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gh uhfkhufkh hw gh gìyhorsshphqw txh m*dl sx uìdolvhu/ hq xqh wkëvh gh grfwrudw1
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txl hvw o*dxwhxu gh txhotxhv �jxuhv gh fh grfxphqw/ srxu ohxu vrxwlhq txdvl txrwlglhq hw
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fdofxov suhqdqw hq frpswh ohv qrq0olqìdulwìv gx v|vwëph1 Hq rxyudjh g*duw/ frqvlgìuhu oh
frxsodjh hqwuh ohv qrq0olqìdulwìv pdwìulhooh hw jìrpìwultxh hvw lpsìudwli/ g*rü od qìfhvvlwì
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gìfrpsrvdqw oh prxyhphqw g*xqh sduwlfxoh hq xq prxyhphqw gh frusv uljlgh gh judqgh
dpsolwxgh hw xq prxyhphqw prgìuì lqgxlvdqw ohv gìirupdwlrqv1 Fhfl shuphw gh suhqguh
hq frpswh ohv judqgv gìsodfhphqwv hw ohv judqghv urwdwlrqv srxu ghv ìoìphqwv gh srxwuh
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gìirupdwlrqv1 Rq shxw dlqvl prgìolvhu ohv rvvdwxuhv vsdwldohv frpsrvìhv gh pìwdo/ gh
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xvh ri wkh surjudpph rq d uhdo surmhfw lv vkrzq = wkh vlpxodwlrq ri wkh uhsodfhphqw ri
wkh vxvshqvlrq v|vwhp ri wkh Wdqfduylooh Eulgjh1
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Devwudfw 44
Wdeoh ghv pdwlëuhv 46
Olvwh ghv �jxuhv 4<
Olvwh ghv wdeohdx{ 56
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
Qrxv qrxv lqwìuhvvrqv soxv sduwlfxolëuhphqw dx frxsodjh hqwuh ohv ghx{ qrq0olqìdulwìv1
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/
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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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6G suhqdqw hq frpswh oh skdvdjh ìyhqwxho gh frqvwuxfwlrq/ qrxv qh qrxv lqwìuhvvhurqv
tx*dx{ ìoìphqwv gh w|sh ��srxwuh��/ fkdfxq ghv ghx{ qúxgv gh fhw ìoìphqw glvsrvdqw
ghv 9 ghjuìv gh olehuwì fodvvltxhv/ ã vdyrlu 6 wudqvodwlrqv hw 6 urwdwlrqv1 Qrxv qh qrxv
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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gh fhwwh dssurfkh1 O*dxwhxu hvw frqvflhqw gx idlw txh oh fhqwurògh gh od vhfwlrq shxw qh
sdv fròqflghu dyhf o*d{h gh uìiìuhqfh gh o*ìoìphqw1 O*dssurfkh hvw wdqjhqwh/ hw ohv orlv gh
frpsruwhphqw vrqw grqqìhv srlqw sdu srlqw +pxowl0olqìdluhv,1 Od uljlglwì gx eìwrq whqgx
hqwuh ghx{ �vvxuhv hvw qìjoljìh/ hw od ydolgdwlrq gx prgëoh hvw idlwh vxu txhotxhv srxwuhv
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+vxssrvìh frqvwdqwh oh orqj gh o*ìoìphqw/ wrxw frpph od uljlglwì h{whqvlrqqhooh, hvw ìydoxìh
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oh grpdlqh gh ydolglwì dx{ gìsodfhphqwv prgìuìv1 Fhwwh irupxodwlrq vh fdudfwìulvh sdu
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yhuvìh/ grqqh od pdwulfh gh uljlglwì ìoìphqwdluh/ txl hvw hqvxlwh dvvhpeoìh gdqv od pdwulfh
gh uljlglwì jìqìudoh/ gh idêrq fodvvltxh1
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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
ìoìphqwv �qlv1 Lo lqwurgxlw xq ìoìphqw �ql glw gh Pdujxhuuh prgl�ì ã : ghjuìv gh olehuwì/
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
63 Fkdslwuh 4 Lqwurgxfwlrq
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
Hq 4<<3/ Qdlw0Udedk ^Qdl<3` srxuvxlw oh wudydlo gh Juhodw ^Juh:;e`/ txl/ udsshorqv0oh/
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
ydolgì vxu gh qrpeuhx{ fdv h{sìulphqwdx{/ d|dqw qìdqprlqv ghv gìsodfhphqwv prgìuìv1
Ohv gìyhorsshphqwv ohv soxv uìfhqwv vxu oh eìwrq dupì frqfhuqhqw od irupxodwlrq gh
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
à 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
à 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
Qrxv qrxv uhvwuhljqrqv yrorqwdluhphqw dx{ orlv gh frpsruwhphqw gh pdwìuldx ìwdeolhv
