq gh o*d{h { gx uhsëuh lqwulqvëtxh sdu udssruw dx uhsëuh orfdo {f|f}f

Srxu od ghx{lëph urwdwlrq/ lo ylhqw =

^Sq ` @

5997

frv+�, 3 vlq+�,

3 4 3

� vlq+�, 3 frv+�,

6::8 +6146<,

O*dqjoh � uhsuìvhqwh od urwdwlrq dxwrxu gh od gluhfwlrq |�/ gdqv oh sodq shushqglfxodluh ã

o*d{h |�1

� @ �dufwdq

#zs

+Of . x,2 . y2

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^Sq` hvw gì�qlh sdu =

+[,E%�+�5f�@ ^Sq` = +[,E%+�5��

+61474,


γy

x

z

I

1

1

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Iljxuh 61: = Srvlwlrq gx uhsëuh lqwulqvëtxh sdu udssruw dx uhsëuh {|�}�

Hq srvdqw =

O @

t+Of . x,

2. y2 .z2 +61475,

d @Of . x

O+61476,

e @y

O+61477,

f @z

O+61478,

rq uhpdutxh/ dsuëv vlpsol�fdwlrq/ txh =

^Sk` ^Sq` @

5997d 3KI

@2nK23@SI@2nK2

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3KSI@2nK2

f 3sd2 . e2

6::8 +61479,

^S�` uhsuìvhqwh od pdwulfh gh wudqvirupdwlrq txl suhqg hq frpswh od urwdwlrq gh o*ìoìphqw

dxwrxu gh o*d{h {/ hw hvw grqqìh sdu =

617 Ohv frrugrqqìhv frqyhfwìhv 9:

^S�` @

5997

4 3 3

3 frv+�, � vlq+�,

3 vlq+�, frv+�,

6::8 +6147:,

rü � hvw od urwdwlrq gh frusv uljlgh dxwrxu gh o*d{h {1 Fhw dqjoh hvw gì�ql hq suhqdqw od

pr|hqqh ghv urwdwlrqv ghv qúxgv L hw M dxwrxu gh o*d{h {f gx uhsëuh orfdo =

� @��%f . ��%f

5+6147;,

^S�` hvw gì�qlh sdu =

+[,E%+�5��@ ^S�` = +[,E%+5�

Od pdwulfh ^Skq� ` shxw grqf v*ìfuluh =

^Skq� ` @ ^Sk` ^Sq` ^S�`

@

5997d �K ULt �n@S t�? �I

@2nK2�3K t�? �n@S ULt �I

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e �3@ ULt �nKS t�? �I@2nK2

�@ t�? �nKS ULt �I@2nK2

fsd2 . e2 vlq �

sd2 . e2 frv �

6::8 +6147<,

Ohv frrugrqqìhv frqyhfwìhv

Hq 6G/ ohv frrugrqqìhv frqyhfwìhv vrqw dx qrpeuh gh 91 Frpph suìfìghpphqw/ rq qrwhud

sdu V? oh yhfwhxu uhsuìvhqwdqw ohv gìsodfhphqwv gdqv oh uhsëuh lqwulqvëtxh {|}1

V? @

3EEEEEEEEEC

h

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rü h uhsuìvhqwh od yduldwlrq gh orqjxhxu gh o*ìoìphqw =

h @ O�Of +61484,

hw �% od urwdwlrq uhodwlyh gh wruvlrq ghv qúxgv M hw L =

�% @ ��% � ��% +61485,

��% / ��+ / ��5 +uhvshfwlyhphqw ��% / ��+ / ��5, vrqw ohv urwdwlrqv gx qúxg L +uhvshfwlyhphqw

M,/ dxwrxu ghv d{hv {/ | hw }1 Vrlw I? oh yhfwhxu uhsuìvhqwdqw ohv h�ruwv lqgìshqgdqwv


Iljxuh 61; = Surmhfwlrq gh o*ìoìphqw gdqv oh sodq {} gx uhsëuh lqwulqvëtxh {|}

fruuhvsrqgdqw ã V? gdqv oh uhsëuh lqwulqvëtxh =

I? @

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617 Ohv frrugrqqìhv frqyhfwìhv 9<

Iljxuh 61< = Surmhfwlrq gh o*ìoìphqw gdqv oh sodq {| gx uhsëuh lqwulqvëtxh {|}

I? hvw xq yhfwhxu grqw ohv frpsrvdqwhv vrqw h{wudlwhv gx yhfwhxu I% = oh yhfwhxu ghv 45

h�ruwv dx{ h{wuìplwìv gh o*ìoìphqw gdqv oh uhsëuh lqwulqvëtxh1 Rq d =

I% @


Q�

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Od uhodwlrq hqwuh I? hw I% hvw gìgxlwh ghv 9 ìtxdwlrqv g*ìtxloleuh gh o*ìoìphqw gdqv oh

uhsëuh lqwulqvëtxh =

I% @|^W ` =I? +61488,

:3 Fkdslwuh 6 Qrq0olqìdulwì jìrpìwultxh

dyhf =

^W ` @

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�4 3 3 3 3 3 4 3 3 3 3 3

3 3 � �u

3 4 3 3 3�u

3 3 3

3�u

3 3 3 4 3 � �u

3 3 3 3

3 3 3 �4 3 3 3 3 3 4 3 3

3 3 � �u

3 3 3 3 3�u

3 4 3

3�u

3 3 3 3 3 � �u

3 3 3 4

6:::::::::8

+61489,

Hq�q/ rq qrwh sdu ^S�` +uhvshfwlyhphqw ^S�`, od pdwulfh +6� 6, ruwkrjrqdoh gh sdvvdjh

gx uhsëuh lqwulqvëtxh dx uhsëuh�h�% > h�+ > h�5

�+uhvshfwlyhphqw

�h�% > h�+ > h�5

�, olì dx qúxg L

+uhvshfwlyhphqw M,1 Rq vxssrvh txh o*ìoìphqw hvw vx!vdpphqw shwlw srxu idluh o*k|srwkëvh

ghv shwlwhv urwdwlrqv gdqv oh uhsëuh lqwulqvëtxh1

+[,E%+5� @ ^S�` = +[,Ee�%e�+e�5 �+6148:,

+[,E%+5� @ ^S� ` = +[,Ee�%e�+ e�5 �+6148;,

Sdu dqdorjlh ã ^S�f ` hw ^S�f `/ hw hq olqìdulvdqw/ rq rewlhqw =

^S�` @

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4 ��5 ��+

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6::8 +6148<,

^S�` @

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6::8 +61493,

Rq uhpdutxh txh ^S�` hw ^S�` qh vrqw sdv ghv pdwulfhv ruwkrjrqdohv/ gx idlw gh ohxu

olqìdulvdwlrq1

Pdwulfh gh uljlglwì gdqv oh uhsëuh lqwulqvëtxh

Od uhodwlrq pdwulflhooh oldqw ohv dffurlvvhphqwv g*h�ruwv dx{ dffurlvvhphqwv ghv gìsodfh0

phqwv ghv qúxgv gh o*ìoìphqw/ gdqv oh uhsëuh lqwulqvëtxh {|}/ hvw od vxlydqwh =

7I? .D? @ ^N?` =7 V? +61494,

rü oh yhfwhxu D? uhsuìvhqwh o*dfwlrq gh od suìfrqwudlqwh/ hw od pdwulfh ^N?` od pdwulfh gh

uljlglwì gh o*ìoìphqw gdqv oh uhsëuh lqwulqvëtxh {|}1 Od irupxodwlrq gh fhv 5 hqwlwìv vhud

gìwdlooìh gdqv oh fkdslwuh 7 frqvdfuì ã od qrq0olqìdulwì pdwìulhooh1

617 Ohv frrugrqqìhv frqyhfwìhv :4

Pdwulfh gh uljlglwì gdqv oh uhsëuh orfdo

Ohv k|srwkëvhv hw qrwdwlrqv qrxv frqgxlvhqw qdwxuhoohphqw ã ìfuluh ohv ghx{ uhodwlrqv oldqw

ohv pdwulfhv ^S�f `/ ^S�f`/ ^S�`/ ^S� `/ hw ^Skq�`1

^S�f ` @ ^Skq�` = ^S�` +61495,

^S�f ` @ ^Skq�` = ^S� ` +61496,

Fhfl hvw od wudgxfwlrq pdwulflhooh gx idlw txh oh prxyhphqw g*xqh sduwlfxoh hvw od frpsr0

vlwlrq g*xq prxyhphqw gh frusv uljlgh pdwìuldolvì sdu ^Skq� ` hw g*xqh gìirupdwlrq sursuh

pdwìuldolvìh sdu ^S�` hq L hw ^S�` hq M 1 Fh srlqw gh yxh uhmrlqw fhoxl gh od ghvfulswlrq

odjudqjlhqqh dfwxdolvìh dssurfkìh1 Qrxv yrxorqv h{wudluh gh fhv uhodwlrqv ohv frrugrqqìhv

frqyhfwìhv/ f*hvw0ã0gluh oh yhfwhxu V?1 Rq shxw ìfuluh/ sxlvtxh ^Skq� ` hvw ruwkrjrqdoh =

^S�` @ ^Skq� `3�

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^S�` @ ^Skq� `3�

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Fhv ghx{ uhodwlrqv pdwulflhoohv vrqw ìtxlydohqwhv ã 5�< ìtxdwlrqv vfdodluhv uholdqw ��% / ��+ /

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q*dyrqv hq idlw txh wurlv lqfrqqxhv = ��% / ��+ hw ��5 1 O*k|srwkëvh ghv shwlwhv urwdwlrqv gdqv

oh uhsëuh lqwulqvëtxh {|} shuphw gh vlpsol�hu/ hw píph gh uhqguh srvvleoh/ o*h{wudfwlrq

ghv frpsrvdqwhv gh V?1 Oh fdofxo gh V? hvw phqì ã o*dlgh g*xq orjlflho gh fdofxo irupho/

Pdsoh Y1

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rx g*ìydoxdwlrqv gh irqfwlrqv wuljrqrpìwultxhv =^S�`�c2 = 56 dgglwlrqv 85 pxowlsolfdwlrqv 9 irqfwlrqv 5; d�hfwdwlrqv

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^S�`�c� = 87 dgglwlrqv 473 pxowlsolfdwlrqv 43 irqfwlrqv 7; d�hfwdwlrqv

^S�`2c� = 85 dgglwlrqv 453 pxowlsolfdwlrqv 43 irqfwlrqv 84 d�hfwdwlrqv

^S�`�c2 = 86 dgglwlrqv 46< pxowlsolfdwlrqv 43 irqfwlrqv 84 d�hfwdwlrqvRq fkrlvlud grqf ohv whuphv ohv prlqv frþwhx{/ ã vdyrlu ^S�`�c2/ ^S�`�c�/hw ^S�`2c�1 Od

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h{dfwh srxu fhv ghx{ pdwulfhv/ q*lpsruwh txho fkrl{ gh wurlv ìtxdwlrqv sdupl ohv qhxi/

derxwludlw dx píph uìvxowdw1 Lfl/ oh fkrl{ gh fhv wurlv ìtxdwlrqv vh mxvwl�h sdu oh idlw


tx*hoohv hqjhqguhurqw ghv fdofxov soxv vlpsohv1 Rq d grqf =

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olqìdulvdqw ohv ìtxdwlrqv uhwhqxhv/ rq rewlhqw =

��% @ ��%f � � +614:5,

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shupxwhqw hw vrqw dgglwlyhv1

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pdwulflhooh od uhodwlrq oldqw ohv dffurlvvhphqwv gh gìsodfhphqwv 7V? ghv qúxgv gdqv oh

uhsëuh lqwulqvëtxh dx{ dffurlvvhphqwv gh gìsodfhphqwv 7V� ghv qúxgv gdqv oh uhsëuh

orfdo1

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rü ^E` uhsuìvhqwh xqh pdwulfh +9� <, gh wudqvirupdwlrq jìrpìwultxh/ sduidlwhphqw dqd0


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C��+

C��5f

C��5

Cx

C��5

Cy

C��5

Cz3 3 3

C��5

C��%f

C��5

C��+f

C��5

C��5f

6:::::::::::::::::::::::::::::::8

+614:<,

Vhorq oh wkìruëph ghv wudydx{ yluwxhov/ oh wudydlo yluwxho gdqv oh uhsëuh lqwulqvëtxh ghv

irufhv I?/ dssoltxìhv dx{ qúxgv/ gþ ã xq gìsodfhphqw yluwxho �V? hvw ìjdo dx wudydlo gdqv

oh uhsëuh orfdo ghv irufhv I� dssoltxìhv dx{ qúxgv/ gþ ã xq gìsodfhphqw yluwxho �V� =

|I?=�V? @|I�=�V� +614;3,

Ohv yduldwlrqv ghv gìsodfhphqwv �V? hw �V� vrqw flqìpdwltxhphqw dgplvvleohv/ sdu frq0

vìtxhqw =|I?= Ê` =�V� @

|I�=�V� +614;4,

Od yduldwlrq �V� hvw duelwudluh/ g*rü =

I� @|Ê` =I? +614;5,

Od uhodwlrq +614;5, hqwuh I� hw I? hvw xqh uhodwlrq h{dfwh/ hw qrq sdv wdqjhqwh1 Hq

gl�ìuhqfldqw/ rq rewlhqw =

�I� @|Ê` =�I? .

|^�E` =I? +614;6,

G*rü =

�I� .D� @|Ê` ^N?` Ê` =�V� .

|^�E` =I? +614;7,

dyhf =

D� @|Ê` =D? +614;8,


Frqvlgìurqv oh whuph |^�E`1 Frpph hq 5G/ rq srvh =

G�& @CE��

CV�&I?� +614;9,

g*rü = �|g Ê` =I?

��@ G�& gV�& +614;:,

Rq shxw grqf ìfuluh =|^�E` =I? @ ^G` =�V� +614;;,

Ilqdohphqw/ lo ylhqw =

�I� .D� @�|Ê` ^N?` Ê` . ^G`

�=�V� +614;<,

Hq qrwdqw =

^N�` @�|Ê` ^N?` Ê` . ^G`

�+614<3,

rq rewlhqw od uhodwlrq vxlydqwh =

�I� .D� @ ^N�` =�V� +614<4,

^N�` fruuhvsrqg ã xqh irupxodwlrq wdqjhqwh gx sureoëph = f*hvw xq rsìudwhxu wdqjhqw

txl vhud xwlolvì frpph rsìudwhxu gh frqyhujhqfh1

Od uhodwlrq oldqw ohv dffurlvvhphqwv g*h�ruwv dx{ dffurlvvhphqwv gh gìsodfhphqwv ghv

qúxgv gh o*ìoìphqw gdqv oh uhsëuh orfdo v*ìfulw =

�Iu .Du @ ^Nu` =�Vu +614<5,

rü ^Nu` hvw od pdwulfh gh uljlglwì v|pìwultxh +45� 45, gh o*ìoìphqw gdqv oh uhsëuh orfdo =

^Nu` @|^Wf` ^N�` ^Wf` +614<6,

hw =

Du @|^Wf` =D� +614<7,

Od pdwulfh ^Wf` hvw gì�qlh sdu +61438,1

Od irupxodwlrq hq judqgv gìsodfhphqwv hvw hqwlëuhphqw frqwhqxh gdqv o*h{suhvvlrq ghv

pdwulfhv Ê` hw ^G`1 Vl o*rq h�hfwxh xq gìyhorsshphqw olplwì ã o*ruguh 4 pdwkìpdwltxh/

rq rewlhqw =

Ê�` @

5999999999997

4�uf

�uf

3 3 3 3 3 3

� �u2f

�w�%fuf

uf3�u2f

w�5f2 � �

uf4

w�%f2 3 3 3

�u2f

�uf3�u2f

�w�%fuf

�w�+f2 � �

uf�w�%f

2 4 3 3 3

3w�+f

3w�+fuf

3 �4 w�5f2 � �

uf

w�+f2 4 �w�5f

2 .�uf

�w�+f2

� �u2f

�w�%fuf

uf3�u2f

3 3 3w�5f2 � �

uf4

w�%f2

�u2f

�uf3�u2f

�w�%fuf

3 3 3 �w�+f2 � �

uf�w�%f

2 4

6:::::::::::8

+614<8,


Oh gìyhorsshphqw pdwkìpdwltxh gh ^G` ã o*ruguh 4 hvw wurs yroxplqhx{ srxu íwuh

h{solflwì lfl1

Gdqv ^E�`/ o*ruguh 4 pdwkìpdwltxh fruuhvsrqg dx vhfrqg ruguh pìfdqltxh1 Rq shxw

ìjdohphqw uhpdutxhu tx*hq jìqìudo/ rq hqwhqg sdu ��judqgv gìsodfhphqwv�� od sulvh hq

frpswh gx udffrxuflvvhphqw gh o*ìoìphqw oruv g*xqh urwdwlrq dxwrxu gh o*d{h | rx }1

Xq h{hpsoh vlpsoh

Hwxglrqv o*h{hpsoh vlpsoh txl vxlw =

I J0

F

PJ

v

uL

0

X

Y

Iljxuh 6143 = Srxwuh frqvroh fkdujìh hq wíwh

Lfl/ x ? 3/ y ? 3/ S ? 3/ I ? 3/ Of A 31 O*h{suhvvlrq gh ^E�` qrxv pëqh ã ìfuluh/ gdqv

fh fdv sduwlfxolhu hq 5 glphqvlrqv =

I�%f @ Q� .y

O2f+P�5 .P�5, @ S +614<9,

I�+f @y

OfQ� � Of � x

O2f+P�5 .P�5, @ I +614<:,

P�5f@ P�5 +614<;,

P�5f@ P�5 @ 3 +614<<,

G*rü =

Q� @ S � y

O2fP�5f

+61533,

IOf � Sy @

�x

Of� 4

�P�5f

+61534,


Vl rq v*hq wlhqw dx{ whuphv g*ruguh 4/ lo ylhqw =

P�5f@ F @ Sy � I +Of . x, +61535,

Gdqv fhw h{hpsoh/ rq shxw gluh txh ^E�` hqwuh gdqv oh fdguh ghv judqgv gìsodfhphqwv1

Oh frgh gh fdofxo yìul�h o*ìtxloleuh ghv qúxgv dyhf o*h{suhvvlrq h{dfwh gh ^E`1 Rq hvw

doruv gdqv oh fdguh ghv judqgv gìsodfhphqwv hw ghv judqghv urwdwlrqv1

Pdwulfh gh uljlglwì gdqv oh uhsëuh �{h olì ã od vwuxfwxuh

Od vwuxfwxuh ìwxglìh hvw uhsìuìh duelwudluhphqw gdqv xq uìiìuhqwlho joredo R[\ ] ru0

wkrqrupì gluhfw1 Srxu shuphwwuh o*dvvhpeodjh gh od pdwulfh gh uljlglwì gh od vwuxfwxuh/ lo

hvw qìfhvvdluh gh iruphu ohv pdwulfhv gh uljlglwì ghv ìoìphqwv gì�qlvvdqw od vwuxfwxuh/ gdqv

oh uhsëuh R[\ ]1

Rq qrwh sdu If oh yhfwhxu frqwhqdqw ohv h�ruwv dssoltxìv dx{ h{wuìplwìv g*xq ìoìphqw/

gdqv oh uhsëuh joredo R[\ ]1

If @


I�f

I�t

I�~

P�f

P�t

P�~

I�f

I�t

I�~

P�f

P�t

P�~


+61536,

Gh od píph idêrq/ rq qrwh Vf oh yhfwhxu frqwhqdqw ohv gìsodfhphqwv dx{ h{wuìplwìv

617 Ohv frrugrqqìhv frqyhfwìhv ::

gh o*ìoìphqw gdqv oh uhsëuh devrox R[\ ]1

Vf @


X�

Y�

Z�

��f

��t

��~

X�

Y�

Z�

��f

��t

��~


+61537,

Oh sdvvdjh gx uhsëuh devrox R[\ ] dx uhsëuh orfdo {f|f}f hvw lqgìshqgdqw ghv h�ruwv

hw ghv gìsodfhphqwv vxelv sdu o*ìoìphqw/ pdlv lo qh gìshqg txh gh o*rulhqwdwlrq lqlwldoh gh

o*ìoìphqw1 Od urwdwlrq vsdwldoh gx uhsëuh devrox R[\ ] grqqdqw oh uhsëuh orfdo {f|f}fg*xq ìoìphqw shxw íwuh gìfrpsrvìh hq wurlv urwdwlrqv �f/ �f hw �f dx vhqv g*Hxohu1 Od

pdwulfh gh sdvvdjh ^sf` hvw od frpsrvìh gh 6 pdwulfhv ruwkrjrqdohv ^skf `/�sqf

�hw�s�f

�1

^sf` @ ^skf `�sqf

� �s�f

�+61538,

dyhf

^skf ` @

5997

frv +�f, � vlq +�f, 3

vlq +�f, frv +�f, 3

3 3 4

6::8 +61539,

�sqf

�@

5997

frv +�f, 3 vlq +�f,

3 4 3

� vlq +�f, 3 frv +�f,

6::8 +6153:,

�s�f

�@

5997

4 3 3

3 frv +�f, � vlq +�f,

3 vlq +�f, frv +�f,

6::8 +6153;,

Od uhodwlrq pdwulflhooh uholdqw ohv gìsodfhphqwv ghv qúxgv Vf gdqv oh uhsëuh devrox hw

ohv gìsodfhphqwv ghv qúxgv Vu gdqv oh uhsëuh orfdo v*ìfulw =

Vf @ ^Sf` =Vu +6153<,

:; Fkdslwuh 6 Qrq0olqìdulwì jìrpìwultxh

dyhf

^Sf` @

599997

^sf` 3

^sf`

^sf`

3 ^sf`

6::::8 +61543,

^Sf` hvw xqh pdwulfh ruwkrjrqdoh =

^Sf`3�

@|^Sf` +61544,

G*rü =

Vu @|^Sf` =Vf +61545,

Hq fh txl frqfhuqh ohv h�ruwv/ rq rewlhqw =

If @ ^Sf` =Iu +61546,

G*rü =

�If .Df @ ^Sf` ^Nu` =�Vu +61547,

dyhf

Df @ ^Sf` =Du +61548,

Dlqvl/ od uhodwlrq pdwulflhooh uholdqw ohv dffurlvvhphqwv g*h�ruwv hw ohv dffurlvvhphqwv

gh gìsodfhphqwv dx{ h{wuìplwìv g*xq ìoìphqw gdqv oh uhsëuh devrox R[\ ] v*ìfulw =

�If .Df @ ^Nf ` =�Vf +61549,

dyhf

^Nf ` @ ^Sf` ^Nu`|^Sf` +6154:,

^Nf ` @ ^Sf`|^Wf`

�|Ê` ^N?` Ê` . ^G`

�^Wf`

|^Sf` +6154;,

rü ^Nf ` uhsuìvhqwh od pdwulfh gh uljlglwì +45� 45, g*xq ìoìphqw gdqv oh uhsëuh devrox

R[\ ]1

Frqfoxvlrq

Od pdwulfh gh uljlglwì gh o*ìoìphqw ^Nf ` gdqv oh uhsëuh devrox v*h{sulph hq irqfwlrq gh

od pdwulfh gh uljlglwì gh o*ìoìphqw ^N?` gdqv oh uhsëuh lqwulqvëtxh gh o*ìoìphqw1 Ohv pd0

wulfhv ^Sf` hw ^Wf` vrqw ghv pdwulfhv frqvwdqwhv/ f*hvw0ã0gluh tx*hoohv qh gìshqghqw sdv

ghv gìsodfhphqwv gh o*ìoìphqw1 Ohv pdwulfhv Ê` hw ^G` suhqqhqw hq frpswh ohv gìsodfh0

phqwv gh o*ìoìphqw1 Od pdwulfh ^N?` wlhqw frpswh/ txdqw ã hooh/ ghv gìirupdwlrqv sursuhv

gh o*ìoìphqw gdqv vrq uhsëuh lqwulqvëtxh1 Frpph rq oh yhuud gdqv fh txl vxlw/ od qrq0

olqìdulwì pdwìulhooh qh v*h{sulph txh gdqv +61494,1 Od irupxodwlrq hq judqgv gìsodfhphqwv

hw judqghv urwdwlrqv hvw hqwlëuhphqw frqwhqxh gdqv o*h{suhvvlrq ghv pdwulfhv Ê` hw ^G`1

618 Prgìolvdwlrq sdu ìoìphqwv �qlv gh w|sh srxwuh ��gìsodfhphqw�� :<

Gx srlqw gh yxh dojrulwkpltxh/ oh vfkìpd lwìudwli v*dssxlhud vxu ghx{ rsìudwhxuv = xq

rsìudwhxu gh frqyhujhqfh hw xq rsìudwhxu g*duuíw +txl fruuhvsrqg ã o*ìtxloleuh gh fkdfxq

ghv qúxgv gh od vwuxfwxuh,1

O*rsìudwhxu gh frqyhujhqfh vhud ^N�` @�|Ê` ^N?` Ê` . ^G`

�hw oh whvw g*duuíw fruuhv0

srqgud ã o*ìtxloleuh ghv qúxgv irupxoì dyhf |Ê`1 ^N�` hvw xq rsìudwhxu wdqjhqw> vrq

ìydoxdwlrq shxw ìyhqwxhoohphqw vh idluh ã o*dlgh gh Ê�` hw ^G�` rx Ê2` hw ^G2`/ txl fruuhv0

srqghqw uhvshfwlyhphqw dx{ gìyhorsshphqwv olplwìv g*ruguh 4 hw 5 gh Ê` hw ^G`1

Sdu frqwuh/ o*ìtxloleuh ghv qúxgv grlw íwuh fdofxoì dyhf I� @|Ê` =I?1 Lfl/ Ê` hvw

ìydoxì gh idêrq h{dfwh1 Rq yrlw lfl txh fh txl hvw lpsruwdqw q*hvw sdv whoohphqw g*ìydoxhu

dyhf suìflvlrq o*rsìudwhxu gh frqyhujhqfh txl yd jxlghu ohv lwìudwlrqv vxffhvvlyhv/ pdlv od

irupdwlrq fruuhfwh gx whuph gh uìvlgx g*ìtxloleuh gh od vwuxfwxuh1

Xqh dssurfkh hq frrugrqqìhv frqyhfwìhv vsdwldohv q*d mdpdlv ìwì h�hfwxìh dxsdudydqw

ã qrwuh frqqdlvvdqfh1

618 Prgìolvdwlrq sdu ìoìphqwv �qlv gh w|sh srxwuh ��gì0

sodfhphqw��

O*dssurfkh ghv ìoìphqwv �qlv gh w|sh ��gìsodfhphqw�� frqvlvwh ã uìvrxguh +6197, hq vxs0

srvdqw txh o*rq shxw h{sulphu oh fkdps yluwxho g*lqfuìphqwv gh gìsodfhphqwv �x� hq

irqfwlrq g*xq fhuwdlq qrpeuh gh ydohxuv glvfuëwhv gh fh fkdps fkrlvlhv frpph lqfrqqxhv1

Od vwuxfwxuh hvw dlqvl glvfuìwlvìh hq ìoìphqwv txl vrqw frqqhfwìv hqwuh hx{ sdu ghv qúxgv1

Ã sduwlu ghv lqfrqqxhv qrgdohv dlqvl fkrlvlhv/ rq yd lqwhusrohu oh fkdps gh gìsodfhphqwv ã

o*dlgh gh irqfwlrqv gh iruph1 Fhv irqfwlrqv vrqw hq jìqìudo ghv sro|q÷phv txh o*rq lghqwl�h

hq h{sulpdqw ohv frqglwlrqv dx{ olplwhv flqìpdwltxhv1

61814 Xq h{hpsoh = od srxwuh 5G gh Ehuqrxool

x

y

U

V

2

2 θ2

UV

11

θ1

a a

Iljxuh 6144 = Hoìphqw �ql gh srxwuh Ehuqrxool

;3 Fkdslwuh 6 Qrq0olqìdulwì jìrpìwultxh

Ohv k|srwkëvhv fodvvltxhv txl vrqw idlwhv vrqw ohv vxlydqwhv =

� od srxwuh hvw xq frusv wudqvyhuvdohphqw uljlgh>

� oh vhfwlrq gurlwh uhvwh sodqh hw shushqglfxodluh ã od �euh pr|hqqh>

� vd vhfwlrq hvw frqvwdqwh>

� dydqw gìirupdwlrq/ o*d{h ghv fhqwuhv gh judylwì hvw xqh gurlwh>

� ohv gìsodfhphqwv vrqw shwlwv = o*dqjoh gh urwdwlrq hvw dvvlploì ã od shqwh1

O*ìoìphqw �ql suìvhqwì srvvëgh 5 qúxgv1 Ohv lqfrqqxhv qrgdohv/ dx qrpeuh gh 9/ vrqw

+X�> Y�> ��, srxu oh qúxg 4 hw +X2> Y2> �2, srxu oh qúxg 51 Oh fkdps gh gìsodfhphqwv

dgrswì hvw oh vxlydqw =

x +{> |, @ X�k� +{, .X2k2 +{,� |kY�k

�

� +{, . ��k�

e +{, . Y2k�

D +{, . �2k�

S +{,l

+6154<,

y +{> |, @ Y�k� +{, . ��ke +{, . Y2kD +{, . �2kS +{, +61553,

dyhf

k� +{, @4� �

5+61554,

k2 +{, @4 . �

5+61555,

k� +{, @�� 6� . 5

7+61556,

ke +{, @�� 2 � � . 4

� d7

+61557,

kD +{, @�� . 6� . 5

7+61558,

kS +{, @�� . �2 � � � 4

� d7

+61559,

� @{

d+6155:,

d @O

5+6155;,

Ohv gìirupdwlrqv txl uìvxowhqw gh fh fkdps gh gìsodfhphqwv vrqw =

%%% +{> |, @Cx +{> |,

C{

@4

O+X2 �X�,� |

%�2%u�

+Y� � Y2, .�S%3uu2

��

.�S%nuu2

��2

&+6155<,

%%+ +{> |, @4

5

�Cx +{> |,

C|.Cy +{> |,

C{

�@ 3 +61563,

%+% +{> |, @4

5

�Cy +{> |,

C{.Cx +{> |,

C|

�@ %%+ +{> |, @ 3 +61564,

%++ +{> |, @Cy +{> |,

C|@ 3 +61565,

618 Prgìolvdwlrq sdu ìoìphqwv �qlv gh w|sh srxwuh ��gìsodfhphqw�� ;4

Od pdwulfh gh uljlglwì pdwìulhooh txh o*rq rewlhqw/ vl o*rq vxssrvh oh pdwìuldx olqìdluh

ìodvwltxh/ hvw od vxlydqwh =

^N` @

59999999997

.�u

3 3 �.�u

3 3

3�2.Uu�

S.Uu2

3 ��2.Uu�

S.Uu2

3S.Uu2

e.Uu

3 �S.Uu2

2.Uu

�.�u

3 3.�u

3 3

3 ��2.Uu�

�S.Uu2

3�2.Uu�

�S.Uu2

3S.Uu2

2.Uu

3 �S.Uu2

e.Uu

6:::::::::8

+61566,

Rq uhwurxyh od pdwulfh gh uljlglwì elhq frqqxh gh o*ìoìphqw gh ��srxwuh lqjìqlhxu��1

61815 Ìoìphqw gh srxwuh fodvvltxh 6G

Hq 6G/ ohv lqfrqqxhv qrgdohv vrqw dx qrpeuh gh 45 =�X�> Y�>Z�> ��% > ��+ > ��5

�srxu oh

qúxg 4 hw�X2> Y2>Z2> �2% > �2+ > �25

�srxu oh qúxg 51 Gdqv oh fdguh ghv fdofxov phqìv gdqv

oh grpdlqh olqìdluh ìodvwltxh hq shwlwv gìsodfhphqwv/ wrxmrxuv hq qìjoljhdqw ohv h�hwv gx

jdxfklvvhphqw/ od pdwulfh gh uljlglwì shxw ìjdohphqw íwuh fdofxoìh ^Gxu:<`1 Lo vx!w gh

grqqhu od prlwlì gx txdgudqw lqiìulhxu gurlw gh od pdwulfh/ ohv dxwuhv whuphv vh gìgxlvdqw

sdu uìflsurflwì hw ìtxloleuh +�jxuh 6145,1

1 2 3 4 5 6 7 8 9 10 11 12

1234567891011121

2

3

4

5

Réciproc i té

Equi

libre

Réc iproc i té

Equi

libre

Iljxuh 6145 = Frqvwuxfwlrq gh od pdwulfh gh uljlglwì

Ohv whuphv qrq qxov gh od sduwlwlrq wuldqjxodluh 4 vrqw =

^N`.c. @HD

O+61567,

^N`HcH @45HL5�

4 . !+�O�

+61568,


^N`bcb @45HL+

+4 . !5,O�

+61569,

^N`�fcH @ �}S ^N`HcH +6156:,

^N`�fcb @ �|S ^N`bcb +6156;,

^N`�fc�f @ |2S ^N`bcb . }2S ^N`HcH .L%J

O+6156<,

^N`��cb @9HL+

+4 . !5,O2

+61573,

^N`��c�f @ |S ^N`��cb +61574,

^N`��c�� @+7 . !5,HL+

+4 . !5,O+61575,

^N`�2cH @�9HL5�

4 . !+�O2

+61576,

^N`�2c�f @ �}S ^N`�2cH +61577,

^N`�2c�2 @

�7 . !+

�HL5�

4 . !+�O

+61578,

O*ìtxloleuh gh o*ìoìphqw vh wudgxlw sdu =

^N`�c� @ � ^N`.c� +61579,

^N`2c� @ � ^N`Hc� +6157:,

^N`�c� @ � ^N`bc� +6157;,

^N`ec� @ � ^N`�fc� +6157<,

^N`Dc� @ � ^N`��c� .O ^N`bc� +61583,

^N`Sc� @ � ^N`�2c� .O ^N`Hc� +61584,

Ohv qrwdwlrqv xwlolvìhv lfl vrqw ohv vxlydqwhv =D = Dluh gh od vhfwlrq gurlwh

H = Prgxoh g*\rxqj

|S/ }S = Frrugrqqìhv gx fhqwuh gh wruvlrq

J = Prgxoh gh wruvlrq1 J @.

2E�nD�

L+/ L5 =U}2gD /

U|2gD

L% = Lqhuwlh gh wruvlrq

D+/ D5 = Vhfwlrqv uìgxlwhv g*h�ruw wudqfkdqw

O = Orqjxhxu gh od srxwuh

!+/ !5 = �2.U5�+Cu2

/ �2.U+�5Cu2

F*hvw fhw ìoìphqw txl hvw lpsodqwì gdqv od yhuvlrq olqìdluh gh SFS1

Rq wurxyh gdqv od olwwìudwxuh xq fhuwdlq qrpeuh g*ìoìphqwv �qlv srvvìgdqw wrxwhv vruwhv

gh ghjuìv gh olehuwì1 Dlqvl/ rq shxw hqulfklu xq ìoìphqw hq dxjphqwdqw vrq qrpeuh gh

qúxgv1 Oruvtxh o*rq yhxw suhqguh hq frpswh oh jdxfklvvhphqw ìyhqwxho gh od vhfwlrq/ rq

619 Prgìolvdwlrq sdu ìoìphqwv �qlv gh w|sh srxwuh ��vhpl0lqyhuvh�� ;6

dmrxwh dx{ 9 ghjuìv gh olehuwì fodvvltxhv xq ghjuì gh olehuwì vxssoìphqwdluh txl hvw o*dqjoh

gh jdxfklvvhphqw/ o*h�ruw dvvrflì ìwdqw oh elprphqw ^GhY;<`1

Ohv fkdpsv gh gìsodfhphqwv qh vrqw sdv ohv vhxov ã srxyrlu íwuh lqwhusroìv = rq shxw

doruv wudydloohu vxu ghv ghjuìv gh olehuwì txh vrqw od whpsìudwxuh/ xq fkdps ìohfwultxh/

pdjqìwltxh111/ od pdwulfh gh uljlglwì gh o*ìoìphqw q*ìwdqw soxv yudlphqw xqh pdwulfh gh

uljlglwì dx vhqv gh od uìvlvwdqfh ghv pdwìuldx{1 F*hvw fh fdudfwëuh jìqìudo gh o*dssurfkh

��gìsodfhphqw�� txl idlw od irufh ghv ìoìphqwv �qlv gh qrv mrxuv1 Wrxw hvw prgìolvdeoh

srxuyx txh o*rq irxuqlvvh xqh irqfwlrq gh iruph dgìtxdwh srxu lqwhusrohu gdqv o*ìoìphqw

oh skìqrpëqh txh o*rq ìwxglh1

619 Prgìolvdwlrq sdu ìoìphqwv �qlv gh w|sh srxwuh ��vhpl0

lqyhuvh��

Fhwwh dssurfkh ^Juh:;e`/ ^Qdl<3` frqvlvwh qrq soxv ã srvwxohu od iruph gx fkdps gh

gìsodfhphqwv/ pdlv od iruph gx fkdps gh frqwudlqwhv jìqìudolvìhv +Q%>P+>P5,1 Rq

v*ìfduwh gx frqfhsw txh o*rq vrxv0hqwhqg hq jìqìudo oruvtxh o*rq sduoh g*ìoìphqwv �qlv1 Gdqv

oh sulqflsh/ frqqdlvvdqw ohv h�ruwv jìqìudx{ dx{ qúxgv gh o*ìoìphqw/ rq yd ohv lqwhusrohu

oh orqj gh o*ìoìphqw1 Rq gìwhuplqhud od pdwulfh gh uljlglwì hw od gìirupdwlrq gh fkdfxqh

ghv vhfwlrqv g*lqwìjudwlrq oh orqj gh o*ìoìphqw1 Fkdtxh pdwulfh gh uljlglwì gh vhfwlrq vhud

lqyhuvìh srxu grqqhu od pdwulfh gh vrxsohvvh gh od vhfwlrq1 Ohv pdwulfhv gh vrxsohvvh

vrqw lqwìjuìhv oh orqj gh o*ìoìphqw srxu iruphu od pdwulfh gh vrxsohvvh gh o*ìoìphqw1 Fhfl

uhylhqw ã lqwìjuhu od frxuexuh oh orqj gh o*ìoìphqw1 Od pdwulfh gh vrxsohvvh gh o*ìoìphqw

hvw hqvxlwh lqyhuvìh/ srxu grqqhu od pdwulfh gh uljlglwì gh o*ìoìphqw1 Elhq txh g*dssduhqfh

frpsoh{h/ fhwwh surfìgxuh hvw dvvh} vlpsoh ã phwwuh hq úxyuh/ hw suìvhqwh gh qrpeuhx{

dydqwdjhv/ qrwdpphqw srxu od sulvh hq frpswh gh o*dfwlrq gh od suìfrqwudlqwh1 Od vxlwh

gx gìyhorsshphqw v*lqvfulw gluhfwhphqw gdqv fh fdguh1

Fkdslwuh 7

Qrq0olqìdulwì pdwìulhooh

Frpph qrxv o*dyrqv yx gdqv oh fkdslwuh 6/ od sulvh hq frpswh gh od qrq0olqìdulwì pdwìulhooh

lqwhuylhqw dx qlyhdx gh od frqvwuxfwlrq gh od pdwulfh ^N?`/ od pdwulfh gh uljlglwì gh o*ìoìphqw

gdqv oh uhsëuh lqwulqvëtxh1 Fhwwh pdwulfh yd íwuh frqvwuxlwh ã sduwlu ghv pdwulfhv gh uljlglwì

ghv vhfwlrqv frqvwlwxwlyhv gh o*ìoìphqw/ txh qrxv qrwhurqv ^Nr`1 Gdqv fh fkdslwuh/ qrxv

doorqv h{solflwhu oh prgh g*rewhqwlrq ghv pdwulfhv gh uljlglwì ghv vhfwlrqv/ dlqvl txh o*dfwlrq

gh od suìfrqwudlqwh vxu od vhfwlrq/ txh qrxv qrwhurqvDr1 Hq�q/ qrxv lqwìjuhurqv oh orqj gh

o*ìoìphqw srxu rewhqlu ^N?` hw D? txl vrqw uhvshfwlyhphqw od pdwulfh gh uljlglwì hw o*dfwlrq

gh od suìfrqwudlqwh vxu o*ìoìphqw gdqv oh uhsëuh lqwulqvëtxh1 Qrxv doorqv grqf gì�qlu gdqv

xq suhplhu whpsv ohv orlv gh frpsruwhphqw qrq olqìdluh srxu ohv pdwìuldx{1

714 Ohv orlv gh frpsruwhphqw

Od vhfwlrq txh o*rq vh sursrvh g*ìwxglhu hvw vrlw krprjëqh/ vrlw frpsrvlwh1 Hooh hvw kr0

prjëqh v*lo v*djlw g*xq sur�oì pìwdooltxh sdu h{hpsoh/ rx frpsrvlwh v*lo v*djlw sdu h{hpsoh

g*xq fdlvvrq hq eìwrq suìfrqwudlqw1 Gdqv oh fdv gh od vhfwlrq frpsrvlwh/ oh pdwìuldx gh

od pdwulfh hvw oh eìwrq/ hw fhoxl ghv �euhv hvw o*dflhu1 Hq fh txl frqfhuqh ohv dupdwxuhv

pìwdooltxhv/ rq glvwlqjxhud ohv dupdwxuhv sdvvlyhv hw ohv dflhuv gh suìfrqwudlqwh1 Qrxv

glvsrvrqv gh ghx{ idploohv gh orlv gh frpsruwhphqw = o*xqh srxu od pdwulfh gh od vhfwlrq/

hw hooh frpsuhqgud hvvhqwlhoohphqw ghv orlv gh frpsruwhphqw gh eìwrq/ hw o*dxwuh srxu ohv

�euhv gh od vhfwlrq/ hw hooh frpsuhqgud h{foxvlyhphqw ghv orlv gh frpsruwhphqw g*dflhu1

71414 Pdwìuldx gh od pdwulfh gh od vhfwlrq

Orl ìodvwltxh olqìdluh

F*hvw od orl od soxv vlpsoh/ gì�qlh vhxohphqw sdu od ydohxu H��f gx prgxoh g*\rxqj lqvwdq0

wdqì gx pdwìuldx1 Vrq gldjudpph hvw suìvhqwì hq �jxuh 7141 Fhshqgdqw/ oh pdwìuldx shxw

;9 Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh

ε

σ

0

arctan(E )i j0

compress iontract ion

Iljxuh 714 = Orl ìodvwltxh olqìdluh

íwuh ylhloolvvdqw/ f*hvw0ã0gluh txh od ydohxu gh H��f gìshqg gh o*åjh gx pdwìuldx ã o*lqvwdqw

gx fkdujhphqw1 Oh frpsruwhphqw gx pdwìuldx hvw oh píph hq wudfwlrq hw hq frpsuhvvlrq1

Orl ìodvwltxh olqìdluh iudjloh

Fhwwh orl hvw ìjdohphqw olqìdluh pdlv vrq gldjudpph hvw eruqì hq frpsuhvvlrq sdu od

gìirupdwlrq %� hw hq wudfwlrq sdu od frqwudlqwh �|1 Vrq gldjudpph hvw uhsuìvhqwì hq �jxuh

7151

Orl elolqìdluh

Fhwwh orl hvw ìjdohphqw olqìdluh/ pdlv sdu prufhdx{1 Hooh hvw fdudfwìulvìh sdu oh prgxoh

g*\rxqj ã o*ruljlqh H��f / od frqwudlqwh �f/ hw ohv frrugrqqìhv +%�> ��, gx srlqw xowlph1 Oh

frpsruwhphqw hvw fhqvì íwuh oh píph hq frpsuhvvlrq hw hq wudfwlrq1 Vrq gldjudpph hvw

suìvhqwì hq �jxuh 7161

714 Ohv orlv gh frpsruwhphqw ;:

ε

σ

0

arctan(E )ij0

εu

σt


Iljxuh 715 = Orl ìodvwltxh olqìdluh iudjloh

;; Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh

ε

σ

0

arctan(E )i j0

ε u

σu

ε 0

Iljxuh 716 = Orl elolqìdluh

714 Ohv orlv gh frpsruwhphqw ;<

Orl gx ESHO<4

ε

σ

0

arctan(E )i j0

εu

ε0

b

b

f cu

bb

f c j


Iljxuh 717 = Orl ESHO<4

Fhwwh orl hvw lvvxh gx uëjohphqw iudqêdlv ESHO<4 ^Esho<4`1 Vrq gldjudpph hvw suìvhqwì

hq �jxuh 7171 Hooh prgìolvh oh frpsruwhphqw hq frpsuhvvlrq gx eìwrq1 F*hvw srxutxrl ohv

qrwdwlrqv gh gìirupdwlrqv hw gh frqwudlqwhv vhurqw lqglfìhv dyhf e1 Hooh hvw edvìh vxu od

irupxoh vxlydqwh/ gxh ã Vdujlq ^Vdu:4`/ txl v*ìfulw/ hq uhsuhqdqw ohv qrwdwlrqv gx ESHO<4 =

�K @ iS�

n�

0K0Kf

�.