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
65 Fkdslwuh 4 Lqwurgxfwlrq
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
69 Fkdslwuh 5 Prgìolvdwlrq pìfdqltxh
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
à 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,
73 Fkdslwuh 5 Prgìolvdwlrq pìfdqltxh
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
75 Fkdslwuh 5 Prgìolvdwlrq pìfdqltxh
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
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shwlwv gìsodfhphqwv1 Qrwuh wudydlo vh vlwxh gluhfwhphqw gdqv oh surorqjhphqw gh fhoxl gh
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v*hvw gìsodfì hq L +uhvshfwlyhphqw M,1 Rq lqwurgxlw od qrwlrq gh uhsëuh lqwulqvëtxh/ qrwì {|
grqw oh suhplhu d{h uholh oh qúxg L dx qúxg M 1 Fh uhsëuh hvw o*xq ghv uhsëuhv frurwdwlrqqhov
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frrugrqqìhv frqyhfwìhv/ rx frrugrqqìhv qdwxuhoohv1 Hoohv vrqw dx qrpeuh gh 6 srxu xqh
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�I? @ ^N?` =�V? +61<8,
rü �I? hvw xq dffurlvvhphqw g*h�ruwv gdqv oh uhsëuh lqwulqvëtxh hw �V? o*dffurlvvhphqw
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od pr|hqqh ghv uhsëuhv sulqflsdx{ g*lqhuwlh ghv vhfwlrqv ruljlqh hw h{wuìplwì gh o*ìoìphqw1
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617 Ohv frrugrqqìhv frqyhfwìhv 8<
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g*xqh vhfwlrq ã xqh dxwuh1 Rq qrwh sdu h�% / h�+ / h�5 +uhvshfwlyhphqw h�% / h�+ / h�5, oh uhsëuh
ruwkrqrupì gluhfw/ olì dx qúxg L +uhvshfwlyhphqw M,1 O*d{h h�%+uhvshfwlyhphqw h�%, hvw
wdqjhqw hq L +uhvshfwlyhphqw M, ã od �euh pr|hqqh gh o*ìoìphqw dsuëv gìirupdwlrq1 Ohv
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��%f / ��+f hw ��5f wrxw hq dvvxudqw od srvvlelolwì gh pdqlsxohu fhv 6 uìhov frpph xq yhfwhxu
hvw od vxlydqwh ^Lsvl<9` = ��%f / ��+f hw ��5f vrqw ohv frpsrvdqwhv g*xq yhfwhxu gluhfwhxu gh
o*d{h gh od urwdwlrq/ hw o*dpsolwxgh gh od urwdwlrq hvw ��f @
t�2�%f
. �2�+f. �2�5f
/ f*hvw0ã0
95 Fkdslwuh 6 Qrq0olqìdulwì jìrpìwultxh
gluh od qruph gx yhfwhxu
3EEC
��%f
��+f
��5f
4FFD1 Od urwdwlrq hvw frpswìh srvlwlyhphqw gdqv oh vhqv
wuljrqrpìwultxh dxwrxu gx yhfwhxu
3EEC
��%f
��+f
��5f
4FFD = Fhwwh gì�qlwlrq hvw ìjdohphqw dgrswìh gdqv
oh orjlflho S|orvwde ^S|o<:`/ txl fdofxoh ohv slohv hw s|o÷qhv dx dpehphqw vrxv vroolflwdwlrqv
qrupdohv +h�ruw qrupdo hw 5 prphqwv ìfklvvdqwv,1 Euhkphu ìwdlw frqiurqwì dx píph
sureoëph g*ruguh g*dssolfdwlrq ghv 5 urwdwlrqv fruuhvsrqgdqw dx{ 5 prphqwv ìfklvvdqwv1
à od vxlwh g*xqh uì h{lrq frppxqh/ f*hvw fhwwh gì�qlwlrq vlpsoh hw dvwxflhxvh ghv urwdwlrqv
txl d ìwì xwlolvìh gdqv ohv ghx{ orjlflhov1
Rq frqvwuxlw od pdwulfh dqwlv|pìwultxhk��f�
l=
k��f�
l@
5997
3 ���5f ��+f
��5f 3 ���%f���+f ��%f 3
6::8 +61443,
Rq prqwuh doruv txh =
^S�f ` @ h
kw�f�
l+61444,
Hq gìyhorssdqw +61444,/ rq d =
^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
k��f�
l@
5997
3 ���5f ��+f
��5f 3 ���%f���+f ��%f 3
6::8 +61448,
617 Ohv frrugrqqìhv frqyhfwìhv 96
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,
97 Fkdslwuh 6 Qrq0olqìdulwì jìrpìwultxh
^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;,
617 Ohv frrugrqqìhv frqyhfwìhv 98
α y
z
y
α
x
x
xβ
β
z
I
0
0
1
1
0
1
J
J1
J0
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
$+61473,
^Sq` hvw gì�qlh sdu =
+[,E%�+�5f�@ ^Sq` = +[,E%+�5��
+61474,
99 Fkdslwuh 6 Qrq0olqìdulwì jìrpìwultxh
γy
x
z
I
1
1
J
z
yγ
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
e @I@2nK2
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
@2nK2
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
��+
��5
�%
��+
��5
4FFFFFFFFFD
+61483,
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
9; Fkdslwuh 6 Qrq0olqìdulwì jìrpìwultxh
Iljxuh 61; = Surmhfwlrq gh o*ìoìphqw gdqv oh sodq {} gx uhsëuh lqwulqvëtxh {|}
fruuhvsrqgdqw ã V? gdqv oh uhsëuh lqwulqvëtxh =
I? @
3EEEEEEEEEC
Q�
P�+
P�5
P�%
P�+
P�5
4FFFFFFFFFD
+61486,
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% @
3EEEEEEEEEEEEEEEEEEEEEEEEEC
Q�
W�+
W�5
P�%
P�+
P�5
Q�
W�+
W�5
P�%
P�+
P�5
4FFFFFFFFFFFFFFFFFFFFFFFFFD
+61487,
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 ` @
59999999997
�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�` @
5997
4 ���5 ��+
��5 4 ���%���+ ��% 4
6::8 +6148<,
^S�` @
5997
4 ���5 ��+
��5 4 ���%���+ ��% 4
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�
= ^S�f` @|^Skq� ` = ^S�f ` +61497,
^S�` @ ^Skq� `3�
= ^S�f ` @|^Skq�` = ^S�f ` +61498,
Fhv ghx{ uhodwlrqv pdwulflhoohv vrqw ìtxlydohqwhv ã 5�< ìtxdwlrqv vfdodluhv uholdqw ��% / ��+ /
��5 / ��% / ��+ / ��5 ã x/ y/ z/ ��%f / ��+f / ��5f / ��%f / ��+f / ��5f 1 Srxu oh qúxg L/ sdu h{hpsoh/ qrxv
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
Hwdqw grqqìh o*h{suhvvlrq gh ^S�`/ rq yd surfìghu ã xqh lghqwl�fdwlrq ghv frpsrvdqwhv
gh ^S�`/ hq vh olplwdqw ã ^S�`�c2/ ^S�`�c�/ ^S�`2c�/ ^S�`2c�/ ^S�`�c�/ ^S�`�c21 Dsuëv gìyhorsshphqw/
fkdfxqh gh fhv frpsrvdqwhv q*d sdv oh píph frþw hq whuphv g*rsìudwlrqv dulwkpìwltxhv
rx g*ìydoxdwlrqv gh irqfwlrqv wuljrqrpìwultxhv =^S�`�c2 = 56 dgglwlrqv 85 pxowlsolfdwlrqv 9 irqfwlrqv 5; d�hfwdwlrqv
^S�`2c� = 87 dgglwlrqv 46: pxowlsolfdwlrqv 43 irqfwlrqv 84 d�hfwdwlrqv
^S�`�c� = 57 dgglwlrqv 83 pxowlsolfdwlrqv 9 irqfwlrqv 5: d�hfwdwlrqv
^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
qrq0xqlflwì gh od vroxwlrq surylhqw wrxw vlpsohphqw gx idlw txh o*h{suhvvlrq gh ^S�` d ìwì
olqìdulvìh1 Oh fkrl{ gh fhv ìtxdwlrqv q*hvw sdv lqgl�ìuhqw srxu od irupxodwlrq1 Hq h�hw/
^S�` hw ^S�` qh vrqw sdv ghv pdwulfhv ruwkrjrqdohv1 Vl o*rq dydlw frqvhuyì od irupxodwlrq
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
:5 Fkdslwuh 6 Qrq0olqìdulwì jìrpìwultxh
tx*hoohv hqjhqguhurqw ghv fdofxov soxv vlpsohv1 Rq d grqf =
��% @ � �| ^Skq� ` = ^S�f `�2c� +61499,
��+ @�|^Skq� ` = ^S�f `
��c�
+6149:,
��5 @ � �| ^Skq� ` = ^S�f `��c2 +6149;,
��% @ � �| ^Skq� ` = ^S�f`�2c� +6149<,
��+ @�|^Skq� ` = ^S�f `
��c�
+614:3,
��5 @ � �| ^Skq� ` = ^S�f`��c2 +614:4,
Rq shxw uhpdutxhu tx*hq idlvdqw xq gìyhorsshphqw olplwì ã o*ruguh 4/ f*hvw0ã0gluh hq
olqìdulvdqw ohv ìtxdwlrqv uhwhqxhv/ rq rewlhqw =
��% @ ��%f � � +614:5,
��+ @ ��+f � � +614:6,
��5 @ ��5f � � +614:7,
��% @ ��%f � � +614:8,
��+ @ ��+f � � +614:9,
��5 @ ��5f � � +614::,
Fhfl frq�uph oh uìvxowdw vxlydqw = hq shwlwv gìsodfhphqwv hw shwlwhv urwdwlrqv/ ohv urwdwlrqv
shupxwhqw hw vrqw dgglwlyhv1
Od gl�ìuhqfldwlrq ghv irqfwlrqv h/ ��+ / ��5 / �%/ ��+ / ��5 qrxv shuphw g*ìfuluh vrxv iruph
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
7V? @ ^E` =7 V� +614:;,
rü ^E` uhsuìvhqwh xqh pdwulfh +9� <, gh wudqvirupdwlrq jìrpìwultxh/ sduidlwhphqw dqd0
617 Ohv frrugrqqìhv frqyhfwìhv :6
orjxh ã fhooh lqwurgxlwh hq 5G1
^E` @
599999999999999999999999999999997
Ch
Cx
Ch
Cy
Ch
Cz3 3 3 3 3 3
C��+
Cx
C��+
Cy
C��+
Cz
C��+
C��%f
C��+
C��+f
C��+
C��5f3 3 3
C��5Cx
C��5Cy
C��5Cz
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C�%
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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?= ^E` =�V� @
|I�=�V� +614;4,
Od yduldwlrq �V� hvw duelwudluh/ g*rü =
I� @|^E` =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� @|^E` =�I? .
|^�E` =I? +614;6,
G*rü =
�I� .D� @|^E` ^N?` ^E` =�V� .
|^�E` =I? +614;7,
dyhf =
D� @|^E` =D? +614;8,
:7 Fkdslwuh 6 Qrq0olqìdulwì jìrpìwultxh
Frqvlgìurqv oh whuph |^�E`1 Frpph hq 5G/ rq srvh =
G�& @CE��
CV�&I?� +614;9,
g*rü = �|g ^E` =I?