�n� � 4

��0K0Kf

�24 . +n � 5,

�0K0Kf

�. n

�

�0K0Kf

�2 +714,

rü iS� hvw od uìvlvwdqfh ã od frpsuhvvlrq gx eìwrq åjì gh m mrxuv/ grqqìh hq PSd sdu =

iS� @

;AA?AA=

�ec.SnfcH��

iS2H vl m 9 5; mrxuv hw iS2H 9 73 PSd�

�cefnfcbD�iS2H vl m 9 5; mrxuv hw iS2H A 73 PSd

iS2H vl m A 5; mrxuv

+715,

%Kf hvw od gìirupdwlrq dx slf gh frqwudlqwh =

%Kf @ 3> 95=433� �

tiS� +716,

<3 Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh

n hvw xq frh!flhqw grqqì sdu =

n @H��f%KfiS�

+717,

H��f hvw oh prgxoh g*\rxqj ã o*ruljlqh =

H��f @ 44333 �

tiS� +718,

n�

hvw xq frh!flhqw shuphwwdqw g*dmxvwhu o*dooxuh gx gìexw gh od eudqfkh ghvfhqgdqwh gx

gldjudpph uìho =

n�

@

;AA?AA=

n � 4 srxu iS� 9 63 PSd

3 srxu iS� A 88 PSd

+n � 4,

�DD3sS�2D

�srxu 63 PSd ? iS� ? 88 PSd

+719,

Oh eìwrq whqgx hvw frpsoëwhphqw qìjoljì1

Sxlvtx*lo v*djlw g*xqh orl uìjohphqwdluh/ rq qrwh o*h{lvwhqfh gh ghx{ frh!flhqwv gh

vìfxulwì = iS� hvw uhpsodfì sdufcHD�KiS� / hw H��f sdu

.��f

�K1 Oh frh!flhqw gh vìfxulwì �K ydxw

4> 68 vl o*rq pëqh xq fdofxo vwuxfwxudo/ hw 4> 8 gdqv oh fdv g*xq fdofxo gh vhfwlrq1

Orl DIUHP<9

O*h{whqvlrq ghv uëjohphqwv gh fdofxo EDHO hw ESHO<4 dx{ eìwrqv grqw od uìvlvwdqfh

dwwhlqw ;3 PSd hvw uhjurxsìh vrxv oh qrp gh orl DIUHP<9 ^Ofsf<9`1 Oh frpsruwhphqw

gx eìwrq vrxv fkdujh gh frxuwh gxuìh hvw vhqvleohphqw oh píph txh srxu od orl ESHO<41

Ohv sdudpëwuhv vrqw lghqwltxhv/ vdxi n�

=

n�

@n

4> 98.iS� � 63

iS� � 69+71:,

Orl HXURFRGHV5

Od orl ghv Hxurfrghv ^Hxu<5` v*dssxlh ìjdohphqw vxu xqh orl gh w|sh Vdujlq1 Hq uhsuhqdqw

ohv qrwdwlrqv ghv Hxurfrghv/ rq d =

�S @ iS

n�

0S0S�

��

0S0S�

�24 . +n � 5,

�0S0S�

� +71;,

rü =

�S hvw od frqwudlqwh gdqv oh eìwrq/ txl gìshqg gh od gìirupdwlrq %S gx eìwrq>

iS hvw od uìvlvwdqfh ã od frpsuhvvlrq gx eìwrq>

%S� hvw od gìirupdwlrq vrxv od frqwudlqwh pd{lpdoh iS =

%S� @ 3> 3355 +71<,

714 Ohv orlv gh frpsruwhphqw <4

n hvw xq frh!flhqw grqqì sdu =

n @4> 4HS6%S�

iS+7143,

HS6 hvw oh prgxoh vìfdqw ã 5; mrxuv1

Frpph rq shxw oh frqvwdwhu/ oh frh!flhqw n�

gh od orl ghv Hxurfrghv hvw qxo1

Orl jìqìudoh gh Vdujlq hw sulvh hq frpswh gx eìwrq whqgx

ε

σ

0

arctan(E )i j0

εu

ε0

b

b

f cu

bb

f c j

Branche descendante

l inéaire


Iljxuh 718 = Orl gh Vdujlq jìqìudoh

Fhwwh orl hvw od píph txh +714,/ vdxi hq fh txl frqfhuqh ohv sdudpëwuhv iS� / %Kf / n/ n�

txl qh vrqw sdv gì�qlv dxwrpdwltxhphqw1 F*hvw srxutxrl hooh uhyíw xq fdudfwëuh jìqìudo1

Vrq gldjudpph hvw suìvhqwì hq �jxuh 7181 Frpph ohv dxwuhv orlv vh edvdqw vxu oh prgëoh

sursrvì sdu Vdujlq/ od sduwlh ghvfhqgdqwh gh od frxueh shxw íwuh dssur{lpìh sdu xqh

eudqfkh ghvfhqgdqwh olqìdluh uholdqw ohv srlqwv�%Kf > iS�

�hw +%K� > iS�,1 Sdu h{hpsoh/ srxu

n @ 5 hw n�

@ 3/ rq uhwurxyh od orl elhq frqqxh sduderoh0uhfwdqjoh gx EDHO<4 ^Edho<4` hw

gx ESHO<41

Od sulvh hq frpswh gx eìwrq whqgx xwlolvh oh prgëoh gìyhorssì sdu Juhodw ^Juh:;e`1

Fh prgëoh hvw fhqvì prgìolvhu od uìvlvwdqfh gx eìwrq hqwuh ghx{ �vvxuhv/ soxv frqqxh

vrxv oh qrp gh ��whqvlrq0vwl�hqlqj��1 Od sulvh hq frpswh gh fh skìqrpëqh q*hvw sdv

qhxwuh srxu oh fdofxo ghv ëfkhv/ hw srxu od uhglvwulexwlrq ghv h�ruwv gdqv ohv vwuxfwxuhv

k|shuvwdwltxhv vxmhwwhv dx{ h�hwv gh od qrq0olqìdulwì jìrpìwultxh1 Oh prgëoh gh Juhodw


vlpxoh oh frpsruwhphqw pr|hq gx eìwrq hqwuh ghx{ �vvxuhv1

ε

σ

0arctan(E )i j

0

εε

ft

ftat

f t j

t ract ion

Iljxuh 719 = Gldjudpph frqwudlqwh0gìirupdwlrq gh od �euh od soxv whqgxh

Fhwwh pìwkrgh frqvlvwh ã uhsìuhu od �euh od soxv whqgxh gh od vhfwlrq/ f*hvw0ã0gluh fhooh

grqw od gìirupdwlrq hq wudfwlrq hvw od soxv lpsruwdqwh hq ydohxu devroxh1 Rq qrwhud %s|fhwwh gìirupdwlrq1 Hqvxlwh/ rq fdofxoh od frqwudlqwh �s| gh od �euh od soxv whqgxh gh od

idêrq vxlydqwh =

�s| @

;AA?AA=

H��f%s| srxu 3 ? %s| ? %|

i|�+0@30s|,

2

E0@30|�2srxu %| ? %s| ? %@

3 srxu %s| A %@

+7144,

rü H��f hvw oh prgxoh g*\rxqj wdqjhqw ã o*ruljlqh>

i|� hvw od uìvlvwdqfh ã od wudfwlrq gx eìwrq>

%| hvw od gìirupdwlrq gh wudfwlrq fruuhvsrqgdqw ã i|� = %| @s|�.��f

>

%@ hvw od olplwh g*ìodvwlflwì gh o*dflhu oh soxv whqgx1

Oh gldjudpph gh �s| hvw suìvhqwì hq �jxuh 7191 Rq d�hfwh dx eìwrq whqgx xq gld0

judpph gh frqwudlqwhv �fwli wuldqjxodluh ã sduwlu gh od �euh qhxwuh1 Gdqv od }rqh whqgxh/

od frqwudlqwh � v*h{sulph hq irqfwlrq gh od gìirupdwlrq % vhorq =

� @�s|

%s|% +7145,


ε

σ

0arctan(E )s

ε

s

ss2 ‰

σl

σe

σr

r

Iljxuh 71: = Orl dflhu EDHO;6

Orl dflhu EDHO;6

Fhwwh orl shuphw gh prgìolvhu oh frpsruwhphqw g*xq dflhu1 Hooh hvw lvvxh gh o*dqflhq

uëjohphqw iudqêdlv gh eìwrq dupì EDHO;6 fdu o*dfwxho/ oh EDHO<4/ d vlpsol�ì ohv orlv

gh frpsruwhphqw ghv dflhuv hq ohv vxssrvdqw ìodvwrsodvwltxhv sduidlwhv1 Vrq gldjudpph

hvw suìvhqwì hq �jxuh 71:1 Od orl EDHO;6 d ìwì gìyhorssìh lqlwldohphqw srxu ohv dflhuv

ghv dupdwxuhv sdvvlyhv1 Hooh hvw xwlolvìh lfl srxu oh pdwìuldx frqvwlwxwli gh od vhfwlrq1 Oh

gldjudpph hvw v|pìwultxh sdu udssruw dx srlqw R/ hw vh gìfrpsrvh hq xqh sduwlh olqìdluh

hw xqh sduwlh qrq olqìdluh =

�r @ Hr%r srxu 3 ? �r ? �, @ 3> :�e

%r @jr.r

. 3> ;56�jrje� 3> :

�Dsrxu �, ? �r ? �o

+7146,

rü Hrf hvw oh prgxoh g*\rxqj gh o*dflhu ã o*ruljlqh

�e od olplwh ìodvwltxh frqyhqwlrqqhooh ã 3/5(/

�, hvw od frqwudlqwh rü v*duuíwh oh gldjudpph olqìdluh/

�o hvw od frqwudlqwh gh uxswxuh1


Rq qrwh dx sdvvdjh txh srxu �, ? �r ? �o/ od orl gh frpsruwhphqw hvw grqqìh ��ã

o*hqyhuv��1 Qrxv lqyhuvrqv od uhodwlrq +7146, sdu xq vfkìpd gh Qhzwrq lwìudwli1

71415 Pdwìuldx ghv �euhv gh od vhfwlrq

Ohv �euhv gh od vhfwlrq vrqw hq idlw ohv dupdwxuhv vrlw sdvvlyhv/ vrlw dfwlyhv1 Oh pdwìuldx

frqvwlwxwli gh fhv dupdwxuhv hvw o*dflhu1 Rq vxssrvhud gdqv fh sdudjudskh txh oh frp0

sruwhphqw gx pdwìuldx hvw oh píph hq wudfwlrq tx*hq frpsuhvvlrq1

Sdu dloohxuv/ rq uhwurxyh ohv orlv gh frpsruwhphqw olqìdluh ìodvwltxh hw olqìdluh ìodvwltxh

iudjloh/ txl qh vxvflwhqw dxfxq frpphqwdluh vxssoìphqwdluh sdu udssruw ã fh txl d ìwì glw

suìfìghpphqw1

Orl ìodvwrsodvwltxh sduidlwh

ε

σ

0arc tan(E )s

s

s1 0 ‰f e

Es

_ _

fe

f eEs

_ _-- 1 0 ‰

-fe

fe_ _γs

Iljxuh 71; = Orl ìodvwrsodvwltxh sduidlwh

Fhwwh orl hvw uhfrppdqgìh sdu oh EDHO<4 hw oh ESHO<4 srxu ohv dupdwxuhv sdvvlyhv1 Vrq

gldjudpph hvw grqqì hq �jxuh 71;/ dyhf ohv qrwdwlrqv gx ESHO<41 F*hvw xq fdv sduwlfxolhu

gh od orl elolqìdluh/ dyhf xq sdolhu sodvwltxh sodw1 Oh uëjohphqw uhfrppdqgh g*h�hfwxhu vxu

fhwwh orl xqh d!qlwì gh udssruw ��r

sdudooëohphqw ã od wdqjhqwh ã o*ruljlqh/ frpph lqgltxì

hq �jxuh 71;1 Oh frh!flhqw �r ydxw 4/481


Orl dflhu ESHO<4

ε

σ

0arctan(E )

p

p

p1 ‰

σl

σe

σr

2 0 ‰

Courbe après af f in i té

Iljxuh 71< = Orl dflhu ESHO<4

Fhwwh orl hvw uhfrppdqgìh srxu ohv �ov wuì�oìv hw ohv wrurqv gh suìfrqwudlqwh1 Vrq gld0

judpph hvw grqqì hq �jxuh 71<1 Hq uhsuhqdqw ohv qrwdwlrqv gx ESHO<4/ rq d =

�R @ HR%R srxu 3 ? �R ? �, @ 3> <�e

%R @jR.R

. 433

�jRje� 3> <

�Dsrxu �, ? �R ? �o

+7147,

rü �R hvw od frqwudlqwh gdqv o*dflhu gh suìfrqwudlqwh/

%R hvw od gìirupdwlrq gdqv o*dflhu gh suìfrqwudlqwh/

HR hvw oh prgxoh g*\rxqj ã o*ruljlqh/

�e hvw od olplwh ìodvwltxh frqyhqwlrqqhooh ã 3/4(/

�, hvw od frqwudlqwh rü v*duuíwh oh gldjudpph olqìdluh/

�o hvw od frqwudlqwh gh uxswxuh1

Frpph srxu od orl ìodvwrsodvwltxh sduidlwh/ oh uëjohphqw uhfrppdqgh g*h�hfwxhu vxu

fhwwh orl xqh d!qlwì gh udssruw ��r

sdudooëohphqw ã od wdqjhqwh ã o*ruljlqh1 Frpph srxu

od orl dflhu EDHO;6/ od uhodwlrq frqwudlqwh0gìirupdwlrq hvw uìvroxh sdu xq vfkìpd lwìudwli


gh Qhzwrq1 Oruvtxh o*d!qlwì gh udssruw ��r

sdudooëohphqw ã od wdqjhqwh ã o*ruljlqh hvw

dssoltxìh/ rq rewlhqw xqh qrxyhooh uhodwlrq hqwuh ��

R hw %�

R txl vrqw uhvshfwlyhphqw ohv

frqwudlqwhv hw ohv gìirupdwlrqv gx gldjudpph prgl�ì sdu o*d!qlwì1 Rq prqwuh txh =

��

R @ HR%�

R srxu 3 ? ��

R ? ��

, @j,�r

%�

R @j�

R

.R. 433

�j�

R�rje

� 3> <

�D

srxu ��

, ? ��

R ? ��

o @jo�r

+7148,

71416 Frqfoxvlrq

Ohv orlv gh frpsruwhphqw gì�qlhv lfl vrqw xwlolvìhv srxu ohv fkdujhphqwv lqvwdqwdqìv1 Srxu

ohv h�hwv gl�ìuìv/ rq vh uìiìuhud dx fkdslwuh 91 Ohv orlv gh frpsruwhphqw lqvwdqwdqìhv vrqw

qrq olqìdluhv ìodvwltxhv/ l=h= ohv fkhplqv gh fkdujhphqw hw gh gìfkdujhphqw vrqw lghqwltxhv1

Fh w|sh gh orl d ìwì lpsrvì sdu ghv frqvlgìudwlrqv uìjohphqwdluhv1 Oh idlw txh ohv orlv

gh frpsruwhphqw vrlhqw sduidlwhphqw uìyhuvleohv q*hvw sdv suìmxglfldeoh gdqv oh fdv gh

fkdujhphqwv prqrwrqhv1 Oh sureoëph vh srvh srxu ohv fkdujhphqwv f|foltxhv/ oruv gh

fdofxov vlvpltxhv sdu h{hpsoh1 Gdqv fh fdv/ gx idlw txh od orl gh frpsruwhphqw qh glvvlsh

sdv g*ìqhujlh/ lo idxgud ìydoxhu dxwuhphqw o*dpruwlvvhphqw1

715 Ìwxgh g*xqh vhfwlrq gurlwh

71514 K|srwkëvhv hw qrwdwlrqv

Od vhfwlrq ìwxglìh dssduwlhqw ã o*ìoìphqw grqw rq yhxw fdofxohu od pdwulfh gh uljlglwì1

Oh uhsëuh gh gì�qlwlrq gh od vhfwlrq d vrq ruljlqh vxu od �euh pr|hqqh gh o*ìoìphqw =

vrq ruljlqh hvw grqf oh fhqwuh gh judylwì J gh od vhfwlrq1 O*d{h J{ hvw wdqjhqw ã od �euh

pr|hqqh gh o*ìoìphqw1 Gdqv od phvxuh rü o*rq vxssrvh txh ohv gìsodfhphqwv gdqv oh uhsëuh

lqwulqvëtxh uhvwhqw prgìuìv/ o*d{h J{ hvw qrupdo ã od vhfwlrq gurlwh1 Ohv d{hv J| hw J}

gìfrxohqw gluhfwhphqw gh fhx{ gx uhsëuh lqwulqvëtxh1 Rq qrwhud F oh fhqwuh gh wruvlrq gh

od vhfwlrq1 Gdqv oh uhsëuh J|}/ ohv frrugrqqìhv gh F vhurqw qrwìhv |S hw }S1

Ohv vroolflwdwlrqv txl v*h{hufhqw vxu od vhfwlrq vrqw gì�qlhv frpph ohv dfwlrqv gh od

sduwlh gurlwh gh od srxwuh vxu od sduwlh jdxfkh1 Gdqv oh uhsëuh lqwulqvëtxh/ ohv h�ruwv I%hw I? uhvshfwhqw ìylghpphqw ohv frqyhqwlrqv wuljrqrpìwultxhv1 Pdlv hq fh txl frqfhuqh

o*ìwxgh gh od vhfwlrq/ qrxv doorqv dgrswhu ohv frqyhqwlrqv lqgltxìhv vxu od �jxuh 71431 Fhv

frqyhqwlrqv yrqw qrxv shuphwwuh g*rewhqlu xqh ìfulwxuh soxv djuìdeoh ghv gìirupdwlrqv/

dyhf od uëjoh vxlydqwh/ sursuh dx fdofxo hq eìwrq dupì = xq udffrxuflvvhphqw gh o*ìoì0

phqw fruuhvsrqg ã xqh gìirupdwlrq % srvlwlyh1 Dlqvl xqh frpsuhvvlrq fruuhvsrqg ã xqh

frqwudlqwh � srvlwlyh1

715 Ìwxgh g*xqh vhfwlrq gurlwh <:

x

y

z

G

T

N

M

Ty

z

z

My

Mx

C

Iljxuh 7143 = Frqyhqwlrq gh vljqh srxu od vhfwlrq

71515 Ohv gìirupdwlrqv qrupdohv

Frpph qrxv o*dyrqv lqgltxì gdqv ohv k|srwkëvhv jìqìudohv/ ohv h�hwv gxv dx jdxfklvvhphqw

vhurqw qìjoljìv1 Fhfl hvw oìjlwlph vl oh sur�o gh od vhfwlrq hvw ihupì/ rx vl ohv frqglwlrqv dx{

olplwhv odlvvhqw oleuh oh jdxfklvvhphqw1 Qrxv idlvrqv ìjdohphqw o*k|srwkëvh txh ohv vhfwlrqv

sodqhv oh uhvwhqw1 Rq shxw doruv ìfuluh =

% +|> }, @ %% . }!+ . |!5 +7149,

rü % +|> }, hvw od gìirupdwlrq qrupdoh hq xq srlqw gh frrugrqqìhv +|> }, gh od vhfwlrq/ hw

%% hvw od gìirupdwlrq dx fhqwuh gh judylwì gh od vhfwlrq1 O*h�ruw dvvrflì hvw o*h�ruw qru0

pdo Q 1 Ohv dqjohv !+ hw !5 uhsuìvhqwhqw ohv urwdwlrqv gh od vhfwlrq dxwrxu ghv d{hv J|

hw J} uhvshfwlyhphqw1 Ohv h�ruwv dvvrflìv vrqw ohv prphqwv ìfklvvdqwv P+ hw P51 Od

qrq0olqìdulwì pdwìulhooh qh vhud sulvh hq frpswh txh sdu od uhodwlrq hqwuh ohv vroolflwdwlrqv

qrupdohv +Q>P+>P5, hw ohv gìirupdwlrqv qrupdohv dvvrflìhv�%%> !+> !5

�1 Ohv gìirupd0

wlrqv qrupdohv vhurqw gìvljqìhv sdu oh yhfwhxu

%? @

3EEC

%%

!+

!5

4FFD +714:,

<; Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh

71516 Ohv gìirupdwlrqv wdqjhqwhv

Sdu dloohxuv/ qrxv vxssrvhurqv txh od uhodwlrq hqwuh ohv vroolflwdwlrqv wdqjhqwhv ã od vhfwlrq

txh vrqw ohv h�ruwv wudqfkdqwv W+/ W5 hw oh prphqw gh wruvlrqP�% hq F g*xqh sduw/ hw ohv

gìirupdwlrqv wdqjhqwhv g*h�ruwv wudqfkdqwv �+/ �5 hw o*dqjoh gh wruvlrq �% g*dxwuh sduw/

hvw olqìdluh1 Oh prphqw gh wruvlrq P�% vh gìgxlw gx prphqw gh wruvlrq P% hq J gh od

idêrq vxlydqwh =

P�% @P% . }SW+ � |SW5 +714;,

Ohv gìirupdwlrqv wdqjhqwhv vhurqw gìvljqìhv sdu oh yhfwhxu

%| @

3EEC

�+

�5

�%

4FFD +714<,

71517 Ohv gìirupdwlrqv ghv dupdwxuhv

Ohv dupdwxuhv ìwdqw lqfolqìhv sdu udssruw ã od vhfwlrq/ ohxu gìirupdwlrq qh vhud sdv od

píph txh fhooh gh od vhfwlrq1 Rq qrwh � o*dqjoh txh idlw od oljqh pr|hqqh gh o*dupdwxuh

dyhf o*d{h J{ +�jxuh 7144,/ hw � o*dqjoh hqwuh od surmhfwlrq gh o*dupdwxuh gdqv oh sodq gh

od vhfwlrq hw o*d{h J|1 Gdqv oh uhsëuh J|}/ rq qrwhud +|s > }s , ohv frrugrqqìhv gx srlqw gh

sdvvdjh gh o*dupdwxuh1 Doruv/ hq vxssrvdqw o*dupdwxuh sduidlwhphqw dgkìuhqwh dx eìwrq/

srxu xq dffurlvvhphqw gh gìirupdwlrqv gh od vhfwlrq =

�%? @

3EEC

�%%

�!+

�!5

4FFD +7153,

�%| @

3EEC

��+

��5

��%

4FFD +7154,

Rq prqwuh txh o*dffurlvvhphqw gh gìirupdwlrq gdqv o*dupdwxuh �%s hvw grqqì sdu =

�%s @ frv2 ��

4 > }s > |s

�=�%?

.vlq� frv��

frv�> vlq�> +|s � |S, vlq� � +}s � }S, frv�

�=�%|+7155,

71518 Ohv vroolflwdwlrqv lqwhuqhv

Ohv ìoìphqwv frqvwlwxwliv gh od vhfwlrq/ txh vrqw od pdwulfh hw ohv �euhv/ yrqw hqjhqguhu

ghv vroolflwdwlrqv txh o*rq yd vìsduhu hq ghx{ fdwìjrulhv = ohv vroolflwdwlrqv qrupdohv hw ohv

vroolflwdwlrqv wdqjhqwhv1 Rq qrwhud =

715 Ìwxgh g*xqh vhfwlrq gurlwh <<

I6? ohv vroolflwdwlrqv qrupdohv gxhv ã od pdwulfh gh od vhfwlrq +gx eìwrq hq jìqìudo,>

I6| ohv vroolflwdwlrqv wdqjhqwhv gxhv ã od pdwulfh gh od vhfwlrq>

Is? ohv vroolflwdwlrqv qrupdohv gxhv dx{ �euhv gh od vhfwlrq +ohv dupdwxuhv pìwdooltxhv/

dfwlyhv rx sdvvlyhv,>

Is| ohv vroolflwdwlrqv wdqjhqwhv gxhv dx{ �euhv gh od vhfwlrq1

Ohv vroolflwdwlrqv qrupdohv gxhv ã od pdwulfh gh od vhfwlrq

Gdqv oh uhsëuh J{|}/ rq shxw ìfuluh =

I6? @

3EEC

Q6

P6+

P65

4FFD +7156,

rüQ6 hvw o*h�ruw qrupdo/P6+ hwP65 ohv prphqwv ìfklvvdqwv/ wrxv wurlv gxv ã od pdwulfh

gh od vhfwlrq1 Rq d doruv =

I6? @

]76

�6 +|> },

3EEC

4

}

|

4FFDgV6 +7157,

rü V6 hvw o*dluh gh od pdwulfh gh od vhfwlrq/ hw �6 +|> }, od frqwudlqwh hq xq srlqw frxudqw

gh od vhfwlrq gh frrugrqqìhv +|> },1

Gdqv oh fdv g*xqh vhfwlrq hq eìwrq/ V6 uhsuìvhqwh od vhfwlrq qhwwh gh eìwrq/ l=h= od

vhfwlrq euxwh lvvxh gx ghvvlq gh fr�udjh uìgxlwh gh od vxuidfh tx*rffxshqw ohv dupdwxuhv

sdvvlyhv rx ohv jdlqhv gh suìfrqwudlqwh1 Fhwwh gìgxfwlrq wlhqw frpswh gh o*dqjoh� h{lvwdqw

hqwuh o*d{h gh o*dupdwxuh hw o*d{h J{1 Vl ds gìvljqh od vhfwlrq gurlwh gh o*dupdwxuh sdvvlyh

rx gh od jdlqh gh suìfrqwudlqwh/ doruv o*dluh gh od surmhfwlrq gh ds vxu oh sodq J|} hvw =ds

frv�+7158,

O*dluh gìgxlwh gh od vhfwlrq euxwh/ txl fruuhvsrqg ã o*hpsodfhphqw gh o*dupdwxuh sdvvlyh

rx gh od jdlqh gh suìfrqwudlqwh/ hvw grqf wrxmrxuv vxsìulhxuh rx ìjdoh ã od vhfwlrq gurlwh

gh o*dupdwxuh sdvvlyh rx gh od jdlqh gh suìfrqwudlqwh1

Vl o*rq qrwh H6 +|> }, oh prgxoh ìodvwltxh orqjlwxglqdo frugh/ f*hvw0ã0gluh oh prgxoh

uholdqw o*dffurlvvhphqw gh frqwudlqwh ã o*dffurlvvhphqw gh gìirupdwlrq/ g*xq srlqw gh fr0

rugrqqìhv +|> }, gh od vhfwlrq/ doruv rq d =

��6 +|> }, @ H6 +|> },�% +|> }, +7159,

rü ��6 +|> }, hvw o*dffurlvvhphqw gh frqwudlqwh qrupdoh dx srlqw frqvlgìuì/ hw �% +|> },

o*dffurlvvhphqw gh gìirupdwlrq qrupdoh1 Hq uhpsodêdqw�% +|> }, sdu vd ydohxu grqqìh hq

+7149,/ rq d =

��6 +|> }, @ H6 +|> },�

4 } |

��%? +715:,

433 Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh

Sect ion dro i tede l 'a rmature

Sect ion de l 'a rmature pro je téedans le p lan de la sect ion

G

x

z y

α

Iljxuh 7144 = Dupdwxuh lqfolqìh sdu udssruw ã od vhfwlrq gurlwh

O*dffurlvvhphqw gh vroolflwdwlrq v*ìfulw =

�I6? @

]76

��6 +|> },

3EEC

4

}

|

4FFDgV6 +715;,

hw o*rq rewlhqw =

�I6? @

]76

H6 +|> },�

4 } |

��%?

3EEC

4

}

|

4FFDgV6

@

]76

H6 +|> },

5997

4 } |

} }2 |}

| |} |2

6::8 gV6=�%? +715<,

Vl o*rq qrwh ^N6? ` od pdwulfh gh uljlglwì frugh uholdqw o*dffurlvvhphqw gh vroolflwdwlrq qru0

pdoh gh od pdwulfh gh od vhfwlrq ã od yduldwlrq gh gìirupdwlrq qrupdoh gh od vhfwlrq/ rq

d =

^N6? ` @

]76

H6 +|> },

5997

4 } |

} }2 |}

| |} |2

6::8 gV6 +7163,

715 Ìwxgh g*xqh vhfwlrq gurlwh 434

hw

�I6? @ ^N6? ` =�%? +7164,

Ohv vroolflwdwlrqv wdqjhqwhv gxhv ã od pdwulfh gh od vhfwlrq

Gdqv oh uhsëuh J{|}/ rq ìfulw ìjdohphqw =

I6| @

3EEC

W6+

W65

P6�%

4FFD +7165,

rü W6+ hw W65 vrqw ohv h�ruwv wudqfkdqwv/ hw P6�%oh prphqw gh wruvlrq dxwrxu gh F/

wrxv wurlv gxv ã od pdwulfh gh od vhfwlrq1 Vl o*rq qrwh ^N6| ` od pdwulfh gh uljlglwì uholdqw

ohv vroolflwdwlrqv wdqjhqwhv gh od pdwulfh gh od vhfwlrq dx{ gìirupdwlrqv wdqjhqwhv gh od

vhfwlrq/ hw hq vxssrvdqw txh fhwwh uhodwlrq hvw olqìdluh/ rq d =

I6| @ ^N6| ` =%| +7166,

dyhf =

^N6| ` @

5997JD+ 3 3

3 JD5 3

3 3 JL%

6::8 +7167,

rü J hvw oh prgxoh gh flvdloohphqw gh Frxorpe = J @.

2E�nD�/

D+ hw D5 vrqw ohv vhfwlrqv uìgxlwhv g*h�ruw wudqfkdqw/

L% hvw o*lqhuwlh gh wruvlrq gh od vhfwlrq1

Frpph od uhodwlrq +7166, hvw olqìdluh/ rq d lppìgldwhphqw =

�I6| @ ^N6| ` =�%| +7168,

Rq shxw doruv ìfuluh =#�I6?

�I6|

$@

%N6? 3

3 N6|

&=

#�%?

�%|

$+7169,

Ohv vroolflwdwlrqv qrupdohv gxhv dx{ �euhv gh od vhfwlrq

Ohv �euhv vrqw vxssrvìhv sduidlwhphqw dgkìuhqwhv ã od vhfwlrq1 Vl who q*hvw sdv oh fdv/ hoohv

qh sduwlflshqw sdv ã od uìvlvwdqfh gh od vhfwlrq hw q*hqwuhqw sdv gdqv oh fdguh gh fh txl yd

vxlyuh1 Gdqv oh uhsëuh J{|}/ rq shxw ìfuluh =

Is? @

3EEC

Qs

Ps+

Ps5

4FFD +716:,


rüQs hvw o*h�ruw qrupdo/Ps+ hwPs5 ohv prphqwv ìfklvvdqwv wrxv wurlv gxv dx{ dupdwxuhv

pìwdooltxhv wudyhuvdqw od vhfwlrq1 Vl o*rq qrwh �s� od frqwudlqwh qrupdoh gh o*dupdwxuh

g*ruguh l dgkìuhqwh hw wudyhuvdqw od vhfwlrq/ lo ylhqw =

Is? @

?s[�'�

�s� =

3EEC

4

}s�

|s�

4FFD = frv��=vs� +716;,

rü qs hvw oh qrpeuh g*dupdwxuhv dgkìuhqwhv hw wudyhuvdqw od vhfwlrq/

+|s� > }s�, vrqw ohv frrugrqqìhv gx srlqw gh sdvvdjh gh o*dupdwxuh g*ruguh l gdqv oh uhsëuh

J|}/

�� hvw o*dqjoh h{lvwdqw hqwuh o*dupdwxuh g*ruguh l hw o*d{h J{/

vs� hvw od vhfwlrq gurlwh gh o*dupdwxuh g*ruguh l1

Vl o*rq qrwh Hs� oh prgxoh ìodvwltxh orqjlwxglqdo frugh/ f*hvw0ã0gluh oh prgxoh uholdqw

o*dffurlvvhphqw gh frqwudlqwh ã o*dffurlvvhphqw gh gìirupdwlrq/ gh o*dupdwxuh g*ruguh l/

doruv =

��s� @ Hs� =�%s� +716<,

rü ��s� hvw o*dffurlvvhphqw gh frqwudlqwh qrupdoh gh o*dupdwxuh g*ruguh l/ hw �%s� o*df0

furlvvhphqw gh gìirupdwlrq qrupdoh gdqv fhwwh píph dupdwxuh1 Rq shxw doruv ìfuluh =

�Is? @

?s[�'�

��s� =

3EEC

4

}s�

|s�

4FFD = frv��=vs� +7173,

@

?s[�'�

Hs� =

3EEC

4

}s�

|s�

4FFD = frv��=vs� =�%s� +7174,

Hq xwlolvdqw od uhodwlrq +7155,/ lo ylhqw =

�Is? @ ^Ns? ` =�%? . ^Ns?| ` =�%| +7175,

dyhf =

^Ns? ` @

?s[�'�

Hs� = frv� ��=vs� =

5997

4 }s� |s�

}s� }2s� |s�}s�

|s� |s�}s� |2s�

6::8 +7176,

hw

^Ns?| ` @

?s[�'�

Hs� = vlq��= frv2 ��=vs� =

3EEC

4

}s�

|s�

4FFD =�

frv�� vlq�� +|s� � |S, vlq�� +}s� � }S, frv��

�

+7177,


rü/ srxu pìprluh/ �� hvw o*dqjoh h{lvwdqw hqwuh od surmhfwlrq gh o*dupdwxuh g*ruguh l gdqv

od sodq J|}/ hw o*d{h J|1 Hq gìyhorssdqw/ rq rewlhqw =

^Ns?| ` @

?s[�'�

Hs� = vlq��= frv2 ��=vs� =

5997

frv�� vlq�� +|s� � |S, vlq�� +}s� � }S, frv��

}s� frv�� }s� vlq�� }s� ++|s� � |S, vlq�� +}s� � }S, frv��,

|s� frv�� |s� vlq�� |s� ++|s� � |S, vlq�� +}s� � }S, frv��,

6::8

+7178,

Ohv vroolflwdwlrqv wdqjhqwhv gxhv dx{ �euhv gh od vhfwlrq

Gdqv oh uhsëuh J{|}/ rq d =

Is| @

3EEC

Ws+

Ws5

Ps�%

4FFD +7179,

rü Ws+ hw Ws5 vrqw ohv h�ruwv wudqfkdqwv/ hwPs�%oh prphqw gh wruvlrq dxwrxu gh F/ wrxv

wurlv gxv dx{ dupdwxuhv pìwdooltxhv dgkìuhqwhv hw wudyhuvdqw od vhfwlrq1 Hq uhsuhqdqw ohv

qrwdwlrqv gx sdudjudskh suìfìghqw/ rq rewlhqw =

Is| @

?s[�'�

�s� =

3EEC

frv��

vlq��

+|s� � |S, vlq�� +}s� � }S, frv��

4FFD = vlq��=vs� +717:,

Rq hq gìgxlw/ gh od píph idêrq txh suìfìghpphqw/ txh o*dffurlvvhphqw gh vroolflwdwlrqv

wdqjhqwhv gþ dx{ �euhv gh od vhfwlrq v*h{sulph vhorq =

�Is| @

?s[�'�

��s� =

3EEC

frv��

vlq��

+|s� � |S, vlq�� +}s� � }S, frv��

4FFD = vlq��=vs�

@

?s[�'�

Hs� = vlq��=vs� =

3EEC

frv��

vlq��

+|s� � |S, vlq�� +}s� � }S, frv��

4FFD =�%s� +717;,

Hq xwlolvdqw od uhodwlrq +7155,/ rq d =

�Is| @ ^Ns|? ` =�%? . ^Ns| ` =�%| +717<,

dyhf =

^Ns|? ` @

?s[�'�

Hs� = frv2 �� vlq��=vs� =

3EEC

frv��

vlq��

+|s� � |S, vlq�� +}s� � }S, frv��

4FFD =�

4 }s� |s�

�

+7183,


Rq uhpdutxh txh =

^Ns|? ` @|^Ns?| ` +7184,

Vl o*rq srvh =

Ys|� @

3EEC

frv��

vlq��

+|s� � |S, vlq�� +}s� � }S, frv��

4FFD +7185,

od pdwulfh ^Ns| ` v*rewlhqw doruv gh od idêrq vxlydqwh =

^Ns| ` @

?s[�'�

Hs� = vlq2 ��= frv��=vs� =Ys|� =

|Ys|� +7186,

Rq shxw doruv ìfuluh = #�Is?

�Is|

$@

%Ns? Ns?|

|Ns?| Ns|

&=

#�%?

�%|

$+7187,

71519 Ohv vroolflwdwlrqv h{whuqhv

Od vhfwlrq hvw vrxplvh ã ghv vroolflwdwlrqv h{wìulhxuhv txh vrqw o*h�ruw qrupdoQ / ohv h�ruwv

wudqfkdqwv W+ hw W5/ oh prphqw gh wruvlrq P�% / hw ohv prphqwv ìfklvvdqwv P+ hw P51

Uhjurxsrqv ohv vroolflwdwlrqv h{wìulhxuhv qrupdohv ã od vhfwlrq gurlwh gdqv oh yhfwhxu Ir? /

hw ohv vroolflwdwlrqv wdqjhqwhv ã od vhfwlrq gurlwh gdqv oh yhfwhxu Ir| 1 Lo ylhqw =

Ir? @

3EEC

Q

P+

P5

4FFD +7188,

hw

Ir| @

3EEC

W+

W5

P�%

4FFD +7189,

Sdu dloohxuv/ od vhfwlrq vxelw o*dfwlrq gh od suìfrqwudlqwh txh o*rq shxw gìfrpsrvhu hq xqh

sduwlh/ qrwìh Dr? / qrupdoh ã od vhfwlrq/ hw xqh sduwlh wdqjhqwh ã od vhfwlrq/ qrwìh Dr| 1 Hq

vh uìiìudqw dx{ uhodwlrqv +7173, hw +717:,/ hw hq uhvshfwdqw ohv frqyhqwlrqv gh vljqh txh o*rq

v*hvw �{ìhv/ rq d =

Dr? @ �?sR[�'�

�sRf�=

3EEC

4

}sR�|sR�

4FFD = frv�R� =vsR� +718:,

hw

Dr| @ �?sR[�'�

�sRf�=

3EEC

frv�R�

vlq�R��|sR� � |S

�vlq�R� �

�}sR� � }S

�frv�R�

4FFD = vlq�R� =vsR� +718;,


rü qsR hvw oh qrpeuh gh fåeohv gh suìfrqwudlqwh whqgxv dx frxuv gh o*rsìudwlrq frxudqwh

hw wudyhuvdqw od vhfwlrq/

�sRf�hvw od whqvlrq frxudqwh gdqv oh fåeoh g*ruguh l/ frpswh whqx ghv shuwhv lqvwdqwdqìhv1

Ohv judqghxuv vrqw lqglfìhv sdu s d�q gh prqwuhu txh vhxov ohv fåeohv gh suìfrqwudlqwh

whqgxv vrqw frqvlgìuìv1

7151: Ìtxloleuh gh od vhfwlrq

O*ìtxloleuh gh od vhfwlrq h{sulph vlpsohphqw oh idlw txh o*dffurlvvhphqw gh vroolflwdwlrqv

h{whuqhv hw o*dfwlrq gh od suìfrqwudlqwh vrqw ìtxloleuìv sdu o*dffurlvvhphqw gh vroolflwdwlrqv

lqwhuqhv/ g*rü = #�Ir?

�Ir|

$.

#Dr?

Dr|

$@

#�I6?

�I6|

$.

#�Is?

�Is|

$+718<,

Fhfl shxw hqfruh vh phwwuh vrxv od iruph =#�Ir?

�Ir|

$.

#Dr?

Dr|

$@ ^Nr` =

#�%?

�%|

$+7193,

rü ^Nr` hvw od pdwulfh gh uljlglwì gh od vhfwlrq/ grqw od irupxodwlrq hvw grqqìh sdu =

^Nr` @

%^N6? ` . ^Ns? ` ^Ns?| `

|^Ns?| ` ^N6| ` . ^Ns| `

&+7194,

Od pdwulfh ^Nr` hvw od pdwulfh gh uljlglwì frugh uholdqw o*dffurlvvhphqw g*h�ruwv h{whuqhv

ã o*dffurlvvhphqw gh gìirupdwlrqv1 Od qrq0olqìdulwì pdwìulhooh lqwhuylhqw gdqv oh fdofxo gh

od pdwulfh ^Nr`/ txl gìshqg gh o*ìwdw gh gìirupdwlrq gh od vhfwlrq/ f*hvw0ã0gluh gh %?1 Od

uìvroxwlrq gh o*ìtxdwlrq +7193, hvw irufìphqw lwìudwlyh1

7151; Irupdwlrq gh od pdwulfh gh uljlglwì gh od vhfwlrq

Oh fdofxo gh ^Nr` xwlolvh od uhodwlrq +7194,1 Ohv ìoìphqwv irupdqw ^Nr` q*rqw sdv wrxv oh

píph frþw hq whpsv gh fdofxo1

Dlqvl/ oh frþw gh od irupdwlrq gh ^N6| `/ xwlolvdqw od uhodwlrq +7167,/ hvw wuëv idleoh fdu

lo uhtxlhuw vhxohphqw 6 pxowlsolfdwlrqv1

Oh frþw gx fdofxo gh ^Ns? `/ ^Ns?| ` hw ^Ns| ` g*dsuëv ohv uhodwlrqv +7176,/ +7178, hw +7186,/

hvw soxv ìohyì fdu lo qìfhvvlwh soxv g*rsìudwlrqv/ pdlv lo uhvwh prghvwh fdu od vrpph dssd0

udlvvdqw gdqv fhv uhodwlrqv hvw glvfuëwh1

Ilqdohphqw/ oh whuph oh soxv frþwhx{ hvw ^N6? `/ fdu/ hq vh uìiìudqw ã +7163,/ rq frq0

vwdwh txh o*ìydoxdwlrq g*xqh lqwìjudoh vxuidfltxh hvw qìfhvvdluh1 Frpph qrxv o*dyrqv lq0

gltxì gdqv ohv k|srwkëvhv jìqìudohv/ fkdtxh vhfwlrq hvw xq frxwrxu sro|jrqdo/ gìfrxsì hq


wudsë}hv gdqv oh uhsëuh Lf|f}f1 Frpph qrxv idlvrqv o*k|srwkëvh txh oh uhsëuh lqwulqvëtxh

vh gìgxlw gx uhsëuh orfdo sdu xq prxyhphqw gh frusv uljlgh/ hw frpph qrxv vxssrvrqv txh

gdqv oh uhsëuh lqwulqvëtxh/ ohv gìsodfhphqwv hw ohv gìirupdwlrqv uhvwhqw prgìuìv/ o*lqwìjud0

wlrq vxuidfltxh gdqv oh uhsëuhJ|} v*dssoltxh ã od jìrpìwulh lvvxh gx gìfrxsdjh wudsì}rògdo

gdqv oh uhsëuh Lf|f}f1

y

z

G

Poin ts de dé f in i t ion du contour

T rapèze é lémen ta i re

Iljxuh 7145 = Vhfwlrq gurlwh

Gdqv od uhodwlrq +7163,/ o*lqwìjudoh vxu od vxuidfh V6 hvw fdofxoìh frpph ìwdqw od vrpph

ghv lqwìjudohv vxu fkdfxq ghv wudsë}hv gx gìfrxsdjh1 Srxu lqwìjuhu vxu xq wudsë}h/ qrxv

dyrqv dgrswì xq vfkìpd g*lqwìjudwlrq gh w|sh Vlpsvrq1 D�q g*rewhqlu ghv ìydoxdwlrqv

fruuhfwhv ghv whuphv gh ^N6? `/ qrxv dyrqv frqvwdwì ghv whpsv gh fdofxo wuëv lpsruwdqwv1 Hq

h�hw/ oruvtxh o*rq xwlolvh ghv orlv qìjoljhdqw oh eìwrq whqgx/ od ydohxu gx prgxoh ìodvwltxh

orqjlwxglqdo frugh H6 hvw glvfrqwlqxh =

H6 @

+H��f oruvtxh % +|> }, whqg yhuv 3n

3 oruvtxh % +|> }, whqg yhuv 33+7195,

rü % +|> }, hvw od gìirupdwlrq orqjlwxglqdoh hq xq srlqw frxudqw gh od vhfwlrq/ rewhqxh dyhf

+7149,1 Od suìflvlrq rewhqxh srxu o*ìydoxdwlrq ghv whuphv gh ^N6? ` gìshqg gluhfwhphqw

gx qrpeuh gh srlqwv g*lqwìjudwlrq vxu oh wudsë}h = soxv rq d gh srlqwv g*lqwìjudwlrq/ prlqv

od }rqh rü H6 hvw glvfrqwlqx hvw lpsruwdqwh/ hw soxv o*huuhxu frpplvh hvw idleoh1

715 Ìwxgh g*xqh vhfwlrq gurlwh 43:

Srxu uìvrxguh fh sureoëph/ qrxv dyrqv plv dx srlqw xq pdloodjh dgdswdwli txl h�hfwxh/

ã sduwlu gx gìfrxsdjh wudsì}rògdo lqlwldo +�jxuh 7145,/ xq dxwuh gìfrxsdjh txl vìsduh

od vhfwlrq hq ghx{ }rqhv = xqh }rqh rü % +|> }, A %S hw xqh }rqh rü % +|> }, ? %S/ hq

gìvljqdqw sdu %S od ydohxu fulwltxh gh od gìirupdwlrq srxu odtxhooh oh prgxoh g*\rxqj

ydulh euxvtxhphqw +�jxuh 7146,1 Od ydohxu gh %S gìshqg gh od orl gx frpsruwhphqw gx

pdwìuldx hw ydxw 3 srxu wrxwhv ohv orlv gh frpsruwhphqw gx pdwìuldx eìwrq/ txl qìjoljhqw

oh eìwrq whqgx1 Gx idlw gh o*k|srwkëvh ghv gìirupdwlrqv sodqhv/ o*ìtxdwlrq % +|> }, @ %S hvw

uhsuìvhqwìh sdu xqh gurlwh gdqv oh uhsëuh J|}1

y

z

G

ε = εc

Po in ts supp lémenta i resdu con tour

Trapèzes é lémenta i ressupp lémenta i res

Iljxuh 7146 = Pdloodjh dgdswdwli

Oh pdloodjh dlqvl dgrswì ydulh hq irqfwlrq gh %?/ grqw od ydohxu prgl�h fhooh gh % +|> },1

Od srvlwlrq gh od gurlwh % +|> }, @ %S gìshqg gh od ydohxu frxudqwh gh %?1 Gh fh idlw/ ohv

srlqwv g*lqwìjudwlrq qh vrqw sdv ohv píphv g*xqh lwìudwlrq ã o*dxwuh1 Rq qh shxw grqf

sdv frqqdñwuh o*klvwrultxh ghv frqwudlqwhv gh fkdfxq ghv srlqwv g*lqwìjudwlrq hw fhfl qrxv

hpsífkh gh srxyrlu xwlolvhu ghv orlv gh frpsruwhphqw k|vwìuìwltxhv/ rx ghv prgëohv ã

hqgrppdjhphqw +Oderughulh ^Ode<4`,1 Hq frqwuhsduwlh/ ohv whpsv gh fdofxo dlqvl txh od

suìflvlrq rewhqxh srxu o*ìydoxdwlrq gh ^N6? ` vrqw wuëv vdwlvidlvdqwv1

43; Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh

7151< Uìvroxwlrq lwìudwlyh gh o*ìtxloleuh gh od vhfwlrq

Ohv ìoìphqwv �qlv fodvvltxhv/ gh w|sh ��gìsodfhphqw��/ rqw xq dydqwdjh sdu udssruw dx{

ìoìphqwv �qlv gh w|sh ��vhpl0lqyhuvh�� txh qrxv xwlolvrqv1 Hq h�hw/ frqqdlvvdqw oh fkdps

gh gìirupdwlrqv/ lov hq gìgxlvhqw gluhfwhphqw oh fkdps ghv frqwudlqwhv hw shxyhqw dlqvl hq

gìgxluh ohv h�ruwv lqwhuqhv gh o*ìoìphqw1 Gdqv qrwuh fdv/ qrxv frqqdlvvrqv od uìsduwlwlrq

ghv h�ruwv oh orqj gh o*ìoìphqw/ hw qrxv yrxorqv hq gìgxluh ohv gìirupdwlrqv fruuhvsrq0

gdqwhv/ hq xwlolvdqw od uhodwlrq +7193,1 Frpph o*rsìudwhxu gh sdvvdjh hqwuh ohv h�ruwv hw

ohv gìirupdwlrqv/ ^Nr`/ gìshqg ghv gìirupdwlrqv/ f*hvw0ã0gluh gh o*lqfrqqxh/ qrxv vrpphv

reoljìv g*xwlolvhu xq vfkìpd lwìudwli gh uìvroxwlrq1 Qrxv xwlolvhurqv od pìwkrgh gh vxe0

vwlwxwlrq/ txl frqvlvwh ã ìydoxhu ^Nr` ã fkdtxh lwìudwlrq hq irqfwlrq gh %?/ hw ã uìvrxguh

+7193,1

Qrxv doorqv frqvlgìuhu xqh vhfwlrq hq ìtxloleuh hw qrxv qrwhurqv =

%r oh yhfwhxu gìirupdwlrq gh od vhfwlrq =

%r @

#%?