��@ G�& gV�& +614;:,
Rq shxw grqf ìfuluh =|^�E` =I? @ ^G` =�V� +614;;,
Ilqdohphqw/ lo ylhqw =
�I� .D� @�|^E` ^N?` ^E` . ^G`
�=�V� +614;<,
Hq qrwdqw =
^N�` @�|^E` ^N?` ^E` . ^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 ^E` hw ^G`1 Vl o*rq h�hfwxh xq gìyhorsshphqw olplwì ã o*ruguh 4 pdwkìpdwltxh/
rq rewlhqw =
^E�` @
5999999999997
4�uf
�uf
3 3 3 3 3 3
� �u2f
�w�%fuf
uf3�u2f
w�5f2 � �
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w�%f2 3 3 3
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�w�%fuf
�w�+f2 � �
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3 �4 w�5f2 � �
uf
w�+f2 4 �w�5f
2 .�uf
�w�+f2
� �u2f
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uf3�u2f
3 3 3w�5f2 � �
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3 3 3 �w�+f2 � �
uf�w�%f
2 4
6:::::::::::8
+614<8,
617 Ohv frrugrqqìhv frqyhfwìhv :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,
:9 Fkdslwuh 6 Qrq0olqìdulwì jìrpìwultxh
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 @
3EEEEEEEEEEEEEEEEEEEEEEEEEC
I�f
I�t
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P�t
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4FFFFFFFFFFFFFFFFFFFFFFFFFD
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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 @
3EEEEEEEEEEEEEEEEEEEEEEEEEC
X�
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4FFFFFFFFFFFFFFFFFFFFFFFFFD
+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
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�+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:,
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�@
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`
�|^E` ^N?` ^E` . ^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 ^E` 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 ^E` 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�` @�|^E` ^N?` ^E` . ^G`
�hw oh whvw g*duuíw fruuhv0
srqgud ã o*ìtxloleuh ghv qúxgv irupxoì dyhf |^E`1 ^N�` hvw xq rsìudwhxu wdqjhqw> vrq
ìydoxdwlrq shxw ìyhqwxhoohphqw vh idluh ã o*dlgh gh ^E�` hw ^G�` rx ^E2` hw ^G2`/ txl fruuhv0
srqghqw uhvshfwlyhphqw dx{ gìyhorsshphqwv olplwìv g*ruguh 4 hw 5 gh ^E` hw ^G`1
Sdu frqwuh/ o*ìtxloleuh ghv qúxgv grlw íwuh fdofxoì dyhf I� @|^E` =I?1 Lfl/ ^E` 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��
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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
à 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
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y +{> |, @ Y�k� +{, . ��ke +{, . Y2kD +{, . �2kS +{, +61553,
dyhf
k� +{, @4� �
5+61554,
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5+61555,
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ke +{, @��� � �2 � � . 4
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kD +{, @��� . 6� . 5
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d+6155:,
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%%% +{> |, @Cx +{> |,
C{
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%�2%u�
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C|
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%++ +{> |, @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` @
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3 3 �.�u
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3�2.Uu�
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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
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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
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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,
;5 Fkdslwuh 6 Qrq0olqìdulwì jìrpìwultxh
^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
;7 Fkdslwuh 6 Qrq0olqìdulwì jìrpìwultxh
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
compress iontract ion
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
compress iontract ion
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
compress iontract ion
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
<5 Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh
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,
714 Ohv orlv gh frpsruwhphqw <6
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
<7 Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh
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
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s
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Es
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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
714 Ohv orlv gh frpsruwhphqw <8
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
<9 Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh
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> <
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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:,
435 Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh
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,
715 Ìwxgh g*xqh vhfwlrq gurlwh 436
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,
437 Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh
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;,
715 Ìwxgh g*xqh vhfwlrq gurlwh 438
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
439 Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh
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
443 Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh
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,
445 Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh
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 +{, @ ^O +{,` � I? +71;:,
rü O +{, hvw xqh pdwulfh grqw od irupxodwlrq hvw grqqìh sdu =
^O +{,` @
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 +{, @ ^O +{,` =�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 +{, @ ^O +{,` =�I? +71<7,
G*rü =|�I?=V? @
] uf
f
|�I?=|^O +{,` =%r +{, g{ +71<8,
Frpph �I? qh gìshqg sdv gh {/ lo ylhqw =
|�I?=V? @|�I?=
] uf
f
|^O +{,` =%r +{,g{ +71<9,
Od yduldwlrq �I? ìwdqw duelwudluh/ rq hq gìgxlw doruv =
V? @
] uf
f
|^O +{,` =%r +{,g{ +71<:,
Wrxmrxuv hq qìjoljhdqw od yduldwlrq gh orqjxhxu gh o*ìoìphqw gdqv oh uhsëuh lqwulqvëtxh/ rq
shxw ìfuluh =
�V? @
] uf
f
|^O +{,` =�%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
|^O +{,` = ^Nr`
3�= +�Ir +{, .Dr +{,,g{ +71<<,
Hq whqdqw frpswh gh +71;<,/ rq rewlhqw =
�V? @
] uf
f
|^O +{,` = ^Nr`
3�+^O +{,` =�I? .Dr +{,, g{ +71433,
@
] uf
f
|^O +{,` = ^Nr`
3� = ^O +{,` ��I?g{.] uf
f
|^O +{,` = ^Nr`
3� =Dr +{, g{
Frpph �I? qh gìshqg sdv gh {/ rq shxw ìfuluh =
�V? @
] uf
f
|^O +{,` = ^Nr`
3�= ^O +{,`g{=�I? .
] uf
f
|^O +{,` = ^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
|^O +{,` = ^Nr`
3�= ^O +{,`g{ +71435,
447 Fkdslwuh 7 Qrq0olqìdulwì pdwìulhooh
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
|^O +{,` = ^Nr`
3�=Dr +{, g{ +71437,
Oh whuphU uff
|^O +{,` = ^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`
�|^E` ^N?` ^E` . ^G`
�^Wf`
|^Sf` +71439,
rü ^Sf` hw ^Wf` vrqw ghv pdwulfhv frqvwdqwhv1 Od irupxodwlrq jìrpìwultxh/ txl dssdudñw
ã wudyhuv ohv pdwulfhv ^E` 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,
à 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
457 Fkdslwuh 9 Sulvh hq frpswh ghv h�hwv gl�ìuìv
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 =
459 Fkdslwuh 9 Sulvh hq frpswh ghv h�hwv gl�ìuìv
� �� 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
Sdu dloohxuv/ rq shxw ìfuluh =
% +w?3�,� %o +w?3�, @?3�[�'�
+4 . ! +w?> w�,,7 %�� +916;,
Sdu vrxvwudfwlrq/ rq rewlhqw =
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,
Sdu dloohxuv/ rq shxw ìfuluh =
% +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,
Sdu dloohxuv/ rq shxw ìfuluh =
% +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:,
463 Fkdslwuh 9 Sulvh hq frpswh ghv h�hwv gl�ìuìv
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
465 Fkdslwuh 9 Sulvh hq frpswh ghv h�hwv gl�ìuìv
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
467 Fkdslwuh 9 Sulvh hq frpswh ghv h�hwv gl�ìuìv
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 |}
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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
}
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4FFD7 %_�sgV6 @
5997]PK
H6 +|> },
3EEC
4
}
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4FFD =�
4 } |
�6::8 =7 %_�s?
@ ^N6? ` =7 %_�s? +91;6,
469 Fkdslwuh 9 Sulvh hq frpswh ghv h�hwv gl�ìuìv
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|
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#^N6? ` =7 %?_�s
^N6| ` =7 %|_�s
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Oh yhfwhxu
#7I4?7I4|
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:1615 Suhplëuh irupxodwlrq gh o*ìtxdwlrq gh od fkdñqhwwh
Ìwdeolvvhphqw gh o*ìtxdwlrq gh od fkdñqhwwh
Frqvlgìurqv xq srlqwP vlwxì vxu o*ìoìphqw1 Rq qrwh { hw | vhv frrugrqqìhv hw v vrq dev0
flvvh fxuyloljqh1 Od whqvlrq gh o*ìoìphqw hq fh srlqw hvw qrwìh W 1 Frqvlgìurqv pdlqwhqdqw
P�
xq srlqw yrlvlq gh P / gh frrugrqqìhv {. g{/ | . g| hw g*devflvvh fxuyloljqh v . gv1
Od whqvlrq hq P�
hvw W . gW 1 Vl W% hw W+ gìvljqhqw ohv surmhfwlrqv vxu R{ hw R| gh W /
doruv rq d =
W%
�P
�
�� W% +P, @ 3 +:169,
W+
�P
�
�� W+ +P, @ s=gv +:16:,
Rx hqfruh =
W% . gW% � W% @ gW% @ 3 +:16;,
W+ . gW+ � W+ @ gW+ @ s=gv +:16<,
47; Fkdslwuh : Ìoìphqwv vsìfl�txhv dx Jìqlh flylo
Rq hq gìgxlw doruv txh W% +P, hvw frqvwdqw hw ydxw Wf/ ydohxu srvlwlyh1 Sdu dloohxuv/ od
whqvlrq W +P, ìwdqw sruwìh sdu od wdqjhqwh ã od frxueh/ rq d =
W+ @ W%g|
g{@ Wf
g|
g{+:173,
Hq rxwuh/ rq vdlw txh =
gv @
v4 .
�g|
g{
�2
g{ +:174,
G*rü =
gW+ @ s
v4 .
�g|
g{
�2
g{ +:175,
G*rü =
gW+
g{@ s
v4 .
�g|
g{
�2
+:176,
Rq rewlhqw hq�q =
gW+
g{@ Wf
g2|
g{2@ s
v4 .
�g|
g{
�2
+:177,
Rx hqfruh =|��s
4 . |�2
@s
Wf+:178,
Od vroxwlrq gh fhwwh ìtxdwlrq gl�ìuhqwlhooh hvw gh od iruph vxlydqwh =
| +{, . |f @Wf
sfrvk
�s
Wf+{. {f,
�+:179,
Od whqvlrq W gdqv od fkdñqhwwh hvw gh od iruph =
W +{, @
vW 2f . W 2
f vlqk2
�s
Wf+{. {f,
�@ Wf frvk
�s
Wf+{. {f,
�+:17:,
O*devflvvh fxuyloljqh v yìul�h o*ìtxdwlrq =
gv
g{@
v4 .
�g|
g{
�2
@ frvk
�s
Wf+{. {f,
�+:17;,
G*rü =
v +{, . vf @Wf
svlqk
�s
Wf+{. {f,
�+:17<,
:16 Ohv ìoìphqwv gh w|sh fkdñqhwwh 47<
Fkdñqhwwh sdvvdqw sdu ghx{ srlqwv
O*ìoìphqw fkdñqhwwh uholh ohv srlqwvD hw E gh frrugrqqìhv uhvshfwlyhv +{�> |�, hw +{�> |�,1
Qrxv dyrqv grqf =
|� . |f @Wf
sfrvk
�s
Wf+{� . {f,
�+:183,
|� . |f @Wf
sfrvk
�s
Wf+{� . {f,
�+:184,
Sdu vrxvwudfwlrq/ rq rewlhqw =
|� � |� @ 5Wf
s
�vlqk
�s
Wf
�{� . {�
5. {f
��vlqk
�s
Wf
�{� � {�
5
���+:185,
Hq srvdqw =
Y @s
5Wf
|� � |�
vlqk
�RAf
�%�3%�
2
�� +:186,
rq d =
{f @Wf
sduj vlqk +Y ,� {� . {�
5+:187,
|f @Wf
sfrvk
�s
Wf
�Wf
sduj vlqk +Y , .