%|

$+7196,

Ir oh yhfwhxu ghv vroolflwdwlrqv h{whuqhv =

Ir @

#Ir?

Ir|

$+7197,

Fhwwh vhfwlrq yd vxelu xq dffurlvvhphqw gh vroolflwdwlrqv h{whuqhv =

�Ir @

#�Ir?

�Ir|

$+7198,

dlqvl txh o*dfwlrq gh od suìfrqwudlqwh frqfrplwdqwh dx fkdujhphqw frqvlgìuì =

Dr @

#Dr?

Dr|

$+7199,

Qrxv fkhufkrqv ã fdofxohu o*dffurlvvhphqw gh gìirupdwlrq gh od vhfwlrq fruuhvsrqgdqw =

�%r @

#�%?

�%|

$+719:,

Hq lqyhuvdqw +7193,/ rq rewlhqw =

�%r @ ^Vr` = +�Ir .Dr, +719;,

rü ^Vr` hvw od pdwulfh gh vrxsohvvh gh od vhfwlrq/ gì�qlh sdu =

^Vr` @ ^Nr`3� +719<,

715 Ìwxgh g*xqh vhfwlrq gurlwh 43<

ε

0

s

s

F

sF

εs

s s

ε + ∆εss

F + F∆

Iljxuh 7147 = Uìvroxwlrq lwìudwlyh gh o*ìtxloleuh gh od vhfwlrq

Qrxv frqvwuxlvrqv xqh vxlwh gh yhfwhxuv gìirupdwlrqv gh od vhfwlrq �%r� gì�qlh sdu =

�%rf @ +3, +71:3,

�%r�n� @ ^Vr`� = +�Ir .Dr, +71:4,

rü ^Vr`� hvw od pdwulfh gh vrxsohvvh gh od vhfwlrq frugh hqwuh ohv ìwdwv gh gìirupdwlrq %r hw

%r.�%r� 1 Rq frqvlgëuh txh od vxlwh gh yhfwhxuv gìirupdwlrq gh od vhfwlrq�%r� d frqyhujì

oruvtxh = ��%r�n� ��%r��2��%r .�%r�n�

��2

? wrovhf +71:5,

rü wrovhf hvw od wroìudqfh gh frqyhujhqfh gh o*ìtxloleuh gh od vhfwlrq/ grqw od ydohxu hvw

�{ìh sdu o*xwlolvdwhxu1 Sdu gìidxw/ qrxv dyrqv =

wrovhf @ 433e +71:6,

715143 Ohv olplwhv

Frpph rq shxw oh frqvwdwhu vxu od �jxuh +7147,/ oruvtxh o*rq slorwh hq irufh/ fh txl hvw qrwuh

fdv/ rq qh shxw sdv iudqfklu oh slf gx gldjudpph h�ruw0gìirupdwlrq1 Hq h�hw/ srxu xq

h�ruw grqqì/ rq shxw wurxyhu soxvlhxuv gìirupdwlrqv fruuhvsrqgdqwhv1 Qrwuh pìwkrgh gh

uìvroxwlrq irxuqlw od suhplëuh gìirupdwlrq vroxwlrq1 Srxu iudqfklu oh slf/ lo hvw qìfhvvdluh


gh slorwhu hq gìirupdwlrq/ hw grqf hq gìsodfhphqw1 Fhfl hvw grqf xqh olplwh srxu qrwuh

ìoìphqw �ql gh w|sh ��vhpl0lqyhuvh��1 Qrxv qrxv hq dffrpprgrqv ìwdqw grqqì oh fkdps

g*dssolfdwlrq gx surjudpph/ txl hvw fhqvì fdofxohu o*ìwdw gh vwuxfwxuhv gh Jìqlh flylo/

vxssrvìhv uhvwhu gdqv oh grpdlqh ìodvwltxh/ ìyhqwxhoohphqw qrq olqìdluh/ pdlv hq dxfxq

fdv gdqv oh grpdlqh srvw0slf1

Sdu dloohxuv/ od uxswxuh hvw gìwhfwìh gëv txh od pdwulfh ^Nr` q*hvw soxv gì�qlh srvlwlyh1

Srxu whvwhu fhfl/ fhuwdlqv dxwhxuv d!uphqw tx*lo vx!w gh yìul�hu oh vljqh gx gìwhuplqdqw

gh ^Nr`/ txl fruuhvsrqg dx surgxlw ghv ydohxuv sursuhv gh ^Nr`1 Gdqv od soxsduw ghv

fdv/ fhod shxw vx!uh1 Pdlv qrxv dyrqv uhqfrqwuì xq fdv rü ghx{ ydohxuv sursuhv gh ^Nr`

vrqw ghyhqxhv qìjdwlyhv dx frxuv gh od píph lwìudwlrq1 Od vhfwlrq ìwdlw grqf uxlqìh/

pdlv oh whvw edvì vxu oh vljqh gx gìwhuplqdqw q*dydlw ulhq gìwhfwì1 Qrxv h�hfwxrqv grqf

v|vwìpdwltxhphqw xqh dqdo|vh vshfwudoh gh ^Nr` ã fkdtxh lwìudwlrq/ hw qrxv whvwrqv oh vljqh

gh od soxv shwlwh ghv ydohxuv sursuhv srxu gìwhfwhu od uxlqh ìyhqwxhooh gh od vhfwlrq1

Od pìwkrgh gh vxevwlwxwlrq shxw v*dyìuhu ohqwh gdqv fhuwdlqv fdv1 F*hvw srxutxrl qrxv

dyrqv lpsodqwì xq dffìoìudwhxu gh frqyhujhqfh gdqv o*dojrulwkph lwìudwli1 Ã sduwlu g*xqh

txlq}dlqh g*lwìudwlrqv/ dx olhx g*h�hfwxhu vlpsohphqw od vxevwlwxwlrq/ qrxv hqfohqfkrqv od

pìwkrgh glwh gh od idxvvh srvlwlrq/ txl frqvlvwh ã frqvwuxluh �%r�n� gh od idêrq vxlydqwh =

�%r�n�� @

�%r�3�� i� +�%r�,��%r�� i��%r�3�

�i� +�%r�,� i�

��%r�3�

� +71:7,

rü i hvw od irqfwlrq yhfwrulhooh txh o*rq yhxw dqqxohu =

i +�%r, @ �%r � ^Nr +%r .�%r,`3�

= +�Ir .Dr, +71:8,

�%r�n�� uhsuìvhqwh od mëph frpsrvdqwh gx yhfwhxu �%r�n� / m yduldqw gh 4 ã 91

Vl oh qrpeuh g*lwìudwlrqv ghylhqw wurs lpsruwdqw vdqv txh o*rq dlw sx frqvwdwhu od

frqyhujhqfh ql od uxswxuh gh od vhfwlrq/ qrxv frqfoxrqv ã xq ìwdw glw gh glyhujhqfh gh od

vhfwlrq/ txl lqwhuurpsw qìdqprlqv oh surfhvvxv lwìudwli1

716 Lqwìjudwlrq oh orqj gh o*ìoìphqw

71614 Uhodwlrq hqwuh ohv vroolflwdwlrqv ghv vhfwlrqv hw ohv h�ruwv ìoìphq0

wdluhv

Gdqv fh sdudjudskh/ qrxv doorqv ìwdeolu od uhodwlrq h{lvwdqw hqwuh Ir/ ohv vroolflwdwlrqv gdqv

xqh vhfwlrq gurlwh hw I?/ ohv h�ruwv dx{ qúxgv gh o*ìoìphqw gdqv oh uhsëuh lqwulqvëtxh1

716 Lqwìjudwlrq oh orqj gh o*ìoìphqw 444

Qrxv dyrqv yx txh od vhfwlrq vxelw ohv h�ruwv h{whuqhv vxlydqwv =

Ir @

#Ir?

Ir|

$@

3EEEEEEEEEC

Q

P+

P5

W+

W5

P�%

4FFFFFFFFFD

+71:9,

Fhv h�ruwv vrqw dssoltxìv vxu od vhfwlrq g*devflvvh uhodwlyh { vxu o*d{h J{ gx uhsëuh lqwulq0

vëtxh1 Rq shxw grqf ìfuluh =

Ir +{, @

#Ir? +{,

Ir| +{,

$@

3EEEEEEEEEC

Q +{,

P+ +{,

P5 +{,

W+ +{,

W5 +{,

P�% +{,

4FFFFFFFFFD

+71::,

Od irupxodwlrq hq ìoìphqwv �qlv gh w|sh ��vhpl0lqyhuvh�� qìfhvvlwh od gì�qlwlrq gh od

irqfwlrq gh iruph srxu ohv h�ruwv1 Qrxv fkrlvlvvrqv ohv irqfwlrqv gh iruph ohv soxv vlpsohv

txl vrlhqw/ f*hvw0ã0gluh txh qrxv doorqv vxssrvhu od glvwulexwlrq g*h�ruw qrupdo frqvwdqwh oh

orqj gh o*ìoìphqw/ hw txh ohv prphqwv ìfklvvdqwv ydulhqw olqìdluhphqw oh orqj gh o*ìoìphqw1

Ohv h�ruwv wudqfkdqwv hw oh prphqw gh wruvlrq vrqw fhqvìv uhvwhu frqvwdqwv gdqv o*ìoìphqw1

Qrxv vxssrvrqv hq rxwuh txh gdqv oh uhsëuh lqwulqvëtxh/ ohv h�hwv gx vhfrqg ruguh vrqw

qìjoljìv1

Dlqvl/ od uhodwlrq h{lvwdqw hqwuh Ir +{, hw I?/ oh yhfwhxu ghv h�ruwv uìgxlw gdqv oh uhsëuh

lqwulqvëtxh hvw od vxlydqwh/ ìwdqw grqqìhv ohv frqyhqwlrqv gh vljqh gh od �jxuh +7143, =

Srxu ohv vroolflwdwlrqv qrupdohv =

Q +{, @ �Q� +71:;,

P+ +{, @

�4� {

OJ

�P�+ �

{

OJP�+ +71:<,

P5 +{, @ ��4� {

OJ

�P�5 .

{

OJP�5 +71;3,

Srxu ohv h�ruwv wudqfkdqwv =

W+ +{, @ �W�+ @P�5 .P�5

O+71;4,

W5 +{, @ �W�5 @ �P�+ .P�+

O+71;5,


Srxu oh prphqw gh wruvlrq =

P% +{, @ �P�% +71;6,

Vdfkdqw txh =

P�% +{, @P% +{,� |S=W5 +{, . }S=W+ +{, +71;7,

lo ylhqw =

P�% +{, @ �P�% � |S=W5 +{, . }S=W+ +{, +71;8,

@ �P�% .|S

OP�+ .

|S

OP�+ .

}S

OP�5 .

}S

OP�5 +71;9,

Wrxw fhfl vh phw pdwulflhoohphqw vrxv od iruph =

Ir +{, @ Ô +{,` � I? +71;:,

rü O +{, hvw xqh pdwulfh grqw od irupxodwlrq hvw grqqìh sdu =

Ô +{,` @

59999999997

�4 3 3 3 3 3

3 4� %uf

3 3 � %uf

3

3 3 �4 . %uf

3 3%uf

3 3�u

3 3�u

3 � �u

3 3 � �u

3

3+Su

5Su

�4 +Su

5Su

6:::::::::8

+71;;,

Hq qìjoljhdqw od yduldwlrq gh orqjxhxu gh o*ìoìphqw gdqv oh uhsëuh lqwulqvëtxh/ rq shxw

ìfuluh =

�Ir +{, @ Ô +{,` =�I? +71;<,

Fhfl uhylhqw ã gluh =

^�O +{,` @ ^3` +71<3,

71615 Uhodwlrq hqwuh ohv gìsodfhphqwv ìoìphqwdluhv hw ohv gìirupdwlrqv

ghv vhfwlrqv

Gdqv fh sdudjudskh/ qrxv doorqv ìwdeolu od uhodwlrq h{lvwdqw hqwuh V?/ ohv gìsodfhphqwv ghv

qúxgv gh o*ìoìphqw gdqv oh uhsëuh lqwulqvëtxh hw %r/ ohv gìirupdwlrqv ghv vhfwlrqv gh fhw

ìoìphqw1 Srxu fhod/ qrxv doorqv frqvlgìuhu �I?/ xqh yduldwlrq vwdwltxhphqw dgplvvleoh ghv

h�ruwv dx{ qúxgv gh o*ìoìphqw/ gdqv oh uhsëuh lqwulqvëtxh1 Oh wudydlo yluwxho dlqvl lqgxlw

hvw =

�z? @|�I?=V? +71<4,

Od yduldwlrq vwdwltxhphqw dgplvvleoh �I? lqgxlw xqh yduldwlrq �Ir ghv vroolflwdwlrqv gdqv

ohv vhfwlrqv gh o*ìoìphqw1 Oh wudydlo yluwxho ghv vhfwlrqv v*ìfulw =

�zr @

] uf

f

|�Ir +{, =%r +{,g{ +71<5,

716 Lqwìjudwlrq oh orqj gh o*ìoìphqw 446

G*dsuëv oh wkìruëph gx wudydlo yluwxho frpsoìphqwdluh/ rq d =

�z? @ �zr +71<6,

Hq frqvlgìudqw +71;<,/ rq rewlhqw =

�Ir +{, @ Ô +{,` =�I? +71<7,

G*rü =|�I?=V? @

] uf

f

|�I?=|Ô +{,` =%r +{, g{ +71<8,

Frpph �I? qh gìshqg sdv gh {/ lo ylhqw =

|�I?=V? @|�I?=

] uf

f

|Ô +{,` =%r +{,g{ +71<9,

Od yduldwlrq �I? ìwdqw duelwudluh/ rq hq gìgxlw doruv =

V? @

] uf

f

|Ô +{,` =%r +{,g{ +71<:,

Wrxmrxuv hq qìjoljhdqw od yduldwlrq gh orqjxhxu gh o*ìoìphqw gdqv oh uhsëuh lqwulqvëtxh/ rq

shxw ìfuluh =

�V? @

] uf

f

|Ô +{,` =�%r +{,g{ +71<;,

71616 Irupdwlrq gh od pdwulfh gh uljlglwì ìoìphqwdluh dg?o gdqv oh uhsëuh

lqwulqvëtxh

Hq uhpsodêdqw +719;, gdqv +71<;,/ lo ylhqw =

�V? @

] uf

f

|Ô +{,` = ^Nr`

3�= +�Ir +{, .Dr +{,,g{ +71<<,

Hq whqdqw frpswh gh +71;<,/ rq rewlhqw =

�V? @

] uf

f

|Ô +{,` = ^Nr`

3�+Ô +{,` =�I? .Dr +{,, g{ +71433,

@

] uf

f

|Ô +{,` = ^Nr`

3� = Ô +{,` ��I?g{.] uf

f

|Ô +{,` = ^Nr`

3� =Dr +{, g{

Frpph �I? qh gìshqg sdv gh {/ rq shxw ìfuluh =

�V? @

] uf

f

|Ô +{,` = ^Nr`

3�= Ô +{,`g{=�I? .

] uf

f

|Ô +{,` = ^Nr`

3�=Dr +{,g{ +71434,

Od pdwulfh gh vrxsohvvh gh o*ìoìphqw gdqv oh uhsëuh lqwulqvëtxh/ qrwìh ^V?`/ dssdudñw qd0

wxuhoohphqw hq srvdqw =

^V?` @

] uf

f

|Ô +{,` = ^Nr`

3�= Ô +{,`g{ +71435,


Rq gì�qlw od pdwulfh gh uljlglwì gh o*ìoìphqw gdqv oh uhsëuh lqwulqvëtxh =

^N?` @ ^V?`3� +71436,

Hq�q/ o*dfwlrq gh od suìfrqwudlqwh gdqv oh uhsëuh lqwulqvëtxh v*ìfulw =

D? @ ^N?` =

] uf

f

|Ô +{,` = ^Nr`

3�=Dr +{, g{ +71437,

Oh whuphU uff

|Ô +{,` = ^Nr`

3�=Dr +{,g{ fruuhvsrqg ã o*lqwìjudwlrq ghv gìirupdwlrqv lq0

gxlwhv sdu od suìfrqwudlqwh/ oh orqj gh o*ìoìphqw1 Dlqvl/ hq lqyhuvdqw od uhodwlrq +71434,/ rq

rewlhqw =

�I? .D? @ ^N?` =�V? +71438,

Od pdwulfh ^N?` dlqvl rewhqxh hvw od pdwulfh frugh uholdqw o*dffurlvvhphqw g*h�ruwv

ã o*dffurlvvhphqw gh gìsodfhphqwv dx{ qúxgv gh o*ìoìphqw gdqv vrq uhsëuh lqwulqvëtxh1

O*rsìudwhxu ^N?` suhqg hq frpswh ã oxl wrxw vhxo od qrq0olqìdulwì pdwìulhooh1 Frpph qrxv

o*dyrqv yx gdqv oh fkdslwuh frqvdfuì ã od qrq0olqìdulwì jìrpìwultxh/ od pdwulfh gh uljlglwì

gh o*ìoìphqw gdqv oh uhsëuh joredo vh phw vrxv od iruph =

^Nf ` @ ^Sf`|^Wf`

�|Ê` ^N?` Ê` . ^G`

�^Wf`

|^Sf` +71439,

rü ^Sf` hw ^Wf` vrqw ghv pdwulfhv frqvwdqwhv1 Od irupxodwlrq jìrpìwultxh/ txl dssdudñw

ã wudyhuv ohv pdwulfhv Ê` hw ^G`/ hvw xqh irupxodwlrq wdqjhqwh1 O*rsìudwhxu txh qrxv

xwlolvrqv hvw k|eulgh = lo hvw frugh hq qrq0olqìdulwì pdwìulhooh hw wdqjhqw hq qrq0olqìdulwì

jìrpìwultxh1

Fkdslwuh 8

Dojrulwkph jìqìudo gh uìvroxwlrq

Qrxv doorqv suìvhqwhu qrwuh dojrulwkph jìqìudo gh uìvroxwlrq1 Qrxv doorqv g*derug gìyhors0

shu oh surfhvvxv plv hq úxyuh srxu wurxyhu o*ìtxloleuh gh od vhfwlrq/ sxlv qrxv gìfulurqv

o*dojrulwkph shuphwwdqw gh yìul�hu o*ìtxloleuh jìqìudo gh od vwuxfwxuh1

814 Ìtxloleuh gh od vhfwlrq

O*dojrulwkph shuphwwdqw od uhfkhufkh gh o*ìtxloleuh gh od vhfwlrq hvw gìfulw vxu od �jxuh

+814,1 Ã sduwlu g*xq ìwdw ìtxloleuì gh od vhfwlrq/ qrxv dssoltxrqv o*lqfuìphqw g*h�ruwv

h{wìulhxuv �Ir/ dyhf ìyhqwxhoohphqw o*dfwlrq gh od suìfrqwudlqwh Dr1 Hq suhqdqw xq df0

furlvvhphqw gh gìirupdwlrqv�%r qxo srxu frpphqfhu/ qrxv fdofxorqv od pdwulfh gh uljlglwì

gh od vhfwlrq ^Nr`1 Qrxv h�hfwxrqv doruv xqh dqdo|vh vshfwudoh gh ^Nr` d�q g*hq gìwhuplqhu

od soxv shwlwh ghv ydohxuv sursuhv/ txh qrxv qrwhurqv �6�?1 Vl �6�? 9 3/ doruv qrxv frq0

foxrqv ã od uxlqh gh od vhfwlrq hw oh surfhvvxv lwìudwli hvw lppìgldwhphqw vwrssì1 Vlqrq/

rq uìvrxw o*ìtxdwlrq g*ìtxloleuh gh od vhfwlrq =

�Ir .Dr @ ^Nr` =�%r +814,

mxvtx*ã frqyhujhqfh gh �%r1 V*lo q*| d sdv frqyhujhqfh/ �%r hvw prgl�ì oh fdv ìfkìdqw ã

o*dlgh gh od pìwkrgh gh od idxvvh srvlwlrq/ sxlv qrxv uììydoxrqv ^Nr` srxu uhfrpphqfhu

oh f|foh/ mxvtx*ã frqyhujhqfh1 Vl dx erxw g*xq qrpeuh g*lwìudwlrqv �{ì ã o*dydqfh/ rq qh

frqvwdwh sdv gh frqyhujhqfh/ qrxv frqfoxrqv doruv ã xq ìwdw gh glyhujhqfh gh od vhfwlrq/

txl lqwhuurpsw oh surfhvvxv1

815 Ìtxloleuh jìqìudo gh od vwuxfwxuh

O*dojrulwkph shuphwwdqw od uhfkhufkh gh o*ìtxloleuh gh od vwuxfwxuh hvw gìfulw vxu od �jxuh

+815,1 Od vwuxfwxuh ìwxglìh hvw srvlwlrqqìh gdqv oh uhsëuh joredo R[\ ]1 Hooh hvw fhqvìh

449 Fkdslwuh 8 Dojrulwkph jìqìudo gh uìvroxwlrq

Etat équilibré: Fs, εs

∆Fs

Αs

∆εs = 0

Formation de la matrice de rigidité de la section Ks

Analyse spectrale de Ks

Test sur λmin

Résolution du système∆Fs + Αs = Ks. ∆εs

Test sur la convergence de ∆εs

Code retour 0Code retour 2

Fin

Méthode de la fausse position

Code retour 1

Rupture de la section

Divergence

λmin > 0

λmin < 0

Oui

Non

Iljxuh 814 = Rujdqljudpph gh o*dojrulwkph gh uhfkhufkh g*ìtxloleuh gh od vhfwlrq

815 Ìtxloleuh jìqìudo gh od vwuxfwxuh 44:

íwuh gdqv xq ìwdw ìtxloleuì vrxv xq fkdujhphqw S gì�ql gdqv oh uhsëuh joredo/ dyhf oh gì0

sodfhphqw fruuhvsrqgdqw V} hw ohv h�ruwv lqwhuqhv I} gdqv oh píph uhsëuh1 Od gìirupdwlrq

gh fkdfxqh ghv vhfwlrqv hvw ìjdohphqw frqqxh1

Od vwuxfwxuh yd vxelu xq dffurlvvhphqw gh fkdujh �S / frxsoì ìyhqwxhoohphqw ã xqh plvh

hq suìfrqwudlqwh qrwìh D}/ hw qrxv ghyrqv wurxyhu o*dffurlvvhphqw gh gìsodfhphqwv �V}fruuhvsrqgdqw1 Srxu fhod/ qrxv doorqv frqvwuxluh xqh vxlwh gh vroxwlrqv �V}� gh od idêrq

vxlydqwh = �V}�n�hvw vroxwlrq gh o*ìtxdwlrq

�S .D} @ ^N}`� =�V}�n� +815,

rü ^N}`� hvw od pdwulfh gh uljlglwì gh od vwuxfwxuh/ rewhqxh dsuëv dvvhpeodjh ghv pdwulfhv

gh uljlglwì ìoìphqwdluhv ^Nf `/ ã o*ìwdsh l1 Od vxlwh hvw lqlwldolvìh dyhf =

�V}f @ +3, +816,

Ã fkdtxh lwìudwlrq/ fkdtxh pdwulfh gh uljlglwì ìoìphqwdluh ^Nf ` hvw uhfdofxoìh gdqv oh

uhsëuh joredo/ sxlv hvw dvvhpeoìh gdqv ^N}`1 Od uhodwlrq +815, hvw lqyhuvìh1 Rq gìfoduh txh

od vxlwh d frqyhujì oruvtxh = ��V}�n� ��V}��

nV} .�V}�n? wroghs +817,

rü wroghs hvw od wroìudqfh gh frqyhujhqfh hq gìsodfhphqw gh od vwuxfwxuh1 wroghs hvw �{ì

sdu gìidxw ã =

wroghs @ 433D +818,

Vl oh qrpeuh g*lwìudwlrqv ghylhqw wurs lpsruwdqw/ od vwuxfwxuh hvw gìfoduìh glyhujhqwh1 Dx

frxuv gh od irupdwlrq ghv pdwulfhv ^Nf `/ xqh uxswxuh rx xqh glyhujhqfh g*xqh vhfwlrq

shxw íwuh revhuyìh1 Fhfl duuíwh lppìgldwhphqw oh surfhvvxv gh fdofxo1 O*lqyhuvlrq gh +815,

v*h�hfwxh dx pr|hq g*xqh uìvroxwlrq gh Fkrohvn|/ fh txl d o*dydqwdjh gh frqwu÷ohu oh vljqh

ghv ydohxuv sursuhv gh ^N}`1 Vl o*xqh ghv ydohxuv sursuhv gh ^N}` hvw qìjdwlyh/ od uìvroxwlrq

gh Fkrohvn| hvw lpsrvvleoh/ hw oh surfhvvxv v*duuíwh/ hq gìwhfwdqw oh qúxg hw oh ghjuì gh

olehuwì frqfhuqì1 Oh vwrfndjh vh idlw fodvvltxhphqw vrxv od oljqh gh flho/ srxu ohv pdwulfhv

uìhoohv v|pìwultxhv1 Sdu dloohxuv/ ohv qúxgv rqw vxel xqh uhqxpìurwdwlrq lqwhuqh d�q gh

plqlplvhu od wdlooh gh od pdwulfh gh uljlglwì ^N}` +yrlu dqqh{h E,1

Xqh irlv txh od vwuxfwxuh d frqyhujì hq gìsodfhphqw/ qrxv iruprqv ohv h�ruwv lqwhuqhv ã od

vwuxfwxuh fruuhvsrqgdqw dx qrxyho ìwdw gh gìsodfhphqw ã o*dlgh gh od uhodwlrq +614;5,/ txl

v*ìfulw lfl =

I� .�I� @|^E +V� .�V�,` = +I? .�I?, +819,

Sdu +61439, hw +61546,/ qrxv iruprqv ohv h�ruwv lqwhuqhv If gh fkdfxq ghv ìoìphqwv gh

od vwuxfwxuh/ hw ohxu dffurlvvhphqw fruuhvsrqgdqw �If gdqv oh uhsëuh joredo1 Gh sdu od

44; Fkdslwuh 8 Dojrulwkph jìqìudo gh uìvroxwlrq

Etat équilibré: P, Fg, Sg, ε

résidu = ∆P

Boucle sur les éléments actifs de la structure

∆Fn

∆Sg = 0

Boucle sur les sections de l'élément

Kn, An K l, Al

KX, AX

Assemblage de Kg

Formation de Ag

Résolution du systèmerésidu + Ag = Kg. ∆Sg

Test sur la convergence de ∆Sg

Cumul Sg, Fg, Fl, Fn, ε

résidu = (P+∆P) - Fg

Test sur la norme du résidu d'équilibre

Fin

∆P = ∆P/2

Recherche d'équilibre des sections

∆P trop faible

Divergence

Oui

Divergence

résidu faible

Divergence ourupture

résidu important

Non

Iljxuh 815 = Rujdqljudpph gh o*dojrulwkph gh uhfkhufkh g*ìtxloleuh gh od vwuxfwxuh

815 Ìtxloleuh jìqìudo gh od vwuxfwxuh 44<

frqqh{lrq ghv ìoìphqwv hqwuh hx{/ qrxv fdofxorqv ohv h�ruwv lqwhuqhv I} gh od vwuxfwxuh

duulydqw ã fkdtxh qúxg/ hw ohxu dffurlvvhphqw �I}1 Rq surfëgh ã xqh uììydoxdwlrq gx

uìvlgx sdu gl�ìuhqfh hqwuh oh fkdujhphqw h{whuqh hw ohv h�ruwv lqwhuqhv gh od vwuxfwxuh gdqv

oh uhsëuh joredo =

u�hvlgx @ S .�S � +I} .�I}, +81:,

Fh qrxyhdx uìvlgx hvw uìlqmhfwì frpph qrxyhdx fkdujhphqw hw oh f|foh uhsuhqg1 Oruvtxh

od qruph gx uìvlgx hvw idleoh/ f*hvw0ã0gluh =

nu�hvlgxn 9 wrouhv=nS .�Sn +81;,

rü wrouhv hvw od wroìudqfh hq uìvlgx g*ìtxloleuh/ oh surfhvvxv v*duuíwh1 Sdu gìidxw/ qrxv

sursrvrqv =

wrouhv @ 433D +81<,

Vl ã xq prphqw grqqì/ qrxv revhuyrqv xqh uxswxuh g*xqh ghv vhfwlrqv/ rx xqh glyhujhqfh

dx qlyhdx ghv vhfwlrqv/ ghv gìsodfhphqwv gh od vwuxfwxuh rx gh vrq uìvlgx g*ìtxloleuh/

oh fkdujhphqw h{whuqh hvw glylvì sdu 51 Vl o*dojrulwkph q*duulyh sdv ã frqyhujhu píph

dsuëv dyrlu glylvì sdu 5 oh fkdujhphqw lqlwldo qgly irlv/ oh surfhvvxv v*duuíwh1 qgly hvw xq

sdudpëwuh/ txl ydxw 43 sdu gìidxw1 Oh surjudpph gìfrxsh dlqvl oh fkdujhphqw h{whuqh hq

xq fkdujhphqw soxv idleoh/ txl/ v*lo | d frqyhujhqfh/ hvw uìdssoltxì1 Sdu fh surfhvvxv/ rq

gìwhuplqh od fkdujh fulwltxh gh od vwuxfwxuh/ frpph xqh iudfwlrq gx fkdujhphqw lqlwldo1

Fhw dojrulwkph shuphw dlqvl g*dyrlu xqh judqgh �delolwì gdqv od uìvroxwlrq gx fkdujh0

phqw/ hw ìylwh ã o*xwlolvdwhxu gh uhfkhufkhu pdqxhoohphqw oh sdv gh fkdujh/ dlqvl txh od

fkdujh fulwltxh1

453 Fkdslwuh 8 Dojrulwkph jìqìudo gh uìvroxwlrq

Fkdslwuh 9

Sulvh hq frpswh ghv h�hwv gl�ìuìv

Oh surjudpph SFS d ìwì frqêx lqlwldohphqw srxu ìwxglhu ohv vwuxfwxuhv hq eìwrq suì0

frqwudlqw1 Oh eìwrq hvw xq pdwìuldx grqw ohv fdudfwìulvwltxhv ìyroxhqw dyhf oh whpsv1 Lo

hvw vxmhw dx uhwudlw dlqvl tx*dx xdjh/ f*hvw0ã0gluh ã xqh yduldwlrq gh gìirupdwlrq ã frq0

wudlqwh frqvwdqwh1 Hq rxwuh/ ohv dupdwxuhv gh suìfrqwudlqwh ìwdqw iruwhphqw whqgxhv/ rq

revhuyh xq skìqrpëqh gh uhod{dwlrq gh o*dflhu/ f*hvw0ã0gluh xqh yduldwlrq gh frqwudlqwh ã

gìirupdwlrq frqvwdqwh1 O*hqvhpeoh gh fhv skìqrpëqhv suryrtxh ohv h�hwv gl�ìuìv g*xq

fkdujhphqw1 O*remhfwli gh fh fkdslwuh q*hvw sdv gh uhirupxohu oh sureoëph gx xdjh +xqh

wkëvh qh vx!udlw sdv,/ pdlv gh idluh oh srlqw vxu ohv gl�ìuhqwhv pìwkrghv g*lqwìjudwlrq

whpsruhooh gx xdjh d�q g*hq fkrlvlu xqh hw gh od frghu1 Hq rxwuh/ qrxv qrxv olplwhurqv

dx{ suhvfulswlrqv uìjohphqwdluhv hq fh txl frqfhuqh ohv orlv gh uhwudlw/ gh xdjh hw gh

uhod{dwlrq1

914 Ioxdjh dyhf orl gh frpsruwhphqw olqìdluh

91414 Od irqfwlrq gh xdjh/ od irqfwlrq gh uhod{dwlrq

Fodvvltxhphqw ^Hvs;9d`/ od gìirupdwlrq gx eìwrq gh gìfrpsrvh hq 6 ìoìphqwv =

% @ %� . %s, . %o +914,

dyhf =

� %� = gìirupdwlrq lqvwdqwdqìh/

� %s, = gìirupdwlrq gh xdjh vrxv frqwudlqwh frqvwdqwh/

� %o = gìirupdwlrq gh uhwudlw1

455 Fkdslwuh 9 Sulvh hq frpswh ghv h�hwv gl�ìuìv

Fhv gìirupdwlrqv/ hq fh txl qrxv frqfhuqh/ fruuhvsrqghqw ã od gìirupdwlrq qrupdoh gh

od vhfwlrq1 F*hvw srxutxrl/ vdxi lqglfdwlrq frqwudluh/ gdqv oh gìyhorsshphqw txl vxlw/ qrxv

vxssrvrqv txh o*rq idlw uìiìuhqfh ã od gìirupdwlrq qrupdoh orqjlwxglqdoh gh od vhfwlrq1

Wrxv ohv prgëohv hpslultxhv rx qrupdolvìv gh xdjh vrqw olqìdluhv1 Fhfl lpsoltxh

tx*dxvvl elhq od gìirupdwlrq lqvwdqwdqìh %� txh od gìirupdwlrq gh xdjh %s, vrqw sursru0

wlrqqhoohv ã od frqwudlqwh dssoltxìh1 Fhfl qrxv shuphw g*ìfuluh =

% +w, @ %�

�w�

�. %s,

�w> w

�

�. %o +w, +915,

% +w, @ ��w�

�M�w> w

�

�. %o +w, +916,

M�w> w

�

�hvw dsshoìh irqfwlrq gh xdjh gx pdwìuldx1 Hooh uhsuìvhqwh od gìirupdwlrq vsì0

fl�txh +sdu xqlwì gh frqwudlqwh, wrwdoh dx whpsv w/ gìirupdwlrq gh uhwudlw gìgxlwh1 w�

hvw

o*åjh gh fkdujhphqw frqvlgìuì frpph xqh yduldeoh1 Od irqfwlrq gh xdjh M�w> w

�

�hvw

fodvvltxhphqw vìsduìh hq vhv frpsrvdqwhv lqvwdqwdqìh hw gl�ìuìh1

M�w> w

�

�@

4

H +w�

,.F

�w> w

�

�+917,

Oh whuph4

H +w�

,uhsuìvhqwh od sduwlh lqvwdqwdqìh gh od gìirupdwlrq1 Qrxv dyrqv g*dloohxuv

od uhodwlrq =

H�w�

�@

4

M +w�

> w�

,+918,

F�w> w

�

�v*dsshooh oh xdjh vsìfl�txh hw uhsuìvhqwh vwulfwhphqw od sduwlh gh gìirupdwlrq

gl�ìuìh> sdu gì�qlwlrq F�w�

> w�

�@ 31 Oh xdjh vsìfl�txh shxw dyrlu ghx{ gì�qlwlrqv =

F�w> w

�

�@

!�w> w

�

�H +w

�

,+919,

rx =

F�w> w

�

�@

!2H

�w> w

�

�H2H

+91:,

Lo | d dlqvl ghx{ gì�qlwlrqv gx frh!flhqw gh xdjh vxlydqw txh o*rq frqvlgëuh fhoxl0fl

frpph =

41 Oh udssruw gh od gìirupdwlrq gh xdjh/ srxu wrxw åjh w/ dsuëv dssolfdwlrq gh od

fkdujh hq w�

/ ã od gìirupdwlrq ìodvwltxh ã o*åjh gh 5; mrxuv +dssurfkh FHE rx ESHO

sdu h{hpsoh,

!2H

�w> w

�

�@

% +w,� %o +w,

%� +5;,� 4 +91;,

M�w> w

�

�@

4

H +w�

,.

!2H

�w> w

�

�H2H

+91<,

914 Ioxdjh dyhf orl gh frpsruwhphqw olqìdluh 456

51 Oh udssruw gh od gìirupdwlrq gh xdjh/ srxu wrxw åjh w/ dsuëv dssolfdwlrq gh od

fkdujh hq w�

/ ã od gìirupdwlrq ìodvwltxh dx prphqw gh od plvh hq fkdujh +dssurfkh

DFL sdu h{hpsoh,

!�w> w

�

�@

% +w,� %o +w,

%� +w�

,� 4 +9143,

M�w> w

�

�@

4

H +w�

,

�4 . !

�w> w

�

��+9144,

Txrl tx*lo hq vrlw/ od gì�qlwlrq gx xdjh vsìfl�txh ìwdqw/ hooh/ xqlyrtxh/ rq d o*ìtxl0

ydohqfh =

!�w> w

�

�@

H�w�

�H2H

!2H

�w> w

�

�+9145,

Srxu lqirupdwlrq/ od irqfwlrq gh xdjh gx ESHO hvw od vxlydqwh =

M�w> w

�

�@

4

H +w�

,.

Ns,

�w�

�i�w� w

�

�H�2H

+9146,

Od irqfwlrq gh uhod{dwlrqU�w> w

�

�wudgxlw pdwkìpdwltxhphqw gdqv oh whpsv o*ìyroxwlrq

vsìfl�txh +f*hvw0ã0gluh srxu xqh gìirupdwlrq xqlwdluh, gh od frqwudlqwh d{ldoh =

� +w, @k%�w�

�� %o

�w�

�lU�w> w

�

�+9147,

91415 Od iruph lqwìjudoh = oh sulqflsh gh vxshusrvlwlrq

Oh sulqflsh gh vxshusrvlwlrq/ gþ ã Erow}pdqq/ hvw ã od edvh gh wrxv ohv dojrulwkphv olqìdluhv

gh fdofxo ghv gìirupdwlrqv2frqwudlqwhv uìvxowdqw g*klvwrultxhv gh frqwudlqwhv2gìirupdwlrqv

yduldeohv1 Lo vh udpëqh/ pdwkìpdwltxhphqw/ ã xqh vrppdwlrq lqwìjudoh1

Srxu oh xdjh =

% +w,� %o +w, @

] |

f

M�w> w

�

�g��w�

�+9148,

Srxu od uhod{dwlrq =

� +w, @

] |

f

U�w> w

�

� kg%�w�

�� g%o

�w�

�l+9149,

91416 Od iruph gl�ìuhqwlhooh

Od uhodwlrq h{lvwdqw hqwuh od frqwudlqwh hw od gìirupdwlrq g*xq pdwìuldx shxw vh phwwuh vrxv

od iruph g*xqh ìtxdwlrq gl�ìuhqwlhooh olqìdluh1 Frpph oh frpsruwhphqw ylvfr0ìodvwltxh vh

gìfrpsrvh hq xqh sduwlh ìodvwltxh hw xqh sduwlh ylvtxhxvh/ ohv ghx{ prgëohv gh edvh xwlolvìv

vrqw oh prgëoh gh Krrnh/ prgìolvì sdu xq uhvvruw/ hw oh prgëoh gh Qhzwrq/ prgìolvì sdu

xq dpruwlvvhxu1


O*ìtxdwlrq frqvwlwxwlyh gx prgëoh gh Krrnh hvw grqf =

� @ H% +914:,

rü H hvw oh prgxoh g*\rxqj gx pdwìuldx1

Oh prgëoh gh Qhzwrq hvw fdudfwìulvì sdu =

� @ � b% +914;,

Hq h�hfwxdqw ghv frpelqdlvrqv/ hq vìulh rx hq sdudooëoh/ gh fhv prgëohv/ rq derxwlw ã ghv

ìtxdwlrqv gl�ìuhqwlhoohv gh od iruph =

�[�'f

d�C��

Cw�@

�[�'f

e�C�%

Cw�+914<,

rü d� hw e� vrqw irqfwlrqv ghv fdudfwìulvwltxhv ukìrorjltxhv gx pdwìuldx1 Rq shxw lqwìjuhu

fhwwh ìtxdwlrq gl�ìuhqwlhooh dqdo|wltxhphqw/ hw o*rq rewlhqw =

U�w> w

�

�@

o[>'f

D>h3k>

�|3|�

�+9153,

M�w> w

�

�@

o[>'f

E>h3q>

�|3|�

�+9154,

rü D> hw E> vrqw ghv frqvwdqwhv g*lqwìjudwlrq/ hw �> hw �> vrqw ghv frqvwdqwhv srvlwlyhv

grqw ohv lqyhuvhv vrqw dsshoìv uhvshfwlyhphqw whpsv gh uhod{dwlrq hw whpsv gh uhwdug1

Srxu srxyrlu xwlolvhu fhwwh irupxodwlrq/ lo yd idoorlu surfìghu ã xq wudydlo g*lghqwl�fdwlrq

ghv frh!flhqwv D>/ E>/ �> hw �>1 O*dydqwdjh gh fhwwh pìwkrgh hvw gh qh sdv dyrlu

ã vwrfnhu wrxw o*klvwrultxh gh frqwudlqwh srxu uìvrxguh oh sureoëph/ pdlv gh frqqdñwuh

vhxohphqw u yduldeohv lqwhuqhv sduidlwhphqw gì�qlhv1 Lo qh idxw sdv fhshqgdqw vrxv0hvwlphu

u1 O*lqfrqyìqlhqw uìvlgh gdqv o*lghqwl�fdwlrq irufìphqw h{sìulphqwdoh ghv sdudpëwuhv1

91417 Od pìwkrgh gx whpsv ìtxlydohqw

Srxu dssoltxhu od pìwkrgh gx whpsv ìtxlydohqw/ rq vxssrvh vlpsohphqw o*h{lvwhqfh g*xq

prgëoh lqfuìphqwdo1 Fh prgëoh frqvlvwh ã gluh txh od ylwhvvh gh xdjh g*xq ìoìphqw gh

yroxph qh gìshqg txh gh od frqwudlqwh � dfwxhooh/ gh od gìirupdwlrq dfwxhooh gh xdjh %s,hw gh o*åjh w gx pdwìuldx =

b%s, @ i +�> %s,> w, +9155,

Fhwwh pìwkrgh shuphw gh qh soxv vwrfnhu o*klvwrultxh gh frqwudlqwh rx gh gìirupdwlrq fdu

wrxw gìshqg gh yduldeohv g*ìwdw1 Oh sulqflsh gh vxshusrvlwlrq hw od pìwkrgh gx whpsv

ìtxlydohqw ìwdqw wuëv gl�ìuhqwv dx qlyhdx gh ohxu ìtxdwlrq frqvwlwxwlyh/ lo hvw fodlu txh

fhv ghx{ pìwkrghv vrqw lqfrpsdwleohv1 Qrxv dyrqv gìflgì gh surjudpphu oh sulqflsh gh

vxshusrvlwlrq/ xqlyhuvhoohphqw uhfrqqx1

915 Ioxdjh dyhf orl gh frpsruwhphqw qrq olqìdluh 458

91418 Oh sureoëph vwuxfwxudo

Od irqfwlrq gh xdjh hvw grqqìh +vrlw sdu xq uëjohphqw/ vrlw sdu ghv phvxuhv h{sìulphq0

wdohv, hw o*rq glvsrvh gh o*klvwrultxh gh gìirupdwlrq rx gh frqwudlqwh1 Rq yhxw fdofxohu od

frqwudlqwh dfwxhooh uìvxowdqw gh fhw klvwrultxh1 Oh sureoëph ghylhqw frpsoh{h fdu o*ìtxd0

wlrq frqvwlwxwlyh

% +w,� %o +w, @

] |

f

M�w> w

�

�g��w�

�+9156,

ghylhqw xqh ìtxdwlrq lqwìjudoh sxlvtxh o*lqfrqqxh �jxuh vrxv oh vljqh lqwìjudoh1 Vl o*rq

uhpsodfh M�w> w

�

�sdu vd ydohxu/ rq rewlhqw =

% +w,� %o +w, @

] |

f

�4 . !