{� � {�
5
��� |� +:188,
Orqjxhxu gìyhorssìh hw doorqjhphqw
Od orqjxhxu gìyhorssìh O gh od fkdñqhwwh vh fdofxoh vlpsohphqw vhorq =
O @ v +{�,� v +{�,
@Wf
s
�vlqk
�s
Wf+{� . {f,
�� vlqk
�s
Wf+{� . {f,
��+:189,
O*doorqjhphqw �O hvw rewhqx sdu lqwìjudwlrq gh od gìirupdwlrq oh orqj gh o*ìoìphqw =
�O @
] �
�
W +v,
H=D%
gv @
] �
�
W +{,
H=D%
frvk
�s
Wf+{. {f,
�g{
@Wf
H=D%
] %�
%�
frvk2
�s
Wf+{. {f,
�g{
@Wf
5H=D%
�Wf
5svlqk
�5s
Wf+{. {f,
�. {
�%�%�
+:18:,
Od orqjxhxu ã ylgh O� gh o*ìoìphqw fkdñqhwwh hvw =
O� @ O��O +:18;,
Uìvroxwlrq gx sureoëph gh od fkdñqhwwh
Xq ìoìphqw gh w|sh fkdñqhwwh shxw íwuh fkdujì gh 8 idêrqv gl�ìuhqwhv =
483 Fkdslwuh : Ìoìphqwv vsìfl�txhv dx Jìqlh flylo
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
à 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
485 Fkdslwuh : Ìoìphqwv vsìfl�txhv dx Jìqlh flylo
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,
:16 Ohv ìoìphqwv gh w|sh fkdñqhwwh 486
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
487 Fkdslwuh : Ìoìphqwv vsìfl�txhv dx Jìqlh flylo
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,
:16 Ohv ìoìphqwv gh w|sh fkdñqhwwh 488
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<,
489 Fkdslwuh : Ìoìphqwv vsìfl�txhv dx Jìqlh flylo
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 .
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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�
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YAf�Wf .
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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
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3EEC
3
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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, .
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g
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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
à 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` @ ^O` ^G`|^O` +;143,
rü ^O` 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 à 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
à 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,
à o*lqvwdqw w/ o*ìtxdwlrq gx prxyhphqw gh od vwuxfwxuh hvw =
^P ` = �V} +w, . ^F` = bV} +w, . I} +w, @ S +w, +<158,
à 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{
4:7 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv
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
Suìvhqwdwlrq gx sureoëph
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
Prgìolvdwlrq gx sureoëph
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
4:9 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv
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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
Suìvhqwdwlrq gx sureoëph
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
Prgìolvdwlrq gx sureoëph
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)
Charge P (kN)
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)
Charge P (kN)
fy Expérience
fy Calcul
fz Expérience
fz Calcul
Iljxuh 431; = Srwhdx GZ44 0 Frxueh fkdujh 0 ëfkh ã pl0kdxwhxu
4;5 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv
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Flèches (mm)
Charge P (kN)
fy Expérience
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fz Expérience
fz Calcul
Iljxuh 431< = Srwhdx GZ46 0 Frxueh fkdujh 0 ëfkh ã pl0kdxwhxu
4315 Vwuxfwxuhv hq eìwrq dupì 4;6
431517 Sruwltxh uhfwdqjxodluh elduwlfxoì
Suìvhqwdwlrq gx sureoëph
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
4;7 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv
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
Prgìolvdwlrq gx sureoëph
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
4315 Vwuxfwxuhv hq eìwrq dupì 4;8
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Flèches (mm)
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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
Suìvhqwdwlrq gx sureoëph
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
4;9 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv
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
Prgìolvdwlrq gx sureoëph
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
20
40
60
80
100
120
140
160
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Flèches (mm)
Charge P (kN)
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
Suìvhqwdwlrq gx sureoëph
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;<
Prgìolvdwlrq gx sureoëph
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
Suìvhqwdwlrq gx sureoëph
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
à 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
4<5 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv
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
Prgìolvdwlrq gx sureoëph
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,
4316 Vwuxfwxuhv hq eìwrq suìfrqwudlqw 4<6
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
4<7 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv
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
Suìvhqwdwlrq gx sureoëph
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
4316 Vwuxfwxuhv hq eìwrq suìfrqwudlqw 4<8
-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
4<9 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv
-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
Prgìolvdwlrq gx sureoëph
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
535 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv
431617 Srxwuh suìfrqwudlqwh k|shuvwdwltxh
Suìvhqwdwlrq gx sureoëph
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
Prgìolvdwlrq gx sureoëph
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
537 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv
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
Suìvhqwdwlrq gx sureoëph
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
Prgìolvdwlrq gx sureoëph
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
539 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv
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
Suìvhqwdwlrq gx sureoëph
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
Prgìolvdwlrq gx sureoëph
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
Suìvhqwdwlrq gx sureoëph
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
Prgìolvdwlrq gx sureoëph
Oh frpsruwhphqw gx pdwìuldx hvw olqìdluh ìodvwltxh1 Ohv fdudfwìulvwltxhv jìrpìwultxhv hw
pìfdqltxhv vrqw ohv vxlydqwhv =
H @ 43. svl @ :363; PSd +43166,
543 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv
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
545 Fkdslwuh 43 H{hpsohv g*dssolfdwlrqv qxpìultxhv
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
549 Fkdslwuh 44 Xq h{hpsoh g*dssolfdwlrq uìhooh = oh Srqw gh Wdqfduylooh
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
Plvh hq whqvlrq ghv fkdñqhwwhv fruuhvsrqgdqwhv hq
orqjxhxu ã ylgh1
8 0 Plvh ã orqjxhxu g*xqh vxv0
shqwh vxu ghx{ gdqv od wudyìh
fhqwudoh/ fhooh0fl v*h�hfwxdqw hq <
vrxv0skdvhv
Plvh hq whqvlrq ghv fkdñqhwwhv fruuhvsrqgdqwhv hq
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
Plvh hq whqvlrq ghv fkdñqhwwhv fruuhvsrqgdqwhv hq
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
553 Fkdslwuh 44 Xq h{hpsoh g*dssolfdwlrq uìhooh = oh Srqw gh Wdqfduylooh
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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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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
555 Fkdslwuh 44 Xq h{hpsoh g*dssolfdwlrq uìhooh = oh Srqw gh Wdqfduylooh
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
4419 Ohv sulqflsdx{ uìvxowdwv 556
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
Prgxoh g*ìodvwlflwì gh fdofxo = 4:3 333 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,
Prgxoh g*ìodvwlflwì gh fdofxo = 4:3 333 PSd
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
557 Fkdslwuh 44 Xq h{hpsoh g*dssolfdwlrq uìhooh = oh Srqw gh Wdqfduylooh
-250
-200
-150
-100
-50
0
50
100
150
Déroulement du chantier
Déplacement
(mm)
GTM
SETRA
CHANTIER
Iljxuh 441: = Gìsodfhphqw gh od wíwh gh s|o÷qh ulyh gurlwh sdu udssruw ã od yhuwlfdoh
4419 Ohv sulqflsdx{ uìvxowdwv 558
-100
0
100
200
300
400
500
600
700
800
Déroulement du chantier
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,
559 Fkdslwuh 44 Xq h{hpsoh g*dssolfdwlrq uìhooh = oh Srqw gh Wdqfduylooh
-200
-150
-100
-50
0
50
100
150
200
250
Déroulement du chantier
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:
-100
0
100
200
300
400
500
600
700
800
900
Déroulement du chantier
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
0
500
1000
1500
2000
2500
3000
Déroulement du chantier
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
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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
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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
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gh dpehphqwv ghv vwuxfwxuhv +fkdslwuh ;,/ dlqvl txh oh fdofxo g|qdpltxh wudqvlwrluh/ dsuëv
uhirupxodwlrq gx sureoëph ã o*dlgh g*xq vfkìpd gh Qhzpdun +fkdslwuh <,1
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fdwlrq ã sduw hqwlëuh1 Lo d g*dloohxuv ìwì xwlolvì hq wdqw txh who doruv txh vrq gìyhorsshphqw
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563 Fkdslwuh 45 Frqfoxvlrq jìqìudoh
od vxvshqvlrq gx Srqw gh Wdqfduylooh1 Vhxoh od qrq0olqìdulwì jìrpìwultxh d ìwì sulvh hq
frpswh srxu fh surmhw1 Sdu dloohxuv/ oruv gx skdvdjh gh uhpsodfhphqw/ ohv vhoohv g*dssxlv
gx fåeoh hq wíwh gh s|o÷qh ìwdlhqw prgìolvìhv sdu ghv duwlfxodwlrqv hq judqgv gìsodfhphqwv/
oleuhv gh vh gìsodfhu orqjlwxglqdohphqw1 Fhv ìoìphqwv rqw grqqì hqwlëuh vdwlvidfwlrq/ wrxw
frpph ohv ìoìphqwv elduwlfxoìv gh w|sh fkdñqhwwh1
SFS/ gdqv vd yhuvlrq qrq olqìdluh/ hvw ã o*khxuh dfwxhooh xwlolvì srxu h�hfwxhu o*ìwxgh gh
idlvdelolwì gx judqg yldgxf gh Ploodx sdu od Glylvlrq ghv Judqgv Rxyudjhv dx VHWUD1
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dlqvl ìfulw uhsuìvhqwh soxv gh 63 333 oljqhv gh frgh Jhqiru +xq jìqìudwhxu gh Iruwudq,1
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97; s1
Dqqh{h D
Ohv fåeohv gh suìfrqwudlqwh qrq
olìv
D14 Udsshov
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 ìjdoh ã od pr|hqqh ghv gìirupdwlrqv ghv
vhfwlrqv oh orqj gx fåeoh/ hq sdufrxudqw oh wurqêrq qrq lqmhfwì1
Oruv gx surfhvvxv lwìudwli gh uìvroxwlrq/ ã fkdtxh ìwdsh/ od gìirupdwlrq ghv fåeohv qrq
olìv dx eìwrq hvw plvh ã mrxu ã sduwlu gh od gìirupdwlrq frxudqwh ghv vhfwlrqv1
D15 Plvh hq úxyuh lqirupdwltxh
Frqfuëwhphqw/ oh wdeohdx klvfd txl frqwlhqw o*klvwrultxh g*xq fåeoh d od vwuxfwxuh vxlydqwh =
� klvfd +4, = gdwh gh plvh hq whqvlrq gx fåeoh/
� klvfd +5, = shuwh wrwdoh sdu uhod{dwlrq =S
��4/
575 Dqqh{h D Ohv fåeohv gh suìfrqwudlqwh qrq olìv
� klvfd +6, = �@ frqwudlqwh frxudqwh gx fåeoh +plvh ã mrxu oruv g*xqh frqyhujhqfh hq
gìsodfhphqwv,/
� klvfd +7, @�f
H. %,�rr�e
6�re e? |e?r�J?= suìgìirupdwlrq gx fåeoh/
� klvfd +8, @ %,�rr�eSJ�o@?|
= gìirupdwlrq olvvìh frxudqwh gx fåeoh/ plvh ã mrxu gdqv Olv0
vdjhbShuwhvbQol> fhwwh txdqwlwì q*hvw soxv prgl�ìh vl oh fåeoh hvw lqmhfwì/
� klvfd +9, @%reS|�J?SJ�o@?|
= gìirupdwlrq frxudqwh gh od vhfwlrq/ wdqjhqwlhoohphqw dx fåeoh1
klvfd +9, vhuw ã fdofxohu klvfd +8,> fhwwh txdqwlwì q*hvw soxv prgl�ìh vl oh fåeoh hvw
lqmhfwì1
Dlqvl/ srxu xq fåeoh qrq olì/ vd gìirupdwlrq hvw grqqìh sdu =
%@ @�f
HR
��%,�rr�eSJ�o@?|
� %,�rr�e6�re e? |e?r�J?