�w> w

�

�� g� �w��H +w

�

,+9157,

Frpph ohv prgëohv olqìdluhv gh xdjh vxssrvhqw od sursruwlrqqdolwì hqwuh od gìirupdwlrq

lqvwdqwdqìh %��w�

�hw od frqwudlqwh dssoltxìh �

�w�

�/ rq d =

g��w�

�H +w

�

,@ g%�

�w�

�+9158,

G*rü =

% +w,� %o +w, @

] |

f

�4 . !

�w> w

�

��g%�

�w�

�+9159,

Dlqvl/ dyhf ohv prgëohv olqìdluhv gh xdjh hw xqh orl gh frpsruwhphqw lqvwdqwdqì olqìdluh/

oh xdjh hvw olqìdluh dxvvl elhq hq frqwudlqwh tx*hq gìirupdwlrq1

915 Ioxdjh dyhf orl gh frpsruwhphqw qrq olqìdluh

Xq sureoëph vh srvh oruvtxh o*rq yhxw wudydloohu dyhf xqh orl gh frpsruwhphqw gx pdwìuldx

qrq olqìdluh1 Oh prgxoh g*\rxqj H qh gìshqg soxv gh w�

vhxohphqw/ pdlv dxvvl gh od

gìirupdwlrq lqvwdqwdqìh gx eìwrq = H�w�

> %�

�1 Lo idxw hq rxwuh fkrlvlu hqwuh xq xdjh

olqìdluh hq frqwudlqwh +fdv gx FHE sdu h{hpsoh, rx xq xdjh olqìdluh hq gìirupdwlrq +fdv

gx ESHO <4 sdu h{hpsoh,1

Srxu hq vdyrlu soxv/ lo hvw lqwìuhvvdqw gh frqvxowhu ohv uëjohphqwv qrq sdv dx fkdslwuh gx

xdjh/ pdlv dx{ duwlfohv frqvdfuìv ã od sulvh hq frpswh gx xdjh srxu oh dpehphqw ghv

srwhdx{1 Oh EDHO <4 lqgltxh +duwlfoh D1717/65, g*rsìuhu vxu oh gldjudpph gh od orl gh

frpsruwhphqw gx eìwrq xqh d!qlwì sdudooëoh ã o*d{h ghv gìirupdwlrqv/ gh udssruw =

4 .

[��!�i +w� m, +915:,

h{suhvvlrq gdqv odtxhooh =


� �� hvw oh udssruw gx prphqw gh suhplhu ruguh gþ ã o*dfwlrq dssoltxìh dx whpsv m dx

prphqw gx suhplhu ruguh/

� !� hvw od ydohxu gh ! srxu xqh fkdujh dssoltxìh dx whpsv m/

� i +w, hvw od orl g*ìyroxwlrq gx xdjh grqqìh gdqv ohv uëjohv ESHO +duwlfoh 514/85,/

� w hvw o*lqvwdqw frqvlgìuì1

Fhfl prqwuh elhq txh o*rq yd vrpphu ohv gìirupdwlrqv srxu xqh frqwudlqwh grqqìh = oh

xdjh hvw grqf olqìdluh hq gìirupdwlrq1 Oh ESHO hw oh EDHO vrqw frkìuhqwv dyhf xq who

prgëoh gh xdjh1

Oh FHE txdqw ã oxl hvw ehdxfrxs prlqv surol{h ã sursrv gh od sulvh hq frpswh gx

xdjh gdqv oh dpehphqw ghv srwhdx{1 Lo glw ^Fhe<3` +duwlfoh 919161617, txh ��wkh h�hfwv

ri fuhhs pd| eh wdnhq lqwr dffrxqw dssur{lpdwho| e| grxeolqj erwk ydoxhv ri jhrphwulfdo

lpshuihfwlrqv��/ fh txl uhylhqw ã idluh xqh d!qlwì gh udssruw 5 sdudooëohphqw ã o*d{h ghv

gìirupdwlrqv/ frpph oh EDHO <4/ fh txl/ vdqv íwuh glw/ vxssrvh txh oh xdjh hvw olqìdluh hq

gìirupdwlrq1 Fhfl hvw hq frpsoëwh frqwudglfwlrq dyhf o*duwlfoh 514191716 wudlwdqw gx xdjh

vhxo/ txl gìfoduh oh xdjh olqìdluh hq frqwudlqwh1

Uhpdutxrqv txh oh prgëoh gh xdjh olqìdluh hq gìirupdwlrq yd gdqv oh vhqv gh od

vìfxulwì fdu lo vxuhvwlph od gìirupdwlrq gh xdjh sdu udssruw ã xq prgëoh gh xdjh olqìdluh

hq frqwudlqwh1

916 Glvfuìwlvdwlrq whpsruhooh

Srxu ìydoxhu o*ìtxdwlrq lqwìjudoh +9157,/ rq yd vxssrvhu txh o*rq d xqh glvfuìwlvdwlrq hq

whpsv gx sureoëph1 Qrxv doorqv fkhufkhu ã ìydoxhu o*dffurlvvhphqw gh frqwudlqwh7�? vh

surgxlvdqw vxu oh sdv gh whpsv 7w? @ w? � w?3�1 Lo h{lvwh soxvlhxuv pdqlëuhv g*lqwìjuhu od

irqfwlrq gh xdjh1

91614 Suhplëuh pìwkrgh = od uëjoh gx uhfwdqjoh

Prgëoh gh xdjh olqìdluh hq frqwudlqwh

Qrxv doorqv idluh o*k|srwkëvh vxlydqwh = o*dffurlvvhphqw gh frqwudlqwh7�� / txl fruuhvsrqgã o*dffurlvvhphqw gh gìirupdwlrq lqvwdqwdqìh 7%�� / vxu oh sdv gh whpsv 7w� @ w� � w�3�/

hvw dssoltxì ã o*lqvwdqw w� = fhfl hvw sduwlfxolëuhphqw dgdswì gdqv oh fdguh g*xq skdvdjh

gh frqvwuxfwlrq1 G*rü =

% +w?,� %o +w?, @

?[�'�

M +w?> w�,7 ��

916 Glvfuìwlvdwlrq whpsruhooh 45:

@ M +w?> w?,7 �? .

?3�[�'�

M +w?> w�,7 �� +915;,

Sdu dloohxuv/ rq shxw ìfuluh =

% +w?3�,� %o +w?3�, @?3�[�'�

M +w?3�> w�,7 �� +915<,

Sdu vrxvwudfwlrq/ rq rewlhqw =

7% +w?,�7%o +w?, @ M +w?> w?,7 �? .

?3�[�'�

^M +w?> w�,� M +w?3�> w�,`7 �� +9163,

Fhfl shxw vh phwwuh vrxv od iruph vxlydqwh =

7% +w?,�7%o +w?, @ 7�?H�?

.7%s, +w?, +9164,

rx hqfruh

7�? @ H�? +7% +w?,�7%o +w?,�7%s, +w?,, +9165,

dyhf =

7% +w?, @ % +w?,� % +w?3�, +9166,

7%o +w?, @ %o +w?,� %o +w?3�, +9167,

H�? @

4

M +w?> w?,@ H +w?, +9168,

7%s, +w?, @

?3�[�'�

^M +w?> w�,� M +w?3�> w�,`7 �� +9169,

Prgëoh gh xdjh olqìdluh hq gìirupdwlrq

Qrxv doorqv idluh o*k|srwkëvh vxlydqwh = o*dffurlvvhphqw gh gìirupdwlrq lqvwdqwdqìh7%�� /txl fruuhvsrqg ã o*dffurlvvhphqw gh frqwudlqwh 7�� / vxu oh sdv gh whpsv 7w� @ w� � w�3�/

hvw dssoltxì ã o*lqvwdqw w� 1 G*rü =

% +w?,� %o +w?, @

?[�'�

+4 . ! +w?> w�,,7 %��

@ 7%�? .

?3�[�'�

+4 . ! +w?> w�,,7 %��

@7�?H +w?,

.

?3�[�'�

+4 . ! +w?> w�,,7 %�� +916:,

45; Fkdslwuh 9 Sulvh hq frpswh ghv h�hwv gl�ìuìv


% +w?3�,� %o +w?3�, @?3�[�'�

+4 . ! +w?> w�,,7 %�� +916;,


7% +w?,�7%o +w?, @ 7�?H +w?,

.

?3�[�'�

^! +w?> w�,� ! +w?3�> w�,`7 %�� +916<,

Fhfl shxw vh phwwuh vrxv od iruph vxlydqwh =

7�? @ H�? +7% +w?,�7%o +w?,�7%s, +w?,, +9173,

dyhf =

7% +w?, @ % +w?,� % +w?3�, +9174,

7%o +w?, @ %o +w?,� %o +w?3�, +9175,

H�? @ H +w?, +9176,

7%s, +w?, @

?3�[�'�

^! +w?> w�,� ! +w?3�> w�,`7 %�� +9177,

91615 Ghx{lëph pìwkrgh = od uëjoh gx wudsë}h

Prgëoh gh xdjh olqìdluh hq frqwudlqwh

Qrxv doorqv idluh o*k|srwkëvh vxlydqwh = o*dffurlvvhphqw gh frqwudlqwh 7�� vxu oh sdv ghwhpsv 7w� @ w� � w�3� hvw dssoltxì srxu prlwlì ã o*lqvwdqw w�3� hw srxu prlwlì ã o*lqvwdqw

w�1 O*huuhxu gh fhwwh pìwkrgh hvw gx vhfrqg ruguh1 G*rü =

% +w?,� %o +w?, @

?[�'�

^M +w?> w�, . M +w?> w�3�,`7��5

@ ^M +w?> w?, . M +w?> w?3�,`7�?5

.

?3�[�'�

^M +w?> w�, . M +w?> w�3�,`7��5

+9178,


% +w?3�,� %o +w?3�, @?3�[�'�

^M +w?3�> w�, . M +w?3�> w�3�,`7��5

+9179,

Sdu vrxvwudfwlrq/ rq rewlhqw gh od píph idêrq txh suìfìghpphqw =

7�? @ H�? +7% +w?,�7%o +w?,�7%s, +w?,, +917:,

916 Glvfuìwlvdwlrq whpsruhooh 45<

dyhf =

7% +w?, @ % +w?,� % +w?3�, +917;,

7%o +w?, @ %o +w?,� %o +w?3�, +917<,

H�? @

5

M +w?> w?, . M +w?> w?3�,+9183,

7%s, +w?, @

?3�[�'�

^M +w?> w�, . M +w?> w�3�,� M +w?3�> w�,� M +w?3�> w�3�,`7��5

+9184,

Prgëoh gh xdjh olqìdluh hq gìirupdwlrq

Qrxv doorqv idluh o*k|srwkëvh vxlydqwh = o*dffurlvvhphqw gh gìirupdwlrq lqvwdqwdqìh7%�� /vxu oh sdv gh whpsv7w� @ w��w�3� hvw dssoltxì srxu prlwlì ã o*lqvwdqw w�3� hw srxu prlwlìã o*lqvwdqw w� 1 G*rü =

% +w?,� %o +w?, @

?[�'�

^5 . ! +w?> w�, . ! +w?> w�3�,`7%��5

@

�4 .

! +w?> w?3�,

5

�7 %�?

.

?3�[�'�

^5 . ! +w?> w�, . ! +w?> w�3�,`7%��5

+9185,

Ru/ frpph rq vxssrvh txh 7%�? hvw dssoltxì srxu prlwlì hq w?3� hw srxu prlwlì hq w?/

o*dffurlvvhphqw gh frqwudlqwh 7�? hvw who txh =

7�? @7%�?

5M +w?> w?,.

7%�?5M +w?> w?3�,

+9186,

grqf =

7%�? @5M +w?> w?3�,

4 .M +w?> w?3�,

M +w?> w?,

7 �? +9187,


% +w?3�,� %o +w?3�, @?3�[�'�

^5 . ! +w?3�> w�, . ! +w?3�> w�3�,`7%��5

+9188,

Sdu vrxvwudfwlrq/ rq rewlhqw gh od píph idêrq txh suìfìghpphqw =

7�? @ H�? +7% +w?,�7%o +w?,�7%s, +w?,, +9189,

dyhf =

7% +w?, @ % +w?,� % +w?3�, +918:,


7%o +w?, @ %o +w?,� %o +w?3�, +918;,

H�? @

H +w?3�, .H +w?, +4 . ! +w?> w?3�,,

+5 . ! +w?> w?3�,, +4 . ! +w?> w?3�,,+918<,

7%s, +w?, @

?3�[�'�

^! +w?> w�, . ! +w?> w�3�,� ! +w?3�> w�,� ! +w?3�> w�3�,`7%��5

+9193,

917 Dqdo|vh

91714 Fdv gx xdjh olqìdluh hq frqwudlqwh

Od gl!fxowì uìvlgh gdqv o*ìydoxdwlrq gh od gìirupdwlrq gl�ìuìh gh xdjh7%s, +w?,/ qrq sdvã fdxvh gh od irqfwlrq gh xdjh M

�w> w

�

�txl hvw xqh grqqìh gx sureoëph/ pdlv ã fdxvh gx

whuph 7��1 Hq h�hw/ od orl gh frpsruwhphqw lqvwdqwdqì gx pdwìuldx ìwdqw qrq olqìdluh/

lo idxw glvfuìwlvhu hq �euhv od vhfwlrq/ vwrfnhu o*klvwrultxh gh frqwudlqwh gh fkdtxh �euh

g*lqwìjudwlrq gh fkdtxh vhfwlrq gh fkdtxh ìoìphqw/ srxu wrxv ohv lqvwdqwv frqvlgìuìv1 Hq

rxwuh/ od suìflvlrq gh o*lqwìjudwlrq ghv irufhv gl�ìuìhv txh oh xdjh yd jìqìuhu hvw olìh ã od

�qhvvh gx pdloodjh gh od vhfwlrq hq �euhv1

Pdlv o*k|srwkëvh gx xdjh olqìdluh hq frqwudlqwh srvh xq sureoëph dvvh} lpsruwdqw olì

ã od sodqìlwì vxssrvìh gh od gìirupdwlrq wrwdoh gh od vhfwlrq1 Hq h�hw/ o*dffurlvvhphqw gh

gìirupdwlrq wrwdoh 7% hvw fhqvì íwuh sodq1 O*dffurlvvhphqw gh gìirupdwlrq gh uhwudlw hvw

ìjdohphqw sodq fdu vhxoh vd frpsrvdqwh vhorq J{ hvw qrq qxooh1 Gh idlw/ oh yhfwhxu 7%�7%o +w?, @ 7%�? .7%s, +w?, hvw sodq1 Oruvtxh o*rq revhuyh ohv ìtxdwlrqv +9169, hw +9184,/

rq yrlw txh oh yhfwhxu7%s, +w?, hvw rewhqx ã sduwlu g*xqh frpelqdlvrq olqìdluh ghv yhfwhxuv7�� / txl vrqw hx{0píphv gìgxlwv ã o*dlgh g*xqh orl gh frpsruwhphqw ìyhqwxhoohphqw qrqolqìdluh1 Dlqvl/ ulhq qh jdudqwlw od sodqìlwì gx yhfwhxu 7%s, +w?,1 Pdlv frpph 7%�? .

7%s, +w?, hvw sodq/ fhfl vxssrvh txh o*dffurlvvhphqw gh gìirupdwlrq lqvwdqwdqìh7%�? q*hvw

sdv sodq/ fh txl sdudñw gl!flohphqw frqfhydeoh1 Qrxv dyrqv oã xqh frqwudglfwlrq gl!floh

ã ohyhu/ ã prlqv g*dgrswhu o*k|srwkëvh gx sdudjudskh vxlydqw1

91715 Fdv gx xdjh olqìdluh hq gìirupdwlrq

Rq shxw lfl udlvrqqhu hq dffurlvvhphqw gh gìirupdwlrq lqvwdqwdqìh 7%�1 O*dydqwdjh txhsurfxuh oh xdjh olqìdluh hq gìirupdwlrq hvw od sodqìlwì gh od gìirupdwlrq lqvwdqwdqìh hw

gh od gìirupdwlrq gh xdjh1 Hq yrlfl od suhxyh sdu uìfxuuhqfh =

Suhplhu sdv =

Rq v*hvw grqqì frpph k|srwkëvh oh idlw txh od gìirupdwlrq wrwdoh gh od vhfwlrq 7% hvw

sodqh1 Dx suhplhu fkdujhphqw gh od vhfwlrq/ od gìirupdwlrq lqvwdqwdqìh hvw od gìirupdwlrq

wrwdoh/ hooh hvw grqf sodqh1

Sdv q =

917 Dqdo|vh 464

Vxssrvrqv txh srxu 4 � m � q � 4 ohv gìirupdwlrqv lqvwdqwdqìhv 7%�� vrlhqw sodqhv1

Frpph 7%s, +w?, hvw xqh frpelqdlvrq olqìdluh ghv +q� 4, 7%�� / 7%s, +w?, hvw sodq1 O*df0furlvvhphqw gh gìirupdwlrq gh uhwudlw7%o +w?, hvw irufìphqw sodq fdu lo hvw frqvwdqw gdqv

od vhfwlrq1 Frpph7% hvw sodq sdu k|srwkëvh/ hw frpph7%�? @7%�7%o +w?,�7%s, +w?,/doruv 7%�? hvw sodq1

Rq d prqwuì txh wrxwhv ohv gìirupdwlrqv/ dxvvl elhq lqvwdqwdqìhv txh gl�ìuìhv/ vrqw sodqhv1

Dlqvl/ dx olhx gh vwrfnhu o*klvwrultxh gh frqwudlqwh gh fkdtxh �euh gh fkdtxh vhfwlrq/ lo

hvw qìfhvvdluh gh qh vwrfnhu txh 9 ydohxuv/ txl fruuhvsrqghqw dx{ 9 frpsrvdqwhv gh +7%�,/oh yhfwhxu gìirupdwlrq lqvwdqwdqìh1 Rq ìfulw wrxw qdwxuhoohphqw od uhodwlrq

+7%, @ +7%�, . +7%_�s , +9194,

rü =

+7%, hvw oh yhfwhxu gìirupdwlrq wrwdoh gh od vhfwlrq/

+7%�, hvw oh yhfwhxu gìirupdwlrq lqvwdqwdqìh gx eìwrq/

+7%_�s , hvw oh yhfwhxu gìirupdwlrq gl�ìuìh gx eìwrq1

Oh fdofxo sursuhphqw glw gh +7%_�s , vh idlw gh od pdqlëuh vxlydqwh =�7%

_�s

�@ +7%o, . +7%s,, +9195,

7%o uhsuìvhqwh o*dffurlvvhphqw gh gìirupdwlrq sodqh gh uhwudlw> vhxoh vd suhplëuh frp0

srvdqwh hvw qrq qxooh1

Dssolfdwlrq gx ESHO <4

Oh ESHO <4 srvh =

7%s, +w?, @?3�[�'�

Ns, +w�, i +w? � w�, +7%�2H +w�,, +9196,

7%�2H +w�, hvw oh yhfwhxu gh yduldwlrq frqyhqwlrqqhooh gh gìirupdwlrq lqvwdqwdqìh/ txl hvw

gì�qlh frpph od yduldwlrq gh gìirupdwlrq fruuhvsrqgdqw ã od yduldwlrq gh frqwudlqwh7��/dssoltxìh ã sduwlu gh o*ìwdw gh frqwudlqwh dqwìulhxu =

��3� @

�3�[&'�

7�& +9197,

hq dgrswdqw frqyhqwlrqqhoohphqw od orl gh gìirupdwlrq gx eìwrq åjì gh 5; mrxuv1 Hq h�hw/

od yduldwlrq frqyhqwlrqqhooh gh gìirupdwlrq lqvwdqwdqìh7%�S� gì�qlh sdu oh uëjohphqw/ hvwsodqh srxu ohv píphv udlvrqv txh suìfìghpphqw1 Rq shxw grqf od udphqhu ã xq yhfwhxu

7%�2H +w�, gh glphqvlrq 91


Ns, +w�, hvw oh frh!flhqw gh xdjh/ ã o*åjh w� =

Ns, +w�, @ nr ^ne . nSn +w�,` +9198,

rü ohv frh!flhqwv nr/ ne/ nS hw n +w�, vrqw gì�qlv gh od idêrq vxlydqwh =

� oh frh!flhqw nr gìshqg gx srxufhqwdjh ghv dupdwxuhv dgkìuhqwhv orqjlwxglqdohv �rhw ydxw �

�n2f4r>

� oh frh!flhqw ne uhsuìvhqwh od olplwh lqiìulhxuh gx frh!flhqw gh xdjh gx eìwrq qrq

dupì oruvtx*lo hvw fkdujì wuëv ylhx{/ hw ydxw 3/73>

� oh frh!flhqw nS gìshqg gh ��/ o*k|jurpìwulh dpeldqwh h{sulpìh hq srxufhqwdjh g*kx0

plglwì uhodwlyh/ hw gx ud|rq pr|hq gh od slëfh u6 h{sulpì hq fp =

nS @453� ��

63.

5

6

433� ��53 . u6

+9199,

� oh frh!flhqw n +w�, gìshqg gx gxuflvvhphqw gx eìwrq ã o*åjh gh plvh hq fkdujh/ hw

ydxw =

n +w�, @433

433 . w�+919:,

rü w� hvw o*åjh gx eìwrq/ h{sulpì hq mrxuv>

� od irqfwlrq i +w? � w�, uhsuìvhqwh od orl g*ìyroxwlrq gx xdjh/ hw ydxw =

i +w? � w�, @

sw? � w�s

w? � w� . 8su6

+919;,

Hq wrxwh uljxhxu/ vhorq oh uëjohphqw/ srxu fdofxohu h{dfwhphqw oh yhfwhxu yduldwlrq

frqyhqwlrqqhooh gh gìirupdwlrq lqvwdqwdqìh 7%�2H +w�,/ lo idxgudlw surfìghu gh od idêrq

vxlydqwh = oruv gh od uhfkhufkh lwìudwlyh gx yhfwhxu 7%� +w�, txl ìtxloleuh o*dffurlvvhphqwgh vroolflwdwlrq vxu od vhfwlrq ã o*lqvwdqw w� / rq pëqh hq sdudooëoh od uhfkhufkh gx yhfwhxu

7%�2H +w�,/ fdofxoì dyhf od orl gh frpsruwhphqw gx pdwìuldx ã 5; mrxuv/ hw ìtxloleudqw oh

píph dffurlvvhphqw gh vroolflwdwlrq1

Hq sudwltxh/ dx olhx gh phqhu fhwwh grxeoh uìvroxwlrq/ lo qrxv sdudñw mxglflhx{ gh idluh

o*dssur{lpdwlrq vxlydqwh/ txl frqvlvwh ã idluh xq udssruw ghv wdqjhqwhv ã o*ruljlqh ghv orlv

gh gìirupdwlrq gx eìwrq ã w� mrxuv hw ã 5; mrxuv =

7%�2H +w�, @H +w�,

H�2H

+7%� +w�,, +919<,

rü 7%� +w�, hvw oh yhfwhxu dffurlvvhphqw gh gìirupdwlrq qrupdoh lqvwdqwdqìh gx eìwrq

ã o*lqvwdqw w�1 Oh fdofxo hvw uljrxuhx{ vl od orl gh frpsruwhphqw hvw olqìdluh/ píph vl oh

pdwìuldx hvw ylhloolvvdqw1 Gh soxv/ oh ESHO <4 lqgltxh txh srxu w� � 5; mrxuv/ H +w�, @ H�2H /

o*dssur{lpdwlrq q*hvw grqf h�hfwlyh txh gh 3 ã 5; mrxuv1 Xq fdofxo udslgh d qìdqprlqv

frq�upì txh fhwwh dssur{lpdwlrq lqgxlw xqh huuhxu idleoh sdu udssruw dx fdofxo uìjohphq0

wdluh1

917 Dqdo|vh 466

91716 Uhod{dwlrq ghv dflhuv gh suìfrqwudlqwh

Od uhod{dwlrq hvw oh skìqrpëqh sk|vltxh gxdo gx xdjh1 Ohv dupdwxuhv gh suìfrqwudlqwh

yrqw vxelu xqh shuwh gh frqwudlqwhv ã gìirupdwlrq frqvwdqwh1 Vhorq oh ESHO<4/ od shuwh

sdu uhod{dwlrq dx whpsv w v*h{sulph hq wrxw srlqw gh o*dupdwxuh sdu od irupxoh vxlydqwh =

��4 +w, @ n��fff

�w

4333

��eE�3>�

h�f>3.cD

&2 �R6| +91:3,

gdqv odtxhooh =

� w hvw h{sulpì hq khxuhv>

� ��fff hvw od ydohxu jdudqwlh gh od uhod{dwlrq lvrwkhuph ã 4333 khxuhv hq srxufhqwdjh

gh od whqvlrq lqlwldoh>

� � @jR6|

sRo}rü �R6| hvw ìjdoh ã od whqvlrq lqlwldoh gh o*dupdwxuh gdqv oh fdv gh od srvw0

whqvlrq/ hw ã od whqvlrq ã o*ruljlqh glplqxìh ghv shuwhv ã od plvh hq whqvlrq gdqv oh

fdv gh od suì0whqvlrq>

� n� @ 9� 433� vl ��fff 9 5> 8

;� 433� vl ��fff A 5> 8

�n2 @ 4> 4 vl ��fff 9 5> 8

4> 8 vl ��fff A ;

4> 58 vl 5> 8 ? ��fff ? ;

91717 Frqfoxvlrq

Vl o*rq xwlolvh xq prgëoh olqìdluh hq frqwudlqwh/ lo q*| d sdv g*dxwuh vroxwlrq txh gh pdloohu od

vhfwlrq hq �euhv hw gh vwrfnhu o*klvwrultxh gh frqwudlqwh gh fkdtxh �euh1 Fhfl qrxv frqgxlw

ã pdqlsxohu ghv gìirupdwlrqv lqvwdqwdqìhv hw gh xdjh qrq sodqhv1 Lo dssdudñw txh oh

prgëoh gh xdjh olqìdluh hq gìirupdwlrq hvw soxv dydqwdjhx{/ hw sduidlwhphqw frpsdwleoh

dyhf oh pdloodjh g|qdpltxh gh od vhfwlrq txh qrxv xwlolvrqv1 Qrxv o*dyrqv grqf fkrlvl/ hw fh

g*dxwdqw soxv tx*lo hvw suìfrqlvì sdu oh uëjohphqw iudqêdlv1 Fhshqgdqw/ oh xdjh v*h�hfwxh

vrxv fkdujhv shupdqhqwhv/ fh txl lpsoltxh txh ohv frqwudlqwhv qh grlyhqw sdv íwuh wurs

lpsruwdqwhv hw vrqw wuëv shx qrq olqìdluhv1 Dlqvl/ od gl�ìuhqfh ghv uìvxowdwv hqwuh xq fdofxo

idlvdqw o*k|srwkëvh gx xdjh olqìdluh hq frqwudlqwh hw xq fdofxo idlvdqw o*k|srwkëvh gx

xdjh olqìdluh hq gìirupdwlrq grlw íwuh idleoh1

Qrxv dyrqv fkrlvl g*xwlolvhu od uëjoh gx uhfwdqjoh fdu gdqv fh fdv/ H�? @ H +w?,/ fh txl

vh wudgxlud sdu oh idlw txh od pdwulfh gh uljlglwì gh od vwuxfwxuh vhud gì�qlh dyhf oh prgxoh

g*ìodvwlflwì xvxho hw tx*lo qh vhud sdv qìfhvvdluh gh od uììydoxhu1


Qrxv qrxv vrpphv frqwhqwìv gh fdofxohu od yduldwlrq frqyhqwlrqqhooh gh gìirupdwlrq

lqvwdqwdqìh hq xwlolvdqw o*dssur{lpdwlrq gx udssruw ghv prgxohv g*\rxqj ã o*ruljlqhH +w�,

H�2H

1

918 Ìtxloleuh gh od vhfwlrq vrxv gìirupdwlrqv gl�ìuìhv

91814 Ohv dupdwxuhv

Gdqv oh uhsëuh g*ìwxgh gh od vhfwlrq/ o*ìtxloleuh ghv dupdwxuhv sdvvlyhv hvw oh píph txh

suìfìghpphqw hw lo hvw grqqì sdu +716<, hw +7187,1

Srxu ohv dupdwxuhv dfwlyhv/ od uhodwlrq hqwuh o*dffurlvvhphqw gh gìirupdwlrq hw gh frq0

wudlqwh q*hvw soxv grqqìh sdu +716<, pdlv hooh ghylhqw =

7�sR� @ HsR�7 %sR� ��4� +91:4,

rü HsR�hvw oh prgxoh g*\rxqj gh o*dflhu gh o*dupdwxuh dfwlyh g*ruguh l hw ��4� od shuwh

sdu uhod{dwlrq gh o*dflhu1 O*ìtxloleuh ghv �euhv gh od vhfwlrq v*rewlhqw gh od píph idêrq

txh gdqv +7187, =#�Is?

�Is|

$@

%^Ns? ` ^Ns?| `

|^Ns?| ` ^Ns| `

&=

#�%?

�%|

$�#7I4?7I4|

$+91:5,

dyhf

7I4? @

?sR[�'�

��4� =

3EEC

4

}sR�|sR�

4FFD = frv�R� =vsR� +91:6,

7I4| @?sR[�'�

��4� =

3EEC

frv�R�

vlq�R��|sR� � |S

�vlq�R� �

�}sR� � }S

�frv�R�

4FFD = vlq�R� =vsR� +91:7,

Oh yhfwhxu

+7I4, @#7I4?7I4|

$+91:8,

uhsuìvhqwh od irufh ìtxlydohqwh ã od shuwh gh suìfrqwudlqwh djlvvdqw vxu od vhfwlrq1

91815 Oh eìwrq

Hq uhsuhqdqw ohv qrwdwlrqv xwlolvìhv srxu o*ìwxgh gh o*ìtxloleuh gh od vhfwlrq/ hw hq whqdqw

frpswh gh fh txh o*rq d glw ã sursrv gx xdjh/ rq d prqwuì txh o*rq srxydlw ìfuluh/ srxu

oh eìwrq frqvwlwxdqw od pdwulfh gh od vhfwlrq/ o*ìjdolwì vxlydqwh =

7�6 @ H�+7%�7%o �7%s,, +91:9,

918 Ìtxloleuh gh od vhfwlrq vrxv gìirupdwlrqv gl�ìuìhv 468

Gdqv xq suhplhu whpsv/ qrxv doorqv wudydloohu vxu ohv dfwlrqv qrupdohv dx{txhoohv hvw

vrxplvh od vhfwlrq1 Gdqv oh wulëguh J{|}/ gx idlw gh o*k|srwkëvh ghv gìirupdwlrqv sodqhv/

qrxv srxyrqv ìfuluh =

7% +|> }, @�

4 } |

�� +7%?, +91::,

dyhf

+7%?, @

3EEC

7%%7!+7!5

4FFD +91:;,

Wrxmrxuv hq lqwìjudqw vxu od vhfwlrq/ o*ìtxloleuh gh od vhfwlrq gh eìwrq v*ìfulw =

7I6? @

]76

7�6 +|> },

3EEC

4

}

|

4FFDgV6

@

]76

H�6 +|> },

5997

4 } |

} }2 |}

| |} |2

6::8gV6=�%?

�]76

H�6 +|> },

3EEC

4

}

|

4FFD +7%o? .7%s,?, gV6 +91:<,

rü 7%o? hw 7%s,? vrqw ohv dffurlvvhphqwv gh gìirupdwlrqv qrupdohv gh uhwudlw hw gh xdjh

uhvshfwlyhphqw1 Sxlvtxh H�6 +|> }, @ H6 +|> },/ rq uhfrqqdñw od pdwulfh ^N6? `/ hw =

7I6? @ ^N6? ` =�%? �]76

H�6 +|> },

3EEC

4

}

|

4FFD7 %_�s?gV6 +91;3,

Vdfkdqw txh 7%_�s

@7%o? .7%s,? hvw sodq dlqvl txh 7%?/ rq shxw doruv ìfuluh =

7%_�s +|> }, @�

4 } |

�� 7%_�s? +91;4,

hw

7%_�s? @

3EEC

%%_�s

!+_�s

!5_�s

4FFD +91;5,

Doruv =

]76

H�6 +|> },

3EEC

4

}

|

4FFD7 %_�sgV6 @

5997]PK

H6 +|> },

3EEC

4

}

|

4FFD =�

4 } |

�6::8 =7 %_�s?

@ ^N6? ` =7 %_�s? +91;6,


Ilqdohphqw/ rq rewlhqw =

7I6? @ ^N6? ` =7 %? � ^N6? ` =7 %_�s? +91;7,

Vl o*rq vxssrvh txh oh pdwìuldx xh gh od píph idêrq vhorq ohv frpsrvdqwhv wdqjhqwhv

ã od vhfwlrq txh vhorq ohv frpsrvdqwhv qrupdohv ã od vhfwlrq/ rq shxw ìfuluh =

7I6| @ ^N6| ` =7 %| � ^N6| ` =7 %_�s| +91;8,

rü 7%_�s| hvw o*dffurlvvhphqw gh gìirupdwlrq wdqjhqwh gl�ìuìh/ frqvwuxlw gh od píph

pdqlëuh txh 7%_�s? =

7%_�s| @7%o| .7%s,| +91;9,

Lo hvw fodlu txh =

7%o| @

3EEC

3

3

3

4FFD +91;:,

Fhfl vh phw vrxv od iruph vxlydqwh =#7I6?

7I6|

$@

%^N6? ` 3

3 ^N6| `

&�#7%?7%|

$�#

^N6? ` =7 %_�s?

^N6| ` =7 %_�s|

$+91;;,

91816 Ìtxloleuh joredo gh od vhfwlrq

O*ìtxloleuh joredo gh od vhfwlrq v*ìfulw grqf/ hq vxssrvdqw txh shqgdqw od skdvh gh xdjh/

lo q*| d dxfxqh plvh hq suìfrqwudlqwh =

#�Ir?

�Ir|

$@ ^Nr` =

#�%?

�%|

$�#7I4?7I4|

$�#

^N6? ` =7 %?_�s

^N6| ` =7 %|_�s

$+91;<,

Fhfl shxw hqfruh v*ìfuluh =#�Ir?

�Ir|

$.

#7I4?7I4|

$.

#^N6? ` =7 %?_�s

^N6| ` =7 %|_�s

$@ ^Nr` =

#�%?

�%|

$+91<3,

Rq frqvwdwh doruv txh o*ìtxdwlrq g*ìtxloleuh hvw gx píph w|sh txh +7193,1 Od gl�ìuhqfh

surylhqw ghv whuphv

#7I4?7I4|

$hw

#^N6? ` =7 %?_�s

^N6| ` =7 %|_�s

$1

Oh yhfwhxu

#7I4?7I4|

$fruuhvsrqg dx{ vroolflwdwlrqv lqgxlwhv sdu od shuwh gh suìfrqwudlqwh

gxh ã od uhod{dwlrq1 Fhv vroolflwdwlrqv vrqw irupìhv ã sduwlu ghv dupdwxuhv gh suìfrqwudlqwh

vhxohv/ hw vrqw dssoltxìhv ã o*hqvhpeoh gh od vhfwlrq1

919 Frqfoxvlrq 46:

Oh yhfwhxu

#^N6? ` =7 %?_�s

^N6| ` =7 %|_�s

$fruuhvsrqg dx{ vroolflwdwlrqv lqgxlwhv sdu od gìirupdwlrq

gh xdjh fdu vhxo oh eìwrq xh/ pdlv o*hqvhpeoh gh od vhfwlrq vxelw fhv vroolflwdwlrqv1

G*xq srlqw gh yxh lqirupdwltxh/ ohv yhfwhxuv

#7I4?7I4|

$hw

#^N6? ` =7 %?_�s

^N6| ` =7 %|_�s

$vrqw

dmrxwìv hw vwrfnìv gdqv oh píph yhfwhxu txh

#Dr?

Dr|

$/ fdu lov rqw oh píph u÷oh g*xq srlqw

gh yxh dojrulwkpltxh1

Rq d grqf xqh sulvh hq frpswh qdwxuhooh gh o*lqwhudfwlrq hqwuh oh xdjh gx eìwrq hw od

uhod{dwlrq ghv dflhuv gh suìfrqwudlqwh/ fdu o*dojrulwkph yd gìwhuplqhu o*dffurlvvhphqw gh

gìirupdwlrq wrwdoh gh od vhfwlrq fruuhvsrqgdqw dx{ h�hwv gl�ìuìv1

Od judqgh gl�ìuhqfh sdu udssruw dx fkdslwuh 7 uìvlgh gdqv oh fdofxo gx prgxoh ìodvwltxh

orqjlwxglqdo frugh H6 +|> },1 Fhoxl0fl hvw fdofxoì ã sduwlu gh od gìirupdwlrq lqvwdqwdqìh/

hw qrq sdv ã sduwlu gh od gìirupdwlrq wrwdoh =

7�6 @ H�67 %� +91<4,

919 Frqfoxvlrq

Od plvh hq úxyuh ghv h�hwv gl�ìuìv dx vhlq gh o*dojrulwkph gh uìvroxwlrq qrxv d frqgxlw ã

suhqguh ghv rswlrqv vlpsohv/ pdlv xqlyhuvhoohphqw uhfrqqxhv1 Oh xdjh gx eìwrq dlqvl txh

od uhod{dwlrq ghv dflhuv/ txl vrqw ghx{ skìqrpëqhv dqwdjrqlvwhv srxu oh frpsruwhphqw gh

od vwuxfwxuh/ vrqw uljrxuhxvhphqw sulv hq frpswh gëv oruv txh od glvfuìwlvdwlrq whpsruhooh

plvh hq úxyuh hvw vx!vdqwh1 Fhwwh glvfuìwlvdwlrq hvw idlwh gh idêrq dxwrpdwltxh sdu oh sur0

judpph1 O*dojrulwkph dlqvl gìfulw hvw sduidlwhphqw frpsdwleoh dyhf o*hqvhpeoh ghv orlv gh

xdjh sursrvìhv sdu oh surjudpph olqìdluh/ hw qrwdpphqw ghv ìyroxwlrqv ghv orlv gh xdjh

sursrvìhv sdu o*DIUHP hq 4<<9 ^Ofsf<9`/ frqfhuqdqw ohv eìwrqv kdxwhv shuirupdqfhv1

46; Fkdslwuh 9 Sulvh hq frpswh ghv h�hwv gl�ìuìv

Fkdslwuh :

Ìoìphqwv vsìfl�txhv dx Jìqlh flylo

:14 Ohv duwlfxodwlrqv

:1414 Shwlwv gìsodfhphqwv = k|srwkëvhv hw qrwdwlrqv

Hq shwlwv gìsodfhphqwv/ oh uhsëuh lqwulqvëtxh hw oh uhsëuh orfdo gh o*ìoìphqw vrqw frqirq0

gxv1 Vrlhqw D hw E ghx{ srlqwv jìrpìwultxhphqw frqirqgxv hw uholìv pìfdqltxhphqw sdu

xqh duwlfxodwlrq1 Rq qrwh Iu� oh wruvhxu ghv irufhv v*dssoltxdqw dx srlqw E hw Vu� ohv

gìsodfhphqwv +dx vhqv odujh, gh E gdqv oh uhsëuh orfdo/ rx lqwulqvëtxh gh o*ìoìphqw1 Rq d

grqf =

Iu� @

3EEEEEEEEEC

I�%

I�+

I�5

P�%

P�+

P�5

4FFFFFFFFFD

+:14,

hw

Vu� @

3EEEEEEEEEC

x�%

x�+

x�5

��%

��+

��5

4FFFFFFFFFD

+:15,

Hq vxssrvdqw qxo oh gìsodfhphqw gx qúxg E/ o*duwlfxodwlrq hvw xq ìoìphqw �fwli/ gh

orqjxhxu qxooh/ hw grqw oh frpsruwhphqw/ gdqv oh uhsëuh orfdo/ rx lqwulqvëtxh gh o*du0

wlfxodwlrq/ hvw gìfulw sdu =

Iu� @ ^N��` =Vu� +:16,

473 Fkdslwuh : Ìoìphqwv vsìfl�txhv dx Jìqlh flylo

dyhf =

^N��` @

59999999997

� 3 3 3 3 3

3 � 3 3 3 3

3 3 � 3 3 3

3 3 3 � 3 3

3 3 3 3 � 3

3 3 3 3 3 �

6:::::::::8

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Od pdwulfh joredoh +45� 45, gh o*duwlfxodwlrq hvw fdofxoìh gh idêrq fodvvltxh ã sduwlu gh

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Xq ìoìphqw gh w|sh fkdñqhwwh shxw íwuh fkdujì gh 8 idêrqv gl�ìuhqwhv =


Whqvlrq krul}rqwdoh �{ìh Vl od whqvlrq krul}rqwdoh Wf hvw �{ìh/ doruv od vroxwlrq gx

sureoëph hvw lppìgldwh1 Rq fdofxoh Y / {f hw |f gluhfwhphqw1

Whqvlrq yhuwlfdoh �{ìh hq D rx hq E Od whqvlrq yhuwlfdoh W+ vh fdofxoh vhorq =

W+ +{, @ Wfg|

g{@ Wf vlqk

�s

Wf+{. {f,

�+:18<,

Vl od whqvlrq yhuwlfdoh hvw �{ìh hq D ã od ydohxu W+� / o*ìtxdwlrq ã uìvrxguh hvw =

i +Wf, @ Wf vlqk

�s

Wf+{� . {f,

�� W+� @ 3 +:193,

Vl od whqvlrq yhuwlfdoh hvw �{ìh hq E ã od ydohxu W+� / o*ìtxdwlrq ã uìvrxguh hvw =

i +Wf, @ Wf vlqk

�s

Wf+{� . {f,

�� W+� @ 3 +:194,

Whqvlrq wdqjhqwh �{ìh hq D rx hq E Vl od whqvlrq wdqjhqwh hvw �{ìh hq D ã od ydohxu

W�/ o*ìtxdwlrq ã uìvrxguh hvw =

i +Wf, @ Wf frvk

�s

Wf+{� . {f,

�� W� @ 3 +:195,

Vl od whqvlrq wdqjhqwh hvw �{ìh hq E ã od ydohxu W�/ o*ìtxdwlrq ã uìvrxguh hvw =

i +Wf, @ Wf frvk

�s

Wf+{� . {f,

�� W� @ 3 +:196,

Whqvlrq frugh �{ìh hq D rx hq E Hq qrwdqw k od sruwìh krul}rqwdoh gh od fkdñqhwwh

hw y vd kdxwhxu/ hw g od orqjxhxu gh vd frugh/ rq d =

k @ {� � {� A 3 +:197,

y @ |� � |� +:198,

g @

sk2 . y2 +:199,

Vl od whqvlrq frugh hvw �{ìh hq D ã od ydohxu W�� / o*ìtxdwlrq ã uìvrxguh hvw =

i +Wf, @k

gWf .

y

gWf vlqk

�s

Wf+{� . {f,

�� W�� @ 3 +:19:,

Vl od whqvlrq frugh hvw �{ìh hq E ã od ydohxu W�� / o*ìtxdwlrq ã uìvrxguh hvw =

i +Wf, @k

gWf .

y

gWf vlqk

�s

Wf+{� . {f,

�� W�� @ 3 +:19;,

:16 Ohv ìoìphqwv gh w|sh fkdñqhwwh 484

Orqjxhxu ã ylgh �{ìh Vl od orqjxhxu ã ylgh hvw �{ìh ã xqh ydohxu O�f/ o*ìtxdwlrq ã

uìvrxguh hvw =

i +Wf, @ O�O�f ��O @ 3 +:19<,

Rx hqfruh =

i +Wf, @Wf

s

�vlqk

�s

Wf+{� . {f,

�� vlqk

�s

Wf+{� . {f,

��O�f

� Wf

5H=D%

3C Af

2Rvlqk

�2RAf

+{� . {f,�. {�

�Af2R vlqk

�2RAf

+{� . {f,�� {�

4D

@ 3 +:1:3,

O*ìtxdwlrq i +Wf, @ 3/ lvvxh gh +:193,/ +:194,/ +:195,/ +:196,/ +:19:,/ +:19;, rx +:1:3,

vhorq oh prgh gh fkdujhphqw/ hvw uìvroxh gh idêrq lwìudwlyh ã o*dlgh g*xq vfkìpd gh Qhzwrq/

d�q gh wurxyhu od whqvlrq krul}rqwdoh Wf txl vdwlvidlw ã od frqglwlrq i +Wf, @ 31

Vl xq ìoìphqw fkdñqhwwh qh vxelw sdv gh fkdujhphqw/ pdlv vl od vwuxfwxuh vxelw xq

fkdujhphqw/ od uìsrqvh gh o*ìoìphqw fkdñqhwwh ã fhwwh vroolflwdwlrq v*h�hfwxh hq pdlqwhqdqw

frqvwdqwh vd orqjxhxu ã ylgh1

Ã fkdtxh frqyhujhqfh hq gìsodfhphqwv/ ohv srvlwlrqv ghv srlqwvD hw E vrqw lqwurgxlwhv

gdqv o*dojrulwkph gh od fkdñqhwwh1 Fh ghuqlhu grqqh hq uhwrxu ohv irufhv dx{ qúxgv D hw

E/ fh txl shuphw g*ìydoxhu oh uìvlgx g*ìtxloleuh hq irufh/ o*ìoìphqw fkdñqhwwh wudydloodqw

gluhfwhphqw gdqv oh uhsëuh joredo gh od vwuxfwxuh1 Hq fh txl frqfhuqh oh prgxoh g*ìodvwlf0

lwì dssduhqw gh o*ìoìphqw/ qrxv xwlolvrqv od irupxodwlrq dssurfkìh fodvvltxh dvvlplodqw od

fkdñqhwwh ã xqh sduderoh1 Fhv ydohxuv/ frqqxhv vrxv oh qrp gh irupxoh gh Huqvw/ vrqw ohv

vxlydqwhv =

H|@?}e?| @H

4 ..��%�R2�2EA�nA��

2eA 2�A 2�

+:1:4,

Hr�eS@?| @H

4 ..��%�R2�2

�2A ��

rxH

4 ..��%�R2�2

�2A ��

+:1:5,

Fhfl hvw olflwh gdqv od phvxuh rü fh prgxoh q*lqwhuylhqw txh srxu frqvwuxluh o*rsìudwhxu

gh frqyhujhqfh/ hw qrq sdv srxu irupxohu oh uìvlgx g*ìtxloleuh/ fh ghuqlhu ìwdqw fdofxoì gh

idêrq h{dfwh1

:1616 Xqh dxwuh irupxodwlrq gx sureoëph

Frpph rq shxw oh frqvwdwhu/ od irupxodwlrq gx sureoëph suìfìghqw q*hvw soxv ydodeoh

oruvtxh Wf ghylhqw idleoh/ f*hvw0ã0gluh oruvtx*lo whqg yhuv 31 Fhfl hvw oh fdv qrwdpphqw

oruvtxh o*ìoìphqw ghylhqw yhuwlfdo/ frpph ohv vxvshqwhv gh srqw vxvshqgx sdu h{hpsoh1 Oh

sdudpëwuh fkrlvl/ ã vdyrlu Wf/ qh slorwh soxv fruuhfwhphqw oh sureoëph1 Lo idxw uhirupxohu

oh sureoëph/ qrq sdv hq irqfwlrq gh {/ pdlv gh v/ o*devflvvh fxuyloljqh1


Ìwdeolvvhphqw ghv ìtxdwlrqv gh od fkdñqhwwh

Rq uhsduw gx sulqflsh txh od whqvlrq W +P, hvw sruwìh sdu od wdqjhqwh ã od frxueh/ g*rü =

W% +v, @ W +v,g{

gv@ Wf +:1:6,

W+ +v, @ W +v,g|

gv+:1:7,

rü { hw | vrqw ghv irqfwlrqv gh v1 Rq rewlhqw doruv =

s @gW+ +v,

gv@gW +v,

gv

g|

gv. W +v,

g2|

gv2+:1:8,

Fhfl v*ìfulw hqfruh =

W +v,{�

+v, @ Wf +:1:9,

W�

+v, |�

+v, . W +v, |��

+v, @ s +:1::,

Fh vrqw ohv ìtxdwlrqv gl�ìuhqwlhoohv txh grlyhqw yìul�hu ohv irqfwlrqv W +v,/ { +v, hw | +v,1

Ohv vroxwlrqv vrqw gh od iruph =

{ +v, . {f @Wf

sduj vlqk

�s

Wf+v. vf,

�+:1:;,

| +v, . |f @

v�Wf

s

�2

. +v. vf,2 +:1:<,

W +v, @

tW 2f . s2 +v. vf,

2 +:1;3,

Frqwudluhphqw ã od suhplëuh irupxodwlrq/ vl Wf whqg yhuv 3/ oh sureoëph d hqfruh xqh

vroxwlrq fdu ohv ìtxdwlrqv ghylhqqhqw =

{ +v, @ {� @ {� +:1;4,

| +v, . |f @ mv. vfm +:1;5,

W +v, @ s mv. vfm +:1;6,

Fkdñqhwwh sdvvdqw sdu ghx{ srlqwv

Ohv frqglwlrqv gh sdvvdjh hq D hw E v*ìfulyhqw =

Hq D =

{� . {f @Wf

sduj vlqk

�s

Wf+vf,

�+:1;7,

|� . |f @

v�Wf

s

�2

. v2f +:1;8,


Hq E =

{� . {f @Wf

sduj vlqk

�s

Wf+vf .O,

�+:1;9,

|� . |f @

v�Wf

s

�2

. +vf .O,2 +:1;:,

rü O hvw od orqjxhxu gìyhorssìh gh od fkdñqhwwh1 Sdu vrxvwudfwlrq/ rq rewlhqw =

{� � {� @ k @Wf

s

�duj vlqk

�s

Wf+vf .O,

�� duj vlqk

�s

Wf+vf,

��+:1;;,

|� � |� @ y @

v�Wf

s

�2

. +vf .O,2 �

v�Wf

s

�2

. v2f +:1;<,

Ohv lqfrqqxhv gx sureoëph vrqw grqf Wf/ od whqvlrq krul}rqwdoh/ vf o*devflvvh fxuyloljqh gh

D/ hw O od orqjxhxu gìyhorssìh gh od fkdñqhwwh1

Doorqjhphqw gh od fkdñqhwwh

Gh od píph idêrq/ rq d =

�O @

] u

f

W +v,

H=D%

gv

@4

H=D%

] u

f

tW 2f . s2 +v. vf,

2gv

@4

5H=D%

57 +v. vf,

tW 2f . s2 +v. vf,

2.

A 2fRduj vlqk

�RAf

+v. vf,�

68u

f

+:1<3,

Uìvroxwlrq gx sureoëph gh od fkdñqhwwh

Oh sureoëph qh vh uìvxph soxv ã xqh ìtxdwlrq hw xqh lqfrqqxh Wf frpph suìfìghpphqw/

pdlv ã 6 ìtxdwlrqv ã 6 lqfrqqxhv txh vrqw Wf/ vf hw O1 Ohv 5 suhplëuhv ìtxdwlrqv vrqw

+:1;;, hw +:1;<, =

i� +Wf> vf> O, @Wf

s

�duj vlqk

�s

Wf+vf .O,

�� duj vlqk

�s

Wf+vf,

�� k @ 3+:1<4,

i2 +Wf> vf> O, @

v�Wf

s

�2

. +vf .O,2 �v�

Wf

s

�2

. v2f � y @ 3 +:1<5,

Od wurlvlëph ìtxdwlrq hvw grqqìh sdu oh prgh gh fkdujhphqw1

Whqvlrq krul}rqwdoh �{ìh Vl od whqvlrq krul}rqwdoh Wf hvw �{ìh/ doruv oh v|vwëph q*d

soxv txh 5 ìtxdwlrqv hw 5 lqfrqqxhv = vf hw O/ txh o*rq wurxyh hq uìvroydqw +:1;;, hw +:1;<,1


Whqvlrq yhuwlfdoh �{ìh hq D rx hq E Od whqvlrq yhuwlfdoh W+ vh fdofxoh vhorq =

W+ +v, @ s +v. vf, +:1<6,

Vl od whqvlrq yhuwlfdoh hvw �{ìh hq D ã od ydohxu W+� / o*ìtxdwlrq ã uìvrxguh hvw =

i +Wf> vf> O, @ s=vf � W+� @ 3 +:1<7,

Vl od whqvlrq yhuwlfdoh hvw �{ìh hq E ã od ydohxu W+� / o*ìtxdwlrq ã uìvrxguh hvw =

i +Wf> vf> O, @ s= +vf .O,� W+� @ 3 +:1<8,

Whqvlrq wdqjhqwh �{ìh hq D rx hq E

Vl od whqvlrq wdqjhqwh hvw �{ìh hq D ã od ydohxu W�/ o*ìtxdwlrq ã uìvrxguh hvw =

i +Wf> vf> O, @

tW 2f . s2 +vf,

2 � W� @ 3 +:1<9,

Vl od whqvlrq wdqjhqwh hvw �{ìh hq E ã od ydohxu W�/ o*ìtxdwlrq ã uìvrxguh hvw =

i +Wf> vf> O, @

tW 2f . s2 +vf .O,

2 � W� @ 3 +:1<:,

Whqvlrq frugh �{ìh hq D rx hq E Vl od whqvlrq frugh hvw �{ìh hq D ã od ydohxu W�� /

o*ìtxdwlrq ã uìvrxguh hvw =

i +Wf> vf> O, @k

gWf .

y

gs=vf � W�� @ 3 +:1<;,

Vl od whqvlrq frugh hvw �{ìh hq E ã od ydohxu W�� / o*ìtxdwlrq ã uìvrxguh hvw =

i +Wf> vf> O, @k

gWf .

y

gs= +vf .O,� W�� @ 3 +:1<<,

Orqjxhxu ã ylgh �{ìh Vl od orqjxhxu ã ylgh hvw �{ìh ã xqh ydohxu O�f / o*ìtxdwlrq ã

uìvrxguh hvw =

i +Wf> vf> O, @ O�O�f ��O @ 3 +:1433,

Fhfl v*ìfulw hqfruh =

i +Wf> vf> O, @ O�O�f

� 4

5H=D%

57 +v. vf,

tW 2f . s2 +v. vf,

2.

A 2fRduj vlqk

�RAf

+v. vf,�

68u

f

@ 3 +:1434,


Rq uìvrxw ohv ìtxdwlrqv

i +Wf> vf> O, @ 3 +:1435,

i� +Wf> vf> O, @ 3 +:1436,

i2 +Wf> vf> O, @ 3 +:1437,

hq irqfwlrq gh Wf/ vf/ hw O/ vdfkdqw txh +:1435, hvw lvvxh gh +:1<7,/ +:1<8,/ +:1<9,/ +:1<:,/

+:1<;,/ +:1<<, rx +:1434, vhorq oh prgh gh fkdujhphqw1 Qrxv xwlolvrqv xq vfkìpd gh w|sh

Qhzwrq/ f*hvw0ã0gluh txh qrxv frqvwuxlvrqv od pdwulfh mdfrelhqqh gx sureoëph =

^M ` @

5997

YsEAfcrfcu�YAf

YsEAfcrfcu�Yrf

YsEAfcrfcu�Yu

Ys�EAfcrfcu�YAf

Ys�EAfcrfcu�Yrf

Ys�EAfcrfcu�Yu

Ys2EAfcrfcu�YAf

Ys2EAfcrfcu�Yrf

Ys2EAfcrfcu�Yu

6::8 +:1438,

hw rq rewlhqw ã fkdtxh sdv o*lqfuìphqw ã dssoltxhu dx yhfwhxu

3EEC

Wf

vf

O

4FFD srxu v*dssurfkhu

gh od vroxwlrq = 3EEC

�Wf

�vf

�O

4FFD @ ^M `

3�

3EEC�i +Wf> vf> O,�i� +Wf> vf> O,�i2 +Wf> vf> O,

4FFD +:1439,

Oruvtxh o*rq d gìwhuplqì oh yhfwhxu

3EEC

Wf

vf

O

4FFD vroxwlrq gx sureoëph/ ohv h�ruwv dx{

h{wuìplwìv gh o*ìoìphqw vh gìgxlvhqw gluhfwhphqw1

Fdofxo gx prgxoh dssduhqw

Qrxv srxyrqv ìjdohphqw fdofxohu oh prgxoh dssduhqw gh o*ìoìphqw fkdñqhwwh/ uholdqw o*df0

furlvvhphqw lq�qlwìvlpdo gh irufh dx plolhx gh o*ìoìphqw oh orqj gh od frugh gþ ã xq

doorqjhphqw lq�qlwìvlpdo gh od frugh g1 Od irufh oh orqj gh od frugh dx plolhx gh o*ìoìphqw

v*h{sulph sdu =

W� @k

gWf .

y

gs=

�vf .

O

5

�+:143:,

Hq dssoltxdqw xqh yduldwlrq lq�qlwìvlpdoh �g gh od orqjxhxu gh od frugh gdqv od gluhfwlrq

gì�qlh sdu fhwwh ghuqlëuh/ rq d =

�k @k

g�g +:143;,

�y @y

g�g +:143<,


G*rü =

W� . �W� @k. �k

g. �g+Wf . �Wf, .

y . �y

g. �gs=

�vf . �vf .

O. �O

5

�

@k

g+Wf . �Wf, .

y

gs=

�vf . �vf .

O. �O

5

�+:1443,

G*rü =

�W� @k

g�Wf .

y

gs=

��vf .

�O

5

�+:1444,

O*doorqjhphqw lq�qlwìvlpdo vh idlw dxwrxu gh od vroxwlrq +Wf> vf> O, yìul�dqw =

i +Wf> vf> O, @ 3 +:1445,

i� +Wf> vf> O, @ 3 +:1446,

i2 +Wf> vf> O, @ 3 +:1447,

rü o*ìtxdwlrq i +Wf> vf> O, @ 3 hvw fhqvìh pdlqwhqlu od orqjxhxu ã ylgh frqvwdqwh/ f*hvw0ã0

gluh gh od iruph gh +:1434,1 Hq h�hfwxdqw xq gìyhorsshphqw olplwì gh i / i� hw i2 dxwrxu

gh +Wf> vf> O,/ rq d =

�i +Wf> vf> O, @YsEAfcrfcu�

YAf�Wf .

YsEAfcrfcu�Yrf

�vf .YsEAfcrfcu�

Yu�O +:1448,

�i� +Wf> vf> O, @Ys�EAfcrfcu�

YAf�Wf .

Ys�EAfcrfcu�Yrf

�vf .Ys�EAfcrfcu�

Yu�O +:1449,

�i2 +Wf> vf> O, @Ys2EAfcrfcu�

YAf�Wf .

Ys2EAfcrfcu�Yrf

�vf .Ys2EAfcrfcu�

Yu�O +:144:,

Rq uhpdutxh doruv txh �i� +Wf> vf> O, fruuhvsrqg ã o*doorqjhphqw krul}rqwdo �k hw �i2 +Wf> vf> O,

ã o*doorqjhphqw yhuwlfdo �y1 O*dffurlvvhphqw �i +Wf> vf> O, grlw íwuh qxo d�q gh pdlqwhqlu

od orqjxhxu ã ylgh frqvwdqwh1 Rq rewlhqw �qdohphqw oh v|vwëph =

^M ` =

3EEC

�Wf

�vf

�O

4FFD @

3EEC

3

�k

�y

4FFD +:144;,

Rq lqyhuvh oh v|vwëph dlqvl rewhqx/ hw hq qrwdqw =

^M `3�

@

3EEC

t�� t�2 t��

t2� t22 t2�

t�� t�2 t��

4FFD +:144<,

Rq d =

�Wf @ t�2=�k. t��=�y +:1453,

�vf @ t22=�k. t2�=�y +:1454,

�O @ t�2=�k. t��=�y +:1455,

:17 Ohv fåeohv gh suìfrqwudlqwh qrq olìv 48:

G*rü =

�W� @k

g+t�2=�k. t��=�y, .

y

gs=

�t22=�k. t2�=�y .

t�2=�k. t��=�y

5

�

@

�t�2

k

g. s

�t22 .

t�2

5

� yg

��k.

�t��

k

g. s

�t2� .

t��

5

� yg

��y

@

�t�2

k

g. s

�t22 .

t�2

5

� yg

�k

g�g.

�t��

k

g. s

�t2� .

t��

5

� yg

�y

g�g +:1456,

Rq rewlhqw hq�q =

H@RR@oe?| @�W�

�g

g

D%

@4

D%

# �t�2

�_. s

�t22 .

^�22

��_

�k.�

t��_. s

�t2� .

^��2

��_

�y

$+:1457,

Fhv gìyhorsshphqwv vxu od fkdñqhwwh q*rqw/ ã qrwuh frqqdlvvdqfh/ mdpdlv ìwì h�hfwxìv1

Od ghx{lëph irupxodwlrq gx sureoëph gh od fkdñqhwwh hvw sduwlfxolëuhphqw lqgltxìh

srxu oh fdofxo ghv h�ruwv dx{ h{wuìplwìv ghv vxvshqwhv1 Od suhplëuh hvw uìvhuyìh dx{

ìoìphqwv fkdñqhwwhv txl qh vrqw sdv surfkhv gh od yhuwlfdoh1 Oh sdvvdjh g*xqh irupxodwlrq

ã o*dxwuh hvw sduidlwhphqw frqwlqx1

:17 Ohv fåeohv gh suìfrqwudlqwh qrq olìv

Fodvvltxhphqw/ gdqv xqh vhfwlrq frpsrvlwh/ oruvtxh ohv fåeohv gh suìfrqwudlqwh vrqw olìv

dx eìwrq/ o*dflhu ghv fåeohv hvw vrolgdluh gh od vhfwlrq = lo qh jolvvh sdv sdu udssruw dx

eìwrq1 Rq shxw grqf gluh txh ohv fåeohv frqwulexhqw ã od uljlglwì gh od vhfwlrq hw lo vrqw

lqwìjuìv gluhfwhphqw gdqv od pdwulfh gh uljlglwì gh od vhfwlrq1 Fhfl suìvhqwh o*dydqwdjh gh

qh sdv dyrlu ã jìuhu ohv vxuwhqvlrqv ghv fåeohv sxlvtx*hoohv vrqw dxwrpdwltxhphqw sulvhv hq

frpswh sdu od pdwulfh gh uljlglwì gh od vhfwlrq frpsrvlwh1

Vl ohv fåeohv qh vrqw sdv olìv dx eìwrq/ lov shxyhqw jolvvhu sdu udssruw ã od vhfwlrq1 Rq

qh shxw grqf soxv frqvlgìuhu txh ohv fåeohv frqwulexhqw ã od uljlglwì gh od vhfwlrq1 Oh eìwrq

vhxo uìvlvwh hw ohv yduldwlrqv gh whqvlrq ghv fåeohv qrq olìv grlyhqw íwuh uìshufxwìhv frpph

ghv h�ruwv h{wìulhxuv ã od vhfwlrq1 Od whqvlrq g*xq fåeoh/ vxu xq wurqêrq qrq lqmhfwì/ hvw

doruv vxssrvìh frqvwdqwh oh orqj gx fåeoh hw hvw fdofxoìh dyhf od pr|hqqh ghv gìirupdwlrqv

ghv vhfwlrqv oh orqj gx fåeoh/ hq sdufrxudqw oh wurqêrq qrq lqmhfwì1

Ã fkdtxh ìwdsh gx surfhvvxv lwìudwli gh uìvroxwlrq/ od gìirupdwlrq ghv fåeohv qrq olìv

dx eìwrq hvw plvh ã mrxu ã sduwlu gh od gìirupdwlrq frxudqwh ghv vhfwlrqv1

Od plvh hq úxyuh lqirupdwltxh idlw o*remhw gh o*dqqh{h D1

48; Fkdslwuh : Ìoìphqwv vsìfl�txhv dx Jìqlh flylo

Fkdslwuh ;

Fdofxo g*lqvwdelolwì

;14 Lqwurgxfwlrq

Gdqv fh fkdslwuh/ qrxv qrxv lqwìuhvvhurqv hvvhqwlhoohphqw dx fdofxo g*lqvwdelolwì olqìdluh1 Oh

fdofxo g*lqvwdelolwì frqvlvwh/ srxu xqh vwuxfwxuh grqqìh vrxplvh ã xqh fkdujh S / ã wurxyhu

oh idfwhxu gh fkdujh fulwltxh txh o*rq qrwhud �/ who txh vrxv od fkdujh �S / od vwuxfwxuh

ghylhqw lqvwdeoh1 Hq jìqìudo/ vhxohv ohv ydohxuv srvlwlyhv gh � vrqw lqwìuhvvdqwhv1

Oh fdofxo g*lqvwdelolwì olqìdluh shuphw gh uìvrxguh fh sureoëph hq oh olqìdulvdqw ã o*r0

uljlqh gdqv oh gldjudpph h�ruw0gìsodfhphqw1 � hvw doruv vroxwlrq gx sureoëph dx{ ydohxuv

sursuhv vxlydqw =

^Nf` ={ @ �� ^Nj` ={ +;14,

rü ^Nf` hvw od pdwulfh gh uljlglwì pdwìulhooh gh od vwuxfwxuh vxssrvdqw oh pdwìuldx olqìdluh/

hw ^Nj` od pdwulfh ghv frqwudlqwhv lqlwldohv gì�qlh sdu +6197,1 Hq fh txl qrxv frqfhuqh/

od pdwulfh ^Nj` hvw fdofxoìh ã sduwlu gh od pdwulfh ^G` gì�qlh sdu +614;9, hq vxssrvdqw

oh gìsodfhphqw qxo1 Dlqvl/ ^Nj` hvw sursruwlrqqhooh dx{ h�ruwv lqwhuqhv gh od vwuxfwxuh

frqvlgìuìh1

Oh fdofxo g*lqvwdelolwì qrq olqìdluh qh shxw vh idluh gh idêrq uljrxuhxvh tx*hq dffurlvvdqw

od fkdujh S 1 Oh fdofxo glyhujh oruvtxh od fkdujh fulwltxh hvw dwwhlqwh1

Fhshqgdqw/ rq shxw hqylvdjhu xq fdofxo txl olqìdulvh oh sureoëph hqwuh 5 ìwdwv gh fkdujh

grqqìv/ dx olhx gh oh olqìdulvhu ã o*ruljlqh frpph f*hvw vrxyhqw oh fdv1

Oh sureoëph dx{ ydohxuv sursuhv gh od iruph

^D` ={ @ � ^E` ={ +;15,

vh uhwurxyh ìjdohphqw gdqv o*dqdo|vh g|qdpltxh ghv vwuxfwxuhv1 Hq h�hw/ ohv prghv gh

yleudwlrq gh od vwuxfwxuh vrqw ohv yhfwhxuv sursuhv {� dvvrflìv dx{ ydohxuv sursuhv �� gx

sureoëph vxlydqw =

^N` ={ @ � ^P ` ={ +;16,

493 Fkdslwuh ; Fdofxo g*lqvwdelolwì

rü ^P ` hvw od pdwulfh ghv pdvvhv gh od vwuxfwxuh/ hw ^N` vd pdwulfh gh uljlglwì1

;15 Uìvroxwlrq gx sureoëph

Hq dqdo|vh g|qdpltxh ghv vwuxfwxuhv/ ohv pdwulfhv ^N` hw ^P ` suìvhqwhqw o*dydqwdjh qrq

qìjoljhdeoh g*íwuh gì�qlhv srvlwlyhv1 Fhfl shuphw g*xwlolvhu fhuwdlqhv pìwkrghv frpph od

pìwkrghv ghv vrxv0hvsdfhv rx od pìwkrgh gh Mdfrel srxu h{wudluh ohv ydohxuv sursuhv hw

ohv yhfwhxuv sursuhv gx sureoëph1

Gdqv oh fdofxo g*lqvwdelolwì/ od pdwulfh ^Nf` hvw fhuwhv gì�qlh srvlwlyh/ pdlv fh q*hvw

sdv oh fdv gh od pdwulfh ghv frqwudlqwhv lqlwldohv ^Nj`1 Srxu uìvrxguh fh sureoëph/ lo h{lvwh

soxvlhxuv whfkqltxhv =

;1514 Oh vkliw olqìdluh

Fhwwh whfkqltxh frqvlvwh ã h�hfwxhu xq gìfdodjh vxu � hq srvdqw =

� @ �f . � +;17,

Oh sureoëph v*ìfulw doruv =

^^Nf` . �f ^Nj`` ={ @ �� ^Nj` ={ +;18,

Sdu xq fkrl{ mxglflhx{ gh �f/ od uhfkhufkh ghv ydohxuv gh � dqqxodqw ghw +^Nf` . � ^Nj`,

hvw soxv dlvìh/ pdlv v*dyëuh qìdqprlqv gìolfdwh1

;1515 Oh vkliw qrq olqìdluh

Lfl/ rq srvh =

� @�

4� ��+;19,

Oh sureoëph v*ìfulw doruv =

^Nf` ={ @ �� ^^Nj`� � ^Nf`` ={ +;1:,

Vl o*rq qrwh �3� od suhplëuh ydohxu sursuh qìjdwlyh gx sureoëph/ hw gh idêrq jìqìudoh �3�od lëph ydohxu sursuh qìjdwlyh hq sduwdqw gh 3/ rq frqvwdwh doruv txh vl � A �

�b3�� doruv

� A 3 hw o*rq d �3� A ��1 Hq rxwuh/ dx{ ydohxuv sursuhv �� srvlwlyhv/ f*hvw0ã0gluh fhoohv txl

qrxv lqwìuhvvhqw/ fruuhvsrqghqw ghv ydohxuv �� whoohv txh 3 9 �� 9�k1

Fhwwh whfkqltxh v*dyëuh ehdxfrxs soxv vwdeoh hq sudwltxh1

;15 Uìvroxwlrq gx sureoëph 494

;1516 Xqh dxwuh dssurfkh

Srxu uìvrxguh oh sureoëph dx{ ydohxuv sursuhv

^^Nf` . � ^Nj`` ={ @ 3 +;1;,

dyhf { 9@ 3/ wrxw hq h{wud|dqw ghv ydohxuv sursuhv � srvlwlyhv/ qrxv qrxv vrpphv edvìv vxu

oh idlw vxlydqw = od pdwulfh ^^Nf` . � ^Nj`` hvw vlqjxolëuh/ grqf vrq gìwhuplqdqw hvw qxo/ rx

hqfruh xqh gh vhv ydohxuv sursuhv/ txl ìwdlw srvlwlyh/ ghylhqw qxooh1 Qrxv fdofxorqv/ srxu

xqh ydohxu grqqìh gh � od pdwulfh

^N` @ ^Nf` . � ^Nj` +;1<,

Fhwwh pdwulfh hvw uìhooh v|pìwultxh = hooh hvw grqf gldjrqdolvdeoh hw shxw vh phwwuh vrxv od

iruph

^N` @ Ô` ^G`|Ô` +;143,

rü Ô` hvw xqh pdwulfh wuldqjxodluh lqiìulhxuh +Orzhu, hw ^G` xqh pdwulfh gldjrqdoh/ frp0

srvìh ghv ydohxuv sursuhv gh ^N`1 Qrxv h�hfwxrqv od idfwrulvdwlrq gh ^N`/ hw qrxv frps0

wrqv oh qrpeuh gh ydohxuv sursuhv qìjdwlyhv frqwhqxhv gdqv ^G`1 Dlqvl/ sdu xqh vlpsoh

glfkrwrplh/ qrxv frqqdlvvrqv ohv ydohxuv gh � txl fkdqjhqw oh qrpeuh gh ydohxuv sursuhv

qìjdwlyhv/ fh txl uhylhqw ã gìwhfwhu ohv ydohxuv gh � txl dqqxohqw oh gìwhuplqdqw gh ^N`1 Hq

sduwdqw gh 3/ rq shxw dlqvl h{wudluh ohv ydohxuv dssurfkìhv ��/111/�R ghv s ydohxuv sursuhv

srvlwlyhv txl uìsrqghqw dx sureoëph srvì1

Fhshqgdqw/ fhwwh whfkqltxh qh shuphw sdv gh fdofxohu ohv yhfwhxuv sursuhv dvvrflìv ã

��/111/�R1 Srxu h{wudluh oh yhfwhxu sursuh {� dvvrflì ã od ydohxu sursuh ��/ qrxv xwlolvrqv

od pìwkrgh gh od sxlvvdqfh lqyhuvh/ hq h�hfwxdqw xq gìfdodjh +vkliw, gh �� sdu udssruw ã

o*ruljlqh gh o*d{h ghv ydohxuv sursuhv1 Rq srvh doruv

� @ �� . �W +;144,

Rq qrwh dx sdvvdjh txh �W hvw shwlw hq ydohxu devroxh1 Rq rewlhqw doruv =

^^^Nf` . �� ^Nj`` . �W ^Nj``{ @ 3 +;145,

Qrxv srvrqv =

^Nf` . �� ^Nj` @ ^N�` +;146,

Rq frqvwuxlw doruv od vxlwh gh yhfwhxuv qrupìv

t&n� @^N�`

3�^Nj` =t&��^N�`

3�^Nj` =t&

�� +;147,

495 Fkdslwuh ; Fdofxo g*lqvwdelolwì

dyhf tf txhofrqtxh srxuyx txh ntfn @ 41 Hq sudwltxh/ od qruph xwlolvìh hvw od qruph

lq�qlh1 Qrwrqv dx sdvvdjh txh oh fdofxo gh ^N�`3� q*hvw sdv qìfhvvdluh = lo vx!w hq idlw

g*lqyhuvhu oh v|vwëph1 Rq vdlw txh

olp&<n"

nt&nnt&n�n @ m�Wm @ 4

m�Wm +;148,

t @ olp&<n"

t& hvw xq yhfwhxu sursuh dvvrflì ã od ydohxu sursuh �W @4

�Wtxl hvw od soxv

judqgh hq prgxoh1 Rq fdofxoh doruv

{� @ ^N�`3�

^Nj` =t +;149,

oh yhfwhxu sursuh dvvrflì ã od ydohxu sursuh ��1

Ohv prghv gh dpehphqw txh o*rq h{wudlw dlqvl vrqw qrupìv dx vhqv gh od qruph lq�qlh1

Lov shxyhqw íwuh uìlqmhfwìv frpph gìirupìh lqlwldoh gh od vwuxfwxuh srxu xq fkdujhphqw

sdv ã sdv/ d�q g*dprufhu o*lqvwdelolwì/ qrq vdqv dyrlu ìwì uìgxlwv hq dpsolwxgh1

;16 Frqfoxvlrq

Od whfkqltxh txh qrxv sursrvrqv d soxvlhxuv dydqwdjhv = hooh shuphw g*h{wudluh ohv ydohxuv

sursuhv hq sdufrxudqw o*d{h ghv ydohxuv sursuhv gh idêrq frqwlqxh ã sduwlu gh 3/ vdqv hq

vdxwhu1 Hq od frxsodqw ã od pìwkrgh gh od sxlvvdqfh lqyhuvh gìfdoìh/ rq rewlhqw ohv yhfwhxuv

sursuhv dvvrflìv1 Ã dxfxq prphqw qrxv qh idlvrqv g*k|srwkëvh vxu od gì�qlh0srvlwlylwì ghv

pdwulfhv pdqlsxoìhv1 Qrxv xwlolvrqv vlpsohphqw oh idlw tx*hoohv vrqw uìhoohv hw v|pìwultxhv/

hw grqf gldjrqdolvdeohv1 Od frqwuh0sduwlh vh sdlh hq whpsv gh fdofxo/ fdu oh qrpeuh gh

idfwrulvdwlrqv hvw lpsruwdqw/ pdlv ohv whpsv gh fdofxo vrqw wrxmrxuv uhvwìv dffhswdeohv/

píph vxu ghv vwuxfwxuhv lpsruwdqwhv1

Fkdslwuh <

Fdofxo g|qdpltxh

<14 Lqwurgxfwlrq

Oh fdofxo g|qdpltxh ghv vwuxfwxuhv hvw oh surorqjhphqw orjltxh gx fdofxo qrq olqìdluh

vwdwltxh1 Oh fdofxo g|qdpltxh shuphw g*dssuìkhqghu fruuhfwhphqw ohv h�hwv gx yhqw hw

ghv vìlvphv vxu ohv vwuxfwxuhv1 Od plvh hq úxyuh g*xq who w|sh gh fdofxo hvw sduidlwhphqw

oìjlwlph gdqv od phvxuh rü ohv vwuxfwxuhv txh o*rq ìwxglh ìwdqw gh soxv hq soxv ìodqfìhv/

hoohv ghylhqqhqw gh idlw gh soxv hq soxv vhqvleohv dx{ h�hwv gx yhqw hw ghv vìlvphv1 Hq

rxwuh/ oh fdofxo ghv vwuxfwxuhv hq }rqh vlvpltxh vhpeoh íwuh xqh suìrffxsdwlrq iruwh gh od

sduw ghv dfwhxuv gx Jìqlh flylo1

Gdqv xq suhplhu whpsv/ qrxv suìvhqwhurqv vxfflqfwhphqw oh fdofxo g|qdpltxh olqìdluh/

sxlv qrxv qrxv lqwìuhvvhurqv dx fdofxo g|qdpltxh whpsruho/ txl frqvlvwh ã lqwìjuhu sdv gh

whpsv sdu sdv gh whpsv/ oh prxyhphqw gh od vwuxfwxuh1

<15 Fdofxo g|qdpltxh olqìdluh

Xqh vwuxfwxuh grqw oh frpsruwhphqw hvw ìodvwltxh olqìdluh/ prgìolvìh ã o*dlgh gh eduuhv hw

frpsruwdqw xqh pdvvh dvvrflìh ã fkdfxq ghv qúxgv/ hvw hq idlw xq rvfloodwhxu pxowlsoh/

f*hvw0ã0gluh xq hqvhpeoh gh pdvvhv uholìhv hqwuh hoohv hw ã xqh edvh �{h sdu ghv uhvvruwv hw

ghv dpruwlvvhxuv1

<1514 Ìtxdwlrq gx prxyhphqw

Gdqv oh uhsëuh joredo/ od vwuxfwxuh hvw fkdujìh dx{ qúxgv sdu oh yhfwhxuS +w,/ txl gìvrupdlv

hvw xqh fkdujh gìshqgdqw gx whpsv1 Srxu xq gìsodfhphqw grqqì V} ghv qúxgv/ ohv pdvvhv

dvvrflìhv ã fkdfxq ghv qúxgv vxelvvhqw ohv h�ruwv vxlydqwv =

497 Fkdslwuh < Fdofxo g|qdpltxh

� ohv irufhv ìodvwltxhv gh udssho Io/ txl fruuhvsrqghqw ã o*rssrvì ghv irufhv lqwhuqhv

gh od vwuxfwxuh I}/ txl shxyhqw v*ìfuluh gdqv oh fdv olqìdluh =

Io @ �I} @ � ^N` =V} +<14,

rü ^N` hvw od pdwulfh gh uljlglwì gh od vwuxfwxuh1

� ohv irufhv g*dpruwlvvhphqw I@/ txl vrqw rssrvìhv hw sursruwlrqqhoohv ã od ylwhvvh ghv

pdvvhv =

I@ @ � ^F` = bV} +<15,

rü bV} hvw od gìulyìh sdu udssruw dx whpsv w gh V} =

bV} @gV}

gw+<16,

bV} hvw hq idlw oh yhfwhxu ylwhvvh ghv qúxgv1

� ohv irufhv h{wìulhxuhv S +w,1

Oh sulqflsh irqgdphqwdo gh od g|qdpltxh fodvvltxh v*ìfulw doruv =

Io . I@ . S +w, @ ^P ` = �V} +<17,

rü ^P ` hvw od pdwulfh ghv pdvvhv gh od vwuxfwxuhv hw �V} od gìulyìh vhfrqgh gh V} sdu udssruw

dx whpsv w =

�V} @g2V}

gw2+<18,

�V} hvw hq idlw oh yhfwhxu dffìoìudwlrq ghv qúxgv1 Gh idêrq soxv fodvvltxh/ fhwwh ìtxdwlrq

vh phw vrxv od iruph vxlydqwh =

^P ` = �V} . ^F` = bV} . ^N` =V} @ S +w, +<19,

<1515 Rvfloodwlrqv oleuhv

Od suhplëuh ìwdsh gx fdofxo g|qdpltxh olqìdluh frqvlvwh ã gìwhuplqhu ohv prghv sursuhv

gh od vwuxfwxuh dlqvl txh ohv iuìtxhqfhv sursuhv1 Srxu fhod/ rq ìolplqh gh o*ìtxdwlrq +<19,

ohv irufhv g*dpruwlvvhphqw dlqvl txh o*h{flwdwlrq h{wìulhxuh S +w,/ hw o*rq rewlhqw o*ìtxdwlrq

uìjlvvdqw ohv rvfloodwlrqv oleuhv gh od vwuxfwxuh =

^P ` = �V} . ^N` =V} @ 3 +<1:,

Xqh vroxwlrq sduwlfxolëuh gh +<1:, hvw grqqìh sdu =

V} @ dG vlq +$w. *, +<1;,

<15 Fdofxo g|qdpltxh olqìdluh 498

rü d/ $ hw * vrqw ghv frqvwdqwhv/ hw G xq yhfwhxu frqvwdqw grqw ohv frrugrqqìhv uhsuìvhq0

whqw ohv gìsodfhphqwv ghv qúxgv1 Oh yhfwhxu dffìoìudwlrq vh phw vrxv od iruph vxlydqwh =

�V} @ �$2dG vlq +$w.*, +<1<,

O*ìtxdwlrq +<1:, v*ìfulw doruv =

�^N`� $2 ^P `

�=G d vlq +$w. *, @ 3 +<143,

Srxu txh fhwwh uhodwlrq vrlw yìul�ìh ã wrxw lqvwdqw w/ lo idxw txh =

�^N`� $2 ^P `

�=G @ 3 +<144,

Rq uhwurxyh oh sureoëph dx ydohxuv sursuhv/ gìmã uhqfrqwuì oruv gx fdofxo g*lqvwdelolwì1

Frpph ^N` hw ^P ` vrqw gì�qlhv srvlwlyhv/ rq shxw xwlolvhu ghv pìwkrghv fodvvltxhv dvvh}

sxlvvdqwhv frpph od pìwkrgh g*lwìudwlrq ghv vrxv0hvsdfhv1 Od uìvroxwlrq gx sureoëph

shuphw g*rewhqlu ohv q ydohxuv ghv sxovdwlrqv $�/ $2/111/ $�/111/ $?/ dlqvl txh ohv prghv

sursuhv qrupdolvìv gh od vwuxfwxuh G�/ G2/111/ G�/111/ G?1

Gdqv oh fdv jìqìudo/ od vroxwlrq hvw rewhqxh hq vxshusrvdqw ohv prxyhphqwv vhorq ohv

q prghv gh yleudwlrq =

V} +w, @

?[�'�

d�G� vlq +$�w. *�, +<145,

Ohv frh!flhqwv d� hw *� vrqw gìwhuplqìv sdu ohv frqglwlrqv dx{ olplwhv1

<1516 Vxshusrvlwlrq prgdoh

Gdqv oh fdguh g*xqh ìwxgh vlvpltxh ã o*dlgh g*xq vshfwuh gh uìsrqvh/ srxu fkdtxh prgh/

rq gìwhuplqh ohv svhxgr0dffìoìudwlrqv ã dssoltxhu dx{ pdvvhv/ hw grqf ohv irufhv vwdwltxhv

ìtxlydohqwhv1 Vrxv o*dfwlrq gh fhv irufhv vwdwltxhv ìtxlydohqwhv/ od vwuxfwxuh vxelw ghv

gìsodfhphqwv hw gìyhorssh ghv h�ruwv lqwhuqhv1 Dsuëv dyrlu vìohfwlrqqì ohv Q prghv hq0

jhqgudqw ghv h�ruwv qrwdeohv gdqv od vwuxfwxuh/ rq surfëgh doruv ã xqh vxshusrvlwlrq ghv

uìsrqvhv prgdohv gh od vwuxfwxuh1 Srxu vlpsol�hu o*ìqrqfì/ rq vxssrvhud txh od yìul�fdwlrq

txh o*rq pëqh hvw h�hfwxìh g*dsuëv od ydohxu g*xq vhxo sdudpëwuh1

Dsshorqv s� od ydohxu pd{lpdoh txh suhqg xq sdudpëwuh s srxu oh prgh gh yleudwlrq

l1 Oruvtxh ohv prghv gh yleudwlrqv vrqw lqgìshqgdqwv/ od ydohxu pd{lpdoh s gx sdudpëwuh

s hvw grqqìh sdu od pr|hqqh txdgudwltxh =

s @

yxxw �[�'�

s2� +<146,

Dlqvl/ rq suhqg hq frpswh od qrq frqfrplwdqfh ghv uìsrqvhv prgdohv pd{lpdohv1

499 Fkdslwuh < Fdofxo g|qdpltxh

Sdu frqwuh/ vl sdu h{hpsoh ohv prghv 4 hw 5 rqw ghv iuìtxhqfhv yrlvlqhv ã prlqv gh 43(

suëv/ rq qh shxw sdv frqvlgìuhu tx*lov vrqw lqgìshqgdqwv/ hw o*rq d =

s @

yxxw+ms�m. ms2m,2 .

�[�'�

s2� +<147,

Hq idlw/ rq uhjurxsh ohv 5 prghv surfkhv hq xq vhxo1

Oh fdofxo g|qdpltxh olqìdluh shuphw dlqvl g*h�hfwxhu od soxsduw ghv yìul�fdwlrqv uìjoh0

phqwdluhv hq yljxhxu/ txh fh vrlw gdqv oh grpdlqh gx yhqw rx gx vìlvph1

<16 Fdofxo g|qdpltxh qrq olqìdluh

Gdqv oh fdguh gx fdofxo qrq olqìdluh ghv vwuxfwxuhv/ o*ìtxdwlrq gx prxyhphqw hvw vhqvleoh0

phqw od píph txh gdqv oh fdguh olqìdluh/ vdxi txh o*rq shug od olqìdulwì gdqv od uhodwlrq

h�ruwv0gìsodfhphqwv1

<1614 Ìtxdwlrq gx prxyhphqw

Ã o*lqvwdqw w/ o*ìtxdwlrq gx prxyhphqw v*ìfulw =

^P ` = �V} . ^F` = bV} . I} +w, @ S +w, +<148,

Oh yhfwhxu ghv h�ruwv lqwhuqhv I} +w, gìshqg gx whpsv sdu oh eldlv gx gìsodfhphqw V} +w,

gh od píph idêrq tx*hq vwdwltxh1 Srxu uìvrxguh +<148,/ rq shxw xwlolvhu ghv pìwkrghv g*lq0

wìjudwlrq gluhfwh ghv v|vwëphv gx vhfrqg ruguh1 Rq shxw flwhu od pìwkrgh ghv gl�ìuhqfhv

�qlhv fhqwudohv +pìwkrgh h{solflwh,/ od pìwkrgh gh Krxerow +lpsolflwh,/ od pìwkrgh gh Qhz0

pdun +lpsolflwh,/ dlqvl txh fhooh gh Zlovrq +lpsolflwh, ^Gkd;7`1 Qrwrqv dx sdvvdjh txh od

pìwkrgh gh Qhzpdun hvw xq fdv sduwlfxolhu gh fhooh gh Zlovrq1 Qrxv dyrqv gìyhorssì od

pìwkrgh g*lqwìjudwlrq gh Qhzpdun fdu hooh suìvhqwdlw gh qrpeuhx{ dydqwdjhv/ dxvvl elhq

gx srlqw gh yxh wkìrultxh txh gx srlqw gh yxh gh od plvh hq úxyuh1

<1615 Od pìwkrgh gh Qhzpdun = dssurfkh lqfuìphqwdoh

Rq yd srxyrlu frqvwuxluh od vroxwlrq gx sureoëph ã o*lqvwdqw w.�w ã sduwlu ghv yhfwhxuv

V} +w,/ bV} +w, hw �V} +w, frqqxv ã o*lqvwdqw w1 Fhwwh pìwkrgh uhsrvh vxu ohv gìyhorsshphqwv

olplwìv vxlydqwv =

bV} +w.�w, @ bV} +w, .�w�+4� d, �V} +w, . d �V} +w.�w,

�+<149,

V} +w.�w, @ V} +w, .�w bV} +w, .�w2

5

�+4� e, �V} +w, . e �V} +w.�w,

�+<14:,

<16 Fdofxo g|qdpltxh qrq olqìdluh 49:

Oruvtxh d @ e @4

5/ fhv dssur{lpdwlrqv frqvlvwhqw ã vxssrvhu o*dffìoìudwlrq frqvwdqwh

gdqv o*lqwhuydooh ^w> w.�w` hw ìjdoh ã vd ydohxu pr|hqqh1 Oruvtxh d @4

5hw e @

4

6/ fhv

dssur{lpdwlrqv frqvlvwhqw ã vxssrvhu xqh dffìoìudwlrq yduldqw olqìdluhphqw vxu o*lqwhuydooh

^w> w.�w`1

Hq srvdqw =

�V} +w, @ V} +w.�w,� V} +w, +<14;,

� bV} +w, @ bV} +w.�w,� bV} +w, +<14<,

� �V} +w, @ �V} +w.�w,� �V} +w, +<153,

ohv ìtxdwlrqv +<149, hw +<14:, v*ìfulyhqw =

� bV} +w, @ �w �V} +w, . d�w� �V} +w, +<154,

�V} +w, @ �w bV} +w, .�w2

5

�V} +w, .�w2

5e� �V} +w, +<155,

Hq lqyhuvdqw fhv ghx{ uhodwlrqv/ rq rewlhqw =

e�w2

5� �V} +w, @

��V} +w,��w bV} +w,� �w2

5

�V} +w,

�+<156,

e�w2

5� bV} +w, @

e� d

5�w� �V} +w,� d�w2 bV} +w, . d�w�V} +w, +<157,

Ã o*lqvwdqw w/ o*ìtxdwlrq gx prxyhphqw gh od vwuxfwxuh hvw =

^P ` = �V} +w, . ^F` = bV} +w, . I} +w, @ S +w, +<158,

Ã o*lqvwdqw w.�w/ hooh v*ìfulw =

^P ` = �V} +w.�w, . ^F` = bV} +w.�w, . I} +w.�w, @ S +w.�w, +<159,

Hq qrwdqw �S +w, o*dffurlvvhphqw gh fkdujh h{wìulhxuh dssoltxìh ã od vwuxfwxuh gdqv o*lq0

whuydooh ^w> w.�w` =

�S +w, @ S +w.�w,� S +w, +<15:,

hw hq idlvdqw od gl�ìuhqfh hqwuh ohv ìtxdwlrqv +<159, hw +<158,/ rq rewlhqw =

^P ` =� �V} +w, . ^F` =� bV} +w, . I} +w.�w,� I} +w, @ �S +w, +<15;,

D�q gh frqvwuxluh xq rsìudwhxu gh frqyhujhqfh/ rq olqìdulvh o*h{suhvvlrq I} +w.�w, �I} +w, hq od uhpsodêdqw sdu ^N}` =�V} +w,/ rü ^N}` hvw od pdwulfh gh uljlglwì wdqjhqwh gh od

vwuxfwxuh/ hw o*rq rewlhqw =

^P ` =� �V} +w, . ^F` =� bV} +w, . ^N}` =�V} +w, @ �S +w, +<15<,

49; Fkdslwuh < Fdofxo g|qdpltxh

Rq uhwurxyh oh píph vfkìpd gh uìvroxwlrq txh gdqv oh fkdslwuh 81 Rq frqvwuxlw xqh vxlwh

gh vroxwlrqv �V}� +w, gh od idêrq vxlydqwh = �V}�n� +w, hvw vroxwlrq gh

^P ` =� �V} +w, . ^F` =� bV} +w, . ^N}� ` =�V}�n� +w, @ �S +w, +<163,

Oruvtxh od vxlwh �V}� +w, d frqyhujì/ rq fdofxoh oh uìvlgx gh od idêrq vxlydqwh =

u�hvlgx @ S +w.�w,� ^P ` = �V} +w.�w,� ^F` = bV} +w.�w,� I} +w.�w, +<164,

hw o*rq uìsëwh oh f|foh mxvtx*ã fh txh oh uìvlgx vrlw idleoh hq qruph1

<1616 Plvh hq úxyuh

Ohv ìtxdwlrqv +<163, hw +<164, qh vrqw sdv h{sorlwdeohv gluhfwhphqw1 Hq lqmhfwdqw +<157, hw

+<156, gdqv +<163,/ rq rewlhqw =�^P ` . d�w ^F` .

e�w2

5^N}`

��V} +w, @

e�w2

5�S +w,

. ^P ` =

��w bV} +w, .