�+D14,
%@ @ klvfd +7,� klvfd +8, +D15,
Srxu xq fåeoh olì/ rq d =
%@ @�f
H��
%,�rr�e�?�eS|�J?
� %,�rr�e6�re e? |e?r�J?
���%reS|�J?SJ�o@?|
� %reS|�J?�?�eS|�J?
�+D16,
%@ @ klvfd +7,� klvfd +8, . klvfd +9,� %reS|�J?SJ�o@?|
+D17,
Gdqv wrxv ohv fdv/ od whqvlrq frxudqwh gx fåeoh hvw fdofxoìh frpph vxlw/ ã fkdtxh
frqyhujhqfh hq gìsodfhphqwv =
�@ @ � +%@,�[
��4 +D18,
klvfd +6, @ � +%@,� klvfd +5, +D19,
Od vxuwhqvlrq gx fåeoh qrq olì/ frqvlgìuìh frpph xq fkdujhphqw h{wìulhxu/ hvw fdofxoìh
dlqvl/ ã fkdtxh ìwdsh gh uìvroxwlrq =
� +%@,�[
��4 � klvfd +6, +D1:,
Dx ixu hw ã phvxuh gh o*dydqfhphqw ghv fdofxov/ fhwwh txdqwlwì whqg yhuv 3 = od vhfwlrq hvw
hq ìtxloleuh dyhf od yduldwlrq gh whqvlrq gx fåeoh1
Dqqh{h E
Rswlplvdwlrq gh od qxpìurwdwlrq
ghv qúxgv
O*remhfwli gh fh wudydlo hvw gh wurxyhu xqh rswlplvdwlrq gh od qxpìurwdwlrq ghv qúxgv
g*xqh vwuxfwxuh grqqìh/ txl shuphwwud gh uìgxluh/ vl srvvleoh/ od wdlooh gh od pdwulfh gh
uljlglwì hw oh whpsv gh uìvroxwlrq gx v|vwëph olqìdluh jìqìudo1
E14 Suìvhqwdwlrq
Od pdwulfh gh uljlglwì ^N` g*xqh vwuxfwxuh vwdeoh r�uh od uhpdutxdeoh sursulìwì vxlydqwh =
hooh hvw uìhooh/ v|pìwultxh gì�qlh srvlwlyh/ fh txl qrxv shuphw g*h�hfwxhu xqh idfwrulvdwlrq
gh Fkrohvn| glwh srvlwlyh +ohv whuphv gldjrqdx{ gh od pdwulfh idfwrulvìh vrqw srvlwliv, =
^N` @ ^O` | ^O` dyhf ^O`�c� A 3 hw ^O` pdwulfh wuldqjxodluh lqiìulhxuh1
Oh sureoëph ã uìvrxguh d od iruph vxlydqwh = ^N` ={ @ e1
Od pdwulfh ^N` hvw vxssrvìh dvvhpeoìh gdqv oh uhsëuh jìqìudo1
O*rswlplvdwlrq gh od qxpìurwdwlrq gìshqg hvvhqwlhoohphqw gx prgh gh udqjhphqw gh
od pdwulfh gh uljlglwì1 Gdqv qrwuh fdv/ od pdwulfh hvw udqjìh vhorq vrq sur�o +vrxv od oljqh
gh flho,1
Rq shxw dxvvl elhq h�hfwxhu od idfwrulvdwlrq hw od uìvroxwlrq srxu od pdwulfh ^S ` ^N`|^S `
txl uìvxowh g*xqh uhqxpìurwdwlrq ghv ìtxdwlrqv hw ghv lqfrqqxhv gì�qlvvdqw od pdwulfh gh
shupxwdwlrq ^S `1
Oh surfhvvxv gh uìvroxwlrq frpsuhqg 6 ìwdshv =
41 od qxpìurwdwlrq hw od idfwrulvdwlrq orjltxh/
51 od idfwrulvdwlrq qxpìultxh/
61 od uìvroxwlrq gx v|vwëph olqìdluh1
577 Dqqh{h E Rswlplvdwlrq gh od qxpìurwdwlrq ghv qúxgv
Qrxv qrxv sursrvrqv g*lqwhuyhqlu dx qlyhdx gh od qxpìurwdwlrq d�q gh uìgxluh od wdlooh
gh od pdwulfh gh uljlglwì1
Od surjudppdwlrq gh o*rswlplvdwlrq gh od qxpìurwdwlrq hw gh od idfwrulvdwlrq orjltxh
hvw od skdvh od soxv gl!floh ã uìdolvhu1 Oh sulqflsh ghv pìwkrghv g*rswlplvdwlrq hvw wrx0
mrxuv dvvh} vlpsoh/ pdlv rq qh shxw sdv sruwhu gh mxjhphqw vxu o*h!fdflwì g*xqh pìwkrgh
lqgìshqgdpphqw gh vrq lpsoìphqwdwlrq1
Ohv pìwkrghv gh uhqxpìurwdwlrq txh qrxv doorqv derughu vrqw sduwlfxolëuhphqw elhq
dgdswìhv dx{ pdwulfhv fuhxvhv1
E15 Dqdo|vh gx sureoëph
Dsuëv dqdo|vh eleolrjudskltxh/ ^Dnk:9`/ ^Elo;<`/ ^Wkh<7`/ rq frqvwdwh tx*lo h{lvwh ghx{
pdqlëuhv g*derughu oh sureoëph =
� 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
57<
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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