�w2

5

�V} +w,

�+<165,

.^F` =

�d� e

5�w� �V} +w, . d�w2 bV} +w,

�

Dlqvl/ o*rsìudwhxu wdqjhqw qh vhud sdv ^N}` frpph hq vwdwltxh pdlv ^N}` gì�ql sdu =

^N}` @ ^P ` . d�w ^F` .e�w2

5^N}` +<166,

Oh uìvlgx q*hvw sdv lqlwldolvì sdu �S pdlv sdu =

e�w2

5�S +w,.^P ` =

��w bV} +w, .

�w2

5

�V} +w,

�.^F` =

�d� e

5�w� �V} +w, . d�w2 bV} +w,

�+<167,

Hq xwlolvdqw ìjdohphqw +<157, hw +<156, gdqv +<164,/ lo ylhqw =

e�w2

5u�hvlgx @

e�w2

5+S +w.�w,� I} +w.�w,,

� ^^P ` . d�w ^F`` =�V} +w,

. ^P ` =

��w bV} +w, . +4� e,

�w2

5

�V} +w,

�

.^F` =

�+5d� e,

�w2

5

bV} +w, .d� e

5�w� �V} +w,

�+<168,

Hq fh txl frqfhuqh ohv frqglwlrqv lqlwldohv/ qrxv vxssrvrqv gh idêrq lpsolflwh txh od vwuxf0

wxuh hvw dx gìexw hq ìtxloleuh vrxv xq fkdujhphqw vwdwltxh qrwì Sr|@|�^�e/ dyhf oh gìsodfh0

phqw V}r|@|�^�e fruuhvsrqgdqw1 Hooh hvw hqvxlwh h{flwìh sdu xq fkdujhphqw h{wìulhxu dxwrxu

gh fhwwh srvlwlrq g*ìtxloleuh vwdwltxh1 Qrxv dyrqv grqf =

V} +3, @ V}r|@|�^�e +<169,

bV} +3, @ 3 +<16:,

<17 Frqfoxvlrq 49<

Dlqvl/ hq xwlolvdqw +<148,/ rq fdofxoh o*dffìoìudwlrq lqlwldoh �V} +3,/ txl hvw gìgxlwh gh od

uìvroxwlrq gh =

^P ` = �V} +3, @ S +3,� Sr|@|�^�e +<16;,

<17 Frqfoxvlrq

Ohv uhfrppdqgdwlrqv gh Qhzpdun srxu oh fkrl{ ghv sdudpëwuhv gh od pìwkrgh vrqw ohv

vxlydqwv =

d @4

5srxu ìolplqhu o*dpruwlvvhphqw dojrulwkpltxh +<16<,

e 5�4

6>4

5

�+<173,

�w

W�5

�4

9>4

8

�+<174,

rü W� hvw od sìulrgh gx prgh gìgxlw gx fdofxo hq rvfloodwlrqv oleuhv txl srvvëgh od sxovdwlrq

sursuh od soxv ìohyìh1

Frpph rq shxw oh frqvwdwhu/ rq uhwurxyh h{dfwhphqw od píph orjltxh hw oh píph

vfkìpd gh uìvroxwlrq txh vrxv fkdujhphqw vwdwltxh1 F*hvw grqf oh píph dojrulwkph gh

uìvroxwlrq txl hvw hpsor|ì1 Hq sudwltxh/ fhod vljql�h txh oh frgh gh fdofxo fruuhvsrqgdqw ã

xqh uìvroxwlrq vwdwltxh d vxelw ghv wudqvirupdwlrqv vrpph wrxwh plqhxuhv d�q gh srxyrlu

phqhu xq fdofxo g|qdpltxh dyhf lqwìjudwlrq whpsruhooh sdv ã sdv1 Od sulvh hq frpswh ghv

qrq0olqìdulwìv pdwìulhooh hw jìrpìwultxh hvw dlqvl wrwdoh1

Od vhxoh lqfrqqxh hq fh txl qrxv frqfhuqh hvw o*ìydoxdwlrq gh o*dpruwlvvhphqw1 Qrxv

sursrvrqv grqf ã o*xwlolvdwhxu gh gì�qlu od pdwulfh g*dpruwlvvhphqw ^F` ã o*dlgh gh ghx{

frh!flhqwv �� hw �2 whov txh =

^F` @ �� ^P ` . �2 ^N}` +<175,

4:3 Fkdslwuh < Fdofxo g|qdpltxh

Fkdslwuh 43

H{hpsohv g*dssolfdwlrqv

qxpìultxhv

4314 Lqwurgxfwlrq

Gdqv fh fkdslwuh/ qrxv doorqv suìvhqwhu xq fhuwdlq qrpeuh g*h{hpsohv txl rqw idlw o*re0

mhw g*ìwxghv h{sìulphqwdohv hw2rx wkìrultxhv1 Qrxv frqiurqwrqv oh uìvxowdw gh qrv fdofxov

h�hfwxìv ã sduwlu gh fh txl suìfëgh/ dx{ uìvxowdwv h{sìulphqwdx{1 Qrxv wudlwhurqv gli0

iìuhqwv w|shv gh vwuxfwxuhv1 Qrxv frpphqfhurqv sdu ìwxglhu ghv vwuxfwxuhv hq eìwrq dupì/

sxlv qrxv ìwxglhurqv ghv vwuxfwxuhv frpsrvìhv gh eìwrq dupì hw suìfrqwudlqw/ dlqvl txh

ghv vwuxfwxuhv pìwdooltxhv1 Qrxv phwwurqv hq úxyuh od qrq0olqìdulwì pdwìulhooh frxsoìh

ìyhqwxhoohphqw dyhf od qrq0olqìdulwì jìrpìwultxh/ hw ìyhqwxhoohphqw od ylvfr0ìodvwlflwì1

4315 Vwuxfwxuhv hq eìwrq dupì

431514 Srxwuh frqwlqxh

Suìvhqwdwlrq gx sureoëph

Oh suhplhu h{hpsoh hvw wluì g*xqh vìulh gh whvwv h�hfwxìv sdu Pdwwrfn ^Pdw8<` vxu ghv

srxwuhv frqwlqxhv k|shuvwdwltxhv hq eìwrq dupì1 Hooh d idlw o*remhw g*ìwxghv qxpìultxhv sdu

Juhodw ^Juh:;e` hw Qdlw0Udedk ^Qdl<3`1 Od srxwuh txh o*rq vh sursrvh g*ìwxglhu frpsruwh

ghx{ wudyìhv ìjdohv gh 45 slhgv/ vrlw 4/;5< p1 Xqh fkdujh srqfwxhooh qrwìhS hvw dssoltxìh

dx plolhx gh od suhplëuh wudyìh/ frpph lqgltxì vxu od �jxuh 4314/ mxvtx*ã od uxlqh frpsoëwh1

Od vhfwlrq wudqvyhuvdoh gh od srxwuh hvw uhfwdqjxodluh/ hw idlw < srxfhv sdu 7 srxfhv/

vrlw 55/;9 fp sdu 43/49 fp1 Od srxwuh ìwxglìh fruuhvsrqg ã od srxwuh qxpìur 6 gh od

suhplëuh vìulh/ hw suìvhqwh od sduwlfxodulwì gh srvvìghu hq E hw hq F oh píph srxufhqwdjh

g*dupdwxuhv sdvvlyhv whqgxhv = 43 dupdwxuhv gh 3/58 srxfh gh gldpëwuh/ vrlw 6/49: fp2

4:5 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv

BC

P

A D

1,829m 1,829m 10,16cm

22,86cm

B C

Iljxuh 4314 = Srxwuh gh Pdwwrfn

g*dflhu/ fh txl fruuhvsrqg ã 4/69( gh od vhfwlrq wrwdoh1 Fh sureoëph qh idlw lqwhuyhqlu txh

od qrq0olqìdulwì gx frpsruwhphqw gx pdwìuldx1

Prgìolvdwlrq gx sureoëph

Od uìvlvwdqfh hq frpsuhvvlrq gx eìwrq phvxuìh vxu ghv ìsurxyhwwhv fxeltxhv hvw gh

5</7 PSd1 Gdqv od prgìolvdwlrq sursrvìh/ od uìvlvwdqfh hq frpsuhvvlrq sulvh hq frpswh

hvw gh 3/;8 irlv od uìvlvwdqfh phvxuìh vxu ghv fxehv1 Oh prgxoh g*\rxqj gx eìwrq ã o*r0

uljlqh hvw ìydoxì ã 58 JSd1 Od orl gh frpsruwhphqw gx eìwrq hvw xqh orl gh w|sh Vdujlq

dyhf sulvh hq frpswh gx eìwrq whqgx +whqvlrq0vwl�hqlqj, vhorq od pìwkrgh gh Juhodw1 Ohv

frrugrqqìhv gx slf gh frpsuhvvlrq vrqw ohv vxlydqwhv =

%Kf @ 3/3357 +4314,

iS� @ 3/;8� 5</7 @ 57/<< PSd +4315,

Od uìvlvwdqfh hq wudfwlrq gx eìwrq hvw sulvh ã 43( gh vd uìvlvwdqfh hq frpsuhvvlrq fruuljìh1

Hq fh txl frqfhuqh o*dflhu/ oh prgxoh g*\rxqj ã o*ruljlqh hvw Hr @ 533 JSd/ od olplwh

ìodvwltxh hvw ie @ 733 PSd/ hw od olplwh gh uxswxuh hvw ìydoxìh ã 868 PSd1 Od orl gh

frpsruwhphqw xwlolvìh hvw xqh orl elolqìdluh dyhf ud�huplvvhphqw1

Uìvxowdwv

Ohv srxwuhv vrqw frqêxhv srxu xqh fkdujh gh vhuylfh qrwìh Sr ìjdoh ã 56 nQ1 Od fkdujh gh

uxswxuh phvxuìh h{sìulphqwdohphqw v*ìoëyh ã :7/54 nQ1 Oh wdeohdx 4314 grqqh od ydohxu ghv

prphqwv vrxv od fkdujh gh vhuylfh Sr vhorq xq fdofxo olqìdluh/ vhorq oh fdofxo qrq olqìdluh

sursrvì/ hw vhorq o*h{sìulhqfh1

4315 Vwuxfwxuhv hq eìwrq dupì 4:6

Fdofxo olqìdluh Fdof1 qrq olq1 H{sìulhqfh Fdof1qol2H{s1

Prphqw hq E ;/85 nQ1p ;/4; nQ1p ;/5< nQ1p 3/<<

Prphqw hq F 6/<< nQ1p 7/9; nQ1p 7/:6 nQ1p 3/<<

Wdeohdx 4314 = Prphqwv hq vhuylfh gdqv ohv vhfwlrqv E hw F

Vrxv fkdujh gh vhuylfh/ rq frqvwdwh od erqqh fruuìodwlrq hqwuh oh fdofxo qrq olqìdluh hw

o*h{sìulhqfh1 O*ìfduw hqwuh oh fdofxo olqìdluh hw o*h{sìulhqfh hvw gìmã vhqvleoh/ vxuwrxw hq fh

txl frqfhuqh oh prphqw ìfklvvdqw hq F txl hvw vrxv0hvwlpì gh 49(1

O*ìyroxwlrq ghv prphqwv hq E hw hq F hvw uhsuìvhqwìh vxu od �jxuh 43151 Rq | d

uhsuìvhqwì ohv uìvxowdwv h{sìulphqwdx{ dlqvl txh fhx{ lvvxv gh qrwuh fdofxo1

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70 80

Charge P (kN)

Moments (kN.m)

Mb Expérience

Mb calculé

Mc Expérience

Mc calculé

Ps

Iljxuh 4315 = Ìyroxwlrq ghv prphqwv hq E hw F hq irqfwlrq gh od fkdujh

Rq frqvwdwh xq erq dffrug hqwuh oh fdofxo hw o*h{sìulhqfh/ hw o*ìydoxdwlrq gh od fkdujh

gh uxlqh fdofxoìh hvw gh :6/5; nQ srxu xqh fkdujh gh uxlqh uìhooh gh :7/54 nQ/ vrlw xq ìfduw

gh 4( vhxohphqw/ gdqv oh vhqv gh od vìfxulwì1

Oh wdeohdx 4315 grqqh ohv ydohxuv ghv prphqwv hq E hw hq F ã od uxswxuh1

Oh fdofxo vxuhvwlph oh prphqw hq wudyìh gh 7( hw vrxv0hvwlph oh prphqw vxu dssxl

gh 44(1 Vxu od �jxuh 4315/ oh fkdqjhphqw gh shqwh vxu ohv gldjudpphv ghv prphqwv dx{


Fdofxo olqìdluh Fdof1 qrq olq1 H{sìulhqfh Fdof1qol2H{s1

Prphqw hq E 5:/7< nQ1p 57/55 nQ1p 56/5; nQ1p 4/37

Prphqw hq F 45/;: nQ1p 4;/93 nQ1p 53/<4 nQ1p 3/;<

Wdeohdx 4315 = Prphqwv xowlphv gdqv ohv vhfwlrqv E hw F

dohqwrxuv gh od fkdujh S @ 93 nQ v*h{soltxh sdu oh irqfwlrqqhphqw hq urwxoh sodvwltxh gh

od vhfwlrq E1 Hqwuh S @ 93 nQ hw S @ :7/54 nQ/ lo v*rsëuh xqh uhglvwulexwlrq ghv prphqwv

sdu urwxoh sodvwltxh hq E/ txh oh surjudpph wudgxlw uhodwlyhphqw elhq juåfh ã o*xwlolvdwlrq

gh o*duwl�fh gh Fduqhlur/ txl frqvlvwh ã prgl�hu orfdohphqw od urwdwlrq gh od vhfwlrq/ hq

dmrxwdqw dx whuph gìgxlw gh o*lqwìjudwlrq ghv frxuexuhv xq whuph orfdo sursruwlrqqho ã od

frxuexuh gdqv od vhfwlrq od soxv vroolflwìh/ frqvhuydqw dlqvl oh sulqflsh gh fdofxo ã sduwlu ghv

orlv frqwudlqwhv0gìirupdwlrqv1 Oh fdofxo olqìdluh hvw sdu dloohxuv wuëv ìorljqì gh od uìdolwì =

lo vxuhvwlph gh 4;( oh prphqw hq E hw vrxv0hvwlph gh 6;( oh prphqw hq F1

431515 Iodpehphqw g*xq srwhdx


Oh dpehphqw gx srwhdx txh qrxv doorqv ìwxglhu hvw wluì g*xqh vìulh lpsruwdqwh g*h{0

sìulhqfhv uìdolvìhv sdu Irxuì ^Irx:;`1 Hooh d idlw o*remhw g*ìwxghv qxpìultxhv sdu Xop

^Xop<7`1 Oh sureoëph txh qrxv ìwxglrqv hvw xq dpehphqw lqvwdqwdqì/ f*hvw0ã0gluh txh

oh fkdujhphqw hvw gh frxuwh gxuìh +53 plqxwhv hqylurq,/ gh vruwh txh oh uhwudlw hw oh xdjh

gx eìwrq q*lqwhuylhqqhqw sdv1 Oh srwhdx frqvlgìuì hvw oh srwhdx LL041

Oh srwhdx h{sìulphqwdo hvw gh vhfwlrq uhfwdqjxodluh/ hw gh kdxwhxu 7/83 p1 Lo hvw

elduwlfxoì ã vhv h{wuìplwìv/ hw hvw fkdujì dyhf xqh irufh d{ldoh h{fhqwuìh gh 4/8 fp1 Oh

sureoëph vh udpëqh ã ìwxglhu xq srwhdx gh 5/58 p gh kdxwhxu/ hqfdvwuì ã vd edvh hw

fkdujì gh od píph idêrq/ frpph lqgltxì vxu od �jxuh 43161 Fhw h{hpsoh phw hq úxyuh od

qrq0olqìdulwì pdwìulhooh g*xqh sduw/ hw vxuwrxw od qrq0olqìdulwì jìrpìwultxh g*dxwuh sduw1

Od vhfwlrq uhfwdqjxodluh gx srwhdx d xqh edvh gh 53 fp srxu xqh kdxwhxu gh 48 fp1

Hooh hvw dupìh sdu 7 dupdwxuhv hq dflhu gh 45 pp gh gldpëwuh1


Oh srwhdx hvw glvfuìwlvì hq 43 ìoìphqwv1 Oh prgxoh g*\rxqj gx eìwrq ã o*ruljlqh hvw ìydoxì

ã 67 JSd1 Od orl gh frpsruwhphqw gx eìwrq hvw xqh orl gh w|sh Vdujlq/ dyhf n @ 5/ hw

vdqv sulvh hq frpswh gx eìwrq whqgx1 Ohv frrugrqqìhv gx slf gh frpsuhvvlrq vrqw ohv

vxlydqwhv =

%Kf @ 3/3356 +4316,

iS� @ 6</3 PSd +4317,

4315 Vwuxfwxuhv hq eìwrq dupì 4:8

P

e=1,5cm

1 5 0 m m

2,25m

1 5 0 m m

2 0 0 m m

2 0 m m

1 5 m m

4 H A 1 2

Iljxuh 4316 = Srwhdx gh Irxuì = jìrpìwulh hw fkdujhphqw

Od uìvlvwdqfh hq wudfwlrq gx eìwrq hvw qìjoljìh1 Fhfl hvw oìjlwlph gx idlw gx idleoh qrpeuh

g*dupdwxuhv sdvvlyhv gdqv od vhfwlrq = oh whqvlrq0vwl�hqlqj hvw qìjoljhdeoh1 O*dflhu d xq

prgxoh g*\rxqj ã o*ruljlqh gh 533 JSd/ vd olplwh ìodvwltxh ìwdqw ìydoxìh ã 7:7 PSd1 Rq

vxssrvh od orl gh frpsruwhphqw gh o*dflhu ìodvwr0sodvwltxh sduidlwh1

Uìvxowdwv

Od frxueh fkdujh0 ëfkh hvw wudfìh vxu od �jxuh 4317/ rü o*rq d uhsuìvhqwì ìjdohphqw od

frxueh prgìolvdqw od qrq0olqìdulwì jìrpìwultxh vhxoh1 Vxu fh gldjudpph/ rq d sruwì od

frxueh uhsuìvhqwdqw oh frpsruwhphqw gh od vwuxfwxuh gdqv o*k|srwkëvh g*xq frpsruwhphqw

olqìdluh ghv pdwìuldx{1 Od iruph frxueìh hvw gxh dx{ vhxov h�hwv gx vhfrqg ruguh/ hw oh

idleoh ìfduw hqwuh fhwwh frxueh hw od frxueh h{sìulphqwdoh prqwuh ohxu lpsruwdqfh gdqv

fh w|sh g*h{sìulhqfh1 Vl o*rq frxsoh oh fdofxo qrq olqìdluh jìrpìwultxh dyhf xq frpsruwh0

phqw qrq olqìdluh ghv pdwìuldx{/ rq frqvwdwh txh od frxueh fdofxoìh hvw wuëv surfkh gh

o*h{sìulhqfh1

Oh wdeohdx 4316 prqwuh txh od fkdujh xowlph hvw vxuhvwlpìh gh 6(1 Qrwrqv txh od


0

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0 5 10 15 20 25 30 35 40

Flèche (mm)

Charge P (kN)

NL géométrique

Expérience

Calcul

Ulm

Iljxuh 4317 = Frxueh fkdujh 0 ëfkh

Fdof1 qrq olq1 H{sìulhqfh Fdof1qol2H{s1

Fkdujh xowlph 79:/33 nQ 786 nQ 4/36

Ioëfkh xowlph 59/6 pp 59/5 pp 4/337

Wdeohdx 4316 = Fkdujh hw ëfkh xowlphv

uxswxuh hvw revhuyìh qrq sdv ã fdxvh gh od uxswxuh gh od vhfwlrq g*hqfdvwuhphqw +g*dloohxuv/

ohv dflhuv vrqw hqfruh gdqv ohxu gldjudpph olqìdluh,/ pdlv sdufh txh od pdwulfh gh uljlglwì

gh od vwuxfwxuh d xqh ydohxu sursuh qìjdwlyh = od vwuxfwxuh hvw lqvwdeoh jìrpìwultxhphqw/

hooh dpeh1 Od ydohxu gh od ëfkh xowlph hvw wuëv surfkh gh od ydohxu h{sìulphqwdoh1

431516 Gìyhuvhphqw gh srwhdx{ hq L


Qrxv doorqv ìwxglhu oh gìyhuvhphqw gh wurlv srwhdx{ hq L/ grqw ohv uìvxowdwv h{sìulphqwdx{

vrqw lvvxv g*xqh vìulh gh 47 hvvdlv h�hfwxìv sdu Ehod}rxjkl ^Eho:5`1 Ohv srwhdx{ vrqw

gìqrppìv GZ4 ã GZ471 Ohv wurlv srwhdx{ ìwxglìv vrqw GZ</ GZ44 hw GZ461 Od

vhfwlrq ghv srwhdx{ hvw lghqwltxh/ hw hvw uhsuìvhqwìh hq �jxuh 43181 Ohv srwhdx{ vrqw

4315 Vwuxfwxuhv hq eìwrq dupì 4::

Srwhdx GZ< Srwhdx GZ44 Srwhdx GZ46

h| 7 pp ; pp 45 pp

h} 0:8 pp 0:8 pp 0:8 pp

Wdeohdx 4317 = H{fhqwuhphqw gx fkdujhphqw

duwlfxoìv ã ohxuv h{wuìplwìv/ od orqjxhxu hqwuh duwlfxodwlrq ìwdqw gh 7/93 p1

31cm

4,60m

A A

C O U P E

17cm

pente 1/10

5 c m

5 c m

15cm5 c m

G

z

y

A A

Iljxuh 4318 = Srwhdx{ GZ 0 Fr�udjh

Oh ihuudloodjh sdvvli hvw frqvwlwxì gh 6 olwv g*dupdwxuhv gh gldpëwuhv !; hw !43 glvsrvìv

orqjlwxglqdohphqw/ frpph lqgltxì vxu od �jxuh 43191

Oh fkdujhphqw hvw frqvwlwxì g*xqh irufh d{ldoh/ txl hvw h{fhqwuìh sdu udssruw dx fhqwuh

gh judylwì gh od vhfwlrq1 Ohv frpsrvdqwhv gx srlqw g*dssolfdwlrq gh od irufh gdqv oh uhsëuh

gh od vhfwlrq vrqw suìvhqwìhv gdqv oh wdeohdx 43171


Oh eìwrq frqvwlwxwli ghv srwhdx{ d idlw o*remhw gh whvwv vxu ìsurxyhwwhv f|olqgultxhv d�q

g*hq gìwhuplqhu ohv fdudfwìulvwltxhv/ txl vrqw uìvxpìhv gdqv oh wdeohdx 43181 Srxu pìprluh/

4:; Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv

C O U P E

4 barres

2 barres

Etr iers 5 t=15φφ 10

φ 8

A A

Iljxuh 4319 = Srwhdx{ GZ 0 Ihuudloodjh sdvvli

iS� hvw od uìvlvwdqfh hq frpsuhvvlrq gx eìwrq/ i|�hvw vd uìvlvwdqfh hq wudfwlrq hw HKf vrq

prgxoh g*\rxqj ã o*ruljlqh1 Oh frpsruwhphqw gx eìwrq hvw prgìolvì sdu od orl gh Vdujlq

dyhf n @ 51 Dlqvl/ od gìirupdwlrq gx slf %Kf fruuhvsrqgdqw ã iS� vh gìgxlw sdu =

%Kf @ 5iS�

HKf

+4318,

Ohv fdudfwìulvwltxhv ghv dflhuv sdvvliv vrqw grqqìhv srxu fkdtxh srwhdx gdqv oh wdeohdx

43191

�e hvw od olplwh g*ìodvwlflwì frqyhqwlrqqhooh gh o*dflhu hw �o od frqwudlqwh gh uxswxuh1

Oh frpsruwhphqw ghv dflhuv hvw prgìolvì sdu od orl gx EDHO;6/ dyhf xqh frqwudlqwh �,

Srwhdx GZ< Srwhdx GZ44 Srwhdx GZ46

iS� 69/7 PSd 65/; PSd 65/5 PSd

i|� 5/9; PSd 5/;3 PSd 5/<9 PSd

HKf 67/3 JSd 65/3 JSd 5</6 JSd

Wdeohdx 4318 = Fdudfwìulvwltxhv pìfdqltxhv gx eìwrq

4315 Vwuxfwxuhv hq eìwrq dupì 4:<

�e +PSd, �o +PSd,

Olw ã Olw Olw ã Olw ã Olw Olw ã

jdxfkh gh fhqwudo gurlwh gh jdxfkh gh fhqwudo gurlwh gh

o*d{h } o*d{h } o*d{h } o*d{h }

Srwhdx GZ< 76;/5 7<8 777/8 934/5 :54/9 99:/8

Srwhdx GZ44 7;4/8 834 773/: :9:/3 :3:/: 9;7

Srwhdx GZ46 798/3 7<: 5<6/3 :45/4 :4</8 756

Wdeohdx 4319 = Fdudfwìulvwltxhv pìfdqltxhv ghv dflhuv

fruuhvsrqgdqw ã od �q gx gldjudpph olqìdluh grqqìh sdu =

�, @ 3> :�e +4319,

Oh prgxoh g*\rxqj gh o*dflhu hvw hvwlpì ã 533 JSd1

Uìvxowdwv

Ohv uìvxowdwv vrqw grqqìv gdqv ohv �jxuhv 431:/ 431; hw 431< vrxv od iruph gh frxuehv

uhsuìvhqwdqw o*ìyroxwlrq ghv ëfkhv hq irqfwlrq gh od fkdujh1 Od ëfkh i+ hvw qìjdwlyh fdu

od frpsrvdqwh gh o*h{fhqwuhphqw vhorq o*d{h J| hvw srvlwlyh1 Lqyhuvhphqw/ od ëfkh i5 hvw

srvlwlyh fdu od frpsrvdqwh gh o*h{fhqwuhphqw vhorq o*d{h J} hvw qìjdwlyh1 Rq frqvwdwh

xqh erqqh fruuìodwlrq hqwuh ohv frxuehv h{sìulphqwdohv hw ohv frxuehv fdofxoìhv/ hq gìslw

gx idlw txh srxu oh fdofxo/ qrxv vxssrvrqv olqìdluh oh frpsruwhphqw gx pdwìuldx ylv0ã0

ylv ghv vroolflwdwlrqv wdqjhqwhv1 O*lqhuwlh gh wruvlrq/ qrwdpphqw/ grlw uhvwhu vx!vdpphqw

lpsruwdqwh dx frxuv gh fhv h{sìulhqfhv srxu txh oh idlw gh od vxssrvhu frqvwdqwh hq gìslw

gh od �vvxudwlrq gh od vhfwlrq/ qh gìjudgh sdv od txdolwì gx fdofxo1 Rq frqvwdwh ìjdohphqw

txh od ëfkh wudqvyhuvdoh i+ dxjphqwh prlqv ylwh txh od ëfkh i5 dx gìexw gx fkdujhphqw1

Fhfl hvw sduidlwhphqw qrupdo/ fdu od vwuxfwxuh hvw txdvlphqw ìodvwltxh/ hw vhxoh od ydohxu gh

o*h{fhqwuhphqw mrxh1 Gëv txh o*rq dwwhlqw ghv fkdujhphqwv soxv iruwv/ ohv h�hwv gx vhfrqg

ruguh hqwuhqw hq mhx hw od ëfkh i+ ghylhqw píph qhwwhphqw soxv lpsruwdqwh txh od ëfkh

yhuwlfdoh i5 ã o*ìwdw xowlph1

Gdqv oh wdeohdx 431:/ qrxv dyrqv uhsruwì od ydohxu ghv fkdujhv xowlphv srxu fkdtxh

hvvdl1Frpph rq shxw oh frqvwdwhu/ ohv uìvxowdwv vrqw vdwlvidlvdqwv/ pdlv vrqw ìwudqjhphqw

prlqv erqv srxu oh srwhdx GZ44/ dyhf xq ìfduw gh <( sdu udssruw ã od uìdolwì1 Qrxv

shqvrqv txh od ydohxu gx prgxoh g*\rxqj sulvh hq frpswh/ ã vdyrlu 65/3 PSd hvw xq shx

idleoh/ frpph oh frq�uph o*dooxuh gh od frxueh gh od ëfkh i5 txl qh srvvëgh sdv od erqqh

shqwh ã o*ruljlqh/ doruv txh fhwwh shqwh hvw wrxmrxuv wuëv �gëoh ã o*h{sìulhqfh srxu ohv dxwuhv

srwhdx{1

4;3 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv

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Flèches (mm)

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fy Expérience

fy Calcul

fz Expérience

fz Calcul

Iljxuh 431: = Srwhdx GZ< 0 Frxueh fkdujh 0 ëfkh ã pl0kdxwhxu

Fdof1 qrq olq1 H{sìulhqfh Fdof1qol2H{s1

Srwhdx GZ< 646/: nQ 656 nQ 3/<:

Srwhdx GZ44 5:6/5 nQ 633 nQ 3/<4

Srwhdx GZ46 57:/5 nQ 56; nQ 4/37

Wdeohdx 431: = Fkdujhv xowlphv ghv srwhdx{ GZ

4315 Vwuxfwxuhv hq eìwrq dupì 4;4

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Flèches (mm)

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fy Expérience

fy Calcul

fz Expérience

fz Calcul

Iljxuh 431; = Srwhdx GZ44 0 Frxueh fkdujh 0 ëfkh ã pl0kdxwhxu


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fy Expérience

fy Calcul

fz Expérience

fz Calcul

Iljxuh 431< = Srwhdx GZ46 0 Frxueh fkdujh 0 ëfkh ã pl0kdxwhxu


431517 Sruwltxh uhfwdqjxodluh elduwlfxoì


Oh sruwltxh txh qrxv qrxv sursrvrqv g*ìwxglhu idlw sduwlh g*xq surjudpph g*h{sìulphqwd0

wlrqv phqì sdu Fudqvwrq ^Fud98`1 O*ìwxgh frqfhuqdlw kxlw sruwltxhv gh píphv glphqvlrqv/

duwlfxoìv hq slhg1

A B

H

P/2 P/2

1

2

3

4

5 6 7 8 9 10 11 12 13 14 15 16

17

18

19

20

4 6 0 m m

1930mm

2 6 4 0 m m

101 ,6mm

76,2

mm

76,2

mm

Y A s

Y A iA i

A s

Iljxuh 43143 = Sruwltxh S: gh Fudqvwrq

Oh sruwltxh ìwxglì hvw oh sruwltxh S: gh od vìulh gh kxlw/ hw vhv glphqvlrqv vrqw uhsruwìhv

vxu od �jxuh 431431 Lo hvw lqwìuhvvdqw gdqv od phvxuh rü lo d idlw o*remhw g*dx prlqv 7 ìwxghv

qxpìultxhv/ sdu Od}dur ^Od}:6`/ Juhodw ^Juh:;e`/ Hvslrq ^Hvs;9d` hw Qdlw0Udedk ^Qdl<3`1

Fh sruwltxh hvw fkdujì hq wíwh gh frorqqh sdu xqh irufh K txl q*hvw sdv lqgìshqgdqwh

gh od irufh S dssoltxìh hq 5 srlqwv vxu od srxwuh fhqwudoh1 Hq h�hw/ vrq lqwhqvlwì hvw


Ìoìphqwv Dr \�r +pp, D� \��+pp,

4/ 53 5 !</8 95 5 !</8 95

5/ 9/ 48/ 4< 7 !</8 95 5 !</8 95

6/ 7/ 8/ 49/ 4:/ 4; 9 !</8 95 5 !</8 95

:/ 47 7 !</8 95 7 !</8 95

;/ 46 5 !</8 95 7 !</8 95

</ 43/ 44/ 45 5 !</8 95 9 !</8 95

Wdeohdx 431; = Fdudfwìulvwltxhv gx ihuudloodjh orqjlwxglqdo

vhqvleohphqw grqqìh sdu =

K @S

43+431:,

Sdu dloohxuv/ elhq txh qrxv frqqdlvvlrqv oh ihuudloodjh dx frlqE hw dx srlqw g*dssolfdwlrq

gh od fkdujhS

5/ od glvsrvlwlrq orqjlwxglqdoh ghv dflhuv q*hvw sdv suìflvìh srxu oh sruwltxh

hq hqwlhu1 Qrxv dyrqv uhwhqx fhooh gh Juhodw/ tx*rqw uhwhqxh ã ohxu wrxu Hvslrq hw Qdlw0

Udedk/ hw txl frqvlvwh ã gìfrxshu oh sruwltxh hq 53 ìoìphqwv/ frpph lqgltxì vxu od �jxuh

43143/ oh ihuudloodjh ìwdqw grqqì gdqv oh wdeohdx 431;1 Oh gldpëwuh !</8 fruuhvsrqg hq idlw

ã ghv dupdwxuhv gh6

;srxfh1


Oh eìwrq frqvwlwxwli gx sruwltxh d xqh uìvlvwdqfh hq frpsuhvvlrq/ ìydoxìh ã sduwlu g*hvvdlv

vxu ìsurxyhwwhv fxeltxhv/ gh 73 PSd hqylurq1 Qrxv dyrqv dgrswì od ydohxu uhwhqxh sdu

Hvslrq/ ã vdyrlu xqh uìvlvwdqfh hq frpsuhvvlrq gh 66/5PSd1 Od gìirupdwlrq fruuhvsrqgdqw

ã fh slf gh frqwudlqwh hvw ìydoxìh ã 3/3351 Frpph rq xwlolvh xqh orl gh Vdujlq dyhf n @ 5/

od ydohxu gx prgxoh g*\rxqj gx eìwrq ã o*ruljlqh hvw gh 66533 PSd1 Od uìvlvwdqfh hq

wudfwlrq hvw sulvh ìjdoh ã 5/7< PSd1

Txdqw dx{ dflhuv/ lov vrqw prgìolvìv ã o*dlgh g*xqh orl ìodvwr0sodvwltxh sduidlwh/ ohxu

prgxoh g*\rxqj hvw sulv ìjdo ã 533333PSd/ hw od frqwudlqwh gh sdolhu sodvwltxh ã 5:;PSd1

Uìvxowdwv

Ohv uìvxowdwv vrqw uhsruwìv vxu od �jxuh 431441 Rq d uhsuìvhqwì o*ìyroxwlrq gh od ëfkh

yhuwlfdoh gh od vhfwlrq pìgldqh �� dlqvl txh fhooh gh od ëfkh krul}rqwdoh gx frlq E �� 1

Rq frqvwdwh xqh erqqh fruuìodwlrq hqwuh o*h{sìulhqfh hw qrwuh prgìolvdwlrq1 Qrv uìvxo0

wdwv vrqw wuëv surfkhv gh fhx{ gh Juhodw hw Qdlw0Udedk/ f*hvw srxutxrl qrxv qh ohv dyrqv

sdv uhsruwìv1 Qrxv vrpphv ìjdohphqw surfkhv g*Hvslrq hq fh txl frqfhuqh �� hw gdqv

xqh prlqguh phvxuh srxu �� 1 Frpph oh uhpdutxh Qdlw0Udedk/ lo vhpeohudlw tx*Hvslrq

dlw qìjoljì od uìvlvwdqfh hq wudfwlrq gx eìwrq gh od srxwuh uholdqw ohv ghx{ frorqqhv1


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Expérience

Calcul

Espion

Expérience

Calcul

Espion

δhB

δvA

δvA

δhB

δhB

δvA

Iljxuh 43144 = Sruwltxh S: gh Fudqvwrq 0 Ìyroxwlrq ghv ëfkhv hq irqfwlrq gh od fkdujh

431518 Fdguh ihupì


Qrxv ìwxglrqv xq fdguh ihupì txl d ìwì whvwì ã Dxvwlq gdqv oh Wh{dv +XVD, sdu Ihujxvvrq

hw Euhhq ^Ihu99`1 Oh fdguh hq txhvwlrq hvw uìiìuhqfì O6/ hw uhsuìvhqwh xqh edlh g*rvvdwxuh

gh eåwlphqw grqw od sduwlh frorqqh hvw ehdxfrxs soxv udlgh txh od sduwlh srxwuh1 Frq0

wudluhphqw ã o*ìwxgh gx sruwltxh elduwlfxoì hq slhg/ od sulvh hq frpswh gh od qrq0olqìdulwì

jìrpìwultxh hvw sulprugldoh gdqv fh fdv1 Fh sruwltxh d idlw o*remhw g*ìwxghv qxpìultxhv gh

od sduw gh Jxqqlq ^Jxq::`/ Hvslrq ^Hvs;9d` hw Qdlw0Udedk ^Qdl<3`1 Oh fkdujhphqw dlqvl

txh ohv glphqvlrqv gx fdguh vrqw uhsuìvhqwìv vxu od �jxuh 431451

Od irufh yhuwlfdoh S hvw uholìh ã od irufh krul}rqwdoh K sdu =

K @S

433+431;,

Oh ihuudloodjh sdvvli ghv frorqqhv hvw v|pìwultxh hw hvw frpsrvì gh 5 dupdwxuhv gh gldpëwuh6

;srxfh/ vrlw </8 pp/ hq sduwlh vxsìulhxuh/ hw gh od píph txdqwlwì g*dflhu hq sduwlh

lqiìulhxuh1 Hq fh txl frqfhuqh ohv srxwuhv/ hoohv vrqw ìjdohphqw ihuudlooìhv v|pìwultxhphqw/

dyhf 7 dupdwxuhv gh4

5srxfh gh gldpëwuh/ vrlw 45/: pp/ hq sduwlh vxsìulhxuh/ hw gh od


2 H

P-HP + H

3 4 m m

3 4 m m

18,3mm18 ,3mm

101 ,6mm

8 1 m m

Colonne

Poutre

2133 ,6mm

2133 ,6mm

152 ,4mm

Iljxuh 43145 = Fdguh O6 gh Ihujxvvrq hw Euhhq

píph txdqwlwì g*dflhu hq sduwlh lqiìulhxuh1


Srxu od prgìolvdwlrq qxpìultxh/ qrxv xwlolvrqv oh píph gìfrxsdjh tx*Hvslrq hw Qdlw0

Udedk/ ã vdyrlu 49 ìoìphqwv gh orqjxhxu lghqwltxh1 Oh eìwrq gx fdguh d xqh uìvlvwdqfh hq

frpsuhvvlrq gh 55/8 PSd/ vrq prgxoh g*\rxqj ã o*ruljlqh hvw gh 55/8 JSd1 Od gìirupdwlrq

fruuhvsrqgdqw dx slf gh frpsuhvvlrq hvw gh 3/5(1 Od uìvlvwdqfh hq wudfwlrq gx eìwrq hvw

qìjoljìh1 Od orl gh frpsruwhphqw gx eìwrq hvw prgìolvìh sdu xqh orl sduderoh0uhfwdqjoh

rewhqxh ã sduwlu g*xqh orl gh Vdujlq dyhf n @ 5 hw n�

@ 31

O*dflhu sdvvli ghv srxwuhv d xq prgxoh g*\rxqj ã o*ruljlqh gh 539 JSd1 Vd orl gh frp0

sruwhphqw hvw vxssrvìh ìodvwr0sodvwltxh sduidlwh/ od frqwudlqwh sodvwltxh ìwdqw gh 743 PSd1

O*dflhu sdvvli ghv frorqqhv d oh píph prgxoh g*\rxqj/ pdlv od frqwudlqwh gx sdolhu sodv0

wltxh hvw sulvh ìjdoh ã 733 PSd1

4316 Vwuxfwxuhv hq eìwrq suìfrqwudlqw 4;:

Uìvxowdwv

Ohv uìvxowdwv vrqw sruwìv vxu od �jxuh 431461 Rq d uhsuìvhqwì od ëfkh krul}rqwdoh gx srlqw

kdxw ghv frorqqhv/ hq irqfwlrq gh od fkdujh S dssoltxìh1 Ohv uìvxowdwv gh Qdlw0Udedk hw

g*Hvslrq vrqw wuëv surfkhv/ f*hvw srxutxrl qrxv q*dyrqv uhsruwì tx*xqh frxueh1 Od fkdujh

olplwh h{sìulphqwdoh hvw gh 473/9 nQ1 Qrxv wurxyrqv xqh fkdujh olplwh gh 476/8 nQ/ vrlw

xqh vxuhvwlpdwlrq gh 5(1

0

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160

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Flèches (mm)

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Expérience

Calcul

Espion

Gunnin

Iljxuh 43146 = Fdguh O6 gh Ihujxvvrq 0 Frxueh fkdujh 0 ëfkh

Qrxv frqvwdwrqv xqh erqqh fruuìodwlrq hqwuh qrwuh prgìolvdwlrq qxpìultxh hw o*h{sìul0

hqfh/ hq uhpdutxdqw tx*Hvslrq rewlhqw od uxswxuh soxv w÷w1 Qrwrqv dx sdvvdjh txh od uxlqh

hvw rewhqxh sdu xqh lqvwdelolwì ghv srwhdx{1

4316 Vwuxfwxuhv hq eìwrq suìfrqwudlqw

431614 Frorqqh suìfrqwudlqwh


O*ìwxgh frqfhuqh xqh frorqqh lvrvwdwltxh suìfrqwudlqwh/ wluìh g*xqh vìulh g*hvvdlv h�hf0

wxìv sdu Durql ^Dur9;`1 Od frorqqh hq txhvwlrq hvw uìiìuhqfìh G273e6 hw o*rq glvsrvh gh

4;; Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv

o*ìyroxwlrq gh od ëfkh hq irqfwlrq gh od fkdujh1 Fhwwh frorqqh d idlw o*remhw g*xqh ìwxgh

qxpìultxh gh od sduw gh Qdlw0Udedk ^Qdl<3`1

P

P

Excentr ic i té

in i t ia le 6,35mm

2052

,3m

me

12,7

mm

1 3 ,0mm

P

e=38 ,5mm

φ 5 m m

76,2

mm

5 1 ,3mm

Iljxuh 43147 = Frorqqh suìfrqwudlqwh G273e6

Od vhfwlrq gh od frorqqh hvw uhfwdqjxodluh/ gh 5/35 srxfhv sdu 6 srxfhv vrlw 84/6�:9/5 pp1Od kdxwhxu gh od frorqqh hvw gh ;3/; srxfhv vrlw 5385/6 pp1 Od frorqqh hvw suìfrqwudlqwh

sdu 7 �ov dgkìuhqwv gh gldpëwuh 3/4<; srxfh vrlw 8/3 pp/ txl vrqw suì0whqgxv/ hw txl vrqw

uhsuìvhqwìv vxu od �jxuh 431471 Ã o*åjh gh 47 mrxuv/ oh eìwrq hvw plv hq suìfrqwudlqwh hq

uhoåfkdqw ohv dqfudjhv ghv �ov/ hw o*hvvdl d olhx ã o*åjh gh 5; mrxuv1 Lo gxuh 6 khxuhv hqylurq1

Rq h{hufh xqh irufh S h{fhqwuìh gh 4/848 srxfhv/ vrlw 6;/8 pp/ dx{ h{wuìplwìv du0

wlfxoìhv gh od srxwuh1 Hq rxwuh/ xq gìidxw jìrpìwultxh lqwhqwlrqqho gh 3/58 srxfh vrlw

9/68 pp grlw íwuh sulv hq frpswh srxu phqhu fruuhfwhphqw o*ìwxgh1 O*dflhu gh suìfrq0

wudlqwh hvw whqgx xqlirupìphqw ã 7;6 PSd1

4316 Vwuxfwxuhv hq eìwrq suìfrqwudlqw 4;<


Od frorqqh hvw glvfuìwlvìh hq 43 ìoìphqwv gh orqjxhxu ìjdoh1 Oh prgxoh g*\rxqj ã o*ruljlqh

gx eìwrq hvw ìydoxì ã 67;33 PSd1 Od orl gh frpsruwhphqw gx eìwrq hvw gh w|sh Vdujlq

dyhf sulvh hq frpswh gx eìwrq whqgx1 Ohv frrugrqqìhv gx slf gh frpsuhvvlrq vrqw ohv

vxlydqwhv =

%Kf @ 3/33559 +431<,

iS� @ 6</63 PSd +43143,

Od uìvlvwdqfh hq wudfwlrq gx eìwrq hvw sulvh ìjdoh ã 43( gh vd uìvlvwdqfh hq frpsuhvvlrq1

O*dflhu txdqw ã oxl suìvhqwh xq prgxoh g*\rxqj ã o*ruljlqh gh 539 JSd1 Vd olplwh

g*ìodvwlflwì frqyhqwlrqqhooh �e hvw ìjdoh ã 4843 PSd1 Vd orl gh frpsruwhphqw hvw prgìolvìh

sdu od orl sursrvìh sdu oh ESHO<41 Oh frpsruwhphqw gh o*dflhu q*hvw soxv olqìdluh dx ghoã

gh 3/<�e1

Uìvxowdwv

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Flèche (mm)

Charge P (kN)

Expérience

Calcul

Nait Rabah

Iljxuh 43148 = Frorqqh suìfrqwudlqwh g*Durql 0 Frxueh fkdujh0 ëfkh ã pl0kdxwhxu

4<3 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv

Od frxueh fkdujh0 ëfkh ã pl0kdxwhxu hvw uhsuìvhqwìh vxu od �jxuh 431481 Od fkdujh ol0

plwh h{sìulphqwdoh hvw gh 46/; nQ1 Gh qrwuh f÷wì/ qrxv dyrqv fdofxoì xqh fkdujh olplwh

gh 46/89 nQ/ vrlw xq ìfduw gh 4/:( gdqv oh vhqv gh od vìfxulwì1 Od frqfrugdqfh hqwuh

o*h{sìulhqfh hw qrwuh fdofxo hvw wuëv erqqh1

431615 Srxwuh suìfrqwudlqwh ã 5 wudyìhv


0 ,375m

0,520m 0,70m

0,40m

0,70m

13,3m

0,132m

Câble 38 φ 7

Iljxuh 43149 = Srxwuh suìfrqwudlqwh FHEWS

Od srxwuh txh qrxv ìwxglrqv d ìwì whvwìh dx FHEWS ã Vdlqw0Uìp|0oëv0Fkhyuhxvh/ hqwuh

4<:9 hw 4<;8 ^Irx;;`1 Hooh frpsruwh ghx{ wudyìhv ìjdohv gh 46/3 p fkdfxqh1 Od orqjxhxu

wrwdoh gh od srxwuh hvw gh 59/93 p/ frpph lqgltxì vxu od �jxuh 431491 Vd vhfwlrq hvw

uhfwdqjxodluh/ gh 73 fp sdu :3 fp1 Od srxwuh hvw suìfrqwudlqwh sdu xq fåeoh ã �ov erxwrqqìv

6;!: EEU0E gh wudfì sduderoltxh1

Od srxwuh d ìwì frxoìh vxu flqwuh1 Ã o*åjh gh 5 mrxuv/ hooh d ìwì gìfr�uìh/ sxlv gìflqwuìh/

4316 Vwuxfwxuhv hq eìwrq suìfrqwudlqw 4<4

suìfrqwudlqwh hw plvh hq fkdujh ã o*åjh gh 6 mrxuv/ frpph lqgltxì vxu od �jxuh 4314:1

Ã 4;5 mrxuv/ ohv ghx{ dssxlv g*h{wuìplwì rqw ìwì vrxohyìv gh 63 pp d�q gh gìfkdujhu

sduwlhoohphqw od vhfwlrq vlwxìh dx gurlw gh o*dssxl fhqwudo/ frpph lqgltxì vxu od �jxuh

4314;1 Od srxwuh d ìwì frqvhuyìh gdqv xq orfdo ã whpsìudwxuh hw k|jurpìwulh frqvwdqwhv

shqgdqw soxv gh qhxi dqv/ gh 4<:9 ã 4<;81

13 ,0m

Q=13 x 13 ,7kN

Iljxuh 4314: = Ghvfulswlrq gx fkdujhphqw

13 ,0m

3 0 m m

Iljxuh 4314; = Gìqlyhoodwlrq g*dssxlv

Fhwwh srxwuh d idlw o*remhw gh soxvlhxuv vlpxodwlrqv qxpìultxhv/ gdqv oh fdguh g*xq

jurxsh gh wudydlo DIUHP ��Prgìolvdwlrq qxpìultxh gx xdjh��/ dqlpì sdu Sdxo Dfnhu


gx OFSF/ hw uìxqlvvdqw ghv uhsuìvhqwdqwv ghv hqwuhsulvhv hw exuhdx{ g*ìwxghv +M1 gh od

Irxfkduglëuh gh HHJ/ U1 Dgholqh gh Erx|jxhv,/ gh o*Dgplqlvwudwlrq ghv Urxwhv +M1 Jxdo

hw S1 Sh|udf gx VHWUD, hw ghv oderudwrluhv ghv Srqwv hw Fkdxvvìhv +U1 H|pdug hw I1 Xop

gx OFSF,1 Mh uhphuflh lfl shuvrqqhoohphqw Prqvlhxu U1 Dgholqh txl p*d idlw o*dpdelolwì

gh ph wudqvphwwuh ohv uìvxowdwv gx jurxsh/ txl vrqw sxeolìv gdqv ^Ofsf<;`1 Ohv orjlflhov

xwlolvìv vrqw vflqgìv hq ghx{ idploohv = ohv orjlflhov xwlolvdqw od pìwkrgh gh vxshusrvlwlrq

+Srqw2Duf/ HHJ/ SFS, hw fhx{ xwlolvdqw od pìwkrgh gx whpsv ìtxlydohqw +ILHI/ VW4,1


Oh surjudpph h{sìulphqwdo hvw gìwdlooì gdqv ^Irx;;`1 Ohv sdudpëwuhv frqfhuqdqw od

jìrpìwulh/ oh fkdujhphqw dlqvl txh od suìfrqwudlqwh rqw ìwì ìydoxìv vdqv sureoëph1 Sdu

frqwuh/ oh jurxsh d uhwhqx od orl g*ìyroxwlrq gx prgxoh g*\rxqj hw od orl gh xdjh sursrvìhv

sdu E1 Irxuì1 Od orl gh uhwudlw hvw fhooh gx ESHO1

Orl gh prgxoh gx eìwrq

H +w, @43

Sw

4; . 63w+43144,

rü H +w, hvw oh prgxoh gx eìwrq h{sulpì hq PSd ã od gdwh w h{sulpìh hq mrxuv1

Orl gh uhwudlw

%o +w, @ %"o +u +wf,� u +w,, +43145,

dyhf =

%"o @ �753> 3 433S +43146,

hw =

u +w, @w

w. 47> 47u6+43147,

Orl gh xdjh

%s, +w� wf> wf, @�

H�2H

Ns, +wf, i +w� wf, +43148,

dyhf =

Ns, +wf, @ nrne . nSn +wf, +43149,

n +wf, @433

433 . wf+4314:,

�r @Dr

E@ 3/4:( +4314;,

nr @4

4 . 53�r@ 3/<9: +4314<,

ne @ 3/7 +43153,


nS @453� ��

63.

5

6

433� ��53� u6

@ 6/3< +43154,

i +w� �, @

sw� �s

w� � . 6su6

+43155,

Uìvxowdwv

Oh surjudpph h{sìulphqwdo grqqh dffëv ã wurlv judqghxuv txl rqw ìwì ìwxglìhv sdu oh

jurxsh =

� od uìdfwlrq g*dssxl fhqwudo +�jxuhv 4314< hw 43153,/

� od gìirupdwlrq gh od �euh vxsìulhxuh gdqv od vhfwlrq I vlwxìh ã 4/6< p gh o*dssxl

fhqwudo +�jxuhv 43154 hw 43155,/

� od irufh gh suìfrqwudlqwh ã pl0wudyìh +�jxuhv 43156 hw 43157,1

290,00

295,00

300,00

305,00

310,00

315,00

320,00

0 20 40 60 80 100 120 140 160 180 200

Temps (jours)

Réaction d'appui

(kN)Expérience

PCP

Pont/Arc

EEG

ST1

FIEF

Iljxuh 4314< = Ìyroxwlrq gh od uìdfwlrq g*dssxl fhqwudo hqwuh 6 hw 4;5 mrxuv

Ohv uìvxowdwv txh qrxv dyrqv rewhqxv vxvflwhqw soxvlhxuv frpphqwdluhv1 G*xqh sduw/

od sulvh hq frpswh ghv h�hwv gl�ìuìv gdqv oh qr|dx gh fdofxo qrq olqìdluh gh SFS hvw

sduidlwhphqw ydolgìh dx yx ghv uìvxowdwv1 Rq frqvwdwh xq ìfduw sduirlv lpsruwdqw hqwuh


270,00

275,00

280,00

285,00

290,00

295,00

300,00

305,00

310,00

0 1000 2000 3000 4000 5000 6000

Temps (jours)

Réaction d'appui

(kN)Expérience

PCP

Pont/Arc

EEG

ST1

FIEF

Iljxuh 43153 = Ìyroxwlrq gh od uìdfwlrq g*dssxl fhqwudo hqwuh 4;5 hw 9333 mrxuv

o*h{sìulhqfh hw ohv fdofxov1 Fhw ìfduw v*h{soltxh sdu od gl!fxowì ã phwwuh dx srlqw xqh orl

gh xdjh fruuhfwh/ hw qrq sdu od gl�ìuhqfh ghv dojrulwkphv xwlolvìv gdqv ohv gl�ìuhqwv sur0

judpphv1 Sdu dloohxuv/ hw f*ìwdlw oã xqh ghv vxusulvhv gx jurxsh gh wudydlo/ rq frqvwdwh txh

od pìwkrgh gx whpsv ìtxlydohqw txl hvw fhqvìh plhx{ uhqguh frpswh g*xq gìfkdujhphqw/

q*dssruwh dxfxqh suìflvlrq vxssoìphqwdluh sdu udssruw ã od pìwkrgh gh vxshusrvlwlrq1

Fhfl qrxv frqiruwh gdqv oh fkrl{ txh qrxv dyrqv idlw/ hw fh g*dxwdqw soxv txh od pìwkr0

gh gh vxshusrvlwlrq hvw uhfrqqxh gdqv ohv uëjohphqwv lqwhuqdwlrqdx{/ ohv Hxurfrghv hq

sduwlfxolhu1

431616 Srxwuh suìfrqwudlqwh lvrvwdwltxh


Od srxwuh txh qrxv doorqv ìwxglhu q*d sdv idlw o*remhw g*h{sìulphqwdwlrq1 Lo v*djlw hq idlw

g*xq frqfrxuv gh orjlflhov rujdqlvì sdu Ì1 Frqwl hw U1 Wdug|/ grqw ohv uìvxowdwv vrqw sxeolìv

gdqv ^Frq<6d` hw ^Frq<6e`1 Od srxwuh hvw suìvhqwìh vxu od �jxuh 431581

Qrxv dyrqv vxssrvì txh od srxwuh ìwdlw frxoìh hq sodfh/ hw qrxv dyrqv phqì ghx{

ìwxghv1 O*xqh vxssrvdlw od suìfrqwudlqwh lqwìulhxuh dx eìwrq/ oh fåeoh gh suìfrqwudlqwh


-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

0 20 40 60 80 100 120 140 160 180 200

Temps (jours)

Déformation totale

(10-6

)

Expérience

PCP

Pont/Arc

ST1

FIEF

Iljxuh 43154 = Ìyroxwlrq gh od gìirupdwlrq gh od �euh vxsìulhxuh gh od vhfwlrq I hqwuh 6

hw 4;5 mrxuv


-1600

-1500

-1400

-1300

-1200

-1100

-1000

0 1000 2000 3000 4000 5000 6000

Temps (jours)

Déformation totale

(10-6

)

Expérience

PCP

Pont/Arc

ST1

FIEF

Iljxuh 43155 = Ìyroxwlrq gh od gìirupdwlrq gh od �euh vxsìulhxuh gh od vhfwlrq I hqwuh 4;5

hw 9333 mrxuv

4316 Vwuxfwxuhv hq eìwrq suìfrqwudlqw 4<:

1,65

1,7

1,75

1,8

1,85

1,9

1,95

0 20 40 60 80 100 120 140 160 180 200

Temps (jours)

Force de

précontrainte

(MN)

PCP

PONT/ARC

ST1

EEG

Iljxuh 43156 = Ìyroxwlrq gh od irufh gh suìfrqwudlqwh ã pl0wudyìh hqwuh 6 hw 4;5 mrxuv

4<; Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv

1,5

1,55

1,6

1,65

1,7

1,75

0 1000 2000 3000 4000 5000 6000

Temps (jours)

Force de

précontrainte

(MN)

PCP

PONT/ARC

EEG

ST1

Iljxuh 43157 = Ìyroxwlrq gh od irufh gh suìfrqwudlqwh ã pl0wudyìh hqwuh 4;5 hw 9333 mrxuv

4316 Vwuxfwxuhv hq eìwrq suìfrqwudlqw 4<<

10 m 10 m 10 m

30 m

2 m

0,62 m

10 m

0,6 m2 m

0,2 m

2 m

0,2 m

F

F

Iljxuh 43158 = Srxwuh suìfrqwudlqwh lvrvwdwltxh

vxelvvdqw doruv ohv píphv gìirupdwlrqv txh od vhfwlrq hq eìwrq1 O*dxwuh vxssrvdlw od

suìfrqwudlqwh h{wìulhxuh dx eìwrq/ oh fåeoh srxydqw doruv jolvvhu sdu udssruw ã od vhfwlrq

gh eìwrq tx*lo wudyhuvh1

Od srxwuh hvw suìfrqwudlqwh sdu xqh sdluh gh fåeohv/ d|dqw fkdfxq xqh vhfwlrq gh

8333 pp2 hw xqh whqvlrq xqliruph gh 9333 nQ dsuëv shuwhv1 Rq qrwh od suìvhqfh gh

ihuudloodjh sdvvli/ vlwxì ã 43 fp gh od �euh lqiìulhxuh ghv wdorqv ghv qhuyxuhv gh od srxwuh/

hw d|dqw xqh vhfwlrq gh 5333 pp2 srxu fkdtxh qhuyxuh1

Od srxwuh hvw frqvwuxlwh hq xqh skdvh vxu flqwuh/ sxlv suìfrqwudlqwh1

Ohv fkdujhphqwv ã suhqguh hq frpswh vrqw oh srlgv sursuh/ dyhf xq srlgv yroxpltxh gh

58 nQ2p� srxu oh eìwrq suìfrqwudlqw1 Xqh fkdujh I hvw dssoltxìh dx gurlw g*xqh vhfwlrq

gh gìyldwlrq gx fåeoh/ frpph lqgltxì vxu od �jxuh 431581 Oh uhwudlw/ oh xdjh gx eìwrq/

533 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv

dlqvl txh od uhod{dwlrq ghv fåeohv gh suìfrqwudlqwh vrqw qìjoljìv1


Od orl gh frpsruwhphqw gx eìwrq hvw lpsrvìh = lo v*djlw g*xqh orl sduderoh0uhfwdqjoh lvvxh

gx uëjohphqw iudqêdlv1 Oh prgxoh g*\rxqj ã o*ruljlqh gx eìwrq ydxw 53333 PSd/ ohv

frrugrqqìhv gx slf gh frpsuhvvlrq vrqw =

%Kf @ 3/335 +43156,

iS� @ 53/33 PSd +43157,

Od gìirupdwlrq xowlph hvw rewhqxh ã 3/68( srxu xqh frqwudlqwh gh 53/33 PSd1 Rq suhqg

hq frpswh xqh uìvlvwdqfh hq wudfwlrq gx eìwrq ìjdoh ã 5/7 PSd1

O*dflhu sdvvli/ txdqw ã oxl/ d xqh orl gh frpsruwhphqw ìodvwr0sodvwltxh sduidlwh/ dyhf

xq prgxoh g*\rxqj ã o*ruljlqh gh 533333 PSd/ hw xq sdolhu sodvwltxh ã 733 PSd1

0

0,5

1

1,5

2

2,5

3

3,5

-50 0 50 100 150 200 250 300 350

Flèches (mm)

Charge F (MN)

Calcul

Ulm

Garcia-Vargas

Franssen

Perisic

Iljxuh 43159 = Frxueh fkdujh0 ëfkh dyhf suìfrqwudlqwh h{whuqh

O*dflhu gh suìfrqwudlqwh vxlw od orl sro|qrpldoh gh ghjuì 8 lvvxh gx ESHO<41 Vrq

prgxoh g*\rxqj ã o*ruljlqh ydxw 4<3333 PSd/ vd olplwh g*ìodvwlflwì frqyhqwlrqqhooh ã 3/5(

ydxw 48:8 PSd/ fh txl fruuhvsrqg ã xqh olplwh gh frpsruwhphqw ìodvwltxh olqìdluh gh

474:/8 PSd/ hw xqh frqwudlqwh gh uxswxuh gh 499</8 PSd1

4316 Vwuxfwxuhv hq eìwrq suìfrqwudlqw 534

Uìvxowdwv

0

0,5

1

1,5

2

2,5

3

3,5

4

-50 0 50 100 150 200 250 300 350 400

Flèches (mm)

Charge F (MN)

Calcul

Ulm

Garcia-Vargas

Franssen

Perisic

Iljxuh 4315: = Frxueh fkdujh0 ëfkh dyhf suìfrqwudlqwh lqwhuqh

Lo ìwdlw ghpdqgì dx{ sduwlflsdqwv gh gìwhuplqhu oh gìsodfhphqw yhuwlfdo gx srlqw vlwxì dx

plolhx gh od srxwuh hq irqfwlrq gh od fkdujh I 1 Dx wrwdo/ 47 sduwlflsdqwv rqw yx ohxuv

uìvxowdwv sxeolìv1 Srxu ghv udlvrqv gh olvlelolwì/ qrxv q*hq dyrqv uhwhqx txh txdwuh = g*xqh

sduw ohv fdofxov iudqêdlv/ uhsuìvhqwìv sdu I1M1 Xop hw Y1 Pdjqdw gx OFSF +Iudqfh,/ M1D1

Jdufld0Ydujdv/ Q1 Pìqh}ëv hw E1 Irxuì gx FHEWS +Iudqfh,/ hw g*dxwuh sduw M1P1 Iudqvvhq

gh o*Xqlyhuvlwì gh Olëjh +Ehojltxh, hw ]1 Shulvlf hw Q1 Rmgurylf gh o*Xqlyhuvlwì gh Ehojudgh

+\rxjrvodylh, fdu ohxuv uìvxowdwv vrqw vljql�fdwlyhphqw gdqv od pr|hqqh gh o*hqvhpeoh ghv

uìvxowdwv suìvhqwìv1

Frpph rq shxw oh frqvwdwhu vxu ohv �jxuhv 43159 hw 4315:/ ohv uìvxowdwv vrqw wuëv vdwlv0

idlvdqwv hw vrqw vlwxìv gdqv od erqqh pr|hqqh ghv gl�ìuhqwv sduwlflsdqwv1 Od gl!fxowì gh

fh whvw uìvlgh hvvhqwlhoohphqw gdqv oh idlw tx*lo q*| d sdv gh uìvxowdwv h{sìulphqwdx{ srxu

wudqfkhu gì�qlwlyhphqw1


431617 Srxwuh suìfrqwudlqwh k|shuvwdwltxh


Fh sureoëph hvw hq idlw lvvx gx píph frqfrxuv gh orjlflhov ^Frq<6d`/ ^Frq<6e`1 Od srxwuh

hvw suìvhqwìh vxu od �jxuh 4315;1

10 m 10 m 10 m

30 m

1,5 m

5 m

2,5 m

0,2 m

1,5 m

0,25 m

F

F

10 m 10 m 10 m 10 m

20 m 20 m

0,2 m

0,2 m

0,582m

Iljxuh 4315; = Srxwuh suìfrqwudlqwh k|shuvwdwltxh

Frpph suìfìghpphqw/ qrxv dyrqv vxssrvì txh od srxwuh ìwdlw frxoìh hq sodfh1 Qrxv

dyrqv ìjdohphqw phqì ghx{ w|shv g*ìwxghv/ o*xqh vxssrvdqw od suìfrqwudlqwh lqwhuqh/

o*dxwuh od vxssrvdqw h{whuqh1

Od srxwuh hvw suìfrqwudlqwh juåfh ã xqh sdluh gh fåeohv/ d|dqw fkdfxq xqh vhfwlrq gh

5583 pp2 hw xqh whqvlrq xqliruph gh 6333 nQ dsuëv shuwhv1 Oh ihuudloodjh sdvvli frpsuhqg/

srxu xqh sduw 6333 pp2 g*dflhu gdqv oh krxuglv vxsìulhxu gx fdlvvrq/ ã 43 fp gh od �euh

4316 Vwuxfwxuhv hq eìwrq suìfrqwudlqw 536

vxsìulhxuh/ hw srxu xqh dxwuh sduw 7333 pp2 g*dflhu gdqv oh krxuglv lqiìulhxu gx fdlvvrq/

ã 43 fp gh od �euh lqiìulhxuh1

Od srxwuh hvw frqvwuxlwh hq xqh vhxoh skdvh vxu flqwuh/ sxlv suìfrqwudlqwh1

Ohv fkdujhphqwv ã suhqguh hq frpswh vrqw oh srlgv sursuh/ dyhf xq srlgv yroxpltxh gh

58 nQ2p� srxu oh eìwrq suìfrqwudlqw1 Xqh fkdujh I hvw dssoltxìh dx gurlw g*xqh vhfwlrq

gh gìyldwlrq gx fåeoh/ frpph lqgltxì vxu od �jxuh 4315;1 Oh uhwudlw/ oh xdjh gx eìwrq/

dlqvl txh od uhod{dwlrq ghv fåeohv gh suìfrqwudlqwh vrqw qìjoljìv1


Ohv orlv gh frpsruwhphqw ghv gl�ìuhqwv pdwìuldx{ vrqw vwulfwhphqw ohv píphv txh fhoohv

gìfulwhv gdqv oh sdudjudskh 4316161

Uìvxowdwv

0

0,5

1

1,5

2

2,5

3

3,5

0 20 40 60 80 100 120 140 160 180

Flèches (mm)

Charge F (MN)

Calcul

Ulm

Garcia-Vargas

Perisic

Iljxuh 4315< = Frxueh fkdujh0 ëfkh dyhf suìfrqwudlqwh lqwhuqh

Gh píph tx*dx sdudjudskh 431616/ qrxv dyrqv ìwxglì oh gìsodfhphqw yhuwlfdo gx srlqw

vlwxì dx plolhx gh od srxwuh hq irqfwlrq gh od fkdujh1 Ohv uìvxowdwv vrqw uhsuìvhqwìv vxu

ohv �jxuhv 4315< hw 431631 Qrxv wlurqv ohv píphv frqfoxvlrqv tx*dx sdudjudskh 431616/ ã


0

0,5

1

1,5

2

2,5

3

0 20 40 60 80 100 120 140 160 180 200 220 240

Flèches (mm)

Charge F (MN)

Calcul

Ulm

Garcia-Vargas

Franssen

Perisic

Iljxuh 43163 = Frxueh fkdujh0 ëfkh dyhf suìfrqwudlqwh h{whuqh

vdyrlu txh ohv uìvxowdwv txh qrxv suìvhqwrqv vrqw gdqv oh ixvhdx ghv uìvxowdwv ghv dxwuhv

sduwlflsdqwv1 Pdokhxuhxvhphqw/ rq qh shxw sdv frqfoxuh gì�qlwlyhphqw gx idlw gh o*devhqfh

gh phvxuhv h{sìulphqwdohv1

4317 Vwuxfwxuhv pìwdooltxhv

Gdqv fhwwh sduwlh/ qrxv vxssrvhurqv oh frpsruwhphqw gx pdwìuldx olqìdluh ìodvwltxh1

Ohv h{hpsohv txl yrqw vxlyuh shuphwwhqw gh ydolghu od irupxodwlrq gh od qrq0olqìdulwì

jìrpìwultxh txh qrxv dyrqv ìwdeolh1

431714 Iodpehphqw g*xq påw hqfdvwuì


Qrxv frqvlgìurqv xq påw hqfdvwuì vrxplv ã xqh fkdujh gh frpsuhvvlrq gh gluhfwlrq frq0

vwdqwh dssoltxìh ã vrq h{wuìplwì oleuh/ frpph lqgltxì vxu od �jxuh 431641 Fh sureoëph d

ìwì uìvrox dqdo|wltxhphqw sdu V1S1 Wlprvkhqnr gdqv ^Wlp99`1 Oh dpehphqw hvw hpsífkì

4317 Vwuxfwxuhv pìwdooltxhv 538

F

v

u

L

Iljxuh 43164 = Påw hqfdvwuì

vxlydqw oh vhqv gh od shwlwh lqhuwlh1 Od irufh fulwltxh g*Hxohu gh fh påw hvw grqqìh sdu =

SSo @�2HL

7O2+43158,

Srxu xq sur�oì hq dflhu gh od jdpph LSH 453/ rq d =

SSo @ 8:3/48 nQ +43159,

Srxu dprufhu oh dpehphqw/ rq frqvlgëuh xqh gìirupìh lqlwldoh vlqxvrògdoh grqw od ëfkh

pd{lpdoh fruuhvsrqg ã 3/6( gh od orqjxhxu1


Od prgìolvdwlrq gx sureoëph hvw h{wuíphphqw vlpsoh fdu oh frpsruwhphqw gx pdwìuldx

hvw olqìdluh ìodvwltxh1 Ohv fdudfwìulvwltxhv jìrpìwultxhv hw pìfdqltxhv vrqw ohv vxlydqwhv =

O @ 4:33 pp +4315:,

H @ 5/4 43D PSd +4315;,

D @ 4653 pp2 +4315<,

L @ 64; 43e ppe +43163,

Oh påw glvsrvh g*xq hqfdvwuhphqw sduidlw ã vd edvh1


Uìvxowdwv

0

1

2

3

4

5

6

7

8

9

10

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

Flèches relatives

Charge relative

P/Pcr

Timoshenko u/L

Timoshenko v/L

4 éléments u/L

4 éléments v/L

10 éléments u/L

10 éléments v/L

Iljxuh 43165 = Påw hqfdvwuì 0 Ìyroxwlrq ghv ëfkhv hq irqfwlrq gx fkdujhphqw

Ohv uìvxowdwv vrqw sruwìv vxu od �jxuh 431651 Lo v*djlw hq idlw ghv ëfkhv orqjlwxglqdoh hw

wudqvyhuvdoh udssruwìhv ã O/ od orqjxhxu lqlwldoh gx påw hq irqfwlrq gx fkdujhphqw h{sulpì

frpph xqh iudfwlrq gh od fkdujh fulwltxh SSo1 Qrxv | dyrqv idlw �jxuhu ohv uìvxowdwv rewhqxv

sdu Wlprvkhqnr1 Qrxv dyrqv xwlolvì ghx{ w|shv gh pdloodjh1 Oh suhplhu hvw uìgxlw hw qh

frpsruwh txh 7 ìoìphqwv1 Oh vhfrqg hq d 431 Rq frqvwdwh txh oh uìvxowdw hvw �qdohphqw

shx vhqvleoh ã od �qhvvh gx pdloodjh hw vxuwrxw tx*lo hvw hq sduidlwh dgìtxdwlrq dyhf od

wkìrulh ìodvwltxh1

Vxu od �jxuh 43166/ qrxv dyrqv ghvvlqì/ vdqv dpsol�fdwlrq/ ohv gìirupìhv gx påw dx

frxuv gh vrq fkdujhphqw1 Fhfl prqwuh txh oh fdofxo vh idlw h�hfwlyhphqw hq judqgv gì0

sodfhphqwv hw hq judqghv urwdwlrqv1

431715 Srxwuh frqvroh vroolflwìh wudqvyhuvdohphqw


Qrxv ìwxglrqv xqh srxwuh frqvroh vroolflwìh sdu vrq srlgv sursuh hw xqh fkdujh wudqvyhuvdoh

�S dssoltxìh ã vrq h{wuìplwì oleuh/ frpph lqgltxì vxu od �jxuh 431671 Od srxwuh d xqh

4317 Vwuxfwxuhv pìwdooltxhv 53:

P/Pcr = 2,5

P/Pcr = 1,7

P/Pcr = 1,3

P/Pcr = 1,2

P/Pcr = 1,1

P/Pcr = 1,07

P/Pcr = 1

P/Pcr = 5P/Pcr = 9

Iljxuh 43166 = Påw hqfdvwuì 0 Gìirupìhv vxffhvvlyhv

53; Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv

Z

X

Y , V

2 ,032 m

p = 12,41 N/m

Z

Y

3 ,15 mm

G 50,8 mm

0,00545 rad.(déformée in i t ia le)

λP

O

Iljxuh 43167 = Srxwuh frqvroh vroolflwìh sdu xqh fkdujh wudqvyhuvdoh

vhfwlrq wudqvyhuvdoh uhfwdqjxodluh plqfh1 Od vroxwlrq dqdo|wltxh gx frpsruwhphqw srvw0

fulwltxh d ìwì ìwdeol sdu Zrrofrfn hw Wudkdlu ^Zrr:7`1 Hq rxwuh/ rq glvsrvh gh uìvxowdwv

h{sìulphqwdx{ mxvtx*dx dpehphqw1 Srxu dprufhu oh gìyhuvhphqw odwìudo/ rq lpsrvh

dydqw fkdujhphqw xqh urwdwlrq gh 3/33878 udg dxwrxu gh o*d{h R[ vxu wrxwh od orqjxhxu

gh od srxwuh1 Fhwwh srxwuh d idlw o*remhw g*xqh ìwxgh qxpìultxh gh od sduw gh gh Ylooh gh

Jr|hw ^GhY;<` ã o*dlgh gx orjlflho ILQHOJ1 Gdqv ^GhY;<`/ gh Ylooh gh Jr|hw lqvlvwh vxu

od vìohfwlylwì gh fhw h{hpsoh ã o*ìjdug gh o*ìoìphqw �ql xwlolvì1


Od srxwuh hvw gìfrpsrvìh hq 43 ìoìphqwv gh píph orqjxhxu1 Oh pdwìuldx d xq frpsruwh0

phqw olqìdluh hw vhv fdudfwìulvwltxhv pìfdqltxhv vrqw ohv vxlydqwhv =

H @ 536;<3 PSd +43164,

� @ 3/6 +43165,

O*hqfdvwuhphqw hvw vxssrvì sduidlw1

Uìvxowdwv

Ohv uìvxowdwv fruuhvsrqghqw sduidlwhphqw ã o*h{sìulhqfh hw ã od wkìrulh/ píph dvvh} orlq

gdqv oh frpsruwhphqw srvw0fulwltxh gh od srxwuh1 Ohv ydohxuv dwwhlqwhv sdu od ëfkh

wudqvyhuvdoh vrqw gh o*ruguh gh od prlwlì gh od orqjxhxu lqlwldoh gh od srxwuh1 Qrxv vrpphv

frpsoëwhphqw gdqv oh grpdlqh ghv judqgv gìsodfhphqwv hw ghv judqghv urwdwlrqv wulglphq0

4317 Vwuxfwxuhv pìwdooltxhv 53<

0

5

10

15

20

25

30

35

0 0,2 0,4 0,6 0,8 1 1,2

Déplacement transversal (m)

Charge (N)

Expérience

Théorie

Calcul

FINELG

Iljxuh 43168 = Srxwuh frqvroh fkdujìh wudqvyhuvdohphqw 0 Ìyroxwlrq gh od ëfkh wudqvyhuvdoh

hq irqfwlrq gx fkdujhphqw

vlrqqhov1 Fhw h{hpsoh ydolgh frpsoëwhphqw qrwuh dssurfkh srxu wudlwhu od qrq0olqìdulwì

jìrpìwultxh1

431716 Duf vxuedlvvì


Lo v*djlw g*ìwxglhu xq duf vxuedlvvì gh iruph flufxodluh/ hqfdvwuì ã vhv ghx{ h{wuìplwìv/ hw

vroolflwì sdu xqh fkdujh frqfhqwuìh hq foì/ frpph lqgltxì vxu od �jxuh 431691 Fhw duf d

idlw o*remhw g*ìwxghv qxpìultxhv gh od sduw gh Vwroduvnl hw Eho|wvfknr ^Vwr;5` hw gh Ylooh

gh Jr|hw ^GhY;<`1 Gdqv ^Vwr;5`/ qrxv wurxyrqv od vroxwlrq wkìrultxh1


Oh frpsruwhphqw gx pdwìuldx hvw olqìdluh ìodvwltxh1 Ohv fdudfwìulvwltxhv jìrpìwultxhv hw

pìfdqltxhv vrqw ohv vxlydqwhv =

H @ 43. svl @ :363; PSd +43166,


P,v

R=100,0 pouces

3 pouces

48,62 pouces

1 pouce

2 pouces

Iljxuh 43169 = Duf vxuedlvvì

D @ 5 lqfk2 @ 45<3 pp2 +43167,

L @56lqfke @ 5::/8 43� ppe +43168,

O*duf glvsrvh gh ghx{ hqfdvwuhphqwv sduidlwv hq slhgv1

Uìvxowdwv

Frpph rq shxw oh frqvwdwhu vxu od �jxuh 4316:/ od fruuìodwlrq hqwuh od wkìrulh hw o*h{sìulhqfh

hvw h{fhoohqwh1 G*dsuëv od vroxwlrq wkìrultxh h{wudlwh gh ^Vwr;5`/ srxu xq gìsodfhphqw gh

6/8 srxfhv/ vrlw ;;/< pp/ gh od fohi/ od fkdujh ã dssoltxhu hvw gh 5<;33 oev/ vrlw 468/4:6 nQ1

G*dsuëv qrwuh fdofxo/ srxu fh píph gìsodfhphqw/ lo idxw xqh fkdujh gh 5<896 oev/ vrlw

467/43 nQ/ fh txl fruuhvsrqg ã xq ìfduw gh 3/;(1

4317 Vwuxfwxuhv pìwdooltxhv 544

0

0,5

1

1,5

2

2,5

3

3,5

4

0 1 2 3 4

Flèche verticale (inch)

Charge P (104 lb)

FINELG

Calcul

Théorie

Iljxuh 4316: = Duf vxuedlvvì 0 Ìyroxwlrq gh od ëfkh ã od fohi hq irqfwlrq gh od fkdujh

Fkdslwuh 44

Xq h{hpsoh g*dssolfdwlrq uìhooh =

oh Srqw gh Wdqfduylooh

4414 Lqwurgxfwlrq

Oh Srqw gh Wdqfduylooh hvw xq rxyudjh vxvshqgx txl hqmdpeh o*hvwxdluh gh od Vhlqh/ ã

sur{lplwì gx Kdyuh1 Vd plvh hq vhuylfh uhprqwh dx 6 mxloohw 4<8<1 Ghsxlv/ gh qrpeuhxvhv

uxswxuhv gh �ov ghv fåeohv rqw ìwì gìwhfwìhv1 Oh 5 mxloohw 4<<8/ od vlwxdwlrq v*hvw djjudyìh

euxvtxhphqw oruv gh od uxswxuh frpsoëwh g*xq wrurq dx gurlw gh od vhoohwwh g*lq h{lrq vlwxìh

vxu od ulyh jdxfkh/ f÷wì dprqw1 Rq surfìgd doruv ã od uìsdudwlrq gx wrurq gìihfwxhx{/ hw

o*rq gìflgd gh uhpsodfhu od vxvshqvlrq frpsoëwh gh o*rxyudjh gdqv ohv phloohxuv gìodlv1 Oh

pdufkì ìwdlw frq�ì dx jurxshphqw JWP0Gxph} hw Edxglq0Fkdwhdxqhxi1 Hq mxloohw 4<<9/

oh VHWUD frpphqêdlw vd plvvlrq gh frqwu÷oh ghv ìwxghv g*h{ìfxwlrq1

4415 Remhw gx frqwu÷oh

Od yìul�fdwlrq frpsuhqdlw ghx{ sduwlhv glvwlqfwhv ^Zdw<:`1 G*xqh sduw/ lo idoodlw yìul�hu ohv

gl�ìuhqwv ìoìphqwv frqvwlwxdqw od qrxyhooh vxvshqvlrq +fåeohv/ vxvshqwhv/ froolhuv/ ìwulhuv/

vhoohv hq wíwh gh s|o÷qhv/ vhoohwwhv g*lq h{lrq gdqv ohv fkdpeuhv g*dqfudjh/ gìyldwhxuv111,1

G*dxwuh sduw/ lo frqyhqdlw gh v*dvvxuhu txh ohv ìoìphqwv dfwxhov vxssruwhudlhqw vdqv grp0

pdjh oh wudqvihuw gh o*dqflhqqh vxvshqvlrq yhuv od qrxyhooh/ hw txh fhx{ g*hqwuh hx{ txl

qh vhudlhqw sdv uhpsodfìv +ã vdyrlu oh wdeolhu/ ohv s|o÷qhv hw ohv wludqwv g*dqfudjh, vxs0

sruwhudlhqw ohv h�ruwv gh od skdvh gh vhuylfh gx srqw1 Od qìfhvvlwì gh od sulvh hq frpswh

ghv h�hwv gx vhfrqg ruguh dlqvl txh gh o*h�hw fkdñqhwwh ghv fåeohv d frqgxlw od Glylvlrq ghv

Judqgv Rxyudjhv gx VHWUD ã phqhu ohv frqwuh0fdofxov ã o*dlgh gh SFS1 Oh jurxshphqw

g*hqwuhsulvhv xwlolvdlw srxu vd sduw oh orjlflho JS6G1

547 Fkdslwuh 44 Xq h{hpsoh g*dssolfdwlrq uìhooh = oh Srqw gh Wdqfduylooh

Iljxuh 4414 = Oh Srqw gh Wdqfduylooh

4416 Ìwxgh gx wudqvihuw gh fkdujhv

Dxsdudydqw/ oh Srqw gh Wdqfduylooh ìwdlw vxvshqgx ã ghx{ fåeohv sruwhxuv1 Fkdfxq g*hqwuh

hx{ ìwdlw irupì gh 89 wrurqv gh :5 pp gh gldpëwuh hq wudyìh fhqwudoh/ hw gh 93 wrurqv hq

wudyìhv gh ulyh1

441614 Oh sulqflsh gx wudqvihuw

Oh sulqflsh gh od uìsdudwlrq hvw gh uhpsodfhu fkdfxq ghv fåeohv sdu ghx{ fåeohv gh soxv

shwlwv gldpëwuhv +yrlu wdeohdx 4416,1 Ohv ghx{ qrxyhdx{ fåeohv vhurqw vlwxìv gh sduw hw

g*dxwuh gh o*dqflhq fåeoh hw ylhqgurqw v*dqfuhu vxu ohv wljhv g*dqfudjh h{lvwdqwhv1

Oh wudqvihuw g*xqh vxvshqvlrq ã o*dxwuh v*h�hfwxh hq 43 skdvhv sulqflsdohv1 Srxu fkdfxqh

g*hqwuh hoohv/ lo frqyhqdlw gh txdqwl�hu ohv gìsodfhphqwv gx wdeolhu/ ghv wíwhv gh s|o÷qhv

hw ghv fåeohv sruwhxuv/ hw gh yìul�hu txh ohv h�ruwv ìwdlhqw dgplvvleohv gdqv oh wdeolhu/

ohv vxvshqwhv/ ohv fåeohv/ ohv vhoohv hq wíwh gh s|o÷qhv hw ohv vhoohwwhv g*lq h{lrq gdqv ohv

fkdpeuhv g*dqfudjhv1

Oh qrpeuh gh fkdujhphqwv ìwdqw dvvh} ìohyì/ srxu olplwhu ohv whpsv gh fdofxo/ qrxv

dyrqv grqf xwlolvì xq prgëoh hq ghx{ glphqvlrqv/ vlpxodqw od v|pìwulh wudqvyhuvdoh gh

o*rxyudjh1

441615 Oh prgëoh 5G

Oh prgëoh 5G frpsruwh 7<3 qúxgv hw 98< ìoìphqwv/ grqw 68: fkdñqhwwhv prgìolvdqw ohv

fåeohv hw vxvshqwhv1

Ohv ghx{ fåeohv dqflhqv vrqw prgìolvìv sdu xq fåeoh xqltxh +gh vhfwlrq ghx{ irlv soxv

4416 Ìwxgh gx wudqvihuw gh fkdujhv 548

Iljxuh 4415 = Prgìolvdwlrq FDR Rsìud gx Srqw gh Wdqfduylooh

judqgh,1 Ohv txdwuh ixwxuv fåeohv vrqw ìjdohphqw prgìolvìv sdu xq vhxo fåeoh1 Lo hq hvw gh

píph srxu ohv vxvshqwhv1

Od jìrpìwulh gh uìiìuhqfh gh o*rxyudjh hvw od jìrpìwulh wkìrultxh whooh tx*hooh d ìwì

gì�qlh oruv gh od frqvwuxfwlrq gx srqw/ hw fhfl d�q txh gdqv od ixwxuh skdvh gh vhuylfh/ ohv

ìoìphqwv h{lvwdqwv gh o*rxyudjh +ohv s|o÷qhv hw oh wdeolhu, qh vrlhqw sdv vroolflwìv gl�ìuhp0

phqw gh fh txl dydlw ìwì suìyx oruv gh ohxu frqfhswlrq1

Qìdqprlqv/ xqh ìwxgh klvwrultxh gh o*rxyudjh ghsxlv vd frqvwuxfwlrq mxvtx*ã qrv mrxuv

qrxv d dssulv txh oh wdeolhu suìvhqwdlw/ vrxv fkdujh shupdqhqwh/ xqh ëfkh gh ;: fp dx

qlyhdx gx qúxg fhqwudo sdu udssruw ã vd srvlwlrq wkìrultxh1 Fhwwh ëfkh v*h{soltxh hq

judqgh sduwlh sdu xq h�hw gh uhod{dwlrq ghv fåeohv dx frxuv gx whpsv1 Qrxv vdyrqv txh

od qrxyhooh vxvshqvlrq q*ìfkdsshud sdv dx píph skìqrpëqh1 Fhoxl0fl vhud soxv idleoh txh

srxu ohv fåeohv h{lvwdqwv gx idlw gx soxv idleoh gldpëwuh ghv wrurqv ìoìphqwdluhv hw gh od

soxv judqgh orqjxhxu gx sdv gh wrurqdjh1 Od ëfkh gx wdeolhu gxh ã od uhod{dwlrq d ìwì

hvwlpìh ã xqh wuhqwdlqh gh fhqwlpëwuhv1 Ã od �q ghv wudydx{/ oh wdeolhu grlw dlqvl suìvhqwhu/

vrxv fkdujh shupdqhqwh/ xqh frqwuh0 ëfkh gh 63 fp sdu udssruw dx sur�o wkìrultxh1

D�q g*ìydoxhu dx plhx{ ohv gìsodfhphqwv hw frqwudlqwhv gdqv od vwuxfwxuh oruv gx

fkdqjhphqw gh vxvshqvlrq/ lo idoodlw whqlu frpswh ghv gìsodfhphqwv hw frqwudlqwhv tx*lo

vxelvvdlw1

Ohv wurlv fkdujhphqwv vxlydqwv shuphwwhqw gh uhwurxyhu od ëfkh gh ;: fp / dlqvl txh


ohv gìsodfhphqwv krul}rqwdx{ phvxuìv hq wíwh ghv s|o÷qhv1

41 Xq doorqjhphqw ghv fåeohv dqflhqv gh od wudyìh fhqwudoh hqwudñqdqw xqh ëfkh gx

wdeolhu gh < fp1

51 Xqh glodwdwlrq wkhupltxh gh od vxvshqvlrq hqwudñqdqw xqh ëfkh gh 97 fp1

61 Xqh glodwdwlrq wkhupltxh ghv fåeohv vlwxìv hqwuh o*dqfudjh hw od vhoohwwh ulyh jdxfkh

hqwudñqdqw xqh ëfkh gh 47 fp1

O*rsìudwlrq 4 vlpxoh xq gìidxw gh uìjodjh lqlwldo/ o*rsìudwlrq 5 vlpxoh od uhod{dwlrq

ghv fåeohv/ hw o*rsìudwlrq 6 vlpxoh xq gìsodfhphqw gx pdvvli g*dqfudjh ulyh jdxfkh/ wurlv

skìqrpëqhv txl rqw mrxì xq u÷oh fhuwdlq gdqv oh jìrpìwulh dfwxhooh gh od vwuxfwxuh1

Hq fh txl frqfhuqh oh uhpsodfhphqw gh od vxvshqvlrq gx srqw gh Wdqfduylooh/ od qrq0

olqìdulwì pdwìuldx q*d sdv ìwì uhwhqxh +h{fhswì srxu xqh ìwxgh gh gìwdlo ghv wljhv g*dq0

fudjh,1 Ohv eduuhv prgìolvdqw ohv s|o÷qhv hw oh wdeolhu/ rqw grqf ìwì gì�qlhv fodvvltxhphqw/

ã o*dlgh gh ohxu vhfwlrq/ lqhuwlh hw frrugrqqìhv gx fhqwuh gh wruvlrq1

Od vxvshqvlrq/ txdqw ã hooh/ d ìwì prgìolvìh sdu ghv ìoìphqwv elduwlfxoìv +od vhfwlrq

qrupdoh hvw ohxu vhxoh fdudfwìulvwltxh, gh w|sh fkdñqhwwh1

441616 Ohv ìwdshv gx wudqvihuw gh fkdujhv

Oh wdeohdx 4414 uhwudfh ohv gl�ìuhqwhv ìwdshv gx wudqvihuw gh fkdujhv/ dyhf ohv rsìudwlrqv

fruuhvsrqgdqwhv txdqw ã od vlpxodwlrq dyhf SFS1

4417 Ìwxghv vsìfl�txhv

441714 Xq prgëoh 5G26G srxu o*ìwxgh ghv s|o÷qhv

Ohv mdpehv gh s|o÷qhv qh vrqw sdv yhuwlfdohv/ pdlv suìvhqwhqw xq oìjhu iuxlw wudqvyhuvdo1

Gh fh idlw/ lo h{lvwh gdqv oh s|o÷qh ghv prphqwv wudqvyhuvdx{ qrq fdofxodeohv vxu oh prgëoh

5G hw txl prgl�hqw srxuwdqw ohv frqwudlqwhv gh idêrq vhqvleoh1 Lo idoodlw grqf prgìolvhu

ohv s|o÷qhv hq 6G srxu frqqdñwuh dyhf suìflvlrq ohv frqwudlqwhv tx*lov vxelvvhqw oruv gx

wudqvihuw gh fkdujhv1 F*hvw srxutxrl xq prgëoh pl{wh 5G26G d ìwì frqvwlwxì/ gdqv ohtxho

ohv vxvshqvlrqv hw oh wdeolhu ìwdlhqw fhx{ gx prgëoh 5G suìfìghqw/ hw rü oh s|o÷qh ìwdlw

prgìolvì hq 6G / frpph rq oh yrlw vxu od �jxuh 44161

441715 Xq prgëoh 6G srxu o*ìwxgh gx uìjodjh �q gh od vxvshqvlrq

Xqh prgìolvdwlrq frpsoëwh hq 6G d ìwì plvh dx srlqw +yrlu �jxuh 4417,1 Hooh hvw elhq

hqwhqgx ehdxfrxs soxv orxugh txh ohv ghx{ suìfìghqwhv/ sxlvtx*hooh frpsruwh 4746 qúxgv

4417 Ìwxghv vsìfl�txhv 54:

Ìwdshv vxu fkdqwlhu Vlpxodwlrq dyhf SFS

3 0 Dqflhq ìwdw Prgl�fdwlrq ghv orqjxhxuv ã ylgh ghv fkdñqhwwhv gh

od wudyìh fhqwudoh1

Prgl�fdwlrq ghv orqjxhxuv ã ylgh gh wrxwhv ohv

fkdñqhwwhv1

Prgl�fdwlrq gh od orqjxhxu ã ylgh gh od fkdñqhwwh

hqwuh od vhoohwwh hw o*dqfudjh ulyh jdxfkh1

4 0 Plvh hq sodfh ghv qrxyhdx{

fåeohv

Dfwlydwlrq ghv fkdñqhwwhv prgìolvdqw ohv qrxyhdx{

fåeohv1

Plvh hq whqvlrq gh fhv fkdñqhwwhv hq orqjxhxu ã ylgh1

5 0 Dffurfkdjh ã whqvlrq qxooh ghv

qrxyhoohv vxvshqwhv

Dfwlydwlrq ghv fkdñqhwwhv prgìolvdqw ohv qrxyhoohv

vxvshqwhv1

Lqfuìphqw gh fkdujh vxu oh wdeolhu srxu whqlu frpswh

gx srlgv ghv qrxyhdx{ froolhuv hw ìwulhuv1

6 0 Plvh ã orqjxhxu ghv vxvshqwhv

ulyh jdxfkh

Plvh hq whqvlrq ghv fkdñqhwwhv fruuhvsrqgdqwhv hq

orqjxhxu ã ylgh1

7 0 Plvh ã orqjxhxu ghv vxvshqwhv

ulyh gurlwh


orqjxhxu ã ylgh1

8 0 Plvh ã orqjxhxu g*xqh vxv0

shqwh vxu ghx{ gdqv od wudyìh

fhqwudoh/ fhooh0fl v*h�hfwxdqw hq <

vrxv0skdvhv


irufh dx qrhxg vxsìulhxu1 Ã fkdtxh vrxv0skdvh/ rq

dxjphqwh gh 46 wrqqhv fhwwh whqvlrq mxvtx*ã dwwhlq0

guh od orqjxhxu ã ylgh �qdoh1

9 0 Plvh ã orqjxhxu ghv dxwuhv

vxvshqwhv gh od wudyìh fhqwudoh


orqjxhxu ã ylgh1

: 0 Vxssuhvvlrq ghv dqflhqqhv

vxvshqwhv gdqv ohv wudyìhv

odwìudohv

Gìvdfwlydwlrq ghv fkdñqhwwhv fruuhvsrqgdqwhv1

; 0 Vxssuhvvlrq ghv dqflhqqhv

vxvshqwhv gdqv od wudyìh fhqwudoh

Gìvdfwlydwlrq ghv fkdñqhwwhv fruuhvsrqgdqw ã xqh

vxvshqwh vxu ghx{1

Gìwhqvlrq ghv dqflhqqhv fkdñqhwwhv uhvwdqwhv sdu

gìfuìphqw gh 5 wrqqhv +h{fhswìhv 5 fkdñqhwwhv gh

sduw hw g*dxwuh gx qrhxg fhqwudo,1

< 0 Vxssuhvvlrq gx qrhxg fhqwudo Gìvdfwlydwlrq ghv eduuhv prgìolvdqw oh qrhxg fhq0

wudo hw ghv 5 fkdñqhwwhv dxwrxu1

Gìfuìphqw gh fkdujh vxu oh wdeolhu vlpxodqw oh uh0

wudlw ghv dqflhqv froolhuv hw ìwulhuv1

43 0 Vxssuhvvlrq gh o*dqflhq fåeoh Gìvdfwlydwlrq ghv fkdñqhwwhv1

Wdeohdx 4414 = Ìwdshv gx wudqvihuw gh fkdujhv

54; Fkdslwuh 44 Xq h{hpsoh g*dssolfdwlrq uìhooh = oh Srqw gh Wdqfduylooh

SETRA--PCP-PH2 05.05.97/16.12.19 TANCARVILLE - MODELE 2D -30.0 0.0 0.0 1/759.6 1/759.6

PYLONE RIVE DROITE

Iljxuh 4416 = Prgëoh 5G26G

hw 5383 ìoìphqwv1 Xq prgëoh 6G ìwdlw fhshqgdqw lqglvshqvdeoh srxu o*ìwxgh gx uìjodjh �q

gh od vxvshqvlrq1

Hq h�hw/ ohv qrxyhoohv vxvshqwhv qh vrqw sdv yhuwlfdohv/ pdlv lqfolqìhv/ sxlvtxh ohxu

srlqw g*dwwdfkh vxu oh wdeolhu q*hvw sdv prgl�ì/ doruv txh ohv ixwxuv fåeohv sruwhxuv vh

wurxyhqw gh sduw hw g*dxwuh gh o*dqflhq fåeoh/ frpph lqgltxì vxu od �jxuh 44181

Gh fh idlw/ o*ìfduwhphqw hqwuh ohv ghx{ fåeohv g*xqh píph qdssh gìshqg gh od orqjxhxu

ghv vxvshqwhv hw gh od whqvlrq gdqv od vxvshqvlrq1 Rq pdñwulvh fhw ìfduwhphqw +srxu ìylwhu

ohv frqwdfwv hqwuh ohv vxvshqwhv, hq o*lpsrvdqw dx{ gl{ sdluhv gh vxvshqwhv fhqwudohv +glwhv

��vsìfldohv��, sdu o*lqwhupìgldluh gh exwrqv uholdqw ohv froolhuv gh fhv sdluhv gh vxvshqwhv1

Oh prgëoh 6G qrxv d shuplv gh gìwhuplqhu od orqjxhxu ã lpsrvhu ã fhv exwrqv srxu

rewhqlu xq ìfduwhphqw plqlpxp gh ;3 fp hqwuh ohv ghx{ fåeohv g*xqh píph qdssh +yrlu

�jxuh 4419,/ gh gìgxluh gh fhw ìfduwhphqw ohv orqjxhxuv ã ylgh ghv vxvshqwhv hq whqdqw

frpswh gh ohxu lqfolqdlvrq/ hw hq�q gh frqqdñwuh od irufh gh frpsuhvvlrq v*h{huêdqw gdqv

ohv exwrqv1

O*xwlolwì gx prgëoh hvw ìjdohphqw gh srxyrlu suhqguh hq frpswh ohv fkdujhphqwv

glvv|pìwultxhv +h�hw g*xq fkdqjhphqw rx g*xqh uxswxuh gh vxvshqwh vxu ohv vxvshqwhv

4418 Txhotxhv srlqwv sduwlfxolhuv gh prgìolvdwlrq 54<

SETRA--PCP-PH2 05.05.97/10.47.49 TANCARVILLE - MODELE 3D -45.0 0.0 -30.0 1/2672.5 1/2672.5

VUE D’ENSEMBLE

Iljxuh 4417 = Prgëoh 6G

yrlvlqhv,1

4418 Txhotxhv srlqwv sduwlfxolhuv gh prgìolvdwlrq

441814 Ohv vhoohv hq wíwh gh s|o÷qh

Shqgdqw ohv skdvhv gh wudqvihuw gh fkdujhv/ ohv qrxyhoohv vhoohv vrqw gìsodêdeohv/ gh idêrq

ã glplqxhu ohv h�ruwv gdqv od vwuxfwxuh1

D�q g*ìylwhu ohv gìsodfhphqwv huudwltxhv +gxv dx wud�f/ qrwdpphqw, hw glvv|pìwultxhv/

fhv vhoohv qh vrqw sdv oleuhv/ pdlv ã gìsodfhphqw krul}rqwdo orqjlwxglqdo frqwu÷oì sdu yìulqv1

Gdqv oh fdofxo/ ohv vhoohv hq wíwh gh s|o÷qhv vrqw oleuhv gh vh gìsodfhu orqjlwxglqdohphqw/

fh txl shuphw gh gìwhuplqhu oh gìsodfhphqw uhodwli hqwuh oh fåeoh hw od wíwh gx s|o÷qh

gdqv fkdtxh skdvh gh wudqvihuw1 Vxu fkdqwlhu/ dydqw fkdtxh skdvh +yrluh vrxv0skdvh,/ rq

gìsodfhud grqf ohv vhoohv gh od ydohxu gx gìsodfhphqw fdofxoì/ sxlv rq ohv eortxhud gdqv

fhwwh srvlwlrq mxvtx*dx gìexw gh od skdvh vxlydqwh1

Gx srlqw gh yxh prgìolvdwlrq/ ohv vhoohv vrqw uhsuìvhqwìhv sdu ghv duwlfxodwlrqv grqw

rq frqqdñw od pdwulfh gh uljlglwì ìoìphqwdluh/ hw txl suhqqhqw hq frpswh o*h{fhqwuhphqw


Nouvel le suspente

Anc ienne suspente

Nouveau câble por teur

Ancien câble por teur

Point d 'accrochage

sur le tabl ier

Iljxuh 4418 = Glvsrvlwlrq wudqvyhuvdoh ghv vxvshqwhv

hqwuh ohv ghx{ h{wuìplwìv gh o*duwlfxodwlrq/ srxu oh fdofxo gh od uhglvwulexwlrq ghv h�ruwv1

Fhwwh sulvh hq frpswh hvw qìfhvvdluh fdu/ vxlydqw ohv skdvhv/ oh gìsodfhphqw uhodwli hqwuh oh

fåeoh hw od wíwh gx s|o÷qh hvw lpsruwdqw/ hw/ frqmxjxì ã o*h�ruw qrupdo tx*lqgxlw oh fåeoh

sruwhxu gdqv oh s|o÷qh/ vrq h�hw +hq prphqw orqjlwxglqdo, q*hvw sdv gx wrxw qìjoljhdeoh1

441815 Ohv elhoohv g*dssxl

Oh wdeolhu uhsrvh/ dx qlyhdx ghv s|o÷qhv/ vxu ghv elhoohwwhv/ fdu od uìdfwlrq g*dssxl shxw

wuëv elhq íwuh gluljìh yhuv oh edv1

Od elhoohwwh d ìwì prgìolvìh whooh txhooh/ sdu xq ìoìphqw elduwlfxoì/ hw qrq sdv sdu xq

dssxl jolvvdqw1 Fh prgëoh/ soxv frpsoh{h g*xq srlqw gh yxh qxpìultxh/ shuphw g*íwuh soxv

surfkh gh od uìdolwì/ ã sursrv gh od gluhfwlrq gh od uìdfwlrq g*dssxl/ qrwdpphqw1

4419 Ohv sulqflsdx{ uìvxowdwv 554

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0,00 200,00 400,00 600,00 800,00 1000,00

Abscisse (m)

Espacement

horizontal

(m)

Calcul GTM

Calcul SETRA

Iljxuh 4419 = Srqw gh Wdqfduylooh 0 Hvsdfhphqw krul}rqwdo ghv fåeohv sruwhxuv

4419 Ohv sulqflsdx{ uìvxowdwv

441914 Ohv gìsodfhphqwv oruv gx wudqvihuw gh fkdujhv

Ohv sulqflsdx{ uìvxowdwv vrqw uhsulv gdqv oh wdeohdx 44151

441915 Ohv gl!fxowìv gh prgìolvdwlrq hw gh fdofxo

Xq sureoëph gh gìidxw gh frqyhujhqfh gh o*dojrulwkph jìqìudo gh uìvroxwlrq v*hvw sduirlv

srvì/ srxu gl�ìuhqwhv udlvrqv =

� Od udlghxu uhodwlyh ghv dssxlv sduidlwv d qìfhvvlwì xq vrlq wrxw sduwlfxolhu txdqw dx

frqglwlrqqhphqw gh od pdwulfh gh uljlglwì1 Hq h�hw/ od vwuxfwxuh ìwxglìh frqwlhqw ghv

ìoìphqwv grqw od udlghxu hvw wuëv judqgh +edvh gh s|o÷qh hqfdvwuìh,/ rx wuëv shwlwh

+vhooh oleuh hq gìsodfhphqw/ vxvshqwh,1 Od glvsdulwì ghv udlghxuv d�hfwh od suìflvlrq

qxpìultxh gx uìvxowdw gh o*lqyhuvlrq gx v|vwëph1

� Ohv urwdwlrqv ghv qúxgv uholìv h{foxvlyhphqw ã ghv ìoìphqwv elduwlfxoìv q*rqw sdv

gh vljql�fdwlrq sk|vltxh = rq shxw ìolplqhu ohv ìtxdwlrqv fruuhvsrqgdqw ã fhv gh0


Skdvh Gìsodfhphqwv oruv gx wudqvihuw gh fkdujhv +p,

ëfkh wdeolhu Gìsodfhphqw krul}rqwdo

S|o÷qh UG Vhooh s|o÷qh UG S|o÷qh UJ Vhooh s|o÷qh UJ

3 04/4;9 3/43: 3/333 03/53: 3/333

4 04/4;: 3/439 03/8:; 03/539 03/9<6

5 04/593 3/448 03/8;; 03/548 03/:33

6 04/586 3/447 03/8;9 03/547 03/:3:

7 04/576 3/446 03/8<: 03/546 03/:37

8 4/684 03/4:; 3/354 3/3<5 03/398

9 4/6:; 03/4;6 3/35; 3/3<: 03/38;

: 3/:46 3/434 03/554 03/4:7 03/5:<

; 3/47< 03/49; 3/494 3/3<< 3/3<;

< 3/345 03/594 3/587 3/4;7 3/4;7

43 3/3:4: 3/333 03/337 3/333 03/334

Wdeohdx 4415 = Gìsodfhphqwv oruv gx wudqvihuw gh fkdujhv

juìv gh olehuwì1 Hq sudwltxh/ fhv urwdwlrqv vrqw eortxìhv d�q g*ìylwhu wrxw lqflghqw

qxpìultxh1

� Hq wíwh ghv s|o÷qhv gh judqgh kdxwhxu/ vxu ohvtxhov v*dssxlhqw ohv vhoohv oleuhv gh vh

gìsodfhu orqjlwxglqdohphqw/ vrqw dssduxhv ghv rvfloodwlrqv hq gìsodfhphqw1 Lo d idoox

ohv gìwhfwhu d�q gh ohv wudlwhu frqyhqdeohphqw1

Sdu dloohxuv/ lo v*hvw dyìuì txh ohv uìvxowdwv hq gìsodfhphqwv ìwdlhqw wuëv vhqvleohv dx

uìjodjh lqlwldo gh od vxvshqvlrq hq whuph gh orqjxhxuv ã ylgh ghv fkdñqhwwhv = xqh prgl�0

fdwlrq h{wuíphphqw idleoh gh od orqjxhxu ã ylgh/ g*xqh ydohxu wuëv lqiìulhxuh ã od suìflvlrq

gh fkdqwlhu/ prgl�h ohv ëfkhv gh idêrq vljql�fdwlyh1 Sdu h{hpsoh/ xqh yduldwlrq uhodwlyh

gh orqjxhxu gh 3/334( gh od vxvshqvlrq lqgxlw xqh yduldwlrq gh ëfkh hq plolhx gh wdeolhu

gh 5 fp1

Ohv uìvxowdwv vrqw suìvhqwìv vxu ohv �jxuhv 441: ã 441441 Frpph rq shxw oh frqvwdwhu/ ohv

ìfduwv hqwuh od prgìolvdwlrq hw o*rxyudjh uìho uhvwhqw wuëv idleohv1 Lo idxw vrxoljqhu oh wudydlo

lpsruwdqw gh plvh dx srlqw gx prgxoh qrq olqìdluh vxu xq rxyudjh dxvvl lpsruwdqw1 Hq

h�hw/ ã o*ìsrtxh/ lo v*djlvvdlw gx prgëoh oh soxv lpsruwdqw mdpdlv wudlwì sdu SFS ghsxlv vd

plvh hq vhuylfh/ oh soxv jurv prgëoh suìfìghqw ìwdqw fhoxl gx Srqw gh Qrupdqglh1 Ghsxlv/

oh prgëoh gx Srqw gh Wdqfduylooh d ìwì vxssodqwì sdu fhoxl gx Srqw gh Ploodx/ hq frxuv

g*ìwxgh dx VHWUD/ dyhf SFS1 Oh fdv whvw txh frqvwlwxh oh uhpsodfhphqw gh od vxvshqvlrq

gx Srqw gh Wdqfduylooh ydolgh gh idêrq luuìixwdeoh od sulvh hq frpswh gh od qrq0olqìdulwì

jìrpìwultxh vxu ghv vwuxfwxuhv dxvvl lpsruwdqwhv hw frpsoh{hv1


ILFKH WHFKQLTXH

Sur�o hq orqj

Wudyxuhv = 4:9/66 p 0 93:/<55 p 4:7/938 p

Ud|rqv gh frxuexuh = 43 333 p 0 48 733 p 0 7 833 p

Sur�o hq wudyhuv Odujhxu hqwuh d{hv gh od vxvshqvlrq dfwxhooh = 49 p

Vxvshqvlrq dqflhqqh

Fåeohv dqflhqv = 5 fåeohv

4 fåeoh hvw irupì gh 89 wrurqv hq wudyìh fhqwudoh

93 wrurqv hq wudyìhv gh ulyh

4 wrurq hvw irupì gh 49< �ov ! 7/: pp

Uìvlvwdqfh ã od uxswxuh = 4 833 PSd

Prgxoh g*ìodvwlflwì gh fdofxo = 4:3 333 PSd

Vxvshqwhv dqflhqqhv = 5 qdsshv gh vxvshqwhv

4 vxvshqwh hvw irupìh gh 593 �ov ! 7/: pp wrurqqìv

vxu xq �o gh ! 8/; pp

Uìvlvwdqfh ã od uxswxuh = 4 833 PSd


Vxvshqvlrq qrxyhooh

Fåeohv qrxyhdx{ = 5�5 fåeohv4 fåeoh hvw irupì gh <3 wrurqv

4 wrurq hvw irupì gh 6: �ov ! 8/87 pp +kruv jdoydqlvdwlrq,

Uìvlvwdqfh ã od uxswxuh = 4 993 PSd +frhi1 gh shuwh 3/<8,


Frqwudlqwh pd{lpdoh g*xq fåeoh hq vhuylfh � 833 PSd

Vxvshqwhv qrxyhoohv = 5�5 qdsshv gh vxvshqwhv4 vxvshqwh d xqh vhfwlrq gh 5 466 pp2

Prgxoh g*ìodvwlflwì gh fdofxo = 488 333 PSd

Uìvlvwdqfh ã od uxswxuh = 4 993 PSd +frhi1 gh shuwh 3/<,

Wudfwlrq pd{lpdoh g*xqh vxvshqwh ã o*HOV � 93 w1

Wdeohdx 4416 = Ilfkh whfkqltxh


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Déplacement

(mm)

GTM

SETRA

CHANTIER

Iljxuh 441: = Gìsodfhphqw gh od wíwh gh s|o÷qh ulyh gurlwh sdu udssruw ã od yhuwlfdoh


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Position

(mm)

GTM

SETRA

CHANTIER

Iljxuh 441; = Srvlwlrq ghv vhoohv ulyh gurlwh sdu udssruw dx s|o÷qh +od srvlwlrq qxooh hvw

fhooh gh od vhooh ã od �q gh o*rsìudwlrq 5,


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Déplacement

(mm)

GTM

SETRA

CHANTIER

Iljxuh 441< = Gìsodfhphqw gh od wíwh gh s|o÷qh ulyh jdxfkh sdu udssruw ã od yhuwlfdoh

4419 Ohv sulqflsdx{ uìvxowdwv 55:

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Position

(mm)

GTM

SETRA

CHANTIER

Iljxuh 44143 = Srvlwlrq ghv vhoohv ulyh jdxfkh sdu udssruw dx s|o÷qh +od srvlwlrq qxooh hvw

fhooh ã od �q gh o*rsìudwlrq 5,

55; Fkdslwuh 44 Xq h{hpsoh g*dssolfdwlrq uìhooh = oh Srqw gh Wdqfduylooh

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Flèche

(mm)

GTM

SETRA

CHANTIER

Iljxuh 44144 = Yduldwlrq gh ëfkh gx wdeolhu dx qrhxg fhqwudo sdu udssruw ã vd srvlwlrq

hq �q g*rsìudwlrq 5

Fkdslwuh 45

Frqfoxvlrq jìqìudoh

Oh exw gh od wkëvh frqvlvwdlw hq o*lqwìjudwlrq ghv irqfwlrqqdolwìv gh fdofxo qrq olqìdluh

gdqv xq orjlflho h{lvwdqw/ SFS/ rulhqwì Rxyudjhv g*Duw hw edvì vxu od wkìrulh gh o*ìodvwlflwì

olqìdluh hw ghv shwlwv gìsodfhphqwv1 Rq shxw hvwlphu o*dyrlu frpsoëwhphqw dwwhlqw1 Hq

h�hw/ oh fkdslwuh 43 gìprqwuh txh qrwuh irupxodwlrq shuphw gh wudlwhu fruuhfwhphqw xq

judqg qrpeuh gh sureoëphv gdqv oh fkdps g*dssolfdwlrq gh SFS1 Hq rxwuh/ oh fkdslwuh 44

surxyh o*dvshfw rsìudwlrqqho h�hfwli gx surjudpph vxu ghv vwuxfwxuhv dxvvl lpsruwdqwhv

hw dx irqfwlrqqhphqw dxvvl frpsoh{h tx*xq srqw vxvshqgx/ who txh oh Srqw gh Wdqfduylooh1

Dx frxuv gh fhwwh ìwxgh/ dsuëv dyrlu ìwdeol xq ìwdw gh od txhvwlrq/ qrxv dyrqv ìwì

dphqìv ã sursrvhu +fkdslwuh 6, xqh h{whqvlrq dx 6G gh od irupxodwlrq lqlwldohphqw sodqh

gxh ã Mhqqlqjv/ hw txl d lqvsluì gh qrpeuhx{ dxwhxuv1 Qrxv dyrqv rswlplvì o*dqdo|vh gh

od qrq0olqìdulwì pdwìulhooh sursrvìh sdu Qdlw0Udedk/ d�q gh od uhqguh frpsdwleoh dyhf ohv

h{ljhqfhv g*xq frgh lqgxvwulho +fkdslwuh 7,1 Qrxv dyrqv gìyhorssì xq dojrulwkph jìqìudo

gh uìvroxwlrq grqw ohv vsìfl�fdwlrqv sulqflsdohv vrqw o*h!fdflwì hw od urexvwhvvh +fkdslwuh 8,1

G*dloohxuv/ lo frpsuhqg soxvlhxuv whfkqltxhv gh slorwdjh doodqw gh od glylvlrq gh od fkdujh

ã od gìwhfwlrq g*rvfloodwlrqv gdqv oh surfhvvxv gh frqyhujhqfh1 Sxlv/ qrxv qrxv vrpphv

lqwìuhvvìv dx{ h�hwv gl�ìuìv +fkdslwuh 9, d�q g*íwuh sduidlwhphqw frqiruph ã o*remhfwli

lqlwldo/ txl ìwdlw gh pdlqwhqlu oh fdofxo ghv h�hwv gl�ìuìv gdqv oh SFS qrq olqìdluh1 Xq

lpsruwdqw wudydlo d ìwì irxuql srxu prgìolvhu fruuhfwhphqw ohv duwlfxodwlrqv hq judqgv

gìsodfhphqwv/ hw srxu jìuhu ohv skdvhv gh whqvlrq0gìwhqvlrq txh o*rq shxw revhuyhu gdqv

ohv vxvshqwhv gh srqwv vxvshqgxv +fkdslwuh :,1 Qrxv qrxv vrpphv doruv dshuêxv txh o*rq

srxydlw sursrvhu gh qrxyhoohv irqfwlrqqdolwìv dx{ xwlolvdwhxuv/ frpph oh fdofxo ghv prghv

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41 od qxpìurwdwlrq hw od idfwrulvdwlrq orjltxh/

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� vrlw rq udlvrqqh vxu od ghpl0odujhxu gh edqgh/ txh o*rq fkhufkh ã plqlplvhu +dojr0

ulwkphv gh Fxwkloo Pdf Nhh hw Fxwkloo Pdf Nhh lqyhuvh,

� vrlw rq udlvrqqh vxu od odujhxu ghv iurqwv = ã sduwlu g*xq suhplhu iurqw gh idleoh odujhxu/

rq hvvdlh gh pdlqwhqlu od odujhxu gx iurqw od soxv shwlwh srvvleoh +Plqlplvdwlrq gh

o*Dffurlvvhphqw gh Iurqw/ dojrulwkphv gh Nlqj/ gh Oìy|,1

E1514 Fxwkloo Pdf Nhh +FP, hw lqyhuvh +UFP,

Oh sulqflsh hvw oh vxlydqw =

41 Fkrlvlu xq vrpphw u g*h{fhqwulflwì pd{lpdoh hw d�hfwhu y� # u1

51 Srxu l @ 4> 5> ===> q wurxyhu wrxv ohv vrpphwv dgmdfhqwv ã y� hw ohv qxpìurwhu gdqv

o*ruguh furlvvdqw gh ohxu ghjuì1

61 Od qxpìurwdwlrq lqyhuvh hvw grqqìh sdu |� @ y?3�n� srxu l yduldqw gh 4 ã q1

E1515 Plqlplvdwlrq gh o*dffurlvvhphqw gh iurqw/ dojrulwkph gh Oìy|/ do0

jrulwkph gh Nlqj

41 Fkrlvlu xq vrpphw u gh ghjuì plqlpdo hw d�hfwhu y� # u1

51 Frqqdlvvdqw ohv vrpphwv y� ã y�3�/ srxu wrxv ohv vrpphwv | qrq qxpìurwìv +Oìy|,

rx srxu wrxv ohv vrpphwv | gx iurqw +Nlqj,/ rq fdofxoh o*dffurlvvhphqw gh iurqw

rx oh iurqw/ hw rq d�hfwh ã y� oh vrpphw txl plqlplvh oh iurqw +Oìy| hw Nlqj, rx

o*dffurlvvhphqw gh iurqw +PDI,1

E16 Lpsoìphqwdwlrq ghv dojrulwkphv 578

G*dsuëv od olwwìudwxuh vsìfldolvìh/ o*dojrulwkph gh Fxwkloo Pdf Nhh hw lqyhuvh vrqw ghv

dojrulwkphv dvvh} udslghv pdlv txl rswlplvhqw pdo o*hqyhorssh gh od pdwulfh1 Ohv phloohxuv

uìvxowdwv vrqw rewhqxv dyhf UFP1 O*dojrulwkph gh Oìy| hvw xq dojrulwkph soxv frþwhx{/

pdlv lo r�uh o*dydqwdjh gh wuëv elhq rswlplvhu o*hqyhorssh gh od pdwulfh1 Vrq whpsv g*h{ì0

fxwlrq hvw dvvh} orqj1 O*dojrulwkph gh Nlqj/ gdqv od phvxuh rü lo uìgxlw o*hqvhpeoh gh

uhfkhufkh gh vrpphwv fdqglgdwv dx qxpìur vxlydqw sdu udssruw ã fhoxl gh Oìy|/ vhpeoh

uhsuìvhqwhu d sulrul xq erq frpsurplv hqwuh oh frþw hw o*h!fdflwì1

E16 Lpsoìphqwdwlrq ghv dojrulwkphv

E1614 Fxwkloo Pdf Nhh +FP, hw lqyhuvh +UFP,

Lo idxw gìwhuplqhu oh vrpphw gh gìsduw1 Srxu fhod/ rq xwlolvh xq dojrulwkph txl qh grqqh

sdv irufìphqw xq vrpphw sìulskìultxh +g*h{fhqwulflwì pd{lpdoh, pdlv txl grqqh xq erq

srlqw gh gìsduw srxu FP hw UFP1

41 Rq fkrlvlw xq vrpphw duelwudluh u1

51 Rq frqvwuxlw od vwuxfwxuh gh qlyhdx{ '+u, d|dqw srxu udflqh u/ fh txl grqqh ohv

gl�ìuhqwv qlyhdx{ Of+u,/ O�+u,/ 111/ O,Eo�+u,1

61 Rq fkrlvlw xq vrpphw y gdqv O,Eo,+u, gh ghjuì plqlpxp1

71 Rq frqvwuxlw od vwuxfwxuh gh qlyhdx{ d|dqw srxu udflqh y1 Vl o+y, A o+u, doruv rq

d�hfwh u# y hw rq uhwrxuqh ã o*ìwdsh 61

81 Oh vrpphw gh gìsduw u hvw glw vrpphw svhxgr0sìulskìultxh/ rx wrsrorjltxhphqw

h{wuíph1

91 Rq d�hfwh y� # u1

:1 Srxu l @ 4> 5> ===> q rq wurxyh wrxv ohv vrpphwv dgmdfhqwv ã y� hw rq ohv qxpìurwh gdqv

o*ruguh furlvvdqw gh ohxu ghjuì txh o*rq dxud fdofxoì dx suìdodeoh1

;1 Rq rewlhqw od qxpìurwdwlrq y�> ===> y? txl hvw fhooh gh Fxwkloo Pdf Nhh

<1 Od qxpìurwdwlrq lqyhuvh hvw grqqìh sdu |� @ y?3�n� srxu l yduldqw gh 4 ã q1

E1615 Dojrulwkph gh Oìy|

41 Rq gìwhuplqh oh ghjuì gh fkdtxh vrpphw hw rq hq vìohfwlrqqh xq gh ghjuì plqlpxp =

y�

579 Dqqh{h E Rswlplvdwlrq gh od qxpìurwdwlrq ghv qúxgv

51 Srxu l yduldqw gh 4 ã q � 4/ rq fkrlvlw xq vrpphw qrq qxpìurwì |1 Rq fdofxoh oh

yrlvlqdjh �+iy�> ===> y�> |j, hw I� +|, @ �+iy�> ===> y�> |j,qiy�> ===> y�> |j1 Rq fkrlvlw oh

vrpphw | txl plqlplvh Fdug +I� +|,, = y�n� # |1

E1616 Dojrulwkph gh Nlqj

F*hvw oh píph txh fhoxl gh Oìy| vdxi txh | hvw sulv gdqv �+iy�> ===> y�j,qiy�> ===> y�j1

E1617 Dojrulwkph gh Plqlplvdwlrq gh o*Dffurlvvhphqw gh Iurqw

41 Rq gìwhuplqh oh ghjuì gh fkdtxh vrpphw hw rq hq vìohfwlrqqh xq gh ghjuì plqlpxp =

y�

51 Srxu l yduldqw gh 4 ã q� 4/ rq fkrlvlw xq vrpphw qrq qxpìurwì |1

61 Vrlw U� oh frpsoìphqwdluh gh �+iy4> ===> ylj,

71 Rq fdofxoh W�+|, @ U� _ �+i|j,

81 O*dffurlvvhphqw gh iurqw hvw grqqì sdu = i�n� � i� @ ��+|, dyhf =

��+|, @ Fdug+W�+|,,� 4 vl | 5 �+iy�> ===> y�j,qiy�> ===> y�j��+|, @ Fdug+W�+|,, vlqrq1

91 Rq fkrlvlw oh vrpphw | txl plqlplvh ��+|, = y�n� # |1

Xqh yduldqwh gh fhw dojrulwkph frqvlvwh ã suhqguh | gdqv �+iy�> ===> y�j,qiy�> ===> y�j1

E17 Ìydoxdwlrq ghv dojrulwkphv

Txdwuh dojrulwkphv rqw ìwì lpsoìphqwìv/ ã vdyrlu =

41 Plqlplvdwlrq gh o*dffurlvvhphqw gh iurqw fodvvltxh +PDIF,>

51 Xqh yduldqwh gh PDIF/ dyhf uhfkhufkh gx qrxyhdx vrpphw gdqv oh iurqw +PDII,>

61 Dojrulwkph gh Oìy| +OHY\,>

71 Dojrulwkph gh Nlqj +NLQJ,1

Fhv dojrulwkphv vrqw uhjurxsìv gdqv xq sdtxhwdjh gh uhqxpìurwdwlrq/ rü o*rq wurxyh

ìjdohphqw ghx{ pìwkrghv gh uhfkhufkh gh suhplhu vrpphw gh uhqxpìurwdwlrq =

41 Vrpphw gh ghjuì plqlpxp +GHJPLQ,>

51 Vrpphw svhxgr0sìulskìultxh +SVHSHU,1

E17 Ìydoxdwlrq ghv dojrulwkphv 57:

srlqw gh gìsduw = GHJPLQ PDIF PDIY NLQJ OHY\

Hqyhorssh �qdoh/ gldjrqdoh frpsulvh 576 693 578 745 578 745 575 469

Whpsv FSX +vxu Frqyh{, 876 v ;: v 653 v 8 49< v

Wdeohdx E14 = Srqw gh Od Urfkh Ehuqdug

srlqw gh gìsduw = SVHSHU PDIF PDIY NLQJ OHY\

Hqyhorssh �qdoh/ gldjrqdoh frpsulvh 47; :49 583 965 583 965 57< 597

Whpsv FSX +vxu Frqyh{, 879 v ;; v 5;8 v 8 474 v

Wdeohdx E15 = Srqw gh Od Urfkh Ehuqdug

Oh fkrl{ ghv pìwkrghv hvw dlqvl ruwkrjrqdo = fkdtxh dojrulwkph gh uhqxpìurwdwlrq

shxw suhqguh srxu vrpphw gh gìsduw xq vrpphw fdofxoì dyhf GHJPLQ rx SVHSHU1 Hq

rxwuh/ oh vrpphw gh gìsduw gh od uhqxpìurwdwlrq shxw íwuh lpsrvì sdu o*xwlolvdwhxu1

Srxu ìydoxhu fhv dojrulwkphv/ qrxv dyrqv h{wudlw od qxpìurwdwlrq gh ghx{ prgëohv gh

grqqìhv lpsruwdqwv = oh Srqw gh Od Urfkh Ehuqdug hw oh Srqw gh Qrupdqglh1 Qrwrqv txh

od qxpìurwdwlrq gh Od Urfkh Ehuqdug ìwdlw ��euxwh��/ vdqv uhfkhufkh sduwlfxolëuh srxu

dpìolruhu od wdlooh gh od pdwulfh gh uljlglwì1 Od qxpìurwdwlrq gx Srqw gh Qrupdqglh/

sdu frqwuh/ txdqw ã hooh/ dydlw ìwì rswlplvìh ��ã od pdlq��/ gh idêrq lqwxlwlyh hw qrq

dxwrpdwltxh1 Fhwwh qxpìurwdwlrq d o*dydqwdjh gh qrxv r�ulu xqh uìiìuhqfh hq pdwlëuh

g*rswlplvdwlrq gh qxpìurwdwlrq1

E1714 Srqw gh Od Urfkh Ehuqdug

Srqw gh Od Urfkh Ehuqdug = :65 qúxgv/ 4347 ìoìphqwv1

Hqyhorssh lqlwldoh/ gldjrqdoh frpsulvh = 5 343 5:91

O*hqyhorssh lqlwldoh fruuhvsrqg dx qrpeuh gh whuphv orjltxhphqw qrq qxov gh od pd0

wulfh gh uljlglwì/ rx hqfruh od orqjxhxu gx yhfwhxu frqwhqdqw od pdwulfh gh uljlglwì1

Rq uhpdutxh wrxw gh vxlwh txh od qxpìurwdwlrq lqlwldoh hvw orlq g*íwuh rswlpdoh/ fdu

oh jdlq uhodwli dyhf PDFI 0 SVHSHU ydxw 46/841 Rq qrwh ìjdohphqw txh o*dojrulwkph gh

Oìy| hvw wuëv hw píph wurs frþwhx{1

PDIY hw NLQJ/ txl uhfkhufkhqw wrxv ghx{ gdqv oh iurqw/ grqqhqw od píph hqyhorssh

�qdoh/ PDIY ìwdqw oh soxv udslgh gh wrxv1 Od phloohxuh hqyhorssh hvw rewhqxh dyhf

PDIF0SVHSHU1

E1715 Srqw gh Qrupdqglh

Srqw gh Qrupdqglh = 4966 qúxgv/ 5347 ìoìphqwv1

Hqyhorssh lqlwldoh/ gldjrqdoh frpsulvh = 868 3;<1

57; Dqqh{h E Rswlplvdwlrq gh od qxpìurwdwlrq ghv qúxgv

srlqw gh gìsduw = GHJPLQ PDIF PDIY NLQJ OHY\

Hqyhorssh �qdoh/ gldjrqdoh frpsulvh 5<8 65< 697 566 :94 ;;< 0

Whpsv FSX +vxu Frqyh{, 93<4 v <4< v 6 645 v dedqgrq = A <33 pq

Wdeohdx E16 = Srqw gh Qrupdqglh

srlqw gh gìsduw = SVHSHU PDIF PDIY NLQJ OHY\

Hqyhorssh �qdoh/ gldjrqdoh frpsulvh 647 994 783 786 :;3 438 0

Whpsv FSX +vxu Frqyh{, 9 3<9 v <63 v 7 66: v 0

Wdeohdx E17 = Srqw gh Qrupdqglh

Ohv whpsv gh fdofxo vrqw qhwwhphqw soxv lpsruwdqwv1 PDIF hw PDIY vhpeohqw

uhsuìvhqwhu ohv phloohxuv frpsurplv1 Rq qrwh dx sdvvdjh txh NLQJ gìjudgh od qxpìur0

wdwlrq lqlwldoh/ txl/ srxu xqh qxpìurwdwlrq ��pdqxhooh�� ìwdlw wuëv erqqh1 OHY\/ txdqw

ã oxl/ q*duulyh sdv ã frqfoxuh dsuëv 48 khxuhv FSX vxu Frqyh{1

Oh srlqw gh gìsduw fdofxoì sdu GHJPLQ grqqh ohv phloohxuv uìvxowdwv1

E18 Frqfoxvlrq

Oh srlqw gh gìsduw gh od qxpìurwdwlrq q*hvw sdv qhxwuh hq fh txl frqfhuqh o*hqyhorssh �qdoh/

g*xqh sduw hw g*dxwuh sduw/ oh fkrl{ hqwuh GHJPLQ hw SVHSHU q*hvw sdv v|vwìpdwltxh1

Ghx{ dojrulwkphv vh gìwdfkhqw qhwwhphqw = PDIF srxu od txdolwì gh vrq rswlplvdwlrq

hw PDIY srxu vd udslglwì1 Hwdqw grqqìh od ylwhvvh vdwlvidlvdqwh gh PDIY/ o*dojrulwkph

UFP q*d sdv ìwì frgì = vrq h!fdflwì hvw qhwwhphqw prlqguh srxu xq jdlq hq ylwhvvh txl

qh vhudlw sdv wuëv lpsruwdqw1

THESE SOUTENUE DEVANT L’INSTITUT NATIONAL DES SCIENCES APPLIQUEES DE LYON

NOM : ROBERTDATE de SOUTENANCE

Prénom : Florent Jean-Claude Germain2 avril 1999

TITRE :

Contribution à l’analyse non linéaire géométrique et matérielle des ossatures spatiales en Génie civil.

Application aux ouvrages d’art.

NATURE : Doctorat Numéro d'ordre : 99ISAL0032

Formation doctorale : Génie civil : Sols, matériaux, structures, physique du bâtiment

Ecole doctorale : MECANIQUE, ENERGETIQUE, GENIE CIVIL, ACOUSTIQUE (MEGA)

Cote B.I.U. - Lyon : T 50/210/19 / et bis CLASSE :

RESUME : En Génie civil, la tendance actuelle à concevoir des structures de plus en plus élancées et le besoind’optimiser les quantités de matériau, conduisent les projeteurs à mener des calculs prenant en compte les non-linéarités du système. En

ouvrage d’art, considérer le couplage entre les non-linéarités matérielle et géométrique est impératif, d’où la nécessité de disposer d’outils

informatiques adaptés au métier et résolvant ce problème. Nous proposons une formulation basée sur les coordonnées convectées spatiales en

décomposant le mouvement d’une particule en un mouvement de corps rigide de grande amplitude et un mouvement modéré induisant les

déformations. Ceci permet de prendre en compte les grands déplacements et les grandes rotations pour des éléments de poutre

tridimensionnels alors que la non-linéarité matérielle repose sur l’hypothèse des petites déformations. On peut ainsi modéliser les ossatures

spatiales composées de métal, de béton armé ou de béton précontraint. Les effets différés, à savoir le retrait et le fluage du béton ainsi que la

relaxation des aciers de précontrainte, sont intégrés dans le schéma général de résolution. Pour simuler la construction des ouvrages d’art et

étudier leur schéma statique, nous développons la prise en compte des grands déplacements pour les articulations ainsi qu’un élément

simulant correctement le comportement d’un cible pesant, même lors des phases où il est détendu. Nous développons le calcul d’instabilité

élastique à partir de la matrice des contraintes initiales issue de notre formulation. Nous proposons l’extension possible de l’algorithme de

résolution du cas de chargement statique au cas dynamique. Nous validons notre approche en confrontant les résultats de nos calculs à des

mesures expérimentales ou à la solution théorique de cas tests représentatifs. Enfin, un exemple d’utilisation du programme sur un projet réel

est présenté : la simulation du remplacement du système de suspension du Pont de Tancarvillc..

MOTS-CLES : Constructions à ossatures, éléments finis, poutre, béton armé, béton précontraint, non-linéarité,

ouvrages d’art, grands déplacements.

Laboratoire(s) de recherche :

Laboratoire GéoMatériaux de I’ENTPE et le SETRA.

Directeur de Thèse : Patrick ROYIS et Claude-Henri LAMARQUE

Président du Jury : Pierre LAREALComposition du jury : Mme COURTADE, MM. ROELANDT, de VILLE de GOYET, ROYIS, LAMARQUE.

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